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+ + + + + + + +
+ + + \ No newline at end of file diff --git a/theories/html/Goedel.codeSysPrf.html b/theories/html/Goedel.codeSysPrf.html new file mode 100644 index 00000000..832f6787 --- /dev/null +++ b/theories/html/Goedel.codeSysPrf.html @@ -0,0 +1,251 @@ + + + + + +Goedel.codeSysPrf + + + + +
+ + + +
+ +

Library Goedel.codeSysPrf

+ +
+
+ +
+TO DO: Define abbreviations and re-indent !!! + +
+
+ +
+From Coq Require Import Ensembles.
+From Coq Require Import List.
+From hydras.Ackermann Require Import checkPrf.
+From hydras.Ackermann Require Import code.
+From hydras.Ackermann Require Import Languages.
+From hydras.Ackermann Require Import folProp.
+From hydras.Ackermann Require Import folProof.
+From hydras.Ackermann Require Import folLogic3.
+From hydras.Ackermann Require Import folReplace.
+From Goedel Require Import PRrepresentable.
+From hydras.Ackermann Require Import expressible.
+From hydras.Ackermann Require Import primRec.
+From Coq Require Import Arith Lia.
+From hydras.Ackermann Require Import PA.
+From hydras.Ackermann Require Import NNtheory.
+From hydras.Ackermann Require Import codeList.
+From hydras.Ackermann Require Import subProp.
+From hydras.Ackermann Require Import ListExt.
+From hydras.Ackermann Require Import cPair.
+From hydras.Ackermann Require Import wellFormed.
+From hydras.Ackermann Require Import prLogic.
+ +
+From hydras Require Import Compat815.
+Import NNnotations.
+ +
+ +
+Section code_SysPrf.
+ +
+Generalizable All Variables.
+Variable L : Language.
+Context `(cL: Lcode L cf cr).
+ +
+Variable codeArityF : nat → nat.
+Variable codeArityR : nat → nat.
+Context (codeArityFIsPR : isPR 1 codeArityF).
+Hypothesis
+  codeArityFIsCorrect1 :
+    ∀ f : Functions L, codeArityF (cf f) = S (arityF L f).
+Hypothesis
+  codeArityFIsCorrect2 :
+    ∀ n : nat, codeArityF n ≠ 0 → ∃ f : Functions L, cf f = n.
+Context (codeArityRIsPR : isPR 1 codeArityR).
+Hypothesis
+  codeArityRIsCorrect1 :
+    ∀ r : Relations L, codeArityR (cr r) = S (arityR L r).
+Hypothesis
+  codeArityRIsCorrect2 :
+    ∀ n : nat, codeArityR n ≠ 0 → ∃ r : Relations L, cr r = n.
+ +
+Import LNN NN.
+Import NNnotations.
+ +
+Section LNN.
+ +
+Variable T : System.
+Hypothesis TextendsNN : Included _ NN T.
+ +
+Variable U : fol.System L.
+Variable fU : Formula. +
+Variable v0 : nat.
+Hypothesis freeVarfU : ∀ v : nat, In v (freeVarF fU) → v = v0.
+Hypothesis
+  expressU1 :
+    ∀ f : fol.Formula L,
+    mem _ U f →
+    SysPrf T
+      (substF fU v0 (natToTerm (codeFormula f))).
+Hypothesis
+  expressU2 :
+    ∀ f : fol.Formula L,
+    ¬ mem _ U f →
+    SysPrf T
+      (notH (substF fU v0 (natToTerm (codeFormula f)))).
+ +
+Definition codeSysPrf : Formula :=
+  let nv := newVar (2 :: 1 :: 0 :: v0 :: nil) in
+  existH nv
+    (andH
+       (substF
+          (substF
+             (primRecFormula 2
+                (proj1_sig
+                   (checkPrfIsPR L cL codeArityF codeArityR
+                      codeArityFIsPR codeArityRIsPR))) 0
+             (Succ (var nv))) 2 (var 0))
+       (forallH (S nv)
+          (impH (LT (var (S nv)) (var nv))
+             (orH
+                (substF
+                   (substF
+                      (substF
+                         (primRecFormula 2 (proj1_sig codeInIsPR)) 2
+                         (var (S nv))) 1 (var nv)) 0 Zero)
+                (substF fU v0 (var (S nv))))))).
+ +
+Lemma codeSysPrfCorrect1 :
+ ∀ (f : fol.Formula L) (A : list (fol.Formula L)) (p : Prf L A f),
+ (∀ g : fol.Formula L, In g A → mem _ U g) →
+ SysPrf T
+   (substF
+      (substF codeSysPrf 0
+         (natToTerm (codeFormula f))) 1
+      (natToTerm (codePrf A f p))).
+ +
+Lemma codeSysPrfCorrect2 :
+  ∀ (f : fol.Formula L) (A : fol.Formulas L),
+    (∃ g : fol.Formula L, In g A ∧ ¬ mem _ U g) →
+    ∀ p : Prf L A f,
+      SysPrf T
+        (notH
+           (substF
+              (substF codeSysPrf 0
+                 (natToTerm (codeFormula f))) 1
+              (natToTerm (codePrf A f p)))).
+ +
+Lemma codeSysPrfCorrect3 :
+  ∀ (f : fol.Formula L) (n : nat),
+    (∀ (A : list (fol.Formula L)) (p : Prf L A f),
+        n ≠ codePrf A f p) →
+    SysPrf T
+      (notH
+         (substF
+            (substF codeSysPrf 0
+               (natToTerm (codeFormula f))) 1
+            (natToTerm n))).
+ +
+Lemma freeVarCodeSysPrf :
+ ∀ v : nat, In v (freeVarF codeSysPrf) → v ≤ 1.
+ +
+Definition codeSysPf : Formula := existH 1 codeSysPrf.
+ +
+Lemma freeVarCodeSysPf :
+ ∀ v : nat, In v (freeVarF codeSysPf) → v = 0.
+ +
+Lemma codeSysPfCorrect :
+ ∀ f : fol.Formula L,
+ folProof.SysPrf L U f →
+ SysPrf T
+   (substF codeSysPf 0
+      (natToTerm (codeFormula f))).
+ +
+Definition codeSysPrfNot :=
+  existH 2
+    (andH (substF codeSysPrf 0 (var 2))
+       (substF
+          (substF (primRecFormula 1 (proj1_sig codeNotIsPR)) 0
+             (var 2)) 1 (var 0))).
+ +
+Lemma freeVarCodeSysPrfN :
+ ∀ v : nat, In v (freeVarF codeSysPrfNot) → v ≤ 1.
+ +
+Lemma codeSysPrfNCorrect1 :
+  ∀ (f : fol.Formula L) (A : fol.Formulas L) (p : Prf L A (notH f)),
+    (∀ g : fol.Formula L, In g A → mem _ U g) →
+    SysPrf T
+      (substF
+         (substF codeSysPrfNot 0
+            (natToTerm (codeFormula f))) 1
+         (natToTerm (codePrf A (notH f) p))).
+ +
+Lemma codeSysPrfNCorrect2 :
+  ∀ (f : fol.Formula L) (A : fol.Formulas L),
+    (∃ g : fol.Formula L, In g A ∧ ¬ mem _ U g) →
+    ∀ p : Prf L A (notH f),
+      SysPrf T
+        (notH
+           (substF
+              (substF codeSysPrfNot 0
+                 (natToTerm (codeFormula f))) 1
+              (natToTerm (codePrf A (notH f) p)))).
+ +
+Lemma codeSysPrfNCorrect3 :
+ ∀ (f : fol.Formula L) (n : nat),
+ (∀ (A : fol.Formulas L) (p : Prf L A (notH f)),
+  n ≠ codePrf A (notH f) p) →
+ SysPrf T
+   (notH
+      (substF
+         (substF codeSysPrfNot 0
+            (natToTerm (codeFormula f))) 1
+         (natToTerm n))).
+ +
+End LNN.
+ +
+End code_SysPrf.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/Goedel.fixPoint.html b/theories/html/Goedel.fixPoint.html new file mode 100644 index 00000000..7c4600d9 --- /dev/null +++ b/theories/html/Goedel.fixPoint.html @@ -0,0 +1,101 @@ + + + + + +Goedel.fixPoint + + + + +
+ + + +
+ +

Library Goedel.fixPoint

+ +
+From hydras.Ackermann Require Import primRec.
+From hydras.Ackermann Require Import cPair.
+From Coq Require Import Arith Lia.
+From hydras.Ackermann Require Import code.
+From hydras.Ackermann Require Import codeSubFormula.
+From hydras.Ackermann Require Import folProp.
+From hydras.Ackermann Require Import folProof.
+From hydras.Ackermann Require Import Languages.
+From hydras.Ackermann Require Import subAll.
+From hydras.Ackermann Require Import subProp.
+From hydras.Ackermann Require Import folLogic3.
+From hydras.Ackermann Require Import folReplace.
+From hydras.Ackermann Require Import LNN.
+From hydras.Ackermann Require Import codeNatToTerm.
+From Goedel Require Import PRrepresentable.
+From hydras.Ackermann Require Import ListExt.
+From Coq Require Import List.
+From hydras.Ackermann Require Import NN.
+From hydras.Ackermann Require Import expressible.
+From hydras Require Import Compat815.
+Import FolNotations.
+Import NNnotations.
+ +
+Definition subStar (a v n : nat) : nat := codeSubFormula a v (codeNatToTerm n).
+ +
+#[export] Instance subStarIsPR : isPR 3 subStar.
+ +
+Section LNN_FixPoint.
+ +
+Let codeFormula := codeFormula (cl:=LcodeLNN).
+ +
+Lemma FixPointLNN :
+ ∀ (A : Formula) (v : nat),
+ {B : Formula |
+   SysPrf NN
+     (B ↔ substF A v (natToTermLNN (codeFormula B)))%fol ∧
+   (∀ x : nat,
+    In x (freeVarF B) ↔
+    In x (List.remove eq_nat_dec v (freeVarF A)))}.
+ +
+End LNN_FixPoint.
+ +
+From hydras.Ackermann Require Import PA.
+From hydras.Ackermann Require Import NN2PA.
+ +
+Section LNT_FixPoint.
+ +
+Let codeFormula := codeFormula (cl:=LcodeLNT).
+ +
+Lemma FixPointLNT (A : Formula) (v : nat):
+  {B : Formula |
+    SysPrf PA
+      (iffH B (substF A v (natToTermLNT (codeFormula B)))) ∧
+      (∀ x : nat,
+          In x (freeVarF B) ↔
+            In x (List.remove eq_nat_dec v (freeVarF A)))}.
+ +
+End LNT_FixPoint.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/Goedel.goedel1.html b/theories/html/Goedel.goedel1.html new file mode 100644 index 00000000..2727755b --- /dev/null +++ b/theories/html/Goedel.goedel1.html @@ -0,0 +1,139 @@ + + + + + +Goedel.goedel1 + + + + +
+ + + +
+ +

Library Goedel.goedel1

+ +
+From Coq Require Import Ensembles.
+From Coq Require Import List.
+From Coq Require Import Arith.
+From hydras.Ackermann Require Import folProp.
+From hydras.Ackermann Require Import folProof.
+From hydras.Ackermann Require Import subProp.
+From hydras.Ackermann Require Import ListExt.
+From Goedel Require Import fixPoint codeSysPrf.
+From hydras.Ackermann Require Import wConsistent.
+From hydras.Ackermann Require Import NN.
+From hydras.Ackermann Require Import code.
+From hydras.Ackermann Require Import checkPrf.
+From hydras Require Import Compat815.
+ +
+From LibHyps Require Export LibHyps.
+From hydras Require Export MoreLibHyps NewNotations.
+ +
+Import NNnotations codeNatToTerm.
+ +
+Definition codeFNN := codeFormula (cl:=LcodeLNN) .
+ +
+Notation reflection f := (natToTerm (codeFNN f)).
+ +
+Section Goedel's_1st_Incompleteness.
+ +
+Variable T : System.
+ +
+Hypothesis extendsNN : Included _ NN T.
+ +
+Variable repT : Formula.
+Variable v0 : nat.
+Hypothesis
+  freeVarRepT : ∀ v : nat, In v (freeVarF repT) → v = v0.
+ +
+Hypothesis
+  expressT1 :
+    ∀ f : Formula, mem _ T f →
+    SysPrf T (substF repT v0 (reflection f)).
+ +
+Hypothesis
+  expressT2 :
+    ∀ f : Formula, ¬ mem _ T f →
+    SysPrf T (¬ (substF repT v0 (reflection f)))%fol.
+ +
+Definition codeSysPrf :=
+  codeSysPrf LNN LcodeLNN codeArityLNTF codeArityLNNR
+    codeArityLNTFIsPR codeArityLNNRIsPR repT v0.
+ +
+Definition codeSysPf :=
+  codeSysPf LNN LcodeLNN codeArityLNTF codeArityLNNR
+    codeArityLNTFIsPR codeArityLNNRIsPR repT v0.
+ +
+Definition G := let (a,_) := FixPointLNN (notH codeSysPf) 0 in a.
+ +
+Definition codeSysPfCorrect :=
+  codeSysPfCorrect LNN LcodeLNN codeArityLNTF
+    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNNRIsPR
+    codeArityLNNRIsCorrect1 T extendsNN T repT v0 freeVarRepT expressT1.
+ +
+Definition codeSysPrfCorrect2 :=
+  codeSysPrfCorrect2 LNN LcodeLNN codeArityLNTF
+    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNNRIsPR
+    codeArityLNNRIsCorrect1 T extendsNN T repT v0 freeVarRepT expressT2.
+ +
+Definition codeSysPrfCorrect3 :=
+  codeSysPrfCorrect3 LNN LcodeLNN codeArityLNTF
+    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1
+    codeArityLNTFIsCorrect2 codeArityLNNRIsPR codeArityLNNRIsCorrect1
+    codeArityLNNRIsCorrect2 T extendsNN.
+ +
+Lemma freeVarG : closed G.
+ +
+Lemma FirstIncompletenessA : SysPrf T G → Inconsistent LNN T.
+ +
+ +
+Lemma FirstIncompletenessB :
+  wConsistent T → ¬ SysPrf T (notH G).
+ +
+Theorem Goedel'sIncompleteness1st :
+ wConsistent T →
+ ∃ f : Formula, independent T f ∧ closed f.
+ +
+End Goedel's_1st_Incompleteness.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/Goedel.goedel2.html b/theories/html/Goedel.goedel2.html new file mode 100644 index 00000000..777add81 --- /dev/null +++ b/theories/html/Goedel.goedel2.html @@ -0,0 +1,159 @@ + + + + + +Goedel.goedel2 + + + + +
+ + + +
+ +

Library Goedel.goedel2

+ +
+From Coq Require Import Ensembles.
+From Coq Require Import List.
+From Coq Require Import Arith.
+From hydras.Ackermann Require Import folProp.
+From hydras.Ackermann Require Import folProof.
+From hydras.Ackermann Require Import folReplace.
+From hydras.Ackermann Require Import folLogic3.
+From hydras.Ackermann Require Import subProp.
+From hydras.Ackermann Require Import ListExt.
+ +
+From Goedel Require Import fixPoint.
+From hydras.Ackermann Require Import NN2PA.
+From Goedel Require Import codeSysPrf.
+From hydras.Ackermann Require Import PAtheory.
+From hydras.Ackermann Require Import code.
+From hydras.Ackermann Require Import checkPrf.
+From hydras.Ackermann Require Import codeNatToTerm.
+From Goedel Require Import rosserPA.
+ +
+Section Goedel's_2nd_Incompleteness.
+ +
+  Variable T : System.
+ +
+  Hypothesis extendsPA : Included _ PA T.
+ +
+  Variable repT : Formula.
+  Variable v0 : nat.
+  Hypothesis
+    freeVarRepT : ∀ v : nat, In v (freeVarF repT) → v = v0.
+  Hypothesis
+    expressT1 :
+    ∀ f : Formula,
+      mem _ T f →
+      SysPrf T (substF repT v0 (natToTerm (codeFormula f))).
+  Hypothesis
+    expressT2 :
+    ∀ f : Formula,
+      ¬ mem _ T f →
+      SysPrf T
+        (notH (substF repT v0 (natToTerm (codeFormula f)))).
+ +
+  Definition codeSysPf :=
+    (codeSysPf LNT LcodeLNT codeArityLNTF codeArityLNTR
+       codeArityLNTFIsPR codeArityLNTRIsPR (LNT2LNN_formula repT) v0).
+ +
+  Section Goedel1PA.
+ +
+ +
+    Definition T' := T' T.
+ +
+    Definition extendsNN := extendsNN T.
+ +
+    Definition Tprf2T'prf := Tprf2T'prf T.
+ +
+    Definition expressT'1 := expressT'1 T repT v0 expressT1.
+ +
+    Definition expressT'2 := expressT'2 T repT v0 expressT2.
+ +
+    Definition freeVarRepT' := freeVarRepT' repT v0 freeVarRepT.
+ +
+    Definition codeSysPfCorrect :=
+      codeSysPfCorrect LNT LcodeLNT codeArityLNTF
+        codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1
+        codeArityLNTRIsPR
+        codeArityLNTRIsCorrect1 T' extendsNN T (LNT2LNN_formula repT) v0
+        freeVarRepT' expressT'1.
+ +
+    Definition G := let (a,_) :=
+                      FixPointLNT (notH (LNN2LNT_formula codeSysPf)) 0
+                    in a.
+ +
+    Definition box (f:Formula) :=
+      (substF (LNN2LNT_formula codeSysPf) 0
+         (natToTerm (codeFormula f))).
+ +
+    Lemma GS : SysPrf T (iffH G (notH (box G))).
+ +
+    Lemma HBL1 : ∀ f, SysPrf T f → SysPrf T (box f).
+ +
+    Lemma FirstIncompletenessA : SysPrf T G → Inconsistent LNT T.
+ +
+  End Goedel1PA.
+ +
+  Definition F : Formula :=
+    (notH (L:= LNT) (forallH 0 (equal (var 0) (var 0)))).
+ +
+  Definition Con := notH (box F).
+ +
+  Hypothesis HBL2 : ∀ f, SysPrf T (impH (box f) (box (box f))).
+  Hypothesis HBL3 : ∀ f g, SysPrf T (impH (box (impH f g)) (impH (box f) (box g))).
+ +
+  Theorem SecondIncompletness :
+    SysPrf T Con → Inconsistent LNT T.
+ +
+End Goedel's_2nd_Incompleteness.
+ +
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/Goedel.rosser.html b/theories/html/Goedel.rosser.html new file mode 100644 index 00000000..bf41e2ef --- /dev/null +++ b/theories/html/Goedel.rosser.html @@ -0,0 +1,207 @@ + + + + + +Goedel.rosser + + + + +
+ + + +
+ +

Library Goedel.rosser

+ +
+From Coq Require Import Ensembles.
+From Coq Require Import List.
+From Coq Require Import Arith Lia.
+From hydras.Ackermann Require Import folProp.
+From hydras.Ackermann Require Import folProof.
+From hydras.Ackermann Require Import folReplace.
+From hydras.Ackermann Require Import folLogic3.
+From hydras.Ackermann Require Import subProp.
+From hydras.Ackermann Require Import ListExt.
+From hydras.Ackermann Require Export Languages.
+From Goedel Require Import fixPoint.
+From Goedel Require Import codeSysPrf.
+From hydras.Ackermann Require Import NNtheory.
+From hydras.Ackermann Require Import code.
+From Goedel Require Import PRrepresentable.
+From hydras.Ackermann Require Import expressible.
+From hydras.Ackermann Require Import checkPrf.
+From hydras.Ackermann Require Import codeNatToTerm.
+Import LNN NN NNnotations.
+From hydras Require Import Compat815.
+ +
+Section Rosser's_Incompleteness.
+ +
+Definition codeFormula := codeFormula (cl:=LcodeLNN).
+ +
+Variable T : System.
+ +
+Hypothesis extendsNN : Included _ NN T.
+ +
+Variable repT : Formula.
+Variable v0 : nat.
+Hypothesis
+  freeVarRepT : ∀ v : nat, In v (freeVarF repT) → v = v0.
+Hypothesis
+  expressT1 :
+    ∀ f : Formula,
+    mem _ T f →
+    SysPrf T (substF repT v0 (natToTerm (codeFormula f))).
+Hypothesis
+  expressT2 :
+    ∀ f : Formula,
+    ¬ mem _ T f →
+    SysPrf T
+      (notH (substF repT v0 (natToTerm (codeFormula f)))).
+ +
+Definition codeSysPrf :=
+  codeSysPrf LNN LcodeLNN codeArityLNTF codeArityLNNR
+    codeArityLNTFIsPR codeArityLNNRIsPR repT v0.
+ +
+Definition codeSysPrfCorrect1 :=
+  codeSysPrfCorrect1 LNN LcodeLNN codeArityLNTF
+    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNNRIsPR
+    codeArityLNNRIsCorrect1 T extendsNN T repT v0 freeVarRepT expressT1.
+ +
+Definition codeSysPrfCorrect2 :=
+  codeSysPrfCorrect2 LNN LcodeLNN codeArityLNTF
+    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNNRIsPR
+    codeArityLNNRIsCorrect1 T extendsNN T repT v0 freeVarRepT expressT2.
+ +
+Definition codeSysPrfCorrect3 :=
+  codeSysPrfCorrect3 LNN LcodeLNN codeArityLNTF
+    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1
+    codeArityLNTFIsCorrect2 codeArityLNNRIsPR codeArityLNNRIsCorrect1
+    codeArityLNNRIsCorrect2 T extendsNN.
+ +
+ +
+Definition codePrf := codePrf (cl:=LcodeLNN).
+ +
+Definition codeSysPrfNot :=
+  codeSysPrfNot LNN LcodeLNN codeArityLNTF
+    codeArityLNNR codeArityLNTFIsPR codeArityLNNRIsPR repT v0.
+ +
+Definition freeVarCodeSysPrfN :=
+  freeVarCodeSysPrfN LNN LcodeLNN codeArityLNTF
+    codeArityLNNR codeArityLNTFIsPR codeArityLNNRIsPR repT v0 freeVarRepT.
+ +
+Definition codeSysPrfNCorrect1 :=
+  codeSysPrfNCorrect1 LNN LcodeLNN codeArityLNTF
+    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNNRIsPR
+    codeArityLNNRIsCorrect1 T extendsNN T repT v0 freeVarRepT expressT1.
+ +
+Definition codeSysPrfNCorrect2 :=
+  codeSysPrfNCorrect2 LNN LcodeLNN codeArityLNTF
+    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNNRIsPR
+    codeArityLNNRIsCorrect1 T extendsNN T repT v0 freeVarRepT expressT2.
+ +
+Definition codeSysPrfNCorrect3 :=
+  codeSysPrfNCorrect3 LNN LcodeLNN codeArityLNTF
+    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1
+    codeArityLNTFIsCorrect2 codeArityLNNRIsPR codeArityLNNRIsCorrect1
+    codeArityLNNRIsCorrect2 T extendsNN
+    repT v0 freeVarRepT.
+ +
+Lemma decideAxioms :
+  (∀ x : Formula, mem _ T x ∨ ¬ mem _ T x) →
+  ∀ x : Formulas,
+    (∀ g : Formula, In g x → mem _ T g) ∨
+      (∃ g : Formula, In g x ∧ ¬ mem _ T g).
+ +
+Lemma searchProof :
+  (∀ x : Formula, mem _ T x ∨ ¬ mem _ T x) →
+  ∀ (a b : Formula) (A : Formulas) (p : Prf LNN A a),
+    (∃ B : Formulas,
+        (∃ q : Prf LNN B b,
+            codePrf _ _ q < S (codePrf _ _ p) ∧
+              (∀ x : Formula, In x B → mem _ T x))) ∨
+      (∀ (B : Formulas) (q : Prf LNN B b),
+          codePrf _ _ q < S (codePrf _ _ p) →
+          ∃ g : Formula, In g B ∧ ¬ mem _ T g).
+ +
+
+ +
+To prove the strong contructive result we need the decidability of T +
+
+ +
+Theorem Rosser'sIncompleteness :
+ (∀ x : Formula, mem _ T x ∨ ¬ mem _ T x) →
+ ∃ f : Formula,
+   (∀ v : nat, ¬ In v (freeVarF f)) ∧
+   (SysPrf T f ∨ SysPrf T (notH f) → Inconsistent LNN T).
+ +
+End Rosser's_Incompleteness.
+ +
+Definition RepresentsInSelf (T:System) :=
+  ∃ rep:Formula, ∃ v:nat,
+    (∀ x : nat, In x (freeVarF rep) → x = v) ∧
+      (∀ f : Formula,
+          mem Formula T f →
+          SysPrf T (substF rep v (natToTerm (codeFormula f)))) ∧
+      (∀ f : Formula,
+          ¬ mem Formula T f →
+          SysPrf T
+            (notH (substF rep v (natToTerm (codeFormula f))))).
+ +
+Definition DecidableSet (A:_)(s:Ensemble A) :=
+  (∀ x : A,
+      mem A s x ∨
+        ¬ mem A s x).
+ +
+Theorem Incompleteness :
+  ∀ T : System,
+    Included Formula NN T →
+    RepresentsInSelf T →
+    DecidableSet Formula T →
+    ∃ f : Formula,
+      (Sentence f) ∧
+        (SysPrf T f ∨ SysPrf T (notH f) → Inconsistent LNN T).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/Goedel.rosserPA.html b/theories/html/Goedel.rosserPA.html new file mode 100644 index 00000000..d8b962b5 --- /dev/null +++ b/theories/html/Goedel.rosserPA.html @@ -0,0 +1,222 @@ + + + + + +Goedel.rosserPA + + + + +
+ + + +
+ +

Library Goedel.rosserPA

+ +
+From Coq Require Import Ensembles.
+From Coq Require Import List.
+From Coq Require Import Arith Lia.
+From hydras.Ackermann Require Import folProp.
+From hydras.Ackermann Require Import folProof.
+From hydras.Ackermann Require Import folReplace.
+From hydras.Ackermann Require Import folLogic3.
+From hydras.Ackermann Require Import subProp.
+From hydras.Ackermann Require Import ListExt.
+From hydras.Ackermann Require Import NNtheory.
+From hydras.Ackermann Require Import NN2PA.
+From Goedel Require Import fixPoint.
+From Goedel Require Import codeSysPrf.
+From hydras.Ackermann Require Import PAtheory.
+From hydras.Ackermann Require Import code.
+From Goedel Require Import PRrepresentable.
+From hydras.Ackermann Require Import expressible.
+From hydras.Ackermann Require Import checkPrf.
+From hydras.Ackermann Require Import codeNatToTerm.
+From hydras Require Import Compat815.
+Import NNnotations.
+ +
+Section Rosser's_Incompleteness.
+ +
+  Definition codeFormula := codeFormula (cl:=LcodeLNT).
+ +
+  Variable T : System.
+  Definition T' : fol.System LNN :=
+    Union _ NN
+      (fun x : fol.Formula LNN ⇒ mem _ T (LNN2LNT_formula x)).
+ +
+Lemma extendsNN : Included _ NN T'.
+ +
+Hypothesis extendsPA : Included _ PA T.
+ +
+Variable repT : Formula.
+Variable v0 : nat.
+Hypothesis
+  freeVarRepT : ∀ v : nat, In v (freeVarF repT) → v = v0.
+Hypothesis
+  expressT1 :
+    ∀ f : Formula,
+    mem _ T f →
+    SysPrf T (substF repT v0 (natToTerm (codeFormula f))).
+Hypothesis
+  expressT2 :
+    ∀ f : Formula,
+    ¬ mem _ T f →
+    SysPrf T
+      (notH (substF repT v0 (natToTerm (codeFormula f)))).
+ +
+Lemma freeVarRepT' :
+ ∀ v : nat, In v (freeVarF (LNT2LNN_formula repT)) → v = v0.
+ +
+Lemma Tprf2T'prf :
+  ∀ f : Formula, SysPrf T f → folProof.SysPrf LNN T' (LNT2LNN_formula f).
+ +
+Lemma expressT'1 :
+ ∀ f : Formula,
+ mem _ T f →
+ folProof.SysPrf LNN T'
+   (substF (LNT2LNN_formula repT) v0
+      (natToTermLNN (codeFormula f))).
+ +
+Lemma expressT'2 :
+  ∀ f : Formula,
+    ¬ mem _ T f →
+    folProof.SysPrf LNN T'
+      (notH
+         (substF (LNT2LNN_formula repT) v0
+            (natToTermLNN (codeFormula f)))).
+ +
+Definition codeSysPrf :=
+  codeSysPrf LNT LcodeLNT codeArityLNTF codeArityLNTR
+    codeArityLNTFIsPR codeArityLNTRIsPR (LNT2LNN_formula repT) v0.
+ +
+Definition codeSysPrfCorrect1 :=
+  codeSysPrfCorrect1 LNT LcodeLNT codeArityLNTF
+    codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNTRIsPR
+    codeArityLNTRIsCorrect1 T' extendsNN T (LNT2LNN_formula repT) v0
+    freeVarRepT' expressT'1.
+ +
+Definition codeSysPrfCorrect2 :=
+  codeSysPrfCorrect2 LNT LcodeLNT codeArityLNTF
+    codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNTRIsPR
+    codeArityLNTRIsCorrect1 T' extendsNN T (LNT2LNN_formula repT) v0
+    freeVarRepT' expressT'2.
+ +
+Definition codeSysPrfCorrect3 :=
+  codeSysPrfCorrect3 LNT LcodeLNT codeArityLNTF
+    codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1
+    codeArityLNTFIsCorrect2 codeArityLNTRIsPR codeArityLNTRIsCorrect1
+    codeArityLNTRIsCorrect2 T'
+    extendsNN.
+ +
+Definition codePrf := codePrf (cl:=LcodeLNT).
+ +
+Definition codeSysPrfNot :=
+  codeSysPrfNot LNT LcodeLNT codeArityLNTF
+    codeArityLNTR codeArityLNTFIsPR codeArityLNTRIsPR
+    (LNT2LNN_formula repT) v0.
+ +
+Definition freeVarCodeSysPrfN :=
+  freeVarCodeSysPrfN LNT LcodeLNT codeArityLNTF
+    codeArityLNTR codeArityLNTFIsPR codeArityLNTRIsPR
+    (LNT2LNN_formula repT) v0 freeVarRepT'.
+ +
+Definition codeSysPrfNCorrect1 :=
+  codeSysPrfNCorrect1 LNT LcodeLNT codeArityLNTF
+    codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNTRIsPR
+    codeArityLNTRIsCorrect1 T' extendsNN T (LNT2LNN_formula repT) v0
+    freeVarRepT' expressT'1.
+ +
+Definition codeSysPrfNCorrect2 :=
+  codeSysPrfNCorrect2 LNT LcodeLNT codeArityLNTF
+    codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNTRIsPR
+    codeArityLNTRIsCorrect1 T' extendsNN T (LNT2LNN_formula repT) v0
+    freeVarRepT' expressT'2.
+ +
+Definition codeSysPrfNCorrect3 :=
+  codeSysPrfNCorrect3 LNT LcodeLNT codeArityLNTF
+    codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1
+    codeArityLNTFIsCorrect2 codeArityLNTRIsPR codeArityLNTRIsCorrect1
+    codeArityLNTRIsCorrect2 T'
+    extendsNN (LNT2LNN_formula repT) v0 freeVarRepT'.
+ +
+Lemma decideAxioms :
+  (∀ x : Formula, mem _ T x ∨ ¬ mem _ T x) →
+  ∀ x : Formulas,
+    (∀ g : Formula, In g x → mem _ T g) ∨
+      (∃ g : Formula, In g x ∧ ¬ mem _ T g).
+ +
+Lemma searchProof :
+  (∀ x : Formula, mem _ T x ∨ ¬ mem _ T x) →
+  ∀ (a b : Formula) (A : Formulas) (p : Prf LNT A a),
+    (∃ B : Formulas,
+        (∃ q : Prf LNT B b,
+            codePrf _ _ q < S (codePrf _ _ p) ∧
+              (∀ x : Formula, In x B → mem _ T x))) ∨
+      (∀ (B : Formulas) (q : Prf LNT B b),
+          codePrf _ _ q < S (codePrf _ _ p) →
+          ∃ g : Formula, In g B ∧ ¬ mem _ T g).
+ +
+Lemma T'prf2Tprf :
+  ∀ f : fol.Formula LNN,
+    folProof.SysPrf LNN T' f → SysPrf T (LNN2LNT_formula f).
+ +
+ +
+Theorem Rosser'sIncompleteness :
+  (∀ x : Formula, mem _ T x ∨ ¬ mem _ T x) →
+  ∃ f : Formula,
+    (∀ v : nat, ¬ In v (freeVarF f)) ∧
+      (SysPrf T f ∨ SysPrf T (notH f) → Inconsistent LNT T).
+ +
+End Rosser's_Incompleteness.
+ +
+From hydras.Ackermann Require Import codePA.
+From hydras.Ackermann Require Import PAconsistent.
+ +
+Theorem PAIncomplete :
+ ∃ f : Formula,
+   (Sentence f) ∧ ~(SysPrf PA f ∨ SysPrf PA (notH f)).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.AM.html b/theories/html/additions.AM.html new file mode 100644 index 00000000..47010312 --- /dev/null +++ b/theories/html/additions.AM.html @@ -0,0 +1,528 @@ + + + + + +additions.AM + + + + +
+ + + +
+ +

Library additions.AM

+ +
+
+ +
+Abstract machine for following an Euclidean addition chain adapted from the old contrib coq-additions +
+ + Work in progress + +
+ + +
+
+From Coq Require Import List PArith Relations Lia.
+Import Morphisms.
+ +
+From additions Require Import Monoid_def Monoid_instances Strategies.
+Import Monoid_def.
+ +
+From Coq Require Import Recdef Wf_nat.
+From additions Require Import More_on_positive Pow Euclidean_Chains
+  Dichotomy BinaryStrat.
+Generalizable All Variables.
+ +
+
+ +
+basic instructions +
+
+Inductive instr : Set :=
+| MUL : instr
+| SQR : instr
+| PUSH : instr
+| SWAP : instr.
+ +
+Definition code := list instr.
+ +
+ +
+Definition mults_squares c :=
+  let fix count (c: code) mults squares :=
+    match c with
+      nil ⇒ (mults, squares)
+    | PUSH :: c' | SWAP :: c' ⇒ count c' mults squares
+    | SQR :: c' ⇒ count c' mults (S squares)
+    | MUL :: c' ⇒ count c' (S mults) squares
+    end in count c 0%nat 0%nat.
+ +
+ +
+Section Semantics.
+ +
+ Variable A : Type.
+ Variable mul : A → A → A.
+ Variable one : A.
+ +
+ Definition stack := list A.
+ Definition config := (A × list A)%type.
+ +
+ Fixpoint exec (c : code) (x:A) (s: stack) : option config :=
+   match c, s with
+     nil, _ ⇒ Some (x,s)
+   | MUL::c, y::s ⇒ exec c (mul x y) s
+   | SQR::c, s ⇒ exec c (mul x x) s
+   | PUSH::c, s ⇒ exec c x (x::s)
+   | SWAP::c, y::z::s ⇒ exec c x (z::y::s)
+   | _,_ ⇒ None
+   end.
+ +
+ Lemma exec_app :
+   ∀ (c c' : code) x s ,
+     exec (c ++ c') x s =
+     match exec c x s with
+     | None ⇒ None
+     | Some (y,s') ⇒ exec c' y s'
+     end.
+ +
+ +
+
+ +
+Main well-formed chains +
+
+Definition F1 : code := nil.
+ +
+
+ +
+raises x to its cube +
+
+ +
+Definition F3 := PUSH::SQR::MUL::nil.
+ +
+
+ +
+chain for raising x to its (2 ^ q)th power +
+
+ +
+Fixpoint F2q_of_nat q := match q with
+                  | 0%nat ⇒ nil
+                  | S p ⇒ SQR:: F2q_of_nat p
+                  end.
+ +
+Definition F2q (p:positive) :=
+  F2q_of_nat (Pos.to_nat p).
+ +
+
+ +
+for computing x^(pq+r) passing by x^p +
+
+ +
+Definition KFF (kpr mq:code) : code :=
+  kpr++(mq++MUL::nil).
+ +
+
+ +
+for computing x^p and x^(pq) +
+
+ +
+Definition FFK (mp mq: code) := mp ++ PUSH :: mq.
+ +
+
+ +
+for computing x^p then x^(pq + r) +
+
+ +
+Definition KFK (kpr mq: code) :=
+  kpr ++ PUSH::SWAP :: (mq ++ MUL :: nil).
+ +
+Definition FK (fn: code) := PUSH::fn.
+ +
+End Semantics.
+ +
+Definition chain_apply c {A:Type}
+           {op:A→A→A}{one:A}{equ: Equiv A}
+           (M: EMonoid op one equ) x
+  := exec _ op c x nil.
+ +
+
+ +
+Example code for 7 via 3 +
+
+Example M7_3 := PUSH::PUSH::SQR::MUL::PUSH::SQR::SWAP::MUL::nil.
+ +
+ +
+
+ +
+Example code for 31 via 7 +
+
+Example C31_7 := KFF M7_3 (F2q 2).
+ +
+ +
+ +
+From Coq Require Import NArith.
+ +
+ +
+
+ +
+24, 3 +
+ + For 99 and 24 +
+
+Example K99_24 := KFK (FFK F3 (F2q 3)) (F2q 2).
+ +
+ +
+
+ +
+Specification and generation of correct chains +
+ + We have to build equivalences between configurations +
+
+ +
+Inductive stack_equiv {A}`{M : @EMonoid A op one equ}:
+  list A → list A → Prop
+  :=
+  stack_equiv0 : stack_equiv nil nil
+  | stack_equivn: ∀ x y s s', x == y → stack_equiv s s' →
+                                  stack_equiv (x::s) (y:: s').
+ +
+Definition config_equiv {A}`{M : @EMonoid A op one equ}
+           (c c' : config A): Prop :=
+  fst c == fst c' ∧ stack_equiv (snd c) (snd c').
+ +
+Inductive result_equiv`{M : @EMonoid A op one equ}: relation (option (config A)):=
+  result_equiv_fail : result_equiv None None
+| result_equiv_success : ∀ x s y s',
+    config_equiv (x,s) (y, s') →
+    result_equiv (Some (x,s)) (Some (y,s')).
+ +
+Definition Fchain_correct (p: positive) (c: code) :=
+  (∀ A op one equ (M: @EMonoid A op one equ) (x:A) s,
+        result_equiv (M:=M) (exec A op c x s)
+                     (Some (Pow.Pos_bpow x p, s))) .
+ +
+Definition Kchain_correct n p c :=
+   (∀ A op one equ (M: @EMonoid A op one equ) (x:A) s,
+        result_equiv (exec A op c x s)
+                     (Some (Pow.Pos_bpow x n, Pow.Pos_bpow x p ::s))).
+ +
+Definition correctness_statement (s: signature) : code → Prop :=
+match s with
+  | gen_F p ⇒ fun c ⇒ Fchain_correct p c
+  | gen_K p d ⇒ fun c ⇒ Kchain_correct (p + d) p c
+end.
+ +
+#[ global ] Instance Stack_equiv_refl {A}`{M : @EMonoid A op one equ} :
+  Reflexive stack_equiv.
+ +
+#[ global ] Instance Stack_equiv_equiv {A}`{M : @EMonoid A op one equ}:
+  Equivalence stack_equiv.
+ +
+#[ global ] Instance result_equiv_equiv `{M : @EMonoid A op one equ}:
+  Equivalence result_equiv.
+ +
+Lemma exec_equiv `{M : @EMonoid A op one equ} :
+  ∀ c x s y s' , config_equiv (x, s) (y, s') →
+                      result_equiv (exec A op c x s) (exec A op c y s').
+ +
+#[ global ] Instance exec_Proper `{M : @EMonoid A op one equ} :
+  Proper (Logic.eq ==> equiv ==> stack_equiv ==> result_equiv) (@exec A op) .
+ +
+Lemma F1_correct : Fchain_correct 1 F1.
+ +
+
+ +
+F3 is correct +
+
+ +
+Lemma F3_correct : Fchain_correct 3 F3.
+ +
+Import Pow.
+ +
+Lemma F2q_correct_0: ∀ `{M : @EMonoid A op one equ} (n:nat) x s,
+    result_equiv (exec A op (F2q_of_nat n) x s)
+                 (Some(Pow.power x (2 ^ n), s)).
+ +
+Lemma F2q_correct_1 (n:nat) : Fchain_correct (Pos.of_nat (2 ^ n))
+                                              (F2q_of_nat n).
+ +
+Remark Pos_to_nat_diff_0 n : Pos.to_nat n ≠ 0%nat.
+ +
+Lemma F2q_correct (n:positive) : Fchain_correct (2 ^ n) (F2q n).
+ +
+Section CompositionProofs.
+ +
+  Section FK.
+    Variable n : positive.
+    Variable cn : code.
+    Hypothesis Hn : Fchain_correct n cn.
+ +
+    Lemma FK_correct : Kchain_correct n 1 (FK cn).
+ +
+    End FK.
+ +
+  Section App.
+    Variables n p: positive.
+    Variables cn cp: code.
+    Hypothesis Hn : Fchain_correct n cn .
+    Hypothesis Hp : Fchain_correct p cp.
+ +
+    Lemma correct_app : Fchain_correct (n × p) (cn ++ cp).
+ +
+  End App.
+ +
+  Section FFK.
+    Variables p q: positive.
+    Variables cp cq: code.
+    Hypothesis Hp : Fchain_correct p cp.
+    Hypothesis Hq : Fchain_correct q cq.
+ +
+    Lemma FFK_correct : Kchain_correct (p × q) p (FFK cp cq).
+ +
+  End FFK.
+ +
+  Section KFK.
+    Variables p q r : positive.
+    Variables kpr mq : code.
+    Hypothesis Hpr : Kchain_correct p r kpr.
+    Hypothesis Hq : Fchain_correct q mq.
+ +
+    Lemma KFK_correct : Kchain_correct (p × q + r) p (KFK kpr mq).
+ +
+    Lemma KFF_correct: Fchain_correct (p × q + r) (KFF kpr mq).
+ +
+  End KFK.
+ +
+  End CompositionProofs.
+ +
+
+ +
+

Euclidean chain generation

+ +
+
+ +
+Definition OK (s: signature)
+  := fun c: code ⇒ correctness_statement s c.
+ +
+Section Gamma.
+ +
+Variable gamma: positive → positive.
+Context (Hgamma : Strategy gamma).
+ +
+Function chain_gen (s:signature) {measure signature_measure}
+  : code :=
+  match s with
+    gen_F i ⇒
+    if pos_eq_dec i 1 then F1 else
+      if pos_eq_dec i 3
+      then F3
+      else
+        match exact_log2 i with
+          Some p ⇒ F2q p
+        | _ ⇒
+          match N.pos_div_eucl i (Npos (gamma i))
+          with
+          | (q, 0%N) ⇒
+            (chain_gen (gen_F (gamma i))) ++
+                                          (chain_gen (gen_F (N2pos q)))
+          | (q,_r) ⇒ KFF (chain_gen
+                             (gen_K (N2pos _r)
+                                    (gamma i - N2pos _r)))
+                          (chain_gen (gen_F (N2pos q)))
+                          
+          end
+        end
+  | gen_K p d ⇒
+    if pos_eq_dec p 1 then FK (chain_gen (gen_F (1 + d)))
+    else
+      match N.pos_div_eucl (p + d) (Npos p) with
+      | (q, 0%N) ⇒ FFK (chain_gen (gen_F p))
+                          (chain_gen (gen_F (N2pos q)))
+      | (q, _r) ⇒ KFK (chain_gen (gen_K (N2pos _r)
+                                        (p - N2pos _r)))
+                      (chain_gen (gen_F (N2pos q)))
+      end
+  end.
+ +
+Defined.
+ +
+Definition make_chain (n:positive) : code :=
+ (chain_gen (gen_F n)).
+ +
+
+ +
+Proof of correctness +
+
+ +
+Lemma chain_gen_OK : ∀ s:signature, OK s (chain_gen s).
+ +
+Section All_OK.
+  Variables (n:positive).
+  Let c := chain_gen (gen_F n).
+ +
+  Lemma L0: Fchain_correct n c.
+ +
+End All_OK.
+ +
+Definition AM_power {A : Type}
+           `(M: @EMonoid A E_op E_one E_eq) (x:A) (n:positive) :=
+  exec A E_op (chain_gen (gen_F n)) x nil.
+ +
+Lemma AM_power_Ok {A : Type}
+            `(M: @EMonoid A E_op E_one E_eq) (x:A) (n:positive):
+    result_equiv (AM_power M x n) (Some (Pos_bpow x n, nil)).
+ +
+End Gamma.
+ +
+Arguments chain_gen gamma {Hgamma} _.
+ +
+ +
+ +
+ +
+Definition F197887 := chain_gen dicho (gen_F 197887).
+ +
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Addition_Chains.html b/theories/html/additions.Addition_Chains.html new file mode 100644 index 00000000..e2a878c8 --- /dev/null +++ b/theories/html/additions.Addition_Chains.html @@ -0,0 +1,1074 @@ + + + + + +additions.Addition_Chains + + + + +
+ + + +
+ +

Library additions.Addition_Chains

+ +
+ +
+
+ +
+

Addition Chains

+ +Pierre Casteran, LaBRI, University of Bordeaux + +
+ + +
+
+ +
+From additions Require Export Monoid_instances Pow.
+From Coq Require Import Relations RelationClasses Lia List.
+From Param Require Import Param.
+ +
+From additions Require Import More_on_positive.
+Generalizable Variables A op one Aeq.
+Infix "==" := Monoid_def.equiv (at level 70) : type_scope.
+Open Scope nat_scope.
+Open Scope M_scope.
+ +
+Generalizable All Variables.
+ +
+
+ +
+

Computations composed of multiplications over type A

+ + (in Continuation Passing Style) +
+
+ +
+Inductive computation {A:Type} : Type :=
+| Return (a : A)
+| Mult (x y : A) (k : A → computation).
+ +
+
+ +
+

Monadic Notation for computations

+ + +
+
+ +
+Notation "z '<---' x 'times' y ';' e2 " :=
+  (Mult x y (fun z ⇒ e2))
+    (right associativity, at level 60).
+ +
+Example comp128 : computation :=
+  x <--- 2 times 2;
+  y <--- x times 2;
+  z <--- y times y ;
+  t <--- 2 times z ;
+  Return t.
+ +
+
+ +
+

Definition

+ + +
+ +An addition chain (in short a chain) is a function that maps + any type and any value of type into a computation on . + +
+
+ +
+Definition chain := ∀ A:Type, A → @computation A.
+ +
+
+ +
+The chain associated with the empty computation + (raising to the first power) +
+
+ +
+Definition C1 : chain := (@Return).
+ +
+Example C3 : chain :=
+  fun (A:Type) (x:A) ⇒
+     x2 <--- x times x;
+     x3 <--- x2 times x;
+     Return x3.
+ +
+
+ +
+Chain associated with the 7-th power +
+
+ +
+Example C7 : chain :=
+ fun (A:Type) (x:A) ⇒
+ x2 <--- x times x;
+ x3 <--- x2 times x;
+ x6 <--- x3 times x3 ;
+ x7 <--- x6 times x;
+ Return x7.
+ +
+
+ +
+

Our Favorite example

+ + +
+ + The chain below is intented to compute 87-th power in any EMonoid. + +
+
+ +
+Example C87 : chain :=
+ fun A (x : A)=>
+  x2 <--- x times x ;
+  x3 <--- x2 times x ;
+  x6 <--- x3 times x3 ;
+  x7 <--- x6 times x ;
+  x10 <--- x7 times x3 ;
+  x20 <--- x10 times x10 ;
+  x40 <--- x20 times x20 ;
+  x80 <--- x40 times x40 ;
+  x87 <--- x80 times x7 ;
+  Return x87.
+ +
+
+ +
+

Chain length (number of mutiplications)

+ + +
+
+ +
+Fixpoint computation_length {A} (a:A) (m : @computation A) : nat :=
+match m with
+  | Mult _ _ k ⇒ S (computation_length a (k a))
+  | _ ⇒ 0%nat
+end.
+ +
+Definition chain_length (c:chain) := computation_length tt (c _ tt).
+ +
+ +
+
+ +
+

Execution of chains

+ + +
+ +Chains are designed for effectively computing powers. + +
+ +First, we define recursively the evaluation of a computation, then +the execution of a chain. + +
+
+ +
+Fixpoint computation_execute {A:Type} (op: Mult_op A)
+         (c : computation) :=
+  match c with
+    | Return x ⇒ x
+    | Mult x y k ⇒ computation_execute op (k (x × y))
+  end.
+ +
+Definition computation_eval `{M:@EMonoid A E_op E_one E_eq}
+           (c : computation) : A :=
+  computation_execute E_op c.
+ +
+Definition chain_execute (c:chain) {A} op (a:A) :=
+  computation_execute op (c A a).
+ +
+Definition chain_apply
+           (c:chain) `{M:@EMonoid A E_op E_one E_eq} a : A :=
+   computation_eval (c A a).
+ +
+Lemma computation_eval_rw `{M:@EMonoid A E_op E_one E_eq} c :
+         computation_eval c = computation_execute E_op c.
+ +
+ +
+ +
+
+ +
+

Chain correctness

+ + +
+ +In this section, we define formally the correctness of a given chain + with respect to some given exponent, +and more generally the correctness of a chain generator, i.e. a function that, +given any positive integer , returns a chain that is correct with respect to . + +
+ + +
+
+ +
+Definition chain_correct_nat (n:nat) (c: chain) := n ≠ 0 ∧
+∀ `(M:@EMonoid A E_op E_one E_eq) (x:A),
+   chain_apply c x == x ^ n.
+ +
+Definition chain_correct (p:positive) (c: chain) :=
+  chain_correct_nat (Pos.to_nat p) c.
+ +
+Definition optimal (p:positive) (c : chain) :=
+  ∀ c', chain_correct p c' →
+             (chain_length c ≤ chain_length c')%nat.
+ +
+
+ +
+A slow tactic for proving a chain's correctness +
+
+ +
+Ltac slow_chain_correct_tac :=
+  match goal with
+      [ |- chain_correct ?p ?c ] ⇒
+      let A := fresh "A" in
+      let op := fresh "op" in
+      let one := fresh "one" in
+      let eqv := fresh "eqv" in
+      let M := fresh "M" in
+      let x := fresh "x"
+      in split;[discriminate |
+                 unfold c, chain_apply, computation_eval; simpl;
+                 intros A op one eq M x; monoid_simpl M; reflexivity]
+  end.
+ +
+Example C7_ok : chain_correct 7 C7.
+ +
+
+ +
+The following proof takes a very long time. Happily, C87's correctness +will be proved more efficiently, using reflection or parametricity. + +
+ + Remove the comment if you can wait ... +
+
+ +
+ +
+
+ +
+

Correctness Proof by Reflection

+ + See chap 16 of Coq'Art +
+
+ +
+
+ +
+Binary trees of multiplications over A +
+
+ +
+Inductive Monoid_Exp (A:Type) : Type :=
+ Mul_node (t t' : Monoid_Exp A) | One_node | A_node (a:A).
+ +
+Arguments Mul_node {A} _ _.
+Arguments One_node {A} .
+Arguments A_node {A} _ .
+ +
+
+ +
+Linearization functions +
+
+ +
+Fixpoint flatten_aux {A:Type}(t fin : Monoid_Exp A)
+  : Monoid_Exp A :=
+  match t with
+    Mul_node t t' ⇒ flatten_aux t (flatten_aux t' fin)
+  | One_node ⇒ fin
+  | x ⇒ Mul_node x fin
+  end.
+ +
+Fixpoint flatten {A:Type} (t: Monoid_Exp A)
+  : Monoid_Exp A :=
+  match t with
+  | Mul_node t t' ⇒ flatten_aux t (flatten t')
+  | One_node ⇒ One_node
+  | X ⇒ Mul_node X One_node
+  end.
+ +
+ +
+
+ +
+Interpretation function +
+
+ +
+Function eval {A:Type} {op one eqv}
+         (M: @EMonoid A op one eqv)
+         (t: Monoid_Exp A) : A :=
+  match t with
+    Mul_node t1 t2 ⇒ (eval M t1 × eval M t2)%M
+  | One_node ⇒ one
+  | A_node a ⇒ a
+end.
+ +
+Lemma flatten_aux_valid `(M: @EMonoid A op one eqv):
+∀ t t', (eval M t × eval M t')%M ==
+             (eval M (flatten_aux t t')).
+ +
+Lemma flatten_valid `(M: @EMonoid A op one eqv):
+  ∀ t , eval M t == eval M (flatten t).
+ +
+Lemma flatten_valid_2 `(M: @EMonoid A op one eqv):
+∀ t t' , eval M (flatten t) == eval M (flatten t') →
+              eval M t == eval M t'.
+ +
+
+ +
+"Quote" tactic +
+
+ +
+Ltac model A op one v :=
+match v with
+| (?x × ?y)%M ⇒ let r1 := model A op one x
+                  with r2 :=(model A op one y)
+                  in constr:(@Mul_node A r1 r2)
+| one ⇒ constr:(@One_node A)
+| ?x ⇒ constr:(@A_node A x)
+end.
+ +
+Ltac monoid_eq_A A op one E_eq M :=
+match goal with
+| [ |- E_eq ?X ?Y ] ⇒
+  let tX := model A op one X with
+      tY := model A op one Y in
+      (change (E_eq (eval M tX) (eval M tY)))
+end.
+ +
+Ltac reflection_correct_tac :=
+match goal with
+[ |- chain_correct ?n ?c ] ⇒
+ split; [try discriminate |
+         let A := fresh "A"
+         in let op := fresh "op"
+         in let one := fresh "one"
+         in let E_eq := fresh "eq"
+         in let M := fresh "M"
+         in let x := fresh "x"
+         in (try unfold c); unfold chain_apply;
+           simpl; red; intros A op one E_eq M x;
+           unfold computation_eval;simpl;
+           monoid_eq_A A op one E_eq M;
+           apply flatten_valid_2;try reflexivity
+        ]
+end.
+ +
+Example C87_ok : chain_correct 87 C87. +
+Example C7_ok' : chain_correct 7 C7.
+ +
+
+ +
+

Correctness and parametricity

+ + +
+ +In this section, we show some tools for proving automatically the +correctness of a given chain, and try to avoid spending time +while proceeding to a lot of setoid rewritings + +
+ +First, we notice that any chain is able to compute its associated exponent: + +
+
+ +
+Definition the_exponent_nat (c:chain) : nat :=
+ chain_apply c (M:=Natplus) 1%nat.
+ +
+Definition the_exponent (c:chain) : positive :=
+  chain_execute c Pos.add 1%positive.
+ +
+ +
+
+ +
+ +
+ +Roughly, if cA is a computation on A and cB a computation on B, +cA and cB are related through (computation_R A B R) if, during their execution, +the corresponding variables of type A and B are always bound to related +(w.r.t. R ) values. + +
+ + +
+
+ +
+ +
+Print computation_R.
+ +
+
+ +
+We say that a chain c is parametric if + c A a and c B b are equivalent with respect to any relation that + contains the pair (a,b). + +
+
+ +
+Definition parametric (c:chain) :=
+  ∀ A B (R: A → B → Type) (a:A) (b:B),
+   R a b → computation_R A B R (c A a) (c B b).
+ +
+
+ +
+ The following statement cannot be proven in Coq (AFAIK) +
+
+ +
+Definition any_chain_parametric : Type :=
+ ∀ c:chain, parametric c.
+ +
+Goal any_chain_parametric.
+ +
+
+ +
+ +
+ + Nevertheless, if we don't want to assume any_chain_parametric, + we can use the following tactic for proving a given that a given chain c + is parametric. + +
+
+ +
+Ltac parametric_tac :=
+ match goal with [ |- parametric ?c] ⇒
+   red ; intros;
+  repeat (right;[assumption | assumption | ]); left; assumption
+end.
+ +
+Example P87 : parametric C87. +
+
+ +
+

computation of compared to computation of

+ + +
+ +The following section is dedicated to prove that, if a chain is parametric, +it computes powers of the form , where is obtained by applying the +function the_exponent_nat to . + +
+ +Basically, we proceed by an induction on the hypothesis +equiv gamma_A gamma_nat l where gamma is a computation on A, +gamma_nat a computation on nat (with respect to the additive monoid on nat), +and l is a list of pairs of the form . + +
+ + +
+
+ +
+Section Refinement_proof.
+ Variable A: Type.
+ Variable E_op : Mult_op A.
+ Variable E_one : A.
+ Variable E_eq : Equiv A.
+ Context (M : EMonoid E_op E_one E_eq).
+ +
+
+ +
+ Let us characterise the lists of pairs of the form , for a given + and . + +
+
+ +
+Definition power_R (a:A) :=
+  fun (x:A)(n:nat) ⇒ n ≠ 0 ∧ x == a ^ n.
+ +
+Remark power_R_Mult : ∀ a x y i j,
+    power_R a x i → power_R a y j →
+    power_R a (x × y) (i+j)%nat.
+ +
+Remark power_R_1 : ∀ a, power_R a a 1%nat.
+ +
+Lemma power_R_is_a_refinement (a:A) :
+  ∀(gamma : @computation A)
+        (gamma_nat : @computation nat),
+    computation_R _ _ (power_R a) gamma gamma_nat →
+     power_R a (computation_eval gamma)
+             (computation_eval (M:= Natplus) gamma_nat).
+ +
+Lemma param_correctness_aux :
+  ∀ (c:chain)(a:A),
+    computation_R A nat (power_R a ) (c A a) (c nat 1%nat) →
+     power_R a (computation_eval (c A a)) (the_exponent_nat c) .
+ +
+End Refinement_proof.
+ +
+
+ +
+

Correctness Theorem

+ + +
+ +From our study of the computation\_R predicate, we infer that any parametric chain +is correct with respect to the number the_exponent_nat c + +
+ + +
+
+ +
+Lemma exponent_nat_neq_0 : ∀ c: chain, parametric c →
+                                             the_exponent_nat c ≠ 0.
+Lemma exponent_pos2nat : ∀ c: chain,
+    parametric c →
+    the_exponent_nat c = Pos.to_nat (the_exponent c).
+ +
+Lemma exponent_pos_of_nat :
+  ∀ c: chain, parametric c →
+                    the_exponent c = Pos.of_nat (the_exponent_nat c).
+ +
+Lemma param_correctness_nat (c: chain) :
+  parametric c →
+  chain_correct_nat (the_exponent_nat c) c.
+ +
+Lemma param_correctness :
+  ∀ (p:positive) (c:chain),
+    p = the_exponent c → parametric c →
+    chain_correct p c.
+ +
+
+ +
+

Remark

+ + +
+ +If we admit that any chain is parametric, we infer the correctness of every chain. + +
+ + + +
+
+ +
+Lemma param_correctness_for_free :
+  any_chain_parametric →
+      ∀ p (c : chain) ,
+p = the_exponent c → chain_correct p c.
+ +
+
+ +
+

Back to 87

+ + +
+ +We present two new-proofs of consistency of C87. +The first one uses the parametric_tac tactic, the other one +using the "free refinement" method + +
+ + +
+
+ +
+Ltac param_chain_correct :=
+  match goal with
+    [|- chain_correct ?p ?c ] ⇒
+    let H := fresh "H" in
+    assert (p = the_exponent c) by reflexivity;
+    apply param_correctness;[trivial | parametric_tac]
+  end.
+ +
+Lemma C87_ok' : chain_correct 87 C87. +
+Lemma L87'' : any_chain_parametric → chain_correct 87 C87.
+ +
+
+ +
+

Correct by construction chains

+ +
+
+ +
+Definition chain_generator := positive → chain.
+ +
+Definition correct_generator (gen : positive → chain) :=
+ ∀ p, chain_correct p (gen p).
+ +
+
+ +
+Computing using a chain generator +
+
+ +
+Definition cpower_pos (g : chain_generator) p
+           `{M:@EMonoid A E_op E_one E_eq} a :=
+  chain_apply (g p) (M:=M) a.
+ +
+Definition cpower (g : chain_generator) n
+           `{M:@EMonoid A E_op E_one E_eq} a :=
+  match n with 0%N ⇒ E_one
+             | Npos p ⇒ cpower_pos g p a
+  end.
+ +
+
+ +
+

Definition

+ +A chain generator is optimal if it returns chains whose length are less or +equal than the chains returned by any correct generator. + +
+
+ +
+Definition optimal_generator (g : positive → chain ) :=
+ ∀ p:positive, optimal p (g p).
+ +
+
+ +
+

Some chain generators

+ + +
+ +We reinterpret the naïve and binary exponentiation algorithms in the framework +of addition chains. + +
+ +Instead of directly computing for some base and exponent , +we build chains that describe the computations associated with both methods. +Not surprisingly, this chain generation will be described in terms of recursive +functions, once the underlying monoid is fixed. + +
+ +

Chains associated to the well-known binary exponentiation algorithm

+ +
+ +

computation of

+ + +
+ +As for the "classical" binary exponentiation algorithm, +we define an auxiliary computation generator for computing the +product of an accumulator with an arbitrary power of some value + +
+ + +
+
+ +
+Fixpoint axp_scheme {A} p : A → A → @computation A :=
+  match p with
+  | xH ⇒ (fun a x ⇒ y <--- a times x ; Return y)
+  | xO q ⇒ (fun a x ⇒ x2 <--- x times x ; axp_scheme q a x2)
+  | xI q ⇒ (fun a x ⇒ ax <--- a times x ;
+            x2 <--- x times x ;
+            axp_scheme q ax x2)
+  end.
+ +
+Fixpoint bin_pow_scheme {A} (p:positive) : A → @computation A:=
+  match p with | xH ⇒ fun x ⇒ Return x
+             | xI q ⇒ fun x ⇒ x2 <--- x times x; axp_scheme q x x2
+             | xO q ⇒ fun x ⇒ x2 <--- x times x ; bin_pow_scheme q x2
+  end.
+ +
+Definition binary_chain (p:positive) : chain :=
+  fun A ⇒ bin_pow_scheme p.
+ +
+ +
+
+ +
+

Correctness of the binary method

+ +
+
+ +
+Section binary_power_proof.
+ +
+Variables (A: Type)
+         (E_op : Mult_op A)
+         (E_one : A)
+         (E_eq: Equiv A).
+ +
+Context (M : EMonoid E_op E_one E_eq).
+ +
+Existing Instance Eop_proper.
+ +
+
+ +
+ +
+ + As for linear chains, we first establish the correctness of the auxiliary + function axp_scheme. + +
+
+ +
+Lemma axp_correct : ∀ p a x,
+    computation_eval (axp_scheme p a x) ==
+    a × x ^ (Pos.to_nat p).
+ +
+Lemma binary_correct :
+  ∀ p x,
+    computation_eval (bin_pow_scheme p (A:=A) x) ==
+    x ^ (Pos.to_nat p).
+ +
+End binary_power_proof.
+ +
+Lemma binary_generator_correct : correct_generator binary_chain.
+ +
+
+ +
+

The binary method is not optimal

+ + +
+ + We use the function chain_length and proofs of correctness + for showing that binary_chain, although correct, is not an optimal + chain generator. + +
+ + Our proof is structured as a proof by contradiction, under the + hypothesis that binary_chain is optimal. + +
+
+ +
+Section non_optimality_proof.
+ +
+ Hypothesis binary_opt : optimal_generator binary_chain.
+ +
+ +
+ +
+ Lemma binary_generator_not_optimal : False. +
+End non_optimality_proof.
+ +
+Open Scope nat_scope.
+Open Scope M_scope.
+ +
+Section S1.
+ Fixpoint clog2 (p:positive) : nat :=
+ match p with xH ⇒ 1%nat
+            | xO xH ⇒ 2%nat
+            | xO p ⇒ S (clog2 p)
+            | xI p ⇒ S (clog2 p)
+  end.
+ +
+Fixpoint exp2 (n:nat) : positive :=
+  match n with O ⇒ 1
+             | S p ⇒ 2 × exp2 p
+ end.
+ Lemma exp2_Plus : ∀ n p, exp2 (n + p) = (exp2 n × exp2 p)%positive.
+ +
+Lemma axp_scheme_length1 :
+  ∀ p, (computation_length tt (axp_scheme p tt tt ) ≤ 2 × clog2 p)%nat.
+ +
+Lemma bin_pow_scheme_length1 :
+  ∀ p, (computation_length tt (bin_pow_scheme p tt ) ≤ 2 × clog2 p)%nat.
+ +
+Lemma binary_chain_length : ∀ p,
+                              (chain_length (binary_chain p) ≤ 2 × clog2 p)%nat.
+ +
+Lemma optimal_upper_bound: ∀ p c, optimal p c →
+   (chain_length c ≤ 2 × clog2 p)%nat.
+ +
+End S1.
+ +
+
+ +
+

How to prove chain correctness in general ?

+ + +
+ + In previous sections, we showed proofs of correctness of two chain generators. + By definition, every chain returned by these generators is correct w.r.t. + the given exponent. + +
+ + The problem with C87 is quite different. This chain + has been given by hand, and we had to prove its correctness. + +
+ + Although the proof script for lemma L87 is quite short, the resulting + proof is quite clumsy, consisting in a long chain of equivalences using + associativity of the multiplication. + +
+ + The reader can easily be convinced of this fact, just by the command +Print L87. + +
+ +In a similar way, it may happen that a correct chain generator is hard +to certify in . In this case, one may chose to prove the +correctness of various chains returned by the generator. + +
+ +In the rest of this section, we investigate some ways of proving +the correctness of given chains. + +
+ + +
+
+ +
+Section S2.
+
+ +
+Difficult and unfinished (conjecture) + +
+ + We would like to prove that for any parametric chain c for p, +c's length is greater or equal than log2(p). + +
+ + +
+
+ +
+Variables A B : Type.
+Variables (a:A)(b:B).
+Let R_true := fun (x:A)(y:B) ⇒ True.
+Lemma L :
+  ∀(gammaA : @computation A)
+        (gammaB : @computation B),
+    computation_R _ _ R_true gammaA gammaB →
+     @computation_length A a gammaA = @computation_length B b gammaB.
+ +
+Lemma L2 : ∀ c : chain, parametric c →
+    computation_length a (c A a) = computation_length b (c B b).
+ +
+End S2.
+ +
+Extraction Language Scheme.
+Recursive Extraction chain_apply.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.BinaryStrat.html b/theories/html/additions.BinaryStrat.html new file mode 100644 index 00000000..96137d96 --- /dev/null +++ b/theories/html/additions.BinaryStrat.html @@ -0,0 +1,69 @@ + + + + + +additions.BinaryStrat + + + + +
+ + + +
+ +

Library additions.BinaryStrat

+ +
+
+ +
+ +
+ + Binary strategy, according to Bergeron, Brlek et al. + +
+ + Let be a positive number. We associate to the half of . + +
+
+ +
+From Coq Require Import Arith NArith Lia.
+From additions Require Import Pow Compatibility More_on_positive.
+From additions Require Export Strategies.
+ +
+Open Scope positive_scope.
+ +
+Definition half (p:positive) :=
+  match p with xH ⇒ xH
+          | xI q | xO q ⇒ q
+  end.
+ +
+Definition two (p:positive) := 2%positive.
+ +
+#[ global ] Instance Binary_strat : Strategy half.
+ +
+#[ global ] Instance Two_strat : Strategy two.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Compatibility.html b/theories/html/additions.Compatibility.html new file mode 100644 index 00000000..6eb50f03 --- /dev/null +++ b/theories/html/additions.Compatibility.html @@ -0,0 +1,295 @@ + + + + + +additions.Compatibility + + + + +
+ + + +
+ +

Library additions.Compatibility

+ +
+ +
+
+ +
+

Compatibility with StdLib exponentiation functions

+ + +
+ +Function for computing powers already exist in 's standard library. We provide +equivalence theorems with functions defined in our module Pow. + +
+ + +
+
+ +
+From additions Require Import Monoid_def Monoid_instances
+       Pow.
+From Coq Require Import Lia ArithRing.
+ +
+
+ +
+

really logarithmic versions of N.pow, Pos.pow and Z.pow

+ + +
+ +We propose three functions that are extensionnaly equivalent to functions +of the standard library. +These functions are defined in the module; we just rename them for readability's sake. + +
+ + +
+
+ +
+Definition N_pow (a n: N) :=Pow.N_bpow a n.
+ +
+Definition Pos_pow (a n : positive) := Pow.Pos_bpow a n.
+ +
+Definition Z_pow (x y : Z) :=
+ match y with
+| 0%Z ⇒ 1%Z
+| Z.pos p ⇒ Pow.N_bpow x (Npos p)
+| Z.neg _ ⇒ 0%Z
+end.
+ +
+
+ +
+

Equivalence between StdLib definitions and ours

+ +
+
+ +
+Section Equivalence.
+Variable (A: Type)
+         (op : Mult_op A)
+         (one : A).
+ +
+Context (M : Monoid op one).
+Open Scope M_scope.
+ +
+Check fun x y:A ⇒ x × y.
+ +
+Ltac monoid_rw :=
+    rewrite (@one_left A op one M) ||
+    rewrite (@one_right A op one M)||
+    rewrite (@op_assoc A op one M).
+ +
+  Ltac monoid_simpl := repeat monoid_rw.
+ +
+ Ltac power_simpl := repeat (monoid_rw || rewrite <- power_of_plus).
+ +
+ Let pos_iter_M x := Pos.iter (op x).
+ +
+
+ +
+ +
+ + During an execution of the binary exponentiation algorithm + through the function binary_power_mult, + the "accumulator" acc is always a power of . Thus, even if the + considered monoid is not abelian, the accumulator commutes with any + other power of . + +
+ + +
+
+ Let is_power_of (x acc:A) := ∃ i, acc = x ^ i.
+ +
+Lemma Pos_iter_op_ok_0 :
+  ∀ p x acc, is_power_of x acc →
+                  pos_iter_M x acc p = binary_power_mult x acc p .
+ +
+Lemma Pos_iter_op_ok:
+  ∀ p x,
+    pos_iter_M x one p = binary_power_mult x one p .
+ +
+Lemma Pos_iter_ok: ∀ p x, N_bpow x (Npos p) = pos_iter_M x one p.
+ +
+End Equivalence.
+ +
+Lemma Pos_pow_power : ∀ n a : positive ,
+                        (a ^ n )%positive = power a (Pos.to_nat n).
+ +
+Lemma Npos_power_compat : ∀ (n:nat)(a:positive),
+                            Npos (power a n) = power (Npos a) n.
+ +
+Lemma N_pow_power : ∀ n a , (a ^ n )%N = power a (N.to_nat n).
+ +
+Lemma N_pow_compat : ∀ n (a:N), (a ^ n )%N = N_pow a n.
+ +
+Lemma Pos_pow_compat : ∀ n (a:positive), (a ^ n )%positive = Pos_pow a n.
+ +
+Lemma nat_power_ok : ∀ a b:nat, (a ^ b)%nat = (a ^ b)%M.
+ +
+  Lemma Pos2Nat_morph : ∀ (n:nat)(a : positive),
+     Pos.to_nat (a ^ n)%M = Pos.to_nat a ^ n.
+ +
+Lemma Z_pow_compat_pos : ∀ (p:positive) (x:Z),
+        Z.pow_pos x p = N_bpow x (Npos p).
+ +
+Lemma Z_pow_compat : ∀ x y: Z, Z_pow x y = (x ^ y)%Z.
+ +
+
+ +
+

Tests

+ + +
+ +Let us chose a big exponent, and a computation that does not create +big numbers. So, the following computation time will be proportional to + the number of recursive calls, hence to the number of multiplications + +
+ + +
+ + Time consuming ! +
+Time Compute (1 ^ 56666667)%Z. +
+ + +
+
+
+ +
+ +
+ +
+Finished transaction in 3.604 secs (3.587u,0.007s) +
+ + +
+
+ +
+
+ +
+ +
+ +
+Finished transaction in 0. secs (0.u,0.s) (successful) +
+ + +
+ +

Adapting lemmas from Standard Library

+ + +
+ +Standard library already contains many lemmas about +N.pow, Pos.pow and Z.pow. By using our extensional equivalence properties, one can + easily prove the same properties for our implementation of the same functions. + +
+ +We just give a simple example of this adaptation. + +
+ + +
+
+Section Adaptation.
+ +
+Lemma N_size_gt : ∀ n : N, (n < N_pow 2 (N.size n))%N.
+ +
+End Adaptation.
+ +
+
+ +
+Comments on Stdlib's exponentiation +
+
+ +
+Print Z.pow.
+ +
+Print Z.pow_pos.
+ +
+ +
+Eval simpl in fun (x:nat) ⇒ Pos.iter S x 12%positive.
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Demo.html b/theories/html/additions.Demo.html new file mode 100644 index 00000000..6f174284 --- /dev/null +++ b/theories/html/additions.Demo.html @@ -0,0 +1,47 @@ + + + + + +additions.Demo + + + + +
+ + + +
+ +

Library additions.Demo

+ +
+Module Alt.
+ +
+  Definition double (n:nat) := 2 × n.
+ +
+End Alt.
+ +
+From Coq Require Import Arith Lia Even.
+ +
+Lemma alt_double_ok n : Nat.double n = Alt.double n.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Demo_power.html b/theories/html/additions.Demo_power.html new file mode 100644 index 00000000..8e45eae2 --- /dev/null +++ b/theories/html/additions.Demo_power.html @@ -0,0 +1,47 @@ + + + + + +additions.Demo_power + + + + +
+ + + +
+ +

Library additions.Demo_power

+ +
+From additions Require Import Monoid_def Monoid_instances Pow.
+Open Scope M_scope.
+ +
+ +
+Import Uint63.
+Search (Z → int).
+Coercion of_Z : Z >-> int.
+ +
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Dichotomy.html b/theories/html/additions.Dichotomy.html new file mode 100644 index 00000000..2eebf686 --- /dev/null +++ b/theories/html/additions.Dichotomy.html @@ -0,0 +1,154 @@ + + + + + +additions.Dichotomy + + + + +
+ + + +
+ +

Library additions.Dichotomy

+ +
+
+ +
+ +
+ + Dichotomic strategy, according to Bergeron, Brlek et al. + +
+ + Let be a positive number. We associate to the integer + . + For instance, if , the result is 10. + +
+ + +
+
+ +
+From Coq Require Import Arith NArith.
+From additions Require Export Strategies Pow Compatibility
+  More_on_positive.
+ +
+Open Scope positive_scope.
+ +
+
+ +
+Computes +
+
+ +
+Function dicho_aux (p:positive) {struct p} : positive :=
+ match p with
+   | 1%positive ⇒ xH
+   | 2%positive | 3%positive ⇒ 2
+   | xO (xO q) | xO (xI q) | xI (xO q) | xI (xI q) ⇒
+                                         xO (dicho_aux q)
+ end.
+ +
+Definition dicho (p:positive) : positive :=
+  N2pos (N.div (Npos p) (Npos (dicho_aux p))).
+ +
+ +
+ +
+ +
+ +
+
+ +
+rewriting lemmas +
+
+ +
+Lemma dicho_aux_xOxO : ∀ p: positive,
+                          dicho_aux (xO (xO p)) = xO (dicho_aux p).
+ +
+Lemma dicho_aux_xOxI : ∀ p: positive,
+                          dicho_aux (xO (xI p)) = xO (dicho_aux p).
+ +
+Lemma dicho_aux_xIxO : ∀ p: positive,
+                          dicho_aux (xI (xO p)) = xO (dicho_aux p).
+ +
+Lemma dicho_aux_xIxI : ∀ p: positive,
+                          dicho_aux (xI (xI p)) = xO (dicho_aux p).
+ +
+Lemma dicho_aux_le_xOXO :
+  ∀ p:positive , xO (dicho_aux p) ≤ p →
+                       xO (dicho_aux (p~0~0)) ≤ p~0~0.
+ +
+Lemma dicho_aux_le_xOXI :
+  ∀ p:positive , xO (dicho_aux p) ≤ p →
+                      xO (dicho_aux (p~0~1)) ≤ p~0~1.
+ +
+Lemma dicho_aux_le_xIXO :
+  ∀ p:positive , xO (dicho_aux p) ≤ p →
+                      xO (dicho_aux (p~1~0)) ≤ p~1~0.
+ +
+Lemma dicho_aux_le_xIXI :
+  ∀ p:positive , xO (dicho_aux p) ≤ p →
+                      xO (dicho_aux (p~1~1)) ≤ p~1~1.
+ +
+#[global] Hint Resolve dicho_aux_le_xOXO dicho_aux_le_xOXI
+             dicho_aux_le_xIXO dicho_aux_le_xIXI : chains.
+ +
+Lemma dicho_aux_lt : ∀ p, 3 < p →
+                               xO (dicho_aux p) ≤ p.
+ +
+Lemma dicho_aux_gt : ∀ p, 3 < p → 1 < dicho_aux p.
+ +
+Lemma dicho_lt : ∀ p:positive, 3 < p → dicho p < p.
+ +
+Lemma dicho_gt : ∀ p:positive, 3 < p → 1 < dicho p.
+ +
+#[ global ] Instance Dicho_strat : Strategy dicho.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Euclidean_Chains.html b/theories/html/additions.Euclidean_Chains.html new file mode 100644 index 00000000..b6abcf62 --- /dev/null +++ b/theories/html/additions.Euclidean_Chains.html @@ -0,0 +1,1248 @@ + + + + + +additions.Euclidean_Chains + + + + +
+ + + +
+ +

Library additions.Euclidean_Chains

+ +
+
+ +
+

Euclidean chains

+ +
+ +

Introduction

+ + +
+ + In this module, we study a way to build efficiently efficient chains. + Our approach is recursive (compositional?). Chains associated with big exponents are built by composition of smaller chains. Thus, the construction of a +small computation may be parameterized by the context in which it will be + used. In other terms, we shall use Continuation Passing Style + +
+ + Euclidean chains are introduced by +
+
+ +
+From Coq Require Import Inverse_Image Inclusion Wf_nat
+  NArith Arith PArith Recdef Wf_nat Lexicographic_Product Lia.
+From additions Require Import Addition_Chains Compatibility
+  More_on_positive Wf_transparent Dichotomy BinaryStrat.
+From Coq Require Import Lia.
+ +
+Generalizable All Variables.
+Import Morphisms.
+Import Monoid_def.
+ +
+Ltac add_op_proper M H :=
+ let h := fresh H in
+   generalize (@Eop_proper _ _ _ _ M); intro h.
+ +
+
+ +
+

CPS chain construction

+ + +
+ + Type of chain continuations +
+
+ +
+Definition Fkont (A:Type) := A → @computation A.
+ +
+Definition Fchain := ∀ A, Fkont A → A → @computation A.
+ +
+
+ +
+F3 k x computes , then executes the computation associated + with k z +
+
+ +
+Definition F3 : Fchain :=
+ fun A k (x:A) ⇒
+  y <--- x times x ;
+  z <--- y times x ;
+  k z.
+ +
+Definition F1 : Fchain :=
+ fun A k (x:A) ⇒ k x.
+ +
+Definition F2 : Fchain :=
+fun A k (x:A) ⇒
+  y <--- x times x ;
+  k y.
+ +
+
+ +
+An Fchain f can be considered as a function that takes as + argument another chain c for continueing the computation. + +
+
+ +
+Definition Fapply (f : Fchain) (c: chain) : chain :=
+ fun (A:Type) (x: A) ⇒ f A (c A) x.
+ +
+Definition Fcompose (f1 f2: Fchain) : Fchain :=
+ fun A k x ⇒ f1 A (fun y ⇒ f2 A k y) x.
+ +
+
+ +
+Any Fchain can be transformed into a plain chain +
+
+ +
+Definition F2C (f : Fchain) : chain :=
+ fun (A:Type) ⇒ f A Return .
+ +
+ +
+
+ +
+Composition of Fchains + +
+ +Fchains are used for building correct exponentiation schemes by composition +of correct components. So, we have to define composition of Fchains. + +
+ + +
+
+ +
+Example F9 := Fcompose F3 F3.
+ +
+ +
+
+ +
+Fchains associated with powers of 2 +
+ + computes then send this value to +
+ + The neutral element for Fcompose +
+
+ +
+Lemma F1_neutral_l : ∀ f, Fcompose F1 f = f.
+ +
+Lemma F1_neutral_r : ∀ f, Fcompose f F1 = f.
+ +
+Fixpoint Fexp2_of_nat (n:nat) : Fchain :=
+ match n with O ⇒ F1
+            | S p ⇒ Fcompose F2 (Fexp2_of_nat p)
+ end.
+ +
+Definition Fexp2 (p:positive) : Fchain :=
+  Fexp2_of_nat (Pos.to_nat p).
+ +
+ +
+ +
+ +
+Remark F9_correct :chain_correct 9 (F2C F9).
+ +
+ +
+
+ +
+A first attempt to define Fchain correctness +
+
+ +
+Module Bad.
+ +
+Definition Fchain_correct (n:nat) (fc : Fchain) :=
+  ∀ A `(M : @EMonoid A op E_one E_equiv) k (a:A),
+    computation_execute op (fc A k a)==
+    computation_execute op (k (a ^ n)).
+ +
+Theorem F3_correct : Fchain_correct 3 F3. End Bad.
+ +
+ +
+
+ +
+Equivalence on computations +
+
+ +
+Definition computation_equiv {A:Type} (op: Mult_op A)
+           (equiv : Equiv A)
+           (c c': @computation A) :=
+   computation_execute op c == computation_execute op c'.
+ +
+#[ global ] Instance Comp_equiv {A:Type} (op: Mult_op A) (equiv : Equiv A):
+  Equiv (@computation A) :=
+  @computation_equiv A op equiv.
+ +
+#[ global ] Instance comp_equiv_equivalence {A:Type} (op: Mult_op A)
+           (equiv : Equiv A) : Equivalence equiv →
+                               Equivalence (computation_equiv op equiv).
+ +
+
+ +
+Fkonts that respect E_equiv +
+
+ +
+Class Fkont_proper
+      `(M : @EMonoid A op E_one E_equiv) (k: Fkont A ) :=
+  Fkont_proper_prf:
+    Proper (equiv ==> computation_equiv op E_equiv) k.
+ +
+#[ global ] Instance Return_proper `(M : @EMonoid A op E_one E_equiv) :
+  Fkont_proper M (@Return A).
+ +
+
+ +
+Fchain correctness (for exponent of type nat +
+
+ +
+Definition Fchain_correct_nat (n:nat) (f : Fchain) :=
+ ∀ A `(M : @EMonoid A op E_one E_equiv) k
+        (Hk :Fkont_proper M k)
+        (a : A) ,
+ computation_execute op (f A k a) ==
+ computation_execute op (k (a ^ n)).
+ +
+Definition Fchain_correct (p:positive) (f : Fchain) :=
+ Fchain_correct_nat (Pos.to_nat p) f.
+ +
+Lemma F1_correct : Fchain_correct 1 F1.
+ +
+Lemma F3_correct : Fchain_correct 3 F3. +
+Lemma F2_correct : Fchain_correct 2 F2.
+ +
+
+ +
+F2C preserves correctness +
+
+ +
+Lemma F2C_correct (p:positive) (fc : Fchain) :
+  Fchain_correct p fc → chain_correct p (F2C fc).
+ +
+Module Bad2.
+ +
+Lemma Fcompose_correct :
+  ∀ f1 f2 n1 n2,
+    Fchain_correct n1 f1 →
+    Fchain_correct n2 f2 →
+    Fchain_correct (n1 × n2) (Fcompose f1 f2).
+ +
+End Bad2.
+ +
+
+ +
+Fisrt attempt to define Fchain_proper +
+
+ +
+Module Bad3.
+ +
+Class Fchain_proper (fc : Fchain) := Fchain_proper_bad_prf :
+ ∀ `(M : @EMonoid A op E_one E_equiv) k ,
+    Fkont_proper M k →
+    ∀ x y, x == y →
+               @computation_equiv _ op E_equiv
+                                  (fc A k x)
+                                  (fc A k y).
+ +
+ +
+#[ global ] Instance Fcompose_proper (f1 f2 : Fchain)
+         (_ : Fchain_proper f1)
+         (_ : Fchain_proper f2) :
+  Fchain_proper (Fcompose f1 f2).
+ +
+End Bad3.
+ +
+
+ +
+Correct definition +
+
+ +
+Definition Fkont_equiv `(M : @EMonoid A op E_one E_equiv)
+ (k k': Fkont A ) :=
+ ∀ x y : A, x == y →
+                 computation_equiv op E_equiv (k x) (k' y).
+ +
+Class Fchain_proper (fc : Fchain) := Fchain_proper_prf :
+ ∀ `(M : @EMonoid A op E_one E_equiv) k k' ,
+    Fkont_proper M k → Fkont_proper M k' →
+    Fkont_equiv M k k' →
+   ∀ x y, x == y →
+               @computation_equiv _ op E_equiv
+                                  (fc A k x)
+                                  (fc A k' y).
+ +
+#[ global ] Instance F1_proper : Fchain_proper F1.
+ +
+#[ global ] Instance F3_proper : Fchain_proper F3.
+ +
+#[ global ] Instance F2_proper : Fchain_proper F2.
+ +
+
+ +
+ Fcompose respects correctness and properness +
+
+ +
+Lemma Fcompose_correct_nat : ∀ fc1 fc2 n1 n2,
+                           Fchain_correct_nat n1 fc1 →
+                           Fchain_correct_nat n2 fc2 →
+                           Fchain_proper fc2 →
+                           Fchain_correct_nat (n1 × n2)%nat
+                                              (Fcompose fc1 fc2).
+ +
+Lemma Fcompose_correct :
+  ∀ fc1 fc2 n1 n2,
+    Fchain_correct n1 fc1 →
+    Fchain_correct n2 fc2 →
+    Fchain_proper fc2 →
+    Fchain_correct (n1 × n2) (Fcompose fc1 fc2).
+ +
+#[ global ] Instance Fcompose_proper (fc1 fc2: Fchain)
+                         (_ : Fchain_proper fc1)
+                         (_ : Fchain_proper fc2) :
+  Fchain_proper (Fcompose fc1 fc2).
+ +
+#[ global ] Instance Fexp2_nat_proper (n:nat) :
+                           Fchain_proper (Fexp2_of_nat n).
+ +
+Lemma Fexp2_nat_correct (n:nat) :
+                           Fchain_correct_nat (2 ^ n)
+                                              (Fexp2_of_nat n).
+ +
+Lemma Fexp2_correct (p:positive) :
+  Fchain_correct (2 ^ p) (Fexp2 p).
+ +
+#[ global ] Instance Fexp2_proper (p:positive) : Fchain_proper (Fexp2 p).
+ +
+
+ +
+

Remark

+ + +
+ +We are now able to build chains for any exponent of the form +, using Fcompose and previous lemmas. + +
+ +Let us look at a simple example +
+
+ +
+#[global] Hint Resolve F1_correct F1_proper
+     F3_correct F3_proper Fcompose_correct Fcompose_proper
+     Fexp2_correct Fexp2_proper : chains.
+ +
+Example F144: {f : Fchain | Fchain_correct 144 f ∧
+                             Fchain_proper f}. +
+ +
+ +
+
+ +
+Bad solution +
+
+ +
+Module Bad4.
+ +
+Definition Fplus (f1 f2 : Fchain) : Fchain :=
+  fun A k x ⇒ f1 A
+                  (fun y ⇒
+                     f2 A
+                        (fun z ⇒ t <--- z times y; k t) x) x.
+ +
+Example F23 := Fplus F3 (Fplus (Fexp2 4) (Fexp2 2)).
+ +
+Lemma F23_ok : chain_correct 23 (F2C F23).
+ +
+ +
+ +
+End Bad4.
+ +
+
+ +
+Continuations with two arguments +
+
+ +
+Definition Kkont A:= A → A → @computation A.
+ +
+
+ +
+CPS chain builders for two exponents +
+
+ +
+Definition Kchain := ∀ A, Kkont A → A → @computation A.
+ +
+ +
+
+ +
+Kchain for and +
+
+ +
+Example k3_1 : Kchain := fun A (k:Kkont A) (x:A) ⇒
+  x2 <--- x times x ;
+  x3 <--- x2 times x ;
+  k x3 x.
+ +
+ +
+
+ +
+Kchain for and +
+
+ +
+Example k7_3 : Kchain := fun A (k:Kkont A) (x:A) ⇒
+  x2 <--- x times x;
+  x3 <--- x2 times x ;
+  x6 <--- x3 times x3 ;
+  x7 <--- x6 times x ;
+  k x7 x3.
+ +
+
+ +
+The Definition of correct chains and proper chains and + continuations are adapted to Kchains +
+
+ +
+Definition Kkont_proper `(M : @EMonoid A op E_one E_equiv)
+           (k : Kkont A) :=
+ Proper (equiv ==> equiv ==> computation_equiv op E_equiv) k .
+ +
+Definition Kkont_equiv `(M : @EMonoid A op E_one E_equiv)
+           (k k': Kkont A ) :=
+ ∀ x y : A, x == y → ∀ z t, z == t →
+         computation_equiv op E_equiv (k x z) (k' y t).
+ +
+
+ +
+A Kchain is correct with respect to two exponents and + if it computes and for every +
+
+ +
+About EMonoid.
+ +
+Definition Kchain_correct_nat (n p : nat) (kc : Kchain) :=
+  ∀ (A : Type) (op : Mult_op A) (E_one : A) (E_equiv : Equiv A)
+          (M : EMonoid op E_one E_equiv)
+          (k : Kkont A),
+    Kkont_proper M k →
+    ∀ (a : A) ,
+      computation_execute op (kc A k a) ==
+      computation_execute op (k (a ^ n) (a ^ p)).
+ +
+Definition Kchain_correct (n p : positive) (kc : Kchain) :=
+  Kchain_correct_nat (Pos.to_nat n) (Pos.to_nat p) kc.
+ +
+Class Kchain_proper (kc : Kchain) :=
+Kchain_proper_prf :
+ ∀ `(M : @EMonoid A op E_one E_equiv) k k' x y ,
+   Kkont_proper M k →
+   Kkont_proper M k' →
+   Kkont_equiv M k k' →
+   E_equiv x y →
+   computation_equiv op E_equiv (kc A k x) (kc A k' y).
+ +
+#[ global ] Instance k7_3_proper : Kchain_proper k7_3.
+ +
+Lemma k7_3_correct : Kchain_correct 7 3 k7_3.
+ +
+
+ +
+conversion between several definitions of correctness +
+
+ +
+Lemma Kchain_correct_conv (kc : Kchain) (n p : nat) :
+  0%nat ≠ n → 0%nat ≠ p →
+  Kchain_correct_nat n p kc →
+  Kchain_correct (Pos.of_nat n) (Pos.of_nat p) kc.
+ +
+
+ +
+

More chain combinators

+ + +
+ + Since we are working with two types of functional chains, we have to define + several ways of composing them. Each of these operators is certified to + preserve correctnes and properness +
+ + Conversion of Kchains into Fchains +
+
+ +
+Definition K2F (knp : Kchain) : Fchain :=
+  fun A (k:Fkont A) ⇒ knp A (fun y _ ⇒ k y).
+ +
+Lemma K2F_correct_nat :
+  ∀ knp n p, Kchain_correct_nat n p knp →
+                 Fchain_correct_nat n (K2F knp).
+ +
+Lemma K2F_correct :
+  ∀ kc n p, Kchain_correct n p kc →
+                 Fchain_correct n (K2F kc).
+ +
+#[ global ] Instance K2F_proper (kc : Kchain)(_ : Kchain_proper kc) :
+  Fchain_proper (K2F kc).
+ +
+
+ +
+ +
+ + Using kbr for computing and , then using Cq for + computing , then sending and to the continuation + +
+
+ +
+Definition KFK (kbr : Kchain) (fq : Fchain) : Kchain :=
+  fun A k a ⇒
+    kbr A (fun xb xr ⇒
+              fq A (fun y ⇒
+                      z <--- y times xr; k z xb) xb) a.
+ +
+Definition KFF (kbr : Kchain) (fq : Fchain) : Fchain :=
+  K2F (KFK kbr fq).
+ +
+Definition FFK (fp fq : Fchain) : Kchain :=
+  fun A k a ⇒ fp A (fun xb ⇒ fq A (fun y ⇒ k y xb) xb) a.
+ +
+Definition FK (f : Fchain) : Kchain :=
+  fun (A : Type) (k : Kkont A) (a : A) ⇒
+    f A (fun y ⇒ k y a) a.
+ +
+Example k17_7 := KFK k7_3 (Fexp2 1).
+ +
+
+ +
+In the following section, we prove that the constructions KFK and KFF + respect properness and correctness +
+
+ +
+Section KFK_proof.
+ Variables b q r: nat.
+ Variable kbr : Kchain.
+ Variable fq : Fchain.
+ Hypothesis Hbr : Kchain_correct_nat b r kbr.
+ Hypothesis Hq : Fchain_correct_nat q fq.
+ Hypothesis Hbr_prop : Kchain_proper kbr.
+ Hypothesis Hq_prop : Fchain_proper fq.
+ +
+ Lemma KFK_correct_nat : Kchain_correct_nat (b × q + r)%nat b (KFK kbr fq).
+
+ +
+simplifying the hypotheses +
+
+ +
+ Lemma KFF_correct_nat : Fchain_correct_nat (b × q + r)%nat (KFF kbr fq).
+ +
+Lemma KFK_proper : Kchain_proper (KFK kbr fq).
+ +
+#[global] Instance KFF_proper : Fchain_proper (KFF kbr fq).
+ +
+End KFK_proof.
+Lemma KFK_correct :
+  ∀ (b q r : positive) (kbr : Kchain) (fq : Fchain),
+    Kchain_correct b r kbr →
+    Fchain_correct q fq →
+    Kchain_proper kbr →
+    Fchain_proper fq →
+    Kchain_correct (b × q + r) b (KFK kbr fq).
+ +
+Check KFK_proper.
+ +
+Check KFK_proper.
+ +
+Lemma KFF_correct :
+  ∀ (b q r : positive) (kbr : Kchain) (fq : Fchain),
+    Kchain_correct b r kbr →
+    Fchain_correct q fq →
+    Kchain_proper kbr →
+    Fchain_proper fq →
+    Fchain_correct (b × q + r) (KFF kbr fq).
+ +
+Lemma FFK_correct_nat :
+  ∀ (p q : nat) (fp fq : Fchain),
+    Fchain_correct_nat p fp →
+    Fchain_correct_nat q fq →
+    Fchain_proper fp →
+    Fchain_proper fq → Kchain_correct_nat (p × q) p (FFK fp fq).
+
+ +
+simplifying the hypotheses +
+
+ +
+Lemma FFK_correct (p q : positive) (fp fq : Fchain):
+    Fchain_correct p fp →
+    Fchain_correct q fq →
+    Fchain_proper fp →
+    Fchain_proper fq →
+    Kchain_correct (p × q ) p (FFK fp fq).
+ +
+#[ global ] Instance FFK_proper
+         (fp fq : Fchain)
+         (_ : Fchain_proper fp)
+         (_ : Fchain_proper fq)
+  : Kchain_proper (FFK fp fq).
+ +
+Lemma FK_correct : ∀ (p: positive) (Fp : Fchain),
+                     Fchain_correct p Fp →
+                     Fchain_proper Fp →
+                     Kchain_correct p 1 (FK Fp).
+ +
+#[ global ] Instance FK_proper (Fp : Fchain) (_ : Fchain_proper Fp):
+  Kchain_proper (FK Fp).
+ +
+#[global] Hint Resolve KFF_correct KFF_proper KFK_correct KFK_proper : chains.
+ +
+Lemma k3_1_correct : Kchain_correct 3 1 k3_1.
+ +
+Lemma k3_1_proper : Kchain_proper k3_1.
+ +
+#[global] Hint Resolve k3_1_correct k3_1_proper : chains.
+ +
+
+ +
+an example of correct chain construction +
+
+ +
+ +
+Definition F87 :=
+ let k7_3 := KFK k3_1 (Fexp2 1) in
+ let k10_7 := KFK k7_3 F1 in
+ KFF k10_7 (Fexp2 3).
+ +
+ +
+ +
+Lemma OK87 : Fchain_correct 87 F87.
+ +
+Ltac compute_chain ch :=
+   let X := fresh "x" in
+   let Y := fresh "y" in
+   let X := constr:(ch) in
+   let Y := (eval vm_compute in X)
+   in exact Y.
+ +
+Definition C87' := ltac:( compute_chain C87 ).
+ +
+Print C87'.
+ +
+Lemma PF87: parametric C87'.
+ +
+
+ +
+

Automatic generation of correct euclidean chains

+ + +
+ +We want to define a function that builds a correct chain +for any positive exponent, using the previously defined +and certified composition operators : Fcompose, KFK, etc. + +
+ +Obviously, we have to define total mutually recursive functions: + +
+ +
    +
  • A function that builds an Fchain for any positive exponent p + +
    • +
  • A function that builds a Kchain for any pair of exponents + (n,p) where + +
    • +
+ +
+ + In Coq, various ways of building functions are available: +
    +
  • Structural mutual recursion with Fixpoint + +
    • +
  • Using Program Fixpoint + +
    • +
  • Using Function + +
    • +
+ +
+ + For simplicity's sake, we chosed to avoid dependent elimination + and used Function with a decreasing measure. + For this purpose, we define a single data-type for associated with + the generation of F- and K-chains. + +
+ +For specifying the computation of a Kchain for and +where , we use the pair of positive numbers , +thus avoiding to propagate the constraint in +our definitions. + +
+
+ +
+ +
+Inductive signature : Type :=
+| gen_F (n:positive)
+ +
+Fchain for the exponent n +
+
+| gen_K (p d: positive)
+ +
+Kchain for the exponents p+d and p
+
+.
+ +
+
+ +
+Unifying statements about chain generation +
+
+ +
+ +
+Definition signature_exponent (s:signature) : positive :=
+ match s with
+| gen_F n ⇒ n
+| gen_K p d ⇒ p + d
+end.
+ +
+Definition kont_type (s: signature)(A:Type) : Type :=
+match s with
+| gen_F _ ⇒ Fkont A
+| gen_K _ _ ⇒ Kkont A
+end.
+ +
+Definition chain_type (s: signature) : Type :=
+ match s with
+| gen_F _ ⇒ Fchain
+| gen_K _ _ ⇒ Kchain
+end.
+ +
+Definition correctness_statement (s: signature) :
+chain_type s → Prop :=
+match s with
+  | gen_F p ⇒ fun ch ⇒ Fchain_correct p ch
+  | gen_K p d ⇒ fun ch ⇒ Kchain_correct (p + d) p ch
+end.
+ +
+Definition proper_statement (s: signature) :
+chain_type s → Prop :=
+match s with
+  | gen_F _ ⇒ fun ch ⇒ Fchain_proper ch
+  | gen_K _ _ ⇒ fun ch ⇒ Kchain_proper ch
+end.
+ +
+
+ +
+

Full correctness

+ +
+
+ +
+Definition OK (s: signature)
+  := fun c: chain_type s ⇒ correctness_statement s c ∧
+                            proper_statement s c.
+ +
+ +
+#[global] Hint Resolve pos_gt_3 : chains.
+ +
+Section Gamma.
+Variable gamma: positive → positive.
+Context (Hgamma : Strategy gamma).
+ +
+Definition signature_measure (s : signature) : nat :=
+match s with
+  | gen_F n ⇒ 2 × Pos.to_nat n
+  | gen_K p d ⇒ 2 × Pos.to_nat (p + d) +1
+end.
+ +
+ +
+ Lemma PO1 :∀ (s : signature) (i : positive),
+  s = gen_F i →
+  ∀ anonymous : i ≠ 1,
+  pos_eq_dec i 1 = right anonymous →
+  ∀ anonymous0 : i ≠ 3,
+  pos_eq_dec i 3 = right anonymous0 →
+  exact_log2 i = None →
+  ∀ q r : N,
+  r = 0%N →
+  N.pos_div_eucl i (N.pos (gamma i)) = (q, 0%N) →
+  (signature_measure (gen_F (N2pos q)) < signature_measure (gen_F i))%nat.
+ +
+ Lemma PO2 : ∀ (s : signature) (i : positive),
+     s = gen_F i →
+     ∀ anonymous : i ≠ 1,
+       pos_eq_dec i 1 = right anonymous →
+       ∀ anonymous0 : i ≠ 3,
+         pos_eq_dec i 3 = right anonymous0 →
+         exact_log2 i = None →
+         ∀ q r : N,
+           r = 0%N →
+           N.pos_div_eucl i (N.pos (gamma i)) = (q, 0%N) →
+           (signature_measure (gen_F (gamma i)) < signature_measure (gen_F i))%nat.
+ +
+ Lemma PO3 : ∀ (s : signature) (i : positive),
+  s = gen_F i →
+  ∀ anonymous : i ≠ 1,
+  pos_eq_dec i 1 = right anonymous →
+  ∀ anonymous0 : i ≠ 3,
+  pos_eq_dec i 3 = right anonymous0 →
+  exact_log2 i = None →
+  ∀ (q r : N) (p : positive),
+  r = N.pos p →
+  N.pos_div_eucl i (N.pos (gamma i)) = (q, N.pos p) →
+  (signature_measure (gen_F (N2pos q)) < signature_measure (gen_F i))%nat.
+ +
+ Lemma PO4 : ∀ (s : signature) (i : positive),
+  s = gen_F i →
+  ∀ anonymous : i ≠ 1,
+  pos_eq_dec i 1 = right anonymous →
+  ∀ anonymous0 : i ≠ 3,
+  pos_eq_dec i 3 = right anonymous0 →
+  exact_log2 i = None →
+  ∀ (q r : N) (p : positive),
+  r = N.pos p →
+  N.pos_div_eucl i (N.pos (gamma i)) = (q, N.pos p) →
+  (signature_measure (gen_K (N2pos (N.pos p)) (gamma i - N2pos (N.pos p))) <
+   signature_measure (gen_F i))%nat.
+ +
+ Lemma PO6: ∀ (s : signature) (p d : positive),
+  s = gen_K p d →
+  ∀ anonymous : p ≠ 1,
+  pos_eq_dec p 1 = right anonymous →
+  ∀ q r : N,
+  r = 0%N →
+  N.pos_div_eucl (p + d) (N.pos p) = (q, 0%N) →
+  (signature_measure (gen_F (N2pos q)) < signature_measure (gen_K p d))%nat.
+ +
+Lemma PO8 :∀ (s : signature) (p d : positive),
+  s = gen_K p d →
+  ∀ anonymous : p ≠ 1,
+  pos_eq_dec p 1 = right anonymous →
+  ∀ (q r : N) (p0 : positive),
+  r = N.pos p0 →
+  N.pos_div_eucl (p + d) (N.pos p) = (q, N.pos p0) →
+  (signature_measure (gen_F (N2pos q)) < signature_measure (gen_K p d))%nat.
+ +
+ Lemma PO9 :∀ (s : signature) (p d : positive),
+  s = gen_K p d →
+  ∀ anonymous : p ≠ 1,
+  pos_eq_dec p 1 = right anonymous →
+  ∀ (q r : N) (p0 : positive),
+  r = N.pos p0 →
+  N.pos_div_eucl (p + d) (N.pos p) = (q, N.pos p0) →
+  (signature_measure (gen_K (N2pos (N.pos p0)) (p - N2pos (N.pos p0))) <
+   signature_measure (gen_K p d))%nat.
+Function chain_gen (s:signature) {measure signature_measure}
+: chain_type s :=
+  match s return chain_type s with
+    | gen_F i ⇒
+      if pos_eq_dec i 1 then F1 else
+        if pos_eq_dec i 3
+        then F3
+        else
+          match exact_log2 i with
+              Some p ⇒ Fexp2 p
+            | _ ⇒
+              match N.pos_div_eucl i (Npos (gamma i))
+              with
+                | (q, 0%N) ⇒
+                  Fcompose (chain_gen (gen_F (gamma i)))
+                            (chain_gen (gen_F (N2pos q)))
+                | (q,_r) ⇒ KFF (chain_gen
+                                   (gen_K (N2pos _r)
+                                          (gamma i - N2pos _r)))
+                                (chain_gen (gen_F (N2pos q)))
+              end end
+    | gen_K p d ⇒
+      if pos_eq_dec p 1 then FK (chain_gen (gen_F (1 + d)))
+      else
+        match N.pos_div_eucl (p + d) (Npos p) with
+          | (q, 0%N) ⇒ FFK (chain_gen (gen_F p))
+                              (chain_gen (gen_F (N2pos q)))
+          | (q, _r) ⇒ KFK (chain_gen (gen_K (N2pos _r)
+                                            (p - N2pos _r)))
+                          (chain_gen (gen_F (N2pos q)))
+        end
+  end.
+ +
+Definition make_chain (n:positive) : chain :=
+ F2C (chain_gen (gen_F n)).
+ +
+Lemma chain_gen_OK : ∀ s:signature,
+    OK s (chain_gen s).
+ +
+Theorem make_chain_correct : ∀ p, chain_correct p (make_chain p).
+ +
+End Gamma.
+Arguments make_chain gamma {_} _ _ _ .
+ +
+ +
+Require Import Int63.Uint63.
+Require Import Monoid_instances.
+ +
+Check cpower (make_chain dicho) 10.
+Module Examples.
+Search (int → Z).
+Search (positive → int).
+ +
+Definition fast_int63_power (x :positive)(n:N) : Z :=
+  to_Z (cpower (make_chain dicho) n (of_pos x)).
+ +
+Definition slow_int31_power (x :positive)(n:N) : Z :=
+  to_Z (power (of_pos x) (N.to_nat n) ).
+ +
+Definition binary_int31_power (x :positive)(n:N) : Z :=
+  to_Z (N_bpow (of_pos x) n).
+ +
+
+ +
+long computations ... +
+
+ +
+Definition big_chain := ltac:(compute_chain (make_chain dicho 45319)).
+ +
+Print big_chain.
+ +
+Arguments big_chain _%type_scope.
+ +
+Remark RM : (1 < 56789)%N.
+ +
+Definition M := Nmod_Monoid 56789%N RM.
+ +
+Definition exp56789 x := chain_apply big_chain (M:=M) x.
+ +
+ +
+Eval cbv iota match delta [big_chain chain_apply computation_eval ] zeta beta in fun x ⇒ chain_apply big_chain (M:=M) x.
+ +
+Definition C87' := ltac:( compute_chain C87 ).
+ +
+ +
+Goal parametric (make_chain dicho 45319).
+ +
+Remark big_correct :chain_correct 45319 (make_chain dicho 45319).
+ +
+Remark big_correct' : chain_correct 453 (make_chain dicho 453).
+ +
+ +
+Remark big_correct''' : chain_correct 453 (make_chain dicho 453).
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+End Examples.
+ +
+Require Import Extraction.
+Locate exp56789.
+Extraction Language OCaml.
+Extraction "bigmod" Examples.exp56789.
+ +
+Recursive Extraction cpower.
+Recursive Extraction make_chain.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Fib2.html b/theories/html/additions.Fib2.html new file mode 100644 index 00000000..3cd6ed10 --- /dev/null +++ b/theories/html/additions.Fib2.html @@ -0,0 +1,153 @@ + + + + + +additions.Fib2 + + + + +
+ + + +
+ +

Library additions.Fib2

+ +
+ +
+From Coq Require Import NArith Ring.
+ +
+From additions Require Import Monoid_instances Euclidean_Chains Pow
+        Strategies Dichotomy BinaryStrat.
+Import Addition_Chains.
+Open Scope N_scope.
+ +
+Fixpoint fib (n:nat) : N :=
+  match n with
+    0%nat | 1%nat ⇒ 1
+  | (S ((S p) as q)) ⇒ fib p + fib q
+  end.
+ +
+ +
+Lemma fib_ind (P:nat→Prop) :
+  P 0%nat → P 1%nat → (∀ n, P n → P (S n) → P(S (S n))) →
+  ∀ n, P n.
+ +
+Lemma fib_SSn : ∀ (n:nat) , fib (S (S n)) = (fib n + fib (S n)).
+ +
+
+ +
+Yves' encoding +
+
+ +
+Definition mul2 (p q : N × N) :=
+  match p, q with
+    (a, b),(c,d) ⇒ (a×c + a×d + b×c, a×c + b×d)
+  end.
+ +
+Lemma neutral_l p : mul2 (0,1) p = p.
+ +
+Lemma neutral_r p : mul2 p (0,1) = p.
+ +
+#[ global ] Instance Mul2 : Monoid mul2 (0,1).
+ +
+ +
+Lemma next_fib (n:nat) : mul2 (1,0) (fib (S n), fib n) =
+                         (fib (S (S n)), fib (S n)).
+ +
+Definition fib_mul2 n := let (a,b) := power (M:=Mul2) (1,0) n
+                         in (a+b).
+ +
+ +
+Lemma fib_mul2_OK_0 (n:nat) :
+  power (M:=Mul2) (1,0) (S (S n)) =
+  (fib (S n), fib n).
+ +
+Lemma fib_mul2_OK n : fib n = fib_mul2 n.
+ +
+ +
+Definition fib_pos n :=
+  let (a,b) := Pos_bpow (M:= Mul2) (1,0) n in
+  (a+b).
+ +
+ +
+ +
+Locate chain_apply.
+About chain_apply.
+ +
+Definition fib_eucl gamma `{Hgamma: Strategy gamma} n :=
+  let c := make_chain gamma n
+  in let r := chain_apply c (M:=Mul2) (1,0) in
+       fst r + snd r.
+ +
+ +
+ +
+From additions Require Import AM.
+ +
+Definition fib_with_chain c :=
+  match chain_apply c Mul2 (1,0) with
+    Some ((a,b), nil) ⇒ Some (a+b) | _ ⇒ None end.
+ +
+Definition c153 := chain_gen dicho (gen_F 153%positive).
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.FirstSteps.html b/theories/html/additions.FirstSteps.html new file mode 100644 index 00000000..51f1cc11 --- /dev/null +++ b/theories/html/additions.FirstSteps.html @@ -0,0 +1,270 @@ + + + + + +additions.FirstSteps + + + + +
+ + + +
+ +

Library additions.FirstSteps

+ +
+
+ +
+Polymorphic versions of exponentiation functions +
+
+ +
+From Coq Require Import Arith ZArith String.
+ +
+
+ +
+ +
+ + Polymorphic exponentiation functions + +
+
+ +
+Section Definitions.
+ +
+ Variables (A : Type)
+           (mult : A → A → A)
+           (one : A).
+ +
+#[local] Infix "*" := mult.
+#[local] Notation "1" := one.
+ +
+
+ +
+ Naive (linear) implementation +
+
+ +
+Fixpoint power (x:A)(n:nat) : A :=
+  match n with
+    | 0%nat ⇒ 1
+    | S p ⇒ x × x ^ p
+  end
+where "x ^ n" := (power x n).
+ +
+
+ +
+Logarithmic implementation (with exponents in N) +
+
+ +
+ +
+Fixpoint binary_power_mult (x a:A)(p:positive) : A
+  :=
+  match p with
+    | xH ⇒ a × x
+    | xO q ⇒ binary_power_mult (x × x) a q
+    | xI q ⇒ binary_power_mult (x × x) (a × x) q
+  end.
+ +
+Fixpoint Pos_bpow (x:A)(p:positive) :=
+ match p with
+  | xH ⇒ x
+  | xO q ⇒ Pos_bpow (x × x) q
+  | xI q ⇒ binary_power_mult (x × x) x q
+end.
+ +
+Definition N_bpow x (n:N) :=
+  match n with
+  | 0%N ⇒ 1
+  | Npos p ⇒ Pos_bpow x p
+  end.
+ +
+ +
+End Definitions.
+ +
+Arguments N_bpow {A}.
+Arguments power {A}.
+ +
+
+ +
+

Examples

+ +
+
+ +
+ +
+ +
+Open Scope string_scope.
+ +
+ +
+ +
+
+ +
+Exponentiation on 2x2 matrices +
+
+ +
+ +
+Module M2.
+Section M2_Definitions.
+ +
+  Variables (A: Type)
+           (zero one : A)
+           (plus mult : A → A → A).
+ +
+  Variable rt : semi_ring_theory zero one plus mult (@eq A).
+  Add Ring Aring : rt.
+ +
+  Notation "0" := zero.
+  Notation "1" := one.
+  Notation "x + y" := (plus x y).
+  Notation "x * y " := (mult x y).
+ +
+  Structure t : Type := mat{c00 : A; c01 : A;
+                            c10 : A; c11 : A}.
+ +
+  Definition Id2 : t := mat 1 0 0 1.
+ +
+ +
+  Definition M2_mult (M M':t) : t :=
+    mat (c00 M × c00 M' + c01 M × c10 M')
+        (c00 M × c01 M' + c01 M × c11 M')
+        (c10 M × c00 M' + c11 M × c10 M')
+        (c10 M × c01 M' + c11 M × c11 M').
+ +
+End M2_Definitions.
+End M2.
+ +
+Import M2.
+ +
+Arguments M2_mult {A} plus mult _ _.
+Arguments mat {A} _ _ _ _.
+Arguments Id2 {A} _ _.
+ +
+Definition fibonacci (n:N) :=
+ c00 N (N_bpow (M2_mult Nplus Nmult) (Id2 0%N 1%N)(mat 1 1 1 0)%N n).
+ +
+ +
+Definition power_t := ∀ (A:Type)
+                             (mult : A → A → A)
+                             (one:A)
+                             (x:A)
+                             (n:N), A.
+ +
+
+ +
+

A wrong definition of correctness

+ +
+
+ +
+Module Bad.
+ +
+  Definition correct_expt_function (f : power_t) : Prop :=
+    ∀ A (mult : A → A → A) (one:A)
+           (x:A) (n:N), power mult one x (N.to_nat n) =
+                        f A mult one x n.
+ +
+  Section CounterExample.
+    Let mul (n p : nat) := n + 2 × p.
+    Let one := 0.
+ +
+
+ +
+With our fake definition, N_bpow is not correct! +
+
+ +
+    Remark mul_not_associative :
+      ∃ n p q, mul n (mul p q) ≠ mul (mul n p) q.
+ +
+    Remark one_not_neutral :
+      ∃ n : nat, mul one n ≠ n.
+ +
+    Lemma correct_exp_too_strong : ¬ correct_expt_function (@N_bpow).
+ +
+  End CounterExample.
+ +
+End Bad.
+ +
+
+ +
+Fibonacci matrices +
+
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Monoid_def.html b/theories/html/additions.Monoid_def.html new file mode 100644 index 00000000..b048ba05 --- /dev/null +++ b/theories/html/additions.Monoid_def.html @@ -0,0 +1,292 @@ + + + + + +additions.Monoid_def + + + + +
+ + + +
+ +

Library additions.Monoid_def

+ +
+From Coq Require Import RelationClasses Relations Morphisms String.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+

The Monoid type class (with Operational Type Classes)

+ +
+
+ +
+Declare Scope M_scope.
+ +
+Class Mult_op (A:Type) := mult_op : A → A → A.
+ +
+Print Mult_op.
+ +
+Print mult_op.
+ +
+ +
+Goal ∀ A (op: Mult_op A), @mult_op A op = op.
+ +
+Delimit Scope M_scope with M.
+Infix "×" := mult_op : M_scope.
+Open Scope M_scope.
+ +
+ +
+Module Demo.
+ +
+  #[local] Instance nat_mult_op : Mult_op nat := Nat.mul.
+ +
+  Set Printing All.
+ +
+  Check 3 × 4.
+ +
+   Unset Printing All.
+ +
+ +
+End Demo.
+ +
+ +
+#[ global ] Instance string_op : Mult_op string := append.
+Open Scope string_scope.
+ +
+Example ex_string : "ab" × "cde" = "abcde".
+ +
+ +
+#[ global ] Instance bool_and_binop : Mult_op bool := andb.
+ +
+Example ex_bool : true × false = false.
+ +
+
+ +
+within M_scope, a term of the form (x * y) is an abbreviation of +(mult_op A op x y) where op : Mult_op A and x, y : A. + +
+
+ +
+Class Monoid {A:Type}(op : Mult_op A)(one : A) : Prop :=
+{
+    op_assoc : ∀ x y z, x × (y × z) = x × y × z;
+    one_left : ∀ x, one × x = x;
+    one_right : ∀ x, x × one = x
+}.
+ +
+
+ +
+

Exercice

+ + +
+ +Define a class for semi-groups, and re-define monoids as semi-groups with a neutral element + +
+ + +
+ +

Monoids and Equivalence Relations

+ + +
+ +In some situations, the previous definition may be too restrictive. +For instance, consider the computation of where + and are positive integers, and . + +
+ +Although it could possible to compute with values of the dependent +type {n:N | n < m}, it looks simpler to compute with numbers of type +N, and consider the multiplication . + +
+ +It is easy to prove that this operation is associative, using library + . Unfortunately, it is not possible to prove the +following proposition, +required for building an instance of Monoid: + +
+ +
+∀ x:N, 1 × x mod m = x. +
+ + +
+ +Thus, we define a more general class, parameterized by an equivalence +relation Aeq on type A, compatible with the multiplication . +The laws of associativity and neutral element +are not expressed as Leibniz equalities but as equivalence statements: + +
+
+ +
+Class Equiv A := equiv : relation A.
+ +
+Infix "==" := equiv (at level 70) : type_scope.
+ +
+ +
+Class EMonoid (A:Type)(E_op : Mult_op A)(E_one : A)
+      (E_eq: Equiv A): Prop :=
+  {
+    Eq_equiv :> Equivalence equiv;
+    Eop_proper : Proper (equiv ==> equiv ==> equiv) E_op;
+    Eop_assoc : ∀ x y z, x × (y × z) == x × y × z;
+    Eone_left : ∀ x, E_one × x == x;
+    Eone_right : ∀ x, x × E_one == x
+  }.
+ +
+#[ global ] Instance Equiv_Equiv (A:Type)(E_op : Mult_op A)(E_one : A)
+      (E_eq: Equiv A)(M :EMonoid E_op E_one E_eq) :
+   Equivalence E_eq.
+Qed.
+ +
+#[ global ] Instance Equiv_Refl (A:Type)(E_op : Mult_op A)(E_one : A)
+      (E_eq: Equiv A)(M :EMonoid E_op E_one E_eq) :
+   Reflexive E_eq.
+Qed.
+ +
+#[ global ] Instance Equiv_Sym (A:Type)(E_op : Mult_op A)(E_one : A)
+      (E_eq: Equiv A)(M :EMonoid E_op E_one E_eq) :
+   Symmetric E_eq.
+Qed.
+ +
+#[ global ] Instance Equiv_Trans (A:Type)(E_op : Mult_op A)(E_one : A)
+      (E_eq: Equiv A)(M :EMonoid E_op E_one E_eq) :
+   Transitive E_eq.
+Qed.
+ +
+Generalizable All Variables.
+ +
+
+ +
+

Coercion from Monoid to EMonoid

+ + +
+ +Every instance of class Monoid can be transformed into an instance of +EMonoid, considering Leibniz' equality eq. + +
+
+ +
+#[global] Instance eq_equiv {A} : Equiv A := eq.
+ +
+#[global] Instance Monoid_EMonoid `(M:@Monoid A op one) :
+        EMonoid op one eq_equiv.
+ +
+
+ +
+We can now register Monoid_EMonoid as a coercion: + +
+
+ +
+Coercion Monoid_EMonoid : Monoid >-> EMonoid.
+ +
+
+ +
+

Commutative Monoids

+ + +
+ +The following type class definitions allow to take advantage of + the possible commutativity of the operation + +
+ + +
+
+ +
+Class Abelian_EMonoid `(M:@EMonoid A op one Aeq ):= {
+  Eop_comm : ∀ x y, op x y == op y x}.
+ +
+Class Abelian_Monoid `(M:Monoid ):= {
+  op_comm : ∀ x y, op x y = op y x}.
+ +
+Ltac add_op_proper M H :=
+ let h := fresh H in
+   generalize (@Eop_proper _ _ _ _ M); intro h.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Monoid_instances.html b/theories/html/additions.Monoid_instances.html new file mode 100644 index 00000000..d2c24b63 --- /dev/null +++ b/theories/html/additions.Monoid_instances.html @@ -0,0 +1,372 @@ + + + + + +additions.Monoid_instances + + + + +
+ + + +
+ +

Library additions.Monoid_instances

+ +
+
+ +
+

Some useful instances of Monoid classes

+ + +
+ +File +
+
+ +
+Require Export Monoid_def.
+Require Import RelationClasses Morphisms.
+ +
+Require Import ZArith PArith.
+Require Import Arith.
+Require Import NArith.
+Require Import Ring63.
+ +
+Open Scope Z_scope.
+ +
+
+ +
+

Multiplicative monoid on Z

+ +
+
+ +
+#[ global ] Instance Z_mult_op : Mult_op Z := Z.mul.
+ +
+#[ global ] Instance ZMult : Monoid Z_mult_op 1. +
+#[ global ] Instance ZMult_Abelian : Abelian_Monoid ZMult.
+ +
+
+ +
+

Multiplicative monoid on nat

+ +
+
+ +
+ +
+#[ global ] Instance nat_mult_op : Mult_op nat | 5 := Nat.mul.
+ +
+#[ global ] Instance Natmult : Monoid nat_mult_op 1%nat | 5.
+ +
+
+ +
+

Additive monoid on nat

+ + +
+ +The following monoid is useful for proving the correctness of complex +exponentiation algorithms. In effect, the -th "power" of is +equal to . See Sect. +
+
+ +
+#[ global ] Instance nat_plus_op : Mult_op nat | 12 := Nat.add.
+ +
+#[ global ] Instance Natplus : Monoid nat_plus_op 0%nat | 12.
+ +
+ +
+Open Scope N_scope.
+ +
+#[ global ] Instance N_mult_op : Mult_op N | 5 := N.mul.
+ +
+#[ global ] Instance NMult : Monoid N_mult_op 1 | 5.
+ +
+Check NMult : EMonoid N.mul 1%N eq.
+ +
+#[ global ] Instance N_plus_op : Mult_op N | 12 := N.add.
+ +
+#[ global ] Instance NPlus : Monoid N_plus_op 0 | 12.
+ +
+
+ +
+Multiplicative Monoid on positive + +
+
+ +
+#[ global ] Instance P_mult_op : Mult_op positive | 5 := Pos.mul .
+ +
+#[ global ] Instance PMult : Monoid P_mult_op xH | 5.
+ +
+Import Uint63.
+Open Scope int63_scope.
+ +
+
+ +
+

Multiplicative monoid on 63-bits integers

+ + +
+ +Cyclic numeric types are good candidates for testing exponentiations +with big exponents, since the size of data is bounded. + +
+ +The type int63 is defined in Standard Library in Module +Coq.Numbers.Cyclic.Int63.Uint63. + +
+
+ +
+#[ global ] Instance int63_mult_op : Mult_op int := mul.
+ +
+#[ global ] Instance Int63mult : Monoid int63_mult_op 1.
+ +
+Module Bad.
+ +
+  Fixpoint int63_from_nat (n:nat) :int :=
+    match n with
+    | O ⇒ 1
+    | S p ⇒ 1 + int63_from_nat p
+    end.
+ +
+  Coercion int63_from_nat : nat >-> int.
+ +
+  Fixpoint fact (n:nat) : int := match n with
+                             O ⇒ 1
+                           | S p ⇒ n × fact p
+                           end.
+ +
+End Bad.
+ +
+Close Scope int63_scope.
+ +
+
+ +
+

Monoid of 2x2 matrices

+ + +
+ +Let be some type, provided with a ring structure. We define the multiplication +of -matrices, the coefficients of which have type . + +
+ + +
+
+ +
+ +
+Section M2_def.
+Variables (A:Type)
+           (zero one : A)
+           (plus mult : A → A → A).
+ +
+ Notation "0" := zero.
+ Notation "1" := one.
+ Notation "x + y" := (plus x y).
+ Notation "x * y " := (mult x y).
+ +
+ Variable rt : semi_ring_theory zero one plus mult (@eq A).
+ Add Ring Aring : rt.
+ +
+Structure M2 : Type := {c00 : A; c01 : A;
+                        c10 : A; c11 : A}.
+ +
+Definition Id2 : M2 := Build_M2 1 0 0 1.
+ +
+Definition M2_mult (m m':M2) : M2 :=
+ Build_M2 (c00 m × c00 m' + c01 m × c10 m')
+          (c00 m × c01 m' + c01 m × c11 m')
+          (c10 m × c00 m' + c11 m × c10 m')
+          (c10 m × c01 m' + c11 m × c11 m').
+ +
+ +
+Lemma M2_eq_intros : ∀ a b c d a' b' c' d',
+  a=a' → b=b' → c=c' → d=d' →
+   Build_M2 a b c d = Build_M2 a' b' c' d'.
+ +
+ +
+#[global] Instance M2_op : Mult_op M2 := M2_mult.
+ +
+#[global] Instance M2_Monoid : Monoid M2_op Id2.
+ +
+End M2_def.
+ +
+Arguments M2_Monoid {A zero one plus mult} rt.
+Arguments Build_M2 {A} _ _ _ _.
+ +
+
+ +
+Matrices over N +
+
+Definition M2N := M2_Monoid Nth.
+ +
+
+ +
+

Integers modulo m

+ + +
+ +The following instance of EMonoid describes the set of integers modulo +, where is some integer greater or equal than . +For simplicity's sake, we represent such values using the type N, +and consider "equivalence modulo " instead of equality. + +
+
+ +
+Section Nmodulo.
+  Variable m : N.
+  Hypothesis m_gt_1 : 1 < m.
+ +
+  Remark m_neq_0 : m ≠ 0.
+ +
+  #[local] Hint Resolve m_neq_0 : chains.
+ +
+  Definition mult_mod (x y : N) := (x × y) mod m.
+  Definition mod_eq (x y: N) := x mod m = y mod m.
+ +
+  Instance mod_equiv : Equiv N := mod_eq.
+ +
+  Instance mod_op : Mult_op N := mult_mod.
+ +
+  Instance mod_Equiv : Equivalence mod_equiv.
+ +
+  #[global] Instance mult_mod_proper :
+    Proper (mod_equiv ==> mod_equiv ==> mod_equiv) mod_op.
+ +
+  #[local] Open Scope M_scope.
+ +
+  Lemma mult_mod_associative : ∀ x y z,
+      x × (y × z) = x × y × z.
+ +
+  Lemma one_mod_neutral_l : ∀ x, 1 × x == x.
+  Lemma one_mod_neutral_r : ∀ x, x × 1 == x.
+ +
+  #[global] Instance Nmod_Monoid : EMonoid mod_op 1 mod_equiv.
+ +
+End Nmodulo.
+ +
+Section S256.
+ +
+  Let mod256 := mod_op 256.
+ +
+  #[local] Existing Instance mod256 | 1.
+ +
+ +
+ +
+ +
+End S256.
+ +
+ +
+ +
+Close Scope N_scope.
+Close Scope positive_scope.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.More_on_positive.html b/theories/html/additions.More_on_positive.html new file mode 100644 index 00000000..38555bc0 --- /dev/null +++ b/theories/html/additions.More_on_positive.html @@ -0,0 +1,287 @@ + + + + + +additions.More_on_positive + + + + +
+ + + +
+ +

Library additions.More_on_positive

+ +
+ +
+From Coq Require Import Arith NArith Lia Recdef.
+From additions Require Import Pow Compatibility Wf_transparent.
+Open Scope positive_scope.
+Import Monoid_def.
+ +
+
+ +
+

Basic lemmas about positive and N

+ +
+
+ +
+Definition pos_eq_dec : ∀ p p':positive, {p = p'}+{p ≠ p'}.
+ +
+Lemma N_0_le_n: ∀ n:N, (0 ≤ n)%N.
+ +
+Lemma Pos_to_nat_neq_0 : ∀ p, Pos.to_nat p ≠ 0%nat.
+ +
+#[global] Hint Resolve Pos_to_nat_neq_0 : chains.
+ +
+
+ +
+

Relationship with nat and N

+ + +
+
+Lemma Npos_diff_zero : ∀ p, N.pos p ≠ 0%N.
+ +
+Lemma Npos_gt_0 : ∀ p, (0 < N.pos p)%N.
+ +
+#[global] Hint Resolve Npos_diff_zero Npos_gt_0 : chains.
+ +
+Lemma pos2N_inj_lt : ∀ n p, (n < p)%positive ↔ (N.pos n < N.pos p)%N.
+ +
+Lemma pos2N_inj_add : ∀ n p, N.pos (n + p) = (N.pos n + N.pos p)%N.
+ +
+Ltac pos2nat_inj_tac :=
+  repeat (rewrite Pos2Nat.inj_add || rewrite Pos2Nat.inj_mul ||
+          rewrite Pos2Nat.inj_lt || rewrite Pos2Nat.inj_le).
+ +
+Lemma Pos2Nat_le_1_p : ∀ p, (1 ≤ Pos.to_nat p)%nat.
+ +
+Lemma N_le_1_pos : ∀ p, (1 ≤ N.pos p)%N.
+ +
+Lemma pos_le_mul : ∀ p q , (p ≤ p × q)%positive.
+ +
+Lemma pos_lt_mul : ∀ p q , (1 < q → p < p × q)%positive.
+ +
+Lemma Pos2Nat_le_n_pn :
+  ∀ p q,
+    (Pos.to_nat p ≤ Pos.to_nat p × Pos.to_nat q)%nat.
+ +
+#[global] Hint Resolve Pos2Nat_le_1_p : chains.
+ +
+
+ +
+

Surjection from N into positive

+ + +
+
+ +
+Definition N2pos (n:N) : positive :=
+ match n with 0%N ⇒ xH | Npos p ⇒ p end.
+ +
+Lemma N2pos_pos :
+  ∀ i, N2pos (Npos i) = i.
+ +
+Lemma N_pos_N2pos : ∀ n, 0%N ≠ n → n = Npos (N2pos n).
+ +
+Lemma N2pos_lt_switch : ∀ n p, (0%N < n)%N →
+                                     ( (N.pos p < n)%N ↔ (p < N2pos n)%positive).
+ +
+Ltac N2pos_simpl x := simpl (N2pos (N.pos x)) in ×.
+ +
+Ltac N2pos_destruct t y :=
+ destruct t as [| y] ; [try (discriminate || contradiction) | N2pos_simpl y].
+ +
+Lemma N2pos_lt_switch2 : ∀ n p, (0%N < n)%N →
+                                      ((N2pos n < p)%positive
+                                       ↔ (n < N.pos p)%N).
+ +
+Lemma pos_lt_wf : well_founded Pos.lt.
+ +
+
+ +
+Partial exact log2 function +
+
+ +
+Fixpoint exact_log2(p:positive) : option positive :=
+match p with
+  | 1%positive | xI _ ⇒ None
+  | 2%positive ⇒ Some xH
+  | xO q ⇒ match exact_log2 q with
+              | Some l ⇒ Some (l+1)%positive
+              | _ ⇒ None
+            end
+end.
+ +
+
+ +
+Compute exact_log2 16. + = Some 4 + : option positive + +
+ +Compute exact_log2 10. += None + : option positive + +
+
+ +
+Lemma exact_log2xOx0 :
+  ∀ p i, exact_log2 (xO p) = Some i →
+              exact_log2 (xO (xO p)) = Some (i+1)%positive.
+ +
+Lemma exact_log2_spec :
+  ∀ p i: positive, exact_log2 p = Some i → p = (2 ^ i)%positive.
+ +
+
+ +
+Another induction principle for positive +
+
+ +
+Lemma positive_4step_ind : ∀ P : positive → Prop,
+   P 1%positive → P 2%positive → P 3%positive →
+  (∀ p, P p → P (xO (xO p)) ∧ P (xI (xO p)) ∧ P (xO (xI p)) ∧
+               P (xI (xI p))) →
+  ∀ p, P p.
+ +
+Lemma pos_gt_3 : ∀ p:positive,
+  p ≠ 1 → p ≠ 3 → exact_log2 p = None → 3 < p.
+ +
+#[global] Hint Resolve pos_gt_3 : chains.
+ +
+
+ +
+

Lemmas on Euclidean division

+ + N.pos_div_eucl (a:positive) (b:N) : N * N + +
+
+ +
+Lemma pos_div_eucl_quotient_pos : ∀ a b q r,
+                                    N.pos_div_eucl a b = (q, r) →
+                                    (b ≤ N.pos a)%N →
+                                    b ≠ 0%N →
+                                    (q ≠ 0%N).
+ +
+Lemma pos_div_eucl_quotient_lt : ∀ a b q r,
+                                   N.pos_div_eucl a b = (q, r) →
+                                   (1 < b)%N →
+                                   (q < N.pos a)%N.
+ +
+Lemma N_pos_div_eucl_divides : ∀ i b q,
+                                 N.pos_div_eucl i (N.pos b) = (q, 0%N) →
+                                 (b × N2pos q)%positive = i.
+ +
+ Lemma N_pos_div_eucl_rest : ∀ i b q r,
+                               N.pos_div_eucl i (N.pos b) = (q, r) →
+                               (0 < r)%N → (0 < q)%N →
+                               (b × N2pos q + N2pos r)%positive = i.
+ +
+ Lemma N_pos_div_eucl_q0 : ∀ i b r,
+                             N.pos_div_eucl i (N.pos b) = (0%N, r) →
+                             i = N2pos r.
+ +
+
+ +
+An auxiliary lemma +
+
+Lemma lt_S_2i :
+∀ i j:nat, (i < j → 2 × i + 1 < 2 × j)%nat.
+ +
+Lemma N_le_mul_pos : ∀ q p, (q ≤ q × N.pos p)%N.
+ +
+Ltac quotient_small div_equation H :=
+  match type of div_equation with
+    (N.pos_div_eucl ?a ?b = (?q,?r)) ⇒
+    assert (H : (q < N.pos a)%N);
+    [apply (pos_div_eucl_quotient_lt _ _ _ _ div_equation); auto|]
+  end.
+ +
+Ltac rest_small div_equation H :=
+  match type of div_equation with
+    (N.pos_div_eucl ?a ?b = (?q,?r)) ⇒
+    let H0 := fresh "H" in
+    assert (H : (r < b)%N);
+    [generalize (N.pos_div_eucl_remainder a b); simpl; intro H0;
+     rewrite div_equation in H0; apply H0 ; try discriminate| ]
+  end.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Naive.html b/theories/html/additions.Naive.html new file mode 100644 index 00000000..be9685dc --- /dev/null +++ b/theories/html/additions.Naive.html @@ -0,0 +1,249 @@ + + + + + +additions.Naive + + + + +
+ + + +
+ +

Library additions.Naive

+ +
+ +
+
+ +
+

Naïve and Monomorphic Functions for Computing Powers.

+ +
+ +

Exponentiation of integers

+ + +
+ +Let us define a function Z.power : Z → nat → Z by structural recursion +on the exponent n: + +
+
+ +
+Require Import ZArith.
+Open Scope Z_scope.
+ +
+Module Z.
+ +
+Fixpoint power (x:Z)(n:nat) :=
+match n with
+| 0%nat ⇒ 1
+| S p ⇒ x × power x p
+end.
+ +
+Compute power 2 10.
+
+ +
+ + +
+ +

Some basic properties of Z.power

+ + +
+ + +
+
+ +
+Lemma power_S : ∀ (x:Z)(n:nat), power x (S n) = x × power x n.
+ +
+Lemma power_of_plus (x:Z) :
+  ∀ n p, power x (n + p) = power x n × power x p.
+ +
+End Z.
+ +
+
+ +
+

Exponentiation modulo m

+ + +
+ + Let m be some natural number. We can compute for any number x +and exponent n the number . + +
+ + Since m and x can be arbitrary large numbers, we give them the type +N of binary natural number. Following our naïve approach, the exponent n +is still of type nat. + +
+ + +
+
+ +
+Module N_mod.
+#[local] Open Scope N_scope.
+ +
+Section m_fixed.
+ +
+  Variable m: N.
+ +
+  Definition mult_mod (x y : N) := (x × y) mod m.
+ +
+  Fixpoint power (x: N) (n : nat) : N :=
+    match n with
+      | 0%nat ⇒ 1 mod m
+      | S p ⇒ mult_mod x (power x p)
+    end.
+End m_fixed.
+End N_mod.
+ +
+Compute N_mod.power 14555553%N 5689%N 27.
+ +
+
+ +
+ +
+   = 9086410
+     : N +
+ + + +
+ +

Exponentiation of matrices

+ + +
+ +Our last example is a definition of where is a matrix +over any scalar type , assuming one can provide with a semi-ring structure. + +
+ + +
+
+Module M2.
+Section Definitions.
+ +
+  Variables (A: Type)
+           (zero one : A)
+           (plus mult : A → A → A).
+ +
+  Variable rt : semi_ring_theory zero one plus mult (@eq A).
+  Add Ring Aring : rt.
+ +
+  Notation "0" := zero.
+  Notation "1" := one.
+  Notation "x + y" := (plus x y).
+  Notation "x * y " := (mult x y).
+ +
+  Structure t : Type := mat{c00 : A; c01 : A;
+                            c10 : A; c11 : A}.
+ +
+  Definition Id2 : t := mat 1 0 0 1.
+ +
+  Definition M2_mult (M M':t) : t :=
+    mat (c00 M × c00 M' + c01 M × c10 M')
+        (c00 M × c01 M' + c01 M × c11 M')
+        (c10 M × c00 M' + c11 M × c10 M')
+        (c10 M × c01 M' + c11 M × c11 M').
+ +
+  Infix "**" := M2_mult (at level 40, left associativity).
+ +
+  Fixpoint power (M : t) (n : nat) : t :=
+    match n with
+      | 0%nat ⇒ Id2
+      | S p ⇒ M ** (power M p)
+    end.
+ +
+ +
+
+ +
+

+ + The ring tactic, applied to inhabitants of type A, allows us to + prove associativity of matrix multiplication, then a law of + distributivity of power upon ** + +
+
+ +
+  Lemma Id2_neutral : ∀ M:t, Id2 ** M = M.
+ +
+  Lemma M2_mult_assoc : ∀ M M' M'':t, M ** (M' ** M'') =
+                                            (M ** M') ** M''.
+ +
+  Lemma power_of_plus (M:t) :
+  ∀ n p, power M (n + p) = power M n ** power M p.
+ +
+End Definitions.
+End M2.
+ +
+Definition fibonacci (n:nat) :=
+  M2.c00 N (M2.power N 0%N 1%N Nplus Nmult (M2.mat _ 1 1 1 0)%N n).
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Pow.html b/theories/html/additions.Pow.html new file mode 100644 index 00000000..16585b84 --- /dev/null +++ b/theories/html/additions.Pow.html @@ -0,0 +1,469 @@ + + + + + +additions.Pow + + + + +
+ + + +
+ +

Library additions.Pow

+ +
+
+ +
+

Exponentiation in a Monoid

+ + +
+ +In this section, we give two polymorphic functions for computing + : the naive ( linear) one and the aforementionned +binary method, that takes less than multiplications. + +
+ + Both functions require an instance of EMonoid. Their code use the + multiplication of the monoid, and sometimes its unity. + Correctness proofs require the "axioms" of monoid structure. + +
+
+ +
+Set Implicit Arguments.
+ +
+Require Export ZArith Div2.
+Require Export Recdef.
+Require Import Relations Morphisms.
+ +
+Require Import Monoid_def.
+Require Import Arith Lia.
+ +
+Open Scope M_scope.
+ +
+
+ +
+

Two functions for computing powers

+ + +
+ + The module defines two functions for exponentiation on any Emonoid + on carrier . + +
+ +
    +
  • The function power has type A → nat → A; it is linear with respect to + the exponent. Its simplicity and the fact that the exponent has type nat + make it adequate for being the reference for any other definition, and for + easily proving laws like . + + +
    + + +
    • +
  • The function Pos_bpow has type A → positive → A and is logarithmic + with respect to its exponent. This function should be used for effective + computations. Its variant N_bpow allows the exponent to be . + +
    • +
+ +
+ + +
+ +

The "naive" reference function

+ +
+
+Generalizable Variables A E_op E_one E_eq.
+ +
+Fixpoint power `{M: @EMonoid A E_op E_one E_eq}
+         (x:A)(n:nat) :=
+  match n with 0%nat ⇒ E_one
+             | S p ⇒ x × x ^ p
+  end
+where "x ^ n" := (power x n) : M_scope.
+ +
+ +
+Lemma power_eq1 `{M: @EMonoid A E_op E_one E_eq}(x:A) :
+  x ^ 0 = E_one.
+ +
+Lemma power_eq2 `{M: @EMonoid A E_op E_one E_eq}(x:A) (n:nat) :
+ x ^ (S n) = x × x ^ n.
+ +
+Lemma power_eq3 `{M: @EMonoid A E_op E_one E_eq}(x:A) :
+ x ^ 1 == x.
+ +
+
+ +
+

The binary exponentiation function (exponents in positive)

+ +
+ +

+ + +
+ +The auxiliary function below computes , where +the "accumulator" acc is intented to be an already computed power of x: + +
+ + +
+
+ +
+Fixpoint binary_power_mult `{M: @EMonoid A E_op E_one E_eq}
+         (x a:A)(p:positive) : A
+  :=
+  match p with
+    | xH ⇒ a × x
+    | xO q ⇒ binary_power_mult (x × x) a q
+    | xI q ⇒ binary_power_mult (x × x) (a × x) q
+  end.
+ +
+
+ +
+

+ + +
+ +The following function decomposes the exponent p + into , then calls binary_power_mult + with and . + +
+ + +
+
+ +
+Fixpoint Pos_bpow `{M: @EMonoid A E_op E_one E_eq}
+         (x:A)(p:positive) :=
+ match p with
+  | xH ⇒ x
+  | xO q ⇒ Pos_bpow (x × x) q
+  | xI q ⇒ binary_power_mult (x × x) x q
+end.
+ +
+
+ +
+*** + It is straightforward to adapt Pos_bpow + for accepting exponents of type N : + +
+
+ +
+Definition N_bpow `{M: @EMonoid A E_op E_one E_eq} x (n:N) :=
+  match n with
+  | 0%N ⇒ E_one
+  | Npos p ⇒ Pos_bpow x p
+  end.
+ +
+Infix "^b" := N_bpow (at level 30, right associativity) : M_scope.
+ +
+
+ +
+

Properties of the power function

+ + +
+ + Taking power as a reference, it remains to prove two kinds of properties +
    +
  • Mathematical properties of exponentiation, i.e the function power, + +
    + + +
    • +
  • proving correctness of functions Pos_bpow and N_bpow + +
    • +
+ +
+ +First, let us consider some Emonoid and define some useful notations and tactics: + +
+
+ +
+Section M_given.
+ +
+  Variables (A:Type) (E_one:A) .
+  Context (E_op : Mult_op A) (E_eq : Equiv A)
+          (M:EMonoid E_op E_one E_eq).
+ +
+#[global] Instance Eop_proper : Proper (equiv ==> equiv ==> equiv) E_op.
+ +
+Ltac monoid_rw :=
+    rewrite Eone_left ||
+    rewrite Eone_right ||
+    rewrite Eop_assoc.
+ +
+Ltac monoid_simpl := repeat monoid_rw.
+ +
+ +
+#[global] Instance power_proper :
+  Proper (equiv ==> eq ==> equiv) power.
+ +
+Lemma power_of_plus :
+  ∀ x n p, x ^ (n + p) == x ^ n × x ^ p.
+ +
+Ltac power_simpl := repeat (monoid_rw || rewrite <- power_of_plus).
+ +
+Lemma power_commute x n p:
+  x ^ n × x ^ p == x ^ p × x ^ n.
+ +
+Lemma power_commute_with_x x n:
+  x × x ^ n == x ^ n × x.
+ +
+Lemma power_of_power x n p:
+  (x ^ n) ^ p == x ^ (p × n).
+ +
+Lemma power_of_power_comm x n p : (x ^ n) ^ p == (x ^ p) ^ n.
+ +
+Lemma sqr_eqn : ∀ x, x ^ 2 == x × x.
+ +
+Ltac factorize := repeat (
+                rewrite <- power_commute_with_x ||
+                rewrite <- power_of_plus ||
+                rewrite <- sqr_eqn ||
+                rewrite <- power_eq2 ||
+                rewrite power_of_power).
+ +
+Lemma power_of_square x n : (x × x) ^ n == x ^ n × x ^ n.
+ +
+
+ +
+

Correctness of the binary algorithm

+ + +
+ +Correctness of the "concrete" functions Pos_bpow and N_bpow +with respect to the more abstract function power is expressed +by extensional equalities, taking into account the conversion between +various representations of natural numbers. + +
+ + +
+
+ +
+Lemma binary_power_mult_ok :
+  ∀ p a x, binary_power_mult x a p == a × x ^ Pos.to_nat p.
+ +
+Lemma Pos_bpow_ok :
+  ∀ p x, Pos_bpow x p == x ^ Pos.to_nat p.
+ +
+#[global] Instance Pos_bpow_proper :
+  Proper (equiv ==> eq ==> equiv) Pos_bpow.
+ +
+Lemma N_bpow_ok :
+  ∀ n x, x ^b n == x ^ N.to_nat n.
+ +
+Lemma N_bpow_ok_R :
+  ∀ n x, x ^b (N.of_nat n) == x ^ n.
+ +
+Lemma Pos_bpow_ok_R :
+  ∀ p x, p ≠ 0 →
+              Pos_bpow x (Pos.of_nat p) == x ^ p.
+ +
+Lemma N_bpow_commute : ∀ x n p,
+                        x ^b n × x ^b p ==
+                        x ^b p × x ^b n.
+ +
+Lemma Pos_bpow_of_plus : ∀ x n p, Pos_bpow x (n + p)%positive ==
+                                       Pos_bpow x n × Pos_bpow x p.
+ +
+Lemma Pos_bpow_of_bpow : ∀ (x:A) n p,
+    Pos_bpow (Pos_bpow x n) p == Pos_bpow x (p × n)%positive.
+ +
+
+ +
+

Remark

+ + +
+ +If we normalize exponentiation functions with a given exponent, we notice +that the obtained functions do not execute the same computations, but it is +hard to visualize why the binary method is more efficient than the naive one. + +
+ + +
+
+ +
+Eval simpl in fun (x:A) ⇒ x ^b 17.
+ +
+Eval simpl in fun x ⇒ x ^ 17.
+ +
+ +
+Definition pow_17 (x:A) :=
+  let x2 := x × x in
+  let x4 := x2 × x2 in
+  let x8 := x4 × x4 in
+  let x16 := x8 × x8 in
+  x16 × x.
+ +
+ +
+Eval cbv zeta beta delta [pow_17] in pow_17.
+ +
+
+ +
+ = fun x : A => + x * x * (x * x) * (x * x * (x * x)) * + (x * x * (x * x) * (x * x * (x * x))) * x + : A -> A + +
+ + +
+ +In order to compare the real computations needed to raise some to its -th +power, we need to make more explicit how intermediate values are used during +some computation. +This is described in the module (see ). + +
+ + +
+ +

Properties of Abelian Monoids

+ + +
+ + Some equalities hold in the restricted context of abelian (a.k.a. commutative) + monoids. + +
+
+ +
+Section Power_of_op.
+ Context {AM:Abelian_EMonoid M}.
+ +
+Theorem power_of_mult :
+   ∀ n x y, (x × y) ^ n == x ^ n × y ^ n.
+ +
+End Power_of_op.
+ +
+End M_given.
+ +
+Infix "^" := power : M_scope.
+ +
+ +
+Ltac monoid_simpl M := generalize (Eop_proper M); intro;
+  repeat ( rewrite (Eone_left ) ||
+    rewrite (Eone_right ) ||
+    rewrite (Eop_assoc )).
+ +
+  Ltac power_simpl M := generalize (Eop_proper M); intro;
+  repeat ( rewrite Eone_left || rewrite Eone_right || rewrite Eop_assoc
+     || rewrite power_of_plus).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Pow_variant.html b/theories/html/additions.Pow_variant.html new file mode 100644 index 00000000..02c0cc0b --- /dev/null +++ b/theories/html/additions.Pow_variant.html @@ -0,0 +1,477 @@ + + + + + +additions.Pow_variant + + + + +
+ + + +
+ +

Library additions.Pow_variant

+ +
+
+ +
+

Exponentiation in a Monoid

+ + +
+ +In this section, we give two polymorphic functions for computing + : the naive ( linear) one and the aforementionned +binary method, that takes less than multiplications. + +
+ + Both functions require an instance of EMonoid. Their code use the multiplication of the monoid, and sometimes its unity. Correctness proofs require +the "axioms" of monoid structure. + +
+
+ +
+Set Implicit Arguments.
+ +
+Require Export ZArith Div2.
+Require Export Recdef.
+Require Import Relations Morphisms.
+ +
+Require Import Monoid_def.
+Require Import Arith Lia.
+ +
+Generalizable Variables A op one Aeq.
+ +
+Open Scope M_scope.
+ +
+
+ +
+

Two functions for computing powers

+ + +
+ + The module defines two functions for exponentiation on any Emonoid + on carrier . + +
+ +
    +
  • The function power has type A → nat → A; it is linear with respect to + the exponent. Its simplicity and the fact that the exponent has type nat + make it adequate for being the reference for any other definition, and for + easily proving laws like . + + +
    + + +
    • +
  • The function Pos_bpow has type A → positive → A and is logarithmic + with respect to its exponent. This function should be used for effective + computations. Its variant N_bpow allows the exponent to be . + +
    • +
+ +
+ + +
+ +

The "naive" reference function

+ +
+
+Generalizable Variables E_op E_one E_eq.
+ +
+Fixpoint power `{M: @EMonoid A E_op E_one E_eq}(x:A)(n:nat) :=
+  match n with 0%nat ⇒ E_one
+             | S p ⇒ x × x ^ p
+  end
+where "x ^ n" := (power x n) : M_scope.
+ +
+Lemma power_eq1 {A:Type} `{M: @EMonoid A E_op E_one E_eq}(x:A) :
+  x ^ 0 = E_one.
+ +
+Lemma power_eq2 {A:Type} `{M: @EMonoid A E_op E_one E_eq}(x:A) (n:nat) :
+ x ^ (S n) = x × x ^ n.
+ +
+Lemma power_eq3 {A:Type} `{M: @EMonoid A E_op E_one E_eq}(x:A) :
+ x ^ 1 == x.
+ +
+
+ +
+

The binary exponentiation function (exponents in positive)

+ +
+ +

+ + +
+ +The auxiliary function below computes , where +the "accumulator" acc is intented to be an already computed power of x: + +
+ + +
+
+ +
+Fixpoint binary_power_mult `{M: @EMonoid A E_op E_one E_eq}
+         (x a:A)(p:positive) : A
+  :=
+  match p with
+    | xH ⇒ a × x
+    | xO q ⇒ binary_power_mult (x × x) a q
+    | xI q ⇒ binary_power_mult (x × x) (a × x) q
+  end.
+ +
+
+ +
+

+ + +
+ +The following function decomposes the exponent p + into , then calls binary_power_mult + with and . + +
+ + +
+
+ +
+Fixpoint Pos_bpow `{M: @EMonoid A E_op E_one E_eq}
+         (x:A)(p:positive) :=
+ match p with
+  | xH ⇒ x
+  | xO q ⇒ Pos_bpow (x × x) q
+  | xI q ⇒ binary_power_mult (x × x) x q
+end.
+ +
+
+ +
+*** + It is straightforward to adapt Pos_bpow + for accepting exponents of type N : + +
+
+ +
+Definition N_bpow `{M: @EMonoid A E_op E_one E_eq} x (n:N) :=
+  match n with
+  | 0%N ⇒ E_one
+  | Npos p ⇒ Pos_bpow x p
+  end.
+ +
+Infix "^b" := N_bpow (at level 30, right associativity) : M_scope.
+ +
+
+ +
+

Properties of the power function

+ + +
+ + Taking power as a reference, it remains to prove two kinds of properties +
    +
  • Mathematical properties of exponentiation, i.e the function power, + +
    + + +
    • +
  • proving correctness of functions Pos_bpow and N_bpow + +
    • +
+ +
+ +First, let us consider some Emonoid and define some useful notations and tactics: + +
+
+ +
+Section M_given.
+ +
+ Variables (A:Type) (E_op : Mult_op A)(E_one:A) (E_eq : Equiv A).
+ Context (M:EMonoid E_op E_one E_eq).
+ +
+#[global] Instance Eop_proper : Proper (equiv ==> equiv ==> equiv) E_op.
+Qed.
+ +
+Ltac monoid_rw :=
+    rewrite Eone_left ||
+    rewrite Eone_right ||
+    rewrite Eop_assoc .
+ +
+Ltac monoid_simpl := repeat monoid_rw.
+ +
+ +
+Section About_power.
+ +
+#[global] Instance power_proper : Proper (equiv ==> eq ==> equiv) power.
+ +
+Lemma power_of_plus : ∀ x n p, x ^ (n + p) == x ^ n × x ^ p.
+ +
+Ltac power_simpl := repeat (monoid_rw || rewrite <- power_of_plus).
+ +
+Lemma power_commute : ∀ x n p,
+                        x ^ n × x ^ p == x ^ p × x ^ n.
+ +
+Lemma power_commute_with_x : ∀ x n, x × x ^ n == x ^ n × x.
+ +
+Lemma power_of_power : ∀ x n p, (x ^ n) ^ p == x ^ (p × n).
+ +
+Lemma sqr_eqn : ∀ x, x ^ 2 == x × x.
+ +
+Ltac factorize := repeat (
+                rewrite <- power_commute_with_x ||
+                rewrite <- power_of_plus ||
+                rewrite <- sqr_eqn ||
+                rewrite <- power_eq2 ||
+                rewrite power_of_power).
+ +
+Lemma power_of_square : ∀ x n, (x × x) ^ n == x ^ n × x ^ n.
+ +
+
+ +
+

Correctness of the binary algorithm

+ + +
+ +Correctness of the "concrete" functions Pos_bpow and N_bpow +with respect to the more abstract function power is expressed +by extensional equalities, taking into account the conversion between +various representations of natural numbers. + +
+ + +
+
+ +
+Lemma binary_power_mult_ok :
+  ∀ p a x, binary_power_mult x a p == a × x ^ Pos.to_nat p.
+ +
+Lemma Pos_bpow_ok :
+  ∀ (p:positive)(x:A), Pos_bpow x p == x ^ Pos.to_nat p.
+ +
+Lemma N_bpow_ok :
+∀ (x:A) (n:N), x ^b n == x ^ N.to_nat n.
+ +
+Lemma N_bpow_ok_R :
+  ∀ (x:A) (n:nat), x ^b (N.of_nat n) == x ^ n.
+ +
+Lemma Pos_bpow_ok_R :
+  ∀ (x:A) (n:nat), n ≠ 0 →
+                      Pos_bpow x (Pos.of_nat n) == x ^ n.
+ +
+End About_power.
+ +
+Lemma N_bpow_commute : ∀ x n p,
+                        x ^b n × x ^b p ==
+                        x ^b p × x ^b n.
+ +
+
+ +
+

Remark

+ + +
+ +Iw we normalize exponentiation functions with a given exponent, we notice +that the obtained functions do not execute the same computations, but it is +hard to visualize why the binary method is more efficient than the na\u00efve one. + +
+ + +
+
+ +
+Eval simpl in fun (x:A) ⇒ x ^b 17.
+
+ +
+[ + = fun x : A ⇒ + x × + (x × x × (x × x) × (x × x × (x × x)) × + (x × x × (x × x) × (x × x × (x × x)))) + : A → A +] + +
+
+Eval simpl in fun x ⇒ x ^ 17.
+ +
+
+ +
+
+= fun x : A ⇒
+       x × (x × (x × (x × (x × (x × (x × (x ×
+           (x × (x × (x × (x × (x × (x × (x × (x × (x × one))))))))))))))))
+     : A → A +
+ + +
+
+ +
+ +
+Definition pow_17 (x:A) :=
+  let x2 := x × x in
+  let x4 := x2 × x2 in
+  let x8 := x4 × x4 in
+  let x16 := x8 × x8 in
+  x16 × x.
+ +
+Eval cbv zeta beta delta [pow_17] in pow_17.
+
+ +
+ = fun x : A => + x * x * (x * x) * (x * x * (x * x)) * + (x * x * (x * x) * (x * x * (x * x))) * x + : A -> A + +
+ + +
+ +In order to compare the real computations needed to raise some to its -th +power, we need to make more explicit how intermediate values are used during +some computation. +This is described in the module (see ). + +
+ + +
+ +

Properties of Abelian Monoids

+ + +
+ + Some equalities hold in the restricted context of abelian (a.k.a. commutative) + monoids. + +
+
+ +
+Section Power_of_op.
+ Context {AM:Abelian_EMonoid M}.
+ +
+Theorem power_of_mult :
+   ∀ n x y, (x × y) ^ n == x ^ n × y ^ n.
+ +
+End Power_of_op.
+ +
+End M_given.
+ +
+Infix "^" := power : M_scope.
+ +
+ +
+Ltac monoid_simpl M := generalize (Eop_proper M); intro;
+  repeat ( rewrite (Eone_left ) ||
+    rewrite (Eone_right ) ||
+    rewrite (Eop_assoc )).
+ +
+  Ltac power_simpl M := generalize (Eop_proper M); intro;
+  repeat ( rewrite Eone_left || rewrite Eone_right || rewrite Eop_assoc
+     || rewrite power_of_plus).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Strategies.html b/theories/html/additions.Strategies.html new file mode 100644 index 00000000..cb439291 --- /dev/null +++ b/theories/html/additions.Strategies.html @@ -0,0 +1,69 @@ + + + + + +additions.Strategies + + + + +
+ + + +
+ +

Library additions.Strategies

+ +
+
+ +
+A strategy is any function of type positive → positive +such that, if , then . + +
+ + +
+
+ +
+Require Import Arith NArith Pow Compatibility More_on_positive.
+Open Scope positive_scope.
+ +
+Open Scope positive_scope.
+Class Strategy (gamma : positive → positive):=
+  {
+  gamma_lt :∀ p:positive, 3 < p → gamma p < p;
+  gamma_gt : ∀ p:positive, 3 < p → 1 < gamma p
+  }.
+ +
+Ltac gamma_bounds gamma i H1 H2 :=
+  assert (H1 : 1 < gamma i) by (apply gamma_gt;auto with chains);
+  assert (H2 : gamma i < i) by (apply gamma_lt; auto with chains).
+ +
+Lemma div_gamma_pos {gamma}{Hgamma : Strategy gamma}
+: ∀ (p:positive) q r,
+    N.pos_div_eucl p (N.pos (gamma p)) = (q, r) →
+    3 < p →
+    (0 < q)%N.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Trace_exercise.html b/theories/html/additions.Trace_exercise.html new file mode 100644 index 00000000..16228079 --- /dev/null +++ b/theories/html/additions.Trace_exercise.html @@ -0,0 +1,99 @@ + + + + + +additions.Trace_exercise + + + + +
+ + + +
+ +

Library additions.Trace_exercise

+ +
+
+ +
+Solution to an exercise +
+
+ +
+From Coq Require Import List.
+Import ListNotations.
+Require Import Addition_Chains PArith.
+ +
+Fixpoint fusion {A} (compare : A → A → comparison)(l1 l2 : list A) :=
+      let fix aux l2 {struct l2}:=
+          match l1,l2 with
+            | nil,_ ⇒ l2
+            | _,nil ⇒ l1
+            | a1::l1',a2::l2' ⇒
+              match compare a1 a2 with
+                Lt ⇒ a2 :: aux l2'
+              | Eq ⇒ a1 :: fusion compare l1' l2'
+              | Gt ⇒ a1 :: fusion compare l1' l2
+              end
+          end
+      in aux l2.
+ +
+Open Scope positive_scope.
+ +
+
+ +
+Traces with full information +
+
+ +
+Inductive info : Set :=
+  Init
+| Add (p q : positive).
+ +
+Definition trace_compare (t t' : positive × info) :=
+  match t, t' with
+    (x,_),(y,_) ⇒ Pos.compare x y end.
+ +
+Definition trace_mult (l l' : list (positive × info)):=
+    match l, l' with
+    nil, _ | _,nil ⇒ l
+  | ((x,_)::l1),((y,_)::l'1) ⇒ (x+y,Add x y):: fusion trace_compare l l'
+  end.
+ +
+Definition chain_trace c :=
+  List.rev (chain_execute c trace_mult ((1,Init)::nil)).
+ +
+Definition exponents c := List.map fst (chain_trace c).
+ +
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.Wf_transparent.html b/theories/html/additions.Wf_transparent.html new file mode 100644 index 00000000..cf65c156 --- /dev/null +++ b/theories/html/additions.Wf_transparent.html @@ -0,0 +1,69 @@ + + + + + +additions.Wf_transparent + + + + +
+ + + +
+ +

Library additions.Wf_transparent

+ +
+
+ +
+Transparent versions of wf_incl and wf_inverse_image + +
+
+ +
+From Coq Require Export Relation_Definitions.
+ +
+Lemma wf_incl_transparent :
+∀ (A : Type) (R1 R2 : A → A → Prop),
+Relation_Definitions.inclusion A R1 R2 → well_founded R2 → well_founded R1.
+ +
+Section Inverse_Image_transp. +
+  Variables A B : Type.
+  Variable R : B → B → Prop.
+  Variable f : A → B.
+ +
+  Let Rof (x y:A) : Prop := R (f x) (f y).
+ +
+  Remark Acc_lemma : ∀ y:B, Acc R y → ∀ x:A, y = f x → Acc Rof x.
+ +
+  Lemma Acc_inverse_image : ∀ x:A, Acc R (f x) → Acc Rof x.
+ +
+  Theorem wf_inverse_image_transparent : well_founded R → well_founded Rof.
+ +
+End Inverse_Image_transp.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/additions.fib.html b/theories/html/additions.fib.html new file mode 100644 index 00000000..ccfcf3b6 --- /dev/null +++ b/theories/html/additions.fib.html @@ -0,0 +1,333 @@ + + + + + +additions.fib + + + + +
+ + + +
+ +

Library additions.fib

+ +
+
+ +
+author Yves Bertot, Inria +
+
+ +
+From Coq Require Import Extraction ZArith Lia.
+ +
+From mathcomp Require Import all_ssreflect all_algebra.
+ +
+Set Implicit Arguments.
+ +
+Fixpoint fib (n : nat) :=
+  if n is S (S p as p') then fib p + fib p' else 1.
+ +
+Fixpoint fibt (n : nat) (acc1 acc2 : nat) : nat :=
+  if n is S p then fibt p (acc2 + acc1) acc1 else acc1.
+ +
+Fixpoint Zfibt (n : nat) (acc1 acc2 : Z) : Z :=
+  if n is S p then Zfibt p (Z.add acc2 acc1) acc1 else acc1.
+ +
+Lemma fibt_aux (n k: nat) : fibt n (fib k.+1) (fib k) = fib (n + k).+1.
+ +
+Lemma fibtP (n : nat) : fibt n 1 0 = fib n.
+ +
+Fixpoint bits p acc : list bool :=
+  match p with
+  | xH ⇒ true :: acc
+  | xI p ⇒ bits p (true :: acc)
+  | xO p ⇒ bits p (false :: acc)
+  end.
+ +
+Lemma bits_cat p a : bits p a = bits p [::] ++ a.
+ +
+Lemma bitsP p :
+  Pos.to_nat p =
+  \sum_(i < size (bits p [::])) nth false (bits p [::])
+      (size (bits p [::]) - 1 - i) × 2 ^ i.
+ +
+Section with_matrices.
+ +
+Variable R : comUnitRingType.
+ +
+Import GRing.Theory.
+Open Scope ring_scope.
+ +
+Definition ZtoR (x : Z) : R :=
+  if (Z.ltb x 0) then
+    -(Z.abs_nat (-x))%:R else (Z.abs_nat x)%:R.
+ +
+Lemma ZtoRD (x y : Z) : ZtoR (x + y) = ZtoR x + ZtoR y.
+ +
+Lemma ZtoRM x y : ZtoR (x × y) = ZtoR x × ZtoR y.
+ +
+Lemma iter_mul (n : nat) (m v : 'M[R]_2) :
+  iter n [eta *%R m] v = m ^+ n × v.
+ +
+Definition fibm : 'M[R]_2 :=
+  \matrix_(i, j) if (val i == 1%N) && (val j == 1%N) then 0%R else 1%R.
+ +
+Lemma fibmP n : fibm ^+ n.+2 =
+  \matrix_(i, j)
+     if (val i == 0%N) && (val j == 0%N) then (fib n.+2)%:R else
+     if ((val i + val j == 1)%N) then (fib n.+1)%:R else
+      (fib n)%:R.
+ +
+ +
+Definition m4lval (l : seq Z) (i j : nat) :=
+  nth Z0 l (j + 2 × i).
+ +
+Definition m4lmx (l : seq Z) : 'M[R]_2 :=
+  \matrix_(i, j) (ZtoR (m4lval l i j)).
+ +
+Definition m4lmul (l1 l2 : seq Z) :=
+  (Z.add (m4lval l1 0 0 × m4lval l2 0 0) (m4lval l1 0 1 × m4lval l2 1 0)) ::
+  (Z.add (m4lval l1 0 0 × m4lval l2 0 1) (m4lval l1 0 1 × m4lval l2 1 1)) ::
+  (Z.add (m4lval l1 1 0 × m4lval l2 0 0) (m4lval l1 1 1 × m4lval l2 1 0)) ::
+  (Z.add (m4lval l1 1 0 × m4lval l2 0 1) (m4lval l1 1 1 × m4lval l2 1 1)) :: nil.
+ +
+Open Scope Z_scope.
+Definition m4lfib :=
+  [:: 1; 1;
+      1; 0].
+ +
+Close Scope Z_scope.
+ +
+ +
+Definition m3lmx (l : seq Z) : 'M[R]_2 :=
+  \matrix_(i, j)
+   if (i == j) then
+     ZtoR (nth Z0 l i)
+   else
+     ZtoR (nth Z0 l 2).
+ +
+Definition m3lmul (l1 l2 : seq Z) :=
+  (Z.add (nth Z0 l1 0 × nth Z0 l2 0) (nth Z0 l1 2 × nth Z0 l2 2)) ::
+  (Z.add (nth Z0 l1 2 × nth Z0 l2 2) (nth Z0 l1 1 × nth Z0 l2 1)) ::
+(Z.add (nth Z0 l1 0 × nth Z0 l2 2) (nth Z0 l1 2 × nth Z0 l2 1)) :: nil.
+ +
+Definition m3lfib := [:: Z.pos xH ; Z0; Z.pos xH].
+ +
+Fixpoint fastexp (m : list Z) (p : positive) : list Z :=
+  match p with
+  | xH ⇒ m
+  | xI p ⇒
+    let r := fastexp m p in
+    m3lmul (m3lmul r r) m
+  | xO p ⇒
+    let r := fastexp m p in
+      m3lmul r r
+  end.
+ +
+Fixpoint fastexp2 (m : list Z) (p : positive) (acc : list Z) : list Z :=
+  match p with
+  | xH ⇒ acc
+  | xO p ⇒ fastexp2 m p (m3lmul acc acc)
+  | xI p ⇒ fastexp2 m p (m3lmul m (m3lmul acc acc))
+  end.
+ +
+Definition fastexp3 {A : Type} (mul : A → A → A)
+  (m : A) :=
+  fix f (l : list bool) (acc : A) : A :=
+  match l with
+  | nil ⇒ acc
+  | true :: l ⇒ f l (mul (mul acc acc) m)
+  | false :: l ⇒ f l (mul acc acc)
+  end.
+ +
+Definition my_pow {A : Type} (mul : A → A → A) (m : A)
+  (p : positive)
+  : A :=
+  fastexp3 mul m (behead (bits p nil)) m.
+ +
+Definition m3lid := [:: Z.pos xH; Z0; Z0; Z.pos xH].
+ +
+Definition slowexp (m : list Z) p :=
+  Pos.iter (m3lmul m) m3lid p.
+ +
+ +
+ +
+ +
+Definition binary_power_mult (mul : list Z → list Z → list Z) :
+  list Z → list Z → positive → list Z :=
+  fix f (x a : list Z) (p : positive) : list Z :=
+  match p with
+  | xH ⇒ mul a x
+  | xO q ⇒ f (mul x x) a q
+  | xI q ⇒ f (mul x x) (mul a x) q
+end.
+ +
+Definition fastexp4 (mul : list Z → list Z → list Z)
+  : list Z → positive → list Z :=
+  fix f (x : list Z) (p : positive) :=
+  match p with
+  | xH ⇒ x
+  | xO q ⇒ f (mul x x) q
+  | xI q ⇒ binary_power_mult mul (mul x x) x q
+  end.
+ +
+Lemma m3lmulP l1 l2:
+  GRing.comm (m3lmx l1) (m3lmx l2) →
+  m3lmx (m3lmul l1 l2) = m3lmx l1 × m3lmx l2.
+ +
+Lemma m3lfibP : m3lmx m3lfib = fibm.
+ +
+Lemma iter_comm {A : Type} (op : A → A → A) (e1 e2 : A)
+  (assoc: associative op)(cm : op e1 e2 = op e2 e1) n :
+  op e1 (iter n (op e1) e2) = op (iter n (op e1) e2) e1.
+ +
+Lemma iter_combine {A : Type} (op : A → A → A) (e1 e2 : A)
+  (assoc: associative op)(cm : op e1 e2 = op e2 e1) n m :
+  op (iter n (op e1) e2) (iter m (op e1) e2) =
+  (iter (n + m) (op e1) (op e2 e2)).
+ +
+Lemma fastexp3P {A : Type} (op : A → A → A) (id e : A) (h l : nat)
+  (v : list bool) (assoc : associative op) (cm : op id e = op e id)
+  (idn : ∀ a, op id a = a):
+  l = (\sum_(i < size v) nth false v (size v - 1 - i) × 2 ^ i)%N →
+  iter (h × 2 ^ size v + l) (op e) id =
+  fastexp3 op e v (iter h (op e) id).
+ +
+Lemma headbits p l : bits p l = true :: behead (bits p l).
+ +
+Lemma my_powP m p :
+  @my_pow 'M[R]_2 *%R m p = m ^ Pos.to_nat p.
+ +
+Lemma my_pow_m3lmul m p :
+  m3lmx (my_pow m3lmul m p) = my_pow *%R (m3lmx m) p.
+ +
+Definition m2lmul : list Z → list Z → list Z :=
+ fun l1 l2 ⇒
+ let a := nth Z0 l1 0 in
+ let b := nth Z0 l1 1 in
+ let c := nth Z0 l2 0 in
+ let d := nth Z0 l2 1 in
+   [:: Z.add (a × (c + d)) (b × c) ; Z.add (a × c) (b × d)].
+ +
+Definition m2lmx (l : list Z) : 'M[R]_2 :=
+  let a := nth Z0 l 0 in
+  let b := nth Z0 l 1 in
+  \matrix_(i, j)
+      if ((val i == 0%N) && (val j == 0%N)) then
+         ZtoR (a + b)
+      else if (i == j) then
+         ZtoR b
+      else ZtoR a.
+ +
+Definition m2lfib := [:: Zpos xH; Z0].
+ +
+Lemma m2lfibP : m2lmx m2lfib = fibm.
+ +
+Lemma m2lmulP m1 m2 :
+  m2lmx (m2lmul m1 m2) = m2lmx m1 × m2lmx m2.
+ +
+Lemma my_pow_m2lmul m p :
+  m2lmx (my_pow m2lmul m p) = my_pow *%R (m2lmx m) p.
+ +
+Lemma fibZ3P p :
+  ZtoR (nth Z0 (my_pow m3lmul m3lfib p) 0) = (fib (Pos.to_nat p))%:R.
+ +
+Lemma fibZ2P p :
+  ZtoR (nth Z0 (my_pow m2lmul m2lfib p) 0 +
+        nth Z0 (my_pow m2lmul m2lfib p) 1) = (fib (Pos.to_nat p))%:R.
+ +
+Definition m3lpow (m : list Z) (n : positive) :=
+  fastexp3 m3lmul m (behead (bits n nil)) m.
+ +
+Definition m2lpow (m : list Z) (n : positive) :=
+  fastexp3 m2lmul m (behead (bits n nil)) m.
+ +
+End with_matrices.
+ +
+Definition bigarg := 30000%positive.
+ +
+Extraction "bigfib" Z.ltb Z.div_eucl Pos.iter Z.log2 fastexp4
+  my_pow m4lmul m3lmul m2lmul m4lfib m3lfib m2lfib bigarg.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/coqdoc.css b/theories/html/coqdoc.css new file mode 100644 index 00000000..48096e55 --- /dev/null +++ b/theories/html/coqdoc.css @@ -0,0 +1,338 @@ +body { padding: 0px 0px; + margin: 0px 0px; + background-color: white } + +#page { display: block; + padding: 0px; + margin: 0px; + padding-bottom: 10px; } + +#header { display: block; + position: relative; + padding: 0; + margin: 0; + vertical-align: middle; + border-bottom-style: solid; + border-width: thin } + +#header h1 { padding: 0; + margin: 0;} + + +/* Contents */ + +#main{ display: block; + padding: 10px; + font-family: sans-serif; + font-size: 100%; + line-height: 100% } + +#main h1 { line-height: 95% } /* allow for multi-line headers */ + +#main a.idref:visited {color : #416DFF; text-decoration : none; } +#main a.idref:link {color : #416DFF; text-decoration : none; } +#main a.idref:hover {text-decoration : none; } +#main a.idref:active {text-decoration : none; } + +#main a.modref:visited {color : #416DFF; text-decoration : none; } +#main a.modref:link {color : #416DFF; text-decoration : none; } +#main a.modref:hover {text-decoration : none; } +#main a.modref:active {text-decoration : none; } + +#main .keyword { color : #cf1d1d } +#main { color: black } + +.section { background-color: rgb(60%,60%,100%); + padding-top: 13px; + padding-bottom: 13px; + padding-left: 3px; + margin-top: 5px; + margin-bottom: 5px; + font-size : 175% } + +h2.section { background-color: rgb(80%,80%,100%); + padding-left: 3px; + padding-top: 12px; + padding-bottom: 10px; + font-size : 130% } + +h3.section { background-color: rgb(90%,90%,100%); + padding-left: 3px; + padding-top: 7px; + padding-bottom: 7px; + font-size : 115% } + +h4.section { +/* + background-color: rgb(80%,80%,80%); + max-width: 20em; + padding-left: 5px; + padding-top: 5px; + padding-bottom: 5px; +*/ + background-color: white; + padding-left: 0px; + padding-top: 0px; + padding-bottom: 0px; + font-size : 100%; + font-weight : bold; + text-decoration : underline; + } + +#main .doc { margin: 0px; + font-family: sans-serif; + font-size: 100%; + line-height: 125%; + max-width: 40em; + color: black; + padding: 10px; + background-color: #90bdff } + +.inlinecode { + display: inline; +/* font-size: 125%; */ + color: #666666; + font-family: monospace } + +.doc .inlinecode { + display: inline; + font-size: 120%; + color: rgb(30%,30%,70%); + font-family: monospace } + +.doc .inlinecode .id { + color: rgb(30%,30%,70%); +} + +.inlinecodenm { + display: inline; + color: #444444; +} + +.doc .code { + display: inline; + font-size: 120%; + color: rgb(30%,30%,70%); + font-family: monospace } + +.comment { + display: inline; + font-family: monospace; + color: rgb(50%,50%,80%); +} + +.code { + display: block; +/* padding-left: 15px; */ + font-size: 110%; + font-family: monospace; + } + +table.infrule { + border: 0px; + margin-left: 50px; + margin-top: 10px; + margin-bottom: 10px; +} + +td.infrule { + font-family: monospace; + text-align: center; +/* color: rgb(35%,35%,70%); */ + padding: 0px; + line-height: 100%; +} + +tr.infrulemiddle hr { + margin: 1px 0 1px 0; +} + +.infrulenamecol { + color: rgb(60%,60%,60%); + font-size: 80%; + padding-left: 1em; + padding-bottom: 0.1em +} + +/* Pied de page */ + +#footer { font-size: 65%; + font-family: sans-serif; } + +/* Identifiers: ) */ + +.id { display: inline; } + +.id[title="constructor"] { + color: rgb(60%,0%,0%); +} + +.id[title="var"] { + color: rgb(40%,0%,40%); +} + +.id[title="variable"] { + color: rgb(40%,0%,40%); +} + +.id[title="definition"] { + color: rgb(0%,40%,0%); +} + +.id[title="abbreviation"] { + color: rgb(0%,40%,0%); +} + +.id[title="lemma"] { + color: rgb(0%,40%,0%); +} + +.id[title="instance"] { + color: rgb(0%,40%,0%); +} + +.id[title="projection"] { + color: rgb(0%,40%,0%); +} + +.id[title="method"] { + color: rgb(0%,40%,0%); +} + +.id[title="inductive"] { + color: rgb(0%,0%,80%); +} + +.id[title="record"] { + color: rgb(0%,0%,80%); +} + +.id[title="class"] { + color: rgb(0%,0%,80%); +} + +.id[title="keyword"] { + color : #cf1d1d; +/* color: black; */ +} + +/* Deprecated rules using the 'type' attribute of (not xhtml valid) */ + +.id[type="constructor"] { + color: rgb(60%,0%,0%); +} + +.id[type="var"] { + color: rgb(40%,0%,40%); +} + +.id[type="variable"] { + color: rgb(40%,0%,40%); +} + +.id[title="binder"] { + color: rgb(40%,0%,40%); +} + +.id[type="definition"] { + color: rgb(0%,40%,0%); +} + +.id[type="abbreviation"] { + color: rgb(0%,40%,0%); +} + +.id[type="lemma"] { + color: rgb(0%,40%,0%); +} + +.id[type="instance"] { + color: rgb(0%,40%,0%); +} + +.id[type="projection"] { + color: rgb(0%,40%,0%); +} + +.id[type="method"] { + color: rgb(0%,40%,0%); +} + +.id[type="inductive"] { + color: rgb(0%,0%,80%); +} + +.id[type="record"] { + color: rgb(0%,0%,80%); +} + +.id[type="class"] { + color: rgb(0%,0%,80%); +} + +.id[type="keyword"] { + color : #cf1d1d; +/* color: black; */ +} + +.inlinecode .id { + color: rgb(0%,0%,0%); +} + + +/* TOC */ + +#toc h2 { + padding: 10px; + background-color: rgb(60%,60%,100%); +} + +#toc li { + padding-bottom: 8px; +} + +/* Index */ + +#index { + margin: 0; + padding: 0; + width: 100%; +} + +#index #frontispiece { + margin: 1em auto; + padding: 1em; + width: 60%; +} + +.booktitle { font-size : 140% } +.authors { font-size : 90%; + line-height: 115%; } +.moreauthors { font-size : 60% } + +#index #entrance { + text-align: center; +} + +#index #entrance .spacer { + margin: 0 30px 0 30px; +} + +#index #footer { + position: absolute; + bottom: 0; +} + +.paragraph { + height: 0.75em; +} + +ul.doclist { + margin-top: 0em; + margin-bottom: 0em; +} + +.code :target { + border: 2px solid #D4D4D4; + background-color: #e5eecc; +} diff --git a/theories/html/gaia_hydras.GCanon.html b/theories/html/gaia_hydras.GCanon.html new file mode 100644 index 00000000..44ba1f7c --- /dev/null +++ b/theories/html/gaia_hydras.GCanon.html @@ -0,0 +1,204 @@ + + + + + +gaia_hydras.GCanon + + + + +
+ + + +
+ +

Library gaia_hydras.GCanon

+ +
+
+ +
+Gaia-compatible canonical sequences + +
+ +(imported from hydras.Epsilon0.Canon) +
+
+ +
+From mathcomp Require Import all_ssreflect zify.
+From hydras Require Import Canon.
+Require Import T1Bridge.
+From hydras Require Import T1.
+ +
+From gaia Require Import ssete9.
+Import CantorOrdinal.
+Set Implicit Arguments.
+ +
+
+ +
+Importation of Ketonen-Solovay's machinery into gaia's world + (work in progress) + +
+ + Note that a lemma Foo may mask Canon.Foo + +
+
+ +
+#[global] Notation hcanon := Epsilon0.Canon.canon.
+ +
+Definition canon (a: T1) (i:nat) : T1 := h2g (hcanon (g2h a) i).
+ +
+Notation canonS a := (fun i ⇒ canon a (S i)).
+ +
+Example ex0: canon (phi0 T1omega) 6 == phi0 (\F 6).
+ +
+Example ex1: canonS (phi0 T1omega) 6 == phi0 (\F 7).
+ +
+
+ +
+rewriting lemmas +
+
+ +
+Lemma gcanon_zero i: canon zero i = zero.
+ +
+Lemma g2h_canon a i: g2h (canon a i) = hcanon (g2h a) i.
+ +
+Lemma canon_succ i a (Ha: T1nf a):
+  canon (T1succ a) i = a.
+ +
+Lemma canonS_LE a n: T1nf a → canon a n.+1 ≤ canon a n.+2.
+ +
+Lemma canon0_phi0_succE a:
+  T1nf a → canon (phi0 (T1succ a)) 0 = zero.
+ +
+Lemma canon0_lt a:
+  T1nf a → a ≠ zero → T1lt (canon a 0) a.
+ +
+Lemma canonS_lt (i : nat) [a : T1]:
+  T1nf a → a ≠ zero → T1lt (canon a i.+1) a.
+ +
+Lemma canon_lt (i : nat) [a : T1]: T1nf a → a ≠ zero → canon a i < a.
+ +
+Lemma canonS_cons_not_zero (i : nat) (a : T1) (n : nat) (b : T1):
+  a ≠ zero → canon (cons a n b) i.+1 ≠ zero.
+ +
+Lemma limit_canonS_not_zero i lambda:
+  T1nf lambda → T1limit lambda → canon lambda i.+1 ≠ zero.
+ +
+Lemma canonS_phi0_succE (i : nat) (gamma : T1):
+  T1nf gamma → canon (phi0 (T1succ gamma)) i.+1 = cons gamma i zero.
+ +
+Lemma canon_SSn_zero (i : nat) (a : T1) (n : nat):
+  T1nf a →
+  canon (cons a n.+1 zero) i = cons a n (canon (phi0 a) i).
+ +
+Lemma canonS_zero_inv (a : T1) (i : nat):
+  canon a i.+1 = zero → a = zero ∨ a = one.
+ +
+Lemma canon_lim1 i (lambda: T1) :
+  T1nf lambda →
+  T1limit lambda → canon (phi0 lambda) i = phi0 (canon lambda i).
+ +
+Lemma canon_tail a (n : nat) b (i : nat):
+  T1nf (cons a n b) →
+  b ≠ zero → canon (cons a n b) i = cons a n (canon b i).
+ +
+Lemma canonS_ocons_succE i n (gamma: T1)(Hnf : T1nf gamma) :
+  canon (cons (T1succ gamma) n.+1 zero) i.+1 =
+    cons (T1succ gamma) n (cons gamma i zero).
+ +
+Lemma canon_lim2 i n (lambda : T1) (Hnf: T1nf lambda) (Hlim: T1limit lambda):
+  canon (cons lambda n.+1 zero) i = cons lambda n (phi0 (canon lambda i)).
+ +
+Lemma canon_lim3 i n a lambda (Ha: T1nf a)
+      (Hlambda: T1nf lambda) (Hlim :T1limit lambda) :
+  canon (cons a n lambda) i = cons a n (canon lambda i).
+ +
+Lemma canon_limit_strong lambda :
+  T1nf lambda → T1limit lambda →
+  ∀ b, T1nf b →
+               T1lt b lambda → {i : nat | b < canon lambda i}.
+ +
+Lemma T1nf_canon a i : T1nf a → T1nf (canon a i).
+ +
+Lemma gcanon_limit_v2 (lambda: T1):
+  T1nf lambda → T1limit lambda → limit_v2 (canon lambda) lambda.
+ +
+Lemma canon_limit_mono lambda i j (Hnf : T1nf lambda)
+        (Hlim : T1limit lambda) (Hij : (i < j)%N) :
+  canon lambda i < canon lambda j.
+ +
+ Lemma canon_limit_of lambda (Hnf : T1nf lambda) (Hlim : T1limit lambda) :
+   limit_of (canon lambda) lambda.
+ +
+
+ +
+

Adaptation of canon to type E0

+ +
+
+ +
+#[program] Definition E0Canon (a: E0) (i: nat): E0 :=
+   @mkE0 (canon (cnf a) i) _.
+ +
+Lemma E0_canon_lt (a: E0) i:
+  cnf a ≠ zero → E0lt (E0Canon a i) a.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.GF_alpha.html b/theories/html/gaia_hydras.GF_alpha.html new file mode 100644 index 00000000..1f7a3909 --- /dev/null +++ b/theories/html/gaia_hydras.GF_alpha.html @@ -0,0 +1,169 @@ + + + + + +gaia_hydras.GF_alpha + + + + +
+ + + +
+ +

Library gaia_hydras.GF_alpha

+ +
+
+ +
+Gaia-compatible F_ alpha fast growing functions + +
+ +(imported from hydras.Epsilon0.F_alpha ) + +
+
+ +
+From mathcomp Require Import all_ssreflect zify.
+From gaia Require Export ssete9.
+From Coq Require Import Logic.Eqdep_dec Arith.
+From hydras Require Import DecPreOrder.
+From hydras Require Import T1 E0.
+From hydras Require Paths.
+ +
+From hydras Require Import primRec.
+From hydras Require Import F_alpha F_omega.
+ +
+From gaia_hydras Require Import T1Bridge GCanon GHprime.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+

Rapidly growing functions

+ +
+
+ +
+Notation hF_ := F_alpha.F_.
+ +
+Definition F_ (alpha : E0) := F_alpha.F_ (E0_g2h alpha).
+ +
+#[program]
+ Definition T1F_ (a : T1)(Hnf : T1nf a == true) (n:nat) : nat:=
+  (F_ (@mkE0 a _) n).
+ +
+
+ +
+Please note that a lemma foo may mask F_alpha.Foo +
+
+ +
+Lemma F_alpha_gt (alpha : E0) (n : nat): (n < F_ alpha n)%N.
+ +
+Lemma F_alpha_mono (alpha: E0): strict_mono (F_ alpha).
+ +
+Lemma F_alpha_dom alpha:
+  dominates_from 1 (F_ (E0_succ alpha)) (F_ alpha).
+ +
+ +
+Lemma F_alpha_Succ_le alpha:
+ fun_le (F_ alpha) (F_ (E0_succ alpha)).
+ +
+Lemma F_alpha_positive (alpha : hE0) (n : nat): (0 < F_alpha.F_ alpha n)%N.
+ +
+Lemma F_zeroE i: F_ E0zero i = i.+1.
+ +
+Lemma F_mono_l alpha beta:
+  E0lt beta alpha → dominates (F_ alpha) (F_ beta).
+ +
+Lemma F_alpha_0_eq (alpha : E0): F_ alpha 0 = 1.
+ +
+Lemma F_succE alpha i :
+  F_ (E0_succ alpha) i = Iterates.iterate (F_ alpha) i.+1 i.
+ +
+Lemma F_limE alpha i:
+  T1limit (cnf alpha) → F_ alpha i = F_ (E0Canon alpha i) i.
+ +
+
+ +
+ numerical examples +
+
+ +
+Lemma LF1 i: F_ (E0fin 1) i = ((2 × i) .+1)%N.
+ +
+Lemma LF2 i: (Exp2.exp2 i × i < F_ (E0fin 2) i)%N.
+ +
+Lemma LF2' i: (1 ≤ i)%N → (Exp2.exp2 i < F_ (E0fin 2) i)%N.
+ +
+Lemma LF3_2:
+  Iterates.dominates_from 2 (F_ (E0fin 3))
+    (fun n : nat ⇒ Iterates.iterate Exp2.exp2 n.+1 n).
+ +
+Definition Canon_plus n alpha beta :=
+  Paths.Canon_plus n (E0_g2h alpha) (E0_g2h beta).
+ +
+Lemma F_restricted_mono_l alpha beta n:
+    Canon_plus n alpha beta → (F_ beta n ≤ F_ alpha n)%N.
+ +
+Lemma H'_F alpha n : (F_ alpha n.+1 ≤ H'_ (E0_phi0 alpha) n.+1)%N.
+ +
+Lemma F_alpha_not_PR_E0 alpha:
+  E0le E0_omega alpha → isPR 1 (F_ alpha) → False.
+ +
+Lemma F_alpha_not_PR alpha (Hnf: T1nf alpha == true):
+  LE T1omega alpha → isPR 1 (@T1F_ alpha Hnf) → False.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.GHessenberg.html b/theories/html/gaia_hydras.GHessenberg.html new file mode 100644 index 00000000..27ad388d --- /dev/null +++ b/theories/html/gaia_hydras.GHessenberg.html @@ -0,0 +1,150 @@ + + + + + +gaia_hydras.GHessenberg + + + + +
+ + + +
+ +

Library gaia_hydras.GHessenberg

+ +
+
+ +
+Hessenberg (commutative) sum +
+
+ +
+From mathcomp Require Import all_ssreflect zify.
+From Coq Require Import Logic.Eqdep_dec.
+From hydras Require Import DecPreOrder ON_Generic.
+From hydras Require Import T1 E0 Hessenberg.
+ +
+From gaia Require Export ssete9.
+From gaia_hydras Require Import T1Bridge.
+ +
+Set Implicit Arguments.
+ +
+Open Scope cantor_scope.
+ +
+#[local] Notation hoplus := Epsilon0.Hessenberg.oplus.
+ +
+Definition oplus alpha beta := h2g (hoplus (g2h alpha) (g2h beta)).
+ +
+Infix "o+" := oplus: cantor_scope.
+ +
+Fixpoint o_finite_mult n alpha :=
+  if n is p.+1 then alpha o+ (o_finite_mult p alpha)
+  else zero.
+ +
+ +
+ +
+Module FixpointDef.
+Fixpoint oplus (alpha beta : T1) : T1 :=
+  let fix oplus_aux beta {struct beta} :=
+      match alpha, beta with
+        | zero, _ ⇒ beta
+        | _, zero ⇒ alpha
+        | cons a1 n1 b1, cons a2 n2 b2 ⇒
+          match compare a1 a2 with
+            | Gt ⇒ cons a1 n1 (oplus b1 beta)
+            | Lt ⇒ cons a2 n2 (oplus_aux b2)
+            | Eq ⇒ cons a1 (S (n1 + n2)%nat) (oplus b1 b2)
+          end
+      end
+  in oplus_aux beta.
+End FixpointDef.
+ +
+Lemma oplusE (a b :T1) :
+  a o+ b =
+    match a, b with
+    | zero, _ ⇒ b
+    | _, zero ⇒ a
+    | cons a1 n1 b1, cons a2 n2 b2 ⇒
+        match compare a1 a2 with
+        | Gt ⇒ cons a1 n1 (b1 o+ b)
+        | Eq ⇒ cons a1 (S (n1 + n2)) (b1 o+ b2)
+        | Lt ⇒ cons a2 n2 (a o+ b2)
+        end
+    end.
+ +
+
+ +
+Equations for oplus +
+
+ +
+Lemma oplus0b: left_id zero oplus.
+ +
+Lemma oplusa0: right_id zero oplus.
+ +
+Lemma oplus_nf (a b : T1) : T1nf a → T1nf b → T1nf (a o+ b).
+ +
+Lemma oplusC (a b: T1): T1nf a → T1nf b → a o+ b = b o+ a.
+ +
+Lemma oplusA (a b c: T1) :
+  T1nf a → T1nf b → T1nf c → a o+ (b o+ c) = a o+ b o+ c.
+ +
+Lemma oplus_lt1 (a b:T1):
+  T1nf a → T1nf b → zero < a → b < b o+ a.
+ +
+ +
+Lemma oplus_lt2 (a b: T1):
+  T1nf a → T1nf b → zero < b → a < b o+ a.
+ +
+Lemma oplus_strict_mono_l (a b c: T1):
+  T1nf a → T1nf b → T1nf c → a < b → a o+ c < b o+ c.
+ +
+Lemma oplus_strict_mono_r (a b c: T1):
+  T1nf a → T1nf b → T1nf c → b < c → a o+ b < a o+ c.
+ +
+Lemma oplus_lt_phi0 (a b c: T1):
+  T1nf a → T1nf b → T1nf c →
+  a < c → b < c → phi0 a o+ phi0 b < phi0 c.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.GHprime.html b/theories/html/gaia_hydras.GHprime.html new file mode 100644 index 00000000..5a133d32 --- /dev/null +++ b/theories/html/gaia_hydras.GHprime.html @@ -0,0 +1,121 @@ + + + + + +gaia_hydras.GHprime + + + + +
+ + + +
+ +

Library gaia_hydras.GHprime

+ +
+ +
+
+ +
+Gaia-compatible H'_ alpha fast growing functions + +
+ +(imported from hydras.Epsilon0.Hprime ) + +
+
+ +
+From mathcomp Require Import all_ssreflect zify.
+From gaia Require Export ssete9.
+From Coq Require Import Logic.Eqdep_dec.
+From hydras Require Import DecPreOrder T1 E0.
+From hydras Require Paths.
+From hydras Require Import Iterates Hprime L_alpha.
+From gaia_hydras Require Import T1Bridge GCanon GPaths.
+ +
+Set Implicit Arguments.
+ +
+Definition H'_ alpha i := Hprime.H'_ (E0_g2h alpha) i.
+ +
+
+ +
+

Equations for H'_

+ +
+
+ +
+Lemma H'_eq1 (i: nat) : H'_ E0zero i = i.
+ +
+Lemma H'_eq2 alpha i :
+  H'_ (E0_succ alpha) i = H'_ alpha (S i).
+ +
+Lemma H'_eq3 alpha i :
+  E0limit alpha → H'_ alpha i = H'_ (E0Canon alpha (S i)) i.
+ +
+
+ +
+

Examples

+ +
+
+ +
+Lemma H'_omega k : H'_ E0_omega k = (2 × k).+1 %nat.
+ +
+Lemma H'_omega_double (k: nat) :
+  H'_ (E0mul E0_omega (E0fin 2)) k = (4 × k + 3)%coq_nat.
+ +
+
+ +
+TODO + import more abstract properties of H' +
+
+ +
+Lemma H'_dom alpha beta :
+  E0lt alpha beta → dominates_strong (H'_ beta) (H'_ alpha).
+ +
+Lemma H'_alpha_mono (alpha : E0) : strict_mono (H'_ alpha).
+ +
+Theorem H'_alpha_gt alpha (Halpha: alpha ≠ E0zero) n :
+  (n < H'_ alpha n)%N.
+ +
+Lemma H'_omega_cube_min :
+∀ k : nat,
+  0 ≠ k → (hyper_exp2 k.+1 ≤ H'_ (E0_phi0 (E0fin 3)) k)%N.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.GHydra.html b/theories/html/gaia_hydras.GHydra.html new file mode 100644 index 00000000..eeac6608 --- /dev/null +++ b/theories/html/gaia_hydras.GHydra.html @@ -0,0 +1,115 @@ + + + + + +gaia_hydras.GHydra + + + + +
+ + + +
+ +

Library gaia_hydras.GHydra

+ +
+From mathcomp Require Import all_ssreflect zify.
+ +
+From Coq Require Import Logic.Eqdep_dec.
+From hydras Require Import DecPreOrder ON_Generic.
+From hydras Require Import T1 E0 Hessenberg Hydra_Theorems Hydra_Definitions
+     Hydra_Termination Battle_length Hydra_Examples
+     Epsilon0_Needed_Free Epsilon0_Needed_Std.
+From gaia Require Export ssete9.
+Require Import T1Bridge GHessenberg GL_alpha GPrelude.
+Set Implicit Arguments.
+ +
+Import Hydra_Definitions.
+ +
+Fixpoint m (h:Hydra) : T1 :=
+  if h is node (hcons _ _ as hs) then ms hs else zero
+with ms (s : Hydrae) : T1 :=
+  if s is hcons h s' then phi0 (m h) o+ ms s' else zero.
+ +
+ +
+Lemma m_ref h : g2h (m h) = (Hydra_Termination.m h).
+ +
+Lemma m_nf h : T1nf (m h).
+ +
+Lemma mVariant: Hvariant nf_Wf free m .
+ +
+Theorem every_battle_terminates : Termination.
+ +
+Definition l_std alpha k := (L_ alpha (S k) - k)%nat.
+ +
+Lemma l_stdE alpha k : l_std alpha k = Battle_length.l_std (E0_g2h alpha) k.
+ +
+Definition T1toH (a: T1) : Hydra := O2H.iota (g2h a).
+ +
+Lemma l_std_ok : ∀ alpha : E0,
+    alpha != E0zero →
+    ∀ k : nat,
+      (1 ≤ k)%N → battle_length standard k (T1toH (cnf alpha))
+                                  (l_std alpha k).
+ +
+Section ImpossibilityProof.
+  Context (b: Battle).
+  Variable mu:T1.
+  Hypothesis nfMu: T1nf mu.
+  Variable m : Hydra → T1.
+  Let mh (h:Hydra) := g2h (m h).
+ +
+  Context (Var : Hvariant nf_Wf b m)
+           (BVar: BoundedVariant Var mu).
+ +
+  #[local] Instance hVar : Hvariant T1_wf b mh.
+ +
+  #[local] Instance bVar : BoundedVariant hVar (g2h mu).
+ +
+  End ImpossibilityProof.
+ +
+ +
+Lemma Impossibility_free mu m (Var: Hvariant nf_Wf free m):
+  ¬ BoundedVariant Var mu.
+ +
+Lemma Impossibility_std mu m (Var: Hvariant nf_Wf standard m):
+  ¬ BoundedVariant Var mu.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.GL_alpha.html b/theories/html/gaia_hydras.GL_alpha.html new file mode 100644 index 00000000..6eb024d6 --- /dev/null +++ b/theories/html/gaia_hydras.GL_alpha.html @@ -0,0 +1,116 @@ + + + + + +gaia_hydras.GL_alpha + + + + +
+ + + +
+ +

Library gaia_hydras.GL_alpha

+ +
+ +
+
+ +
+Gaia-compatible L_ alpha fast growing functions + +
+ +(imported from hydras.Epsilon0.Hprime ) + +
+
+ +
+From mathcomp Require Import all_ssreflect zify.
+From gaia Require Export ssete9.
+From Coq Require Import Logic.Eqdep_dec.
+From hydras Require Import DecPreOrder.
+From hydras Require Import T1 E0.
+From hydras Require Paths L_alpha.
+From hydras Require Import L_alpha.
+From gaia_hydras Require Import T1Bridge GCanon GPaths GHprime.
+ +
+Set Implicit Arguments.
+ +
+Definition L_ (alpha:E0) (i:nat): nat :=
+ L_alpha.L_ (E0_g2h alpha) i.
+ +
+Lemma L_zeroE i : L_ E0zero i = i.
+ +
+Lemma L_eq2 alpha i :
+  E0is_succ alpha → L_ alpha i = L_ (E0_pred alpha) (S i).
+ +
+Lemma L_succE alpha i : L_ (E0_succ alpha) i = L_ alpha i.+1.
+ +
+Lemma L_limE alpha i :
+  E0limit alpha → L_ alpha i = L_ (E0Canon alpha i) (S i).
+ +
+Lemma L_finite : ∀ i k :nat, L_ (E0fin i) k = (i+k)%nat.
+ +
+Lemma L_omega : ∀ k, L_ E0_omega k = S (2 × k)%nat.
+ +
+Lemma L_ge_id alpha : ∀ i, (i ≤ L_ alpha i)%N.
+ +
+Lemma L_ge_S alpha i:
+  alpha ≠ E0zero → (i < L_ alpha i)%N.
+ +
+Definition L_spec (alpha:T1) (f: nat → nat):=
+  Large_Sets.L_spec (g2h alpha) f.
+ +
+Lemma L_spec0 (f:nat → nat) : L_spec zero f ↔ ∀ k, f k.+1 = k.+1.
+ +
+Lemma L_spec1 (a:T1) (f:nat → nat) :
+  a != zero →
+  L_spec a f ↔
+    (∀ k,
+        Large_Sets.mlarge (g2h a)
+                          ( MoreLists.interval (S k)
+                                               (Nat.pred (f (S k))))) .
+ +
+Lemma L_pos_inv a f :
+ a != zero → L_spec a f →
+                        ∀ k, (S k < f (S k))%N.
+ +
+Theorem L_correct a : L_spec (cnf a) (L_ a).
+ +
+Theorem H'_L_ a i: (H'_ a i ≤ L_ a i.+1)%N.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.GLarge_Sets.html b/theories/html/gaia_hydras.GLarge_Sets.html new file mode 100644 index 00000000..b228244a --- /dev/null +++ b/theories/html/gaia_hydras.GLarge_Sets.html @@ -0,0 +1,88 @@ + + + + + +gaia_hydras.GLarge_Sets + + + + +
+ + + +
+ +

Library gaia_hydras.GLarge_Sets

+ +
+
+ +
+Gaia-compatible large sequences + +
+ +(imported from hydras.Epsilon0.Large_Sets ) + +
+
+ +
+From mathcomp Require Import all_ssreflect zify.
+From hydras Require Import T1.
+From hydras Require Import Canon Paths Large_Sets.
+From gaia_hydras Require Import T1Bridge GCanon GPaths.
+ +
+From gaia Require Import ssete9.
+Import CantorOrdinal.
+ +
+Notation hmlarge := mlarge.
+Notation hmlargeS := mlargeS.
+Definition mlarge a s := hmlarge (g2h a) s.
+Definition mlargeS a s := hmlargeS (g2h a) s.
+ +
+Notation hlarge := large.
+Notation hlargeS := largeS.
+Definition large a A := hlarge (g2h a) A.
+Definition largeS a A := hlargeS (g2h a) A.
+ +
+Notation hL_spec := L_spec.
+Definition L_spec a f := L_spec (g2h a) f.
+ +
+Lemma mlarge_unicity a k l l' :
+  mlarge a (index_iota k.+1 l.+1) →
+  mlarge a (index_iota k.+1 l'.+1) →
+  l = l'.
+ +
+Lemma L_fin_ok i : L_spec (\F i) (L_fin i).
+ +
+Theorem Theorem_4_5 (a: T1)(Ha : T1nf a)
+        (A B : seq nat)
+        (HA : Sorted.Sorted Peano.lt A)
+        (HB : Sorted.Sorted Peano.lt B)
+        (HAB : List.incl A B) :
+  largeS a A → largeS a B.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.GPaths.html b/theories/html/gaia_hydras.GPaths.html new file mode 100644 index 00000000..b838a946 --- /dev/null +++ b/theories/html/gaia_hydras.GPaths.html @@ -0,0 +1,250 @@ + + + + + +gaia_hydras.GPaths + + + + +
+ + + +
+ +

Library gaia_hydras.GPaths

+ +
+
+ +
+Gaia-compatible accessibility in epsilon0 + +
+ +(imported from hydras.Epsilon0.Paths) +
+
+ +
+From mathcomp Require Import all_ssreflect zify.
+From hydras Require Import Canon Paths.
+From gaia_hydras Require Import T1Bridge GCanon.
+ +
+From hydras Require Import T1.
+From gaia Require Import ssete9.
+Import CantorOrdinal.
+ +
+
+ +
+Importation of Ketonen-Solovay's machinery into gaia's world + (work in progress) + +
+ + small transition step associated with canonical sequences +
+
+ +
+#[global] Notation htransition := Epsilon0.Paths.transition.
+#[global] Notation hbounded_transitionS := Paths.bounded_transitionS.
+ +
+Definition transition i (a b: T1) :=
+  [/\ i != 0 , a!= zero & b == canon a i].
+ +
+Definition bounded_transitionS n (a b: T1) :=
+  ∃ i, (i ≤ n)%N ∧ transition (S i) a b.
+ +
+Definition transitionb i (a b: T1) :=
+  [&& i != 0 , a!= zero & b == canon a i].
+ +
+Lemma transitionP i a b : reflect (transition i a b )
+                                  (transitionb i a b).
+ +
+
+ +
+TODO : define path_to as a boolean function +
+
+#[global] Notation hpath_to := path_to.
+#[global] Notation hpath := path.
+#[global] Notation hpathS from s to := (path_toS to s from).
+#[global] Notation hconst_pathS := const_pathS.
+#[global] Notation hconst_path := const_path.
+#[global] Notation hgnawS := gnawS.
+#[global] Notation hgnaw := gnaw.
+#[global] Notation hstandard_gnaw := standard_gnaw.
+ +
+
+ +
+path_to b s alpha : b is accessible from alpha with trace s +
+
+ +
+Definition path_to (to: T1)(s: seq nat) (from:T1) : Prop :=
+  hpath_to (g2h to) s (g2h from).
+ +
+Notation path from s to := (path_to to s from).
+ +
+Definition acc_from a b := ∃ s, path a s b.
+ +
+ +
+Definition const_path i a b := hconst_path i (g2h a) (g2h b).
+ +
+Definition standard_path i a j b :=
+   Paths.standard_path i (g2h a) j (g2h b).
+ +
+Definition standard_gnaw i a l := h2g (hstandard_gnaw i (g2h a) l).
+ +
+Definition gnaw a s := h2g (hgnaw (g2h a) s).
+Definition gnawS a s := h2g (hgnawS (g2h a) s).
+ +
+Definition pathS (from: T1)(s: seq nat) (to: T1) : Prop :=
+  hpathS (g2h from) s (g2h to).
+ +
+Fixpoint path_tob (b: T1) (s: seq nat) (a:T1): bool :=
+  match s with
+  | [::] ⇒ false
+  | [:: i] ⇒ (i != 0) && transitionb i a b
+  | i :: s ⇒ (i != 0) && path_tob b s (canon a i)
+  end.
+ +
+
+ +
+SSreflect's iota was already defined in Prelude +
+ + To simplify +
+
+ +
+Lemma path_to_inv1 to i from : path_to to [:: i] from →
+                               i ≠ 0 ∧ transition i from to.
+ +
+Lemma path_to_iff1 to i from :
+  T1nf from → i ≠ 0 → from ≠ zero →
+  path_to to [:: i] from ↔ to = canon from i ∧ T1nf from.
+ +
+Lemma iota_adapt i l: iota i l = MoreLists.iota_from i l.
+ +
+Lemma standard_gnaw_iota_from i a l :
+  i ≠ 0 → standard_gnaw i a l = gnaw a (iota i l).
+ +
+Lemma interval_def i j: MoreLists.interval i j = index_iota i (S j).
+ +
+Lemma path_to_LT b s a:
+  path_to b s a → T1nf a → T1nf b → b < a.
+ +
+Lemma LT_path_to (a b : T1) :
+  T1nf a → T1nf b → b < a → {s : list nat | path_to b s a}.
+ +
+Theorem KS_thm_2_4 (lambda : T1) :
+  T1nf lambda → T1limit lambda →
+  ∀ i j, (i < j)%N →
+              const_path 1 (canon lambda (S j))
+                         (canon lambda (S i)).
+ +
+Lemma Cor12 (a: T1) : T1nf a →
+  ∀ b i n, T1nf b → b < a → (i < n)%N →
+                const_path i.+1 a b → const_path n.+1 a b.
+ +
+Lemma Lemma2_6_1 (a:T1) :
+T1nf a → ∀ b, T1nf b → b < a → {n:nat | const_path n.+1 a b}.
+ +
+Lemma constant_to_standard_path (a b : T1) (i : nat):
+  T1nf a → const_path i.+1 a b → zero < a →
+  {j:nat | standard_path i.+1 a j b}.
+ +
+Theorem LT_to_standard_path (a b : T1) :
+  T1nf a → T1nf b → b < a →
+  {n : nat & {j:nat | standard_path n.+1 a j b}}.
+ +
+
+ +
+

Adaptation to E0

+ +
+
+ +
+Notation hCanon_plus := Canon_plus.
+Definition Canon_plus i a b :=
+  hCanon_plus i (E0_g2h a) (E0_g2h b).
+ +
+
+ +
+

Examples

+ +
+
+Example ex_path1 : path (T1omega × (\F 2)) [:: 2; 2; 2] T1omega.
+ +
+Example ex_path2: path (T1omega × \F 2) [:: 3; 4; 5; 6] T1omega.
+ +
+Example ex_path3: path (T1omega × \F 2) (index_iota 3 15) zero.
+ +
+Example ex_path4: path (T1omega × \F 2) (List.repeat 3 8) zero.
+ +
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.GPrelude.html b/theories/html/gaia_hydras.GPrelude.html new file mode 100644 index 00000000..27c58c88 --- /dev/null +++ b/theories/html/gaia_hydras.GPrelude.html @@ -0,0 +1,44 @@ + + + + + +gaia_hydras.GPrelude + + + + +
+ + + +
+ +

Library gaia_hydras.GPrelude

+ +
+
+ +
+Learning ssreflect: +
+
+ +
+From mathcomp Require Import all_ssreflect zify.
+Lemma diffP {T:eqType}(i j:T): reflect (i ≠ j) (i != j).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.GaiaToHydra.html b/theories/html/gaia_hydras.GaiaToHydra.html new file mode 100644 index 00000000..346da598 --- /dev/null +++ b/theories/html/gaia_hydras.GaiaToHydra.html @@ -0,0 +1,60 @@ + + + + + +gaia_hydras.GaiaToHydra + + + + +
+ + + +
+ +

Library gaia_hydras.GaiaToHydra

+ +
+ +
+From mathcomp Require Import all_ssreflect zify.
+From gaia_hydras Require Import T1Bridge .
+From hydras Require Import T1.
+ +
+From gaia Require Import ssete9.
+Import Epsilon0.T1.
+ +
+Locate T1.
+ +
+ +
+Lemma hmultA : associative T1mul.
+ +
+Example Ex1 (a: T1): T1omega × (a × T1omega) = T1omega × a × T1omega.
+ +
+Lemma hmult_dist : right_distributive T1mul T1add.
+ +
+Close Scope t1_scope.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.HydraGaia_Examples.html b/theories/html/gaia_hydras.HydraGaia_Examples.html new file mode 100644 index 00000000..9b4fcf68 --- /dev/null +++ b/theories/html/gaia_hydras.HydraGaia_Examples.html @@ -0,0 +1,99 @@ + + + + + +gaia_hydras.HydraGaia_Examples + + + + +
+ + + +
+ +

Library gaia_hydras.HydraGaia_Examples

+ +
+From mathcomp Require Import all_ssreflect zify.
+ +
+From Coq Require Import Logic.Eqdep_dec.
+From hydras Require Import DecPreOrder ON_Generic.
+From hydras Require Import T1 E0 Hessenberg.
+From gaia Require Export ssete9.
+ +
+From gaia_hydras Require Import T1Bridge GCanon GHessenberg
+  GLarge_Sets.
+Set Implicit Arguments.
+ +
+Check \F 42.
+ +
+ +
+ +
+Example α : T1 :=
+ T1omega + phi0 T1omega × \F 3 + phi0 (\F 5) × \F 4 + T1omega × T1omega.
+ +
+Example β : T1 := phi0 (phi0 (\F 2)).
+ +
+ +
+ +
+ +
+ +
+ +
+
+ +
+Hessenberg's sum +
+
+ +
+Print oplus.
+ +
+ +
+ +
+Check T1lt (\F 42) T1omega.
+ +
+Check (\F 42 < T1omega)%ca.
+ +
+Check \F 42 < T1omega. +
+ +
+Check T1.lt (T1.T1nat 42) T1.T1omega.
+ +
+Check T1.lt (\F 42)%t1 T1.T1omega.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.ON_gfinite.html b/theories/html/gaia_hydras.ON_gfinite.html new file mode 100644 index 00000000..3a4789e7 --- /dev/null +++ b/theories/html/gaia_hydras.ON_gfinite.html @@ -0,0 +1,86 @@ + + + + + +gaia_hydras.ON_gfinite + + + + +
+ + + +
+ +

Library gaia_hydras.ON_gfinite

+ +
+
+ +
+ordinals a la mathcomp +
+
+From hydras Require Import DecPreOrder ON_Generic.
+From mathcomp Require Import all_ssreflect zify.
+From gaia Require Export ssete9.
+From Coq Require Import Logic.Eqdep_dec.
+ +
+Set Implicit Arguments.
+ +
+Definition finord_lt (n:nat) (alpha beta: 'I_n): bool :=
+  (alpha < beta)%N.
+ +
+#[global] Instance finord_compare (n:nat) : Compare ('I_n) :=
+  fun alpha beta ⇒ Nat.compare alpha beta.
+ +
+Lemma finord_compare_correct (n:nat) (alpha beta : 'I_n) :
+  CompSpec eq (@finord_lt n) alpha beta (compare alpha beta).
+ +
+#[global] Instance finord_sto n : StrictOrder (@finord_lt n).
+ +
+#[global] Instance finord_comp n :
+  Comparable (@finord_lt n) (@finord_compare n).
+ +
+Lemma finord_lt_wf n : well_founded (@finord_lt n).
+ +
+#[global] Instance finord_ON n : ON (@finord_lt n) (@finord_compare n).
+ +
+
+ +
+Examples +
+
+ +
+#[program] Example o_33_of_42: 'I_42 := @Ordinal 42 33 _.
+ +
+#[program] Example o_36_of_42: 'I_42 := @Ordinal 42 36 _.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.T1Bridge.html b/theories/html/gaia_hydras.T1Bridge.html new file mode 100644 index 00000000..1a567ca9 --- /dev/null +++ b/theories/html/gaia_hydras.T1Bridge.html @@ -0,0 +1,682 @@ + + + + + +gaia_hydras.T1Bridge + + + + +
+ + + +
+ +

Library gaia_hydras.T1Bridge

+ +
+
+ +
+

Bridge between Hydra-battle's and Gaia's T1 (Experimental)

+ + +
+ +This library introduces tools for making definitions and lemmas from + Hydra battles's ordinals compatible with + Gaia's ssete9 library. + +
+
+ +
+From mathcomp Require Import all_ssreflect zify.
+ +
+From Coq Require Import Logic.Eqdep_dec.
+From hydras Require Import DecPreOrder ON_Generic T1 E0.
+From gaia Require Export ssete9.
+Set Implicit Arguments.
+ +
+
+ +
+The name T1 denotes Gaia's data type. We use T1.T1 or hT1 for Hydra battles ordinal terms. + +
+
+ +
+#[global] Notation hT1 := T1.T1.
+#[global] Notation T1 := ssete9.CantorOrdinal.T1.
+ +
+
+ +
+We rename Hydra battle's total order on hT1 +
+
+ +
+ +
+
+ +
+Restrictions to terms in normal form +
+
+ +
+#[global] Notation LT := (restrict T1nf T1lt).
+#[global] Notation LE := (restrict T1nf T1le).
+ +
+ +
+
+ +
+Translation functions +
+
+ +
+Fixpoint h2g (a : hT1) : T1 :=
+  if a is T1.cons a n b then cons (h2g a) n (h2g b) else zero.
+ +
+Fixpoint g2h (a : T1) : hT1 :=
+  if a is cons a n b then T1.cons (g2h a) n (g2h b) else T1.zero.
+ +
+Lemma h2g_g2hK : cancel g2h h2g.
+ +
+Lemma g2h_h2gK : cancel h2g g2h.
+ +
+Lemma g2h_eqE (a b: T1): g2h a = g2h b ↔ a = b.
+ +
+Lemma h2g_eqE (a b :hT1): h2g a = h2g b ↔ a = b.
+ +
+Lemma g2h_diffE (a b : T1) : g2h a ≠ g2h b ↔ a ≠ b.
+ +
+Lemma h2g_diffE (a b : hT1) : h2g a ≠ h2g b ↔ a ≠ b.
+ +
+
+ +
+Pretty printing +
+
+Definition T1pp (a:T1) : ppT1 := pp (g2h a).
+ +
+
+ +
+

refinement of constants, functions, etc.

+ +
+
+ +
+ +
+Definition refines0 (x:hT1)(y:T1) := y = h2g x.
+ +
+Definition refines1 (f:hT1 → hT1) (f': T1 → T1) :=
+  ∀ x: hT1, f' (h2g x) = h2g (f x).
+ +
+Definition refines2 (f:hT1 → hT1 → hT1) (f': T1 → T1 → T1 ) :=
+  ∀ x y : hT1, f' (h2g x) (h2g y) = h2g (f x y).
+ +
+Definition refinesPred (hP: hT1 → Prop) (gP: T1 → Prop) :=
+  ∀ x : hT1, hP x ↔ gP (h2g x).
+ +
+Definition refinesRel (hR: hT1 → hT1 → Prop)
+  (gR: T1 → T1 → Prop) :=
+  ∀ x y : hT1, hR x y ↔ gR (h2g x) (h2g y).
+ +
+ +
+Lemma refines1_R f f' :
+  refines1 f f' ↔ ∀ y: T1, f (g2h y) = g2h (f' y).
+ +
+Lemma refines2_R f f' :
+  refines2 f f' ↔ ∀ y z: T1, f (g2h y) (g2h z) = g2h (f' y z).
+ +
+
+ +
+

Refinements of usual constants

+ +
+
+ +
+ +
+Lemma zero_ref : refines0 T1.zero zero.
+ +
+Lemma one_ref : refines0 T1.one one.
+ +
+Lemma Finite_ref (n:nat) : refines0 (T1.T1nat n) (\F n).
+ +
+Lemma omega_ref : refines0 T1.T1omega T1omega.
+ +
+
+ +
+

unary functions and predicates

+ +
+
+ +
+Lemma succ_ref: refines1 T1.succ T1succ.
+Lemma phi0_ref x: refines0 (T1.phi0 x) (phi0 (h2g x)). +
+Lemma g2h_phi0 a : g2h (phi0 a) = T1.phi0 (g2h a).
+ +
+Lemma ap_ref : refinesPred Epsilon0.T1.ap T1ap.
+ +
+
+ +
+

Equality and comparison

+ +
+
+ +
+Lemma T1eq_refl (a: T1) : T1eq a a.
+ +
+Lemma T1eqE a b: T1eq a b → g2h a = g2h b.
+ +
+Lemma T1eq_h2g (a b : hT1) : T1eq (h2g a) (h2g b) → a = b.
+ +
+Lemma compare_ref (x y: hT1) :
+  match T1.compare_T1 x y with
+  | Datatypes.Lt ⇒ T1lt (h2g x) (h2g y)
+  | Datatypes.Eq ⇒ h2g x = h2g y
+  | Datatypes.Gt ⇒ T1lt (h2g y) (h2g x)
+  end.
+ +
+Lemma decide_hltE (a b : hT1):
+  bool_decide (T1.lt a b) = (h2g a < h2g b).
+ +
+Lemma lt_ref : refinesRel T1.lt T1lt.
+ +
+Lemma le_ref : refinesRel T1.le T1le.
+ +
+
+ +
+

Ordinal addition

+ +
+
+ +
+Lemma plus_ref : refines2 T1.T1add T1add.
+ +
+
+ +
+

Ordinal multiplication

+ +
+
+ +
+Section Proof_of_mult_ref.
+ +
+  Lemma T1mul_a0E (c : T1) : c × zero = zero.
+ +
+  Lemma T1mul_cons_cons_E n b n' b' :
+    cons zero n b × cons zero n' b' =
+      cons zero (n × n' + n + n') b'.
+ +
+ +
+  Lemma T1mulE4 a n b n' b' :
+    a != zero → (cons a n b) × (cons zero n' b') =
+                   cons a (n × n' + n + n') b.
+ +
+  Lemma multE4 a n b n' b' :
+    a ≠ Epsilon0.T1.zero →
+    Epsilon0.T1.T1mul (T1.cons a n b)
+      (T1.cons Epsilon0.T1.zero n' b') =
+      T1.cons a (n × n' + n + n') b.
+ +
+  Lemma T1mulE5 a n b a' n' b' :
+    a' != zero →
+    (cons a n b) × (cons a' n' b') =
+      cons (a + a') n' (T1mul (cons a n b) b').
+ +
+  Lemma multE5 a n b a' n' b' :
+    a' ≠ T1.zero →
+    Epsilon0.T1.T1mul (T1.cons a n b) (T1.cons a' n' b') =
+      T1.cons (T1.T1add a a') n' (T1.T1mul (T1.cons a n b) b').
+ +
+  Lemma h2g_cons a n b : h2g (T1.cons a n b)= cons (h2g a) n (h2g b).
+ +
+  Lemma g2h_cons a n b : g2h (cons a n b) = T1.cons (g2h a) n (g2h b).
+ +
+  Lemma h2g_zero : h2g T1.zero = zero.
+ +
+  Lemma g2h_zero : g2h zero = T1.zero.
+ +
+  Lemma mult_ref0 : refines2 T1.T1mul T1mul.
+ +
+End Proof_of_mult_ref.
+ +
+Lemma mult_ref : refines2 T1.T1mul T1mul.
+ +
+Lemma g2h_multE (a b : T1) : g2h (a × b) = T1.T1mul (g2h a) (g2h b).
+ +
+ +
+Lemma g2h_plusE (a b: T1) : g2h (a + b) = T1.T1add (g2h a) (g2h b).
+ +
+
+ +
+

Ordinal terms in normal form

+ +
+
+ +
+ +
+Lemma nf_ref (a: hT1) : T1.nf_b a = T1nf (h2g a).
+ +
+Lemma LT_ref : refinesRel T1.LT LT.
+ +
+Lemma LE_ref : refinesRel T1.LE LE.
+ +
+
+ +
+Limits, successors, etc +
+
+ +
+Lemma T1limit_ref (a:Epsilon0.T1.T1) : T1.T1limit a = T1limit (h2g a).
+ +
+Lemma T1is_succ_ref (a:Epsilon0.T1.T1): T1.T1is_succ a = T1is_succ (h2g a).
+ +
+ +
+Lemma hnf_g2h a : T1.nf (g2h a) = T1nf a.
+ +
+Lemma g2h_succ a : g2h (T1succ a) = T1.succ (g2h a).
+ +
+Lemma hlt_iff a b: T1.lt a b ↔ h2g a < h2g b.
+ +
+ +
+Lemma T1lt_iff a b: T1nf a → T1nf b →
+                          a < b ↔ g2h a t1< g2h b.
+ +
+Lemma T1le_iff (a b: T1):
+  a ≤ b ↔ T1.le (g2h a) (g2h b).
+ +
+
+ +
+

Well formed ordinals as a data type

+ +
+
+#[global] Notation hE0 := E0.E0.
+#[global] Notation hcnf := E0.cnf.
+#[global] Notation hE0lt := E0.E0lt.
+#[global] Notation hE0le := E0.E0le.
+#[global] Notation hE0zero := E0.E0zero.
+#[global] Notation hE0omega := E0.E0_omega.
+#[global] Notation hE0phi0 := E0.E0_phi0.
+#[global] Notation hE0fin := E0.E0fin.
+#[global] Notation hE0limit := E0.E0limit.
+#[global] Notation hE0is_succ := E0.E0is_succ.
+ +
+Record E0 := mkE0 { cnf : T1 ; _ : T1nf cnf == true}.
+Coercion cnf: E0 >-> T1.
+ +
+Canonical e0Sub := [subType for cnf].
+ +
+Check fun (x:E0) ⇒ val x.
+ +
+Remark omeganf : T1nf T1omega == true.
+ +
+Check (Sub T1omega).
+ +
+Check (Sub T1omega omeganf : e0Sub).
+ +
+Check (Sub T1omega omeganf : E0).
+ +
+Definition E0eqb (a b: E0):= cnf a == cnf b.
+ +
+Lemma gE0_eq_intro a b : cnf a = cnf b → a = b.
+ +
+Definition E0_eq_mixin : Equality.mixin_of E0.
+ +
+Definition E0_eqtype := Equality.Pack E0_eq_mixin.
+Canonical Structure E0_eqtype.
+ +
+ +
+Definition ppE0 (a: E0) := T1pp (cnf a).
+ +
+Definition E0lt (a b: E0) := cnf a < cnf b.
+Definition E0le (a b: E0) := cnf a ≤ cnf b.
+ +
+#[global, program] Definition E0zero: E0 := @mkE0 zero _.
+ +
+#[global, program]
+ Definition E0_succ (a: E0): E0 := @mkE0 (T1succ (cnf a)) _.
+ +
+#[global, program]
+ Definition E0_pred (a:E0) : E0:= @mkE0 (T1pred (cnf a)) _.
+ +
+Fixpoint E0fin (n:nat) : E0 :=
+  if n is p.+1 then E0_succ (E0fin p) else E0zero.
+ +
+#[program] Definition E0_omega: E0 := @mkE0 T1omega _.
+ +
+#[program] Definition E0_phi0 (a: E0) : E0 := @mkE0 (phi0 (cnf a)) _.
+ +
+#[program] Definition E0plus (a beta: E0) : E0 :=
+  @mkE0 (T1add (cnf a) (cnf beta)) _.
+ +
+#[program] Definition E0mul (a beta: E0) : E0 :=
+  @mkE0 (T1mul (cnf a) (cnf beta)) _.
+ +
+Lemma E0fin_cnf (n:nat) : cnf (E0fin n) = \F n.
+ +
+#[program] Definition E0_h2g (a: hE0): E0:= @mkE0 (h2g (E0.cnf a)) _.
+ +
+#[program] Definition E0_g2h (a: E0): hE0 := @E0.mkord (g2h (cnf a)) _.
+ +
+ +
+Definition E0limit a := hE0limit (E0_g2h a).
+ +
+Definition E0is_succ a := hE0is_succ (E0_g2h a).
+ +
+Lemma E0_h2g_nf (a:hE0) : T1nf (cnf (E0_h2g a)).
+ +
+Lemma gE0lt_iff a beta : E0lt a beta ↔ E0_g2h a o< E0_g2h beta.
+ +
+Lemma gE0le_iff a beta : E0le a beta ↔ E0_g2h a o≤ E0_g2h beta.
+ +
+Lemma E0_h2g_g2hK : cancel E0_g2h E0_h2g.
+ +
+Lemma E0_g2h_h2gK : cancel E0_h2g E0_g2h.
+ +
+Lemma g2h_E0_succ a : E0_g2h (E0_succ a)= E0.E0_succ (E0_g2h a).
+ +
+Lemma E0is_succ_succ a : E0is_succ (E0_succ a).
+ +
+ +
+Lemma E0is_succE a: E0is_succ a → {beta: E0 | a = E0_succ beta}.
+ +
+Lemma E0_eqE (x y: E0) : x = y ↔ (E0_g2h x = E0_g2h y).
+ +
+Lemma E0_diffE (x y: E0) : x ≠ y ↔ (E0_g2h x ≠ E0_g2h y).
+ +
+ +
+Lemma E0_pred_succK x : E0_pred (E0_succ x) = x.
+ +
+Lemma g2h_E0zero : E0_g2h E0zero = E0.E0zero.
+ +
+Lemma E0g2h_Fin i: E0_g2h (E0fin i) = E0.E0fin i.
+ +
+Lemma E0g2h_phi0 a : E0_g2h (E0_phi0 a) = E0.E0_phi0 (E0_g2h a).
+ +
+Lemma E0g2h_mulE (a b: E0): E0_g2h (E0mul a b) = E0.E0mul (E0_g2h a) (E0_g2h b).
+ +
+Lemma E0g2h_plusE (a b: E0): E0_g2h (E0plus a b)= E0.E0add (E0_g2h a) (E0_g2h b).
+ +
+Lemma E0g2h_omegaE : E0_g2h E0_omega = hE0omega.
+ +
+From Coq Require Import Relations Basics
+     Wellfounded.Inverse_Image Wellfounded.Inclusion.
+ +
+
+ +
+TODO: simplify this proof !!! +
+
+ +
+Lemma gE0lt_wf : well_founded E0lt.
+ +
+Lemma L1' (a: T1) : T1omega × (a × T1omega) = T1omega × a × T1omega.
+ +
+
+ +
+Sequences and limits +
+
+ +
+Definition g2h_seq (s: nat→ T1) n := g2h (s n).
+Definition h2g_seq (s: nat → hT1) n := h2g (s n).
+ +
+Definition gstrict_lub (s : nat → T1) (lambda : T1) :=
+  (∀ i : nat, LT (s i) lambda) ∧
+    (∀ a : T1, (∀ i : nat, LE (s i) a) → LE lambda a).
+ +
+Lemma strict_lub_ref (s:nat→ hT1) (lambda: hT1) :
+  strict_lub s lambda ↔ gstrict_lub (h2g_seq s) (h2g lambda).
+ +
+Search ( _ × ?beta = ?beta)%ca.
+ +
+Search ( _ × ?beta = ?beta)%t1.
+ +
+#[global] Instance T1compare : Compare T1:=
+  fun a beta ⇒ compare (g2h a) (g2h beta).
+ +
+ +
+Lemma compare_g2h (a beta : T1):
+  compare (g2h a) (g2h beta) = compare a beta .
+ +
+ Lemma compare_h2g (a beta: hT1) :
+   compare (h2g a) (h2g beta) =compare a beta .
+ +
+
+ +
+

Make E0 an ordinal notation

+ +
+
+ +
+Lemma T1compare_correct (a b: T1):
+  CompSpec eq T1lt a b (compare a b).
+ +
+#[global] Instance E0compare: Compare E0 :=
+  fun (alpha beta: E0) ⇒ T1compare (cnf alpha) (cnf beta).
+ +
+Lemma E0compare_correct (alpha beta : E0) :
+  CompSpec eq E0lt alpha beta (compare alpha beta).
+ +
+#[global] Instance E0_sto : StrictOrder E0lt.
+ +
+#[global] Instance E0_comp : Comparable E0lt compare.
+ +
+ +
+#[global] Instance Epsilon0 : ON E0lt compare.
+ +
+Locate T1omega.
+ +
+
+ +
+

Abstract properties of arithmetic functions (with SSreflect inequalities)

+ + +
+
+ +
+Definition strict_mono (f: nat → nat) :=
+  ∀ n p, (n< p)%N → (f n < f p)%N.
+ +
+Definition dominates_from (n : nat) (g f : nat → nat) :=
+  ∀ p : nat, (n ≤ p)%N → (f p < g p)%N.
+ +
+Definition dominates g f := ∃ n : nat, dominates_from n g f .
+ +
+Definition dominates_strong g f := {n : nat | dominates_from n g f}.
+ +
+Definition fun_le f g := ∀ n:nat, (f n ≤ g n)%N.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.T1Choice.html b/theories/html/gaia_hydras.T1Choice.html new file mode 100644 index 00000000..ee14084e --- /dev/null +++ b/theories/html/gaia_hydras.T1Choice.html @@ -0,0 +1,216 @@ + + + + + +gaia_hydras.T1Choice + + + + +
+ + + +
+ +

Library gaia_hydras.T1Choice

+ +
+
+ +
+Experimental !!!! This file is a draft !!! +
+
+ +
+ +
+From mathcomp Require Import all_ssreflect zify.
+From Coq Require Import Logic.Eqdep_dec.
+From hydras Require Import DecPreOrder ON_Generic T1 E0.
+From gaia Require Export ssete9.
+From gaia_hydras Require Import T1Bridge.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+ Type T1 vs generic trees +
+
+ +
+Fixpoint T12Tree (a: T1): GenTree.tree nat :=
+  if a is cons b n c
+  then GenTree.Node n [:: T12Tree b; T12Tree c]
+  else GenTree.Leaf 0.
+ +
+Fixpoint Tree2T1 (t: GenTree.tree nat): option T1 :=
+  match t with
+  | GenTree.Leaf 0 ⇒ Some zero
+  | GenTree.Node n [:: t1; t2] ⇒
+      match Tree2T1 t1, Tree2T1 t2 with
+      | Some b, Some c ⇒ Some (cons b n c)
+      | _, _ ⇒ None
+      end
+  | _ ⇒ None
+  end.
+ +
+Lemma TreeT1K : pcancel T12Tree Tree2T1.
+ +
+
+ +
+to remove (useless) +
+
+Lemma T12Tree_inj: injective T12Tree.
+ +
+Definition T1mixin :
+  Countable.mixin_of T1 := PcanCountMixin TreeT1K.
+ +
+Canonical T1Choice :=
+  Eval hnf in ChoiceType T1 (CountChoiceMixin T1mixin).
+ +
+Example ex_pos: ∃ alpha: T1, zero != alpha.
+ +
+Example some_pos: T1 := xchoose ex_pos.
+ +
+Example some_pos' : T1 := choose (fun p : T1 ⇒ zero != p)
+                                 T1omega.
+ +
+Goal (zero: T1Choice) != some_pos'.
+ +
+
+ +
+ = EqType T1 (EqMixin (@T1eqP)) + : eqType +
+
+ +
+ +
+
+ +
+ = Choice.Pack + {| + Choice.base := EqMixin (@T1eqP); + Choice.mixin := PcanChoiceMixin (pcan_pickleK TreeT1K) + |} + : choiceType + +
+ + +
+
+ +
+Definition T1_le_Mixin := leOrderMixin T1Choice.
+ +
+Definition T1min a b := if T1lt a b then a else b.
+Definition T1max a b := if T1lt a b then b else a.
+ +
+Lemma T1ltE x y : T1lt x y = (y != x) && T1le x y.
+ +
+Lemma T1minE x y : T1min x y = (if T1lt x y then x else y).
+ +
+Lemma T1maxE x y : T1max x y = (if T1lt x y then y else x).
+ +
+Lemma T1le_asym: ssrbool.antisymmetric T1le.
+ +
+Definition T1leOrderMixin : leOrderMixin T1Choice :=
+  LeOrderMixin T1ltE T1minE T1maxE T1le_asym T1le_trans T1le_total.
+ +
+Canonical T1orderType :=
+  @OrderOfChoiceType tt T1Choice T1leOrderMixin.
+ +
+Goal @Order.le tt T1orderType T1omega T1omega.
+ +
+Check T1omega: T1orderType.
+ +
+Goal ((T1omega:T1orderType) ≤ (T1omega:T1orderType))%O.
+ +
+Notation "x <= y" := (@Order.le _ T1orderType x y).
+ +
+Notation "x < y" := (@Order.lt _ T1orderType x y).
+ +
+About Order.le.
+ +
+Goal @Order.le tt T1orderType T1omega T1omega.
+Import Order.POrderTheory.
+ +
+Goal T1omega ≤ T1omega.
+Set Printing All.
+ +
+Goal T1omega < T1succ T1omega.
+ +
+ +
+Goal ~~ (<%O (T1omega:T1orderType) (T1omega:T1orderType)).
+ +
+Goal ~~(T1omega < T1omega).
+ +
+Goal ~~ @Order.lt tt (T1orderType) T1omega T1omega.
+ +
+Goal T1omega ≤ T1omega.
+ +
+ +
+Print E0.
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.T2Bridge.html b/theories/html/gaia_hydras.T2Bridge.html new file mode 100644 index 00000000..4442c65b --- /dev/null +++ b/theories/html/gaia_hydras.T2Bridge.html @@ -0,0 +1,114 @@ + + + + + +gaia_hydras.T2Bridge + + + + +
+ + + +
+ +

Library gaia_hydras.T2Bridge

+ +
+
+ +
+

Bridge between Hydra-battle's and Gaia's T1 (Experimental)

+ + +
+
+ +
+From mathcomp Require Import all_ssreflect zify.
+From Coq Require Import Logic.Eqdep_dec.
+From hydras Require Import DecPreOrder ON_Generic T2 Gamma0.
+ +
+From gaia Require Export ssete9.
+Import Gamma0.
+ +
+Set Implicit Arguments.
+ +
+ +
+
+ +
+Hydra-Battles' type for ordinal terms below Gamma00 +
+
+Check Gamma0.T2.T2.
+Print Gamma0.T2.T2.
+#[global] Notation hT2 := hydras.Gamma0.T2.T2.
+ +
+
+ +
+Gaia's type for ordinal terms below epsilon0 +
+
+#[global] Notation T2 := ssete9.Gamma0.T2.
+ +
+#[global] Notation hcons := gcons.
+#[global] Notation hzero := hydras.Gamma0.T2.zero.
+ +
+Fixpoint h2g (alpha : hT2) : T2 :=
+  match alpha with
+    hzero ⇒ zero
+  | hcons a b n c ⇒ cons (h2g a) (h2g b) n (h2g c)
+  end.
+ +
+Fixpoint g2h (alpha : T2) : hT2 :=
+  match alpha with
+    zero ⇒ hzero
+  | cons a b n c ⇒ hcons (g2h a)(g2h b) n (g2h c)
+  end.
+ +
+Lemma h2g_g2hK : cancel g2h h2g.
+ +
+Lemma g2h_h2gK : cancel h2g g2h.
+ +
+Lemma g2h_eqE (a b: T2) : g2h a = g2h b ↔ a = b.
+ +
+Lemma h2g_eqE (a b :hT2) : h2g a = h2g b ↔ a = b.
+ +
+ +
+Check (fun a b : hT2 ⇒ compare a b).
+ +
+About T2lt.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.nfwfgaia.html b/theories/html/gaia_hydras.nfwfgaia.html new file mode 100644 index 00000000..99c626d2 --- /dev/null +++ b/theories/html/gaia_hydras.nfwfgaia.html @@ -0,0 +1,3901 @@ + + + + + +gaia_hydras.nfwfgaia + + + + +
+ + + +
+ +

Library gaia_hydras.nfwfgaia

+ +
+
+ +
+Copy of gaia's ssete9.v (for experimenting Alectryon documentation) Changes from original proof script are signaled as additions/changes or inside Alectryon snippets + +
+ +

Ordinals in Pure Coq

+ + Copyright INRIA (2013-2013) Marelle Team (José Grimm). + After a work of Castéran + +
+
+ +
+From mathcomp
+  Require Import ssreflect ssrfun ssrbool eqtype ssrnat.
+From mathcomp Require Import fintype bigop.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+

prelude

+ +
+ + Useful lemmas missing in ssreflect +
+
+ +
+Lemma if_simpl (p: bool): (if p then true else false) = p.
+ +
+Lemma ltn_simpl1 n n': ((n' + n).+1 < n) = false.
+ +
+Lemma eqn_simpl1 n n': ((n' + n).+1 == n) = false.
+ +
+Lemma ltn_simpl2 n n' n'':
+  (n × n' + n + n' < n × n'' + n + n'') = (n' < n'').
+ +
+Lemma eqn_simpl2 n n' n'':
+   (n × n' + n + n' == n × n'' + n + n'') = (n' == n'').
+ +
+Lemma ltn_add_le m1 m2 n1 n2: m1 < n1 → m2 ≤ n2 → m1 + m2 < n1 + n2.
+ +
+Lemma ltn_add_el m1 m2 n1 n2: m1 ≤ n1 → m2 < n2 → m1 + m2 < n1 + n2.
+ +
+Lemma ltn_add_ll m1 m2 n1 n2: m1 < n1 → m2 < n2 → m1 + m2 < n1 + n2.
+ +
+
+ +
+The ssreflect comparison on nat is WF +
+
+ +
+Lemma lt_wf: well_founded (fun (a b:nat) ⇒ a < b).
+ +
+
+ +
+

Example 1

+ +
+ + An example of function defined by transfinite induction using Fix +
+
+Module Wf_ex.
+ +
+Definition f_spec f n :=
+   if n is m.+2 then (f (f m.+1).+1 ).+1 else 0.
+ +
+Lemma f_spec_simp f n: (∀ n, f n = f_spec f n) → f n = n.-1.
+ +
+Lemma f0 n p: p ≤ n → p.+2 ≤ n.+2.
+ +
+Definition f1 :
+  ∀ x, (∀ z, z < x → {y:nat |y ≤ z.-1}) →
+  {y:nat | y ≤ x.-1}.
+ +
+Definition f2 := Fix lt_wf _ f1.
+Definition f (x:nat): nat := sval (f2 x).
+ +
+Lemma f_eqn x: f2 x = f1 (fun y _ ⇒ f2 y).
+ +
+Lemma f_correct n: f n = f_spec f n.
+ +
+End Wf_ex.
+ +
+
+ +
+

Example 2

+ +
+ + Second example, + f(n) = 1 + \sum(i < n) f(i) +
+
+ +
+Module Wfsum.
+ +
+Definition psum (f: nat → nat) n := \sum_(i< n) (f i).
+Definition f_spec f:= ∀ n, f n = (psum f n).+1.
+ +
+Lemma f_spec_simp f n: f_spec f → f n = 2 ^ n.
+ +
+Lemma psum_exten n f g :
+   (∀ k, k < n → f k = g k) → (psum f n).+1 = (psum g n).+1.
+ +
+Lemma lt_dec n m: {n <m} + {~~ (n < m) }.
+ +
+Definition extension (n : nat) (p : ∀ k : nat, k < n → nat) k :=
+  match lt_dec k n with
+    | left x ⇒ p k x
+    | _ ⇒ 0 end.
+ +
+Definition f1 (n : nat) (h : ∀ z : nat, z < n → nat) :=
+    (psum (extension h) n).+1.
+ +
+Definition f2 := Fix lt_wf _ f1.
+ +
+Lemma f_eqn x: f2 x = f1 (n:=x) (fun y _ ⇒ f2 y).
+ +
+Definition f (x:nat): nat := f2 x.
+ +
+Lemma f_correct: f_spec f.
+ +
+End Wfsum.
+ +
+
+ +
+

Example 3

+ +
+ + We consider here only even numbers, show that comparison is WF, define a + function by transfinite induction and show it is correct. +
+
+ +
+Module Wf_ex3.
+ +
+Definition lte n m := [&& ~~ odd n, ~~ odd m & n < m].
+ +
+Lemma lte_wf: well_founded lte.
+ +
+Definition f_spec f n :=
+   if n is m.+4 then (f (f (m.+2)).*2.+2 ).+1 else 0.
+ +
+Lemma f_spec_simp f n: ~~ odd n → (∀ n, ~~odd n → f n = f_spec f n)
+   → f n = (n.-1)./2.
+ +
+Lemma f_spec_simp1 f n: (∀ n, ~~odd n → f n = f_spec f n)
+   → f (n.*2.+2) = n.
+ +
+Lemma f_spec_simp2 f n: (∀ n, f n = f_spec f n) → f(n.*2.+3) = n.
+ +
+Lemma f0a y n: odd n = false → odd n.+2 ∨ y ≤ (n.+2)./2.-1 →
+   y ≤ n./2 ∧ lte (y.*2).+2 n.+4.
+ +
+Lemma f0b a b: odd a.*2.+2 ∨ b ≤ (a.*2.+2)./2.-1 → b ≤ a.
+ +
+Lemma f0c n: odd n = false → lte n.+2 n.+4.
+ +
+Lemma odd_dec n : {odd n} + {odd n = false}.
+ +
+Definition f1 :
+  ∀ x, (∀ z, lte z x → {y:nat | odd z ∨ y ≤ (z./2).-1}) →
+  {y:nat | odd x ∨ y ≤ (x./2).-1}.
+ +
+Definition f2 := Fix lte_wf _ f1.
+Definition f (x:nat): nat := sval (f2 x).
+ +
+Lemma f_eqn x: f2 x = f1 (fun y _ ⇒ f2 y).
+ +
+Lemma f_correct n: ~~odd n → f n = f_spec f n.
+ +
+End Wf_ex3.
+ +
+
+ +
+

More on accessiblity

+ +
+ + We show that there is no striclty decreasing function with domain nat +
+
+Section Sequences.
+ +
+Variable A : Set.
+Variable R : A → A → Prop.
+ +
+Lemma acc_rec a b: R a b → Acc R b → Acc R a.
+ +
+Hypothesis W : well_founded R.
+ +
+Theorem not_decreasing :
+  ¬ (∃ f : nat → A, (∀ i:nat, R (f i.+1) (f i))).
+ +
+End Sequences.
+ +
+
+ +
+We show here an induction principle; we could use it for ordinals in NF +form. +
+
+ +
+Section restricted_recursion.
+ +
+Variables (A:Type)(P:A→Prop)(R:A→A→Prop).
+ +
+Definition restrict a b := [/\ P a, R a b & P b].
+ +
+Definition well_founded_P := ∀ a, P a → Acc restrict a.
+ +
+Lemma P_well_founded_induction_type :
+       well_founded_P →
+       ∀ Q : A → Type,
+       (∀ x : A, P x → (∀ y : A, P y → R y x → Q y) → Q x) →
+       ∀ a : A, P a → Q a.
+End restricted_recursion.
+Module CantorOrdinal.
+ +
+ +
+
+ +
+

The type T1

+ +
+ + This type represents all ordinals less that + ε 0 , + via the +Cantor Normal Form. More exactly cons a n b represents + ω A * (n.+1) + B + if a represents A and b represents B. +
+
+ +
+ +
+Inductive T1 : Set :=
+  zero : T1
+| cons : T1 → nat → T1 → T1.
+ +
+ +
+
+ +
+

Equality

+ +
+ + we define a boolean equality, the use the mechanism of canonical + structures provided by ssreflect +
+
+ +
+Fixpoint T1eq x y {struct x} :=
+  match x, y with
+  | zero, zero ⇒ true
+  | cons a n b, cons a' n' b' ⇒ [&& T1eq a a', n== n' & T1eq b b' ]
+  | _, _ ⇒ false
+end.
+ +
+Lemma T1eqP : Equality.axiom T1eq.
+ +
+Canonical T1_eqMixin := EqMixin T1eqP.
+Canonical T1_eqType := Eval hnf in EqType T1 T1_eqMixin.
+ +
+Arguments T1eqP {x y}.
+ +
+Lemma T1eqE a n b a' n' b':
+  (cons a n b == cons a' n' b') = [&& a == a', n== n' & b == b' ].
+ +
+Declare Scope cantor_scope.
+Delimit Scope cantor_scope with ca.
+Open Scope cantor_scope.
+ +
+
+ +
+Some definitions +
    +
  • φ0(x) is cons x 0 zero , + it represents ω x + +
    • +
  • one is φ0(0) + +
    • +
  • omega is φ0(1) + +
    • +
  • bad is an example of an ordinal not in normal form + +
    • +
  • fun n := \F n is the canonical injection of nat into T1 + +
    • +
  • the log of cons a n b is a + +
    • +
  • an ordinal is AP if it is in the image of φ0. + +
    • +
+ +
+
+ +
+Definition phi0 a := cons a 0 zero.
+Definition one := cons zero 0 zero.
+Definition T1omega := phi0 (phi0 zero).
+Definition T1bad := cons zero 0 T1omega.
+Definition T1nat (n:nat) : T1 :=
+  if n is p.+1 then cons zero p zero else zero.
+Definition T1log a := if a is cons a _ _ then a else zero.
+Definition T1ap x := if x is cons a n b then ((n==0) && (b==zero)) else false.
+ +
+Notation "\F n" := (T1nat n)(at level 29) : cantor_scope.
+ +
+Lemma T1F_inj: injective T1nat.
+ +
+Lemma T1phi0_zero : phi0 zero = \F 1.
+Lemma T1phi0_zero' : phi0 zero = one.
+Lemma T1log_phi0 x : T1log (phi0 x) = x.
+Lemma T1ap_phi0 x: T1ap (phi0 x).
+ +
+
+ +
+

Order on T1

+ +
+ + We give here a recursion definition of comparison. +Essentially, φ0(x) is strictly increasing, + +
+
+ +
+Fixpoint T1lt x y {struct x} :=
+  if x is cons a n b then
+    if y is cons a' n' b' then
+      if a < a' then true
+      else if a == a' then
+         if (n < n')%N then true
+         else if (n == n') then b < b' else false
+         else false
+      else false
+  else if y is cons a' n' b' then true else false
+where "x < y" := (T1lt x y) : cantor_scope.
+ +
+Definition T1le (x y :T1) := (x == y) || (x < y).
+Notation "x <= y" := (T1le x y) : cantor_scope.
+Notation "x >= y" := (y ≤ x) (only parsing) : cantor_scope.
+Notation "x > y" := (y < x) (only parsing) : cantor_scope.
+ +
+ +
+Lemma T1lenn x: x ≤ x.
+ +
+#[local] Hint Resolve T1lenn : core.
+ +
+Lemma T1ltnn x: (x < x) = false.
+ +
+Lemma T1lt_ne a b : a < b → (a == b) = false.
+ +
+Lemma T1lt_ne' a b : a < b → (b == a) = false.
+ +
+Lemma T1ltW a b : (a < b) → (a ≤ b).
+ +
+Lemma T1le_eqVlt a b : (a ≤ b) = (a == b) || (a < b).
+ +
+Lemma T1lt_neAle a b : (a < b) = (a != b) && (a ≤ b).
+ +
+Lemma T1ltn0 x: (x < zero) = false.
+Lemma T1le0n x: zero ≤ x.
+Lemma T1len0 x: (x ≤ zero) = (x == zero).
+Lemma T1lt0n x: (zero < x) = (x != zero).
+ +
+Lemma T1ge1 x: (one ≤ x) = (x != zero).
+ +
+Lemma T1lt1 x: (x < one) = (x==zero).
+ +
+Lemma T1nat_inc n p : (n < p)%N = (\F n < \F p).
+ +
+
+ +
+This is an alternative version of less-or-equal +
+
+ +
+Lemma T1le_consE a n b a' n' b':
+ (cons a n b ≤ cons a' n' b') =
+    if a < a' then true
+      else if a == a' then
+         if (n < n')%N then true
+         else if (n == n') then b ≤ b' else false
+         else false.
+ +
+
+ +
+We have exactly one of: a is less than, greater than, or equal to b +
+
+ +
+Lemma T1lt_trichotomy a b: [|| (a< b), (a==b) | (b < a)].
+ +
+Lemma T1lt_anti b a: a < b → (b < a) = false.
+ +
+Lemma T1leNgt a b: (a ≤ b) = ~~ (b < a).
+ +
+Lemma T1ltNge a b: (a < b) = ~~ (b ≤ a).
+ +
+Lemma T1eq_le m n : (m == n) = ((m ≤ n) && (n ≤ m)).
+ +
+Lemma T1le_total m n : (m ≤ n) || (n ≤ m).
+ +
+
+ +
+The next three definitions are similar to to those defined in ssrnat. +we shall use T1ltgtP a lot. + +
+
+ +
+CoInductive T1ltn_xor_geq m n : bool → bool → Set :=
+  | T1LtnNotGeq of m < n : T1ltn_xor_geq m n false true
+  | T1GeqNotLtn of n ≤ m : T1ltn_xor_geq m n true false.
+ +
+CoInductive T1leq_xor_gtn m n : bool → bool → Set :=
+  | T1GeqNotGtn of m ≤ n : T1leq_xor_gtn m n true false
+  | T1GtnNotLeq of n < m : T1leq_xor_gtn m n false true.
+ +
+CoInductive compare_T1 m n : bool → bool → bool → Set :=
+  | CompareT1Lt of m < n : compare_T1 m n true false false
+  | CompareT1Gt of m > n : compare_T1 m n false true false
+  | CompareT1Eq of m = n : compare_T1 m n false false true.
+ +
+Lemma T1leP x y : T1leq_xor_gtn x y (x ≤ y) (y < x).
+ +
+Lemma T1ltP m n : T1ltn_xor_geq m n (n ≤ m) (m < n).
+ +
+Lemma T1ltgtP m n : compare_T1 m n (m < n) (n < m) (m == n).
+ +
+
+ +
+We show here transitivity of comparison, using T1ltgtP . +
+
+ +
+Lemma T1lt_trans b a c: a < b → b < c → a < c.
+ +
+Lemma T1lt_le_trans b a c: a < b → b ≤ c → a < c.
+ +
+Lemma T1le_lt_trans b a c: a ≤ b → b < c → a < c.
+ +
+Lemma T1le_trans b a c: a ≤ b → b ≤ c → a ≤ c.
+ +
+
+ +
+The following lemma implies +x < ω x, so all ordinals are less than + ε 0 +
+
+ +
+Lemma head_lt_cons a n b: a < cons a n b.
+ +
+Lemma T1lt_cons_le a n b a' n' b': (cons a n b < cons a' n' b') → (a ≤ a').
+ +
+Lemma T1le_cons_le a n b a' n' b': (cons a n b ≤ cons a' n' b') → (a ≤ a').
+ +
+Lemma phi0_lt a b: (phi0 a < phi0 b) = (a < b).
+ +
+Lemma phi0_le a b: (phi0 a ≤ phi0 b) = (a ≤ b).
+ +
+Lemma phi0_lt1 a n b a': (cons a n b < phi0 a') = (a < a').
+ +
+
+ +
+

Normal form

+ +
+ + There exists a strictly infinite decreasing sequence of ordinals, +so the order is not well founded +
+
+ +
+Theorem lt_not_wf : ¬ (well_founded T1lt).
+ +
+
+ +
+We say that cons a n b is NF if +)b <φ0(a). +If b is cons a' n' b', this says that b is less than b'. +If a is zero, this says that b=0. + +
+
+ +
+Fixpoint T1nf x :=
+  if x is cons a _ b then [&& T1nf a, T1nf b & b < phi0 a ]
+  else true.
+ +
+Lemma T1nf_cons0 a n: T1nf a → T1nf (cons a n zero).
+ +
+Lemma T1nf_cons_cons a n a' n' b' : T1nf (cons a n (cons a' n' b')) → a' < a.
+ +
+Lemma T1nf_consa a n b: T1nf (cons a n b) → T1nf a.
+ +
+Lemma T1nf_consb a n b: T1nf (cons a n b) → T1nf b.
+ +
+Lemma T1nf_finite1 n b: T1nf (cons zero n b) = (b == zero).
+ +
+Lemma T1nf_finite n b: T1nf (cons zero n b) → (b = zero).
+ +
+Lemma T1nfCE: ~~(T1nf T1bad).
+ +
+
+ +
+We show here that the restriction of T1lt to NF ordinals is well-founded, + then prove two induction principles. Note that nf_Wf' says every + NF x is accessible by the relation: u<v, u and v NF. + If x is not NF it is trivially accessible. The proof is a bit tricky +
+
+ +
+Section AddLocalNotation.
+  #[local] Notation LT :=
+    (@restrict T1 (fun x : T1 ⇒ is_true (T1nf x))
+               (fun x y : T1 ⇒ is_true (x < y))).
+ +
+  Lemma nf_Wf : well_founded (restrict T1nf T1lt). +
+  Lemma nf_Wf' : well_founded_P T1nf T1lt.
+ +
+  Lemma T1transfinite_induction P:
+    (∀ x, T1nf x → (∀ y, T1nf y → y < x → P y) → P x) →
+    ∀ a, T1nf a → P a.
+ +
+  Lemma T1transfinite_induction_Q (P: T1 → Type) (Q: T1 → Prop):
+    (∀ x:T1, Q x → T1nf x →
+                  (∀ y:T1, Q y → T1nf y → y < x → P y) → P x) →
+    ∀ a, T1nf a → Q a → P a.
+ +
+  Lemma T1nf_rect (P : T1 → Type):
+    P zero →
+    (∀ n: nat, P (cons zero n zero)) →
+    (∀ a n b n' b', T1nf (cons a n b) →
+                         P (cons a n b) →
+                         b' < phi0 (cons a n b) →
+                         T1nf b' →
+                         P b' →
+                         P (cons (cons a n b) n' b')) →
+    ∀ a, T1nf a → P a.
+ +
+End AddLocalNotation.
+ +
+
+ +
+

Successor

+ +
+ + We say that cons a n b is +
    +
  • limit if a is non-zero, b is limit or zero + +
    • +
  • finite if a is zero + +
    • +
  • a successor if a is zero or b is a successor + +
    • +
+ and define its +
    +
  • successor as \F (n+2) or cons a n (succ b) + +
    • +
  • predecessor as \F n or cons a n (pred b) + +
    • +
  • split u,v as cons a n x, y if b split as x,y and a is non-zero; + and as 0,n+1 if a is zero + +
    • +
+Note that if a=0, the quantity b is ignored; but when x is NF, +then b is zero. +
+
+ +
+Fixpoint T1limit x :=
+  if x is cons a n b then
+    if a==zero then false else (b== zero) || T1limit b
+  else false.
+ +
+Definition T1finite x := if x is cons a n b then a == zero else true.
+ +
+Fixpoint T1is_succ x :=
+  if x is cons a n b then (a==zero) || T1is_succ b else false.
+ +
+Fixpoint T1succ (c:T1) : T1 :=
+  if c is cons a n b
+  then if a == zero then cons zero n.+1 zero else cons a n (T1succ b)
+  else one.
+ +
+Fixpoint T1pred (c:T1) : T1 :=
+  if c is cons a n b then
+    if (a==zero) then \F n else (cons a n (T1pred b))
+  else zero.
+ +
+Fixpoint T1split x:=
+  if x is cons a n b then
+    if a==zero then (zero, n.+1) else
+      let: (x, y) := T1split b in (cons a n x,y)
+  else (zero,0).
+ +
+Lemma split_limit x: ((T1split x).2 == 0) = ((x==zero) || T1limit x).
+ +
+Lemma split_is_succ x: ((T1split x).2 != 0) = (T1is_succ x).
+ +
+Lemma split_finite x: ((T1split x).1 == zero) = T1finite x.
+ +
+Lemma split_succ x: let:(y,n):= T1split x in T1split (T1succ x) = (y,n.+1).
+ +
+Lemma split_pred x: let:(y,n):= T1split x in T1split (T1pred x) = (y,n.-1).
+ +
+Lemma split_le x : (T1split x).1 ≤ x.
+ +
+Lemma nf_split x : T1nf x → T1nf (T1split x).1.
+ +
+Lemma T1finite1 n: T1finite (\F n).
+ +
+Lemma T1finite2 x: T1finite x → T1nf x → x = \F ((T1split x).2).
+ +
+Lemma T1finite2CE: T1finite T1bad ∧ ∀ n, T1bad ≠ \F n.
+ +
+Lemma T1finite_succ x: T1finite x → T1finite (T1succ x).
+ +
+Lemma T1succ_nat n: T1succ (\F n) = \F (n.+1).
+ +
+Lemma nf_omega : T1nf T1omega.
+Lemma nf_finite n: T1nf (\F n).
+Lemma nf_phi0 a: T1nf (phi0 a) = T1nf a.
+Lemma nf_log a: T1nf a → T1nf (T1log a).
+ +
+
+ +
+An ordinal is zero, limit or a successor, exclusively. When we split x +the first component is zero or limit, the second is a natural number +
+
+ +
+Lemma limit_pr1 x: (x == zero) (+) (T1limit x (+) T1is_succ x).
+ +
+Lemma split_limit1 x (y:= (T1split x).1): (y == zero) || (T1limit y).
+ +
+
+ +
+If x is limit, if y is less than x, so is the successor of y +
+
+ +
+Lemma limit_pr x y: T1limit x → y < x → T1succ y < x.
+ +
+Lemma pred_le a: T1pred a ≤ a.
+ +
+Lemma pred_lt a: T1is_succ a → T1pred a < a.
+ +
+Lemma succ_lt a: a < T1succ a.
+ +
+Lemma nf_succ a: T1nf a → T1nf (T1succ a).
+ +
+Lemma nf_pred a: T1nf a → T1nf (T1pred a).
+ +
+Lemma succ_pred x: T1nf x → T1is_succ x → x = T1succ (T1pred x).
+ +
+Lemma succ_predCE: T1is_succ T1bad ∧ ∀ y, T1bad ≠ T1succ y.
+ +
+Lemma succ_p1 x: T1is_succ (T1succ x).
+ +
+Lemma pred_succ x: T1nf x → T1pred (T1succ x) = x.
+ +
+Lemma pred_succ_CE: T1pred (T1succ T1bad) ≠ T1bad.
+ +
+Lemma succ_inj x y: T1nf x → T1nf y → (T1succ x == T1succ y) = (x==y).
+ +
+Lemma succ_injCE: one ≠ T1bad ∧ (T1succ one = T1succ T1bad).
+ +
+Lemma lt_succ_succ x y: T1succ x < T1succ y → x < y.
+ +
+Lemma le_succ_succ x y: x ≤ y → T1succ x ≤ T1succ y.
+ +
+Lemma lt_succ2CE: one < T1bad ∧ T1bad < T1succ one.
+ +
+Lemma lt_succ_succE x y:
+  T1nf x → T1nf y → (T1succ x < T1succ y) = (x < y).
+ +
+
+ +
+Some properties of comparison and successor +
+
+ +
+Lemma le_succ_succE x y:
+  T1nf x → T1nf y → (T1succ x ≤ T1succ y) = (x ≤ y).
+ +
+Lemma lt_succ_le_1 a b : T1succ a ≤ b → a < b.
+ +
+Lemma lt_succ_le_2 a b: T1nf a → a < T1succ b → a ≤ b.
+ +
+Lemma lt_succ_le_3 a b: T1nf a → (a < T1succ b) = (a ≤ b).
+ +
+Lemma lt_succ_le_4 a b: T1nf b → (a < b) = (T1succ a ≤ b).
+ +
+Lemma phi0_log a: a < phi0 (T1succ (T1log a)).
+ +
+Lemma tail_lt_cons a n b: b < phi0 a → b < cons a n b.
+ +
+
+ +
+

Addition

+ +
+ + The definition of addition and subtraction given here are straightforward +given our interpretation of cons +
+
+ +
+Fixpoint T1add x y :=
+  if x is cons a n b then
+    if y is cons a' n' b' then
+      if a < a' then cons a' n' b'
+      else if a' < a then (cons a n (b + (cons a' n' b')))
+           else (cons a (n+n').+1 b')
+    else x
+  else y
+where "a + b" := (T1add a b) : cantor_scope.
+ +
+Fixpoint T1sub x y :=
+  if x is cons a n b then
+    if y is cons a' n' b' then
+      if x < y then zero
+      else if a' < a then cons a n b
+           else if (n < n')%N then zero
+                else if (a==zero) then
+                       if (n ==n') then zero else cons zero ((n-n').-1) zero
+                     else if (n == n') then b - b' else cons a (n-n').-1 b
+    else x
+  else zero
+where "x - y" := (T1sub x y):cantor_scope.
+ +
+
+ +
+Easy properties +
+
+ +
+Lemma succ_is_add_one a: T1succ a = a + one.
+ +
+Lemma add1Nfin a: ~~ T1finite a → one + a = a.
+ +
+Lemma sub1Nfin a: ~~ T1finite a → a - one = a.
+ +
+Lemma sub1a x: x != zero → T1nf x → x = one + (x - one).
+ +
+Lemma sub1b x: T1nf x → x = (one + x) - one.
+ +
+Lemma sub_1aCE (a:= cons zero 0 T1bad) : one + (a - one) != a.
+ +
+Lemma sub_1bCE (a:= cons zero 0 T1bad) : (one + a - one) != a.
+ +
+Lemma T1add0n : left_id zero T1add.
+Lemma T1addn0: right_id zero T1add.
+ +
+Lemma T1subn0 x: x - zero = x.
+ +
+Lemma T1subnn x: x - x = zero.
+ +
+Lemma add_int n m : \F n + \F m = \F (n +m)%N.
+ +
+Lemma sub_int n m : \F n - \F m = \F (n -m)%N.
+ +
+Lemma add_fin_omega n: \F n + T1omega = T1omega.
+ +
+Lemma fooCE (x:= T1bad):
+  ~~T1limit x /\(∀ u v, T1limit u → x ≠ u + \F v.+1).
+ +
+Lemma split_add x: let: (y,n) :=T1split x in T1nf x →
+                                             (x == y + \F n) && ((y==zero) || T1limit y ).
+ +
+Lemma add_to_cons a n b:
+  b < phi0 a → cons a n zero + b = cons a n b.
+ +
+Lemma addC_CE (a := one) (b := T1omega):
+  [/\ T1nf a, T1nf b & a + b ≠ b + a].
+ +
+
+ +
+We say that x is AP is the sum of two ordinals less than x is +less than x. This conditionq holds if x has the form +)φ0(a); the converse is true when x is non-zero. +We may also assume everything NF. + +
+
+ +
+Lemma ap_pr0 a (x := phi0 a) b c:
+  b < x → c < x → b + c < x.
+ +
+Lemma ap_pr1 c:
+  (∀ a b, a < c → b < c → a + b < c) →
+  (c== zero) || T1ap c.
+ +
+Lemma ap_pr2 c:
+  T1nf c → c ≠ zero →
+  (∀ a b, T1nf a → T1nf b → a < c → b < c → a + b < c) →
+  T1ap c.
+ +
+Lemma ap_pr2CE (c := cons T1bad 1 zero):
+  (∀ a b, T1nf a → T1nf b → a < c → b < c → a + b < c).
+ +
+
+ +
+Alternate definition of an AP: if a<x then a+x=x. + +
+
+ +
+Lemma add_simpl1 a n b n' b': a != zero →
+                              cons a n b + cons zero n' b' = cons a n (b + cons zero n' b').
+ +
+Lemma add_simpl2 n b a' n' b': a' != zero →
+                                cons zero n b + cons a' n' b' = cons a' n' b'.
+ +
+Lemma ap_pr3 a b (x := phi0 a): b < x → b + x = x.
+ +
+Lemma ap_pr4 x: (∀ b, b < x → b + x = x) → (x == zero) || T1ap x.
+ +
+
+ +
+It follows tthat the sum of two NF ordinals is NF +
+
+ +
+Lemma nf_add a b: T1nf a → T1nf b → T1nf (a + b).
+ +
+
+ +
+Results anbout addition subtraction comparison +
+
+ +
+Lemma T1add_eq0 m n: (m + n == zero) = (m == zero) && (n == zero).
+ +
+Lemma add_le1 a b: a ≤ a + b.
+ +
+Lemma add_le2 a b: b ≤ a + b.
+ +
+Lemma minus_lt a b: a < b → a - b = zero.
+ +
+Lemma minus_le a b: a ≤ b → a - b = zero.
+ +
+Lemma T1sub0 a: a - zero = a.
+ +
+Lemma nf_sub a b: T1nf a → T1nf b → T1nf (a - b).
+ +
+Lemma sub_le1 a b : T1nf a → (a - b) ≤ a.
+ +
+Lemma sub_pr a b: T1nf b → (a + b) - a = b.
+ +
+Lemma add_inj a b c : T1nf b → T1nf c → a + b = a + c → b = c.
+ +
+Lemma sub_pr1 a b: T1nf b → a ≤ b → b = a + (b - a).
+ +
+Lemma sub_pr1CE: (one ≤ T1bad) && (T1bad != one + (T1bad - one)).
+ +
+Lemma sub_pr1r a b: T1nf a → a - b = zero → a ≤ b.
+ +
+Lemma omega_minus_one : T1omega - one = T1omega.
+ +
+Lemma sub_nz a b: T1nf b → a < b → (b - a) != zero.
+ +
+Lemma sub_nzCE (a := one) (b := (cons zero 0 one)):
+  (a < b) && (b-a == zero).
+ +
+
+ +
+Associativity of addition +
+
+ +
+Lemma T1addA c1 c2 c3: c1 + (c2 + c3) = (c1 + c2) + c3.
+ +
+Lemma T1addS a b : (a + T1succ b) = T1succ (a+ b).
+ +
+Lemma T1le_add2l p m n : (p + m ≤ p + n) = (m ≤ n).
+ +
+Lemma T1lt_add2l p m n : (p + m < p + n) = (m < n).
+ +
+Lemma T1lt_add2r p m n : (m + p < n + p ) → (m < n).
+ +
+Lemma T1le_add2r p m n : (m ≤n) → (m + p ≤ n + p).
+ +
+Lemma T1eq_add2l p m n : (p + m == p + n) = (m == n).
+ +
+Lemma add_le3 a b: a = a + b → b = zero.
+ +
+Lemma add_le4 a b: b != zero → a < a + b.
+ +
+Lemma sub_pr1rCE (a := T1bad) (b := one) : (a - b == zero) && (b < a).
+ +
+
+ +
+

Limits

+ +
+ + A limit ordinal is the supremum of a sequence of ordinals. We first show + that some sequences are unbounded. We then show that, if the sequence is + bounded, there is a least upper bound, more preciselly, if a property is + satisfied for some NF ordinal, it is satisfied for a least NF aordinal. + This requires teh excluded middel principle. + +
+
+ +
+Fixpoint omega_tower (n:nat) : T1 :=
+  if n is p.+1 then phi0 (omega_tower p) else one.
+ +
+Lemma omega_tower_nf n: T1nf (omega_tower n).
+ +
+Lemma omega_tower_unbounded x: ¬ (∀ n, (omega_tower n) < x).
+ +
+Definition ex_middle:=
+  ∀ (P: T1 → Prop), let Q := ∃ x, (T1nf x ∧ P x) in Q ∨ ¬Q.
+ +
+Lemma ex_middle_pick (P: T1 → Prop): ex_middle →
+                                       (∃ x, (T1nf x ∧ P x)) ∨ (∀ x, T1nf x → ¬ (P x)).
+ +
+Lemma min_exists (P: T1 → Prop) x: ex_middle →
+                                    T1nf x → (P x) →
+                                    ∃ y, T1nf y ∧ P y ∧ ∀ z, T1nf z → P z → y ≤ z.
+ +
+
+ +
+

Definition

+ +
+ + We say that x is the limit of f, or the supremum of the f(i) if +f(i)<x (we could use less-or-equal here as f will be strictly increasing) +and if moreover, every ordinal less than x is bounded by some f(i). +Note that x is then the least upper bound. The trouble is that each f(i) +may be NF and x is not. Thus , we give an alternate definition. +Trouble is: a function may have more then one limit (at most one of them +being NF). + +
+
+ +
+Notation Tf := (nat → T1).
+ +
+Definition limit_v1 (f: Tf) x :=
+  (∀ n, f n < x) ∧ (∀ y, y < x → (∃ n, y ≤ f n)).
+ +
+Definition limit_v2 (f: Tf) x :=
+  (∀ n, f n < x) ∧ (∀ y, T1nf y → y < x → (∃ n, y ≤ f n)).
+ +
+Lemma limit_unique1 (f: Tf) x x' :limit_v1 f x → limit_v1 f x' →
+                                  x = x'.
+ +
+Lemma limit_unique2 (f: Tf) x x' : limit_v2 f x → limit_v2 f x' →
+                                   T1nf x → T1nf x'→ x = x'.
+ +
+Definition omega_plus_n n := cons one 0 (cons zero n zero).
+ +
+Lemma nf_omega_plus_n n : T1nf ( omega_plus_n n).
+ +
+Lemma limit_CE1: limit_v1 omega_plus_n (cons one 0 T1omega).
+ +
+Lemma limit_CE2: limit_v2 omega_plus_n (cons one 1 zero).
+ +
+Lemma limit_CE3: limit_v2 omega_plus_n (cons one 0 T1omega).
+ +
+
+ +
+

The normal form

+ To each ordinal, one can associate another ordinal that is NF. +However, this is in general incompatible with other operations +
+
+ +
+Fixpoint toNF x :=
+  if x is cons a n b then (cons (toNF a) n zero) + toNF b else zero.
+ +
+Lemma nf_toNF x: T1nf (toNF x).
+ +
+Lemma toNF_nz x : toNF x = zero → x = zero.
+ +
+Lemma toNF_nf x: T1nf x → toNF x = x.
+ +
+Lemma toNF_mon x : x ≤ toNF x.
+ +
+Lemma toNF_ex1 x: toNF (cons zero 0 x) = one + toNF x.
+ +
+Lemma toNF_ex2: toNF (cons one 0 T1omega) = cons one 1 zero.
+ +
+Lemma toNF_succ (x := cons zero 0 one): toNF (T1succ x) != T1succ (toNF x).
+ +
+Lemma toNF_pred (x := cons zero 0 one): toNF (T1pred x) != T1pred (toNF x).
+ +
+
+ +
+

Realizing the limit

+ This is a simplification of the code given for the type T3 below. +We define a function F(x); so that for any limit ordinal x, if f= F(x), +then f is stictly increasing (of type nat → T1), and its limit is x. + +
+
+ +
+Lemma fincP (f: Tf) :
+  (∀ n, f n < f n.+1) →
+  (∀ n m, (n < m)%N → f n < f m).
+ +
+Definition limit_of (f: Tf) x :=
+  [/\ (∀ n m, (n < m)%N → f n < f m), limit_v2 f x & T1nf x].
+ +
+Lemma limit_unique f x y: limit_of f x → limit_of f y → x = y.
+ +
+Lemma limit_lub f x y: limit_of f x → (∀ n, f n ≤ y) → T1nf y →
+                       x ≤ y.
+ +
+Definition phi1 a (f:Tf) := fun n ⇒ a + f n.
+Definition phi2 (f:Tf) := fun n ⇒ phi0 (f n).
+Definition phi3 a:= fun n ⇒ cons a n zero.
+ +
+Lemma limit1 a b f: T1nf a → limit_of f b → limit_of (phi1 a f) (a + b).
+ +
+Lemma limit2 b f: limit_of f b → limit_of (phi2 f) (phi0 b).
+ +
+Lemma limit3 a: T1nf a → limit_of (phi3 a) (phi0 (T1succ a)).
+ +
+
+ +
+

Normal functions

+ We say that f:T2 → T2 is a normal function if it is striclly increasing +and the suppremum of all f(y), for y<x is f(x) whenever x is limit. +Everything is assumed NF. +
+
+ +
+Fixpoint limit_fct x :=
+  if x is cons a n b then
+    if (b==zero) then
+      if(a==zero) then phi3 a
+      else if (T1is_succ a)
+           then if (n==0) then phi3 (T1pred a) else
+                  phi1 (cons a n.-1 zero) (phi3 (T1pred a))
+           else if(n==0) then (phi2 (limit_fct a))
+                else phi1 (cons a n.-1 zero) (phi2 (limit_fct a))
+    else phi1 (cons a n zero) (limit_fct b)
+  else phi3 zero.
+ +
+Lemma limit_prop x: T1nf x → T1limit x → limit_of (limit_fct x) x.
+ +
+Definition sup (f: T1→ T1) x z :=
+  [/\ T1nf z,
+   (∀ y, T1nf y → y < x → f y ≤ z) &
+   (∀ z', T1nf z' → z' < z → ∃ y,
+                   [&& T1nf y, y < x & z' < f y])].
+ +
+Definition normal f:=
+  [/\ ∀ x, T1nf x → T1nf (f x),
+     (∀ x y, T1nf x → T1nf y → x < y → f x < f y)&
+     (∀ x, T1nf x → T1limit x → sup f x (f x)) ].
+ +
+Lemma sup_unique f x z z': sup f x z → sup f x z' → z = z'.
+ +
+Lemma sup_Oalpha_zero: sup id zero zero.
+ +
+Lemma sup_Oalpha_succ x: T1nf x → sup id (T1succ x) x.
+ +
+Lemma sup_Oalpha_limit x: T1nf x → T1limit x → sup id x x.
+ +
+
+ +
+ Identity is normal, composition of normal functions is normal, +addition is normal when the firtst argument is fixed. A normal function maps limit ordinals to limit ordinls +
+
+ +
+Lemma normal_id: normal id.
+ +
+Lemma normal_limit f x: normal f → T1nf x → T1limit x → T1limit (f x).
+ +
+Lemma add_normal a: T1nf a → normal (T1add a).
+ +
+Lemma normal_compose f g:
+  normal f → normal g → normal (f \o g).
+ +
+
+ +
+

multiplication

+ +
+ + There is a unique way to define multiplication (for NF arguments) +compatible with our interpretation of cons. In the case where a and a' +are zero, we could use zero or b instead of b'. With the current +implementation, multiplication is associative, and there is a distributivity +law +
+
+ +
+Fixpoint T1mul (c1 c2 : T1) {struct c2}:T1 :=
+  if c2 is cons a' n' b' then
+    if c1 is cons a n b then
+       if((a==zero) && (a' == zero)) then cons zero (n×n' + n + n')%N b'
+       else if(a'==zero) then cons a (n×n' + n + n')%N b
+       else cons (a + a') n' ((cons a n b) × b')
+     else zero
+  else zero
+where "c1 * c2" := (T1mul c1 c2) : cantor_scope.
+ +
+Lemma mul_na n b x: (cons zero n b) × x = (cons zero n zero) × x.
+ +
+Lemma T1muln0 x: x × zero = zero.
+ +
+Lemma T1mul0n x: zero × x = zero.
+ +
+Lemma mul_int n m : \F n × \F m = \F (n ×m)%N.
+ +
+Lemma mul_phi0 a b: phi0 (a + b) = phi0 a × phi0 b.
+ +
+Lemma T1mul_eq0 x y: (x × y == zero) = (x== zero) || (y == zero).
+ +
+Lemma T1mul_eq1 a b: T1nf a → (a× b == one) = ((a == one) && (b == one)).
+ +
+Lemma mul_distr: right_distributive T1mul T1add.
+ +
+Lemma mulA: associative T1mul.
+ +
+
+ +
+Note that in some case the product of x and one is not x +
+
+ +
+Lemma T1muln1 x: T1nf x → x × one = x.
+ +
+Lemma T1mul1n x: one × x = x.
+ +
+Lemma T1mul1nCE (x := T1bad): x × one ≠ x.
+ +
+Lemma T1muln1_CE x:
+  (x == x × one) =
+    (if x is cons a n b then ((a != zero) || (b== zero)) else true).
+ +
+Lemma mul_succ x y: T1nf x → x × (T1succ y) = x × y + x.
+ +
+Lemma T1lt_mul2l x y z: x != zero → T1nf z → ((x ×y < x ×z) = (y < z)).
+ +
+Lemma mulnf0 a n b: a != zero → b < phi0 a → (cons zero n zero) × b < phi0 a.
+ +
+Lemma nf_mul a b: T1nf a → T1nf b → T1nf (a × b).
+ +
+Lemma T1lt_mul2r x y z: (y × x < z × x) → (y < z).
+ +
+Lemma T1le_mul2l x y z : x != zero → T1nf y →
+    (x ×y ≤ x ×z) = (y ≤ z).
+ +
+Lemma T1le_mul2r x y z: (y ≤ z) → (y × x ≤ z × x).
+ +
+Lemma T1eq_mul2l p m n : p != zero → T1nf m → T1nf n →
+   (p × m == p × n) = (m == n).
+ +
+Lemma T1le_pmulr x a: T1nf a → x != zero → a ≤ a × x.
+ +
+Lemma T1le_pmulrCE (x:= \F1 ) (a:=T1bad) : (a ≤ a × x) = false.
+ +
+Lemma T1le_pmulrl x a: x != zero → a ≤ x × a.
+ +
+Lemma T1le_mulCE (m1:= one) (m2:= T1bad) (n1 := \F1) (n2 := one) :
+   (m1 ≤ n1) && (m2 ≤ n2) && ( m1 × m2 ≤ n1 × n2) == false.
+ +
+Lemma T1le_mul m1 m2 n1 n2 : T1nf m2 → m1 ≤ n1 → m2 ≤ n2 →
+   m1 × m2 ≤ n1 × n2.
+ +
+
+ +
+

Preparation of the exponention

+ +The prouct of an integer and omega is omega. This holds in fact for any limit +ordinals. We give here a formula for the product of omega and x, and show +that this is a limit ordinal. The converse holds. +
+
+ +
+Lemma mul_fin_omega n: (\F n.+1) × T1omega = T1omega.
+ +
+Lemma mul_int_limit n y: T1limit y → \F n.+1 × y = y.
+ +
+Lemma T1mul_omega a n b:
+   T1omega × (cons a n b) =
+   if (a== zero) then cons one n zero else cons (one + a) n (T1omega × b).
+ +
+Lemma mul_omega_limit x: x != zero → T1limit (T1omega × x).
+ +
+Fixpoint T1div_by_omega x :=
+  if x is cons a n b then cons (a - one) n (T1div_by_omega b) else zero.
+ +
+Lemma div_by_omega_pr x: T1nf x → ((x==zero) || T1limit x)
+  → T1omega × (T1div_by_omega x) = x.
+ +
+
+ +
+We show here every ordinal x is the product of omega and y, +to which an integer is added. We study the behaviour +of this decomposition and multiplication +
+
+ +
+Lemma nf_div_by_omega x: T1limit x → T1nf x → T1nf (T1div_by_omega x).
+ +
+Lemma nf_revCE u v: T1bad ≠ T1omega × u + \F v.
+ +
+Lemma add_simpl3 x y: y != zero →
+  x + x × (T1omega × y) = x × (T1omega × y).
+ +
+Lemma plus_int_Ox n x: x != zero → \F n + T1omega × x = T1omega × x.
+ +
+Lemma nf_rev x (u := (T1div_by_omega (T1split x).1)) (v:= (T1split x).2):
+ T1nf x → T1nf u ∧ x = T1omega × u + \F v.
+ +
+Lemma nf_rev_unique u v (x:= T1omega ×u + \F v): T1nf u →
+    u = T1div_by_omega (T1split x).1 ∧ v = (T1split x).2.
+ +
+Lemma nf_rev_sum x y
+  (u := T1div_by_omega (T1split x).1) (n:= (T1split x).2)
+  (v := T1div_by_omega (T1split y).1) (m:= (T1split y).2)
+  (w := T1div_by_omega (T1split (x+y)).1) (p:= (T1split (x+y)).2):
+  T1nf x → T1nf y →
+  if (v==zero) then (w = u ∧ p = (n + m)%N) else (w = u+v ∧ p = m).
+ +
+Lemma mul_sum_omega a n: a != zero →
+   (T1omega × a + \F n) × T1omega = (T1omega × a) × T1omega.
+ +
+Lemma nf_rev_prod x y
+  (u := T1div_by_omega (T1split x).1) (n:= (T1split x).2)
+  (v := T1div_by_omega (T1split y).1) (m:= (T1split y).2)
+  (w := T1div_by_omega (T1split (x×y)).1) (p:= (T1split (x×y)).2):
+  T1nf x → T1nf y →
+  if (u== zero)
+     then if (n == 0) then (w = zero ∧ p = 0)
+     else (w = v ∧ p = (n×m)%N)
+  else if (m==0) then (w = u × T1omega ×v ∧ p = 0)
+  else (w = u × T1omega ×v + u × \F m ∧ p = n).
+ +
+
+ +
+

Normality of multiplication

+ +If a is a non-zero ordinal, the multiplication by a is normal. +This means, if b is limit, the supremum of all a ×c for c<b is a×b. +We show this for omega, and for some special cases. +This is equivalent to existence of ordinal division. + +
+
+ +
+Lemma mul_omega_pr1 a: a != zero → T1nf a →
+  sup (T1mul a) T1omega (a × T1omega).
+ +
+Lemma mul_omega2_pr1 a (u:= cons one 1 zero): a != zero → T1nf a →
+  sup (T1mul a) u (a × u).
+ +
+Lemma mul_omega_pr3 a b c: a != zero → c != zero →
+   T1nf a → T1nf b → T1nf c →
+  sup (T1mul a) c (a × c) →
+  sup (T1mul a) (b+c) (a × (b + c)).
+ +
+ +
+ +
+
+ +
+

Exponentiation

+ +
+ + In order to compute a ^b , we first write b as the sum of a limit +ordinal and an integer n. Computing a ^n is trivial. The limit ordinal +is omega times c; if a is at least one, then a ^omega = omega ^d +for some d, and the result is a ^ (omega × c) = omega ^(d×c)=phi0(d×c) . +This leads to the following definitions. + +
+
+ +
+Fixpoint exp_F a n :=
+  if n is p.+1 then a × (exp_F a p) else one.
+ +
+Definition exp_O a b :=
+  if (a==zero) then if (b== zero) then one else a
+  else if (a== one) then one
+  else if (T1finite a) then (phi0 b)
+  else phi0 ((T1log a) × T1omega × b).
+ +
+Definition T1exp a b:=
+  (exp_O a (T1div_by_omega (T1split b).1)) × (exp_F a ( (T1split b).2)).
+ +
+Notation "a ^ b" := (T1exp a b) : cantor_scope.
+ +
+
+ +
+Properties of exp_O +
+
+ +
+Lemma expO_mul1 a b: (exp_O a b) × (one) = exp_O a b.
+ +
+Lemma nf_expO a b: T1nf a → T1nf b → T1nf (exp_O a b).
+ +
+Lemma expO_n0 x: exp_O x zero = one.
+ +
+Lemma expO_1n n: exp_O (one) n = one.
+ +
+Lemma expO_eq0 a b: (exp_O a b == zero) = ((a== zero) && (b != zero)).
+ +
+Lemma expO_eq1 a b: (exp_O a b == one) = ((a== one) || (b == zero)).
+ +
+Lemma expO_add z u v: exp_O z u × exp_O z v = exp_O z (u + v).
+ +
+
+ +
+Properties of exp_F +
+
+Lemma nf_expF a n: T1nf a → T1nf (exp_F a n).
+ +
+Lemma expF_add a n m: (exp_F a n) × (exp_F a m) = exp_F a (n + m).
+ +
+Lemma expF_mul a n m: exp_F a (n × m) = exp_F (exp_F a n) m.
+ +
+Lemma expF_1n n: exp_F (one) n = one.
+ +
+Lemma expF_eq0 a n: (exp_F a n == zero) = ((a== zero) && (n != 0)).
+ +
+Lemma expF_eq1 a n: T1nf a → (exp_F a n == one) = ((a== one) || (n == 0)).
+ +
+
+ +
+Properties of exp +
+
+ +
+Lemma nf_exp a b: T1nf a → T1nf b → T1nf (a ^b).
+ +
+Lemma exp00: zero ^zero = one.
+ +
+Lemma expx0 x: x ^zero = one.
+ +
+Lemma expx_pnat x n b: x ^ (cons zero n b) = exp_F x n.+1.
+ +
+Lemma expx_nat x n: x ^ \F n = exp_F x n.
+ +
+Lemma expx1 x: T1nf x → x ^ one = x.
+ +
+Lemma expx1CE: T1bad ^ one = one.
+ +
+Lemma exp2omega n: (\F n.+2)^ T1omega = T1omega.
+ +
+Lemma exp1x x: one ^ x = one.
+ +
+Lemma exp_eq0 x y: x^y == zero = ((x==zero) && (y != zero)).
+ +
+Lemma exp0nz x: x != zero → zero ^ x = zero.
+ +
+Lemma exp_eq1 x y: T1nf x → T1nf y →
+  (x^y == one) = ((x== one) || (y == zero)).
+ +
+Lemma exp_int a b: (\F a) ^ (\F b) = \F (a ^b%N).
+ +
+Lemma exp_consCE1 (x := \F 2) (a := zero) (n := 0)(b := T1omega):
+   x ^(cons a n b) != x ^(cons a n zero) × x ^b.
+ +
+Lemma pow_omega x: T1nf x → T1omega ^x = phi0 x.
+ +
+
+ +
+

Existence and uniqueness of the Cantor Normal Form

+ +
+
+ +
+Lemma cantor_exists a n b: T1nf (cons a n b) →
+    cons a n b = (T1omega^a) × (\F n.+1) + b.
+ +
+Lemma cantor_unique a n b a' n' b':
+  T1nf (cons a n b) → T1nf (cons a' n' b') →
+  (T1omega^a) × (\F n.+1) + b = (T1omega^a') × (\F n'.+1) + b' →
+  (a=a' ∧ n = n' ∧ b = b').
+ +
+Lemma cantor_CE1 : T1omega ^ T1bad != phi0 T1bad.
+ +
+Lemma cantorCE2: cons zero 0 T1omega != (T1omega^ zero) × (one) + T1omega.
+ +
+Lemma cantorCE3: cons T1bad 0 zero != (T1omega^ T1bad) × (one) + zero.
+ +
+Lemma T1log_prod a b: a != zero → b != zero →
+    T1log(a × b) = T1log a + T1log b.
+ +
+Lemma T1log_exp0 x n: T1nf x → T1log (exp_F x n) = (T1log x) × (\F n).
+ +
+Lemma T1log_exp1 z x: T1nf z → T1nf x → ~~ T1finite z →
+   T1log (z ^ x) = (T1log z) × x.
+ +
+Lemma T1log_exp2 z u v: (z == zero) = false → (z == one) = false →
+   T1finite z → T1nf u → T1log (z ^ (T1omega × u + \F v)) = u.
+ +
+Lemma exp_FO z n v: v != zero → exp_F z n × exp_O z v = exp_O z v.
+ +
+Lemma exp_FO1 z v n m: T1nf z → T1nf v → v != zero → n != 0 →
+  exp_O z (v × \F n) × exp_F z m = exp_F (exp_O z v × exp_F z m) n.
+ +
+Lemma exp_FO2 z m u: T1nf z → m != 0 → exp_O (exp_F z m) u = exp_O z u.
+ +
+Lemma exp_FO3 z x u (w := T1div_by_omega (T1split x).1):
+   T1nf z → T1nf w → (w == zero) = false → (z == zero) = false →
+  exp_O (z ^ x) u = phi0( T1log (z ^x) × T1omega × u).
+ +
+
+ +
+

Basic Properties

+ +
+
+ +
+Lemma exp_sum x y z: T1nf x → T1nf y → z ^(x+y) = z ^x × z ^y.
+ +
+Lemma exp_prod x y z: T1nf z → T1nf x → T1nf y → z ^(x ×y) = (z ^x) ^y.
+ +
+Lemma pow_mon1 x y z: T1nf x → T1nf y → T1nf z → x != zero →
+   y ≤ z → x ^y ≤ x ^z.
+ +
+Lemma pow_mon2 x y z: T1nf x → T1nf y → T1nf z → x != zero → x != one →
+   y < z → x ^y < x ^z.
+ +
+Lemma T1le_pmull x a: x != zero → a ≤ x × a.
+ +
+Lemma pow_mon3 x y z: T1nf x → x ≤ y → x ^z ≤ y ^z.
+ +
+End CantorOrdinal.
+ +
+Export CantorOrdinal.
+ +
+
+ +
+

The type T2

+ +
+
+ +
+Module Gamma0.
+ +
+
+ +
+

Definition and Equality

+ +
+ + This is like T1 with one more argument +
+
+ +
+Inductive T2 : Set :=
+  zero : T2
+| cons : T2 → T2 → nat → T2 → T2.
+ +
+Declare Scope g0_scope.
+Delimit Scope g0_scope with g0.
+Open Scope g0_scope.
+ +
+Fixpoint T2eq x y {struct x} :=
+  match x, y with
+  | zero, zero ⇒ true
+  | cons a b n c, cons a' b' n' c' ⇒
+      [&& T2eq a a', T2eq b b', n== n' & T2eq c c' ]
+  | _, _ ⇒ false
+end.
+ +
+Lemma T2eqP : Equality.axiom T2eq.
+ +
+Canonical T2_eqMixin := EqMixin T2eqP.
+Canonical T2_eqType := Eval hnf in EqType T2 T2_eqMixin.
+ +
+Arguments T2eqP {x y}.
+ +
+Lemma T2eqE a b n d a' b' n' d':
+  (cons a b n d == cons a' b' n' d') =
+      [&& a == a', b == b', n== n' & d == d' ].
+ +
+
+ +
+ We write psi a b instead of cons a b 0 0; we introduce omega and +epsilon0. We consider also the size, this is really the depth of the object +
+
+ +
+Notation "[ x , y ]" := (cons x y 0 zero) (at level 0) :g0_scope.
+ +
+Definition T2nat p := if p is n.+1 then cons zero zero n zero else zero.
+ +
+Notation "\F n" := (T2nat n)(at level 29) : g0_scope.
+ +
+Definition psi a b := [a, b].
+ +
+Definition one := [zero, zero].
+ +
+Definition omega := [zero, one].
+ +
+Definition epsilon0 := [one,zero].
+ +
+Fixpoint T1T2 (c: T1) : T2 :=
+  if c is CantorOrdinal.cons a n b then cons zero (T1T2 a) n (T1T2 b)
+  else zero.
+ +
+Fixpoint size x :=
+  if x is cons a b n c then
+     (maxn (size a) (maxn (size b) (size c))).+1
+  else 0.
+ +
+Lemma size_prop1 a b n c (l:= size (cons a b n c)):
+   [/\ size a < l, size b < l, size c < l & size [a, b] ≤ l]%N.
+ +
+Lemma size_prop a b n c a' b' n' c'
+   (l := (size (cons a b n c) + size (cons a' b' n' c'))%N) :
+   (size c + size c' < l)%N ∧ (size [a, b] + size b' < l)%N ∧
+   (size a' + size a < l)%N ∧ (size b + size b' < l)%N ∧
+   (size b + size [a', b'] < l)%N ∧ (size a + size a' < l)%N.
+ +
+
+ +
+

Order

+ Comparing ordinals is complicated. We are looking for the fixpoint of +some complicated expression F(a,b), in which the psi-parts of a and b +appear as arguments of F. Thus, a definition by induction is impossible. +However if l is the some of the size if the arguments of F, then all calls +of F have a smaller value. Thus, we use a definition by induction on l. +All proofs will be by induction on l as well. + +
+
+ +
+Definition lt_rec f x y :=
+ if x is cons a b n c then
+   if y is cons a' b' n' c' then
+     if ( ((f a a') && (f b ([a', b'])))
+         || ((a == a') && (f b b'))
+         || ((f a' a) && (f ([a, b]) b'))
+         || (((f a' a) && ([a, b] == b'))))
+     then true
+     else if ((a== a') && (b==b')) then
+       if (n < n')%N then true
+       else if (n == n') then (f c c') else false
+       else false
+   else false
+ else if y is cons a' b' n' c' then true else false.
+ +
+Fixpoint T2lta k:=
+ if k is k.+1 then lt_rec (T2lta k) else fun x y ⇒ false.
+ +
+Definition T2lt a b := T2lta ((size a) + size b).+1 a b.
+Definition T2le (x y :T2) := (x == y) || (T2lt x y).
+Notation "x < y" := (T2lt x y) : g0_scope.
+Notation "x <= y" := (T2le x y) : g0_scope.
+Notation "x >= y" := (y ≤ x) (only parsing) : g0_scope.
+Notation "x > y" := (y < x) (only parsing) : g0_scope.
+ +
+
+ +
+Main result: Our comparison is a fix-point +
+
+ +
+Lemma T2ltE x y : x < y = lt_rec T2lt x y.
+ +
+
+ +
+This is how we compare two psi-terms +
+
+Definition lt_psi a b a' b':=
+      ((a < a') && (b < [a', b']))
+   || ((a == a') && (b < b'))
+   || ((a' < a) && ([a, b] < b'))
+   || ((a' < a) && ([a, b] == b')).
+ +
+Lemma T2lt_psi a b a' b': [a,b] < [a', b'] = lt_psi a b a' b'.
+ +
+Lemma T2lt_consE a b n c a' b' n' c' :
+   cons a b n c < cons a' b' n' c' =
+     if (lt_psi a b a' b') then true
+     else if ((a== a') && (b==b')) then
+       if (n < n')%N then true
+       else if (n == n') then (c < c') else false
+       else false.
+ +
+
+ +
+Less-or-equal +
+
+ +
+Lemma T2le_consE a b n c a' b' n' c' :
+   cons a b n c ≤ cons a' b' n' c' =
+     if (lt_psi a b a' b') then true
+     else if ((a== a') && (b==b')) then
+       if (n < n')%N then true
+       else if (n == n') then (c ≤ c') else false
+       else false.
+ +
+Lemma T2ltn0 x: (x < zero) = false.
+Lemma T2lt0n x: (zero < x) = (x != zero).
+Lemma T2le0n x: zero ≤ x.
+Lemma T2len0 x: (x ≤ zero) = (x == zero).
+Lemma omega_lt_epsilon0: omega < epsilon0.
+ +
+Lemma T2ltnn x: (x < x) = false.
+ +
+Lemma T2lt_ne a b : a < b → (a == b) = false.
+ +
+Lemma T2lt_ne' a b : a < b → (b == a) = false.
+ +
+Lemma T2ltW a b : (a < b) → (a ≤ b).
+ +
+Lemma T2le_eqVlt a b : (a ≤ b) = (a == b) || (a < b).
+ +
+Lemma T2lt_neAle a b : (a < b) = (a != b) && (a ≤ b).
+ +
+Lemma T2lenn x: x ≤ x.
+ +
+#[local] Hint Resolve T2lenn : core.
+ +
+Lemma T2ge1 x: (one ≤ x) = (x != zero).
+ +
+Lemma T2lt1 x: (x < one) = (x==zero).
+ +
+Lemma T2nat_inc n p : (n < p)%N = (\F n < \F p).
+ +
+Lemma psi_lt1 a b c n a' b':
+   cons a b n c < [a', b'] = ([a, b] < [a', b']).
+ +
+Lemma psi_lt2 a b n c n' c': cons a b n' c' < cons a b n c =
+   (if (n' < n)%N then true else if n' == n then c' < c else false).
+ +
+Lemma T1T2_inj n p : (n == p) = (T1T2 n == T1T2 p).
+ +
+Lemma T1T2_inc n p : (n < p)%ca = (T1T2 n < T1T2 p)%g0.
+ +
+
+ +
+First two non-trivial results +
+
+ +
+Lemma T2lt_anti b a: a < b → (b < a) = false.
+ +
+Lemma T2lt_trichotomy a b: [|| (a< b), (a==b) | (b < a)].
+ +
+
+ +
+what follows is the same as for T1 +
+
+ +
+Lemma T2leNgt a b: (a ≤ b) = ~~ (b < a).
+ +
+Lemma T2ltNge a b: (a < b) = ~~ (b ≤ a).
+ +
+Lemma T2eq_le m n : (m == n) = ((m ≤ n) && (n ≤ m)).
+ +
+Lemma T2le_total m n : (m ≤ n) || (n ≤ m).
+ +
+CoInductive T2ltn_xor_geq m n : bool → bool → Set :=
+  | T2LtnNotGeq of m < n : T2ltn_xor_geq m n false true
+  | T2GeqNotLtn of n ≤ m : T2ltn_xor_geq m n true false.
+ +
+CoInductive T2leq_xor_gtn m n : bool → bool → Set :=
+  | T2GeqNotGtn of m ≤ n : T2leq_xor_gtn m n true false
+  | T2GtnNotLeq of n < m : T2leq_xor_gtn m n false true.
+ +
+CoInductive compare_T2 m n : bool → bool → bool → Set :=
+  | CompareT2Lt of m < n : compare_T2 m n true false false
+  | CompareT2Gt of m > n : compare_T2 m n false true false
+  | CompareT2Eq of m = n : compare_T2 m n false false true.
+ +
+Lemma T2leP x y : T2leq_xor_gtn x y (x ≤ y) (y < x).
+ +
+Lemma T2ltP m n : T2ltn_xor_geq m n (n ≤ m) (m < n).
+ +
+Lemma T2ltgtP m n : compare_T2 m n (m < n) (n < m) (m == n).
+ +
+Lemma T2lt_psi_aux a b a' b': a < a' → b < [a', b'] → [a,b] < [a',b'].
+ +
+Lemma T2gt1 x: (one < x) = ((x != zero) && (x != one)).
+ +
+
+ +
+Second non-trivial result +
+
+ +
+Theorem T2lt_trans b a c: a < b → b < c → a < c.
+ +
+Lemma T2lt_le_trans b a c: a < b → b ≤ c → a < c.
+ +
+Lemma T2le_lt_trans b a c: a ≤ b → b < c → a < c.
+ +
+Lemma T2le_trans b a c: a ≤ b → b ≤ c → a ≤ c.
+ +
+Lemma T2le_psi1 a b n c: [a, b] ≤ cons a b n c.
+ +
+Lemma T2lt_psi_b a b: b < [a,b].
+ +
+Lemma T2lt_psi_a a b: a < [a,b].
+ +
+
+ +
+

Normal form

+ Same as in T1. +TODO:: show that compraison is well-founded for NF + +
+
+ +
+Fixpoint T2nf x :=
+  if x is cons a b n c then [&& T2nf a, T2nf b, T2nf c & c < [a,b] ]
+  else true.
+ +
+Lemma T2nf_cons_cons a b n a' b' n' c':
+  T2nf(cons a b n (cons a' b' n' c')) =
+   [&& [a', b'] < [a, b], T2nf a, T2nf b & T2nf(cons a' b' n' c') ].
+ +
+Lemma nf_psi a b n: T2nf (cons a b n zero) = T2nf a && T2nf b.
+ +
+Lemma T2nf_consE a b n c:
+    T2nf (cons a b n c) = [&& T2nf a, T2nf b, T2nf c & c < [a,b] ].
+ +
+Lemma nf_omega : T2nf omega.
+Lemma nf_one : T2nf one.
+Lemma nf_finite n: T2nf (\F n).
+ +
+
+ +
+

Successor and predecessor

+ Same as for T1 +
+
+ +
+Lemma lt_tail a b n c: T2nf (cons a b n c) → c < cons a b n c.
+ +
+Lemma T1T2range1 x: T1T2 x < epsilon0.
+ +
+Lemma T1T2range2 x: T2nf x → x < epsilon0 → {y: T1 | x = T1T2 y}.
+ +
+Definition T2finite x:=
+  if x is cons a b n c then ([a,b]==one) else true.
+ +
+Fixpoint T2limit x :=
+  if x is cons a b n c then
+    if ([a,b]==one) then false else (c== zero) || T2limit c
+  else false.
+ +
+Fixpoint T2split x:=
+ if x is cons a b n c then
+      if ([a,b]==one) then (zero, n.+1) else
+     let: (x, y) := T2split c in (cons a b n x,y)
+   else (zero,0).
+ +
+Lemma T2nf_finite a b n c: [a,b]==one → T2nf (cons a b n c) → c = zero.
+ +
+Lemma split_finite x: ((T2split x).1 == zero) = T2finite x.
+ +
+Lemma T2finite2 x: T2finite x → T2nf x → x = \F ((T2split x).2).
+ +
+Lemma omega_least_inf1 x: T2finite x → x < omega.
+ +
+Lemma omega_least_inf2 x: ~~ T2finite x → omega ≤ x.
+ +
+Lemma split_limit x: ((T2split x).2 == 0) = ((x==zero) || T2limit x).
+ +
+Fixpoint T2is_succ x :=
+  if x is cons a b n c then ([a,b]==one) || T2is_succ c else false.
+ +
+Fixpoint T2succ x :=
+  if x is cons a b n c
+     then if ([a,b]==one) then \F n.+2 else cons a b n (T2succ c)
+  else one.
+ +
+Fixpoint T2pred x :=
+  if x is cons a b n c then
+     if ([a,b]==one) then \F n else (cons a b n (T2pred c))
+  else zero.
+ +
+Lemma split_is_succ x: ((T2split x).2 != 0) = (T2is_succ x).
+ +
+Lemma split_succ x: let:(y,n):= T2split x in T2split (T2succ x) = (y,n.+1).
+ +
+Lemma split_pred x: let:(y,n):= T2split x in T2split (T2pred x) = (y,n.-1).
+ +
+Lemma split_le x : (T2split x).1 ≤ x.
+ +
+Lemma nf_split x : T2nf x → T2nf (T2split x).1.
+ +
+Lemma T2finite_succ x: T2finite x → T2finite (T2succ x).
+ +
+Lemma T1succ_nat n: T2succ (\F n) = \F (n.+1).
+ +
+Lemma limit_pr1 x: (x == zero) (+) (T2limit x (+) T2is_succ x).
+ +
+Lemma limit_pr x y: T2limit x → y < x → T2succ y < x.
+ +
+Lemma T2le_psi_b a b : T2succ b ≤ [a,b].
+ +
+Lemma pred_le a: T2pred a ≤ a.
+ +
+Lemma pred_lt a: T2is_succ a → T2pred a < a.
+ +
+Lemma succ_lt a: a < T2succ a.
+ +
+Lemma nf_succ a: T2nf a → T2nf (T2succ a).
+ +
+Lemma nf_pred a: T2nf a → T2nf (T2pred a).
+ +
+Lemma succ_pred x: T2nf x → T2is_succ x → x = T2succ (T2pred x).
+ +
+Lemma succ_p1 x: T2is_succ (T2succ x).
+ +
+Lemma pred_succ x: T2nf x → T2pred (T2succ x) = x.
+ +
+Lemma succ_inj x y: T2nf x → T2nf y → (T2succ x == T2succ y) = (x==y).
+ +
+Lemma lt_succ_succ x y: T2succ x < T2succ y → x < y.
+ +
+Lemma le_succ_succ x y: x ≤ y → T2succ x ≤ T2succ y.
+ +
+Lemma lt_succ_succE x y:
+  T2nf x → T2nf y → (T2succ x < T2succ y) = (x < y).
+ +
+Lemma le_succ_succE x y:
+  T2nf x → T2nf y → (T2succ x ≤ T2succ y) = (x ≤ y).
+ +
+Lemma lt_succ_le_1 a b : T2succ a ≤ b → a < b.
+ +
+Lemma lt_succ_le_2 a b: T2nf a → a < T2succ b → a ≤ b.
+ +
+Lemma lt_succ_le_3 a b: T2nf a → (a < T2succ b) = (a ≤ b).
+ +
+Lemma lt_succ_le_4 a b: T2nf b → (a < b) = (T2succ a ≤ b).
+ +
+Lemma succ_nz x: T2succ x != zero.
+ +
+Lemma succ_psi a b: [a, b] != one → T2succ [a,b] = cons a b 0 one.
+ +
+Lemma succ_psi_lt x a b : [a, b] != one → x < [a,b] → T2succ x < [a,b].
+ +
+Lemma succ_psi_lt2 a b x: [a, b] != one → ([a, b] ≤ T2succ x) = ([a, b] ≤ x).
+ +
+
+ +
+

Addition

+ same as for T1 +
+
+ +
+Fixpoint T2add x y :=
+  if x is cons a b n c then
+    if y is cons a' b' n' c' then
+       if [a,b] < [a',b'] then y
+       else if [a',b'] < [a,b] then cons a b n (c + y)
+       else cons a b (n+n').+1 c'
+    else x
+  else y
+ where "x + y" := (T2add x y) : g0_scope.
+ +
+Fixpoint T2sub x y :=
+  if x is cons a b n c then
+     if y is cons a' b' n' c' then
+           if (x < y) then zero
+           else if ([a',b'] < [a,b]) then x
+           else if (n<n')%N then zero
+           else if ([a,b]==one) then
+             if (n==n')%N then zero else cons zero zero ((n-n').-1) zero
+           else if(n==n') then c - c' else cons a b (n - n').-1 c
+     else x
+  else zero
+where "a - b" := (T2sub a b) : g0_scope.
+ +
+Lemma T2subn0 x: x - zero = x.
+ +
+Lemma T2sub0n x: zero - x = zero.
+ +
+Lemma minus_lt a b: a < b → a - b = zero.
+ +
+Lemma T2subnn x: x - x = zero.
+ +
+Lemma minus_le a b: a ≤ b → a - b = zero.
+ +
+Lemma nf_sub a b: T2nf a → T2nf b → T2nf (a - b).
+ +
+Lemma sub_int n m : \F n - \F m = \F (n -m)%N.
+ +
+Lemma succ_is_add_one a: T2succ a = a + one.
+ +
+Lemma add1Nfin a: ~~ T2finite a → one + a = a.
+ +
+Lemma sub1Nfin a: ~~ T2finite a → a - one = a.
+ +
+Lemma sub1a x: x != zero → T2nf x → x = one + (x - one).
+ +
+Lemma sub1b x: T2nf x → x = (one + x) - one.
+ +
+Lemma T2add0n: left_id zero T2add.
+Lemma T2addn0: right_id zero T2add.
+ +
+Lemma add_int n m : \F n + \F m = \F (n +m)%N.
+ +
+Lemma add_fin_omega n: \F n + omega = omega.
+ +
+Lemma split_add x: let: (y,n) :=T2split x in T2nf x →
+   (x == y + \F n) && ((y==zero) || T2limit y ).
+ +
+Lemma add_to_cons a b n c:
+  c < [a,b] → cons a b n zero + c = cons a b n c.
+ +
+Lemma nf_add a b: T2nf a → T2nf b → T2nf (a + b).
+ +
+Lemma T2add_eq0 m n: (m + n == zero) = (m == zero) && (n == zero).
+ +
+Lemma add_le1 a b: a ≤ a + b.
+ +
+Lemma add_le2 a b: b ≤ a + b.
+ +
+Lemma sub_le1 a b : T2nf a → (a - b) ≤ a.
+ +
+Lemma sub_pr a b: T2nf b → (a + b) - a = b.
+ +
+Lemma add_inj a b c : T2nf b → T2nf c → a + b = a + c → b = c.
+ +
+Lemma sub_pr1 a b: T2nf b → a ≤ b → b = a + (b - a).
+ +
+Lemma omega_minus_one : omega - one = omega.
+ +
+Lemma sub_nz a b: T2nf b → a < b → (b - a) != zero.
+ +
+Lemma T2addA c1 c2 c3: c1 + (c2 + c3) = (c1 + c2) + c3.
+ +
+Lemma T2le_add2l p m n : (p + m ≤ p + n) = (m ≤ n).
+ +
+Lemma T2lt_add2l p m n : (p + m < p + n) = (m < n).
+ +
+Lemma T2lt_add2r p m n : (m + p < n + p ) → (m < n).
+ +
+Lemma T2le_add2r p m n : (m ≤n) → (m + p ≤ n + p).
+ +
+Lemma T2eq_add2l p m n : (p + m == p + n) = (m == n).
+ +
+Lemma add_le3 a b: a = a + b → b = zero.
+ +
+Lemma add_le4 a b: b != zero → a < a + b.
+ +
+Lemma sub_pr1r a b: T2nf a → a - b = zero → a ≤ b.
+ +
+Definition T2ap x :=
+  if x is cons a b n c then ((n==0) && (c==zero)) else false.
+ +
+Lemma ap_pr0 a b (x := [a,b]) u v:
+  u < x → v < x → u + v < x.
+ +
+Lemma ap_limit x: T2ap x → (x == one) || (T2limit x).
+ +
+Lemma ap_pr1 c:
+   (∀ a b, a < c → b < c → a + b < c) →
+   (c== zero) || T2ap c.
+ +
+Lemma ap_pr2 c:
+   T2nf c → c ≠ zero →
+   (∀ a b, T2nf a → T2nf b → a < c → b < c → a + b < c) →
+   T2ap c.
+ +
+Lemma ap_pr3 a b y (x := [a,b]): y < x → y + x = x.
+ +
+Lemma ap_pr4 x: (∀ b, b < x → b + x = x) → (x == zero) || T2ap x.
+ +
+
+ +
+

The function phi

+ We consider he some funciton phi +
+
+ +
+Definition T2_pr1 x:= if x is cons a b n c then a else zero.
+Definition T2_pr2 x:= if x is cons a b n c then b else zero.
+Definition T2finite1 x:=
+  if x is cons a b n c then [&& a == zero, b== zero & c == zero] else false.
+ +
+Definition phi a b :=
+   if b is cons u v n k then
+     if ((n==0) && (k==zero)) then
+        if (a < u) then b else [a,b]
+     else if ((n==0) && (T2finite1 k) && (a <u))
+       then [a, cons u v 0 (T2pred k) ]
+     else [a,b]
+   else [a,b].
+ +
+Lemma phi_ap x y : (phi x y) = [T2_pr1 (phi x y), T2_pr2 (phi x y)].
+ +
+Lemma phi_le1 a b: a ≤ T2_pr1 (phi a b).
+ +
+Lemma phi_le2 a b: T2_pr2 (phi a b) ≤ b.
+ +
+Lemma phi_le3 a b: a < T2_pr1 (phi a b) → (phi a b) = b.
+ +
+Lemma phi_fix1 a u v: a < u → phi a [u,v] = [u, v].
+ +
+Lemma phi_fix2 a b (u:= T2_pr1 b) (v:= T2_pr2 b):
+  phi a b = b → b = [u,v] ∧ a < u.
+ +
+Lemma phi_succ a u v n: a < u →
+  phi a (cons u v 0 (\F n.+1)) = [a, cons u v 0 (\F n)].
+ +
+
+ +
+phi a b is either b, psi a b or psi a (b-1). +
+
+ +
+Lemma phi_cases a b:
+    {phi a b = b} + {phi a b = [a, b]} +
+    { phi a b = [a, T2pred b] ∧ b = T2succ (T2pred b) }.
+ +
+Lemma nf_phi x y : T2nf x → T2nf y → T2nf (phi x y).
+ +
+Lemma phi_principalR a b: { c:T2 | [a, b] = phi zero c}.
+ +
+Theorem phi_spec1 a b c: c < a → phi c (phi a b) = phi a b.
+ +
+Lemma phi_spec2 a x:
+    T2nf a → T2nf x → (∀ c, T2nf c → c < a → phi c x = x) →
+    a ≤ T2_pr1 x.
+ +
+Lemma phi_spec3 a x:
+  T2nf a → T2nf x → (∀ c, T2nf c → c < a → phi c x = x) →
+  a != zero → {b : T2 | x = phi a b}.
+ +
+Lemma phi_spec4a u v: u != zero → phi zero [u,v] = [u, v].
+ +
+Lemma phi_spec4b x: phi zero x = x →
+  x = [T2_pr1 x, T2_pr2 x] ∧ T2_pr1 x != zero.
+ +
+Lemma phi_spec4c x: T2nf x → phi zero x = x →
+  { b: T2 | x = phi one b }.
+ +
+Lemma no_critical a: a < phi a zero.
+ +
+Lemma phi_ab_le1 a b: b ≤ phi a b.
+ +
+Lemma phi_ab_le2 a b:a < phi a b.
+ +
+Lemma phi_inv1 a b: phi a (T2succ b) = [a,b] →
+   { n: nat | (b = cons (T2_pr1 b) (T2_pr2 b) 0 (\F n) ∧ a < T2_pr1 b) }.
+ +
+
+ +
+Monotonicity is non-trivial +
+
+ +
+Lemma phi_mono_a a b b': T2nf b → b < b' → phi a b < phi a b'.
+ +
+Lemma phi_mono_b a b b': T2nf b → b ≤ b' → phi a b ≤ phi a b'.
+ +
+Lemma phi_mono_c a b b': T2nf b → T2nf b' → (phi a b < phi a b') = (b < b').
+ +
+Lemma phi_inj a b b': T2nf b → T2nf b' → phi a b = phi a b' → b = b'.
+ +
+Lemma phi_inj1 a b b': T2nf b → T2nf b' → (phi a b == phi a b') = (b == b').
+ +
+
+ +
+Two lemmas for equal or less-than +
+
+ +
+Lemma phi_eqE a b a' b': T2nf a → T2nf a' → T2nf b → T2nf b' →
+   (phi a b == phi a' b') =
+    (if a < a' then b == phi a' b'
+     else if a' < a then phi a b == b' else b== b').
+ +
+Lemma phi_ltE a b a' b': T2nf a → T2nf a' → T2nf b → T2nf b' →
+   (phi a b < phi a' b') =
+    (if a < a' then b < phi a' b'
+     else if a' < a then phi a b < b' else b < b').
+ +
+
+ +
+Every x is uniquely a phi with some conditions +
+
+ +
+Lemma phi_inv0 a b a' b':
+  phi a b = phi a' b' → b < phi a b → b' < phi a' b' → a = a'.
+ +
+Lemma phi_inv2 a b a' b':
+  phi a b = phi a' b' → b < phi a b → b' < phi a' b' → b = b'.
+ +
+Lemma phi_inv3 x:
+ T2ap x → { a: T2 & { b: T2 |
+    [/\ x = phi a b, b < x, (size a < size x)%N & (size b < size x)%N ] }}.
+ +
+
+ +
+Expression psi in terms of phi +
+
+Definition psi_phi_aux a b :=
+  let (b', n) := T2split b in if phi a b' == b' then (T2succ b) else b.
+ +
+Definition psi_phi a b := phi a (psi_phi_aux a b).
+ +
+Lemma psi_phi1 a b (c:= psi_phi_aux a b): c < phi a c.
+ +
+End Gamma0.
+ +
+
+ +
+

Ackermann

+ +
+
+ +
+ +
+Module Ackermann.
+Declare Scope ak_scope.
+Delimit Scope ak_scope with ak.
+Open Scope ak_scope.
+ +
+Inductive T3 : Set :=
+  zero : T3
+| cons : T3 → T3 → T3 → nat → T3 → T3.
+ +
+Fixpoint T3eq x y {struct x} :=
+  match x, y with
+  | zero, zero ⇒ true
+  | cons a b c n d, cons a' b' c' n' d' ⇒
+      [&& T3eq a a', T3eq b b', T3eq c c', n== n' & T3eq d d' ]
+  | _, _ ⇒ false
+end.
+ +
+Lemma T3eqP : Equality.axiom T3eq.
+ +
+Canonical T3_eqMixin := EqMixin T3eqP.
+Canonical T3_eqType := Eval hnf in EqType T3 T3_eqMixin.
+ +
+Arguments T3eqP {x y}.
+ +
+Lemma T3eqE a b c n d a' b' c' n' d':
+  (cons a b c n d == cons a' b' c' n' d') =
+      [&& a == a', b == b', c == c', n== n' & d == d' ].
+ +
+Notation "[ x , y , z ]" := (cons x y z 0 zero) (at level 0) :ak_scope.
+Definition T3nat p := if p is n.+1 then cons zero zero zero n zero else zero.
+Notation "\F n" := (T3nat n)(at level 29) : ak_scope.
+ +
+Fixpoint size x :=
+  if x is cons a b c n d then
+     (maxn (size a) (maxn (size b) (maxn (size c) (size d)))).+1
+  else 0.
+ +
+Lemma size_a a b c n d: (size a < size (cons a b c n d))%N.
+ +
+Lemma size_b a b c n d: (size b < size (cons a b c n d))%N.
+ +
+Lemma size_c a b c n d: (size c < size (cons a b c n d))%N.
+ +
+Lemma size_d a b c n d: (size d < size (cons a b c n d))%N.
+ +
+Lemma size_psi a b c n d: (size [a, b, c] ≤ size (cons a b c n d))%N.
+ +
+Lemma size_prop1 a b c n d (l:= size (cons a b c n d)):
+   [&& size a < l, size b < l, size c < l, size d < l
+   & size [a, b, c] ≤ l]%N.
+ +
+Lemma size_prop a b c n d a' b' c' n' d'
+   (l := ((size (cons a b c n d) + size (cons a' b' c' n' d')))%N) :
+  [&& (size a' + size a < l), (size b + size b' < l),
+   (size c + size c' < l), (size d + size d' < l),
+   (size a + size a' < l), (size b' + size b < l),
+   (size [a, b, c] + size b' < l),(size b + size [a', b', c'] < l),
+   (size [a, b, c] + size c' < l) &(size c + size [a', b', c'] < l)]%N.
+ +
+
+ +
+

Comparison

+ +
+
+ +
+Definition lt_psi_rec f a b c a' b' c' (x := [a,b,c])(x':= [a', b', c']):=
+  [|| [&& a==a', b==b' & f c c'],
+      [&& a==a', f b b' & f c x'],
+      [&& a==a', f b' b & f x c'],
+      [&& a==a', f b' b & x == c'],
+      [&& f a a', f b x' & f c x'],
+      ((f a' a) && f x b'),
+      ((f a' a) && (x == b')),
+      ((f a' a) && f x c') |
+      ((f a' a) && (x == c'))].
+ +
+Definition lt_rec f x y :=
+ if x is cons a b c n d then
+   if y is cons a' b' c' n' d' then
+     if (lt_psi_rec f a b c a' b' c')
+     then true
+     else if ((a== a') && (b==b') && (c==c')) then
+       if (n < n')%N then true
+       else if (n == n') then (f d d') else false
+       else false
+   else false
+ else if y is cons a' b' c' n' d' then true else false.
+ +
+Fixpoint T3lta k {struct k}:=
+ if k is k.+1 then lt_rec (T3lta k) else fun x y ⇒ false.
+ +
+Definition T3lt a b := T3lta ((size a) + size b).+1 a b.
+Definition T3le (x y :T3) := (x == y) || (T3lt x y).
+Notation "x < y" := (T3lt x y) : ak_scope.
+Notation "x <= y" := (T3le x y) : ak_scope.
+Notation "x >= y" := (y ≤ x) (only parsing) : ak_scope.
+Notation "x > y" := (y < x) (only parsing) : ak_scope.
+ +
+Lemma T3ltE x y : x < y = lt_rec T3lt x y.
+ +
+Definition lt_psi (a b c a' b' c': T3):=
+ [|| [&& a==a', b==b' & c < c'],
+      [&& a==a', b < b' & c < [a',b',c']],
+      [&& a==a', b' < b & [a,b,c] < c'],
+      [&& a==a', b' < b & [a,b,c] == c'],
+      [&& a < a', b < [a',b',c'] & c < [a',b',c']],
+      ((a' < a) && ([a,b,c] < b')),
+      ((a' < a) && ([a,b,c] == b')),
+      ((a' < a) && ([a,b,c] < c')) |
+      ((a' < a) && ([a,b,c] == c'))].
+ +
+Lemma T3lt_psi a b c a' b' c': [a,b,c] < [a', b',c'] = lt_psi a b c a' b' c'.
+ +
+Lemma T3lt_consE a b c n d a' b' c' n' d' :
+   cons a b c n d < cons a' b' c' n' d' =
+     if ([a, b, c] < [a', b', c']) then true
+     else if ([a, b, c] == [a', b', c']) then
+       if (n < n')%N then true
+       else if (n == n') then (d < d') else false
+       else false.
+ +
+Lemma T3ltn0 x: (x < zero) = false.
+Lemma T3lt0n x: (zero < x) = (x != zero).
+ +
+Lemma T3ltnn x: (x < x) = false.
+ +
+Lemma T3lt_ne a b : a < b → (a == b) = false.
+ +
+Lemma T3lt_ne' a b : a < b → (b == a) = false.
+ +
+Lemma T3ltW a b : (a < b) → (a ≤ b).
+ +
+Lemma T3le_eqVlt a b : (a ≤ b) = (a == b) || (a < b).
+ +
+Lemma T3lt_neAle a b : (a < b) = (a != b) && (a ≤ b).
+ +
+Definition one := [zero,zero,zero].
+Definition omega := [zero,zero, one].
+Definition epsilon0 := [zero, one, zero].
+Definition T3bad := cons zero zero zero 0 one.
+ +
+Lemma T3le0n x: zero ≤ x.
+Lemma T3len0 x: (x ≤ zero) = (x == zero).
+ +
+Lemma T3ge1 x: (one ≤ x) = (x != zero).
+ +
+Lemma T3lt1 x: (x < one) = (x==zero).
+ +
+Lemma T3lcp0_pr x y: x < y → (y==zero) = false.
+ +
+Lemma finite_ltP n p : (n < p)%N = (\F n < \F p).
+ +
+Lemma T3lt_anti b a: a < b → (b < a) = false.
+ +
+Lemma T3lt_trichotomy a b: [|| (a< b), (a==b) | (b < a)].
+ +
+Lemma T3lenn x: x ≤ x.
+ +
+#[local] Hint Resolve T3lenn : core.
+ +
+Lemma T3leNgt a b: (a ≤ b) = ~~ (b < a).
+ +
+Lemma T3ltNge a b: (a < b) = ~~ (b ≤ a).
+ +
+Lemma T3eq_le m n : (m == n) = ((m ≤ n) && (n ≤ m)).
+ +
+CoInductive T3ltn_xor_geq m n : bool → bool → Set :=
+  | T3LtnNotGeq of m < n : T3ltn_xor_geq m n false true
+  | T3GeqNotLtn of n ≤ m : T3ltn_xor_geq m n true false.
+ +
+CoInductive T3leq_xor_gtn m n : bool → bool → Set :=
+  | T3GeqNotGtn of m ≤ n : T3leq_xor_gtn m n true false
+  | T3GtnNotLeq of n < m : T3leq_xor_gtn m n false true.
+ +
+CoInductive compare_T3 m n : bool → bool → bool → Set :=
+  | CompareT3Lt of m < n : compare_T3 m n true false false
+  | CompareT3Gt of m > n : compare_T3 m n false true false
+  | CompareT3Eq of m = n : compare_T3 m n false false true.
+ +
+Lemma T3leP x y : T3leq_xor_gtn x y (x ≤ y) (y < x).
+ +
+Lemma T3ltP m n : T3ltn_xor_geq m n (n ≤ m) (m < n).
+ +
+Lemma T3ltgtP m n : compare_T3 m n (m < n) (n < m) (m == n).
+ +
+Lemma T3le_consE a b c n d a' b' c' n' d' :
+   cons a b c n d ≤ cons a' b' c' n' d' =
+     if ([a, b, c] < [a', b', c']) then true
+     else if ([a, b, c] == [a', b', c']) then
+       if (n < n')%N then true
+       else if (n == n') then (d ≤ d') else false
+       else false.
+ +
+Lemma T3lt_psi' a b c a' b' c': [a, b, c] < [a', b', c' ] =
+  [|| [&& a==a', b==b' & c < c'],
+      [&& a==a', b < b' & c < [a', b', c'] ],
+      [&& a==a', b' <b & [a,b,c] ≤ c'],
+      [&& a < a', b < [a', b', c'] & c < [a', b', c']],
+      ((a' < a) && ([a,b,c] ≤ b')) |
+      ((a' < a) && ([a,b,c] ≤ c'))].
+ +
+Theorem T3lt_trans b a c: a < b → b < c → a < c.
+ +
+Lemma T3lt_le_trans b a c: a < b → b ≤ c → a < c.
+ +
+Lemma T3le_lt_trans b a c: a ≤ b → b < c → a < c.
+ +
+Lemma T3le_trans b a c: a ≤ b → b ≤ c → a ≤ c.
+ +
+Lemma T3le_anti : antisymmetric T3le.
+ +
+Lemma T3le_total m n : (m ≤ n) || (n ≤ m).
+ +
+Lemma T3le_psi a b c n d: [a,b,c] ≤ cons a b c n d.
+ +
+Lemma T3lt_psi_bc a b c: ((b < [a,b,c]) && (c < [a, b, c])).
+ +
+Lemma psi_lt1 a b c d n a' b' c':
+   cons a b c n d < [a', b', c'] = ([a, b,c] < [a', b', c']).
+ +
+Lemma psi_lt2 a b c n d n' d': cons a b c n' d' < cons a b c n d =
+   (if (n' < n)%N then true else if n' == n then d' < d else false).
+ +
+Lemma T3lt_psi_b a b c: b < [a,b,c].
+ +
+Lemma T3lt_psi_c a b c: c < [a,b,c].
+ +
+Lemma T3lt_psi_a a b c: a < [a,b,c].
+ +
+
+ +
+

Normal form

+ +
+
+ +
+Fixpoint T3nf x :=
+  if x is cons a b c _ d
+  then [&& T3nf a, T3nf b, T3nf c, T3nf d & d < [a,b,c] ]
+  else true.
+ +
+Lemma nf_0: T3nf zero.
+ +
+Lemma nf_psi a b c: T3nf [a, b, c] = [&& T3nf a, T3nf b & T3nf c].
+ +
+Lemma nf_int n: T3nf (\F n).
+ +
+Lemma nf_cons_cons a b c n a' b' c' n' d':
+  T3nf (cons a b c n (cons a' b' c' n' d')) =
+   [&& [a', b',c'] < [a, b,c], T3nf [a,b, c] &
+    T3nf (cons a' b' c' n' d') ].
+ +
+Lemma nf_consE a b c n d:
+    T3nf (cons a b c n d) = [&& T3nf [a,b,c], T3nf d & d < [a,b,c] ].
+ +
+Lemma nf_Wf : well_founded_P T3nf T3lt.
+ +
+Theorem lt_not_wf : ¬ (well_founded T3lt).
+ +
+
+ +
+

Successor Predecessor

+ +
+
+ +
+Fixpoint T1_T3 (c:CantorOrdinal.T1) : T3 :=
+  if c is CantorOrdinal.cons a n b then cons zero zero (T1_T3 a) n (T1_T3 b)
+  else zero.
+ +
+Lemma lt_epsilon0 a b c n d :
+  cons a b c n d < epsilon0 = [&& a==zero, b == zero & c < epsilon0 ].
+ +
+Lemma T1T3_lt_epsilon0 x: T1_T3 x < epsilon0.
+ +
+Delimit Scope cantor_scope with ca.
+Notation "x < y" := (CantorOrdinal.T1lt x y) : cantor_scope.
+ +
+Lemma T1T3_inc x y: (x <y)%ca→ (T1_T3 x) < (T1_T3 y).
+ +
+Lemma TT1T3_inj: injective T1_T3.
+ +
+Lemma T1T3_surj x: T3nf x → x < epsilon0 → ∃ y, x = T1_T3 y.
+ +
+Definition all_zero a b c :=[&& a==zero, b==zero & c== zero].
+ +
+Fixpoint T3limit x :=
+  if x is cons a b c n d then
+    if (all_zero a b c) then false else (d== zero) || T3limit d
+  else false.
+ +
+Definition T3finite x :=
+   if x is cons a b c n d then all_zero a b c else true.
+ +
+Fixpoint T3split x:=
+ if x is cons a b c n d then
+      if all_zero a b c then (zero, n.+1) else
+     let: (x, y) := T3split d in (cons a b c n x,y)
+   else (zero,0).
+ +
+Lemma all_zeroE a b c: all_zero a b c = ([a,b,c] == one).
+ +
+Lemma T3nf_finite a b c n d: all_zero a b c → T3nf (cons a b c n d) →
+    d = zero.
+ +
+Lemma split_finite x: ((T3split x).1 == zero) = T3finite x.
+ +
+Lemma T3finite1 n: T3finite (\F n).
+ +
+Lemma T3finite2 x: T3finite x → T3nf x → x = \F ((T3split x).2).
+ +
+Lemma T3gt1 x: (one < x) = ((x != zero) && (x != one)).
+ +
+Lemma omega_least_inf1 x: T3finite x → x < omega.
+ +
+Lemma omega_least_inf2 x: ~~ T3finite x → omega ≤ x.
+ +
+Lemma lt_omega1 c n d a' b' c' n' d' :
+   cons zero zero c n d < cons a' b' c' n' d' =
+     if ((a'== zero) && (b'==zero)) then
+       ((c < c') || ((c==c') && ((n < n')%N || ((n==n') && (d < d')))))
+    else (c < [a', b', c']).
+ +
+Lemma lt_omega2 c a' b' c' :
+   ([zero, zero, c] < [a', b', c']) =
+     if ((a'== zero) && (b'==zero)) then c < c' else (c < [a', b', c']).
+ +
+Lemma split_limit x: ((T3split x).2 == 0) = ((x==zero) || T3limit x).
+ +
+Fixpoint T3is_succ x :=
+  if x is cons a b c n d then (all_zero a b c) || T3is_succ d else false.
+ +
+Fixpoint T3succ x :=
+  if x is cons a b c n d
+     then if all_zero a b c then \F n.+2 else cons a b c n (T3succ d)
+  else one.
+ +
+Fixpoint T3pred x :=
+  if x is cons a b c n d then
+     if all_zero a b c then \F n else (cons a b c n (T3pred d))
+  else zero.
+ +
+Lemma split_is_succ x: ((T3split x).2 != 0) = (T3is_succ x).
+ +
+Lemma split_succ x: let:(y,n):= T3split x in T3split (T3succ x) = (y,n.+1).
+ +
+Lemma split_pred x: let:(y,n):= T3split x in T3split (T3pred x) = (y,n.-1).
+ +
+Lemma split_le x : (T3split x).1 ≤ x.
+ +
+Lemma nf_split x : T3nf x → T3nf (T3split x).1.
+ +
+Lemma T3finite_succ x: T3finite x → T3finite (T3succ x).
+ +
+Lemma T1succ_nat n: T3succ (\F n) = \F (n.+1).
+ +
+Lemma nf_omega : T3nf omega.
+Lemma nf_finite n: T3nf (\F n).
+ +
+Lemma limit_pr1 x: (x == zero) (+) (T3limit x (+) T3is_succ x).
+ +
+Lemma limit_pr x y: T3limit x → y < x → T3succ y < x.
+ +
+Lemma pred_le a: T3pred a ≤ a. +
+Lemma pred_lt a: T3is_succ a → T3pred a < a.
+ +
+Lemma succ_lt a: a < T3succ a.
+ +
+Lemma nf_succ a: T3nf a → T3nf (T3succ a).
+ +
+Lemma nf_pred a: T3nf a → T3nf (T3pred a).
+ +
+Lemma succ_pred x: T3nf x → T3is_succ x → x = T3succ (T3pred x).
+ +
+Lemma succ_p1 x: T3is_succ (T3succ x).
+ +
+Lemma pred_succ x: T3nf x → T3pred (T3succ x) = x.
+ +
+Lemma succ_inj x y: T3nf x → T3nf y → (T3succ x == T3succ y) = (x==y).
+ +
+Lemma lt_succ_succ x y: T3succ x < T3succ y → x < y.
+ +
+Lemma le_succ_succ x y: x ≤ y → T3succ x ≤ T3succ y.
+ +
+Lemma lt_succ_succE x y:
+  T3nf x → T3nf y → (T3succ x < T3succ y) = (x < y).
+ +
+Lemma le_succ_succE x y:
+  T3nf x → T3nf y → (T3succ x ≤ T3succ y) = (x ≤ y).
+ +
+Lemma lt_succ_le_1 a b : T3succ a ≤ b → a < b.
+ +
+Lemma lt_succ_le_2 a b: T3nf a → a < T3succ b → a ≤ b. +
+Lemma lt_succ_le_3 a b: T3nf a → (a < T3succ b) = (a ≤ b).
+ +
+Lemma lt_succ_le_4 a b: T3nf b → (a < b) = (T3succ a ≤ b).
+ +
+Lemma succ_nz x: T3succ x != zero.
+ +
+Lemma succ_psi a b c: [a, b, c] != one → T3succ [a,b,c] = cons a b c 0 one.
+ +
+Lemma succ_psi_lt x a b c: [a, b, c] != one →
+   x < [a,b,c] → T3succ x < [a,b,c].
+ +
+Lemma succ_psi_lt2 a b c x: [a, b, c] != one →
+  ([a, b, c] ≤ T3succ x) = ([a, b, c] ≤ x).
+ +
+
+ +
+

Addition

+ +
+
+ +
+Fixpoint T3add x y :=
+  if x is cons a b c n d then
+    if y is cons a' b' c' n' d' then
+       if [a,b,c] < [a',b',c'] then y
+       else if [a',b',c'] < [a,b,c] then cons a b c n (d + y)
+       else cons a b c (n+n').+1 d'
+    else x
+  else y
+ where "x + y" := (T3add x y) : ak_scope.
+ +
+Fixpoint T3sub x y :=
+  if x is cons a b c n d then
+     if y is cons a' b' c' n' d' then
+           if (x < y) then zero
+           else if ([a',b',c'] < [a,b,c]) then x
+           else if (n<n')%N then zero
+           else if ([a,b,c] == one) then
+             if (n==n')%N then zero else cons zero zero zero ((n-n').-1) zero
+           else if(n==n') then d - d' else cons a b c (n - n').-1 d
+     else x
+  else zero
+where "a - b" := (T3sub a b) : ak_scope.
+ +
+Lemma T3subn0 x: x - zero = x.
+ +
+Lemma T3sub0n x: zero - x = zero.
+ +
+Lemma minus_lt a b: a < b → a - b = zero.
+ +
+Lemma T3subnn x: x - x = zero.
+ +
+Lemma minus_le a b: a ≤ b → a - b = zero.
+ +
+Lemma nf_sub a b: T3nf a → T3nf b → T3nf (a - b).
+ +
+Lemma sub_int n m : \F n - \F m = \F (n -m)%N.
+ +
+Lemma succ_is_add_one a: T3succ a = a + one.
+ +
+Lemma add1Nfin a: ~~ T3finite a → one + a = a.
+ +
+Lemma sub1Nfin a: ~~ T3finite a → a - one = a.
+ +
+Lemma sub1a x: x != zero → T3nf x → x = one + (x - one).
+ +
+Lemma sub1b x: T3nf x → x = (one + x) - one.
+ +
+Lemma sub_1aCE (a:= T3bad) : one + (a - one) != a.
+ +
+Lemma sub_1bCE (a:= T3bad) : (one + a - one) != a.
+ +
+Lemma T3add0n : left_id zero T3add.
+Lemma T3addn0: right_id zero T3add.
+ +
+Lemma add_int n m : \F n + \F m = \F (n +m)%N.
+ +
+Lemma add_fin_omega n: \F n + omega = omega.
+ +
+Lemma fooCE (x:= T3bad):
+   ~~T3limit x /\(∀ u v, T3limit u → x ≠ u + \F v.+1).
+ +
+Lemma split_add x: let: (y,n) :=T3split x in T3nf x →
+   (x == y + \F n) && ((y==zero) || T3limit y ).
+ +
+Lemma add_to_cons a b c n d:
+  d < [ a,b,c] → cons a b c n zero + d = cons a b c n d.
+ +
+Lemma addC_CE (a := one) (b := omega):
+  [&& T3nf a, T3nf b & a + b != b + a].
+ +
+Lemma nf_add a b: T3nf a → T3nf b → T3nf (a + b).
+ +
+Lemma T3add_eq0 m n: (m + n == zero) = (m == zero) && (n == zero).
+ +
+Lemma add_le1 a b: a ≤ a + b.
+ +
+Lemma add_le2 a b: b ≤ a + b.
+ +
+Lemma sub_le1 a b : T3nf a → (a - b) ≤ a.
+ +
+Lemma sub_pr a b: T3nf b → (a + b) - a = b.
+ +
+Lemma add_inj a b c : T3nf b → T3nf c → a + b = a + c → b = c.
+ +
+Lemma sub_pr1 a b: T3nf b → a ≤ b → b = a + (b - a).
+ +
+Lemma omega_minus_one : omega - one = omega.
+ +
+Lemma sub_nz a b: T3nf b → a < b → (b - a) != zero.
+ +
+Lemma T3addA c1 c2 c3: c1 + (c2 + c3) = (c1 + c2) + c3.
+ +
+Lemma T3le_add2l p m n : (p + m ≤ p + n) = (m ≤ n).
+ +
+Lemma T3lt_add2l p m n : (p + m < p + n) = (m < n).
+ +
+Lemma T3lt_add2r p m n : (m + p < n + p ) → (m < n).
+ +
+Lemma T3le_add2r p m n : (m ≤n) → (m + p ≤ n + p).
+ +
+Lemma T3eq_add2l p m n : (p + m == p + n) = (m == n).
+ +
+Lemma add_le3 a b: a = a + b → b = zero.
+ +
+Lemma add_le4 a b: b != zero → a < a + b.
+ +
+Lemma sub_pr1r a b: T3nf a → a - b = zero → a ≤ b.
+ +
+Lemma sub_pr1rCE (a := T3bad) (b := one) : (a - b == zero) && (b < a).
+ +
+Lemma T3addS a b : (a + T3succ b) = T3succ (a+ b).
+ +
+
+ +
+

limit

+ +
+
+Notation Tf := (nat → T3).
+ +
+Definition limit_of (f: Tf) x :=
+  [/\ (∀ n m, (n < m)%N → f n < f m),
+      (∀ n, f n < x) &
+      (∀ y, T3nf y → y < x → (∃ n, y ≤ f n))].
+ +
+Lemma fincP (f: Tf) :
+  (∀ n, f n < f n.+1) →
+  (∀ n m, (n < m)%N → f n < f m).
+ +
+Definition limit12_hyp a b c:=
+   if c is cons a1 b1 c1 n1 d1 then
+       (n1 == 0) && (d1 == zero) &&
+       ( ((a == a1) && (b < b1)) || ((a < a1) && (b < c)))
+   else false.
+ +
+Definition phi0:= fun _ :nat ⇒ zero.
+Definition phi1 a (f:Tf) := fun n ⇒ a + f n.
+Definition phi5 (f:Tf) := fun n ⇒ [f n, zero,zero].
+Definition phi12a a b (f:Tf) := fun n ⇒ [a,b,f n].
+ +
+Lemma limit1 a b f:
+   T3nf a → (limit_of f b) → (limit_of (phi1 a f) (a + b)).
+ +
+Lemma limit5 f x: (limit_of f x) → (limit_of (phi5 f) [x,zero,zero]).
+ +
+Lemma limit12a f a b c: ~~ (limit12_hyp a b c) →
+   (limit_of f c) → (limit_of (phi12a a b f)[a, b, c]).
+ +
+Fixpoint phi3 x n := if n is n.+1 then phi3 x n + x else x.
+ +
+Lemma phi3v a b c k: phi3 [a,b,c] k = cons a b c k zero.
+ +
+Lemma limit3 x: limit_of (phi3 [zero,zero,x]) [zero, zero, T3succ x].
+ +
+Lemma limit2: limit_of (phi3 one) omega.
+ +
+Lemma limit12b1 x: (limit12_hyp zero zero x) →
+  limit_of (phi3 x) [zero, zero, x].
+ +
+Fixpoint phi4 x n :=
+   if n is n.+1 then [x, phi4 x n, phi4 x n] else [x,zero,zero].
+ +
+Lemma limit4 x: limit_of (phi4 x) [T3succ x, zero, zero].
+ +
+Fixpoint phi8 x y n :=
+   if n is n.+1 then [x, phi8 x y n, phi8 x y n] else [T3succ x,zero,y].
+ +
+Lemma limit8 x y: limit_of (phi8 x y) [T3succ x, zero, T3succ y].
+ +
+Fixpoint phi12b2 x y n :=
+   if n is n.+1 then [x, phi12b2 x y n, phi12b2 x y n] else y.
+ +
+Lemma limit12b2 x y: (limit12_hyp (T3succ x) zero y) →
+    limit_of (phi12b2 x y) [T3succ x, zero, y].
+ +
+Fixpoint phi6 x y n :=
+   if n is n.+1 then [x, y, phi6 x y n] else [x,y,zero].
+ +
+Fixpoint phi10 x y z n :=
+   if n is n.+1 then [x, y, phi10 x y z n] else [x,T3succ y,z].
+ +
+Fixpoint phi12b4 x y z n :=
+   if n is n.+1 then [x, y, phi12b4 x y z n] else z.
+ +
+Lemma limit6 x y:
+    limit_of (phi6 x y) [x,T3succ y, zero].
+ +
+Lemma limit10 x y z:
+    limit_of (phi10 x y z) [x,T3succ y, T3succ z].
+ +
+Lemma limit12b4 x y z: (limit12_hyp x (T3succ y) z) →
+    limit_of (phi12b4 x y z) [x,T3succ y,z].
+ +
+Fixpoint phi7 x y f n :=
+  if n is n.+1 then [x, f n, phi7 x y f n] else y.
+ +
+Fixpoint phi9 x y f n :=
+  if n is n.+1 then [f n, phi9 x y f n, phi9 x y f n] else [x, zero,y].
+ +
+Fixpoint phi11 x y z f n :=
+  if n is n.+1 then [x,f n, phi11 x y z f n ] else [x, y,z].
+ +
+Fixpoint phi12b3 y f n :=
+  if n is n.+1 then [f n, phi12b3 y f n , phi12b3 y f n] else y.
+ +
+Fixpoint phi12b5 x z f n :=
+  if n is n.+1 then [x,f n, phi12b5 x z f n ] else z.
+ +
+Lemma limit7 x y f: (limit_of f y) →
+  (limit_of ( phi7 x y f) [x,y,zero]).
+ +
+Lemma limit9 x y f: (limit_of f x) →
+  (limit_of (phi9 x y f) [x,zero, T3succ y]).
+ +
+Lemma limit11 x y z f: (limit_of f y) →
+  (limit_of (phi11 x y z f) [x, y, T3succ z]).
+ +
+Lemma limit12b3 x y f: (limit_of f x) → (limit12_hyp x zero y) →
+  (limit_of (phi12b3 y f) [x, zero, y]).
+ +
+Lemma limit12b5 x y z f: (limit_of f y) → (limit12_hyp x y z) →
+  (limit_of (phi12b5 x z f) [x,y,z]).
+ +
+Definition phi_rec_psi f a b c :=
+  if (c==zero) then
+    if(b==zero) then
+       if(a==zero) then phi0
+       else if(T3is_succ a) then phi4 (T3pred a)
+       else phi5 (f a)
+    else if(T3is_succ b) then phi6 a (T3pred b)
+    else phi7 a b (f b)
+  else if(T3is_succ c) then
+    if(b==zero) then
+      if(a==zero) then phi3 [zero,zero, T3pred c]
+      else if (T3is_succ a) then phi8 (T3pred a) (T3pred c)
+      else phi9 a (T3pred c) (f a)
+    else if(T3is_succ b) then phi10 a (T3pred b) (T3pred c)
+    else phi11 a b (T3pred c) (f b)
+  else if (limit12_hyp a b c) then
+    if(b==zero) then
+      if(a==zero) then phi3 c
+      else if(T3is_succ a) then phi12b2 (T3pred a) c
+      else phi12b3 c (f a)
+    else if (T3is_succ b) then phi12b4 a (T3pred b) c
+    else phi12b5 a c (f b)
+  else phi12a a b (f c).
+ +
+Definition phi_rec f (x: T3) :=
+  if x is cons a b c n d then
+  if (d==zero) then
+    if n is n.+1 then phi1 (cons a b c n zero) (phi_rec_psi f a b c)
+     else phi_rec_psi f a b c
+  else phi1 (cons a b c n zero) (f d)
+  else phi0.
+ +
+Fixpoint phia k := if k is k.+1 then phi_rec (phia k) else (fun x ⇒phi0).
+Definition phi x := phia (size x).+1 x.
+ +
+Lemma phiE x : phi x = phi_rec phi x.
+ +
+Lemma phiE_1 a b c n:
+  phi (cons a b c n.+1 zero) = phi1 (cons a b c n zero) (phi [a, b, c]).
+ +
+Lemma phiE_2 a b c n d: d != zero →
+  phi (cons a b c n d) = phi1 (cons a b c n zero) (phi d).
+ +
+Lemma phiE_3 a b c: phi ([a,b,c]) = phi_rec_psi phi a b c.
+ +
+Lemma phiL x: T3nf x → T3limit x → limit_of (phi x) x.
+ +
+Lemma conc1 (x:= [zero,zero, epsilon0]): limit_of (phi3 epsilon0) x.
+ +
+
+ +
+

additive principal

+ +
+
+ +
+Definition T3ap x :=
+  if x is cons a b c n d then ((n==0) && (d==zero)) else false.
+ +
+Lemma ap_pr0 a b c (x := [a,b,c]) u v:
+  u < x → v < x → u + v < x.
+ +
+Lemma ap_limit x: T3ap x → (x == one) || (T3limit x).
+ +
+Lemma ap_pr1 c:
+   (∀ a b, a < c → b < c → a + b < c) →
+   (c== zero) || T3ap c.
+ +
+Lemma ap_pr2 c:
+   T3nf c → c ≠ zero →
+   (∀ a b, T3nf a → T3nf b → a < c → b < c → a + b < c) →
+   T3ap c.
+ +
+Lemma ap_pr3 a b c y (x := [a,b,c]): y < x → y + x = x.
+ +
+Lemma ap_pr4 x: (∀ b, b < x → b + x = x) → (x == zero) || T3ap x.
+ +
+Lemma ap_pr2CE (c := cons zero zero T3bad 1 zero):
+   (∀ a b, T3nf a → T3nf b → a < c → b < c → a + b < c).
+ +
+Definition psi_succ x :=
+  if x is cons a b c _ _ then
+   if ((a==zero) && (b==zero)) then [zero,zero, T3succ c] else [zero,zero, x]
+  else zero.
+ +
+Lemma psi_succ_pr1 a b c: [a,b,c] < psi_succ ([a,b,c]).
+ +
+Lemma succ_psi1 a b c (x:= [a, b, c]): ((a != zero) || (b != zero)) →
+    (∀ a' b' c', x < [a',b',c'] → (psi_succ x) ≤ [a',b',c']).
+ +
+Lemma succ_psi2 u (x := [zero,zero,u]) :
+   (∀ a' b' c', T3nf c' → x < [a',b',c'] → (psi_succ x) ≤ [a',b',c']).
+ +
+Lemma succ_prCE (u:= one) (v := T3bad): (u < v) && (v < T3succ u).
+ +
+Lemma succ_psiCE (z := [zero,zero, T3bad]):
+   (omega < z) && (z < (psi_succ omega)) && ~~(T3nf z).
+ +
+Definition sup_of (f: T3→ T3) x z :=
+  [/\ T3nf z,
+      (∀ y, T3nf y → y < x → f y ≤ z) &
+      (∀ z', T3nf z' → z' < z → ∃ y,
+          [&& T3nf y, y < x & z' < f y])].
+ +
+Definition normal f:=
+  [/\ ∀ x, T3nf x → T3nf (f x),
+      (∀ x y, T3nf x → T3nf y → x < y → f x < f y)&
+      (∀ x, T3nf x → T3limit x → sup_of f x (f x)) ].
+ +
+Lemma sup_unique f x z z': sup_of f x z → sup_of f x z' → z = z'.
+ +
+Lemma sup_Oalpha_zero: sup_of id zero zero.
+ +
+Lemma sup_Oalpha_limit x: T3nf x → T3limit x → sup_of id x x.
+ +
+Lemma sup_Oalpha_succ x: T3nf x → sup_of id (T3succ x) x.
+ +
+Lemma normal_id: normal id.
+ +
+Lemma normal_limit f x: normal f → T3nf x → T3limit x → T3limit (f x).
+ +
+Lemma normal_compose f g:
+   normal f → normal g → normal (f \o g).
+ +
+Lemma add_normal a: T3nf a → normal (T3add a).
+ +
+End Ackermann.
+ +
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/gaia_hydras.onType.html b/theories/html/gaia_hydras.onType.html new file mode 100644 index 00000000..76b04484 --- /dev/null +++ b/theories/html/gaia_hydras.onType.html @@ -0,0 +1,289 @@ + + + + + +gaia_hydras.onType + + + + +
+ + + +
+ +

Library gaia_hydras.onType

+ +
+
+ +
+Ordinal Notations (Experimental !!!!) Warning: This file is a draft !!! We try to adapt to Gaia the ON class of ordinal notations + (defined as a naïve (pre Stdpp) type class). + +
+ + An Ordinal Notation is just a well-founded ordered type, with + a trichotomic comparison + +
+ + Notions of ordinal arithmetics should be defined in substructures + +
+
+ +
+From mathcomp Require Import all_ssreflect zify.
+From mathcomp Require Import fintype.
+From Coq Require Import Logic.Eqdep_dec.
+From Coq Require Import Wellfounded.Inclusion Wf_nat Inverse_Image.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+

Preliminaries

+ restriction of a decidable relation +
+
+ +
+Definition restrict {A} (p: A → bool) (r: A → A → bool):=
+  fun x y ⇒ [&& (p x), (p y) & r x y].
+ +
+
+ +
+

Complements on orderType

+ +
+
+ +
+Section MoreOrderType.
+  Variable disp: unit.
+  Variable T: orderType disp.
+ +
+ +
+  Definition limit_v2 (f: nat → T)(x: T) :=
+    (∀ (n:nat), (f n < x)%O) ∧
+      (∀ y : T, (y < x)%O → ∃ n : nat, (y ≤ f n)%O).
+ +
+ +
+  Definition limit_of (f: nat → T) x :=
+    (∀ n m : nat, (n < m)%nat → (f n < f m)%O) ∧ limit_v2 f x.
+ +
+  Definition is_successor_of (x y: T):=
+    (x < y)%O ∧ ∀ z, (x < z)%O → ~~ (z < y)%O.
+ +
+  Section Succ_no_limit.
+    Variables (x y: T) (s: nat → T).
+    Hypothesis Hsucc : is_successor_of x y.
+    Hypothesis Hlim : limit_of s y.
+ +
+    Remark R0: ∃ (n:nat), (x ≤ (s n))%O.
+ +
+    Remark R1: ∃ z, (x < z < y)%O.
+ +
+    Lemma F: False.
+ +
+  End Succ_no_limit.
+ +
+End MoreOrderType.
+ +
+
+ +
+Type for ordinal notations +
+
+ +
+Module ONDef.
+  Record mixin_of disp (T: orderType disp) :=
+    Mixin {
+        _ : @well_founded T <%O;
+      }.
+ +
+  Section Packing.
+    Variable disp: unit.
+    Structure pack_type : Type :=
+      Pack{ type: orderType disp; _ : mixin_of type}.
+ +
+ +
+    Variable cT: pack_type.
+ +
+    Definition on_struct: mixin_of cT :=
+      let: Pack _ c := cT return mixin_of cT in c.
+ +
+  End Packing.
+ +
+  Module Exports.
+ +
+    Notation on := (pack_type ).
+    Notation ONMixin := Mixin.
+    Notation ON T m := (@Pack T m).
+    Coercion type : pack_type >-> orderType.
+ +
+    Section Lemmas.
+      Variable disp: unit.
+      Variable U: @on disp.
+ +
+      Definition tricho (a b: U) := if (a == b)%O then Eq
+                                    else if (a < b)%O then Lt
+                                         else Gt.
+ +
+      Lemma trichoP (a b: U) :
+        CompareSpec (a == b) (a < b)%O (b < a)%O (tricho a b).
+ +
+ +
+      Lemma wf : @well_founded U <%O.
+ +
+    End Lemmas.
+  End Exports.
+End ONDef.
+ +
+Export ONDef.Exports.
+ +
+
+ +
+

First instances of ON

+ +
+ +

The Ordinal notation for 'I_i

+ +
+
+ +
+Lemma wf_ltn: well_founded (fun n ⇒ [eta ltn n]).
+ +
+Section onFiniteDef.
+  Variable i: nat.
+ +
+ +
+  Lemma I_i_wf: @well_founded 'I_i (<%O).
+ +
+Definition onFiniteMixin := ONMixin I_i_wf.
+Canonical onFiniteType := ON _ _ onFiniteMixin.
+ +
+End onFiniteDef.
+ +
+
+ +
+Tests +Definition O12_33: onFiniteType 33. +by exists 12. +Defined. + +
+ +Compute tricho O12_33 O12_33. + +
+ +Goal (O12_33 <= O12_33) +
+ +

An ordinal notation for omega

+ +
+
+ +
+Section onOmegaDef.
+ +
+ Lemma omega_lt_wf : @well_founded Order.NatOrder.orderType <%O.
+ +
+ +
+Definition onOmegaMixin := ONMixin omega_lt_wf.
+Canonical onOmegaType := ON _ _ onOmegaMixin.
+ +
+End onOmegaDef.
+ +
+Example om12 : onOmegaType := 12.
+Example om67 : onOmegaType := 67.
+ +
+ +
+
+ +
+To do : Notation for epsilon0 +
+
+ +
+From gaia_hydras Require Import T1Bridge.
+ +
+Section ONEpsilon0Def.
+ +
+ +
+ +
+End ONEpsilon0Def.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.Deduction.html b/theories/html/hydras.Ackermann.Deduction.html new file mode 100644 index 00000000..e4d5af87 --- /dev/null +++ b/theories/html/hydras.Ackermann.Deduction.html @@ -0,0 +1,82 @@ + + + + + +hydras.Ackermann.Deduction + + + + +
+ + + +
+ +

Library hydras.Ackermann.Deduction

+ +
+
+ +
+ FOL's deduction Lemma + +
+ + Original file by Russel O'Connor + +
+ + +
+
+ +
+From Coq Require Import Ensembles List.
+ +
+From hydras.Ackermann Require Import folProof folProp.
+Import FolNotations.
+ +
+Section Deduction_Theorem.
+ +
+Variable L : Language.
+Let Formula := Formula L.
+Let Formulas := Formulas L.
+Let System := System L.
+Let Term := Term L.
+Let Terms := Terms L.
+Let Prf := Prf L.
+Let SysPrf := SysPrf L.
+ +
+Let list_incl (X: Ensemble Formula) (l: list Formula) :=
+    ∀ x, List.In x l → mem _ X x.
+ +
+Lemma SysPrf_rephrase X F : SysPrf X F ↔ ∃ s (_ : Prf s F), list_incl X s.
+ +
+Theorem DeductionTheorem :
+  ∀ (T : System) (f g : Formula)
+         (prf : SysPrf (Ensembles.Add _ T g) f),
+    SysPrf T (g → f)%fol.
+ +
+End Deduction_Theorem.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.LNN.html b/theories/html/hydras.Ackermann.LNN.html new file mode 100644 index 00000000..3f328da1 --- /dev/null +++ b/theories/html/hydras.Ackermann.LNN.html @@ -0,0 +1,395 @@ + + + + + +hydras.Ackermann.LNN + + + + +
+ + + +
+ +

Library hydras.Ackermann.LNN

+ +
+
+ +
+ LNN.v : Language of Natural Numbers (LNT+ <) + +
+ +Original version by Russel O'Connor + +
+ + +
+
+ +
+From Coq Require Import Arith Ensembles List.
+ +
+From hydras.Ackermann Require Export Languages folProof
+  folProp folLogic3.
+From LibHyps Require Import LibHyps.
+From hydras Require Import MoreLibHyps NewNotations.
+ +
+
+ +
+

Instantiations of a few generic constructs

+ + +
+ + To do perhaps these redefinitions should be deprecated, because they cause some issues + in statements which mix LNN and LNT terms and formulas + +
+
+ +
+Definition Formula := Formula LNN.
+Definition Formulas := Formulas LNN.
+Definition System := System LNN.
+Definition Sentence := Sentence LNN.
+Definition Term := Term LNN.
+Definition Terms := Terms LNN.
+Definition SysPrf := SysPrf LNN.
+ +
+#[local] Arguments apply _ _ _ : clear implicits.
+#[local] Arguments atomic _ _ _ : clear implicits.
+ +
+ +
+
+ +
+

Notations (Experimental and unstable)

+ +
+
+ +
+ Module NNnotations.
+ Export FolNotations.
+ +
+Definition Plus (x y : Term) : Term :=
+  apply LNN Plus_ (Tcons x (Tcons y (Tnil))).
+ +
+Definition Times (x y : Term) : Term :=
+  apply LNN Times_ (Tcons x (Tcons y (Tnil))).
+ +
+Definition Succ (x : Term) : Term :=
+  apply LNN Succ_ (Tcons x (Tnil)).
+ +
+Definition Zero : Term := apply LNN Zero_ (Tnil).
+ +
+Definition LT (x y : Term) : Formula :=
+  atomic LNN LT_ (Tcons x (Tcons y (Tnil))).
+ +
+ +
+Notation S_ t := (apply LNN Succ_ (Tcons t (Tnil))).
+ +
+Infix "+" := Plus: fol_scope.
+Notation "n + p" := (Plus n p) (in custom fol at level 50, left associativity).
+ +
+Check f[ ∀ 1, {var 1} + {var 1} = {var 1} ]f.
+Infix "×" := Times: fol_scope.
+Notation "n * p" := (Times n p) (in custom fol at level 40, left associativity).
+ +
+Infix "<" := LT: fol_scope.
+End NNnotations.
+ +
+Import NNnotations.
+ +
+Lemma LNN_eqdec : language_decidable LNN.
+ +
+Section Free_Variables.
+ +
+Lemma freeVarPlus (x y : Term) :
+ freeVarT (Plus x y) = freeVarT x ++ freeVarT y.
+ +
+Lemma freeVarTimes (x y : Term):
+ freeVarT (Times x y) = freeVarT x ++ freeVarT y.
+ +
+Lemma freeVarSucc (x : Term):
+  freeVarT (Succ x) = freeVarT x.
+ +
+Lemma freeVarZero : freeVarT Zero = nil.
+ +
+Lemma freeVarLT (x y : Term) :
+ freeVarF (LT x y) = freeVarT x ++ freeVarT y.
+ +
+End Free_Variables.
+ +
+
+ +
+

Basic and derived properties

+ +
+
+Section Logic.
+ +
+Lemma Axm (T : System) (f : Formula): mem _ T f → SysPrf T f.
+ +
+Lemma sysExtend (T U : System) (f : Formula):
+ Included _ T U → SysPrf T f → SysPrf U f.
+ +
+Lemma sysWeaken (T : System) (f g : Formula):
+  SysPrf T f → SysPrf (Ensembles.Add _ T g) f.
+ +
+Lemma impI (T : System) (f g : Formula):
+ SysPrf (Ensembles.Add _ T g) f → SysPrf T (g → f)%fol.
+ +
+Lemma impE (T : System) (f g : Formula):
+  SysPrf T (g → f)%fol → SysPrf T g → SysPrf T f.
+ +
+Lemma contradiction (T : System) (f g : Formula):
+  SysPrf T f → SysPrf T (¬ f)%fol → SysPrf T g.
+ +
+Lemma nnE (T : System) (f : Formula):
+  SysPrf T (~~ f)%fol → SysPrf T f.
+ +
+Lemma noMiddle (T : System) (f : Formula): SysPrf T (¬ f ∨ f)%fol.
+ +
+Lemma nnI (T : System) (f : Formula):
+  SysPrf T f → SysPrf T (¬ ¬ f)%fol.
+ +
+Lemma cp1 (T : System) (f g : Formula) :
+ SysPrf T (¬ f → ¬ g)%fol → SysPrf T (impH g f).
+ +
+Lemma cp2 (T : System) (f g : Formula):
+ SysPrf T (g → f)%fol → SysPrf T (¬ f → ¬ g)%fol.
+ +
+Lemma orI1 (T : System) (f g : Formula):
+  SysPrf T f → SysPrf T (f ∨ g)%fol.
+ +
+Lemma orI2 (T : System) (f g : Formula):
+  SysPrf T g → SysPrf T (f ∨ g)%fol.
+ +
+Lemma orE (T : System) (f g h : Formula):
+  SysPrf T (f ∨ g)%fol →
+  SysPrf T (f → h)%fol → SysPrf T (g → h)%fol → SysPrf T h.
+ +
+Lemma orSys (T : System) (f g h : Formula) :
+ SysPrf (Ensembles.Add _ T f) h → SysPrf (Ensembles.Add _ T g) h →
+ SysPrf (Ensembles.Add _ T (f ∨ g)%fol) h.
+ +
+Lemma andI (T : System) (f g : Formula):
+ SysPrf T f → SysPrf T g → SysPrf T (f ∧ g)%fol.
+ +
+Lemma andE1 (T : System) (f g : Formula):
+  SysPrf T (f ∧ g)%fol → SysPrf T f.
+ +
+Lemma andE2 (T : System) (f g : Formula):
+  SysPrf T (f ∧ g)%fol → SysPrf T g.
+ +
+Lemma iffI (T : System) (f g : Formula):
+ SysPrf T (f → g)%fol → SysPrf T (g → f)%fol →
+ SysPrf T (f ↔ g)%fol.
+ +
+Lemma iffE1 (T : System) (f g : Formula):
+ SysPrf T (f ↔ g)%fol → SysPrf T (f → g)%fol.
+ +
+Lemma iffE2 (T : System) (f g : Formula):
+ SysPrf T (f ↔ g)%fol → SysPrf T (g → f)%fol.
+ +
+Lemma forallI (T : System) (f : Formula) (v : nat):
+ ¬ In_freeVarSys LNN v T → SysPrf T f → SysPrf T (allH v, f)%fol.
+ +
+Lemma forallE (T : System) (f : Formula) (v : nat) (t : Term):
+ SysPrf T (allH v, f)%fol → SysPrf T (substF f v t).
+ +
+Lemma forallSimp (T : System) (f : Formula) (v : nat) :
+ SysPrf T (allH v, f)%fol → SysPrf T f.
+ +
+Lemma existI (T : System) (f : Formula) (v : nat) (t : Term):
+ SysPrf T (substF f v t) → SysPrf T (exH v, f)%fol.
+ +
+Lemma existE (T : System) (f g : Formula) (v : nat):
+  ¬ In_freeVarSys LNN v T →
+  ¬ In v (freeVarF g) →
+  SysPrf T (exH v, f)%fol → SysPrf T (f → g)%fol →
+  SysPrf T g.
+ +
+Lemma existSimp (T : System) (f : Formula) (v : nat):
+  SysPrf T f → SysPrf T (exH v, f)%fol.
+ +
+Lemma existSys (T : System) (f g : Formula) (v : nat):
+  ¬ In_freeVarSys LNN v T →
+  ¬ In v (freeVarF g) →
+  SysPrf (SetAdds T f) g →
+  SysPrf (SetAdds T (exH v, f)%fol) g.
+ +
+Lemma absurd1 (T : System) (f : Formula):
+ SysPrf T (f → ¬ f)%fol → SysPrf T (¬ f)%fol.
+ +
+Lemma nImp (T : System) (f g : Formula):
+ SysPrf T (f ∧ ¬ g)%fol → SysPrf T (¬ (f → g))%fol.
+ +
+Lemma nOr (T : System) (f g : Formula):
+ SysPrf T (¬ f ∧ ¬ g)%fol → SysPrf T (¬ (f ∨ g))%fol.
+ +
+Lemma nAnd (T : System) (f g : Formula):
+ SysPrf T (¬ f ∨ ¬ g)%fol → SysPrf T (¬ (f ∧ g))%fol.
+ +
+Lemma nForall (T : System) (f : Formula) (v : nat):
+ SysPrf T (exH v, ¬ f)%fol → SysPrf T (¬ (allH v, f))%fol.
+ +
+Lemma nExist (T : System) (f : Formula) (v : nat):
+ SysPrf T (allH v, ¬ f)%fol → SysPrf T (¬ (exH v, f))%fol.
+ +
+Lemma impRefl (T : System) (f : Formula): SysPrf T (f → f)%fol.
+ +
+Lemma impTrans (T : System) (f g h : Formula):
+ SysPrf T (f → g)%fol → SysPrf T (g → h)%fol → SysPrf T (f → h)%fol.
+ +
+Lemma orSym (T : System) (f g : Formula):
+  SysPrf T (f ∨ g)%fol → SysPrf T (g ∨ f)%fol.
+ +
+Lemma andSym (T : System) (f g : Formula):
+SysPrf T (f ∧ g)%fol → SysPrf T (g ∧ f)%fol.
+ +
+Lemma iffRefl (T : System) (f : Formula): SysPrf T (f ↔ f)%fol.
+ +
+Lemma iffSym (T : System) (f g : Formula):
+  SysPrf T (f ↔ g)%fol → SysPrf T (g ↔ f)%fol.
+ +
+Lemma iffTrans (T : System) (f g h : Formula):
+  SysPrf T (f ↔ g)%fol → SysPrf T (g ↔ h)%fol →
+  SysPrf T (f ↔ h)%fol.
+ +
+Lemma eqRefl (T : System) (a : Term): SysPrf T (a = a)%fol.
+ +
+Lemma eqSym (T : System) (a b : Term):
+ SysPrf T (a = b)%fol → SysPrf T (b = a)%fol.
+ +
+Lemma eqTrans (T : System) (a b c : Term):
+SysPrf T (a = b)%fol → SysPrf T (b = c)%fol → SysPrf T (a = c)%fol.
+ +
+Lemma eqPlus (T : System) (a b c d : Term):
+  SysPrf T (a = b)%fol → SysPrf T (c = d)%fol →
+  SysPrf T (a + c = b + d)%fol.
+ +
+Lemma eqTimes (T : System) (a b c d : Term):
+ SysPrf T (a = b)%fol → SysPrf T (c = d)%fol →
+ SysPrf T (a × c = b × d)%fol.
+ +
+Lemma eqSucc (T : System) (a b : Term):
+  SysPrf T (a = b)%fol → SysPrf T (Succ a = Succ b)%fol.
+ +
+Lemma eqLT (T : System) (a b c d : Term):
+ SysPrf T (a = b)%fol → SysPrf T (c = d)%fol →
+ SysPrf T (a < c ↔ b < d)%fol.
+ +
+End Logic.
+ +
+Fixpoint natToTerm (n : nat) : Term :=
+  match n with
+  | O ⇒ Zero
+  | S m ⇒ Succ (natToTerm m)
+  end.
+ +
+#[global] Notation n2t i := (natToTerm i).
+ +
+Lemma closedNatToTerm :
+ ∀ a v : nat, ¬ In v (freeVarT (natToTerm a)).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.LNN2LNT.html b/theories/html/hydras.Ackermann.LNN2LNT.html new file mode 100644 index 00000000..fba54782 --- /dev/null +++ b/theories/html/hydras.Ackermann.LNN2LNT.html @@ -0,0 +1,318 @@ + + + + + +hydras.Ackermann.LNN2LNT + + + + +
+ + + +
+ +

Library hydras.Ackermann.LNN2LNT

+ +
+
+ +
+LNN2LNT: + +
+ + Translation of LNN-formulas and proofs into LNT by replacement of + (t < t')%fol subformulas with (∃ v, t + Succ v = t')%nt. + +
+ + Original file by Russel O'Connor + +
+ + +
+
+ +
+ +
+
+ +
+

Translation of terms

+ +
+
+ +
+Fixpoint LNN2LNT_term (t : fol.Term LNN) : fol.Term LNT:=
+  match t with
+  | var v ⇒ var v
+  | apply f ts ⇒ apply LNT f (LNN2LNT_terms _ ts)
+  end
+ 
+ with LNN2LNT_terms (n : nat) (ts : fol.Terms LNN n) {struct ts} :
+ Terms n :=
+  match ts in (fol.Terms _ n0) return (Terms n0) with
+  | Tnil ⇒ @Tnil LNT
+  | Tcons m s ss ⇒ Tcons (LNN2LNT_term s) (LNN2LNT_terms m ss)
+  end.
+ +
+ +
+
+ +
+

Inverse translation

+ +
+
+ +
+Fixpoint LNT2LNN_term (t : Term) : fol.Term LNN :=
+  match t with
+  | var v ⇒ var v
+  | apply f ts ⇒ apply LNN f (LNT2LNN_terms _ ts)
+  end
+ 
+ with LNT2LNN_terms (n : nat) (ts : Terms n) {struct ts} :
+ fol.Terms LNN n :=
+  match ts in (fol.Terms _ n0) return (fol.Terms LNN n0) with
+  | Tnil ⇒ @Tnil LNN
+  | Tcons m s ss ⇒ Tcons (LNT2LNN_term s) (LNT2LNN_terms m ss)
+  end.
+ +
+Lemma LNT2LNN_natToTerm (n: nat) :
+  LNT2LNN_term (natToTerm n) = natToTermLNN n.
+ +
+Lemma LNT2LNT_term (t : Term): LNN2LNT_term (LNT2LNN_term t) = t.
+ +
+
+ +
+

Translation of formulas

+ +
+ +

Translation of (v#0 < v#1)%fol

+ +
+
+ +
+Definition LTFormula := (exH 2, v#0 + LNT.Succ v#2 = v#1)%nt.
+ +
+
+ +
+

Translation of (t < t')%fol

+ +
+ + The function translateLT is defined by an interactive proof (omitted). + It is specified by the lemma translateLT1 + +
+
+ +
+Definition translateLT (ts : fol.Terms LNN (arityR LNN LT_)) : Formula.
+ +
+Lemma translateLT1 :
+ ∀ a a0 b0,
+ translateLT (Tcons a (Tcons a0 b0)) =
+ subAllFormula LNT LTFormula
+   (fun H : nat ⇒
+    nat_rec (fun _ : nat ⇒ fol.Term LNT) (LNN2LNT_term a)
+      (fun (H0 : nat) (_ : fol.Term LNT) ⇒
+       nat_rec (fun _ : nat ⇒ fol.Term LNT) (LNN2LNT_term a0)
+         (fun (H1 : nat) (_ : fol.Term LNT) ⇒ var H1) H0) H).
+ +
+
+ +
+

Translation of any LNN-formula

+ + +
+ + The translation of any LNN-formula is straigthforward, except for + the special case of t_1 < t_2 (handled by translateLT ) + +
+ + +
+
+ +
+Fixpoint LNN2LNT_formula (f : fol.Formula LNN) : Formula :=
+  match f with
+  | (t1 = t2)%fol ⇒ (LNN2LNT_term t1 = LNN2LNT_term t2)%nt
+  | atomic r ts ⇒
+      match
+        r as l return (fol.Terms LNN (arityR LNN l) → Formula)
+      with
+      | LT_ ⇒ fun ts : fol.Terms LNN (arityR LNN LT_) ⇒ translateLT ts
+      end ts
+  | (A → B)%fol ⇒ (LNN2LNT_formula A → LNN2LNT_formula B)%nt
+  | (¬ A)%fol ⇒ (¬ LNN2LNT_formula A)%nt
+  | (allH v, A)%fol ⇒ (allH v, LNN2LNT_formula A)%nt
+  end.
+ +
+
+ +
+

Helpful rewriting lemmas

+ +
+
+ +
+Lemma LNN2LNT_or (a b : fol.Formula LNN):
+  LNN2LNT_formula (orH a b) =
+    orH (LNN2LNT_formula a) (LNN2LNT_formula b).
+ +
+Lemma LNN2LNT_and (a b : fol.Formula LNN):
+ LNN2LNT_formula (andH a b) =
+ andH (LNN2LNT_formula a) (LNN2LNT_formula b).
+ +
+Lemma LNN2LNT_iff (a b : fol.Formula LNN):
+ LNN2LNT_formula (iffH a b) =
+   iffH (LNN2LNT_formula a) (LNN2LNT_formula b).
+ +
+Lemma LNN2LNT_exist (v : nat) (a : fol.Formula LNN) :
+ LNN2LNT_formula (existH v a) = existH v (LNN2LNT_formula a).
+ +
+Lemma LNN2LNT_freeVarF (f : fol.Formula LNN) (v : nat):
+  In v (freeVarF (LNN2LNT_formula f)) ↔ In v (freeVarF f).
+ +
+Lemma LNN2LNT_freeVarF1 (f : fol.Formula LNN) (v : nat):
+ In v (freeVarF (LNN2LNT_formula f)) →
+ In v (freeVarF f).
+ +
+Lemma LNN2LNT_freeVarF2 (f : fol.Formula LNN) (v : nat):
+  In v (freeVarF f) →
+  In v (freeVarF (LNN2LNT_formula f)).
+ +
+Lemma LNN2LNT_subFormula
+ (T : System) (f : fol.Formula LNN) (v : nat) (s : fol.Term LNN):
+ SysPrf T
+   (iffH (LNN2LNT_formula (substF f v s))
+      (substF (LNN2LNT_formula f) v (LNN2LNT_term s))).
+ +
+
+ +
+

Inverse translation

+ +
+
+ +
+Fixpoint LNT2LNN_formula (f : Formula) : fol.Formula LNN :=
+  match f with
+  | (t1 = t2)%nt ⇒ (LNT2LNN_term t1 = LNT2LNN_term t2)%fol
+  | atomic r ts ⇒ match r with
+                   end
+  | (A → B)%nt ⇒ (LNT2LNN_formula A → LNT2LNN_formula B)%fol
+  | (¬ A)%nt ⇒ (¬ LNT2LNN_formula A)%fol
+  | (allH v, A)%nt ⇒ (allH v, LNT2LNN_formula A)%fol
+  end.
+ +
+
+ +
+

Commutation lemmas

+ +
+
+ +
+Lemma LNT2LNT_formula (f : Formula):
+ LNN2LNT_formula (LNT2LNN_formula f) = f.
+ +
+Lemma LNT2LNN_subTerm (t : Term) (v : nat) (s : Term):
+  LNT2LNN_term (substT t v s) =
+    substT (LNT2LNN_term t) v (LNT2LNN_term s).
+ +
+Lemma LNT2LNN_freeVarT ( t : Term):
+  freeVarT (LNT2LNN_term t) = freeVarT t.
+ +
+Lemma LNT2LNN_freeVarF (f : Formula):
+  freeVarF (LNT2LNN_formula f) = freeVarF f.
+ +
+Lemma LNT2LNN_subFormula :
+  ∀ (f : Formula) (v : nat) (s : Term),
+    LNT2LNN_formula (substF f v s) =
+      substF (LNT2LNN_formula f) v (LNT2LNN_term s).
+ +
+
+ +
+

Proof translation

+ +
+
+ +
+ +
+
+ +
+ If the translation of every axiom of a LNN-system U is provable in a closed + LNT-system V, + then the translation of any LNN-formula in U is provable in V. + +
+
+ +
+Lemma translateProof (U : fol.System LNN) (V : System):
+    ClosedSystem LNT V →
+    (∀ f : fol.Formula LNN,
+        mem (fol.Formula LNN) U f → SysPrf V (LNN2LNT_formula f)) →
+    ∀ f : fol.Formula LNN,
+      folProof.SysPrf LNN U f → SysPrf V (LNN2LNT_formula f).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.LNT.html b/theories/html/hydras.Ackermann.LNT.html new file mode 100644 index 00000000..c6d3acb8 --- /dev/null +++ b/theories/html/hydras.Ackermann.LNT.html @@ -0,0 +1,450 @@ + + + + + +hydras.Ackermann.LNT + + + + +
+ + + +
+ +

Library hydras.Ackermann.LNT

+ +
+
+ +
+ LNT.v : Language of Number Theory + +
+ +Original version by Russel O'Connor + +
+ + +
+
+ +
+From Coq Require Import Arith Ensembles List.
+ +
+From hydras.Ackermann
+ Require Import Languages folProof folProp folLogic3.
+ +
+
+ +
+

Instantiations of a few generic constructs

+ + +
+ + To do perhaps these redefinitions should be deprecated, because they cause some issues + in statements which mix LNN and LNT terms and formulas + +
+
+ +
+Definition Formula := Formula LNT.
+Definition Formulas := Formulas LNT.
+Definition System := System LNT.
+Definition Sentence := Sentence LNT.
+Definition Term := Term LNT.
+Definition Terms := Terms LNT.
+Definition SysPrf := SysPrf LNT.
+#[local] Arguments apply _ _ _ : clear implicits.
+ +
+Module LNT.
+ +
+Definition Plus (x y : Term) : Term :=
+  apply LNT Plus_ (Tcons x (Tcons y (Tnil))).
+ +
+Definition Times (x y : Term) : Term :=
+  apply LNT Times_ (Tcons x (Tcons y (Tnil))).
+ +
+Definition Succ (x : Term) : Term :=
+  apply LNT Succ_ (Tcons x (Tnil)).
+ +
+Definition Zero : Term := apply LNT Zero_ (Tnil).
+ +
+End LNT.
+Export LNT.
+ +
+
+ +
+

Notations for LNT formulas: experimental and unstable

+ +
+
+ +
+Declare Scope nt_scope.
+Delimit Scope nt_scope with nt.
+ +
+Infix "=" := (equal _): nt_scope.
+Infix "∨" := (orH): nt_scope.
+Infix "∧" := (andH):nt_scope.
+Infix "→" := (impH): nt_scope.
+Notation "~ A" := (@notH _ A): nt_scope.
+Notation "A <-> B" := (@iffH _ A B): nt_scope.
+ +
+Notation "t = u" := (@equal _ t u): nt_scope.
+Notation "t <> u" := (¬ t = u)%nt : nt_scope.
+Notation "'v#' i" := (var i) (at level 3, format "'v#' i", i at level 0) : nt_scope.
+ +
+Notation "'exH' x .. y , p" := (existH x .. (existH y p) ..)
+  (x at level 0, y at level 0, at level 200, right associativity) : nt_scope.
+ +
+Notation "'allH' x .. y , p" := (forallH x .. (forallH y p) ..)
+  (x at level 0, y at level 0, at level 200, right associativity) : nt_scope.
+ +
+Infix "+" := Plus :nt_scope.
+Infix "×" := Times :nt_scope.
+Notation S_ t := (apply LNT Succ_ (Tcons t (Tnil))).
+ +
+
+ +
+

Notations for printing computed formulas/terms with derived connectives

+ +
+
+ +
+Reserved Notation "x '\/'' y" (at level 85, right associativity).
+Reserved Notation "x '/\'' y" (at level 80, right associativity).
+Reserved Notation "x '<->'' y" (at level 95, no associativity).
+Reserved Notation "x '<->''' y" (at level 95, no associativity).
+ +
+Notation "x \/' y" := (¬ x → y)%nt : nt_scope.
+Notation "x /\' y" := (¬ (~ x \/' ¬ y))%nt : nt_scope.
+Notation "x <->'' y" := ((x → y) ∧ (y → x))%nt: nt_scope.
+Notation "x <->' y" := (¬ (~ (x → y) \/' ~(y → x)))%nt : nt_scope.
+Notation exH' v A := (¬ (forallH v (¬ A)))%nt.
+ +
+
+ +
+

Examples

+ +
+
+ +
+Section Examples.
+Variable f : Formula.
+Check (allH 0 1 , (f → v#0 = v#0 → v#1 = v#1))%nt.
+ +
+Check (exH 0 1 , (v#0 = v#0 → v#1 = v#1))%nt.
+End Examples.
+ +
+
+ +
+

Basic properties

+ + +
+
+ +
+Lemma LNT_eqdec : language_decidable LNT.
+ +
+
+ +
+

List of free variables of a formula

+ +
+
+ +
+Section Free_Variables.
+ +
+Lemma freeVarPlus (x y: Term) :
+ freeVarT (Plus x y) = freeVarT x ++ freeVarT y.
+ +
+Lemma freeVarTimes (x y : Term) :
+ freeVarT (Times x y) = freeVarT x ++ freeVarT y.
+ +
+Lemma freeVarSucc (x : Term):
+  freeVarT (S_ x)%nt = freeVarT x.
+ +
+Lemma freeVarZero : freeVarT Zero = nil.
+ +
+End Free_Variables.
+ +
+
+ +
+

Basic and derived proof rules

+ +
+
+ +
+Section Logic.
+ +
+Lemma Axm (T : System) (f : Formula) : mem _ T f → SysPrf T f.
+ +
+Lemma sysExtend (T U : System) (f : Formula):
+  Included _ T U → SysPrf T f → SysPrf U f.
+ +
+Lemma sysWeaken (T : System) (f g : Formula):
+  SysPrf T f → SysPrf (Ensembles.Add _ T g) f.
+ +
+Lemma impI (T : System) (f g : Formula):
+  SysPrf (Ensembles.Add _ T g) f → SysPrf T (g → f)%nt.
+ +
+Lemma impE (T : System) (f g : Formula):
+ SysPrf T (g → f)%nt → SysPrf T g → SysPrf T f.
+ +
+Lemma contradiction (T : System) (f g : Formula):
+ SysPrf T f → SysPrf T (¬ f)%nt → SysPrf T g.
+ +
+Lemma nnE (T : System) (f : Formula):
+  SysPrf T (¬ ¬ f)%nt → SysPrf T f.
+ +
+Lemma noMiddle (T : System) (f : Formula): SysPrf T (¬ f ∨ f)%nt.
+ +
+Lemma nnI (T : System) (f : Formula):
+  SysPrf T f → SysPrf T (¬ ¬ f)%nt.
+ +
+Lemma cp1 (T : System) (f g : Formula):
+ SysPrf T (¬ f → ¬ g)%nt → SysPrf T (g → f)%nt.
+ +
+Lemma cp2 (T : System) (f g : Formula):
+ SysPrf T (g → f)%nt → SysPrf T (¬ f → ¬ g)%nt.
+ +
+Lemma orI1 (T : System) (f g : Formula):
+  SysPrf T f → SysPrf T (f ∨ g)%nt.
+ +
+Lemma orI2 (T : System) (f g : Formula):
+  SysPrf T g → SysPrf T (f ∨ g)%nt.
+ +
+Lemma orE (T : System) (f g h : Formula):
+  SysPrf T (f ∨ g)%nt →
+  SysPrf T (f → h)%nt → SysPrf T (g → h)%nt →
+  SysPrf T h.
+ +
+Lemma orSys (T : System) (f g h : Formula):
+ SysPrf (Ensembles.Add _ T f) h →
+ SysPrf (Ensembles.Add _ T g) h →
+ SysPrf (Ensembles.Add _ T (f ∨ g)%nt) h.
+ +
+Lemma andI (T : System) (f g : Formula) :
+ SysPrf T f → SysPrf T g → SysPrf T (f ∧ g)%nt.
+ +
+Lemma andE1 (T : System) (f g : Formula) :
+  SysPrf T (f ∧ g)%nt → SysPrf T f.
+ +
+Lemma andE2 (T : System) (f g : Formula) :
+  SysPrf T (f ∧ g)%nt → SysPrf T g.
+ +
+Lemma iffI (T : System) (f g : Formula) :
+ SysPrf T (f → g)%nt → SysPrf T (g → f)%nt → SysPrf T (f ↔ g)%nt.
+ +
+Lemma iffE1 (T : System) (f g : Formula):
+ SysPrf T (f ↔ g)%nt → SysPrf T (f → g)%nt.
+ +
+Lemma iffE2 (T : System) (f g : Formula) :
+ SysPrf T (f ↔ g)%nt → SysPrf T (g → f)%nt.
+ +
+Lemma forallI (T : System) (f : Formula) (v : nat):
+ ¬ In_freeVarSys LNT v T → SysPrf T f → SysPrf T (allH v, f)%nt.
+ +
+Lemma forallE (T : System) (f : Formula) (v : nat) (t : Term) :
+ SysPrf T (allH v, f)%nt → SysPrf T (substF f v t).
+ +
+Lemma forallSimp (T : System) (f : Formula) (v : nat):
+ SysPrf T (allH v, f)%nt → SysPrf T f.
+ +
+Lemma existI (T : System) (f : Formula) (v : nat) (t : Term):
+ SysPrf T (substF f v t) → SysPrf T (exH v, f)%nt.
+ +
+Lemma existE (T : System) (f g : Formula) (v : nat):
+  ¬ In_freeVarSys LNT v T →
+  ¬ In v (freeVarF g) →
+  SysPrf T (exH v, f)%nt → SysPrf T (f → g)%nt → SysPrf T g.
+ +
+Lemma existSimp (T : System) (f : Formula) (v : nat):
+ SysPrf T f → SysPrf T (exH v, f)%nt.
+ +
+Lemma existSys (T : System) (f g : Formula) (v : nat):
+  ¬ In_freeVarSys LNT v T →
+  ¬ In v (freeVarF g) →
+  SysPrf (Ensembles.Add _ T f) g →
+  SysPrf (Ensembles.Add _ T (exH v, f)%nt) g.
+ +
+Lemma absurd1 (T : System) (f : Formula):
+ SysPrf T (f → ¬ f)%nt → SysPrf T (¬ f)%nt.
+ +
+Lemma nImp (T : System) (f g : Formula):
+ SysPrf T (f ∧ ¬g)%nt → SysPrf T (¬ (f → g))%nt.
+ +
+Lemma nOr (T : System) (f g : Formula):
+ SysPrf T (¬ f ∧ ¬g)%nt → SysPrf T (¬ (f ∨ g))%nt.
+ +
+Lemma nAnd (T : System) (f g : Formula):
+ SysPrf T (¬ f ∨ ¬ g)%nt → SysPrf T (¬ (f ∧ g))%nt.
+ +
+Lemma nForall (T : System) (f : Formula) (v : nat) :
+ SysPrf T (exH v, ¬ f)%nt → SysPrf T (¬ (allH v, f))%nt.
+ +
+Lemma nExist (T : System) (f : Formula) (v : nat):
+ SysPrf T (allH v, ¬ f)%nt → SysPrf T (¬ (exH v, f))%nt.
+ +
+Lemma impRefl (T : System) (f : Formula): SysPrf T (f → f)%nt.
+ +
+Lemma impTrans (T : System) (f g h : Formula):
+ SysPrf T (f → g)%nt → SysPrf T (g → h)%nt → SysPrf T (f → h)%nt.
+ +
+Lemma orSym (T : System) (f g : Formula):
+ SysPrf T (f ∨ g)%nt → SysPrf T (g ∨ f)%nt.
+ +
+Lemma andSym (T : System) (f g : Formula) :
+ SysPrf T (f ∧ g)%nt → SysPrf T (g ∧ f)%nt.
+ +
+Lemma iffRefl (T : System) (f : Formula) : SysPrf T (f ↔ f)%nt.
+ +
+Lemma iffSym (T : System) (f g : Formula):
+  SysPrf T (f ↔ g)%nt → SysPrf T (g ↔ f)%nt.
+ +
+Lemma iffTrans (T : System) (f g h : Formula):
+ SysPrf T (f ↔ g)%nt → SysPrf T (g ↔ h)%nt → SysPrf T (f ↔ h)%nt.
+ +
+Lemma eqRefl (T : System) (a : Term): SysPrf T (a = a)%nt.
+ +
+Lemma eqSym (T : System) (a b : Term):
+ SysPrf T (a = b)%nt → SysPrf T (b = a)%nt.
+ +
+Lemma eqTrans (T : System) (a b c : Term):
+  SysPrf T (a = b)%nt → SysPrf T (b = c)%nt → SysPrf T (a = c)%nt.
+ +
+Lemma eqPlus (T : System) (a b c d : Term):
+  SysPrf T (a = b)%nt → SysPrf T (c = d)%nt →
+  SysPrf T (a + c = b + d)%nt.
+ +
+Lemma eqTimes (T : System) (a b c d : Term):
+  SysPrf T (a = b)%nt → SysPrf T (c = d)%nt →
+  SysPrf T (a × c = b × d)%nt.
+ +
+Lemma eqSucc (T : System) (a b : Term):
+ SysPrf T (a = b)%nt → SysPrf T (Succ a = Succ b)%nt.
+ +
+End Logic.
+ +
+
+ +
+Conversion from nat +
+
+Fixpoint natToTerm (n : nat) : Term :=
+  match n with
+  | O ⇒ Zero
+  | S m ⇒ S_ (natToTerm m)
+  end.
+ +
+Lemma closedNatToTerm :
+ ∀ a v : nat, ¬ In v (freeVarT (natToTerm a)).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.Languages.html b/theories/html/hydras.Ackermann.Languages.html new file mode 100644 index 00000000..8de98a06 --- /dev/null +++ b/theories/html/hydras.Ackermann.Languages.html @@ -0,0 +1,234 @@ + + + + + +hydras.Ackermann.Languages + + + + +
+ + + +
+ +

Library hydras.Ackermann.Languages

+ +
+
+ +
+Languages : Definitions of LNT and LNN + +
+ + Original development by Russel O'Connor +
+ + TO do : reorganize Alectryon snippets +
+
+ +
+From Coq Require Import Arith List.
+From hydras.Ackermann Require Import fol primRec code.
+ +
+
+ +
+

Language of Number Theory: LNT

+ +
+
+ +
+Inductive LNTFunction : Set :=
+  | Plus_ : LNTFunction
+  | Times_ : LNTFunction
+  | Succ_ : LNTFunction
+  | Zero_ : LNTFunction.
+ +
+Inductive LNTRelation : Set :=.
+ +
+Definition LNTFunctionArity (x : LNTFunction) : nat :=
+  match x with
+  | Plus_ ⇒ 2
+  | Times_ ⇒ 2
+  | Succ_ ⇒ 1
+  | Zero_ ⇒ 0
+  end.
+ +
+ +
+Definition LNTRelationR (x : LNTRelation) : nat :=
+  match x with bot ⇒ LNTRelation_rec (fun _ ⇒ nat) bot end.
+ +
+Definition LNT : Language := language LNTRelation LNTFunction LNTRelationR LNTFunctionArity.
+ +
+
+ +
+

Language of Natural Numbers: LNN

+ + Its functions are also LNT's + +
+
+ +
+ +
+Inductive LNNRelation : Set :=
+    LT_ : LNNRelation.
+ +
+Definition LNNArityR (x : LNNRelation) : nat :=
+ match x with LT_ ⇒ 2 end.
+ +
+Definition LNNArityF (f : LNTFunction) :=
+     LNTFunctionArity f.
+ +
+Definition LNN : Language := language LNNRelation LNTFunction
+                               LNNArityR LNNArityF.
+ +
+ +
+
+ +
+

Goedel encoding for LNT

+ +
+ + This function is also used as encoding for LNN +
+
+ +
+Definition codeLNTFunction (f : LNTFunction) : nat :=
+  match f with
+  | Plus_ ⇒ 0
+  | Times_ ⇒ 1
+  | Succ_ ⇒ 2
+  | Zero_ ⇒ 3
+  end.
+ +
+Definition codeLNTRelation (R : LNTRelation) : nat :=
+  match R return nat with
+  end.
+ +
+Lemma codeLNTFunctionInj :
+ ∀ f g : LNTFunction, codeLNTFunction f = codeLNTFunction g → f = g.
+ +
+Lemma codeLNTRelationInj :
+ ∀ R S : LNTRelation, codeLNTRelation R = codeLNTRelation S → R = S.
+ +
+ +
+Definition codeArityLNTR (r : nat) := 0.
+ +
+#[export]Instance codeArityLNTRIsPR : isPR 1 codeArityLNTR.
+ +
+Lemma codeArityLNTRIsCorrect1 :
+ ∀ r : Relations LNT,
+ codeArityLNTR (codeLNTRelation r) = S (arityR LNT r).
+ +
+Lemma codeArityLNTRIsCorrect2 :
+ ∀ n : nat,
+ codeArityLNTR n ≠ 0 → ∃ r : Relations LNT, codeLNTRelation r = n.
+ +
+Definition codeArityLNTF (f : nat) :=
+  switchPR f
+    (switchPR (pred f)
+       (switchPR (pred (pred f)) (switchPR (pred (pred (pred f))) 0 1) 2) 3)
+    3.
+ +
+#[export]Instance codeArityLNTFIsPR : isPR 1 codeArityLNTF.
+ +
+Lemma codeArityLNTFIsCorrect1 :
+ ∀ f : Functions LNT,
+ codeArityLNTF (codeLNTFunction f) = S (arityF LNT f).
+ +
+Lemma codeArityLNNFIsCorrect1 :
+ ∀ f : Functions LNN,
+ codeArityLNTF (codeLNTFunction f) = S (arityF LNN f).
+ +
+Lemma codeArityLNTFIsCorrect2 :
+ ∀ n : nat,
+ codeArityLNTF n ≠ 0 →
+ ∃ f : Functions LNT, codeLNTFunction f = n.
+ +
+
+ +
+

Goedel encoding for LNN

+ +
+
+ +
+Definition codeLNNRelation (R : LNNRelation) : nat := 0.
+ +
+Lemma codeLNNRelationInj :
+ ∀ R S : LNNRelation, codeLNNRelation R = codeLNNRelation S → R = S.
+ +
+Definition codeArityLNNR (r : nat) := switchPR r 0 3.
+ +
+#[export]Instance codeArityLNNRIsPR : isPR 1 codeArityLNNR.
+ +
+Lemma codeArityLNNRIsCorrect1 :
+  ∀ r : Relations LNN,
+    codeArityLNNR (codeLNNRelation r) = S (arityR LNN r).
+ +
+Lemma codeArityLNNRIsCorrect2 :
+ ∀ n : nat,
+ codeArityLNNR n ≠ 0 → ∃ r : Relations LNN, codeLNNRelation r = n.
+ +
+Lemma codeArityLNNFIsCorrect2 :
+  ∀ n : nat,
+    codeArityLNTF n ≠ 0 → ∃ f : Functions LNN, codeLNTFunction f = n.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.ListExt.html b/theories/html/hydras.Ackermann.ListExt.html new file mode 100644 index 00000000..31acf1a4 --- /dev/null +++ b/theories/html/hydras.Ackermann.ListExt.html @@ -0,0 +1,76 @@ + + + + + +hydras.Ackermann.ListExt + + + + +
+ + + +
+ +

Library hydras.Ackermann.ListExt

+ +
+
+ +
+ ListExt.v: + +
+ + The most part of the original contents by Russel have been + replaced by lemmas from Coq.List. + We decided to keep in_remove_neq (originally In_list_remove2 + since it appears in many composed tactics. + +
+
+ +
+From Coq Require Import Lists.List.
+ +
+Section List_Remove.
+ +
+Variable A : Set.
+Hypothesis Aeq_dec : ∀ a b : A, {a = b} + {a ≠ b}.
+ +
+Lemma in_remove_neq:
+ ∀ (a b : A) (l : list A), In a (List.remove Aeq_dec b l) →
+                                a ≠ b.
+ +
+End List_Remove.
+ +
+
+ +
+Old name in Russel's development +
+
+ +
+#[deprecated(note="use ListExt.in_remove_neq instead")]
+ Notation In_list_remove2 := in_remove_neq (only parsing).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.NN.html b/theories/html/hydras.Ackermann.NN.html new file mode 100644 index 00000000..91dfc198 --- /dev/null +++ b/theories/html/hydras.Ackermann.NN.html @@ -0,0 +1,153 @@ + + + + + +hydras.Ackermann.NN + + + + +
+ + + +
+ +

Library hydras.Ackermann.NN

+ +
+ +
+
+ +
+ NN.v : Natural Numbers: Axioms and basic properties + +
+ +Original version by Russel O'Connor + +
+ + +
+
+ +
+From Coq Require Import Arith Lia Ensembles List.
+ +
+From hydras.Ackermann Require Import folProp subAll folLogic3 Languages.
+From hydras.Ackermann Require Export LNN.
+From hydras Require Import Compat815 NewNotations.
+Import NNnotations.
+
+ +
+

Axioms of NN

+ +
+
+Section NN.
+ +
+Definition NN1 := (allH 0, S_ v#0 ≠ Zero)%fol.
+ +
+Definition NN2 := (allH 1 0, S_ v#0 = S_ v#1 → v#0 = v#1)%fol.
+ +
+Definition NN3 := (allH 0, v#0 + Zero = v#0)%fol.
+ +
+Definition NN4 := (allH 1 0, v#0 + S_ v#1 = S_ (v#0 + v#1))%fol.
+ +
+Definition NN5 := (allH 0, v#0 × Zero = Zero)%fol.
+ +
+Definition NN6 := (allH 1 0, v#0 × S_ v#1 = v#0 × v#1 + v#0)%fol.
+ +
+Definition NN7 := (allH 0, ¬ v#0 < Zero)%fol.
+ +
+Definition NN8 :=
+ (allH 1 0, v#0 < Succ(v#1) → v#0 < v#1 ∨ v#0 = v#1)%fol.
+ +
+Definition NN9 := (allH 1 0, v#0 < v#1 ∨ v#0 = v#1 ∨ v#1 < v#0)%fol.
+ +
+Definition NN := SetEnum NN1 NN2 NN3 NN4 NN5 NN6 NN7 NN8 NN9.
+ +
+
+ +
+

Properties of the system NN

+ +
+
+ +
+Lemma closedNN1 : ClosedSystem LNN NN.
+ +
+Lemma closedNN : ∀ v : nat, ¬ In_freeVarSys LNN v NN.
+ +
+
+ +
+

Generic instantiation of axioms

+ +
+
+ +
+Lemma nn1 (a : Term) : SysPrf NN (S_ a ≠ Zero)%fol.
+ +
+Lemma nn2 (a b : Term): SysPrf NN (S_ a = S_ b → a = b)%fol.
+ +
+Lemma nn3 (a : Term): SysPrf NN (a + Zero = a)%fol.
+ +
+Lemma nn4 (a b : Term) : SysPrf NN (a + S_ b = S_ (a + b))%fol.
+ +
+Lemma nn5 ( a : Term) : SysPrf NN (a × Zero = Zero)%fol.
+ +
+Lemma nn6 (a b : Term):
+  SysPrf NN (a × Succ b = a × b + a)%fol.
+ +
+Lemma nn7 (a : Term): SysPrf NN (¬ (a <Zero))%fol.
+ +
+Lemma nn8 (a b : Term) : SysPrf NN (a < Succ b → a < b ∨ a = b)%fol.
+ +
+Lemma nn9 (a b : Term): SysPrf NN (a < b ∨ a = b ∨ b < a)%fol.
+ +
+End NN.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.NN2PA.html b/theories/html/hydras.Ackermann.NN2PA.html new file mode 100644 index 00000000..9573a6e4 --- /dev/null +++ b/theories/html/hydras.Ackermann.NN2PA.html @@ -0,0 +1,83 @@ + + + + + +hydras.Ackermann.NN2PA + + + + +
+ + + +
+ +

Library hydras.Ackermann.NN2PA

+ +
+
+ +
+ NN2PA.v : + +
+ +Original version by Russel O'Connor + +
+ + +
+
+ +
+From Coq Require Import Ensembles List Arith.
+ +
+From hydras.Ackermann
+  Require Import folProp folProof subProp folLogic3 folReplace NN
+  PAtheory.
+From hydras.Ackermann Require Export LNN2LNT.
+From hydras.Ackermann Require Import subAll ListExt.
+Import NNnotations.
+ +
+Lemma NN2PA (f : fol.Formula LNN):
+  folProof.SysPrf LNN NN f → SysPrf PA (LNN2LNT_formula f).
+ +
+
+ +
+If F[x\0], F[x\1] ... F[x\m-1] are provable in PA, + then v_x <' m → F is also provable (where a <' b is the translation of a < b into PA). + +
+ + More precisely: + +
+ + +
+
+Lemma PAboundedLT :
+  ∀ (m : nat) (F : LNT.Formula) (x : nat),
+    (∀ n : nat,
+        n < m → SysPrf PA (substF F x (natToTerm n))) →
+    SysPrf PA (LNN2LNT_formula (v#x < LNN.natToTerm m)%fol → F)%nt.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.NNtheory.html b/theories/html/hydras.Ackermann.NNtheory.html new file mode 100644 index 00000000..c38b25b1 --- /dev/null +++ b/theories/html/hydras.Ackermann.NNtheory.html @@ -0,0 +1,72 @@ + + + + + +hydras.Ackermann.NNtheory + + + + +
+ + + +
+ +

Library hydras.Ackermann.NNtheory

+ +
+From Coq Require Import Arith Lia.
+ +
+From hydras.Ackermann Require Import folLogic3 folProp subProp.
+From hydras.Ackermann Require Import folLogic3 folProp subProp.
+From hydras.Ackermann Require Export NN.
+From hydras Require Import Compat815.
+Import NNnotations.
+ +
+Lemma natNE (a b : nat) :
+  a ≠ b → SysPrf NN (natToTerm a ≠ natToTerm b)%fol.
+ +
+Lemma natLE (a b : nat):
+ b ≤ a → SysPrf NN (¬ (natToTerm a < natToTerm b))%fol.
+ +
+Lemma natLT (a b : nat):
+  a < b → SysPrf NN (natToTerm a < natToTerm b)%fol.
+ +
+Lemma natPlus (a b : nat):
+ SysPrf NN (natToTerm a + natToTerm b = natToTerm (a + b)%nat)%fol.
+ +
+Lemma natTimes (a b : nat):
+ SysPrf NN (natToTerm a ×natToTerm b = natToTerm (a × b)%nat)%fol.
+ +
+Lemma boundedLT (m : nat) (a : Formula) (x : nat):
+  (∀ n : nat,
+      n < m → SysPrf NN (substF a x (natToTerm n))) →
+  SysPrf NN (v#x < natToTerm m → a)%fol.
+ +
+Lemma nnPlusNotNeeded (n:nat) :
+ SysPrf NN
+ (v#1 < natToTerm n ∨ v#1 = natToTerm n →
+  v#1 < S_ (natToTerm n))%fol.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.NewNotations.html b/theories/html/hydras.Ackermann.NewNotations.html new file mode 100644 index 00000000..5c13840d --- /dev/null +++ b/theories/html/hydras.Ackermann.NewNotations.html @@ -0,0 +1,44 @@ + + + + + +hydras.Ackermann.NewNotations + + + + +
+ + + +
+ +

Library hydras.Ackermann.NewNotations

+ +
+From Coq Require Import Ensembles.
+From hydras.Ackermann Require Import folProp.
+ +
+Notation "'SetAdds' E0 x1 .. xn" :=
+  (Ensembles.Add _ (.. (Ensembles.Add _ E0 x1) .. ) xn)
+    (at level 0, E0 at level 0, x1 at level 0, xn at level 0).
+ +
+Notation "'SetEnum' x0 x1 .. xn" :=
+  (Ensembles.Add _ (.. (Ensembles.Add _ (Singleton _ x0) x1) .. ) xn)
+    (at level 0, x0 at level 0, x1 at level 0, xn at level 0).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.PA.html b/theories/html/hydras.Ackermann.PA.html new file mode 100644 index 00000000..3d881dfb --- /dev/null +++ b/theories/html/hydras.Ackermann.PA.html @@ -0,0 +1,133 @@ + + + + + +hydras.Ackermann.PA + + + + +
+ + + +
+ +

Library hydras.Ackermann.PA

+ +
+
+ +
+ PA : Peano Arithmetic + +
+ + Original development by Russel O'Connor + Bullets and abbreviations by Pierre Casteran + +
+
+ +
+From Coq Require Import Arith Lia Ensembles Decidable.
+From hydras.Ackermann Require Import folProp subAll folLogic3.
+From hydras.Ackermann Require Export Languages LNT.
+From hydras.Ackermann Require Import NewNotations.
+ +
+Section PA.
+ +
+Definition PA1 := (allH 0, Succ v#0 ≠ Zero)%nt.
+ +
+Definition PA2 := (allH 1 0, Succ v#0 = Succ v#1 → v#0 = v#1)%nt.
+ +
+Definition PA3 := (allH 0, v#0 + Zero = v#0)%nt.
+ +
+Definition PA4 := (allH 1 0, v#0 + Succ v#1 = Succ (v#0 + v#1))%nt.
+ +
+Definition PA5 := (allH 0, v#0 × Zero = Zero)%nt.
+ +
+Definition PA6 := (allH 1 0, v#0 × Succ v#1 = v#0 × v#1 + v#0)%nt.
+ +
+Definition PA7 (f : Formula) (v : nat) : Formula :=
+   let f_0 := substF f v Zero%nt in
+   let f_Sv := substF f v (Succ v#v)%nt in
+   close _ (f_0 → (allH v, f → f_Sv) → allH v, f)%nt.
+ +
+Definition InductionSchema (f : Formula) : Prop :=
+  ∃ g : Formula, (∃ v : nat, f = PA7 g v).
+ +
+Definition PA := SetAdds InductionSchema PA1 PA2 PA3 PA4 PA5 PA6.
+ +
+Definition open :=
+  Formula_rec LNT (fun _ ⇒ Formula) (fun t t0 : Term ⇒ (t = t0)%nt)
+    (fun (r : Relations LNT)
+         (ts : Terms (arityR LNT r)) ⇒ atomic r ts)
+    (fun (f : Formula) _ (f0 : Formula) _ ⇒ (f → f0)%nt)
+    (fun (f : Formula) _ ⇒ (¬ f)%nt)
+    (fun (n : nat) _ (recf : Formula) ⇒ recf).
+ +
+Lemma PAdec : ∀ x : Formula, decidable (In _ PA x).
+ +
+Lemma closedPA1 : ClosedSystem LNT PA.
+ +
+Lemma closedPA : ∀ v : nat, ¬ In_freeVarSys LNT v PA.
+ +
+Lemma pa1 (a : Term): SysPrf PA (Succ a ≠ Zero)%nt.
+ +
+Lemma pa2 (a b : Term): SysPrf PA (Succ a = Succ b → a = b)%nt.
+ +
+Lemma pa3 (a : Term): SysPrf PA (a + Zero = a)%nt.
+ +
+Lemma pa4 ( a b : Term) : SysPrf PA (a + Succ b = Succ (a + b))%nt.
+ +
+Lemma pa5 (a : Term): SysPrf PA (a × Zero = Zero)%nt.
+ +
+Lemma pa6 (a b : Term) : SysPrf PA (a × Succ b = a × b + a)%nt.
+ +
+Lemma induct (f : Formula) (v : nat):
+  let f_0 := substF f v Zero
+  in let f_Sv := substF f v (Succ (v#v))%nt
+     in SysPrf PA f_0 →
+         SysPrf PA (allH v, f → f_Sv)%nt
+         → SysPrf PA (allH v, f)%nt.
+ +
+End PA.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.PAconsistent.html b/theories/html/hydras.Ackermann.PAconsistent.html new file mode 100644 index 00000000..387cd15a --- /dev/null +++ b/theories/html/hydras.Ackermann.PAconsistent.html @@ -0,0 +1,61 @@ + + + + + +hydras.Ackermann.PAconsistent + + + + +
+ + + +
+ +

Library hydras.Ackermann.PAconsistent

+ +
+
+ +
+PAconsistent.v + +
+ + Original file by Russel O' Connor +
+
+ +
+From Coq Require Import Arith.
+From hydras.Ackermann Require Import folProof folProp PA model.
+ +
+Definition natModel : Model LNT :=
+  model LNT nat
+    (fun f : Functions LNT ⇒
+     match f return (naryFunc nat (arityF LNT f)) with
+     | Languages.Plus_ ⇒ fun x y : nat ⇒ y + x
+     | Languages.Times_ ⇒ fun x y : nat ⇒ y × x
+     | Languages.Succ_ ⇒ S
+     | Languages.Zero_ ⇒ 0
+     end)
+    (fun r : Relations LNT ⇒ match r with end).
+ +
+Theorem PAconsistent : Consistent LNT PA.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.PAtheory.html b/theories/html/hydras.Ackermann.PAtheory.html new file mode 100644 index 00000000..8142abe8 --- /dev/null +++ b/theories/html/hydras.Ackermann.PAtheory.html @@ -0,0 +1,80 @@ + + + + + +hydras.Ackermann.PAtheory + + + + +
+ + + +
+ +

Library hydras.Ackermann.PAtheory

+ +
+
+ +
+ PAtheory.v (Peano Arithmetic) + +
+ + Original file by Russel O'Connor + +
+ + +
+
+ +
+From Coq Require Import Ensembles List Arith Lia.
+ +
+From hydras.Ackermann Require Import subAll folReplace folProp folLogic3 NN LNN2LNT.
+From hydras.Ackermann Require Export PA.
+From hydras.Ackermann Require Import NewNotations.
+Import NNnotations.
+ +
+#[local] Arguments apply _ _ _ : clear implicits.
+ +
+Lemma paZeroOrSucc (t : Term):
+ let nv := newVar (0 :: freeVarT t) in
+ SysPrf PA (t = LNT.Zero ∨ exH nv, t = LNT.Succ (v#nv))%nt.
+           Opaque eq_nat_dec.
+           Transparent eq_nat_dec.
+          Opaque eq_nat_dec.
+          Transparent eq_nat_dec.
+ +
+Lemma paPlusSym : ∀ a b : Term,
+    SysPrf PA (a + b = b + a)%nt.
+ +
+Lemma NN72PA : SysPrf PA (LNN2LNT_formula NN7).
+ +
+Lemma NN82PA : SysPrf PA (LNN2LNT_formula NN8).
+ +
+Lemma NN92PA : SysPrf PA (LNN2LNT_formula NN9).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.cPair.html b/theories/html/hydras.Ackermann.cPair.html new file mode 100644 index 00000000..c04c78c4 --- /dev/null +++ b/theories/html/hydras.Ackermann.cPair.html @@ -0,0 +1,396 @@ + + + + + +hydras.Ackermann.cPair + + + + +
+ + + +
+ +

Library hydras.Ackermann.cPair

+ +
+
+ +
+Cantor's pairing function + (Originally Russel O'Connors goedel contrib +
+
+ +
+From Coq Require Import Arith Lists.List Lia.
+From hydras Require Import extEqualNat primRec.
+From hydras Require Export Compat815 ssrnat_extracts.
+Import PRNotations.
+Import Nat.
+ +
+
+ +
+

Bijection from nat × nat to nat

+ +
+ +

Preliminary definitions

+ +
+ + Sum of all natural numbers upto n +
+
+ +
+Fixpoint sumToN (n : nat): nat :=
+  match n with
+    0 ⇒ 0
+  | S p ⇒ S p + sumToN p
+  end.
+ +
+Lemma sumToN1 n : n ≤ sumToN n.
+ +
+Lemma sumToN2 b a : a ≤ b → sumToN a ≤ sumToN b.
+ +
+#[export] Instance sumToNIsPR : isPR 1 sumToN.
+ +
+
+ +
+

Definition and properties of cPair

+ +
+ + The cPair function defined below is slightly different + from the "usual" Cantor pairing function shown in a big part + of the litterature (and Coq's standard library). + Since both versions are equivalent upto a swap of the + arguments a and b, we still use Russel O'Connors notations +
+
+ +
+Definition cPair (a b : nat) := a + sumToN (a + b).
+ +
+ +
+
+ +
+Usual definition (e.g. wikipedia) +
+
+ +
+ Definition xPair a b := b + sumToN ( b + a).
+ +
+ Lemma xPairDef a b : xPair a b = cPair b a.
+ +
+#[export] Instance cPairIsPR : isPR 2 cPair.
+ +
+Section CPair_Injectivity.
+ +
+  Remark cPairInjHelp :
+    ∀ a b c d : nat, cPair a b = cPair c d → a + b = c + d.
+ +
+  Lemma cPairInj1 a b c d: cPair a b = cPair c d → a = c.
+ +
+  Lemma cPairInj2 a b c d : cPair a b = cPair c d → b = d.
+ +
+End CPair_Injectivity.
+ +
+Section CPair_projections.
+ +
+ +
+  Let searchXY (a : nat) :=
+        boundedSearch (fun a y : nat ⇒ ltBool a (sumToN y.+1)) a.
+ +
+  Definition cPairPi1 (a : nat) := a - sumToN (searchXY a).
+  Definition cPairPi2 (a : nat) := searchXY a - cPairPi1 a.
+ +
+Lemma cPairProjectionsHelp (a b: nat):
+  b < sumToN a.+1 → sumToN a ≤ b → searchXY b = a.
+ +
+Lemma cPairProjections a: cPair (cPairPi1 a) (cPairPi2 a) = a.
+ +
+#[local] Instance searchXYIsPR : isPR 1 searchXY.
+ +
+#[export] Instance cPairPi1IsPR : isPR 1 cPairPi1.
+ +
+Lemma cPairProjections1 (a b : nat): cPairPi1 (cPair a b) = a.
+ +
+Lemma cPairProjections2 (a b : nat): cPairPi2 (cPair a b) = b.
+ +
+#[export] Instance cPairPi2IsPR : isPR 1 cPairPi2.
+ +
+End CPair_projections.
+ +
+Section CPair_Order.
+ +
+  Lemma cPairLe1 (a b : nat) : a ≤ cPair a b.
+ +
+  Lemma cPairLe1A (a : nat) : cPairPi1 a ≤ a.
+ +
+  Lemma cPairLe2 (a b : nat) : b ≤ cPair a b.
+ +
+  Lemma cPairLe2A (a: nat): cPairPi2 a ≤ a.
+ +
+  Lemma cPairLe3 (a b c d : nat): a ≤ b → c ≤ d → cPair a c ≤ cPair b d.
+ +
+  Lemma cPairLt1 (a b : nat): a < cPair a (S b).
+ +
+  Lemma cPairLt2 (a b : nat): b < cPair (S a) b.
+ +
+End CPair_Order.
+ +
+Require Import Extraction.
+Section code_nat_list.
+ +
+Fixpoint codeList (l : list nat) : nat :=
+  match l with
+  | nil ⇒ 0
+  | n :: l' ⇒ S (cPair n (codeList l'))
+  end.
+ +
+Lemma codeListInj (l m : list nat): codeList l = codeList m → l = m.
+ +
+Definition codeNth (n m:nat) : nat :=
+  let X := nat_rec (fun _ : nat ⇒ nat)
+             m
+             (fun _ Hrecn : nat ⇒ cPairPi2 (pred Hrecn)) n
+  in cPairPi1 (pred X).
+ +
+
+ +
+drops n first elements from l +
+
+Let drop (n : nat) : ∀ (l : list nat), list nat.
+ +
+Lemma codeNthCorrect :
+  ∀ (n : nat) (l : list nat), codeNth n (codeList l) = nth n l 0.
+ +
+#[export] Instance codeNthIsPR : isPR 2 codeNth.
+ +
+End code_nat_list.
+ +
+Extraction codeNth.
+Print codeNth.
+ +
+Section Strong_Recursion.
+Definition evalStrongRecHelp (n : nat) (f : naryFunc n.+2) :
+  naryFunc n.+1 :=
+  evalPrimRecFunc n (evalComposeFunc n 0 (Vector.nil _) (codeList nil))
+    (evalComposeFunc n.+2 2
+       (Vector.cons _ f _
+          (Vector.cons _ (evalProjFunc n.+2 n
+                            (Nat.lt_lt_succ_r _ _
+                               (Nat.lt_succ_diag_r _))) _
+             (Vector.nil _)))
+       (fun a b : nat ⇒ S (cPair a b))).
+ +
+Definition evalStrongRec (n : nat) (f : naryFunc n.+2):
+  naryFunc n.+1 :=
+  evalComposeFunc n.+1 1
+    (Vector.cons _
+       (fun z : nat ⇒ evalStrongRecHelp n f z.+1) _ (Vector.nil _))
+    (fun z : nat ⇒ cPairPi1 z.-1).
+ +
+#[export] Instance
+ evalStrongRecIsPR (n : nat) (f : naryFunc n.+2):
+  isPR _ f → isPR _ (evalStrongRec n f).
+ +
+Lemma computeEvalStrongRecHelp :
+  ∀ (n : nat) (f : naryFunc n.+2) (c : nat),
+    evalStrongRecHelp n f c.+1 =
+      compose2 n (evalStrongRecHelp n f c)
+        (fun a0 : nat ⇒
+           evalComposeFunc n 2
+             (Vector.cons (naryFunc n) (f c a0) 1
+                (Vector.cons (naryFunc n) (evalConstFunc n a0) 0
+                   (Vector.nil (naryFunc n))))
+             (fun a1 b0 : nat ⇒ S (cPair a1 b0))).
+ +
+Fixpoint listValues (f : naryFunc 2) (n : nat) : list nat :=
+  match n with
+    0 ⇒ nil
+  | S m ⇒ evalStrongRec _ f m :: listValues f m
+  end.
+ +
+Lemma evalStrongRecHelp1 :
+ ∀ (f : naryFunc 2) (n m : nat),
+   m < n →
+   codeNth (n - m.+1) (evalStrongRecHelp _ f n) = evalStrongRec _ f m.
+ +
+Lemma evalStrongRecHelpParam :
+  ∀ (a n c : nat) (f : naryFunc a.+3),
+    extEqual a (evalStrongRecHelp (S a) f n c)
+      (evalStrongRecHelp a (fun x y : nat ⇒ f x y c) n).
+ +
+Lemma evalStrongRecHelp2 :
+  ∀ (a : nat) (f : naryFunc a.+2) (n m : nat),
+    m < n →
+    extEqual _
+      (evalComposeFunc _ 1
+         (Vector.cons _ (evalStrongRecHelp _ f n) 0 (Vector.nil _))
+         (fun b : nat ⇒ codeNth (n - m.+1) b)) (evalStrongRec _ f m).
+ +
+#[export] Instance callIsPR (g : nat → nat) (H : isPR 1 g) :
+   isPR 2 (fun a recs : nat ⇒ codeNth (a - S (g a)) recs).
+ +
+End Strong_Recursion.
+ +
+#[export] Instance div2IsPR : isPR 1 div2.
+ +
+Lemma cPairLemma1 :
+  ∀ a b : nat, (a + b) × S (a + b) + 2 × a = 2 × cPair a b.
+ +
+
+ +
+abbreviations a la Lisp (to improve) +
+
+ +
+Module LispAbbreviations.
+  #[global] Notation car := cPairPi1.
+  #[global] Notation cdr := cPairPi2.
+  #[global] Notation caar n := (cPairPi1 (cPairPi1 n)).
+  #[global] Notation cadr n := (cPairPi1 (cPairPi2 n)).
+  #[global] Notation caddr n := (cPairPi1 (cPairPi2 (cPairPi2 n))).
+ +
+  #[global] Notation cddr n := (cPairPi2 (cPairPi2 n)).
+  #[global] Notation cdddr n := (cPairPi2 (cPairPi2 (cPairPi2 n))).
+  #[global] Notation cddddr n :=
+    (cPairPi2 (cPairPi2 (cPairPi2 (cPairPi2 n)))).
+ +
+End LispAbbreviations.
+ +
+
+ +
+

Triples

+ + (moved from codeSubFormula.v) + +
+
+ +
+Definition cTriple (a b c : nat) : nat := cPair a (cPair b c).
+ +
+Definition cTriplePi1 (n : nat) : nat := cPairPi1 n.
+ +
+Definition cTriplePi2 (n : nat) : nat := cPairPi1 (cPairPi2 n).
+ +
+Definition cTriplePi3 (n : nat) : nat := cPairPi2 (cPairPi2 n).
+ +
+#[export] Instance cTripleIsPR : isPR 3 cTriple.
+ +
+#[export] Instance cTriplePi1IsPR : isPR 1 cTriplePi1.
+ +
+#[export] Instance cTriplePi2IsPR : isPR 1 cTriplePi2.
+ +
+#[export] Instance cTriplePi3IsPR : isPR 1 cTriplePi3.
+ +
+Lemma cTripleProj1 (a b c : nat) : cTriplePi1 (cTriple a b c) = a.
+ +
+Lemma cTripleProj2 (a b c : nat) : cTriplePi2 (cTriple a b c) = b.
+ +
+Lemma cTripleProj3 (a b c : nat) : cTriplePi3 (cTriple a b c) = c.
+ +
+Lemma cTripleProj (a: nat) :
+ cTriple (cTriplePi1 a) (cTriplePi2 a) (cTriplePi3 a) = a.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.checkPrf.html b/theories/html/hydras.Ackermann.checkPrf.html new file mode 100644 index 00000000..6fcf4f21 --- /dev/null +++ b/theories/html/hydras.Ackermann.checkPrf.html @@ -0,0 +1,472 @@ + + + + + +hydras.Ackermann.checkPrf + + + + +
+ + + +
+ +

Library hydras.Ackermann.checkPrf

+ +
+
+ +
+checkPrf.v + +
+ + Original file by Russel O'Connor + +
+ + +
+
+ +
+From Coq Require Import Arith Lia.
+From hydras.Ackermann Require Import primRec codeFreeVar
+  codeSubFormula cPair code folProp extEqualNat wellFormed
+  folProof prLogic.
+From hydras Require Import Compat815.
+From LibHyps Require Export LibHyps.
+From hydras Require Export MoreLibHyps NewNotations.
+Import LispAbbreviations.
+Import FolNotations.
+ +
+Section Check_Proof.
+ +
+Generalizable All Variables.
+Variable L : Language.
+Context `(cL: Lcode L cf cr).
+ +
+Variable codeArityF : nat → nat.
+Variable codeArityR : nat → nat.
+ +
+ +
+ +
+Context (codeArityFIsPR : isPR 1 codeArityF).
+ +
+Hypothesis codeArityFIsCorrect1 :
+  ∀ f : Functions L, codeArityF (cf f) = S (arityF L f).
+ +
+Hypothesis codeArityFIsCorrect2 :
+    ∀ n : nat, codeArityF n ≠ 0 →
+                    ∃ f : Functions L, cf f = n.
+ +
+Context (codeArityRIsPR : isPR 1 codeArityR).
+ +
+Hypothesis
+  codeArityRIsCorrect1 :
+    ∀ r : Relations L, codeArityR (cr r) = S (arityR L r).
+ +
+Hypothesis
+  codeArityRIsCorrect2 :
+    ∀ n : nat, codeArityR n ≠ 0 →
+                    ∃ r : Relations L, cr r = n.
+ +
+Hypothesis codeFInj : ∀ f g : Functions L,
+    cf f = cf g → f = g.
+ +
+Hypothesis codeRInj :
+  ∀ R S : Relations L, cr R = cr S → R = S.
+ +
+ +
+Let Term := Term L.
+Let Terms := Terms L.
+Let Formula := Formula L.
+Let wellFormedTerm := wellFormedTerm codeArityF.
+Let wellFormedFormula := wellFormedFormula codeArityF codeArityR.
+Let Prf := Prf L.
+ +
+
+ +
+The wellFormedness requirement isn't really neccesary, +because any proof that used ``extra symbols'' could be +turned into a proof that only used symbols from the +axioms and the conclusion. + +
+ +However making this assumption makes the proof easier +
+ + p is (cPair (formula) (proof of the Formula)) +
+
+ +
+Definition checkPrfAXM (p recs : nat) :=
+  switchPR (charFunction 2 Nat.eqb (cddr p) (car p))
+    (S (S (cPair (car p) 0))) 0.
+ +
+#[export] Instance checkPrfAXMIsPR : isPR 2 checkPrfAXM.
+ +
+Definition checkPrfMP (p recs : nat) :=
+  switchPR
+    (wellFormedFormula (car (cdr (cdr (cdr p)))) ×
+     (charFunction 2 Nat.eqb (car (car (cdr (cdr p))))
+        (codeImp (car (cdr (cdr (cdr p)))) (car p)) ×
+      (codeNth (p - S (caddr p)) recs ×
+       codeNth (p - S (cdddr p)) recs)))
+    (S
+       (codeApp
+          (pred (codeNth (p - S (caddr p)) recs))
+          (pred (codeNth (p - S (cdr (cdr (cdr p)))) recs))))
+    0.
+ +
+#[export] Instance checkPrfMPIsPR : isPR 2 checkPrfMP.
+ +
+Definition checkPrfGEN (p recs : nat) :=
+  switchPR
+    (charFunction 2 Nat.eqb
+       (cPair 3
+          (cPair (car (cdr (cdr p)))
+             (car (cdr (cdr (cdr p))))))
+       (car p) ×
+     (codeNth (p - S (cdr (cdr (cdr p)))) recs ×
+      (1 -
+       codeIn (car (cdr (cdr p)))
+         (codeFreeVarListFormula
+            (pred (codeNth (p - S (cdr (cdr (cdr p)))) recs))))))
+    (codeNth (p - S (cdr (cdr (cdr p)))) recs) 0.
+ +
+#[export] Instance checkPrfGENIsPR : isPR 2 checkPrfGEN.
+ +
+Definition checkPrfIMP1 (p recs : nat) :=
+  let A := car (cdr (cdr p)) in
+  let B := cdr (cdr (cdr p)) in
+  charFunction 2 Nat.eqb (cPair 1 (cPair A (cPair 1 (cPair B A))))
+    (car p).
+ +
+#[export] Instance checkPrfIMP1IsPR : isPR 2 checkPrfIMP1.
+ +
+Definition checkPrfIMP2 (p recs : nat) :=
+  let A := car (cdr (cdr p)) in
+  let B := car (cdr (cdr (cdr p))) in
+  let C := cdr (cdr (cdr (cdr p))) in
+  charFunction 2 Nat.eqb
+    (cPair 1
+       (cPair (cPair 1 (cPair A (cPair 1 (cPair B C))))
+          (cPair 1 (cPair (cPair 1 (cPair A B)) (cPair 1 (cPair A C))))))
+    (car p).
+ +
+#[export] Instance checkPrfIMP2IsPR : isPR 2 checkPrfIMP2.
+ +
+Definition checkPrfCP (p recs : nat) :=
+  let A := car (cdr (cdr p)) in
+  let B := cdr (cdr (cdr p)) in
+  charFunction 2 Nat.eqb
+    (cPair 1
+       (cPair (cPair 1 (cPair (cPair 2 A) (cPair 2 B))) (cPair 1 (cPair B A))))
+    (car p).
+ +
+#[export] Instance checkPrfCPIsPR : isPR 2 checkPrfCP.
+ +
+Definition checkPrfFA1 (p recs : nat) :=
+  let A := car (cdr (cdr p)) in
+  let v := car (cdr (cdr (cdr p))) in
+  let t := cdr (cdr (cdr (cdr p))) in
+  wellFormedTerm t ×
+  charFunction 2 Nat.eqb
+    (cPair 1 (cPair (cPair 3 (cPair v A)) (codeSubFormula A v t)))
+    (car p).
+ +
+#[export] Instance checkPrfFA1IsPR : isPR 2 checkPrfFA1.
+ +
+Definition checkPrfFA2 (p recs : nat) :=
+  let A := car (cdr (cdr p)) in
+  let v := cdr (cdr (cdr p)) in
+  (1 - codeIn v (codeFreeVarFormula A)) ×
+  charFunction 2 Nat.eqb (cPair 1 (cPair A (cPair 3 (cPair v A))))
+    (car p).
+ +
+#[export] Instance checkPrfFA2IsPR : isPR 2 checkPrfFA2.
+ +
+Definition checkPrfFA3 (p recs : nat) :=
+  let A := car (cdr (cdr p)) in
+  let B := car (cdr (cdr (cdr p))) in
+  let v := cdr (cdr (cdr (cdr p))) in
+  charFunction 2 Nat.eqb
+    (cPair 1
+       (cPair (cPair 3 (cPair v (cPair 1 (cPair A B))))
+          (cPair 1 (cPair (cPair 3 (cPair v A)) (cPair 3 (cPair v B))))))
+    (car p).
+ +
+#[export] Instance checkPrfFA3IsPR : isPR 2 checkPrfFA3.
+ +
+Definition checkPrfEQ1 (p recs : nat) :=
+  charFunction 2 Nat.eqb (cdr (cdr p)) 0 ×
+  charFunction 2 Nat.eqb
+    (codeFormula (equal (var 0) (var 0)))
+    (car p).
+ +
+Lemma checkPrfEQnIsPR (n: nat):
+  isPR 2
+    (fun p recs : nat ⇒
+       charFunction 2 Nat.eqb (cdr (cdr p)) 0 ×
+         charFunction 2 Nat.eqb n (car p)).
+ +
+#[export] Instance checkPrfEQ1IsPR : isPR 2 checkPrfEQ1.
+ +
+Definition checkPrfEQ2 (p recs : nat) :=
+  charFunction 2 Nat.eqb (cdr (cdr p)) 0 ×
+    charFunction 2 Nat.eqb
+      (codeFormula (v#0 = v#1 → v#1 = v#0)%fol) (car p).
+ +
+#[export] Instance checkPrfEQ2IsPR : isPR 2 checkPrfEQ2.
+ +
+Definition checkPrfEQ3 (p recs : nat) :=
+  charFunction 2 Nat.eqb (cdr (cdr p)) 0 ×
+    charFunction 2 Nat.eqb
+      (codeFormula
+         (v#0 = v#1 → v#1 = v#2 → v#0 = v#2)%fol)
+      (car p).
+ +
+#[export] Instance checkPrfEQ3IsPR : isPR 2 checkPrfEQ3.
+ +
+Definition codeAxmEqHelp (n f : nat) : nat :=
+  nat_rec (fun _ ⇒ nat) f
+    (fun m rec : nat ⇒
+       cPair 1
+         (cPair (cPair 0 (cPair (cPair 0 (m + m)) (cPair 0 (S (m + m))))) rec))
+    n.
+ +
+#[export] Instance codeAxmEqHelpIsPR : isPR 2 codeAxmEqHelp.
+ +
+Definition codeNVars1 (n : nat) : nat :=
+  nat_rec (fun _ ⇒ nat) 0
+    (fun m rec : nat ⇒ S (cPair (cPair 0 (m + m)) rec)) n.
+ +
+#[export] Instance codeNVars1IsPR : isPR 1 codeNVars1.
+ +
+Definition codeNVars2 (n : nat) : nat :=
+  nat_rec (fun _ ⇒ nat) 0
+    (fun m rec : nat ⇒ S (cPair (cPair 0 (S (m + m))) rec)) n.
+ +
+#[export] Instance codeNVars2IsPR : isPR 1 codeNVars2.
+ +
+Lemma codeNVarsCorrect (n: nat) :
+ codeNVars1 n = codeTerms (fst (nVars L n)) ∧
+ codeNVars2 n = codeTerms (snd (nVars L n)).
+ +
+Definition checkPrfEQ4 (p recs : nat) :=
+  let r := cdr (cdr p) in
+  let A := cPair (S (S (S (S r)))) (codeNVars1 (pred (codeArityR r))) in
+  let B := cPair (S (S (S (S r)))) (codeNVars2 (pred (codeArityR r))) in
+  notZero (codeArityR r) ×
+    charFunction 2 Nat.eqb (codeAxmEqHelp (pred (codeArityR r)) (codeIff A B))
+      (car p).
+ +
+#[export] Instance codeOrIsPR : isPR 2 codeOr.
+ +
+#[export] Instance codeAndIsPR : isPR 2 codeAnd.
+ +
+#[export] Instance codeIffIsPR : isPR 2 codeIff.
+ +
+#[export] Instance checkPrfEQ4IsPR : isPR 2 checkPrfEQ4.
+ +
+Definition checkPrfEQ5 (p recs : nat) :=
+  let f := cdr (cdr p) in
+  notZero (codeArityF f) ×
+    charFunction 2 Nat.eqb
+      (codeAxmEqHelp (pred (codeArityF f))
+         (cPair 0
+            (cPair (cPair (S f) (codeNVars1 (pred (codeArityF f))))
+               (cPair (S f) (codeNVars2 (pred (codeArityF f)))))))
+      (car p).
+ +
+#[export] Instance checkPrfEQ5IsPR : isPR 2 checkPrfEQ5.
+ +
+Definition checkPrfHelp : nat → nat :=
+  evalStrongRec 0
+    (fun p recs : nat ⇒
+     let type := car (cdr p) in
+     switchPR type
+       (switchPR (pred type)
+          (switchPR (pred (pred type))
+             (switchPR (pred (pred (pred type)))
+                (switchPR (pred (pred (pred (pred type))))
+                   (switchPR (pred (pred (pred (pred (pred type)))))
+                      (switchPR
+                         (pred (pred (pred (pred (pred (pred type))))))
+                         (switchPR
+                            (pred
+                               (pred (pred (pred (pred (pred (pred type)))))))
+                            (switchPR
+                               (pred
+                                  (pred
+                                     (pred
+                                        (pred
+                                           (pred (pred (pred (pred type))))))))
+                               (switchPR
+                                  (pred
+                                     (pred
+                                        (pred
+                                           (pred
+                                              (pred
+                                                 (pred
+                                                  (pred (pred (pred type)))))))))
+                                  (switchPR
+                                     (pred
+                                        (pred
+                                           (pred
+                                              (pred
+                                                 (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                     (pred (pred type))))))))))
+                                     (switchPR
+                                        (pred
+                                           (pred
+                                              (pred
+                                                 (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                     (pred (pred type)))))))))))
+                                        (switchPR
+                                           (pred
+                                              (pred
+                                                 (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred (pred (pred type))))))))))))
+                                           (switchPR
+                                              (pred
+                                                 (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred
+                                                  (pred (pred (pred type)))))))))))))
+                                              0 (checkPrfEQ5 p recs))
+                                           (checkPrfEQ4 p recs))
+                                        (checkPrfEQ3 p recs))
+                                     (checkPrfEQ2 p recs))
+                                  (checkPrfEQ1 p recs))
+                               (checkPrfFA3 p recs))
+                            (checkPrfFA2 p recs)) (checkPrfFA1 p recs))
+                      (checkPrfCP p recs)) (checkPrfIMP2 p recs))
+                (checkPrfIMP1 p recs)) (checkPrfGEN p recs))
+          (checkPrfMP p recs)) (checkPrfAXM p recs)).
+ +
+#[local] Instance checkPrfHelpIsPR : isPR 1 checkPrfHelp.
+ +
+Definition checkPrf (f p : nat) : nat :=
+  switchPR (wellFormedFormula f) (checkPrfHelp (cPair f p)) 0.
+ +
+#[export] Instance checkPrfIsPR : isPR 2 checkPrf.
+ +
+Lemma checkPrfCorrect1 (l : list Formula) (f : Formula) (p : Prf l f):
+ checkPrf (codeFormula f) (codePrf l f p)
+ = S (codeList (map codeFormula l)).
+ +
+Lemma checkPrfCorrect2 (n m : nat):
+ checkPrf n m ≠ 0 →
+ ∃ f : Formula,
+   codeFormula f = n ∧
+   (∃ l : list Formula,
+      (∃ p : Prf l f, codePrf l f p = m)).
+ +
+End Check_Proof.
+ +
+Arguments codePrf {L cf cr cl} _.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.code.html b/theories/html/hydras.Ackermann.code.html new file mode 100644 index 00000000..d871655b --- /dev/null +++ b/theories/html/hydras.Ackermann.code.html @@ -0,0 +1,213 @@ + + + + + +hydras.Ackermann.code + + + + +
+ + + +
+ +

Library hydras.Ackermann.code

+ +
+
+ +
+

Encoding terms, formulas and proofs

+ + +
+ +Original script by Russel O'Connor + +
+ + +
+
+ +
+From Coq Require Import Arith.
+From hydras.Ackermann Require Import fol folProof cPair.
+ +
+Section Code_Term_Formula_Proof.
+ +
+Variable L : Language.
+ Section LcodeDef.
+   Variable cF : Functions L → nat.
+   Variable cR : Relations L → nat.
+   Class Lcode : Prop :=
+     { codeFInj :
+       ∀ f g : Functions L, cF f = cF g → f = g;
+       codeRInj :
+       ∀ R S : Relations L, cR R = cR S → R = S
+     }.
+  End LcodeDef.
+ +
+ Definition codeF {cf : Functions L → nat} {cr : Relations L → nat} (c: Lcode cf cr) := cf.
+ +
+ Definition codeR {cf : Functions L → nat} {cr : Relations L → nat} (c: Lcode cf cr) := cr.
+ +
+Let Formula := Formula L.
+Let Formulas := Formulas L.
+Let System := System L.
+Let Term := Term L.
+Let Terms := Terms L.
+Let Prf := Prf L.
+Let SysPrf := SysPrf L.
+ +
+Generalizable All Variables.
+Section codeTermFormDef.
+ +
+  Context `(cl : Lcode cf cr).
+Fixpoint codeTerm (t : Term) : nat :=
+  match t with
+  | var n ⇒ cPair 0 n
+  | apply f ts ⇒ cPair (S (codeF cl f)) (codeTerms _ ts)
+  end
+ 
+ with codeTerms (n : nat) (ts : Terms n) {struct ts} : nat :=
+  match ts with
+  | Tnil ⇒ 0
+  | Tcons n t ss ⇒ S (cPair (codeTerm t) (codeTerms n ss))
+  end.
+ +
+Lemma codeTermInj :
+  ∀ t s : Term, codeTerm t = codeTerm s → t = s.
+ +
+Lemma codeTermsInj :
+ ∀ (n : nat) (ts ss : Terms n),
+ codeTerms n ts = codeTerms n ss → ts = ss.
+ +
+Fixpoint codeFormula (f : Formula) : nat :=
+  match f with
+  | equal t1 t2 ⇒ cPair 0 (cPair (codeTerm t1) (codeTerm t2))
+  | impH f1 f2 ⇒ cPair 1 (cPair (codeFormula f1) (codeFormula f2))
+  | notH f1 ⇒ cPair 2 (codeFormula f1)
+  | forallH n f1 ⇒ cPair 3 (cPair n (codeFormula f1))
+  | atomic R ts ⇒ cPair (4 + codeR cl R) (codeTerms _ ts)
+  end.
+ +
+Lemma codeFormulaInj :
+  ∀ f g : Formula, codeFormula f = codeFormula g → f = g.
+ +
+Fixpoint codePrf (Z : Formulas) (f : Formula) (prf : Prf Z f) {struct prf} :
+ nat :=
+  match prf with
+  | AXM A ⇒ cPair 0 (codeFormula A)
+  | MP Axm1 Axm2 A B rec1 rec2 ⇒
+      cPair 1
+        (cPair
+           (cPair (cPair 1 (cPair (codeFormula A) (codeFormula B)))
+              (codePrf _ _ rec1)) (cPair (codeFormula A) (codePrf _ _ rec2)))
+  | GEN Axm A v _ rec ⇒
+      cPair 2 (cPair v (cPair (codeFormula A) (codePrf _ _ rec)))
+  | IMP1 A B ⇒ cPair 3 (cPair (codeFormula A) (codeFormula B))
+  | IMP2 A B C ⇒
+      cPair 4 (cPair (codeFormula A) (cPair (codeFormula B) (codeFormula C)))
+  | CP A B ⇒ cPair 5 (cPair (codeFormula A) (codeFormula B))
+  | FA1 A v t ⇒ cPair 6 (cPair (codeFormula A) (cPair v (codeTerm t)))
+  | FA2 A v _ ⇒ cPair 7 (cPair (codeFormula A) v)
+  | FA3 A B v ⇒ cPair 8 (cPair (codeFormula A) (cPair (codeFormula B) v))
+  | EQ1 ⇒ cPair 9 0
+  | EQ2 ⇒ cPair 10 0
+  | EQ3 ⇒ cPair 11 0
+  | EQ4 r ⇒ cPair 12 (codeR cl r)
+  | EQ5 f ⇒ cPair 13 (codeF cl f)
+  end.
+ +
+Lemma codePrfInjAxm :
+ ∀ (a b : Formula) (A B : Formulas) (p : Prf A a) (q : Prf B b),
+ codePrf A a p = codePrf B b q → A = B.
+ +
+Definition codeImp (a b : nat) := cPair 1 (cPair a b).
+ +
+Lemma codeImpCorrect :
+ ∀ a b : Formula,
+ codeImp (codeFormula a) (codeFormula b) = codeFormula (impH a b).
+ +
+Definition codeNot (a : nat) := cPair 2 a.
+ +
+Lemma codeNotCorrect :
+ ∀ a : Formula, codeNot (codeFormula a) = codeFormula (notH a).
+ +
+Definition codeForall (n a : nat) := cPair 3 (cPair n a).
+ +
+Lemma codeForallCorrect :
+ ∀ (n : nat) (a : Formula),
+ codeForall n (codeFormula a) = codeFormula (forallH n a).
+ +
+Definition codeOr (a b : nat) := codeImp (codeNot a) b.
+ +
+Lemma codeOrCorrect :
+ ∀ a b : Formula,
+ codeOr (codeFormula a) (codeFormula b) = codeFormula (orH a b).
+ +
+Definition codeAnd (a b : nat) := codeNot (codeOr (codeNot a) (codeNot b)).
+ +
+Lemma codeAndCorrect :
+ ∀ a b : Formula,
+ codeAnd (codeFormula a) (codeFormula b) = codeFormula (andH a b).
+ +
+Definition codeIff (a b : nat) := codeAnd (codeImp a b) (codeImp b a).
+ +
+Lemma codeIffCorrect :
+ ∀ a b : Formula,
+ codeIff (codeFormula a) (codeFormula b) = codeFormula (iffH a b).
+ +
+End codeTermFormDef.
+ +
+End Code_Term_Formula_Proof.
+ +
+Arguments codeTerm {L} {cf cr cl} _.
+Arguments codeTerms {L} {cf cr cl n} _.
+Arguments codeFormula {L} {cf cr cl} _.
+Arguments codePrf {L} {cf cr cl} _ _ _.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.codeFreeVar.html b/theories/html/hydras.Ackermann.codeFreeVar.html new file mode 100644 index 00000000..b1a3e070 --- /dev/null +++ b/theories/html/hydras.Ackermann.codeFreeVar.html @@ -0,0 +1,150 @@ + + + + + +hydras.Ackermann.codeFreeVar + + + + +
+ + + +
+ +

Library hydras.Ackermann.codeFreeVar

+ +
+
+ +
+ codeFreeVar.v + +
+ + Original content by Russel O'Connor +
+
+ +
+From hydras.Ackermann Require Import primRec cPair.
+From Coq Require Import List Arith.
+From hydras.Ackermann Require Import ListExt.
+From hydras.Ackermann Require Export codeList.
+From hydras Require Import folProp code Compat815.
+Import LispAbbreviations.
+Import PRNotations.
+ +
+Section Code_Free_Vars.
+ +
+Generalizable All Variables.
+Variable L : Language.
+Context `(cL: Lcode L cf cr).
+ +
+Let Formula := Formula L.
+Let Formulas := Formulas L.
+Let System := System L.
+Let Term := Term L.
+Let Terms := Terms L.
+ +
+Definition codeFreeVarTermTerms : nat → nat :=
+  evalStrongRec 0
+    (fun t recs : nat ⇒
+     cPair
+       (switchPR (car t) (cdr (codeNth (t - S (cdr t)) recs))
+          (S (cPair (cdr t) 0)))
+       (switchPR t
+          (codeApp (car (codeNth (t - S (car (pred t))) recs))
+             (cdr (codeNth (t - S (cdr (pred t))) recs))) 0)).
+ +
+Definition codeFreeVarTerm (t : nat) : nat :=
+  car (codeFreeVarTermTerms t).
+ +
+Definition codeFreeVarTerms (t : nat) : nat :=
+  cdr (codeFreeVarTermTerms t).
+ +
+Lemma codeFreeVarTermCorrect (t: Term) :
+ codeFreeVarTerm (codeTerm t) = codeList (freeVarT t).
+ +
+Lemma codeFreeVarTermsCorrect (n : nat) (ts : Terms n):
+ codeFreeVarTerms (codeTerms ts) = codeList (freeVarTs ts).
+ +
+#[export]
+  Instance codeFreeVarTermTermsIsPR : isPR 1 codeFreeVarTermTerms.
+ +
+Lemma codeFreeVarTermIsPR : isPR 1 codeFreeVarTerm.
+ +
+#[export]
+  Instance codeFreeVarTermsIsPR : isPR 1 codeFreeVarTerms.
+ +
+Definition codeFreeVarFormula : nat → nat :=
+  evalStrongRec 0
+    (fun f recs : nat ⇒
+     switchPR (car f)
+       (switchPR (pred (car f))
+          (switchPR (pred (pred (car f)))
+             (switchPR (pred (pred (pred (car f))))
+                (codeFreeVarTerms (cdr f))
+                (codeListRemove (cadr f)
+                   (codeNth (f - S (cddr f)) recs)))
+             (codeNth (f - S (cdr f)) recs))
+          (codeApp (codeNth (f - S (cadr f)) recs)
+             (codeNth (f - S (cddr f)) recs)))
+       (codeApp (codeFreeVarTerm (cadr f))
+          (codeFreeVarTerm (cddr f)))).
+ +
+Lemma codeFreeVarFormulaCorrect (f : Formula) :
+  codeFreeVarFormula (codeFormula f) =
+    codeList (freeVarF f).
+ +
+#[export]
+  Instance codeFreeVarFormulaIsPR : isPR 1 codeFreeVarFormula.
+ +
+Definition codeFreeVarListFormula : nat → nat :=
+  evalStrongRec 0
+    (fun l recs : nat ⇒
+     switchPR l
+       (codeApp (codeFreeVarFormula (car (pred l)))
+          (codeNth (l - S (cdr (pred l))) recs)) 0).
+ +
+Lemma codeFreeVarListFormulaCorrect (l : list Formula):
+ codeFreeVarListFormula (codeList (map (codeFormula) l)) =
+ codeList (freeVarListFormula L l).
+ +
+#[export]
+  Instance codeFreeVarListFormulaIsPR : isPR 1 codeFreeVarListFormula.
+ +
+End Code_Free_Vars.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.codeList.html b/theories/html/hydras.Ackermann.codeList.html new file mode 100644 index 00000000..bcf6c2ea --- /dev/null +++ b/theories/html/hydras.Ackermann.codeList.html @@ -0,0 +1,141 @@ + + + + + +hydras.Ackermann.codeList + + + + +
+ + + +
+ +

Library hydras.Ackermann.codeList

+ +
+
+ +
+ codeList.v + +
+ + Original script by Russel O'Connor + +
+
+ +
+From hydras.Ackermann Require Import primRec cPair.
+From Coq Require Export Lists.List.
+From hydras.Ackermann Require Import ListExt.
+From Coq Require Import Arith.
+From Coq Require Vector.
+From hydras Require Import extEqualNat Compat815.
+ +
+Definition codeLength : nat → nat :=
+  evalStrongRec 0
+    (fun n Hrecs : nat ⇒
+     switchPR n (S (codeNth (n - S (cPairPi2 (pred n))) Hrecs)) 0).
+ +
+Lemma codeLengthCorrect :
+ ∀ l : list nat, codeLength (codeList l) = length l.
+ +
+#[export] Instance codeLengthIsPR : isPR 1 codeLength.
+ +
+Definition codeApp : nat → nat → nat :=
+  evalStrongRec 1
+    (fun n Hrecs p1 : nat ⇒
+       switchPR n
+         (S
+            (cPair (cPairPi1 (pred n))
+               (codeNth (n - S (cPairPi2 (pred n))) Hrecs))) p1).
+ +
+Lemma codeAppCorrect (l1 l2 : list nat):
+ codeApp (codeList l1) (codeList l2) = codeList (l1 ++ l2).
+ +
+#[export] Instance codeAppIsPR : isPR 2 codeApp.
+ +
+Definition codeListRemove (a l : nat) : nat :=
+  evalStrongRec 1
+    (fun n Hrecs p1 : nat ⇒
+     switchPR n
+       (switchPR (charFunction 2 Nat.eqb (cPairPi1 (pred n)) p1)
+          (codeNth (n - S (cPairPi2 (pred n))) Hrecs)
+          (S
+             (cPair (cPairPi1 (pred n))
+                (codeNth (n - S (cPairPi2 (pred n))) Hrecs))))
+       (codeList nil)) l a.
+ +
+Lemma codeListRemoveCorrect (a : nat) (l : list nat):
+ codeListRemove a (codeList l) =
+   codeList (List.remove eq_nat_dec a l).
+ +
+#[export] Instance codeListRemoveIsPR : isPR 2 codeListRemove.
+ +
+Definition codeIn (a l : nat) : nat :=
+  evalStrongRec 1
+    (fun n Hrecs p1 : nat ⇒
+     switchPR n
+       (switchPR (charFunction 2 Nat.eqb (cPairPi1 (pred n)) p1) 1
+          (codeNth (n - S (cPairPi2 (pred n))) Hrecs)) 0) l a.
+ +
+Lemma codeInCorrect (a : nat) (l : list nat) :
+ codeIn a (codeList l) =
+ match In_dec eq_nat_dec a l with
+ | left _ ⇒ 1
+ | right _ ⇒ 0
+ end.
+ +
+#[export] Instance codeInIsPR : isPR 2 codeIn.
+ +
+Definition codeNoDup : nat → nat :=
+  evalStrongRec 0
+    (fun l recs : nat ⇒
+       switchPR l
+         (switchPR
+            (codeIn (cPairPi1 (pred l))
+               (codeNth (l - S (cPairPi2 (pred l))) recs))
+            (codeNth (l - S (cPairPi2 (pred l))) recs)
+            (S
+               (cPair (cPairPi1 (pred l))
+                  (codeNth (l - S (cPairPi2 (pred l))) recs)))) 0).
+ +
+Lemma codeNoDupCorrect (l : list nat) :
+ codeNoDup (codeList l) = codeList (List.nodup eq_nat_dec l).
+ +
+#[export] Instance codeNoDupIsPR : isPR 1 codeNoDup.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.codeNatToTerm.html b/theories/html/hydras.Ackermann.codeNatToTerm.html new file mode 100644 index 00000000..34a69e40 --- /dev/null +++ b/theories/html/hydras.Ackermann.codeNatToTerm.html @@ -0,0 +1,66 @@ + + + + + +hydras.Ackermann.codeNatToTerm + + + + +
+ + + +
+ +

Library hydras.Ackermann.codeNatToTerm

+ +
+From hydras.Ackermann Require Import primRec cPair code folProp folProof
+                      Languages.
+From Coq Require Import Arith.
+From hydras.Ackermann Require LNN LNT.
+ +
+Definition natToTermLNN := LNN.natToTerm.
+ +
+Definition natToTermLNT := LNT.natToTerm.
+ +
+Fixpoint codeNatToTerm (n: nat) : nat :=
+  match n with
+    0 ⇒ cPair 4 0
+  | S p ⇒ cPair 3 (S (cPair (codeNatToTerm p) 0))
+  end.
+ +
+#[global] Instance LcodeLNN : Lcode LNN codeLNTFunction codeLNNRelation.
+ +
+#[global] Instance LcodeLNT : Lcode LNT codeLNTFunction codeLNTRelation.
+ +
+Lemma codeNatToTermCorrectLNN n :
+ codeNatToTerm n = codeTerm (natToTermLNN n).
+ +
+Lemma codeNatToTermCorrectLNT n :
+ codeNatToTerm n = codeTerm (natToTermLNT n).
+ +
+#[export] Instance codeNatToTermIsPR : isPR 1 codeNatToTerm.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.codePA.html b/theories/html/hydras.Ackermann.codePA.html new file mode 100644 index 00000000..495f27e9 --- /dev/null +++ b/theories/html/hydras.Ackermann.codePA.html @@ -0,0 +1,186 @@ + + + + + +hydras.Ackermann.codePA + + + + +
+ + + +
+ +

Library hydras.Ackermann.codePA

+ +
+
+ +
+ codePA.v + +
+ + Original script by Russel O'Connor + +
+
+ +
+From Coq Require Import Ensembles List Arith Lia.
+From Coq Require Vector.
+Import Bool.
+ +
+From hydras
+  Require Import primRec cPair folProp code codeList codeFreeVar
+  extEqualNat prLogic codeNatToTerm Compat815.
+ +
+Import LispAbbreviations.
+ +
+Section close.
+Generalizable All Variables.
+Variable L : Language.
+Context `(cL: Lcode L cf cr).
+ +
+ +
+Let Formula := Formula L .
+Let codeFormula := codeFormula (cl:=cL).
+ +
+Definition codeCloseList : nat → nat → nat :=
+  evalStrongRec 1
+    (fun l recs f : nat ⇒
+     switchPR l
+       (cPair 3
+          (cPair (car (pred l))
+             (codeNth (l - S (cdr (pred l))) recs))) f).
+ +
+Lemma codeCloseListCorrect (l : list nat) (f : Formula):
+ codeCloseList (codeList l) (codeFormula f) =
+   codeFormula (closeList L l f).
+ +
+#[export] Instance codeCloseListIsPR : isPR 2 codeCloseList.
+ +
+Definition codeClose (f : nat) : nat :=
+  codeCloseList (codeNoDup (codeFreeVarFormula f)) f.
+ +
+Lemma codeCloseCorrect (f : Formula) :
+  codeClose (codeFormula f) = codeFormula (close L f).
+ +
+#[export] Instance codeCloseIsPR : isPR 1 codeClose.
+ +
+End close.
+ +
+From hydras.Ackermann Require Import PA codeSubFormula.
+ +
+Section Code_PA.
+ +
+Let codeTerm := codeTerm (cl :=LcodeLNT).
+Let codeFormula := codeFormula (cl:=LcodeLNT).
+Let codeFormulaInj := codeFormulaInj LNT LcodeLNT.
+ +
+Definition codeOpen : nat → nat :=
+  evalStrongRec 0
+    (fun f recs : nat ⇒
+     switchPR (car f)
+       (switchPR (pred (car f))
+          (switchPR (pred (pred (car f)))
+             (switchPR (pred (pred (pred (car f)))) f
+                (codeNth (f - S (cdr (cdr f))) recs)) f) f) f).
+ +
+Lemma codeOpenCorrect (f : Formula):
+  codeOpen (codeFormula f) = codeFormula (open f).
+ +
+#[export] Instance codeOpenIsPR : isPR 1 codeOpen.
+ +
+Definition codeInductionSchema (f : nat) : bool :=
+  let n :=
+    car (cdr (cddddr (codeOpen f))) in
+  let g := cdr (cdr (cddddr (codeOpen f))) in
+  Nat.eqb
+    (codeClose
+       (codeImp (codeSubFormula g n (codeTerm Zero))
+          (codeImp
+             (codeForall n
+                (codeImp g (codeSubFormula g n
+                              (codeTerm (Succ (var n))))))
+             (codeForall n g)))) f.
+ +
+Lemma codeInductionSchemaCorrect1 (f : Formula) :
+ InductionSchema f → codeInductionSchema (codeFormula f) = true.
+ +
+Lemma codeInductionSchemaCorrect2 (f : Formula):
+ codeInductionSchema (codeFormula f) = true → InductionSchema f.
+        Opaque cPairPi1.
+        Opaque cPairPi2.
+ +
+Lemma codeInductionSchemaCorrect3 (f : Formula):
+ ¬ InductionSchema f → codeInductionSchema (codeFormula f) = false.
+ +
+#[local] Instance codeInductionSchemaIsPR : isPRrel 1 codeInductionSchema.
+ +
+Definition codePA : nat → bool :=
+  orRel 1 (Nat.eqb (codeFormula PA6))
+    (orRel 1 (Nat.eqb (codeFormula PA5))
+       (orRel 1 (Nat.eqb (codeFormula PA4))
+          (orRel 1 (Nat.eqb (codeFormula PA3))
+             (orRel 1 (Nat.eqb (codeFormula PA2))
+                (orRel 1 (Nat.eqb (codeFormula PA1))
+                   codeInductionSchema))))).
+ +
+Lemma codePAcorrect1 (f : Formula):
+ codePA (codeFormula f) = true → mem Formula PA f.
+ +
+Lemma codePAcorrect2 (f : Formula):
+ ¬ mem Formula PA f → codePA (codeFormula f) = false.
+ +
+Lemma codePAcorrect3 (f : Formula):
+  mem Formula PA f → codePA (codeFormula f) = true.
+ +
+#[export] Instance codePAIsPR : isPRrel 1 codePA.
+ +
+End Code_PA.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.codeSubFormula.html b/theories/html/hydras.Ackermann.codeSubFormula.html new file mode 100644 index 00000000..63023844 --- /dev/null +++ b/theories/html/hydras.Ackermann.codeSubFormula.html @@ -0,0 +1,535 @@ + + + + + +hydras.Ackermann.codeSubFormula + + + + +
+ + + +
+ +

Library hydras.Ackermann.codeSubFormula

+ +
+
+ +
+codeSubFormula: + +
+ + Original file by Russel O'Connor + +
+ + +
+
+ +
+From Coq Require Import Arith Vector Lia.
+From Coq Require Vector.
+ +
+From hydras.Ackermann
+  Require Import primRec cPair folProp code extEqualNat codeSubTerm
+  codeFreeVar.
+ +
+Import LispAbbreviations.
+Import PRNotations.
+ +
+From LibHyps Require Export LibHyps.
+From hydras Require Export MoreLibHyps NewNotations Compat815.
+Import Bool.
+ +
+Section Code_Substitute_Formula.
+ +
+Generalizable All Variables.
+Variable L : Language.
+Context `(cL : Lcode L cf cr).
+ +
+Notation Formula := (Formula L) (only parsing).
+Notation Formulas := (Formulas L) (only parsing).
+Notation System := (System L) (only parsing).
+Notation Term := (Term L) (only parsing).
+Notation Terms := (Terms L) (only parsing).
+Let codeFormula := codeFormula (cl := cL).
+Let codeTerm := codeTerm (cl := cL).
+ +
+Definition codeNewVar (l : nat) : nat :=
+  evalStrongRec 0
+    (fun n Hrecs : nat ⇒
+     switchPR n
+       (Nat.max (S (car (pred n)))
+          (codeNth (n - S (cdr (pred n))) Hrecs)) 0) l.
+ +
+Lemma codeNewVarCorrect (l : list nat) :
+  codeNewVar (codeList l) = newVar l.
+ +
+#[export] Instance codeNewVarIsPR : isPR 1 codeNewVar.
+ +
+ +
+Definition checkSubFormulaTrace : nat → nat :=
+  evalStrongRec 0
+    (fun trace recs : nat ⇒
+     let v := cTriplePi1 (cTriplePi1 trace) in
+     let s := cTriplePi2 (cTriplePi1 trace) in
+     let input := cTriplePi3 (cTriplePi1 trace) in
+     let output := cTriplePi2 trace in
+     let rest := cTriplePi3 trace in
+     let type := cPairPi1 input in
+     switchPR type
+       (switchPR (pred type)
+          (switchPR (pred (pred type))
+             (switchPR (pred (pred (pred type)))
+                (charFunction 2 Nat.eqb output
+                   (cPair type (codeSubTerms (cdr input) v s)))
+                (switchPR
+                   (charFunction 2 Nat.eqb v (car (cdr input)))
+                   (charFunction 2 Nat.eqb input output)
+                   (switchPR
+                      (codeIn (car (cdr input)) (codeFreeVarTerm s))
+                      (let nv :=
+                         codeNewVar
+                           (S
+                              (cPair v
+                                 (codeApp (codeFreeVarTerm s)
+                                    (codeFreeVarFormula
+                                       (cdr (cdr input)))))) in
+                       charFunction 0
+                         (Nat.eqb output
+                            (cPair 3 (cPair nv (cTriplePi2 (cdr rest)))) &&
+                          (Nat.eqb (cTriple v s (cTriplePi2 (car rest)))
+                             (cTriplePi1 (cdr rest)) &&
+                           Nat.eqb
+                             (cTriple (car (cdr input))
+                                (cPair 0 nv) (cdr (cdr input)))
+                             (cTriplePi1 (car rest)))) ×
+                       (codeNth (trace - S (car rest)) recs ×
+                        codeNth (trace - S (cdr rest)) recs))
+                      (charFunction 0
+                         (Nat.eqb output
+                            (cPair 3
+                               (cPair (car (cdr input))
+                                  (cTriplePi2 rest))) &&
+                          Nat.eqb (cTriple v s (cdr (cdr input)))
+                            (cTriplePi1 rest)) ×
+                       codeNth (trace - S rest) recs))))
+             (charFunction 0
+                (Nat.eqb output (cPair 2 (cTriplePi2 rest)) &&
+                 Nat.eqb (cTriple v s (cdr input)) (cTriplePi1 rest)) ×
+              codeNth (trace - S rest) recs))
+          (charFunction 0
+             (Nat.eqb output
+                (cPair 1
+                   (cPair (cTriplePi2 (car rest))
+                      (cTriplePi2 (cdr rest)))) &&
+              (Nat.eqb (cTriple v s (car (cdr input)))
+                 (cTriplePi1 (car rest)) &&
+               Nat.eqb (cTriple v s (cdr (cdr input)))
+                 (cTriplePi1 (cdr rest)))) ×
+           (codeNth (trace - S (car rest)) recs ×
+            codeNth (trace - S (cdr rest)) recs)))
+       (charFunction 2 Nat.eqb output
+          (cPair 0
+             (cPair (codeSubTerm (car (cdr input)) v s)
+                (codeSubTerm (cdr (cdr input)) v s))))).
+ +
+Definition makeTraceImp (f1 : fol.Formula L)
+  (f1rec : nat × fol.Term L → nat) (f2 : fol.Formula L)
+  (f2rec : nat × fol.Term L → nat) (p : nat × fol.Term L) : nat :=
+  let v := fst p in
+  let s := snd p in
+  cTriple (cTriple v (codeTerm s) (codeFormula (impH f1 f2)))
+    (codeFormula (substF (impH f1 f2) v s))
+    (cPair (f1rec p) (f2rec p)).
+ +
+Definition makeTraceNot
+  (f : fol.Formula L) (frec : nat × fol.Term L → nat)
+  (p : nat × fol.Term L) : nat :=
+  let v := fst p in
+  let s := snd p in
+  cTriple (cTriple v (codeTerm s) (codeFormula (notH f)))
+    (codeFormula (substF (notH f) v s))
+    (frec p).
+ +
+Definition makeTraceForall (n : nat) (f : fol.Formula L)
+  (rec : ∀ b : fol.Formula L,
+      lt_depth L b (forallH n f) → nat × fol.Term L → nat)
+  (p : nat × fol.Term L) : nat.
+ +
+Definition makeTrace : fol.Formula L → nat × fol.Term L → nat :=
+  Formula_depth_rec2 L (fun _ : fol.Formula L ⇒ nat × fol.Term L → nat)
+    (fun (t t0 : fol.Term L) (p : nat × fol.Term L) ⇒
+     let v := fst p in
+     let s := snd p in
+     cTriple (cTriple v (codeTerm s) (codeFormula (equal t t0)))
+       (codeFormula (substF (equal t t0) v s)) 0)
+    (fun (r : Relations L) t (p : nat × fol.Term L) ⇒
+     let v := fst p in
+     let s := snd p in
+     cTriple (cTriple v (codeTerm s) (codeFormula (atomic r t)))
+       (codeFormula (substF (atomic r t) v s)) 0)
+    makeTraceImp
+    makeTraceNot makeTraceForall.
+ +
+Lemma makeTraceImpNice (f2 g : fol.Formula L)
+  (z1 z2 : nat × fol.Term L → nat):
+  (∀ q : nat × fol.Term L, z1 q = z2 q) →
+  ∀ z3 z4 : nat × fol.Term L → nat,
+    (∀ q : nat × fol.Term L, z3 q = z4 q) →
+    ∀ q : nat × fol.Term L,
+      makeTraceImp f2 z1 g z3 q = makeTraceImp f2 z2 g z4 q.
+ +
+Lemma makeTraceNotNice (f2 : fol.Formula L)
+  (z1 z2 : nat × fol.Term L → nat) :
+ (∀ q : nat × fol.Term L, z1 q = z2 q) →
+ ∀ q : nat × fol.Term L,
+   makeTraceNot f2 z1 q = makeTraceNot f2 z2 q.
+ +
+Lemma makeTraceForallNice
+  (v0 : nat) (a : fol.Formula L)
+  (z1 z2 : ∀ b : fol.Formula L,
+      lt_depth L b (forallH v0 a) → nat × fol.Term L → nat):
+  (∀ (b : fol.Formula L) (q : lt_depth L b (forallH v0 a))
+          (r : nat × fol.Term L), z1 b q r = z2 b q r) →
+  ∀ q : nat × fol.Term L,
+    makeTraceForall v0 a z1 q = makeTraceForall v0 a z2 q.
+ +
+Remark makeTrace1 (f : fol.Formula L) (v : nat) (s : fol.Term L):
+ cTriplePi1 (makeTrace f (v, s)) = cTriple v (codeTerm s) (codeFormula f).
+ +
+Remark makeTrace2 (f : fol.Formula L) (v : nat) (s : fol.Term L):
+ cTriplePi2 (makeTrace f (v, s)) = codeFormula (substF f v s).
+ +
+Lemma makeTraceCorrect :
+ ∀ (f : fol.Formula L) (v : nat) (s : fol.Term L),
+ checkSubFormulaTrace (makeTrace f (v, s)) = 1.
+ +
+Lemma checkTraceCorrect :
+ ∀ (f : fol.Formula L) (v : nat) (s : fol.Term L) (n m : nat),
+ checkSubFormulaTrace (cTriple (cTriple v (codeTerm s) (codeFormula f)) n m) ≠
+ 0 → codeFormula (substF f v s) = n.
+ +
+Lemma switch5IsPR :
+ ∀ (f1 f2 f3 f4 f5 : nat → nat → nat) (g : nat → nat),
+ isPR 2 f1 →
+ isPR 2 f2 →
+ isPR 2 f3 →
+ isPR 2 f4 →
+ isPR 2 f5 →
+ isPR 1 g →
+ isPR 2
+   (fun n recs : nat ⇒
+    switchPR (g n)
+      (switchPR (pred (g n))
+         (switchPR (pred (pred (g n)))
+            (switchPR (pred (pred (pred (g n)))) (f1 n recs) (f2 n recs))
+            (f3 n recs)) (f4 n recs)) (f5 n recs)).
+ +
+#[export] Instance checkTraceIsPR : isPR 1 checkSubFormulaTrace.
+ +
+Definition ReplaceTermTermsTerm : nat → nat → nat :=
+  evalStrongRec 1
+    (fun t recs s : nat ⇒
+     cPair
+       (switchPR (car t)
+          (cPair (car t) (cdr (codeNth (t - S (cdr t)) recs)))
+          (cPair 0 s))
+       (switchPR t
+          (S
+             (cPair (car (codeNth (t - S (car (pred t))) recs))
+                (cdr (codeNth (t - S (cdr (pred t))) recs)))) 0)).
+ +
+#[local] Instance ReplaceTermTermsTermIsPR : isPR 2 ReplaceTermTermsTerm.
+ +
+Definition ReplaceTermTerm (t s : nat) : nat :=
+  car (ReplaceTermTermsTerm t s).
+ +
+Definition ReplaceTermsTerm (t s : nat) : nat :=
+  cdr (ReplaceTermTermsTerm t s).
+ +
+#[export] Instance ReplaceTermTermIsPR : isPR 2 ReplaceTermTerm.
+ +
+#[export] Instance ReplaceTermsTermIsPR : isPR 2 ReplaceTermsTerm.
+ +
+Remark ReplaceTermTermsTermMonotone :
+ ∀ a s1 s2 : nat,
+ s1 ≤ s2 →
+ ReplaceTermTerm a s1 ≤ ReplaceTermTerm a s2 ∧
+ ReplaceTermsTerm a s1 ≤ ReplaceTermsTerm a s2.
+ +
+Lemma ReplaceTermTermMonotone :
+  ∀ a s1 s2 : nat,
+    s1 ≤ s2 → ReplaceTermTerm a s1 ≤ ReplaceTermTerm a s2.
+ +
+Lemma ReplaceTermsTermMonotone :
+ ∀ a s1 s2 : nat,
+ s1 ≤ s2 → ReplaceTermsTerm a s1 ≤ ReplaceTermsTerm a s2.
+ +
+Remark maxLemma :
+ ∀ a b c d : nat, a ≤ b → c ≤ d → Nat.max a c ≤ Nat.max b d.
+ +
+Remark maxLemma2 :
+  ∀ a b : list nat,
+    fold_right Nat.max 0 a ≤ fold_right Nat.max 0 (a ++ b).
+ +
+Remark maxLemma3 :
+  ∀ a b : list nat,
+    fold_right Nat.max 0 b ≤ fold_right Nat.max 0 (a ++ b).
+ +
+Remark maxApp :
+ ∀ a b : list nat,
+ {fold_right Nat.max 0 (a ++ b) = fold_right Nat.max 0 a} +
+ {fold_right Nat.max 0 (a ++ b) = fold_right Nat.max 0 b}.
+ +
+Lemma boundSubTerm :
+ ∀ (t : fol.Term L) (v : nat) (s : fol.Term L),
+ codeTerm (substT t v s) ≤
+ ReplaceTermTerm (codeTerm t)
+   (fold_right Nat.max 0 (codeTerm s :: freeVarT t)).
+ +
+Lemma boundSubTerms :
+  ∀ (n : nat) (ts : fol.Terms L n) (v : nat) (s : fol.Term L),
+    codeTerms (substTs ts v s) ≤
+      ReplaceTermsTerm (codeTerms ts)
+        (fold_right Nat.max 0 (codeTerm s :: freeVarTs ts)).
+ +
+Lemma ReplaceTermTermSub :
+ ∀ (t : fol.Term L) (v w s2 : nat),
+ ReplaceTermTerm (codeTerm (substT t v (var w))) s2 =
+ ReplaceTermTerm (codeTerm t) s2.
+ +
+Lemma ReplaceTermsTermSub :
+ ∀ (n : nat) (ts : fol.Terms L n) (v w s2 : nat),
+ ReplaceTermsTerm (codeTerms (substTs ts v (var w))) s2 =
+ ReplaceTermsTerm (codeTerms ts) s2.
+ +
+Definition ReplaceFormulaTerm : nat → nat → nat :=
+  evalStrongRec 1
+    (fun f recs s : nat ⇒
+     switchPR (car f)
+       (switchPR (pred (car f))
+          (switchPR (pred (pred (car f)))
+             (switchPR (pred (pred (pred (car f))))
+                (cPair (car f) (ReplaceTermsTerm (cdr f) s))
+                (cPair 3
+                   (cPair s (codeNth (f - S (cdr (cdr f))) recs))))
+             (cPair 2 (codeNth (f - S (cdr f)) recs)))
+          (cPair 1
+             (cPair (codeNth (f - S (car (cdr f))) recs)
+                (codeNth (f - S (cdr (cdr f))) recs))))
+       (cPair 0
+          (cPair (ReplaceTermTerm (car (cdr f)) s)
+             (ReplaceTermTerm (cdr (cdr f)) s)))).
+ +
+#[export] Instance ReplaceFormulaTermIsPR : isPR 2 ReplaceFormulaTerm.
+ +
+Lemma ReplaceFormulaTermMonotone :
+  ∀ a s1 s2 : nat,
+    s1 ≤ s2 → ReplaceFormulaTerm a s1 ≤ ReplaceFormulaTerm a s2.
+ +
+Fixpoint varFormula (f : fol.Formula L) : list nat :=
+  match f with
+  | equal t s ⇒ freeVarT t ++ freeVarT s
+  | atomic r ts ⇒ freeVarTs ts
+  | impH A B ⇒ varFormula A ++ varFormula B
+  | notH A ⇒ varFormula A
+  | forallH v A ⇒ v :: varFormula A
+  end.
+ +
+Lemma ReplaceFormulaTermSub :
+  ∀ (f : fol.Formula L) (v w s2 : nat),
+    ReplaceFormulaTerm (codeFormula (substF f v (var w))) s2 =
+      ReplaceFormulaTerm (codeFormula f) s2.
+ +
+Remark codeTermFreeVar :
+ ∀ s : fol.Term L, fold_right Nat.max 0 (freeVarT s) ≤ codeTerm s.
+ +
+Remark maxVarFreeVar :
+ ∀ f : fol.Formula L,
+   fold_right Nat.max 0 (freeVarF f) ≤
+     fold_right Nat.max 0 (varFormula f).
+ +
+Remark maxSubTerm (t : fol.Term L) (v : nat) (s : fol.Term L):
+  fold_right Nat.max 0 (freeVarT (substT t v s)) ≤
+    fold_right Nat.max 0 (freeVarT s ++ freeVarT t).
+ +
+Remark maxSubTerms (n : nat) (ts : fol.Terms L n) (v : nat) (s : fol.Term L):
+  fold_right Nat.max 0 (freeVarTs (substTs ts v s)) ≤
+    fold_right Nat.max 0 (freeVarT s ++ freeVarTs ts).
+ +
+
+ +
+3 ^ n +
+
+Definition pow3 : nat → nat :=
+  nat_rec (fun _ ⇒ nat) 1 (fun _ rec : nat ⇒ rec + (rec + rec)).
+ +
+#[export] Instance pow3IsPR : isPR 1 pow3.
+ +
+Lemma pow3Monotone : ∀ a b : nat, a ≤ b → pow3 a ≤ pow3 b.
+ +
+Lemma pow3Min : ∀ a : nat, 1 ≤ pow3 a.
+ +
+Remark mapListLemma :
+  ∀ l : list nat,
+    fold_right Nat.max 0 (map S l) ≤ S (fold_right Nat.max 0 l).
+ +
+Remark boundSubFormulaHelp2 (a : fol.Formula L) (v0 : nat) (s : fol.Term L):
+  newVar (v0 :: freeVarT s ++ freeVarF a) ≤
+    S
+      (fold_right Nat.max 0
+         (v0 :: fold_right Nat.max 0 (freeVarT s) :: varFormula a)).
+ +
+Remark boundSubFormulaHelp1 :
+  ∀ (b a : fol.Formula L) (v0 v : nat) (s : fol.Term L),
+    fold_right Nat.max 0
+      (varFormula
+         (substF b v
+            (var (newVar (v0 :: freeVarT s ++ freeVarF a))))) ≤
+      pow3 (depth L b) + pow3 (depth L b) +
+        Nat.max v0
+          (Nat.max (fold_right Nat.max 0 (freeVarT s))
+             (Nat.max v
+                (Nat.max (fold_right Nat.max 0 (varFormula b))
+                   (fold_right Nat.max 0 (varFormula a))))).
+ +
+Remark boundSubFormulaHelp :
+ ∀ (f : fol.Formula L) (v : nat) (s : fol.Term L),
+ codeFormula (substF f v s) ≤
+ ReplaceFormulaTerm (codeFormula f)
+   (Nat.max (codeTerm s)
+      (pow3 (depth L f) +
+       fold_right Nat.max 0 (v :: freeVarT s ++ varFormula f))).
+ +
+Definition boundComputation (d p1 p2 : nat) : nat :=
+  nat_rec (fun _ ⇒ nat) (cTriple p1 p2 0)
+    (fun _ rec : nat ⇒ cTriple p1 p2 (cPair rec rec)) d.
+ +
+#[export] Instance boundComputationIsPR : isPR 3 boundComputation.
+ +
+Lemma boundComputationMonotone :
+ ∀ a1 a2 b1 b2 c1 c2 : nat,
+ a1 ≤ a2 →
+ b1 ≤ b2 →
+ c1 ≤ c2 → boundComputation a1 b1 c1 ≤ boundComputation a2 b2 c2.
+ +
+Lemma boundMakeTrace :
+ ∀ (f : fol.Formula L) (v : nat) (s : fol.Term L),
+ let C :=
+   Nat.max (codeTerm s)
+     (cPair 0
+        (pow3 (depth L f) +
+         fold_right Nat.max 0 (v :: freeVarT s ++ varFormula f))) in
+ makeTrace f (v, s) ≤
+ boundComputation (depth L f)
+   (cTriple C C (ReplaceFormulaTerm (codeFormula f) C))
+   (ReplaceFormulaTerm (codeFormula f) C).
+ +
+Definition codeSubFormula (f v s : nat) : nat :=
+  let C := cPair 0 (pow3 f + (v + (s + f))) in
+  car
+    (boundedSearch
+       (fun p x : nat ⇒
+        ltBool 0 (checkSubFormulaTrace (cPair (car p) x)))
+       (cPair (cTriple v s f)
+          (S
+             (boundComputation f (cTriple C C (ReplaceFormulaTerm f C))
+                (ReplaceFormulaTerm f C))))).
+ +
+Lemma codeSubFormulaCorrect (f : Formula) (v : nat) (s : Term):
+ codeSubFormula (codeFormula f) v (codeTerm s) =
+ codeFormula (substF f v s).
+ +
+#[export] Instance codeSubFormulaIsPR : isPR 3 codeSubFormula.
+ +
+End Code_Substitute_Formula.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.codeSubTerm.html b/theories/html/hydras.Ackermann.codeSubTerm.html new file mode 100644 index 00000000..c0c48431 --- /dev/null +++ b/theories/html/hydras.Ackermann.codeSubTerm.html @@ -0,0 +1,100 @@ + + + + + +hydras.Ackermann.codeSubTerm + + + + +
+ + + +
+ +

Library hydras.Ackermann.codeSubTerm

+ +
+From hydras.Ackermann Require Import primRec cPair folProp code
+                        extEqualNat.
+From Coq Require Import Arith.
+From Coq Require Vector.
+From hydras Require Import Compat815.
+Import LispAbbreviations.
+Require Import NewNotations.
+Import PRNotations.
+ +
+Section Code_Substitute_Term.
+ +
+Generalizable All Variables.
+Variable L : Language.
+Context `(cL : Lcode L cf cr).
+ +
+Let Formula := Formula L.
+Let Formulas := Formulas L.
+Let System := System L.
+Let Term := Term L.
+Let Terms := Terms L.
+ +
+Definition codeSubTermTerms : nat → nat → nat → nat :=
+  evalStrongRec 2
+    (fun t recs v s : nat ⇒
+       cPair
+         (switchPR (car t)
+            (cPair (car t) (cdr (codeNth (t - S (cdr t)) recs)))
+            (switchPR (charFunction 2 Nat.eqb (cdr t) v) s t))
+         (switchPR t
+            (S
+               (cPair (car (codeNth (t - S (car (pred t))) recs))
+                  (cdr (codeNth (t - S (cdr (pred t))) recs)))) 0)).
+ +
+Definition codeSubTerm (t s v : nat) : nat :=
+  car (codeSubTermTerms t s v).
+ +
+Definition codeSubTerms (t s v : nat) : nat :=
+  cdr (codeSubTermTerms t s v).
+ +
+Lemma codeSubTermCorrect :
+  ∀ (t : Term) (v : nat) (s : Term),
+    codeSubTerm (codeTerm t) v (codeTerm s) =
+      codeTerm (substT t v s).
+ +
+Lemma codeSubTermsCorrect :
+  ∀ (n : nat) (ts : Terms n) (v : nat) (s : Term),
+    codeSubTerms (codeTerms ts) v (codeTerm s) =
+      codeTerms (substTs ts v s).
+ +
+#[export] Instance codeSubTermTermsIsPR : isPR 3 codeSubTermTerms.
+ +
+#[export] Instance codeSubTermIsPR : isPR 3 codeSubTerm.
+ +
+Lemma codeSubTermsIsPR : isPR 3 codeSubTerms.
+ +
+End Code_Substitute_Term.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.expressible.html b/theories/html/hydras.Ackermann.expressible.html new file mode 100644 index 00000000..8db9fd94 --- /dev/null +++ b/theories/html/hydras.Ackermann.expressible.html @@ -0,0 +1,181 @@ + + + + + +hydras.Ackermann.expressible + + + + +
+ + + +
+ +

Library hydras.Ackermann.expressible

+ +
+
+ +
+ expressible.v + +
+ + Original file by Russel O'Connor + +
+
+ +
+From Coq Require Import Arith List.
+From hydras.Ackermann
+  Require Import ListExt folProp subProp extEqualNat Languages LNN
+  NewNotations.
+Import NNnotations.
+ +
+#[local] Arguments apply _ _ _ : clear implicits.
+ +
+Section RepresentableExpressible.
+ +
+Variable T : System.
+ +
+Hypothesis closedT1: (ClosedSystem LNN T).
+ +
+Remark closedT : ∀ v : nat, ¬ In_freeVarSys LNN v T.
+ +
+Fixpoint RepresentableHalf1 (n : nat) :
+  naryFunc n → Formula → Prop :=
+  match n return (naryFunc n → Formula → Prop) with
+  | O ⇒
+      fun (f : naryFunc 0) (A : Formula) ⇒
+      SysPrf T (A → v#0 = natToTerm f)%fol
+  | S m ⇒
+      fun (f : naryFunc (S m)) (A : Formula) ⇒
+      ∀ a : nat,
+      RepresentableHalf1 m (f a)
+        (substF A (S m) (natToTerm a))
+  end.
+ +
+Fixpoint RepresentableHalf2 (n : nat) : naryFunc n → Formula → Prop :=
+  match n return (naryFunc n → Formula → Prop) with
+  | O ⇒
+      fun (f : naryFunc 0) (A : Formula) ⇒
+      SysPrf T (v#0 = natToTerm f → A)%fol
+  | S m ⇒
+      fun (f : naryFunc (S m)) (A : Formula) ⇒
+      ∀ a : nat,
+      RepresentableHalf2 m (f a)
+        (substF A (S m) (natToTerm a))
+  end.
+ +
+Lemma RepresentableHalf1Alternate :
+ ∀ (n : nat) (f : naryFunc n) (A B : Formula),
+ SysPrf T (impH A B) → RepresentableHalf1 n f B →
+ RepresentableHalf1 n f A.
+ +
+Lemma RepresentableHalf2Alternate :
+ ∀ (n : nat) (f : naryFunc n) (A B : Formula),
+   SysPrf T (A → B)%fol → RepresentableHalf2 n f A →
+   RepresentableHalf2 n f B.
+ +
+Fixpoint RepresentableHelp (n : nat) : naryFunc n → Formula → Prop :=
+  match n return (naryFunc n → Formula → Prop) with
+  | O ⇒
+      fun (f : naryFunc 0) (A : Formula) ⇒
+      SysPrf T (A ↔ v#0 = natToTerm f)%fol
+  | S m ⇒
+      fun (f : naryFunc (S m)) (A : Formula) ⇒
+      ∀ a : nat,
+      RepresentableHelp m (f a)
+        (substF A (S m) (natToTerm a))
+  end.
+ +
+Lemma RepresentableHalfHelp :
+ ∀ (n : nat) (f : naryFunc n) (A : Formula),
+ RepresentableHalf1 n f A →
+ RepresentableHalf2 n f A → RepresentableHelp n f A.
+ +
+Definition Representable (n : nat) (f : naryFunc n)
+  (A : Formula) : Prop :=
+  (∀ v : nat, In v (freeVarF A) → v ≤ n) ∧
+  RepresentableHelp n f A.
+ +
+Lemma RepresentableAlternate :
+ ∀ (n : nat) (f : naryFunc n) (A B : Formula),
+ SysPrf T (iffH A B) → RepresentableHelp n f A →
+ RepresentableHelp n f B.
+ +
+Lemma Representable_ext :
+ ∀ (n : nat) (f g : naryFunc n) (A : Formula),
+   extEqual n f g → RepresentableHelp n f A →
+   RepresentableHelp n g A.
+ +
+Fixpoint ExpressibleHelp (n : nat) : naryRel n → Formula → Prop :=
+  match n return (naryRel n → Formula → Prop) with
+  | O ⇒
+      fun (R : naryRel 0) (A : Formula) ⇒
+      match R with
+      | true ⇒ SysPrf T A
+      | false ⇒ SysPrf T ( ¬ A)%fol
+      end
+  | S m ⇒
+      fun (R : naryRel (S m)) (A : Formula) ⇒
+      ∀ a : nat,
+      ExpressibleHelp m (R a)
+        (substF A (S m) (natToTerm a))
+  end.
+ +
+Definition Expressible (n : nat) (R : naryRel n) (A : Formula) : Prop :=
+  (∀ v : nat, In v (freeVarF A) →
+                   v ≤ n ∧ v ≠ 0) ∧
+  ExpressibleHelp n R A.
+ +
+Lemma expressibleAlternate :
+  ∀ (n : nat) (R : naryRel n) (A B : Formula),
+    SysPrf T (iffH A B) → ExpressibleHelp n R A →
+    ExpressibleHelp n R B.
+ +
+Hypothesis nn1: SysPrf T (natToTerm 1 ≠ natToTerm 0)%fol.
+ +
+Lemma Representable2Expressible :
+ ∀ (n : nat) (R : naryRel n) (A : Formula),
+ Representable n (charFunction n R) A →
+ Expressible n R (substF A 0 (natToTerm 1)).
+ +
+End RepresentableExpressible.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.extEqualNat.html b/theories/html/hydras.Ackermann.extEqualNat.html new file mode 100644 index 00000000..6f672028 --- /dev/null +++ b/theories/html/hydras.Ackermann.extEqualNat.html @@ -0,0 +1,99 @@ + + + + + +hydras.Ackermann.extEqualNat + + + + +
+ + + +
+ +

Library hydras.Ackermann.extEqualNat

+ +
+
+ +
+extEqualNat: + +
+ + Original script by Russel O'Connor + +
+
+ +
+From Coq Require Import Arith.
+ +
+ +
+Fixpoint naryFunc (n : nat) : Set :=
+  match n with
+  | O ⇒ nat
+  | S n ⇒ nat → naryFunc n
+  end.
+ +
+ +
+Fixpoint naryRel (n : nat) : Set :=
+  match n with
+  | O ⇒ bool
+  | S n ⇒ nat → naryRel n
+  end.
+ +
+ +
+Fixpoint extEqual (n : nat) : ∀ (a b : naryFunc n), Prop :=
+  match n with
+    0 ⇒ fun a b ⇒ a = b
+  | S p ⇒ fun a b ⇒ ∀ c, extEqual p (a c) (b c)
+  end.
+ +
+ +
+Fixpoint charFunction (n : nat) : naryRel n → naryFunc n :=
+  match n return (naryRel n → naryFunc n) with
+  | O ⇒ fun R : bool ⇒ match R with
+                         | true ⇒ 1
+                         | false ⇒ 0
+                         end
+  | S m ⇒ fun (R : naryRel (S m)) (a : nat) ⇒ charFunction m (R a)
+  end.
+ +
+Lemma extEqualRefl n a: extEqual n a a.
+ +
+Lemma extEqualSym :
+  ∀ (n : nat) (a b : naryFunc n), extEqual n a b → extEqual n b a.
+ +
+Lemma extEqualTrans :
+ ∀ (n : nat) (a b c : naryFunc n),
+ extEqual n a b → extEqual n b c → extEqual n a c.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.fol.html b/theories/html/hydras.Ackermann.fol.html new file mode 100644 index 00000000..2a9474bb --- /dev/null +++ b/theories/html/hydras.Ackermann.fol.html @@ -0,0 +1,457 @@ + + + + + +hydras.Ackermann.fol + + + + +
+ + + +
+ +

Library hydras.Ackermann.fol

+ +
+
+ +
+First Order Logic + +
+ + Original author: Russell O'Connor + +
+ +This file is Public Domain + +
+ + +
+
+ +
+ +
+
+ +
+

First Order Formulas over a language

+ +
+
+ +
+Record Language : Type := language
+ { Relations : Set;
+   Functions : Set;
+   arityR : Relations → nat;
+   arityF : Functions → nat}.
+ +
+ +
+Section First_Order_Logic.
+ +
+Variable L : Language.
+ +
+Inductive Term : Set :=
+  | var : nat → Term
+  | apply : ∀ f : Functions L, Terms (arityF L f) → Term
+with Terms : nat → Set :=
+  | Tnil : Terms 0
+  | Tcons : ∀ n : nat, Term → Terms n → Terms (S n).
+ +
+Scheme Term_Terms_ind := Induction for Term Sort Prop
+  with Terms_Term_ind := Induction for Terms Sort Prop.
+ +
+Scheme Term_Terms_rec := Minimality for Term Sort Set
+  with Terms_Term_rec := Minimality for Terms Sort Set.
+Scheme Term_Terms_rec_full := Induction for Term Sort Set
+  with Terms_Term_rec_full := Induction for Terms Sort Set.
+Inductive Formula : Set :=
+  | equal : Term → Term → Formula
+  | atomic : ∀ r : Relations L, Terms (arityR L r) → Formula
+  | impH : Formula → Formula → Formula
+  | notH : Formula → Formula
+  | forallH : nat → Formula → Formula.
+ +
+Definition Formulas := list Formula.
+Definition System := Ensemble Formula.
+Definition mem := Ensembles.In.
+ +
+
+ +
+

Extensions of the basic language of formulas

+ +
+
+ +
+Definition orH (A B : Formula) := impH (notH A) B.
+Definition andH (A B : Formula) := notH (orH (notH A) (notH B)).
+Definition iffH (A B : Formula) := andH (impH A B) (impH B A).
+Definition existH (x : nat) (A : Formula) := notH (forallH x (notH A)).
+ +
+ +
+Definition ifThenElseH (A B C : Formula) :=
+  andH (impH A B) (impH (notH A) C).
+ +
+
+ +
+

Decidability of equality between terms, formulas, ...

+ +
+
+ +
+Section Formula_Decidability.
+ +
+Definition language_decidable :=
+  ((∀ x y : Functions L, {x = y} + {x ≠ y}) ×
+   (∀ x y : Relations L, {x = y} + {x ≠ y}))%type.
+ +
+Hypothesis language_eqdec : language_decidable.
+ +
+ +
+Lemma term_eqdec : ∀ x y : Term, {x = y} + {x ≠ y}.
+ +
+Lemma terms_eqdec n (x y : Terms n): {x = y} + {x ≠ y}.
+ +
+ +
+Lemma formula_eqdec : ∀ x y : Formula, {x = y} + {x ≠ y}.
+ +
+End Formula_Decidability.
+ +
+
+ +
+

Depth Induction

+ + +
+ + Many functions on term and formulas are not structurally recursive (e.g. because of substitution of variables with terms). + In which case, we may use some notion of depth as a measure. +
+
+ +
+Section Formula_Depth_Induction.
+ +
+Fixpoint depth (A : Formula) : nat :=
+  match A with
+  | equal _ _ ⇒ 0
+  | atomic _ _ ⇒ 0
+  | impH A B ⇒ S (Nat.max (depth A) (depth B))
+  | notH A ⇒ S (depth A)
+  | forallH _ A ⇒ S (depth A)
+  end.
+ +
+Definition lt_depth (A B : Formula) : Prop := depth A < depth B.
+ +
+ +
+Definition Formula_depth_ind :
+  ∀ P : Formula → Prop,
+  (∀ a : Formula, (∀ b : Formula, lt_depth b a → P b) → P a) →
+  ∀ a : Formula, P a.
+ +
+Lemma Formula_depth_ind2 :
+ ∀ P : Formula → Prop,
+ (∀ t t0 : Term, P (equal t t0)) →
+ (∀ (r : Relations L)
+         (t : Terms (arityR L r)),
+     P (atomic r t)) →
+ (∀ f : Formula, P f → ∀ f0 : Formula, P f0 → P (impH f f0)) →
+ (∀ f : Formula, P f → P (notH f)) →
+ (∀ (v : nat) (a : Formula),
+  (∀ b : Formula, lt_depth b (forallH v a) → P b) → P (forallH v a)) →
+ ∀ f : Formula, P f.
+ +
+End Formula_Depth_Induction.
+ +
+End First_Order_Logic.
+ +
+
+ +
+

Implicit arguments and notations

+ +
+
+ +
+ +
+
+ +
+ +
+ + In Russel O'Connor's work, the abstract syntax of first-order terms and formulas is available in three versions: + +
+ +
    +
  • A generic one (for any Language L) + +
    • +
  • An instantiation for the language of number theory LNT + +
    • +
  • An instantiation for the language of natural numbers LNN + +
    • +
+ +
+ +In the current version, we propose to use three notation scopes: fol_scope, nt_scope and nn_scope, in order to make clear the relationship between the three sets of formulas. + +
+ + fol_scope is defined in this file, lnt_scope in +LNT.v, and +lnn_scope in +LNN.v, + +
+ +

Implicit arguments

+ +The original code of this library contains some redefinitions like + +
+ +
+Definition Formula := Formula LNN.
+
+ +
+ +We plan to use systematically implicit arguments and avoid such redefinitions, which make more complex formula and term displaying, e.g. in goals or results of computation. + +
+
+ +
+Arguments impH {L} _ _.
+Arguments notH {L} _.
+Arguments forallH {L} _ _.
+Arguments orH {L} _ _.
+Arguments andH {L} _ _.
+Arguments iffH {L} _ _.
+Arguments equal {L} _ _.
+Arguments existH {L} _ _.
+Arguments var {L} _.
+Arguments atomic {L} _ _.
+Arguments apply {L} _ _.
+Arguments ifThenElseH {L} _ _ _.
+Arguments Tnil {L}.
+Arguments Tcons {L} {n} _ _.
+ +
+Notation "'f[' A ']f'" := A (A custom fol at level 200).
+ +
+Notation "x ∨ y" := (orH x y) (in custom fol at level 85, x custom fol, y custom fol, right associativity).
+ +
+Notation "x ∧ y" := (andH x y) (in custom fol at level 80, x custom fol, y custom fol, right associativity).
+ +
+Notation "x -> y" := (impH x y) (in custom fol at level 99, right associativity, y at level 200).
+ +
+Notation "x <-> y" := (iffH x y) (in custom fol at level 99, right associativity, y at level 200).
+ +
+Notation "~ x" := (notH x) (in custom fol at level 75, right associativity).
+ +
+Notation "( x )" := x (in custom fol, x custom fol at level 200).
+ +
+Notation "x = y" := (equal x y) (in custom fol at level 70, no associativity).
+ +
+Notation "x <> y" := (notH (equal x y)) (in custom fol at level 70, no associativity).
+ +
+Notation "{ x }" := x (in custom fol, x constr).
+ +
+Notation "'∀' i ',' f" := (forallH i f) (in custom fol at level 200, i constr, f custom fol at level 200).
+ +
+Notation "'∃' i , f" := (existH i f)
+  (in custom fol at level 200, i constr, f custom fol at level 200).
+ +
+Check f[ ∀ 3, ∃ 4, {equal (var 3) (var 4)} ∨ {var 3} = {var 4} ]f.
+ +
+Check f[ ¬ { equal (var 3) (var 4) } ]f.
+ +
+Check f[ {var 3} ≠ {var 4} ]f.
+ +
+Check f[ ¬ ( {var 3} = {var 4} ) ]f.
+ +
+
+ +
+

The fol_scope notation scope

+ +
+
+ +
+Module FolNotations.
+Declare Scope fol_scope.
+Delimit Scope fol_scope with fol.
+ +
+Infix "=" := (equal _): fol_scope.
+Infix "∨" := (orH): fol_scope.
+Infix "∧" := (andH):fol_scope.
+Infix "→" := (impH): fol_scope.
+Notation "~ A" := (@notH _ A): fol_scope.
+Notation "A <-> B" := (@iffH _ A B): fol_scope.
+ +
+Notation "'v#' i" := (var i) (at level 3, format "'v#' i", i at level 0) : fol_scope.
+Notation "'exH' x .. y , p" := (existH x .. (existH y p) ..)
+  (x at level 0, y at level 0, at level 200, right associativity) : fol_scope.
+Notation "'allH' x .. y , p" := (forallH x .. (forallH y p) ..)
+  (x at level 0, y at level 0, at level 200, right associativity) : fol_scope.
+ +
+Notation "t = u" := (@equal _ t u): fol_scope.
+Notation "t <> u" := (¬ t = u)%fol : fol_scope.
+ +
+ +
+
+ +
+the following notations are used when some computation + expands a disjunction, conjuction, etc. + in terms of implication and negation +
+
+ +
+Reserved Notation "x '\/'' y" (at level 85, right associativity).
+Reserved Notation "x '/\'' y" (at level 80, right associativity).
+Reserved Notation "x '<->'' y" (at level 95, no associativity).
+Reserved Notation "x '<->''' y" (at level 95, no associativity).
+ +
+Notation "x \/' y" := (¬ x → y)%fol : fol_scope.
+Notation "x /\' y" := (¬ (~ x \/' ¬ y))%fol : fol_scope.
+Notation "x <->'' y" := ((x → y) ∧ (y → x))%fol: fol_scope.
+Notation "x <->' y" := (¬ (~ (x → y) \/' ¬ (y → x)))%fol : fol_scope.
+Notation exH' v A := (¬ (forallH v (¬ A)))%fol.
+ +
+End FolNotations.
+ +
+Import FolNotations.
+Check (v#5)%fol.
+ +
+
+ +
+

Examples

+ +
+
+ +
+Section LExamples.
+Variable L: Language.
+Variables P Q : Formula L.
+ +
+Let ex1 : Formula L := (P ∧ Q)%fol.
+ +
+Let ex2 : Formula L := (¬ (~ ¬P → ¬Q))%fol.
+ +
+Let ex3 : Formula L:= (¬ (~P ∨ ¬Q))%fol.
+ +
+ +
+End LExamples.
+ +
+Section Correctness.
+ Variable L: Language.
+ Variables P Q R : Formula L.
+ +
+ Goal (P ∨ Q)%fol = (P \/' Q)%fol.
+ +
+Goal (P ∧ Q)%fol = (P /\' Q)%fol.
+ +
+End Correctness.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.folLogic.html b/theories/html/hydras.Ackermann.folLogic.html new file mode 100644 index 00000000..ca232a6a --- /dev/null +++ b/theories/html/hydras.Ackermann.folLogic.html @@ -0,0 +1,255 @@ + + + + + +hydras.Ackermann.folLogic + + + + +
+ + + +
+ +

Library hydras.Ackermann.folLogic

+ +
+
+ +
+folLogic.v: + +
+ + Original script by Russel O'Connor +
+
+ +
+From Coq Require Import Ensembles List.
+ +
+From hydras.Ackermann Require Import ListExt folProof folProp Deduction.
+Import FolNotations.
+ +
+Section Logic_Rules.
+ +
+Variable L : Language.
+Let Formula := Formula L.
+Let Formulas := Formulas L.
+Let System := System L.
+Let Term := Term L.
+Let Terms := Terms L.
+Let Prf := Prf L.
+Let SysPrf := SysPrf L.
+Arguments Ensembles.Add {U} _ _.
+ +
+Lemma Axm T f: mem _ T f → SysPrf T f.
+ +
+Lemma sysExtend (T U : System) (f : Formula):
+ Included _ T U → SysPrf T f → SysPrf U f.
+ +
+Lemma sysWeaken (T : System) (f g : Formula):
+  SysPrf T f → SysPrf (Ensembles.Add T g) f.
+ +
+Lemma impI (T : System) (f g : Formula):
+ SysPrf (Ensembles.Add T g) f → SysPrf T (g → f)%fol.
+ +
+Lemma impE (T : System) (f g : Formula):
+ SysPrf T (g → f)%fol → SysPrf T g → SysPrf T f.
+ +
+Lemma contradiction (T : System) (f g : Formula):
+ SysPrf T f → SysPrf T (¬ f)%fol → SysPrf T g.
+ +
+Lemma nnE (T : System) (f : Formula): SysPrf T (¬ ¬ f)%fol → SysPrf T f.
+ +
+Lemma nnI (T : System) (f : Formula): SysPrf T f → SysPrf T (¬ ¬ f)%fol.
+ +
+
+ +
+ contraposition +
+
+ +
+Lemma cp1 (T : System) (f g : Formula) :
+ SysPrf T (¬ f → ¬ g)%fol → SysPrf T (g → f)%fol.
+ +
+Lemma cp2 (T : System) (f g : Formula):
+ SysPrf T (g → f)%fol → SysPrf T (¬f → ¬g)%fol.
+ +
+Lemma orI1 (T : System) (f g : Formula): SysPrf T f → SysPrf T (f ∨ g)%fol.
+ +
+Lemma orI2 (T : System) (f g : Formula): SysPrf T g → SysPrf T (f ∨ g)%fol.
+ +
+Lemma noMiddle (T : System) (f : Formula): SysPrf T (¬ f ∨ f)%fol.
+ +
+Lemma orE (T : System) (f g h : Formula):
+ SysPrf T (f ∨ g)%fol →
+ SysPrf T (f → h)%fol → SysPrf T (g → h)%fol → SysPrf T h.
+ +
+Lemma orSys (T : System) (f g h : Formula):
+ SysPrf (Ensembles.Add T f) h → SysPrf (Ensembles.Add T g) h →
+ SysPrf (Ensembles.Add T (f ∨ g)%fol) h.
+ +
+Lemma andI (T : System) (f g : Formula):
+ SysPrf T f → SysPrf T g → SysPrf T (f ∧ g)%fol.
+ +
+Lemma andE1 (T : System) (f g : Formula):
+  SysPrf T (f ∧ g)%fol → SysPrf T f.
+ +
+Lemma andE2 (T : System) (f g : Formula):
+  SysPrf T (f ∧ g)%fol → SysPrf T g.
+ +
+Lemma iffI (T : System) (f g : Formula) :
+ SysPrf T (f → g)%fol → SysPrf T (g → f)%fol → SysPrf T (f ↔ g)%fol.
+ +
+Lemma iffE1 (T : System) (f g : Formula):
+  SysPrf T (f ↔ g)%fol → SysPrf T (f → g)%fol.
+ +
+Lemma iffE2 (T : System) (f g : Formula):
+  SysPrf T (f ↔ g)%fol → SysPrf T (g → f)%fol.
+ +
+Lemma forallI (T : System) (f : Formula) (v : nat):
+ ¬ In_freeVarSys L v T → SysPrf T f → SysPrf T (allH v, f)%fol.
+ +
+Lemma forallE (T : System) (f : Formula) (v : nat) (t : Term):
+ SysPrf T (allH v, f)%fol → SysPrf T (substF f v t).
+ +
+Lemma forallSimp (T : System) (f : Formula) (v : nat):
+  SysPrf T (allH v, f)%fol → SysPrf T f.
+ +
+Lemma existI (T : System) (f : Formula) (v : nat) (t : Term):
+  SysPrf T (substF f v t) → SysPrf T (exH v, f)%fol.
+ +
+Lemma existE (T : System) (f g : Formula) (v : nat):
+ ¬ In_freeVarSys L v T →
+ ¬ In v (freeVarF g) →
+ SysPrf T (exH v, f)%fol → SysPrf T (f → g)%fol → SysPrf T g.
+ +
+Lemma existSimp (T : System) (f : Formula) (v : nat) :
+ SysPrf T f → SysPrf T (exH v, f)%fol.
+ +
+Lemma existSys (T : System) (f g : Formula) (v : nat):
+ ¬ In_freeVarSys L v T →
+ ¬ In v (freeVarF g) →
+ SysPrf (Ensembles.Add T f) g →
+ SysPrf (Ensembles.Add T (exH v, f)%fol) g.
+ +
+Section Not_Rules.
+ +
+Lemma absurd1 (T : System) (f : Formula) :
+ SysPrf T (f → ¬f)%fol → SysPrf T (¬f)%fol.
+ +
+Lemma nImp (T : System) (f g : Formula) :
+ SysPrf T (f ∧ ¬g)%fol → SysPrf T (¬ (f → g))%fol.
+ +
+Lemma nOr (T : System) (f g : Formula):
+ SysPrf T (¬ f ∧ ¬ g)%fol → SysPrf T (¬ (f ∨ g))%fol.
+ +
+Lemma nAnd (T : System) (f g : Formula):
+ SysPrf T (¬ f ∨ ¬ g)%fol → SysPrf T (¬ (f ∧ g))%fol.
+ +
+Lemma nForall (T : System) (f : Formula) (v : nat) :
+ SysPrf T (exH v, ¬ f)%fol → SysPrf T (¬ (allH v, f))%fol.
+ +
+Lemma nExist (T : System) (f : Formula) (v : nat):
+ SysPrf T (allH v, ¬f)%fol → SysPrf T (¬ (exH v, f))%fol.
+ +
+End Not_Rules.
+ +
+Section Other_Rules.
+ +
+Lemma impRefl (T : System) (f : Formula): SysPrf T (f → f)%fol.
+ +
+Lemma impTrans (T : System) (f g h : Formula):
+ SysPrf T (f → g)%fol → SysPrf T (g → h)%fol → SysPrf T (f → h)%fol.
+ +
+Lemma orSym (T : System) (f g : Formula):
+ SysPrf T (f ∨ g)%fol → SysPrf T (g ∨ f)%fol.
+ +
+Lemma andSym (T : System) (f g : Formula):
+  SysPrf T (f ∧ g)%fol → SysPrf T (g ∧ f)%fol.
+ +
+Lemma iffRefl (T : System) (f : Formula) : SysPrf T (f ↔ f)%fol.
+ +
+Lemma iffSym (T : System) (f g : Formula) :
+  SysPrf T (f ↔ g)%fol → SysPrf T (g ↔ f)%fol.
+ +
+Lemma iffTrans (T : System) (f g h : Formula):
+ SysPrf T (f ↔ g)%fol → SysPrf T (g ↔ h)%fol → SysPrf T (f ↔ h)%fol.
+ +
+End Other_Rules.
+ +
+Lemma openClosed (T : System) (f : Formula):
+ SysPrf T (close L f) → SysPrf T f.
+ +
+End Logic_Rules.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.folLogic2.html b/theories/html/hydras.Ackermann.folLogic2.html new file mode 100644 index 00000000..b78ad5c3 --- /dev/null +++ b/theories/html/hydras.Ackermann.folLogic2.html @@ -0,0 +1,107 @@ + + + + + +hydras.Ackermann.folLogic2 + + + + +
+ + + +
+ +

Library hydras.Ackermann.folLogic2

+ +
+
+ +
+folLogic2.v + +
+ + Original script by Russel O'Connor + +
+ + +
+
+ +
+From Coq Require Import Ensembles List Arith.
+ +
+From hydras.Ackermann Require Import ListExt folProp folProof
+  folLogic subProp folReplace.
+Import FolNotations.
+ +
+Section More_Logic_Rules.
+ +
+Variable L : Language.
+Let Formula := Formula L.
+Let Formulas := Formulas L.
+Let System := System L.
+Let Term := Term L.
+Let Terms := Terms L.
+Let Prf := Prf L.
+Let SysPrf := SysPrf L.
+ +
+Lemma rebindForall (T : System) (a b : nat) (f : Formula):
+  ¬ In b (freeVarF (forallH a f)) →
+  SysPrf T ((allH a, f) ↔
+              (allH b, (substF f a v#b)))%fol.
+ +
+Lemma rebindExist (T : System) (a b : nat) (f : Formula):
+  ¬ In b (freeVarF (existH a f)) →
+  SysPrf T (iffH (existH a f) (existH b (substF f a (var b)))).
+ +
+Lemma subSubTerm (t : Term) (v1 v2 : nat) (s1 s2 : Term):
+  v1 ≠ v2 →
+  ¬ In v1 (freeVarT s2) →
+  substT (substT t v1 s1) v2 s2 =
+    substT (substT t v2 s2) v1 (substT s1 v2 s2).
+ +
+Lemma subSubTerms (n : nat) (ts : Terms n) (v1 v2 : nat) (s1 s2 : Term):
+  v1 ≠ v2 →
+  ¬ In v1 (freeVarT s2) →
+  substTs (substTs ts v1 s1) v2 s2 =
+    substTs (substTs ts v2 s2) v1 (substT s1 v2 s2).
+ +
+Lemma subSubFormula (f : Formula) (v1 v2 : nat) (s1 s2 : Term):
+ v1 ≠ v2 →
+ ¬ In v1 (freeVarT s2) →
+ ∀ T : System,
+ SysPrf T
+   (iffH (substF (substF f v1 s1) v2 s2)
+      (substF (substF f v2 s2) v1
+         (substT s1 v2 s2))).
+ +
+End More_Logic_Rules.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.folLogic3.html b/theories/html/hydras.Ackermann.folLogic3.html new file mode 100644 index 00000000..c32b2a64 --- /dev/null +++ b/theories/html/hydras.Ackermann.folLogic3.html @@ -0,0 +1,163 @@ + + + + + +hydras.Ackermann.folLogic3 + + + + +
+ + + +
+ +

Library hydras.Ackermann.folLogic3

+ +
+
+ +
+folLogic2.3 + +
+ + Original script by Russel O'Connor + +
+ + +
+
+ +
+From Coq Require Import Ensembles List Arith Lia.
+From LibHyps Require Export LibHyps.
+ +
+From hydras.Ackermann Require Import ListExt folProp folProof.
+From hydras.Ackermann Require Export folLogic folLogic2.
+From hydras.Ackermann Require Import subProp folReplace subAll misc.
+From hydras Require Import Compat815.
+From hydras Require Export MoreLibHyps.
+Import FolNotations.
+ +
+Section Equality_Logic_Rules.
+ +
+Variable L : Language.
+Notation Formula := (Formula L) (only parsing).
+Notation Formulas := (Formulas L) (only parsing).
+Notation System := (System L) (only parsing).
+Notation Term := (Term L) (only parsing).
+Notation Terms := (Terms L) (only parsing).
+ +
+Let Prf := Prf L.
+Let SysPrf := SysPrf L.
+ +
+Lemma eqRefl (T : fol.System L) (a : fol.Term L):
+  SysPrf T (a = a)%fol.
+ +
+Lemma eqSym (T : fol.System L) (a b : fol.Term L):
+  SysPrf T (a = b)%fol → SysPrf T (b = a)%fol.
+ +
+Lemma eqTrans (T : fol.System L) (a b c : fol.Term L):
+ SysPrf T (a = b)%fol → SysPrf T (b = c)%fol →
+ SysPrf T (a = c)%fol.
+ +
+Fixpoint PairwiseEqual (T : fol.System L) (n : nat) {struct n} :
+ fol.Terms L n → fol.Terms L n → Prop :=
+  match n return (fol.Terms L n → fol.Terms L n → Prop) with
+  | O ⇒ fun ts ss : fol.Terms L 0 ⇒ True
+  | S x ⇒
+      fun ts ss : fol.Terms L (S x) ⇒
+      let (a, b) := proj1_sig (consTerms L x ts) in
+      let (c, d) := proj1_sig (consTerms L x ss) in
+      SysPrf T (a = c)%fol ∧ PairwiseEqual T x b d
+  end.
+ +
+
+ +
+TODO : specify and document +
+
+ +
+Let termsMap (m : nat) (ts ss : fol.Terms L m) : nat → fol.Term L.
+ +
+Remark subAllnVars1 (a : nat) (ts ss : fol.Terms L a):
+ ts = subAllTerms L _ (fst (nVars L a)) (termsMap a ts ss).
+ +
+Remark subAllnVars2 (a : nat) (ts ss : fol.Terms L a):
+ ss = subAllTerms L _ (snd (nVars L a)) (termsMap a ts ss).
+    Opaque eq_nat_dec.
+    Transparent eq_nat_dec.
+ +
+Remark addPairwiseEquals (T : fol.System L) (n : nat) (ts ss : fol.Terms L n):
+  PairwiseEqual T n ts ss →
+  ∀ m0 : nat → fol.Term L,
+    (∀ q : nat, q < n + n → m0 q = termsMap n ts ss q) →
+    ∀ f0 : fol.Formula L,
+      SysPrf T
+        (subAllFormula L
+           (nat_rec (fun _ : nat ⇒ fol.Formula L) f0
+              (fun (n : nat) (Hrecn : fol.Formula L) ⇒
+                 (v#(n + n) = v#(S (n + n)) → Hrecn)%fol
+                 )
+              n) m0) → SysPrf T (subAllFormula L f0 m0).
+ +
+Lemma equalRelation (T : fol.System L) (r : Relations L) (ts ss : fol.Terms L _):
+ PairwiseEqual T _ ts ss → SysPrf T (atomic r ts ↔ atomic r ss)%fol.
+ +
+Lemma equalFunction (T : fol.System L) (f : Functions L) (ts ss : fol.Terms L _):
+ PairwiseEqual T _ ts ss → SysPrf T (apply f ts = apply f ss)%fol.
+ +
+Lemma subWithEqualsTerm (a b t : fol.Term L) (v : nat)
+  (T : fol.System L):
+ SysPrf T (a = b)%fol →
+ SysPrf T (substT t v a = substT t v b)%fol.
+ +
+Lemma subWithEqualsTerms (a b : fol.Term L) (n : nat) (ts : fol.Terms L n)
+  (v : nat) (T : fol.System L):
+  SysPrf T (a = b)%fol →
+  PairwiseEqual T _ (substTs ts v a) (substTs ts v b).
+ +
+Lemma subWithEquals :
+  ∀ (f : fol.Formula L) (v : nat) (a b : fol.Term L) (T : fol.System L),
+    SysPrf T (a = b)%fol →
+    SysPrf T
+      (substF f v a → substF f v b)%fol.
+ +
+End Equality_Logic_Rules.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.folProof.html b/theories/html/hydras.Ackermann.folProof.html new file mode 100644 index 00000000..f9e2211b --- /dev/null +++ b/theories/html/hydras.Ackermann.folProof.html @@ -0,0 +1,155 @@ + + + + + +hydras.Ackermann.folProof + + + + +
+ + + +
+ +

Library hydras.Ackermann.folProof

+ +
+
+ +
+folProof.v + +
+ + Original script by Russel O'Connor + +
+ + +
+
+ +
+From Coq Require Import Ensembles Lists.List Arith.
+Import ListNotations.
+ +
+From hydras.Ackermann Require Export fol.
+From hydras.Ackermann Require Import folProp.
+Import FolNotations.
+ +
+Section ProofH.
+ +
+Variable L : Language.
+ +
+Let Formula := Formula L.
+Let Formulas := Formulas L.
+Let System := System L.
+Let Term := Term L.
+Let Terms := Terms L.
+ +
+Fixpoint nVars (n: nat) : Terms n × Terms n:=
+  match n with
+    0 ⇒ (Tnil, Tnil)
+  | S n0 ⇒
+      (let (a,b) := nVars n0 in
+       (Tcons (var (n0 + n0)) a,
+         Tcons (var (S (n0 + n0))) b))
+  end.
+ +
+Section Example.
+End Example.
+ +
+Definition AxmEq4 (R : Relations L) : Formula.
+ +
+
+ +
+TODO : An example in PA +
+
+ +
+Definition AxmEq5 (f : Functions L) : Formula.
+ +
+Inductive Prf : Formulas → Formula → Set :=
+| AXM : ∀ A : Formula, Prf [A] A
+| MP :
+  ∀ (Hyp1 Hyp2 : Formulas) (A B : Formula),
+    Prf Hyp1 (A → B)%fol → Prf Hyp2 A → Prf (Hyp1 ++ Hyp2) B
+| GEN :
+  ∀ (Hyp : Formulas) (A : Formula) (v : nat),
+    ¬ In v (freeVarListFormula L Hyp) → Prf Hyp A →
+    Prf Hyp (allH v, A)%fol
+| IMP1 : ∀ A B : Formula, Prf [] (A → B → A)%fol
+| IMP2 :
+  ∀ A B C : Formula,
+    Prf [] ((A → B → C) → (A → B) → A → C)%fol
+| CP :
+  ∀ A B : Formula,
+    Prf [] ((¬ A → ¬ B) → B → A)%fol
+| FA1 :
+  ∀ (A : Formula) (v : nat) (t : Term),
+    Prf [] ((allH v, A) → substF A v t)%fol
+| FA2 :
+  ∀ (A : Formula) (v : nat),
+    ¬ In v (freeVarF A) → Prf [] (A → allH v, A)%fol
+| FA3 :
+  ∀ (A B : Formula) (v : nat),
+    Prf []
+      ((allH v, A → B) → (allH v, A) → allH v, B)%fol
+| EQ1 : Prf [] (v#0 = v#0)%fol
+| EQ2 : Prf [] (v#0 = v#1 → v#1 = v#0)%fol
+| EQ3 : Prf [] (v#0 = v#1 → v#1 = v#2 → v#0 = v#2)%fol
+| EQ4 : ∀ R : Relations L, Prf [] (AxmEq4 R)
+| EQ5 : ∀ f : Functions L, Prf [] (AxmEq5 f).
+ +
+Definition SysPrf (T : System) (f : Formula) :=
+  ∃ Hyp : Formulas,
+    (∃ prf : Prf Hyp f,
+       (∀ g : Formula, In g Hyp → mem _ T g)).
+ +
+Definition Inconsistent (T : System) := ∀ f : Formula, SysPrf T f.
+ +
+Definition Consistent (T : System) := ∃ f : Formula, ¬ SysPrf T f.
+ +
+Definition independent T f := ¬ SysPrf T f ∧ ¬ SysPrf T (¬ f)%fol.
+ +
+End ProofH.
+ +
+Arguments independent {L} _ _.
+ +
+Notation undecidable := (independent) (only parsing).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.folProp.html b/theories/html/hydras.Ackermann.folProp.html new file mode 100644 index 00000000..70a78450 --- /dev/null +++ b/theories/html/hydras.Ackermann.folProp.html @@ -0,0 +1,574 @@ + + + + + +hydras.Ackermann.folProp + + + + +
+ + + +
+ +

Library hydras.Ackermann.folProp

+ +
+From Coq Require Import Wf_nat Arith Lists.List Peano_dec.
+ +
+From hydras.Ackermann Require Import ListExt.
+From hydras.Ackermann Require Export fol.
+Import FolNotations.
+ +
+Section Fol_Properties.
+ +
+Variable L : Language.
+ +
+Notation Formula := (Formula L) (only parsing).
+Notation Formulas := (Formulas L) (only parsing).
+Notation System := (System L) (only parsing).
+Notation Term := (Term L) (only parsing).
+Notation Terms := (Terms L) (only parsing).
+ +
+Let lt_depth := lt_depth L.
+ +
+Section Free_Variables.
+Fixpoint freeVarT (s : fol.Term L) : list nat :=
+  match s with
+  | var v ⇒ v :: nil
+  | apply f ts ⇒ freeVarTs (arityF L f) ts
+  end
+with freeVarTs (n : nat) (ss : fol.Terms L n) {struct ss} : list nat :=
+       match ss with
+       | Tnil ⇒ nil (A:=nat)
+       | Tcons m t ts ⇒ freeVarT t ++ freeVarTs m ts
+       end.
+ +
+Lemma freeVarTApply :
+  ∀ (f : Functions L) (ts : fol.Terms L _),
+    freeVarT (apply f ts) = freeVarTs _ ts.
+ +
+Fixpoint freeVarF (A : fol.Formula L) : list nat :=
+  match A with
+  | equal t s ⇒ freeVarT t ++ freeVarT s
+  | atomic r ts ⇒ freeVarTs _ ts
+  | impH A B ⇒ freeVarF A ++ freeVarF B
+  | notH A ⇒ freeVarF A
+  | forallH v A ⇒ remove eq_nat_dec v (freeVarF A)
+  end.
+ +
+Definition ClosedSystem (T : fol.System L) :=
+  ∀ (v : nat) (f : fol.Formula L),
+    mem _ T f → ¬ In v (freeVarF f).
+ +
+Definition closed (a : fol.Formula L):=
+  ∀ v: nat, ¬ In v (freeVarF a).
+ +
+Fixpoint closeList (l: list nat)(a : fol.Formula L) :=
+ match l with
+   nil ⇒ a
+| cons v l ⇒ f[ ∀ v, {closeList l a} ]f
+end.
+ +
+Definition close (x : fol.Formula L) : fol.Formula L :=
+  closeList (nodup eq_nat_dec (freeVarF x)) x.
+ +
+Lemma freeVarClosedList1 :
+  ∀ (l : list nat) (v : nat) (x : fol.Formula L),
+    In v l → ¬ In v (freeVarF (closeList l x)).
+ +
+Lemma freeVarClosedList2 :
+  ∀ (l : list nat) (v : nat) (x : fol.Formula L),
+    In v (freeVarF (closeList l x)) →
+    In v (freeVarF x).
+ +
+Lemma freeVarClosed :
+  ∀ (x : fol.Formula L) (v : nat), ¬ In v (freeVarF (close x)).
+ +
+Fixpoint freeVarListFormula (l : fol.Formulas L) : list nat :=
+  match l with
+  | nil ⇒ nil (A:=nat)
+  | f :: l ⇒ freeVarF f ++ freeVarListFormula l
+  end.
+ +
+Lemma freeVarListFormulaApp :
+  ∀ a b : fol.Formulas L,
+    freeVarListFormula (a ++ b) =
+      freeVarListFormula a ++ freeVarListFormula b.
+ +
+Lemma In_freeVarListFormula :
+  ∀ (v : nat) (f : fol.Formula L) (F : fol.Formulas L),
+    In v (freeVarF f) → In f F → In v (freeVarListFormula F).
+ +
+Lemma In_freeVarListFormulaE :
+  ∀ (v : nat) (F : fol.Formulas L),
+    In v (freeVarListFormula F) →
+    ∃ f : fol.Formula L, In v (freeVarF f) ∧ In f F.
+ +
+Definition In_freeVarSys (v : nat) (T : fol.System L) :=
+  ∃ f : fol.Formula L, In v (freeVarF f) ∧ mem _ T f.
+ +
+Lemma notInFreeVarSys :
+  ∀ x, ¬ In_freeVarSys x (Ensembles.Empty_set (fol.Formula L)).
+ +
+End Free_Variables.
+ +
+Section Substitution.
+ +
+ +
+Fixpoint substT (s : fol.Term L) (x : nat)
+  (t : fol.Term L) {struct s} : fol.Term L :=
+  match s with
+  | var v ⇒
+      match eq_nat_dec x v with
+      | left _ ⇒ t
+      | right _ ⇒ var v
+      end
+  | apply f ts ⇒ apply f (substTs _ ts x t)
+  end
+with substTs (n : nat) (ss : fol.Terms L n)
+       (x : nat) (t : fol.Term L) {struct ss} : fol.Terms L n :=
+       match ss in (fol.Terms _ n0) return (fol.Terms L n0) with
+       | Tnil ⇒ Tnil
+       | Tcons m s ts ⇒
+           Tcons (substT s x t) (substTs m ts x t)
+       end.
+ +
+Lemma subTermVar1 :
+  ∀ (v : nat) (s : fol.Term L), substT (var v) v s = s.
+ +
+Lemma subTermVar2 :
+  ∀ (v x : nat) (s : fol.Term L),
+    v ≠ x → substT (var x) v s = var x.
+ +
+Lemma subTermApply :
+  ∀ (f : Functions L) (ts : fol.Terms L (arityF L f))
+         (v : nat) (s : fol.Term L),
+    substT (apply f ts) v s = apply f (substTs _ ts v s).
+ +
+Definition newVar (l : list nat) : nat :=
+  fold_right Nat.max 0 (map S l).
+ +
+Lemma newVar2 : ∀ (l : list nat) (n : nat), In n l → n < newVar l.
+ +
+Lemma newVar1 : ∀ l : list nat, ¬ In (newVar l) l.
+ +
+Definition substituteFormulaImp (f : fol.Formula L)
+  (frec : nat × fol.Term L → {y : fol.Formula L | depth L y = depth L f})
+  (g : fol.Formula L)
+  (grec : nat × fol.Term L → {y : fol.Formula L | depth L y = depth L g})
+  (p : nat × fol.Term L) :
+  {y : fol.Formula L | depth L y = depth L (impH f g)} :=
+  match frec p with
+  | exist f' prf1 ⇒
+      match grec p with
+      | exist g' prf2 ⇒
+          exist
+            (fun y : fol.Formula L ⇒
+               depth L y = S (Nat.max (depth L f) (depth L g)))
+            (impH f' g')
+            (eq_ind_r
+               (fun n : nat ⇒
+                  S (Nat.max n (depth L g')) =
+                    S (Nat.max (depth L f) (depth L g)))
+               (eq_ind_r
+                  (fun n : nat ⇒
+                     S (Nat.max (depth L f) n) =
+                       S (Nat.max (depth L f) (depth L g)))
+                  (refl_equal (S (Nat.max (depth L f) (depth L g))))
+                  prf2) prf1)
+      end
+  end.
+ +
+Remark substituteFormulaImpNice :
+  ∀ (f g : fol.Formula L)
+         (z1 z2 : nat × fol.Term L →
+                  {y : fol.Formula L | depth L y = depth L f}),
+    (∀ q : nat × fol.Term L, z1 q = z2 q) →
+    ∀
+      z3 z4 : nat × fol.Term L →
+              {y : fol.Formula L | depth L y = depth L g},
+      (∀ q : nat × fol.Term L, z3 q = z4 q) →
+      ∀ q : nat × fol.Term L,
+        substituteFormulaImp f z1 g z3 q =
+          substituteFormulaImp f z2 g z4 q.
+ +
+Definition substituteFormulaNot (f : fol.Formula L)
+  (frec : nat × fol.Term L →
+          {y : fol.Formula L | depth L y = depth L f})
+  (p : nat × fol.Term L) :
+  {y : fol.Formula L | depth L y = depth L (notH f)} :=
+  match frec p with
+  | exist f' prf1 ⇒
+      exist (fun y : fol.Formula L ⇒ depth L y = S (depth L f))
+        (notH f')
+        (eq_ind_r (fun n : nat ⇒ S n = S (depth L f))
+           (refl_equal (S (depth L f))) prf1)
+  end.
+ +
+Remark substituteFormulaNotNice :
+  ∀ (f : fol.Formula L)
+         (z1 z2 : nat × fol.Term L →
+                  {y : fol.Formula L | depth L y = depth L f}),
+    (∀ q : nat × fol.Term L, z1 q = z2 q) →
+    ∀ q : nat × fol.Term L,
+      substituteFormulaNot f z1 q = substituteFormulaNot f z2 q.
+ +
+Definition substituteFormulaForall (n : nat) (f : fol.Formula L)
+  (frec : ∀ b : fol.Formula L,
+      lt_depth b (forallH n f) →
+      nat × fol.Term L → {y : fol.Formula L | depth L y = depth L b})
+  (p : nat × fol.Term L) :
+  {y : fol.Formula L | depth L y = depth L (forallH n f)} :=
+  match p with
+  | (v, s) ⇒
+      match eq_nat_dec n v with
+      | left _ ⇒
+          exist (fun y : fol.Formula L ⇒ depth L y = S (depth L f))
+            (forallH n f) (refl_equal (depth L (forallH n f)))
+      | right _ ⇒
+          match In_dec eq_nat_dec n (freeVarT s) with
+          | left _ ⇒
+              let nv := newVar (v :: freeVarT s ++ freeVarF f) in
+              match frec f (depthForall L f n) (n, var nv) with
+              | exist f' prf1 ⇒
+                  match
+                    frec f'
+                      (eqDepth L f' f (forallH n f)
+                         (sym_eq prf1) (depthForall L f n)) p
+                  with
+                  | exist f'' prf2 ⇒
+                      exist
+                        (fun y : fol.Formula L ⇒ depth L y = S (depth L f))
+                        (forallH nv f'')
+                        (eq_ind_r (fun n : nat ⇒ S n = S (depth L f))
+                           (refl_equal (S (depth L f)))
+                           (trans_eq prf2 prf1))
+                  end
+              end
+          | right _ ⇒
+              match frec f (depthForall L f n) p with
+              | exist f' prf1 ⇒
+                  exist (fun y : fol.Formula L ⇒ depth L y = S (depth L f))
+                    (forallH n f')
+                    (eq_ind_r (fun n : nat ⇒ S n = S (depth L f))
+                       (refl_equal (S (depth L f))) prf1)
+              end
+          end
+      end
+  end.
+ +
+Remark substituteFormulaForallNice :
+  ∀ (v : nat) (a : fol.Formula L)
+         (z1 z2 : ∀ b : fol.Formula L,
+             lt_depth b (forallH v a) →
+             nat × fol.Term L → {y : fol.Formula L | depth L y = depth L b}),
+    (∀ (b : fol.Formula L) (q : lt_depth b (forallH v a))
+            (r : nat × fol.Term L), z1 b q r = z2 b q r) →
+    ∀ q : nat × fol.Term L,
+      substituteFormulaForall v a z1 q = substituteFormulaForall v a z2 q.
+Definition substituteFormulaHelp (f : fol.Formula L)
+  (v : nat) (s : fol.Term L) :
+  {y : fol.Formula L | depth L y = depth L f}.
+ +
+Definition substF (f : fol.Formula L) (v : nat) (s : fol.Term L) :
+  fol.Formula L := proj1_sig (substituteFormulaHelp f v s).
+ +
+Lemma subFormulaEqual :
+  ∀ (t1 t2 : fol.Term L) (v : nat) (s : fol.Term L),
+    substF (t1 = t2)%fol v s =
+      (substT t1 v s = substT t2 v s)%fol.
+ +
+Lemma subFormulaRelation :
+  ∀ (r : Relations L) (ts : fol.Terms L (arityR L r))
+         (v : nat) (s : fol.Term L),
+    substF (atomic r ts) v s =
+      atomic r (substTs (arityR L r) ts v s).
+ +
+Lemma subFormulaImp :
+  ∀ (f1 f2 : fol.Formula L) (v : nat) (s : fol.Term L),
+    substF (f1 → f2)%fol v s =
+      (substF f1 v s → substF f2 v s)%fol.
+ +
+Lemma subFormulaNot :
+  ∀ (f : fol.Formula L) (v : nat) (s : fol.Term L),
+    substF (¬ f)%fol v s = (¬ substF f v s)%fol.
+ +
+Lemma subFormulaForall :
+  ∀ (f : fol.Formula L) (x v : nat) (s : fol.Term L),
+    let nv := newVar (v :: freeVarT s ++ freeVarF f) in
+    substF (allH x, f)%fol v s =
+      match eq_nat_dec x v with
+      | left _ ⇒ forallH x f
+      | right _ ⇒
+          match In_dec eq_nat_dec x (freeVarT s) with
+          | right _ ⇒ (allH x, substF f v s)%fol
+          | left _ ⇒ (allH nv, substF (substF f x (v# nv) ) v s)%fol
+
+          end
+      end.
+ +
+Section Extensions.
+ +
+Lemma subFormulaOr :
+  ∀ (f1 f2 : fol.Formula L) (v : nat) (s : fol.Term L),
+    substF (f1 ∨ f2)%fol v s =
+      (substF f1 v s ∨ substF f2 v s)%fol.
+ +
+Lemma subFormulaAnd :
+  ∀ (f1 f2 : fol.Formula L) (v : nat) (s : fol.Term L),
+    substF (f1 ∧ f2)%fol v s =
+      (substF f1 v s ∧ substF f2 v s)%fol.
+ +
+Lemma subFormulaExist :
+  ∀ (f : fol.Formula L) (x v : nat) (s : fol.Term L),
+    let nv := newVar (v :: freeVarT s ++ freeVarF f) in
+    substF (existH x f) v s =
+      match eq_nat_dec x v with
+      | left _ ⇒ existH x f
+      | right _ ⇒
+          match In_dec eq_nat_dec x (freeVarT s) with
+          | right _ ⇒ existH x (substF f v s)
+          | left _ ⇒
+              existH nv (substF
+                           (substF f x (var nv)) v s)
+          end
+      end.
+ +
+Lemma subFormulaIff :
+  ∀ (f1 f2 : fol.Formula L) (v : nat) (s : fol.Term L),
+    substF (iffH f1 f2) v s =
+      iffH (substF f1 v s) (substF f2 v s).
+ +
+Lemma subFormulaIfThenElse :
+  ∀ (f1 f2 f3 : fol.Formula L) (v : nat) (s : fol.Term L),
+    substF (ifThenElseH f1 f2 f3) v s =
+      ifThenElseH (substF f1 v s) (substF f2 v s)
+        (substF f3 v s).
+ +
+End Extensions.
+ +
+Lemma subFormulaDepth :
+  ∀ (f : fol.Formula L) (v : nat) (s : fol.Term L),
+    depth L (substF f v s) = depth L f.
+ +
+Section Substitution_Properties.
+ +
+Lemma subTermId :
+  ∀ (t : fol.Term L) (v : nat), substT t v (var v) = t.
+ +
+Lemma subTermsId :
+  ∀ (n : nat) (ts : fol.Terms L n) (v : nat),
+    substTs n ts v (var v) = ts.
+ +
+Lemma subFormulaId :
+  ∀ (f : fol.Formula L) (v : nat), substF f v (var v) = f.
+ +
+Lemma subFormulaForall2 :
+  ∀ (f : fol.Formula L) (x v : nat) (s : fol.Term L),
+  ∃ nv : nat,
+    ¬ In nv (freeVarT s) ∧
+      nv ≠ v ∧
+      ¬ In nv (remove eq_nat_dec x (freeVarF f)) ∧
+      substF (forallH x f) v s =
+        match eq_nat_dec x v with
+        | left _ ⇒ forallH x f
+        | right _ ⇒
+            forallH nv (substF (substF f x (var nv)) v s)
+        end.
+ +
+Lemma subFormulaExist2 :
+  ∀ (f : fol.Formula L) (x v : nat) (s : fol.Term L),
+  ∃ nv : nat,
+    ¬ In nv (freeVarT s) ∧
+      nv ≠ v ∧
+      ¬ In nv (remove eq_nat_dec x (freeVarF f)) ∧
+      substF (existH x f) v s =
+        match eq_nat_dec x v with
+        | left _ ⇒ existH x f
+        | right _ ⇒
+            existH nv (substF (substF f x (var nv)) v s)
+        end.
+ +
+Lemma substExHC (A : Formula) (v x : nat)(t: Term):
+    v ≠ x → ¬ In v (freeVarT t) →
+       substF (existH v A) x t =
+       existH v (substF A x t).
+ +
+End Substitution_Properties.
+ +
+End Substitution.
+ +
+Definition Sentence (f:Formula) :=
+   (∀ v : nat, ¬ In v (freeVarF f)).
+ +
+End Fol_Properties.
+ +
+Arguments closed {L} _.
+ +
+#[global] Arguments substF {L} _ _.
+#[global] Arguments substT {L} _ _.
+#[global] Arguments substTs {L n} _ _ _ .
+
+ +
+compatibility with older names +
+
+ +
+#[deprecated(note="use substF")]
+ Notation substituteFormula := substF (only parsing).
+ +
+#[deprecated(note="use substT")]
+ Notation substituteTerm := substT (only parsing).
+ +
+#[deprecated(note="use substTs")]
+ Notation substituteTerms := substTs (only parsing).
+ +
+#[deprecated(note="use freeVarF")]
+ Notation freeVarFormula := freeVarF (only parsing).
+ +
+#[deprecated(note="use freeVarT")]
+ Notation freeVarTerm := freeVarT (only parsing).
+ +
+#[deprecated(note="use freeVarTs")]
+ Notation freeVarTerms := freeVarTs (only parsing).
+ +
+About substF.
+Search substF.
+ +
+
+ +
+to replace with a single recursive custom notation ? +
+
+ +
+#[global] Notation substF2 e v1 t1 v2 t2 :=
+  (substF (substF e v1 t1) v2 t2).
+ +
+#[global] Notation substF3 e v1 t1 v2 t2 v3 t3 :=
+  (substF (substF2 e v1 t1 v2 t2) v3 t3).
+ +
+#[global] Notation substF4 e v1 t1 v2 t2 v3 t3 v4 t4 :=
+  (substF (substF3 e v1 t1 v2 t2 v3 t3) v4 t4).
+ +
+#[global] Notation substF5 e v1 t1 v2 t2 v3 t3 v4 t4 v5 t5 :=
+  (substF (substF4 e v1 t1 v2 t2 v3 t3 v4 t4) v5 t5).
+ +
+#[global] Notation substF6 e v1 t1 v2 t2 v3 t3 v4 t4 v5 t5 v6 t6 :=
+  (substF (substF5 e v1 t1 v2 t2 v3 t3 v4 t4 v5 t5) v6 t6).
+ +
+#[global] Notation substF7 e v1 t1 v2 t2 v3 t3 v4 t4 v5 t5 v6 t6 v7 t7:=
+  (substF (substF6 e v1 t1 v2 t2 v3 t3 v4 t4 v5 t5 v6 t6) v7 t7).
+ +
+#[global] Notation
+  substF8 e v1 t1 v2 t2 v3 t3 v4 t4 v5 t5 v6 t6 v7 t7 v8 t8 :=
+  (substF2 (substF6 e v1 t1 v2 t2 v3 t3 v4 t4 v5 t5 v6 t6)
+     v7 t7 v8 t8).
+ +
+#[global] Notation
+  substF9 e v1 t1 v2 t2 v3 t3 v4 t4 v5 t5 v6 t6 v7 t7 v8 t8 v9 t9:=
+  (substF3 (substF6 e v1 t1 v2 t2 v3 t3 v4 t4 v5 t5 v6 t6)
+     v7 t7 v8 t8 v9 t9).
+ +
+#[global] Arguments freeVarF {L} _.
+#[global] Arguments freeVarT {L} _.
+#[global] Arguments freeVarTs {L n} _.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.folReplace.html b/theories/html/hydras.Ackermann.folReplace.html new file mode 100644 index 00000000..f050472a --- /dev/null +++ b/theories/html/hydras.Ackermann.folReplace.html @@ -0,0 +1,117 @@ + + + + + +hydras.Ackermann.folReplace + + + + +
+ + + +
+ +

Library hydras.Ackermann.folReplace

+ +
+From Coq Require Import Ensembles Lists.List Peano_dec.
+ +
+From hydras.Ackermann
+  Require Import ListExt folProof folLogic folProp.
+ +
+Section Replacement.
+ +
+Variable L : Language.
+Let Formula := Formula L.
+Let Formulas := Formulas L.
+Let System := System L.
+Let Term := Term L.
+Let Terms := Terms L.
+Let SysPrf := SysPrf L.
+ +
+Lemma reduceImp :
+  ∀ (f1 f2 f3 f4 : Formula) (T : System),
+    SysPrf T (iffH f1 f3) →
+    SysPrf T (iffH f2 f4) →
+    SysPrf T (iffH (impH f1 f2) (impH f3 f4)).
+ +
+Lemma reduceNot (f1 f2 : Formula) (T : System):
+ SysPrf T (iffH f1 f2) → SysPrf T (iffH (notH f1) (notH f2)).
+ +
+Lemma impForall (f1 f2 : Formula) (v : nat) (T : System):
+  ¬ In_freeVarSys _ v T →
+  SysPrf T (impH f1 f2) → SysPrf T (impH (forallH v f1) (forallH v f2)).
+ +
+Lemma reduceForall (f1 f2 : Formula) (v : nat) (T : System):
+ ¬ In_freeVarSys _ v T →
+ SysPrf T (iffH f1 f2) → SysPrf T (iffH (forallH v f1) (forallH v f2)).
+ +
+Lemma reduceOr (f1 f2 f3 f4 : Formula) (T : System):
+ SysPrf T (iffH f1 f3) →
+ SysPrf T (iffH f2 f4) → SysPrf T (iffH (orH f1 f2) (orH f3 f4)).
+ +
+Lemma reduceAnd (f1 f2 f3 f4 : Formula) (T : System):
+    SysPrf T (iffH f1 f3) →
+    SysPrf T (iffH f2 f4) → SysPrf T (iffH (andH f1 f2) (andH f3 f4)).
+ +
+Lemma iffExist (f1 f2 : Formula) (v : nat) (T : System):
+ ¬ In_freeVarSys _ v T →
+ SysPrf T (impH f1 f2) → SysPrf T (impH (existH v f1) (existH v f2)).
+ +
+Lemma reduceExist (f1 f2 : Formula) (v : nat) (T : System):
+  ¬ In_freeVarSys _ v T →
+  SysPrf T (iffH f1 f2) → SysPrf T (iffH (existH v f1) (existH v f2)).
+ +
+Lemma reduceIff (f1 f2 f3 f4 : Formula) (T : System):
+ SysPrf T (iffH f1 f3) →
+ SysPrf T (iffH f2 f4) → SysPrf T (iffH (iffH f1 f2) (iffH f3 f4)).
+ +
+Lemma reduceIfThenElse (f1 f2 f3 f4 f5 f6 : Formula) (T : System):
+ SysPrf T (iffH f1 f4) →
+ SysPrf T (iffH f2 f5) →
+ SysPrf T (iffH f3 f6) →
+ SysPrf T (iffH (ifThenElseH f1 f2 f3) (ifThenElseH f4 f5 f6)).
+ +
+Lemma reduceSub (T : System) (v : nat) (s : Term) (f g : Formula):
+ ¬ In_freeVarSys L v T →
+ SysPrf T (iffH f g) →
+ SysPrf T (iffH (substF f v s) (substF g v s)).
+ +
+Lemma reduceCloseList (l : list nat) (f1 f2 : Formula) (T : System):
+ (∀ v : nat, In v l → ¬ In_freeVarSys _ v T) →
+ SysPrf T (iffH f1 f2) →
+ SysPrf T (iffH (closeList L l f1) (closeList L l f2)).
+ +
+End Replacement.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.misc.html b/theories/html/hydras.Ackermann.misc.html new file mode 100644 index 00000000..7b3f2b23 --- /dev/null +++ b/theories/html/hydras.Ackermann.misc.html @@ -0,0 +1,38 @@ + + + + + +hydras.Ackermann.misc + + + + +
+ + + +
+ +

Library hydras.Ackermann.misc

+ +
+From Coq Require Export Eqdep_dec.
+ +
+#[global] Set Asymmetric Patterns.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.model.html b/theories/html/hydras.Ackermann.model.html new file mode 100644 index 00000000..b46a5c7a --- /dev/null +++ b/theories/html/hydras.Ackermann.model.html @@ -0,0 +1,223 @@ + + + + + +hydras.Ackermann.model + + + + +
+ + + +
+ +

Library hydras.Ackermann.model

+ +
+
+ +
+model.v : + +
+ + Original script by Russel O'Connor + +
+
+ +
+From Coq Require Import Ensembles List Vector Arith.
+ +
+From hydras.Ackermann Require Import ListExt folProof folProp.
+From Coq Require Import Peano_dec.
+From hydras.Ackermann Require Import misc.
+Import FolNotations.
+From hydras.Ackermann Require Import NewNotations.
+ +
+Section Model_Theory.
+ +
+Variable L : Language.
+ +
+Fixpoint naryFunc (A : Set) (n : nat) {struct n} : Set :=
+  match n with
+  | O ⇒ A
+  | S m ⇒ A → naryFunc A m
+  end.
+ +
+Fixpoint naryRel (A : Set) (n : nat) {struct n} : Type :=
+  match n with
+  | O ⇒ Prop
+  | S m ⇒ A → naryRel A m
+  end.
+ +
+Record Model : Type := model
+  {U : Set;
+   func : ∀ f : Functions L, naryFunc U (arityF L f);
+   rel : ∀ r : Relations L, naryRel U (arityR L r)}.
+ +
+Variable M : Model.
+ +
+Fixpoint interpTerm (value : nat → U M) (t : Term L) {struct t} :
+ U M :=
+  match t with
+  | var v ⇒ value v
+  | apply f ts ⇒ interpTerms _ (func M f) value ts
+  end
+ 
+ with interpTerms (m : nat) (f : naryFunc (U M) m)
+ (value : nat → U M) (ts : Terms L m) {struct ts} :
+ U M :=
+  match ts in (Terms _ n) return (naryFunc (U M) n → U M) with
+  | Tnil ⇒ fun f ⇒ f
+  | Tcons m t ts ⇒ fun f ⇒ interpTerms m (f (interpTerm value t)) value ts
+  end f.
+ +
+Fixpoint interpRels (m : nat) (r : naryRel (U M) m)
+ (value : nat → U M) (ts : Terms L m) {struct ts} : Prop :=
+  match ts in (Terms _ n) return (naryRel (U M) n → Prop) with
+  | Tnil ⇒ fun r ⇒ r
+  | Tcons m t ts ⇒ fun r ⇒ interpRels m (r (interpTerm value t)) value ts
+  end r.
+ +
+Definition updateValue (value : nat → U M) (n : nat)
+  (v : U M) (x : nat) : U M :=
+  match eq_nat_dec n x with
+  | left _ ⇒ v
+  | right _ ⇒ value x
+  end.
+ +
+Fixpoint interpFormula (value : nat → U M) (f : Formula L) {struct f} :
+ Prop :=
+  match f with
+  | equal t s ⇒ interpTerm value t = interpTerm value s
+  | atomic r ts ⇒ interpRels _ (rel M r) value ts
+  | impH A B ⇒ interpFormula value A → interpFormula value B
+  | notH A ⇒ interpFormula value A → False
+  | forallH v A ⇒ ∀ x : U M, interpFormula (updateValue value v x) A
+  end.
+ +
+Lemma freeVarInterpTerm (v1 v2 : nat → U M) (t : Term L):
+ (∀ x : nat, List.In x (freeVarT t) → v1 x = v2 x) →
+ interpTerm v1 t = interpTerm v2 t.
+ +
+Lemma freeVarInterpRel (v1 v2 : nat → U M) (n : nat)
+  (ts : Terms L n) (r : naryRel (U M) n):
+  (∀ x : nat, List.In x (freeVarTs ts) → v1 x = v2 x) →
+  interpRels n r v1 ts → interpRels n r v2 ts.
+ +
+Lemma freeVarInterpFormula (v1 v2 : nat → U M) (g : Formula L):
+ (∀ x : nat, List.In x (freeVarF g) → v1 x = v2 x) →
+ interpFormula v1 g → interpFormula v2 g.
+ +
+Lemma subInterpTerm (value : nat → U M) (t : Term L) (v : nat) (s : Term L):
+ interpTerm (updateValue value v (interpTerm value s)) t =
+ interpTerm value (substT t v s).
+ +
+Lemma subInterpRel (value : nat → U M) (n : nat) (ts : Terms L n)
+  (v : nat) (s : Term L) (r : naryRel (U M) n):
+  interpRels n r (updateValue value v (interpTerm value s)) ts ↔
+    interpRels n r value (substTs ts v s).
+ +
+Lemma subInterpFormula :
+ ∀ (value : nat → U M) (f : Formula L) (v : nat) (s : Term L),
+ interpFormula (updateValue value v (interpTerm value s)) f ↔
+ interpFormula value (substF f v s).
+ +
+Lemma subInterpFormula1 (value : nat → U M) (f : Formula L) (v : nat) (s : Term L):
+ interpFormula (updateValue value v (interpTerm value s)) f →
+ interpFormula value (substF f v s).
+ +
+Lemma subInterpFormula2 (value : nat → U M) (f : Formula L) (v : nat) (s : Term L):
+  interpFormula value (substF f v s) →
+  interpFormula (updateValue value v (interpTerm value s)) f.
+ +
+Fixpoint nnHelp (f : Formula L) : Formula L :=
+  match f with
+  | equal t s ⇒ equal t s
+  | atomic r ts ⇒ atomic r ts
+  | impH A B ⇒ impH (nnHelp A) (nnHelp B)
+  | notH A ⇒ notH (nnHelp A)
+  | forallH v A ⇒ (allH v, ¬ ¬ nnHelp A)%fol
+  end.
+ +
+Definition nnTranslate (f : Formula L) : Formula L :=
+  notH (notH (nnHelp f)).
+ +
+Lemma freeVarNNHelp (f : Formula L): freeVarF f = freeVarF (nnHelp f).
+ +
+Lemma subNNHelp :
+ ∀ (f : Formula L) (v : nat) (s : Term L),
+ substF (nnHelp f) v s = nnHelp (substF f v s).
+ +
+Section Consistent_Theory.
+ +
+  Variable T : System L.
+ +
+  Fixpoint interpTermsVector (value : nat → U M) (n : nat)
+    (ts : Terms L n) {struct ts} : Vector.t (U M) n :=
+    match ts in (Terms _ n) return (Vector.t (U M) n) with
+    | Tnil ⇒ Vector.nil (U M)
+    | Tcons m t ts ⇒
+        Vector.cons (U M) (interpTerm value t) m (interpTermsVector value m ts)
+    end.
+ +
+Lemma preserveValue (value : nat → U M):
+ (∀ f : Formula L,
+  mem _ T f → interpFormula value (nnTranslate f)) →
+ ∀ g : Formula L, SysPrf L T g → interpFormula value (nnTranslate g).
+ +
+Lemma ModelConsistent (value : nat → U M):
+  (∀ f : Formula L,
+      mem _ T f → interpFormula value (nnTranslate f)) →
+  Consistent L T.
+ +
+End Consistent_Theory.
+ +
+End Model_Theory.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.prLogic.html b/theories/html/hydras.Ackermann.prLogic.html new file mode 100644 index 00000000..363c0f05 --- /dev/null +++ b/theories/html/hydras.Ackermann.prLogic.html @@ -0,0 +1,60 @@ + + + + + +hydras.Ackermann.prLogic + + + + +
+ + + +
+ +

Library hydras.Ackermann.prLogic

+ +
+
+ +
+prLogic.v + +
+ + Original script by Russel O'Connor + +
+ + Although this module doesn't depend on FOL, the following lemmas + are JUST helpers to prove that the encoding of basic FOL connectives are PR +
+
+ +
+From hydras.Ackermann Require Import primRec code cPair.
+From Coq Require Import Arith.
+ +
+Lemma codeForallIsPR : isPR 2 (fun a b : nat ⇒ cPair 3 (cPair a b)).
+ +
+#[export] Instance codeNotIsPR : isPR 1 codeNot.
+ +
+#[export] Instance codeImpIsPR : isPR 2 codeImp.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.primRec.html b/theories/html/hydras.Ackermann.primRec.html new file mode 100644 index 00000000..62b663ea --- /dev/null +++ b/theories/html/hydras.Ackermann.primRec.html @@ -0,0 +1,938 @@ + + + + + +hydras.Ackermann.primRec + + + + +
+ + + +
+ +

Library hydras.Ackermann.primRec

+ +
+
+ +
+Primitive Recursive functions Russel O'Connor +
+
+ +
+From Coq Require Import Arith Peano_dec Compare_dec List Eqdep_dec Utf8.
+From hydras Require Import extEqualNat misc Compat815.
+From Coq Require Vector Bool EqNat.
+From Coq Require Import Lia.
+ +
+
+ +
+

Definitions

+ +
+ + PrimRec n : data type of primitive recursive functions of arity n + PrimRec n m : m-tuples of PrimRec n +
+
+ +
+ +
+Inductive PrimRec : nat → Set :=
+  | succFunc : PrimRec 1
+  | zeroFunc : PrimRec 0
+  | projFunc : ∀ n m : nat, m < n → PrimRec n
+  | composeFunc :
+      ∀ (n m : nat) (g : PrimRecs n m) (h : PrimRec m),
+        PrimRec n
+  | primRecFunc :
+      ∀ (n : nat) (g : PrimRec n) (h : PrimRec (S (S n))),
+        PrimRec (S n)
+with PrimRecs : nat → nat → Set :=
+  | PRnil : ∀ n : nat, PrimRecs n 0
+  | PRcons : ∀ n m : nat, PrimRec n → PrimRecs n m →
+                               PrimRecs n (S m).
+ +
+ +
+  Notation "f '=x=' g" := (extEqual _ f g) (at level 70, no associativity).
+ +
+ +
+  Module PRNotations.
+    Declare Scope pr_scope.
+    Delimit Scope pr_scope with pr.
+    Notation "h :: t" := (PRcons _ _ h t) (at level 60, right associativity)
+        : pr_scope.
+    Notation "[ x ]" := (PRcons _ _ x (PRnil _)) : pr_scope.
+ +
+    Notation "[ x ; y ; .. ; z ]" :=
+      (PRcons _ _ x (PRcons _ _ y .. (PRcons _ _ z (PRnil _)) ..)) : pr_scope.
+ +
+    Notation PRcomp f v := (composeFunc _ _ v f).
+ +
+    Notation PRrec f0 fS := (primRecFunc _ f0 fS).
+ +
+
+ +
+Popular projections +
+
+    Notation pi1_1 := (projFunc 1 0 (le_n 1)).
+ +
+    Notation pi1_2 := (projFunc 2 1 (le_n 2)).
+    Notation pi2_2 := (projFunc 2 0 (le_S 1 1 (le_n 1))).
+ +
+    Notation pi1_3 := (projFunc 3 2 (le_n 3)).
+    Notation pi2_3 := (projFunc 3 1 (le_S 2 2 (le_n 2))).
+    Notation pi3_3 := (projFunc 3 0 (le_S 1 2 (le_S 1 1 (le_n 1)))).
+ +
+  End PRNotations.
+ +
+ +
+Import PRNotations.
+ +
+Scheme PrimRec_PrimRecs_rec := Induction for PrimRec Sort Set
+  with PrimRecs_PrimRec_rec := Induction for PrimRecs Sort Set.
+ +
+Arguments PrimRec_PrimRecs_rec P P0 : rename.
+Arguments PrimRecs_PrimRec_rec P P0 : rename.
+ +
+ +
+Scheme PrimRec_PrimRecs_ind := Induction for PrimRec Sort Prop
+    with PrimRecs_PrimRec_ind := Induction for PrimRecs Sort Prop.
+ +
+Arguments PrimRec_PrimRecs_ind P P0 : rename.
+Arguments PrimRecs_PrimRec_ind P P0 : rename.
+ +
+Check PrimRec_PrimRecs_ind.
+ +
+
+ +
+

Semantics

+ +
+ +

Constants

+ +
+
+ +
+Fixpoint evalConstFunc (n m : nat) {struct n} : naryFunc n :=
+  match n return (naryFunc n) with
+  | O ⇒ m
+  | S n' ⇒ fun _ ⇒ evalConstFunc n' m
+  end.
+ +
+
+ +
+

Projections

+ + The parameters are numbered in opposite order. + So proj(2,0)(a,b) = b. +
+
+ +
+Fixpoint evalProjFunc (n : nat) :
+  ∀ m : nat, m < n → naryFunc n :=
+  match n return (∀ m : nat, m < n → naryFunc n) with
+  | O ⇒ fun (m : nat) (l : m < 0) ⇒ False_rec _ (Nat.nlt_0_r _ l)
+  | S n' ⇒
+      fun (m : nat) (l : m < S n') ⇒
+      match eq_nat_dec m n' with
+      | left _ ⇒ fun a : nat ⇒ evalConstFunc _ a
+      | right l1 ⇒
+          fun _ ⇒
+          evalProjFunc n' m
+            match Compat815.le_lt_or_eq _ _
+                    (Compat815.lt_n_Sm_le _ _ l) with
+            | or_introl l2 ⇒ l2
+            | or_intror l2 ⇒ False_ind _ (l1 l2)
+            end
+      end
+  end.
+ +
+
+ +
+Irrelevance of the proof that m < n +
+
+ +
+Lemma evalProjFuncInd :
+ ∀ (n m : nat) (p1 p2 : m < n),
+ evalProjFunc n m p1 = evalProjFunc n m p2.
+ +
+
+ +
+ Applies an m-ary function to the vector l +
+
+ +
+ +
+Fixpoint evalList (m : nat) (l : Vector.t nat m) {struct l} :
+  naryFunc m → nat :=
+  match l in (Vector.t _ m) return (naryFunc m → nat) with
+  | Vector.nil ⇒ fun x : naryFunc 0 ⇒ x
+  | Vector.cons a n l' ⇒
+    fun x : naryFunc (S n) ⇒ evalList n l' (x a)
+  end.
+ +
+ +
+Fixpoint evalOneParamList (n m a : nat) (l : Vector.t (naryFunc (S n)) m)
+ {struct l} : Vector.t (naryFunc n) m :=
+  match l in (Vector.t _ m) return (Vector.t (naryFunc n) m) with
+  | Vector.nil ⇒ Vector.nil (naryFunc n)
+  | Vector.cons f m' l' ⇒
+      Vector.cons _ (f a) m' (evalOneParamList n m' a l')
+  end.
+ +
+
+ +
+

Function composition

+ +
+
+ +
+Fixpoint evalComposeFunc (n : nat) :
+  ∀ m : nat,
+    Vector.t (naryFunc n) m → naryFunc m → naryFunc n :=
+  match
+    n
+    return
+    (∀ m : nat,
+        Vector.t (naryFunc n) m → naryFunc m → naryFunc n)
+  with
+  | O ⇒ evalList
+  | S n' ⇒
+      fun (m : nat) (l : Vector.t (naryFunc (S n')) m)
+          (f : naryFunc m) (a : nat) ⇒
+        evalComposeFunc n' m (evalOneParamList _ _ a l) f
+  end.
+ +
+Fixpoint compose2 (n : nat) :
+  naryFunc n → naryFunc (S n) → naryFunc n :=
+  match n return (naryFunc n → naryFunc (S n) → naryFunc n) with
+  | O ⇒ fun (a : nat) (g : nat → nat) ⇒ g a
+  | S n' ⇒
+      fun (f : naryFunc (S n')) (g : naryFunc (S (S n'))) (a : nat) ⇒
+        compose2 n' (f a) (fun x : nat ⇒ g x a)
+  end.
+ +
+
+ +
+

Primitive recursion

+ +
+
+ +
+Fixpoint evalPrimRecFunc (n : nat) (g : naryFunc n)
+ (h : naryFunc (S (S n))) (a : nat) {struct a} : naryFunc n :=
+  match a with
+  | O ⇒ g
+  | S a' ⇒ compose2 _ (evalPrimRecFunc n g h a') (h a')
+  end.
+ +
+
+ +
+ The interpretation function +
+
+ +
+ +
+Fixpoint evalPrimRec (n : nat) (f : PrimRec n) {struct f} :
+ naryFunc n :=
+  match f in (PrimRec n) return (naryFunc n) with
+  | succFunc ⇒ S
+  | zeroFunc ⇒ 0
+  | projFunc n m pf ⇒ evalProjFunc n m pf
+  | composeFunc n m l f ⇒
+      evalComposeFunc n m (evalPrimRecs _ _ l) (evalPrimRec _ f)
+  | primRecFunc n g h ⇒
+      evalPrimRecFunc n (evalPrimRec _ g) (evalPrimRec _ h)
+  end
+ 
+ with evalPrimRecs (n m : nat) (fs : PrimRecs n m) {struct fs} :
+ Vector.t (naryFunc n) m :=
+  match fs in (PrimRecs n m) return (Vector.t (naryFunc n) m) with
+  | PRnil a ⇒ Vector.nil (naryFunc a)
+  | PRcons a b g gs ⇒
+    Vector.cons _ (evalPrimRec _ g) _ (evalPrimRecs _ _ gs)
+  end.
+ +
+Notation PReval f := (evalPrimRec _ f).
+Notation PRevalN fs := (evalPrimRecs _ _ fs).
+ +
+ +
+Definition extEqualVectorGeneral (n m : nat) (l : Vector.t (naryFunc n) m) :
+  ∀ (m' : nat) (l' : Vector.t (naryFunc n) m'), Prop.
+Defined.
+ +
+
+ +
+Every element of l is extensionally equal to the element of l' at the same position +
+
+ +
+Definition extEqualVector n:
+  ∀ m (l l' : Vector.t (naryFunc n) m), Prop.
+ +
+Lemma extEqualVectorRefl (n m: nat):
+  ∀ (l : Vector.t (naryFunc n) m), extEqualVector n m l l.
+ +
+Lemma extEqualOneParamList :
+  ∀ (n m : nat) (l1 l2 : Vector.t (naryFunc (S n)) m) (c : nat),
+    extEqualVector (S n) m l1 l2 →
+    extEqualVector n m (evalOneParamList n m c l1)
+      (evalOneParamList n m c l2).
+ +
+Lemma extEqualCompose :
+  ∀ (n m : nat) (l1 l2 : Vector.t (naryFunc n) m)
+         (f1 f2 : naryFunc m),
+    extEqualVector n m l1 l2 →
+    f1 =x= f2 →
+    evalComposeFunc n m l1 f1 =x= evalComposeFunc n m l2 f2.
+ +
+Lemma extEqualCompose2 :
+  ∀ (n : nat) (f1 f2 : naryFunc n),
+     f1 =x= f2 →
+    ∀ g1 g2 : naryFunc (S n),
+       g1 =x= g2 →
+       compose2 n f1 g1 =x= compose2 n f2 g2.
+ +
+Lemma extEqualPrimRec :
+  ∀ (n : nat) (g1 g2 : naryFunc n) (h1 h2 : naryFunc (S (S n))),
+     g1 =x= g2 → h1 =x= h2 →
+       (evalPrimRecFunc n g1 h1: naryFunc (S n)) =x=
+         evalPrimRecFunc n g2 h2.
+ +
+
+ +
+

Predicates "to be primitive recursive"

+ +
+
+ +
+Class isPR (n : nat) (f : naryFunc n) : Set :=
+  is_pr : {p : PrimRec n | extEqual n (PReval p) f}.
+ +
+Definition fun2PR {n:nat}(f: naryFunc n)
+  {p: isPR _ f}: PrimRec n := proj1_sig p.
+ +
+Class isPRrel (n : nat) (R : naryRel n) : Set :=
+  is_pr_rel: isPR n (charFunction n R).
+ +
+#[export] Instance succIsPR : isPR 1 S.
+ +
+#[export] Instance const0_NIsPR (n:nat): isPR 0 n.
+ +
+#[export] Instance const1_NIsPR n: isPR 1 (fun _ ⇒ n).
+ +
+
+ +
+

Usual projections (in curried form) are primitive recursive

+ +
+
+ +
+#[export] Instance idIsPR : isPR 1 (fun x : nat ⇒ x).
+ +
+#[export] Instance pi1_2IsPR : isPR 2 (fun a b : nat ⇒ a).
+ +
+#[export] Instance pi2_2IsPR : isPR 2 (fun a b : nat ⇒ b).
+ +
+#[export] Instance pi1_3IsPR : isPR 3 (fun a b c : nat ⇒ a).
+ +
+#[export] Instance pi2_3IsPR : isPR 3 (fun a b c : nat ⇒ b).
+ +
+#[export] Instance pi3_3IsPR : isPR 3 (fun a b c : nat ⇒ c).
+ +
+#[export] Instance pi1_4IsPR : isPR 4 (fun a b c d : nat ⇒ a).
+ +
+#[export] Instance pi2_4IsPR : isPR 4 (fun a b c d : nat ⇒ b).
+ +
+#[export] Instance pi3_4IsPR : isPR 4 (fun a b c d : nat ⇒ c).
+ +
+#[export] Instance pi4_4IsPR : isPR 4 (fun a b c d : nat ⇒ d).
+ +
+
+ +
+

Composition lemmas

+ +
+
+ +
+#[export] Instance filter01IsPR (g : nat → nat) (H : isPR 1 g):
+  isPR 2 (fun a b : nat ⇒ g b).
+ +
+#[export] Instance filter10IsPR (g : nat → nat) (H: isPR 1 g):
+  isPR 2 (fun a b : nat ⇒ g a).
+ +
+#[export] Instance filter100IsPR(g : nat → nat)(H: isPR 1 g) :
+  isPR 3 (fun a b c : nat ⇒ g a).
+ +
+#[export] Instance filter010IsPR (g : nat → nat)(H: isPR 1 g):
+  isPR 3 (fun a b c : nat ⇒ g b).
+ +
+#[export] Instance filter001IsPR (g:nat → nat)(H: isPR 1 g) :
+ isPR 3 (fun a b c : nat ⇒ g c).
+ +
+#[export] Instance filter011IsPR (g: nat → nat → nat)(H: isPR 2 g):
+   isPR 3 (fun a b c : nat ⇒ g b c).
+ +
+#[export] Instance filter110IsPR(g : nat → nat → nat) :
+  isPR 2 g → isPR 3 (fun a b c : nat ⇒ g a b).
+ +
+#[export] Instance filter101IsPR :
+  ∀ g : nat → nat → nat, isPR 2 g →
+                                isPR 3 (fun a b c : nat ⇒ g a c).
+ +
+#[export] Instance filter0011IsPR (g : nat → nat → nat) :
+  isPR 2 g → isPR 4 (fun a b c d : nat ⇒ g c d).
+ +
+#[export] Instance filter1000IsPR (g : nat → nat):
+  isPR 1 g → isPR 4 (fun a b c d : nat ⇒ g a).
+ +
+#[export] Instance filter1011IsPR (g : nat → nat → nat → nat) :
+ isPR 3 g → isPR 4 (fun a b c d : nat ⇒ g a c d).
+ +
+#[export] Instance filter1100IsPR (g : nat → nat → nat) :
+  isPR 2 g → isPR 4 (fun a b c d : nat ⇒ g a b).
+ +
+#[export] Instance compose1_1IsPR (f : nat → nat):
+  isPR 1 f →
+  ∀ g : nat → nat, isPR 1 g → isPR 1 (fun x : nat ⇒ g (f x)).
+ +
+#[export] Instance compose1_2IsPR :
+  ∀ f : nat → nat,
+    isPR 1 f →
+    ∀ f' : nat → nat,
+      isPR 1 f' →
+      ∀ g : nat → nat → nat,
+        isPR 2 g → isPR 1 (fun x : nat ⇒ g (f x) (f' x)).
+ +
+#[export] Instance compose1_3IsPR :
+ ∀ f1 : nat → nat,
+ isPR 1 f1 →
+ ∀ f2 : nat → nat,
+ isPR 1 f2 →
+ ∀ f3 : nat → nat,
+ isPR 1 f3 →
+ ∀ g : nat → nat → nat → nat,
+ isPR 3 g → isPR 1 (fun x : nat ⇒ g (f1 x) (f2 x) (f3 x)).
+ +
+#[export] Instance compose2_1IsPR :
+  ∀ f : nat → nat → nat,
+    isPR 2 f →
+    ∀ g : nat → nat, isPR 1 g →
+                           isPR 2 (fun x y : nat ⇒ g (f x y)).
+ +
+#[export] Instance compose2_2IsPR :
+ ∀ f : nat → nat → nat,
+ isPR 2 f →
+ ∀ g : nat → nat → nat,
+ isPR 2 g →
+ ∀ h : nat → nat → nat,
+ isPR 2 h → isPR 2 (fun x y : nat ⇒ h (f x y) (g x y)).
+ +
+#[export] Instance compose2_3IsPR :
+  ∀ f1 : nat → nat → nat,
+    isPR 2 f1 →
+    ∀ f2 : nat → nat → nat,
+      isPR 2 f2 →
+      ∀ f3 : nat → nat → nat,
+        isPR 2 f3 →
+        ∀ g : nat → nat → nat → nat,
+          isPR 3 g →
+          isPR 2 (fun x y : nat ⇒ g (f1 x y) (f2 x y) (f3 x y)).
+ +
+#[export] Instance compose2_4IsPR :
+ ∀ f1 : nat → nat → nat,
+ isPR 2 f1 →
+ ∀ f2 : nat → nat → nat,
+ isPR 2 f2 →
+ ∀ f3 : nat → nat → nat,
+ isPR 2 f3 →
+ ∀ f4 : nat → nat → nat,
+ isPR 2 f4 →
+ ∀ g : nat → nat → nat → nat → nat,
+ isPR 4 g →
+ isPR 2 (fun x y : nat ⇒ g (f1 x y) (f2 x y) (f3 x y) (f4 x y)).
+ +
+#[export] Instance compose3_1IsPR :
+  ∀ f : nat → nat → nat → nat,
+    isPR 3 f →
+    ∀ g : nat → nat, isPR 1 g →
+                           isPR 3 (fun x y z : nat ⇒ g (f x y z)).
+ +
+#[export] Instance compose3_2IsPR :
+  ∀ f1 : nat → nat → nat → nat,
+    isPR 3 f1 →
+    ∀ f2 : nat → nat → nat → nat,
+      isPR 3 f2 →
+      ∀ g : nat → nat → nat,
+        isPR 2 g → isPR 3 (fun x y z : nat ⇒ g (f1 x y z) (f2 x y z)).
+ +
+#[export] Instance compose3_3IsPR :
+ ∀ f1 : nat → nat → nat → nat,
+ isPR 3 f1 →
+ ∀ f2 : nat → nat → nat → nat,
+ isPR 3 f2 →
+ ∀ f3 : nat → nat → nat → nat,
+ isPR 3 f3 →
+ ∀ g : nat → nat → nat → nat,
+ isPR 3 g → isPR 3 (fun x y z : nat ⇒ g (f1 x y z) (f2 x y z) (f3 x y z)).
+ +
+#[export] Instance compose4_2IsPR :
+ ∀ f1 : nat → nat → nat → nat → nat,
+ isPR 4 f1 →
+ ∀ f2 : nat → nat → nat → nat → nat,
+ isPR 4 f2 →
+ ∀ g : nat → nat → nat,
+ isPR 2 g → isPR 4 (fun w x y z : nat ⇒ g (f1 w x y z) (f2 w x y z)).
+ +
+#[export] Instance compose4_3IsPR :
+ ∀ f1 : nat → nat → nat → nat → nat,
+ isPR 4 f1 →
+ ∀ f2 : nat → nat → nat → nat → nat,
+ isPR 4 f2 →
+ ∀ f3 : nat → nat → nat → nat → nat,
+ isPR 4 f3 →
+ ∀ g : nat → nat → nat → nat,
+ isPR 3 g →
+ isPR 4 (fun w x y z : nat ⇒ g (f1 w x y z) (f2 w x y z) (f3 w x y z)).
+ +
+#[export] Instance swapIsPR :
+  ∀ f : nat → nat → nat, isPR 2 f → isPR 2 (fun x y : nat ⇒ f y x).
+ +
+#[export] Instance indIsPR :
+  ∀ f : nat → nat → nat,
+    isPR 2 f →
+    ∀ g : nat,
+      isPR 1
+           (fun a : nat ⇒ nat_rec (fun _ : nat ⇒ nat) g
+                             (fun x y : nat ⇒ f x y) a).
+ +
+#[export] Instance ind1ParamIsPR :
+  ∀ f : nat → nat → nat → nat,
+    isPR 3 f →
+    ∀ g : nat → nat,
+      isPR 1 g →
+      isPR 2
+           (fun a b : nat ⇒
+              nat_rec (fun n : nat ⇒ nat) (g b) (fun x y : nat ⇒ f x y b) a).
+ +
+#[export] Instance ind2ParamIsPR :
+  ∀ f : nat → nat → nat → nat → nat,
+    isPR 4 f →
+    ∀ g : nat → nat → nat,
+      isPR 2 g →
+      isPR 3
+        (fun a b c : nat ⇒
+           nat_rec (fun n : nat ⇒ nat) (g b c) (fun x y : nat ⇒ f x y b c) a).
+ +
+#[export] Instance plusIndIsPR : isPR 3 (fun n fn b : nat ⇒ S fn).
+ +
+#[export] Instance plusIsPR : isPR 2 plus.
+ +
+#[export] Instance multIndIsPR : isPR 3 (fun n fn b : nat ⇒ fn + b).
+ +
+#[export] Instance multIsPR : isPR 2 mult.
+ +
+#[export] Instance predIsPR : isPR 1 pred.
+ +
+#[export] Instance minusIndIsPR : isPR 3 (fun n fn b : nat ⇒ pred fn).
+ +
+#[export] Instance minusIsPR : isPR 2 minus.
+ +
+Definition notZero (a : nat) :=
+  nat_rec (fun n : nat ⇒ nat) 0 (fun x y : nat ⇒ 1) a.
+ +
+#[export] Instance notZeroIsPR : isPR 1 notZero.
+ +
+Definition ltBool (a b : nat) : bool :=
+  if le_lt_dec b a then false else true.
+ +
+Lemma ltBoolTrue : ∀ a b : nat, ltBool a b = true → a < b.
+ +
+Lemma ltBoolFalse : ∀ a b : nat, ltBool a b = false → ¬ a < b.
+ +
+#[export] Instance ltIsPR : isPRrel 2 ltBool.
+ +
+#[export] Instance maxIsPR : isPR 2 max.
+ +
+#[export] Instance gtIsPR : isPRrel 2 (fun a b : nat ⇒ ltBool b a).
+ +
+Remark replaceCompose2 :
+ ∀ (n : nat) (a b a' b' : naryFunc n) (c c' : naryFunc 2),
+ extEqual n a a' →
+ extEqual n b b' →
+ extEqual 2 c c' →
+ extEqual
+   n
+   (evalComposeFunc _ _
+                    (Vector.cons _ a _
+                                 (Vector.cons _ b _
+                                              (Vector.nil (naryFunc n)))) c)
+   (evalComposeFunc _ _
+                    (Vector.cons _ a' _
+                                 (Vector.cons _ b' _
+                                              (Vector.nil (naryFunc n)))) c').
+ +
+Definition orRel (n : nat) (R S : naryRel n) : naryRel n.
+ +
+Lemma orRelPR :
+  ∀ (n : nat) (R R' : naryRel n),
+    isPRrel n R → isPRrel n R' → isPRrel n (orRel n R R').
+ +
+Definition andRel (n : nat) (R S : naryRel n) : naryRel n.
+ +
+Lemma andRelPR :
+  ∀ (n : nat) (R R' : naryRel n),
+    isPRrel n R → isPRrel n R' → isPRrel n (andRel n R R').
+ +
+Definition notRel (n : nat) (R : naryRel n) : naryRel n.
+ +
+Lemma notRelPR (n : nat) (R : naryRel n):
+  isPRrel n R → isPRrel n (notRel n R).
+ +
+Fixpoint bodd (n : nat) : bool :=
+  match n with
+  | O ⇒ false
+  | S n' ⇒ negb (bodd n')
+  end.
+ +
+#[export] Instance boddIsPR : isPRrel 1 bodd.
+ +
+Lemma nat_eqb_false (a b: nat) : a ≠ b → Nat.eqb a b = false.
+ +
+#[local] Instance neqIsPR : isPRrel 2 (fun a b : nat ⇒ negb (Nat.eqb a b)).
+ +
+#[export] Instance eqIsPR : isPRrel 2 Nat.eqb.
+ +
+Definition leBool (a b : nat) : bool :=
+  if le_lt_dec a b then true else false.
+ +
+#[export] Instance leIsPR : isPRrel 2 leBool.
+ +
+Section Ignore_Params.
+ +
+  Fixpoint ignoreParams (n m : nat) (f : naryFunc n) {struct m} :
+    naryFunc (m + n) :=
+    match m return (naryFunc (m + n)) with
+    | O ⇒ f
+    | S x ⇒ fun _ ⇒ ignoreParams n x f
+    end.
+ +
+  Definition projectionListPR (n m : nat) (p : m ≤ n) : PrimRecs n m.
+ +
+  Definition projectionList (n m : nat) (p : m ≤ n) :
+    Vector.t (naryFunc n) m := evalPrimRecs n m (projectionListPR n m p).
+ +
+  Lemma projectionListInd :
+    ∀ (n m : nat) (p1 p2 : m ≤ n),
+      projectionList n m p1 = projectionList n m p2.
+ +
+  Lemma projectionListApplyParam :
+    ∀ (m n c : nat) (p1 : m ≤ n) (p2 : m ≤ S n),
+      extEqualVector _ _ (projectionList n m p1)
+                     (evalOneParamList n m c (projectionList (S n) m p2)).
+ +
+  Lemma projectionListId :
+    ∀ (n : nat) (f : naryFunc n) (p : n ≤ n),
+      extEqual n f (evalComposeFunc n n (projectionList n n p) f).
+ +
+  #[export] Instance ignoreParamsIsPR :
+    ∀ (n m : nat) (f : naryFunc n),
+      isPR _ f → isPR _ (ignoreParams n m f).
+ +
+End Ignore_Params.
+ +
+Lemma reduce1stCompose :
+  ∀ (c n m : nat) (v : Vector.t (naryFunc n) m) (g : naryFunc (S m)),
+    extEqual n
+      (evalComposeFunc n _ (Vector.cons (naryFunc n)
+                              (evalConstFunc n c) _ v) g)
+      (evalComposeFunc n _ v (g c)).
+ +
+Lemma reduce2ndCompose :
+  ∀ (c n m : nat) (v : Vector.t (naryFunc n) m) (n0 : naryFunc n)
+         (g : naryFunc (S (S m))),
+    extEqual n
+      (evalComposeFunc n _
+         (Vector.cons
+            (naryFunc n) n0 _
+            (Vector.cons (naryFunc n)
+               (evalConstFunc n c) _ v))
+         g)
+      (evalComposeFunc n _ (Vector.cons (naryFunc n) n0 _ v)
+         (fun x : nat ⇒ g x c)).
+ +
+Lemma evalPrimRecParam :
+  ∀ (n c d : nat) (g : naryFunc (S n)) (h : naryFunc (S (S (S n)))),
+     evalPrimRecFunc n (g d) (fun x y : nat ⇒ h x y d) c =x=
+      evalPrimRecFunc (S n) g h c d.
+ +
+#[export] Instance compose2IsPR :
+  ∀ (n : nat) (f : naryFunc n) (p : isPR n f) (g : naryFunc (S n))
+         (q : isPR (S n) g), isPR n (compose2 n f g).
+ +
+#[export] Instance compose1_NIsPR :
+  ∀ (n : nat) (g : naryFunc (S n)),
+    isPR (S n) g →
+    ∀ f : naryFunc 1, isPR 1 f →
+                           isPR (S n) (fun x : nat ⇒ g (f x)).
+ +
+Definition switchPR : naryFunc 3 :=
+  fun n x y ⇒ match n with 0 ⇒ y | _ ⇒ x end.
+ +
+#[export] Instance switchIsPR : isPR 3 switchPR.
+ +
+Fixpoint boundedSearchHelp (P : naryRel 1) (b : nat) {struct b} : nat :=
+  match b with
+  | O ⇒ 0
+  | S b' ⇒ let q := boundedSearchHelp P b'
+            in
+      match eq_nat_dec q b' with
+      | left _ ⇒ match P b' with
+                  | true ⇒ b'
+                  | false ⇒ S b'
+                  end
+      | right _ ⇒ q
+      end
+  end.
+ +
+
+ +
+If there exists some x ≤ b such that (P b x), returns x + Otherwise, returns b + +
+
+ +
+Definition boundedSearch (P : naryRel 2) (b : nat) : nat :=
+  boundedSearchHelp (P b) b.
+ +
+Module Examples.
+ +
+  Definition sqrtHelp (n: nat) :=
+    boundedSearch (fun _ x ⇒ Nat.eqb (x × x) n) (S (n / 2)).
+ +
+  Definition exact_sqrt (n: nat) :=
+    let q := sqrtHelp n
+    in if Nat.eqb (q × q) n then Some q else None.
+ +
+ +
+  End Examples.
+ +
+  Lemma boundedSearch1 :
+    ∀ (P : naryRel 2) (b x : nat), x < boundedSearch P b →
+                                        P b x = false.
+ +
+Lemma boundedSearch2 :
+  ∀ (P : naryRel 2) (b : nat),
+    boundedSearch P b = b ∨ P b (boundedSearch P b) = true.
+ +
+#[export] Instance boundSearchIsPR :
+  ∀ P : naryRel 2,
+    isPRrel 2 P → isPR 1 (boundedSearch P).
+ +
+Definition iterate (g : nat → nat) :=
+  nat_rec (fun _ ⇒ nat → nat) (fun x : nat ⇒ x)
+          (fun (_ : nat) (rec : nat → nat) (x : nat) ⇒ g (rec x)).
+ +
+#[export] Instance iterateIsPR :
+  ∀ g : nat → nat, isPR 1 g →
+                         ∀ n : nat, isPR 1 (iterate g n).
+ +
+#[export] Instance isPRTrans (n:nat) (f g : naryFunc n):
+  f =x= g → isPR n f → isPR n g.
+ +
+#[export] Instance isPRextEqual (n:nat) (f g : naryFunc n):
+  isPR n f → f =x= g → isPR n g.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.subAll.html b/theories/html/hydras.Ackermann.subAll.html new file mode 100644 index 00000000..00111639 --- /dev/null +++ b/theories/html/hydras.Ackermann.subAll.html @@ -0,0 +1,329 @@ + + + + + +hydras.Ackermann.subAll + + + + +
+ + + +
+ +

Library hydras.Ackermann.subAll

+ +
+
+ +
+ subAll.v + +
+ + Original script by Russel O'Connor + +
+
+ +
+From Coq Require Import Ensembles List Arith Peano_dec.
+ +
+From hydras.Ackermann Require Import ListExt folProof folLogic
+  folLogic2 folProp.
+From hydras Require Import folReplace subProp Compat815.
+From Coq Require Import Lia.
+Import FolNotations.
+ +
+From LibHyps Require Export LibHyps.
+From hydras Require Export MoreLibHyps.
+ +
+Section SubAllVars.
+ +
+Variable L : Language.
+Notation Formula := (Formula L) (only parsing).
+Notation Formulas := (Formulas L) (only parsing).
+Notation System := (System L) (only parsing).
+Notation Term := (Term L) (only parsing).
+Notation Terms := (Terms L) (only parsing).
+Notation SysPrf := (SysPrf L) (only parsing).
+ +
+Fixpoint subAllTerm (t : fol.Term L) : (nat → fol.Term L) → fol.Term L :=
+  match t return ((nat → fol.Term L) → fol.Term L) with
+  | var x ⇒ fun m ⇒ m x
+  | apply f ts ⇒ fun m ⇒ apply f (subAllTerms _ ts m)
+  end
+ 
+ with subAllTerms (n : nat) (ts : fol.Terms L n) {struct ts} :
+ (nat → fol.Term L) → fol.Terms L n :=
+  match
+    ts in (fol.Terms _ n) return ((nat → fol.Term L) → fol.Terms L n)
+  with
+  | Tnil ⇒ fun _ ⇒ Tnil
+  | Tcons n' t ss ⇒
+      fun m ⇒ Tcons (subAllTerm t m) (subAllTerms _ ss m)
+  end.
+ +
+Lemma subAllTerm_ext (t : fol.Term L) :
+  ∀ (m1 m2 : nat → fol.Term L),
+    (∀ m : nat, In m (freeVarT t) → m1 m = m2 m) →
+    subAllTerm t m1 = subAllTerm t m2.
+ +
+Lemma subAllTerms_ext (n : nat) (ts : fol.Terms L n) (m1 m2 : nat → fol.Term L):
+ (∀ m : nat, In m (freeVarTs ts) → m1 m = m2 m) →
+ subAllTerms n ts m1 = subAllTerms n ts m2.
+ +
+Fixpoint freeVarMap (l : list nat) : (nat → fol.Term L) → list nat :=
+  match l with
+  | nil ⇒ fun _ ⇒ nil
+  | a :: l' ⇒ fun m ⇒ freeVarT (m a) ++ freeVarMap l' m
+  end.
+ +
+Lemma freeVarMap_ext (l : list nat) (f1 f2 : nat → fol.Term L):
+ (∀ m : nat, In m l → f1 m = f2 m) →
+ freeVarMap l f1 = freeVarMap l f2.
+ +
+Lemma freeVarMap1 (l : list nat) (m : nat → fol.Term L) (v n : nat):
+ In v (freeVarT (m n)) → In n l → In v (freeVarMap l m).
+ +
+Fixpoint subAllFormula (f : Formula) (m : (nat → Term)) {struct f} : Formula :=
+  match f with
+  | (t = s)%fol ⇒ equal (subAllTerm t m) (subAllTerm s m)
+  | atomic r ts ⇒ atomic r (subAllTerms _ ts m)
+  | (f → g)%fol ⇒
+      (subAllFormula f m → subAllFormula g m)%fol
+  | (¬ f)%fol ⇒ (¬ subAllFormula f m)%fol
+  | (allH n, f)%fol ⇒
+      let nv :=
+        newVar
+          (freeVarF f ++
+           freeVarMap (freeVarF (allH n, f)%fol) m) in
+      (allH nv,
+        (subAllFormula f
+           (fun v : nat ⇒
+            match eq_nat_dec v n with
+            | left _ ⇒ var nv
+            | right _ ⇒ m v
+            end)))%fol
+  end.
+ +
+Lemma subAllFormula_ext :
+ ∀ (f : fol.Formula L) (m1 m2 : nat → fol.Term L),
+ (∀ m : nat, In m (freeVarF f) → m1 m = m2 m) →
+ subAllFormula f m1 = subAllFormula f m2.
+ +
+Lemma freeVarSubAllTerm1 (t : fol.Term L) (m : nat → fol.Term L) (v : nat):
+ In v (freeVarT (subAllTerm t m)) →
+ ∃ n : nat, In n (freeVarT t) ∧ In v (freeVarT (m n)).
+ +
+Lemma freeVarSubAllTerms1 (n : nat) (ts : fol.Terms L n)
+  (m : nat → fol.Term L) (v : nat):
+ In v (freeVarTs (subAllTerms n ts m)) →
+ ∃ a : nat, In a (freeVarTs ts) ∧ In v (freeVarT (m a)).
+ +
+Lemma freeVarSubAllTerm2 (t : fol.Term L) (m : nat → fol.Term L) (v n : nat):
+ In n (freeVarT t) →
+ In v (freeVarT (m n)) → In v (freeVarT (subAllTerm t m)).
+ +
+Lemma freeVarSubAllTerms2 (a : nat) (ts : fol.Terms L a)
+  (m : nat → fol.Term L)
+  (v n : nat):
+  In n (freeVarTs ts) →
+  In v (freeVarT (m n)) → In v (freeVarTs (subAllTerms a ts m)).
+ +
+Lemma freeVarSubAllFormula1 :
+ ∀ (f : fol.Formula L) (m : nat → fol.Term L) (v : nat),
+ In v (freeVarF (subAllFormula f m)) →
+ ∃ n : nat, In n (freeVarF f) ∧ In v (freeVarT (m n)).
+ +
+Lemma freeVarSubAllFormula2 :
+ ∀ (f : fol.Formula L) (m : nat → fol.Term L) (v n : nat),
+ In n (freeVarF f) →
+ In v (freeVarT (m n)) →
+ In v (freeVarF (subAllFormula f m)).
+ +
+Lemma subSubAllTerm (t : fol.Term L) (m : nat → fol.Term L)
+  (v : nat) (s : fol.Term L):
+ substT (subAllTerm t m) v s =
+ subAllTerm t (fun n : nat ⇒ substT (m n) v s).
+ +
+Lemma subSubAllTerms (n : nat) (ts : fol.Terms L n) (m : nat → fol.Term L)
+  (v : nat) (s : fol.Term L) :
+  substTs (subAllTerms n ts m) v s =
+    subAllTerms n ts (fun n : nat ⇒ substT (m n) v s).
+ +
+Lemma subSubAllFormula :
+  ∀ (T : fol.System L) (f : fol.Formula L) (m : nat → fol.Term L)
+         (v : nat) (s : fol.Term L),
+    folProof.SysPrf L T
+      (iffH (substF (subAllFormula f m) v s)
+         (subAllFormula f (fun n : nat ⇒ substT (m n) v s))).
+ +
+Lemma subAllTermId (t : fol.Term L):
+  subAllTerm t (fun x : nat ⇒ var x) = t.
+ +
+Lemma subAllTermsId (n : nat) (ts : fol.Terms L n):
+  subAllTerms n ts (fun x : nat ⇒ var x) = ts.
+ +
+Lemma subAllFormulaId (T : fol.System L) (f : fol.Formula L):
+ folProof.SysPrf L T
+   (iffH (subAllFormula f (fun x : nat ⇒ var x)) f).
+ +
+Lemma subAllSubAllTerm (t : fol.Term L) :
+ ∀ (m1 m2 : nat → fol.Term L),
+ subAllTerm (subAllTerm t m1) m2 =
+ subAllTerm t (fun n : nat ⇒ subAllTerm (m1 n) m2).
+ +
+Lemma subAllSubAllTerms (n : nat) (ts : fol.Terms L n) (m1 m2 : nat → fol.Term L):
+ subAllTerms n (subAllTerms n ts m1) m2 =
+ subAllTerms n ts (fun n : nat ⇒ subAllTerm (m1 n) m2).
+ +
+Lemma subAllSubAllFormula (T : fol.System L) (f : fol.Formula L):
+ ∀ (m1 m2 : nat → fol.Term L),
+ folProof.SysPrf L T
+   (iffH (subAllFormula (subAllFormula f m1) m2)
+      (subAllFormula f (fun n : nat ⇒ subAllTerm (m1 n) m2))).
+ +
+Section subAllCloseFrom.
+ +
+Fixpoint closeFrom (a n : nat) (f : fol.Formula L) {struct n} :
+ fol.Formula L :=
+  match n with
+  | O ⇒ f
+  | S m ⇒ forallH (a + m) (closeFrom a m f)
+  end.
+ +
+Opaque le_lt_dec.
+ +
+Lemma liftCloseFrom (n : nat) :
+ ∀(f : fol.Formula L) (T : fol.System L) (m : nat),
+ (∀ v : nat, In v (freeVarF f) → v < m) →
+ n ≤ m →
+ folProof.SysPrf L T (closeFrom 0 n f) →
+ folProof.SysPrf L T
+   (closeFrom m n
+      (subAllFormula f
+         (fun x : nat ⇒
+          match le_lt_dec n x with
+          | left _ ⇒ var x
+          | right _ ⇒ var (m + x)
+          end))).
+ +
+Lemma subAllCloseFrom1 :
+  ∀ (n m : nat) (map : nat → fol.Term L) (f : fol.Formula L)
+         (T : fol.System L),
+    (∀ v : nat,
+        v < n → ∀ w : nat, In w (freeVarT (map (m + v))) → w < m) →
+    folProof.SysPrf L T (closeFrom m n f) →
+    folProof.SysPrf L T
+      (subAllFormula f
+         (fun x : nat ⇒
+            match le_lt_dec m x with
+            | left _ ⇒
+                match le_lt_dec (m + n) x with
+                | left _ ⇒ var x
+                | right _ ⇒ map x
+                end
+            | right _ ⇒ var x
+            end)).
+ +
+Lemma subAllCloseFrom :
+  ∀ (n : nat) (m : nat → fol.Term L) (f : fol.Formula L)
+         (T : fol.System L),
+    folProof.SysPrf L T (closeFrom 0 n f) →
+    folProof.SysPrf L T
+      (subAllFormula f
+         (fun x : nat ⇒
+            match le_lt_dec n x with
+            | left _ ⇒ var x
+            | right _ ⇒ m x
+            end)).
+ +
+Lemma reduceSubAll :
+  ∀ (T : fol.System L) (map : nat → fol.Term L) (A B : fol.Formula L),
+    (∀ v : nat, ¬ In_freeVarSys L v T) →
+    folProof.SysPrf L T (iffH A B) →
+    folProof.SysPrf L T (iffH (subAllFormula A map) (subAllFormula B map)).
+ +
+End subAllCloseFrom.
+ +
+Lemma subToSubAll (T : fol.System L) (A : fol.Formula L) (v : nat)
+  (s : fol.Term L):
+ folProof.SysPrf L T
+   (iffH (substF A v s)
+      (subAllFormula A
+         (fun x : nat ⇒
+          match eq_nat_dec v x with
+          | left _ ⇒ s
+          | right _ ⇒ var x
+          end))).
+ +
+Lemma subAllSubFormula :
+  ∀ (T : fol.System L) (A : fol.Formula L) (v : nat)
+         (s : fol.Term L) (map : nat → fol.Term L),
+    folProof.SysPrf L T
+      (iffH (subAllFormula (substF A v s) map)
+         (subAllFormula A
+            (fun x : nat ⇒
+               match eq_nat_dec v x with
+               | left _ ⇒ subAllTerm s map
+               | right _ ⇒ map x
+               end))).
+ +
+End SubAllVars.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.subProp.html b/theories/html/hydras.Ackermann.subProp.html new file mode 100644 index 00000000..613f221f --- /dev/null +++ b/theories/html/hydras.Ackermann.subProp.html @@ -0,0 +1,218 @@ + + + + + +hydras.Ackermann.subProp + + + + +
+ + + +
+ +

Library hydras.Ackermann.subProp

+ +
+From Coq Require Import Ensembles List Peano_dec.
+From hydras.Ackermann Require Import ListExt.
+From Coq Require Import Arith.
+ +
+From hydras.Ackermann
+  Require Import folProof folLogic folProp folReplace.
+ +
+From LibHyps Require Export LibHyps.
+From hydras Require Export MoreLibHyps NewNotations.
+ +
+Section Substitution_Properties.
+ +
+Variable L : Language.
+Let Formula := Formula L.
+Let Formulas := Formulas L.
+Let System := System L.
+Let Term := Term L.
+Let Terms := Terms L.
+Let SysPrf := SysPrf L.
+ +
+Lemma freeVarSubTerm1 (t : Term):
+  ∀ (v : nat) (s : Term) (x : nat),
+    In x (freeVarT t) →
+    v ≠ x → In x (freeVarT (substT t v s)).
+ +
+Lemma freeVarSubTerms1 (n : nat) (ts : Terms n) (v : nat) (s : Term) (x : nat):
+  In x (freeVarTs ts) →
+  v ≠ x → In x (freeVarTs (substTs ts v s)).
+ +
+Lemma freeVarSubFormula1 (f : Formula):
+ ∀ (v : nat) (s : Term) (x : nat),
+ v ≠ x →
+ In x (freeVarF f) →
+ In x (freeVarF (substF f v s)).
+ +
+Lemma freeVarSubTerm2 (t : Term) :
+  ∀ (v : nat) (s : Term) (x : nat),
+    In x (freeVarT s) →
+    In v (freeVarT t) → In x (freeVarT (substT t v s)).
+ +
+Lemma freeVarSubTerms2 (n : nat) (ts : Terms n) (v : nat) (s : Term) (x : nat):
+ In x (freeVarT s) →
+ In v (freeVarTs ts) →
+ In x (freeVarTs (substTs ts v s)).
+ +
+Lemma freeVarSubFormula2 (f : Formula):
+  ∀ (v : nat) (s : Term) (x : nat),
+    In x (freeVarT s) →
+    In v (freeVarF f) →
+    In x (freeVarF (substF f v s)).
+ +
+Lemma freeVarSubTerm3 (t : Term):
+  ∀ (v : nat) (s : Term) (x : nat),
+    In x (freeVarT (substT t v s)) →
+    In x (List.remove eq_nat_dec v (freeVarT t)) ∨
+      In x (freeVarT s).
+ +
+Lemma freeVarSubTerms3 (n : nat) (ts : fol.Terms L n) (v : nat)
+  (s : Term) (x : nat):
+  In x (freeVarTs (substTs ts v s)) →
+  In x (List.remove eq_nat_dec v (freeVarTs ts)) ∨
+    In x (freeVarT s).
+ +
+Lemma freeVarSubFormula3 (f : Formula):
+ ∀ (v : nat) (s : Term) (x : nat),
+ In x (freeVarF (substF f v s)) →
+ In x (List.remove eq_nat_dec v (freeVarF f)) ∨
+ In x (freeVarT s).
+ +
+Lemma freeVarSubTerm4 (t : Term) :
+ ∀ (v : nat) (s : Term) (x : nat),
+ In x (freeVarT (substT t v s)) →
+ ¬ In v (freeVarT t) → In x (freeVarT t).
+ +
+Lemma freeVarSubTerms4 (n : nat) (ts : Terms n) (v : nat)
+  (s : Term) (x : nat):
+  In x (freeVarTs (substTs ts v s)) →
+  ¬ In v (freeVarTs ts) → In x (freeVarTs ts).
+ +
+Lemma freeVarSubFormula4 (f : Formula) :
+ ∀ (v : nat) (s : Term) (x : nat),
+ In x (freeVarF (substF f v s)) →
+ ¬ In v (freeVarF f) → In x (freeVarF f).
+ +
+Lemma subTermNil (t : Term) (v : nat) (s : Term):
+ ¬ In v (freeVarT t) → substT t v s = t.
+ +
+Lemma subTermTrans (t : Term) (v1 v2 : nat) (s : Term):
+ ¬ In v2 (List.remove eq_nat_dec v1 (freeVarT t)) →
+ substT (substT t v1 (var v2)) v2 s = substT t v1 s.
+ +
+Lemma subTermExch (t : Term) (v1 v2 : nat) (s1 s2 : Term):
+ v1 ≠ v2 →
+ ¬ In v2 (freeVarT s1) →
+ ¬ In v1 (freeVarT s2) →
+ substT (substT t v1 s1) v2 s2 =
+ substT (substT t v2 s2) v1 s1.
+ +
+Lemma subTermsNil (n : nat) (ts : Terms n) (v : nat) (s : Term):
+ ¬ In v (freeVarTs ts) → substTs ts v s = ts.
+ +
+Lemma subTermsTrans (n : nat) (ts : Terms n) (v1 v2 : nat) (s : Term):
+ ¬ In v2 (List.remove eq_nat_dec v1 (freeVarTs ts)) →
+ substTs (substTs ts v1 (var v2)) v2 s =
+ substTs ts v1 s.
+ +
+Lemma subTermsExch (n : nat) (ts : Terms n) (v1 v2 : nat)
+  (s1 s2 : Term):
+  v1 ≠ v2 →
+  ¬ In v2 (freeVarT s1) →
+  ¬ In v1 (freeVarT s2) →
+  substTs (substTs ts v1 s1) v2 s2 =
+    substTs (substTs ts v2 s2) v1 s1.
+ +
+Remark subFormulaNTEHelp (f g : Formula) (v : nat) (s : Term):
+ SysPrf (Ensembles.Add _ (Empty_set Formula) f) g →
+ SysPrf (Ensembles.Add _ (Empty_set Formula) (substF f v s))
+   (substF g v s).
+ +
+Remark subFormulaNTE (f : Formula):
+  ∀ (T : System),
+    (∀ (v : nat) (s : Term),
+        ¬ In v (freeVarF f) →
+        SysPrf T (iffH (substF f v s) f)) ∧
+      (∀ (v1 v2 : nat) (s : Term),
+          ¬ In v2 (List.remove eq_nat_dec v1 (freeVarF f)) →
+          SysPrf T
+            (iffH (substF (substF f v1 (var v2)) v2 s)
+               (substF f v1 s))) ∧
+      (∀ (v1 v2 : nat) (s1 s2 : Term),
+          v1 ≠ v2 →
+          ¬ In v2 (freeVarT s1) →
+          ¬ In v1 (freeVarT s2) →
+          SysPrf T
+            (iffH (substF (substF f v1 s1) v2 s2)
+               (substF (substF f v2 s2) v1 s1))).
+ +
+Lemma subFormulaNil :
+  ∀ (f : Formula) (T : System) (v : nat) (s : Term),
+    ¬ In v (freeVarF f) → SysPrf T (iffH (substF f v s) f).
+ +
+Lemma subFormulaTrans :
+  ∀ (f : Formula) (T : System) (v1 v2 : nat) (s : Term),
+    ¬ In v2 (List.remove eq_nat_dec v1 (freeVarF f)) →
+    SysPrf T
+      (iffH (substF (substF f v1 (var v2)) v2 s)
+         (substF f v1 s)).
+ +
+Lemma subFormulaExch :
+ ∀ (f : Formula) (T : System) (v1 v2 : nat) (s1 s2 : Term),
+ v1 ≠ v2 →
+ ¬ In v2 (freeVarT s1) →
+ ¬ In v1 (freeVarT s2) →
+ SysPrf T
+   (iffH (substF (substF f v1 s1) v2 s2)
+      (substF (substF f v2 s2) v1 s1)).
+ +
+End Substitution_Properties.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.wConsistent.html b/theories/html/hydras.Ackermann.wConsistent.html new file mode 100644 index 00000000..c250bb44 --- /dev/null +++ b/theories/html/hydras.Ackermann.wConsistent.html @@ -0,0 +1,59 @@ + + + + + +hydras.Ackermann.wConsistent + + + + +
+ + + +
+ +

Library hydras.Ackermann.wConsistent

+ +
+From hydras.Ackermann Require Import folProof folProp LNN.
+From Coq Require Import Arith List.
+ +
+Import NNnotations.
+ +
+Definition wConsistent (T : System) :=
+  ∀ (f : Formula) (v : nat),
+  (∀ x : nat, In x (freeVarF f) → v = x) →
+  SysPrf T (existH v (notH f)) →
+  ∃ n : nat, ¬ SysPrf T (substF f v (natToTerm n)).
+ +
+Lemma wCon2Con : ∀ T : System, wConsistent T → Consistent LNN T.
+ +
+Definition wInconsistent (T : System) :=
+  ∃ f : Formula,
+    (∃ v : nat,
+       (∀ x : nat, In x (freeVarF f) → v = x) ∧
+       SysPrf T (existH v (notH f)) ∧
+       (∀ n : nat, SysPrf T (substF f v (natToTerm n)))).
+ +
+Lemma notCon2wNotCon :
+ ∀ T : System, Inconsistent LNN T → wInconsistent T.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Ackermann.wellFormed.html b/theories/html/hydras.Ackermann.wellFormed.html new file mode 100644 index 00000000..cb29dba3 --- /dev/null +++ b/theories/html/hydras.Ackermann.wellFormed.html @@ -0,0 +1,182 @@ + + + + + +hydras.Ackermann.wellFormed + + + + +
+ + + +
+ +

Library hydras.Ackermann.wellFormed

+ +
+ +
+From hydras.Ackermann Require Import primRec cPair.
+From Coq Require Import Arith Lia.
+From hydras.Ackermann Require Import code folProp
+  extEqualNat codeList.
+From hydras Require Import Compat815.
+Import LispAbbreviations.
+ +
+Section Well_Formed_Term.
+ +
+Variable L : Language.
+Generalizable All Variables.
+Context `(cL : Lcode L cf cr).
+ +
+Variable codeArityF : nat → nat.
+Context (codeArityFIsPR : isPR 1 codeArityF).
+Hypothesis
+  codeArityFIsCorrect1 :
+    ∀ f : Functions L, codeArityF (cf f) = S (arityF L f).
+Hypothesis
+  codeArityFIsCorrect2 :
+    ∀ n : nat, codeArityF n ≠ 0 → ∃ f : Functions L, cf f = n.
+ +
+Let Term := Term L.
+Let Terms := Terms L.
+ +
+Definition wellFormedTermTerms : nat → nat :=
+  evalStrongRec 0
+    (fun t recs : nat ⇒
+     cPair
+       (switchPR (cPairPi1 t)
+          (charFunction 2 Nat.eqb (codeArityF (pred (cPairPi1 t)))
+             (S (codeLength (cPairPi2 t))) ×
+           cPairPi2 (codeNth (t - S (cPairPi2 t)) recs)) 1)
+       (switchPR t
+          (cPairPi1 (codeNth (t - S (cPairPi1 (pred t))) recs) ×
+           cPairPi2 (codeNth (t - S (cPairPi2 (pred t))) recs)) 1)).
+ +
+Definition wellFormedTerm (t : nat) : nat := cPairPi1 (wellFormedTermTerms t).
+ +
+Definition wellFormedTerms (ts : nat) : nat :=
+  cPairPi2 (wellFormedTermTerms ts).
+ +
+Lemma lengthTerms :
+ ∀ (n : nat) (ts : Terms n), codeLength (codeTerms ts) = n.
+ +
+Lemma wellFormedTermCorrect1 :
+  ∀ t : Term, wellFormedTerm (codeTerm t) = 1.
+ +
+Lemma wellFormedTermsCorrect1 (n : nat) (ts : Terms n):
+  wellFormedTerms (codeTerms ts) = 1.
+ +
+Lemma multLemma1 : ∀ a b : nat, a × b ≠ 0 → a ≠ 0.
+ +
+Lemma multLemma2 : ∀ a b : nat, a × b ≠ 0 → b ≠ 0.
+ +
+Remark wellFormedTermTermsCorrect2 :
+ ∀ n : nat,
+ (wellFormedTerm n ≠ 0 → ∃ t : Term, codeTerm t = n) ∧
+ (wellFormedTerms n ≠ 0 →
+  ∃ m : nat, (∃ ts : Terms m, codeTerms ts = n)).
+ +
+Lemma wellFormedTermCorrect2 :
+  ∀ n : nat,
+    wellFormedTerm n ≠ 0 → ∃ t : Term, codeTerm t = n.
+ +
+Lemma wellFormedTermsCorrect2 :
+  ∀ n : nat,
+    wellFormedTerms n ≠ 0 →
+    ∃ m : nat, (∃ ts : Terms m, codeTerms ts = n).
+ +
+#[local] Instance wellFormedTermTermsIsPR :
+  isPR 1 wellFormedTermTerms.
+ +
+#[export] Instance wellFormedTermIsPR : isPR 1 wellFormedTerm.
+ +
+#[export] Instance wellFormedTermsIsPR : isPR 1 wellFormedTerms.
+ +
+Section Well_Formed_Formula.
+ +
+Variable codeArityR : nat → nat.
+Context (codeArityRIsPR : isPR 1 codeArityR).
+Hypothesis
+  codeArityRIsCorrect1 :
+    ∀ r : Relations L, codeArityR (cr r) = S (arityR L r).
+Hypothesis
+  codeArityRIsCorrect2 :
+    ∀ n : nat, codeArityR n ≠ 0 → ∃ r : Relations L, cr r = n.
+ +
+Let Formula := Formula L.
+ +
+Definition wellFormedFormula : nat → nat :=
+  evalStrongRec 0
+    (fun f recs : nat ⇒
+     switchPR (cPairPi1 f)
+       (switchPR (pred (cPairPi1 f))
+          (switchPR (pred (pred (cPairPi1 f)))
+             (switchPR (pred (pred (pred (cPairPi1 f))))
+                (charFunction 2 Nat.eqb
+                   (codeArityR (pred (pred (pred (pred (cPairPi1 f))))))
+                   (S (codeLength (cPairPi2 f))) ×
+                 wellFormedTerms (cPairPi2 f))
+                (codeNth (f - S (cPairPi2 (cPairPi2 f))) recs))
+             (codeNth (f - S (cPairPi2 f)) recs))
+          (codeNth (f - S (cPairPi1 (cPairPi2 f))) recs ×
+           codeNth (f - S (cPairPi2 (cPairPi2 f))) recs))
+       (wellFormedTerm (cPairPi1 (cPairPi2 f)) ×
+        wellFormedTerm (cPairPi2 (cPairPi2 f)))).
+ +
+Lemma wellFormedFormulaCorrect1 :
+ ∀ f : Formula, wellFormedFormula (codeFormula f) = 1.
+ +
+Lemma wellFormedFormulaCorrect2 :
+  ∀ n : nat,
+    wellFormedFormula n ≠ 0 →
+    ∃ f : Formula, codeFormula f = n.
+ +
+#[export] Instance wellFormedFormulaIsPR : isPR 1 wellFormedFormula.
+ +
+End Well_Formed_Formula.
+ +
+End Well_Formed_Term.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.Canon.html b/theories/html/hydras.Epsilon0.Canon.html new file mode 100644 index 00000000..3a141e3c --- /dev/null +++ b/theories/html/hydras.Epsilon0.Canon.html @@ -0,0 +1,430 @@ + + + + + +hydras.Epsilon0.Canon + + + + +
+ + + +
+ +

Library hydras.Epsilon0.Canon

+ +
+ +
+
+ +
+ Canonical Sequences of ordinals below epsilon0 + +
+ +Pierre Casteran, + LaBRI and University of Bordeaux. + +
+ + After J. Ketonen and R. Solovay's paper + " Rapidly Growing Ramsey Functions" in + Annals of mathematics, Mar. 1981 + +
+ + +
+
+ +
+From Coq Require Export Arith Lia.
+Import Relations Relation_Operators.
+ +
+From hydras.Prelude Require Import DecPreOrder.
+From hydras.Epsilon0 Require Export T1 E0.
+ +
+Set Implicit Arguments.
+ +
+Open Scope t1_scope.
+ +
+
+ +
+

Definitions

+ +
+
+ +
+Fixpoint canon alpha (i:nat) : T1 :=
+  match alpha with
+  | zero ⇒ zero
+  | cons zero 0 zero ⇒ zero
+  | cons zero (S k) zero ⇒ FS k
+  | cons gamma 0 zero ⇒ (match T1.pred gamma with
+                            Some gamma' ⇒
+                              match i with
+                              | 0 ⇒ zero
+                              | S j ⇒ cons gamma' j zero
+                              end
+                          | None ⇒ cons (canon gamma i) 0 zero
+                          end)
+  
+  | cons gamma (S n) zero ⇒
+     (match T1.pred gamma with
+       Some gamma' ⇒
+       (match i with
+         0 ⇒ cons gamma n zero
+       | S j ⇒ cons gamma n (cons gamma' j zero)
+       end)
+      | None ⇒ cons gamma n (cons (canon gamma i) 0 zero)
+      end)
+   | cons alpha n beta ⇒ cons alpha n (canon beta i)
+end.
+ +
+Section Canon_examples.
+ +
+#[local] Open Scope ppT1_scope.
+ +
+ +
+Close Scope ppT1_scope.
+ +
+ +
+ +
+ +
+Goal canon (T1omega ^ T1omega) 10 = phi0 10. +
+End Canon_examples.
+ +
+ +
+
+ +
+

Properties of canonical sequences

+ +
+
+ +
+Lemma canon_zero i : canon zero i = zero.
+ +
+Lemma canon_tail :
+  ∀ alpha n beta i, nf (cons alpha n beta) →
+                          beta ≠ 0 →
+                          canon (cons alpha n beta) i =
+                          cons alpha n (canon beta i).
+ +
+Lemma canonS_lim1 : ∀ i lambda,
+    nf lambda → T1limit lambda
+    → canon (cons lambda 0 zero) (S i) =
+         T1.phi0 (canon lambda (S i)).
+ +
+Lemma canon_lim1 : ∀ i lambda, nf lambda → T1limit lambda
+                                    → canon (cons lambda 0 zero) i =
+                                       T1.phi0 (canon lambda i).
+ +
+
+ +
+Here +
+
+ +
+Lemma canonS_lim2 i n lambda:
+    nf lambda → T1limit lambda
+    → canon (cons lambda (S n) zero) (S i) =
+       cons lambda n (T1.phi0 (canon lambda (S i))).
+ +
+Lemma canon0_lim2 n lambda:
+    nf lambda → T1limit lambda
+    → canon (cons lambda (S n) zero) 0 =
+       cons lambda n (T1.phi0 (canon lambda 0)).
+ +
+Lemma canon_lim2 i n lambda :
+    nf lambda → T1limit lambda
+    → canon (cons lambda (S n) zero) i =
+       cons lambda n (T1.phi0 (canon lambda i)).
+ +
+Lemma canon_lim3 i n alpha lambda :
+  nf alpha → nf lambda → T1limit lambda →
+  canon (cons alpha n lambda) i = cons alpha n (canon lambda i).
+ +
+ +
+Lemma canon_succ i alpha :
+  nf alpha → canon (succ alpha) i = alpha. Lemma canonS_succ i alpha : nf alpha →
+                            canon (succ alpha) (S i) = alpha.
+ +
+Lemma canon0_succ alpha : nf alpha → canon (succ alpha) 0 = alpha.
+ +
+Lemma canonS_phi0_succ_eqn i gamma:
+  nf gamma →
+  canon (T1.phi0 (succ gamma)) (S i) = cons gamma i zero.
+ +
+Lemma canon0_phi0_succ_eqn gamma:
+  nf gamma →
+  canon (T1.phi0 (succ gamma)) 0 = zero.
+ +
+Lemma canonS_cons_succ_eqn2 i n gamma :
+    nf gamma →
+    canon (cons (T1.succ gamma) (S n) zero) (S i) =
+    cons (T1.succ gamma) n (cons gamma i zero).
+ +
+Lemma canon0_cons_succ_eqn2 n gamma :
+    nf gamma →
+    canon (cons (T1.succ gamma) (S n) zero) 0=
+    cons (T1.succ gamma) n zero.
+ +
+Lemma canon_SSn_zero (i:nat) :
+  ∀ alpha n ,
+    nf alpha →
+    canon (cons alpha (S n) zero) i =
+    cons alpha n (canon (cons alpha 0 zero) i).
+ +
+ #[deprecated(note="use canon_SSn_zero")]
+  Notation canonSSn := canon_SSn_zero (only parsing).
+ +
+ +
+Lemma canonS_zero_inv (alpha:T1) (i:nat) :
+  canon alpha (S i) = zero → alpha = zero ∨ alpha = one.
+ +
+
+ +
+

Canonical sequences and the order LT

+ +
+
+ +
+Lemma canonS_LT i alpha :
+  nf alpha → alpha ≠ zero →
+  canon alpha (S i) t1< alpha.
+ +
+Lemma canon0_LT alpha :
+  nf alpha → alpha ≠ zero →
+  canon alpha 0 t1< alpha.
+ +
+Lemma nf_canon i alpha: nf alpha → nf (canon alpha i).
+Lemma canon_LT i alpha : nf alpha → alpha ≠ zero →
+                          canon alpha i t1< alpha.
+ +
+ +
+ +
+Lemma canon_lt : ∀ i alpha, nf alpha → alpha ≠ zero →
+                              T1.lt (canon alpha i) alpha.
+ +
+Lemma canonS_cons_not_zero : ∀ i alpha n beta,
+    alpha ≠ zero → canon (cons alpha n beta) (S i) ≠ zero.
+ +
+
+ +
+

Canonical sequences of limit ordinals

+ +
+
+ +
+Lemma T1limit_canonS_not_zero i lambda:
+  nf lambda → T1limit lambda → canon lambda (S i) ≠ zero.
+ +
+
+ +
+ +
+ +Let lambda be a limit ordinal. For any beta < lambda, we can +compute some item of the canonical sequence of lambda which is +greater than beta. + +
+ + It is a constructive way of expressing that lambda is the +limit of its own canonical sequence + +
+
+ +
+ +
+Lemma canonS_limit_strong lambda :
+  nf lambda → T1limit lambda →
+  ∀ beta, beta t1< lambda → {i:nat | beta t1< canon lambda (S i)}.
+ +
+Lemma canon_limit_strong lambda :
+  nf lambda →
+  T1limit lambda →
+  ∀ beta, beta t1< lambda →
+                {i:nat | beta t1< canon lambda i}.
+ +
+ +
+Lemma canonS_limit_lub (lambda : T1) :
+  nf lambda → T1limit lambda → strict_lub (fun i ⇒ canon lambda( S i)) lambda. +
+Lemma canon_limit_mono alpha i j : nf alpha → T1limit alpha →
+                                    i < j →
+                                    canon alpha i t1< canon alpha j.
+ +
+Lemma canonS_limit_mono alpha i j : nf alpha → T1limit alpha →
+                                    i < j →
+                                    canon alpha (S i) t1< canon alpha (S j).
+ +
+Lemma canonS_LE alpha n :
+    nf alpha → canon alpha (S n) t1≤ canon alpha (S (S n)).
+ +
+ +
+ +
+
+ +
+exercise after Guillaume Melquiond +
+
+ +
+ From hydras Require Import Fuel.
+ +
+Fixpoint approx alpha beta fuel i :=
+  match fuel with
+          FO ⇒ None
+        | Fuel.FS f ⇒
+          let gamma := canon alpha (S i) in
+          if decide (lt beta gamma)
+          then Some (i,gamma)
+          else approx alpha beta (f tt) (S i)
+        end.
+ +
+Lemma approx_ok alpha beta :
+  ∀ fuel i j gamma, approx alpha beta fuel i = Some (j,gamma) →
+                         gamma = canon alpha (S j) ∧ lt beta gamma.
+ +
+ +
+
+ +
+

Adaptation to E0 (well formed terms of type T1 )

+ +
+
+ +
+Open Scope E0_scope.
+ +
+#[program] Definition Canon (alpha:E0)(i:nat): E0 :=
+  @mkord (@canon (cnf alpha) i) _.
+ +
+Lemma Canon_Succ beta n: Canon (E0_succ beta) (S n) = beta.
+ +
+Lemma Canon_Omega k : Canon E0_omega k = E0fin k.
+ +
+#[global] Hint Rewrite Canon_Omega : E0_rw.
+ +
+Lemma CanonSSn (i:nat) :
+  ∀ alpha n , alpha ≠ E0zero →
+                    Canon (Cons alpha (S n) E0zero) (S i) =
+                    Cons alpha n (Canon (E0_phi0 alpha) (S i)).
+ +
+Lemma CanonS_phi0_lim alpha k : E0limit alpha →
+                                Canon (E0_phi0 alpha) (S k) =
+                                E0_phi0 (Canon alpha (S k)).
+ +
+Lemma CanonS_lt : ∀ i alpha, alpha ≠ E0zero →
+                                  Canon alpha (S i) o< alpha.
+ +
+Lemma Canon_lt : ∀ i alpha, alpha ≠ E0zero → Canon alpha i o< alpha.
+ +
+Lemma Canon_of_limit_not_null : ∀ i alpha, E0limit alpha →
+                                       Canon alpha (S i) ≠ E0zero.
+ +
+#[global]
+  Hint Resolve CanonS_lt Canon_lt Canon_of_limit_not_null : E0.
+ +
+Lemma CanonS_phi0_Succ alpha i : Canon (E0_phi0 (E0_succ alpha)) (S i) =
+                                 Omega_term alpha i.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.E0.html b/theories/html/hydras.Epsilon0.E0.html new file mode 100644 index 00000000..7d0c5346 --- /dev/null +++ b/theories/html/hydras.Epsilon0.E0.html @@ -0,0 +1,559 @@ + + + + + +hydras.Epsilon0.E0 + + + + +
+ + + +
+ +

Library hydras.Epsilon0.E0

+ +
+
+ +
+ +
+ + Class of ordinals less than epsilon0 + +
+ + E0 is a class type for ordinal terms proved to be in normal form. + +
+
+ +
+From Coq Require Import Arith Max Bool Lia Compare_dec Relations Ensembles
+     ArithRing Wellfounded Operators_Properties Logic.Eqdep_dec.
+ +
+From hydras Require Import Prelude.More_Arith Prelude.Restriction
+           Prelude.DecPreOrder ON_Generic OrdNotations.
+ +
+From hydras.Epsilon0 Require Export T1 Hessenberg.
+ +
+Set Implicit Arguments.
+ +
+Create HintDb E0.
+Create HintDb E0_rw.
+ +
+ +
+Declare Scope E0_scope.
+Delimit Scope E0_scope with e0.
+Open Scope E0_scope.
+ +
+ +
+Declare Scope E0_scope.
+ +
+ +
+Class E0 : Type := mkord {cnf : T1; cnf_ok : nf cnf}.
+ +
+Arguments cnf : clear implicits.
+ +
+#[export] Hint Resolve cnf_ok : E0.
+ +
+ +
+
+ +
+

Lifting functions from T1 to E0

+ +
+
+ +
+ +
+Definition E0lt (alpha beta : E0) := T1.LT (@cnf alpha) (@cnf beta).
+Definition E0le := leq E0lt.
+ +
+Infix "o<" := E0lt : E0_scope.
+Infix "o<=" := E0le : E0_scope.
+ +
+ +
+#[export] Instance E0zero : E0 := @mkord zero refl_equal.
+ +
+#[export] Instance E0_omega : E0 := @mkord T1omega refl_equal.
+ +
+ +
+#[deprecated(note="use E0zero")]
+ Notation Zero := E0zero (only parsing).
+ +
+#[deprecated(note="use E0_omega")]
+ Notation _Omega := E0_omega (only parsing).
+ +
+#[global, program] Instance E0_succ (alpha : E0) : E0 :=
+@mkord (T1.succ (@cnf alpha)) _.
+ +
+#[deprecated(note="use E0_succ")]
+ Notation Succ := E0_succ (only parsing).
+ +
+Definition E0limit (alpha : E0) : bool := T1limit (@cnf alpha).
+ +
+#[deprecated(note="use E0limit")]
+ Notation Limitb := E0limit (only parsing).
+ +
+Definition E0is_succ (alpha : E0) : bool :=
+  T1is_succ (@cnf alpha).
+ +
+#[deprecated(note="use E0is_succ")]
+ Notation E0succb := E0is_succ (only parsing).
+ +
+#[global, program] Instance E0one : E0:= @mkord (T1.succ zero) _.
+ +
+ +
+ +
+#[global, program] Instance E0add (alpha beta : E0) : E0 :=
+  @mkord (T1add (@cnf alpha) (@cnf beta))_ .
+ +
+Infix "+" := E0add : E0_scope.
+ +
+ +
+#[deprecated(note="use E0add")]
+ Notation Plus := E0add (only parsing).
+ +
+ +
+Check E0_omega + E0_omega.
+ +
+ +
+#[global, program] Instance E0_phi0 (alpha: E0) : E0 :=
+ @mkord (T1.phi0 (cnf alpha)) _.
+ +
+Notation "'E0_omega^'" := E0_phi0 (only parsing) : E0_scope.
+ +
+#[global, program] Instance Omega_term (alpha: E0) (n: nat) : E0 :=
+   @mkord (cons (cnf alpha) n zero) _.
+ +
+#[global] Instance Cons (alpha : E0) (n: nat) (beta: E0) : E0
+  := (Omega_term alpha n + beta)%e0.
+ +
+#[global, program] Instance E0finS (i:nat) : E0:= @mkord (FS i)%t1 _.
+ +
+#[global, program] Instance E0fin (i:nat) : E0:=
+  match i with
+    0 ⇒ E0zero
+  | S i ⇒ E0finS i
+  end.
+ +
+#[deprecated(note="use E0fin")]
+ Notation Fin := E0fin (only parsing).
+ +
+Coercion E0fin : nat >-> E0.
+ +
+#[global, program] Instance E0mul (alpha beta : E0) : E0 :=
+  @mkord (cnf alpha × cnf beta)%t1 _.
+ +
+#[deprecated(note="use E0mul")]
+ Notation Mult := E0mul (only parsing).
+ +
+Infix "×" := E0mul : E0_scope.
+ +
+#[global, program] Instance Mult_i (alpha: E0) (n: nat) : E0 :=
+  @mkord (cnf alpha × n)%t1 _.
+ +
+
+ +
+

Lemmas

+ +
+
+ +
+ +
+
+ +
+

On equality on type E0

+ +
+
+ +
+ +
+Lemma nf_proof_unicity :
+  ∀ (alpha:T1) (H H': nf alpha), H = H'.
+ +
+Lemma E0_eq_intro : ∀ alpha beta : E0,
+    cnf alpha = cnf beta → alpha = beta.
+ +
+ +
+Corollary E0_eq_iff (alpha beta: E0) :
+  alpha = beta ↔ cnf alpha = cnf beta.
+ +
+Remark Le_iff : ∀ alpha beta,
+    E0le alpha beta ↔ T1.LE (@cnf alpha) (@cnf beta).
+ +
+Lemma Succb_Succ alpha : E0is_succ alpha → {beta : E0 | alpha = E0_succ beta}.
+ +
+#[export] Hint Resolve E0_eq_intro : E0.
+ +
+Ltac orefl := (apply E0_eq_intro; try reflexivity).
+ +
+Ltac ochange alpha beta := (replace alpha with beta; [| orefl]).
+ +
+Lemma E0_eq_dec : ∀ alpha beta: E0,
+    {alpha = beta}+{alpha ≠ beta}.
+ +
+
+ +
+

Adaptation to E0 of lemmas about T1

+ +
+
+ +
+Lemma alpha_plus_zero alpha : alpha + E0zero = alpha.
+ +
+#[export] Hint Rewrite alpha_plus_zero : E0_rw.
+ +
+Lemma cnf_phi0 (alpha : E0) :
+  cnf (E0_phi0 alpha) = T1.phi0 (cnf alpha).
+ +
+Lemma cnf_Succ (alpha : E0) :
+  cnf (E0_succ alpha) = succ (cnf alpha).
+ +
+Lemma cnf_Omega_term (alpha: E0) (n: nat) :
+  cnf (Omega_term alpha n) = omega_term (cnf alpha) n.
+ +
+Lemma T1limit_Omega_term alpha i : alpha ≠ E0zero →
+                                  E0limit (Omega_term alpha i).
+ +
+Lemma T1limit_phi0 alpha : alpha ≠ E0zero →
+                           E0limit (E0_phi0 alpha).
+ +
+#[export] Hint Resolve T1limit_phi0 : E0.
+ +
+Definition Zero_Limit_Succ_dec (alpha : E0) :
+  {alpha = E0zero} + {E0limit alpha = true} +
+  {beta : E0 | alpha = E0_succ beta}.
+Defined.
+ +
+Definition E0_pred (alpha: E0) : E0 :=
+  match Zero_Limit_Succ_dec alpha with
+      inl _ ⇒ alpha
+    | inr (exist _ beta _) ⇒ beta
+  end.
+ +
+Tactic Notation "mko" constr(alpha) := refine (@mkord alpha eq_refl).
+ +
+#[global] Instance Two : E0 := ltac:(mko (\F 2)).
+ +
+#[global] Instance Omega_2 : E0 :=ltac:(mko (T1omega × T1omega)%t1).
+ +
+#[global] Instance E0_sto : StrictOrder E0lt.
+ +
+#[ global ] Instance compare_E0 : Compare E0 :=
+ fun (alpha beta:E0) ⇒ compare (cnf alpha) (cnf beta).
+ +
+Lemma compare_correct (alpha beta : E0) :
+  CompSpec eq E0lt alpha beta (compare alpha beta).
+ +
+Lemma E0lt_wf : well_founded E0lt.
+ +
+#[export] Hint Resolve E0lt_wf : E0.
+ +
+Lemma Lt_Succ_Le (alpha beta: E0): beta o< alpha → E0_succ beta o≤ alpha.
+ +
+Lemma E0_pred_of_Succ (alpha:E0) : E0_pred (E0_succ alpha) = alpha.
+ +
+#[export] Hint Rewrite E0_pred_of_Succ: E0_rw.
+ +
+Lemma Succ_inj alpha beta : E0_succ alpha = E0_succ beta → alpha = beta.
+ +
+Lemma E0_pred_Lt (alpha : E0) : alpha ≠ E0zero → E0limit alpha = false →
+                       E0_pred alpha o< alpha.
+ +
+#[export] Hint Resolve E0_pred_Lt : E0.
+ +
+Lemma Succ_Succb (alpha : E0) : E0is_succ (E0_succ alpha).
+ +
+#[export] Hint Resolve Succ_Succb : E0.
+ +
+Ltac ord_eq alpha beta := assert (alpha = beta);
+      [apply E0_eq_intro ; try reflexivity|].
+ +
+Lemma FinS_eq i : E0finS i = E0fin (S i).
+ +
+Lemma mult_fin_rw alpha (i:nat) : alpha × i = Mult_i alpha i.
+ +
+Lemma FinS_Succ_eq : ∀ i, E0finS i = E0_succ (E0fin i).
+ +
+#[export] Hint Rewrite FinS_Succ_eq : E0_rw.
+ +
+Lemma Limit_not_Zero alpha : E0limit alpha → alpha ≠ E0zero.
+ +
+Lemma Succ_not_Zero alpha: E0is_succ alpha → alpha ≠ E0zero.
+ +
+Lemma Lt_Succ : ∀ alpha, E0lt alpha (E0_succ alpha).
+ +
+Lemma Succ_not_T1limit alpha : E0is_succ alpha → ¬ E0limit alpha .
+ +
+#[export]
+  Hint Resolve Limit_not_Zero Succ_not_Zero Lt_Succ Succ_not_T1limit : E0.
+ +
+Lemma lt_Succ_inv : ∀ alpha beta, beta o< alpha ↔
+                                       E0_succ beta o≤ alpha.
+ +
+Lemma lt_Succ_le_2 (alpha beta: E0):
+    alpha o< E0_succ beta → alpha o≤ beta.
+ +
+Lemma Succ_lt_T1limit alpha beta:
+    E0limit alpha → beta o< alpha → E0_succ beta o< alpha.
+ +
+Lemma le_lt_eq_dec : ∀ alpha beta, alpha o≤ beta →
+                                        {alpha o< beta} + {alpha = beta}.
+ +
+ +
+ +
+#[global] Instance E0_comp: Comparable E0lt compare.
+ +
+#[global] Instance Epsilon0 : ON E0lt compare.
+ +
+ +
+Definition Zero_limit_succ_dec : ZeroLimitSucc_dec .
+ +
+ +
+ +
+Qed.
+ +
+
+ +
+

Rewriting lemmas

+ +
+
+ +
+Lemma Succ_rw : ∀ alpha, cnf (E0_succ alpha) = T1.succ (cnf alpha).
+ +
+Lemma Plus_rw : ∀ alpha beta, cnf (alpha + beta) =
+                                   (cnf alpha + cnf beta)%t1.
+ +
+Lemma Lt_trans alpha beta gamma :
+  alpha o< beta → beta o< gamma → alpha o< gamma.
+ +
+Lemma Le_trans alpha beta gamma :
+  alpha o≤ beta → beta o≤ gamma → alpha o≤ gamma.
+ +
+Lemma Omega_term_plus alpha beta i :
+  alpha ≠ E0zero → (beta o< E0_phi0 alpha)%e0 →
+  cnf (Omega_term alpha i + beta)%e0 = cons (cnf alpha) i (cnf beta).
+ +
+Lemma cnf_Cons (alpha beta: E0) n : alpha ≠ E0zero → beta o< E0_phi0 alpha →
+                                     cnf (Cons alpha n beta) =
+                                     cons (cnf alpha) n (cnf beta).
+ +
+Lemma T1limit_plus alpha beta i:
+  (beta o< E0_phi0 alpha)%e0 → E0limit beta →
+  E0limit (Omega_term alpha i + beta)%e0.
+ +
+Lemma Succ_of_cons alpha gamma i : alpha ≠ E0zero → gamma o< E0_phi0 alpha →
+                                cnf (E0_succ (Omega_term alpha i + gamma)%e0) =
+                                cnf (Omega_term alpha i + E0_succ gamma)%e0.
+ +
+#[global, program] Instance Oplus (alpha beta : E0) : E0 :=
+@mkord (oplus (cnf alpha) (cnf beta)) _.
+ +
+Infix "O+" := Oplus (at level 50, left associativity): E0_scope.
+ +
+#[global] Instance Oplus_assoc : Assoc eq Oplus.
+ +
+Lemma oPlus_rw (alpha beta : E0) :
+  cnf (alpha O+ beta)%e0 = (cnf alpha o+ cnf beta)%t1.
+ +
+Example L_3_plus_omega : 3 + E0_omega = E0_omega.
+ +
+ +
+Lemma succ_correct alpha beta : cnf beta = succ (cnf alpha) ↔
+                                Successor beta alpha.
+ +
+Lemma Le_refl alpha : alpha o≤ alpha.
+ +
+Lemma Lt_Le_incl alpha beta : alpha o< beta → alpha o≤ beta.
+ +
+Lemma E0_Lt_irrefl (alpha : E0) : ¬ alpha o< alpha.
+ +
+Lemma E0_Lt_Succ_inv (alpha beta: E0):
+  alpha o< E0_succ beta → alpha o< beta ∨ alpha = beta.
+ +
+Lemma E0_not_Lt_zero alpha : ¬ alpha o< E0zero.
+ +
+Lemma lt_omega_inv: ∀ alpha:E0, alpha o< E0_omega →
+                                      ∃ (i:nat), alpha = E0fin i.
+ +
+Lemma E0_lt_eq_lt alpha beta : alpha o< beta ∨ alpha = beta ∨ beta o< alpha.
+ +
+Lemma E0_lt_ge alpha beta : alpha o< beta ∨ beta o≤ alpha.
+ +
+Lemma Limit_gt_Zero (alpha: E0) : E0limit alpha → E0zero o< alpha.
+ +
+Lemma phi0_mono alpha beta : alpha o< beta → E0_phi0 alpha o< E0_phi0 beta.
+ +
+#[global] Instance plus_assoc : Assoc eq E0add .
+ +
+Theorem mult_plus_distr_l (alpha beta gamma: E0) :
+  alpha × (beta + gamma) = alpha × beta + alpha × gamma.
+ +
+Example Ex42: E0_omega + 42 + E0_omega^2 = E0_omega^2.
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.Epsilon0.html b/theories/html/hydras.Epsilon0.Epsilon0.html new file mode 100644 index 00000000..982cfe24 --- /dev/null +++ b/theories/html/hydras.Epsilon0.Epsilon0.html @@ -0,0 +1,33 @@ + + + + + +hydras.Epsilon0.Epsilon0 + + + + +
+ + + +
+ +

Library hydras.Epsilon0.Epsilon0

+ +
+From hydras Require Export T1 Canon Paths Large_Sets.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.Epsilon0rpo.html b/theories/html/hydras.Epsilon0.Epsilon0rpo.html new file mode 100644 index 00000000..2143a731 --- /dev/null +++ b/theories/html/hydras.Epsilon0.Epsilon0rpo.html @@ -0,0 +1,336 @@ + + + + + +hydras.Epsilon0.Epsilon0rpo + + + + +
+ + + +
+ +

Library hydras.Epsilon0.Epsilon0rpo

+ +
+
+ +
+ Pierre Casteran + LaBRI, University of Bordeaux + +
+ + Evelyne Contejean + LRI. + +
+ + Cantor "pre" Normal form + After Manolios and Vroon work on ACL2 +
+
+ +
+From Coq Require Import Arith Lia Compare_dec Relations
+  Wellfounded Wf_nat List Bool Eqdep_dec Ensembles ArithRing Logic.
+ +
+From hydras Require Import More_Arith Restriction
+     DecPreOrder rpo.term rpo.rpo Epsilon0.T1.
+ +
+Set Implicit Arguments.
+ +
+Create HintDb rpo.
+ +
+Module Alt.
+ +
+Module Eps0_sig <: Signature.
+ +
+  Inductive symb0 : Set := nat_0 | nat_S | ord_zero | ord_cons.
+  Definition symb := symb0.
+ +
+  Lemma eq_symbol_dec : ∀ f1 f2 : symb, {f1 = f2} + {f1 ≠ f2}.
+ +
+
+ +
+The arity of a symbol contains also the information about built-in theories as in CiME +
+
+ +
+  Inductive arity_type : Set :=
+  | AC : arity_type
+  | C : arity_type
+  | Free : nat → arity_type.
+ +
+  Definition arity : symb → arity_type :=
+    fun f ⇒ match f with
+             | nat_0 ⇒ Free 0
+             | ord_zero ⇒ Free 0
+             | nat_S ⇒ Free 1
+             | ord_cons ⇒ Free 3
+             end.
+End Eps0_sig.
+ +
+
+ +
+

Module Type Variables.

+ + There are almost no assumptions, except a decidable equality. +
+
+ +
+Module Vars <: Variables.
+ +
+  Inductive empty_set : Set := .
+  Definition var := empty_set.
+ +
+  Lemma eq_variable_dec : ∀ v1 v2 : var, {v1 = v2} + {v1 ≠ v2}.
+ +
+End Vars.
+ +
+Module Eps0_prec <: Precedence.
+ +
+  Definition A : Set := Eps0_sig.symb.
+  Import Eps0_sig.
+ +
+  Definition prec : relation A :=
+    fun f g ⇒ match f, g with
+               | nat_0, nat_S ⇒ True
+               | nat_0, ord_zero ⇒ True
+               | nat_0, ord_cons ⇒ True
+               | ord_zero, nat_S ⇒ True
+               | ord_zero, ord_cons ⇒ True
+               | nat_S, ord_cons ⇒ True
+               | _, _ ⇒ False
+               end.
+ +
+  Inductive status_type : Set :=
+  | Lex : status_type
+  | Mul : status_type.
+ +
+  Definition status : A → status_type := fun f ⇒ Lex.
+ +
+  Lemma prec_dec : ∀ a1 a2 : A, {prec a1 a2} + {¬ prec a1 a2}.
+ +
+  Lemma prec_antisym : ∀ s, prec s s → False.
+ +
+  Lemma prec_transitive : transitive A prec.
+ +
+End Eps0_prec.
+ +
+Module Eps0_alg <: Term := term.Make (Eps0_sig) (Vars).
+Module Eps0_rpo <: RPO := rpo.Make (Eps0_alg) (Eps0_prec).
+ +
+Import Eps0_alg Eps0_rpo Eps0_sig.
+ +
+Fixpoint nat_2_term (n:nat) : term :=
+  match n with 0 ⇒ (Term nat_0 nil)
+             | S p ⇒ Term nat_S ((nat_2_term p)::nil)
+  end.
+ +
+
+ +
+

+ +Every (representation of a) natural number is less than + any non zero ordinal +
+
+ +
+Lemma nat_lt_cons : ∀ (n:nat) a p b ,
+    rpo (nat_2_term n) (Term ord_cons (a::p::b::nil)).
+ +
+Theorem rpo_trans (t t1 t2: term): rpo t t1 → rpo t1 t2 → rpo t t2.
+ +
+Fixpoint T1_2_term (a:T1) : term :=
+match a with
+| zero ⇒ Term ord_zero nil
+| cons a n b ⇒
+    Term ord_cons (T1_2_term a :: nat_2_term n ::T1_2_term b::nil)
+end.
+ +
+Fixpoint T1_size (o:T1):nat :=
+ match o with zero ⇒ 0
+            | cons a n b ⇒ S (T1_size a + n + T1_size b)%nat
+         end.
+ +
+Lemma T1_size1 a n b: T1_size zero < T1_size (cons a n b).
+ +
+Lemma T1_size2 a n b: T1_size a < T1_size (cons a n b).
+ +
+Lemma T1_size3 a n b: T1_size b < T1_size (cons a n b).
+ +
+#[global] Hint Resolve T1_size2 T1_size3 : rpo.
+ +
+
+ +
+let us recall subterm properties on T1 +
+
+ +
+Lemma lt_subterm1 a a' n' b': lt a a' → lt a (cons a' n' b').
+ +
+Lemma lt_subterm2 a a' n n' b b':
+  lt a a' →
+  nf (cons a n b) →
+  nf (cons a' n' b') →
+  lt b (cons a' n' b').
+ +
+#[global] Hint Resolve nat_lt_cons : rpo.
+#[global] Hint Resolve lt_subterm2 lt_subterm1 : rpo.
+#[global] Hint Resolve T1_size3 T1_size2 T1_size1 : rpo.
+ +
+Lemma nat_2_term_mono (n n': nat):
+  n < n' → rpo (nat_2_term n) (nat_2_term n').
+ +
+ +
+Theorem lt_inc_rpo_0 : ∀ n,
+    ∀ o' o, (T1_size o + T1_size o' ≤ n)%nat→
+                 lt o o' → nf o → nf o' →
+                 rpo (T1_2_term o) (T1_2_term o').
+ +
+Remark R1 : Acc P.prec nat_0.
+ +
+#[global] Hint Resolve R1 : rpo.
+ +
+Remark R2 : Acc P.prec ord_zero.
+ +
+#[global] Hint Resolve R2 : rpo.
+ +
+Remark R3 : Acc P.prec nat_S.
+ +
+#[global] Hint Resolve R3 : rpo.
+ +
+Remark R4 : Acc P.prec ord_cons.
+ +
+#[global] Hint Resolve R4 : rpo.
+ +
+Theorem well_founded_rpo : well_founded rpo.
+ +
+Section well_founded.
+ +
+  Let R := restrict nf lt.
+ +
+  #[local] Hint Unfold restrict R : rpo.
+ +
+  Lemma R_inc_rpo (o o':T1) :
+    R o o' → rpo (T1_2_term o) (T1_2_term o').
+ +
+ +
+  Lemma nf_Wf : well_founded_restriction _ nf lt.
+ +
+
+ +
+For compatibility with T1 +
+
+  Lemma nf_Acc : ∀ alpha, nf alpha → Acc LT alpha.
+ +
+End well_founded.
+ +
+Definition transfinite_recursor_lt :
+  ∀ (P:T1 → Type),
+    (∀ x:T1, nf x →
+                  (∀ y:T1, nf y → lt y x → P y) → P x) →
+    ∀ a, nf a → P a.
+ +
+Definition transfinite_recursor :
+  ∀ (P:T1 → Type),
+    (∀ x:T1, nf x →
+                  (∀ y:T1, y t1< x → P y) → P x) →
+    ∀ a, nf a → P a.
+ +
+Ltac transfinite_induction_lt a :=
+  pattern a; apply transfinite_recursor_lt ;[ | try assumption].
+ +
+Ltac transfinite_induction a :=
+  pattern a; apply transfinite_recursor;[ | try assumption].
+ +
+End Alt.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.F_alpha.html b/theories/html/hydras.Epsilon0.F_alpha.html new file mode 100644 index 00000000..9bb762c9 --- /dev/null +++ b/theories/html/hydras.Epsilon0.F_alpha.html @@ -0,0 +1,497 @@ + + + + + +hydras.Epsilon0.F_alpha + + + + +
+ + + +
+ +

Library hydras.Epsilon0.F_alpha

+ +
+
+ +
+

The Wainer hierarchy of rapidly growing functions (variant)

+ + +
+ + After Wainer, Ketonen, Solovay, etc . + +
+
+ +
+From hydras Require Import Iterates Simple_LexProd Exp2.
+From hydras Require Import E0 Canon Paths primRec Hprime.
+Import RelationClasses Relations.
+ +
+From Coq Require Import ArithRing Lia.
+Require Import Compat815.
+ +
+From Equations Require Import Equations.
+ +
+From hydras Require Import primRec.
+ +
+
+ +
+For masking primRec's iterate +
+
+ +
+Import Prelude.Iterates.
+ +
+
+ +
+

Definition, using coq-equations

+ + +
+ +The following definition is not accepted by the equations plug-in. + +
+ + +
+
+ +
+#[global] Instance Olt : WellFounded E0lt := E0lt_wf.
+ +
+ +
+
+ +
+ +
+ + Indeed, we define the -th iterate of F_ alpha by well-founded + recursion on the pair (alpha,n), then F_ alpha as the first iterate + of the defined function. + +
+
+ +
+Definition call_lt (c c' : E0 × nat) :=
+  lexico E0lt (Peano.lt) c c'.
+ +
+Lemma call_lt_wf : well_founded call_lt.
+ +
+#[ global ] Instance WF : WellFounded call_lt := call_lt_wf.
+ +
+ +
+Equations F_star (c: E0 × nat) (i:nat) : nat by wf c call_lt :=
+  F_star (alpha, 0) i := i;
+  F_star (alpha, 1) i
+    with E0_eq_dec alpha E0zero :=
+    { | left _zero ⇒ S i ;
+      | right _nonzero
+          with Utils.dec (E0limit alpha) :=
+          { | left _limit ⇒ F_star (Canon alpha i,1) i ;
+            | right _notlimit ⇒
+              F_star (E0_pred alpha, S i) i}};
+  F_star (alpha,(S (S n))) i :=
+    F_star (alpha, 1) (F_star (alpha, (S n)) i).
+ +
+ +
+ +
+ +
+ +
+Definition F_ alpha i := F_star (alpha, 1) i.
+ +
+
+ +
+

We get the "usual" equations for F_

+ +
+ +

Relations between F_star and F_

+ +
+
+ +
+Lemma F_star_zero_eqn : ∀ alpha i, F_star (alpha, 0) i = i.
+ +
+Lemma Fstar_S : ∀ alpha n i, F_star (alpha, S (S n)) i =
+                                  F_ alpha (F_star (alpha, S n) i).
+ +
+Lemma F_eq2 : ∀ alpha i,
+    E0is_succ alpha →
+    F_ alpha i = F_star (E0_pred alpha, S i) i.
+ +
+Lemma F_star_Succ: ∀ alpha n i,
+    F_star (alpha, S n) i =
+    F_ alpha (F_star (alpha, n) i).
+ +
+Lemma F_star_iterate : ∀ alpha n i,
+    F_star (alpha, n) i = iterate (F_ alpha) n i.
+ +
+
+ +
+

Usual equations for F_

+ +
+
+ +
+ +
+Lemma F_zero_eqn : ∀ i, F_ E0zero i = S i. +
+Lemma F_lim_eqn : ∀ alpha i,
+    E0limit alpha →
+    F_ alpha i = F_ (Canon alpha i) i. +
+Lemma F_succ_eqn : ∀ alpha i,
+    F_ (E0_succ alpha) i = iterate (F_ alpha) (S i) i. +
+
+ +
+

First steps of the hierarchy

+ +
+ + performs an induction only on the occ1-th and occ2_th occurrences of n +
+
+ +
+Tactic Notation "undiag2" constr(n) integer(occ1) integer(occ2) :=
+  let n' := fresh "n" in
+  generalize n at occ1 occ2; intro n'; induction n'.
+ +
+ +
+Lemma LF1 : ∀ i, F_ 1 i = S (2 × i). +
+Lemma LF2 : ∀ i, exp2 i × i < F_ 2 i. +
+Corollary LF2' : ∀ i, 1 ≤ i → exp2 i < F_ 2 i.
+ +
+Lemma F_alpha_0_eq : ∀ alpha: E0, F_ alpha 0 = 1.
+       Search (Canon ?a _ o< ?a).
+Qed.
+ +
+
+ +
+Properties of F_ alpha +
+
+ +
+ +
+Theorem F_alpha_mono alpha : strict_mono (F_ alpha). +
+Theorem F_alpha_gt alpha : ∀ n, n < F_ alpha n. +
+Corollary F_alpha_positive alpha : ∀ n, 0 < F_ alpha n. +
+Theorem F_alpha_Succ_le alpha : F_ alpha <<= F_ (E0_succ alpha).
+ +
+Theorem F_alpha_dom alpha :
+  dominates_from 1 (F_ (E0_succ alpha)) (F_ alpha). +
+Theorem F_restricted_mono_l alpha :
+  ∀ beta n, Canon_plus n alpha beta →
+                 F_ beta n ≤ F_ alpha n. +
+ #[deprecated(note="use F_alpha_gt")]
+  Notation F_alpha_ge_S := StrictOrder_Transitive (only parsing).
+ +
+ +
+Lemma LF2_0 : dominates_from 0 (F_ 2) (fun i ⇒ exp2 i × i).
+ +
+Lemma LF3_2 : dominates_from 2 (F_ 3) (fun n ⇒ iterate exp2 (S n) n).
+ +
+
+ +
+From Ketonen and Solovay, page 284, op. cit. +
+
+ +
+ +
+Section F_monotony_l.
+ +
+  Variables alpha beta : E0.
+  Hypothesis H'_beta_alpha : E0lt beta alpha.
+ +
+ +
+ +
+  Lemma F_mono_l_0 : ∀ n,
+      Canon_plus (S n) alpha beta →
+      ∀ i, (S n < i → F_ beta i < F_ alpha i)%nat.
+ +
+ +
+  Lemma F_mono_l: dominates (F_ alpha) (F_ beta). +
+End F_monotony_l.
+ +
+
+ +
+

Comparison with the Hardy hierarchy

+ + +
+ + (F_ alpha (S n) ≤ H'_ (Phi0 alpha) (S n)) + +
+
+ +
+Section H'_F.
+ +
+  Let P (alpha: E0) :=
+        ∀ n, (F_ alpha (S n) ≤ H'_ (E0_phi0 alpha) (S n))%nat.
+ +
+ Variable alpha: E0.
+ +
+ Hypothesis IHalpha : ∀ beta, beta o< alpha → P beta.
+ +
+ Lemma HF0 : P E0zero.
+ +
+ Lemma HFsucc : E0is_succ alpha → P alpha.
+ +
+
+ +
+The following proof is far from being trivial. + It uses some lemmas from the Ketonen-Solovay machinery +
+
+ +
+  Lemma HFLim : E0limit alpha → P alpha.
+ +
+End H'_F.
+ +
+ +
+Lemma H'_F alpha : ∀ n, F_ alpha (S n) ≤ H'_ (E0_phi0 alpha) (S n).
+ +
+
+ +
+

A variant (Lob-Wainer hierarchy)

+ + +
+
+ +
+Equations f_star (c: E0 × nat) (i:nat) : nat by wf c call_lt :=
+  f_star (alpha, 0) i := i;
+  f_star (alpha, 1) i
+    with E0_eq_dec alpha E0zero :=
+    { | left _zero ⇒ S i ;
+      | right _nonzero
+          with Utils.dec (E0limit alpha) :=
+          { | left _limit ⇒ f_star (Canon alpha i,1) i ;
+            | right _successor ⇒
+              f_star (E0_pred alpha, i) i}};
+  f_star (alpha,(S (S n))) i :=
+    f_star (alpha, 1) (f_star (alpha, (S n)) i).
+ +
+ +
+ +
+ +
+ +
+
+ +
+ Finally, f_ alpha is defined as its first iterate ! +
+
+ +
+Definition f_ alpha i := f_star (alpha, 1) i.
+ +
+
+ +
+

We get the "usual" equations for F_

+ +
+ +

Relations between F_star and F_

+ +
+
+ +
+Lemma f_star_zero_eqn : ∀ alpha i, f_star (alpha, 0) i = i.
+ +
+Lemma fstar_S : ∀ alpha n i, f_star (alpha, S (S n)) i =
+                                  f_ alpha (f_star (alpha, S n) i).
+ +
+Lemma f_eq2 : ∀ alpha i,
+    E0is_succ alpha →
+    f_ alpha i = f_star (E0_pred alpha, i) i.
+ +
+Lemma f_star_Succ: ∀ alpha n i,
+    f_star (alpha, S n) i =
+    f_ alpha (f_star (alpha, n) i).
+ +
+Lemma f_star_iterate : ∀ alpha n i,
+    f_star (alpha, n) i = iterate (f_ alpha) n i.
+ +
+
+ +
+

Usual equations for f_

+ +
+
+ +
+Lemma f_zero_eqn : ∀ i, f_ E0zero i = S i.
+ +
+Lemma f_lim_eqn : ∀ alpha i, E0limit alpha →
+                                   f_ alpha i = f_ (Canon alpha i) i.
+ +
+Lemma f_succ_eqn : ∀ alpha i,
+    f_ (E0_succ alpha) i = iterate (f_ alpha) i i.
+ +
+Lemma id_le_f_alpha (alpha: E0) : ∀ i, i ≤ f_ alpha i.
+ +
+Section Properties_of_f_alpha.
+ +
+Record Q (alpha:E0) : Prop :=
+    mkQ {
+        QA : strict_mono (f_ alpha);
+        QD : dominates_from 2 (f_ (E0_succ alpha)) (f_ alpha);
+        QE : ∀ beta n, Canon_plus n alpha beta →
+                            f_ beta n ≤ f_ alpha n}.
+ +
+Section The_induction.
+ +
+  Lemma QA0 : strict_mono (f_ E0zero).
+ +
+  Lemma QD0 : dominates_from 2 (f_ (E0_succ E0zero)) (f_ E0zero).
+ +
+ +
+
+ +
+TODO : Study the equality F alpha i = Nat.pred (f alpha (S i)) +
+
+ +
+ +
+End The_induction.
+ +
+End Properties_of_f_alpha.
+ +
+Print Assumptions F_zero_eqn.
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.F_omega.html b/theories/html/hydras.Epsilon0.F_omega.html new file mode 100644 index 00000000..1324d362 --- /dev/null +++ b/theories/html/hydras.Epsilon0.F_omega.html @@ -0,0 +1,217 @@ + + + + + +hydras.Epsilon0.F_omega + + + + +
+ + + +
+ +

Library hydras.Epsilon0.F_omega

+ +
+
+ +
+ About F_ omega +
+
+ +
+From hydras Require Import Iterates F_alpha E0 Ack AckNotPR.
+From Coq Require Import ArithRing Lia Arith.
+Import Exp2.
+Open Scope nat_scope.
+ +
+
+ +
+

Rewriting lemmas

+ +
+
+ +
+Lemma Ack_iterate_rw n p : Ack (S n) (S p) = iterate (Ack n) (S (S p)) 1.
+ +
+Lemma F_iterate_rw n p : F_ (S n) (S p) = iterate (F_ n) (S (S p)) (S p).
+ +
+
+ +
+

Comparison between Ack n and F_ n

+ +
+
+ +
+Lemma L02 p: 2 ≤ p → 2 × p + 3 ≤ exp2 p × p.
+ +
+Section inductive_step.
+  Variable n: nat.
+  Hypothesis Hn : ∀ p, 2 ≤ p → Ack n p ≤ F_ n p.
+ +
+  Lemma L: ∀ p, 2 ≤ p → Ack (S n) p ≤ F_ (S n) p.
+ +
+End inductive_step.
+ +
+Lemma L2 : ∀ n, 2 ≤ n → ∀ p, 2 ≤ p →
+                                         Ack n p ≤ F_ n p.
+ +
+Lemma F_vs_Ack n : 2 ≤ n → Ack n n ≤ F_ E0_omega n. +
+
+ +
+

F_ omega is not primitive recursive

+ +
+
+ +
+Import primRec extEqualNat.
+ +
+Section F_omega_notPR.
+ +
+  Context (F_omega_PR : isPR 1 (F_ E0_omega)).
+ +
+  Lemma F_omega_not_PR : False.
+ +
+End F_omega_notPR.
+ +
+
+ +
+

F_ alpha is not primitive recursive, for omega ≤ alpha

+ +
+
+ +
+Section F_alpha_notPR.
+ +
+  Variable alpha: E0.
+  Section case_lt.
+ +
+    Hypothesis Halpha : E0_omega o< alpha.
+ +
+    Remark R5: ∃ N, ∀ k, N ≤ k → F_ E0_omega k < F_ alpha k.
+ +
+    Context (H: isPR 1 (F_ alpha)).
+ +
+    Remark R00 : F_ alpha >> fun n ⇒ Ack n n.
+ +
+    Lemma FF : False.
+ +
+  End case_lt.
+ +
+ +
+  Hypothesis H : E0_omega o≤ alpha.
+  Context (H0: isPR 1 (F_ alpha)).
+ +
+  Lemma F_alpha_not_PR: False.
+ +
+End F_alpha_notPR.
+ +
+
+ +
+

On the other hand, F_ n is PR for any n:nat

+ + +
+
+ +
+#[export] Instance F_0_isPR : isPR 1 (F_ 0).
+ +
+Section step.
+  Variable n: nat.
+  Hypothesis Hn : isPR 1 (F_ n).
+ +
+  Let F := fun a b ⇒ nat_rec (fun _ ⇒ nat) a (fun x y ⇒ F_ n y) b.
+ +
+  Remark L00: ∀ i, F_ (E0_succ n) i = F i (S i) .
+ +
+  #[local] Instance R01 : isPR 2 F.
+ +
+  #[local] Instance R02 : isPR 1 (fun i ⇒ F i (S i)).
+ +
+  #[local] Instance R03 : isPR 1 (F_ (S n)).
+ +
+End step.
+ +
+#[export] Instance F_n_PR (n:nat) : isPR 1 (F_ n).
+ +
+
+ +
+ Keep in mind that isPR is of sort Set, so not and iff + cannot be used +
+
+ +
+Lemma F_alpha_PR_inv: ∀ alpha, isPR 1 (F_ alpha) → alpha o< E0_omega.
+ +
+Lemma F_alpha_notPR_inv: ∀ alpha,
+    (isPR 1 (F_ alpha) → False) → E0_omega o≤ alpha.
+ +
+Check F_alpha_not_PR.
+Check F_alpha_PR_inv.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.Hessenberg.html b/theories/html/hydras.Epsilon0.Hessenberg.html new file mode 100644 index 00000000..2c51cb8f --- /dev/null +++ b/theories/html/hydras.Epsilon0.Hessenberg.html @@ -0,0 +1,333 @@ + + + + + +hydras.Epsilon0.Hessenberg + + + + +
+ + + +
+ +

Library hydras.Epsilon0.Hessenberg

+ +
+
+ +
+

Hessenberg sum of ordinals terms (commutative and strictly monotonous)

+ + +
+ +Pierre Castéran, Labri, University of Bordeaux +
+
+ +
+From Coq Require Import Arith ArithRing Lia.
+From hydras Require Import Prelude.More_Arith Prelude.Merge_Sort
+     Epsilon0.T1.
+ +
+Set Implicit Arguments.
+ +
+ +
+Fixpoint oplus (alpha beta : T1) : T1 :=
+  let fix oplus_aux beta {struct beta} :=
+      match alpha, beta with
+        | zero, _ ⇒ beta
+        | _, zero ⇒ alpha
+        | cons a1 n1 b1, cons a2 n2 b2 ⇒
+          match compare a1 a2 with
+            | Gt ⇒ cons a1 n1 (oplus b1 beta)
+            | Lt ⇒ cons a2 n2 (oplus_aux b2)
+            | Eq ⇒ cons a1 (S (n1 + n2)%nat) (oplus b1 b2)
+          end
+      end
+  in oplus_aux beta.
+ +
+Infix "o+" := oplus (at level 50, left associativity).
+ +
+ +
+ +
+Fixpoint o_finite_mult n alpha :=
+match n with
+    0 ⇒ T1.zero
+  | S p ⇒ alpha o+ (o_finite_mult p alpha)
+                 end.
+ +
+Open Scope t1_scope.
+ +
+ +
+ +
+Lemma oplus_0_r (alpha : T1) : alpha o+ zero = alpha.
+ +
+Lemma oplus_0_l (beta : T1): zero o+ beta = beta.
+ +
+ +
+Lemma oplus_compare_Lt:
+  ∀ a n b a' n' b',
+  compare a a' = Lt →
+  cons a n b o+ cons a' n' b' = cons a' n' (cons a n b o+ b').
+ +
+Lemma oplus_lt_rw :
+  ∀ a n b a' n' b', T1.lt a a' →
+                         cons a n b o+ cons a' n' b'=
+                         cons a' n' (cons a n b o+ b').
+ +
+Lemma oplus_eq_rw :
+  ∀ a n b n' b',
+    cons a n b o+ cons a n' b'=
+    cons a (S (n+n')%nat) (b o+ b').
+ +
+Lemma oplus_gt_rw :
+  ∀ a n b a' n' b',
+  T1.lt a' a →
+    cons a n b o+ cons a' n' b' = cons a n (b o+ cons a' n' b').
+ +
+Lemma oplus_compare_Gt :
+  ∀ a n b a' n' b', compare a a' = Gt →
+                         cons a n b o+ cons a' n' b' =
+                         cons a n (b o+ cons a' n' b').
+ +
+ +
+Lemma oplus_eqn :
+  ∀ a b,
+    a o+ b =
+    match a, b with
+        T1.zero, _ ⇒ b
+      | _, T1.zero ⇒ a
+      | cons a1 n1 b1, cons a2 n2 b2 ⇒
+        match compare a1 a2 with
+            Gt ⇒ cons a1 n1 (b1 o+ b)
+          | Eq ⇒ cons a1 (S (n1 + n2)) (b1 o+ b2)
+          | Lt ⇒ cons a2 n2 (a o+ b2)
+        end
+    end.
+ +
+Lemma oplus_cons_cons :
+  ∀ a n b a' n' b',
+    (cons a n b) o+ (cons a' n' b') =
+    match compare a a' with
+          | Gt ⇒ cons a n (b o+ (cons a' n' b') )
+          | Eq ⇒ cons a (S (n + n')%nat) (b o+ b')
+          | Lt ⇒ cons a' n' (cons a n b o+ b')
+    end.
+ +
+Lemma lt_a_phi0_b_oplus : ∀ gamma alpha beta,
+                        alpha <_phi0 gamma →
+                        beta <_phi0 gamma →
+                        alpha o+ beta <_phi0 gamma.
+ +
+Lemma oplus_bounded_phi0 alpha beta gamma :
+  nf alpha → nf beta → nf gamma →
+  lt alpha (phi0 gamma) →
+  lt beta (phi0 gamma) →
+  lt (alpha o+ beta) (phi0 gamma).
+ +
+Section Proof_of_plus_nf.
+ +
+  Lemma oplus_nf_0 (gamma : T1):
+    nf gamma → ∀ alpha beta, nf alpha → nf beta →
+                               T1.lt alpha gamma →
+                               T1.lt beta gamma →
+                               nf (alpha o+ beta).
+ +
+  Lemma oplus_nf (alpha beta : T1) :
+      nf alpha → nf beta → nf (alpha o+ beta).
+ +
+End Proof_of_plus_nf.
+ +
+Lemma o_finite_mult_nf : ∀ a n, nf a → nf (o_finite_mult n a).
+ +
+Section Proof_of_oplus_comm.
+  Lemma oplus_comm_0 (gamma: T1):
+    nf gamma →
+    ∀ alpha beta, nf alpha → nf beta →
+                        T1.lt alpha gamma →
+                        T1.lt beta gamma →
+                        alpha o+ beta = beta o+ alpha.
+ +
+  Lemma oplus_comm (alpha beta: T1):
+   nf alpha → nf beta → alpha o+ beta = beta o+ alpha.
+ +
+End Proof_of_oplus_comm.
+ +
+Lemma oplus_lt_rw2 : ∀ a n b x, nf (cons a n b) → nf x →
+                                     x <_phi0 a →
+                                     cons a n b o+ x =
+                                     cons a n (b o+ x).
+ +
+Section Proof_of_oplus_assoc.
+ +
+  Ltac ass_rw Hrec alpha a b c :=
+    match goal with |- context Gamma [oplus ?a (oplus ?b ?c)] ⇒
+                    erewrite Hrec with alpha a b c end.
+ +
+  Ltac ass_rw_rev Hrec alpha a b c :=
+    match goal with |- context Gamma [oplus (oplus ?a ?b) ?c] ⇒
+                          erewrite <- Hrec with alpha a b c end.
+  Lemma oplus_assoc_0 (alpha: T1):
+      nf alpha →
+      ∀ a b c, nf a → nf b → nf c →
+                      T1.lt a alpha →
+                      T1.lt b alpha → T1.lt c alpha →
+                      a o+ (b o+ c) = (a o+ b) o+ c.
+ +
+  Lemma oplus_assoc (alpha beta gamma:T1) :
+    nf alpha → nf beta → nf gamma →
+    alpha o+ (beta o+ gamma) =
+    alpha o+ beta o+ gamma.
+ +
+End Proof_of_oplus_assoc.
+ +
+Section Proof_of_oplus_lt1.
+  Variables a1 a2: T1.
+  Variable n : nat.
+  Hypothesis H0 : nf (cons a1 n a2).
+ +
+  Lemma oplus_lt_0 : ∀ b, nf b → T1.lt b (b o+ (cons a1 n a2)).
+ +
+End Proof_of_oplus_lt1.
+ +
+Lemma oplus_lt1 : ∀ a b, nf a → nf b → T1.lt T1.zero a →
+                              T1.lt b (b o+ a).
+ +
+Lemma oplus_lt2 : ∀ a b, nf a → nf b → T1.lt T1.zero b →
+                              T1.lt a (b o+ a).
+ +
+Lemma oplus_le1 : ∀ a b, nf a → nf b → leq lt a (a o+ b).
+ +
+Lemma oplus_le2 : ∀ a b, nf a → nf b → leq lt b (a o+ b).
+ +
+Lemma oplus_strict_mono_0 :
+  ∀ alpha, nf alpha →
+                ∀ a (Ha:nf a) b (Hb: nf b) c (Hc : nf c),
+                  lt a alpha → lt c alpha → lt b c →
+                  lt (a o+ b) (a o+ c).
+ +
+Lemma oplus_strict_mono_r : ∀ a b c, nf a → nf b → nf c →
+                                          lt b c → lt (a o+ b) (a o+ c).
+ +
+Lemma oplus_strict_mono_l : ∀ a b c, nf a → nf b → nf c →
+                                          T1.lt a b →
+                                          T1.lt (a o+ c) (b o+ c).
+ +
+ +
+Lemma oplus_strict_mono_LT_l (alpha beta gamma : T1) :
+  nf gamma → alpha t1< beta →
+  alpha o+ gamma t1< beta o+ gamma. +
+Lemma oplus_strict_mono_LT_r (alpha beta gamma : T1) :
+  nf alpha → beta t1< gamma →
+  alpha o+ beta t1< alpha o+ gamma. +
+Lemma oplus_strict_mono_bi : ∀ a b c d ,
+    nf a → nf b → nf c → nf d →
+    T1.lt a b → T1.lt c d → T1.lt (a o+ c) (b o+ d).
+ +
+Lemma oplus_of_phi0_0 : ∀ a b,
+                          phi0 a o+ phi0 b =
+                          match compare (phi0 a) (phi0 b)
+                          with Lt ⇒ cons b 0 (cons a 0 T1.zero)
+                            | Eq ⇒ cons a 1 T1.zero
+                            | Gt ⇒ cons a 0 (cons b 0 T1.zero)
+                          end.
+ +
+Lemma oplus_of_phi0 : ∀ a b,
+                        phi0 a o+ phi0 b =
+                        match compare a b
+                        with Lt ⇒ cons b 0 (cons a 0 T1.zero)
+                          | Eq ⇒ cons a 1 T1.zero
+                          | Gt ⇒ cons a 0 (cons b 0 T1.zero)
+                        end.
+ +
+Lemma o_finite_mult_rw : ∀ a n, o_finite_mult (S n) (phi0 a) =
+                                     cons a n T1.zero.
+ +
+Lemma o_finite_mult_lt_phi0_1 : ∀ a b n,
+                                  T1.lt a b →
+                                  T1.lt (o_finite_mult n (phi0 a)) (phi0 b).
+ +
+Lemma o_finite_mult_mono : ∀ a b n, nf a → nf b → T1.lt a b →
+                                          T1.lt (o_finite_mult (S n) a)
+                                             (o_finite_mult (S n) b).
+ +
+Lemma oplus_lt_phi0 : ∀ a b c, nf a → nf b → nf c →
+                                     T1.lt a c → T1.lt b c →
+                                     T1.lt (phi0 a o+ phi0 b) (phi0 c).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.Hprime.html b/theories/html/hydras.Epsilon0.Hprime.html new file mode 100644 index 00000000..0d60e494 --- /dev/null +++ b/theories/html/hydras.Epsilon0.Hprime.html @@ -0,0 +1,577 @@ + + + + + +hydras.Epsilon0.Hprime + + + + +
+ + + +
+ +

Library hydras.Epsilon0.Hprime

+ +
+
+ +
+

A hierarchy of rapidly growing functions

+ + After Hardy, Wainer, Ketonen, Solovay, etc . + +
+ + Pierre Casteran, LaBRI, University of Bordeaux +
+ + Remark + +
+ +Some notations (mainly names of fast-growing functions) of our development + may differ slightly from the litterature. Although this fact does not affect +our proofs, we are preparing a future version where the names + F_ alpha, f_alpha, H _alpha, etc., are fully consistent with the + cited articles. + This change will be provisionally made in a branch called Hardy + (in reference to the Hardy hierarchy). + +
+ + +
+
+ +
+From Coq Require Import ArithRing Lia.
+From hydras Require Import E0 Canon Paths.
+From hydras Require Import Exp2 Iterates Simple_LexProd.
+Import RelationClasses Relations.
+From hydras Require Import Compat815.
+From Equations Require Import Equations.
+ +
+ Open Scope E0_scope.
+ +
+
+ +
+

A variant of the Hardy Wainer hierarchy

+ +
+
+ +
+#[ global ] Instance Olt : WellFounded E0lt := E0lt_wf.
+ +
+ +
+Equations H'_ (alpha: E0) (i:nat) : nat by wf alpha E0lt :=
+  H'_ alpha i with E0_eq_dec alpha E0zero :=
+    { | left _zero ⇒ i ;
+      | right _nonzero
+          with Utils.dec (E0limit alpha) :=
+          { | left _limit ⇒ H'_ (Canon alpha (S i)) i ;
+            | right _successor ⇒ H'_ (E0_pred alpha) (S i)}}.
+ +
+Solve All Obligations with auto with E0.
+ +
+
+ +
+Paraphrases of the equations for H' +
+
+ +
+Lemma H'_eq1 : ∀ i, H'_ E0zero i = i.
+ +
+ +
+Lemma H'_eq2_0 alpha i :
+  E0is_succ alpha →
+  H'_ alpha i = H'_ (E0_pred alpha) (S i).
+ +
+Lemma H'_eq3 alpha i :
+  E0limit alpha → H'_ alpha i = H'_ (Canon alpha (S i)) i.
+ +
+Lemma H'_eq2 alpha i :
+  H'_ (E0_succ alpha) i = H'_ alpha (S i).
+ +
+#[local] Hint Rewrite H'_eq1 H'_eq2 : H'_rw.
+ +
+Ltac lim_rw alpha := (assert (E0limit alpha) by auto with E0);
+                     rewrite (H'_eq3 alpha); auto with E0.
+ +
+
+ +
+Examples : First steps of the hierarchy +
+
+ +
+ +
+Lemma H'_Fin : ∀ i k : nat, H'_ (E0fin i) k = (i+k)%nat.
+ +
+ +
+Lemma H'_omega : ∀ k, H'_ E0_omega k = S (2 × k)%nat.
+ +
+ +
+Lemma H'_Plus_Fin alpha : ∀ i k : nat,
+    H'_ (alpha + i)%e0 k = H'_ alpha (i + k)%nat.
+ +
+Lemma H'_omega_double k :
+  H'_ (E0_omega × 2)%e0 k = (4 × k + 3)%nat.
+ +
+Lemma H'_omega_3 k : H'_ (E0_omega × 3)%e0 k = (8 × k + 7)%nat.
+ +
+Lemma H'_omega_4 k : H'_ (E0_omega × 4)%e0 k = (16 × k + 15)%nat.
+ +
+Lemma H'_omega_i (i:nat) : ∀ k,
+    H'_ (E0_omega × i)%e0 k = (exp2 i × k + Nat.pred (exp2 i))%nat.
+ +
+
+ +
+Let us note that the square of omega can be expressed through the + Phi0 function +
+
+ +
+Remark Phi0_def : E0_phi0 2 = ltac:(mko (T1omega × T1omega)%t1).
+ +
+ +
+Lemma H'_omega_sqr : ∀ k,
+    H'_ (E0_phi0 2)%e0 k = (exp2 (S k ) × (S k) - 1)%nat. +
+
+ +
+Composition lemma +
+
+ +
+Section H'_cons.
+Variable alpha : E0.
+Variable i : nat.
+ +
+Lemma H'_cons : ∀ beta, (beta o< E0_phi0 alpha)%e0 →
+                             ∀ k, H'_ (Cons alpha i beta) k=
+                                        H'_ (Omega_term alpha i) (H'_ beta k).
+ +
+ +
+Lemma H'_Omega_term_1 : alpha ≠ E0zero → ∀ k,
+    H'_ (Omega_term alpha (S i)) k =
+    H'_ (Omega_term alpha i) (H'_ (E0_phi0 alpha) k). +
+End H'_cons.
+ +
+Lemma H'_Omega_term_0 (alpha : E0) :
+alpha ≠ E0zero → ∀ i k,
+  H'_ (Omega_term alpha i) k = iterate (H'_ (E0_phi0 alpha)) (S i) k.
+ +
+Lemma H'_Fin_iterate : ∀ i k,
+    H'_ (E0fin (S i)) k = iterate (H'_ (E0fin 1)) (S i) k.
+ +
+ +
+Lemma H'_Omega_term (alpha : E0) :
+  ∀ i k,
+    H'_ (Omega_term alpha i) k =
+    iterate (H'_ (E0_phi0 alpha)) (S i) k.
+ +
+ +
+Definition H'_succ_fun f k := iterate f (S k) k.
+ +
+Lemma H'_Phi0_succ_1 alpha : alpha ≠ E0zero → ∀ k,
+      H'_ (E0_phi0 (E0_succ alpha)) k = H'_succ_fun (H'_ (E0_phi0 alpha)) k.
+ +
+Lemma H'_Phi0_succ_0 : ∀ k,
+    H'_ (E0_phi0 (E0_succ E0zero)) k = H'_succ_fun (H'_ (E0_phi0 E0zero)) k.
+ +
+Lemma H'_Phi0_succ alpha : ∀ k,
+    H'_ (E0_phi0 (E0_succ alpha)) k = H'_succ_fun (H'_ (E0_phi0 alpha)) k.
+ +
+ +
+Lemma H'_Phi0_Si : ∀ i k,
+    H'_ (E0_phi0 (S i)) k = iterate H'_succ_fun i (H'_ E0_omega) k.
+ +
+Lemma H'_omega_cube : ∀ k,
+    H'_ (E0_phi0 3)%e0 k = iterate (H'_ (E0_phi0 2)) (S k) k.
+ +
+Section H'_omega_cube_3.
+ +
+ +
+  Let f k := (exp2 (S k) × (S k) - 1)%nat.
+ +
+  Remark R0 k : H'_ (E0_phi0 3)%e0 k = iterate f (S k) k.
+ +
+ +
+   Fact F0 : H'_ (E0_phi0 3) 3 = f (f (f (f 3))).
+ +
+ +
+
+ +
+f (f 3)) +
+
+ +
+  Let N := (exp2 64 × 64 - 1)%nat.
+ +
+ +
+  Remark N_simpl: N = exp2 70 - 1.
+ +
+ +
+  Fact F1 : H'_ (E0_phi0 3) 3 = f (f N).
+ +
+  Fact F1_simpl :
+    H'_ (E0_phi0 3) 3 =
+    (exp2 (exp2 (S N) × S N) × (exp2 (S N) × S N) - 1)%nat.
+  Fact F2 : H'_ (E0_phi0 3 + 3) 0 = f (f N).
+ +
+  Remark f_minoration n p : 0 < n → n ≤ p → exp2 n ≤ f p.
+ +
+  Fact F3 : (exp2 (exp2 N) ≤ H'_ (E0_phi0 3 + 3) 0).
+ +
+End H'_omega_cube_3.
+ +
+ +
+Lemma H'_Phi0_omega :
+  ∀ k, H'_ (E0_phi0 E0_omega) k =
+            iterate H'_succ_fun k (H'_ E0_omega) k. +
+ +
+Lemma H'_Phi0_omega_exact_formula k :
+  H'_ (E0_phi0 E0_omega) k =
+    let F f i := iterate f (S i) i
+    in let g k := S (2 × k)%nat
+       in iterate F k g k.
+ +
+Lemma H'_omega_sqr_min : ∀ k, 0 ≠ k →
+                                    (exp2 (S k) ≤ H'_ (E0_phi0 2) k)%nat.
+ +
+Lemma H'_omega_cube_min k :
+  0 ≠ k → (hyper_exp2 (1 + k) ≤ H'_ (E0_phi0 3) k)%nat.
+ +
+Remark H'_non_mono1 :
+  ¬ (∀ alpha beta k,
+        (alpha o≤ beta)%e0 →
+        (H'_ alpha k ≤ H'_ beta k)%nat).
+ +
+ +
+Section Proof_of_Abstract_Properties.
+ +
+  Record P (alpha:E0) : Prop :=
+    mkP {
+        PA : strict_mono (H'_ alpha);
+        PB : alpha ≠ E0zero → ∀ n, (n < H'_ alpha n)%nat;
+        PC : H'_ alpha <<= H'_ (E0_succ alpha);
+        PD : dominates_from 1 (H'_ (E0_succ alpha)) (H'_ alpha);
+        PE : ∀ beta n, Canon_plus n alpha beta →
+                            (H'_ beta n ≤ H'_ alpha n)%nat}.
+ +
+  Section The_induction.
+ +
+ +
+    Lemma PA_Zero : strict_mono (H'_ E0zero).
+ +
+    Lemma PD_Zero : dominates_from 1 (H'_ (E0_succ E0zero)) (H'_ E0zero).
+ +
+    #[local] Hint Resolve PD_Zero PA_Zero : E0.
+ +
+    Lemma PC_Zero : H'_ E0zero <<= H'_ (E0_succ E0zero).
+ +
+    #[local] Hint Resolve PC_Zero : core.
+ +
+    Lemma PZero : P E0zero.
+ +
+    Variable alpha : E0.
+    Hypothesis Halpha : ∀ beta, E0lt beta alpha → P beta.
+ +
+    Section alpha_Succ.
+      Variable beta: E0.
+      Hypothesis alpha_def : alpha = E0_succ beta.
+ +
+      Remark PA_Succ : strict_mono (H'_ alpha).
+ +
+      Remark RB : alpha ≠ E0zero →∀ n, (n < H'_ alpha n)%nat.
+ +
+      Remark RD : dominates_from 1 (H'_ (E0_succ alpha)) (H'_ alpha).
+ +
+      Remark RE : ∀ beta n, Canon_plus n alpha beta →
+                                 (H'_ beta n ≤ H'_ alpha n)%nat.
+ +
+      Remark RC : H'_ alpha <<= H'_ (E0_succ alpha).
+ +
+      Remark RP : P alpha.
+ +
+    End alpha_Succ.
+ +
+    Section alpha_limit.
+      Hypothesis Hlim : E0limit alpha.
+ +
+      Remark RBlim : ∀ n, (n < H'_ alpha n)%nat.
+ +
+      Remark RAlim : strict_mono (H'_ alpha).
+ +
+      Remark RClim : H'_ alpha <<= H'_ (E0_succ alpha).
+ +
+      Remark RDlim : dominates_from 1 (H'_ (E0_succ alpha)) (H'_ alpha).
+ +
+      Remark RElim : ∀ beta n, Canon_plus n alpha beta →
+                                    (H'_ beta n ≤ H'_ alpha n)%nat.
+ +
+    End alpha_limit.
+ +
+    Lemma P_alpha_0 : P alpha.
+ +
+  End The_induction.
+ +
+ +
+  Theorem P_alpha : ∀ alpha, P alpha. +
+End Proof_of_Abstract_Properties.
+ +
+ +
+Section Abstract_Properties.
+ +
+  Variable alpha : E0.
+ +
+  Theorem H'_alpha_mono : strict_mono (H'_ alpha).
+ +
+  Theorem H'_alpha_gt : alpha ≠ E0zero → ∀ n, (n < H'_ alpha n)%nat.
+ +
+  Theorem H'_alpha_Succ_le : H'_ alpha <<= H'_ (E0_succ alpha).
+ +
+  Theorem H'_alpha_dom : dominates_from 1 (H'_ (E0_succ alpha)) (H'_ alpha).
+ +
+
+ +
+H'_ is not mononotonous in alpha in general. + Nevertheless, this lemma may help (from KS) +
+
+ +
+  Theorem H'_restricted_mono_l : ∀ beta n, Canon_plus n alpha beta →
+                                                H'_ beta n ≤ H'_ alpha n.
+ +
+  Lemma H'_alpha_ge_id : id <<= H'_ alpha.
+ +
+  Lemma H'_alpha_mono_weak : ∀ k l, k ≤ l →
+                                         H'_ alpha k ≤ H'_ alpha l.
+ +
+End Abstract_Properties.
+ +
+
+ +
+

Monotony of H' w.r.t. its first argument

+ + +
+ + Although Lemma H'_non_mono1 tells us that H' is not monotonous + with respect to its argument alpha, we prove that, if alpha<beta, then + H'_ alpha k < H'_ beta k for large enough k. + For that purpose, we apply a few lemmas from the Ketonen-Solovay article. + +
+ + +
+
+ +
+Lemma H'_mono_l_0 alpha beta :
+  alpha o< beta →
+  {n : nat | ∀ p, n ≤ p → H'_ alpha (S p) ≤ H'_ beta (S p)}.
+ +
+Lemma H'_mono_l_1 alpha beta :
+  alpha o≤ beta →
+  {n : nat | ∀ p, n ≤ p → H'_ alpha (S p) ≤ H'_ beta (S p)}.
+ +
+Section Proof_of_H'_mono_l.
+ +
+  Variables alpha beta: E0.
+  Hypothesis H_alpha_beta: alpha o< beta.
+ +
+  Section Succ_case.
+    Variable gamma: E0.
+    Hypothesis Hgamma : beta = E0_succ gamma.
+ +
+    Remark R1 : alpha o≤ gamma.
+ +
+    Remark R2 : {n : nat | ∀ p, n ≤ p → H'_ alpha (S p) ≤ H'_ gamma (S p)}.
+ +
+    Remark R3 : {n: nat | ∀ p, n ≤ p →
+                                    H'_ alpha (S p) < H'_ beta (S p)}.
+ +
+  End Succ_case.
+ +
+  Section Limit_case.
+    Hypothesis Hbeta: E0limit beta.
+ +
+    Remark R4 : E0_succ alpha o< beta.
+ +
+    Remark R5 : {n: nat | ∀ p, n ≤ p →
+                                     H'_ alpha (S p) < H'_ beta (S p)}.
+ +
+  End Limit_case.
+ +
+  Lemma H'_mono_l : {n: nat | ∀ p, n ≤ p →
+                                        H'_ alpha (S p) < H'_ beta (S p)}.
+ +
+ +
+  Theorem H'_dom : dominates_strong (H'_ beta) (H'_ alpha). +
+End Proof_of_H'_mono_l.
+ +
+About H'_dom.
+ +
+ +
+Goal
+  (0 < H'_ (ltac:(mko (T1omega × T1omega × T1omega)%t1)) 12)%nat.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.L_alpha.html b/theories/html/hydras.Epsilon0.L_alpha.html new file mode 100644 index 00000000..face1bf3 --- /dev/null +++ b/theories/html/hydras.Epsilon0.L_alpha.html @@ -0,0 +1,197 @@ + + + + + +hydras.Epsilon0.L_alpha + + + + +
+ + + +
+ +

Library hydras.Epsilon0.L_alpha

+ +
+
+ +
+

Computing the length of paths from alpha to zero

+ + +
+ + After Wainer, Ketonen, Solovay, etc . + +
+ + Pierre Casteran, LaBRI, University of Bordeaux +
+
+ +
+From hydras Require Import Hprime E0 Canon Paths
+     Large_Sets.
+From hydras Require Import Simple_LexProd Iterates .
+From Coq Require Import ArithRing Lia.
+ +
+From Equations Require Import Equations.
+Import RelationClasses Relations.
+ +
+#[global] Instance Olt : WellFounded E0lt := E0lt_wf.
+#[global] Hint Resolve Olt : E0.
+ +
+
+ +
+Using Coq-Equations for building a function which satisfies + Large_sets.L_spec +
+
+ +
+Equations L_ (alpha: E0) (i:nat) : nat by wf alpha E0lt :=
+  L_ alpha i with E0_eq_dec alpha E0zero :=
+    { | left _zero ⇒ i ;
+      | right _nonzero
+          with Utils.dec (E0limit alpha) :=
+          { | left _limit ⇒ L_ (Canon alpha i) (S i) ;
+            | right _successor ⇒ L_ (E0_pred alpha) (S i)}}.
+ +
+Solve All Obligations with auto with E0.
+ +
+ +
+ +
+About L__equation_1.
+ +
+
+ +
+Paraphrase of equations generated by coq-equations +
+
+ +
+ +
+Lemma L_zero_eqn : ∀ i, L_ E0zero i = i.
+ +
+Lemma L_eq2 alpha i :
+  E0is_succ alpha → L_ alpha i = L_ (E0_pred alpha) (S i).
+ +
+Lemma L_succ_eqn alpha i :
+  L_ (E0_succ alpha) i = L_ alpha (S i).
+ +
+#[export] Hint Rewrite L_zero_eqn L_succ_eqn : L_rw.
+ +
+Lemma L_lim_eqn alpha i :
+  E0limit alpha →
+  L_ alpha i = L_ (Canon alpha i) (S i).
+ +
+ +
+Lemma L_finite : ∀ i k :nat, L_ i k = (i+k)%nat. +
+Lemma L_omega : ∀ k, L_ E0_omega k = S (2 × k)%nat. +
+Lemma L_ge_id alpha : ∀ i, i ≤ L_ alpha i.
+ +
+ +
+Lemma L_ge_S alpha :
+  alpha ≠ E0zero → S <<= L_ alpha. +
+Lemma L_succ_ok beta f :
+  nf beta → S <<= f → L_spec beta f →
+  L_spec (succ beta) (fun k ⇒ f (S k)).
+ +
+
+ +
+L_ is correct w.r.t. its specification +
+
+ +
+Section L_correct_proof.
+ +
+  Let P alpha := L_spec (cnf alpha) (L_ alpha).
+ +
+  Lemma L_ok0 : P E0zero.
+ +
+  Lemma L_ok_succ beta : P beta → P (E0_succ beta).
+ +
+  Lemma L_ok_lim alpha :
+    (∀ beta, (beta o< alpha)%e0 → P beta) →
+    E0limit alpha → P alpha.
+ +
+ +
+ +
+  Lemma L_ok (alpha: E0) : P alpha.
+ +
+ +
+End L_correct_proof.
+ +
+ +
+Theorem L_correct alpha : L_spec (cnf alpha) (L_ alpha). +
+
+ +
+Comparison with Hardy's function H +
+
+ +
+ +
+Theorem H'_L_ alpha :
+  ∀ i:nat, (H'_ alpha i ≤ L_ alpha (S i))%nat. +
+From Coq Require Import Extraction.
+ +
+Recursive Extraction L_.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.Large_Sets.html b/theories/html/hydras.Epsilon0.Large_Sets.html new file mode 100644 index 00000000..65977a45 --- /dev/null +++ b/theories/html/hydras.Epsilon0.Large_Sets.html @@ -0,0 +1,729 @@ + + + + + +hydras.Epsilon0.Large_Sets + + + + +
+ + + +
+ +

Library hydras.Epsilon0.Large_Sets

+ +
+ +
+ +
+ +
+From hydras Require Import E0 Canon Paths.
+From hydras Require Import MoreLists Iterates Exp2.
+From Coq Require Import Lia Compare_dec.
+ +
+Import Relation_Operators Sorted Operators_Properties.
+Open Scope t1_scope.
+ +
+
+ +
+

minimal large sequences

+ +
+
+ +
+Definition mlarge alpha (s: list nat) := path_to zero s alpha.
+ +
+ +
+Definition mlargeS alpha s := path_toS zero s alpha.
+ +
+
+ +
+

Last step of a minimal path

+ +
+ + TODO : compare with K & S's H_alpha (p. 297) +
+
+ +
+ +
+Inductive L_spec : T1 → (nat → nat) → Prop :=
+  L_spec0 :
+    ∀ f, (∀ k, f (S k) = S k) → L_spec zero f
+| L_spec1 : ∀ alpha f,
+    alpha ≠ zero →
+    (∀ k,
+        mlarge alpha (interval (S k) (Nat.pred (f (S k))))) →
+    L_spec alpha f.
+ +
+ +
+
+ +
+Test functions +
    +
  • If f is correct w.r.t. L_spec, "Compute L_test alpha f k" should return (one=one). + +
    • +
  • If f k is too small, returns (alpha = one) (with one < alpha) + +
    • +
  • If f k is too big, returns (zero = one) + +
    • +
+
+
+ +
+Definition L_test (alpha:T1) f k :=
+  gnaw alpha (interval k (f k - 2)%nat) = one.
+ +
+ +
+
+ +
+

Paths starting with a finite ordinal (fin n)

+ +
+
+ +
+Lemma gnaw_finite_1_iota :
+  ∀ n i, gnaw (S n) (iota_from (S i) n) = 1.
+ +
+Lemma gnaw_finite_1 : ∀ n i, gnaw (S n) (interval (S i) (n + i)%nat) = 1.
+ +
+Lemma gnaw_finite : ∀ n i, gnaw (S n) (iota_from (S i) (S n)) = 0.
+ +
+Lemma gnaw_n_R : ∀ s n, ¬ In 0 s →
+                              (n < List.length s)%nat →
+                              gnaw (S n) s = zero.
+ +
+
+ +
+

Properties of m-largeness and L-functions

+ +
+
+ +
+ +
+Lemma mlarge_unicity alpha k l l' :
+  mlarge alpha (interval (S k) l) →
+  mlarge alpha (interval (S k) l') →
+  l = l'. +
+Lemma mlargeS_iff alpha x s : s ≠nil →
+                              mlargeS alpha (x::s) ↔
+                              gnawS alpha (but_last x s) = one.
+ +
+Lemma mlarge_unshift alpha s :
+   ¬In 0 s → mlarge alpha s ↔ mlargeS alpha (unshift s).
+ +
+ +
+Lemma mlarge_iff alpha x (s:list nat) :
+  s ≠ nil → ¬ In 0 (x::s) →
+  mlarge alpha (x::s) ↔ gnaw alpha (but_last x s) = one. +
+
+ +
+

About the length of mlarge intervals

+ +
+
+ +
+Lemma L_spec_inv2 alpha f :
+  L_spec alpha f → alpha ≠ zero →
+  (∀ k, mlarge alpha (interval (S k) (Nat.pred (f (S k))))).
+ +
+Lemma L_spec_compat alpha (f g : nat → nat) :
+  L_spec alpha f → (∀ n, f n = g n) → L_spec alpha g.
+ +
+
+ +
+

Properties of L_spec

+ + +
+ + +
+
+ +
+Lemma L_zero_inv f : L_spec zero f → ∀ k, f (S k) = S k. +
+Lemma L_pos_inv alpha f : alpha ≠ zero → L_spec alpha f →
+                        ∀ k, (S k < f (S k))%nat.
+ +
+ +
+Lemma L_spec_unicity alpha f g :
+  L_spec alpha f → L_spec alpha g → ∀ k, f (S k) = g (S k). +
+
+ +
+Composition lemmas for computing L_ alpha +
+
+ +
+ +
+Section succ.
+   Variables (beta : T1) (f : nat → nat).
+ +
+   Hypotheses (Hbeta : nf beta)
+              (f_mono : strict_mono f)
+              (f_Sle : S <<= f)
+              (f_ok : L_spec beta f).
+ +
+   Definition L_succ := fun k ⇒ f (S k).
+ +
+ +
+   Lemma L_succ_mono : strict_mono L_succ. +
+   Lemma L_succ_Sle : S <<= L_succ. +
+   Lemma L_succ_ok : L_spec (succ beta) L_succ. End succ.
+ +
+ +
+ +
+Section lim.
+  Variables (lambda : T1)
+            (Hnf : nf lambda)
+            (Hlim : T1limit lambda)
+            (f : nat → nat → nat)
+            (H : ∀ k, L_spec (canon lambda (S k)) (f (S k))).
+ +
+  Remark canon_not_null : ∀ k, canon lambda (S k) ≠ zero.
+ +
+  Definition L_lim k := f k (S k).
+ +
+  Lemma L_lim_ok : L_spec lambda L_lim. +
+End lim.
+ +
+
+ +
+

Finite ordinals

+ +
+
+ +
+ +
+Definition L_fin i := fun k ⇒ (i + k)%nat.
+ +
+Lemma L_finS_S_le i : S <<= L_fin (S i).
+ +
+Lemma L_fin_smono i: strict_mono (L_fin i).
+ +
+Lemma L_S_succ_rw i : ∀ k, L_fin (S i) k = L_succ (L_fin i) k.
+ +
+Lemma L_fin_ok i : L_spec (\F i) (L_fin i).
+ +
+Lemma mlarge_FS :
+  ∀ i k, mlarge (FS i) (interval (S k) (S (i+k)%nat)).
+ +
+ +
+Definition L_omega k := S (2 × k)%nat.
+ +
+ +
+Lemma L_omega_Sle : S <<= L_omega.
+ +
+Lemma L_omega_smono : strict_mono L_omega.
+ +
+Lemma mlarge_omega k : mlarge T1omega (interval (S k) (2 × (S k))%nat).
+ +
+Lemma L_omega_ok : L_spec T1omega L_omega.
+ +
+ +
+Lemma path_to_omega_mult (i k:nat) :
+  path_to (T1omega × i)
+          (interval (S k) (2 × (S k))%nat)
+          (T1omega × (S i)). +
+ +
+Lemma omega_mult_mlarge_0 i : ∀ k,
+    mlarge (T1omega × (S i))
+            (interval (S k)
+                      (Nat.pred (iterate (fun p ⇒ S (2 × p)%nat)
+                                         (S i)
+                                         (S k)))). +
+ +
+Definition L_omega_mult i (x:nat) := iterate L_omega i x.
+ +
+ +
+ +
+Example Ex : L_omega_mult 8 5 = 1535.
+ +
+Lemma L_omega_mult_Sle i : S <<= L_omega_mult (S i).
+ +
+Lemma L_omega_mult_ok (i: nat) :
+  L_spec (T1omega × i) (L_omega_mult i). +
+ +
+ +
+Lemma L_omega_mult_eqn (i : nat) :
+  ∀ (k : nat),
+    (0 < k)%nat → L_omega_mult i k = (exp2 i × S k - 1)%nat. +
+
+ +
+

omega × omega

+ +
+
+ +
+ +
+Definition L_omega_square k :=
+  iterate (fun z ⇒ S (2 × z)%nat) k (S k).
+ +
+ +
+Remark L_omega_square_eqn1 k : L_omega_square k =
+                                L_omega_mult k (S k).
+ +
+
+ +
+

Closed formula

+ +
+
+ +
+ +
+Lemma L_omega_square_eqn k :
+  (0 < k)%nat →
+  L_omega_square k = (exp2 k × (k + 2) - 1)%nat. +
+Lemma L_omega_square_Sle : S <<= L_omega_square.
+ +
+Lemma L_omega_square_smono : strict_mono L_omega_square.
+ +
+ +
+Lemma L_omega_square_ok:
+  L_spec (T1omega × T1omega) L_omega_square. +
+ +
+Section phi0_mult.
+ +
+  Variables (alpha : T1) (f : nat → nat).
+ +
+  Hypotheses (Halpha : nf alpha)
+             (f_mono : strict_mono f)
+             (f_Sle : S <<= f)
+             (f_ok : L_spec (T1.phi0 alpha) f).
+ +
+  Definition L_phi0_mult i := iterate f i.
+ +
+ Remark f_ok_inv :
+   ∀ k, mlarge (T1.phi0 alpha) (interval (S k) (Nat.pred (f (S k)))).
+ +
+Lemma L_phi0_mult_ok i:
+  L_spec (T1.cons alpha i zero) (L_phi0_mult (S i)).
+ +
+Lemma L_phi0_mult_smono i: strict_mono (L_phi0_mult i).
+ +
+ +
+Lemma L_phi0_mult_Sle i: S <<= L_phi0_mult (S i).
+ +
+End phi0_mult.
+ +
+ +
+ +
+Definition L_omega_square_times i := iterate L_omega_square i.
+ +
+Lemma L_omega_square_times_ok i : L_spec (T1.cons 2 i zero)
+                                         (L_omega_square_times (S i)).
+ +
+ +
+Definition L_omega_cube := L_lim L_omega_square_times .
+ +
+Lemma L_omega_cube_ok : L_spec (T1.phi0 3) L_omega_cube.
+ +
+Lemma L_omega_cube_eqn i :
+  L_omega_cube i = L_omega_square_times i (S i).
+ +
+Lemma exp2_k_mult_pos k:
+  (0 < k → 4 ≤ exp2 k × (k + 2))%nat.
+ +
+Example omega_square_thrice_eqn : ∀ k,
+    (0 < k)%nat →
+    let M := (exp2 k × (k + 2) - 1)%nat in
+    let N := exp2 M in
+    let P := (N × (M + 2) - 1)%nat in
+    L_omega_square_times 3 k =
+    (exp2 P × (P + 2) - 1)%nat.
+ +
+ +
+ +
+Lemma L_omega_cube_3_eq:
+  let N := exp2 95 in
+  let P := (N × 97 - 1)%nat in
+  L_omega_cube 3 = (exp2 P × (P + 2) - 1)%nat. +
+ +
+
+ +
+

Plain large sequences (not necessarily minimal)

+ + +
+ +Note : Some names of lemmas and theorem like Theorem_4_5 or +Lemma_4_5_2 refer to sections in Ketonen and Solovay's article. + +
+ + +
+ +

large sequences (as in KS ) (may contain zeros)

+ +
+
+ +
+ +
+Definition largeb (alpha : T1) (s: list nat) :=
+  match (gnaw alpha s)
+          with zero ⇒ true | _ ⇒ false end.
+ +
+Definition large (alpha : T1) (s : list nat) : Prop :=
+  largeb alpha s.
+ +
+ +
+Definition largeSb (alpha : T1) (s: list nat) :=
+  match gnawS alpha s with
+    | T1.zero ⇒ true
+    | _ ⇒ false
+  end.
+ +
+Definition largeS (alpha : T1) (s : list nat) : Prop :=
+  largeSb alpha s.
+ +
+Definition Large alpha s := large (@cnf alpha) s.
+ +
+Lemma large_iff (alpha : T1) (s : list nat) :
+  large alpha s ↔ gnaw alpha s = 0.
+ +
+Lemma largeSb_b (alpha : T1) (s: list nat) :
+  largeSb alpha s = largeb alpha (shift s).
+ +
+Lemma largeb_Sb alpha s :
+  largeb alpha s = largeSb alpha (unshift s).
+ +
+Lemma largeS_iff (alpha : T1) (s : list nat) :
+  largeS alpha s ↔ gnawS alpha s = T1.zero.
+ +
+Section Lemma_4_4_Proof.
+  Variables alpha beta : T1.
+  Hypothesis Halpha : nf alpha.
+  Hypothesis Hbeta : nf beta.
+  Variable n : nat.
+  Variable s : list nat.
+ +
+  Hypothesis Hs : sorted_ge n s.
+  Hypothesis H : const_pathS_eps n alpha beta.
+  Hypothesis H0 : largeS alpha s.
+ +
+  Lemma Lemma4_4 : largeS beta s.
+ +
+End Lemma_4_4_Proof.
+ +
+Lemma Lemma_4_4_2 :
+  ∀ (s : list nat) (n : nat) (alpha beta : T1),
+    nf alpha → nf beta → sorted_ge n s →
+    const_pathS_eps n alpha beta → gnawS beta s t1≤ gnawS alpha s.
+ +
+ +
+Lemma Lemma_4_5_1 n alpha:
+  nf alpha →
+  ∀ s t ,
+    ptwise_le s t →
+    sorted_ge n s →
+    sorted_ge n t →
+    const_pathS_eps (simple_last n t)
+                    (gnawS alpha t) (gnawS alpha s).
+ +
+Section Proof_of_4_5_2.
+  Variables (A B1 B2 : list nat).
+  Variable alpha : T1.
+  Hypothesis Halpha : nf alpha.
+  Hypothesis HA : sorted_ge 0 A.
+  Hypothesis HB : sorted_ge 0 (B1 ++ B2).
+  Hypothesis HAB1 : ptwise_le B1 A.
+  Hypothesis HlargeA : largeS alpha A .
+ +
+  Remark R1 : gnawS alpha A = T1.zero.
+ +
+  Remark R2 : gnawS alpha B1 = T1.zero.
+ +
+  Lemma Lemma_4_5_2: gnawS alpha (B1 ++ B2) = zero.
+ +
+End Proof_of_4_5_2.
+ +
+Theorem Theorem_4_5 (alpha: T1)(Halpha : nf alpha)
+        (A B : list nat)
+        (HA : Sorted Peano.lt A)
+        (HB : Sorted Peano.lt B)
+        (HAB : incl A B) :
+  largeS alpha A → largeS alpha B.
+ +
+Lemma gnaw_last_step alpha s i :
+  gnaw alpha s = 1 → gnaw alpha (s ++ S i :: nil) = 0.
+ +
+
+ +
+ For alpha in class E0 +
+
+ +
+Definition LargeS (alpha : E0) s := largeS (@cnf alpha) s.
+ +
+Definition Gnaw alpha s := gnaw (@cnf alpha) s.
+Definition GnawS alpha s := gnawS (@cnf alpha) s.
+ +
+Lemma Gnaw_GnawS s alpha :
+  GnawS alpha s = Gnaw alpha (shift s).
+ +
+Lemma GnawS_Gnaw (s:list nat) :
+  ∀ alpha, Gnaw alpha s = GnawS alpha (unshift s).
+ +
+Lemma GnawS_omega : ∀ i s, GnawS E0_omega (i::s) = GnawS (E0finS i) s.
+ +
+Lemma Gnaw_omega i s : Gnaw E0_omega (S i::s) = Gnaw (E0finS i) s.
+ +
+Definition Largeb (alpha: E0) (s: list nat) :=
+  largeb (@cnf alpha) s.
+ +
+Definition LargeSb (alpha: E0) (s: list nat) :=
+  largeSb (@cnf alpha) s.
+ +
+Lemma LargeSb_b (alpha : E0) (s: list nat) :
+  LargeSb alpha s = Largeb alpha (List.map S s).
+ +
+Lemma Largeb_Sb alpha s :
+  Largeb alpha s = LargeSb alpha (unshift s).
+ +
+Lemma largeb_finite :
+  ∀ n i, largeb (S n) (iota_from (S i) (S n)) = true.
+ +
+Lemma largeb_n (n:nat): ∀ s, ¬ In 0 s →
+                                    large n s →
+                                    (n ≤ List.length s)%nat.
+ +
+Lemma largeb_n_R : ∀ s n, ¬ In 0 s →
+                                (n < List.length s)%nat →
+                                largeb (S n) s = true.
+ +
+Lemma large_n_iff : ∀ s (n:nat) , ¬ In 0 s →
+                                large n s ↔ (n ≤ List.length s)%nat.
+ +
+Example ex3 : ¬ large 156 (interval 100 254).
+ +
+
+ +
+Gnawing omega omega-large intervals +
+
+ +
+Lemma gnaw_omega_n_SSn :
+  ∀ n, gnaw T1omega (iota_from (S n) (S (S n))) = zero.
+ +
+Lemma gnaw_omega_1 (n:nat) :
+  gnaw T1omega (interval (S n) (S n + n)%nat) = 1.
+ +
+Example omega_ex1 : gnaw T1omega (interval 7 13) = 1.
+Qed.
+ +
+Example omega_ex2 : gnaw T1omega (interval 1000 1999) = 1.
+Qed.
+ +
+Lemma large_omega_1 : ∀ s n, ¬ In 0 (n::s) →
+                                     gnaw T1omega (n::s) = 0 →
+                                     (n ≤ List.length s)%nat.
+ +
+Lemma large_omega_2 : ∀ s n, ¬In 0 (n::s) →
+                                    (n ≤ List.length s)%nat →
+                                    gnaw T1omega (n::s) = zero.
+ +
+Lemma large_omega_iff : ∀ s n, ¬ In 0 (n::s) →
+                                     large T1omega (n::s) ↔
+                                     (n ≤ List.length s)%nat.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.Large_Sets_Examples.html b/theories/html/hydras.Epsilon0.Large_Sets_Examples.html new file mode 100644 index 00000000..4dd71b7a --- /dev/null +++ b/theories/html/hydras.Epsilon0.Large_Sets_Examples.html @@ -0,0 +1,172 @@ + + + + + +hydras.Epsilon0.Large_Sets_Examples + + + + +
+ + + +
+ +

Library hydras.Epsilon0.Large_Sets_Examples

+ +
+From hydras Require Import E0 Canon Paths MoreLists Large_Sets.
+ +
+ +
+Example ex_pathS1 : pathS T1omega (2::3::4::nil) 1.
+ +
+Example ex_pathS2 : pathS T1omega (interval 3 6) 1.
+ +
+Example omega_omega_1_4 : gnaw (T1omega ^ T1omega) (interval 1 4) = 0.
+ +
+Example omega_omega_1_3 : gnaw (T1omega ^ T1omega) (interval 1 3) = 1.
+ +
+Inductive answer : Set := Ok | Too_far | Remaining (rest : ppT1).
+ +
+Definition large_set_check alpha i j :=
+  let beta := gnaw alpha (interval i (Nat.pred j))
+  in match beta with
+     | one ⇒ Ok
+     | zero ⇒ Too_far
+     | _ ⇒ Remaining (pp (canon beta j))
+     end.
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+#[global] Hint Resolve iota_from_lt_not_In: core.
+ +
+Example Ex1 : mlarge (T1omega × T1omega) (interval 6 510).
+ +
+Example Ex2 : large (T1omega × T1omega) (interval 6 700).
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.Paths.html b/theories/html/hydras.Epsilon0.Paths.html new file mode 100644 index 00000000..974c2a09 --- /dev/null +++ b/theories/html/hydras.Epsilon0.Paths.html @@ -0,0 +1,1382 @@ + + + + + +hydras.Epsilon0.Paths + + + + +
+ + + +
+ +

Library hydras.Epsilon0.Paths

+ +
+
+ +
+* + +
+ + Transition systems associated with canonical sequences + +
+ + P. Casteran, University of Bordeaux and Labri + +
+ + After J. Ketonen and R. Solovay's paper + " Rapidly Growing Ramsey Functions" in + Annals of mathematics, Mar. 1981 + +
+ + +
+ + TODO: Check wether the predicates ...PathS... are still useful +
+
+ +
+From hydras Require Import DecPreOrder Canon MoreLists First_toggle OrdNotations.
+Import Relations Relation_Operators.
+From Coq Require Import Lia.
+ +
+Set Implicit Arguments.
+Open Scope t1_scope.
+ +
+
+ +
+

relations associated with canonical sequences

+ +
+
+ +
+ +
+Definition transition_S i : relation T1 :=
+  fun alpha beta ⇒ alpha ≠ zero ∧ beta = canon alpha (S i).
+ +
+Definition transition i : relation T1 :=
+  match i with 0 ⇒ fun _ _ ⇒ False | S j ⇒ transition_S j end.
+ +
+ +
+Definition bounded_transitionS (n:nat) alpha beta :=
+  ∃ i:nat, (i ≤ n)%nat ∧ transition_S i alpha beta.
+ +
+
+ +
+

Paths inside epsilon_0

+ + +
+ + In this module, we study paths; i.e. sequences generated by the relations +associated with the canonS i functions. In module O2H we show how pathes are + related to hydra battles. Thus, the various classes of battles are described by predicates on pathes. + +
+ + path_toS beta s alpha : + pathS alpha s beta : + +
+ + beta is reachable from alpha through the sequences s of indices + ( zeros are ignored) + +
+ + Note that only beta can be equal to zero + +
+
+ +
+ +
+Inductive path_to (beta: T1) : list nat → T1 → Prop :=
+| path_to_1 : ∀ (i:nat) alpha ,
+    i ≠ 0 → transition i alpha beta → path_to beta (i::nil) alpha
+| path_to_cons : ∀ i alpha s gamma,
+    i ≠ 0 → transition i alpha gamma →
+    path_to beta s gamma → path_to beta (i::s) alpha.
+ +
+Definition path alpha s beta := path_to beta s alpha.
+ +
+ +
+
+ +
+tries to solve a goal of the form (path_to beta s alpha) +
+
+ +
+Ltac path_tac :=
+  repeat (match goal with
+          | [ |- path_to ?b (?x :: ?y :: ?s) ?a ] ⇒
+            (eright; [discriminate
+                    | split;[discriminate | reflexivity]
+                    | cbn])
+          | [ |- path_to ?b (?x :: nil) ?a ] ⇒
+                   (eleft; [discriminate |
+                            split;cbn ; [discriminate | reflexivity]])
+                         
+                 end).
+Example ex_path1: path_to T1omega (2::2::2::nil) (T1omega × 2).
+ +
+Example ex_path2: path_to T1omega (3::4::5::6::nil) (T1omega × 2).
+ +
+Example ex_path3: path_to zero (interval 3 14) (T1omega × 2).
+ +
+Example ex_path4: path_to zero (List.repeat 3 8) (T1omega × 2).
+ +
+
+ +
+path_toS beta s alpha : + pathS alpha s beta : + +
+ + beta is reachable from alpha through the sequence of indices + (shift s) + +
+ + Note that only beta can be equal to zero + +
+
+ +
+Lemma path_to_not_nil alpha s beta : path_to beta s alpha → s ≠ nil.
+ +
+Inductive path_toS (beta: T1) : list nat → T1 → Prop :=
+| path_toS_1 : ∀ (i:nat) alpha, transition_S i alpha beta →
+                                      path_toS beta (i::nil) alpha
+| path_toS_cons : ∀ i alpha s gamma,
+    transition_S i alpha gamma →
+    path_toS beta s gamma →
+    path_toS beta (i::s) alpha.
+ +
+Definition pathS alpha s beta := path_toS beta s alpha.
+ +
+
+ +
+

Accessibility (without traces)

+ +
+
+ +
+Definition acc_from alpha beta := ∃ s, path_to beta s alpha.
+ +
+
+ +
+

following a path

+ +
+ + useful helper ? +
+
+Fixpoint gnawS (alpha : T1) (s: list nat) :=
+  match s with
+    | nil ⇒ alpha
+    | (i::s') ⇒ gnawS (canon alpha (S i)) s'
+  end.
+ +
+ +
+Fixpoint gnaw (alpha : T1) (s: list nat) :=
+  match s with
+    | nil ⇒ alpha
+    | (0::s') ⇒ gnaw alpha s'
+    | (S i :: s') ⇒ gnaw (canon alpha (S i)) s'
+  end.
+ +
+ +
+ +
+Fixpoint standard_gnaw (i:nat)(alpha : T1)(l:nat): T1 :=
+  match l with
+  | 0 ⇒ alpha
+  | S m ⇒ standard_gnaw (S i) (canon alpha i) m
+  end.
+ +
+ +
+
+ +
+alpha ---> beta in KP + n + +
+
+ +
+Definition KP_arrowS n := clos_trans_1n T1 (bounded_transitionS n).
+ +
+
+ +
+

Paths with constant index

+ +
+
+ +
+ +
+Definition const_pathS i :=
+  clos_trans_1n T1 (fun alpha beta ⇒ alpha ≠ zero ∧
+                                      beta = canon alpha (S i)).
+ +
+Definition const_path i alpha beta :=
+  match i with
+    0 ⇒ False
+  | S j ⇒ const_pathS j alpha beta
+  end.
+ +
+ +
+ +
+Lemma const_pathSE i alpha beta: const_pathS i alpha beta ↔
+                                  const_path (S i) alpha beta.
+ +
+Definition const_pathS_eps i := clos_refl _ (const_path (S i)).
+ +
+
+ +
+

standard paths

+ +
+
+ +
+ +
+Inductive standard_path_toS (j:nat)( beta : T1): nat → T1 → Prop :=
+  stdS_1 : ∀ i alpha,
+    alpha ≠ zero →
+    beta = canon alpha (S i) → j = i →
+    standard_path_toS j beta i alpha
+| stdS_S : ∀ i alpha,
+    standard_path_toS j beta (S i) (canon alpha (S i)) →
+    standard_path_toS j beta i alpha.
+ +
+Definition standard_pathS i alpha j beta :=
+  standard_path_toS j beta i alpha.
+ +
+ +
+Inductive standard_path_to (j:nat)(beta : T1): nat → T1 → Prop :=
+| std_1 : ∀ i alpha,
+    alpha ≠ zero →
+    beta = canon alpha i → j = i → i ≠ 0 →
+    standard_path_to j beta i alpha
+| std_S : ∀ i alpha,
+    standard_path_to j beta (S i) (canon alpha i) →
+    standard_path_to j beta i alpha.
+ +
+Definition standard_path i alpha j beta :=
+  standard_path_to j beta i alpha.
+ +
+Lemma path_to_interval_inv_le alpha beta i j :
+  path_to beta (interval i j) alpha →
+  (i ≤ j)%nat.
+ +
+
+ +
+

Bridge Lemmas

+ +
+
+ +
+Lemma gnawS_gnaw s alpha :
+  gnawS alpha s = gnaw alpha (shift s).
+ +
+Lemma gnaw_gnawS s alpha :
+  gnaw alpha s = gnawS alpha (unshift s).
+ +
+Lemma path_toS_path_to alpha s beta :
+  path_toS beta s alpha ↔
+  path_to beta (shift s) alpha.
+ +
+Lemma path_to_path_toS :
+  ∀ alpha s beta, path_to beta s alpha →
+                        path_toS beta (unshift s) alpha.
+ +
+Lemma path_to_path_toS_iff :
+  ∀ alpha s beta, ¬ In 0 s → path_to beta s alpha ↔
+                        path_toS beta (unshift s) alpha.
+ +
+Lemma path_toS_nf beta s alpha: path_toS beta s alpha → nf alpha → nf beta.
+ +
+Lemma path_acc_from alpha s beta:
+    path alpha s beta → acc_from alpha beta.
+ +
+Lemma path_toS_gnawS : ∀ s alpha beta, path_toS beta s alpha →
+                                              beta = gnawS alpha s.
+ +
+
+ +
+

Composition/decomposition of paths

+ +
+
+ +
+Lemma path_toS_app beta t gamma :
+    path_toS gamma t beta → ∀ alpha s, path_toS beta s alpha →
+                                             path_toS gamma (s++t)%list alpha.
+ +
+Lemma path_to_app beta t gamma :
+    path_to gamma t beta → ∀ alpha s, path_to beta s alpha →
+                                             path_to gamma (s++t)%list alpha.
+ +
+Lemma path_toS_decompose gamma s alpha :
+  ∀ beta t u, s = t++u →
+                    path_toS beta t alpha → path_toS gamma u beta →
+                    path_toS gamma s alpha.
+ +
+Lemma path_to_decompose gamma alpha beta s t u :
+  s = t++ u →
+  path_to beta t alpha → path_to gamma u beta →
+  path_to gamma s alpha.
+ +
+Ltac path_decompose x :=
+  match goal with |- path_to ?a (interval ?from ?to) ?b ⇒
+                  rewrite (interval_app from x to);
+                  [eapply path_to_app | try lia | try lia]
+  end.
+ +
+Lemma path_toS_appR : ∀ u gamma alpha,
+    path_toS gamma u alpha →
+    ∀ s t, s ≠ nil → t ≠ nil → u = s ++ t →
+                ∃ beta,
+                  path_toS beta s alpha ∧ path_toS gamma t beta.
+ +
+Lemma path_to_appR : ∀ u gamma alpha,
+    path_to gamma u alpha →
+    ∀ s t, s ≠ nil → t ≠ nil → u = s ++ t → ¬ In 0 u →
+                ∃ beta,
+                  path_to beta s alpha ∧ path_to gamma t beta.
+ +
+Lemma path_toS_zero_but_last : ∀ x u alpha,
+    path_toS zero (x::u) alpha → u ≠ nil →
+    path_toS one (but_last x u) alpha.
+ +
+Lemma path_toS_zero_one : ∀ x alpha,
+    path_toS zero (x::nil) alpha → alpha = one.
+ +
+Lemma path_toS_zero_inv x u alpha :
+  path_toS zero (x::u) alpha → {alpha = one ∧ u = nil} +
+                                {u ≠ nil ∧ path_toS one
+                                                      (but_last x u)
+                                                      alpha}.
+ +
+Lemma path_toS_zero alpha s : ¬ path_toS alpha s zero.
+ +
+Lemma path_to_zero alpha s: ¬ path_to alpha s zero.
+ +
+
+ +
+todo : use this lemma to remove useless hypotheses +
+
+ +
+Lemma path_to_not_In_zero : ∀ alpha s beta,
+    path_to beta s alpha → ¬ In 0 s.
+ +
+Lemma path_toS_tail alpha s beta :
+    path_toS beta s alpha →
+    ∀ gamma n, nf (T1.cons gamma n alpha) →
+                    path_toS (T1.cons gamma n beta) s (T1.cons gamma n alpha).
+ +
+Lemma path_to_tail alpha s beta :
+  path_to beta s alpha →
+  ∀ gamma n, nf (T1.cons gamma n alpha) →
+                  path_to (T1.cons gamma n beta) s (T1.cons gamma n alpha).
+ +
+Lemma path_toS_mult alpha s i : nf alpha →
+    path_toS zero s (T1.phi0 alpha) →
+    path_toS (T1.cons alpha i zero) s (T1.cons alpha (S i) zero).
+ +
+Lemma path_to_mult alpha s i :
+  nf alpha →
+  path_to zero s (T1.phi0 alpha) →
+  path_to (T1.cons alpha i zero) s (T1.cons alpha (S i) zero).
+ +
+
+ +
+

Properties of gnaw

+ +
+
+ +
+Lemma gnaw_nf : ∀ s alpha, nf alpha → nf (gnaw alpha s).
+ +
+Lemma gnaw_zero : ∀ s, gnaw 0 s = 0.
+ +
+Lemma gnawS_zero : ∀ s, gnawS zero s = zero.
+ +
+Lemma gnawS_nf : ∀ s alpha, nf alpha → nf (gnawS alpha s).
+ +
+Lemma gnaw_succ: ∀ alpha i s, nf alpha →
+                                    gnaw (T1.succ alpha) (S i::s) =
+                                    gnaw alpha s.
+ +
+Lemma gnaw_rw i s alpha : gnaw alpha (S i::s) = gnaw (canon alpha (S i)) s.
+ +
+Lemma gnawS_to_path_toS : ∀ s alpha beta,
+    beta = gnawS alpha s → beta ≠ zero → s ≠ nil →
+    path_toS beta s alpha.
+ +
+Lemma gnaw_to_path_to : ∀ s alpha beta,
+    beta = gnaw alpha s → ¬ In 0 s → beta ≠ zero → s ≠ nil →
+    path_to beta s alpha.
+ +
+Lemma gnaw_app : ∀ s s' alpha,
+                   gnaw alpha (s ++ s') = gnaw (gnaw alpha s) s'.
+ +
+Lemma gnawS_app : ∀ s s' alpha,
+                   gnawS alpha (s ++ s') = gnawS (gnawS alpha s) s'.
+ +
+Lemma gnaws_rw i s alpha : gnawS alpha (i::s) = gnawS (canon alpha (S i)) s.
+ +
+Lemma gnawS_lim1 (i:nat)(s: list nat) (lambda : T1) :
+  nf lambda → T1limit lambda →
+  gnawS (T1.cons lambda 0 T1.zero) (i::s) =
+  gnawS (T1.cons (canon lambda (S i)) 0 T1.zero) s.
+ +
+Lemma gnawS_lim2 (i n:nat)(s: list nat) (lambda : T1) :
+  nf lambda → T1limit lambda→
+  gnawS (T1.cons lambda (S n) T1.zero) (i::s) =
+  gnawS (T1.cons lambda n (T1.cons (canon lambda (S i)) 0 T1.zero)) s.
+ +
+Lemma gnawS_succ_eqn1 i s gamma :
+  nf gamma → gnawS (T1.cons (T1.succ gamma) 0 T1.zero) (i::s) =
+              gnawS (T1.cons gamma i T1.zero) s.
+ +
+Lemma gnawS_succ_eqn2 :
+  ∀ i n s gamma, nf gamma →
+                      gnawS (T1.cons (T1.succ gamma) (S n) zero) (i::s) =
+                      gnawS (T1.cons (T1.succ gamma) n
+                                     (T1.cons gamma i T1.zero)) s.
+ +
+Lemma gnawS_tail :
+  ∀ i s alpha n beta,
+                  nf (T1.cons alpha n beta) →
+                  beta ≠ T1.zero →
+                  gnawS (T1.cons alpha n beta) (i::s) =
+                  gnawS (T1.cons alpha n (canon beta (S i))) s.
+ +
+Lemma gnawS_SSn (i:nat) s :
+  ∀ alpha n ,
+    nf alpha →
+    gnawS (T1.cons alpha (S n) T1.zero) (i::s) =
+    gnawS (T1.cons alpha n (canon (T1.cons alpha 0 zero) (S i))) s.
+ +
+Lemma gnawS_cut1 : ∀ s alpha n beta,
+                    nf (T1.cons alpha n beta) →
+                    gnawS (T1.cons alpha n beta) s = T1.zero →
+                    ∃ s1 s2, s = s1 ++ s2 ∧
+                                  gnawS beta s1 = zero ∧
+                                  gnawS (T1.cons alpha n T1.zero) s2 = zero.
+ +
+Lemma gnawS_cut2 : ∀ s alpha n ,
+                   nf alpha →
+                   gnawS (T1.cons alpha (S n) T1.zero) s = zero →
+                   ∃ s1 s2, s = s1 ++ s2 ∧
+                                 gnawS (T1.phi0 alpha) s1 = zero ∧
+                                 gnawS (T1.cons alpha n T1.zero) s2 = zero.
+ +
+Lemma path_to_gnaw : ∀ s alpha beta,
+    path_to beta s alpha →
+    beta = gnaw alpha s.
+ +
+Lemma gnawS_path_toS : ∀ s alpha , s ≠ nil → gnawS alpha s ≠ zero →
+                                        path_toS (gnawS alpha s) s alpha.
+ +
+Lemma gnaw_path_to : ∀ s alpha , s ≠ nil → ¬ In 0 s →
+                                      gnaw alpha s ≠ zero →
+                                      path_to (gnaw alpha s) s alpha.
+ +
+
+ +
+

Properties of acc_from

+ +
+
+ +
+Lemma acc_from_trans : ∀ alpha beta gamma,
+    acc_from alpha beta → acc_from beta gamma → acc_from alpha gamma.
+ +
+Lemma LT_path_toS (alpha beta : T1) :
+  beta t1< alpha → {s : list nat | path_toS beta s alpha}.
+ +
+ +
+Lemma LT_path_to (alpha beta : T1) :
+  beta t1< alpha → {s : list nat | path_to beta s alpha}. +
+ +
+ +
+Lemma LT_acc_from (alpha beta : T1) :
+  beta t1< alpha → acc_from alpha beta.
+ +
+Lemma path_toS_LT beta s alpha:
+  path_toS beta s alpha → nf alpha → beta t1< alpha.
+ +
+Lemma path_to_LT beta s alpha :
+  path_to beta s alpha → nf alpha → beta t1< alpha.
+ +
+Lemma acc_from_LT (alpha beta : T1) :
+  acc_from alpha beta → nf alpha → beta t1< alpha.
+ +
+
+ +
+

Paths with constant index: lemmas

+ +
+
+ +
+Lemma const_pathS_repeat i : ∀ alpha beta,
+    const_path (S i) alpha beta →
+    ∃ n:nat, path_toS beta (repeat i (S n)) alpha.
+ +
+Lemma const_pathS_repeatR i : ∀ n alpha beta,
+    path_toS beta (repeat i (S n)) alpha →
+    const_path (S i) alpha beta.
+ +
+Lemma const_pathS_nf : ∀ n alpha beta,
+    const_pathS n alpha beta →
+    nf alpha → nf beta.
+ +
+Lemma const_pathS_zero n alpha : ¬ const_path (S n) zero alpha.
+ +
+Lemma const_pathS_LT i alpha beta:
+                                     nf alpha →
+                                     const_path (S i) alpha beta →
+                                     beta t1< alpha.
+ +
+Lemma const_path_LT i :
+  ∀ alpha beta,
+      nf alpha → const_path i alpha beta →
+    beta t1< alpha.
+ +
+Lemma const_pathS_LE i : ∀ alpha beta,
+    nf alpha → const_path (S i) alpha beta → beta t1≤ alpha.
+ +
+Lemma const_pathS_inv : ∀ n alpha beta,
+    const_path (S n) alpha beta →
+    beta = canon alpha (S n) ∨
+    const_path (S n) (canon alpha (S n)) beta.
+ +
+Lemma const_pathS_inv_strong : ∀ n alpha beta,
+    const_path (S n) alpha beta →
+    {beta = canon alpha (S n)} +
+    {const_path (S n) (canon alpha (S n)) beta}.
+ +
+Lemma const_pathS_trans : ∀ n alpha beta gamma,
+    const_path (S n) alpha beta →
+    const_path (S n) beta gamma →
+    const_path (S n) alpha gamma.
+ +
+Lemma const_pathS_eps_trans : ∀ i alpha beta gamma,
+    const_pathS_eps i alpha beta →
+    const_pathS_eps i beta gamma →
+    const_pathS_eps i alpha gamma .
+ +
+Lemma const_pathS_eps_LE_2 : ∀ n alpha beta,
+    nf alpha → nf beta →
+    const_pathS_eps n alpha beta →
+    beta t1≤ alpha.
+ +
+Lemma Proposition_2_3a:
+  ∀ n alpha , nf alpha → alpha ≠ zero →
+                   ∀ beta gamma, LT gamma beta →
+                                        const_pathS n alpha gamma →
+                                        const_pathS n alpha beta →
+                                        const_pathS n beta gamma.
+ +
+Lemma KS_thm_2_4_lemma1 : ∀ i alpha n beta beta',
+    nf (T1.cons alpha n beta) →
+    beta ≠ zero →
+    const_pathS i beta beta' →
+    const_pathS i (T1.cons alpha n beta)
+            (T1.cons alpha n beta').
+ +
+Lemma const_pathS_first_step : ∀ i alpha beta,
+    const_pathS i alpha beta →
+    {beta = canon alpha (S i) } +
+    {const_pathS i (canon alpha (S i)) beta}.
+ +
+Lemma KS_thm_2_4_lemma1' : ∀ i alpha n beta ,
+    nf alpha → alpha ≠ zero →
+    const_pathS i (T1.phi0 alpha) beta →
+    const_pathS i (T1.cons alpha (S n) zero)
+                (T1.cons alpha n beta).
+ +
+Lemma KS_thm_2_4_lemma2 (n:nat)(alpha:T1) :
+  nf alpha → alpha ≠ zero → const_pathS n alpha zero.
+ +
+Lemma KS_thm_2_4_lemma3_0 : ∀ i alpha n,
+    nf alpha →
+    const_pathS i (T1.cons alpha (S n) zero)
+            (T1.cons alpha n zero).
+ +
+ +
+Lemma KS_thm_2_4_lemma3 : ∀ i n p alpha ,
+    nf alpha → n < p →
+    const_pathS i (T1.cons alpha p zero)
+            (T1.cons alpha n zero).
+ +
+Lemma KS_thm_2_4_lemma4 : ∀ i alpha,
+    nf alpha →
+    const_pathS i (T1.phi0 (succ alpha)) (T1.phi0 alpha).
+ +
+Lemma KS_thm_2_4_lemma5 : ∀ i alpha beta,
+    const_pathS i alpha beta → nf alpha →
+    alpha ≠ zero →
+    const_pathS i (T1.phi0 alpha) (T1.phi0 beta).
+ +
+ +
+Theorem KS_thm_2_4 (lambda : T1) :
+  nf lambda → T1limit lambda →
+  ∀ i j, (i < j)%nat →
+              const_path 1 (canon lambda (S j)) (canon lambda (S i)).
+ +
+
+ +
+ Corollary 12 of KS +
+
+ +
+ +
+Corollary Cor12 (alpha : T1) : nf alpha →
+                ∀ beta i n, beta t1< alpha →
+                                 i < n →
+                                 const_path (S i) alpha beta →
+                                 const_path (S n) alpha beta.
+ +
+ +
+Corollary Cor12_1 (alpha : T1) :
+  nf alpha →
+  ∀ beta i n, beta t1< alpha →
+                   i ≤ n →
+                   const_path (S i) alpha beta →
+                   const_path (S n) alpha beta. +
+Corollary Cor12_2 (alpha : T1) :
+    nf alpha →
+    ∀ beta i n,
+      i ≤ n →
+      const_pathS_eps i alpha beta →
+      const_pathS_eps n alpha beta.
+ +
+Corollary Cor12_3 (alpha : T1) :
+  nf alpha →
+  ∀ beta i n, beta t1< alpha →
+                   i ≤ n →
+                   const_path i alpha beta →
+                   const_path n alpha beta.
+ +
+Lemma const_pathS_eps_zero n alpha :
+   const_pathS_eps n zero alpha → alpha = zero.
+ +
+ +
+ +
+Lemma Lemma2_6_1 (alpha : T1) :
+  nf alpha →
+  ∀ beta, beta t1< alpha →
+               {n:nat | const_path (S n) alpha beta}.
+ +
+Lemma small_lemma (i:nat) (beta : T1) : ∀ alpha,
+      const_pathS i alpha beta →
+      nf alpha →
+      beta t1≤ canon alpha (S i).
+ +
+Lemma L2_6_2 (p: nat) :
+  ∀ alpha, nf alpha →
+                ∀ beta, const_pathS p alpha beta →
+                             T1.succ beta t1< alpha →
+                             const_pathS (S p) alpha (T1.succ beta).
+ +
+Section Lemma_4_3_Proof.
+  Variables alpha beta : T1.
+  Hypothesis H00 : alpha ≠ zero.
+  Hypothesis nf1 : nf alpha.
+  Hypothesis nf2 : nf beta.
+  Variables n0 n1 n2: nat.
+  Hypothesis H0: (n0 ≤n2)%nat.
+  Hypothesis H1: (n1 ≤n2)%nat.
+  Hypothesis H4: const_pathS_eps n0 alpha beta.
+ +
+  Remark R4_3_1: const_pathS_eps n2 alpha beta.
+ +
+  Remark R4_3_2 : const_pathS_eps n2 beta (canon beta (S n1)).
+ +
+ +
+  Remark R4_3_3: const_pathS_eps n2 alpha (canon beta (S n1)).
+ +
+  Remark R4_3_4 : (canon beta (S n1) t1≤ canon alpha (S n2))%t1.
+ +
+  Lemma Lemma_4_3_0 :
+    const_pathS_eps n2 (canon alpha (S n2)) (canon beta (S n1)).
+ +
+ +
+End Lemma_4_3_Proof.
+ +
+Lemma Lemma_4_3 :
+  ∀ alpha beta : T1,
+    nf alpha →
+    nf beta →
+    ∀ n0 n1 n2 : nat,
+      (n0 ≤ n2)%nat →
+      (n1 ≤ n2)%nat →
+      const_pathS_eps n0 alpha beta →
+      const_pathS_eps n2 (canon alpha (S n2)) (canon beta (S n1)).
+ +
+Lemma const_pathS_LT' (i : nat) (alpha beta : T1) (H:nf alpha):
+  const_pathS i alpha beta → alpha = zero ∨ beta t1< alpha.
+ +
+Lemma Lemma_4_4_1 : ∀ s n alpha beta (Ha : nf alpha)
+                           (Hb : nf beta),
+                      sorted_ge n s →
+                      const_pathS_eps n alpha beta →
+                      const_pathS_eps (simple_last n s)
+                              (gnawS alpha s) (gnawS beta s).
+ +
+Lemma KP_arrowS_zero n beta : KP_arrowS n zero beta → beta = zero.
+ +
+ +
+Lemma KP_5_iii n alpha beta: nf alpha → nf beta →
+                              const_pathS n alpha beta ↔
+                              KP_arrowS n alpha beta.
+ +
+Lemma Lemma_4_4_0 :
+  ∀ n p alpha beta (Halpha : alpha ≠ T1.zero),
+    nf alpha → nf beta → (n ≤ p)%nat →
+    const_pathS_eps n alpha beta →
+    const_pathS_eps p (canon alpha (S p)) (canon beta (S p)).
+ +
+Lemma standard_path_shift i alpha j beta :
+  standard_pathS i alpha j beta →
+  standard_path (S i) alpha (S j) beta.
+ +
+Lemma standard_path_unshift_0 i alpha j beta :
+   standard_path i alpha j beta → ∀ k l,
+      i = S k → j = S l →
+  standard_pathS k alpha l beta.
+ +
+Lemma standard_path_unshift i alpha j beta :
+  standard_path (S i) alpha (S j) beta →
+  standard_pathS i alpha j beta.
+ +
+Lemma standard_path_toS_le_inv : ∀ j beta i alpha,
+    standard_path_toS j beta i alpha → (i ≤ j)%nat.
+ +
+Lemma standard_path_to_le_inv : ∀ j beta i alpha,
+    standard_path_to j beta i alpha → (i ≤ j)%nat.
+ +
+Lemma standard_path_toS_zero : ∀ j beta i alpha,
+    standard_path_toS j beta i alpha → alpha ≠ zero.
+ +
+Lemma standard_path_origin : ∀ j beta i alpha,
+   standard_path_to j beta i alpha → alpha ≠ zero.
+ +
+Lemma standard_pathS_path_toS :
+  ∀ i alpha j beta, standard_pathS i alpha j beta →
+                         (i ≤ j)%nat →
+                         path_toS beta (interval i j) alpha.
+ +
+Lemma standard_path_path_to i alpha j beta :
+  standard_path i alpha j beta → i ≠ 0 →
+  (i ≤ j)%nat →
+  path_to beta (interval i j) alpha.
+ +
+Lemma standard_gnaw_zero l i : standard_gnaw i zero l = zero.
+ +
+Lemma standard_gnaw_iota_from i alpha l :
+  i ≠ 0 →
+  standard_gnaw i alpha l = gnaw alpha (iota_from i l).
+ +
+Lemma standard_gnaw_plus : ∀ (t1 t2 i:nat)(alpha : T1) ,
+    standard_gnaw i alpha (t1 + t2)%nat =
+    standard_gnaw (t1 +i)%nat (standard_gnaw i alpha t1) t2.
+ +
+Lemma standard_path_LE : ∀ j beta i alpha,
+    standard_path i alpha j beta → (i ≤ j)%nat.
+ +
+Lemma standard_path_to_nf : ∀ i alpha j beta,
+    standard_path_to j beta i alpha → nf alpha → nf beta.
+ +
+Lemma standard_path_zero : ∀ i j alpha beta,
+    standard_path i alpha j beta →
+    alpha = zero →
+    beta = zero.
+ +
+Lemma standard_path_lt2 : ∀ i alpha j beta,
+    (0 < i)%nat → zero t1< alpha →
+    standard_path_to j beta i alpha →
+    beta t1< alpha.
+ +
+Lemma standard_path_compose j beta k gamma :
+  standard_path (S j) beta k gamma →
+  ∀ i alpha, standard_path i alpha j beta→
+                   standard_path i alpha k gamma.
+ +
+Lemma standard_gnaw_S_zero : ∀ i n alpha ,
+      standard_gnaw n alpha i = zero → standard_gnaw n alpha (S i)=zero.
+ +
+Lemma standard_gnaw_to_path : ∀ t alpha i,
+    alpha ≠ zero → i ≠ 0 →
+    standard_gnaw i alpha t ≠ zero →
+    standard_path i alpha (t + i)%nat
+                  (standard_gnaw i alpha (S t)).
+ +
+Lemma standard_gnaw_nf l : ∀ i alpha,
+  nf alpha → nf (standard_gnaw i alpha l).
+ +
+Lemma standard_path_equiv_1 :
+  ∀ i j alpha beta, standard_path i alpha j beta →
+                         beta = standard_gnaw i alpha (S j - i)%nat.
+ +
+Lemma standard_path_equiv_2 :
+  ∀ l i alpha, i ≠ 0 →
+                    let beta := standard_gnaw i alpha (S l)
+                    in beta ≠ zero →
+                       standard_path i alpha (l + i)%nat beta.
+ +
+Lemma path_to_S_iota_from :
+  ∀ l i alpha beta,
+    path_toS beta (iota_from i (S l)) alpha →
+    standard_path_toS (l + i)%nat beta i alpha.
+ +
+Lemma path_to_S_standard_pathS :
+  ∀ i j alpha beta, (i ≤ j)%nat →
+    path_toS beta (interval i j) alpha →
+    standard_pathS i alpha j beta.
+ +
+Lemma path_toS_standardS_equiv i j alpha beta :
+  (i ≤ j)%nat → path_toS beta (interval i j) alpha ↔
+    standard_pathS i alpha j beta.
+ +
+Lemma path_to_standard_equiv i j alpha beta :
+  i ≠ 0 → (i ≤ j)%nat → path_to beta (interval i j) alpha ↔
+    standard_path i alpha j beta.
+ +
+Lemma standard_pathS_app i alpha j beta k gamma :
+  (i ≤ j)%nat → (j ≤ k)%nat →
+  standard_pathS i alpha j beta → standard_pathS (S j) beta k gamma →
+  standard_pathS i alpha k gamma.
+ +
+Lemma cons_standard_pathS : ∀ alpha n beta ,
+    nf (T1.cons alpha n beta) → ∀ gamma i j,
+    standard_pathS i beta j gamma → gamma ≠ 0 →
+    standard_pathS i (T1.cons alpha n beta) j (T1.cons alpha n gamma).
+ +
+Lemma cons_standard_path : ∀ alpha n beta ,
+    nf (T1.cons alpha n beta) → ∀ gamma i j,
+    standard_path i beta j gamma → gamma ≠ 0 → i ≠ 0 →
+    standard_path i (T1.cons alpha n beta) j (T1.cons alpha n gamma).
+ +
+Lemma flatten : ∀ (i:nat) alpha (j:nat) beta,
+     standard_path i alpha j beta →
+    nf alpha → zero t1< alpha → (0 < i)%nat →
+    const_path j alpha beta.
+ +
+Lemma standard_gnaw_to_zero :
+  ∀ alpha , nf alpha → ∀ i,
+      {p:nat | standard_gnaw i alpha p = zero}.
+ +
+ +
+Lemma standard_path_to_zero:
+  ∀ alpha i, nf alpha → alpha ≠ zero →
+                   {j: nat | standard_path (S i) alpha j zero}. +
+Lemma standard_gnaw_2_zero :
+  ∀ alpha i, nf alpha →
+                   {t: nat | standard_gnaw i alpha t = zero }.
+ +
+
+ +
+This section is dedicated to the proof of Lemma p 300 of KS + We associate to any path from alpha to beta with a constant index + a path from alpha to beta with a sequence of indices n+1, n+2, n+3, ... +
+
+ +
+ +
+Section Constant_to_standard_Proof.
+  Variables (alpha beta: T1) (n : nat).
+  Hypotheses (Halpha: nf alpha) (Hpos : zero t1< beta)
+             (Hpa : const_pathS n alpha beta).
+ +
+  Remark Rem0 : beta t1< alpha.
+ +
+  Remark Rem1 : {k:nat &
+                   {gamma: T1 |
+                    standard_path (S n) alpha k gamma ∧
+                    gamma t1< alpha}}.
+ +
+  Remark Rem01 : zero t1< alpha.
+ +
+  Remark Rem02 : alpha ≠ zero.
+ +
+  Remark Rem2 : {t: nat | standard_gnaw (S n) alpha t t1< beta}.
+ +
+  Let t := proj1_sig Rem2.
+ +
+ +
+ +
+  Remark Rem03 : (standard_gnaw (S n) alpha t t1< beta)%t1.
+ +
+  Let P (i:nat) := compare (standard_gnaw (S n) alpha i) beta ≠ Datatypes.Lt.
+ +
+  Remark Rem04 : P 0.
+ +
+  Remark Rem05 : ¬ P t.
+ +
+  Remark Rem06 : (0 < t)%nat.
+ +
+  Instance P_dec i : Decision (P i).
+ +
+  Let l_def := first_toggle P P_dec 0 t Rem06 Rem04 Rem05.
+ +
+  Let l := proj1_sig l_def.
+ +
+  Let gamma := standard_gnaw (S n) alpha l.
+ +
+  Remark Rem08 :
+    (0 ≤ l)%nat ∧
+    (l < t)%nat ∧
+    (∀ i : nat,
+        (0 ≤ i)%nat →
+        (i ≤ l)%nat → P i) ∧ ¬ P (S l).
+ +
+  Remark Rem09 : (l < t)%nat.
+ +
+  Remark Rem10 : P l.
+ +
+  Remark Rem11 : ¬ P (S l).
+ +
+  Remark R12 : const_pathS (l+n) alpha beta.
+ +
+  Remark R13 : (0 < l)%nat → gamma ≠ zero →
+               standard_path (S n) alpha ((l + n)) gamma.
+ +
+  Remark R14 : l = 0 → gamma = alpha.
+ +
+  Remark R15 : gamma ≠ zero →
+               (0 < l)%nat → const_path (n + l) alpha gamma.
+ +
+  Let m := Nat.pred (n + l).
+ +
+  Lemma m_def : (0 < l)%nat → (n + l = S m)%nat.
+ +
+  Remark R16 : (0 < l)%nat → const_path (n + l) alpha beta.
+ +
+   Remark R17 : (0 < l)%nat → const_pathS (n+l) alpha beta.
+ +
+  Remark R18 : gamma ≠ zero → (0 < l)%nat → const_pathS (n+l) alpha gamma.
+ +
+  Remark R19 : beta t1≤ gamma.
+ +
+  Remark R20 : gamma ≠ zero → (0 < l)%nat → beta t1< gamma →
+               const_pathS (n+l) gamma beta.
+ +
+  Remark R21 : gamma ≠ zero → (0 < l)%nat →
+               gamma = beta ∨ const_pathS (n+l) gamma beta.
+ +
+  Let delta := canon gamma (S (n + l))%nat.
+ +
+  Remark R22 : delta = standard_gnaw (S n) alpha (S l).
+ +
+  Remark R23 : const_pathS (n+l) gamma beta →
+               {beta = delta}+{const_pathS (n + l) delta beta}.
+ +
+  Remark R24 : const_pathS (n+l) gamma beta → beta t1≤ delta.
+ +
+   Remark R25 : delta t1< beta.
+ +
+   Remark R26 : ¬ const_pathS (n+l) gamma beta.
+ +
+   Remark R27 : gamma ≠ zero → (0 < l)%nat → gamma = beta.
+ +
+   Remark R28 : l = 0 → (beta t1≤ delta)%t1.
+ +
+  Remark R29 : l ≠ 0.
+ +
+  Remark R30 : gamma ≠ zero → gamma = beta.
+ +
+  Remark R31_0 : gamma ≠ zero → beta = standard_gnaw (S n) alpha l.
+ +
+ +
+  Lemma gamma_positive : gamma ≠ zero.
+ +
+  Remark R31 : beta = standard_gnaw (S n) alpha l.
+ +
+  Lemma constant_to_standard_0 :
+    {l : nat | standard_gnaw (S n) alpha l = beta}.
+ +
+End Constant_to_standard_Proof.
+ +
+ +
+Lemma constant_to_standard (alpha beta : T1) (n : nat):
+    nf alpha → const_pathS n alpha beta →
+    {l : nat | standard_gnaw (S n) alpha l = beta}. +
+ +
+Lemma constant_to_standard_path
+  (alpha beta : T1) (i : nat):
+  nf alpha → const_pathS i alpha beta → zero t1< alpha →
+  {j:nat | standard_path (S i) alpha j beta}. +
+Corollary LT_to_standard_path (alpha beta : T1) :
+  beta t1< alpha →
+  {n : nat & {j:nat | standard_path (S n) alpha j beta}}.
+ +
+
+ +
+

Adaptation to E0

+ +
+
+ +
+Definition Canon_plus (i:nat)( alpha beta:E0)
+  := const_path i (cnf alpha) (cnf beta).
+ +
+Lemma Canon_plus_inv n alpha beta :
+  Canon_plus (S n) alpha beta →
+  beta = Canon alpha (S n) ∨
+  Canon_plus (S n) (Canon alpha (S n)) beta.
+ +
+Theorem KS_thm_2_4_E0 :
+  ∀ lambda,
+    E0limit lambda →
+    ∀ i j, (i < j)%nat →
+                Canon_plus 1 (Canon lambda (S j))
+                       (Canon lambda (S i)).
+ +
+Corollary Cor12_E0 : ∀ alpha beta i n,
+    E0lt beta alpha → (i ≤ n)%nat →
+    Canon_plus (S i) alpha beta →
+    Canon_plus (S n) alpha beta.
+ +
+Lemma Canon_mono1 alpha i j : E0limit alpha → (i< j)% nat →
+                              (Canon alpha i o< Canon alpha j)%e0.
+ +
+Lemma CanonS_plus_1 alpha beta k i :
+  beta ≠ E0zero → alpha ≠ E0zero →
+  (beta o< E0_phi0 alpha)%e0 →
+  (Canon (Omega_term alpha i + beta)%e0 (S k) =
+   (Omega_term alpha i + (Canon beta (S k)))%e0).
+ +
+Lemma CanonS_Phi0_Succ_eqn i gamma:
+  Canon (E0_phi0 (E0_succ gamma)) (S i) = Omega_term gamma i.
+ +
+Lemma Lemma2_6_1_E0 (alpha beta: E0) :
+    (beta o< alpha)%e0 →
+    {n:nat | Canon_plus (S n) alpha beta}.
+ +
+
+ +
+Lemmas used by F_alpha +
+
+ +
+Lemma Canon_plus_first_step: ∀ i alpha beta,
+    Canon_plus (S i) (E0_succ alpha) beta →
+    alpha = beta ∨ Canon_plus (S i) alpha beta.
+ +
+Lemma Canon_plus_first_step_lim:
+  ∀ i alpha beta, E0limit alpha →
+                       Canon_plus (S i) alpha beta →
+                       beta = Canon alpha (S i) ∨
+                       Canon_plus (S i) (Canon alpha (S i)) beta.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Epsilon0.T1.html b/theories/html/hydras.Epsilon0.T1.html new file mode 100644 index 00000000..f7b82b2a --- /dev/null +++ b/theories/html/hydras.Epsilon0.T1.html @@ -0,0 +1,2392 @@ + + + + + +hydras.Epsilon0.T1 + + + + +
+ + + +
+ +

Library hydras.Epsilon0.T1

+ +
+
+ +
+ +
+ + A type for ordinals terms in Cantor Normal Form + +
+ +After Manolios and Vroon's work on ACL2 + +
+ + +
+
+ +
+From Coq Require Import Arith Max Bool Lia Compare_dec Relations Ensembles
+     Wellfounded Bool RelationClasses Operators_Properties ArithRing
+     Logic.Eqdep_dec.
+ +
+From Coq Require PArith.
+From hydras Require Import More_Arith Restriction DecPreOrder.
+From hydras Require Import OrdNotations.
+From hydras Require Export Prelude.Comparable.
+From hydras Require Export STDPP_compat.
+ +
+Create HintDb T1.
+ +
+Set Implicit Arguments.
+ +
+Declare Scope t1_scope.
+Delimit Scope t1_scope with t1.
+Open Scope t1_scope.
+ +
+Coercion is_true: bool >-> Sortclass.
+ +
+
+ +
+

Definitions

+ + +
+ + +
+ +

A type of terms (not necessarily in normal form)

+ + +
+ +cons a n b is intended to represent + the ordinal omega^a × (S n) + b + +
+ +Note that T1 contains terms which are not in Cantor normal form. +This issue is solved later which the help of the predicate nf + +
+ + +
+
+ +
+ +
+Inductive T1 : Set :=
+| zero
+| cons (alpha : T1) (n : nat) (beta : T1) .
+ +
+ +
+
+ +
+Basic functions and predicates on T1 + +
+
+ +
+ +
+
+ +
+The (S n)-th ordinal +
+
+Notation FS n := (cons zero n zero).
+ +
+Notation one := (FS 0).
+ +
+
+ +
+the n-th (finite) ordinal +
+
+ +
+Definition T1nat (n:nat) := match n with 0 ⇒ zero | S p ⇒ FS p end.
+ +
+Notation "\F n" := (T1nat n) (at level 29): t1_scope.
+ +
+Coercion T1nat : nat >-> T1.
+ +
+Example ten : T1 := 10.
+ +
+#[deprecated(note="use T1nat" )]
+ Notation fin := T1nat (only parsing).
+ +
+ +
+Notation T1omega := (cons (cons zero 0 zero) 0 zero).
+ +
+ +
+#[deprecated(note="use T1omega")]
+ Notation omega := T1omega (only parsing).
+ +
+ +
+
+ +
+Successor and limits (syntactic definitions) +
+
+ +
+ +
+Fixpoint T1is_succ alpha :=
+  match alpha with
+    | zero ⇒ false
+    | cons zero _ _ ⇒ true
+    | cons _alpha _n beta ⇒ T1is_succ beta
+  end.
+ +
+Fixpoint T1limit alpha :=
+  match alpha with
+    | zero ⇒ false
+    | cons zero _ _ ⇒ false
+    | cons _ _ zero ⇒ true
+    | cons _ _ beta ⇒ T1limit beta
+  end.
+ +
+ +
+ +
+ +
+ +
+#[deprecated(note="use T1is_succ")]
+  Notation succb := T1is_succ (only parsing).
+ +
+#[deprecated(note="use T1limit")]
+  Notation limitb := T1limit (only parsing).
+ +
+
+ +
+Exponential of base omega +
+
+ +
+ +
+Notation phi0 alpha := (cons alpha 0 zero).
+ +
+ +
+
+ +
+multiples of phi0 alpha +
+
+ +
+Definition omega_term (alpha:T1)(k:nat) :=
+  cons alpha k zero.
+ +
+
+ +
+ omega towers + +
+
+ +
+Fixpoint omega_tower (height:nat) : T1 :=
+  match height with
+  | 0 ⇒ one
+  | S h ⇒ phi0 (omega_tower h)
+  end.
+ +
+ +
+
+ +
+Additive principal ordinals + +
+
+ +
+Inductive ap : T1 → Prop :=
+  ap_intro : ∀ a, ap (phi0 a).
+ +
+
+ +
+

A linear strict order on T1

+ + +
+
+ +
+#[global] Instance compare_T1 : Compare T1 :=
+ fix cmp (alpha beta:T1) :=
+  match alpha, beta with
+  | zero, zero ⇒ Eq
+  | zero, cons a' n' b' ⇒ Lt
+  | _ , zero ⇒ Gt
+  | (cons a n b),(cons a' n' b') ⇒
+      (match cmp a a' with
+       | Lt ⇒ Lt
+       | Gt ⇒ Gt
+       | Eq ⇒ (match n ?= n' with
+                | Eq ⇒ cmp b b'
+                | comp ⇒ comp
+                end)
+       end)
+  end.
+ +
+Definition lt (alpha beta : T1) : Prop :=
+  compare alpha beta = Lt.
+ +
+Notation le := (leq lt).
+ +
+Example E1 : lt (cons T1omega 56 zero) (omega_tower 3).
+ +
+Example E2 : ¬ lt (omega_tower 3) (cons T1omega 5 (omega_tower 3))%t1.
+ +
+
+ +
+

Properties of compare

+ +
+
+ +
+Lemma compare_cons :
+ ∀ a n b a' n' b',
+ compare (cons a n b) (cons a' n' b') =
+ match compare a a' with
+ | Lt ⇒ Lt
+ | Gt ⇒ Gt
+ | Eq ⇒ (match n ?= n' with
+        | Eq ⇒ compare b b'
+        | comp ⇒ comp
+        end)
+ end.
+ +
+ +
+Lemma compare_rev :
+  ∀ (alpha beta : T1),
+  compare beta alpha = CompOpp (compare alpha beta). +
+Lemma compare_reflect :
+  ∀ alpha beta,
+    match compare alpha beta with
+    | Lt ⇒ lt alpha beta
+    | Eq ⇒ alpha = beta
+    | Gt ⇒ lt beta alpha
+    end.
+ +
+Lemma compare_correct (alpha beta: T1):
+  CompSpec eq lt alpha beta (compare alpha beta).
+ +
+
+ +
+

Properties of Eq

+ +
+
+ +
+Lemma compare_refl :
+  ∀ alpha : T1, compare alpha alpha = Eq.
+ +
+Lemma compare_eq_iff (a b : T1) :
+  compare a b = Eq ↔ a = b.
+ +
+
+ +
+

Properties of lt

+ +
+
+ +
+Theorem not_lt_zero alpha : ¬ lt alpha zero.
+ +
+#[global] Hint Resolve not_lt_zero : T1.
+ +
+Lemma compare_lt_impl a b :
+  compare a b = Lt → lt a b.
+ +
+Lemma compare_lt_iff a b :
+  compare a b = Lt ↔ lt a b.
+ +
+
+ +
+

Properties of lt inv

+ +
+
+Inductive lt_cases (a b : T1) (n :nat) (a' b':T1) (n':nat) : Type :=
+  | lt_left (H : lt a a')
+  | lt_middle (H : a = a')(H1 : (n < n')%nat)
+  | lt_right (H : a = a')(H1 : n = n')(H2 : lt b b').
+ +
+Lemma lt_inv_strong :
+  ∀ a n b a' n' b',
+  lt (cons a n b) (cons a' n' b') →
+  lt_cases a b n a' b' n'.
+ +
+Theorem lt_irrefl (alpha: T1):
+  ¬ lt alpha alpha.
+ +
+Theorem lt_inv :
+  ∀ a n b a' n' b',
+  lt (cons a n b) (cons a' n' b') →
+  lt a a' ∨
+  a = a' ∧ (n < n')%nat ∨
+  a = a' ∧ n = n' ∧ lt b b'.
+ +
+Lemma lt_inv_coeff:
+  ∀ a n n' b b',
+  lt (cons a n b) (cons a n' b') → n ≤ n'.
+ +
+Lemma lt_inv_coeff_dec :
+  ∀ a n n' b b',
+  lt (cons a n b) (cons a n' b') →
+  {(n < n')%nat} + { n = n' ∧ lt b b'}.
+ +
+Lemma lt_inv_tail :
+  ∀ a n b b',
+  lt (cons a n b) (cons a n b') → lt b b'.
+ +
+Lemma head_lt :
+  ∀ alpha alpha' n n' beta beta',
+       lt alpha alpha' →
+       lt (cons alpha n beta) (cons alpha' n' beta').
+ +
+Lemma coeff_lt :
+  ∀ alpha n n' beta beta',
+  (n < n')%nat → lt (cons alpha n beta) (cons alpha n' beta').
+ +
+Lemma tail_lt :
+  ∀ alpha n beta beta',
+  lt beta beta' →
+  lt (cons alpha n beta) (cons alpha n beta').
+ +
+Lemma compare_fin_rw (n n1: nat) :
+  compare (T1nat n) (T1nat n1) = (n ?= n1).
+ +
+Lemma lt_fin_iff (i j : nat): lt (T1nat i) (T1nat j) ↔ Nat.lt i j.
+ +
+Theorem lt_trans:
+  ∀ alpha beta gamma: T1,
+  lt alpha beta → lt beta gamma → lt alpha gamma.
+ +
+ +
+#[global] Instance t1_strorder: StrictOrder lt. +
+#[global] Instance: Comparable lt compare. +
+ +
+Lemma lt_inv_head :
+  ∀ a n b a' n' b',
+    lt (cons a n b) (cons a' n' b') → leq lt a a'.
+ +
+
+ +
+

The predicate "to be in normal form"

+ + +
+ +Cantor normal form needs the exponents of omega to be + in strict decreasing order +
+
+ +
+#[ global ] Instance lt_dec : RelDecision lt :=
+fun alpha beta ⇒ decide (compare alpha beta = Lt).
+ +
+Fixpoint nf_b (alpha : T1) : bool :=
+  match alpha with
+  | zero ⇒ true
+  | cons a n zero ⇒ nf_b a
+  | cons a n ((cons a' n' b') as b) ⇒
+      (nf_b a && nf_b b && (bool_decide (lt a' a)))%bool
+  end.
+ +
+Definition nf alpha :Prop := nf_b alpha.
+ +
+ +
+Example bad_term: T1 := cons 1 1 (cons T1omega 2 zero).
+ +
+ +
+
+ +
+epsilon0 as a set +
+
+ +
+Definition epsilon_0 : Ensemble T1 := nf.
+ +
+
+ +
+

Arithmetic functions

+ + +
+ +

Successor

+ +
+
+ +
+Fixpoint succ (a: T1) : T1 :=
+  match a with
+  | zero ⇒ T1nat 1
+  | cons zero n _ ⇒ cons zero (S n) zero
+  | cons b n c ⇒ cons b n (succ c)
+  end.
+ +
+
+ +
+

Predecessor (partial function

+ +
+
+ +
+Fixpoint pred (c:T1) : option T1 :=
+  match c with zero ⇒ None
+  | cons zero 0 _ ⇒ Some zero
+  | cons zero (S n) _ ⇒ Some (cons zero n zero)
+  | cons a n b ⇒
+      match (pred b) with
+      | None ⇒ None
+      | Some c ⇒ Some (cons a n c)
+      end
+  end.
+ +
+
+ +
+

Addition

+ +
+
+ +
+ +
+Fixpoint T1add (a b : T1) :T1 :=
+  match a, b with
+  | zero, y ⇒ y
+  | x, zero ⇒ x
+  | cons a n b, cons a' n' b' ⇒
+      (match compare a a' with
+       | Lt ⇒ cons a' n' b'
+       | Gt ⇒ (cons a n (T1add b (cons a' n' b')))
+       | Eq ⇒ (cons a (S (n+n')) b')
+       end)
+  end
+where "alpha + beta" := (T1add alpha beta) : t1_scope.
+ +
+#[deprecated(note="use T1add")]
+  Notation plus := T1add (only parsing).
+ +
+
+ +
+

multiplication

+ +
+
+ +
+ +
+Fixpoint T1mul (a b : T1) :T1 :=
+  match a, b with
+  | zero, _ ⇒ zero
+  | _, zero ⇒ zero
+  | cons zero n _, cons zero n' b' ⇒
+      cons zero (Peano.pred((S n) × (S n'))) b'
+  | cons a n b, cons zero n' _ ⇒
+      cons a (Peano.pred ((S n) × (S n'))) b
+  | cons a n b, cons a' n' b' ⇒
+      cons (a + a') n' ((cons a n b) × b')
+  end
+where "a * b" := (T1mul a b) : t1_scope.
+ +
+#[deprecated(note="use T1mul")]
+  Notation mult := T1mul (only parsing).
+ +
+
+ +
+

Substraction (used as a helper for exponentiation)

+ +
+
+ +
+Fixpoint minus (alpha beta : T1) :T1 :=
+  match alpha,beta with
+ | zero, y ⇒ zero
+ | cons zero n _, cons zero n' _ ⇒
+     if (le_lt_dec n n')
+     then zero
+     else cons zero (Peano.pred (n-n')) zero
+ | cons zero n _, zero ⇒ cons zero n zero
+ | cons zero _ _, _ ⇒ zero
+ | cons a n b, zero ⇒ cons a n b
+ | cons a n b, cons a' n' b' ⇒
+     (match compare a a' with
+      | Lt ⇒ zero
+      | Gt ⇒ cons a n b
+      | Eq ⇒ (match Nat.compare n n' with
+               | Lt ⇒ zero
+               | Gt ⇒ (cons a (Peano.pred (n - n')) b')
+               | Eq ⇒ b - b'
+               end)
+       end)
+ end
+ where "alpha - beta" := (minus alpha beta):t1_scope.
+ +
+
+ +
+

exponentiation functions

+ + +
+
+ +
+Fixpoint exp_F (alpha : T1)(n : nat) :T1 :=
+ match n with
+ | 0 ⇒ FS 0
+ | S p ⇒ alpha × (exp_F alpha p)
+ end.
+ +
+Fixpoint exp (alpha beta : T1) :T1 :=
+  match alpha ,beta with
+ | _, zero ⇒ cons zero 0 zero
+ | cons zero 0 _ , _ ⇒ cons zero 0 zero
+ | zero, _ ⇒ zero
+ | x , cons zero n' _ ⇒ exp_F x (S n')
+ | cons zero n _, cons (cons zero 0 zero) n' b' ⇒
+        ((cons (cons zero n' zero) 0 zero) ×
+                ((cons zero n zero) ^ b'))
+ | cons zero n _, cons a' n' b' ⇒
+            (omega_term
+                    (omega_term (a' - 1) n')
+                    0) ×
+                 ((cons zero n zero) ^ b')
+ | cons a n b, cons a' n' b' ⇒
+    ((omega_term (a × (cons a' n' zero))
+                            0) ×
+            ((cons a n b) ^ b'))
+end
+where "alpha ^ beta " := (exp alpha beta) : t1_scope.
+ +
+
+ +
+

Lemmas

+ +
+
+ +
+Lemma compare_of_phi0 alpha beta:
+  compare (phi0 alpha) (phi0 beta) = compare alpha beta.
+ +
+Lemma zero_lt : ∀ alpha n beta, lt zero (cons alpha n beta).
+ +
+#[global] Hint Resolve zero_lt head_lt coeff_lt tail_lt : T1.
+ +
+Open Scope t1_scope.
+ +
+Lemma zero_nf : nf zero.
+ +
+Lemma single_nf :
+  ∀ a n, nf a → nf (cons a n zero).
+ +
+Lemma cons_nf :
+  ∀ a n a' n' b,
+  lt a' a →
+  nf a →
+  nf(cons a' n' b)→
+  nf(cons a n (cons a' n' b)).
+ +
+#[global] Hint Resolve zero_nf single_nf cons_nf: T1.
+ +
+Lemma nf_inv1 :
+  ∀ a n b, nf (cons a n b) → nf a.
+ +
+Lemma nf_inv2 :
+  ∀ a n b, nf (cons a n b) → nf b.
+ +
+Lemma nf_inv3 :
+  ∀ a n a' n' b',
+  nf (cons a n (cons a' n' b')) → lt a' a.
+ +
+Lemma nf_cons_inv a n b : nf (cons a n b) → nf a ∧ nf b ∧ lt b (phi0 a).
+ +
+Lemma nf_cons_iff a n b : nf (cons a n b) ↔ nf a ∧ nf b ∧ lt b (phi0 a).
+ +
+
+ +
+already in Stdlib ? +
+
+ +
+Lemma bool_eq_iff (b b':bool) : (b = b') ↔ (b ↔ b').
+ +
+Lemma nf_b_cons_eq a n b : nf_b (cons a n b) =
+                           nf_b a && nf_b b && bool_decide (lt b (phi0 a)).
+ +
+ +
+Ltac nf_decomp H :=
+  let nf0 := fresh "nf"
+  in let nf1 := fresh "nf"
+     in let Hlp := fresh "lt"
+     in
+     match type of H with
+     | nf (cons ?t ?n zero) ⇒ assert (nf0:= nf_inv1 H)
+     | nf (cons ?t1 ?n (cons ?t2 ?p ?t3))
+       ⇒ assert (nf0 := nf_inv1 H); assert(nf1 := nf_inv2 H);
+          assert (lt := nf_inv3 H)
+     | nf (cons ?t1 ?n ?t2) ⇒ assert (nf0 := nf_inv1 H); assert(nf1 := nf_inv2 H)
+     end.
+ +
+
+ +
+ lt alpha (phi0 beta) +
+
+Inductive lt_a_phi0_b : T1 → T1 → Prop :=
+| lt_a_phi0_b_z : ∀ alpha, lt_a_phi0_b zero alpha
+| lt_a_phi0_b_c : ∀ alpha alpha' n' beta',
+                  lt alpha' alpha →
+                  lt_a_phi0_b (cons alpha' n' beta') alpha.
+ +
+#[global] Hint Constructors lt_a_phi0_b : T1.
+ +
+Reserved Notation "x '<_phi0' y" (at level 70, no associativity).
+Infix "<_phi0" := lt_a_phi0_b.
+ +
+ +
+Definition get_decomposition : ∀ c:T1, lt zero c →
+                           {a:T1 & {n:nat & {b:T1 | c = cons a n b}}}.
+ +
+Ltac decomp_exhib H a n b e:=
+ let Ha := fresh in
+ let Hn := fresh in
+ let Hb := fresh in
+  match type of H
+  with lt zero ?c ⇒
+    case (get_decomposition H);
+     intros a Ha;
+     case Ha;intros n Hn; case Hn; intros b e;
+     clear Ha Hn
+  end.
+ +
+Lemma nf_FS : ∀ n:nat, nf (FS n).
+ +
+Lemma nf_fin : ∀ n:nat, nf (T1nat n).
+ +
+
+ +
+

Successors, limits and zero

+ +
+
+ +
+Lemma succ_not_zero : ∀ alpha, succ alpha ≠ zero.
+ +
+Lemma succ_is_succ : ∀ alpha, T1is_succ (succ alpha).
+ +
+
+ +
+

Second part on lt and le

+ +
+
+ +
+Lemma finite_lt :
+  ∀ n p : nat, (n < p)%nat → lt (FS n) (FS p).
+ +
+Lemma finite_ltR :
+  ∀ n p : nat,
+  lt (FS n) (FS p) → (n < p)%nat.
+ +
+Lemma le_eq_lt_dec alpha beta:
+  leq lt alpha beta →
+  {alpha = beta} + {lt alpha beta}.
+ +
+Lemma lt_succ (alpha : T1) : lt alpha (succ alpha).
+ +
+Lemma lt_a_phi0_a :
+  ∀ a, lt a (phi0 a).
+ +
+Lemma phi0_lt :
+  ∀ a b, lt a b → lt (phi0 a) (phi0 b).
+ +
+Lemma phi0_ltR :
+  ∀ a b, lt (phi0 a) (phi0 b) → lt a b.
+ +
+Lemma nf_of_finite :
+  ∀ n b,
+  nf (cons zero n b) → b = zero.
+ +
+Theorem zero_le :
+  ∀ a, leq lt zero a.
+ +
+Theorem le_inv :
+  ∀ a n b a' n' b',
+  leq lt (cons a n b) (cons a' n' b') →
+  lt a a' ∨
+  a = a' ∧ (n < n')%nat ∨
+  a = a' ∧ n = n' ∧ leq lt b b'.
+ +
+Arguments le_inv [a n b a' n' b'] _.
+ +
+Lemma lt_a_phi0_b_inv :
+  ∀ a n b a', lt (cons a n b) (phi0 a') → lt a a'.
+ +
+Theorem le_zero_inv :
+  ∀ a, leq lt a zero → a = zero.
+ +
+Theorem le_tail :
+  ∀ a n b b',
+  leq lt b b' →
+  leq lt (cons a n b) (cons a n b').
+ +
+#[global] Hint Resolve zero_le le_tail : T1.
+ +
+Theorem le_phi0 :
+  ∀ a n b, leq lt (phi0 a) (cons a n b).
+ +
+Lemma head_lt_cons :
+  ∀ a n b, lt a (cons a n b).
+ +
+#[export] Hint Resolve head_lt_cons: T1.
+ +
+Definition T1_eq_dec (alpha beta : T1):
+{alpha = beta} + {alpha ≠ beta}.
+ +
+Definition lt_eq_lt_dec :
+  ∀ alpha beta : T1,
+  {lt alpha beta} + {alpha = beta} + {lt beta alpha}.
+ +
+Definition lt_le_dec (alpha beta : T1) :
+  {lt alpha beta} + {leq lt beta alpha}.
+ +
+#[ global ] Instance epsilon0_pre_order : TotalPreOrder (leq lt).
+ +
+#[ global ] Instance epsilon0_dec : RelDecision (leq lt).
+ +
+Ltac auto_nf :=
+  match goal with
+    |- nf ?alpha ⇒
+    ( repeat (apply cons_nf || apply single_nf || apply zero_nf))
+    || (eapply nf_inv1 || eapply nf_inv2); eauto
+  end.
+ +
+Lemma nf_tail_lt_nf b b':
+  lt b' b → nf b' →
+  ∀ a n, nf (cons a n b) → nf (cons a n b').
+ +
+Lemma tail_lt_cons :
+  ∀ b n a,
+  nf (cons a n b) → lt b (cons a n b).
+ +
+Lemma lt_a_phi0_b_inv1 :
+  ∀ a n b a', cons a n b <_phi0 a'→ lt a a'.
+ +
+Lemma nf_intro :
+  ∀ a n b, nf a → nf b → b <_phi0 a →
+                nf (cons a n b).
+ +
+Lemma nf_intro' :
+  ∀ a n b,
+  nf a →
+  nf b →
+  lt b (cons a 0 zero) →
+  nf (cons a n b).
+ +
+Lemma lt_a_phi0_b_intro :
+  ∀ a n b, nf (cons a n b) → b <_phi0 a.
+ +
+Lemma nf_coeff_irrelevance :
+  ∀ a b n p, nf (cons a n b) → nf (cons a p b).
+ +
+Lemma lt_a_phi0_b_phi0 :
+  ∀ a b, b <_phi0 a → lt b ( phi0 a).
+ +
+Lemma lt_a_phi0_b_phi0R :
+  ∀ a b, lt b (phi0 a) → b <_phi0 a.
+ +
+Lemma lt_a_phi0_b_def :
+  ∀ a b, b <_phi0 a ↔ lt b (phi0 a).
+ +
+Lemma lt_a_phi0_b_iff :
+  ∀ a b, nf a → nf b →
+              (b <_phi0 a ↔ ∀ n, nf (cons a n b)).
+ +
+Lemma nf_omega_tower : ∀ n, nf (omega_tower n).
+ +
+
+ +
+A strong induction scheme for nf +
+
+ +
+Definition nf_rect : ∀ P : T1 → Type,
+    P zero →
+    (∀ n: nat, P (cons zero n zero)) →
+    (∀ a n b n' b', nf (cons a n b) →
+                         P (cons a n b)→
+                         lt b' (phi0 (cons a n b)) →
+                         nf b' →
+                         P b' →
+                         P (cons (cons a n b) n' b')) →
+    ∀ a: T1, nf a → P a.
+ +
+
+ +
+

Properties of compare: second part

+ +
+
+ +
+Theorem compare_reflectR ( alpha beta : T1) :
+  (match lt_eq_lt_dec alpha beta with
+   | inleft (left _) ⇒ Lt
+   | inleft (right _) ⇒ Eq
+   | inright _ ⇒ Gt
+   end)
+  = compare alpha beta.
+ +
+
+ +
+

Properties of max

+ +
+
+Lemma max_nf (alpha beta : T1) :
+  nf alpha → nf beta → nf (max alpha beta).
+ +
+
+ +
+

Restriction of lt and le to terms in normal form

+ +
+
+Reserved Notation "x 't1<' y" (at level 70, no associativity).
+Reserved Notation "x 't1<=' y" (at level 70, no associativity).
+Reserved Notation "x 't1>=' y" (at level 70, no associativity).
+Reserved Notation "x 't1>' y" (at level 70, no associativity).
+ +
+Reserved Notation "x 't1<=' y 't1<=' z" (at level 70, y at next level).
+Reserved Notation "x 't1<=' y 't1<' z" (at level 70, y at next level).
+Reserved Notation "x 't1<' y 't1<' z" (at level 70, y at next level).
+Reserved Notation "x 't1<' y 't1<=' z" (at level 70, y at next level).
+ +
+ +
+Definition LT := restrict nf lt.
+Infix "t1<" := LT : t1_scope.
+ +
+Definition LE := restrict nf (leq lt).
+Infix "t1<=" := LE : t1_scope.
+ +
+ +
+Notation "alpha t1< beta t1< gamma" :=
+  (LT alpha beta ∧ LT beta gamma) : t1_scope.
+ +
+Definition Elements alpha : Ensemble T1 :=
+  fun beta ⇒ beta t1< alpha.
+ +
+Coercion Elements : T1 >-> Ensemble.
+ +
+Lemma LT_nf_l : ∀ alpha beta , alpha t1< beta → nf alpha.
+ +
+Lemma LT_nf_r : ∀ alpha beta , alpha t1< beta → nf beta.
+ +
+Lemma LT_lt alpha beta : alpha t1< beta → lt alpha beta.
+ +
+Lemma LE_nf_l : ∀ alpha beta , alpha t1≤ beta → nf alpha.
+ +
+Lemma LE_nf_r : ∀ alpha beta , alpha t1≤ beta → nf beta.
+ +
+Lemma LE_le alpha beta : alpha t1≤ beta → leq lt alpha beta.
+ +
+#[global] Hint Resolve LT_nf_r LT_nf_l LT_lt LE_nf_r LE_nf_l LE_le : T1.
+ +
+Lemma not_zero_gt_0 (alpha:T1) : alpha ≠ zero → lt zero alpha.
+ +
+Lemma not_zero_lt (alpha: T1): nf alpha → alpha ≠ zero →
+                               zero t1< alpha.
+ +
+Lemma LE_zero : ∀ alpha, nf alpha → zero t1≤ alpha.
+ +
+Lemma LE_refl : ∀ alpha, nf alpha → alpha t1≤ alpha.
+ +
+Lemma LT_trans : ∀ a b c:T1, a t1< b → b t1< c → a t1< c.
+ +
+Theorem LE_trans (alpha beta gamma: T1):
+          alpha t1≤ beta → beta t1≤ gamma → alpha t1≤ gamma.
+ +
+Lemma LE_antisym (alpha beta : T1): alpha t1≤ beta →
+                                     beta t1≤ alpha →
+                                     alpha = beta.
+ +
+Lemma LT1 : ∀ alpha n beta, nf (cons alpha n beta) →
+                                 zero t1< cons alpha n beta.
+ +
+Lemma LT2 : ∀ alpha alpha' n n' beta beta',
+    nf (cons alpha n beta) →
+    nf (cons alpha' n' beta') →
+    alpha t1< alpha' →
+    cons alpha n beta t1< cons alpha' n' beta'.
+ +
+Lemma LT3 : ∀ alpha n n' beta beta',
+    nf (cons alpha n beta) →
+    nf (cons alpha n' beta') →
+    n < n' →
+    cons alpha n beta t1< cons alpha n' beta'.
+ +
+Lemma LT4 : ∀ alpha n beta beta',
+    nf (cons alpha n beta) →
+    nf (cons alpha n beta') →
+    beta t1< beta' →
+    cons alpha n beta t1< cons alpha n beta'.
+ +
+#[global] Hint Resolve LT1 LT2 LT3 LT4: T1.
+ +
+Lemma LT_irrefl (alpha : T1) :
+  ¬ alpha t1< alpha.
+ +
+Lemma LE_LT_trans :
+  ∀ alpha beta gamma,
+  alpha t1≤ beta → beta t1< gamma → alpha t1< gamma.
+ +
+Lemma LT_LE_trans (alpha beta gamma : T1) : alpha t1< beta →
+                                            beta t1≤ gamma →
+                                            alpha t1< gamma.
+ +
+Lemma not_LT_zero :
+  ∀ alpha, ¬ alpha t1< zero.
+ +
+#[ global ] Instance LT_St : StrictOrder LT.
+ +
+Lemma nf_cons_LT :
+  ∀ (a : T1) (n : nat) (a' : T1) (n' : nat) (b : T1),
+  a' t1< a →
+  nf a → nf (cons a' n' b) → nf (cons a n (cons a' n' b)).
+ +
+#[global] Hint Resolve nf_cons_LT: T1.
+ +
+#[global] Hint Resolve nf_inv1 nf_inv2 nf_inv3 : T1.
+ +
+Lemma head_LT_cons :
+  ∀ alpha n beta,
+  nf (cons alpha n beta) →
+  alpha t1< cons alpha n beta.
+ +
+Lemma tail_LT_cons :
+  ∀ alpha n beta,
+  nf (cons alpha n beta) →
+  beta t1< cons alpha n beta.
+ +
+Lemma LT_inv : ∀ a n b a' n' b',
+    cons a n b t1< cons a' n' b' →
+    a t1< a' ∨
+    a = a' ∧ (n < n' ∨ n = n' ∧ b t1< b').
+ +
+Inductive LT_cases (a b : T1) (n :nat) (a' b':T1) (n':nat) : Type :=
+| LT_left (H : a t1< a')
+| LT_middle (H : a = a')(H1 : n < n')
+| LT_right (H : a = a')(H1 : n = n')(H2 : b t1< b').
+ +
+Lemma LT_inv_strong :
+  ∀ a b n a' b' n',
+  cons a n b t1< cons a' n' b' → LT_cases a b n a' b' n'.
+ +
+Lemma remove_first_sumand :
+  ∀ a n b b',
+    cons a n b t1< cons a n b' → b t1< b'.
+ +
+Lemma LT_cons_0 :
+  ∀ a n b a' b',
+  cons a n b t1< cons a' 0 b' → a t1< a' ∨ n = 0 ∧ a = a' ∧ b t1< b'.
+ +
+Lemma LE_phi0 :
+  ∀ a n b, nf (cons a n b) → phi0 a t1≤ cons a n b.
+ +
+Lemma nf_tail_lt alpha n beta gamma :
+  nf (cons alpha n beta) → gamma t1< beta →
+  nf (cons alpha n gamma).
+ +
+
+ +
+

Well foundedness of LT

+ +
+
+ +
+Module Direct_proof.
+  Section well_foundedness_proof.
+    #[local] Hint Unfold restrict LT: T1.
+ +
+    Lemma Acc_zero : Acc LT zero.
+ +
+ +
+ +
+    Section First_attempt.
+ +
+      Lemma wf_LT : ∀ alpha: T1, nf alpha → Acc LT alpha.
+    End First_attempt.
+ +
+
+ +
+

Strong accessibility (inspired by Tait's proof)

+ +
+
+ +
+    Let Acc_strong (alpha:T1) :=
+      ∀ n beta,
+        nf (cons alpha n beta) → Acc LT (cons alpha n beta).
+ +
+    Remark acc_impl {A} {R: A → A → Prop} (a b:A) :
+      R a b → Acc R b → Acc R a.
+ +
+ +
+    Lemma Acc_strong_stronger : ∀ alpha,
+        nf alpha → Acc_strong alpha → Acc LT alpha.
+ +
+ +
+ +
+    Lemma Acc_implies_Acc_strong : ∀ alpha,
+        Acc LT alpha → Acc_strong alpha. +
+ +
+
+ +
+

A (last) structural induction

+ +
+
+ +
+ +
+    Theorem nf_Acc (alpha : T1): nf alpha → Acc LT alpha. +
+ +
+  End well_foundedness_proof.
+End Direct_proof.
+ +
+Definition nf_Acc := Direct_proof.nf_Acc.
+ +
+Corollary nf_Wf : well_founded_restriction _ nf lt.
+ +
+ +
+Corollary T1_wf : well_founded LT. +
+Definition transfinite_recursor_lt :
+  ∀ (P:T1 → Type),
+    (∀ x:T1,
+        (∀ y:T1, nf x → nf y → lt y x → P y) → P x) →
+    ∀ alpha: T1, P alpha.
+ +
+ +
+Definition transfinite_recursor := well_founded_induction_type T1_wf.
+ +
+Check transfinite_recursor.
+ +
+ +
+Import Direct_proof.
+ +
+Ltac transfinite_induction_lt alpha :=
+  pattern alpha; apply transfinite_recursor_lt.
+ +
+ +
+Ltac transfinite_induction alpha :=
+  pattern alpha; apply transfinite_recursor.
+ +
+ +
+
+ +
+

Properties of successor

+ +
+
+ +
+ +
+Lemma succ_nf : ∀ alpha : T1 , nf alpha → nf (succ alpha).
+ +
+
+ +
+

properties of addition

+ +
+
+ +
+Lemma plus_zero alpha: zero + alpha = alpha.
+ +
+Lemma plus_zero_r alpha: alpha + zero = alpha.
+ +
+ +
+Lemma succ_is_plus_one (a : T1) : succ a = a + 1.
+ +
+Lemma succ_of_plus_finite :
+  ∀ a (H: nf a) (i:nat) , succ (a + i) = a + S i.
+ +
+
+ +
+

plus and LT

+ +
+
+ +
+Lemma plus_cons_cons_rw1 :
+  ∀ a n b a' n' b',
+  lt a a' →
+  cons a n b + cons a' n' b' = cons a' n' b'.
+ +
+Lemma plus_cons_cons_rw2 :
+  ∀ a n b n' b',
+  nf (cons a n b) →
+  nf (cons a n' b') →
+  cons a n b + cons a n' b' = cons a (S (n + n')) b'.
+ +
+Lemma plus_cons_cons_rw3 :
+  ∀ a n b a' n' b',
+  lt a' a →
+  nf (cons a n b) →
+  nf (cons a' n' b') →
+  cons a n b + cons a' n' b'=
+  cons a n (b + (cons a' n' b')).
+ +
+
+ +
+

On additive principal ordinals

+ +
+
+ +
+Lemma ap_plus : ∀ a,
+    ap a →
+    ∀ b c,
+      nf b → nf c → lt b a → lt c a → lt (b + c) a.
+ +
+Lemma ap_plusR :
+  ∀ c,
+  nf c →
+  c ≠ zero →
+  (∀ a b, nf a → nf b → lt a c →lt b c → lt (a + b) c) →
+  ap c.
+ +
+
+ +
+Technical lemma for proving plus_nf +
+
+ +
+Lemma plus_nf0 :
+  ∀ a, nf a →
+  ∀ b c,
+    lt b (phi0 a) →
+    lt c (phi0 a) →
+    nf b → nf c →
+    nf (b + c).
+ +
+ +
+Lemma plus_nf:
+  ∀ a, nf a → ∀ b, nf b → nf (a + b). +
+ +
+Lemma omega_term_plus_rw:
+  ∀ a n b,
+    nf (cons a n b) →
+    omega_term a n + b = cons a n b.
+ +
+ +
+Lemma plus_is_zero alpha beta :
+  nf alpha → nf beta →
+  alpha + beta = zero → alpha = zero ∧ beta = zero. +
+ +
+Lemma T1add_not_monotonous_l :
+  ∃ a b c : T1, a t1< b ∧ a + c = b + c.
+ +
+Lemma T1mul_not_monotonous :
+  ∃ a b c : T1, c ≠ zero ∧ a t1< b ∧ a × c = b × c.
+ +
+
+ +
+

monotonicity of succ

+ +
+
+ +
+Lemma succ_strict_mono :
+  ∀ a b,
+    lt a b → nf a → nf b →
+    lt (succ a) (succ b).
+ +
+Lemma succ_strict_mono_LT :
+  ∀ alpha beta,
+  alpha t1< beta → succ alpha t1< succ beta.
+ +
+Lemma succ_mono :
+  ∀ a b,
+  nf a → nf b →
+  leq lt a b → leq lt (succ a) (succ b).
+ +
+Lemma lt_succ_le_R :
+  ∀ a, nf a → ∀ b, nf b →
+  leq lt (succ a) b → lt a b .
+ +
+Lemma le_lt_LT alpha beta :
+  nf alpha → nf beta → leq lt alpha beta → leq LT alpha beta.
+ +
+Lemma LT_succ_LE_R :
+  ∀ alpha beta,
+    nf alpha →
+    succ alpha t1≤ beta → alpha t1< beta.
+ +
+Lemma lt_succ_le_2 :
+  ∀ a,
+  nf a → ∀ b, nf b →
+  lt a (succ b) → leq lt a b.
+ +
+
+ +
+TODO: bulletize this proof ! +
+
+ +
+Lemma lt_succ_le :
+  ∀ a,
+    nf a → ∀ b, nf b →
+                      lt a b → leq lt (succ a) b.
+ +
+Lemma LT_succ_LE :
+  ∀ alpha beta ,
+  alpha t1< beta → succ alpha t1≤ beta.
+ +
+Lemma LT_succ_LE_2:
+  ∀ alpha beta : T1, nf beta →
+    alpha t1< succ beta → alpha t1≤ beta.
+ +
+Lemma succ_strict_monoR :
+  ∀ alpha beta,
+    nf alpha → nf beta →
+      lt (succ alpha) (succ beta) → lt alpha beta.
+ +
+Lemma succ_monomorphism :
+  ∀ alpha (H:nf alpha) beta (H' : nf beta),
+    lt alpha beta ↔ lt (succ alpha) (succ beta).
+ +
+Lemma succ_injective :
+  ∀ alpha (H:nf alpha) beta (H' : nf beta),
+    succ alpha = succ beta → alpha = beta.
+ +
+Lemma succ_compatS :
+  ∀ n:nat, succ (FS n) = FS (S n).
+ +
+Lemma succ_compat (n:nat) :
+  succ (T1nat n) = FS n.
+ +
+
+ +
+

monotonicity of phi0

+ +
+
+ +
+Lemma phi0_mono_strict :
+  ∀ a b, lt a b → lt (phi0 a) (phi0 b).
+ +
+Lemma phi0_mono_strict_LT :
+  ∀ alpha beta,
+    alpha t1< beta → phi0 alpha t1< phi0 beta.
+ +
+Lemma phi0_mono :
+  ∀ a b, leq lt a b → leq lt (phi0 a) ( phi0 b).
+ +
+Lemma plus_left_absorb :
+  ∀ a n b c,
+  lt c (phi0 a) → c + cons a n b = cons a n b.
+ +
+Lemma plus_compat:
+  ∀ n p, FS n + FS p = FS (S n + p).
+ +
+
+ +
+

Multiplication

+ +
+
+ +
+Lemma mult_fin_omega :
+  ∀ n: nat,
+    FS n × T1omega = T1omega.
+ +
+Lemma phi0_plus_mult :
+  ∀ a b, nf a → nf b → phi0 (a + b) = phi0 a × phi0 b.
+ +
+Lemma mult_compat :
+  ∀ n p, FS n × FS p = FS (n × p + n + p)%nat.
+ +
+Lemma mult_a_0 :
+  ∀ a, a × zero = zero.
+ +
+Lemma mult_1_a :
+  ∀ a, nf a → 1 × a = a.
+ +
+Lemma mult_a_1 :
+  ∀ a, nf a → a × 1 = a.
+ +
+Lemma mult_nf_fin alpha n: nf alpha → nf (alpha × T1nat n).
+ +
+
+ +
+

About minus

+ +
+
+ +
+Lemma minus_lt :
+  ∀ a b, lt a b → a - b = zero.
+ +
+Lemma minus_a_a : ∀ a, a - a = zero.
+ +
+Lemma minus_le : ∀ a b, leq lt a b → a - b = zero.
+ +
+
+ +
+

Exponential

+ +
+
+ +
+Lemma exp_fin_omega : ∀ n, FS (S n) ^ T1omega = T1omega.
+ +
+
+ +
+

Relations between cons, phi0 and +

+ + +
+ + The next three lemmas express the consistency between + the intuitive meaning given to the constructor cons and + its derivates : phi0, omega-term, and the arithmetic + operations on ordinals which belong to epsilon0 +
+
+ +
+Lemma phi0_eq_bad : ∀ alpha, T1omega ^ alpha = phi0 alpha.
+ +
+Lemma phi0_eq : ∀ alpha, nf alpha → T1omega ^ alpha = phi0 alpha.
+ +
+Lemma omega_term_def :
+  ∀ a n, nf a → omega_term a n = T1omega ^ a × FS n.
+ +
+Lemma cons_def :
+  ∀ a n b,
+  nf(cons a n b) → cons a n b = T1omega ^ a × FS n + b.
+ +
+Theorem unique_decomposition :
+  ∀ a n b a' n' b',
+    nf (cons a n b) → nf (cons a' n' b') →
+    T1omega ^ a × FS n + b =
+    T1omega ^ a' × FS n' + b' →
+    a = a' ∧ n = n' ∧ b = b'.
+ +
+Theorem Cantor_normal_form :
+  ∀ o, lt zero o → nf o →
+            {a:T1 & {n: nat & {b : T1 |
+                                o = T1omega ^ a × FS n + b ∧
+                                nf (cons a n b) }}}.
+ +
+ +
+Lemma LT_one alpha :
+  alpha t1< one → alpha = zero. +
+Lemma lt_omega_inv :
+  ∀ alpha,
+  alpha t1< T1omega → alpha = zero ∨ ∃ n, alpha = FS n.
+ +
+Ltac T1_inversion H :=
+  match type of H with lt _ zero ⇒ destruct (not_lt_zero H)
+                  | Nat.lt _ 0 ⇒ destruct (Nat.nlt_0_r _ H)
+                  | Nat.lt ?x ?x ⇒ destruct (Nat.lt_irrefl _ H)
+                  | lt ?x ?x ⇒ destruct (lt_irrefl H)
+                  | lt (cons _ _ _) (cons _ _ _) ⇒
+                    destruct (lt_inv H)
+                  | nf (cons zero ?n ?y) ⇒ let e := fresh "e" in
+                                             generalize (nf_of_finite H);
+                                             intros e
+  end.
+ +
+Lemma LT_of_finite :
+  ∀ alpha n, alpha t1< FS n → alpha = zero ∨
+                                  ∃ p, p < n ∧ alpha = FS p.
+ +
+Lemma nf_omega : nf T1omega.
+ +
+ +
+Theorem nf_phi0 alpha : nf alpha → nf (phi0 alpha).
+ +
+#[global] Hint Resolve nf_phi0 : T1.
+ +
+Definition omega_omega := phi0 T1omega.
+ +
+Lemma nf_omega_omega : nf omega_omega.
+ +
+Lemma mult_0_a : ∀ a, zero × a = zero.
+ +
+Lemma mult_Sn_add (alpha : T1) n :
+  nf alpha →
+  alpha × (FS (S n)) = alpha × FS n + alpha.
+ +
+Lemma cases_for_mult (alpha : T1) :
+  nf alpha →
+  alpha = zero ∨
+  (∃ n : nat, alpha = FS n) ∨
+  (∃ a n, a ≠ zero ∧ alpha = cons a n zero) ∨
+  (∃ a n b, a ≠ zero ∧ b ≠ zero ∧ alpha = cons a n b).
+ +
+Lemma L03 alpha n beta p :
+  alpha ≠ zero →
+  (cons alpha n beta × FS p) = cons alpha (p + n × S p) beta.
+ +
+Lemma L05 a n b c p d :
+  a ≠ zero → c ≠ zero →
+  (cons a n b × cons c p d) =
+  cons (a + c) p (cons a n b × d).
+ +
+Lemma nf_LT_iff :
+  ∀ alpha n beta, nf (cons alpha n beta) ↔
+                       nf alpha ∧ nf beta
+                       ∧ beta t1< phi0 alpha.
+ +
+Lemma lt_plus_l:
+  ∀ {a b c : T1} {n:nat}, lt a (a + cons b n c).
+ +
+Lemma lt_plus_r:
+  ∀ {a b c : T1} {n:nat}, ¬ lt (a + cons b n c) a.
+ +
+Lemma reduce_lt_plus:
+  ∀ a b c: T1,
+  lt (a+ b) (a + c) ↔ lt b c.
+ +
+Lemma plus_smono_LT_r (alpha:T1) :
+  ∀ beta gamma, nf alpha → beta t1< gamma → alpha + beta t1< alpha + gamma.
+ +
+Lemma LT_add (alpha beta : T1): nf alpha → nf beta → beta ≠ zero →
+                                alpha t1< alpha + beta.
+ +
+Section Proof_of_mult_nf.
+ +
+  Variable alpha : T1.
+  Hypothesis Halpha : nf alpha.
+ +
+  Let P (beta : T1) :=
+    nf beta → nf (alpha × beta) ∧
+               (alpha ≠ zero →
+                ∀ gamma, gamma t1< beta →
+                              alpha × gamma t1< alpha × beta).
+  Section Induction.
+ +
+    Variable beta : T1.
+    Hypothesis Hbeta : nf beta.
+    Hypothesis IHbeta : ∀ delta, delta t1< beta → P delta.
+ +
+    Lemma L1 : alpha = zero → P beta.
+ +
+    Lemma L2 : beta = zero → P beta.
+ +
+    Lemma L3 n p : alpha = FS n → beta = FS p → P beta.
+ +
+    Lemma L4 : ∀ a n b p, a ≠ zero →
+                               alpha = cons a n b → beta = FS p →
+                               P beta.
+ +
+    Lemma L5 a n b c p : a ≠ zero → c ≠ zero →
+                         (cons a n b) × (cons c p zero) =
+                         cons (a + c) p zero.
+ +
+    Lemma L6 n c p d : c ≠ zero → FS n × cons c p d = cons c p (FS n × d).
+ +
+    Lemma L7 n c p : c ≠ zero → FS n × cons c p zero = cons c p zero.
+ +
+    Lemma L8 n c p : alpha = FS n → beta = cons c p zero → c ≠ zero →
+                      P beta.
+ +
+    Lemma L9 : ∀ n c, nf c → c ≠ zero → FS n × c ≠ zero.
+ +
+    Lemma L10 : ∀ a n b c, nf c → nf (cons a n b) →
+                                a ≠zero → c ≠ zero →
+                                cons a n b × c ≠ zero.
+ +
+    Lemma L11 n c p d :
+      alpha = FS n → beta = cons c p d → c ≠ zero →
+      d ≠ zero → P beta.
+ +
+    Lemma L12 : ∀ a n b c p d , a ≠ zero → c ≠ zero →
+                                     alpha = cons a n b →
+                                     beta = cons c p d →
+                                     P beta.
+ +
+    Lemma L13 : P beta.
+ +
+  End Induction.
+ +
+  Lemma L14 beta : nf beta → P beta.
+ +
+End Proof_of_mult_nf.
+ +
+Theorem mult_nf alpha beta : nf alpha → nf beta →
+                             nf (alpha × beta).
+ +
+Theorem mult_mono alpha beta gamma : nf alpha → alpha ≠ zero →
+                                     beta t1< gamma → alpha × beta t1< alpha × gamma.
+ +
+Lemma nf_exp_F alpha n : nf alpha → nf (exp_F alpha n).
+ +
+Lemma exp_F_eq alpha n : nf alpha → (exp_F alpha n = alpha ^ n)%t1.
+ +
+Lemma T1limit_cases : ∀ alpha n beta,
+    nf (cons alpha n beta) →
+    T1limit (cons alpha n beta) →
+    { alpha ≠ zero ∧ beta = zero} +
+    {alpha ≠ zero ∧ T1limit beta }.
+ +
+Lemma pred_of_succ : ∀ beta, nf beta → pred (succ beta) = Some beta.
+ +
+Lemma pred_of_limit : ∀ alpha, nf alpha →
+                                    T1limit alpha →
+                                    pred alpha = None.
+ +
+Definition zero_limit_succ_dec :
+  ∀ alpha, nf alpha →
+                ({alpha = zero} + {T1limit alpha }) +
+                {beta : T1 | nf beta ∧ alpha = succ beta} .
+ +
+Lemma pred_of_limitR : ∀ alpha, nf alpha → alpha ≠ zero →
+                                     pred alpha = None → T1limit alpha.
+ +
+Lemma pred_LT : ∀ alpha beta, nf alpha → pred alpha = Some beta →
+                                   beta t1< alpha .
+ +
+Lemma pred_nf : ∀ alpha beta, nf alpha → pred alpha = Some beta →
+                                   nf beta.
+ +
+Lemma T1limit_succ : ∀ alpha, nf alpha → ¬ T1limit (succ alpha) .
+ +
+Lemma LT_succ : ∀ alpha, nf alpha → alpha t1< succ alpha.
+ +
+Lemma T1limit_not_zero : ∀ alpha, nf alpha → T1limit alpha →
+                                        alpha ≠ zero.
+ +
+#[global] Hint Resolve T1limit_not_zero : T1.
+ +
+Lemma T1limit_succ_tail :
+  ∀ alpha n beta, nf beta → ¬ T1limit (cons alpha n (succ beta)).
+ +
+Lemma succ_not_limit : ∀ alpha:T1, T1is_succ alpha → T1limit alpha = false.
+ +
+Lemma T1is_succ_def alpha (Halpha : nf alpha) :
+  T1is_succ alpha → {beta | nf beta ∧ alpha = succ beta}.
+ +
+Lemma T1is_succ_iff alpha (Halpha : nf alpha) :
+  T1is_succ alpha ↔ ∃ beta : T1, nf beta ∧ alpha = succ beta.
+ +
+Lemma LE_r : ∀ alpha beta, alpha t1< beta → alpha t1≤ beta.
+ +
+Lemma LE_LT_eq_dec :
+  ∀ alpha beta, alpha t1≤ beta →
+  {alpha t1< beta} + {alpha = beta}.
+ +
+Lemma LT_eq_LT_dec : ∀ alpha beta,
+    nf alpha → nf beta →
+    {alpha t1< beta} + {alpha = beta} + {beta t1< alpha}.
+ +
+Lemma lt_cons_phi0_inv alpha n beta gamma :
+  cons alpha n beta t1< phi0 gamma ↔ beta t1< phi0 alpha ∧ alpha t1< gamma.
+ +
+Lemma nf_LT_right : ∀ alpha n beta beta',
+    nf (cons alpha n beta) →
+    beta' t1< beta →
+    nf (cons alpha n beta').
+ +
+Lemma eq_succ_LT : ∀ alpha beta, nf beta → alpha = succ beta →
+                                      beta t1< alpha.
+ +
+Lemma eq_succ_lt : ∀ alpha beta, nf beta → alpha = succ beta →
+                                      lt beta alpha.
+ +
+Definition strict_lub (s : nat → T1) (lambda : T1) :=
+  (∀ i, s i t1< lambda) ∧
+  (∀ alpha, (∀ i, s i t1≤ alpha) → lambda t1≤ alpha).
+ +
+Definition strict_lub_lub : ∀ s l alpha, strict_lub s l →
+                                               (∀ i, s i t1≤ alpha) →
+                                               l t1≤ alpha.
+ +
+Definition strict_lub_maj : ∀ s l , strict_lub s l →
+                                          ∀ i, s i t1< l.
+ +
+Lemma strict_lub_unique : ∀ s l l', strict_lub s l →
+                                         strict_lub s l' →
+                                         l = l'.
+ +
+Lemma strict_lub_T1limit : ∀ (alpha :T1)(s : nat → T1),
+    nf alpha → strict_lub s alpha → T1limit alpha.
+ +
+Lemma lt_one (alpha:T1) : lt alpha one → alpha = zero.
+ +
+Lemma omega_limit : strict_lub T1nat T1omega.
+ +
+Lemma LT_succ_LT_eq_dec :
+  ∀ alpha beta, nf alpha → nf beta →
+                     alpha t1< succ beta → {alpha t1< beta} + {alpha = beta}.
+ +
+Lemma lt_succ_le_2':
+  ∀ a : T1, nf a → ∀ b : T1, nf b → a t1< succ b →
+                                        a t1< b ∨ a = b.
+ +
+Lemma succ_lt_limit alpha (Halpha : nf alpha)(H : T1limit alpha ):
+  ∀ beta, beta t1< alpha → succ beta t1< alpha.
+ +
+Lemma succ_cons alpha n beta : beta ≠ zero → nf (cons alpha n beta) →
+                                succ (cons alpha n beta) =
+                                cons alpha n (succ beta).
+ +
+Lemma succ_cons' alpha i beta : alpha ≠ zero → nf (cons alpha i beta) →
+                               succ (cons alpha i beta) =
+                               cons alpha i (succ beta).
+ +
+Lemma succ_rw1 : ∀ alpha n beta, alpha ≠ zero →
+                                      succ (cons alpha n beta)=
+                                      cons alpha n (succ beta).
+ +
+Lemma plus_cons_cons_eqn a n b a' n' b':
+  (cons a n b) + (cons a' n' b') =
+  match compare a a' with
+  | Eq ⇒ cons a (S (n + n')) b'
+  | Lt ⇒ cons a' n' b'
+  | Gt ⇒ cons a n (T1add b (cons a' n' b'))
+  end.
+ +
+Lemma T1addA (x y z :T1) : x + (y + z) = (x + y) + z.
+ +
+#[global] Instance T1addAssoc : Assoc eq T1add.
+ +
+Section Proof_of_dist.
+ +
+   Let P (b: T1) :=
+     ∀ a c, nf a → nf b → nf c →
+    a × (b + c) = a × b + a × c.
+ +
+ +
+ #[local] Ltac rewrite_ind Hind b :=
+    pose proof (Hind b) as ->; [ | try apply tail_LT_cons| | | ];
+    eauto with T1.
+ +
+ Lemma L0 : ∀ p, P p.
+ +
+  Theorem mult_plus_distr_l (a b c: T1) :
+    nf a → nf b → nf c →
+    a × (b + c) = a × b + a × c.
+ +
+End Proof_of_dist.
+ +
+
+ +
+

An abstract syntax for ordinals in Cantor normal form

+ +
+
+ +
+Declare Scope ppT1_scope.
+Delimit Scope ppT1_scope with pT1.
+ +
+Inductive ppT1 :=
+| PP_fin (_ : nat)
+| PP_add (_ _ : ppT1)
+| PP_mult (_ : ppT1) (_ : nat)
+| PP_exp (_ _ : ppT1)
+| PP_omega.
+ +
+Coercion PP_fin : nat >-> ppT1.
+ +
+Notation "alpha + beta" := (PP_add alpha beta) : ppT1_scope.
+ +
+Notation "alpha * n" := (PP_mult alpha n) : ppT1_scope.
+ +
+Notation "alpha ^ beta" := (PP_exp alpha beta) : ppT1_scope.
+ +
+Notation ω := PP_omega.
+ +
+Check (ω ^ ω × 2 + 1)%pT1.
+ +
+ +
+Fixpoint pp0 (alpha : T1) : ppT1 :=
+  match alpha with
+  | zero ⇒ PP_fin 0
+  | cons zero n zero ⇒ PP_fin (S n)
+  | cons one 0 zero ⇒ ω
+  | cons one 0 beta ⇒ (ω + pp0 beta)%pT1
+  | cons one n zero ⇒ (ω × (S n))%pT1
+  | cons one n beta ⇒ (ω × (S n) + pp0 beta)%pT1
+  | cons alpha 0 zero ⇒ (ω ^ pp0 alpha)%pT1
+  | cons alpha 0 beta ⇒ (ω ^ pp0 alpha + pp0 beta)%pT1
+  | cons alpha n zero ⇒ (ω ^ pp0 alpha × (S n))%pT1
+  | cons alpha n beta ⇒ (ω ^ pp0 alpha × (S n) + pp0 beta)%pT1
+  end.
+ +
+Fixpoint eval_pp (e : ppT1) : T1 :=
+  match e with
+    PP_fin 0 ⇒ zero
+  | PP_fin n ⇒ n
+  | PP_add e f ⇒ ( (eval_pp e) + (eval_pp f))%t1
+  | PP_mult e n ⇒ ( (eval_pp e) × (S n))%t1
+  | PP_exp e f ⇒ ((eval_pp e) ^ (eval_pp f))%t1
+  | ω ⇒ T1omega
+  end.
+ +
+ +
+ +
+ +
+Fixpoint reassoc (exp : ppT1) (fuel :nat) : ppT1 :=
+  match exp, fuel with
+  | exp, 0 ⇒ exp
+  | PP_add e (PP_add f g), S n ⇒
+    reassoc (PP_add (PP_add (reassoc e n) (reassoc f n))
+                   (reassoc g n)) n
+  | PP_add e f , S n ⇒ PP_add (reassoc e n) (reassoc f n)
+  | PP_mult e i , S n⇒ PP_mult (reassoc e n) i
+  | PP_exp e f , S n ⇒ PP_exp (reassoc e n) (reassoc f n)
+  | e, _⇒ e
+  end.
+ +
+Fixpoint pp_size (exp : ppT1) : nat :=
+  match exp with
+    PP_add e f | PP_exp e f ⇒ (S ((pp_size e) + (pp_size f)))%nat
+  | PP_mult e _ ⇒ S (pp_size e)
+  | _ ⇒ 1%nat
+  end.
+ +
+Definition pp (e: T1) : ppT1 := let t := pp0 e in reassoc t (pp_size t).
+ +
+ +
+ +
+ +
+Eval simpl in fun n:nat ⇒
+                 (pp (T1omega ^ (T1omega ^ T1omega × n + T1omega ^ n + T1omega + 1)))%t1 .
+ +
+Ltac is_closed alpha :=
+  match alpha with
+    zero ⇒ idtac
+  | 0 ⇒ idtac
+  | S ?n ⇒ is_closed n
+  |cons ?a ?n ?b ⇒ is_closed a ; is_closed n ; is_closed b
+  | ?other ⇒ fail
+  end.
+ +
+Ltac pp0tac alpha :=
+  match alpha with
+  | zero ⇒ exact 0
+  | cons zero ?n zero ⇒ exact (S n)
+  | cons one 0 zero ⇒ exact T1omega
+  | cons one 0 ?beta ⇒ exact (T1omega + ltac :(pp0tac beta))%pT1
+  | cons one ?n zero ⇒ exact (T1omega × (S n))%pT1
+  | cons one ?n ?beta ⇒ exact (T1omega × (S n) + ltac: (pp0tac beta))%pT1
+  | cons ?alpha 0 zero ⇒ exact (T1omega ^ ltac: (pp0tac alpha))%pT1
+  | cons ?alpha 0 ?beta ⇒
+    exact (T1omega ^ ltac :(pp0tac alpha) + ltac: (pp0tac beta))%pT1
+  | cons ?alpha ?n zero ⇒
+    exact (T1omega ^ ltac: (pp0tac alpha) × (S n))%pT1
+  | cons ?alpha ?n ?beta ⇒
+    exact (T1omega ^ ltac: (pp0tac alpha) × (S n) +
+                   ltac : (pp0tac beta)%pT1)
+  end.
+ +
+Ltac pptac term :=
+  let t := eval cbn in term
+    in tryif is_closed t then exact (pp t)
+      else exact term.
+ +
+Section essai.
+  Variable n : nat.
+ +
+ +
+ +
+End essai.
+ +
+Check (phi0 (phi0 (FS 6))).
+ +
+ +
+ +
+ +
+
+ +
+

Examples

+ +
+
+ +
+ +
+ +
+Example Ex1 : 42 + T1omega = T1omega.
+ +
+Example Ex2 : T1omega t1< T1omega + 42.
+ +
+Example Ex3 : 5 × T1omega = T1omega.
+ +
+Example Ex4 : T1omega t1< T1omega × 5.
+ +
+ +
+Example Ex5 : T1limit (T1omega ^ (T1omega + 5)).
+ +
+ +
+ +
+Example alpha_0 : T1 :=
+  cons (cons (cons zero 0 zero)
+               0
+               zero)
+        0
+        (cons (cons zero 2 zero)
+               4
+               (cons zero 1 zero)).
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+Example alpha_0_eq : alpha_0 = phi0 T1omega +
+                               phi0 3 × 5 + 2.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Gamma0.Gamma0.html b/theories/html/hydras.Gamma0.Gamma0.html new file mode 100644 index 00000000..652b0c1c --- /dev/null +++ b/theories/html/hydras.Gamma0.Gamma0.html @@ -0,0 +1,1661 @@ + + + + + +hydras.Gamma0.Gamma0 + + + + +
+ + + +
+ +

Library hydras.Gamma0.Gamma0

+ +
+
+ +
+

A notation system for ordinals less than Gamma0

+ +
+ + Pierre Casteran + LaBRI, Université Bordeaux 1, and LaBRI (CNRS, UMR 5800) + with a contribution by Evelyne Contejean + +
+
+ +
+From Coq Require Import Arith List Lia Compare_dec Relations
+     Wellfounded RelationClasses.
+ +
+From hydras Require Import Epsilon0.
+From hydras Require Import More_Arith Restriction.
+From hydras Require Import ON_Generic.
+From hydras Require Import rpo.term rpo.rpo.
+From hydras Require Import T2.
+From hydras Require Import Compat815.
+Import Datatypes.
+ +
+Set Implicit Arguments.
+ +
+Lemma nf_a : ∀ a b n c, nf (gcons a b n c) → nf a.
+ +
+Lemma nf_b : ∀ a b n c, nf (gcons a b n c) → nf b.
+ +
+Lemma nf_c : ∀ a b n c, nf (gcons a b n c) → nf c.
+ +
+#[global] Hint Resolve nf_a nf_b nf_c : T2.
+ +
+Ltac nf_inv := ((eapply nf_a; eassumption)||
+                (eapply nf_b; eassumption)||
+                (eapply nf_c; eassumption)).
+ +
+Lemma zero_lt_succ : ∀ alpha, zero t2< succ alpha.
+ +
+Lemma not_lt_zero : ∀ alpha, ¬ alpha t2< zero.
+ +
+Lemma lt_irr : ∀ alpha, ¬ alpha t2< alpha.
+ +
+Ltac lt_clean :=
+  try (match goal with
+         [ineq : lt ?a zero |- _ ] ⇒ case (not_lt_zero ineq);auto
+       |[ineq : Peano.lt ?a 0 |- _ ] ⇒ case (Nat.nlt_0_r a);auto
+       |[ref : lt ?a ?a |- _] ⇒ case (lt_irr ref);auto
+       |[ref : Peano.lt ?a ?a |- _] ⇒ case (lt_irr ref);auto
+       end).
+ +
+Lemma le_zero_alpha : ∀ alpha, zero t2≤ alpha.
+ +
+Lemma psi_le_cons : ∀ alpha beta n gamma,
+    [alpha, beta] t2≤ gcons alpha beta n gamma.
+ +
+#[global] Hint Resolve psi_le_cons le_zero_alpha: T2.
+ +
+Lemma le_psi_term_le alpha beta: alpha t2≤ beta →
+                                          psi_term alpha t2≤ psi_term beta.
+ +
+Lemma le_inv_nc : ∀ a b n c n' c',
+    gcons a b n c t2≤ gcons a b n' c' → (n<n')%nat ∨ n=n' ∧ c t2≤ c'.
+ +
+Lemma lt_than_psi : ∀ a b n c a' b',
+    gcons a b n c t2< [a',b'] →
+    [a,b]t2<[a',b'].
+ +
+
+ +
+ in order to establish trichotomy, we first use a measure on pair of + terms +
+
+ +
+Section lemmas_on_length.
+  Open Scope nat_scope.
+ +
+  Lemma tricho_lt_2 : ∀ a1 a2 b1 b2 n1 n2 r1 r2,
+      t2_length a1 + t2_length a2 <
+      t2_length (gcons a1 b1 n1 r1) +
+      t2_length (gcons a2 b2 n2 r2).
+ +
+  Lemma tricho_lt_2' : ∀ a1 a2 b1 b2 n1 n2 r1 r2,
+      t2_length b1 + t2_length (gcons a2 b2 0 zero) <
+      t2_length (gcons a1 b1 n1 r1) +
+      t2_length (gcons a2 b2 n2 r2).
+ +
+  Lemma tricho_lt_3 : ∀ a1 a2 b1 b2 n1 n2 r1 r2,
+      t2_length b1 + t2_length b2 <
+      t2_length (gcons a1 b1 n1 r1) + t2_length (gcons a2 b2 n2 r2).
+ +
+  Lemma tricho_lt_4 : ∀ a1 a2 b1 b2 n1 n2 r1 r2,
+      t2_length a2 + t2_length a1 <
+      t2_length (gcons a1 b1 n1 r1) +
+      t2_length (gcons a2 b2 n2 r2).
+ +
+  Lemma tricho_lt_4' : ∀ a1 a2 b1 b2 n1 n2 c1 c2,
+      t2_length (gcons a1 b1 0 c1) + t2_length b2 <
+      t2_length (gcons a1 b1 n1 c1) +
+      t2_length (gcons a2 b2 n2 c2).
+ +
+  Lemma tricho_lt_5 : ∀ a1 a2 b1 n1 n2 c1 c2,
+      t2_length a2 + t2_length a1 <
+      t2_length (gcons a1 b1 n1 c1) +
+      t2_length (gcons a2 (gcons a1 b1 0 zero) n2 c2).
+ +
+  Lemma tricho_lt_7 : ∀ a1 b1 n1 c1 c2,
+      t2_length c1 + t2_length c2 <
+      t2_length (gcons a1 b1 n1 c1) +
+      t2_length (gcons a1 b1 n1 c2).
+ +
+End lemmas_on_length.
+ +
+#[global] Hint Resolve tricho_lt_7 tricho_lt_5 tricho_lt_4 tricho_lt_4' tricho_lt_3 tricho_lt_2 tricho_lt_2 : T2.
+ +
+Open Scope T2_scope.
+ +
+ +
+Lemma tricho_aux (l: nat) : ∀ t t': T2,
+      t2_length t + t2_length t' < l →
+      {t t2< t'} + {t = t'} + {t' t2< t}. +
+Definition lt_eq_lt_dec (t t': T2) : {t t2< t'}+{t = t'}+{t' t2< t}.
+ +
+ +
+Definition lt_ge_dec : ∀ t t', {t t2< t'}+{t' t2≤ t}.
+Defined.
+ +
+ +
+#[ global ] Instance compare_T2 : Compare T2 :=
+fun (t1 t2 : T2) ⇒
+  match lt_eq_lt_dec t1 t2 with
+  | inleft (left _) ⇒ Lt
+  | inleft (right _) ⇒ Eq
+  | inright _ ⇒ Gt
+  end.
+ +
+ +
+ +
+Fixpoint nfb (alpha : T2) : bool :=
+  match alpha with
+    zero ⇒ true
+  | gcons a b n zero ⇒ andb (nfb a) (nfb b)
+  | gcons a b n ((gcons a' b' n' c') as c) ⇒
+    match compare [a', b'] [a, b] with
+           Lt ⇒ andb (nfb a) (andb (nfb b) (nfb c))
+           | _ ⇒ false
+           end
+end.
+ +
+ +
+ +
+ +
+ +
+Ltac tricho t t' Hname := case (lt_eq_lt_dec t t');
+                          [intros [Hname|Hname] | intro Hname].
+Tactic Notation "tricho" constr(t) constr(t') ident(Hname) := tricho t t' Hname.
+ +
+Section trans_proof.
+  Variables a1 b1 c1 a2 b2 c2 a3 b3 c3:T2.
+  Variables n1 n2 n3:nat.
+ +
+  Hypothesis H12 : gcons a1 b1 n1 c1 t2< gcons a2 b2 n2 c2.
+  Hypothesis H23 : gcons a2 b2 n2 c2 t2< gcons a3 b3 n3 c3.
+ +
+  Hypothesis induc : ∀ t t' t'',
+      (t2_length t + t2_length t' +
+       t2_length t'' <
+       t2_length (gcons a1 b1 n1 c1) +
+       t2_length (gcons a2 b2 n2 c2) +
+       t2_length (gcons a3 b3 n3 c3))%nat →
+      lt t t' → lt t' t'' → lt t t''.
+ +
+  Lemma trans_aux : gcons a1 b1 n1 c1 t2< gcons a3 b3 n3 c3.
+ +
+End trans_proof.
+ +
+Lemma lt_trans0 : ∀ n,
+    ∀ t1 t2 t3,
+      (t2_length t1 + t2_length t2 + t2_length t3 < n)%nat →
+      lt t1 t2 → lt t2 t3 → lt t1 t3.
+ +
+Theorem lt_trans :
+  ∀ t1 t2 t3, t1 t2< t2 → t2 t2< t3 → t1 t2< t3.
+ +
+Theorem le_lt_trans alpha beta gamma: alpha t2≤ beta →
+                                       beta t2< gamma →
+                                       alpha t2< gamma.
+ +
+Theorem lt_le_trans alpha beta gamma :
+  alpha t2< beta → beta t2≤ gamma → alpha t2< gamma.
+ +
+Theorem le_trans : ∀ alpha beta gamma, alpha t2≤ beta →
+                                            beta t2≤ gamma →
+                                            alpha t2≤ gamma.
+ +
+Lemma psi_lt_head : ∀ alpha beta n gamma alpha' beta' n' gamma',
+    [alpha, beta] t2< [alpha', beta'] →
+    gcons alpha beta n gamma t2< gcons alpha' beta' n' gamma'.
+ +
+Lemma nf_inv_tail : ∀ a b n c , nf (gcons a b n c) →
+                                     c t2< [a,b].
+ +
+Theorem lt_beta_psi : ∀ beta alpha, beta t2< [alpha, beta].
+ +
+Lemma lt_beta_cons : ∀ alpha beta n gamma,
+    beta t2< gcons alpha beta n gamma.
+ +
+Theorem lt_alpha_psi : ∀ alpha beta, alpha t2< [alpha, beta].
+ +
+Lemma lt_alpha_cons : ∀ alpha beta n gamma,
+    alpha t2< gcons alpha beta n gamma.
+ +
+#[global] Hint Resolve lt_beta_cons lt_alpha_cons : T2.
+ +
+Lemma le_cons_tail alpha beta n gamma gamma':
+  gamma t2≤ gamma' →
+  gcons alpha beta n gamma t2≤ gcons alpha beta n gamma'.
+ +
+
+ +
+

terms in normal form

+ +
+
+ +
+Lemma nf_fin_inv : ∀ gamma n, nf (gcons zero zero n gamma) →
+                                      gamma = zero.
+ +
+Lemma lt_tail0: ∀ c, nf c → c ≠ zero → gtail c t2< c.
+ +
+Lemma lt_tail: ∀ a b n c, nf (gcons a b n c) → c t2< gcons a b n c.
+ +
+Lemma le_one_cons : ∀ a b n c, one t2≤ gcons a b n c.
+ +
+#[global] Hint Resolve le_one_cons : T2.
+ +
+Lemma fin_lt_omega : ∀ n, fin n t2< omega.
+ +
+Lemma omega_lt_epsilon0 : omega t2< epsilon0.
+ +
+Lemma omega_lt_epsilon : ∀ alpha, omega t2< epsilon alpha.
+ +
+Lemma lt_one_inv : ∀ alpha, alpha t2< one → alpha = zero.
+ +
+Lemma lt_cons_omega_inv : ∀ alpha beta n gamma,
+    gcons alpha beta n gamma t2< omega →
+    nf (gcons alpha beta n gamma) →
+    alpha = zero ∧ beta = zero ∧ gamma = zero.
+ +
+Lemma lt_omega_inv alpha : nf alpha → alpha t2< omega →
+                            {n:nat | alpha = fin n}.
+ +
+Lemma lt_omega_is_fin alpha: nf alpha → alpha t2< omega →
+                                         is_finite alpha.
+ +
+Theorem lt_compat (n p : nat): fin n t2< fin p →
+                                (n < p)%nat.
+ +
+Theorem lt_compatR (n p : nat): (n <p)%nat →
+                                 fin n t2< fin p.
+ +
+Lemma finite_is_finite : ∀ n, is_finite (fin n).
+ +
+Lemma is_finite_finite : ∀ alpha, is_finite alpha →
+                                       {n : nat | alpha = fin n}.
+ +
+Lemma compare_reflect alpha beta : match compare alpha beta with
+                                     | Lt ⇒ alpha t2< beta
+                                     | Eq ⇒ alpha = beta
+                                     | Gt ⇒ beta t2< alpha
+                                     end.
+ +
+Lemma compare_correct alpha beta :
+  CompareSpec (alpha = beta) (lt alpha beta) (lt beta alpha)
+              (compare alpha beta).
+ +
+Lemma compare_Lt : ∀ alpha beta, compare alpha beta = Lt →
+                                         alpha t2< beta.
+ +
+
+ +
+ Compare with the proof of T2.Ex6 +
+
+ +
+Example Ex6 : gcons 1 0 12 omega t2< [0,[2,1]].
+ +
+Lemma compare_Eq : ∀ (alpha beta : T2), compare alpha beta = Eq →
+                                         alpha = beta.
+ +
+Lemma compare_Gt : ∀ alpha beta, compare alpha beta = Gt →
+                                         beta t2< alpha.
+ +
+Arguments compare_Gt [alpha beta].
+Arguments compare_Lt [alpha beta].
+Arguments compare_Eq [alpha beta].
+ +
+#[global] Hint Resolve compare_Eq compare_Lt compare_Gt : T2.
+ +
+Lemma compare_rw_lt alpha beta : alpha t2< beta →
+                                  compare alpha beta = Lt.
+ +
+Lemma compare_rw_eq (alpha beta : T2): alpha = beta →
+                                  compare alpha beta = Eq.
+ +
+Lemma compare_rw_gt alpha beta: beta t2< alpha →
+                                  compare alpha beta = Gt.
+ +
+
+ +
+ plus is defined here, because it requires decidable comparison +
+
+ +
+ +
+Fixpoint plus (t1 t2 : T2) {struct t1} : T2 :=
+  match t1,t2 with
+  | zero, y ⇒ y
+  | x, zero ⇒ x
+  | gcons a b n c, gcons a' b' n' c' ⇒
+     (match compare (gcons a b 0 zero)
+                    (gcons a' b' 0 zero) with
+      | Lt ⇒ gcons a' b' n' c'
+      | Gt ⇒ gcons a b n (c + gcons a' b' n' c')
+      | Eq ⇒ gcons a b (S(n+n')) c'
+      end)
+  end
+where "alpha + beta" := (plus alpha beta): T2_scope.
+ +
+Example Ex7 : 3 + epsilon0 = epsilon0. +
+ +
+Lemma plus_alpha_0 (alpha: T2): alpha + zero = alpha.
+ +
+Lemma lt_succ (alpha: T2): alpha t2< succ alpha.
+ +
+Theorem lt_succ_le alpha : ∀ beta, alpha t2< beta → nf beta →
+                                  succ alpha t2≤ beta.
+ +
+Lemma succ_lt_le : ∀ a b, nf a → nf b → a t2< succ b → a t2≤ b.
+ +
+Lemma succ_of_cons : ∀ a b n c, zero t2< a ∨ zero t2< b →
+                                     succ (gcons a b n c)= gcons a b n (succ c).
+ +
+
+ +
+

Well-foundedness (with rpo) (Evelyne Contejean)

+ +
+
+ +
+Inductive subterm : T2 → T2 → Prop :=
+| subterm_a : ∀ a b n c, subterm a (gcons a b n c)
+| subterm_b : ∀ a b n c, subterm b (gcons a b n c)
+| subterm_c : ∀ a b n c, subterm c (gcons a b n c)
+| subterm_trans : ∀ t t1 t2, subterm t t1 → subterm t1 t2 →
+                                  subterm t t2.
+ +
+Lemma nf_subterm alpha beta : subterm alpha beta →
+                              nf beta → nf alpha.
+ +
+Theorem subterm_lt alpha beta: subterm alpha beta → nf beta →
+                                alpha t2< beta.
+ +
+Ltac subtermtac :=
+  match goal with
+    [|- subterm ?t1 (gcons ?t1 ?t2 ?n ?t3)] ⇒
+    constructor 1
+  | [|- subterm ?t2 (gcons ?t1 ?t2 ?n ?t3)] ⇒
+    constructor 2
+  | [|- subterm ?t3 (gcons ?t1 ?t2 ?n ?t3)] ⇒
+    constructor 3
+  | [|- subterm ?t4 (gcons ?t1 ?t2 ?n ?t3)] ⇒
+    ((constructor 4 with t1; subtermtac) ||
+     (constructor 4 with t2; subtermtac) ||
+     (constructor 4 with t3; subtermtac))
+  end.
+ +
+Module Gamma0_sig <: Signature.
+ +
+  Inductive symb0 : Set := nat_0 | nat_S | ord_zero | ord_psi | ord_cons.
+ +
+  Definition symb := symb0.
+ +
+  Lemma eq_symbol_dec : ∀ f1 f2 : symb, {f1 = f2} + {f1 ≠ f2}.
+ +
+
+ +
+The arity of a symbol contains also the information about built-in theories as in CiME +
+
+ +
+ +
+  Inductive arity_type : Set :=
+  | AC : arity_type
+  | C : arity_type
+  | Free : nat → arity_type.
+ +
+  Definition arity : symb → arity_type :=
+    fun f ⇒ match f with
+             | nat_0 ⇒ Free 0
+             | ord_zero ⇒ Free 0
+             | nat_S ⇒ Free 1
+             | ord_psi ⇒ Free 2
+             | ord_cons ⇒ Free 3
+             end.
+ +
+End Gamma0_sig.
+ +
+
+ +
+

Module Type Variables.

+ + There are almost no assumptions, except a decidable equality. +
+
+ +
+Module Vars <: Variables.
+ +
+  Inductive empty_set : Set := .
+  Definition var := empty_set.
+ +
+  Lemma eq_variable_dec : ∀ v1 v2 : var, {v1 = v2} + {v1 ≠ v2}.
+ +
+End Vars.
+ +
+Module Gamma0_prec <: Precedence.
+ +
+  Definition A : Set := Gamma0_sig.symb.
+  Import Gamma0_sig.
+ +
+  Definition prec : relation A :=
+    fun f g ⇒ match f, g with
+               | nat_0, nat_S ⇒ True
+               | nat_0, ord_zero ⇒ True
+               | nat_0, ord_cons ⇒ True
+               | nat_0, ord_psi ⇒ True
+               | ord_zero, nat_S ⇒ True
+               | ord_zero, ord_cons ⇒ True
+               | ord_zero, ord_psi ⇒ True
+               | nat_S, ord_cons ⇒ True
+               | nat_S, ord_psi ⇒ True
+               | ord_cons, ord_psi ⇒ True
+               | _, _ ⇒ False
+               end.
+ +
+  Inductive status_type : Set :=
+  | Lex : status_type
+  | Mul : status_type.
+ +
+  Definition status : A → status_type := fun f ⇒ Lex.
+ +
+  Lemma prec_dec : ∀ a1 a2 : A, {prec a1 a2} + {¬ prec a1 a2}.
+ +
+  Lemma prec_antisym : ∀ s, prec s s → False.
+ +
+  Lemma prec_transitive : transitive prec.
+ +
+End Gamma0_prec.
+ +
+Module Gamma0_alg <: Term := term.Make (Gamma0_sig) (Vars).
+Module Gamma0_rpo <: RPO := rpo.Make (Gamma0_alg) (Gamma0_prec).
+ +
+Import Gamma0_alg Gamma0_rpo Gamma0_sig.
+ +
+Fixpoint nat_2_term (n:nat) : term :=
+  match n with 0 ⇒ (Term nat_0 nil)
+          | S p ⇒ Term nat_S ((nat_2_term p)::nil)
+  end.
+ +
+
+ +
+

Every (representation of a) natural number is less than

+ + a non zero ordinal +
+
+ +
+Lemma nat_lt_cons : ∀ (n:nat) t p c ,
+    rpo (nat_2_term n)
+        (Term ord_cons (t::p::c::nil)).
+ +
+Lemma nat_lt_psi : ∀ (n:nat) a b, rpo (nat_2_term n)
+                                           (Term ord_psi (a::b::nil)).
+ +
+Theorem rpo_trans : ∀ t t1 t2, rpo t t1 → rpo t1 t2 → rpo t t2.
+ +
+Fixpoint T2_2_term (a:T2) : term :=
+  match a with
+    zero ⇒ Term ord_zero nil
+  |gcons a b 0 zero ⇒ Term ord_psi (T2_2_term a :: T2_2_term b ::nil)
+  |gcons a b n c ⇒
+   Term ord_cons (Term ord_psi
+                       (T2_2_term a :: T2_2_term b ::nil) ::nat_2_term n ::
+                       T2_2_term c::nil)
+  end.
+ +
+Fixpoint T2_size (o:T2):nat :=
+  match o with zero ⇒ 0
+          | gcons a b n c ⇒ S (T2_size a + T2_size b + n + T2_size c)%nat
+  end.
+ +
+Lemma T2_size1 : ∀ a b n c, (T2_size zero < T2_size (gcons a b n c))%nat.
+ +
+Lemma T2_size2 : ∀ a b n c , (T2_size a < T2_size (gcons a b n c))%nat.
+ +
+Lemma T2_size3 : ∀ a b n c , (T2_size b < T2_size (gcons a b n c))%nat.
+ +
+Lemma T2_size4 : ∀ a b n c , (T2_size c < T2_size (gcons a b n c))%nat.
+ +
+#[global] Hint Resolve T2_size1 T2_size2 T2_size3 T2_size4 : T2.
+ +
+
+ +
+let us recall subterm properties on T2 +
+
+ +
+Lemma lt_subterm1 : ∀ a a' n' b' c', a t2< a' →
+                                            a t2< gcons a' b' n' c'.
+ +
+#[global] Hint Resolve nat_lt_cons lt_subterm1 : T2.
+ +
+Lemma nat_2_term_mono : ∀ n n', (n < n')%nat →
+                                     rpo (nat_2_term n) (nat_2_term n').
+ +
+Lemma T2_size_psi : ∀ a b n c ,
+    (T2_size [a,b] ≤ T2_size (gcons a b n c))%nat.
+ +
+ +
+Lemma rpo_2_2 : ∀ ta1 ta2 tb1 tb2 ,
+    rpo ta1 ta2 →
+    rpo tb1 (Term ord_psi (ta2:: tb2::nil)) →
+    rpo (Term ord_psi (ta1:: tb1 ::nil))
+        (Term ord_psi (ta2:: tb2 ::nil)).
+ +
+Lemma rpo_2_3 : ∀ ta1 ta2 tb1 tb2 n1 tc1,
+    rpo ta1 ta2 →
+    rpo tb1 (Term ord_psi (ta2:: tb2::nil)) →
+    rpo tc1 (Term ord_psi (ta1:: tb1::nil)) →
+    rpo (Term ord_cons ((Term ord_psi (ta1:: tb1 ::nil))
+                          ::(nat_2_term n1)::tc1::nil))
+        (Term ord_psi (ta2:: tb2 ::nil)).
+ +
+Lemma rpo_2_1 : ∀ ta1 ta2 tb1 tb2 n1 n2 tc1 tc2,
+    rpo ta1 ta2 →
+    rpo tb1 (Term ord_psi (ta2:: tb2::nil)) →
+    rpo tc1 (Term ord_psi (ta1:: tb1::nil)) →
+    rpo (Term ord_cons
+              ((Term ord_psi (ta1:: tb1 ::nil))::(nat_2_term n1) ::tc1::nil))
+        (Term ord_cons ((Term ord_psi (ta2:: tb2 ::nil))
+                          ::(nat_2_term n2) ::tc2::nil)).
+ +
+Lemma rpo_2_4 : ∀ ta1 ta2 tb1 tb2 n2 tc2,
+    rpo ta1 ta2 →
+    rpo tb1 (Term ord_psi (ta2:: tb2::nil)) →
+    rpo (Term ord_psi (ta1:: tb1 ::nil))
+        (Term ord_cons
+              ((Term ord_psi
+                     (ta2:: tb2 ::nil)):: (nat_2_term n2) ::tc2::nil)).
+ +
+Lemma rpo_3_2 : ∀ ta1 tb1 tb2 ,
+    rpo tb1 tb2 →
+    rpo (Term ord_psi (ta1:: tb1 ::nil))
+        (Term ord_psi (ta1:: tb2 ::nil)).
+ +
+Lemma rpo_3_3 : ∀ ta1 tb1 tb2 n1 tc1,
+    rpo tb1 tb2 →
+    rpo tc1 (Term ord_psi (ta1:: tb1 ::nil)) →
+    rpo (Term ord_cons
+              ((Term ord_psi (ta1:: tb1 ::nil))::(nat_2_term n1) ::tc1::nil))
+        (Term ord_psi (ta1:: tb2 ::nil)).
+ +
+Lemma rpo_3_1 : ∀ ta1 tb1 tb2 n1 n2 tc1 tc2,
+    rpo tb1 tb2 →
+    rpo tc1 (Term ord_psi (ta1:: tb1::nil)) →
+    rpo (Term ord_cons
+              ((Term ord_psi (ta1:: tb1 ::nil))::(nat_2_term n1) ::tc1::nil))
+        (Term ord_cons
+              ((Term ord_psi (ta1:: tb2 ::nil))::(nat_2_term n2) ::tc2::nil)).
+ +
+Lemma rpo_3_4 : ∀ ta1 tb1 tb2 n2 tc2,
+    rpo tb1 tb2 →
+    rpo (Term ord_psi (ta1:: tb1 ::nil))
+        (Term ord_cons
+              ((Term ord_psi (ta1:: tb2 ::nil))::
+                                               (nat_2_term n2) ::tc2::nil)).
+ +
+Lemma rpo_4_2 : ∀ ta1 ta2 tb1 tb2 ,
+    rpo (Term ord_psi (ta1:: tb1 ::nil)) tb2 →
+    rpo (Term ord_psi (ta1:: tb1 ::nil))
+        (Term ord_psi (ta2:: tb2 ::nil)).
+ +
+Lemma rpo_4_3 : ∀ ta1 ta2 tb1 tb2 n1 tc1,
+    rpo (Term ord_psi (ta1:: tb1 ::nil)) tb2 →
+    rpo tc1 (Term ord_psi (ta1:: tb1 ::nil)) →
+    rpo (Term ord_cons
+              ((Term ord_psi (ta1:: tb1 ::nil))::
+                                               (nat_2_term n1) ::tc1::nil))
+        (Term ord_psi (ta2:: tb2 ::nil)).
+ +
+Lemma rpo_4_1 : ∀ ta1 ta2 tb1 tb2 n1 n2 tc1 tc2,
+    rpo (Term ord_psi (ta1:: tb1 ::nil)) tb2 →
+    rpo tc1 (Term ord_psi (ta1:: tb1 ::nil)) →
+    rpo
+      (Term ord_cons
+            ((Term ord_psi (ta1:: tb1 ::nil))::
+                                             (nat_2_term n1) ::tc1::nil))
+      (Term ord_cons
+            ((Term ord_psi (ta2:: tb2 ::nil))::
+                                             (nat_2_term n2) ::tc2::nil)).
+ +
+Lemma rpo_4_4 : ∀ ta1 ta2 tb1 tb2 n2 tc2,
+    rpo (Term ord_psi (ta1:: tb1 ::nil)) tb2 →
+    rpo (Term ord_psi (ta1:: tb1 ::nil))
+        (Term ord_cons
+              ((Term ord_psi (ta2:: tb2 ::nil))::
+                                               (nat_2_term n2) ::tc2::nil)).
+ +
+Lemma rpo_5_2 :
+  ∀ ta1 ta2 tb1 ,
+    rpo (Term ord_psi (ta1:: tb1 ::nil))
+        (Term ord_psi (ta2:: (Term ord_psi (ta1::tb1::nil)) ::nil)).
+ +
+Lemma rpo_5_3 : ∀ ta1 ta2 tb1 n1 tc1,
+    rpo tc1 (Term ord_psi (ta1:: tb1 ::nil)) →
+    rpo
+      (Term ord_cons
+            ((Term ord_psi (ta1:: tb1 ::nil))::
+                                             (nat_2_term n1) ::tc1::nil))
+      (Term ord_psi (ta2:: (Term ord_psi (ta1:: tb1 ::nil)) ::nil)).
+ +
+Lemma rpo_5_1 : ∀ ta1 ta2 tb1 n1 n2 tc1 tc2,
+    rpo tc1 (Term ord_psi (ta1:: tb1 ::nil)) →
+    rpo
+      (Term ord_cons
+            ((Term ord_psi (ta1:: tb1 ::nil))::
+                                             (nat_2_term n1) ::tc1::nil))
+      (Term ord_cons
+            ((Term ord_psi (ta2::
+                               (Term ord_psi (ta1:: tb1 ::nil))
+                               ::nil))::
+                                      (nat_2_term n2) ::tc2::nil)).
+ +
+Lemma rpo_5_4 : ∀ ta1 ta2 tb1 n2 tc2,
+    rpo (Term ord_psi (ta1:: tb1 ::nil))
+        (Term ord_cons
+              ((Term ord_psi (ta2::
+                                 (Term ord_psi (ta1:: tb1 ::nil))
+                                 ::nil))::
+                                        (nat_2_term n2) ::tc2::nil)).
+ +
+Lemma rpo_6_1 : ∀ ta1 tb1 n1 n2 tc1 tc2,
+    rpo tc1 (Term ord_psi (ta1:: tb1 ::nil)) →
+    (n1 < n2)%nat →
+    rpo
+      (Term ord_cons
+            ((Term ord_psi (ta1:: tb1 ::nil))::
+                                             (nat_2_term n1) ::tc1::nil))
+      (Term ord_cons
+            ((Term ord_psi (ta1:: tb1 ::nil)):: (nat_2_term n2) ::tc2::nil)).
+ +
+Lemma rpo_6_4 : ∀ ta1 tb1 n2 tc2,
+    (0 < n2)%nat →
+    rpo (Term ord_psi (ta1:: tb1 ::nil))
+        (Term ord_cons
+              ((Term ord_psi (ta1:: tb1 ::nil))::
+                                               (nat_2_term n2) ::tc2::nil)).
+ +
+Lemma rpo_7_1 : ∀ ta1 tb1 n1 tc1 tc2,
+    rpo tc1 (Term ord_psi (ta1:: tb1 ::nil)) →
+    rpo tc1 tc2 →
+    rpo (Term ord_cons
+              ((Term ord_psi (ta1:: tb1 ::nil))::
+                                               (nat_2_term n1) ::tc1::nil))
+        (Term ord_cons
+              ((Term ord_psi (ta1:: tb1 ::nil))::
+                                               (nat_2_term n1) ::tc2::nil)).
+ +
+
+ +
+

inclusion of the order lt in the rpo

+ +
+
+ +
+Section lt_incl_rpo.
+  Variable s :nat.
+  Variables (a1 b1 c1 a2 b2 c2:T2)(n1 n2:nat).
+ +
+  Hypothesis Hsize :
+    ((T2_size (gcons a1 b1 n1 c1) + T2_size (gcons a2 b2 n2 c2)) = S s)%nat.
+ +
+  Hypothesis Hrec : ∀ o' o, (T2_size o + T2_size o' ≤ s)%nat→
+                                   o t2< o' → nf o → nf o' →
+                                   rpo (T2_2_term o) (T2_2_term o').
+ +
+  Hypothesis nf1 : nf (gcons a1 b1 n1 c1).
+  Hypothesis nf2 : nf (gcons a2 b2 n2 c2).
+ +
+  Remark nf_a1 : nf a1.
+ +
+  Remark nf_a2 : nf a2.
+ +
+  Remark nf_b1 : nf b1.
+ +
+  Remark nf_b2 : nf b2.
+ +
+  #[local] Hint Resolve nf1 nf2 nf_a1 nf_a2 nf_b1 nf_b2 : T2.
+ +
+  Remark nf_c1 : nf c1.
+ +
+  Remark nf_c2 : nf c2.
+ +
+  #[local] Hint Resolve nf_c1 nf_c2 : T2.
+ +
+  Hypothesis H : gcons a1 b1 n1 c1 t2< gcons a2 b2 n2 c2.
+ +
+  Lemma cons_rw : ∀ a b n c,
+      (n=0 ∧ c=zero ∧
+       (T2_2_term (gcons a b n c)=
+        (Term ord_psi
+              ((T2_2_term a)::(T2_2_term b)::nil)))) ∨
+      (T2_2_term (gcons a b n c)=
+       Term ord_cons
+            ((Term ord_psi ((T2_2_term a)::(T2_2_term b)::nil))
+               ::(nat_2_term n)::(T2_2_term c)::nil)).
+ +
+ +
+  Lemma lt_rpo_cons_cons : rpo (T2_2_term (gcons a1 b1 n1 c1))
+                               (T2_2_term (gcons a2 b2 n2 c2)).
+ +
+End lt_incl_rpo.
+ +
+Lemma lt_inc_rpo_0 : ∀ n,
+    ∀ o' o, (T2_size o + T2_size o' ≤ n)%nat→
+                 o t2< o' → nf o → nf o' →
+                 rpo (T2_2_term o) (T2_2_term o').
+ +
+Remark R1 : Acc P.prec nat_0.
+ +
+#[global] Hint Resolve R1 : T2.
+ +
+Remark R2 : Acc P.prec ord_zero.
+ +
+#[global] Hint Resolve R2 : T2.
+ +
+Remark R3 : Acc P.prec nat_S.
+ +
+#[global] Hint Resolve R3 : T2.
+ +
+Remark R4 : Acc P.prec ord_cons.
+ +
+#[global] Hint Resolve R4 : T2.
+ +
+Remark R5 : Acc P.prec ord_psi.
+ +
+#[global] Hint Resolve R5 : T2.
+ +
+Theorem well_founded_rpo : well_founded rpo.
+ +
+Section well_founded.
+ +
+  Let R := restrict nf lt.
+ +
+  #[local] Hint Unfold restrict R : T2.
+ +
+  Lemma R_inc_rpo : ∀ o o', R o o' → rpo (T2_2_term o) (T2_2_term o').
+ +
+ +
+  Lemma nf_Wf : well_founded_restriction _ nf lt.
+ +
+End well_founded.
+ +
+Definition transfinite_induction :
+  ∀ (P:T2 → Type),
+    (∀ x:T2, nf x →
+                  (∀ y:T2, nf y → y t2< x → P y) → P x) →
+    ∀ a, nf a → P a.
+ +
+Definition transfinite_induction_Q :
+  ∀ (P : T2 → Type) (Q : T2 → Prop),
+    (∀ x:T2, Q x → nf x →
+                  (∀ y:T2, Q y → nf y → y t2< x → P y) → P x) →
+    ∀ a, nf a → Q a → P a.
+ +
+
+ +
+

the Veblen function phi

+ +
+ + See Schutte.Critical.phi +
+
+ +
+ +
+Definition phi (alpha beta : T2) : T2 :=
+  match beta with
+    zero ⇒ [alpha, beta]
+  | [b1, b2] ⇒
+    (match compare alpha b1
+     with Datatypes.Lt ⇒ [b1, b2 ]
+     | _ ⇒ [alpha,[b1, b2]]
+ end)
+  | gcons b1 b2 0 (gcons zero zero n zero) ⇒
+    (match compare alpha b1
+     with Datatypes.Lt ⇒
+           [alpha, (gcons b1 b2 0 (fin n))]
+     | _ ⇒ [alpha, (gcons b1 b2 0 (fin (S n)))]
+     end)
+  | any_beta ⇒ [alpha, any_beta]
+  end.
+ +
+Example Ex8: phi 1 (succ epsilon0) = [1, [1,0] + 1]. +
+ +
+
+ +
+ All epsilons are fixpoints of phi 0 +
+
+ +
+ +
+ +
+Theorem epsilon_fxp : ∀ beta, phi zero (epsilon beta) =
+                                   epsilon beta.
+ +
+Theorem epsilon0_fxp : epsilon0 = phi zero epsilon0.
+ +
+ +
+ +
+ +
+Theorem phi_of_psi : ∀ a b1 b2,
+    phi a [b1, b2] =
+    if (lt_ge_dec a b1)
+    then [b1, b2]
+    else [a ,[b1, b2]]. +
+Lemma phi_to_psi : ∀ alpha beta,
+    {alpha' : T2 & {beta' : T2 | phi alpha beta = [alpha', beta']}}.
+ +
+Lemma phi_principal : ∀ alpha beta, ap (phi alpha beta).
+ +
+Theorem phi_alpha_zero : ∀ alpha, phi alpha zero = [alpha, zero].
+ +
+Theorem phi_of_psi_succ : ∀ a b1 b2 n,
+    phi a (gcons b1 b2 0 (fin (S n))) =
+    if lt_ge_dec a b1
+    then [a, (gcons b1 b2 0 (fin n))]
+    else [a ,(gcons b1 b2 0 (fin (S n)))].
+ +
+  Lemma phi_cases_aux : ∀ P : T2 → Type,
+      P zero →
+      (∀ b1 b2, nf b1 → nf b2 → P [b1, b2]) →
+      (∀ b1 b2 n, nf b1 → nf b2 →
+                       P (gcons b1 b2 0 (fin (S n)))) →
+      (∀ b1 b2 n c, nf (gcons b1 b2 n c) →
+                         omega t2≤ c ∨ (0 < n)%nat →
+                         P (gcons b1 b2 n c)) →
+      ∀ alpha, nf alpha → P alpha.
+ +
+Theorem phi_cases' :
+  ∀ a b, nf b →
+              {b1 :T2 & {b2:T2 | b = [b1, b2] ∧
+                                 a t2< b1 ∧ phi a b = b}} +
+              {phi a b = [a, b]} +
+              {b1 :T2 & {b2:T2 &
+                            {n: nat | b = gcons b1 b2 0 (fin (S n)) ∧
+                                      a t2< b1 ∧
+                                      phi a b =
+                                      [a, (gcons b1 b2 0 (fin n))]}}}.
+ +
+Theorem phi_cases : ∀ a b, nf b →
+                                {phi a b = b} +
+                                {phi a b = [a, b]} +
+                                {b': T2 | nf b' ∧ phi a b = [a, b']
+                                          ∧ succ b' = b}.
+ +
+Theorem phi_nf : ∀ alpha beta, nf alpha →
+                                    nf beta →
+                                    nf (phi alpha beta).
+ +
+Lemma phi_of_any_cons : ∀ alpha beta1 beta2 n gamma,
+     omega t2≤ gamma ∨ (0 < n)%nat →
+    phi alpha (gcons beta1 beta2 n gamma) =
+    [alpha, (gcons beta1 beta2 n gamma)].
+ +
+Lemma phi_fix alpha beta :
+  phi alpha beta = beta →
+  {beta1 : T2 & {beta2 : T2 | beta = [beta1, beta2]
+                              ∧ alpha t2< beta1}}.
+ +
+Lemma phi_le : ∀ alpha beta alpha' beta',
+    nf beta →
+    phi alpha beta = [alpha', beta'] → alpha t2≤ alpha'.
+ +
+Lemma phi_le_ge :
+  ∀ alpha beta, nf alpha → nf beta →
+                     {alpha':T2 &
+                             {beta':T2
+                             | phi alpha beta = [alpha' ,beta'] ∧
+                               alpha t2≤ alpha' ∧
+                               beta' t2≤ beta}}.
+ +
+
+ +
+ phi alpha beta is a common fixpoint of all the phi gamma, +for any gamma t2< alpha as specified by Schutte +
+
+ +
+Theorem phi_spec1 : ∀ alpha beta gamma,
+    nf alpha → nf beta → nf gamma →
+    gamma t2< alpha →
+    phi gamma (phi alpha beta) = phi alpha beta.
+ +
+Theorem phi_principalR alpha beta :
+  nf alpha → nf beta →
+  {gamma:T2 | [alpha, beta] = phi zero gamma}.
+ +
+Lemma phi_alpha_zero_gt_alpha : ∀ alpha, alpha t2< phi alpha zero.
+ +
+Theorem le_b_phi_ab : ∀ a b, nf a → nf b → b t2≤ phi a b.
+ +
+Lemma phi_of_psi_plus_fin : ∀ a b1 b2 n,
+    a t2< b1 → phi a (gcons b1 b2 0 (fin n)) t2<
+              [a ,gcons b1 b2 0 (fin n)].
+ +
+Lemma phi_mono_r : ∀ a b c, nf a → nf b → nf c →
+                                 b t2< c → phi a b t2< phi a c.
+ +
+Lemma phi_mono_weak_r : ∀ a b c, nf a → nf b → nf c →
+                                      b t2≤ c → phi a b t2≤ phi a c.
+ +
+Lemma phi_inj_r : ∀ a b c, nf a → nf b → nf c →
+                                phi a b = phi a c → b = c.
+ +
+Lemma lt_a_phi_ab : ∀ a b, nf a → nf b → a t2< phi a b.
+ +
+ +
+Lemma zero_not_lim : ¬ is_limit zero.
+ +
+Lemma F_not_lim : ∀ n, ¬ is_limit (fin n).
+ +
+Lemma succb_not_lim alpha: is_successor alpha → ¬ is_limit alpha.
+ +
+Lemma is_limit_not_succ alpha: is_limit alpha → ¬ is_successor alpha.
+ +
+
+ +
+ limit_plus_fin alpha n beta means : + beta = alpha + fin n and (alpha is limit or alpha = zero) + +
+
+ +
+Inductive limit_plus_fin : T2 → nat → T2 → Prop :=
+  limit_plus_F_0 : ∀ p, limit_plus_fin zero p (fin p)
+ |limit_plus_F_cons : ∀ beta1 beta2 n gamma0 gamma p,
+     zero t2< beta1 ∨ zero t2< beta2 →
+     limit_plus_fin gamma0 p gamma →
+     limit_plus_fin (gcons beta1 beta2 n gamma0)
+                  p (gcons beta1 beta2 n gamma).
+ +
+Lemma limit_plus_fin_plus : ∀ alpha alpha' p,
+    limit_plus_fin alpha p alpha' →
+    nf alpha →
+    alpha' = alpha + fin p.
+ +
+Lemma limit_plus_fin_lim : ∀ alpha alpha' p,
+                      limit_plus_fin alpha p alpha' →
+                      nf alpha →
+                      is_limit alpha ∨ alpha=zero.
+ +
+Lemma limit_plus_fin_inv0 alpha beta:
+  limit_plus_fin alpha 0 beta →
+  nf alpha → alpha = beta.
+ +
+Lemma is_limit_cons_inv : ∀ b1 b2 n c, nf (gcons b1 b2 n c) →
+                          is_limit (gcons b1 b2 n c) → is_limit c ∨ c = zero.
+ +
+ +
+Lemma is_limit_intro : ∀ b1 b2 n , nf b1 → nf b2 →
+                       zero t2< b1 ∨ zero t2< b2 →
+                       is_limit (gcons b1 b2 n zero).
+ +
+Lemma lt_epsilon0_ok : ∀ alpha, nf alpha → lt_epsilon0 alpha →
+                                     alpha t2< epsilon0.
+ +
+Derive Inversion_clear lt_01 with (∀ (a b:T2),
+                gcons a b 0 zero t2< epsilon0) Sort Prop.
+ +
+Derive Inversion_clear lt_02 with (∀ (a b c:T2)(n:nat),
+                gcons a b n c t2< epsilon0) Sort Prop.
+ +
+Lemma psi_lt_epsilon0 : ∀ a b, [a, b] t2< epsilon0 →
+                                    a = zero ∧ b t2< epsilon0.
+ +
+Lemma cons_lt_epsilon0 : ∀ a b n c, gcons a b n c t2< epsilon0 →
+                                         nf (gcons a b n c) →
+                                         a = zero ∧ b t2< epsilon0 ∧ c t2< epsilon0.
+ +
+Lemma lt_epsilon0_okR: ∀ alpha, nf alpha → alpha t2< epsilon0 →
+                                     lt_epsilon0 alpha.
+ +
+Lemma T1_to_T2_inj : ∀ alpha beta : T1,
+    T1_to_T2 alpha = T1_to_T2 beta → alpha = beta.
+ +
+ +
+Lemma T1_to_T2_lt : ∀ c, lt_epsilon0 (T1_to_T2 c).
+ +
+Definition T1_to_T2_R : ∀ a, lt_epsilon0 a → {c:T1 | T1_to_T2 c = a}.
+ +
+Lemma T1_to_T2_mono : ∀ alpha beta, T1.lt alpha beta →
+                                    T1_to_T2 alpha t2< T1_to_T2 beta.
+ +
+Lemma T1_to_T2_monoR : ∀ c c', lt (T1_to_T2 c) (T1_to_T2 c') → T1.lt c c'.
+ +
+Lemma lt_epsilon0_trans : ∀ a, lt_epsilon0 a → nf a →
+     ∀ b, lt b a → nf b → lt_epsilon0 b.
+ +
+Lemma nf_coeff_irrelevance : ∀ a b n n' c, nf (gcons a b n c) →
+                                                nf (gcons a b n' c).
+ +
+Lemma psi_principal : ∀ a b c d, nf c → c t2< [a, b]
+                                           → d t2< [a, b] →
+                                          c + d t2< [a, b].
+ +
+Lemma nf_intro : ∀ a b n c, nf a → nf b →
+                                  c t2< [a,b ] → nf c → nf (gcons a b n c).
+ +
+Lemma plus_nf : ∀ alpha, nf alpha → ∀ beta, nf beta →
+                                                       nf (alpha + beta).
+ +
+Lemma succ_as_plus : ∀ alpha, nf alpha → alpha + one = succ alpha.
+ +
+Lemma succ_nf : ∀ alpha, nf alpha → nf (succ alpha).
+ +
+Lemma lt_epsilon0_succ : ∀ a, lt_epsilon0 a → lt_epsilon0 (succ a).
+ +
+Check epsilon0.
+Theorem epsilon0_as_lub : ∀ b, nf b →
+                                    (∀ a, lt_epsilon0 a → lt a b) →
+                                    le epsilon0 b.
+Locate epsilon0.
+ +
+Definition lub (P: T2 → Prop)(x:T2) :=
+  nf x ∧
+  (∀ y, P y → nf y → y t2≤ x) ∧
+  (∀ y, (∀ x, P x → nf x → x t2≤ y) → nf y →
+                                    x t2≤ y).
+ +
+Theorem lub_unicity : ∀ P l l', lub P l → lub P l' → l = l'.
+ +
+Theorem lub_mono : ∀ (P Q :T2 → Prop) l l',
+    (∀ o, nf o → P o → Q o) →
+    lub P l → lub Q l' → l t2≤ l'.
+ +
+Lemma succ_limit_dec : ∀ a, nf a →
+         {a = zero} +{is_successor a}+{is_limit a}.
+ +
+ +
+Lemma le_plus_r : ∀ alpha beta, nf alpha → nf beta →
+                                     alpha t2≤ alpha + beta.
+ +
+Lemma le_plus_l : ∀ alpha beta, nf alpha → nf beta →
+                                     alpha t2≤ beta + alpha.
+ +
+Lemma plus_mono_r : ∀ alpha , nf alpha → ∀ beta gamma, nf beta →
+       nf gamma → beta t2< gamma → alpha + beta t2< alpha + gamma.
+ +
+Lemma plus_mono_l_weak:
+  ∀ o, nf o →
+    ∀ alpha, nf alpha → alpha t2< o →
+                   ∀ beta,
+                     nf beta → beta t2< o →
+                     ∀ gamma , nf gamma →
+                                    alpha t2< beta →
+                                    alpha + gamma t2≤ beta + gamma.
+ +
+Remark R_pred_Sn : ∀ n, pred (fin (S n)) = Some (fin n).
+ +
+Lemma pred_of_cons : ∀ a b n c,
+                       zero t2< a ∨ zero t2< b →
+                       pred (gcons a b n c) = match pred c with
+                                             Some c' ⇒
+                                               Some (gcons a b n c')
+                                            | None ⇒ None
+                                            end.
+ +
+Lemma pred_of_cons' : ∀ a b n ,
+                       zero t2< a ∨ zero t2< b →
+                       pred (gcons a b n zero) = None.
+ +
+Lemma is_limit_ab : ∀ alpha beta n gamma,
+    is_limit (gcons alpha beta n gamma)
+    → zero t2< alpha ∨ zero t2< beta.
+ +
+Lemma pred_of_limit : ∀ alpha,
+    is_limit alpha → nf alpha → pred alpha = None.
+ +
+Lemma pred_of_succ : ∀ alpha, nf alpha →
+            pred (succ alpha) = Some alpha.
+ +
+Lemma limit_plus_fin_ok : ∀ alpha, is_limit alpha →
+                           ∀ n, limit_plus_fin alpha n (alpha + fin n).
+ +
+Section phi_to_psi.
+ Variable alpha : T2.
+ +
+ Lemma phi_to_psi_1 : ∀ beta1 beta2 n,
+     alpha t2< beta1 →
+     [alpha, (gcons beta1 beta2 0 (fin n))] =
+     phi alpha (gcons beta1 beta2 0 (fin (S n))).
+ +
+ Lemma phi_to_psi_2 : ∀ beta1 beta2 n,
+     beta1 t2≤ alpha →
+     [alpha, (gcons beta1 beta2 0 (fin n))] =
+     phi alpha (gcons beta1 beta2 0 (fin n)).
+ +
+ Lemma phi_to_psi_3 : ∀ beta1 beta2 ,
+                             beta1 t2≤ alpha →
+                             [alpha, [beta1, beta2]] =
+                             phi alpha [beta1, beta2].
+ +
+Lemma phi_to_psi_4 : [alpha, zero] = phi alpha zero.
+ +
+Lemma phi_to_psi_5 :
+   ∀ beta1 beta2 n gamma, omega t2≤ gamma ∨ (0 < n)%nat →
+           [alpha,gcons beta1 beta2 n gamma] =
+           phi alpha (gcons beta1 beta2 n gamma).
+ +
+Lemma phi_to_psi_6 : ∀ beta, nf beta →
+                                  phi alpha beta = beta →
+                                  [alpha, beta] = phi alpha (succ beta).
+ +
+ +
+ +
+Lemma phi_psi : ∀ beta gamma n,
+    nf gamma →
+    limit_plus_fin beta n gamma →
+    phi alpha beta = beta →
+    [alpha, gamma] = phi alpha (succ gamma). +
+Theorem th_14_5 : ∀ alpha1 beta1 alpha2 beta2,
+                   nf alpha1 → nf beta1 → nf alpha2 → nf beta2 →
+                   phi alpha1 beta1 = phi alpha2 beta2 →
+                   {alpha1 t2< alpha2 ∧ beta1 = phi alpha2 beta2} +
+                   {alpha1 = alpha2 ∧ beta1 = beta2} +
+                   {alpha2 t2< alpha1 ∧ phi alpha1 beta1 = beta2}.
+ +
+Lemma lt_not_gt : ∀ a b, a t2< b → ¬ (b t2< a).
+ +
+Lemma phi_mono_RR : ∀ a b c, nf a → nf b → nf c →
+              phi a b t2< phi a c → b t2< c.
+ +
+Theorem th_14_6 : ∀ alpha1 beta1 alpha2 beta2,
+                   nf alpha1 → nf beta1 → nf alpha2 → nf beta2 →
+                   phi alpha1 beta1 t2< phi alpha2 beta2 →
+                   {alpha1 t2< alpha2 ∧ beta1 t2< phi alpha2 beta2} +
+                   {alpha1 = alpha2 ∧ beta1 t2< beta2} +
+                   {alpha2 t2< alpha1 ∧ phi alpha1 beta1 t2< beta2}.
+ +
+Definition moser_lepper (beta0 beta:T2)(n:nat) :=
+ limit_plus_fin beta0 n beta ∧ phi alpha beta0 = beta0.
+ +
+Lemma ml_psi : ∀ beta0 beta n,
+    moser_lepper beta0 beta n →
+    {t1 : T2 & {t2: T2| beta0 =
+                        [t1,t2] ∧ alpha t2< t1}}.
+ +
+Lemma ml_1 : ∀ beta0 beta n, moser_lepper beta0 beta n →
+                                  nf beta → nf beta0 →
+                               [alpha, beta] = phi alpha (succ beta).
+ +
+End phi_to_psi.
+ +
+ +
+ +
+Example Ex9 : [zero, epsilon 2 + 4] = phi 0 (epsilon 2 + 5). +
+ +
+Example Ex10 : phi omega [epsilon0, 5] = [epsilon0, 5]. +
+Declare Scope g0_scope.
+ +
+ +
+Module G0.
+ +
+  Definition LT := restrict nf lt.
+ +
+ +
+  Lemma Lt_wf : well_founded LT.
+ +
+
+ +
+

Temporary compatibility nf/nfb

+ +
+
+ +
+  Lemma zero_nfb : nfb zero.
+ +
+  Lemma nfb_a a b n c: nfb (gcons a b n c) → nfb a.
+ +
+  Lemma nfb_equiv gamma : nfb gamma = true ↔ nf gamma.
+ +
+  Lemma nfb_proof_unicity :
+    ∀ (alpha:T2) (H H': nfb alpha), H = H'.
+ +
+ +
+  Class G0 := mkg0 {vnf : T2; vnf_ok : nfb vnf}.
+ +
+  Definition lt (alpha beta : G0) := T2.lt (@vnf alpha) (@vnf beta).
+ +
+  #[ global ] Instance compare_G0 : Compare G0 :=
+    fun alpha beta ⇒ compare (@vnf alpha) (@vnf beta).
+ +
+ +
+  Lemma lt_LT_incl alpha beta : lt alpha beta → LT (@vnf alpha) (@vnf beta).
+ +
+ +
+  #[ global ] Instance lt_sto : StrictOrder lt. +
+ +
+  Lemma lt_wf : well_founded lt. +
+Lemma compare_correct alpha beta :
+  CompareSpec (alpha = beta) (lt alpha beta) (lt beta alpha)
+              (compare alpha beta).
+ +
+  #[ global, program ] Instance zero : G0 := @mkg0 T2.zero _.
+ +
+  #[ global, program] Instance Omega : G0 := @mkg0 omega _.
+ +
+  Notation omega := Omega.
+ +
+  Definition le := clos_refl G0 lt.
+ +
+ +
+  #[ global ] Instance Gamma0_comp: Comparable lt compare. +
+  #[ global ] Instance Gamma0: ON lt compare. +
+  #[ global ] Instance Finite (n:nat) : G0.
+ +
+  #[ global ] Instance Plus (alpha beta : G0) : G0.
+ +
+  Infix "+" := Plus : g0_scope.
+ +
+  #[ global ] Instance Phi (alpha beta : G0) : G0.
+ +
+  Notation phi := Phi.
+  Notation phi0 := (Phi zero).
+  Notation "'omega^'" := phi0 (only parsing) : g0_scope.
+ +
+  Coercion Finite : nat >-> G0.
+ +
+  #[local] Open Scope g0_scope.
+ +
+  Example Ex42 : omega + 42 + omega^ 2 = omega^ 2.
+ +
+End G0.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Gamma0.T2.html b/theories/html/hydras.Gamma0.T2.html new file mode 100644 index 00000000..bc209574 --- /dev/null +++ b/theories/html/hydras.Gamma0.T2.html @@ -0,0 +1,438 @@ + + + + + +hydras.Gamma0.T2 + + + + +
+ + + +
+ +

Library hydras.Gamma0.T2

+ +
+ +
+
+ +
+Data-type for Veblen normal form + (ordinals below Gamma0) +
+
+ +
+From Coq Require Import Arith Compare_dec Relations Wellfounded Lia.
+From hydras Require Import More_Arith Restriction T1 OrdNotations Compat815.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+

Definitions

+ +
+ + gcons alpha beta n gamma is : psi(alpha,beta)*(S n)+ gamma +
+
+ +
+Declare Scope T2_scope.
+Delimit Scope T2_scope with t2.
+ +
+Open Scope T2_scope.
+ +
+Inductive T2 : Set :=
+| zero : T2
+| gcons : T2 → T2 → nat → T2 → T2.
+ +
+ +
+Notation "[ alpha , beta ]" := (gcons alpha beta 0 zero)
+                                 (at level 0): T2_scope.
+ +
+Definition psi alpha beta := [alpha, beta].
+ +
+Definition psi_term alpha :=
+  match alpha with zero ⇒ zero
+                 | gcons a b n c ⇒ [a, b]
+  end.
+ +
+Lemma psi_eq : ∀ a b, psi a b = [a,b].
+ +
+Ltac fold_psi := rewrite <- psi_eq.
+Ltac fold_psis := repeat fold_psi.
+ +
+ +
+Notation one := [zero,zero].
+ +
+Notation FS n := (gcons zero zero n zero).
+ +
+Definition fin (n:nat) := match n with 0 ⇒ zero | S p ⇒ FS p end.
+ +
+Coercion fin : nat >-> T2.
+ +
+Notation omega := [zero,one].
+ +
+Notation epsilon0 := [one,zero].
+ +
+Definition epsilon alpha := [one, alpha].
+ +
+ +
+Inductive is_finite: T2 → Set :=
+ zero_finite : is_finite zero
+|succ_finite : ∀ n, is_finite (gcons zero zero n zero).
+ +
+#[global] Hint Constructors is_finite : T2.
+ +
+ +
+ +
+Fixpoint T1_to_T2 (alpha :T1) : T2 :=
+  match alpha with
+  | T1.zero ⇒ zero
+  | T1.cons a n b ⇒ gcons zero (T1_to_T2 a) n (T1_to_T2 b)
+  end.
+ +
+ +
+
+ +
+additive principals +
+
+ +
+Inductive ap : T2 → Prop :=
+ap_intro : ∀ alpha beta, ap [alpha, beta].
+ +
+
+ +
+strict order on terms +
+
+ +
+Reserved Notation "x 't2<' y" (at level 70, no associativity).
+Reserved Notation "x 't2<=' y" (at level 70, no associativity).
+Reserved Notation "x 't2>=' y" (at level 70, no associativity).
+Reserved Notation "x 't2>' y" (at level 70, no associativity).
+ +
+Reserved Notation "x 't2<=' y 't2<=' z" (at level 70, y at next level).
+Reserved Notation "x 't2<=' y 't2<' z" (at level 70, y at next level).
+Reserved Notation "x 't2<' y 't2<' z" (at level 70, y at next level).
+Reserved Notation "x 't2<' y 't2<=' z" (at level 70, y at next level).
+ +
+ +
+Inductive lt : T2 → T2 → Prop :=
+|
+ lt_1 : ∀ alpha beta n gamma, zero t2< gcons alpha beta n gamma
+|
+ lt_2 : ∀ alpha1 alpha2 beta1 beta2 n1 n2 gamma1 gamma2,
+    alpha1 t2< alpha2 →
+    beta1 t2< gcons alpha2 beta2 0 zero →
+    gcons alpha1 beta1 n1 gamma1 t2<
+    gcons alpha2 beta2 n2 gamma2
+|
+ lt_3 : ∀ alpha1 beta1 beta2 n1 n2 gamma1 gamma2,
+    beta1 t2< beta2 →
+    gcons alpha1 beta1 n1 gamma1 t2<
+    gcons alpha1 beta2 n2 gamma2
+          
+|
+lt_4 : ∀ alpha1 alpha2 beta1 beta2 n1 n2 gamma1 gamma2,
+    alpha2 t2< alpha1 →
+    [alpha1, beta1] t2< beta2 →
+    gcons alpha1 beta1 n1 gamma1 t2<
+    gcons alpha2 beta2 n2 gamma2
+          
+|
+lt_5 : ∀ alpha1 alpha2 beta1 n1 n2 gamma1 gamma2,
+    alpha2 t2< alpha1 →
+    gcons alpha1 beta1 n1 gamma1 t2<
+    gcons alpha2 [alpha1, beta1] n2 gamma2
+          
+|
+lt_6 : ∀ alpha1 beta1 n1 n2 gamma1 gamma2,
+    (n1 < n2)%nat →
+    gcons alpha1 beta1 n1 gamma1 t2<
+    gcons alpha1 beta1 n2 gamma2
+
+|
+lt_7 : ∀ alpha1 beta1 n1 gamma1 gamma2,
+    gamma1 t2< gamma2 →
+    gcons alpha1 beta1 n1 gamma1 t2<
+    gcons alpha1 beta1 n1 gamma2
+                                           
+where "o1 t2< o2" := (lt o1 o2): T2_scope.
+ +
+ +
+#[global] Hint Constructors lt : T2.
+ +
+Definition le t t' := t = t' ∨ t t2< t'.
+ +
+#[global] Hint Unfold le : T2.
+ +
+Notation "o1 t2<= o2" := (le o1 o2): T2_scope.
+ +
+
+ +
+

Examples

+ +
+
+ +
+ +
+ +
+Example Ex1: 0 t2< epsilon0.
+ +
+Example Ex2: omega t2< epsilon0.
+ +
+Example Ex3: gcons omega 8 12 56 t2< gcons omega 8 12 57.
+ +
+Example Ex4: epsilon0 t2< [2,1].
+ +
+Example Ex5 : [2,1] t2< [2,3].
+ +
+Example Ex6 : gcons 1 0 12 omega t2< [0,[2,1]].
+ +
+Example Ex7 : gcons 2 1 42 epsilon0 t2< [1, [2,1]].
+ +
+ +
+ +
+Definition gtail c := match c with
+                      | zero ⇒ zero
+                      | gcons a b n c ⇒ c
+                      end.
+ +
+ +
+ +
+Inductive nf : T2 → Prop :=
+| zero_nf : nf zero
+| single_nf : ∀ a b n,
+    nf a →
+    nf b → nf (gcons a b n zero)
+| gcons_nf : ∀ a b n a' b' n' c',
+    [a', b'] t2< [a, b] →
+    nf a → nf b →
+    nf(gcons a' b' n' c')→
+    nf(gcons a b n (gcons a' b' n' c')).
+ +
+#[global] Hint Constructors nf : T2.
+ +
+ +
+ +
+Lemma nf_fin i : nf (fin i).
+ +
+Lemma nf_omega : nf omega.
+ +
+Lemma nf_epsilon0 : nf epsilon0.
+ +
+Lemma nf_epsilon : ∀ alpha, nf alpha → nf (epsilon alpha).
+ +
+Example Ex8: nf (gcons 2 1 42 epsilon0).
+ +
+ +
+Inductive is_successor : T2 → Prop :=
+  finite_succ : ∀ n , is_successor (gcons zero zero n zero)
+ |cons_succ : ∀ a b n c, nf (gcons a b n c) → is_successor c →
+                              is_successor (gcons a b n c).
+ +
+Inductive is_limit : T2 → Prop :=
+| is_limit_0 : ∀ alpha beta n, zero t2< alpha ∨ zero t2< beta →
+                                   nf alpha → nf beta →
+                                   is_limit (gcons alpha beta n zero)
+| is_limit_cons : ∀ alpha beta n gamma,
+    is_limit gamma →
+    nf (gcons alpha beta n gamma) →
+    is_limit (gcons alpha beta n gamma).
+ +
+ +
+Fixpoint succ (a:T2) : T2 :=
+ match a with zero ⇒ one
+             | gcons zero zero n c ⇒ fin (S (S n))
+             | gcons a b n c ⇒ gcons a b n (succ c)
+ end.
+ +
+Fixpoint pred (a:T2) : option T2 :=
+ match a with zero ⇒ None
+             | gcons zero zero 0 zero ⇒ Some zero
+             | gcons zero zero (S n) zero ⇒
+                  Some (gcons zero zero n zero)
+             | gcons a b n c ⇒ (match (pred c) with
+                                   Some c' ⇒ Some (gcons a b n c')
+                                  | None ⇒ None
+                                  end)
+ end.
+ +
+Inductive lt_epsilon0 : T2 → Prop :=
+  zero_lt_e0 : lt_epsilon0 zero
+| gcons_lt_e0 : ∀ b n c, lt_epsilon0 b →
+                                lt_epsilon0 c →
+                                lt_epsilon0 (gcons zero b n c).
+ +
+ +
+
+ +
+

Length (as in Schutte)

+ +
+
+ +
+Section on_length.
+ +
+ Open Scope nat_scope.
+ +
+ +
+ +
+Fixpoint nbterms (t:T2) : nat :=
+  match t with zero ⇒ 0
+             | gcons a b n v ⇒ (S n) + nbterms v
+  end.
+ +
+Fixpoint t2_length (t:T2) : nat :=
+  match t with zero ⇒ 0
+             | gcons a b n v ⇒
+                 nbterms (gcons a b n v) +
+                 2 × (Nat.max (t2_length a)
+                              (Nat.max (t2_length b) (t2_length_aux v)))
+  end
+with t2_length_aux (t:T2) : nat :=
+ match t with zero ⇒ 0
+            | gcons a b n v ⇒
+               Nat.max (t2_length a) (Nat.max (t2_length b) (t2_length_aux v))
+ end.
+ +
+ +
+ +
+Lemma length_a : ∀ a b n v, t2_length a <
+                                 t2_length (gcons a b n v).
+ +
+Lemma length_b : ∀ a b n v, t2_length b <
+                                 t2_length (gcons a b n v).
+ +
+Lemma length_c : ∀ a b n v, t2_length v <
+                                 t2_length (gcons a b n v).
+ +
+Lemma length_n : ∀ a b r n p, n < p →
+                        t2_length (gcons a b n r) <
+                        t2_length (gcons a b p r).
+ +
+Lemma length_psi : ∀ a b n c,
+                      t2_length [a, b] ≤ t2_length (gcons a b n c).
+ +
+Lemma length_ab : ∀ a b,
+    t2_length a + t2_length b ≤ t2_length (gcons a b 0 zero).
+ +
+Lemma length_abnc : ∀ a b n c,
+   t2_length a + t2_length b ≤ t2_length (gcons a b n c).
+ +
+End on_length.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.Battle_length.html b/theories/html/hydras.Hydra.Battle_length.html new file mode 100644 index 00000000..d1ab34bb --- /dev/null +++ b/theories/html/hydras.Hydra.Battle_length.html @@ -0,0 +1,119 @@ + + + + + +hydras.Hydra.Battle_length + + + + +
+ + + +
+ +

Library hydras.Hydra.Battle_length

+ +
+
+ +
+ +
+ + "Computing" standard battle length + +
+ +Pierre Castéran, LaBRI, University of Bordeaux + +
+ + +
+
+ +
+From hydras Require Import L_alpha.
+From hydras Require Import Hydra_Lemmas Epsilon0_Needed_Free
+     Epsilon0_Needed_Std Hydra_Termination O2H.
+Import E0 Large_Sets Hprime Paths MoreLists.
+ +
+Section Battle_length.
+ +
+  Variable alpha : E0.
+  Hypothesis Halpha : alpha ≠ E0zero.
+  Variable k : nat.
+  Hypothesis Hk : (1 ≤ k)%nat.
+  Let l := L_ alpha (S k).
+ +
+  Remark R0 : (S k < l)%nat.
+ +
+ +
+  Remark R2 : mlarge (cnf alpha) (interval (S k) (Nat.pred l)).
+ +
+ +
+  Remark R3 : path_toS zero
+                       (unshift (interval (S k) (Nat.pred l)))
+                       (cnf alpha).
+ +
+  Remark R4 :
+    trace_to (O2H.iota zero)
+             (unshift (interval (S k) (Nat.pred l)))
+             (O2H.iota (cnf alpha)).
+ +
+  Lemma R5 : trace_to (O2H.iota zero)
+                      (interval k (Nat.pred (Nat.pred l)))
+                      (O2H.iota (cnf alpha)).
+ +
+  Remark R6 : S (Nat.pred (Nat.pred l)) = (l-1)%nat.
+ +
+  Lemma L06: rounds standard k (iota (cnf alpha)) (l-1)%nat
+                     (iota zero).
+ +
+ +
+  Lemma battle_length_std:
+    battle_length standard k (iota (cnf alpha)) (l-k)%nat. +
+End Battle_length.
+ +
+ +
+Definition l_std alpha k := (L_ alpha (S k) - k)%nat.
+ +
+Lemma l_std_ok : ∀ alpha : E0,
+    alpha ≠ E0zero →
+    ∀ k : nat,
+      1 ≤ k → battle_length standard k (iota (cnf alpha))
+                              (l_std alpha k). +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.BigBattle.html b/theories/html/hydras.Hydra.BigBattle.html new file mode 100644 index 00000000..a119f0cd --- /dev/null +++ b/theories/html/hydras.Hydra.BigBattle.html @@ -0,0 +1,501 @@ + + + + + +hydras.Hydra.BigBattle + + + + +
+ + + +
+ +

Library hydras.Hydra.BigBattle

+ +
+
+ +
+

A long hydra battle

+ + +
+ + Pierre Castéran, LaBRI, Univ Bordeaux + +
+ + +
+
+ +
+From Coq Require Import Arith Relations Lia.
+From hydras Require Import Hydra_Definitions Hydra_Lemmas Iterates Exp2.
+ +
+
+ +
+Let us consider a small hydra hinit +
+
+ +
+#[local] Notation h3 := (hyd_mult head 3).
+Definition hinit := hyd3 h3 head head.
+ +
+
+ +
+ +
+ +the first steps of a standard battle (Hercules chops off the rightmost head) + +
+
+ +
+Lemma L_0_2 : rounds standard 0 hinit 2 (hyd1 h3).
+ +
+
+ +
+In the next round, there is a replication of h3 into 3 copies of h2 + +
+ +More generally, all the future hydras will be composed of several copies +of h2, then several copies of h1, then several heads +
+
+ +
+ +
+Notation h2 := (hyd_mult head 2).
+Notation h1 := (hyd1 head).
+ +
+Notation hyd a b c :=
+  (node (hcons_mult h2 a
+             (hcons_mult h1 b
+                         (hcons_mult head c hnil)))).
+ +
+ +
+Lemma L_2_3 : rounds standard 2 (hyd1 h3) 3 (hyd 3 0 0).
+ +
+Lemma L_0_3 : rounds standard 0 hinit 3 (hyd 3 0 0).
+ +
+
+ +
+From now on, we abstract the configurations of the battle + as tuples (round number, n2, n1, nh) where n2 (resp. n1, nh) is the number of daughters of type h2 resp. h1, heads +
+
+ +
+ +
+Record state : Type :=
+  mks {round : nat ; n2 : nat ; n1 : nat ; nh : nat}.
+ +
+ +
+ +
+Definition (s : state) :=
+  match s with
+  | mks round a b (S c) ⇒ mks (S round) a b c
+  | mks round a (S b) 0 ⇒ mks (S round) a b (S round)
+  | mks round (S a) 0 0 ⇒ mks (S round) a (S round) 0
+  | _ ⇒ s
+  end.
+ +
+ +
+
+ +
+returns the state at the n-th round +
+
+ +
+ +
+Definition test n := iterate next (n-3) (mks 3 3 0 0).
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+Definition doubleS (n : nat) := 2 × (S n).
+ +
+ +
+ +
+ +
+
+ +
+OK, instead of tests based on the function next, we consider now + relations (which allow us to consider accessibility predicates) +
+ + steps i a b c j a' b' c' has almost the same meaning as + iterate test (j - i) (mks t a b c) = (mks t' a' b' c') + +
+ + +
+
+ +
+ +
+Inductive one_step (i: nat) :
+  nat → nat → nat → nat → nat → nat → Prop :=
+| step1: ∀ a b c, one_step i a b (S c) a b c
+| step2: ∀ a b , one_step i a (S b) 0 a b (S i)
+| step3: ∀ a, one_step i (S a) 0 0 a (S i) 0.
+ +
+ +
+ +
+Lemma step_rounds : ∀ i a b c a' b' c',
+    one_step i a b c a' b' c' →
+    rounds standard i (hyd a b c) (S i) (hyd a' b' c'). +
+ +
+ +
+Inductive steps : nat → nat → nat → nat →
+                  nat → nat → nat → nat → Prop :=
+| steps1 : ∀ i a b c a' b' c',
+    one_step i a b c a' b' c' → steps i a b c (S i) a' b' c'
+| steps_S : ∀ i a b c j a' b' c' k a'' b'' c'',
+    steps i a b c j a' b' c' →
+    steps j a' b' c' k a'' b'' c'' →
+    steps i a b c k a'' b'' c''.
+ +
+
+ +
+ reachability (for i > 0) +
+
+ +
+Definition reachable (i a b c : nat) : Prop :=
+  steps 3 3 0 0 i a b c.
+ +
+ +
+ +
+Lemma steps_rounds : ∀ i a b c j a' b' c',
+    steps i a b c j a' b' c' →
+    rounds standard i (hyd a b c) j (hyd a' b' c'). +
+ +
+
+ +
+ From now on, we play again the same tests as above, but instead of plain uses of Compute, we prove and register lemmas that we will be used later +
+
+ +
+ +
+Lemma L4 : reachable 4 2 4 0.
+ +
+
+ +
+Now we prove some laws we observed in our test phase +
+
+ +
+ +
+Lemma LS : ∀ c a b i, steps i a b (S c) (i + S c) a b 0.
+ +
+
+ +
+ The law that relates two consecutive events with (nh = 0) +
+
+ +
+Lemma doubleS_law : ∀ a b i, steps i a (S b) 0 (doubleS i) a b 0.
+ +
+Lemma reachable_S : ∀ i a b, reachable i a (S b) 0 →
+                                   reachable (doubleS i) a b 0.
+ +
+ +
+ +
+ +
+Lemma L10 : reachable 10 2 3 0.
+ +
+Lemma L22 : reachable 22 2 2 0.
+ +
+Lemma L46 : reachable 46 2 1 0.
+ +
+Lemma L94 : reachable 94 2 0 0.
+ +
+Lemma L95 : reachable 95 1 95 0.
+ +
+ +
+ +
+Lemma L0_95 : rounds standard 3 (hyd 3 0 0) 95 (hyd 1 95 0).
+ +
+ +
+
+ +
+No more tests ! we are going to build bigger transitions +
+
+ +
+ +
+ +
+Lemma Bigstep : ∀ b i a , reachable i a b 0 →
+                               reachable (iterate doubleS b i) a 0 0.
+ +
+ +
+ +
+ +
+
+ +
+From all the lemmas above, we now get a pretty big step +
+
+ +
+ +
+Definition M := iterate doubleS 95 95.
+ +
+Lemma L2_95 : reachable M 1 0 0. +
+
+ +
+the next step creates (M+1) copies of h1 +
+
+ +
+ +
+ +
+Lemma L2_95_S : reachable (S M) 0 (S M) 0.
+ +
+ +
+ +
+Definition N := iterate doubleS (S M) (S M).
+ +
+Theorem SuperbigStep : reachable N 0 0 0. +
+ +
+
+ +
+We can ow apply our study based on abstract states to "real" hydras +
+ + transforming our relation one_step into standard battles +
+ + We get now statements about hydras and battles +
+
+ +
+ +
+ +
+Lemma Almost_done :
+  rounds standard 3 (hyd 3 0 0) N (hyd 0 0 0).
+ +
+Theorem Done :
+  rounds standard 0 hinit N head.
+ +
+ +
+ +
+
+ +
+The natural number N is expressed in terms of the (iterate doubleS) + function. The rest of this file is dedicated to get an intuition of how it is huge . + Our idea is to get a minoration by exponentials of base 2 + +
+ + +
+
+ +
+Lemma minoration_0 : ∀ n, 2 × n ≤ doubleS n. +
+Lemma minoration_1 : ∀ n x, exp2 n × x ≤ iterate doubleS n x. +
+Lemma minoration_2 : exp2 95 × 95 ≤ M. +
+Lemma minoration_3 : exp2 (S M) × S M ≤ N. +
+Lemma exp2_mono1 : ∀ n p, n ≤ p → exp2 n ≤ exp2 p .
+ +
+Lemma minoration : exp2 (exp2 95 × 95) ≤ N. +
+
+ +
+Hence, the length of the battle is greater or equal than + exp2 (exp2 95 * 95), a number whose representation in base 10 has + at least 10 ^ 30 digits ! +
+ + from OCAML + +
+ +let exp2 x = 2.0 ** x;; + +
+ +let _ = exp2 95.0 *. 95.0;; + +
+ +float = 3.76333771942755604e+30 + +
+ +N >= 2 ** 3.7e+30 + +
+ +log10 N >= 3.7e+30 * log10 2 >= 1.0e+30 + +
+ +N >= 10 ** (10 ** 30) + +
+ + +
+
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.Epsilon0_Needed_Free.html b/theories/html/hydras.Hydra.Epsilon0_Needed_Free.html new file mode 100644 index 00000000..727f7395 --- /dev/null +++ b/theories/html/hydras.Hydra.Epsilon0_Needed_Free.html @@ -0,0 +1,133 @@ + + + + + +hydras.Hydra.Epsilon0_Needed_Free + + + + +
+ + + +
+ +

Library hydras.Hydra.Epsilon0_Needed_Free

+ +
+ +
+
+ +
+ Pierre Castéran, LaBRI and University of Bordeaux +
+
+ +
+From hydras Require Import Epsilon0_Needed_Generic.
+Import Hydra_Lemmas Epsilon0 Canon Paths Relation_Operators.
+Import O2H.
+ +
+
+ +
+We generalize the results of libraries Omega_Small and Omega2_Small: + We prove that no ordinal strictly less than epsilon0 can be used as a variant + for proving the termination of all hydra battles. +We use the so-called "Ketonen-Solovay machinery" for building +a proof that shares the same structure as for the libraries above, but is much +longer + +
+ +

Bounded variants

+ +
+
+ +
+Open Scope t1_scope.
+ +
+Section Impossibility_Proof.
+ +
+  Context (mu: T1)
+          (Hmu: nf mu)
+          (m : Hydra → T1)
+          (Var : Hvariant T1_wf free m)
+          (Hy : BoundedVariant Var mu).
+ +
+  Let big_h := big_h mu.
+  Let small_h := small_h mu m.
+ +
+  #[local] Hint Resolve nf_m : hydra.
+ +
+ +
+  Lemma m_ge : m big_h t1≤ m small_h.
+ +
+ +
+
+ +
+

Proof of the inequality m small_h t1< m big_h

+ + +
+
+ +
+ +
+  Lemma m_variant_LT : ∀ h h', h -+-> h' → m h' t1< m h.
+ +
+ +
+  Lemma big_to_small : big_h -+-> small_h.
+ +
+ +
+  Lemma m_lt : m small_h t1< m big_h.
+ +
+ +
+ +
+  Fact self_lt_free : m big_h t1< m big_h .
+ +
+  Theorem Impossibility_free : False.
+ +
+End Impossibility_Proof.
+ +
+ +
+Check Impossibility_free.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.Epsilon0_Needed_Generic.html b/theories/html/hydras.Hydra.Epsilon0_Needed_Generic.html new file mode 100644 index 00000000..26d37db4 --- /dev/null +++ b/theories/html/hydras.Hydra.Epsilon0_Needed_Generic.html @@ -0,0 +1,98 @@ + + + + + +hydras.Hydra.Epsilon0_Needed_Generic + + + + +
+ + + +
+ +

Library hydras.Hydra.Epsilon0_Needed_Generic

+ +
+ +
+
+ +
+ Pierre Castéran, LaBRI and University of Bordeaux +
+ + Technical definitions and lemmas about variants bounded by some ordinal + les than epsilon0 + +
+
+ +
+From hydras Require Import Hydra_Lemmas Epsilon0 Canon Paths O2H.
+From Coq Require Import Relation_Operators.
+ +
+Open Scope t1_scope.
+ +
+Section Bounded.
+ +
+  Context (B: Battle)
+          (mu: T1)
+          (Hmu: nf mu)
+          (m : Hydra → T1)
+          (Var : Hvariant T1_wf B m)
+          (Hy : BoundedVariant Var mu).
+ +
+  Hypothesis m_decrease : ∀ i h h', round_n i h h'→ m h' t1< m h.
+ +
+  Lemma nf_m : ∀ h, nf (m h).
+ +
+  #[local] Hint Resolve Rem0 : hydra.
+ +
+ +
+  Lemma mu_positive : mu ≠ T1.zero.
+ +
+  Lemma m_ge_0 alpha: nf alpha → alpha t1≤ m (iota alpha).
+ +
+ +
+ Definition big_h := iota mu.
+ Definition beta_h := m big_h.
+ Definition small_h := iota beta_h.
+ +
+ Lemma mu_beta_h : acc_from mu beta_h.
+ +
+ Corollary m_ge_generic : m big_h t1≤ m small_h.
+ +
+End Bounded.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.Epsilon0_Needed_Std.html b/theories/html/hydras.Hydra.Epsilon0_Needed_Std.html new file mode 100644 index 00000000..d282a2ec --- /dev/null +++ b/theories/html/hydras.Hydra.Epsilon0_Needed_Std.html @@ -0,0 +1,157 @@ + + + + + +hydras.Hydra.Epsilon0_Needed_Std + + + + +
+ + + +
+ +

Library hydras.Hydra.Epsilon0_Needed_Std

+ +
+
+ +
+ Pierre Castéran, LaBRI and University of Bordeaux + +
+ + We prove that the impossibility result of Epsilon0_Needed_Free still holds for "standard" battles +
+
+ +
+From hydras Require Import Epsilon0_Needed_Generic.
+Import Hydra_Lemmas Epsilon0 Canon Paths Relation_Operators O2H.
+Import Hydra_Definitions.
+ +
+Open Scope t1_scope.
+ +
+Section Impossibility_Proof.
+ +
+  Context (mu: T1)
+          (m : Hydra → T1)
+          (Var : Hvariant T1_wf standard m)
+          (Hy : BoundedVariant Var mu).
+ +
+  Let big_h := big_h mu.
+  Let small_h := small_h mu m.
+ +
+ +
+ +
+  Lemma m_ge : m big_h t1≤ m small_h. +
+ +
+
+ +
+

Proof of the inequality m small_h t1< m big_h

+ +
+ + The measure is strictly decreasing along any round +
+
+ +
+  Lemma Lvar : ∀ h h0 i,
+      h ≠ head →
+      battle_rel standard i h h0 → m h0 t1< m h.
+ +
+
+ +
+Application to standard battles +
+
+ +
+  Lemma m_decrease : ∀ j h0 i h,
+      h ≠ head → rounds standard i h j h0 → m h0 t1< m h.
+ +
+
+ +
+Conversion of ordinal inequalities into standard battles +
+
+ +
+ +
+ +
+ +
+  Lemma LT_to_standard_battle :
+    ∀ alpha beta,
+      beta t1< alpha →
+      ∃ n i, rounds standard n (iota alpha) i (iota beta). +
+  Remark Rem3 : beta_h mu m t1< mu.
+ +
+  Remark Rem4 : ∃ n i,
+      rounds standard n (iota mu) i (iota (beta_h mu m)).
+ +
+  Remark Hmu : nf mu.
+ +
+ +
+  Lemma m_lt : m small_h t1< m big_h. +
+
+ +
+End of the proof +
+
+ +
+ +
+ +
+  Fact self_lt_standard : m big_h t1< m big_h.
+ +
+  Theorem Impossibility_std: False.
+ +
+End Impossibility_Proof.
+Check Impossibility_std.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.Hydra_Definitions.html b/theories/html/hydras.Hydra.Hydra_Definitions.html new file mode 100644 index 00000000..d39a5d47 --- /dev/null +++ b/theories/html/hydras.Hydra.Hydra_Definitions.html @@ -0,0 +1,858 @@ + + + + + +hydras.Hydra.Hydra_Definitions + + + + +
+ + + +
+ +

Library hydras.Hydra.Hydra_Definitions

+ +
+
+ +
+Pierre Casteran, LaBRI, University of Bordeaux +
+ +

Data types

+ + +
+ + An hydra is a finitely branched tree. We use the auxiliary type Hydrae for + representing finite sequences of hydras. + +
+
+ +
+From Coq Require Export Relations.
+From hydras Require Import T1 Epsilon0.
+From Coq Require Import List.
+Import ListNotations.
+ +
+ +
+Inductive Hydra : Set :=
+| node : Hydrae → Hydra
+with Hydrae : Set :=
+| hnil : Hydrae
+| hcons : Hydra → Hydrae → Hydrae.
+ +
+ +
+
+ +
+Alternative representation (discarded) +
+
+ +
+ +
+Module Alt.
+  Inductive Hydra : Set :=
+  | hnode (daughters : list Hydra).
+End Alt.
+ +
+ +
+
+ +
+ Mutual induction scheme for types Hydra and Hydrae + +
+
+ +
+Scheme Hydra_rect2 := Induction for Hydra Sort Type
+with Hydrae_rect2 := Induction for Hydrae Sort Type.
+ +
+Arguments Hydra_rect2 P P0 : rename.
+Arguments Hydrae_rect2 P P0 : rename.
+ +
+
+ +
+All elements of s satisfy P +
+
+Fixpoint h_forall (P: Hydra → Prop) (s: Hydrae) :=
+  match s with
+  | hnil ⇒ True
+  | hcons h s' ⇒ P h ∧ h_forall P s'
+  end.
+ +
+Lemma h_eq_dec : ∀ h h':Hydra, {h = h'}+{h ≠ h'}
+with hs_eq_dec : ∀ l l':Hydrae, {l = l'}+{l ≠ l'}.
+ +
+
+ +
+For list usage +
+
+ +
+Fixpoint hs2l (s: Hydrae): list Hydra :=
+match s with
+  hnil ⇒ []
+| hcons h s' ⇒ h :: hs2l s'
+end.
+ +
+Fixpoint l2hs (l: list Hydra): Hydrae :=
+match l with
+  [] ⇒ hnil
+| h :: l' ⇒ hcons h (l2hs l')
+end.
+ +
+Lemma hs2lK (s: Hydrae) : l2hs (hs2l s) = s.
+ +
+Lemma l2hsK (l: list Hydra) : hs2l (l2hs l) = l.
+ +
+
+ +
+ Number of nodes (a.k.a. size) + +
+
+ +
+Fixpoint hsize (h:Hydra) : nat :=
+  match h with
+  | node l ⇒ S (lhsize l)
+  end
+with lhsize l : nat :=
+       match l with
+       | hnil ⇒ 0
+       | hcons h hs ⇒ hsize h + lhsize hs
+       end.
+ +
+
+ +
+

height (length of longest branch)

+ + +
+
+ +
+Fixpoint height (h:Hydra) : nat :=
+  match h with
+  | node l ⇒ lheight l
+  end
+with lheight l : nat :=
+       match l with
+       | hnil ⇒ 0
+       | hcons h hs ⇒ Nat.max (S (height h)) (lheight hs)
+       end.
+ +
+
+ +
+

Abbreviations

+ + +
+ +

Heads : A head is just a node without daughters

+ + +
+
+ +
+
+ +
+

Hydra with 0, 1, 2 or 3 daughters

+ +
+
+ +
+Notation head := (node hnil).
+Notation hyd1 h := (node (hcons h hnil)).
+Notation hyd2 h h' := (node (hcons h (hcons h' hnil))).
+Notation hyd3 h h' h'' := (node (hcons h (hcons h' (hcons h'' hnil)))).
+ +
+
+ +
+

Adds n copies of the same hydra h at the right of s

+ + +
+
+ +
+Fixpoint hcons_mult (h:Hydra)(n:nat)(s:Hydrae):Hydrae :=
+  match n with
+  | O ⇒ s
+  | S p ⇒ hcons h (hcons_mult h p s)
+  end.
+ +
+
+ +
+

Hydra with n equal daughters

+ +
+
+ +
+Definition hyd_mult h n := node (hcons_mult h n hnil).
+ +
+
+ +
+

Managing sequences of hydras

+ + +
+ +

Appending

+ + +
+
+ +
+Fixpoint hy_app (s1 s2: Hydrae) : Hydrae :=
+  match s1 with
+    hnil ⇒ s2
+  | hcons h1 s1' ⇒ hcons h1 (hy_app s1' s2)
+  end.
+ +
+
+ +
+

Adds a head to the right of s

+ + +
+
+ +
+Fixpoint add_head_r (s: Hydrae) :=
+  match s with
+    hnil ⇒ hcons head hnil
+  | hcons h s' ⇒ hcons h (add_head_r s')
+  end.
+ +
+
+ +
+

Adds i heads to the right of s

+ + +
+
+ +
+Fixpoint add_head_r_plus (s:Hydrae) (i:nat) :=
+  match i with
+    0 ⇒ s
+  | S j ⇒ add_head_r (add_head_r_plus s j)
+  end.
+ +
+
+ +
+

adds i heads to the root of h (to the right of its daughters)

+ + +
+
+ +
+Definition add_r h i :=
+  match h with node s ⇒ node (add_head_r_plus s i) end.
+ +
+
+ +
+

Hydra Battles

+ + +
+ +

Relation associated with a single round.

+ + +
+ + We represent the rules of Hydra Battles through a binary relation: + round_n n on the Hydra type. +The proposition round_n n h h' holds if h' is obtained from h by a single round of a battle, and n is the expected replication factor (irrelevant if the chopped head is at distance 1 from the foot). + +
+ +The proposition round_n n h h' holds + +
+ +
    +
  • if h' is obtained from h by chopping off one head of height 1. This situation is described by the proposition R1 h h'. + +
    + + +
    • +
  • Or h' is obtained from h through a beheading at distance from the foot, with creation of n copies of some dub-hydra. + +
    • +
+ +
+ + +
+ +Before giving the definition in Coq of round_n, we need to define several auxiliary relations. + +
+ +

S0

+ + +
+ + The proposition S0 s s' holds if the sequence s' is obtained by removing one head from s + +
+
+ +
+Inductive S0 : relation Hydrae :=
+| S0_first : ∀ s, S0 (hcons head s) s
+| S0_rest : ∀ h s s', S0 s s' → S0 (hcons h s) (hcons h s').
+ +
+
+ +
+

R1

+ + +
+ + The proposition R1 h h' holds if h' is obtained by removing one head from h at distance 1 from the foot + +
+ + +
+
+ +
+Inductive R1 : relation Hydra :=
+| R1_intro : ∀ s s', S0 s s' → R1 (node s) (node s').
+ +
+
+ +
+

S1

+ + +
+ + S1 n s s' holds if s' is obtained from s by replacing some member h of s by copies of h, where R1 h h' holds. + Thus, n is the number of new replicas of h'. + +
+ + +
+
+ +
+ +
+Inductive S1 (n:nat) : relation Hydrae :=
+| S1_first : ∀ s h h' ,
+              R1 h h' →
+              S1 n (hcons h s) (hcons_mult h' (S n) s)
+| S1_next : ∀ h s s',
+                 S1 n s s' →
+                 S1 n (hcons h s) (hcons h s').
+ +
+ +
+
+ +
+

R2

+ + +
+ +The proposition R2 n h h' holds if some sub-hydra h0 of h has been replaced by h'0, + where R1 n h0 h'0. The S2 relation is just a helper for R2. + +
+ + +
+
+ +
+Inductive R2 (n:nat) : relation Hydra :=
+| R2_intro : ∀ s s', S1 n s s' → R2 n (node s) (node s')
+| R2_intro_2 : ∀ s s', S2 n s s' → R2 n (node s) (node s')
+
+with S2 (n:nat) : relation Hydrae :=
+     | S2_first : ∀ h h' s ,
+         R2 n h h'→ S2 n (hcons h s) (hcons h' s)
+     | S2_next : ∀ h r r',
+         S2 n r r' → S2 n (hcons h r) (hcons h r').
+ +
+
+ +
+

round_n

+ + +
+ + Let n be an expected replication number. The relation round_n n h h' + holds: +
    +
  • if h' is obtained from h by removing one head of height 1 (and the factor n is irrelevant) + +
    • +
  • or h' is obtained from h by removing one head of height greater or equal than 2, and this beheading was made with a relocation factor n. + +
    • +
+ +
+
+ +
+Definition round_n n h h' := R1 h h' ∨ R2 n h h'.
+ +
+
+ +
+Transition system associated with battles +
+ +

Binary relation associated with a battle round

+ +
+
+ +
+Definition round h h' := ∃ n, round_n n h h'.
+Infix "-1->" := round (at level 60).
+ +
+
+ +
+

transitive closures of round

+ + +
+
+ +
+Definition round_plus := clos_trans_1n Hydra round.
+Definition round_star h h' := h = h' ∨ round_plus h h'.
+ +
+Infix "-+->" := round_plus (at level 60).
+Infix "-*->" := round_star (at level 60).
+ +
+
+ +
+

Experimental tactics for interactive battles

+ +
+ +

removes the i-th daughter (should be a head)

+ +
+
+ +
+Ltac chop_off i :=
+  match goal with |- S0 ?h ?h' ⇒
+                  match i with
+                    | O ⇒ eapply S0_first
+                    | S ?j ⇒ eapply S0_rest; chop_off j
+                  end
+                  | |- round ?h ?h' ⇒
+                     ∃ 0; left; split; chop_off i
+                  | |- round_n ?n ?h ?h' ⇒
+                     left ; split; chop_off i
+  end.
+ +
+
+ +
+ Calls the R2 relation +
+
+ +
+Ltac h_search_n n := match goal with
+                       |- round ?h ?h' ⇒ ∃ n; eright
+                    end.
+ +
+Ltac h_search := match goal with
+                    | |- round_n ?p ?h ?h' ⇒ eright
+                    end.
+ +
+Ltac s2_nth n :=
+  match n with
+    | 0 ⇒ eleft
+    | S ?p ⇒ eright ; s2_nth p
+    end.
+ +
+
+ +
+Moves to the i-th daugther +
+
+ +
+Ltac r2_up i := match goal with
+                    |- R2 ?n ?h ?h' ⇒ eright; s2_nth i
+end.
+ +
+Ltac S1_nth i :=
+  match goal with
+      |- S1 ?n ?h ?h' ⇒
+      match i with
+        | 0 ⇒ eleft
+        | S ?j ⇒ eright ; S1_nth j
+      end
+  end.
+ +
+
+ +
+chops off a head which is the j-th daughter of the + i-th dauchter + +
+
+ +
+Ltac r2_d2 i j :=
+ match goal with
+      |- R2 ?n ?h ?h' ⇒ eleft; S1_nth i; split; chop_off j
+ end.
+ +
+
+ +
+End of the battle +
+
+ +
+Ltac stop :=
+  match goal with
+      |- round_star ?h ?h' ⇒ left; reflexivity
+  end.
+ +
+
+ +
+Still fighting +
+
+ +
+Lemma round_star_intro h h'' : ∀ h',
+    h -1-> h' → h' -*-> h'' → h -*-> h''.
+ +
+Ltac forward :=
+  match goal with
+      |- round_star ?h ?h' ⇒ eapply round_star_intro
+  end.
+ +
+
+ +
+

Traces

+ +
+
+ +
+Inductive trace_to dest : list nat → Hydra → Prop :=
+  trace_to1 : ∀ n h, round_n n h dest → trace_to dest (n:: nil) h
+| trace_toS : ∀ n t h h', round_n n h h' → trace_to dest t h' →
+                                 trace_to dest (n:: t) h.
+ +
+Definition trace t h h' := trace_to h' t h.
+ +
+
+ +
+ +
+ + Let be an hydra and and expected replication factor. The next step of + the current battle may be specified by the following type +
+
+ +
+Definition round_spec h n :=
+  {h' : Hydra | round_n n h h'} + {h = head}.
+ +
+
+ +
+

Classes of battles

+ + +
+ +In an hydra battle, the interaction between the two players may depend on the time (number of rounds) elapsed since the beginning of the fight. + +
+ +Let us formalize this dependence through a relation linking the number of the current round, and the hydra before and after the considered round. + +
+ + +
+
+ +
+Definition round_t := nat → Hydra → Hydra → Prop.
+ +
+Class Battle :=
+  { battle_rel : round_t;
+    battle_ok : ∀ i h h', battle_rel i h h' → round h h'}.
+ +
+Arguments battle_rel : clear implicits.
+ +
+
+ +
+In the current state of this development, we will consider two instances of class Battle: + +
+ +
    +
  • In free, we hydra may chose any replication factor at any time. + +
    + + +
    • +
  • In standard the replication factor at round number i is just i + +
    • +
+ +
+
+ +
+#[ global, refine ] Instance free : Battle
+  := Build_Battle (fun _ h h' ⇒ round h h') _.
+ +
+ +
+#[ global, refine] Instance standard : Battle :=
+  Build_Battle round_n _.
+ +
+
+ +
+ The following relation allows us to consider sequences of rounds in a given class of battles + +
+ + The proposition rounds b i h j h' holds if there is a battle of class B that + starts with hydra h at round i and ends with hydra h' at round j + +
+
+ +
+Inductive rounds (B:Battle)
+  : nat → Hydra → nat → Hydra → Prop :=
+  rounds_1 : ∀ i h h',
+    battle_rel B i h h' → rounds B i h (S i) h'
+| rounds_n : ∀ i h j h' h'',
+    battle_rel B i h h'' →
+    rounds B (S i) h'' j h' →
+    rounds B i h j h'.
+ +
+
+ +
+number of steps leading to the hydra's death +
+
+ +
+Definition battle_length B k h l :=
+    rounds B k h (Nat.pred (k + l)%nat) head.
+ +
+#[deprecated(note="use rounds")]
+  Notation battle := rounds (only parsing).
+ +
+
+ +
+

Uniform termination

+ +
+
+ +
+ +
+Definition Termination := well_founded (transp _ round).
+ +
+ +
+Definition B_termination (B: Battle) :=
+  well_founded (fun h' h ⇒ ∃ i:nat, battle_rel B i h h' ).
+ +
+
+ +
+

Variants for proving termination

+ + +
+
+ +
+ +
+Class Hvariant {A:Type}{Lt:relation A}
+      (Wf: well_founded Lt)(B : Battle)
+      (m: Hydra → A): Prop :=
+  {variant_decr: ∀ i h h',
+      h ≠ head → battle_rel B i h h' → Lt (m h') (m h)}.
+ +
+ +
+
+ +
+Variant bounded by some ordinal alpha < epsilon0

Strictly Bounded variants

+ +
+
+ +
+ +
+Class BoundedVariant {A:Type}{Lt:relation A}
+      {Wf: well_founded Lt}{B : Battle}
+      {m: Hydra → A} (Var: Hvariant Wf B m)(mu:A):=
+  { m_bounded: ∀ h, Lt (m h ) mu }.
+ +
+ +
+ +
+
+ +
+

Liveness

+ + +
+ + For every reachable configuration (i,h), either h is a head, or + there exists a beheading leading to some configuration (S i, h'). + +
+
+ +
+Definition Alive (B : Battle) :=
+  ∀ i h, h ≠ head → {h' : Hydra | battle_rel B i h h'}.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.Hydra_Examples.html b/theories/html/hydras.Hydra.Hydra_Examples.html new file mode 100644 index 00000000..227baca5 --- /dev/null +++ b/theories/html/hydras.Hydra.Hydra_Examples.html @@ -0,0 +1,168 @@ + + + + + +hydras.Hydra.Hydra_Examples + + + + +
+ + + +
+ +

Library hydras.Hydra.Hydra_Examples

+ +
+ +
+From Coq Require Import Arith Lia.
+From hydras Require Import Hydra_Lemmas More_Arith.
+Open Scope nat_scope.
+ +
+Check Hydra_rect2.
+ +
+Module Examples.
+ +
+  Example ex1 : ∀ h h', R1 (hyd3 h head h') (hyd2 h h').
+ +
+  Example Hy := hyd3 head
+                     (hyd2
+                        (hyd1
+                           (hyd2 head head))
+                        head)
+                     head.
+ +
+ +
+ +
+  Example Hy' := hyd2 head
+                      (hyd2
+                         (hyd1
+                            (hyd2 head head))
+                         head).
+ +
+ +
+  Example ex4: round Hy Hy'.
+ +
+  Example Hy'' :=
+    hyd2 head
+         (hyd2
+            (hyd_mult (hyd1 head) 5)
+            head).
+ +
+  Example Hy'H'' : round Hy' Hy''.
+ +
+  Example R2_example: R2 4 Hy' Hy''.
+ +
+move to 2nd sub-hydra (0-based indices)
+
+
+ +
+move to first sub-hydra
+
+
+ +
+we're at distance 2 from the to-be-chopped-off head + let's go to the first daughter, + then chop-off the leftmost head
+
+ +
+  Example Exx : {h' | round Hy' h'}.
+ +
+ +
+  Example Ex5 : {h' | Hy -*-> h'}.
+ +
+  Example Hy''' :=
+    node (hcons head
+                (hcons_mult (hyd1 (hyd_mult (hyd1 head) 5)) 3 hnil)).
+ +
+ +
+ +
+  Example hsize_bigger : (hsize Hy'' < hsize Hy''')%nat.
+ +
+ +
+  Example height_not_strictly_decreasing : height Hy'' = height Hy'''.
+ +
+ +
+  Example Hy_1 : R1 Hy Hy'. +
+ +
+ +
+  Example Hy_2 : R2 4 Hy' Hy''.
+ +
+  Example ex_2 :{Hy'' | R2 4 Hy' Hy''}.
+ +
+ +
+  Example Hy_3 : R2 2 Hy'' Hy'''.
+ +
+  Example Hy''_Hy''' : Hy'' -1-> Hy'''.
+ +
+  Example Hy_Hy''' : Hy -+-> Hy'''.
+ +
+End Examples.
+ +
+ +
+Check Hydra_ind.
+ +
+ +
+Module Bad.
+  Lemma height_lt_size (h:Hydra) : height h < hsize h. End Bad.
+ +
+ +
+ Lemma height_lt_size (h:Hydra) : height h < hsize h.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.Hydra_Extraction.html b/theories/html/hydras.Hydra.Hydra_Extraction.html new file mode 100644 index 00000000..23efd715 --- /dev/null +++ b/theories/html/hydras.Hydra.Hydra_Extraction.html @@ -0,0 +1,37 @@ + + + + + +hydras.Hydra.Hydra_Extraction + + + + +
+ + + +
+ +

Library hydras.Hydra.Hydra_Extraction

+ +
+From hydras Require Import Hydra_Lemmas.
+ +
+Extraction Language OCaml.
+Recursive Extraction classic_battle next_round.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.Hydra_Lemmas.html b/theories/html/hydras.Hydra.Hydra_Lemmas.html new file mode 100644 index 00000000..d3198a4e --- /dev/null +++ b/theories/html/hydras.Hydra.Hydra_Lemmas.html @@ -0,0 +1,331 @@ + + + + + +hydras.Hydra.Hydra_Lemmas + + + + +
+ + + +
+ +

Library hydras.Hydra.Hydra_Lemmas

+ +
+
+ +
+ Pierre Casteran + LaBRI, Université de Bordeaux + +
+
+ +
+From Coq Require Import Arith Lia List Relation_Operators
+     Operators_Properties Peano_dec Wellfounded.Inverse_Image
+     Wellfounded.Inclusion.
+ +
+From hydras Require Import More_Arith Epsilon0 Hessenberg
+     Simple_LexProd MoreLibHyps.
+ +
+From hydras Require Export Hydra_Definitions.
+Import Relations.
+Open Scope nat_scope.
+ +
+
+ +
+

Properties of the hydra data type

+ + +
+
+ +
+Lemma add_r_0 : ∀ h: Hydra, add_r h 0 = h.
+ +
+Lemma hy_app_assoc :
+  ∀ (s1 s2 s3 : Hydrae) ,
+    hy_app s1 (hy_app s2 s3) = hy_app (hy_app s1 s2) s3.
+ +
+Lemma hcons_mult_app : ∀ (h: Hydra) (n :nat) (s s': Hydrae),
+    hcons_mult h n (hy_app s s') =
+    hy_app (hcons_mult h n s) s'.
+ +
+Lemma hcons_mult_comm :
+  ∀ i h s,
+    hcons_mult h i (hcons h s) = hcons h (hcons_mult h i s).
+ +
+ +
+Lemma R1_iff s s' : R1 (node s) (node s') ↔ S0 s s'.
+ +
+Lemma R2_iff i s s' : R2 i (node s) (node s') ↔
+                      S1 i s s' ∨ S2 i s s'.
+ +
+Lemma S2_iff : ∀ n h h' s s',
+    S2 n (hcons h s) (hcons h' s') ↔
+    R2 n h h' ∧ s = s' ∨
+                 h = h' ∧ S2 n s s'.
+ +
+Lemma S0_app : ∀ s1 s2 s3, S0 s2 s3 →
+                                S0 (hy_app s1 s2)
+                                   (hy_app s1 s3).
+ +
+Lemma R1_app : ∀ s1 s2 s3,
+    R1 (node s2) (node s3) →
+    R1 (node (hy_app s1 s2)) (node (hy_app s1 s3)).
+ +
+Lemma S1_app n : ∀ s1 s2 s3,
+    S1 n s2 s3 → S1 n (hy_app s1 s2) (hy_app s1 s3).
+ +
+ +
+
+ +
+Induction Schemes for R2 +
+
+ +
+Scheme R2_ind2 := Induction for R2 Sort Prop
+                  with
+                  S2_ind2 := Induction for S2 Sort Prop.
+Arguments R2_ind2 n P P0 : rename.
+Arguments S2_ind2 n P P0 : rename.
+ +
+Lemma S2_app : ∀ n s1 s2 ,
+    S2 n s1 s2 →
+    ∀ l, S2 n (hy_app l s1) (hy_app l s2).
+ +
+Lemma R2_app n : ∀ s1 s2 ,
+    R2 n (node s1) (node s2) →
+    ∀ l, R2 n (node (hy_app l s1))
+                  (node (hy_app l s2)).
+ +
+Lemma hcons_mult_S0 : ∀ h i s s' ,
+                             S0 s s' →
+                             S0 (hcons_mult h i s)
+                                (hcons_mult h i s').
+ +
+Lemma hcons_mult_S1 n : ∀ h i s s' , S1 n s s' →
+                                         S1 n (hcons_mult h i s)
+                                              (hcons_mult h i s').
+ +
+Lemma hcons_mult_S2 n : ∀ h i s s' , S2 n s s' →
+                                            S2 n (hcons_mult h i s)
+                                               (hcons_mult h i s').
+ +
+
+ +
+

Properties of single rounds

+ +
+
+ +
+Lemma head_no_round_n : ∀ i h, ¬ round_n i head h.
+ +
+Lemma head_no_round : ∀ h, ¬ round head h.
+ +
+
+ +
+

Properties of battle

+ +
+
+ +
+Lemma rounds_trans {B:Battle} :
+  ∀ i h j h', rounds B i h j h' →
+                   ∀ k h0, rounds B k h0 i h →
+                                 rounds B k h0 j h'. +
+
+ +
+

Properties of standard and free battle classes

+ +
+
+ +
+Lemma standard_incl_free : ∀ i h h',
+     battle_rel standard i h h'→ battle_rel free i h h' .
+ +
+Lemma standard_battle_head :
+  ∀ i h j h',
+    @rounds standard i h j h' → h ≠ head.
+ +
+
+ +
+

Generic properties of round_plus

+ +
+
+ +
+Lemma round_plus_trans : ∀ h h' h'', h -+-> h' →
+                                      h' -+-> h'' →
+                                      h -+-> h''.
+ +
+Lemma S0_remove_r : ∀ l , S0 (add_head_r l) l.
+ +
+Lemma S0_remove_r_i : ∀ i l, S0 (add_head_r_plus l (S i))
+                                     (add_head_r_plus l i).
+ +
+Lemma R1_remove_r_i : ∀ i h,
+                       R1 (add_r h (S i)) (add_r h i).
+ +
+Lemma round_n_remove_h : ∀ i j h,
+    round_n j (add_r h (S i)) (add_r h i).
+ +
+Lemma remove_heads_r : ∀ i h j, rounds standard
+                                       j
+                                       (add_r h (S i))
+                                       (S i+j)%nat
+                                       h.
+ +
+Lemma remove_heads_r_free: ∀ i h j, rounds free
+                                       j
+                                       (add_r h (S i))
+                                       (S i+j)%nat
+                                       h.
+ +
+Lemma variant_mono_free {A:Type} {Lt: relation A}{Tr : Transitive Lt}
+      {Wf: well_founded Lt} m {V: Hvariant Wf free m}:
+  ∀ i h j h', rounds free i h j h' → Lt (m h') (m h).
+ +
+Lemma m_strict_mono {A:Type} {Lt: relation A}{St: StrictOrder Lt}
+      {Wf: well_founded Lt} m (V: @Hvariant _ _ Wf free m){h h':Hydra}:
+  h -+-> h' → Lt (m h') (m h).
+ +
+ +
+
+ +
+companion lemmas for R1 and R2 +
+
+ +
+Lemma round_n_inv i h h' : round_n i h h' → R1 h h' ∨ R2 i h h'.
+ +
+Lemma rounds_free_equiv1 : ∀ i j h h',
+    rounds free i h j h' →
+    h -+-> h'.
+ +
+Lemma rounds_free_equiv2 : ∀ h h',
+     h -+-> h' →
+    ∀ i, ∃ j, rounds free i h j h'.
+ +
+Lemma Termination_strong (B:Battle) :
+   Termination → B_termination B.
+ +
+Lemma Hvariant_Termination {A : Type} {lt : relation A}
+      (Hwf : well_founded lt) (m : Hydra → A) :
+  @Hvariant _ _ Hwf free m → Termination.
+ +
+ +
+
+ +
+If the hydra is not reduced to a head, there exists at + least one head that Hercules can chop off +
+
+ +
+Definition next_round_dec n (h: Hydra) :
+  (h = head) + {h' : Hydra & {R1 h h'} + {R2 n h h'}}.
+ +
+chose a lowest head +
+
+ +
+ +
+Definition next_round n :
+ ∀ h , round_spec h n.
+ +
+Definition next_step (f : ∀ n h, round_spec h n) n h :=
+  match f n h with
+      inright _ ⇒ head
+    | inleft (exist _ h' _) ⇒ h'
+  end.
+ +
+Definition classic_battle f t h :=
+  let fix go n t h :=
+      match h, t with head, _ ⇒ h
+                   | _, 0 ⇒ h
+                | _, S t' ⇒ go (S n) t' (next_step f n h)
+      end
+  in go 1 t h.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.Hydra_Termination.html b/theories/html/hydras.Hydra.Hydra_Termination.html new file mode 100644 index 00000000..bad611a9 --- /dev/null +++ b/theories/html/hydras.Hydra.Hydra_Termination.html @@ -0,0 +1,173 @@ + + + + + +hydras.Hydra.Hydra_Termination + + + + +
+ + + +
+ +

Library hydras.Hydra.Hydra_Termination

+ +
+
+ +
+

Proof of termination of all hydra battles

+ +
+ + Pierre Casteran, Univ. Bordeaux, LaBRI, UMR 5800 +
+
+ +
+ +
+From hydras Require Export Hydra_Lemmas.
+From hydras Require Import E0 Hessenberg.
+Import ON_Generic.
+ +
+
+ +
+

Converting any hydra into an ordinal less than

+ + epsilon0 + +
+ + +
+
+ +
+Fixpoint m (h:Hydra) : T1 :=
+  match h with head ⇒ T1.zero
+             | node hs ⇒ ms hs
+end
+with ms (s:Hydrae) : T1 :=
+  match s with hnil ⇒ T1.zero
+              | hcons h s' ⇒ T1.phi0 (m h) o+ ms s'
+ end.
+ +
+ +
+Lemma ms_eqn2 : ∀ h s, ms (hcons h s) = T1.phi0 (m h) o+ ms s.
+ +
+Lemma o_finite_mult_S_rw :
+  ∀ n a, o_finite_mult (S n) a = a o+ o_finite_mult n a.
+ +
+
+ +
+ The functions m and ms return well formed ordinals (less than epsilon0) + +
+
+ +
+Lemma m_nf : ∀ h, nf (m h).
+ +
+Lemma ms_nf : ∀ s, nf (ms s).
+ +
+#[global] Hint Resolve m_nf nf_phi0 ms_nf : T1.
+ +
+Lemma ms_eqn3 : ∀ h n s, ms (hcons_mult h n s) =
+                                o_finite_mult n (T1.phi0 (m h)) o+ ms s.
+ +
+
+ +
+

Monotonicity lemmas for S0, R1, etc .

+ +
+
+ +
+Open Scope t1_scope.
+ +
+Lemma S0_decr_0 :
+  ∀ s s', S0 s s' → T1.lt (ms s') (ms s).
+ +
+Lemma S0_decr:
+  ∀ s s', S0 s s' → ms s' t1< ms s.
+ +
+Lemma R1_decr_0 : ∀ h h',
+                  R1 h h' → T1.lt (m h') (m h).
+ +
+ +
+Lemma R1_decr :
+  ∀ h h', R1 h h' → m h' t1< m h. +
+Lemma S1_decr_0 n:
+  ∀ s s', S1 n s s' → T1.lt (ms s') (ms s).
+ +
+ +
+Lemma S1_decr n:
+  ∀ s s', S1 n s s' → ms s' t1< ms s. +
+ +
+Lemma m_ms : ∀ s, m (node s) = ms s.
+ +
+Lemma R2_decr_0 n : ∀ h h', R2 n h h' → T1.lt (m h') (m h).
+ +
+ +
+Lemma R2_decr n : ∀ h h', R2 n h h' → m h' t1< m h. +
+ +
+Lemma round_decr : ∀ h h', h -1-> h' → m h' t1< m h.
+ +
+#[ global, program ] Instance var (h:Hydra) : E0:= @mkord (m h) _.
+ +
+#[global] Instance HVariant_0 : Hvariant T1_wf free m.
+ +
+ +
+#[global] Instance HVariant : Hvariant Epsilon0 free var.
+ +
+Theorem every_battle_terminates : Termination.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.Hydra_Theorems.html b/theories/html/hydras.Hydra.Hydra_Theorems.html new file mode 100644 index 00000000..9a241902 --- /dev/null +++ b/theories/html/hydras.Hydra.Hydra_Theorems.html @@ -0,0 +1,240 @@ + + + + + +hydras.Hydra.Hydra_Theorems + + + + +
+ + + +
+ +

Library hydras.Hydra.Hydra_Theorems

+ +
+
+ +
+Pierre Castéran, Univ. Bordeaux and LaBRI +
+
+ +
+From hydras Require Import Hydra_Lemmas Epsilon0_Needed_Free
+     Epsilon0_Needed_Std Hydra_Termination L_alpha Battle_length Ack.
+Import E0 Large_Sets Hprime Paths MoreLists O2H Hydra_Definitions Iterates.
+Export Hydra_Definitions.
+ +
+
+ +
+

Liveness

+ + +
+ + If the hydra is not reduced to one head, then Hercules can chop off + some head + +
+
+ +
+Corollary Alive_free : Alive free. +
+Corollary Alive_standard : Alive standard. +
+ +
+
+ +
+

Termination of free battles

+ + +
+
+ +
+Theorem Variant_LT_free_0 : @Hvariant _ _ T1_wf free Hydra_Termination.m.
+ +
+Theorem Variant_lt_free: @Hvariant _ _ E0lt_wf free Hydra_Termination.var.
+ +
+Theorem Variant_LT_standard : @Hvariant _ _ T1_wf standard Hydra_Termination.m.
+ +
+Theorem Variant_lt_standard : @Hvariant _ _ E0lt_wf standard Hydra_Termination.var.
+ +
+ +
+
+ +
+

Impossibility theorems

+ + +
+ + Termination of free battles cannot be proven with a variant from hydras into a segment with + +
+ + +
+ + Impossiblity to define a variant bounded by some ordinal less than + epsilon0 +
+
+ +
+Check Impossibility_free.
+ +
+About battle_length_std.
+ +
+Open Scope nat_scope.
+ +
+ +
+Theorem battle_length_std_Hardy (alpha : E0) :
+  alpha ≠ E0zero →
+  ∀ k , 1 ≤ k →
+             ∃ l: nat,
+               H'_ alpha k - k ≤ l ∧
+               battle_length standard k (iota (cnf alpha)) l.
+ +
+ +
+
+ +
+

Battle length is not PR

+ +
+
+ +
+From hydras Require Import primRec F_alpha AckNotPR
+  PrimRecExamples F_omega.
+Import E0.
+ +
+ +
+Section battle_length_notPR.
+ +
+  Context (H: ∀ alpha, isPR 1 (l_std alpha)).
+ +
+ +
+
+ +
+A counter example +
+
+ +
+ +
+  Let alpha := E0_phi0 E0_omega.
+  Let h := iota (cnf alpha).
+ +
+ +
+
+ +
+let us get rid of the substraction ... +
+
+ +
+ +
+  Let m k := L_ alpha (S k).
+ +
+  Remark m_eqn : ∀ k, m k = (l_std alpha k + k)%nat. +
+ +
+  #[local] Instance mIsPR : isPR 1 m. +
+ +
+ +
+  Remark m_ge_F_omega k: F_ E0_omega (S k) ≤ m (S k). +
+
+ +
+We compare m with the Ackermann function +
+
+ +
+  Remark m_ge_Ack: ∀ n, 2 ≤ n → Ack (S n) (S n) ≤ m (S n).
+ +
+  Remark m_dominates_Ack_from_3 : ∀ n, 3 ≤ n → Ack n n ≤ m n.
+ +
+ +
+  Remark m_dominates_Ack :
+    dominates (fun n ⇒ S (m n)) (fun n ⇒ Ack.Ack n n). +
+ +
+ Lemma SmNotPR : isPR 1 (fun n ⇒ S (m n)) → False. +
+ +
+ +
+  Theorem LNotPR : False.
+ +
+End battle_length_notPR.
+ +
+Check l_std_ok.
+ +
+Check LNotPR.
+ +
+Search L_ F_.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.KP_example.html b/theories/html/hydras.Hydra.KP_example.html new file mode 100644 index 00000000..35136a27 --- /dev/null +++ b/theories/html/hydras.Hydra.KP_example.html @@ -0,0 +1,152 @@ + + + + + +hydras.Hydra.KP_example + + + + +
+ + + +
+ +

Library hydras.Hydra.KP_example

+ +
+
+ +
+Pierre Casteran, Univ. Bordeaux and LaBRI +
+
+ +
+ From hydras Require Import Hydra_Definitions Hydra_Lemmas.
+ +
+
+ +
+The hydra from page 286 of KP +
+
+ +
+Section KP.
+
+ +
+initial state +
+
+ +
+Definition h0 : Hydra :=
+  hyd2 (hyd2 (hyd_mult head 3)
+             (hyd1 (hyd1 head)))
+       (hyd2 head
+             (hyd2 head head)).
+ +
+Fact F1 : hsize h0 = 14.
+ +
+Fact F2 : height h0 = 4.
+ +
+
+ +
+ after stage 1 +
+
+ +
+Definition h1 :=
+  hyd2 (hyd3 (hyd_mult head 2)
+             (hyd_mult head 2)
+             (hyd1 (hyd1 head)))
+       (hyd2 head
+             (hyd2 head head)).
+ +
+ +
+
+ +
+After stage 2 +
+
+ +
+Notation hyd4 h1 h2 h3 h4 :=
+  (node (hcons h1 (hcons h2 (hcons h3 (hcons h4 hnil))))).
+ +
+Let h' := hyd1 (hyd_mult head 2).
+ +
+Definition h2 :=
+                  hyd4 (hyd3 (hyd_mult head 2)
+                             (hyd_mult head 2)
+                             (hyd1 (hyd1 head)))
+                       h' h' h'.
+ +
+
+ +
+After stage 3 +
+
+ +
+Definition h3 := hyd4
+                   (node (hcons_mult (hyd1 head) 4
+                                     (hcons (hyd2 head head)
+                                            (hcons (hyd1 (hyd1 head))
+                                                   hnil))))
+                   h' h' h'.
+ +
+Fact h0_h1 : round_n 1 h0 h1.
+ +
+Fact h1_h2 : round_n 2 h1 h2.
+ +
+Fact h2_h3 : round_n 3 h2 h3.
+ +
+Lemma battle_example : rounds standard 1 h0 4 h3.
+ +
+
+ +
+= 28 +
+
+ +
+End KP.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.O2H.html b/theories/html/hydras.Hydra.O2H.html new file mode 100644 index 00000000..507ce445 --- /dev/null +++ b/theories/html/hydras.Hydra.O2H.html @@ -0,0 +1,256 @@ + + + + + +hydras.Hydra.O2H + + + + +
+ + + +
+ +

Library hydras.Hydra.O2H

+ +
+
+ +
+Injection from ordinals (less than epsilon0) in CNF into hydras. + +
+ + Pierre Castéran, LaBRI and Univ. Bordeaux + +
+ +We define a function iota : T1 → Hydra such that if alpha < beta, then there exists a battle from iota beta to iota alpha. + +
+ +Note that iota is not a bijection, but is sufficient for proving +impossibility lemmas or minoration of battle lengths (see Hydra_Theorems). + +
+
+ +
+From Coq Require Import Relation_Operators.
+ +
+From hydras Require Import Hydra_Lemmas Epsilon0 Canon Paths .
+Import Hydra_Definitions.
+ +
+
+ +
+Let us transform any ordinal term into an hydra +
+
+ +
+ +
+Fixpoint iota (a : T1) : Hydra :=
+  match a with
+  | zero ⇒ head
+  | cons c n b ⇒ node (hcons_mult (iota c) (S n) (iotas b))
+  end
+with iotas (a : T1) : Hydrae :=
+       match a with
+       | zero ⇒ hnil
+       | cons a0 n b ⇒ hcons_mult (iota a0) (S n) (iotas b)
+       end.
+ +
+
+ +
+ We now prove a lot of technical lemmas that relate Hydras and +ordinals. +
+
+ +
+Lemma iota_iotas : ∀ alpha, nf alpha →
+                                 node (iotas alpha) = iota alpha .
+ +
+Lemma iotas_succ : ∀ alpha, nf alpha →
+                                 iotas (T1.succ alpha) =
+                                 hy_app (iotas alpha) (hcons head hnil).
+ +
+Lemma hy_app_l_nil : ∀ l, hy_app l hnil = l.
+ +
+Lemma iota_succ_R1 : ∀ o, nf o → R1 (iota (T1.succ o)) (iota o).
+ +
+Lemma iota_succ_round_n : ∀ i alpha,
+    nf alpha → round_n i (iota (T1.succ alpha)) (iota alpha).
+ +
+Lemma iota_succ_round : ∀ o, nf o → iota (T1.succ o) -1-> iota o.
+ +
+Lemma iota_rw1 :
+  ∀ i alpha, nf alpha → T1limit alpha = true →
+                   iota (canon (cons alpha 0 zero) (S i)) =
+                   hyd1 (iota (canon alpha (S i))).
+ +
+Lemma iota_rw2 :
+  ∀ i n alpha, nf alpha → T1limit alpha = true →
+                     iota (canon (cons alpha (S n) zero) (S i)) =
+                     node (hcons_mult (iota alpha) (S n)
+                                      (hcons
+                                         (iota (canon alpha (S i))) hnil)).
+ +
+Lemma iota_rw3 :
+  ∀ i alpha, nf alpha →
+                    iota (canon (cons (T1.succ alpha) 0 zero) (S i)) =
+                    node (hcons_mult (iota alpha) (S i) hnil).
+ +
+Lemma iota_rw4 :
+  ∀ i n alpha , nf alpha →
+                       iota (canon (cons (T1.succ alpha) (S n) zero) (S i)) =
+                       node (hcons_mult (iota (T1.succ alpha)) (S n)
+                                        (hcons_mult (iota alpha) (S i) hnil)).
+ +
+Lemma iota_tail :
+  ∀ i alpha n beta,
+    nf (cons alpha n beta) →
+    beta ≠ zero →
+    iota (canon (cons alpha n beta) (S i)) =
+    node (hcons_mult (iota alpha) (S n) (iotas (canon beta (S i)))).
+ +
+Lemma R1_hcons : ∀ h s1 s2, R1 (node s1) (node s2) →
+                                 R1 (node (hcons h s1)) (node (hcons h s2)).
+ +
+Lemma R1_hcons_mult : ∀ n h s1 s2,
+    R1 (node s1) (node s2) →
+    R1 (node (hcons_mult h n s1))
+       (node (hcons_mult h n s2)).
+ +
+Lemma R1_R2 : ∀ h h', R1 h h' → R2 0 (hyd1 h) (hyd1 h').
+ +
+ +
+Inductive mem_head : Hydrae → Prop :=
+  hh_1 : ∀ s, mem_head (hcons head s)
+| hh_2 : ∀ h s, mem_head s → mem_head (hcons h s).
+ +
+Lemma S0_mem_head : ∀ s s', S0 s s' → mem_head s.
+ +
+Lemma mem_head_mult_inv : ∀ n h s, mem_head (hcons_mult h n s) →
+                                        h = head ∨ mem_head s.
+ +
+Lemma R1_mem_head : ∀ l l', R1 (node l) (node l') → mem_head l.
+ +
+Lemma limit_no_head : ∀ alpha, nf alpha → T1limit alpha = true →
+                                    ¬ mem_head (iotas alpha).
+ +
+Lemma limit_no_R1 : ∀ alpha, nf alpha →
+                                  T1limit alpha = true →
+                                  ∀ h', ¬ R1 (iota alpha) h'.
+ +
+Lemma iota_of_succ :
+  ∀ beta, nf beta →
+               iota (T1.succ beta) =
+               match (iota beta) with
+                 node s ⇒ node (hy_app s (hcons head hnil))
+               end.
+ +
+ +
+Lemma canonS_iota_i : ∀ i alpha , nf alpha → alpha ≠ zero →
+                                round_n i (iota alpha) (iota (canon alpha (S i))).
+ +
+ +
+Lemma canonS_iota i a :
+  nf a → a ≠ 0 → iota a -1-> iota (canon a (S i)). +
+  Lemma canonS_rel_rounds : ∀ n alpha beta,
+      nf alpha → nf beta →
+      alpha ≠ zero →
+      beta = canon alpha (S n) →
+      iota alpha -+-> iota beta.
+ +
+Lemma trace_to_round_plus h' : ∀ t h, trace_to h' t h → round_plus h h'.
+ +
+Import MoreLists.
+ +
+Lemma trace_to_std i h j h': (i ≤ j)%nat →
+                             trace_to h' (interval i j) h →
+                             rounds standard i h (S j) h'.
+ +
+Lemma path_toS_trace alpha s beta :
+  path_toS beta s alpha → nf alpha → trace_to (iota beta) s (iota alpha).
+ +
+  Lemma path_toS_round_plus alpha s beta :
+    path_toS beta s alpha → nf alpha →
+    iota alpha -+-> iota beta.
+ +
+ +
+  Lemma path_to_round_plus a s b :
+    path_to b s a → nf a → iota a -+-> iota b. +
+ +
+  Lemma acc_from_to_round_plus alpha beta :
+    nf alpha → nf beta → alpha ≠ 0 →
+    acc_from alpha beta → iota alpha -+-> iota beta.
+ +
+
+ +
+Any strict inequality on T1 can be converted into a (free) battle +
+
+ +
+ +
+  Lemma LT_to_round_plus a b : b t1< a → iota a -+-> iota b.
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.Omega2_Small.html b/theories/html/hydras.Hydra.Omega2_Small.html new file mode 100644 index 00000000..a2afad1f --- /dev/null +++ b/theories/html/hydras.Hydra.Omega2_Small.html @@ -0,0 +1,200 @@ + + + + + +hydras.Hydra.Omega2_Small + + + + +
+ + + +
+ +

Library hydras.Hydra.Omega2_Small

+ +
+
+ +
+Pierre Castéran, University of Bordeaux and LaBRI +
+
+ +
+From Coq Require Import Arith Peano_dec Lia Relations Relation_Operators.
+From hydras Require Import Hydra_Lemmas Simple_LexProd ON_Omega2.
+Import ON_Generic.
+ +
+
+ +
+There is no measure into omega^2 for proving termination +of all hydra battles +
+
+ +
+ +
+Section Impossibility_Proof.
+ +
+
+ +
+Let us assume there is a variant from Hydra into omega^2 + for proving the termination of all hydra battles +
+
+ +
+ +
+  Variable m : Hydra → rep Omega2.
+  Context
+    (Hvar: Hvariant Omega2 free m).
+ +
+ +
+ +
+  Let big_h := hyd1 (hyd2 head head).
+ +
+ +
+
+ +
+To every pair of natural numbers we associate an hydra + with branches of length 2 and branches of length 1 +
+
+ +
+ +
+  Let iota (p: ON_Omega2.t) :=
+    node (hcons_mult (hyd1 head) (fst p)
+                     (hcons_mult head (snd p) hnil)).
+ +
+ +
+ +
+  Let small_h := iota (m big_h).
+ +
+
+ +
+

Proof of the inequality m small_h o< m big_h

+ + +
+
+ +
+  #[local] Hint Constructors R1 S1 S2 : hydra.
+ +
+  Lemma m_big_h_not_null : m big_h ≠ zero.
+ +
+ +
+  Lemma big_to_small : big_h -+-> small_h. +
+ +
+ +
+  Corollary m_lt : m small_h o< m big_h.
+ +
+ +
+
+ +
+

Proof of the inequality m big_h o≤ m small_h

+ +
+ +

Let us decompose any inequality p o< q into elementary steps

+ +
+
+ +
+ +
+  Inductive step : t → t → Prop :=
+  | succ_step : ∀ i j, step (i, S j) (i, j)
+  | limit_step : ∀ i j, step (S i, 0) (i, j).
+ +
+ +
+  Lemma succ_rounds : ∀ i j, iota (i,S j) -+-> iota (i, j).
+ +
+  Lemma limit_rounds_0 :
+    ∀ i j, round_n j (iota (S i, 0)) (iota (i, S j)).
+ +
+  Lemma limit_rounds : ∀ i j, iota (S i, 0) -+-> iota (i, j).
+ +
+  #[local] Hint Constructors step clos_trans_1n : hydra.
+  #[local] Hint Resolve lex_1 lex_2: hydra.
+  #[local] Hint Unfold lt : hydra.
+ +
+ +
+  Lemma step_to_battle : ∀ p q, step p q → iota p -+-> iota q. +
+  #[local] Hint Resolve step_to_battle : hydra.
+ +
+ +
+ +
+  Lemma m_ge : m big_h o≤ m small_h. +
+ +
+ +
+ +
+  Theorem Impossible : False.
+ +
+End Impossibility_Proof.
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Hydra.Omega_Small.html b/theories/html/hydras.Hydra.Omega_Small.html new file mode 100644 index 00000000..ff14d33a --- /dev/null +++ b/theories/html/hydras.Hydra.Omega_Small.html @@ -0,0 +1,147 @@ + + + + + +hydras.Hydra.Omega_Small + + + + +
+ + + +
+ +

Library hydras.Hydra.Omega_Small

+ +
+
+ +
+Pierre Castéran, University of Bordeaux and LaBRI +
+
+ +
+From Coq Require Import Arith Lia.
+From hydras Require Import Hydra_Lemmas ON_Generic ON_Omega.
+Open Scope nat_scope.
+ +
+#[global] Instance height_var : Hvariant Omega free height.
+Abort.
+ +
+Lemma height_bad : ¬ Hvariant Omega free height.
+ +
+
+ +
+There is no measure into omega for proving termination +of all hydra battles +
+
+ +
+ +
+Section Impossibility_Proof.
+ +
+
+ +
+Let us assume there exists a variant from Hydra into nat + for proving the termination of all hydra battles + Omega is an ordinal notation for the least infinite ordinal + omega, whose members are the natural numbers. + +
+
+ +
+  Variable m : Hydra → nat.
+  Context (Hvar : Hvariant Omega free m).
+ +
+ +
+ +
+  Let iota (i: nat) := hyd_mult head (S i).
+ +
+ +
+ +
+  Let big_h := hyd1 (hyd1 head).
+ +
+ +
+ +
+  Let small_h := iota (m big_h).
+ +
+  Fact big_to_small: ∀ i, battle_rel free i big_h small_h. +
+ +
+  #[local] Hint Resolve big_to_small : hydra.
+ +
+ +
+ +
+  Lemma m_lt : m small_h < m big_h.
+ +
+ +
+ +
+
+ +
+ In order to find a contradiction, we prove the inequality + m big_h <= m small_h, i.e. m big_h <= m (iota (m h)) + +
+ + For that purpose, we prove the inequality i <= m (iota i) for any i + +
+
+ +
+  Lemma round_S: ∀ i n, battle_rel free n (iota (S i)) (iota i).
+ +
+  Lemma m_ge : m big_h ≤ m small_h.
+ +
+  Theorem Contradiction : False.
+ +
+End Impossibility_Proof.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.MoreAck.Ack.html b/theories/html/hydras.MoreAck.Ack.html new file mode 100644 index 00000000..4c274707 --- /dev/null +++ b/theories/html/hydras.MoreAck.Ack.html @@ -0,0 +1,420 @@ + + + + + +hydras.MoreAck.Ack + + + + +
+ + + +
+ +

Library hydras.MoreAck.Ack

+ +
+From hydras Require Export Iterates Exp2.
+From Coq Require Import Lia.
+From Coq Require Import Program.Wf Arith.
+From hydras Require Import ssrnat_extracts.
+ +
+
+ +
+The famous Ackermann function +
+ + The following definition fails, because Coq cannot guess a + decreasing argument. + +
+
+ +
+ +
+Fail
+  Fixpoint Ack (m n : nat) : nat :=
+  match m, n with
+  | 0, n ⇒ S n
+  | m.+1, 0 ⇒ Ack m 1
+  | m0.+1, p.+1 ⇒ Ack m0 (Ack m p)
+  end.
+ +
+ +
+
+ +
+Definition (with inner fixpoint) +
+
+ +
+Module Alt.
+ +
+  Fixpoint Ack (m n : nat) : nat :=
+    match m with
+    | O ⇒ n.+1
+    | p.+1 ⇒ let fix Ackm (n : nat) :=
+                 match n with
+                 | O ⇒ Ack p 1
+                 | S q ⇒ Ack p (Ackm q)
+                 end
+             in Ackm n
+    end.
+ +
+ +
+End Alt.
+ +
+
+ +
+Definition using the iterate functional: +
+   iterate : forall {A : Type}, (A -> A) -> nat -> A -> A
+
+ +
+ +Allows to infer monotony properties of Ack (S m) from Ack m. + +
+ + +
+
+ +
+Fixpoint Ack (m:nat) : nat → nat :=
+  match m with
+  | 0 ⇒ S
+  | n.+1 ⇒ fun k ⇒ iterate (Ack n) k.+1 1
+  end.
+ +
+ +
+ +
+
+ +
+Using the lexicographic ordering + (post by Anton Trunov in stackoverflow (May 2018)) +
+
+ +
+   Definition lex_nat (ab1 ab2 : nat × nat) : Prop :=
+    match ab1, ab2 with
+    | (a1, b1), (a2, b2) ⇒
+      (a1 < a2) ∨ ((a1 = a2) ∧ (b1 < b2))
+    end.
+ +
+ +
+
+ +
+this is defined in stdlib, but unfortunately it is opaque +
+
+  Lemma lt_wf_ind :
+    ∀ n (P:nat → Prop), (∀ n, (∀ m, m < n → P m) → P n) → P n.
+ +
+
+ +
+this is defined in stdlib, but unfortunately it is opaque too +
+
+  Lemma lt_wf_double_ind :
+    ∀ P:nat → nat → Prop,
+      (∀ n m,
+          (∀ p (q:nat), p < n → P p q) →
+          (∀ p, p < m → P n p) → P n m) → ∀ n m, P n m.
+ +
+ +
+  Lemma lex_nat_wf : well_founded lex_nat.
+ +
+ +
+ +
+Module Alt2.
+ +
+  Program Fixpoint Ack (ab : nat × nat) {wf lex_nat ab} : nat :=
+    match ab with
+    | (0, b) ⇒ b.+1
+    | (a.+1, 0) ⇒ Ack (a, 1)
+    | (a.+1, b.+1) ⇒ Ack (a, Ack (a.+1, b))
+    end.
+  Example test1 : Ack (1, 2) = 4 := refl_equal.
+ +
+  Example test2 : Ack (3, 4) = 125 := refl_equal.
+ +
+End Alt2.
+ +
+
+ +
+With the Equations plug-in +
+
+ +
+From Equations Require Import Equations.
+ +
+#[ global ] Instance Lex_nat_wf : WellFounded lex_nat.
+Defined.
+ +
+Module Alt3.
+ +
+  Equations ack (p : nat × nat) : nat by wf p lex_nat :=
+    ack (0, n) := n.+1 ;
+    ack (m.+1, 0) := ack (m, 1);
+    ack (m.+1, n.+1) := ack (m, ack (m.+1, n)).
+ +
+End Alt3.
+ +
+
+ +
+

Exercise

+ + +
+ + Prove that the four definitions of the Ackermann function + Ack , Alt.Ack, Alt2.Ack, and Alt3.ack are extensionnally equal + +
+ +

Usual equations

+ +
+
+ +
+ +
+ +
+Lemma Ack_0 : Ack 0 = S.
+ +
+Lemma Ack_S_0 m : Ack m.+1 0 = Ack m 1.
+ +
+Lemma Ack_S_S : ∀ m p,
+    Ack m.+1 p.+1 = Ack m (Ack m.+1 p).
+ +
+ +
+
+ +
+

First values

+ +
+
+ +
+ +
+Lemma Ack_1_n n : Ack 1 n = n.+2. +
+ +
+Lemma Ack_2_n n: Ack 2 n = 2 × n + 3. +
+ +
+Lemma Ack_3_n n: Ack 3 n = exp2 n.+3 - 3. +
+ +
+Lemma Ack_4_n n : Ack 4 n = hyper_exp2 n.+3 - 3. +
+
+ +
+

monotony properties

+ + +
+ + We prove simultaneously 3 properties of Ack n by induction on m: +
    +
  • Ack m is strictly monotonous, + +
    • +
  • Ack m n > n + +
    • +
  • Ack m n ≤ Ack (S m) n + +
    • +
+ +
+
+ +
+Section Ack_Properties.
+ +
+  Let P (m:nat) := strict_mono (Ack m) ∧
+                   S <<= (Ack m) ∧
+                   (∀ n, Ack m n ≤ Ack m.+1 n).
+ +
+  Remark P0 : P 0.
+ +
+  Section Induc_step.
+    Variable m:nat.
+    Hypothesis Hm : P m.
+ +
+    Remark Rem1 : strict_mono (Ack m.+1).
+ +
+    Remark Rem2 : S <<= Ack m.+1.
+ +
+    Remark Ack_m_mono_weak : ∀ n p, n ≤ p →
+                                         Ack m n ≤ Ack m p.
+ +
+    Remark Rem3 : ∀ n, Ack m.+1 n ≤ Ack m.+2 n.
+ +
+    Lemma L5: P m.+1.
+ +
+  End Induc_step.
+ +
+  Lemma Ack_properties : ∀ m, P m.
+ +
+End Ack_Properties.
+ +
+Lemma le_S_Ack m : fun_le S (Ack m).
+ +
+Lemma Ack_strict_mono m : strict_mono (Ack m).
+ +
+Lemma Ack_mono_l m n : m ≤ n → ∀ p, Ack m p ≤ Ack n p.
+ +
+Lemma Ack_mono_r m: ∀ n p, n ≤ p → Ack m n ≤ Ack m p.
+ +
+
+ +
+

Bounding nested calls of Ack

+ + +
+ + The following inequality is applied in the proof that Ack is not primitive + recursive, allowing to eliminate patterns of the form Ack (Ack _ _ _). + +
+ +
+ Lemma nested_Ack_bound : 
+    forall k m n, Ack k (Ack m n) <= Ack (2 + max k m) n.
+
+ +
+
+ +
+Section Proof_of_nested_Ack_bound.
+ +
+  Remark R0 (m:nat): ∀ n, 2 + n ≤ Ack (2 + m) n.
+ +
+  Remark R1 : ∀ m n, Ack m.+1 n.+1 ≤ Ack (2 + m) n.
+ +
+  Remark R2 (m n:nat) : Ack m (Ack m n) ≤ Ack m.+1 n.+1.
+ +
+  Remark R3 (m n:nat) : Ack m (Ack m n) ≤ Ack m.+2 n.
+ +
+ +
+  Lemma nested_Ack_bound k m n :
+    Ack k (Ack m n) ≤ Ack (2 + max k m) n. +
+End Proof_of_nested_Ack_bound.
+ +
+Lemma Ack_Sn_plus : ∀ n p, n.+1 + p < Ack n.+1 p.
+ +
+
+ +
+

Ack is (partially) strictly monotonous in its first argument

+ +
+
+ +
+Remark R5 m n : Ack m n.+1 < Ack m.+1 n.+1.
+ +
+ +
+Lemma Ack_strict_mono_l : ∀ n m p, n < m →
+                                        Ack n p.+1 < Ack m p.+1. +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.MoreAck.AckNotPR.html b/theories/html/hydras.MoreAck.AckNotPR.html new file mode 100644 index 00000000..7314f20e --- /dev/null +++ b/theories/html/hydras.MoreAck.AckNotPR.html @@ -0,0 +1,351 @@ + + + + + +hydras.MoreAck.AckNotPR + + + + +
+ + + +
+ +

Library hydras.MoreAck.AckNotPR

+ +
+
+ +
+

Proof that Ack is not primitive recursive

+ + +
+ + After +planetmath page +and + Bruno Salvy's lecture. + +
+
+ +
+From hydras Require Import primRec Ack MoreVectors.
+From Coq Require Import Arith ArithRing List Lia Compare_dec.
+Import extEqualNat VectorNotations Vector.
+ +
+
+ +
+ Uncurried apply : + v_apply f (x1::x2:: ... ::xn::nil) is f x1 x2 ... xn + +
+ + +
+
+ +
+ +
+Notation "'v_apply' f v" := (evalList _ v f) (at level 10, f at level 9).
+ +
+Check [4].
+Example Ex2 : ∀ (f: naryFunc 2) x y,
+    v_apply f [x;y] = f x y.
+ +
+Example Ex4 : ∀ (f: naryFunc 4) x y z t,
+    v_apply f [x;y;z;t] = f x y z t.
+ +
+ +
+ +
+
+ +
+

Comparing an n-ary and a binary functions

+ +
+
+ +
+Definition majorized {n} (f: naryFunc n) (A: naryFunc 2) :=
+  ∃ (q:nat),
+    ∀ (v: t nat n), v_apply f v ≤ A q (max_v v).
+ +
+Definition majorizedPR {n} (x: PrimRec n) A :=
+  majorized (evalPrimRec n x) A.
+ +
+
+ +
+For vectors of functions +
+
+ +
+Definition majorizedS {n m} (fs : Vector.t (naryFunc n) m)
+           (A : naryFunc 2):=
+  ∃ N, ∀ (v: t nat n),
+      max_v (Vector.map (fun f ⇒ v_apply f v) fs) ≤ A N (max_v v).
+ +
+Definition majorizedSPR {n m} (x : PrimRecs n m) :=
+  majorizedS (evalPrimRecs _ _ x).
+ +
+ +
+Section evalList.
+ +
+ +
+  Lemma evalList_Const : ∀ n (v:t nat n) x,
+    v_apply (evalConstFunc n x) v = x.
+ +
+  Lemma proj_le_max : ∀ n, ∀ v : t nat n, ∀ k (H: k < n),
+          v_apply (evalProjFunc n k H) v ≤ max_v v.
+ +
+  Lemma evalListComp : ∀ n (v: t nat n) m (gs: t (naryFunc n) m)
+                              (h: naryFunc m),
+      v_apply (evalComposeFunc _ _ gs h) v =
+      v_apply h (map (fun g ⇒ v_apply g v) gs).
+  Lemma evalListCompose2 : ∀ n (v: t nat n) (f: naryFunc n)
+                                  (g : naryFunc (S n)),
+      v_apply (compose2 n f g) v =
+      v_apply g ((evalList n v f) :: v).
+ +
+  Lemma evalListPrimrec_0 : ∀ n (v: t nat n) (f : naryFunc n)
+                                   (g : naryFunc (S (S n))),
+      v_apply (evalPrimRecFunc _ f g) (cons 0 v)
+      = v_apply f v.
+ +
+  Lemma evalListPrimrec_S : ∀ n (v: t nat n) (f : naryFunc n)
+                                   (g : naryFunc (S (S n))) a,
+      v_apply (evalPrimRecFunc _ f g) (cons (S a) v)
+      = v_apply g
+                (a :: v_apply (evalPrimRecFunc n f g) (a :: v) :: v).
+ +
+End evalList.
+ +
+
+ +
+

Every primitive recursive function is majorized by Ack

+ +
+ +

Base cases

+ +
+
+ +
+ +
+Lemma majorSucc : majorizedPR succFunc Ack. +
+ +
+Lemma majorZero : majorizedPR zeroFunc Ack. +
+ +
+Lemma majorProjection (n m:nat)(H: m < n): majorizedPR (projFunc n m H) Ack. +
+
+ +
+

The general case is proved by induction on x

+ +
+
+ +
+ +
+ +
+Lemma majorAnyPR: ∀ n (x: PrimRec n), majorizedPR x Ack. +
+
+ +
+Let us specialize Lemma majorAnyPR to unary and binary functions + +
+
+ +
+ +
+Lemma majorPR1 (f: naryFunc 1)(Hf : isPR 1 f)
+  : ∃ (n:nat), ∀ x, f x ≤ Ack n x. +
+ +
+Lemma majorPR2 (f: naryFunc 2)(Hf : isPR 2 f)
+  : ∃ (n:nat), ∀ x y, f x y ≤ Ack n (max x y). +
+ +
+Lemma majorPR2_strict (f: naryFunc 2)(Hf : isPR 2 f):
+    ∃ n:nat,
+      ∀ x y, 2 ≤ x → 2 ≤ y → f x y < Ack n (max x y). +
+
+ +
+

Now, let us assume that Ack is PR

+ +
+
+ +
+ +
+Section Impossibility_Proof.
+ +
+  Context (HAck : isPR 2 Ack).
+ +
+  Lemma Ack_not_PR : False. +
+End Impossibility_Proof.
+ +
+ +
+
+ +
+

Any function which dominates (diagonalized) Ack fails to be PR

+ +
+
+ +
+ +
+ +
+Section dom_AckNotPR.
+ +
+  Variable f : nat → nat.
+  Hypothesis Hf : dominates f (fun n ⇒ Ack n n).
+ +
+  Lemma dom_AckNotPR: isPR 1 f → False.
+ +
+End dom_AckNotPR.
+ +
+ +
+ +
+
+ +
+

Nevertheless, for any n, Ack n is primitive recursive.

+ +
+
+ +
+Lemma AckSn_as_iterate (n:nat) : extEqual 1 (Ack (S n))
+                                          (fun k ⇒ iterate (Ack n) (S k) 1).
+ +
+Lemma AckSn_as_PRiterate (n:nat):
+  extEqual 1 (Ack (S n)) (fun k ⇒ primRec.iterate (Ack n) (S k) 1).
+ +
+Lemma iterate_nat_rec (g: nat→nat) (n:nat) x :
+  iterate g n x = nat_rec (fun _ ⇒ nat) x (fun x y ⇒ g y) n.
+ +
+Section Proof_of_Ackn_PR.
+ +
+  Section S_step.
+    Variable n:nat.
+    Context (IHn: isPR 1 (Ack n)).
+ +
+    Remark R1 : extEqual 1 (Ack (S n))
+                         (fun a : nat ⇒
+                            nat_rec (fun _ : nat ⇒ nat) 1
+                                    (fun _ y : nat ⇒ Ack n y)
+                                    (S a)).
+ +
+    #[local] Instance R2 : isPR 1
+                     (fun a : nat ⇒
+                        nat_rec (fun _ : nat ⇒ nat) 1
+                                (fun _ y : nat ⇒ Ack n y)
+                                (S a)).
+ +
+    #[export] Instance iSPR_Ack_Sn : isPR 1 (Ack (S n)).
+ +
+  End S_step.
+ +
+ +
+ +
+ +
+  #[export] Instance Ackn_IsPR (n: nat) : isPR 1 (Ack n).
+  Proof.
+    induction n.
+ +
+ +
+    - cbn; apply succIsPR.
+    - apply iSPR_Ack_Sn; auto.
+  Qed.
+ +
+End Proof_of_Ackn_PR.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.MoreAck.BadSubst.html b/theories/html/hydras.MoreAck.BadSubst.html new file mode 100644 index 00000000..b713225d --- /dev/null +++ b/theories/html/hydras.MoreAck.BadSubst.html @@ -0,0 +1,98 @@ + + + + + +hydras.MoreAck.BadSubst + + + + +
+ + + +
+ +

Library hydras.MoreAck.BadSubst

+ +
+From hydras Require Import fol folProp folProof Languages folLogic primRec.
+From Coq Require Import Arith.
+ +
+Import FolNotations.
+ +
+
+ +
+

Preliminary lemmas

+ +
+
+ +
+#[local] Arguments Ensembles.In {_} .
+#[local] Arguments Ensembles.Add {_} .
+#[local] Arguments atomic _ _ _ : clear implicits.
+#[local] Arguments apply _ _ _ : clear implicits.
+ +
+Module BadSubst.
+ +
+  Fixpoint substF L (F : Formula L) v (t: Term L) :=
+    match F with
+  | equal t1 t2 ⇒ equal (substT t1 v t) (substT t2 v t)
+  | atomic r s ⇒ atomic L r (substTs s v t)
+  | impH G H ⇒ impH (substF L G v t) (substF L H v t)
+  | notH G ⇒ notH (substF L G v t)
+  | forallH w G ⇒ if Nat.eq_dec w v then F else forallH w (substF L G v t)
+  end.
+ +
+End BadSubst.
+ +
+From Coq Require Import List.
+Import ListNotations.
+ +
+Module BadSubstF2.
+ +
+ +
+End BadSubstF2.
+ +
+From hydras Require Import FolExamples.
+Import Toy.
+ +
+Section BadExample.
+ +
+  Let F := (allH 1, exH 2, v#1 ≠ f v#2)%fol.
+  Let F1: Formula L := (exH 2, v#1 ≠ f v#2)%fol.
+ +
+ +
+End BadExample.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.MoreAck.FolExamples.html b/theories/html/hydras.MoreAck.FolExamples.html new file mode 100644 index 00000000..a41361a8 --- /dev/null +++ b/theories/html/hydras.MoreAck.FolExamples.html @@ -0,0 +1,588 @@ + + + + + +hydras.MoreAck.FolExamples + + + + +
+ + + +
+ +

Library hydras.MoreAck.FolExamples

+ +
+
+ +
+ Use of FOL notations (Experimental) +
+
+ +
+From Coq Require Import Arith Lists.List.
+ +
+From hydras.Ackermann
+  Require Import fol folProp folProof Languages folLogic
+  folLogic2 folLogic3 subAll Deduction primRec.
+Import ListNotations.
+Import FolNotations.
+ +
+
+ +
+

Preliminary lemmas

+ +
+
+ +
+#[local] Arguments Ensembles.In {_} .
+#[local] Arguments Ensembles.Add {_} .
+#[local] Arguments atomic _ _ _ : clear implicits.
+#[local] Arguments apply _ _ _ : clear implicits.
+ +
+About Term_Terms_rec_full.
+About Formula_rect.
+ +
+About Formula_depth_rec.
+ +
+
+ +
+depth-order vs structural order +
+
+ +
+Section depth_rec_demo.
+Variable L: Language.
+Variable P: fol.Formula L → Prop.
+Variable a: fol.Formula L.
+Goal P a.
+Goal P a.
+ +
+End depth_rec_demo.
+Import Ensembles.
+ +
+Lemma In_add1 {T:Type}(a:T)(S: Ensemble T):
+  In (Add S a) a.
+ +
+Lemma In_add2 {T:Type}(a b:T)(S : Ensemble T):
+  In S a → In (Add S b) a.
+ +
+#[local] Hint Unfold mem: core.
+#[local] Hint Resolve In_add1 In_add2 AXM: core.
+ +
+
+ +
+fol_scope allows us to read and write FOL terms and formulas + in a syntax close to Coq's +
+
+ +
+Remark R1 L (t: Term L): (equal t t) = (t = t)%fol.
+ +
+
+ +
+A small variation on MP (without appending contexts) +
+
+ +
+Lemma MPSys L (G: System L) (A B: Formula L) :
+  SysPrf L G (A → B)%fol → SysPrf L G A → SysPrf L G B.
+ +
+
+ +
+A small language +
+
+ +
+Module Toy.
+  Inductive Rel: Set := A_ | B_ | C_ | P_ | Q_ | R_.
+  Inductive Fun : Set := a_ | b_ | f_ | g_ | h_.
+ +
+  Definition arityR (x : Rel): nat :=
+    match x with
+       P_ | Q_ ⇒ 1 | R_ ⇒ 2 | _ ⇒ 0
+    end.
+ +
+ Definition arityF (x : Fun): nat :=
+    match x with f_ | g_ ⇒ 1 | h_ ⇒ 2 | _ ⇒ 0 end.
+ +
+ Definition L := language Rel Fun arityR arityF.
+ +
+Notation a := (apply L a_ Tnil).
+Notation b := (apply L b_ Tnil).
+Notation f t := (apply L f_ (Tcons t Tnil)).
+Notation g t := (apply L g_ (Tcons t Tnil)).
+Notation h t1 t2 := (apply L h_ (Tcons t1 (Tcons t2 Tnil))).
+ +
+Example t0 : Term L := a.
+ +
+Example t1 : Term L := f t0.
+ +
+Example t2 : Term L := h t1 t0.
+ +
+Example t3 : Term L := h (f (var 0)) (g (var 1)).
+ +
+ +
+Goal t0 = a.
+Goal t1 = f a.
+Goal t2 = h (f a) a.
+ +
+ +
+
+ +
+Abreviations for the toy language L +
+
+ +
+  Notation A := (atomic L A_ Tnil).
+  Notation B := (atomic L B_ Tnil).
+  Notation C := (atomic L C_ Tnil).
+  Notation P t := (atomic L P_ (Tcons t Tnil)).
+  Notation Q t := (atomic L Q_ (Tcons t Tnil)).
+  Notation R t1 t2 := (@atomic L R_ (Tcons t1 (Tcons t2 Tnil))).
+ +
+Example F1 : Formula L := R a b.
+ +
+Example F2 : Formula L :=
+  forallH 0 (forallH 1
+               (impH (R (var 0) (var 1)) (R (var 1) (var 0)))).
+ +
+Example F3 : Formula L :=
+  forallH 0 (orH (equal (var 0) a)
+               (existH 1 (equal (var 0) (f (var 1))))).
+ +
+Example F4: Formula L :=
+  orH (forallH 1 (equal (var 0) (var 1)))
+    (existH 0 (existH 1 (notH (equal (var 0) (var 1))))).
+ +
+Example F5: Formula L := (v#0 = a ∨ v#0 = f v#1)%fol.
+ +
+Example F6: Formula L:= (allH 0, exH 1, v#0 = f v#1 ∧ v#0 ≠ v#1)%fol.
+ +
+ +
+ +
+ +
+ +
+Print F1.
+ +
+Print F2.
+ +
+Print F3.
+ +
+ +
+
+ +
+The following computation expands some derived connectives and + quantifiers. Within fol_scope, we print them with a + similar syntax (with primed symbols) +
+
+ +
+Section PrimedSymbols.
+ +
+ +
+Goal (F3 ∧ F1)%fol = (~(~ ¬ F3 → ¬ F1))%fol.
+ +
+Print F6.
+ +
+ +
+#[local] Unset Printing Notations.
+Print F6.
+ +
+End PrimedSymbols.
+ +
+Goal forallH 1 (equal (var 1) a) ≠ forallH 0 (equal (var 0) a).
+ +
+
+ +
+a +
+
+Goal apply L a_ Tnil = a.
+ +
+
+ +
+f a +
+
+Goal apply L f_ (Tcons (apply L a_ Tnil) Tnil) = f a.
+ +
+
+ +
+f (f v1) +
+
+Goal apply L f_
+         (Tcons (apply L f_
+                       (Tcons (var 1) Tnil))
+            Tnil ) = (f (f (var 1))).
+ +
+ +
+Definition Ldec : language_decidable L.
+ +
+
+ +
+Formula_eqdec is opaque ! +
+
+ +
+Check (f a)%fol.
+ +
+Goal lt_depth L (v#0 = v#1 ∨ exH 2, v#1 = f v#2)%fol
+                (v#0 = v#1 ∧ exH 2, v#1 = f v#2)%fol. +
+Goal lt_depth _ F1 F2.
+ +
+ +
+Check subAllFormula.
+ +
+ +
+ +
+Section OnSubstF.
+  Let F : Formula L := (exH 2, v#1 ≠ f v#2)%fol.
+ +
+ +
+End OnSubstF.
+ +
+Example PrfEx1: Prf L [ (A → B → C)%fol] (A → B → C)%fol.
+ +
+Lemma PrfEx2: Prf L [A → B → C; A; A → B; A]%fol C.
+ +
+Print PrfEx2.
+ +
+Lemma MP' f g H1 H2 H: H = H1 ++ H2 → Prf L H1 (f → g)%fol →
+                        Prf L H2 f → Prf L H g.
+ +
+
+ +
+Cuts the current list of hypotheses as (G++?H), then applies MP +
+
+Ltac cutMP G :=
+  match goal with
+ |- Prf ?L ?H ?F ⇒ eapply MP' with (H1 := G);
+[simpl; reflexivity | try apply AXM | try apply AXM ] end.
+ +
+Example PrfEx2': Prf L [A → B → C; A; A → B; A]%fol C. Section ProofOfEx3.
+ +
+#[local] Arguments MP {L Hyp1 Hyp2 A B} _ _.
+ +
+Example PrfEx3 : Prf L [] (A → A)%fol. +
+Print PrfEx3.
+ +
+End ProofOfEx3.
+ +
+
+ +
+Rule of contradiction +
+
+ +
+Example PrfEx4 (A B: Formula L): Prf L [] (¬B → B → A)%fol.
+ +
+ +
+About PrfEx4.
+ +
+
+ +
+

Universal quantifier

+ +
+
+ +
+Example PrfEx5 : Prf L [] ((allH 1 2, R v#1 v#2) → allH 2, R a v#2)%fol.
+ +
+Example PrfEx6 : Prf L [] (R v#1 v#1 → allH 0, R v#1 v#1)%fol.
+ +
+Example PrfContrex7 :
+  Prf L [] (R v#1 v#1 → allH 1, R v#1 v#1)%fol. +
+Example PrfEx8 : Prf L [] ((allH 0, P v#0 → Q v#0) →
+                 (allH 0, P v#0) →
+                 (allH 0, Q v#0))%fol.
+ +
+ +
+Lemma eq_refl (t:Term L): Prf L nil (t = t)%fol.
+ +
+About EQ4.
+Check @EQ4 L (R_).
+ +
+ +
+Example PrfEx9: Prf L [] (v#0 = v#1 → P v#0 ↔ P v#1)%fol. +
+ +
+Example PrfEx10:
+  Prf L [] (v#2 = v#3 → v#0 = v#1 → R v#2 v#0 ↔ R v#3 v#1)%fol. +
+ +
+Example PrfContrex9: Prf L [] (v#1 = v#0 → P v#1 ↔ P v#0)%fol. +
+ +
+Example PrfEx11:
+  Prf L [] (v#2 = v#3 → v#0 = v#1 → h v#2 v#0 = h v#3 v#1)%fol. +
+ +
+ +
+Lemma ded1: ∀ (A: Formula L), (SysPrf L (Empty_set _) A) →
+                            ∃ pf : Prf L (nil) A, True.
+ +
+Lemma ded2: SysPrf L (Empty_set _) (A → A)%fol.
+Search SysPrf.
+Search Add.
+Qed.
+ +
+Lemma ded3 : ∃ pf : Prf L (nil) (A → A)%fol , True.
+ +
+#[local] Arguments Ensembles.In {_} .
+#[local] Arguments Ensembles.Add {_} .
+Import Ensembles.
+ +
+#[local] Hint Unfold mem: sets.
+ +
+#[local] Hint Resolve In_add1 In_add2: sets.
+ +
+Fixpoint Adds {A:Type}(X: Ensemble A)(l: list A) :=
+  match l with
+    nil ⇒ X
+  | x::l ⇒ Add (Adds X l) x
+  end.
+ +
+Example SysPrfEx2 : SysPrf L
+                      (fun x ⇒ List.In x [A; A→B; A → B → C]%fol)
+                      C. +
+ +
+Search SysPrf (?A ∧ ?B)%fol notH.
+ +
+Search (SysPrf ?L ?T (?A ∧ ?B)%fol → SysPrf ?L ?T ?B).
+ +
+Search (SysPrf _ _ (¬ ¬ _)%fol).
+ +
+Search SysPrf (?a = ?b )%fol substF.
+ +
+Search SysPrf (exH ?v, _)%fol (allH ?v, _)%fol.
+ +
+Section PeirceProof.
+Arguments Add {U}.
+Arguments Empty_set {U}.
+ +
+Definition Peirce : Formula L := (((A → B) → A) → A)%fol.
+ +
+Lemma peirce : SysPrf L Empty_set Peirce.
+End PeirceProof.
+ +
+Section Drinkers_theorem.
+ +
+ Lemma D0 : ∀ i,
+      SysPrf _ (Empty_set _)
+        ( ¬ forallH i (P (v#i)) → exH i, (¬ (P (v#i))))%fol.
+ +
+  Lemma D01 T i : SysPrf _ T
+                    (¬ forallH i (P (v#i)) →
+                      exH i, (¬ (P (v#i))))%fol.
+ +
+  Let f : Formula L :=
+        (exH 0, (P (v#0) → forallH 1 (P (v#1))))%fol.
+ +
+  Theorem drinkers_thm : SysPrf L (Empty_set _) f.
+ +
+End Drinkers_theorem.
+ +
+End Toy.
+ +
+ +
+ +
+ +
+
+ +
+v1 + 0 +
+
+Example t1_0: Term LNN :=
+ apply LNN Plus_
+   (Tcons (var 1)
+     (Tcons (apply LNN Zero_ Tnil) Tnil )).
+Print t1_0.
+ +
+
+ +
+forall v0, v0 = 0 \/ exists v1, v0 = S v1 +
+
+Example f1 : Formula LNN :=
+  forallH 0
+    (orH (equal (var 0) (apply LNN Zero_ Tnil ))
+       (existH 1 (equal (var 0)
+                    (apply LNN Succ_
+                       (Tcons (var 1) Tnil))))).
+ +
+Example f2 : Formula LNN :=
+  (existH 1 (equal (var 0)
+               (apply LNN Succ_
+                  (Tcons (var 1) Tnil )))).
+ +
+Example f3 := (orH (equal (var 0) (apply LNN Zero_ Tnil))
+                 (existH 1 (equal (var 0)
+                              (apply LNN Succ_
+                                 (Tcons (var 1) Tnil))))).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.MoreAck.Iterate_compat.html b/theories/html/hydras.MoreAck.Iterate_compat.html new file mode 100644 index 00000000..dccaeffd --- /dev/null +++ b/theories/html/hydras.MoreAck.Iterate_compat.html @@ -0,0 +1,54 @@ + + + + + +hydras.MoreAck.Iterate_compat + + + + +
+ + + +
+ +

Library hydras.MoreAck.Iterate_compat

+ +
+
+ +
+Compatibility between primRec.iterate and Iterates.iterate +
+
+ +
+From hydras Require primRec Iterates.
+From hydras.Ackermann Require Import extEqualNat.
+ +
+ +
+Lemma iterate_compat (f: nat → nat) n x :
+  Iterates.iterate f n x = primRec.iterate f n x.
+ +
+Lemma iterate_extEqual (f: nat → nat) n :
+  extEqual 1 (Iterates.iterate f n) (primRec.iterate f n).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.MoreAck.LNN_Examples.html b/theories/html/hydras.MoreAck.LNN_Examples.html new file mode 100644 index 00000000..3232e619 --- /dev/null +++ b/theories/html/hydras.MoreAck.LNN_Examples.html @@ -0,0 +1,170 @@ + + + + + +hydras.MoreAck.LNN_Examples + + + + +
+ + + +
+ +

Library hydras.MoreAck.LNN_Examples

+ +
+
+ +
+Experimental +
+
+ +
+From Coq Require Import Arith Lists.List.
+Require Import fol folProp Languages LNN folProof.
+Import FolNotations.
+Import NNnotations.
+ +
+Section bare_syntax.
+ +
+Definition f0 : Formula LNN :=
+      forallH 0
+        (orH
+           (equal (var 0) Zero)
+           (existH 1 (equal (var 0)
+                          (apply
+                             (Languages.Succ_ : Functions LNN)
+                             (Tcons (var 1) (@Tnil _)))))).
+ +
+Import LNN.
+Print f0.
+Check Zero.
+Set Printing All.
+Unset Printing All.
+ +
+Print f0.
+End bare_syntax.
+ +
+Print f0.
+Goal f0 = (allH 0, v#0 = Zero ∨ exH 1, v#0 = Succ v#1)%fol.
+ +
+Locate Plus.
+Locate "_ + _".
+ Example t1_0 : Term _ := Plus (S_ (var 1))%fol Zero.
+Print t1_0.
+Check S_ Zero.
+ +
+Print t1_0.
+ +
+Section Examples.
+ +
+Let t1: Term LNN := Plus (var 1) Zero.
+ +
+ +
+
+ +
+forall v0, v0 = 0 \/ exists v1, v0 = S v1 +
+
+Let f1 : Formula LNN :=
+  (forallH 0
+    (v#0 = Zero ∨
+          existH 1 (v#0 = Succ (v#1))))%fol.
+Print f1.
+ +
+ +
+ +
+Print Relations.
+ +
+Let f2 : Formula LNN :=
+   (existH 2 (LT Zero (v#2) ∧ natToTerm 4 = Plus (v#2) (v#2)))%fol.
+ +
+Let f2' : Formula LNN :=
+   (existH 2 (Zero < v#2 ∧ natToTerm 4 = Plus (v#2) (v#2)))%fol.
+ +
+Let f3 := (v#0 = Zero ∨ existH 1 (v#0 = Succ (v#1)))%fol.
+ +
+Let f4 := (v#0 = v#1 + v#1 ↔ v#0 = v#1 × (natToTerm 2))%fol.
+ +
+Print f4.
+ +
+Check (Plus Zero Zero)%fol.
+ +
+ +
+ +
+ +
+ +
+ +
+Locate LT.
+ +
+End Examples.
+ +
+ +
+ +
+ +
+ +
+Check GEN LNN nil (v#0 = v#0)%fol 1.
+ +
+ +
+ +
+ +
+Goal Prf LNN nil
+         (forallH 0 (v#0 = v#0))%fol →
+       Prf LNN nil (Zero = Zero)%fol.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.MoreAck.PrimRecExamples.html b/theories/html/hydras.MoreAck.PrimRecExamples.html new file mode 100644 index 00000000..bf9edd1a --- /dev/null +++ b/theories/html/hydras.MoreAck.PrimRecExamples.html @@ -0,0 +1,481 @@ + + + + + +hydras.MoreAck.PrimRecExamples + + + + +
+ + + +
+ +

Library hydras.MoreAck.PrimRecExamples

+ +
+From Coq Require Import Arith ArithRing List Vector Utf8.
+Import VectorNotations.
+ +
+From hydras Require Import primRec cPair.
+Import extEqualNat PRNotations.
+ +
+ +
+Check leBool : naryRel 2.
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+Check plus: naryFunc 2.
+ +
+Check 42: naryFunc 0.
+ +
+Check (fun n p q : nat ⇒ n × p + q): naryFunc 3.
+ +
+ +
+ +
+ +
+Example extEqual_ex1: (Nat.mul: naryFunc 2) =x= fun x y ⇒ y × x + x - x. +
+
+ +
+

Examples of terms of type PrimRec n and their interpretation

+ +
+
+ +
+ +
+Example Ex1 : PReval zeroFunc = 0.
+ +
+Example Ex2 a : PReval succFunc a = a.+1.
+ +
+Example Ex3 a b c d e f: ∀ (H: 2 < 6),
+    PReval (projFunc 6 2 H) a b c d e f = d.
+ +
+Example Ex4 : extEqual 4 (fun a b c d: nat ⇒ 0)
+                         (PReval (PRcomp zeroFunc (PRnil _))).
+ +
+Section Composition.
+ +
+  Variables (x y z : PrimRec 2) (t: PrimRec 3).
+  Let u : PrimRec 2 := composeFunc 2 3 [x; y; z]%pr t.
+  Let f := PReval x.
+  Let g := PReval y.
+  Let h := PReval z.
+  Let k := PReval t.
+ Goal ∀ n p, PReval u n p = k (f n p) (g n p) (h n p).
+ +
+  Let v := (PRcomp t [ pi1_2 ; pi1_2; pi2_2 ])%pr.
+ +
+  Eval simpl in PReval v.
+ +
+End Composition.
+ +
+Section Primitive_recursion.
+ Variables (x: PrimRec 2)(y: PrimRec 4).
+ Let z := (primRecFunc _ x y).
+ Let g := PReval x.
+ Let h := PReval y.
+ Let f := PReval z.
+ Goal ∀ a b, f 0 a b = g a b.
+ +
+ Goal ∀ n a b, f n.+1 a b = h n (f n a b) a b.
+ +
+End Primitive_recursion.
+ +
+ +
+Section compose2Examples.
+Variables (f: naryFunc 1) (g: naryFunc 2) (h: naryFunc 3)
+  (f': naryFunc 4) (g': naryFunc 5).
+ +
+Eval simpl in compose2 1 f g.
+ +
+Eval simpl in compose2 2 g h.
+ +
+Eval simpl in compose2 _ f' g'.
+End compose2Examples.
+ +
+Module MoreExamples.
+ +
+
+ +
+The unary constant function which returns 0 +
+
+Definition cst0 : PrimRec 1 := (PRcomp zeroFunc (PRnil _))%pr.
+ +
+
+ +
+The unary constant function which returns i +
+
+Fixpoint cst (i: nat) : PrimRec 1 :=
+  match i with
+    0 ⇒ cst0
+  | S j ⇒ (PRcomp succFunc [cst j])%pr
+end.
+ +
+ +
+
+ +
+Addition +
+
+Definition plus : PrimRec 2 :=
+  (PRrec pi1_1 (PRcomp succFunc [pi2_3]))%pr.
+ +
+
+ +
+Multiplication +
+
+Definition mult : PrimRec 2 :=
+  PRrec cst0
+    (PRcomp plus [pi2_3; pi3_3]%pr).
+ +
+
+ +
+Factorial function +
+
+Definition fact : PrimRec 1 :=
+  (PRrec
+     (PRcomp succFunc [zeroFunc])
+     (PRcomp mult [pi2_2; PRcomp succFunc [pi1_2]]))%pr.
+ +
+End MoreExamples.
+ +
+Import MoreExamples.
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+Lemma cst0_correct : (PReval cst0) =x= (fun _ ⇒ 0).
+ +
+Lemma cst_correct (k:nat) : PReval (cst k) =x= (fun _ ⇒ k).
+ +
+Lemma plus_correct:
+  PReval plus =x= Nat.add.
+ +
+Remark mult_eqn1 n p:
+    PReval mult (S n) p =
+      PReval plus (PReval mult n p) p.
+ +
+Lemma mult_correct: PReval mult =x= Nat.mul.
+ +
+Lemma fact_correct : PReval fact =x= Coq.Arith.Factorial.fact.
+ +
+
+ +
+

Understanding some constructions ...

+ + +
+ + These lemmas are trivial and theoretically useless, but they may help to + make the construction more concrete +
+
+ +
+Definition PRcompose1 (g f : PrimRec 1) : PrimRec 1 :=
+  PRcomp g [f]%pr.
+ +
+Goal ∀ f g x, evalPrimRec 1 (PRcompose1 g f) x =
+                     evalPrimRec 1 g (evalPrimRec 1 f x).
+ +
+Remark compose2_0 (a:nat) (g: nat → nat) : compose2 0 a g = g a.
+ +
+Remark compose2_1 (f: nat → nat) (g : nat → nat → nat) x
+  : compose2 1 f g x = g (f x) x.
+ +
+Remark compose2_2 (f: naryFunc 2) (g : naryFunc 3) x y
+  : compose2 2 f g x y = g (f x y) x y.
+ +
+Remark compose2_3 (f: naryFunc 3) (g : naryFunc 4) x y z
+  : compose2 3 f g x y z = g (f x y z) x y z.
+ +
+Remark PrimRec_0_0 (a:nat)(g : nat → nat → nat) :
+  evalPrimRecFunc 0 a g 0 = a.
+ +
+Remark PrimRec_0_S (a : nat) (g : nat → nat → nat) (i:nat):
+  let phi := evalPrimRecFunc 0 a g
+  in phi (S i) = g i (phi i).
+ +
+Remark PrimRec_1_0 (f : nat→nat)(g : nat → nat → nat→ nat) :
+  ∀ x, evalPrimRecFunc 1 f g 0 x = f x.
+ +
+Remark PrimRec_1_S (f: nat→nat)
+       (g : nat → nat → nat → nat) (i:nat):
+  let phi := evalPrimRecFunc 1 f g
+  in ∀ x, phi (S i) x = g i (phi i x) x.
+ +
+Remark PrimRec_2_0 (f : naryFunc 2) (g : naryFunc 4) :
+  ∀ x y, evalPrimRecFunc 2 f g 0 x y = f x y.
+ +
+Remark PrimRec_2_S (f: naryFunc 2) (g : naryFunc 4) (i:nat) :
+  let phi := evalPrimRecFunc 2 f g
+  in ∀ x y, phi (S i) x y = g i (phi i x y) x y.
+ +
+
+ +
+

First proofs of isPR statements

+ + +
+ + The module Alt presents proofs of lemmas already proven in primRec.v + We just hope that such a redundancy will help the reader to get familiar + with the various patterns allowed by that library. + +
+
+ +
+Module Alt.
+ +
+ +
+#[export] Instance zeroIsPR : isPR 0 0. +
+ +
+ +
+ +
+#[export] Instance succIsPR : isPR 1 S.
+ +
+#[export] Instance addIsPR : isPR 2 Nat.add.
+ +
+ +
+ +
+ +
+ +
+#[export] Instance pi2_5IsPR : isPR 5 (fun a b c d e ⇒ b).
+ +
+ +
+ +
+Check composeFunc 0 1.
+ +
+ +
+ +
+Remark compose_01 (x:PrimRec 0) (t : PrimRec 1) :
+    PReval (PRcomp t [x])%pr = PReval t (PReval x).
+ +
+ +
+ +
+#[export] Instance const0_NIsPR n : isPR 0 n. +
+Search (isPR 2 (fun _ _ ⇒ nat_rec _ _ _ _)).
+ +
+Check isPRextEqual.
+ +
+ +
+ +
+Check filter010IsPR.
+ +
+ +
+ +
+Definition add' x y :=
+  nat_rec (fun n : nat ⇒ nat)
+    y
+    (fun z t ⇒ S t)
+    x.
+ +
+Lemma add'_ok:
+  extEqual 2 add' Nat.add.
+ +
+ +
+#[export] Instance addIsPR' : isPR 2 Nat.add. +
+ +
+Definition xpred := primRecFunc 0 zeroFunc pi1_2.
+ +
+ +
+#[export] Instance predIsPR : isPR 1 Nat.pred. +
+End Alt.
+ +
+ +
+Definition double (n:nat) := 2 × n.
+ +
+#[export] Instance doubleIsPR : isPR 1 double. +
+
+ +
+using cPair +
+
+ +
+ +
+Section Exs.
+Let f: naryFunc 2 := fun x y ⇒ x + pred (cPairPi1 y).
+ +
+ +
+  Let ffib : naryFunc 2 :=
+        fun c A ⇒
+          match c with
+            0 | 1 ⇒ 1
+          | _ ⇒ codeNth 0 A + codeNth 1 A
+          end.
+  Let fdiv2 : naryFunc 2 :=
+        fun (n acc: nat) ⇒
+          match n with
+            0 | 1 ⇒ 0
+          | _ ⇒ S (codeNth 1 acc)
+          end.
+ +
+ +
+ +
+ +
+ +
+End Exs.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.MoreAck.expressibleExamples.html b/theories/html/hydras.MoreAck.expressibleExamples.html new file mode 100644 index 00000000..4d3ad03c --- /dev/null +++ b/theories/html/hydras.MoreAck.expressibleExamples.html @@ -0,0 +1,89 @@ + + + + + +hydras.MoreAck.expressibleExamples + + + + +
+ + + +
+ +

Library hydras.MoreAck.expressibleExamples

+ +
+From hydras.Ackermann Require Import expressible.
+ +
+From hydras.Ackermann Require Import NN NewNotations NNtheory.
+Import NNnotations.
+From Coq Require Import Lia.
+ +
+Goal ∀ n, RepresentableHalf1 NN 0 n (v#0 = natToTerm n)%fol.
+ +
+Lemma L1: RepresentableHalf1 NN 1 (fun n ⇒ n + n)
+  (v#0 = v#1 + v#1)%fol.
+ +
+Lemma L2: RepresentableHalf2 NN 1 (fun n ⇒ n + n)
+  (v#0 = v#1 + v#1)%fol.
+ +
+Lemma L3: Representable NN 1 (fun n ⇒ n + n) (v#0 = v#1 + v#1)%fol.
+ +
+Lemma L4 : Representable NN 1 (fun n ⇒ n × 2) (v#0 = v#1 + v#1)%fol.
+ +
+
+ +
+Mind the order of variables v#1 ... v#n + f v#n ... v#1 = v#0 + +
+ + +
+ + v#2 is bound to n +
+
+ +
+Lemma L5: RepresentableHalf1 NN 2 (fun n p ⇒ n) (v#0 = v#2)%fol.
+ Search substT .     Search (SysPrf _ (?A → ?A)%fol).
+   Search freeVarT (S_ _).
+ +
+Qed.
+ +
+
+ +
+v#1 is bound to p +
+
+Lemma L6: RepresentableHalf1 NN 2 (fun n p ⇒ p) (v#0 = v#1)%fol.
+ Search substT .
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.OrdinalNotations.Example_3PlusOmega.html b/theories/html/hydras.OrdinalNotations.Example_3PlusOmega.html new file mode 100644 index 00000000..693cd941 --- /dev/null +++ b/theories/html/hydras.OrdinalNotations.Example_3PlusOmega.html @@ -0,0 +1,70 @@ + + + + + +hydras.OrdinalNotations.Example_3PlusOmega + + + + +
+ + + +
+ +

Library hydras.OrdinalNotations.Example_3PlusOmega

+ +
+
+ +
+A proof of isomorhism between ordinal notations for 3+omega and omega +
+ + Pierre Castéran, Univ. Bordeaux and LaBRI +
+
+ +
+From hydras Require Import ON_plus ON_Finite ON_Omega.
+Import ON_Generic.
+From Coq Require Import Compare_dec Lia Logic.Eqdep_dec.
+ +
+Definition O3O := ON_plus (FinOrd 3) Omega.
+#[ global ] Existing Instance O3O.
+ +
+Arguments ON_t : clear implicits.
+Arguments ON_t {A lt cmp} _.
+ +
+Program Definition f (z: ON_t O3O) : ON_t Omega :=
+  match z with inl i ⇒ i | inr j ⇒ 3+j end.
+ +
+Program Definition g (a : ON_t Omega) : ON_t O3O :=
+  match (le_lt_dec 3 a) with
+    left _ ⇒ inr (a - 3)
+  | right _ ⇒ inl a
+  end.
+ +
+#[ global ] Instance L_3_plus_omega : ON_Iso _ _ f g.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.OrdinalNotations.ON_Finite.html b/theories/html/hydras.OrdinalNotations.ON_Finite.html new file mode 100644 index 00000000..fb6ee64d --- /dev/null +++ b/theories/html/hydras.OrdinalNotations.ON_Finite.html @@ -0,0 +1,185 @@ + + + + + +hydras.OrdinalNotations.ON_Finite + + + + +
+ + + +
+ +

Library hydras.OrdinalNotations.ON_Finite

+ +
+
+ +
+ A notation system for finite ordinals Pierre Castéran, Univ. Bordeaux and LaBRI +
+
+ +
+From Coq Require Import Arith Relations Lia Logic.Eqdep_dec Ensembles
+        Wellfounded.Inverse_Image Wellfounded.Inclusion
+        RelationClasses.
+From hydras Require Import ON_Generic.
+ +
+From Coq Require Wf_nat.
+ +
+Coercion is_true: bool >-> Sortclass.
+ +
+ +
+Definition t (n:nat) := {i:nat | Nat.ltb i n}.
+ +
+Definition lt {n:nat} : relation (t n) :=
+  fun alpha beta ⇒ Nat.ltb (proj1_sig alpha) (proj1_sig beta).
+ +
+ +
+Open Scope ON_scope.
+ +
+ +
+Lemma t0_empty (alpha: t 0): False.
+ +
+ +
+Definition bad : t 10. Abort.
+ +
+ +
+#[global] Instance compare_fin {n:nat} : Compare (t n) :=
+  fun (alpha beta : t n) ⇒ Nat.compare (proj1_sig alpha) (proj1_sig beta).
+ +
+Lemma compare_correct {n} (alpha beta : t n) :
+  CompSpec eq lt alpha beta (compare alpha beta).
+ +
+Lemma compare_reflect {n:nat} (alpha beta : t n) :
+  match (compare alpha beta)
+  with
+    Lt ⇒ lt alpha beta
+  | Eq ⇒ alpha = beta
+  | Gt ⇒ lt beta alpha
+  end.
+ +
+ +
+Lemma lt_wf (n:nat) : well_founded (@lt n). +
+ +
+#[global] Instance sto n : StrictOrder (@lt n). +
+ +
+#[global] Instance comp n: Comparable (@lt n) compare. +
+#[global] Instance FinOrd n : ON (@lt n) compare. +
+ +
+Definition Zero_limit_succ_dec (n:nat) : ZeroLimitSucc_dec (on := FinOrd n).
+ +
+Lemma sig_eq_intro {n:nat} (x y : t n) :
+  proj1_sig x = proj1_sig y → x = y.
+ +
+ +
+Section Inclusion_ij.
+ +
+  Variables i j : nat.
+  Hypothesis Hij : i < j.
+ +
+  Remark Ltb_ij : Nat.ltb i j.
+ +
+ +
+  #[program] Definition iota_ij (alpha: t i) : t j := alpha.
+ +
+ +
+   Let b : t j := exist _ i Ltb_ij.
+ +
+  #[global]
+   Instance F_incl_ij: SubON (FinOrd i) (FinOrd j) b iota_ij.
+ +
+  Lemma iota_compare_commute alpha beta:
+    compare alpha beta =
+    compare (iota_ij alpha) (iota_ij beta).
+ +
+  Lemma iota_mono : ∀ alpha beta,
+      lt alpha beta ↔
+      lt (iota_ij alpha) (iota_ij beta).
+ +
+End Inclusion_ij.
+ +
+Arguments iota_ij {i j}.
+ +
+ +
+Program Example alpha1 : t 7 := 2.
+ +
+Program Example beta1 : t 7 := 5.
+ +
+Example i1 : lt alpha1 beta1. +
+ +
+ +
+Program Example gamma1 : t 8 := 7.
+ +
+ +
+ +
+Example i2 : lt (iota_ij (le_n 8) alpha1) gamma1.
+ +
+Example Ex1 : In (bigO beta1) alpha1.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.OrdinalNotations.ON_Generic.html b/theories/html/hydras.OrdinalNotations.ON_Generic.html new file mode 100644 index 00000000..a6a5d501 --- /dev/null +++ b/theories/html/hydras.OrdinalNotations.ON_Generic.html @@ -0,0 +1,360 @@ + + + + + +hydras.OrdinalNotations.ON_Generic + + + + +
+ + + +
+ +

Library hydras.OrdinalNotations.ON_Generic

+ +
+
+ +
+Pierre Castéran, Univ. Bordeaux and LaBRI +
+ + This module defines a type class of ordinal notations, i.e._, data types + with a well-founded strict order and a compare function +
+
+ +
+From Coq Require Import RelationClasses Relation_Operators Ensembles.
+From hydras Require Import OrdNotations Schutte_basics.
+From Coq Require Export Wellfounded.Inverse_Image Wellfounded.Inclusion.
+Import Relation_Definitions.
+From hydras Require Export MoreOrders.
+From hydras Require Export Comparable.
+ +
+Generalizable All Variables.
+Declare Scope ON_scope.
+Delimit Scope ON_scope with on.
+#[local] Open Scope ON_scope.
+ +
+
+ +
+ Ordinal notation system on type A : + +
+
+ +
+Class ON {A:Type} (lt: relation A) (cmp: Compare A) :=
+  {
+  ON_comp :> Comparable lt cmp;
+  ON_wf : well_founded lt;
+  }.
+ +
+#[global] Existing Instance ON_comp.
+Coercion ON_wf : ON >-> well_founded.
+ +
+Definition rep {A:Type} {lt: relation A} {cmp: Compare A}
+                (on : ON lt cmp) := A.
+ +
+#[global] Coercion rep : ON >-> Sortclass.
+ +
+Section Definitions.
+ +
+  Context {A:Type} {lt : relation A} {cmp : Compare A} {on: ON lt cmp}.
+ +
+  #[using="All"] Definition ON_t := A.
+ +
+  #[using="All"] Definition ON_compare : A → A → comparison := compare.
+ +
+  #[using="All"] Definition ON_lt := lt.
+ +
+  #[using="All"] Definition ON_le: relation A := leq lt.
+ +
+  #[using="All"]
+    Definition measure_lt {B : Type} (m : B → A) : relation B :=
+    fun x y ⇒ ON_lt (m x) (m y).
+ +
+  #[using="All"]
+    Lemma wf_measure {B : Type} (m : B → A) :
+    well_founded (measure_lt m).
+ +
+ +
+  #[using="All"]
+   Definition ZeroLimitSucc_dec :=
+    ∀ alpha,
+      {Least alpha} +
+      {Limit alpha} +
+      {beta: A | Successor alpha beta}.
+ +
+
+ +
+The segment called O alpha in Schutte's book +
+
+ +
+  #[using="All"]
+   Definition bigO (a: A) : Ensemble A := fun x: A ⇒ lt x a.
+ +
+End Definitions.
+ +
+Infix "o<" := ON_lt : ON_scope.
+Infix "o<=" := ON_le : ON_scope.
+Infix "o?=" := ON_compare (at level 70) : ON_scope.
+ +
+ +
+#[global] Hint Resolve wf_measure : core.
+ +
+
+ +
+The segment associated with nA is isomorphic to + the segment of ordinals strictly less than b +
+
+ +
+ +
+Class SubON
+       `(OA: @ON A ltA compareA)
+       `(OB: @ON B ltB compareB)
+       (alpha: B) (iota: A → B):=
+  {
+    SubON_compare: ∀ x y : A,
+      compareB (iota x) (iota y) = compareA x y;
+    SubON_incl : ∀ x, ltB (iota x) alpha;
+    SubON_onto : ∀ y, ltB y alpha → ∃ x:A, iota x = y}.
+ +
+ +
+
+ +
+OA and OB are order-isomporphic +
+
+ +
+ +
+Class ON_Iso
+       `(OA : @ON A ltA compareA)
+       `(OB : @ON B ltB compareB)
+       (f : A → B) (g : B → A):=
+  {
+    iso_compare :∀ x y : A,
+      compareB (f x) (f y) = compareA x y;
+  iso_inv1 : ∀ a, g (f a)= a;
+  iso_inv2 : ∀ b, f (g b) = b
+  }.
+ +
+ +
+
+ +
+OA is an ordinal notation for alpha (in Schutte's model) +
+
+ +
+ +
+Class ON_correct `(alpha : Ord)
+     `(OA : @ON A ltA compareA)
+      (iota : A → Ord) :=
+  { ON_correct_inj : ∀ a, lt (iota a) alpha;
+    ON_correct_onto : ∀ beta, lt beta alpha →
+                                ∃ b, iota b = beta;
+    On_compare_spec : ∀ a b:A,
+        match compareA a b with
+        | Datatypes.Lt ⇒ lt (iota a) (iota b)
+        | Datatypes.Eq ⇒ iota a = iota b
+        | Datatypes.Gt ⇒ lt (iota b) (iota a)
+        end
+  }.
+ +
+ +
+
+ +
+

Relative correctness of a constant or a function

+ +
+
+ +
+Definition SubON_same_cst `{OA : @ON A ltA compareA}
+       `{OB : @ON B ltB compareB}
+       {iota : A → B}
+       {alpha: B}
+       {_ : SubON OA OB alpha iota}
+       (a : A)
+       (b : B)
+  := iota a = b.
+ +
+Definition SubON_same_fun `{OA : @ON A ltA compareA}
+       `{OB : @ON B ltB compareB}
+       {iota : A → B}
+       {alpha: B}
+       {_ : SubON OA OB alpha iota}
+       (f : A → A)
+       (g : B → B)
+  := ∀ x, iota (f x) = g (iota x).
+ +
+ +
+Definition SubON_same_op `{OA : @ON A ltA compareA}
+       `{OB : @ON B ltB compareB}
+       {iota : A → B} {alpha: B}
+       {_ : SubON OA OB alpha iota}
+       (f : A → A → A)
+       (g : B → B → B)
+  := ∀ x y, iota (f x y) = g (iota x) (iota y).
+ +
+ +
+
+ +
+Correctness w.r.t. Schutte's model +
+
+ +
+Definition ON_cst_ok {alpha: Ord} `{OA : @ON A ltA compareA}
+       `{OB : @ON B ltB compareB}
+       {iota : A → Ord}
+       {_ : ON_correct alpha OA iota}
+       (a: A)
+       (b: Ord)
+  := iota a = b.
+ +
+Definition ON_fun_ok {alpha: Ord} `{OA : @ON A ltA compareA}
+       `{OB : @ON B ltB compareB}
+       {iota : A → Ord}
+       {_ : ON_correct alpha OA iota}
+       (f : A → A)
+       (g : Ord → Ord)
+  :=
+    ∀ x, iota (f x) = g (iota x).
+ +
+Definition ON_op_ok {alpha: Ord} `{OA : @ON A ltA compareA}
+       `{OB : @ON B ltB compareB}
+       {iota : A → Ord}
+       {_ : ON_correct alpha OA iota}
+       (f : A → A → A)
+       (g : Ord → Ord → Ord)
+  :=
+    ∀ x y, iota (f x y) = g (iota x) (iota y).
+ +
+Definition Iso_same_cst `{OA : @ON A ltA compareA}
+       `{OB : @ON B ltB compareB}
+       {f : A → B} {g : B → A}
+       {_ : ON_Iso OA OB f g}
+       (a : A)
+       (b : B)
+  := f a = b.
+ +
+Definition Iso_same_fun `{OA : @ON A ltA compareA}
+           `{OB : @ON B ltB compareB}
+           {f : A → B} {g : B → A}
+           {_ : ON_Iso OA OB f g}
+           (fA : A → A)
+           (fB : B → B)
+  :=
+    ∀ x, f (fA x) = fB (f x).
+ +
+Definition Iso_same_op `{OA : @ON A ltA compareA}
+           `{OB : @ON B ltB compareB}
+           {f : A → B} {g : B → A}
+           {_ : ON_Iso OA OB f g}
+           (opA : A → A → A)
+           (opB : B → B → B)
+     
+  :=
+  ∀ x y, f (opA x y) = opB (f x) (f y).
+ +
+Section SubON_properties.
+ +
+  Context `{OA : @ON A ltA compareA}
+          `{OB : @ON B ltB compareB}
+          (f : A → B)
+          (alpha : B)
+          (Su : SubON OA OB alpha f).
+ +
+  Lemma SubON_mono a b : ltA a b ↔ ltB (f a) (f b).
+ +
+  Lemma SubON_inj : ∀ a b, f a = f b → a = b.
+ +
+  Lemma SubON_successor : ∀ a b, Successor a b ↔ Successor (f a) (f b).
+ +
+  Lemma SubON_limit : ∀ a , Limit a ↔ Limit (f a).
+ +
+  Lemma SubON_least : ∀ a , Least a ↔ Least (f a).
+ +
+End SubON_properties.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.OrdinalNotations.ON_O.html b/theories/html/hydras.OrdinalNotations.ON_O.html new file mode 100644 index 00000000..b6ba8809 --- /dev/null +++ b/theories/html/hydras.OrdinalNotations.ON_O.html @@ -0,0 +1,118 @@ + + + + + +hydras.OrdinalNotations.ON_O + + + + +
+ + + +
+ +

Library hydras.OrdinalNotations.ON_O

+ +
+
+ +
+Ordinal Notation for a segment O alpha Pierre Castéran, Unviv. Bordeaux and LaBRI +
+
+ +
+From Coq Require Import Arith Relations Lia Logic.Eqdep_dec Ensembles
+        Wellfounded.Inverse_Image Wellfounded.Inclusion
+        RelationClasses.
+From hydras Require Import ON_Generic.
+ +
+Coercion is_true: bool >-> Sortclass.
+Generalizable Variables A Lt comp.
+ +
+Section OA_given.
+ +
+  Context {A:Type}
+          {Lt Le: relation A}
+          {cmpA: Compare A}
+          (OA : ON Lt cmpA).
+ +
+
+ +
+The type of ordinals less than a +
+
+ +
+Definition t (a:A) := {b:A | compare b a = Datatypes.Lt}.
+ +
+Definition lt {a:A} : relation (t a) :=
+  fun alpha beta ⇒ ON_lt (proj1_sig alpha) (proj1_sig beta).
+ +
+Definition le {a:A} : relation (t a) :=
+  clos_refl (t a) lt.
+ +
+#[global]
+Instance compare_O {a:A} : Compare (t a) :=
+fun (alpha beta : t a) ⇒
+  compare (proj1_sig alpha) (proj1_sig beta).
+ +
+Lemma compare_correct {a} (alpha beta : t a) :
+  CompSpec eq lt alpha beta (compare alpha beta).
+ +
+Lemma compare_reflect {a:A} (alpha beta : t a) :
+  match (compare alpha beta)
+  with
+    Datatypes.Lt ⇒ lt alpha beta
+  | Eq ⇒ alpha = beta
+  | Gt ⇒ lt beta alpha
+  end.
+ +
+Lemma lt_wf (a:A) : well_founded (@lt a).
+ +
+#[global] Instance sto a : StrictOrder (@lt a).
+ +
+#[global] Instance ON_O_comp (a:A) : Comparable (@lt a) compare .
+ +
+
+ +
+We have now an ordinal notation +
+
+ +
+#[global] Instance ON_O (a:A) : ON (@lt a) compare .
+ +
+End OA_given.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.OrdinalNotations.ON_Omega.html b/theories/html/hydras.OrdinalNotations.ON_Omega.html new file mode 100644 index 00000000..508dc02d --- /dev/null +++ b/theories/html/hydras.OrdinalNotations.ON_Omega.html @@ -0,0 +1,76 @@ + + + + + +hydras.OrdinalNotations.ON_Omega + + + + +
+ + + +
+ +

Library hydras.OrdinalNotations.ON_Omega

+ +
+
+ +
+ A notation system for the ordinal omega Pierre Castéran, Univ. Bordeaux and LaBRI +
+
+ +
+From Coq Require Import Arith Compare_dec Lia.
+From hydras Require Import ON_Generic ON_Finite.
+From hydras Require Import Schutte.
+ +
+Import Relations RelationClasses.
+ +
+#[global]
+Instance compare_nat : Compare nat := Nat.compare.
+ +
+#[global] Instance Omega_comp : Comparable Peano.lt compare_nat.
+ +
+#[global] Instance Omega : ON Peano.lt compare_nat.
+ +
+ +
+#[local] Open Scope ON_scope.
+ +
+ +
+Definition Zero_limit_succ_dec : ZeroLimitSucc_dec.
+Defined.
+ +
+#[global] Instance FinOrd_Omega (i:nat) :
+  SubON (FinOrd i) Omega i (fun alpha ⇒ proj1_sig alpha).
+ +
+#[ global ] Instance omega_ok : ON_correct Schutte_basics.omega Omega finite.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.OrdinalNotations.ON_Omega2.html b/theories/html/hydras.OrdinalNotations.ON_Omega2.html new file mode 100644 index 00000000..890557a8 --- /dev/null +++ b/theories/html/hydras.OrdinalNotations.ON_Omega2.html @@ -0,0 +1,383 @@ + + + + + +hydras.OrdinalNotations.ON_Omega2 + + + + +
+ + + +
+ +

Library hydras.OrdinalNotations.ON_Omega2

+ +
+ +
+
+ +
+ A notation system for the ordinal omega^2 +
+ + Pierre Castéran, Univ. Bordeaux and LaBRI +
+
+ +
+From Coq Require Import Arith Compare_dec Lia.
+From hydras Require Import Simple_LexProd ON_Generic
+        ON_mult ON_Omega Compat815.
+ +
+Import Relations ON_Generic.
+ +
+Open Scope ON_scope.
+ +
+ +
+#[ global ] Instance compare_omega2 : Compare ON_mult.t :=
+ compare_mult compare_nat compare_nat.
+ +
+#[ global ] Instance Omega2: ON _ compare_omega2 := ON_mult Omega Omega.
+ +
+Definition t := ON_t.
+ +
+Open Scope ON_scope.
+ +
+ +
+Notation omega := (1,0).
+Definition zero: t := (0,0).
+ +
+Definition fin (i:nat) : t := (0,i).
+Coercion fin : nat >-> t.
+ +
+ +
+Example ex1 : (5,8) o< (5,10).
+ +
+ +
+ +
+Lemma omega_is_limit : Limit omega.
+ +
+ +
+Definition succ (alpha : t) := (fst alpha, S (snd alpha)).
+ +
+ +
+Lemma eq_dec (alpha beta: t) : {alpha = beta} + {alpha ≠ beta}.
+ +
+Lemma le_intror :
+  ∀ i j k:nat, (j ≤ k)%nat → (i,j) o≤ (i,k).
+ +
+Lemma le_0 : ∀ p: t, (0,0) o≤ p.
+ +
+ +
+Lemma lt_succ_le alpha beta :
+  alpha o< beta ↔ succ alpha o≤ beta. +
+ +
+Lemma lt_succ alpha : alpha o< succ alpha.
+ +
+ +
+ +
+#[global] Hint Constructors clos_refl lexico : O2.
+#[global] Hint Unfold lt le : O2.
+ +
+Lemma compare_reflect alpha beta :
+  match (ON_compare alpha beta)
+  with
+    Lt ⇒ alpha o< beta
+  | Eq ⇒ alpha = beta
+  | Gt ⇒ beta o< alpha
+  end.
+ +
+Lemma compare_correct alpha beta :
+    CompSpec eq ON_lt alpha beta (ON_compare alpha beta).
+ +
+Lemma zero_le alpha : zero o≤ alpha.
+ +
+Lemma Least_zero alpha : Least alpha ↔ alpha = zero.
+ +
+Definition Zero_limit_succ_dec : ZeroLimitSucc_dec.
+Defined.
+ +
+Lemma lt_eq_lt_dec alpha beta :
+  {alpha o< beta} + {alpha = beta} + {beta o< alpha}.
+ +
+Lemma Successor_inv : ∀ i j k l : nat, Successor (k, l) (i, j) →
+                                            i = k ∧ l = S j.
+ +
+Corollary Successor_not (i j k : nat) : ¬ Successor (k,0) (i,j).
+ +
+Lemma Successor_succ : ∀ alpha, Successor (succ alpha) alpha.
+ +
+ +
+Lemma succ_ok alpha beta :
+  Successor beta alpha ↔ beta = succ alpha. +
+ +
+Definition succb (alpha: t): bool
+  := match alpha with
+       (_, S _) ⇒ true
+     | _ ⇒ false
+     end.
+ +
+Definition limitb (alpha : t): bool :=
+  match alpha with
+    (S _, 0) ⇒ true
+  | _ ⇒ false
+  end.
+ +
+Lemma Omega_limit_limitb alpha s : Omega_limit s alpha →
+                                   limitb alpha.
+ +
+
+ +
+Canonical sequences +
+
+ +
+Definition canon alpha i :=
+  match alpha with
+    (0,0) ⇒ (0,0)
+  | (_, S p) ⇒ (0,p)
+  | (S n, 0) ⇒ (n, i)
+  end.
+ +
+Lemma limitb_limit alpha :
+  limitb alpha → Omega_limit (canon alpha) alpha.
+ +
+ Example Ex1 : limitb omega.
+ +
+ +
+Lemma limit_is_lub_0 : ∀ i alpha, (∀ j, (i,j) o< alpha) ↔
+                                 (S i, 0) o≤ alpha.
+ +
+Lemma limit_is_lub beta :
+  limitb beta → ∀ alpha,
+    (∀ i, canon beta i o< alpha) ↔ beta o≤ alpha.
+ +
+ Definition zero_limit_succ_dec :
+  ∀ alpha,
+                ({alpha = zero} + {limitb alpha }) +
+                {beta : t | alpha = succ beta} .
+ +
+Definition plus (alpha beta : t) : t :=
+  match alpha,beta with
+  | (0, b), (0, b') ⇒ (0, b + b')
+  | (0,0), y ⇒ y
+  | x, (0,0) ⇒ x
+  | (0, _b), (S n', b') ⇒ (S n', b')
+  | (S n, b), (S n', b') ⇒ (S n + S n', b')
+  | (S n, b), (0, b') ⇒ (S n, b + b')
+   end.
+ +
+Infix "+" := plus : ON_scope.
+ +
+Lemma plus_compat (n p: nat) :
+  fin (n + p )%nat = fin n + fin p. +
+ +
+ +
+ +
+ +
+Example non_commutativity_of_plus: omega + 3 ≠ 3 + omega. +
+ +
+
+ +
+multiplication of an ordinal by a natural number +
+
+ +
+Definition mult_fin_r (alpha : t) (p : nat): t :=
+  match alpha, p with
+  | (0,0), _ ⇒ zero
+  | _, 0 ⇒ zero
+  | (0, n), p ⇒ (0, n × p)
+  | (n, b), n' ⇒ ( n × n', b)
+  end.
+Infix "×" := mult_fin_r : ON_scope.
+ +
+
+ +
+multiplication of a natural number by an ordinal +
+
+ +
+Definition mult_fin_l (n:nat)(alpha : t) : t :=
+  match n, alpha with
+  | 0, _ ⇒ zero
+  | _, (0,0) ⇒ zero
+  | n , (0,n') ⇒ (0, (n×n')%nat)
+  | n, (n',p') ⇒ (n', (n × p')%nat)
+  end.
+ +
+ +
+ +
+Example e1 : (omega × 7 + 15) × 3 = omega × 21 + 15.
+ +
+Example e2 : mult_fin_l 3 (omega × 7 + 15) = omega × 7 + 45.
+ +
+ +
+#[global] Instance plus_assoc: Assoc eq plus.
+ +
+Lemma succ_is_plus_1 alpha : alpha + 1 = succ alpha.
+ +
+Lemma lt_omega alpha : alpha o< omega ↔ ∃ n:nat, alpha = fin n.
+ +
+Lemma decompose (i j : nat): (i,j) = omega × i + j.
+ +
+ +
+Lemma unique_decomposition (alpha: t) :
+  ∃! i j: nat, alpha = omega × i + j.
+ +
+
+ +
+

Additive principal ordinals

+ +
+
+ +
+Definition ap (alpha : t) :=
+  alpha ≠ zero ∧
+  (∀ beta gamma, beta o< alpha → gamma o< alpha →
+                       beta + gamma o< alpha).
+ +
+Lemma omega_ap : ap omega.
+ +
+Lemma ap_cases alpha : ap alpha → alpha = 1 ∨ alpha = omega.
+ +
+ +
+ +
+ +
+ +
+ +
+Open Scope ON_scope.
+ +
+Example L_3_plus_omega : 3 + omega = omega.
+ +
+ +
+From Equations Require Import Equations.
+Section A_def.
+ +
+Let m (x : nat × nat): t := omega × fst x + snd x.
+ +
+#[ local ] Instance WF : WellFounded (measure_lt m):=
+  wf_measure m.
+ +
+Equations A (p : nat × nat) : nat by wf p (measure_lt m):=
+    A (0, j) := S j;
+    A (S i, 0) := A(i, 1);
+    A (S i, S j) := A(i, A(S i, j)).
+ +
+ +
+End A_def.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.OrdinalNotations.ON_Omega_plus_omega.html b/theories/html/hydras.OrdinalNotations.ON_Omega_plus_omega.html new file mode 100644 index 00000000..fab6d8cf --- /dev/null +++ b/theories/html/hydras.OrdinalNotations.ON_Omega_plus_omega.html @@ -0,0 +1,221 @@ + + + + + +hydras.OrdinalNotations.ON_Omega_plus_omega + + + + +
+ + + +
+ +

Library hydras.OrdinalNotations.ON_Omega_plus_omega

+ +
+
+ +
+ A notation system for the ordinal omega + omega Pierre Castéran, Univ. Bordeaux and LaBRI +
+
+ +
+From Coq Require Import Arith Compare_dec Lia.
+From hydras Require Import Comparable Simple_LexProd ON_Generic
+        ON_plus ON_Omega Compat815.
+ +
+Import Relations.
+Declare Scope opo_scope.
+Delimit Scope opo_scope with opo.
+Open Scope ON_scope.
+Open Scope opo_scope.
+ +
+ +
+#[global] Instance compare_nat_nat : Compare t :=
+ compare_plus compare_nat compare_nat.
+ +
+#[global] Instance Omega_plus_Omega: ON _ compare_nat_nat :=
+ ON_plus Omega Omega.
+ +
+Definition t := ON_t.
+ +
+ +
+Example ex1 : inl 7 o< inr 8.
+ +
+ +
+Open Scope opo_scope.
+ +
+Definition fin (i:nat) : t := inl i.
+Coercion fin : nat >-> t.
+ +
+Notation omega := (inr 0:ON_t).
+ +
+ +
+Example ex2 : inl 7 o< omega.
+ +
+Lemma omega_is_limit : Limit omega.
+ +
+Lemma limit_is_omega alpha : Limit alpha → alpha = omega.
+ +
+ +
+Lemma limit_iff (alpha : t) : Limit alpha ↔ alpha = omega. +
+Lemma Successor_inv1 : ∀ i j, Successor (inl j) (inl i) → j = S i.
+ +
+Lemma Successor_inv2 : ∀ i j, Successor (inr j) (inr i) → j = S i.
+ +
+Lemma Successor_inv3 : ∀ i j, ¬ Successor (inr j) (inl i).
+ +
+Lemma Successor_inv4 : ∀ i j, ¬ Successor (inl j) (inr i).
+ +
+ +
+Definition succ (alpha : t) :=
+  match alpha with
+    inl n ⇒ inl (S n)
+  | inr n ⇒ inr (S n)
+  end.
+ +
+ +
+Lemma Successor_succ alpha : Successor (succ alpha) alpha.
+ +
+ +
+Lemma succ_correct alpha beta :
+  Successor beta alpha ↔ beta = succ alpha. +
+ +
+ +
+Definition succb (alpha: t) : bool
+  := match alpha with
+     | inr (S _) | inl (S _) ⇒ true
+     | _ ⇒ false
+     end.
+ +
+Lemma succb_correct (alpha: t) :
+  succb alpha ↔ ∃ beta: t, alpha = succ beta. +
+ +
+Lemma omega_not_succ : ∀ alpha, ¬ Successor omega alpha.
+ +
+Lemma Least_is_0 (alpha:t) : Least alpha ↔ alpha = 0.
+ +
+Lemma ZLS_dec (alpha : t) :
+  {alpha = 0} +
+  {Limit alpha} +
+  {beta : t | Successor alpha beta}.
+ +
+ +
+Definition Zero_limit_succ_dec : ON_Generic.ZeroLimitSucc_dec.
+ +
+Definition limitb (alpha: t) := match ON_compare alpha omega
+                                with Eq ⇒ true | _ ⇒ false end.
+ +
+Lemma eq_dec (alpha beta: t) : {alpha = beta} + {alpha ≠ beta}.
+ +
+Lemma le_introl :
+  ∀ i j :nat, (i≤ j)%nat → inl i o≤ inl j.
+ +
+Lemma le_intror :
+  ∀ i j :nat, (i≤ j)%nat → inr i o≤ inr j.
+ +
+Lemma le_0 : ∀ p: t, fin 0 o≤ p.
+ +
+Lemma le_lt_trans : ∀ p q r, p o≤ q → q o< r → p o< r.
+ +
+Lemma lt_le_trans : ∀ p q r, p o< q → q o≤ r → p o< r.
+ +
+Lemma lt_succ_le alpha beta : alpha o< beta ↔ succ alpha o≤ beta.
+ +
+Lemma lt_succ alpha : alpha o< succ alpha.
+ +
+Lemma lt_omega alpha :
+  alpha o< omega ↔ ∃ n:nat, alpha = fin n.
+ +
+Lemma Omega_as_lub :
+   ∀ alpha,
+     (∀ i, fin i o< alpha) ↔ omega o≤ alpha.
+ +
+ +
+ #[global] Instance Incl : SubON Omega Omega_plus_Omega omega fin. +
+ +
+Section NotIncl.
+  Context (i : nat)
+          (f : t → nat)
+          (Hyp : SubON Omega_plus_Omega Omega i f).
+ +
+  Remark R1: ∀ n: nat, ¬ Limit n.
+ +
+Lemma ExNotIncl: False.
+ +
+End NotIncl.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.OrdinalNotations.ON_mult.html b/theories/html/hydras.OrdinalNotations.ON_mult.html new file mode 100644 index 00000000..feba0760 --- /dev/null +++ b/theories/html/hydras.OrdinalNotations.ON_mult.html @@ -0,0 +1,151 @@ + + + + + +hydras.OrdinalNotations.ON_mult + + + + +
+ + + +
+ +

Library hydras.OrdinalNotations.ON_mult

+ +
+ +
+
+ +
+Product of ordinal notations +
+ + Pierre Castéran, Univ. Bordeaux and LaBRI +
+
+ +
+ +
+From Coq Require Import Arith Compare_dec Lia
+     Relation_Operators RelationClasses.
+From hydras Require Import Comparable Simple_LexProd ON_Generic.
+ +
+Import Relations.
+Generalizable All Variables.
+Coercion is_true: bool >-> Sortclass.
+ +
+
+ +
+

Definitions

+ +
+ + The product of two notation systems NA and NB is defined as the + lexicographic product of the order on NB by the order on NA + (in this order ! ) +
+
+ +
+ +
+Section Defs.
+ +
+  Context `(ltA: relation A)
+          (cmpA : Compare A)
+          (NA: ON ltA cmpA)
+          `(ltB : relation B)
+          (cmpB : Compare B)
+          (NB: ON ltB cmpB).
+ +
+  Definition t := (B × A)%type.
+  Definition lt : relation t := lexico ltB ltA.
+  Definition le := clos_refl _ lt.
+ +
+  #[global] Instance compare_mult : Compare t :=
+    fun (alpha beta: t) ⇒
+    match compare (fst alpha) (fst beta) with
+    | Eq ⇒ compare (snd alpha) (snd beta)
+    | c ⇒ c
+    end.
+ +
+  #[local] Hint Constructors clos_refl lexico: core.
+  #[local] Hint Unfold lt le : core.
+ +
+
+ +
+

Properties

+ +
+
+ +
+  Instance lt_strorder : StrictOrder lt.
+ +
+  Lemma lt_wf : well_founded lt.
+ +
+  Lemma compare_reflect alpha beta :
+    match (compare alpha beta)
+    with
+      Lt ⇒ lt alpha beta
+    | Eq ⇒ alpha = beta
+    | Gt ⇒ lt beta alpha
+    end.
+ +
+  Lemma compare_correct alpha beta :
+    CompSpec eq lt alpha beta (compare alpha beta).
+ +
+ +
+  #[global] Instance mult_comp: Comparable lt compare_mult.
+ +
+  #[global] Instance ON_mult: ON lt compare_mult.
+ +
+  Lemma lt_eq_lt_dec alpha beta :
+    {lt alpha beta} + {alpha = beta} + {lt beta alpha}.
+ +
+ +
+End Defs.
+ +
+Arguments lt_eq_lt_dec {A ltA cmpA} _ {B ltB cmpB} _.
+Arguments ON_mult {A ltA cmpA} _ {B ltB cmpB}.
+Arguments lt_strorder {A} {ltA} {cmpA} _ {B} {ltB} {cmpB} _.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.OrdinalNotations.ON_plus.html b/theories/html/hydras.OrdinalNotations.ON_plus.html new file mode 100644 index 00000000..2d6c1304 --- /dev/null +++ b/theories/html/hydras.OrdinalNotations.ON_plus.html @@ -0,0 +1,148 @@ + + + + + +hydras.OrdinalNotations.ON_plus + + + + +
+ + + +
+ +

Library hydras.OrdinalNotations.ON_plus

+ +
+ +
+
+ +
+The sum of two ordinal notations +
+ + Pierre Castéran, Univ. Bordeaux and LaBRI +
+
+ +
+ +
+From Coq Require Import Arith Compare_dec Lia
+     Relation_Operators RelationClasses Disjoint_Union.
+From hydras Require Import ON_Generic MoreOrders.
+ +
+Import Relations.
+Generalizable All Variables.
+Coercion is_true: bool >-> Sortclass.
+ +
+Section Defs.
+ +
+  Context `(ltA: relation A)
+          (cmpA : Compare A)
+          (NA: ON ltA cmpA).
+  Context `(ltB: relation B)
+          (cmpB : Compare B)
+          (NB: ON ltB cmpB).
+ +
+  Definition t := (A + B)%type.
+  Arguments inl {A B} _.
+  Arguments inr {A B} _.
+ +
+  Definition lt : relation t := le_AsB _ _ ltA ltB.
+ +
+#[global] Instance compare_plus : Compare t :=
+fun (alpha beta: t) ⇒
+   match alpha, beta with
+     inl _, inr _ ⇒ Lt
+   | inl a, inl a' ⇒ compare a a'
+   | inr b, inr b' ⇒ compare b b'
+   | inr _, inl _ ⇒ Gt
+  end.
+ +
+Definition le := clos_refl _ lt.
+ +
+#[local] Hint Unfold lt le : core.
+#[local] Hint Constructors le_AsB: core.
+ +
+Instance lt_strorder : StrictOrder lt.
+ +
+Lemma compare_reflect alpha beta :
+  match (compare alpha beta)
+  with
+    Lt ⇒ lt alpha beta
+  | Eq ⇒ alpha = beta
+  | Gt ⇒ lt beta alpha
+  end.
+ +
+ +
+Lemma compare_correct alpha beta :
+    CompSpec eq lt alpha beta (compare alpha beta). +
+ +
+#[global] Instance plus_comp : Comparable lt compare_plus.
+ +
+ +
+ +
+Lemma lt_wf : well_founded lt.
+ +
+ +
+ +
+ +
+ +
+#[global] Instance ON_plus : ON lt compare_plus.
+ +
+ +
+ +
+Lemma lt_eq_lt_dec alpha beta :
+  {lt alpha beta} + {alpha = beta} + {lt beta alpha}.
+ +
+End Defs.
+ +
+Arguments lt_eq_lt_dec {A ltA cmpA} _ {B ltB cmpB} _.
+Arguments ON_plus {A ltA cmpA} _ {B ltB cmpB}.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.OrdinalNotations.OmegaOmega.html b/theories/html/hydras.OrdinalNotations.OmegaOmega.html new file mode 100644 index 00000000..20c3e7bc --- /dev/null +++ b/theories/html/hydras.OrdinalNotations.OmegaOmega.html @@ -0,0 +1,481 @@ + + + + + +hydras.OrdinalNotations.OmegaOmega + + + + +
+ + + +
+ +

Library hydras.OrdinalNotations.OmegaOmega

+ +
+
+ +
+

An implementation of ωω

+ +
+ + New implementation as a refinement of epsilon0 +
+
+ +
+From hydras Require Import T1 E0 Comparable.
+From Coq Require Import Arith Logic.Eqdep_dec Peano_dec
+  List Bool
+  Recdef Lia Wellfounded.Inverse_Image
+  Wellfounded.Inclusion RelationClasses .
+ +
+
+ +
+

Representation by lists of pairs of integers

+ +
+
+ +
+Module LO.
+ +
+  Definition t := list (nat×nat).
+ +
+  Definition zero : t := nil.
+ +
+
+ +
+omega^ i * S n + alpha +
+
+ +
+  Notation cons i n alpha := ((i,n)::alpha).
+ +
+
+ +
+Finite ordinals +
+
+ +
+  Notation FS n := (cons 0 n zero: t).
+ +
+  Definition fin (n:nat): t := match n with 0 ⇒ zero | S p ⇒ FS p end.
+  Coercion fin : nat >-> t.
+ +
+
+ +
+omega ^i +
+
+ +
+  Notation phi0 i := (cons i 0 nil).
+ +
+  Notation omega := (phi0 1).
+ +
+ +
+
+ +
+data refinement +
+
+ +
+  Fixpoint refine (a : t) : T1.T1 :=
+    match a with
+      nil ⇒ T1.zero
+    | cons i n b ⇒ T1.cons (\F i)%t1 n (refine b)
+    end.
+ +
+  Inductive ap : t → Prop :=
+    ap_intro : ∀ a, ap (phi0 a).
+ +
+
+ +
+Successor and limits (syntactic definitions) +
+
+ +
+  Fixpoint succb (a : t) :=
+    match a with
+      nil ⇒ false
+    | cons 0 _ _ ⇒ true
+    | cons _ _ b ⇒ succb b
+    end.
+ +
+  Fixpoint limitb (a : t) :=
+    match a with
+      nil ⇒ false
+    | cons 0 _ _ ⇒ false
+    | cons _ _ nil ⇒ true
+    | cons _ _ b ⇒ limitb b
+    end.
+ +
+  Lemma succb_ref (a:t): succb a → T1is_succ (refine a).
+ +
+  Lemma limitb_ref (a:t): limitb a → T1limit (refine a).
+ +
+  #[ global ] Instance compare_oo : Compare t :=
+  fix cmp (a b : t) :=
+    match a, b with
+    | nil, nil ⇒ Eq
+    | nil, cons a' n' b' ⇒ Datatypes.Lt
+    | _ , nil ⇒ Gt
+    | (cons a n b),(cons a' n' b') ⇒
+      (match Nat.compare a a' with
+       | Datatypes.Lt ⇒ Datatypes.Lt
+       | Gt ⇒ Gt
+       | Eq ⇒ (match Nat.compare n n' with
+                | Eq ⇒ cmp b b'
+                | comp ⇒ comp
+                end)
+       end)
+    end.
+ +
+  Lemma compare_ref (a b : t) :
+    compare a b = compare (refine a) (refine b).
+ +
+  Definition lt (a b : t) : Prop :=
+    compare a b = Datatypes.Lt.
+ +
+  Lemma compare_rev :
+    ∀ (a b : t),
+      compare b a = CompOpp (compare a b).
+ +
+ +
+  Lemma compare_reflect :
+    ∀ a b,
+      match compare a b with
+      | Datatypes.Lt ⇒ lt a b
+      | Eq ⇒ a = b
+      | Gt ⇒ lt b a
+      end.
+ +
+  Lemma compare_correct (a b: t):
+    CompSpec eq lt a b (compare a b).
+ +
+  Lemma lt_ref (a b : t) :
+    lt a b ↔ T1.lt (refine a) (refine b).
+ +
+  Lemma eq_ref (a b : t) : a = b ↔ refine a = refine b.
+ +
+  Lemma lt_irrefl (a : t): ¬ lt a a.
+ +
+  Lemma lt_trans (a b c : t): lt a b → lt b c → lt a c.
+ +
+  #[global] Instance lo_strorder: StrictOrder lt.
+ +
+  #[global] Instance lo_comparable : Comparable lt compare.
+ +
+  Fixpoint nf_b (alpha : t) : bool :=
+    match alpha with
+    | nil ⇒ true
+    | cons a n nil ⇒ true
+    | cons a n ((cons a' n' b') as b) ⇒
+      (nf_b b && Nat.ltb a' a)%bool
+    end.
+ +
+  Definition nf alpha :Prop := nf_b alpha.
+
+ +
+refinements of T1's lemmas +
+
+ +
+  Lemma zero_nf : nf zero.
+ +
+  Lemma fin_nf (i:nat) : nf (fin i).
+ +
+  Lemma single_nf :
+    ∀ i n, nf (cons i n zero).
+ +
+  Lemma cons_nf :
+    ∀ i n j n' b,
+      Nat.lt j i →
+      nf(cons j n' b)→
+      nf(cons i n (cons j n' b)).
+ +
+  #[local] Hint Resolve zero_nf single_nf cons_nf : core.
+ +
+  Lemma nf_inv2 :
+    ∀ i n b, nf (cons i n b) → nf b.
+ +
+  Lemma nf_inv3 :
+    ∀ i n j n' b',
+      nf (cons i n (cons j n' b')) → Nat.lt j i.
+ +
+  Lemma nf_ref: ∀ a, T1.nf (refine a) ↔ nf a.
+ +
+  Declare Scope lo_scope.
+  Delimit Scope lo_scope with lo.
+  Open Scope lo_scope.
+ +
+ +
+  Fixpoint succ (a : t) : t :=
+    match a with
+    | nil ⇒ fin 1
+    | cons 0 n _ ⇒ cons 0 (S n) nil
+    | cons a n b ⇒ cons a n (succ b)
+    end.
+ +
+  Fixpoint plus (a b : t) :t :=
+    match a, b with
+    | nil, y ⇒ y
+    | x, nil ⇒ x
+    | cons a n b, cons a' n' b' ⇒
+       (match Nat.compare a a' with
+        | Datatypes.Lt ⇒ cons a' n' b'
+        | Gt ⇒ (cons a n (plus b (cons a' n' b')))
+        | Eq ⇒ (cons a (S (n+n')) b')
+        end)
+    end
+  where "a + b" := (plus a b) : lo_scope.
+ +
+  Fixpoint mult (a b : t) : t :=
+    match a, b with
+    | nil, _ ⇒ zero
+    | _, nil ⇒ zero
+    | cons 0 n _, cons 0 n' b' ⇒
+       cons 0 (Peano.pred((S n) × (S n'))) b'
+    | cons a n b, cons 0 n' _ ⇒
+       cons a (Peano.pred ((S n) × (S n'))) b
+    | cons a n b, cons a' n' b' ⇒
+       cons (a + a')%nat n' ((cons a n b) × b')
+    end
+  where "a * b" := (mult a b) : lo_scope.
+ +
+ +
+ +
+  Lemma phi0_ref (i:nat) : refine (phi0 i) = T1.phi0 (\F i).
+ +
+ +
+  Lemma phi0_nf (i:nat) : nf (phi0 i).
+ +
+  Lemma succ_ref (alpha : t) :
+    refine (succ alpha) = T1.succ (refine alpha).
+ +
+  Lemma succ_nf alpha : nf alpha → nf (succ alpha).
+ +
+  Lemma plus_ref : ∀ alpha beta: t,
+      refine (alpha + beta) = T1.T1add (refine alpha) (refine beta).
+ +
+  Lemma plus_nf alpha beta : nf alpha → nf beta → nf (alpha + beta).
+ +
+  Lemma mult_ref : ∀ alpha beta: t,
+      refine (alpha × beta) =
+      T1.T1mul (refine alpha) (refine beta).
+ +
+  Lemma mult_nf alpha beta : nf alpha → nf beta → nf (alpha × beta).
+ +
+ +
+  #[global] Instance plus_assoc: Assoc eq plus.
+ +
+  Lemma mult_plus_distr_l (a b c: t) : nf a → nf b → nf c →
+                                       a × (b + c) = a × b + a × c.
+ +
+End LO.
+ +
+Declare Scope OO_scope.
+ +
+Delimit Scope OO_scope with oo.
+Open Scope OO_scope.
+Import LO.
+ +
+
+ +
+

well formed ordinal representation

+ +
+
+ +
+Module OO.
+  Class OO : Type := mkord {data: LO.t ; data_ok : LO.nf data}.
+ +
+  Arguments data : clear implicits.
+  #[local] Hint Resolve data_ok : core.
+ +
+  Definition lt (alpha beta: OO) := LO.lt (data alpha) (data beta).
+  Definition le := leq lt.
+  #[ global ] Instance compare_OO : Compare OO :=
+    fun (alpha beta: OO) ⇒ LO.compare_oo (data alpha) (data beta).
+ +
+  #[ global ] Instance Zero : OO := @mkord nil refl_equal.
+ +
+  #[ global ] Instance _Omega : OO.
+ +
+  #[ global ] Instance Fin (i:nat) : OO.
+ +
+  Notation omega := _Omega.
+ +
+  #[ global ] Instance Succ (alpha : OO) : OO.
+ +
+  #[ global ] Instance plus (alpha beta : OO) : OO.
+ +
+  Infix "+" := plus : OO_scope.
+ +
+  #[ global ] Instance mult (alpha beta : OO) : OO.
+ +
+  Infix "×" := mult : OO_scope.
+ +
+  #[ global ] Instance phi0 (i : nat): OO.
+ +
+  Notation "'omega^'" := phi0 (only parsing) : OO_scope.
+ +
+  Infix "×" := mult : OO_scope.
+ +
+  #[ global ] Instance embed (alpha: OO) : E0.E0.
+ +
+  Lemma lt_embed (alpha beta : OO): lt alpha beta →
+                                    E0lt (embed alpha) (embed beta).
+ +
+  #[ global ] Instance oo_str : StrictOrder lt.
+  Qed.
+ +
+  Lemma nf_proof_unicity :
+    ∀ (alpha:t) (H H': nf alpha), H = H'.
+ +
+  Lemma OO_eq_intro : ∀ alpha beta : OO,
+      data alpha = data beta → alpha = beta.
+ +
+  #[ global ] Instance OO_comp : Comparable lt compare.
+ +
+  Lemma lt_wf : well_founded lt.
+ +
+  Import ON_Generic.
+ +
+  #[ global ] Instance ON_OO : ON lt compare.
+ +
+End OO.
+ +
+Import OO.
+#[local] Open Scope OO_scope.
+ +
+Check phi0 7.
+ +
+#[global] Coercion Fin : nat >-> OO.
+ +
+Example Ex42: omega + 42 + omega^ 2 = omega^ 2. Qed.
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.Comparable.html b/theories/html/hydras.Prelude.Comparable.html new file mode 100644 index 00000000..2001ab81 --- /dev/null +++ b/theories/html/hydras.Prelude.Comparable.html @@ -0,0 +1,296 @@ + + + + + +hydras.Prelude.Comparable + + + + +
+ + + +
+ +

Library hydras.Prelude.Comparable

+ +
+From Coq Require Import Relations RelationClasses Setoid.
+From hydras Require Export MoreOrders STDPP_compat.
+ +
+Class Compare (A:Type) := compare : A → A → comparison.
+ +
+Class Comparable {A:Type} (lt: relation A) (cmp : Compare A) :=
+{
+  comparable_sto :> StrictOrder lt;
+  comparable_comp_spec : ∀ (a b : A), CompSpec eq lt a b (compare a b)
+}.
+ +
+#[export] Hint Mode Compare ! : typeclass_instances.
+#[export] Hint Mode Comparable ! - - : typeclass_instances.
+ +
+Section Comparable.
+ +
+  Context {A: Type}
+          {lt: relation A}
+          {cmp: Compare A} `{!Comparable lt cmp}.
+  #[local] Notation le := (leq lt).
+ +
+  #[deprecated(note="use StrictOrder_Transitive")]
+   Notation lt_trans := StrictOrder_Transitive (only parsing).
+ +
+  #[deprecated(note="use StrictOrder_Irreflexive")]
+   Notation lt_irrefl := StrictOrder_Irreflexive (only parsing).
+ +
+  Lemma lt_not_gt (a b: A): lt a b → ¬lt b a.
+ +
+  Lemma lt_not_ge (a b: A): lt a b → ¬ le b a.
+ +
+  Lemma compare_lt_iff (a b: A):
+    compare a b = Lt ↔ lt a b.
+ +
+  Lemma compare_lt_trans (a b c: A):
+    compare a b = Lt → compare b c = Lt → compare a c = Lt.
+ +
+  Lemma compare_lt_irrefl (a: A): ¬compare a a = Lt.
+ +
+  Lemma compare_eq_iff (a b: A): compare a b = Eq ↔ a = b.
+ +
+  Lemma compare_refl (a: A):
+    compare a a = Eq.
+ +
+  Lemma compare_eq_trans (a b c: A):
+    compare a b = Eq → compare b c = Eq → compare a c = Eq.
+ +
+  Lemma compare_gt_iff (a b: A):
+    compare a b = Gt ↔ lt b a.
+ +
+  Lemma compare_gt_irrefl (a: A):
+    ¬compare a a = Gt.
+ +
+  Lemma compare_gt_trans (a b c: A):
+    compare a b = Gt → compare b c = Gt → compare a c = Gt.
+ +
+  Lemma compare_lt_not_gt (a b: A):
+    compare a b = Lt → ¬ compare a b = Gt.
+ +
+  Lemma compare_gt_not_lt (a b: A):
+    compare a b = Gt → ¬ compare a b = Lt.
+ +
+  Lemma le_refl (a: A): le a a.
+ +
+  Lemma compare_le_iff_refl (a: A):
+    le a a ↔ compare a a = Eq.
+ +
+  Lemma compare_le_iff (a b: A):
+    le a b ↔ compare a b = Lt ∨ compare a b = Eq.
+ +
+  Lemma compare_ge_iff (a b: A):
+    le b a ↔ compare a b = Gt ∨ compare a b = Eq.
+ +
+  Lemma le_trans (a b c: A):
+    le a b → le b c → le a c.
+ +
+  Lemma le_lt_trans (a b c: A):
+    le a b → lt b c → lt a c.
+ +
+  Lemma lt_le_trans (a b c: A):
+    lt a b → le b c → lt a c.
+ +
+  Lemma lt_incl_le (a b: A):
+    lt a b → le a b.
+ +
+  Lemma le_not_gt (a b: A):
+    le a b → ¬ lt b a.
+ +
+  #[using="All"]
+  Definition max (a b : A): A :=
+  match compare a b with
+  | Gt ⇒ a
+  | _ ⇒ b
+  end.
+ +
+  Lemma max_dec (a b: A):
+    max a b = a ∨ max a b = b.
+ +
+  Lemma max_comm (a b: A):
+    max a b = max b a.
+ +
+  Lemma max_ge_a (a b: A):
+    le b a ↔ max a b = a.
+ +
+  Lemma max_ge_b (a b: A):
+    le a b ↔ max a b = b.
+ +
+  Lemma max_refl (a: A): max a a = a.
+ +
+  Lemma le_max_a (a b: A): le a (max a b).
+ +
+  Lemma le_max_b (a b: A): le b (max a b).
+ +
+  #[global] Instance max_assoc : Assoc eq max.
+ +
+  #[using="All"]
+  Definition min (a b :A): A :=
+  match compare a b with
+  | Lt ⇒ a
+  | _ ⇒ b
+  end.
+ +
+  Lemma min_max_iff (a b: A):
+    min a b = a ↔ max a b = b.
+ +
+  Lemma min_comm (a b: A):
+    min a b = min b a.
+ +
+  Lemma min_dec (a b: A):
+    min a b = a ∨ min a b = b.
+ +
+  Lemma min_le_ad (a b: A):
+    le a b ↔ min a b = a.
+ +
+  Lemma min_le_b (a b: A):
+    le b a ↔ min a b = b.
+ +
+  Lemma min_refl (a:A):
+    min a a = a.
+ +
+ +
+  Lemma le_min_a (a b: A):
+    le (min a b) a.
+ +
+  Lemma le_min_bd (a b: A):
+    le (min a b) b.
+ +
+  #[global] Instance min_assoc: Assoc eq min.
+ +
+ +
+  Lemma compare_trans (a b c: A) (comp_res: comparison):
+    compare a b = comp_res → compare b c = comp_res → compare a c = comp_res.
+ +
+  Lemma compare_reflect (a b: A):
+    match compare a b with
+    | Lt ⇒ lt a b
+    | Eq ⇒ a = b
+    | Gt ⇒ lt b a
+    end.
+ +
+  Lemma lt_eq_lt:
+    ∀ alpha beta, lt alpha beta ∨ alpha = beta ∨ lt beta alpha.
+ +
+  Definition lt_eq_lt_dec
+             (alpha beta : A) :
+    {lt alpha beta} + {alpha = beta} + {lt beta alpha}.
+  Defined.
+ +
+  Lemma LimitNotSucc
+        (alpha: A) :
+    Limit alpha → ∀ beta, ¬ Successor alpha beta.
+ +
+End Comparable.
+ +
+#[local] Ltac compare_trans H1 H2 intropattern :=
+  lazymatch type of (H1, H2) with
+  | ((?compare ?a ?b = ?comp_res) × (?compare ?b ?c = ?comp_res))%type ⇒
+    assert (compare a c = comp_res) as intropattern by
+          (apply compare_trans with b;
+           [ exact H1 | exact H2 ])
+  | ((?compare ?a ?b = ?comp_res) × (?compare ?b ?c = Eq))%type ⇒
+    assert (compare a c = comp_res) as intropattern by
+          (assert (b = c) as → by (apply compare_eq_iff; exact H2);
+           exact H1)
+  | ((?compare ?a ?b = Eq) × (?compare ?b ?c = ?comp_res))%type ⇒
+    assert (compare a c = comp_res) as intropattern by
+          (assert (a = b) as → by (apply compare_eq_iff; exact H1);
+           exact H2)
+  | ((?compare _ _ = _) × (?compare _ _ = _))%type ⇒ fail "Not a supported case."
+  | _ ⇒ fail "Did not find hypotheses talking about compare: did you declare an instance of Comparable?"
+  end.
+ +
+Tactic Notation "compare" "trans" constr(H1) constr(H2) "as" simple_intropattern(intropattern) :=
+  compare_trans H1 H2 intropattern.
+ +
+Ltac compare_destruct_eqn a b H :=
+  destruct (compare a b) eqn: H;
+  [ apply compare_eq_iff in H as <-
+  | apply compare_lt_iff in H
+  | apply compare_gt_iff in H
+  ].
+ +
+Tactic Notation "compare" "destruct" constr(a) constr(b) "as" ident(H) :=
+  compare_destruct_eqn a b H.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.Compat815.html b/theories/html/hydras.Prelude.Compat815.html new file mode 100644 index 00000000..fdd72fee --- /dev/null +++ b/theories/html/hydras.Prelude.Compat815.html @@ -0,0 +1,109 @@ + + + + + +hydras.Prelude.Compat815 + + + + +
+ + + +
+ +

Library hydras.Prelude.Compat815

+ +
+
+ +
+provisionally fixes some compatibilty issues 8.15 -> 8.16 +
+
+ +
+From Coq Require Import Arith Lia.
+Import Nat.
+ +
+Module Compat815.
+ +
+  Lemma le_n_0_eq : ∀ n : nat, n ≤ 0 → 0 = n.
+ +
+  Lemma le_lt_or_eq :
+    ∀ n m : nat, n ≤ m → n < m ∨ n = m.
+ +
+  Lemma lt_n_Sm_le:
+      ∀ n m : nat, n < S m → n ≤ m.
+ +
+  Definition ind_0_1_SS (P : nat → Prop) (H0 : P 0) (H1 : P 1)
+  (H2 : ∀ n : nat, P n → P (S (S n))) : ∀ n, P n :=
+
+fix ind_0_1_SS (n : nat) : P n :=
+  match n as n0 return (P n0) with
+  | 0 ⇒ H0
+  | S n0 ⇒
+      (fun n1 : nat ⇒
+       match n1 as n2 return (P (S n2)) with
+       | 0 ⇒ H1
+       | S n2 ⇒ (fun n3 : nat ⇒ H2 n3 (ind_0_1_SS n3)) n2
+       end) n0
+  end.
+ +
+  Lemma lt_S_n (n m :nat) : S n < S m → n < m.
+ +
+ +
+  Lemma lt_n_S : ∀ n m : nat, n < m → S n < S m.
+ +
+  Lemma le_lt_n_Sm : ∀ n m : nat, n ≤ m → n < S m.
+ +
+ Lemma lt_not_le : ∀ n m : nat, n < m → ¬ m ≤ n.
+ +
+ Lemma le_plus_r : ∀ n m : nat, m ≤ n + m.
+ +
+Lemma mult_O_le : ∀ n m : nat, m = 0 ∨ n ≤ m × n.
+ +
+Lemma plus_Snm_nSm
+     : ∀ n m : nat, S n + m = n + S m.
+ +
+Lemma n_SSSn : ∀ n : nat, n ≠ S (S (S n)).
+ +
+Lemma n_SSn : ∀ n : nat, n ≠ S (S n).
+ +
+Lemma le_not_lt : ∀ n m : nat, n ≤ m → ¬ m < n.
+ +
+End Compat815.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.DecPreOrder.html b/theories/html/hydras.Prelude.DecPreOrder.html new file mode 100644 index 00000000..0af73f62 --- /dev/null +++ b/theories/html/hydras.Prelude.DecPreOrder.html @@ -0,0 +1,180 @@ + + + + + +hydras.Prelude.DecPreOrder + + + + +
+ + + +
+ +

Library hydras.Prelude.DecPreOrder

+ +
+
+ +
+ +
+ + Decidable, Total Pre-orders + +
+ + +
+
+ +
+From Coq Require Export Relations RelationClasses Setoid.
+ +
+Class Total {A:Type}(R: relation A) :=
+  totalness : ∀ a b:A, R a b ∨ R b a.
+ +
+From hydras Require Export STDPP_compat.
+ +
+#[ global ] Instance comparison_eq_dec : EqDecision comparison.
+ +
+#[ global ] Instance Total_Reflexive{A:Type}{R: relation A}(rt : Total R):
+  Reflexive R.
+ +
+Module Semibundled.
+ +
+Class TotalDec {A:Type}(R: relation A):=
+  { total_dec :> Total R ;
+    dec_dec :> RelDecision R}.
+ +
+
+ +
+A class of decidable total pre-orders +
+
+ +
+Class TotalDecPreOrder {A:Type}(le: relation A) :=
+  {
+    total_dec_pre_order :> PreOrder le;
+    total_dec_total :> TotalDec le
+  }.
+ +
+End Semibundled.
+ +
+
+ +
+when applied to a pre-order relation le, returns the equivalence and + the strict order associated to le +
+
+ +
+Definition preorder_equiv {A:Type}{le:relation A}
+           `{P:@PreOrder A le }(a b : A) :=
+  le a b ∧ le b a.
+ +
+Definition lt {A:Type}{le:relation A} `{P:@PreOrder A le }(a b : A) :=
+  le a b ∧ ¬ le b a.
+ +
+
+ +
+A class of total pre-orders +
+
+ +
+Class TotalPreOrder {A} (R : relation A) : Prop := {
+  total_pre_order_pre :> PreOrder R;
+  total_pre_order_total :> Total R
+}.
+ +
+
+ +
+Properties of decidable total pre-orders +
+
+Lemma lt_irreflexive {A:Type}{le:relation A}{P:PreOrder le}:
+  ∀ a, ¬ lt a a.
+ +
+Lemma lt_not_ge {A:Type}{le:relation A}{P:PreOrder le}:
+  ∀ a b, lt a b → ¬ le b a.
+ +
+Lemma not_le_ge {A} {le:relation A} {P0 : TotalPreOrder le} :
+  ∀ a b, ¬ le a b → le b a.
+ +
+Lemma le_not_gt {A:Type}{le:relation A}{P:PreOrder le}:
+  ∀ a b, le a b → ¬ lt b a.
+ +
+Lemma lt_not_equiv {A:Type}{le:relation A}{P:PreOrder le}:
+  ∀ a b, lt a b → ¬ preorder_equiv a b.
+ +
+Lemma equiv_not_lt {A:Type}{le:relation A}{P:PreOrder le}:
+  ∀ a b, preorder_equiv a b → ¬ lt a b.
+ +
+Lemma lt_le_trans {A:Type}{le:relation A}{P:PreOrder le}:
+ ∀ a b c, lt a b → le b c → lt a c.
+ +
+Lemma le_lt_trans {A:Type}{le:relation A}{P:PreOrder le}:
+ ∀ a b c, le a b → lt b c → lt a c.
+ +
+Lemma le_lt_weak {A:Type}{le:relation A}{P:PreOrder le}:
+ ∀ a b, lt a b → le a b.
+ +
+#[ global ] Instance lt_transitive {A:Type}{le:relation A}{P:PreOrder le}:
+  Transitive lt.
+ +
+#[ global ] Instance equiv_equiv {A:Type}{le:relation A} `{P:@PreOrder A le }: Equivalence preorder_equiv.
+ +
+Lemma le_lt_equiv_dec {A:Type} {le : relation A}
+ {P0: TotalPreOrder le} {dec : RelDecision le} (a b:A) :
+ le a b → {lt a b}+{preorder_equiv a b}.
+ +
+Lemma not_le_gt {A:Type}(le : relation A)
+      (P0: TotalPreOrder le) (a b:A) : ¬ le a b → lt b a.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.DecPreOrder_Instances.html b/theories/html/hydras.Prelude.DecPreOrder_Instances.html new file mode 100644 index 00000000..51333e3d --- /dev/null +++ b/theories/html/hydras.Prelude.DecPreOrder_Instances.html @@ -0,0 +1,191 @@ + + + + + +hydras.Prelude.DecPreOrder_Instances + + + + +
+ + + +
+ +

Library hydras.Prelude.DecPreOrder_Instances

+ +
+
+ +
+Pierre Castéran, Univ. Bordeaux and LaBRI +
+
+ +
+From hydras Require Export DecPreOrder.
+From Coq Require Import Arith ZArith List Sets.Finite_sets Sets.Ensembles.
+ +
+#[ global ] Instance Nat_le_dec : RelDecision le := le_dec.
+ +
+#[ global ] Instance Nat_le_TO : TotalPreOrder le.
+ +
+#[ global ] Instance Z_le_dec : RelDecision Z.le := Z_le_dec.
+ +
+#[ global ] Instance Z_le_TO : TotalPreOrder Z.le.
+Qed.
+ +
+Import DecPreOrder.
+ +
+
+ +
+Pre-order associated with an inverse function +
+
+ +
+#[ global ] Instance Inverse_fun {A B:Type}{f : A → B}
+         {leB: relation B}{PB: PreOrder leB}:
+                     PreOrder (fun x y ⇒ leB (f x) (f y)).
+ +
+#[ global ] Instance Total_Inverse_fun {A B:Type}{f : A → B}
+         {leB: relation B}{TB: TotalPreOrder leB} :
+                     TotalPreOrder (fun x y ⇒ leB (f x) (f y)).
+ +
+#[ global ] Instance RelDecision_Inverse_fun {A B:Type}{f : A → B}
+ {leB: relation B} {RDB : RelDecision leB} :
+  RelDecision (fun x y ⇒ leB (f x) (f y)) :=
+  fun x y ⇒ decide_rel leB (f x) (f y).
+ +
+
+ +
+Pre-order associated with inclusion +
+
+ +
+#[ global ] Instance PrO_Included {U:Type}: PreOrder (Included U).
+ +
+Arguments Included {U} _ _.
+ +
+Definition Included_s {U} (A B : Ensemble U) :=
+  Included A B ∧ ¬ Included B A.
+ +
+Infix "<:" := Included (at level 30).
+Infix "'<s'" := Included_s (at level 30).
+ +
+
+ +
+Example + Lists (pre-ordered by their length) +
+
+ +
+#[ global ] Instance List_length {A:Type}:
+   TotalPreOrder (fun l l' : list A ⇒ List.length l ≤ List.length l')%nat.
+Defined.
+ +
+
+ +
+Non dependent lexicographic product +
+
+ +
+Section lexprod.
+ Variables (A B : Type)
+           (leA : relation A)
+           (leB : relation B).
+ Context (TA : TotalPreOrder leA)
+         (TB : TotalPreOrder leB)
+         (DA : RelDecision leA)
+         (DB : RelDecision leB).
+ +
+ Definition PA := @total_pre_order_pre A leA TA.
+ Definition PB := @total_pre_order_pre B leB TB.
+ +
+ +
+Notation "x '<=A' y" := (leA x y) (at level 70, no associativity).
+Notation "x '<A' y" := (@lt A leA PA x y) (at level 70, no associativity).
+Notation "x '<=B' y" := (leB x y) (at level 70, no associativity).
+Notation "x '<B' y" := (@lt B leB PB x y ) (at level 70, no associativity).
+Notation "x =A= y" := (@preorder_equiv A leA PA x y) (at level 70, no associativity).
+Notation "x =B= y" := (@preorder_equiv B leB PB x y) (at level 70, no associativity).
+ +
+Inductive lex_prod (p p':A×B): Prop :=
+| lex1 : (fst p) <A (fst p') → lex_prod p p'
+| lex2 : (fst p) =A= (fst p') → (snd p) ≤B (snd p') → lex_prod p p'.
+ +
+Notation "x '<=lex' y" := (lex_prod x y) (at level 70, no associativity).
+Notation "x '<lex' y" := (@lt (A×B) lex_prod _ x y) (at level 70, no associativity).
+ +
+ #[global] Instance PO_lex_prod : PreOrder lex_prod.
+ +
+Lemma lex_of_equiv (a a':A)(b b':B) :
+     a =A= a' → b =B= b' → preorder_equiv (a,b) (a',b').
+ +
+Lemma lex_inv_left (a a':A)(b b':B) :
+  ((a,b) ≤lex (a',b')) → a ≤A a'.
+ +
+Lemma lex_inv_right (a a':A)(b b':B) :
+  ((a,b) ≤lex (a',b')) → a =A= a' → b ≤B b'.
+ +
+Lemma lex_strict_intro_right (a a':A)(b b':B) :
+  a =A= a' → b <B b' → (a,b) <lex (a',b').
+ +
+Lemma lex_strict_intro_left (a a':A)(b b':B) :
+  a <A a' → (a,b) <lex (a',b').
+ +
+#[global] Instance To_lex_prod : TotalPreOrder lex_prod.
+#[global] Instance lex_prod_dec : RelDecision lex_prod.
+ +
+End lexprod.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.Exp2.html b/theories/html/hydras.Prelude.Exp2.html new file mode 100644 index 00000000..d6ba75c6 --- /dev/null +++ b/theories/html/hydras.Prelude.Exp2.html @@ -0,0 +1,56 @@ + + + + + +hydras.Prelude.Exp2 + + + + +
+ + + +
+ +

Library hydras.Prelude.Exp2

+ +
+From Coq Require Import Arith Lia.
+ +
+Fixpoint exp2 (n:nat) : nat :=
+  match n with
+    0 ⇒ 1
+  | S i ⇒ 2 × exp2 i
+  end.
+ +
+Lemma exp2_positive : ∀ i, 0 < exp2 i.
+ +
+Lemma exp2_not_zero i : exp2 i ≠ 0.
+ +
+Lemma exp2_gt_id : ∀ n, n < exp2 n.
+ +
+Lemma exp2S : ∀ n, exp2 (S n) = 2 × exp2 n.
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.First_toggle.html b/theories/html/hydras.Prelude.First_toggle.html new file mode 100644 index 00000000..393b6265 --- /dev/null +++ b/theories/html/hydras.Prelude.First_toggle.html @@ -0,0 +1,75 @@ + + + + + +hydras.Prelude.First_toggle + + + + +
+ + + +
+ +

Library hydras.Prelude.First_toggle

+ +
+
+ +
+ +
+ + First_toggle + +
+ +Computes the first l between n and p (excluded) such that + P l = true and P (S l) = false. + +
+ + Pierre Castéran, Univ. Bordeaux and LaBRI +
+
+ +
+From Coq Require Import Arith Lia Inclusion Inverse_Image.
+From hydras Require Import DecPreOrder.
+ +
+Section Hypos.
+  Context (P : nat → Prop) `{Pdec: ∀ n, Decision (P n)} (n p : nat).
+ +
+  Hypotheses (Hnp : n < p) (Hn : P n) (Hp : ¬ P p).
+ +
+  Let spec := {l : nat | n ≤ l ∧ l < p ∧
+                          (∀ i: nat, n ≤ i → i ≤ l → P i) ∧
+               (¬ P (S l ))}.
+ +
+  Definition first_toggle : spec.
+ +
+End Hypos.
+ +
+Arguments first_toggle : clear implicits.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.Fuel.html b/theories/html/hydras.Prelude.Fuel.html new file mode 100644 index 00000000..175d320c --- /dev/null +++ b/theories/html/hydras.Prelude.Fuel.html @@ -0,0 +1,117 @@ + + + + + +hydras.Prelude.Fuel + + + + +
+ + + +
+ +

Library hydras.Prelude.Fuel

+ +
+ +
+ +
+From Coq Require Import FunInd Recdef Wf_nat Lia.
+ +
+Function zero (n:nat) {wf lt n} : nat :=
+  match n with
+    0 ⇒ 0
+  | S n' as p ⇒ zero (Nat.div2 p)
+  end.
+ +
+ +
+About lt_wf.
+ +
+
+ +
+Let's see whqat happens if lt_wf was opaque +
+
+ +
+Module OpaqueWf.
+ +
+ Lemma lt_wf : well_founded lt.
+ +
+  Function zero (n:nat) {wf lt n} : nat :=
+    match n with
+      0 ⇒ 0
+    | _ ⇒ zero (Nat.div2 n)
+    end.
+ +
+ +
+
+ +
+From Init.Wf +
+
+ +
+ +
+ +
+About Acc_intro_generator.
+ +
+ +
+Function zero' (n:nat) {wf lt n} : nat :=
+  match n with
+    0 ⇒ 0
+  | _ ⇒ zero' (Nat.div2 n)
+  end.
+ +
+ +
+End OpaqueWf.
+ +
+ +
+Inductive fuel :=
+ | FO : fuel
+ | FS : (unit → fuel) → fuel.
+ +
+ +
+Fixpoint foo (n:nat) x :=
+ match n with
+ | S n ⇒ FS (fun _ ⇒ foo n (foo n x))
+ | O ⇒ x
+ end.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.Iterates.html b/theories/html/hydras.Prelude.Iterates.html new file mode 100644 index 00000000..b1c8f5df --- /dev/null +++ b/theories/html/hydras.Prelude.Iterates.html @@ -0,0 +1,390 @@ + + + + + +hydras.Prelude.Iterates + + + + +
+ + + +
+ +

Library hydras.Prelude.Iterates

+ +
+
+ +
+ Iteration of a function (similar to Nat.iter) + Abstract Properties + +
+ +Experimental use of LibHyps + +
+ + +
+
+ +
+Open Scope nat_scope.
+From Coq Require Import RelationClasses Relations Arith Max Lia.
+From hydras Require Import Exp2.
+ +
+From LibHyps Require Import LibHyps.
+From hydras Require Import MoreLibHyps.
+Ltac rename_hyp n th ::= rename_short n th.
+ +
+ +
+Fixpoint iterate {A:Type}(f : A → A) (n: nat)(x:A) :=
+  match n with
+  | 0 ⇒ x
+  | S p ⇒ f (iterate f p x)
+  end. +
+ +
+Lemma iterate_comm {A: Type} f n (x:A)
+  : iterate f n (f x) = f (iterate f n x).
+ +
+
+ +
+Compatibility with Ackermann Library's definition +
+
+ +
+Lemma iterate_compat {f : nat → nat}(n:nat)(x:nat):
+  iterate f n x = nat_rec
+                    (fun _ ⇒ nat → nat)
+                    (fun x : nat ⇒ x)
+                    (fun (_ : nat) (rec : nat → nat) (x : nat) ⇒ f (rec x))
+                    n x.
+ +
+Lemma iterate_compat2 {A} (f : A → A) n :
+  ∀ x, iterate f n x = Nat.iter n f x.
+ +
+
+ +
+TODO : move to more generic libraries +
+
+ +
+Lemma iterate_compat3 f x n :
+  iterate f n x = nat_rec (fun _ : nat ⇒ nat) x (fun _ y : nat ⇒ f y) n.
+ +
+
+ +
+

Abstract properties of arithmetic functions

+ +
+
+ +
+ +
+Definition strict_mono f := ∀ n p, n < p → f n < f p.
+ +
+Definition dominates_from n g f := ∀ p, n ≤ p → f p < g p.
+ +
+Definition fun_le f g := ∀ n:nat, f n ≤ g n.
+Infix "<<=" := fun_le (at level 60).
+ +
+Definition dominates g f := ∃ n : nat, dominates_from n g f .
+Infix ">>" := dominates (at level 60).
+ +
+Definition dominates_strong g f := {n : nat | dominates_from n g f}.
+Infix ">>s" := dominates_strong (at level 60).
+ +
+ +
+Lemma S_pred_rw (f : nat → nat) : S <<= f →
+                                   ∀ x, S (Nat.pred (f x)) = f x.
+ +
+Lemma fun_le_trans f g h : f <<= g → g <<= h → f <<= h.
+ +
+Lemma mono_le f (Hf : strict_mono f) : ∀ n, n ≤ f n.
+ +
+Lemma mono_injective f (Hf : strict_mono f) :
+  ∀ n p , f n = f p → n = p.
+ +
+Lemma mono_weak f (H: strict_mono f) :
+  ∀ n p, n ≤ p → f n ≤ f p.
+ +
+Lemma dominates_from_trans :
+  ∀ f g h i j, dominates_from i g f →
+                    dominates_from j h g →
+                    dominates_from (Nat.max i j) h f .
+ +
+Lemma dominates_trans f g h :
+  dominates g f → dominates h g → dominates h f.
+ +
+Lemma dominates_trans_strong : ∀ f g h,
+    dominates_strong g f →
+    dominates_strong h g →
+    dominates_strong h f.
+ +
+
+ +
+

Abstract properties of iterate

+ +
+
+ +
+Lemma iterate_S_eqn {A:Type}(f : A → A) (n: nat)(x:A):
+  iterate f (S n) x = f (iterate f n x).
+ +
+Lemma iterate_S_eqn2 {A:Type}(f : A → A) (n: nat)(x:A):
+  iterate f (S n) x = (iterate f n (f x)).
+ +
+Lemma iterate_rw {A} {f : A → A} n :
+  ∀ x, iterate f (S n) x = iterate f n (f x).
+ +
+Lemma iterate_ext {A:Type}(f g: A → A) (H: ∀ x, f x = g x):
+  ∀ n x, iterate f n x = iterate g n x.
+ +
+ +
+Lemma iterate_le f (Hf : strict_mono f) :
+  ∀ i j, i ≤ j → ∀ z, iterate f i z ≤ iterate f j z.
+ +
+Lemma iterate_lt f (Hf : strict_mono f)(Hf': fun_le S f):
+  ∀ i j, i < j → ∀ z, iterate f i z < iterate f j z.
+ +
+Lemma iterate_lt_from f k:
+   strict_mono f →
+    ( ∀ n, k ≤ n → n < f n) →
+    ∀ i j : nat, i < j →
+                       ∀ z : nat, k ≤ z →
+                                       iterate f i z < iterate f j z.
+ +
+ +
+Lemma iterate_le_n_Sn (f: nat → nat):
+  (∀ x, x ≤ f x) →
+  ∀ n x, iterate f n x ≤ iterate f (S n) x. +
+Lemma iterate_le_np_le (f: nat → nat):
+  (∀ x, (x ≤ f x)%nat) →
+  ∀ n p x, (n ≤ p → iterate f n x ≤ iterate f p x)%nat.
+ +
+Lemma iterate_mono2 (f: nat → nat):
+  (∀ x y, x ≤ y → f x ≤ f y)%nat →
+  ∀ n x y, (x ≤ y → iterate f n x ≤ iterate f n y)%nat.
+ +
+Lemma iterate_mono f (Hf : strict_mono f) (Hf' : S <<= f):
+  ∀ n, strict_mono (iterate f n).
+ +
+Lemma iterate_ge : ∀ f , S <<= f →
+                              ∀ j n, j ≤ iterate f n j.
+ +
+Lemma iterate_Sge f j : S <<= f → S <<= iterate f (S j).
+ +
+Lemma iterate_ge' : ∀ f, id <<= f →
+                               ∀ n j, 0 < n → j ≤ iterate f n j.
+ +
+Lemma iterate_ge'' f : id <<= f → strict_mono f → ∀ i k,
+      k ≤ Nat.pred (iterate (fun z ⇒ S (f z)) (S i) k).
+ +
+Lemma strict_mono_iterate_S f :
+  strict_mono f → id <<= f →
+  ∀ i, strict_mono
+               (fun k ⇒ Nat.pred (iterate (fun z ⇒ S (f z)) (S i) k)).
+ +
+Lemma iterate_mono_1 (f g: nat → nat) (k:nat) (Hf: strict_mono f)
+      (Hf' : S <<= f)
+      (H : ∀ n, k ≤ n → f n ≤ g n) :
+  ∀ i n, k ≤ n → iterate f i n ≤ iterate g i n.
+ +
+Lemma iterate_dom_prop :
+  ∀ f g i (Hgt : S <<= f)
+         (Hm : strict_mono f) (Hm': strict_mono g),
+    dominates_from i g f →
+    ∀ k, 0 < k → dominates_from i (iterate g k) (iterate f k).
+ +
+Lemma dominates_from_le i j g f : i ≤ j →
+                                   dominates_from i g f →
+                                   dominates_from j g f .
+ +
+Lemma smono_Sle f : f 0 ≠ 0 → strict_mono f → S <<= f.
+ +
+
+ +
+

Second-order iterate

+ +
+
+ +
+Lemma iterate_ext2 {A:Type} (f g : (A → A) → A → A)
+      (h i : A→A) : (∀ x, h x = i x) →
+                     (∀ h' i', (∀ x, h' x = i' x) →
+                                     ∀ x, f h' x = g i' x) →
+                     ∀ n x, iterate f n h x = iterate g n i x.
+ +
+Lemma iterate2_mono (f : (nat→nat)->(nat→nat)):
+   (∀ g, strict_mono g → S <<= g → strict_mono (f g))->
+   (∀ g, strict_mono g → S <<= g → S <<= (f g))->
+   ∀ k g x y, strict_mono g → S <<= g →
+                     (x < y)%nat →
+                     (iterate f k g x < iterate f k g y)%nat.
+ +
+Lemma iterate2_mono_weak (f : (nat→nat)->(nat→nat)):
+   (∀ g, strict_mono g → S <<= g → strict_mono (f g))->
+   (∀ g, strict_mono g → S <<= g → S <<= (f g))->
+   ∀ k g x y, strict_mono g → S <<= g →
+                     (x ≤ y)%nat →
+                     (iterate f k g x ≤ iterate f k g y)%nat.
+ +
+Lemma iterate2_mono3 (phi : (nat→nat)->(nat→nat)) :
+  (∀ g, strict_mono g → S <<= g →
+             strict_mono (phi g) ∧ S <<= phi g)->
+  (∀ (f g : nat → nat), strict_mono f → S <<= f →
+                              strict_mono g → S <<= g →
+                              ((∀ x, f x ≤ g x) →
+                               ∀ x, phi f x ≤ phi g x)) →
+  ∀ g h, strict_mono g → S <<= g → strict_mono h → S <<= h →
+             (∀ x, g x ≤ h x) →
+  ∀ k x y, x ≤ y →
+                 iterate phi k g x ≤ iterate phi k h y.
+ +
+Lemma iterate2_mono2 (phi psi : (nat→nat)->(nat→nat)):
+  (∀ g, strict_mono g → S <<= g → strict_mono (phi g))->
+  (∀ g, strict_mono g → S <<= g → S <<= (phi g))->
+  (∀ g, strict_mono g → S <<= g → strict_mono (psi g))->
+  (∀ g, strict_mono g → S <<= g → S <<= (psi g))->
+  (∀ g x, strict_mono g → fun_le S g → phi g x ≤ psi g x) →
+  (∀ f g, strict_mono f → strict_mono g → S <<= f → S <<= g →
+               (∀ x, f x ≤ g x) → (∀ x, psi f x ≤ psi g x)) →
+  ∀ k g x y, strict_mono g → S <<= g →
+                    (x ≤ y)%nat →
+                    (iterate phi k g x ≤ iterate psi k g y)%nat.
+ +
+
+ +
+

Exponential and hyper exponential of base 2

+ +
+
+ +
+Lemma exp2_ge_S : S <<= exp2.
+ +
+Lemma exp2_mono : strict_mono exp2.
+ +
+Lemma exp2_mono_weak : ∀ n p, n≤ p → exp2 n ≤ exp2 p.
+ +
+Lemma exp2_as_iterate n : exp2 n = iterate (fun i ⇒ 2 × i)%nat n 1.
+ +
+Definition hyper_exp2 k := iterate exp2 k 1.
+ +
+Lemma hyper_exp2_S : ∀ n, hyper_exp2 (S n) = exp2 (hyper_exp2 n).
+ +
+Lemma iterate_ge_from : ∀ f i, dominates_from i f id →
+                               ∀ j, i ≤ j →
+                                          ∀ n,
+                                            j ≤ iterate f n j.
+ +
+Lemma dominates_iterate :
+  ∀ i f,
+    dominates_from i f id →
+    strict_mono f →
+    ∀ n,
+      {j:nat | i ≤ j ∧ dominates_from j (iterate f (S n)) id}.
+ +
+Corollary iterate_gt_diag' :
+  ∀ i f,
+    dominates_from i f id →
+    strict_mono f →
+    ∀ n, 0 < n →
+              {j:nat | i≤ j ∧ dominates_from j (iterate f n) id}.
+ +
+Corollary iterate_ge_diag' :
+  ∀ i f,
+    dominates_from i f id→
+    strict_mono f →
+    ∀ n,
+      {j:nat | i≤ j ∧ ∀ k, j≤ k → k ≤ iterate f n k}.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.LibHyps_Experiments.html b/theories/html/hydras.Prelude.LibHyps_Experiments.html new file mode 100644 index 00000000..c7192956 --- /dev/null +++ b/theories/html/hydras.Prelude.LibHyps_Experiments.html @@ -0,0 +1,57 @@ + + + + + +hydras.Prelude.LibHyps_Experiments + + + + +
+ + + +
+ +

Library hydras.Prelude.LibHyps_Experiments

+ +
+From LibHyps Require Export LibHyps.
+From hydras Require Export MoreLibHyps.
+ +
+ +
+From Coq Require Import List.
+Import ListNotations.
+#[local] Open Scope autonaming_scope.
+ +
+Ltac rename_hyp n th ::= rename_short n th.
+ +
+Goal ∀ n p , n ≤ p → ∀ q, p ≤ q → n ≤ q.
+ +
+Goal ∀ n p , n ≤ p → ∀ q, p ≤ q → n ≤ q.
+ +
+From Coq Require Import Arith.
+Parameters f g h : nat → nat.
+Parameter P : nat→nat→nat→ Prop.
+Goal ∀ x y , f (g (h x)) = h (g (f y)) → x = y → x < y →
+                   P x y x → f y ≠ f x.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.Merge_Sort.html b/theories/html/hydras.Prelude.Merge_Sort.html new file mode 100644 index 00000000..7668b56b --- /dev/null +++ b/theories/html/hydras.Prelude.Merge_Sort.html @@ -0,0 +1,292 @@ + + + + + +hydras.Prelude.Merge_Sort + + + + +
+ + + +
+ +

Library hydras.Prelude.Merge_Sort

+ +
+
+ +
+P. Casteran, S. Salvati +
+ + Maybe already done in StdLib ???? +
+
+ +
+From Coq Require Import List Wf_nat Recdef Compare_dec Arith Peano_dec
+     Lia RelationClasses Sorting.Sorted Sorting.Permutation.
+ +
+From hydras Require Import DecPreOrder DecPreOrder_Instances
+     Sort_spec.
+Import Relations Morphisms.
+ +
+Section Generic.
+  Variables (A:Type).
+ +
+  Section Splitting.
+ +
+
+ +
+A function that splits any list in two lists of (almost) the same length + +
+
+ +
+    Function split (l:list A):=
+      match l with
+        | nil
+        | _::nil ⇒ (l,nil)
+        | a::b::l ⇒
+          let (l1,l2):= split l in (a::l1,b::l2)
+      end.
+ +
+    Function split'_aux (l l':list A) :=
+      match l,l' with
+      | x::l,_::_::l' ⇒
+        let (l1,l2) := split'_aux l l' in
+        (x::l1, l2)
+      | _,_ ⇒ (nil,l)
+      end.
+ +
+    Function split' (l:list A) := split'_aux l l.
+ +
+ +
+ +
+
+ +
+Applying split to a list l returns a pair of two strictly shorter lists +
+
+ +
+    Lemma split_decr:
+      ∀ l1 l2 a b, l1 = a::b::l2 →
+                        length (fst(split l1)) < length l1 ∧
+                        length (snd(split l1)) < length l1.
+ +
+    Lemma split_permutation:
+      ∀ l, Permutation ((fst(split l))++snd(split l)) l.
+ +
+ +
+    Lemma split'_aux_length_preserve:
+      ∀ l l',
+        length(fst(split'_aux l l')) + length(snd (split'_aux l l')) =
+        length l.
+ +
+    Lemma split'_aux_length_fst:
+      ∀ l l',
+        length(fst(split'_aux l l')) =
+ min (Nat.div2(length l')) (length l).
+ +
+ +
+ +
+ +
+    Lemma split'_decr:
+      ∀ l1 l2 a b, l1 = a::b::l2 →
+                        length (fst(split' l1)) < length l1 ∧
+                        length (snd(split' l1)) < length l1.
+ +
+    Lemma split'_aux_eq:
+      ∀ l l', ((fst(split'_aux l l'))++snd(split'_aux l l')) = l.
+ +
+    Lemma split'_permutation:
+      ∀ l, Permutation ((fst(split' l))++snd(split' l)) l.
+  End Splitting.
+ +
+  Section Merging.
+ +
+    Fixpoint merge (leb : A → A →bool) l1 l2:=
+      let fix merge_aux l2 {struct l2}:=
+          match l1,l2 with
+            | nil,_ ⇒ l2
+            | _,nil ⇒ l1
+            | a1::l1',a2::l2' ⇒
+              if leb a1 a2 then a1 :: merge leb l1' l2 else a2 :: merge_aux l2'
+          end
+      in merge_aux l2.
+ +
+    Variable le : relation A.
+ +
+    Context (le_total : TotalPreOrder le).
+    Context (le_dec : RelDecision le).
+ +
+    Notation "a <= b" := (le a b).
+ +
+    Lemma merge_rect:
+      ∀ P: list A → list A → list A → Type,
+        (∀ l:list A, P nil l l) →
+        (∀ l: list A, P l nil l) →
+        (∀ a1 l1 a2 l2,
+           a1 ≤ a2 →
+           P l1 (a2::l2) (merge (fun x y ⇒ bool_decide (x ≤ y)) l1 (a2::l2)) →
+           P (a1::l1) (a2::l2) (a1::(merge (fun x y ⇒ bool_decide (x ≤ y))l1 (a2::l2)))) →
+        (∀ a1 l1 a2 l2,
+           ¬ le a1 a2 →
+           P (a1::l1) l2 (merge (fun x y ⇒ bool_decide (x ≤ y))(a1::l1) l2) →
+           P (a1::l1) (a2::l2) (a2::(merge (fun x y ⇒ bool_decide (x ≤ y)) (a1::l1) l2))) →
+        ∀ l1 l2, P l1 l2 (merge (fun x y ⇒ bool_decide (x ≤ y)) l1 l2).
+ +
+    Definition merge_ind ( P: list A → list A → list A → Prop) :=
+      merge_rect P.
+ +
+    Definition merge_rec( P: list A → list A → list A → Set) :=
+      merge_rect P.
+ +
+    Ltac induction_merge l1 l2:= pattern (merge (fun x y ⇒ bool_decide (x ≤ y)) l1 l2 ); apply merge_rect.
+ +
+    Lemma merge_equation:
+      ∀ l1 l2,
+        merge (fun x y ⇒ bool_decide (x ≤ y)) l1 l2 =
+        match l1,l2 with
+          | nil,_ ⇒ l2
+          | _,nil ⇒ l1
+          | a1::l1',a2::l2' ⇒
+            if decide (a1 ≤ a2) then a1 :: merge (fun x y ⇒ bool_decide (x ≤ y)) l1' l2
+            else a2 :: merge (fun x y ⇒ bool_decide (x ≤ y)) l1 l2'
+        end.
+ +
+    Section Correctness.
+ +
+ +
+ +
+ +
+ +
+      Lemma merge_Forall:
+        ∀ (f:A→Prop) l1 l2,
+          List.Forall f l1 →
+          List.Forall f l2 →
+          List.Forall f (merge (fun x y ⇒ bool_decide (x ≤ y)) l1 l2) .
+ +
+      Lemma merge_LocallySorted:
+        ∀ l1 l2, LocallySorted le l1 → LocallySorted le l2 →
+                      LocallySorted le (merge (fun x y ⇒ bool_decide (x ≤ y)) l1 l2).
+ +
+      Lemma merge_permutation:
+        ∀ l1 l2, Permutation (l1++l2) (merge (fun x y ⇒ bool_decide (x ≤ y)) l1 l2).
+ +
+      Section merge_sort.
+        Variable split: list A → (list A × list A).
+ +
+        Hypothesis split_decr:
+          ∀ l1 l2 a b, l1 = a::b::l2 →
+                        length (fst(split l1)) < length l1 ∧
+                        length (snd(split l1)) < length l1.
+ +
+        Hypothesis split_permutation:
+          ∀ l, Permutation ((fst(split l))++snd(split l)) l.
+ +
+ +
+        Function merge_sort (leb : A → A →bool) (l:list A) {measure length l} :=
+          match l with
+          | nil | _ :: nil ⇒ l
+          | _::_::_ ⇒
+            let (l1,l2) := split l in
+            merge leb (merge_sort leb l1) (merge_sort leb l2)
+          end.
+        Defined.
+ +
+        Theorem merge_sort_correct: sort_correct A le (merge_sort (fun x y ⇒ bool_decide (x ≤ y))).
+      End merge_sort.
+    End Correctness.
+ +
+ +
+    Definition sp_merge_sort:= merge_sort split split_decr.
+ +
+    Definition stable_merge_sort := merge_sort split' split'_decr.
+ +
+  End Merging.
+ +
+End Generic.
+ +
+Check (sp_merge_sort: sort_fun_t).
+Check (stable_merge_sort: sort_fun_t).
+ +
+Theorem sp_mergesort_OK : sort_spec sp_merge_sort.
+ +
+Theorem stable_mergesort_OK : sort_spec stable_merge_sort.
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.MoreDecidable.html b/theories/html/hydras.Prelude.MoreDecidable.html new file mode 100644 index 00000000..7d0cc34a --- /dev/null +++ b/theories/html/hydras.Prelude.MoreDecidable.html @@ -0,0 +1,51 @@ + + + + + +hydras.Prelude.MoreDecidable + + + + +
+ + + +
+ +

Library hydras.Prelude.MoreDecidable

+ +
+ +
+From Coq Require Export Decidable Arith Lia.
+ +
+Remark boundedCheck :
+ ∀ P : nat → Prop,
+ (∀ x : nat, decidable (P x)) →
+ ∀ c : nat,
+ (∀ d : nat, d < c → ¬ P d) ∨ (∃ d : nat, d < c ∧ P d).
+ +
+Remark smallestExists :
+ ∀ P : nat → Prop,
+ (∀ x : nat, decidable (P x)) →
+ ∀ c : nat,
+ P c → ∃ a : nat, P a ∧ (∀ b : nat, b < a → ¬ P b).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.MoreLibHyps.html b/theories/html/hydras.Prelude.MoreLibHyps.html new file mode 100644 index 00000000..37e66ba5 --- /dev/null +++ b/theories/html/hydras.Prelude.MoreLibHyps.html @@ -0,0 +1,59 @@ + + + + + +hydras.Prelude.MoreLibHyps + + + + +
+ + + +
+ +

Library hydras.Prelude.MoreLibHyps

+ +
+From LibHyps Require Import LibHyps.
+ +
+Tactic Notation (at level 4) tactic4(Tac) "/" "dr" :=
+  Tac ; {< fun h ⇒ try generalize dependent h }.
+ +
+Tactic Notation (at level 4) tactic4(Tac) "/" "r?" :=
+  Tac ; {< fun h ⇒ try revert h }.
+ +
+From Coq Require Import List.
+Import ListNotations.
+#[local] Open Scope autonaming_scope.
+ +
+ +
+Ltac old_rename := rename_hyp_default.
+ +
+Ltac rename_short n th :=
+  match th with
+  | (?f ?x ?y) ⇒ name ((f # 1) ++ (x # 1))
+  | (?f ?x) ⇒ name ((f # 1))
+  | _ ⇒ old_rename n th
+  end.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.MoreLists.html b/theories/html/hydras.Prelude.MoreLists.html new file mode 100644 index 00000000..e9108978 --- /dev/null +++ b/theories/html/hydras.Prelude.MoreLists.html @@ -0,0 +1,417 @@ + + + + + +hydras.Prelude.MoreLists + + + + +
+ + + +
+ +

Library hydras.Prelude.MoreLists

+ +
+
+ +
+Complements on lists + +
+ + Pierre Castéran, Univ. Bordeaux and LaBRI +
+
+ +
+From Coq Require Export List Arith Relations Lia.
+From Coq Require Import Sorting.Sorted Compare_dec Sorting.Sorted.
+ +
+
+ +
+

Sets of natural numbers as lists

+ +
+ +

Definitions

+ +
+ + numbers from i to i+n-1 +
+
+ +
+Fixpoint iota_from i n :=
+  match n with 0 ⇒ nil
+            | S p ⇒ i :: iota_from (S i) p
+  end.
+ +
+Definition interval i j := iota_from i (S j - i).
+ +
+Definition bounded_by (n:nat)(s: list nat) :=
+  List.Forall (fun i ⇒ i≤n)%nat s.
+ +
+ +
+Definition shift (l: list nat) := List.map S l.
+ +
+Fixpoint unshift (l : list nat) : list nat :=
+  match l with
+      nil ⇒ nil
+    | 0 :: l' ⇒ unshift l'
+    | S i :: l' ⇒ i :: unshift l'
+  end.
+ +
+
+ +
+sorted list of elements greater or equal than n +
+
+ +
+Inductive sorted_ge (n: nat) : list nat → Prop :=
+| sorted_ge_nil : sorted_ge n nil
+| sorted_ge_one : ∀ p, n≤p → sorted_ge n (p::nil)
+| sorted_ge_cons: ∀ p q s,
+    n ≤ p → p < q → sorted_ge p (q::s) →
+    sorted_ge n (p::q::s).
+ +
+
+ +
+simpler than StdLib's last +
+
+ +
+Fixpoint simple_last {A} (x:A) s :=
+  match s with
+      nil ⇒ x
+    | i::s' ⇒ simple_last i s'
+  end.
+ +
+
+ +
+the list (x::s) without its last item +
+
+ +
+Fixpoint but_last {A:Type}(x:A) (s : list A) :=
+  match s with
+  | nil ⇒ nil
+  | y::s' ⇒ x :: but_last y s'
+  end.
+ +
+Lemma but_last_iota_from : ∀ l i,
+      but_last i (iota_from (S i) (S l)) = i::iota_from (S i) l.
+ +
+Lemma interval_length i j : length (interval i j) = S j - i.
+ +
+Lemma but_last_interval i j:
+  i < j →
+  but_last i (interval (S i) (S j)) = i:: interval (S i) j.
+ +
+Lemma but_last_shift' s : ∀ x,
+     but_last (S x) (shift s) = shift (but_last x s).
+ +
+Lemma unshift_but_last s : ∀ x,
+    ¬ In 0 (x::s) →
+    unshift (but_last x s) = but_last (Nat.pred x) (unshift s).
+ +
+Lemma unshift_app : ∀ s t, unshift (s ++ t) = unshift s ++ unshift t.
+ +
+Lemma unshift_not_nil : ∀ s, ¬ In 0 s → s ≠ nil → unshift s ≠ nil.
+ +
+Lemma but_last_app {A} : ∀ s (x:A),
+  but_last x s ++ simple_last x s :: nil = x::s.
+ +
+Lemma but_last_iota_from' j : ∀ i, but_last i (iota_from (S i) (S j)) =
+                                       iota_from i (S j).
+ +
+Definition ptwise_le: list nat → list nat → Prop := Forall2 le.
+ +
+
+ +
+

Lemmas

+ +
+
+ +
+Lemma empty_interval i j : (j < i)%nat → interval i j = nil.
+ +
+Lemma shift_iota_from : ∀ i l, shift (iota_from i l) = iota_from (S i) l.
+ +
+Lemma shift_interval (i j: nat): shift (interval i j) = interval (S i) (S j).
+ +
+Lemma unshift_iota_from : ∀ i l, unshift (iota_from (S i) l) =
+                                       iota_from i l.
+ +
+Lemma unshift_interval (i j: nat): unshift (interval (S i) (S j)) =
+                                   interval i j.
+ +
+Lemma unshift_interval_pred (i j:nat) : 0 < j →
+  unshift (interval (S i) j) = interval i (Nat.pred j).
+ +
+Lemma shift_no_zero l : ¬ In 0 (shift l).
+ +
+Lemma shift_unshift l : unshift (shift l) = l.
+ +
+Lemma unshift_shift l (H: ¬ In 0 l): shift (unshift l) = l.
+ +
+Lemma unshift_pred : ∀ (s: list nat), ¬ In 0 s →
+                                            unshift s = List.map Nat.pred s.
+ +
+ +
+Lemma sorted_ge_Forall (n:nat) : ∀ l, sorted_ge n l →
+                                           Forall (fun x ⇒ n ≤ x) l.
+ +
+Lemma sorted_ge_trans n p l : n ≤ p → sorted_ge p l → sorted_ge n l.
+ +
+Lemma sorted_ge_not_In (n:nat) : ∀ l, sorted_ge (S n) l →
+                                           ¬ In n l.
+ +
+#[global] Hint Constructors sorted_ge : lists.
+ +
+Lemma sorted_inv_gt : ∀ n p s, sorted_ge n (p::s) →
+                                    (p < n)%nat → False.
+ +
+Lemma iota_from_app n p :
+  iota_from n (S p) = iota_from n p ++ (n+p::nil)%nat.
+ +
+Lemma iota_from_plus : ∀ k i j, iota_from i (k+j) =
+                                     iota_from i k ++ iota_from (k+i) j.
+ +
+Lemma interval_not_empty : ∀ i j, i ≤ j → interval i j ≠ nil.
+ +
+Lemma interval_not_empty_iff (n p : nat) :
+  interval n p ≠ nil ↔ (n ≤ p)%nat.
+ +
+Lemma interval_singleton (i:nat) : interval i i = i::nil.
+ +
+Lemma interval_app (i j k:nat):
+  (i ≤ j)%nat → (j ≤ k)%nat →
+  interval i k = interval i j ++ interval (S j) k.
+ +
+Lemma iota_from_unroll i l : iota_from i (S l) = i :: iota_from (S i) l.
+ +
+Lemma interval_unroll : ∀ i j:nat , (i < j)%nat →
+                                       interval i j = i :: interval (S i) j.
+ +
+Lemma iota_from_sorted_ge : ∀ p n q : nat,
+    (q ≤ n)%nat →
+    sorted_ge q (iota_from n p).
+ +
+Lemma interval_sorted_ge: ∀ p n q : nat, (q ≤ n)%nat →
+                                                sorted_ge q (interval n p).
+ +
+Lemma iota_from_lt_not_In i j l : i < j → ¬ In i (iota_from j l).
+ +
+Lemma interval_lt_not_In :
+  ∀ i j k, i < j → ¬ In i (interval j k).
+ +
+Section Forall2_right_induction.
+ +
+Inductive Forall2R {A B: Type} (R: A → B → Prop) : list A → list B → Prop :=
+  Forall2R_nil : Forall2R R nil nil
+| Forall2R_last : ∀ l l' x y l1 l'1, Forall2R R l l' →
+                                          R x y →
+                                          l1 = l++(x::nil) →
+                                          l'1 = l' ++ (y::nil) →
+                                          Forall2R R l1 l'1.
+ +
+Remark Forall2R_cons {A B: Type} (R: A → B → Prop):
+  ∀ l l', Forall2R R l l' → ∀ x y, R x y → Forall2R R (x::l) (y:: l').
+ +
+ +
+Remark Forall2_R {A B: Type} (R: A → B → Prop) :
+  ∀ l l', Forall2 R l l' → Forall2R R l l'.
+ +
+Remark Forall2_RR {A B: Type} (R: A → B → Prop) :
+  ∀ l l', Forall2R R l l' → Forall2 R l l'.
+ +
+Lemma Forall2R_iff {A B: Type} (R: A → B → Prop) :
+  ∀ l l', Forall2R R l l' ↔ Forall2 R l l'.
+ +
+Lemma Forall2_indR {A B : Type} (R : A → B → Prop) (P : list A → list B → Prop):
+P nil nil →
+(∀ (l : list A) (l' : list B) (x : A) (y : B)
+   (l1 : list A) (l'1 : list B),
+ Forall2 R l l' → P l l' →
+ R x y → l1 = l ++ x :: nil → l'1 = l' ++ y :: nil → P l1 l'1) →
+  ∀ (l : list A) (l0 : list B), Forall2 R l l0 → P l l0.
+ +
+End Forall2_right_induction.
+ +
+Lemma sorted_le : ∀ i j X, i ≤ j →
+                                sorted_ge j X →
+                                sorted_ge i X.
+ +
+ +
+Lemma sorted_tail : ∀ i j X, sorted_ge i (j::X) →
+                                  sorted_ge i X.
+ +
+Lemma sorted_tail' : ∀ i j X, sorted_ge i (j::X) →
+                                   sorted_ge j X.
+ +
+Lemma sorted_head : ∀ n m s, sorted_ge n (m::s)
+                                  → n≤m.
+ +
+Lemma Sorted_mono {A:Type}(R S : relation A)
+      (Hincl : ∀ x y, R x y → S x y):
+  ∀ l, Sorted R l → Sorted S l.
+ +
+Lemma sorted_ge_iff0 : ∀ l n, sorted_ge n l ↔
+                                    LocallySorted Peano.lt l ∧
+                                    List.Forall (fun i ⇒ n ≤ i) l.
+ +
+Lemma sorted_ge_iff : ∀ l n, sorted_ge n l ↔
+                                    Sorted lt l ∧
+                                    List.Forall (fun i ⇒ n ≤ i) l.
+ +
+Lemma sorted_ge_prefix :
+  ∀ l1 n l2, sorted_ge n (l1 ++ l2) → sorted_ge n l1.
+ +
+Lemma sorted_In : ∀ i X, Sorted lt (i::X) → ∀ j, In j X → i < j.
+ +
+Lemma sorted_not_in_tail : ∀ i j X, Sorted lt (i::X) → j≤i →
+                                           ¬ (In j X).
+ +
+Remark simple_last_correct {A}: ∀ s (x:A), simple_last x s = last s x.
+ +
+Lemma In_sorted_ge_inv : ∀ x y s,
+                             In x (y::s) →
+                             sorted_ge y s →
+                             y < x ∧ In x s ∨ y = x.
+ +
+Lemma incl_inv : ∀ x y l1 l2, Sorted lt (x::l1) →
+                                    Sorted lt (y::l2) →
+                                    incl (x::l1)(y::l2) →
+                                    y ≤ x ∧ incl l1 l2.
+ +
+Lemma incl_decomp : ∀ l1 l2, Sorted lt l1 →
+                                   Sorted lt l2 →
+                                   incl l1 l2 →
+                                   ∃ l3 l4, l2 = l3 ++ l4 ∧
+                                                  ptwise_le l3 l1.
+ +
+  Lemma simple_last_app {A}: ∀ l l1 (x y:A), simple_last x (l++(y::l1)) =
+                                     simple_last y l1 .
+ +
+Lemma simple_last_app1 {A}: ∀ l (x y:A), simple_last x (l++(y::nil)) = y.
+ +
+Lemma sorted_max_1 : ∀ s n, sorted_ge n s →
+                                 (n ≤ simple_last n s)%nat.
+ +
+Lemma sorted_cut :∀ l1 n x l2, sorted_ge n (l1++(x::l2)) →
+                                    simple_last n l1 ≤ x.
+ +
+Lemma sorted_max_2 : ∀ s n, sorted_ge n s →
+                                 Forall (fun i ⇒
+                                           (i ≤ simple_last n s)%nat)
+                                        s.
+ +
+Lemma sorted_ge_suffix :
+  ∀ l1 n l2, sorted_ge n (l1 ++ l2) → sorted_ge (simple_last n l1) l2.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.MoreOrders.html b/theories/html/hydras.Prelude.MoreOrders.html new file mode 100644 index 00000000..9acd9f5a --- /dev/null +++ b/theories/html/hydras.Prelude.MoreOrders.html @@ -0,0 +1,115 @@ + + + + + +hydras.Prelude.MoreOrders + + + + +
+ + + +
+ +

Library hydras.Prelude.MoreOrders

+ +
+
+ +
+Complements on strict orders +
+
+ +
+From Coq Require Import RelationClasses Relation_Operators Ensembles Setoid.
+Import Relation_Definitions.
+Set Implicit Arguments.
+ +
+Definition leq {A:Type}(lt : relation A): relation A :=
+  clos_refl A lt.
+ +
+Section Defs.
+ +
+  Variables (A : Type)
+            (lt: relation A).
+ +
+  #[local] Infix "<" := lt.
+  #[local] Infix "<=" := (leq lt).
+ +
+  Definition Least {sto : StrictOrder lt} (x : A):=
+    ∀ y, x ≤ y.
+ +
+  Definition Successor {sto : StrictOrder lt} (y x : A):=
+    x < y ∧ (∀ z, x < z → z < y → False).
+ +
+  Definition Limit {sto : StrictOrder lt} (x:A) :=
+    (∃ w:A, w < x) ∧
+    (∀ y:A, y < x → ∃ z:A, y < z ∧ z < x).
+ +
+  Definition Omega_limit
+              {sto : StrictOrder lt} (s: nat → A) (x:A) :=
+    (∀ i: nat, s i < x) ∧
+    (∀ y, y < x → ∃ i:nat, y < s i).
+ +
+  Definition Omega_limit_type
+              `{lt : relation A}
+              {sto : StrictOrder lt}
+              (s: nat → A) (x:A) : Type :=
+    ((∀ i: nat, s i < x) ×
+     (∀ y, y < x → {i:nat | y < s i}))%type.
+ +
+  Lemma Omega_limit_not_Succ
+        {sto : StrictOrder lt} (s: nat → A) (x:A) :
+    Omega_limit s x →
+    ∀ y, ¬ Successor x y.
+ +
+  Lemma Least_not_Succ {sto : StrictOrder lt} (x:A) :
+    Least x → ∀ z, ¬ Successor x z.
+ +
+  Lemma Omega_limit_Limit
+        {sto : StrictOrder lt} (s: nat → A) (x:A) :
+    Omega_limit s x → Limit x.
+ +
+  Lemma Least_not_Limit {sto : StrictOrder lt} x :
+    Least x → ¬ Limit x.
+ +
+End Defs.
+ +
+Lemma le_lt_eq {A}{lt: relation A}:
+  ∀ a b, leq lt a b ↔ lt a b ∨ a = b.
+ +
+#[ global ] Instance leq_trans {A:Type}{lt: relation A}{sto: StrictOrder lt}:
+  Transitive (leq lt).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.MoreVectors.html b/theories/html/hydras.Prelude.MoreVectors.html new file mode 100644 index 00000000..4f82e08b --- /dev/null +++ b/theories/html/hydras.Prelude.MoreVectors.html @@ -0,0 +1,236 @@ + + + + + +hydras.Prelude.MoreVectors + + + + +
+ + + +
+ +

Library hydras.Prelude.MoreVectors

+ +
+From Coq Require Export Bool Arith Vector Lia.
+Import Vector VectorNotations.
+ +
+
+ +
+generalities on vectors +
+
+ +
+Arguments cons {A} _ {n} _ .
+Arguments nil {A}.
+ +
+Definition Vid {A : Type} {n:nat} : t A n → t A n :=
+  match n with
+    0 ⇒ fun v ⇒ []
+  | S p ⇒ fun v ⇒ hd v :: tl v
+  end.
+ +
+ +
+Lemma Vid_eq : ∀ (n:nat) (A:Type)(v:t A n), v = Vid v.
+ +
+Theorem t_0_nil : ∀ (A:Type) (v:t A 0), v = nil.
+ +
+Theorem decomp :
+  ∀ (A : Type) (n : nat) (v : t A (S n)),
+    v = cons (hd v) (tl v).
+ +
+Lemma decomp1 {A : Type} (v : t A 1):
+  v = cons (hd v) (nil).
+ +
+Definition vector_double_rect :
+  ∀ (A:Type) (X: ∀ (n:nat),(t A n)->(t A n) → Type),
+    X 0 nil nil →
+    (∀ n (v1 v2 : t A n) a b, X n v1 v2 →
+                                   X (S n) (cons a v1) (cons b v2)) →
+    ∀ n (v1 v2 : t A n), X n v1 v2.
+Defined.
+ +
+Definition vector_triple_rect :
+  ∀ (A:Type)
+         (X: ∀ (n:nat),
+             t A n → t A n → t A n →Type),
+    X 0 nil nil nil →
+    (∀ n (v1 v2 v3: t A n) a b c, X n v1 v2 v3→
+                                       X (S n) (cons a v1) (cons b v2)(cons c v3)) →
+    ∀ n (v1 v2 v3: t A n), X n v1 v2 v3.
+Defined.
+ +
+Fixpoint vector_nth (A:Type)(n:nat)(p:nat)(v:t A p){struct v}
+  : option A :=
+  match n,v with
+    _ , nil ⇒ None
+  | 0 , cons b _ ⇒ Some b
+  | S n', @cons _ _ p' v' ⇒ vector_nth A n' p' v'
+  end.
+ +
+Arguments vector_nth {A } n {p}.
+ +
+Lemma Forall_inv {A :Type}(P: A → Prop)(n:nat)
+  a v : Vector.Forall P (n:= S n) (Vector.cons a v) →
+          P a ∧ Vector.Forall P v .
+ +
+Lemma Forall2_inv {A B:Type}(P: A → B → Prop)(n:nat)
+  a b v w : Vector.Forall2 P (n:=S n)
+              (Vector.cons a v)
+              (Vector.cons b w) →
+            P a b ∧ Vector.Forall2 P v w.
+ +
+
+ +
+Computes the vector f(from), f(from+1), f(from+2),...,f (from+n-1) +
+
+ +
+Fixpoint vect_from_fun {A} (f : nat → A) n from : t A n :=
+  match n return t A n with
+    0%nat ⇒ nil
+  | S p ⇒ cons (f from) (vect_from_fun f p (S from))
+  end.
+ +
+
+ +
+On vector decomposition +
+
+ +
+Notation vfst := Vector.hd.
+Notation vsnd v := (Vector.hd (Vector.tl v)).
+Notation vthird v := (Vector.hd (Vector.tl (Vector.tl v))).
+Notation vfourth v := (Vector.hd (Vector.tl (Vector.tl (Vector.tl v)))).
+ +
+Lemma decomp2 {A} : ∀ v : Vector.t A 2,
+    v = [Vector.hd v; Vector.hd (Vector.tl v)].
+ +
+Lemma decompos2 {A} : ∀ v: Vector.t A 2, {a : A & {b : A | v = [a;b]}}.
+ +
+Definition match2 {A B:Type} (f : A → A → B) (v: Vector.t A 2): B :=
+  match (decompos2 v )
+  with existT _ x Hx ⇒
+         match Hx with exist _ y _ ⇒ f x y end end.
+ +
+Definition Vec2_proj {A} (P2 : A → A → Prop) : Vector.t A 2 → Prop.
+Defined.
+ +
+ +
+Ltac vdec2 v a b :=
+  let x := fresh a in
+  let y :=fresh b in
+  let tmp := fresh "tmp" in
+  let e :=fresh "e" in
+  destruct (decompos2 v) as [x tmp]; destruct tmp as [y e]; subst v.
+ +
+Lemma In_cases {A:Type}{n:nat} (v: t A (S n)) :
+  ∀ x, In x v →
+            x = hd v ∨ In x (tl v).
+ +
+Lemma Forall_and {A:Type}(P: A → Prop) {n:nat} (v: t A (S n)) :
+  Forall P v → P (hd v) ∧ Forall P (tl v).
+ +
+
+ +
+For V8.11 +
+
+ +
+Lemma Forall_forall {A:Type}(P: A → Prop) :
+  ∀ {n:nat} (v: t A n) ,
+    Forall P v ↔ (∀ a, In a v → P a).
+ +
+
+ +
+Vectors of natural numbers +
+ +

Maximum of a vector of nat

+ +
+
+ +
+ +
+Fixpoint max_v {n:nat} (v: Vector.t nat n) : nat :=
+  match v with
+  | nil ⇒ 0
+  | cons x t ⇒ max x (max_v t)
+  end.
+ +
+ +
+ +
+Lemma max_v_2 : ∀ x y, max_v (x::y::nil) = max x y. +
+ +
+Lemma max_v_lub : ∀ n (v: t nat n) y,
+    (Forall (fun x ⇒ x ≤ y) v) →
+    max_v v ≤ y. +
+ +
+Lemma max_v_ge : ∀ n (v: t nat n) y,
+    In y v → y ≤ max_v v. +
+Lemma max_v_tl {n:nat}(v: Vector.t nat (S n)) :
+  max_v (Vector.tl v) ≤ max_v v.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.More_Arith.html b/theories/html/hydras.Prelude.More_Arith.html new file mode 100644 index 00000000..ae4f2805 --- /dev/null +++ b/theories/html/hydras.Prelude.More_Arith.html @@ -0,0 +1,177 @@ + + + + + +hydras.Prelude.More_Arith + + + + +
+ + + +
+ +

Library hydras.Prelude.More_Arith

+ +
+ +
+
+ +
+Note by Pierre: + +
+ + Some lemmas of this file are possibly in Standard Library +
+
+ +
+From Coq Require Import Arith Lia .
+Import Nat.
+ +
+Section Arith_lemmas.
+ +
+ +
+Lemma nat_double_or_s_double :
+  ∀ n, {∃ p, n = double p} + {∃ p, n = S (double p)}.
+ +
+Lemma div2_double_is_id :∀ n : nat, div2 (double n) = n.
+ +
+Lemma double_S (n:nat) : double (S n) = S (S (double n)).
+ +
+Lemma double_plus (x y: nat): double (x + y)= double x + double y.
+ +
+Lemma double_inj :
+  ∀ (m n : nat), double m = double n → m = n.
+ +
+Lemma double_is_even :
+  ∀ n : nat, Nat.Even (double n).
+ +
+Lemma not_double_is_s_double :
+  ∀ (m n : nat), S (double m) ≠ double n.
+ +
+Lemma div2_of_Even n: Nat.Even n → double (div2 n) = n.
+ +
+Lemma even_prod :
+  ∀ p q, Nat.Even ((p + q + 1) × (p + q)).
+ +
+Lemma plus_2 :
+  ∀ n, S (S n) = n + 2.
+ +
+Lemma div2_incr :
+  ∀ n m, n ≤ m → div2 n ≤ div2 m.
+ +
+Lemma div2_even_plus :
+  ∀ n m, Even n → div2 n + m = div2 (n + (double m)).
+ +
+Lemma mult_lt_lt :
+  ∀ p p' k, p × k < p' × k → p < p'.
+ +
+Lemma minus_semi_assoc :
+  ∀ a b c, b > c → a + (b - c) = (a + b) - c.
+ +
+Lemma div_not_qlt :
+  ∀ (a b : nat) (q q' r r' : nat),
+    a = q × b + r → a = q' × b + r' → b > r → b > r' → ¬ q < q'.
+ +
+Lemma div_eucl_unique :
+  ∀ (a b : nat) (q q' r r' : nat),
+    a = q × b + r → a = q' × b + r' → b > r → b > r' → q = q' ∧ r = r'.
+ +
+Lemma max_le_plus (n p: nat) : Nat.max n p ≤ n + p.
+ +
+Lemma max_le_regR : ∀ n p q, p ≤ q → max p n ≤ max q n.
+ +
+Lemma max_le_regL : ∀ n p q, p ≤ q → max n p ≤ max n q.
+ +
+Lemma lt_lt_Sn : ∀ a b c, a < b → b < S c → a < c.
+ +
+End Arith_lemmas.
+ +
+
+ +
+From Cantor contrib +
+
+ +
+Notation power := Nat.pow (only parsing).
+ +
+Lemma power_of_1 : ∀ p, power 1 p = 1.
+ +
+Goal ∀ a b, 0 < power (S a) b.
+ +
+Lemma pred_of_power : ∀ b e, pred (power (S b) (S e)) =
+                                  (power (S b) e)*b +
+                                  pred (power (S b) e).
+ +
+ +
+ +
+Lemma get_predecessor : ∀ (n:nat), 0 < n → {p:nat | n = S p}.
+ +
+Ltac pred_exhib H name :=
+  match type of H
+       with O < ?n ⇒
+         case (get_predecessor H); intro name; intro
+  end.
+ +
+ +
+Lemma Euc1 : ∀ b q q' r r', 0 < b →
+                                  q×b + r = q'×b + r' →
+                                  r < b → r' < b → q = q'.
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.OrdNotations.html b/theories/html/hydras.Prelude.OrdNotations.html new file mode 100644 index 00000000..cc89b016 --- /dev/null +++ b/theories/html/hydras.Prelude.OrdNotations.html @@ -0,0 +1,46 @@ + + + + + +hydras.Prelude.OrdNotations + + + + +
+ + + +
+ +

Library hydras.Prelude.OrdNotations

+ +
+ +
+Reserved Notation "x 'o<' y" (at level 70, no associativity).
+Reserved Notation "x 'o<=' y" (at level 70, no associativity).
+Reserved Notation "x 'o>=' y" (at level 70, no associativity).
+Reserved Notation "x 'o>' y" (at level 70, no associativity).
+ +
+Reserved Notation "x 'o<=' y 'o<=' z" (at level 70, y at next level).
+Reserved Notation "x 'o<=' y 'o<' z" (at level 70, y at next level).
+Reserved Notation "x 'o<' y 'o<' z" (at level 70, y at next level).
+Reserved Notation "x 'o<' y 'o<=' z" (at level 70, y at next level).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.Restriction.html b/theories/html/hydras.Prelude.Restriction.html new file mode 100644 index 00000000..685fcd9b --- /dev/null +++ b/theories/html/hydras.Prelude.Restriction.html @@ -0,0 +1,130 @@ + + + + + +hydras.Prelude.Restriction + + + + +
+ + + +
+ +

Library hydras.Prelude.Restriction

+ +
+
+ +
+ Pierre Casteran + LaBRI, University of Bordeaux and LaBRI + +
+ + Restriction of a binary relation + +
+
+ +
+From Coq Require Import Wellfounded Ensembles Relations.
+ +
+ +
+Definition restrict {A:Type}(E: Ensemble A)(R: relation A) :=
+  fun a b ⇒ E a ∧ R a b ∧ E b.
+ +
+ +
+Section restricted_recursion.
+ +
+  Variables (A:Type)(E:A→Prop)(R:A→A→Prop).
+ +
+ +
+  Definition well_founded_restriction :=
+    ∀ (a:A), E a → Acc (restrict E R) a.
+ +
+
+ +
+Induction principle for the restriction of R to E +
+
+ +
+  Definition well_founded_restriction_rect :
+    well_founded_restriction →
+    ∀ X : A → Type,
+      (∀ x : A, E x → (∀ y : A, In _ E y → R y x → X y) → X x) →
+      ∀ a : A, In _ E a → X a.
+ +
+End restricted_recursion.
+ +
+Section Fix.
+  Variables (A:Type)(E: Ensemble A)(R: relation A).
+  Hypothesis Hwf : well_founded_restriction A E R.
+ +
+ +
+  Variable P : A → Type.
+  Variable F : (∀ x : A, E x → (∀ y : A, In _ E y → R y x → P y) → P x).
+ +
+  Lemma restriction_fwd : ∀ x y (H: restrict E R x y), E x.
+ +
+  Definition restrict_build x y (Hx: E x)(Hy : E y)(H : R x y) :=
+    conj Hx (conj H Hy) .
+ +
+  Fixpoint FixR_F (x:A)(Hx : E x)(a: Acc (restrict E R)x ) : P x :=
+    F _ Hx (fun (y:A) (h0 : E y) (h : R y x) ⇒
+              FixR_F y (restriction_fwd y x
+                                        (restrict_build y x h0 Hx h))
+                     (Acc_inv a (restrict_build y x h0 Hx h))).
+  Lemma FixR_F_eq :
+    ∀ (x:A)(Hx : E x) (r: Acc (restrict E R) x) ,
+      F _ Hx (fun (y : A)( h0 : E y) (h : R y x) ⇒
+                 FixR_F y (restriction_fwd y x (restrict_build y x h0 Hx h))
+                        (Acc_inv r (restrict_build y x h0 Hx h)))=
+      FixR_F x Hx r.
+ +
+  Hypothesis Rwf : well_founded_restriction A E R.
+ +
+  Definition FixR (x:A)(H:E x) := FixR_F x H (Rwf x H).
+ +
+End Fix.
+ +
+Arguments FixR [A E R P] _ _ _ _.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.STDPP_compat.html b/theories/html/hydras.Prelude.STDPP_compat.html new file mode 100644 index 00000000..11e138d6 --- /dev/null +++ b/theories/html/hydras.Prelude.STDPP_compat.html @@ -0,0 +1,75 @@ + + + + + +hydras.Prelude.STDPP_compat + + + + +
+ + + +
+ +

Library hydras.Prelude.STDPP_compat

+ +
+From Coq Require Import Relations.
+ +
+Class Assoc {A} (R : relation A) (f : A → A → A) : Prop :=
+  assoc x y z : R (f x (f y z)) (f (f x y) z).
+ +
+
+ +
+Decision typeclasses following Spitters and van der Weegen +
+
+ +
+Class Decision (P : Prop) := decide : {P} + {¬P}.
+ +
+#[export] Hint Mode Decision ! : typeclass_instances.
+Arguments decide _ {_} : simpl never, assert.
+ +
+Class RelDecision {A B} (R : A → B → Prop) :=
+  decide_rel x y :> Decision (R x y).
+ +
+#[export] Hint Mode RelDecision ! ! ! : typeclass_instances.
+ +
+Arguments decide_rel {_ _} _ {_} _ _ : simpl never, assert.
+ +
+Notation EqDecision A := (RelDecision (@eq A)).
+ +
+Definition bool_decide (P : Prop) {dec : Decision P} : bool :=
+  if dec then true else false.
+ +
+Lemma bool_decide_eq_true (P : Prop) `{Decision P} : bool_decide P = true ↔ P.
+ +
+Lemma bool_decide_eq_false (P : Prop) `{Decision P} : bool_decide P = false ↔ ¬P.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.Simple_LexProd.html b/theories/html/hydras.Prelude.Simple_LexProd.html new file mode 100644 index 00000000..9cb6e313 --- /dev/null +++ b/theories/html/hydras.Prelude.Simple_LexProd.html @@ -0,0 +1,97 @@ + + + + + +hydras.Prelude.Simple_LexProd + + + + +
+ + + +
+ +

Library hydras.Prelude.Simple_LexProd

+ +
+
+ +
+Non dependent lexicographic product Pierre Castéran, Univ. Bordeaux and LaBRI +
+
+ +
+From Coq Require Import Relations Wellfounded.Lexicographic_Product
+        Wellfounded.Inverse_Image Wellfounded.Inclusion Setoid.
+From Coq Require Export RelationClasses.
+ +
+Section Definitions.
+ +
+  Variables (A B : Type)
+            (ltA : relation A)
+            (ltB : relation B).
+ +
+  Hypothesis wfA : well_founded ltA.
+  Hypothesis wfB : well_founded ltB.
+ +
+  Let pair2sig (p: A × B) := existT (fun _ : A ⇒ B) (fst p) (snd p).
+ +
+ +
+  Inductive lexico : relation (A × B) :=
+    lex_1 : ∀ a a' b b', ltA a a' → lexico (a,b) (a',b')
+  | lex_2 : ∀ a b b', ltB b b' → lexico (a,b) (a,b') .
+ +
+ +
+  #[global] Instance Trans_lex {SA : StrictOrder ltA}
+         {SB : StrictOrder ltB} :Transitive lexico.
+ +
+  #[global] Instance Strict_lex {SA : StrictOrder ltA}
+         {SB : StrictOrder ltB} : StrictOrder lexico.
+ +
+ +
+  Lemma wf_lexico : well_founded lexico.
+ +
+ +
+End Definitions.
+ +
+Arguments lexico {A B} _ _ _ _.
+Arguments wf_lexico {A B ltA ltB} _ _ _.
+ +
+Example Ex1 : lexico lt lt (3,5) (4,2).
+ +
+Example Ex2 : lexico lt lt (3,5) (3,6).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.Sort_spec.html b/theories/html/hydras.Prelude.Sort_spec.html new file mode 100644 index 00000000..51fe098d --- /dev/null +++ b/theories/html/hydras.Prelude.Sort_spec.html @@ -0,0 +1,214 @@ + + + + + +hydras.Prelude.Sort_spec + + + + +
+ + + +
+ +

Library hydras.Prelude.Sort_spec

+ +
+
+ +
+Formal specification of list sorting functions +
+
+ +
+From Coq Require Export List RelationClasses Relations Sorting.Permutation
+     Sorting.Sorted.
+From hydras Require Export DecPreOrder.
+From hydras Require Import DecPreOrder_Instances.
+ +
+Definition sort_fun_t := ∀ A, (A → A → bool) → list A → list A.
+ +
+ +
+Section R_given.
+ +
+  Variables (A: Type)(R : relation A).
+ +
+ +
+  Lemma LocallySorted_cons:
+        ∀ l a b, LocallySorted R (b::l) ∧ R a b ↔
+                  LocallySorted R (a::b::l).
+ +
+Definition sort_rel (l l': list A) :=
+  LocallySorted R l' ∧ Permutation l l'.
+ +
+Definition sort_correct (f: list A → list A) : Prop :=
+  ∀ l:list A, sort_rel l (f l).
+ +
+
+ +
+already defined in DecPreOrder ? +
+
+ +
+Definition equiv (x y : A) := R x y ∧ R y x.
+ +
+#[global] Instance equiv_equiv (P: PreOrder R): Equivalence equiv.
+ +
+
+ +
+Abstract properties TODO: look into StdLib's Order whether some lemmas are already + proved +
+
+ +
+Lemma forall_weak (H: Transitive R):
+  ∀ (l:list A) (a b:A), R a b →
+                         List.Forall (R b) l →
+                         List.Forall (R a) l.
+ +
+ +
+ +
+
+ +
+To remove when it compiles again +
+
+ +
+     Lemma LocallySorted_cons' (Htrans : Transitive R):
+        ∀ l a, List.Forall (R a) l ∧ LocallySorted R l ↔
+              LocallySorted R (a::l).
+ +
+      Lemma LocallySorted_trans (Htrans : Transitive R):
+        ∀ l a x, LocallySorted R (a::l) → R x a →
+                 LocallySorted R (x::l).
+ +
+ +
+      Lemma LocallySorted_inv_In (Htrans : Transitive R):
+        ∀ l x, LocallySorted R (x::l) → ∀ y, In y l → R x y.
+ +
+End R_given.
+ +
+Arguments LocallySorted {A} _ _.
+#[global] Hint Constructors LocallySorted : lists.
+ +
+
+ +
+A sort must work on any decidable total pre-order +
+
+ +
+Definition sort_spec (f : sort_fun_t) :=
+ ∀ (A:Type) (le:relation A) (P:TotalPreOrder le) (dec:RelDecision le) (l:list A),
+ let l' := f A (fun x y ⇒ bool_decide (le x y)) l in Permutation l l' ∧ LocallySorted le l'.
+ +
+
+ +
+A prototype for using TotalDecPreOrder type class +
+
+ +
+Definition sort (f:sort_fun_t)
+ {A} {le : relation A} {P: TotalPreOrder le} {dec:RelDecision le}
+ (l: list A) :=
+  f A (fun x y ⇒ bool_decide (le x y)) l.
+ +
+
+ +
+stability +
+
+ +
+Inductive marked {A B : Type}(leA : relation A)(leB : relation B)
+          {pA : PreOrder leA}
+          {pB : PreOrder leB}:
+list (A × B) → Prop :=
+| marked0 : marked leA leB nil
+| marked1 : ∀ a b l, marked leA leB l →
+            (∀ a' b', In (a',b') l →
+                           leA a a' → lt b b') →
+            marked leA leB ((a,b)::l).
+ +
+Definition stable (f : sort_fun_t) : Prop :=
+  ∀ (A B : Type) leA leB
+    (PA : TotalPreOrder leA) (PB : TotalPreOrder leB)
+    (dec : RelDecision leA) (l : list (A × B)),
+         marked leA leB l →
+         marked leA leB (sort f (P:= Total_Inverse_fun
+                                (f:= fst : A × B → A))
+                      l).
+ +
+
+ +
+for testing only +
+
+ +
+Fixpoint mark {A}(l:list A)(from:nat) : list (A × nat) :=
+match l with
+| nil ⇒ nil
+| (a::l') ⇒ (a,from)::mark l' (S from)
+end.
+ +
+Definition stable_test (f : sort_fun_t)
+      {A} {le : relation A}{P: TotalPreOrder le} {dec:RelDecision le}
+           (l: list A) : list (A × nat) :=
+    let m := mark l 0 in
+      sort f (P:= @Total_Inverse_fun (A × nat) A fst le P) m.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.WfVariant.html b/theories/html/hydras.Prelude.WfVariant.html new file mode 100644 index 00000000..7fa98694 --- /dev/null +++ b/theories/html/hydras.Prelude.WfVariant.html @@ -0,0 +1,79 @@ + + + + + +hydras.Prelude.WfVariant + + + + +
+ + + +
+ +

Library hydras.Prelude.WfVariant

+ +
+
+ +
+Pierre Casteran, LaBRI, University of Bordeaux Definition of termination variants +
+
+ +
+Set Implicit Arguments.
+ +
+From Coq Require Import Relations Basics
+     Wellfounded.Inverse_Image Wellfounded.Inclusion.
+ +
+Section Variants.
+ Variable E: Type.
+ Variable tr : relation E. +
+ Definition terminates := well_founded (flip tr).
+ +
+ Variables (T: Type)
+           (lt : relation T)
+           (m : E → T).
+ +
+ Infix "<" := lt.
+ +
+ Class WfVariant :=
+   {
+     wf : well_founded lt;
+     decr : ∀ x y, tr x y → m y < m x
+   }.
+ +
+Lemma Variant_termination (Var : WfVariant ) : terminates .
+ +
+End Variants.
+ +
+ Arguments decr {E tr T lt m} _ _ _ _.
+ Arguments wf {E tr T lt m} _ _ .
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Prelude.ssrnat_extracts.html b/theories/html/hydras.Prelude.ssrnat_extracts.html new file mode 100644 index 00000000..11160b9a --- /dev/null +++ b/theories/html/hydras.Prelude.ssrnat_extracts.html @@ -0,0 +1,61 @@ + + + + + +hydras.Prelude.ssrnat_extracts + + + + +
+ + + +
+ +

Library hydras.Prelude.ssrnat_extracts

+ +
+
+ +
+

Notations adapted from Mathcomp's ssrnat

+ +
+
+ +
+Notation "n .+1" := (S n) (at level 2, left associativity,
+  format "n .+1") : nat_scope.
+Notation "n .+2" := n.+1.+1 (at level 2, left associativity,
+  format "n .+2") : nat_scope.
+Notation "n .+3" := n.+2.+1 (at level 2, left associativity,
+  format "n .+3") : nat_scope.
+Notation "n .+4" := n.+2.+2 (at level 2, left associativity,
+  format "n .+4") : nat_scope.
+Notation "n .+5" := n.+2.+3 (at level 2, left associativity,
+  format "n .+5") : nat_scope.
+ +
+Notation "n .-1" := (pred n) (at level 2, left associativity,
+  format "n .-1") : nat_scope.
+Notation "n .-2" := n.-1.-1 (at level 2, left associativity,
+  format "n .-2") : nat_scope.
+Notation "n .-3" := n.-2.-1 (at level 2, left associativity,
+  format "n .-3") : nat_scope.
+Notation "n .-4" := n.-3.-1 (at level 2, left associativity,
+  format "n .-4") : nat_scope.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.AP.html b/theories/html/hydras.Schutte.AP.html new file mode 100644 index 00000000..822c3774 --- /dev/null +++ b/theories/html/hydras.Schutte.AP.html @@ -0,0 +1,330 @@ + + + + + +hydras.Schutte.AP + + + + +
+ + + +
+ +

Library hydras.Schutte.AP

+ +
+ +
+
+ +
+

Additive principal ordinals

+ +
+ + Pierre Casteran, LaBRI, Universite de Bordeaux +
+ + +
+ +In this library, we define the exponential of basis omega, also called phi0. + +
+ +In fact, ω α , written phi0 alpha in Coq, + is defined as the alpha-th additive principal ordinal. + +
+ + +
+
+ +
+ +
+
+ +
+

Main Definitions

+ + +
+ +

Additive principal ordinals

+ + +
+
+ +
+ +
+Definition AP : Ensemble Ord :=
+  fun alpha ⇒
+    zero < alpha ∧
+    (∀ beta, beta < alpha → beta + alpha = alpha).
+ +
+ +
+
+ +
+

Exponential of basis omega

+ + +
+
+ +
+ +
+Definition _phi0 := ord AP.
+ +
+Notation phi0 := _phi0.
+ +
+Notation "'omega^'" := phi0 (only parsing) : schutte_scope.
+ +
+ +
+
+ +
+

Omega-towers

+ + +
+
+ +
+ +
+Fixpoint omega_tower (i : nat) : Ord :=
+  match i with
+    0 ⇒ 1
+  | S j ⇒ phi0 (omega_tower j)
+  end.
+ +
+ +
+
+ +
+

The limit ordinal epsilon0

+ + +
+
+ +
+Definition epsilon0 := omega_limit omega_tower.
+ +
+ +
+
+ +
+

Proofs, proofs, proofs ...

+ +
+ +

About additive principals

+ +
+
+ +
+Lemma AP_one : In AP 1.
+ +
+Lemma least_AP : least_member lt AP 1.
+ +
+Lemma AP_omega : In AP omega.
+ +
+#[global] Hint Resolve zero_lt_omega : schutte.
+ +
+Lemma AP_finite_eq_one : ∀ n: nat, AP n → n = 1.
+ +
+
+ +
+Thus, omega is the second additive principal +
+
+ +
+Lemma omega_second_AP :
+  least_member lt
+               (fun alpha ⇒ 1 < alpha ∧ In AP alpha)
+               omega.
+ +
+ +
+Lemma AP_plus_closed (alpha beta gamma : Ord):
+  In AP alpha → beta < alpha → gamma < alpha →
+  beta + gamma < alpha. +
+ +
+ +
+Lemma AP_mult_Sn_closed (alpha beta: Ord) :
+  AP alpha → beta < alpha → ∀ n, mult_Sn beta n < alpha.
+ +
+Lemma AP_mult_fin_r_closed (alpha beta: Ord) :
+  AP alpha → beta < alpha → ∀ n, beta × n < alpha.
+ +
+ +
+ +
+Theorem AP_unbounded : Unbounded AP. +
+ +
+ +
+Theorem AP_closed : Closed AP. +
+Theorem AP_o_segment : the_ordering_segment AP = ordinal. +
+
+ +
+

Properties of phi0

+ +
+
+ +
+Theorem normal_phi0 : normal phi0 AP.
+ +
+Lemma phi0_ordering : ordering_function phi0 ordinal AP.
+ +
+ +
+ +
+Lemma phi0_elim : ∀ P : (Ord→Ord)->Prop,
+    (∀ f: Ord→Ord,
+        ordering_function f ordinal AP → P f) →
+    P phi0.
+ +
+Lemma AP_phi0 (alpha : Ord) : In AP (phi0 alpha). +
+Lemma phi0_zero : phi0 zero = 1. +
+Lemma phi0_mono (alpha beta : Ord) :
+  alpha < beta → phi0 alpha < phi0 beta. +
+Lemma phi0_mono_weak (alpha beta : Ord) :
+  alpha ≤ beta → phi0 alpha ≤ phi0 beta. +
+Lemma phi0_mono_R (alpha beta : Ord) :
+  phi0 alpha < phi0 beta → alpha < beta. +
+Lemma phi0_mono_R_weak (alpha beta: Ord):
+    phi0 alpha ≤ phi0 beta → alpha ≤ beta. +
+Lemma phi0_inj (alpha beta : Ord) :
+  phi0 alpha = phi0 beta → alpha = beta. +
+Lemma phi0_positive (alpha : Ord): zero < phi0 alpha. +
+Lemma plus_lt_phi0 (ksi alpha: Ord):
+    ksi < phi0 alpha → ksi + phi0 alpha = phi0 alpha. +
+Lemma phi0_alpha_phi0_beta (alpha beta: Ord) :
+  alpha < beta → phi0 alpha + phi0 beta = phi0 beta. +
+Lemma phi0_sup : ∀ U: Ensemble Ord,
+    Inhabited U →
+    Countable U →
+    phi0 (|_| U) = |_| (image U phi0). +
+Lemma phi0_of_limit (alpha : Ord) :
+  is_limit alpha →
+  phi0 alpha = |_| (image (members alpha) phi0). +
+Lemma AP_to_phi0 (alpha : Ord) :
+  AP alpha → ∃ beta, alpha = phi0 beta. +
+Lemma AP_plus_AP (alpha beta gamma : Ord) :
+  zero < beta →
+  phi0 alpha + beta = phi0 gamma →
+  alpha < gamma ∧ beta = phi0 gamma. +
+Lemma is_limit_phi0 (alpha : Ord) :
+  zero < alpha → is_limit (phi0 alpha). +
+Lemma omega_eqn : omega = phi0 1. +
+Lemma le_phi0 (alpha : Ord) : alpha ≤ phi0 alpha. +
+
+ +
+

Properties of epsilon0

+ +
+
+ +
+Lemma epsilon0_fxp : phi0 epsilon0 = epsilon0.
+ +
+Lemma epsilon0_AP : AP epsilon0.
+ +
+Lemma omega_tower_mono (i : nat) : omega_tower i < omega_tower (S i).
+ +
+ +
+Lemma lt_phi0 (alpha : Ord):
+  alpha < epsilon0 → alpha < phi0 alpha. +
+ +
+Theorem epsilon0_lfp : least_fixpoint lt phi0 epsilon0. +
+Lemma phi0_lt_epsilon0 (alpha : Ord) :
+  alpha < epsilon0 → phi0 alpha < epsilon0.
+ +
+Lemma phi0_lt_epsilon0_R (alpha : Ord):
+  phi0 alpha < epsilon0 → alpha < epsilon0.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.Addition.html b/theories/html/hydras.Schutte.Addition.html new file mode 100644 index 00000000..b92adbc8 --- /dev/null +++ b/theories/html/hydras.Schutte.Addition.html @@ -0,0 +1,272 @@ + + + + + +hydras.Schutte.Addition + + + + +
+ + + +
+ +

Library hydras.Schutte.Addition

+ +
+ +
+From Coq Require Import Arith Logic.Epsilon Ensembles.
+From ZornsLemma Require Import CountableTypes.
+From hydras Require Export Schutte_basics Ordering_Functions
+     PartialFun Countable MoreEpsilonIota.
+Set Implicit Arguments.
+From hydras Require Export STDPP_compat.
+ +
+
+ +
+

Definitions

+ +
+ +

addition, multiplication by a positive integer

+ +
+
+ +
+ +
+Definition plus alpha := ord (ge alpha).
+Infix "+" := plus : schutte_scope.
+ +
+ +
+
+ +
+returns alpha × (S n) + +
+
+ +
+ +
+Fixpoint mult_Sn (alpha:Ord)(n:nat){struct n} :Ord :=
+ match n with 0 ⇒ alpha
+            | S p ⇒ mult_Sn alpha p + alpha
+ end.
+ +
+Definition mult_fin_r alpha n :=
+  match n with
+      0 ⇒ zero
+    | S p ⇒ mult_Sn alpha p
+  end.
+ +
+Infix "×" := mult_fin_r : schutte_scope.
+ +
+ +
+
+ +
+

Proofs, proofs, proofs

+ + +
+
+ +
+Lemma Unbounded_ge (alpha : Ord) : Unbounded (ge alpha).
+ +
+Lemma ge_o_segment (alpha : Ord) :
+ the_ordering_segment (ge alpha) = ordinal.
+ +
+Lemma plus_ordering (alpha : Ord) :
+  ordering_function (plus alpha)
+                    ordinal
+                    (ge alpha).
+ +
+Lemma plus_elim (alpha : Ord) :
+  ∀ P : (Ord→Ord)->Prop,
+    (∀ f: Ord→Ord,
+        ordering_function f ordinal (ge alpha)→ P f) →
+    P (plus alpha).
+ +
+Lemma normal_plus_alpha (alpha : Ord) :
+  normal (plus alpha) (ge alpha).
+ +
+
+ +
+

Basic properties of addition

+ + +
+
+ +
+ +
+Lemma alpha_plus_zero (alpha: Ord): alpha + zero = alpha. +
+Remark ge_zero : (ge zero : Ensemble Ord) = ordinal.
+ +
+ +
+Lemma zero_plus_alpha (alpha : Ord): zero + alpha = alpha.
+ +
+Lemma le_plus_l (alpha beta : Ord) : alpha ≤ alpha + beta. +
+Lemma le_plus_r (alpha beta : Ord) : beta ≤ alpha + beta. +
+Lemma plus_mono_r (alpha beta gamma : Ord) :
+  beta < gamma → alpha + beta < alpha + gamma. +
+Lemma plus_of_succ (alpha beta : Ord) :
+  alpha + (succ beta) = succ (alpha + beta). +
+ +
+Lemma plus_mono_r_weak (alpha beta gamma : Ord) :
+  beta ≤ gamma → alpha + beta ≤ alpha + gamma.
+ +
+Lemma plus_reg_r (alpha beta gamma : Ord) :
+  alpha + beta = alpha + gamma → beta = gamma.
+ +
+Lemma succ_is_plus_1 alpha : succ alpha = alpha + 1.
+ +
+Lemma alpha_plus_sup (alpha : Ord) (A : Ensemble Ord) :
+    Inhabited A →
+    Countable A →
+    alpha + |_| A = |_| (image A (plus alpha)).
+ +
+Lemma plus_limit (alpha beta : Ord)
+  : is_limit beta →
+    alpha + beta = |_| (image (members beta) (plus alpha)).
+ +
+Lemma plus_FF : ∀ i j, F (i + j) = F i + F j.
+ +
+Lemma one_plus_omega : 1 + omega = omega.
+ +
+Lemma minus_exists (alpha beta : Ord) :
+  alpha ≤ beta →
+  ∃ gamma, alpha + gamma = beta.
+ +
+Section proof_of_associativity.
+  Variables alpha beta : Ord.
+ +
+  Lemma plus_assoc1 (gamma : Ord) :
+    alpha + beta ≤ alpha + (beta + gamma) .
+ +
+  Lemma plus_assoc2 (gamma : Ord) :
+    alpha + beta ≤ gamma →
+    ∃ khi, gamma = alpha + (beta + khi).
+ +
+  Let f_alpha_beta := plus (alpha + beta).
+ +
+  Let g_alpha_beta gamma := alpha + (beta + gamma).
+ +
+  Remark of_g : ordering_function g_alpha_beta ordinal (ge (alpha+beta)).
+ +
+  Lemma of_u : fun_equiv f_alpha_beta g_alpha_beta ordinal ordinal.
+ +
+  Lemma plus_assoc3 (gamma : Ord) :
+    f_alpha_beta gamma = g_alpha_beta gamma.
+ +
+  Lemma plus_assoc' (gamma : Ord) :
+    alpha + (beta + gamma) = (alpha + beta) + gamma.
+ +
+End proof_of_associativity.
+ +
+#[global] Instance plus_assoc: Assoc eq plus.
+ +
+Lemma one_plus_infinite (alpha : Ord) :
+  omega ≤ alpha → 1 + alpha = alpha.
+ +
+ +
+Lemma finite_plus_infinite (n : nat) (alpha : Ord) :
+  omega ≤ alpha → n + alpha = alpha. +
+Example L_3_plus_omega : 3 + omega = omega.
+ +
+Lemma plus_mono_weak_l : ∀ alpha beta gamma,
+                          alpha ≤ beta → alpha + gamma ≤ beta + gamma.
+ +
+Lemma plus_mono_bi : ∀ alpha beta gamma delta,
+                        alpha ≤ gamma →
+                        beta < delta →
+                        alpha + beta < gamma + delta.
+ +
+Lemma mult_fin_r_one : ∀ n, (F 1) × S n = F (S n).
+ +
+Lemma mult_fin_r_mono : ∀ alpha beta , alpha < beta →
+   ∀ n, alpha × S n < beta × S n.
+ +
+Lemma le_a_mult_Sn_a : ∀ alpha n, ordinal alpha →
+                                       alpha ≤ alpha × S n.
+ +
+Lemma mult_Sn_mono2 : ∀ a, zero < a →
+                         ∀ n p, (n < p)%nat → a × S n < a × S p.
+ +
+Lemma mult_Sn_mono3 : ∀ alpha, zero < alpha →
+                         ∀ n p, (n < p)%nat → alpha × S n + alpha
+                                                    ≤ alpha × S p.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.CNF.html b/theories/html/hydras.Schutte.CNF.html new file mode 100644 index 00000000..5aff7077 --- /dev/null +++ b/theories/html/hydras.Schutte.CNF.html @@ -0,0 +1,258 @@ + + + + + +hydras.Schutte.CNF + + + + +
+ + + +
+ +

Library hydras.Schutte.CNF

+ +
+ +
+
+ +
+ Cantor normal form +
+ + Pierre Casteran, Labri and Univ Bordeaux. +
+
+ +
+From Coq Require Import Arith List Sorting.Sorted
+     Logic.Epsilon Ensembles.
+From hydras Require Export Schutte_basics Ordering_Functions
+     PartialFun Countable Schutte.Addition AP.
+From Coq Require Export Classical.
+Import ListNotations.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+A Cantor normal form for a countable ordinal alpha is just a sorted +list l (in decreasing order) such that alpha is equal to the sum of the terms phi0 beta, for every term beta in l. + +
+ +Note that, if alpha is greater or equal than epsilon0, the members of l +are less or equal than alpha. + +
+ +For instance, the Cantor Normal Form of epsilon0 is just epsilon0 :: nil. + +
+ + +
+
+ +
+Definition cnf_t := list Ord.
+ +
+Fixpoint eval (l : cnf_t) : Ord :=
+  match l with
+  | nil ⇒ zero
+  | beta :: l' ⇒ phi0 beta + eval l'
+  end.
+ +
+Definition sorted (l: cnf_t) :=
+  LocallySorted (fun alpha beta ⇒ beta ≤ alpha) l.
+ +
+Definition is_cnf_of (alpha : Ord)(l : cnf_t) : Prop :=
+  sorted l ∧ alpha = eval l.
+ +
+Definition exponents_lt (alpha: Ord) :=
+  Forall (fun beta ⇒ beta < alpha).
+ +
+Definition exponents_le (alpha : Ord) :=
+  Forall (fun beta ⇒ beta ≤ alpha).
+ +
+ +
+Lemma exponents_lt_eval alpha l: exponents_lt alpha l →
+                                 eval l < phi0 alpha.
+ +
+Lemma sorted_tail alpha l: sorted (cons alpha l) → sorted l.
+ +
+Lemma nf_bounded beta l alpha :
+  alpha ≤ phi0 beta → is_cnf_of alpha l →
+  exponents_le beta l.
+ +
+Lemma cnf_of_ap (alpha : Ord) :
+  In AP alpha → ∃ l, is_cnf_of alpha l.
+ +
+Lemma sorted_lt_lt (beta: Ord) : ∀ l alpha,
+    sorted (cons alpha l) → alpha < beta →
+    eval (cons alpha l) < phi0 beta.
+ +
+Lemma sorted_lt_lt_2 l alpha :
+  sorted (cons alpha l) →
+  eval (cons alpha l) < phi0 (succ alpha).
+ +
+Lemma cnf_head_eq alpha beta ol ol':
+  sorted (cons alpha ol) →
+  sorted (cons beta ol') →
+  eval (cons alpha ol) = eval (cons beta ol') →
+  alpha = beta.
+ +
+Lemma cnf_eq alpha beta ol ol':
+  sorted (cons alpha ol) →
+  sorted (cons beta ol') →
+  eval (cons alpha ol) = eval (cons beta ol') →
+  alpha = beta ∧ eval ol = eval ol'.
+ +
+Lemma cnf_plus1 (ol : cnf_t) :
+  sorted ol → ∀ alpha,
+    ∃ ol', is_cnf_of (phi0 alpha + eval ol) ol'.
+ +
+Lemma cnf_plus2 : ∀ ol, sorted ol →
+                             ∀ ol', sorted ol' →
+                                         ∃ ol'', is_cnf_of (eval ol + eval ol') ol''.
+ +
+Lemma cnf_plus : ∀ ol alpha,
+    is_cnf_of alpha ol → ∀ ol' beta, is_cnf_of beta ol' →
+                                           ∃ ol'', is_cnf_of (alpha + beta) ol''.
+ +
+ +
+
+ +
+

Every countable ordinal has (at least) a Cantor normal form

+ + +
+ + (Proof by transfinite induction) + +
+
+ +
+ +
+Theorem cnf_exists (alpha : Ord) :
+  ∃ l: cnf_t, is_cnf_of alpha l. +
+Lemma sorted_lt l alpha: sorted (alpha::l) →
+                          eval l < phi0 alpha + eval l.
+ +
+
+ +
+

Unicity of cnf

+ + +
+ +(Proof by induction on lists) + +
+ + +
+
+ +
+ +
+Lemma cnf_unicity l alpha:
+  is_cnf_of alpha l →
+  ∀ l', is_cnf_of alpha l' → l = l'. +
+
+ +
+

The main result

+ +
+
+ +
+ +
+Theorem cnf_exists_unique (alpha:Ord) :
+  ∃! l: cnf_t, is_cnf_of alpha l.
+ +
+
+ +
+

Cantor Normal Form and the ordinal epsilon0

+ +
+
+ +
+ +
+Lemma cnf_lt_epsilon0 : ∀ l alpha,
+    is_cnf_of alpha l →
+    alpha < epsilon0 →
+    Forall (fun beta ⇒ beta < alpha) l. +
+ +
+
+ +
+The normal form of epsilon0 is just phi0 epsilon0 + +
+
+ +
+ +
+ +
+Lemma cnf_of_epsilon0 : is_cnf_of epsilon0 [epsilon0].
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.Correctness_E0.html b/theories/html/hydras.Schutte.Correctness_E0.html new file mode 100644 index 00000000..1733a461 --- /dev/null +++ b/theories/html/hydras.Schutte.Correctness_E0.html @@ -0,0 +1,175 @@ + + + + + +hydras.Schutte.Correctness_E0 + + + + +
+ + + +
+ +

Library hydras.Schutte.Correctness_E0

+ +
+
+ +
+ Injection from the set of ordinal terms in Cantor normal form + into the set of Schutte's countable ordinal numbers stricly less than + epsilon0. + +
+ + Pierre Castéran, Univ. Bordeaux and LaBRI + +
+ + This is intented to be a validation of main constructions and functions + designed for the type T1. + +
+ + +
+
+ +
+ +
+From hydras Require Import Epsilon0.Epsilon0 ON_Generic.
+From hydras Require Import Schutte_basics Schutte.Addition AP CNF.
+ +
+Import List PartialFun Ensembles.
+ +
+ +
+Fixpoint inject (t:T1) : Ord :=
+  match t with
+  | T1.zero ⇒ zero
+  | T1.cons a n b ⇒ AP._phi0 (inject a) × S n + inject b
+  end.
+ +
+ +
+Lemma inject_of_finite_pos : ∀ n, inject (\F (S n)) = F (S n).
+ +
+ +
+ +
+Theorem inject_of_zero : inject T1.zero = zero.
+ +
+Theorem inject_of_finite (n : nat):
+  inject (\F n) = n.
+ +
+Theorem inject_of_omega :
+  inject T1omega = Schutte_basics._omega. +
+Theorem inject_of_phi0 (alpha : T1):
+  inject (T1.phi0 alpha) = AP._phi0 (inject alpha). +
+ +
+ +
+Theorem inject_mono (beta gamma : T1) :
+  T1.lt beta gamma →
+  T1.nf beta → T1.nf gamma →
+  inject beta < inject gamma.
+ +
+Theorem inject_injective (beta gamma : T1) : nf beta → nf gamma →
+                                             inject beta = inject gamma → beta = gamma.
+ +
+Theorem inject_monoR (beta gamma : T1) :
+  T1.nf beta → T1.nf gamma →
+  inject beta < inject gamma →
+  (beta t1< gamma)%t1.
+ +
+ +
+Theorem inject_lt_epsilon0 (alpha : T1):
+  inject alpha < epsilon0. +
+ +
+Lemma inject_rw (a b: T1) n : inject (T1.cons a n b) =
+                              mult_Sn (AP._phi0 (inject a)) n + inject b.
+ +
+ +
+Theorem inject_plus (alpha beta : T1):
+  nf alpha → nf beta →
+  inject (alpha + beta)%t1 = inject alpha + inject beta. +
+ +
+Theorem inject_mult_fin_r (alpha : T1) :
+  nf alpha →
+  ∀ n:nat,
+    inject (alpha × n)%t1 = inject alpha × n. +
+Lemma inject_lt_epsilon0_ex_cnf (alpha : Ord) :
+  ∀ (H : alpha < epsilon0)
+         (l: list Ord), is_cnf_of alpha l →
+                         ∃ t: T1, nf t ∧ inject t = eval l.
+ +
+Theorem inject_lt_epsilon0_ex (alpha : Ord) (H : alpha < epsilon0) :
+  ∃ t: T1, nf t ∧ inject t = alpha.
+ +
+Theorem inject_lt_epsilon0_ex_unique (alpha : Ord) (H : alpha < epsilon0) :
+  ∃! t: T1, nf t ∧ inject t = alpha.
+ +
+ +
+Theorem embedding : fun_bijection (nf: Ensemble T1)
+                                  (members epsilon0)
+                                  inject. +
+ +
+#[ global ] Instance Epsilon0_correct :
+  ON_correct epsilon0 Epsilon0 (fun alpha ⇒ inject (cnf alpha)). +
+ +
+
+ +
+Correctness of E0.plus +
+
+ +
+Theorem E0_plus_correct : ON_op_ok E0add plus.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.Countable.html b/theories/html/hydras.Schutte.Countable.html new file mode 100644 index 00000000..9ec0c2e5 --- /dev/null +++ b/theories/html/hydras.Schutte.Countable.html @@ -0,0 +1,155 @@ + + + + + +hydras.Schutte.Countable + + + + +
+ + + +
+ +

Library hydras.Schutte.Countable

+ +
+
+ +
+A formalization of denumerable sets. by Florian Hatat and Stéphane Desarzens +
+
+ +
+From Coq Require Import Ensembles Arith ArithRing
+     Wellfounded Relations Wf_nat Finite_sets
+     Logic.Epsilon Sets.Image Lia.
+ +
+From ZornsLemma Require Import Classical_Wf CountableTypes Families.
+ +
+From hydras Require Import MoreEpsilonIota PartialFun GRelations
+     Prelude.More_Arith.
+ +
+Import Nat.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+A is countable if there exists an injection from A to + Full_set nat. +
+
+ +
+Section Countable.
+ +
+  Section Definitions.
+    Variable U : Type.
+    Variable A : Ensemble U.
+ +
+
+ +
+Predicate for relations which number the elements of A. + +
+ + These relations map each element of A to at least one integer, but they + are not required to be functional (injectivity is only needed to ensure that + A is countable). +
+
+ +
+    Definition rel_numbers (R: GRelation U nat) := rel_injection A Full_set R.
+ +
+
+ +
+Predicate for relations which enumerate A. +
+
+    Definition rel_enumerates (R : GRelation nat U) := rel_surjection Full_set A R.
+ +
+    Theorem countable_surj :
+      Countable A ↔ ∃ R, rel_enumerates R.
+ +
+  End Definitions.
+ +
+  Variable U : Type.
+ +
+  Section Countable_seq_range.
+ +
+    Definition seq_range (f : nat → U) : Ensemble U :=
+      image Full_set f.
+ +
+    Lemma seq_range_countable :
+      ∀ f, Countable (seq_range f).
+ +
+  End Countable_seq_range.
+ +
+  Section Countable_bijection.
+ +
+    Variable V : Type.
+ +
+    Variable A : Ensemble U.
+    Variable B : Ensemble V.
+    Variable g : U → V.
+ +
+    Hypothesis g_bij : fun_bijection A B g.
+ +
+    Lemma countable_bij_fun :
+      Countable A → Countable B.
+ +
+    Lemma countable_bij_funR :
+      Countable B → Countable A.
+ +
+  End Countable_bijection.
+ +
+End Countable.
+ +
+Lemma countable_image : ∀ (U V:Type)(DA : Ensemble U)(f:U→V),
+    Countable DA → Countable (image DA f).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.Critical.html b/theories/html/hydras.Schutte.Critical.html new file mode 100644 index 00000000..61e13f95 --- /dev/null +++ b/theories/html/hydras.Schutte.Critical.html @@ -0,0 +1,320 @@ + + + + + +hydras.Schutte.Critical + + + + +
+ + + +
+ +

Library hydras.Schutte.Critical

+ +
+ +
+
+ +
+ +
+ + We adapt Schutte's definition of critical ordinals : + +
+ +
    +
  • Cr(zero) = AP (the set of additive principal ordinals ) + +
    + + +
    • +
  • if zero < alpha, then + Cr(alpha) is the intersection of all the sets of fixpoints + of the ordering functions of Cr(beta) for beta < alpha. + +
    • +
+ +
+
+ +
+From Coq Require Import Arith Logic.Epsilon Ensembles Classical.
+From ZornsLemma Require Import CountableTypes.
+From hydras Require Export Schutte_basics Ordering_Functions
+     Countable Schutte.Addition AP CNF Well_Orders MoreEpsilonIota.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+ Let us define a functional, the fixpoint of which we shall consider +
+
+ +
+ +
+Definition Cr_fun : ∀ alpha : Ord,
+    (∀ beta : Ord, beta < alpha → Ensemble Ord) →
+    Ensemble Ord
+  :=
+    fun (alpha: Ord)
+        (Cr : ∀ beta, beta < alpha → Ensemble Ord)
+        (x : Ord) ⇒ (
+        (alpha = zero ∧ AP x) ∨
+        (zero < alpha ∧
+         ∀ beta (H:beta < alpha),
+           In (the_ordering_segment (Cr beta H)) x ∧
+           ord (Cr beta H) x = x)).
+ +
+Definition Cr (alpha : Ord) : Ensemble Ord :=
+  (Fix all_ord_acc
+        (fun (_:Ord) ⇒ Ensemble Ord) Cr_fun) alpha.
+ +
+ +
+ +
+Definition strongly_critical alpha := In (Cr alpha) alpha.
+ +
+Definition maximal_critical alpha : Ensemble Ord :=
+  fun gamma ⇒
+    In (Cr alpha) gamma ∧
+    ∀ xi, alpha < xi → ¬ In (Cr xi) gamma.
+ +
+Definition Gamma0 := the_least strongly_critical.
+ +
+ +
+
+ +
+ See Gamma0.Gamma0.phi +
+
+ +
+ +
+Definition phi (alpha : Ord) : Ord → Ord := ord (Cr alpha).
+ +
+Definition A_ (alpha : Ord) : Ensemble Ord := the_ordering_segment (Cr alpha).
+ +
+ +
+Lemma Cr_extensional :
+  ∀ (x:Ord)
+         (f g : ∀ y : Ord, y < x → (fun _ : Ord ⇒ Ensemble Ord) y),
+    (∀ (y : Ord) (p : y < x), f y p = g y p) →
+    ((Cr_fun f :Ensemble Ord) = (Cr_fun g :Ensemble Ord)).
+ +
+Lemma Cr_equation (alpha : Ord) :
+  Cr alpha =
+  Cr_fun
+    (fun (y : Ord) (h : y < alpha) ⇒ Cr y).
+ +
+Lemma Cr_inv (alpha x : Ord):
+  Cr alpha x →
+  ((alpha = zero ∧ (Cr alpha x ↔ AP x)) ∨
+   (zero < alpha ∧
+    (∀ beta (H:beta < alpha),
+        A_ beta x ∧ ord (Cr beta ) x = x))).
+ +
+Lemma Cr_zero : ∀ x, AP x → Cr zero x.
+ +
+Lemma Cr_pos : ∀ alpha,
+    zero < alpha →
+    ∀ x : Ord ,
+      (∀ beta (H:beta < alpha),
+          A_ beta x ∧ ord (Cr beta) x = x) →
+      Cr alpha x.
+ +
+Lemma Cr_zero_inv : ∀ x, Cr zero x → AP x.
+ +
+ +
+Lemma Cr_zero_AP : Cr zero = AP. +
+Lemma Cr_pos_inv (alpha : Ord) :
+  zero < alpha →
+  ∀ x,
+    Cr alpha x →
+    (∀ beta (H:beta < alpha), In (A_ beta) x ∧ phi beta x = x).
+ +
+Lemma Cr_pos_iff (alpha : Ord) :
+  zero < alpha →
+  ∀ x,
+    (Cr alpha x ↔
+     (∀ beta (H:beta < alpha), In (A_ beta) x ∧ phi beta x = x)).
+ +
+Lemma A_Cr (alpha beta:Ord) : In (A_ alpha) beta → phi alpha beta = beta →
+                             In (Cr alpha) beta.
+ +
+Lemma Cr_lt : ∀ alpha beta,
+    beta < alpha → ∀ x, Cr alpha x → Cr beta x.
+ +
+Lemma Cr_incl (alpha beta : Ord) (H :beta ≤ alpha) :
+  Included (Cr alpha) (Cr beta).
+ +
+Lemma phi0_well_named : ∀ alpha, phi0 alpha = phi 0 alpha.
+ +
+Lemma Cr_1_iff (alpha : Ord):
+  Cr 1 alpha ↔ AP alpha ∧ phi0 alpha = alpha.
+ +
+ +
+Lemma epsilon0_Cr1 : In (Cr 1) epsilon0. +
+ +
+
+ +
+Lemma 5, p 82 of Schutte's book +
+
+ +
+ +
+Section Proof_of_Lemma5.
+  Let P (alpha:Ord) := Unbounded (Cr alpha) ∧ Closed (Cr alpha).
+ +
+  Lemma Lemma5_0 : P zero.
+ +
+  Section Alpha_positive.
+    Variable alpha : Ord.
+    Hypothesis alpha_pos : zero < alpha.
+    Hypothesis IHalpha : ∀ xi, xi < alpha → P xi.
+ +
+    Section Proof_unbounded.
+      Variable beta : Ord.
+ +
+      Fixpoint gamma_ (n:nat) : Ord :=
+        match n with
+          O ⇒ succ beta
+        | S n ⇒ sup
+                   (fun (y : Ord) ⇒
+                      ∃ xi: Ord, xi < alpha ∧
+                                      y = phi xi (gamma_ n))
+        end.
+ +
+      Let gamma := omega_limit gamma_.
+ +
+      Lemma Lemma5_01 : beta < gamma.
+ +
+      Lemma Lemma5_02 : ∀ xi, xi < alpha →
+                                    phi xi gamma = gamma.
+ +
+      Lemma Lemma5_03 : In (Cr alpha) gamma.
+ +
+      Remark A_full : ∀ xi, xi < alpha → A_ xi = ordinal.
+ +
+      Lemma Lemma5_04 : ∃ gamma, In (Cr alpha) gamma ∧ beta < gamma.
+ +
+    End Proof_unbounded.
+ +
+    Lemma Lemma5_1 : Unbounded (Cr alpha).
+ +
+    Section closedness.
+      Variable M : Ensemble Ord.
+      Hypothesis HM : Inhabited M.
+      Hypothesis CM : Countable M.
+      Hypothesis IM : Included M (Cr alpha).
+ +
+      Lemma Lemma5_2 : ∀ xi eta, xi < alpha →
+                                        In M eta →
+                                        phi xi eta = eta.
+        Check (Cr_pos_iff alpha_pos).
+      Qed.
+ +
+      Lemma Lemma5_7 : In (Cr alpha) (sup M).
+ +
+    End closedness.
+ +
+    Lemma induct_step : P alpha.
+ +
+  End Alpha_positive.
+ +
+  Lemma Lemma5 : ∀ alpha, P alpha.
+ +
+End Proof_of_Lemma5.
+ +
+ +
+Theorem Unbounded_Cr alpha : Unbounded (Cr alpha). +
+Theorem Closed_Cr alpha : Closed (Cr alpha). +
+ +
+Theorem Th13_8 alpha : normal (phi alpha) (Cr alpha).
+ +
+Corollary Th13_8_1 alpha : A_ alpha = ordinal.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.GRelations.html b/theories/html/hydras.Schutte.GRelations.html new file mode 100644 index 00000000..54baeb9d --- /dev/null +++ b/theories/html/hydras.Schutte.GRelations.html @@ -0,0 +1,155 @@ + + + + + +hydras.Schutte.GRelations + + + + +
+ + + +
+ +

Library hydras.Schutte.GRelations

+ +
+
+ +
+General relations +
+ + by Florian Hatat, ENS-Lyon +
+
+ +
+From Coq Require Import Ensembles Classical Lia Arith.
+From hydras Require Import PartialFun.
+ +
+Section General_Relations.
+ +
+  Section Definitions.
+    Variables A B : Type.
+ +
+
+ +
+domain +
+
+ +
+    Variable DA : Ensemble A.
+ +
+
+ +
+codomain +
+
+ +
+    Variable DB : Ensemble B.
+ +
+    Definition GRelation := A → B → Prop.
+ +
+    Variable R : GRelation.
+ +
+    Inductive rel_injection : Prop :=
+     rel_inj_i : rel_domain DA R →
+          rel_codomain DA DB R →
+          rel_inj DA R →
+          rel_injection.
+ +
+    Inductive rel_surjection : Prop :=
+     rel_surj_i : rel_codomain DA DB R →
+          rel_functional DA R →
+          rel_onto DA DB R →
+          rel_surjection.
+  End Definitions.
+ +
+ Arguments rel_injection [A B].
+ Arguments rel_surjection [A B].
+ +
+  Variables A B : Type.
+  Variable DA : Ensemble A.
+  Variable DB : Ensemble B.
+ +
+  Section injection2surjection.
+    Variable R : GRelation A B.
+    Hypothesis R_inj : rel_injection DA DB R.
+ +
+    Lemma R_inv_surj : rel_surjection DB DA (rel_inv DA DB R).
+ +
+  End injection2surjection.
+ +
+  Section surjection2injection.
+    Variable R : GRelation A B.
+    Hypothesis R_surj : rel_surjection DA DB R.
+ +
+    Definition R_inv := rel_inv DA DB R.
+ +
+    Lemma R_inv_inj : rel_injection DB DA (rel_inv DA DB R).
+  End surjection2injection.
+ +
+  Section elagage.
+    Section to_nat_elagage.
+      Variable R : GRelation A nat.
+ +
+      Definition R_nat_elaguee (x : A) (n : nat) : Prop :=
+        R x n ∧ (∀ p, R x p → n ≤ p).
+ +
+      Lemma R_nat_elaguee_fun :
+        rel_functional DA R_nat_elaguee.
+ +
+      Lemma R_nat_elaguee_domain :
+        ∀ y n, R y n → ∃ p, R_nat_elaguee y p.
+ +
+    End to_nat_elagage.
+  End elagage.
+ +
+End General_Relations.
+ +
+Arguments rel_injection {A B}.
+Arguments rel_surjection {A B}.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.Lub.html b/theories/html/hydras.Schutte.Lub.html new file mode 100644 index 00000000..ac3c5411 --- /dev/null +++ b/theories/html/hydras.Schutte.Lub.html @@ -0,0 +1,63 @@ + + + + + +hydras.Schutte.Lub + + + + +
+ + + +
+ +

Library hydras.Schutte.Lub

+ +
+ +
+Set Implicit Arguments.
+ +
+From Coq Require Import Relations Ensembles.
+ +
+ +
+Definition upper_bound (M:Type)
+                       (D: Ensemble M)
+                       (lt: relation M)
+                       (X:Ensemble M)
+                       (a:M) :=
+  ∀ x, In _ D x → In _ X x → x = a ∨ lt x a.
+ +
+Definition is_lub (M:Type)
+                  (D : Ensemble M)
+                  (lt : relation M)
+                  (X:Ensemble M)
+                  (a:M) :=
+   In _ D a ∧ upper_bound D lt X a ∧
+   (∀ y, In _ D y →
+              upper_bound D lt X y →
+              y = a ∨ lt a y).
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.MoreEpsilonIota.html b/theories/html/hydras.Schutte.MoreEpsilonIota.html new file mode 100644 index 00000000..a699019b --- /dev/null +++ b/theories/html/hydras.Schutte.MoreEpsilonIota.html @@ -0,0 +1,163 @@ + + + + + +hydras.Schutte.MoreEpsilonIota + + + + +
+ + + +
+ +

Library hydras.Schutte.MoreEpsilonIota

+ +
+
+ +
+ +
+ + Complements to Coq.Logic.Epsilon + +
+ + Pierre Casteran, + LaBRI, University of Bordeaux + +
+
+ +
+From Coq Require Export Ensembles Logic.Epsilon.
+ +
+Print epsilon_statement.
+ +
+Print epsilon.
+ +
+Check constructive_indefinite_description.
+ +
+Check iota_statement.
+ +
+Print iota.
+ +
+Print iota_spec.
+ +
+Set Implicit Arguments.
+Arguments In {U} _ _.
+ +
+ Lemma epsilon_ind {A:Type} (inh : inhabited A) (P Q : A → Prop):
+   (ex P ) → (∀ a, P a → Q a) → Q (epsilon inh P).
+ +
+Theorem epsilon_equiv {A:Type}(a: inhabited A)(P:A→Prop):
+  (ex P)<-> P (epsilon a P).
+ +
+Ltac epsilon_elim_aux :=
+  match goal with [ |- (?P (epsilon (A:=?X) ?a ?Q))] ⇒
+           apply epsilon_ind; auto
+  end.
+ +
+Ltac epsilon_elim := epsilon_elim_aux ||
+  match goal with
+  [ |- (?P (?k ?d))] ⇒
+   (let v := eval cbv zeta beta delta [k] in (k d) in
+     (match v with (epsilon ?w ?d) ⇒ change (P v); epsilon_elim_aux end))
+  | [ |- (?P (?k ?arg ?arg1))] ⇒
+   (let v := eval cbv zeta beta delta [k] in (k arg arg1) in
+     (match v with (epsilon ?w ?d) ⇒ change (P v); epsilon_elim_aux end))
+  | [ |- (?P ?k)] ⇒
+   (let v := eval cbv zeta beta delta [k] in k in
+     (match v with (epsilon ?w ?d) ⇒ change (P v); epsilon_elim_aux end))
+  end.
+ +
+Section On_Iota.
+  Variables (A:Type)(P: A → Prop).
+  Hypothesis inhA : inhabited A.
+  Hypothesis unique_P : ∃ ! x, P x.
+ +
+  Lemma iota_spec_1 : unique P (iota inhA P).
+ +
+  Lemma iota_eq : ∀ a, P a → a = iota inhA P.
+ +
+  Lemma iota_ind (Q:A → Prop) :
+    (∀ a, unique P a → Q a) → Q (iota inhA P).
+ +
+End On_Iota.
+ +
+Ltac iota_elim :=
+  match goal with |- context c [(iota ?b ?P)] ⇒
+  apply iota_ind end.
+ +
+ +
+Class InH (A: Type) : Prop :=
+   InHWit : inhabited A.
+ +
+Definition some {A:Type} {H : InH A} (P: A → Prop)
+  := epsilon (@InHWit A H) P.
+ +
+Definition the {A:Type} {H : InH A} (P: A → Prop)
+  := iota (@InHWit A H) P.
+ +
+#[ global ] Instance inhNat : InH nat.
+Qed.
+ +
+
+ +
+A small example +
+
+ +
+ +
+Definition some_pos := some (fun i ⇒ 0 < i).
+ +
+Example Ex1 : 1 ≤ some_pos.
+    unfold some_pos, some; epsilon_elim.
+    ∃ 42; auto with arith.
+Qed.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.Ordering_Functions.html b/theories/html/hydras.Schutte.Ordering_Functions.html new file mode 100644 index 00000000..a47f2e13 --- /dev/null +++ b/theories/html/hydras.Schutte.Ordering_Functions.html @@ -0,0 +1,519 @@ + + + + + +hydras.Schutte.Ordering_Functions + + + + +
+ + + +
+ +

Library hydras.Schutte.Ordering_Functions

+ +
+
+ +
+Ordering functions (after Schutte) +
+ + Pierre Casteran, LaBRI, University of Bordeaux + +
+ +Every subset A of Ord can be enumerated in an unique way + by a segment of Ord. + +
+ +Thus it makes sense to consider the alpha-th element of A + +
+ +This module shows the construction of the ordering function of A, following +Schutte's definitions. + +
+ + +
+
+ +
+From ZornsLemma Require Import CountableTypes.
+From hydras Require Export Schutte_basics.
+Import Ensembles Well_Orders Countable PartialFun.
+Import Classical MoreEpsilonIota Epsilon.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+

Main definitions

+ +
+
+ +
+ +
+Definition segment (A: Ensemble Ord) :=
+  ∀ alpha beta, In A alpha → beta < alpha → In A beta.
+ +
+Definition proper_segment (A: Ensemble Ord) :=
+  segment A ∧ ¬ Same_set A ordinal.
+ +
+ +
+ +
+Class ordering_function (f : Ord → Ord)
+           (A B : Ensemble Ord) : Prop :=
+  Build_OF {
+      OF_segment : segment A;
+      OF_total : ∀ a, In A a → In B (f a);
+      OF_onto : ∀ b, In B b → ∃ a, In A a ∧ f a = b;
+      OF_mono : ∀ a b, In A a → In A b → a < b → f a < f b
+    }.
+ +
+Definition ordering_segment (A B : Ensemble Ord) :=
+  ∃ f : Ord → Ord, ordering_function f A B.
+ +
+ +
+ +
+Definition the_ordering_segment (B : Ensemble Ord) :=
+  the (fun x ⇒ ordering_segment x B).
+ +
+Definition ord (B : Ensemble Ord) :=
+  some (fun f ⇒ ordering_function f (the_ordering_segment B) B).
+ +
+Definition proper_segment_of (B : Ensemble Ord)(beta : Ord): Ensemble Ord :=
+  fun alpha ⇒ In B alpha ∧ alpha < beta ∧ In B beta.
+ +
+Definition normal (f : Ord → Ord)(B : Ensemble Ord): Prop :=
+ ordering_function f ordinal B ∧ continuous f ordinal B.
+ +
+Definition fun_equiv (f g : Ord → Ord)(A B : Ensemble Ord) :=
+  Same_set A B ∧ ∀ a, In A a → f a = g a.
+ +
+
+ +
+

Properties of segments

+ +
+
+ +
+Lemma ordinal_segment : segment ordinal.
+ +
+Lemma members_proper (alpha : Ord) :
+  proper_segment (members alpha).
+ +
+Lemma proper_members (A: Ensemble Ord) (H : proper_segment A) :
+   ∃ a: Ord, Same_set A (members a).
+ +
+Lemma countable_segment_proper : ∀ A : Ensemble Ord,
+           segment A → Countable A → proper_segment A.
+ +
+Lemma ordering_function_In f A B a :
+   ordering_function f A B → In A a → In B (f a).
+ +
+Lemma ordering_function_mono (f : Ord → Ord) (A B: Ensemble Ord) :
+  ordering_function f A B →
+  ∀ alpha beta,
+    In A alpha → In A beta → alpha < beta → f alpha < f beta.
+ +
+#[global] Hint Resolve ordering_function_mono : schutte.
+ +
+Lemma ordering_function_mono_weak (f : Ord → Ord) (A B: Ensemble Ord) :
+ ordering_function f A B →
+   ∀ a b, In A a → In A b → a ≤ b → f a ≤ f b.
+ +
+#[global] Hint Resolve ordering_function_mono_weak : schutte.
+ +
+Lemma ordering_function_monoR : ∀ f A B, ordering_function f A B →
+   ∀ a b, In A a → In A b → f a < f b → a < b.
+ +
+#[global] Hint Resolve ordering_function_monoR : schutte.
+ +
+Lemma Ordering_bijection : ∀ f A B, ordering_function f A B →
+                                         fun_bijection A B f.
+ +
+Lemma ordering_function_mono_weakR :
+  ∀ f A B, ordering_function f A B →
+                ∀ a b, In A a → In A b → f a ≤ f b → a ≤ b.
+ +
+#[global] Hint Resolve ordering_function_mono_weakR : schutte.
+ +
+Lemma ordering_function_seg : ∀ A B, ordering_segment A B →
+                                          segment A.
+ +
+Lemma empty_ordering : ∀ B, (∀ b, ¬ B b) →
+                                 ordering_function (fun o ⇒ o)
+                                                   (members zero)
+                                                   B.
+ +
+Lemma segment_lt : ∀ A a b, segment A → A a → b < a → A b.
+ +
+Theorem segment_unbounded : ∀ A:Ensemble Ord, segment A →
+                                        Unbounded A →
+                                        A = ordinal.
+ +
+ +
+ +
+Theorem ordering_le : ∀ f A B,
+    ordering_function f A B →
+    ∀ alpha, In A alpha → alpha ≤ f alpha. +
+ +
+Section ordering_function_unicity.
+ +
+ Variables B A1 A2 : Ensemble Ord.
+ +
+ Variables f1 f2 : Ord → Ord.
+ Hypothesis O1 : ordering_function f1 A1 B.
+ Hypothesis O2 : ordering_function f2 A2 B.
+ +
+ +
+Theorem ordering_function_unicity : fun_equiv f1 f2 A1 A2.
+ +
+End ordering_function_unicity.
+ +
+Lemma ordering_function_seg_unicity : ∀ A1 A2 B,
+           ordering_segment A1 B →
+           ordering_segment A2 B → A1 = A2.
+ +
+
+ +
+Let us build now an ordering function, and the associated ordering segment + of any subset B composed of ordinals +
+
+ +
+Lemma proper_of_proper : ∀ B beta beta',
+                           ordinal beta → In B beta →
+                           In (proper_segment_of B beta) beta' →
+                           proper_segment_of B beta' =
+                           proper_segment_of (proper_segment_of B beta) beta'.
+ +
+Section building_ordering_function_1.
+ Variable B : Ensemble Ord.
+ +
+ Hypothesis H_B : ∀ beta, In B beta →
+                               ∃! A : Ensemble Ord,
+                                    ordering_segment A
+                                             (proper_segment_of B beta).
+ +
+ Section beta_fixed.
+ +
+ Variable beta : Ord.
+ Hypothesis beta_B : In B beta.
+ +
+
+ +
+Let us build an ordering function for (proper_segment_of B beta) +
+
+ +
+ Definition _A := the (fun E ⇒
+                          ordering_segment E (proper_segment_of B beta)).
+ +
+ Definition _f := some (fun f ⇒
+                           ordering_function f _A
+                                             (proper_segment_of B beta)).
+ +
+ Lemma of_beta' : ordering_function _f _A (proper_segment_of B beta).
+ +
+ Remark Bbeta_denum : Countable (proper_segment_of B beta).
+ +
+ #[local] Hint Resolve of_beta': schutte.
+ +
+Remark A_denum : Countable _A.
+ +
+Lemma Proper_A : proper_segment _A.
+ +
+Lemma g_def1 : ∃ g_beta: Ord, ordinal g_beta ∧ _A = members g_beta.
+ +
+Lemma g_unic : ∀ g_beta g_beta', ordinal g_beta →
+                                      ordinal g_beta' →
+                                      _A = members g_beta →
+                                      _A = members g_beta' →
+                                      g_beta = g_beta'.
+ +
+Definition g := iota inh_Ord (fun o ⇒ ordinal o ∧ _A = members o).
+ +
+End beta_fixed.
+ +
+Lemma g_def : ∀ beta, In B beta → _A beta = members (g beta ).
+ +
+Lemma g_lemma :
+  ∀ beta, In B beta →
+       ordering_function (_f beta) (members (g beta))
+                                       (proper_segment_of B beta).
+ +
+Lemma g_mono : ∀ beta1 beta2, In B beta1 → In B beta2 →
+                                   beta1 < beta2 →
+                                   g beta1 < g beta2.
+ +
+Lemma L3a : segment (image B g).
+ +
+ Lemma g_bij : fun_bijection B (image B g) g.
+ +
+ #[local] Hint Resolve g_bij : schutte.
+ +
+Let g_1 := inv_fun inh_Ord B (image B g) g.
+ +
+Lemma g_1_bij : fun_bijection (image B g) B g_1.
+ +
+#[local] Hint Resolve g_1_bij : schutte.
+ +
+Lemma g_1_of : ordering_function g_1 (image B g) B.
+ +
+Lemma image_B_g_seg : ordering_segment (image B g) B.
+ +
+
+ +
+Corresponds to Lemma 3 of Schutte's chapter : + It is used twice in the building of ordering function for any subset B of ordinal +
+
+ +
+Lemma L3_u : ∃! S, ordering_segment S B.
+ +
+End building_ordering_function_1.
+ +
+
+ +
+For any set B, we build by transfinite induction the ordering segment of + B and the (unique upto extensionnality) ordering function of B + +
+
+ +
+Section building_ordering_function_by_induction.
+ +
+ Variable B : Ensemble Ord.
+ +
+ Lemma ordering_segments_of_B (beta : Ord) :
+  In B beta →
+  ∃! A : Ensemble Ord,
+    ordering_segment A (proper_segment_of B beta).
+ +
+ Theorem ordering_segment_ex_unique : ∃! S, ordering_segment S B.
+ +
+Theorem ordering_function_ex : ∃ ! S, ∃ f, ordering_function f S B.
+ +
+Lemma ord_ok :
+  ordering_function (ord B) (the_ordering_segment B) B.
+ +
+Lemma segment_the_ordering_segment :
+  segment (the_ordering_segment B).
+ +
+Lemma ord_eq (A : Ensemble Ord) (f : Ord → Ord) :
+  ordering_function f A B →
+  fun_equiv f (ord B) A (the_ordering_segment B).
+ +
+End building_ordering_function_by_induction.
+ +
+ +
+About ordering_function_ex.
+About ordering_function_unicity.
+ +
+ +
+Lemma of_image : ∀ f A B, ordering_function f A B →
+                               ordering_function f A (image A f).
+ +
+Section Th13_5.
+ Variables (A B : Ensemble Ord).
+ Variable f : Ord → Ord.
+ Hypothesis f_ord : ordering_function f A B.
+ +
+ Section recto.
+ +
+ Hypothesis f_cont : continuous f A B.
+ +
+ Lemma Th_13_5_1 : Closed B.
+ +
+End recto.
+ +
+Section verso.
+ Hypothesis B_closed : Closed B.
+ +
+ +
+ Lemma Th_13_5_2 : continuous f A B.
+ +
+End verso.
+ +
+End Th13_5.
+ +
+Theorem TH_13_6 (B : Ensemble Ord)(f : Ord → Ord) :
+  normal f B → Closed B ∧ Unbounded B.
+ +
+Lemma ordering_unbounded_unbounded :
+  ∀ A B f, ordering_function f A B →
+                 (Unbounded B ↔ Unbounded A).
+ +
+Theorem TH_13_6R (A B : Ensemble Ord) (f : Ord → Ord) :
+  ordering_function f A B →
+  Closed B → Unbounded B → normal f B.
+ +
+
+ +
+If f is the ordering function of B, then f 0 is the least element of + B +
+
+ +
+Lemma ordering_function_least_least :
+ ∀ B f , ordering_function f ordinal B →
+     least_member lt B (f zero).
+ +
+Lemma segment_lt_closed : ∀ A a b, segment A →
+                                          In A b →
+                                          a < b →
+                                          In A a.
+ +
+Lemma th_In A alpha : In (the_ordering_segment A) alpha →
+                             In A (ord A alpha).
+ +
+ +
+ +
+About Th_13_5_2.
+ +
+ +
+Arguments ord : clear implicits.
+Arguments the_ordering_segment : clear implicits.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.PartialFun.html b/theories/html/hydras.Schutte.PartialFun.html new file mode 100644 index 00000000..5d302635 --- /dev/null +++ b/theories/html/hydras.Schutte.PartialFun.html @@ -0,0 +1,266 @@ + + + + + +hydras.Schutte.PartialFun + + + + +
+ + + +
+ +

Library hydras.Schutte.PartialFun

+ +
+ +
+
+ +
+

Partial functions

+ +
+ + Pierre Casteran, Univ. Bordeaux and LaBRI +
+ + We study the relationship between two representations of partial + functions : relational presentation, and with the iota description operator +
+
+ +
+From Coq Require Import ClassicalDescription Ensembles Image
+  ProofIrrelevance.
+Import ProofIrrelevanceFacts.
+From ZornsLemma Require Import EnsemblesImplicit FunctionProperties ImageImplicit.
+From hydras Require Import MoreEpsilonIota.
+ +
+Set Implicit Arguments.
+ +
+Section AB_given.
+  Variables (A B : Type)
+            (Ha : inhabited A)
+            (Hb:inhabited B)
+            (DA: Ensemble A)
+            (DB : Ensemble B).
+ +
+
+ +
+

Transformation of a functional relation into a function

+ +
+
+ +
+Definition iota_fun (R:A→B→ Prop) : A → B :=
+  fun a ⇒ iota Hb (fun b':B ⇒ In DA a ∧ R a b' ∧ In DB b').
+ +
+Lemma iota_fun_e (R:A→B→ Prop):
+  ∀ (a:A), In DA a →
+   (∃ ! b, R a b ∧ In DB b) →
+   unique (fun b ⇒ (R a b ∧ In DB b)) (iota_fun R a).
+ +
+Lemma iota_fun_ind (P:A → Prop)
+      (Q R:A→B→ Prop):
+  ∀ (a x:A), In DA a → a = x →
+   (∃ ! b, R a b ∧ In DB b) →
+   P a →
+   (∀ b, unique (fun b ⇒ R a b ∧ In DB b) b → Q a b) →
+   Q x (iota_fun R a).
+ +
+Section f_given.
+ +
+  Variable f : A → B.
+ +
+
+ +
+relational representation +
+
+ +
+  Variable Rf : A → B → Prop.
+ +
+
+ +
+abstract properties of a function (relational representation ) +
+
+ +
+  Definition rel_domain := ∀ a, In DA a → ∃ b, Rf a b .
+  Definition rel_codomain := ∀ a b, In DA a → Rf a b → In DB b.
+  Definition rel_functional := ∀ a b b', In DA a →
+                              Rf a b → Rf a b' → b = b'.
+  Definition rel_onto := ∀ b, In DB b → ∃ a, In DA a ∧ Rf a b.
+  Definition rel_inj := ∀ a a' b, In DA a →
+                                       In DA a' →
+                                       Rf a b →
+                                       Rf a' b →
+                                       a = a'.
+ +
+  Inductive rel_injection : Prop :=
+   rel_inj_i : rel_domain → rel_codomain →
+        rel_functional →
+        rel_inj →
+        rel_injection.
+ +
+  Inductive rel_bijection : Prop :=
+   rel_bij_i : rel_domain → rel_codomain →
+        rel_functional →
+        rel_onto →
+        rel_inj →
+        rel_bijection.
+ +
+
+ +
+Abstract properties of f: A->B (wrt. domain DA and codomain DB +
+
+ +
+  Definition fun_codomain := ∀ a, In DA a → In DB (f a).
+  Definition fun_onto := ∀ b, In DB b → ∃ a, In DA a ∧ f a = b.
+  Definition fun_inj := ∀ a a' , In DA a → In DA a' → f a = f a' →
+                 a = a'.
+ +
+  Definition image := fun b ⇒ ∃ a, In DA a ∧ f a = b.
+ +
+  Inductive fun_injection : Prop :=
+   fun_inj_i : fun_codomain →
+                            fun_inj → fun_injection.
+ +
+  Inductive fun_bijection : Prop :=
+   fun_bij_i : fun_codomain → fun_onto →
+                            fun_inj → fun_bijection.
+ +
+  Definition rel_inv := fun b a ⇒ In DA a ∧ In DB b ∧ Rf a b.
+ +
+End f_given.
+ +
+
+ +
+Conversion from a relational definition : A->B->Prop into a partial + function of type A->B +
+
+ +
+ Section rel_to_fun.
+ +
+ Variables
+           (Rf : A → B → Prop).
+ +
+ Definition r2i := iota_fun Rf.
+ +
+End rel_to_fun.
+End AB_given.
+ +
+Section inversion_of_bijection.
+  Variables (A B : Type)
+           (inhA : inhabited A)
+           (DA : Ensemble A)
+           (DB : Ensemble B)
+           (f : A → B).
+ +
+ Let inv_spec := fun y x ⇒ In DA x ∧ In DB y ∧ f x = y.
+ +
+ Definition inv_fun := r2i inhA DB DA inv_spec.
+ +
+ Hypothesis f_b : fun_bijection DA DB f.
+ +
+Lemma inv_compose :
+  ∀ x, DA x → inv_fun (f x) = x.
+ +
+Lemma inv_composeR : ∀ b, DB b → f (inv_fun b) = b.
+ +
+Lemma inv_fun_bij : fun_bijection DB DA inv_fun.
+ +
+End inversion_of_bijection.
+ +
+Lemma image_as_Im {A B : Type} (U : Ensemble A) (f : A → B) :
+  image U f = Im U f.
+ +
+
+ +
+convert a fun_bijection to a bijection in the sense of the + ZornsLemma library. +
+
+Definition fun_restr {U V : Type} (f : U → V)
+  {A : Ensemble U} {B : Ensemble V}
+  (Hf : fun_codomain A B f) :
+  { x : U | In A x } → { y : V | In B y } :=
+  fun p : { x : U | In A x } ⇒
+    exist (fun y : V ⇒ In B y)
+      (f (proj1_sig p))
+      (Hf (proj1_sig p) (proj2_sig p)).
+ +
+Lemma fun_bijection_codomain
+  {U V : Type} {A : Ensemble U} {B : Ensemble V}
+  (g : U → V) (g_bij : fun_bijection A B g) :
+  fun_codomain A B g.
+ +
+Lemma fun_bijection_is_ZL_bijection
+  {U V : Type} {A : Ensemble U} {B : Ensemble V}
+  (g : U → V) (g_bij : fun_bijection A B g) :
+  FunctionProperties.bijective
+    (fun_restr (fun_bijection_codomain g_bij)).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.Schutte.html b/theories/html/hydras.Schutte.Schutte.html new file mode 100644 index 00000000..83938974 --- /dev/null +++ b/theories/html/hydras.Schutte.Schutte.html @@ -0,0 +1,162 @@ + + + + + +hydras.Schutte.Schutte + + + + +
+ + + +
+ +

Library hydras.Schutte.Schutte

+ +
+ +
+
+ +
+ +
+ + Axiomatic definition of countable ordinal numbers (after Kurt Schutte's + "Proof Theory" + +
+ + Pierre Casteran (LaBRI, University of Bordeaux) + with contributions by Florian Hatat (formerly student at ENS Lyon) + +
+ + +
+
+ +
+From hydras Require Export Schutte_basics Ordering_Functions
+        Addition AP CNF Critical Correctness_E0.
+ +
+
+ +
+

Warning

+ + +
+ +This directory contains an adaptation to Coq of the mathematical +presentation of the set of countable ordinal numbers by K. Schutte. + +
+ +In order to respect as most as possible the style of that presentation, +we chosed to work in classical logic augmented by Hilbert's [epsilon] +operator. + +
+ +So, all the construction herein is powered by six axioms, three of them are +Schutte's axioms, and the other ones allow us to work in that "classical" framework, still using the Coq proof assistant and its libraries. + +
+ +

Schutte's Axioms

+ + +
+ + We consider a type ON (Ordinal numbers), well-ordered by some relation + lt, and such that every subset X of Ord is countable iff X is bounded. + +
+ +
+
+Axiom AX1 : WO lt.
+
+Axiom AX2 : ∀ X: Ensemble Ord, (∃ a, (∀ y, X y → y < a)) →
+                                   countable X.
+
+
+Axiom AX3 : ∀ X : Ensemble Ord,
+              countable X →
+              ∃ a, ∀ y, X y → y < a.
+ +
+ + +
+ +

Classical logic and Hilbert style

+ + +
+ +

From Coq.Logic.Classical

+ + +
+ +
+Axiom classic : ∀ P:Prop, P ∨ ¬ P. +
+ + +
+ +

From Coq.Logic.Epsilon

+ + +
+ +
+
+Axiom epsilon_statement :
+  ∀ (A : Type) (P : A→Prop), inhabited A →
+    { x : A | (∃ x, P x) → P x }.
+ +
+ + +
+ +

Needed for epsilon to work properly.

+ + +
+ +
+Axiom inh_Ord : inhabited Ord.
+ +
+ + +
+
+Example Ex42: omega + 42 + omega^2 = omega^2.
+  Check AP_plus_closed. +
+Qed.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.Schutte_basics.html b/theories/html/hydras.Schutte.Schutte_basics.html new file mode 100644 index 00000000..bf27873b --- /dev/null +++ b/theories/html/hydras.Schutte.Schutte_basics.html @@ -0,0 +1,824 @@ + + + + + +hydras.Schutte.Schutte_basics + + + + +
+ + + +
+ +

Library hydras.Schutte.Schutte_basics

+ +
+ +
+
+ +
+ Ordinals of the first and second class, + After Kurt Schuttes book : Proof theory + +
+ + +
+ + Pierre Casteran, Univ. Bordeaux and LaBRI +
+
+ +
+From Coq Require Import Relations Classical Classical_sets.
+From ZornsLemma Require Import CountableTypes.
+From hydras Require Import Well_Orders Lub Countable.
+ +
+Import Compare_dec Coq.Sets.Image PartialFun MoreEpsilonIota.
+ +
+Declare Scope schutte_scope.
+ +
+Set Implicit Arguments.
+ +
+#[global] Hint Unfold In : schutte.
+ +
+Delimit Scope schutte_scope with sch.
+Open Scope schutte_scope.
+ +
+
+ +
+

Definitions

+ +
+ + The type of countable ordinal numbers +
+
+ +
+ +
+Parameter Ord : Type.
+Parameter lt : relation Ord.
+Infix "<" := lt : schutte_scope.
+ +
+Notation ordinal := (@Full_set Ord).
+Definition big0 alpha : Ensemble Ord := fun beta ⇒ beta < alpha.
+ +
+ +
+ +
+#[global] Hint Resolve ordinal_ok : schutte.
+ +
+
+ +
+

The three axioms by Schutte

+ +
+ + First Schutte's axiom : Ord is well-ordered wrt lt +
+
+ +
+ +
+Axiom AX1 : WO lt.
+ +
+#[global] Hint Resolve AX1 : schutte.
+ +
+#[global] Instance WO_ord : WO lt := AX1.
+ +
+
+ +
+Stuff for using Coq.Logic.Epsilon +
+
+ +
+Axiom inh_Ord : inhabited Ord.
+ +
+#[ global ] Instance InH_Ord : InH Ord. +
+#[ global ] Instance Inh_OSets : InH (Ensemble Ord).
+ +
+#[ global ] Instance Inh_Ord_Ord : InH (Ord → Ord).
+ +
+#[global] Hint Resolve AX1 Inh_Ord_Ord Inh_OSets inh_Ord : schutte.
+ +
+Definition le := Le lt.
+ +
+Infix "≤" := le : schutte_scope.
+ +
+
+ +
+Second and third axioms from Schutte + +
+ +A subset X of Ord is bounded iff X is countable +
+
+ +
+ +
+Axiom AX2 :
+  ∀ X: Ensemble Ord,
+    (∃ a, (∀ y, In X y → y < a)) →
+    Countable X.
+ +
+Axiom AX3 :
+  ∀ X : Ensemble Ord,
+    Countable X →
+    ∃ a, ∀ y, In X y → y < a.
+ +
+ +
+
+ +
+

First definitions

+ + +
+
+ +
+Definition ge alpha : Ensemble Ord := fun beta ⇒ alpha ≤ beta.
+ +
+Definition Unbounded (X : Ensemble Ord) :=
+  ∀ x: Ord, ∃ y, In X y ∧ x < y.
+ +
+ +
+Definition zero := the_least ordinal.
+ +
+ +
+Definition succ (alpha : Ord)
+  := the_least (fun beta ⇒ alpha < beta).
+ +
+ +
+
+ +
+Finite ordinals +
+
+ +
+ +
+Fixpoint finite (i:nat) : Ord :=
+  match i with 0 ⇒ zero
+            | S i ⇒ succ (finite i)
+  end.
+ +
+Notation F i := (finite i).
+ +
+Coercion finite : nat >-> Ord.
+ +
+ +
+Definition is_finite := seq_range finite.
+ +
+
+ +
+

Limits

+ +
+
+ +
+ +
+Definition sup_spec X lambda := is_lub ordinal lt X lambda.
+ +
+Definition sup (X: Ensemble Ord) : Ord := the (sup_spec X).
+ +
+Notation "'|_|' X" := (sup X) (at level 29) : schutte_scope.
+ +
+ +
+
+ +
+ Limit of omega-sequences +
+
+ +
+ +
+Definition omega_limit (s:nat→Ord) : Ord
+  := |_| (seq_range s).
+ +
+Definition _omega := omega_limit finite.
+ +
+Notation omega := (_omega).
+ +
+ +
+
+ +
+Successor and limit ordinals +
+
+ +
+Definition is_succ (alpha:Ord)
+  := ∃ beta, alpha = succ beta.
+ +
+Definition is_limit (alpha:Ord)
+  := alpha ≠ zero ∧ ¬ is_succ alpha.
+ +
+
+ +
+Ordinals considered as sets +
+
+ +
+Definition members (a:Ord) := (fun b ⇒ b < a).
+ +
+Definition set_eq (X Y: Ord → Prop) := ∀ a, (X a ↔ Y a).
+ +
+
+ +
+Induction (after Schutte) +
+
+ +
+Definition progressive (P : Ord → Prop) : Prop :=
+  ∀ a, (∀ b, b < a → P b) → P a.
+ +
+ +
+Definition Closed (B : Ensemble Ord) : Prop :=
+  ∀ M, Included M B → Inhabited M → Countable M →
+                            In B (|_| M).
+ +
+ +
+Definition continuous (f:Ord→Ord)(A B : Ensemble Ord) : Prop :=
+  fun_codomain A B f ∧
+  Closed A ∧
+  (∀ U, Included U A → Inhabited U →
+             Countable U → |_| (image U f) = f (|_| U)).
+ +
+
+ +
+

Basic properties

+ + +
+ + +
+
+ +
+Lemma Unbounded_not_countable (X: Ensemble Ord) :
+    Unbounded X → not (Countable X).
+ +
+Lemma countable_not_Unbounded : ∀ X,
+    Countable X → not (Unbounded X).
+ +
+Lemma Progressive_Acc: progressive (Acc lt).
+ +
+Theorem transfinite_induction (P: Ord→Prop) : progressive P →
+                                 ∀ a, P a.
+ +
+
+ +
+

Properties of le and lt

+ +
+
+ +
+Lemma le_refl : ∀ alpha, alpha ≤ alpha.
+ +
+#[global] Hint Resolve le_refl : schutte.
+ +
+Lemma eq_le : ∀ a b : Ord, a = b → a ≤ b.
+ +
+Lemma lt_le : ∀ a b: Ord, a < b → a ≤ b.
+ +
+Lemma lt_trans : ∀ a b c : Ord, a < b → b < c → a < c.
+ +
+Lemma le_lt_trans : ∀ a b c, a ≤ b → b < c → a < c.
+ +
+Lemma lt_le_trans : ∀ a b c, a < b → b ≤ c → a < c.
+ +
+Lemma le_trans : ∀ a b c, a ≤ b → b ≤ c → a ≤ c.
+ +
+Lemma lt_irrefl : ∀ a, ¬ (a < a).
+ +
+Lemma le_antisym : ∀ a b, a ≤ b → b ≤ a → a = b.
+ +
+Lemma le_eq_or_lt : ∀ a b, a ≤ b → a = b ∨ a < b.
+ +
+Lemma le_not_gt : ∀ a b, a ≤ b → ¬ b < a.
+ +
+#[global] Hint Resolve eq_le lt_le lt_trans le_trans le_lt_trans
+     lt_le_trans lt_irrefl le_not_gt:
+  schutte.
+ +
+Lemma le_disj : ∀ alpha beta, alpha ≤ beta →
+                                   alpha = beta ∨ alpha < beta.
+ +
+Lemma all_ord_acc : ∀ alpha : Ord, Acc lt alpha.
+ +
+Lemma trichotomy : ∀ a b : Ord ,
+                               a < b ∨ a = b ∨ b < a.
+ +
+ +
+Lemma lt_or_ge : ∀ a b: Ord, a < b ∨ b ≤ a.
+ +
+Lemma not_gt_le : ∀ a b, ¬ b < a → a ≤ b.
+ +
+#[global] Hint Unfold Included : schutte.
+ +
+
+ +
+

Global properties

+ +
+
+ +
+Theorem Non_denum : ¬ Countable ordinal.
+ +
+Lemma Inh_ord : Inhabited ordinal.
+ +
+Theorem unbounded : ∀ alpha, ∃ beta, alpha < beta.
+ +
+Lemma the_least_ok : ∀ X : Ensemble Ord,
+    Inhabited X → ∀ a, In X a → the_least X ≤ a.
+ +
+
+ +
+

About zero

+ +
+
+ +
+ +
+Lemma zero_le (alpha : Ord) : zero ≤ alpha. +
+ +
+Lemma not_lt_zero : ∀ alpha, ¬ alpha < zero.
+ +
+Lemma zero_or_greater : ∀ alpha : Ord,
+    alpha = zero ∨ ∃ beta, beta < alpha.
+ +
+Lemma zero_or_positive : ∀ alpha, alpha = zero ∨ zero < alpha.
+ +
+
+ +
+

Properties of successor

+ +
+
+ +
+ +
+Definition succ_spec (alpha:Ord) :=
+  least_member lt (fun z ⇒ alpha < z).
+ +
+ +
+Lemma succ_ok :
+  ∀ alpha, succ_spec alpha (succ alpha). +
+ +
+ +
+Lemma lt_succ (alpha : Ord): alpha < succ alpha.
+ +
+#[global] Hint Resolve lt_succ : schutte. +
+Lemma lt_succ_le (alpha beta : Ord):
+  alpha < beta → succ alpha ≤ beta.
+ +
+Lemma lt_succ_le_2 (alpha beta : Ord):
+  alpha < succ beta → alpha ≤ beta. +
+Lemma succ_mono (alpha beta : Ord):
+  alpha < beta → succ alpha < succ beta. +
+Arguments succ_mono [ alpha beta].
+ +
+Lemma succ_monoR (alpha beta : Ord) :
+ succ alpha < succ beta → alpha < beta. +
+Lemma succ_injection (alpha beta : Ord) :
+  succ alpha = succ beta → alpha = beta. +
+Lemma succ_zero_diff (alpha : Ord): succ alpha ≠ zero. +
+Lemma zero_lt_succ : ∀ alpha, zero < succ alpha. +
+Lemma lt_succ_lt (alpha beta : Ord) :
+  is_limit beta → alpha < beta → succ alpha < beta. +
+ +
+#[global] Hint Resolve zero_lt_succ zero_le : schutte.
+ +
+
+ +
+Less than finite is finite ... +
+
+ +
+Lemma finite_lt_inv : ∀ i o,
+    o < F i →
+    ∃ j:nat , (j<i)%nat ∧ o = F j.
+ +
+Lemma finite_mono : ∀ i j, (i<j)%nat → F i < F j.
+ +
+#[global] Hint Resolve finite_mono : schutte.
+ +
+Lemma finite_inj : ∀ i j, F i = F j → i = j.
+ +
+
+ +
+

Building limits

+ +
+
+ +
+Lemma sup_exists : ∀ X, Countable X →
+                             ex (sup_spec X).
+ +
+Lemma sup_unicity : ∀ X l l',
+                      (Countable X ∧ ∀ x, X x → ordinal x) →
+                      sup_spec X l → sup_spec X l' → l = l'.
+ +
+Lemma sup_spec_unicity (X:Ensemble Ord) (HX : Countable X) :
+  ∃! u, sup_spec X u.
+ +
+Lemma sup_ok1 (X: Ensemble Ord)(HX : Countable X) :
+  sup_spec X (sup X).
+ +
+Lemma sup_upper_bound (X : Ensemble Ord) (alpha : Ord):
+  Countable X → In X alpha → alpha ≤ |_| X.
+ +
+Lemma sup_least_upper_bound (X : Ensemble Ord) (alpha : Ord) :
+  Countable X → (∀ y, In X y → y ≤ alpha) → sup X ≤ alpha.
+ +
+Lemma sup_of_leq (alpha : Ord) :
+    alpha = |_| (fun x : Ord ⇒ x ≤ alpha).
+ +
+Lemma sup_mono (X Y : Ensemble Ord) :
+    Countable X →
+    Countable Y →
+    (∀ x, In X x → ∃ y, In Y y ∧ x ≤ y) →
+|_| X ≤ |_| Y.
+ +
+Lemma sup_eq_intro (X Y : Ensemble Ord):
+  Countable X → Countable Y →
+  Included X Y → Included Y X →
+|_| X = |_| Y.
+ +
+Lemma lt_sup_exists_leq (X: Ensemble Ord) :
+  Countable X →
+  ∀ y, y < sup X →
+            ∃ x, In X x ∧ y ≤ x.
+ +
+Lemma lt_sup_exists_lt (X : Ensemble Ord) :
+  Countable X →
+  ∀ y, y < sup X → ∃ x, In X x ∧ y < x.
+ +
+Lemma members_eq (alpha beta : Ord) :
+  members alpha = members beta → alpha = beta.
+ +
+Lemma sup_members_succ (alpha : Ord) :
+  sup (members alpha) < alpha → alpha = succ (|_| (members alpha)).
+ +
+Lemma sup_members_not_succ (alpha beta : Ord) :
+  alpha = sup (members alpha) → alpha ≠ succ beta.
+ +
+
+ +
+

Sequences of ordinals

+ +
+
+ +
+Definition seq_mono (s:nat → Ord) :=
+  ∀ i j, (i < j)%nat → s i < s j.
+ +
+Lemma seq_mono_intro (s: nat → Ord) :
+  (∀ i, s i < s (S i)) → seq_mono s.
+ +
+Lemma seq_mono_inj (s : nat → Ord) :
+  seq_mono s → injective s.
+ +
+#[global] Hint Resolve Countable.seq_range_countable seq_mono_intro : schutte.
+ +
+#[global] Hint Constructors Full_set: core.
+ +
+Lemma lt_omega_limit (s : nat → Ord) :
+  seq_mono s → ∀ i, s i < omega_limit s.
+ +
+Lemma omega_limit_least (s : nat → Ord) :
+    seq_mono s →
+    ∀ y : Ord,
+      (∀ i, s i < y) →
+      omega_limit s ≤ y.
+ +
+Lemma lt_omega_limit_lt_exists_lt (alpha : Ord) (s : nat → Ord) :
+  (∀ i, s i < s (S i)) →
+  alpha < omega_limit s →
+  ∃ j, alpha < s j.
+ +
+Lemma omega_limit_least_gt (alpha : Ord) (s : nat → Ord)
+      (Hmono : ∀ i, s i < s (S i))
+      (H : alpha < omega_limit s) :
+  ∃ i, least_member Peano.lt (fun j ⇒ alpha < s j) i.
+ +
+
+ +
+

Properties of omega

+ +
+
+ +
+Lemma finite_lt_omega (i : nat) : i < omega.
+ +
+Lemma zero_lt_omega : zero < omega.
+ +
+Lemma lt_omega_finite (alpha : Ord) :
+  alpha < omega → ∃ i:nat, alpha = i.
+ +
+Lemma is_limit_omega : is_limit omega.
+ +
+
+ +
+

About zero, is_succ and is_limit

+ +
+
+ +
+Lemma not_is_succ_zero : ¬ is_succ zero.
+ +
+Lemma not_is_limit_zero : ¬ is_limit zero.
+ +
+Lemma not_is_limit_succ (alpha : Ord) : is_succ alpha → ¬ is_limit alpha.
+ +
+Lemma not_is_succ_limit (alpha : Ord) : is_limit alpha → ¬ is_succ alpha.
+ +
+
+ +
+

About members

+ +
+
+ +
+Lemma countable_members (alpha : Ord): Countable (members alpha).
+ +
+#[global] Hint Resolve countable_members : schutte.
+ +
+Lemma le_sup_members (alpha : Ord) : |_| (members alpha) ≤ alpha.
+ +
+Lemma is_limit_sup_members (alpha : Ord) :
+  is_limit alpha → alpha = |_| (members alpha).
+ +
+Lemma sup_members_disj (alpha : Ord) :
+  alpha = sup (members alpha) →
+  alpha = zero ∨ is_limit alpha.
+ +
+Theorem classification (alpha : Ord) :
+  alpha = zero ∨ is_succ alpha ∨ is_limit alpha.
+ +
+
+ +
+

Extensional equalities on sets of ordinals

+ +
+
+ +
+Lemma members_omega : Same_set (members omega) is_finite.
+ +
+Lemma Not_Unbounded_bounded (X: Ensemble Ord):
+   ¬ Unbounded X →
+   ∃ beta : Ord, ∀ x, In X x → x < beta.
+ +
+Lemma Not_Unbounded_countable (X : Ensemble Ord) :
+  ¬ Unbounded X → Countable X.
+ +
+Lemma not_countable_unbounded (X : Ensemble Ord):
+    ¬ Countable X → Unbounded X.
+ +
+Lemma le_alpha_zero (alpha : Ord) :
+  alpha ≤ zero → alpha = zero.
+ +
+Lemma finite_not_limit (i: nat): ¬ is_limit i.
+ +
+ +
+Definition zero: Ord := iota inh_Ord (least_member lt ordinal).
+ +
+Lemma zero_le (alpha : Ord) : zero ≤ alpha.
+ +
+Module Bad.
+ +
+  Definition bottom := the_least Empty_set.
+ +
+  Lemma le_zero_bottom : zero ≤ bottom.
+ +
+  Lemma bottom_eq : bottom = bottom.
+ +
+  Lemma le_bottom_zero : bottom ≤ zero. +
+End Bad.
+ +
+End iota_demo.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.Schutte.Well_Orders.html b/theories/html/hydras.Schutte.Well_Orders.html new file mode 100644 index 00000000..a6e6bbb2 --- /dev/null +++ b/theories/html/hydras.Schutte.Well_Orders.html @@ -0,0 +1,173 @@ + + + + + +hydras.Schutte.Well_Orders + + + + +
+ + + +
+ +

Library hydras.Schutte.Well_Orders

+ +
+
+ +
+ Well ordered sets (after Schutte) +
+ + Pierre Casteran LaBRI, Universite de Bordeaux +
+
+ +
+From Coq Require Import Relations Classical Classical_sets RelationClasses
+     Wf_nat.
+ +
+From hydras Require Import PartialFun.
+Import MoreEpsilonIota.
+ +
+Arguments In [U].
+Arguments Included [U].
+ +
+Set Implicit Arguments.
+#[global] Hint Unfold In : core.
+ +
+Section the_context.
+ +
+  Variables (M:Type)
+            (Lt : relation M).
+ +
+  Definition Le (a b:M) := a = b ∨ Lt a b.
+ +
+  Definition least_member (X:Ensemble M) (a:M) :=
+    In X a ∧ ∀ x, In X x → Le a x.
+ +
+  Definition least_fixpoint (f : M → M) (x:M) :=
+    f x = x ∧ ∀ y: M, f y = y → Le x y.
+ +
+
+ +
+Well Ordering +
+
+ +
+  Class WO : Type:=
+    {
+    Lt_trans : Transitive Lt;
+    Lt_irreflexive : ∀ a:M, ¬ Lt a a;
+    well_order : ∀ (X:Ensemble M)(a:M),
+        In X a →
+        ∃ a0:M, least_member X a0
+    }.
+ +
+ +
+  Section About_WO.
+    Context (Wo : WO).
+ +
+    Lemma Lt_connect : ∀ a b, Lt a b ∨ a = b ∨ Lt b a.
+ +
+    Lemma Le_refl : ∀ x:M, Le x x.
+ +
+    Lemma Le_antisym : ∀ a b, Le a b → Le b a → a = b.
+ +
+    #[global] Instance Le_trans : Transitive Le.
+ +
+    Lemma Le_Lt_trans : ∀ x y z, Le x y → Lt y z → Lt x z.
+ +
+    Lemma Lt_Le_trans : ∀ x y z, Lt x y → Le y z → Lt x z.
+ +
+    Lemma Lt_not_Gt : ∀ x y, Lt x y → ¬ Lt y x.
+ +
+    Lemma least_member_lower_bound : ∀ X a,
+        least_member X a → ∀ b, In X b → Le a b.
+ +
+    Lemma least_member_glb :
+      ∀ X a,
+        least_member X a →
+        ∀ b, (∀ c, In X c → Le b c) →
+                  Le b a.
+ +
+    Theorem least_member_unicity : ∀ X a b,
+        least_member X a → least_member X b → a = b.
+ +
+    Theorem least_member_ex_unique :
+      ∀ X x
+               (inhX: In X x),
+      ∃! a, least_member X a.
+ +
+    Theorem least_member_of_eq : ∀ (X Y : Ensemble M) a b ,
+        Included X Y → Included Y X →
+        least_member X a →
+        least_member Y b →
+        a = b.
+ +
+  End About_WO.
+ +
+End the_context.
+ +
+ +
+Definition the_least {M: Type} {Lt}
+           {inh : InH M} {WO: WO Lt} (X: Ensemble M) : M :=
+  the (least_member Lt X ).
+ +
+ +
+Lemma the_least_unicity {M: Type} {Lt}
+       {inh : InH M} {WO: WO Lt} (X: Ensemble M)
+       (HX: Inhabited X)
+  : ∃! l , least_member Lt X l.
+ +
+#[ global ] Instance WO_nat : WO Peano.lt.
+Qed.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.rpo.closure.html b/theories/html/hydras.rpo.closure.html new file mode 100644 index 00000000..396d8d09 --- /dev/null +++ b/theories/html/hydras.rpo.closure.html @@ -0,0 +1,73 @@ + + + + + +hydras.rpo.closure + + + + +
+ + + +
+ +

Library hydras.rpo.closure

+ +
+
+ +
+ by Evelyne Contejean, LRI +
+
+ +
+ +
+Set Implicit Arguments.
+ +
+From Coq Require Import Relations.
+ +
+Inductive trans_clos (A : Set) (R : relation A) : relation A:=
+  | t_step : ∀ x y, R x y → trans_clos R x y
+  | t_trans : ∀ x y z, R x y → trans_clos R y z → trans_clos R x z.
+ +
+Lemma trans_clos_is_trans :
+  ∀ (A :Set) (R : relation A) a b c,
+  trans_clos R a b → trans_clos R b c → trans_clos R a c.
+ +
+Lemma acc_trans :
+ ∀ (A : Set) (R : relation A) a, Acc R a → Acc (trans_clos R) a.
+ +
+Lemma wf_trans :
+  ∀ (A : Set) (R : relation A) , well_founded R →
+                                      well_founded (trans_clos R).
+ +
+Lemma inv_trans :
+  ∀ (A : Set) (R : relation A) (P : A → Prop),
+  (∀ a b, P a → R a b → P b) →
+  ∀ a b, P a → trans_clos R a b → P b.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.rpo.decidable_set.html b/theories/html/hydras.rpo.decidable_set.html new file mode 100644 index 00000000..fadd9501 --- /dev/null +++ b/theories/html/hydras.rpo.decidable_set.html @@ -0,0 +1,48 @@ + + + + + +hydras.rpo.decidable_set + + + + +
+ + + +
+ +

Library hydras.rpo.decidable_set

+ +
+
+ +
+ by Evelyne Contejean +
+
+ +
+Module Type S.
+ +
+Parameter A : Set.
+Axiom eq_A_dec : ∀ a1 a2 : A, {a1 = a2} + {a1 ≠ a2}.
+ +
+End S.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.rpo.dickson.html b/theories/html/hydras.rpo.dickson.html new file mode 100644 index 00000000..149dd566 --- /dev/null +++ b/theories/html/hydras.rpo.dickson.html @@ -0,0 +1,190 @@ + + + + + +hydras.rpo.dickson + + + + +
+ + + +
+ +

Library hydras.rpo.dickson

+ +
+
+ +
+ by Evelyne Contejean +
+ +

Dickson Lemma: the multiset extension of a well-founded ordering is well-founded.

+ + +
+
+ +
+Set Implicit Arguments.
+ +
+From Coq Require Import Relations List Setoid Multiset.
+From hydras Require Import closure more_list list_permut.
+ +
+Module Make (DS1 : decidable_set.S).
+ +
+Module DS := DS1.
+Module LP := list_permut.Make (DS1).
+Import LP.
+ +
+
+ +
+

Definition of the multiset extension of a relation.

+ +
+
+Inductive multiset_extension_step (R : relation elt) : list elt → list elt → Prop :=
+  | rmv_case :
+     ∀ l1 l2 l la a, (∀ b, In b la → R b a) →
+      list_permut l1 (la ++ l) → list_permut l2 (a :: l) →
+      multiset_extension_step R l1 l2.
+ +
+
+ +
+If n << {a} U m , then + either, there exists n' such that n = {a} U n' and n' << m, + or, there exists k, such that n = k U m, and k << {a}. +
+
+ +
+Lemma two_cases :
+ ∀ R a m n,
+ multiset_extension_step R n (a :: m) →
+ (∃ n', list_permut n (a :: n') ∧
+             multiset_extension_step R n' m) ∨
+ (∃ k, (∀ b, In b k → R b a) ∧
+            list_permut n (k ++ m)).
+ +
+
+ +
+multiset_extension_step is compatible with permutation. +
+
+Lemma list_permut_multiset_extension_step_1 :
+  ∀ R l1 l2 l, list_permut l1 l2 →
+  multiset_extension_step R l1 l → multiset_extension_step R l2 l.
+ +
+Lemma list_permut_multiset_extension_step_2 :
+  ∀ R l1 l2 l, list_permut l1 l2 →
+  multiset_extension_step R l l1 → multiset_extension_step R l l2.
+ +
+Lemma context_multiset_extension_step_app1 :
+  ∀ R l1 l2 l, multiset_extension_step R l1 l2 →
+                         multiset_extension_step R (l ++ l1) (l ++ l2).
+ +
+Lemma context_trans_clos_multiset_extension_step_app1 :
+  ∀ R l1 l2 l, trans_clos (multiset_extension_step R) l1 l2 →
+                         trans_clos (multiset_extension_step R) (l ++ l1) (l ++ l2).
+ +
+Lemma context_multiset_extension_step_app2 :
+  ∀ R l1 l2 l, multiset_extension_step R l1 l2 →
+                         multiset_extension_step R (l1 ++ l) (l2 ++ l).
+ +
+Add Parametric Morphism R : (@multiset_extension_step R) with signature (list_permut ==> list_permut ==> iff) as mult_morph.
+ +
+
+ +
+

Accessibility lemmata.

+ +
+
+Lemma list_permut_acc :
+  ∀ R l1 l2, list_permut l2 l1 →
+  Acc (multiset_extension_step R) l1 → Acc (multiset_extension_step R) l2.
+ +
+Add Parametric Morphism R : (Acc (multiset_extension_step R)) with signature (list_permut ==> iff) as acc_morph.
+ +
+Lemma dickson_aux1 :
+∀ (R : relation elt) a,
+ (∀ b, R b a →
+  ∀ m, Acc (multiset_extension_step R) m →
+            Acc (multiset_extension_step R) (b :: m)) →
+ ∀ m, Acc (multiset_extension_step R) m →
+ (∀ m', (multiset_extension_step R) m' m →
+             Acc (multiset_extension_step R) (a :: m')) →
+ Acc (multiset_extension_step R) (a :: m).
+ +
+Lemma dickson_aux2 :
+∀ R m,
+  Acc (multiset_extension_step R) m →
+  ∀ a, (∀ b, R b a →
+             ∀ m, Acc (multiset_extension_step R) m →
+                       Acc (multiset_extension_step R) (b :: m)) →
+   Acc (multiset_extension_step R) (a :: m).
+ +
+Lemma dickson_aux3 :
+∀ R a, Acc R a → ∀ m, Acc (multiset_extension_step R) m →
+Acc (multiset_extension_step R) (a :: m).
+ +
+
+ +
+Main lemma. +
+
+Lemma dickson :
+  ∀ R, well_founded R → well_founded (multiset_extension_step R).
+ +
+Lemma multiset_closure :
+  ∀ R p q, transitive _ R →
+  closure.trans_clos (multiset_extension_step R) p q →
+        ∃ p', ∃ q', ∃ pq,
+        list_permut p (p' ++ pq) ∧
+        list_permut q (q' ++ pq) ∧
+        q' ≠ nil ∧
+        (∀ a, In a p' → ∃ b, In b q' ∧ R a b).
+ +
+End Make.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.rpo.list_permut.html b/theories/html/hydras.rpo.list_permut.html new file mode 100644 index 00000000..2152e2f9 --- /dev/null +++ b/theories/html/hydras.rpo.list_permut.html @@ -0,0 +1,356 @@ + + + + + +hydras.rpo.list_permut + + + + +
+ + + +
+ +

Library hydras.rpo.list_permut

+ +
+
+ +
+by Evelyne Contejean, LRI +
+ +

Permutation over lists, and finite multisets.

+ +
+
+ +
+Set Implicit Arguments.
+ +
+From Coq Require Import List Multiset Arith Setoid.
+From hydras Require Import decidable_set more_list.
+ +
+Module Type Permut.
+ +
+Declare Module DS : decidable_set.S.
+ +
+Definition elt := DS.A.
+Definition eq_elt_dec := DS.eq_A_dec.
+ +
+Fixpoint list_to_multiset (l : list elt) {struct l} : multiset elt :=
+  match l with
+  | nil ⇒ EmptyBag elt
+  | h :: tl ⇒
+      munion (SingletonBag _ eq_elt_dec h) (list_to_multiset tl)
+  end.
+ +
+Definition list_permut (l1 l2:list elt) : Prop :=
+  meq (list_to_multiset l1) (list_to_multiset l2).
+ +
+End Permut.
+ +
+
+ +
+

Definition of permutation over lists.

+ +
+
+Module Make (DS1 : decidable_set.S) <: Permut with Module DS:= DS1.
+ +
+Module DS := DS1.
+Import DS1.
+ +
+Definition elt := DS.A.
+Definition eq_elt_dec : ∀ t1 t2 : elt, {t1 = t2} + {t1 ≠ t2} := DS.eq_A_dec.
+ +
+Fixpoint list_to_multiset (l : list elt) {struct l} : multiset elt :=
+  match l with
+  | nil ⇒ EmptyBag elt
+  | h :: tl ⇒
+      munion (SingletonBag _ eq_elt_dec h) (list_to_multiset tl)
+  end.
+ +
+Definition list_permut (l1 l2:list elt) : Prop :=
+  meq (list_to_multiset l1) (list_to_multiset l2).
+ +
+
+ +
+Properties over the multiplicity. +
+
+Lemma multiplicity_app :
+ ∀ (l1 l2:list elt) (t : elt),
+   multiplicity (list_to_multiset (l1 ++ l2)) t =
+   multiplicity (list_to_multiset l1) t + multiplicity (list_to_multiset l2) t.
+ +
+Lemma out_mult_O :
+  ∀ (t : elt) (l:list elt), ¬ In t l → multiplicity (list_to_multiset l) t = 0.
+ +
+Lemma in_mult_S :
+ ∀ (t : elt) (l : list elt), In t l → multiplicity (list_to_multiset l) t ≥ 1.
+ +
+
+ +
+

Permutation is a equivalence relation.

+ +Reflexivity. +
+
+Theorem list_permut_refl :
+ ∀ (l : list elt), list_permut l l.
+ +
+
+ +
+Symetry. +
+
+Theorem list_permut_sym :
+ ∀ l1 l2 : list elt, list_permut l1 l2 → list_permut l2 l1.
+ +
+#[global] Hint Immediate list_permut_refl : core.
+#[global] Hint Resolve list_permut_sym : core.
+ +
+
+ +
+Transitivity. +
+
+Theorem list_permut_trans :
+  ∀ l1 l2 l3 : list elt, list_permut l1 l2 → list_permut l2 l3 → list_permut l1 l3.
+ +
+Add Relation (list elt) list_permut
+reflexivity proved by list_permut_refl
+symmetry proved by list_permut_sym
+transitivity proved by list_permut_trans as LP.
+ +
+
+ +
+Permutation of an empty list. +
+
+Lemma list_permut_nil :
+ ∀ l, list_permut l nil → l = nil.
+ +
+
+ +
+

Compatibility Properties.

+ + Permutation is compatible with In. +
+
+Lemma in_permut_in :
+  ∀ l1 l2 e, In e l1 → list_permut l1 l2 → In e l2.
+Add Morphism (In (A :=elt)) with signature (eq ==> list_permut ==> iff) as in_morph.
+ +
+Lemma cons_permut_in :
+  ∀ l1 l2 e, list_permut (e :: l1) l2 → In e l2.
+ +
+
+ +
+Permutation is compatible with adding an element. +
+
+Lemma context_list_permut_cons :
+  ∀ e l1 l2, list_permut l1 l2 → list_permut (e :: l1) (e :: l2).
+ +
+Add Morphism (List.cons (A:=elt)) with signature (eq ==> list_permut ==> list_permut)
+ as add_elt_morph.
+ +
+Lemma list_permut_add_inside :
+∀ a l1 l2 l3 l4,
+  list_permut (l1 ++ l2) (l3 ++ l4) →
+  list_permut (l1 ++ a :: l2) (l3 ++ a :: l4).
+ +
+Lemma list_permut_add_cons_inside :
+∀ a l l1 l2,
+  list_permut l (l1 ++ l2) →
+  list_permut (a :: l) (l1 ++ a :: l2).
+ +
+
+ +
+Permutation is compatible with append. +
+
+Lemma context_list_permut_app1 :
+  ∀ l l1 l2, list_permut l1 l2 → list_permut (l ++ l1) (l ++ l2).
+ +
+Lemma context_list_permut_app2 :
+  ∀ l l1 l2, list_permut l1 l2 → list_permut (l1 ++ l) (l2 ++ l).
+ +
+Add Morphism (List.app (A:=elt))
+with signature (list_permut ==> list_permut ==> list_permut) as app_morph.
+ +
+Lemma list_permut_app_app :
+ ∀ l1 l2, list_permut (l1 ++ l2) (l2 ++ l1).
+ +
+
+ +
+Permutation is compatible with removal of common elements +
+
+Lemma remove_context_list_permut_cons :
+  ∀ e l1 l2, list_permut (e :: l1) (e :: l2) → list_permut l1 l2.
+ +
+Lemma remove_context_list_permut_app2 :
+  ∀ l l1 l2, list_permut (l1 ++ l) (l2 ++ l) → list_permut l1 l2.
+ +
+Lemma list_permut_remove_hd :
+  ∀ l l1 l2 a,
+  list_permut (a :: l) (l1 ++ a :: l2) → list_permut l (l1 ++ l2).
+ +
+
+ +
+Permutation is compatible with length. +
+
+Lemma list_permut_length :
+ ∀ l1 l2, list_permut l1 l2 → length l1 = length l2.
+ +
+Add Morphism (length (A:=elt)) with signature (list_permut ==> eq) as length_morph.
+ +
+
+ +
+Permutation is compatible with size. +
+
+Lemma list_permut_size :
+  ∀ size l1 l2, list_permut l1 l2 → list_size size l1 = list_size size l2.
+ +
+Add Parametric Morphism size : (@list_size elt size) with signature (list_permut ==> @eq nat) as list_size_morph.
+ +
+
+ +
+Permutation is compatible with map. +
+
+Lemma list_permut_map :
+  ∀ f l1 l2, list_permut l1 l2 → list_permut (map f l1) (map f l2).
+ +
+Add Parametric Morphism f : (map f) with signature list_permut ==> list_permut as map_morph.
+ +
+
+ +
+

Permutation for short lists.

+ +
+
+ +
+Lemma list_permut_length_1:
+ ∀ a b, list_permut (a :: nil) (b :: nil) → a = b.
+ +
+Lemma list_permut_length_2 :
+ ∀ a1 b1 a2 b2, list_permut (a1 :: b1 :: nil) (a2 :: b2 :: nil) →
+ (a1=a2 ∧ b1=b2) ∨ (a1=b2 ∧ a2=b1).
+ +
+
+ +
+

Link with AC syntactic decomposition.

+ +
+
+Lemma ac_syntactic_aux :
+ ∀ (l1 l2 l3 l4 : list elt),
+ list_permut (l1 ++ l2) (l3 ++ l4) →
+ (∃ u1, ∃ u2, ∃ u3, ∃ u4,
+ list_permut l1 (u1 ++ u2) ∧
+ list_permut l2 (u3 ++ u4) ∧
+ list_permut l3 (u1 ++ u3) ∧
+ list_permut l4 (u2 ++ u4)).
+ +
+Lemma ac_syntactic :
+ ∀ (l1 l2 l3 l4 : list elt),
+ list_permut (l2 ++ l1) (l4 ++ l3) →
+ (∃ u1, ∃ u2, ∃ u3, ∃ u4,
+ list_permut l1 (u1 ++ u2) ∧
+ list_permut l2 (u3 ++ u4) ∧
+ list_permut l3 (u1 ++ u3) ∧
+ list_permut l4 (u2 ++ u4)).
+ +
+Lemma list_permut_dec : ∀ l1 l2, {list_permut l1 l2}+{¬list_permut l1 l2}.
+ +
+End Make.
+ +
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.rpo.list_set.html b/theories/html/hydras.rpo.list_set.html new file mode 100644 index 00000000..5fee66c0 --- /dev/null +++ b/theories/html/hydras.rpo.list_set.html @@ -0,0 +1,475 @@ + + + + + +hydras.rpo.list_set + + + + +
+ + + +
+ +

Library hydras.rpo.list_set

+ +
+
+ +
+To do : Is it a clone of ListSet ? +
+ + Evelyne Contejean, LRI +
+ +

Sets built with lists

+ +
+
+ +
+Set Implicit Arguments.
+ +
+From Coq Require Import List Arith Lia.
+ +
+From hydras Require Import more_list list_permut.
+ +
+Module Type S.
+ +
+Declare Module DS : decidable_set.S.
+ +
+Definition elt := DS.A.
+Definition eq_elt_dec := DS.eq_A_dec.
+ +
+Fixpoint without_red (l : list elt) {struct l} : Prop :=
+  match l with
+  | nil ⇒ True
+  | e :: le ⇒ if (In_dec eq_elt_dec e le) then False else without_red le
+  end.
+ +
+Record t : Set :=
+  mk_set
+  {
+     support : list elt;
+     is_a_set : without_red support
+  }.
+ +
+Definition cardinal s := List.length s.(support).
+ +
+Definition subset s1 s2 : Prop :=
+  ∀ e, In e s1.(support) → In e s2.(support).
+ +
+Axiom cardinal_subset :
+  ∀ s1 s2, subset s1 s2 → cardinal s1 ≤ cardinal s2.
+ +
+End S.
+ +
+
+ +
+

Definition of sets using lists.

+ +
+
+Module Make (DS1 : decidable_set.S) <: S with Module DS:= DS1.
+ +
+Module DS := DS1.
+Import DS1.
+Module LP := list_permut.Make (DS1).
+ +
+Definition elt := DS.A.
+Definition eq_elt_dec : ∀ t1 t2 : elt, {t1 = t2} + {t1 ≠ t2} := DS.eq_A_dec.
+ +
+Fixpoint without_red (l : list elt) {struct l} : Prop :=
+  match l with
+  | nil ⇒ True
+  | e :: le ⇒ if (In_dec eq_elt_dec e le) then False else without_red le
+  end.
+ +
+Record t : Set :=
+  mk_set
+  {
+     support : list elt;
+     is_a_set : without_red support
+  }.
+ +
+Definition mem e s := In e s.(support).
+ +
+Lemma mem_dec : ∀ e s, {mem e s}+{¬mem e s}.
+ +
+Lemma add_prf :
+  ∀ e l, without_red l → ¬In e l → without_red (e :: l).
+ +
+Definition add e s :=
+  match In_dec eq_elt_dec e s.(support) with
+  | left _ ⇒ s
+  | right R ⇒ mk_set (e :: s.(support)) (add_prf e s.(support) s.(is_a_set) R)
+  end.
+ +
+Lemma add_1 : ∀ e s, mem e (add e s).
+ +
+Lemma add_2 : ∀ e e' s, mem e s → mem e (add e' s).
+ +
+Lemma add_12 : ∀ e e' s, mem e (add e' s) → e = e' ∨ mem e s.
+ +
+Fixpoint filter_aux (P : elt → Prop) (P_dec : ∀ e, {P e}+{¬ P e})
+   (l : list elt) {struct l} : list elt :=
+  match l with
+  | nil ⇒ nil
+  | e :: le ⇒
+           if (P_dec e)
+           then e :: (filter_aux P P_dec le)
+           else filter_aux P P_dec le
+   end.
+ +
+Lemma included_filter_aux :
+∀ P P_dec e l, In e (filter_aux P P_dec l) → In e l.
+ +
+Lemma without_red_filter_aux :
+  ∀ P P_dec l, without_red l → without_red (filter_aux P P_dec l).
+ +
+Definition filter P P_dec s :=
+   mk_set (filter_aux P P_dec s.(support))
+               (without_red_filter_aux P P_dec _ s.(is_a_set)).
+ +
+Lemma filter_1_list :
+  ∀ (P : elt → Prop) P_dec l e, In e l → P e → In e (filter_aux P P_dec l).
+ +
+Lemma filter_1 :
+  ∀ (P : elt → Prop) P_dec s e, mem e s → P e → mem e (filter P P_dec s).
+ +
+Lemma filter_2_list :
+ ∀ (P : elt → Prop) P_dec l e, In e (filter_aux P P_dec l) →
+                In e l ∧ P e.
+ +
+Lemma filter_2 :
+ ∀ (P : elt → Prop) P_dec s e, mem e (filter P P_dec s) →
+                mem e s ∧ P e.
+ +
+Fixpoint remove_red (l : list elt) : list elt :=
+  match l with
+  | nil ⇒ nil
+  | e :: le ⇒
+           if (In_dec eq_elt_dec e le)
+           then remove_red le
+           else e :: (remove_red le)
+   end.
+ +
+Lemma included_remove_red :
+∀ e l, In e (remove_red l) → In e l.
+ +
+Lemma remove_red_included : ∀ e l, In e l → In e (remove_red l).
+ +
+Lemma without_red_remove_red : ∀ l, without_red (remove_red l).
+ +
+Lemma without_red_remove :
+  ∀ e l1 l2, without_red (l1 ++ e :: l2) → without_red (l1 ++ l2).
+ +
+Lemma without_red_add :
+  ∀ e l1 l2, without_red (l1 ++ l2) → ¬In e (l1 ++ l2) →
+  without_red (l1 ++ e :: l2).
+ +
+Lemma without_red_nil : without_red nil.
+ +
+Definition empty : t :=
+  mk_set nil without_red_nil.
+ +
+Lemma without_red_singleton : ∀ e : elt, without_red (e :: nil).
+ +
+Definition singleton (e : elt) : t :=
+  mk_set (e :: nil) (without_red_singleton e).
+ +
+Definition make_set (l : list elt) : t :=
+  mk_set (remove_red l) (without_red_remove_red l).
+ +
+Fixpoint add_without_red (acc l : list elt) {struct l} : list elt :=
+  match l with
+  | nil ⇒ acc
+  | e :: le ⇒
+     if (In_dec eq_elt_dec e acc)
+     then add_without_red acc le
+     else add_without_red (e :: acc) le
+  end.
+ +
+Lemma without_red_add_without_red :
+  ∀ l1 l2, without_red l1 → without_red (add_without_red l1 l2).
+ +
+Definition union s1 s2 :=
+  mk_set (add_without_red s1.(support) s2.(support))
+               (without_red_add_without_red s1.(support) s2.(support) s1.(is_a_set)).
+ +
+Lemma union_1_aux :
+∀ (l1 l2 : list elt) (e : elt), In e l1 → In e (add_without_red l1 l2).
+ +
+Lemma union_1 : ∀ s1 s2 e, mem e s1 → mem e (union s1 s2).
+ +
+Lemma union_2_aux :
+∀ (l1 l2 : list elt) (e : elt), In e l2 → In e (add_without_red l1 l2).
+ +
+Lemma union_2 : ∀ s1 s2 e, mem e s2 → mem e (union s1 s2).
+ +
+Lemma union_12_aux :
+∀ (l1 l2 : list elt) (e : elt), In e (add_without_red l1 l2) → In e l1 ∨ In e l2.
+ +
+Lemma union_12 :
+  ∀ s1 s2 e, mem e (union s1 s2) → mem e s1 ∨ mem e s2.
+ +
+Fixpoint remove_not_common (acc l1 l2 : list elt) {struct l2} : list elt :=
+  match l2 with
+  | nil ⇒ acc
+  | e :: l ⇒
+      if In_dec eq_elt_dec e l1
+      then remove_not_common (e :: acc) l1 l
+      else remove_not_common acc l1 l
+  end.
+ +
+Lemma without_red_remove_not_common_aux :
+  ∀ acc l1 l2, (∀ e, In e acc ∧ In e l2 → False) →
+                           without_red acc → without_red l1 → without_red l2 →
+                           without_red (remove_not_common acc l1 l2).
+ +
+Lemma without_red_remove_not_common :
+  ∀ l1 l2, without_red l1 → without_red l2 →
+                    without_red (remove_not_common nil l1 l2).
+ +
+Definition inter s1 s2 :=
+  mk_set (remove_not_common nil s1.(support) s2.(support))
+               (without_red_remove_not_common _ _ s1.(is_a_set) s2.(is_a_set)).
+ +
+Lemma inter_1_aux :
+  ∀ acc l1 l2 e, In e (remove_not_common acc l1 l2) → In e acc ∨ In e l1.
+ +
+Lemma inter_1 : ∀ s1 s2 e, mem e (inter s1 s2) → mem e s1.
+ +
+Lemma inter_2_aux :
+  ∀ acc l1 l2 e, In e (remove_not_common acc l1 l2) → In e acc ∨ In e l2.
+ +
+Lemma inter_2 : ∀ s1 s2 e, mem e (inter s1 s2) → mem e s2.
+ +
+Lemma inter_12_aux :
+  ∀ acc l1 l2 e, In e l1 → In e l2 → In e (remove_not_common acc l1 l2).
+ +
+Lemma inter_12 :
+  ∀ s1 s2 e, mem e s1 → mem e s2 → mem e (inter s1 s2).
+ +
+Definition subset s1 s2 : Prop :=
+  ∀ e, mem e s1 → mem e s2.
+ +
+Lemma subset_dec : ∀ s1 s2, {subset s1 s2} + {¬ subset s1 s2}.
+ +
+Lemma subset_union_1 :
+  ∀ s1 s2, subset s1 (union s1 s2).
+ +
+Lemma subset_union_2 :
+  ∀ s1 s2, subset s2 (union s1 s2).
+ +
+Lemma subset_inter_1 :
+  ∀ s1 s2, subset (inter s1 s2) s1.
+ +
+Lemma subset_inter_2 :
+  ∀ s1 s2, subset (inter s1 s2) s2.
+ +
+Definition eq_set s1 s2 :=
+  ∀ e, mem e s1 ↔ mem e s2.
+ +
+Lemma eq_set_dec : ∀ s1 s2, {eq_set s1 s2} + {¬eq_set s1 s2}.
+ +
+Lemma eq_set_refl : ∀ s, eq_set s s.
+ +
+Lemma eq_set_sym :
+  ∀ s1 s2, eq_set s1 s2 → eq_set s2 s1.
+ +
+Lemma eq_set_trans :
+  ∀ s1 s2 s3, eq_set s1 s2 → eq_set s2 s3 → eq_set s1 s3.
+ +
+Lemma add_comm :
+  ∀ e1 e2 s, eq_set (add e1 (add e2 s)) (add e2 (add e1 s)).
+ +
+Lemma union_empty_1 :
+  ∀ s, eq_set s (union empty s).
+ +
+Lemma union_empty_2 :
+  ∀ s, eq_set s (union s empty).
+ +
+Lemma union_comm :
+  ∀ s1 s2, eq_set (union s1 s2) (union s2 s1).
+ +
+Lemma union_assoc :
+  ∀ s1 s2 s3, eq_set (union s1 (union s2 s3)) (union (union s1 s2) s3).
+ +
+Lemma filter_union :
+  ∀ P P_dec s1 s2,
+  eq_set (filter P P_dec (union s1 s2)) (union (filter P P_dec s1) (filter P P_dec s2)).
+ +
+Lemma subset_filter :
+  ∀ (P1 P2 : elt → Prop) P1_dec P2_dec s1 s2, subset s1 s2 →
+  (∀ e, P1 e → P2 e) → subset (filter P1 P1_dec s1) (filter P2 P2_dec s2).
+ +
+Lemma subset_compat_1 :
+  ∀ s1 s1' s2, eq_set s1 s1' → subset s1 s2 → subset s1' s2.
+ +
+Lemma subset_compat_2 :
+  ∀ s1 s2 s2', eq_set s2 s2' → subset s1 s2 → subset s1 s2'.
+ +
+Lemma subset_compat :
+   ∀ s1 s1' s2 s2', eq_set s1 s1' → eq_set s2 s2' →
+                                    subset s1 s2 → subset s1' s2'.
+ +
+Lemma union_compat_subset_1 :
+  ∀ s1 s2 s, subset s1 s2 → subset (union s1 s) (union s2 s).
+ +
+Lemma union_compat_subset_2 :
+  ∀ s1 s2 s, subset s1 s2 → subset (union s s1) (union s s2).
+ +
+Lemma union_compat_eq_set :
+  ∀ s1 s1' s2 s2', eq_set s1 s1' → eq_set s2 s2' →
+    eq_set (union s1 s2) (union s1' s2').
+ +
+Lemma subset_subset_union :
+  ∀ s1 s2 s, subset s1 s → subset s2 s → subset (union s1 s2) s.
+ +
+Definition cardinal s := List.length s.(support).
+ +
+Lemma cardinal_subset :
+  ∀ s1 s2, subset s1 s2 → cardinal s1 ≤ cardinal s2.
+ +
+Lemma cardinal_union_1 :
+  ∀ s1 s2, cardinal s1 ≤ cardinal (union s1 s2).
+ +
+Lemma cardinal_union_2 :
+  ∀ s1 s2, cardinal s2 ≤ cardinal (union s1 s2).
+ +
+Lemma cardinal_union_inter_12 :
+  ∀ s1 s2, cardinal (union s1 s2) + cardinal (inter s1 s2) = cardinal s1 + cardinal s2.
+ +
+Lemma cardinal_union:
+  ∀ s1 s2, cardinal (union s1 s2) = cardinal s1 + cardinal s2 -cardinal (inter s1 s2).
+ +
+Lemma cardinal_eq_set : ∀ s1 s2, eq_set s1 s2 → cardinal s1 = cardinal s2.
+ +
+Lemma subset_cardinal_not_eq_not_eq_set :
+ ∀ s1 s2 e, subset s1 s2 → ¬mem e s1 → mem e s2 →
+  cardinal s1 < cardinal s2.
+ +
+Lemma eq_set_list_permut_support :
+  ∀ s1 s2, eq_set s1 s2 →
+                 LP.list_permut s1.(support) s2.(support).
+ +
+Lemma without_red_permut :
+ ∀ l1 l2, without_red l1 → LP.list_permut l1 l2 → without_red l2.
+ +
+End Make.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.rpo.more_list.html b/theories/html/hydras.rpo.more_list.html new file mode 100644 index 00000000..a0ed48ff --- /dev/null +++ b/theories/html/hydras.rpo.more_list.html @@ -0,0 +1,442 @@ + + + + + +hydras.rpo.more_list + + + + +
+ + + +
+ +

Library hydras.rpo.more_list

+ +
+
+ +
+by Evelyne Contejean, LRI +
+ +

Some additional properties for the Coq lists.

+ +
+
+ +
+Set Implicit Arguments.
+ +
+From Coq Require Import List Arith.
+ +
+
+ +
+

Relations between length, map, append, In and nth.

+ +
+
+ +
+Lemma map_map :
+  ∀ (A B C : Set) (l : (list A)) (f : B → C) (g : A →B),
+  map f (map g l) = map (fun x ⇒ f (g x)) l.
+ +
+Lemma list_app_length :
+ ∀ A, ∀ l1 l2 : list A, length (l1 ++ l2) = length l1 + length l2.
+ +
+Lemma length_map :
+ ∀ (A B : Set) (f : A → B) (l : list A), length (map f l) = length l.
+ +
+Lemma map_app :
+ ∀ (A B : Set) (f : A → B) l1 l2, map f (l1 ++ l2) = (map f l1) ++ (map f l2).
+ +
+Lemma in_in_map :
+  ∀ (A B : Set) (f : A → B) a l, In a l → In (f a) (map f l).
+ +
+Lemma in_map_in :
+  ∀ (A B : Set) (f : A → B) b l, In b (map f l) →
+  ∃ a, In a l ∧ f a = b.
+ +
+Lemma nth_error_map :
+  ∀ (A B : Set) (f : A → B) (l : list A) i,
+  match nth_error (map f l) i with
+  | Some f_li ⇒
+           match nth_error l i with
+            | Some li ⇒ f_li = f li
+            | None ⇒ False
+            end
+  | None ⇒
+            match nth_error l i with
+            | Some li ⇒ False
+            | None ⇒ True
+            end
+end.
+ +
+
+ +
+

A measure on lists based on a measure on elements.

+ +
+
+ +
+Fixpoint list_size (A : Set) (size : A → nat) (l : list A) {struct l} : nat :=
+  match l with
+  | nil ⇒ 0
+  | h :: tl ⇒ size h + list_size size tl
+  end.
+ +
+Lemma list_size_tl_compat :
+  ∀ (A : Set) (size : A → nat) a b l, size a < size b →
+    list_size size (a :: l) < list_size size (b :: l).
+ +
+Lemma list_size_app:
+ ∀ (A : Set) (size : A → nat) l1 l2,
+ list_size size (l1 ++ l2) = list_size size l1 + list_size size l2.
+ +
+Lemma list_size_fold :
+  ∀ (A : Set) (size : A → nat) l n,
+  fold_left (fun (size_acc : nat) (a : A) ⇒ size_acc + size a) l n =
+  n + list_size size l.
+ +
+Lemma list_size_size_eq :
+  ∀ (A : Set) (size1 : A → nat) (size2 : A → nat) l,
+ (∀ a, In a l → size1 a = size2 a) → list_size size1 l = list_size size2 l.
+ +
+
+ +
+

Induction principles for list.

+ + Induction on the length. +
+
+Definition list_rec2 :
+  ∀ A, ∀ P : list A → Type,
+    (∀ (n:nat) (l : list A), length l ≤ n → P l) →
+    ∀ l : list A, P l.
+ +
+Definition o_length (A : Set) (l1 l2 : list A) : Prop := length l1 < length l2.
+ +
+Theorem well_founded_length : ∀ A, well_founded (o_length (A := A)).
+ +
+
+ +
+Induction on the the size. +
+
+Definition list_rec3 (A : Set) (size : A → nat) :
+  ∀ P : list A → Type,
+    (∀ (n:nat) (l : list A), list_size size l ≤ n → P l) →
+    ∀ l : list A, P l.
+ +
+
+ +
+

How to remove an element in a list, whenever it is present.

+ +
+
+Fixpoint split_list (A : Set)
+  (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+  (l : list A) (t : A) {struct l} : list A × list A :=
+  match l with
+  | nil ⇒ (nil, nil)
+  | a :: l' ⇒
+      if eqA t a
+      then (nil, l')
+      else let (l1,l2) := split_list eqA l' t in (a :: l1, l2)
+  end.
+ +
+Lemma split_list_app_cons :
+ ∀ (A : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2}) t l,
+   In t l → let (l1, l2) := split_list eqA l t in l = l1 ++ t :: l2.
+ +
+Fixpoint remove (A : Set) (eqA : ∀ a1 a2 : A, {a1=a2}+{a1≠a2})
+  (a : A) (l : list A) {struct l} : (option (list A)) :=
+  match l with
+  | nil ⇒ None
+  | h :: tl ⇒
+    if eqA a h
+    then Some tl
+    else
+      match remove eqA a tl with
+      | Some rmv ⇒ Some (h :: rmv)
+      | None ⇒ None
+      end
+  end.
+ +
+Lemma in_remove :
+  ∀ (A : Set) (eqA : ∀ a1 a2 : A, {a1=a2}+{a1≠a2}) a l,
+  match remove eqA a l with
+  | None ⇒ ¬In a l
+  | Some l' ⇒ In a l ∧ let (l1, l2) := split_list eqA l a in l' = l1 ++ l2
+  end.
+ +
+Fixpoint remove_list (A : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+(la l : list A) {struct l} : option (list A) :=
+  match la with
+  | nil ⇒ Some l
+  | a :: la' ⇒
+        match l with
+        | nil ⇒ None
+        | b :: l' ⇒
+           if eqA a b
+           then remove_list eqA la' l'
+           else
+             match remove_list eqA la l' with
+             | None ⇒ None
+             | Some rmv ⇒ Some (b :: rmv)
+             end
+        end
+  end.
+ +
+
+ +
+

Iterators.

+ +
+
+Fixpoint fold_left2 (A B C : Set) (f : A → B → C → A) (a : A) (l1 : list B) (l2 : list C)
+  {struct l1} : option A :=
+  match l1, l2 with
+  | nil, nil ⇒ Some a
+  | b :: t1, c :: t2 ⇒ fold_left2 f (f a b c) t1 t2
+  | _, _ ⇒ None
+  end.
+ +
+
+ +
+

more properties on the nth element.

+ +
+
+Lemma nth_error_ok_in :
+  ∀ (A : Set) n (l : list A) (a : A),
+  nth_error l n = Some a → In a l.
+ +
+
+ +
+

Association lists.

+ +

find.

+ +
+
+Fixpoint find (A B : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+(a : A) (l : list (A × B)) {struct l} : option (B) :=
+ match l with
+ | nil ⇒ None
+ | (a1,b1) :: l ⇒
+     if eqA a a1
+     then Some b1
+     else find eqA a l
+  end.
+ +
+Lemma find_not_mem :
+  ∀ (A B : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+  (a : A) (b : B) (l : list (A × B)) (dom : list A),
+  ¬In a dom → (∀ a', In a' dom → find eqA a' ((a,b) :: l) = find eqA a' l).
+ +
+
+ +
+

number of occurences of the first element of a pair.

+ +
+
+Fixpoint nb_occ (A B : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+  (a : A) (l : list (A × B)) {struct l} : nat :=
+  match l with
+  | nil ⇒ 0
+  | (a',_) :: tl ⇒
+     if (eqA a a') then S (nb_occ eqA a tl) else nb_occ eqA a tl
+  end.
+ +
+Lemma none_nb_occ_O :
+  ∀ (A B : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+  (a : A) (l : list (A × B)),
+  find eqA a l = None → nb_occ eqA a l = 0.
+ +
+Lemma some_nb_occ_Sn :
+  ∀ (A B : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+  (a : A) (l : list (A × B)) b,
+  find eqA a l = Some b → 1 ≤ nb_occ eqA a l.
+ +
+Lemma nb_occ_app :
+  ∀ (A B : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+  a (l1 l2 : list (A × B)),
+  nb_occ eqA a (l1++l2) = nb_occ eqA a l1 + nb_occ eqA a l2.
+ +
+Lemma reduce_assoc_list :
+  ∀ (A B : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2}),
+  ∀ (l : list (A × B)), ∃ l',
+ (∀ a, nb_occ eqA a l' ≤ 1) ∧ (∀ a, find eqA a l = find eqA a l').
+ +
+
+ +
+map_without_repetition applies a function to the elements of a list, +but only a single time when there are several consecutive occurences of the +same element. Moreover, the function is supposed to return an option as a result, +in order to simulate exceptions, and the abnormal results are discarted. + +
+
+ +
+Fixpoint map_without_repetition (A B : Set)
+  (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+  (f : A → option B) (l : list A) {struct l} : list B :=
+    match l with
+    | nil ⇒ (nil : list B)
+    | h :: nil ⇒
+      match f h with
+      | None ⇒ nil
+      | Some f_h ⇒ f_h :: nil
+      end
+    | h1 :: ((h2 :: tl) as l1) ⇒
+    if (eqA h1 h2)
+    then map_without_repetition eqA f l1
+    else
+      match f h1 with
+      | None ⇒ map_without_repetition eqA f l1
+      | Some f_h1 ⇒ f_h1 :: (map_without_repetition eqA f l1)
+      end
+end.
+ +
+Lemma prop_map_without_repetition :
+ ∀ (A B : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+  (P : B → Prop) f l,
+  (∀ a, In a l →
+   match f a with
+   | None ⇒ True
+   | Some f_a ⇒ P f_a
+   end) →
+   (∀ b, In b (map_without_repetition eqA f l) → P b).
+ +
+Lemma exists_map_without_repetition :
+  ∀ (A B : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+  (P : B → Prop) f l,
+  (∃ a, In a l ∧ match f a with
+                        | None ⇒ False
+                        | Some f_a ⇒ P f_a
+                        end) →
+  (∃ b, In b (map_without_repetition eqA f l) ∧ P b).
+ +
+
+ +
+map12_without_repetition is similar to map_without_repetition, but the +applied function returns two optional results instead of one. + +
+
+ +
+Fixpoint map12_without_repetition (A B : Set)
+  (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+  (f : A → option B × option B) (l : list A) {struct l} : list B :=
+    match l with
+    | nil ⇒ (nil : list B)
+    | h :: nil ⇒
+      match f h with
+      | (None, None) ⇒ nil
+      | (Some f_h1, None) ⇒ f_h1 :: nil
+      | (None, Some f_h1) ⇒ f_h1 :: nil
+      | (Some f_h1, Some f_h2) ⇒ f_h1 :: f_h2 :: nil
+      end
+    | h :: ((h' :: tl) as l1) ⇒
+    if (eqA h h')
+    then map12_without_repetition eqA f l1
+    else
+      match f h with
+      | (None, None) ⇒ map12_without_repetition eqA f l1
+      | (Some f_h1, None) ⇒ f_h1 :: (map12_without_repetition eqA f l1)
+      | (None, Some f_h1) ⇒ f_h1 :: (map12_without_repetition eqA f l1)
+      | (Some f_h1, Some f_h2) ⇒ f_h2 :: f_h1 :: (map12_without_repetition eqA f l1)
+      end
+end.
+ +
+Lemma prop_map12_without_repetition :
+  ∀ (A B : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+  (P : B → Prop) f l,
+  (∀ a, In a l →
+   match f a with
+   | (None, None) ⇒ True
+   | (Some f1_a, None) ⇒ P f1_a
+   | (None, Some f2_a) ⇒ P f2_a
+   | (Some f1_a, Some f2_a) ⇒ P f1_a ∧ P f2_a
+   end) →
+ (∀ b, In b (map12_without_repetition eqA f l) → P b).
+ +
+Lemma exists_map12_without_repetition :
+  ∀ (A B : Set) (eqA : ∀ (a1 a2 : A), {a1=a2}+{a1≠a2})
+  (P : B → Prop) f l,
+  ((∃ a, In a l ∧ match f a with
+                        | (None, None) ⇒ False
+                        | (None, Some f2_a) ⇒ P f2_a
+                        | (Some f1_a, None) ⇒ P f1_a
+                        | (Some f1_a, Some f2_a) ⇒ P f1_a ∨ P f2_a
+                        end) →
+  (∃ b, In b (map12_without_repetition eqA f l) ∧ P b)).
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.rpo.rpo.html b/theories/html/hydras.rpo.rpo.html new file mode 100644 index 00000000..5466e69d --- /dev/null +++ b/theories/html/hydras.rpo.rpo.html @@ -0,0 +1,676 @@ + + + + + +hydras.rpo.rpo + + + + +
+ + + +
+ +

Library hydras.rpo.rpo

+ +
+
+ +
+by Evelyne Contejean, LRI +
+
+ +
+From Coq Require Import List Relations Wellfounded Arith Wf_nat Lia.
+From hydras Require Import more_list list_permut dickson term.
+ +
+
+ +
+A non-dependent version of lexicographic extension. + +
+
+ +
+Definition lex (A B : Set)
+  (eq_A_dec : ∀ a1 a2, {a1=a2}+{a1≠a2})
+  (o1 : relation A) (o2 : relation B) (s t : _ × _) :=
+  match s, t with (s1,s2), (t1,t2) ⇒
+   if eq_A_dec s1 t1 then o2 s2 t2 else o1 s1 t1
+  end.
+ +
+
+ +
+Transitivity of lexicographic extension. + +
+
+ +
+Lemma lex_trans :
+ ∀ (A B : Set) eq_A_dec o1 o2,
+ antisymmetric A o1 → transitive A o1 → transitive B o2 →
+ transitive _ (lex _ _ eq_A_dec o1 o2).
+ +
+
+ +
+Well-foundedness of lexicographic extension. + +
+
+ +
+Lemma wf_lex :
+  ∀ A B eq_A_dec o1 o2, well_founded o1 → well_founded o2 →
+                             well_founded (lex A B eq_A_dec o1 o2).
+ +
+
+ +
+

Module Type Precedence,

+ +

Definition of a precedence.

+ + +
+
+ +
+Module Type Precedence.
+Parameter A : Set.
+Parameter prec : relation A.
+ +
+Inductive status_type : Set :=
+  | Lex : status_type
+  | Mul : status_type.
+ +
+Parameter status : A → status_type.
+ +
+Axiom prec_dec : ∀ a1 a2 : A, {prec a1 a2} + {¬ prec a1 a2}.
+Axiom prec_antisym : ∀ s, prec s s → False.
+Axiom prec_transitive : transitive A prec.
+ +
+End Precedence.
+ +
+
+ +
+

Module Type RPO,

+ +

Definition of RPO from a precedence on symbols.

+ + +
+
+ +
+Module Type RPO.
+ +
+Declare Module T : term.Term.
+Declare Module P : Precedence with Definition A:= T.symbol.
+ +
+Import T.
+Import P.
+Declare Module LP : list_permut.Permut with Definition DS.A:=term.
+Import LP.
+ +
+
+ +
+

Definition of rpo.

+ + +
+ + +
+
+Inductive rpo : term → term → Prop :=
+  | Subterm : ∀ f l t s, In s l → rpo_eq t s → rpo t (Term f l)
+  | Top_gt :
+       ∀ f g l l', prec g f →
+       (∀ s', In s' l' → rpo s' (Term f l)) →
+       rpo (Term g l') (Term f l)
+  | Top_eq_lex :
+        ∀ f l l', status f = Lex → rpo_lex l' l →
+        (∀ s', In s' l' → rpo s' (Term f l)) →
+        rpo (Term f l') (Term f l)
+
+  | Top_eq_mul :
+        ∀ f l l', status f = Mul → rpo_mul l' l →
+        rpo (Term f l') (Term f l)
+
+with rpo_eq : term → term → Prop :=
+  | Eq : ∀ t, rpo_eq t t
+  | Lt : ∀ s t, rpo s t → rpo_eq s t
+
+with rpo_lex : list term → list term → Prop :=
+  | List_gt :
+      ∀ s t l l', rpo s t → length l = length l' →
+      rpo_lex (s :: l) (t :: l')
+  | List_eq : ∀ s l l', rpo_lex l l' →
+                             rpo_lex (s :: l) (s :: l')
+
+with rpo_mul : list term → list term → Prop :=
+  | List_mul :
+       ∀ a lg ls lc l l',
+       list_permut l' (ls ++ lc) →
+       list_permut l (a :: lg ++ lc) →
+       (∀ b, In b ls → ∃ a', In a' (a :: lg) ∧ rpo b a') →
+       rpo_mul l' l.
+ +
+
+ +
+

rpo is a preorder, and its reflexive closure is an ordering.

+ + +
+
+ +
+Axiom rpo_closure :
+  ∀ s t u,
+  (rpo t s → rpo u t → rpo u s) ∧
+  (rpo s t → rpo t s → False) ∧
+  (rpo s s → False) ∧
+  (rpo_eq s t → rpo_eq t s → s = t).
+ +
+Axiom rpo_trans : ∀ s t u, rpo t s → rpo u t → rpo u s.
+ +
+
+ +
+

Main theorem: when the precedence is well-founded, so is the rpo.

+ + +
+ + +
+
+Axiom wf_rpo : well_founded prec → well_founded rpo.
+ +
+
+ +
+

RPO is compatible with the instanciation by a substitution.

+ + +
+
+ +
+Axiom rpo_subst :
+  ∀ t s, rpo s t →
+  ∀ sigma, rpo (apply_subst sigma s) (apply_subst sigma t).
+ +
+
+ +
+

RPO is compatible with adding context.

+ + +
+
+ +
+Axiom rpo_add_context :
+ ∀ p ctx s t, rpo s t → is_a_pos ctx p = true →
+  rpo (replace_at_pos ctx s p) (replace_at_pos ctx t p).
+ +
+End RPO.
+ +
+Module Make (T1: term.Term)
+                    (P1 : Precedence with Definition A := T1.symbol)
+<: RPO. +
+Module T := T1.
+Module P := P1.
+ +
+Import T.
+Import P.
+ +
+Module LP := list_permut.Make (Term_eq_dec).
+Import LP.
+ +
+
+ +
+

Definition of size-based well-founded orderings for induction.

+ + +
+
+ +
+Definition o_size s t := size s < size t.
+ +
+Lemma wf_size : well_founded o_size.
+ +
+Definition size2 s := match s with (s1,s2) ⇒ (size s1, size s2) end.
+Definition o_size2 s t := lex _ _ eq_nat_dec lt lt (size2 s) (size2 t).
+Lemma wf_size2 : well_founded o_size2.
+ +
+Definition size3 s := match s with (s1,s2) ⇒ (size s1, size2 s2) end.
+Definition o_size3 s t :=
+  lex _ _ eq_nat_dec lt (lex _ _ eq_nat_dec lt lt) (size3 s) (size3 t).
+Lemma wf_size3 : well_founded o_size3.
+ +
+Lemma lex1 :
+ ∀ s f l t1 u1 t2 u2, In s l → o_size3 (s,(t1,u1)) (Term f l,(t2,u2)).
+ +
+Lemma lex1_bis :
+ ∀ a f l t1 u1 t2 u2, o_size3 (Term f l,(t1,u1)) (Term f (a::l),(t2,u2)).
+ +
+Lemma lex2 :
+  ∀ t f l s u1 u2, In t l → o_size3 (s,(t,u1)) (s,(Term f l, u2)).
+ +
+Lemma lex3 :
+  ∀ u f l s t, In u l → o_size3 (s,(t,u)) (s,(t,Term f l)).
+ +
+Lemma o_size3_trans : transitive _ o_size3.
+ +
+
+ +
+

Definition of rpo.

+ + +
+
+ +
+Inductive rpo : term → term → Prop :=
+  | Subterm : ∀ f l t s, In s l → rpo_eq t s → rpo t (Term f l)
+  | Top_gt :
+       ∀ f g l l', prec g f →
+       (∀ s', In s' l' → rpo s' (Term f l)) →
+       rpo (Term g l') (Term f l)
+  | Top_eq_lex :
+        ∀ f l l', status f = Lex → rpo_lex l' l →
+        (∀ s', In s' l' → rpo s' (Term f l)) →
+        rpo (Term f l') (Term f l)
+
+  | Top_eq_mul :
+        ∀ f l l', status f = Mul → rpo_mul l' l →
+        rpo (Term f l') (Term f l)
+
+with rpo_eq : term → term → Prop :=
+  | Eq : ∀ t, rpo_eq t t
+  | Lt : ∀ s t, rpo s t → rpo_eq s t
+
+with rpo_lex : list term → list term → Prop :=
+  | List_gt :
+      ∀ s t l l', rpo s t → length l = length l' →
+      rpo_lex (s :: l) (t :: l')
+  | List_eq : ∀ s l l', rpo_lex l l' → rpo_lex (s :: l) (s :: l')
+
+with rpo_mul : list term → list term → Prop :=
+  | List_mul :
+       ∀ a lg ls lc l l',
+       list_permut l' (ls ++ lc) →
+       list_permut l (a :: lg ++ lc) →
+       (∀ b, In b ls → ∃ a', In a' (a :: lg) ∧ rpo b a') →
+       rpo_mul l' l.
+ +
+Lemma rpo_lex_same_length :
+  ∀ l l', rpo_lex l l' → length l = length l'.
+ +
+Lemma rpo_subterm :
+ ∀ s t, rpo t s → ∀ tj, direct_subterm tj t → rpo tj s.
+ +
+
+ +
+

rpo is a preorder, and its reflexive closure is an ordering.

+ + +
+
+ +
+Lemma rpo_closure :
+  ∀ s t u,
+  (rpo t s → rpo u t → rpo u s) ∧
+  (rpo s t → rpo t s → False) ∧
+  (rpo s s → False) ∧
+  (rpo_eq s t → rpo_eq t s → s = t).
+ +
+Lemma rpo_trans : ∀ s t u, rpo t s → rpo u t → rpo u s.
+ +
+Record SN_term : Set :=
+  mk_sn
+  {
+    tt : term;
+    sn : Acc rpo tt
+    }.
+ +
+
+ +
+

Well-foundedness of rpo.

+ +How to build a built a list of pairs (terms, proof of accessibility) from +a global of accessibility on the list. + +
+
+ +
+Definition build_list_of_SN_terms :
+ ∀ l (proof : ∀ t, In t l → Acc rpo t), list SN_term.
+ +
+
+ +
+Projection on the first element of the pairs after building the +pairs as above is the identity. + +
+
+ +
+Lemma projection_list_of_SN_terms :
+  ∀ l proof, map tt (build_list_of_SN_terms l proof) = l.
+ +
+Lemma in_sn_sn :
+ ∀ l s, In s (map tt l) → Acc rpo s.
+ +
+
+ +
+Definition of rpo on accessible terms. + +
+
+ +
+Definition rpo_rest := fun s t ⇒ rpo (tt s) (tt t).
+ +
+
+ +
+Extension of rpo_lex to the accessible terms. + +
+
+ +
+Inductive rpo_lex_rest : list SN_term → list SN_term → Prop :=
+  | List_gt_rest :
+       ∀ s t l l', rpo_rest s t → length l = length l' →
+       rpo_lex_rest (s :: l) (t :: l')
+  | List_eq_rest : ∀ s t l l', tt s = tt t → rpo_lex_rest l l' →
+        rpo_lex_rest (s :: l) (t :: l').
+ +
+
+ +
+A triviality: rpo on accessible terms is well-founded. + +
+ + +
+
+Lemma wf_on_rest : well_founded rpo_rest.
+ +
+Lemma rpo_lex_rest_same_length :
+  ∀ l l', rpo_lex_rest l l' → length l = length l'.
+ +
+
+ +
+Proof of accessibility does not actually matter, provided at + least one exists. + +
+
+ +
+Lemma acc_lex_drop_proof :
+  ∀ s t l, tt s = tt t → Acc rpo_lex_rest (s::l) → Acc rpo_lex_rest (t::l).
+ +
+
+ +
+Lexicographic extension of rpo on accessible terms lists is well-founded. + +
+
+ +
+Lemma wf_on_lex_rest : well_founded rpo_lex_rest.
+ +
+
+ +
+Extension of rpo_mul to the accessible terms. + +
+
+ +
+Inductive rpo_mul_rest : list SN_term → list SN_term → Prop :=
+  | List_mul_rest :
+       ∀ a lg ls lc l l',
+       list_permut (map tt l') (map tt (ls ++ lc)) →
+       list_permut (map tt l) (map tt (a :: lg ++ lc)) →
+       (∀ b, In b ls → ∃ a', In a' (a :: lg) ∧ rpo_rest b a') →
+       rpo_mul_rest l' l.
+ +
+
+ +
+Definition of a finer grain for multiset extension. + +
+
+ +
+Inductive rpo_mul_rest_step : list SN_term → list SN_term → Prop :=
+  | List_mul_rest_step :
+       ∀ a ls lc l l',
+       list_permut (map tt l') (map tt (ls ++ lc)) →
+       list_permut (map tt l) (map tt (a :: lc)) →
+       (∀ b, In b ls → rpo_rest b a) →
+       rpo_mul_rest_step l' l.
+ +
+
+ +
+The plain multiset extension is in the transitive closure of +the finer grain extension. + +
+
+ +
+Lemma rpo_mul_trans_clos :
+  inclusion _ rpo_mul_rest (clos_trans _ rpo_mul_rest_step).
+ +
+
+ +
+Splitting in two disjoint cases. + +
+
+ +
+Lemma two_cases_rpo :
+ ∀ a m n,
+ rpo_mul_rest_step n (a :: m) →
+ (∃ n', list_permut (map tt n) (map tt (a :: n')) ∧
+             rpo_mul_rest_step n' m) ∨
+ (∃ k, (∀ b, In b k → rpo_rest b a) ∧
+            list_permut (map tt n) (map tt (k ++ m))).
+ +
+Lemma list_permut_map_acc :
+ ∀ l l', list_permut (map tt l) (map tt l') →
+ Acc rpo_mul_rest_step l → Acc rpo_mul_rest_step l'.
+ +
+
+ +
+Multiset extension of rpo on accessible terms lists is well-founded. + +
+
+ +
+Lemma wf_on_mul_rest : well_founded rpo_mul_rest.
+ +
+
+ +
+Another definition of rpo, only on scheme of accessible terms. + +
+
+ +
+Definition rpo_term : relation (symbol × list SN_term) :=
+ fun f_l g_l' ⇒
+  match f_l with
+  | (f,l) ⇒
+  match g_l' with
+  | (g,l') ⇒
+    if F.eq_symbol_dec f g
+    then
+      match status f with
+      | Lex ⇒ rpo_lex_rest l l'
+      | Mul ⇒ rpo_mul_rest l l'
+      end
+    else prec f g
+  end
+  end.
+ +
+Lemma wf_rpo_term : well_founded prec → well_founded rpo_term.
+ +
+Lemma acc_build :
+  well_founded prec → ∀ f l,
+  Acc rpo (Term f (map (fun sn_tt ⇒ tt sn_tt) l)).
+ +
+
+ +
+

Main theorem: when the precedence is well-founded, so is the rpo.

+ + +
+
+ +
+Lemma wf_rpo : well_founded prec → well_founded rpo.
+ +
+
+ +
+

RPO is compatible with the instanciation by a substitution.

+ + +
+
+ +
+Lemma rpo_subst :
+  ∀ t s, rpo s t →
+  ∀ sigma, rpo (apply_subst sigma s) (apply_subst sigma t).
+ +
+
+ +
+

RPO is compatible with adding context.

+ + +
+
+ +
+Lemma rpo_add_context :
+ ∀ p ctx s t, rpo s t → is_a_pos ctx p = true →
+  rpo (replace_at_pos ctx s p) (replace_at_pos ctx t p).
+ +
+End Make.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.rpo.term.html b/theories/html/hydras.rpo.term.html new file mode 100644 index 00000000..323eac53 --- /dev/null +++ b/theories/html/hydras.rpo.term.html @@ -0,0 +1,916 @@ + + + + + +hydras.rpo.term + + + + +
+ + + +
+ +

Library hydras.rpo.term

+ +
+
+ +
+by Evelyne Contejean, LRI +
+ +

Term algebra defined as functor from a Module Signature and a Module Variable.

+ +
+
+ +
+From Coq Require Import List Arith.
+From hydras Require Import more_list list_permut list_set.
+ +
+Set Implicit Arguments.
+ +
+
+ +
+

Module Type Signature.

+ + There are almost no assumptions, except a decidable equality +and an arity function. +
+
+Module Type Signature.
+ +
+  Parameter symb : Set.
+  Axiom eq_symbol_dec : ∀ f1 f2 : symb, {f1 = f2} + {f1 ≠ f2}.
+ +
+
+ +
+The arity of a symbol contains also the information about built-in theories as in CiME +
+
+  Inductive arity_type : Set :=
+  | AC : arity_type
+  | C : arity_type
+  | Free : nat → arity_type.
+ +
+  Parameter arity : symb → arity_type.
+End Signature.
+ +
+
+ +
+

Module Type Variables.

+ + There are almost no assumptions, except a decidable equality. +
+
+Module Type Variables.
+ +
+  Parameter var : Set.
+  Axiom eq_variable_dec : ∀ v1 v2 : var, {v1 = v2} + {v1 ≠ v2}.
+ +
+End Variables.
+ +
+
+ +
+

Module Type Term built from a signature and a set of variables.

+ +
+
+Module Type Term.
+ +
+  Declare Module F : Signature.
+  Declare Module X : Variables.
+ +
+  Definition symbol := F.symb.
+  Definition variable := X.var.
+ +
+  Import F.
+  Import X.
+ +
+ +
+  Ltac destruct_arity f n Af :=
+    generalize (refl_equal (arity f)); pattern f at 1; destruct (arity f) as [ | | n]; intro Af.
+ +
+
+ +
+Definition of terms. +Arity is not taken into account, and terms may be hill-formed. +
+
+  Inductive term : Set :=
+  | Var : variable → term
+  | Term : symbol → list term → term.
+ +
+  Definition direct_subterm t1 t2 : Prop :=
+    match t2 with
+    | Var _ ⇒ False
+    | Term _ l ⇒ In t1 l
+    end.
+ +
+
+ +
+Definition and a few properties of the size of a term. +
+
+  Fixpoint size (t:term) : nat :=
+    match t with
+    | Var v ⇒ 1
+    | Term f l ⇒ 1 + fold_left (fun size_acc e ⇒ size_acc + size e) l 0
+    end.
+ +
+  Axiom size_unfold :
+    ∀ t, size t = match t with
+                       | Var _ ⇒ 1
+                       | Term f l ⇒ 1 + list_size size l
+                       end.
+ +
+  Axiom size_ge_one : ∀ t, 1 ≤ size t.
+ +
+  Axiom size_direct_subterm :
+    ∀ t1 t2, direct_subterm t1 t2 → size t1 < size t2.
+ +
+
+ +
+

Recursion on terms.

+ +
+
+  Section Recursion.
+    Variable P : term → Type.
+    Variable Pl : list term → Type.
+ +
+    Axiom term_rec2 : (∀ n t, size t ≤ n → P t) → ∀ t, P t.
+    Axiom term_rec3 :
+      (∀ v, P (Var v)) → (∀ f l, (∀ t, In t l → P t) → P (Term f l)) → ∀ t, P t.
+    Axiom term_rec4 :
+      (∀ v, P (Var v)) → (∀ f l, Pl l → P (Term f l)) →
+      (∀ l, (∀ t, In t l → P t) → Pl l) → ∀ t, P t.
+  End Recursion.
+ +
+
+ +
+

Double recursion on terms.

+ +
+
+  Section DoubleRecursion.
+    Variable P2 : term → term → Type.
+    Variable Pl2 : list term → list term → Type.
+ +
+    Axiom term_rec7 :
+      (∀ v1 t2, P2 (Var v1) t2) →
+      (∀ t1 v2, P2 t1 (Var v2)) →
+      (∀ f1 f2 l1 l2, Pl2 l1 l2 → P2 (Term f1 l1) (Term f2 l2)) →
+      (∀ l1 l2, (∀ t1 t2, In t1 l1 → In t2 l2 → P2 t1 t2) → Pl2 l1 l2) →
+      ∀ t1 t2, P2 t1 t2.
+ +
+    Axiom term_rec8 :
+      (∀ v1 t2, P2 (Var v1) t2) →
+      (∀ t1 v2, P2 t1 (Var v2)) →
+      (∀ f1 f2 l1 l2, Pl2 l1 l2 → P2 (Term f1 l1) (Term f2 l2)) →
+      (∀ l1 l2, (∀ t1 t2, In t1 l1 → In t2 l2 → P2 t1 t2) → Pl2 l1 l2) →
+      ∀ l1 l2, Pl2 l1 l2.
+  End DoubleRecursion.
+ +
+
+ +
+

Equality on terms is decidable.

+ +
+
+  Axiom eq_term_dec : ∀ t1 t2:term, {t1 = t2} + {t1 ≠ t2}.
+  Declare Module Term_eq_dec : decidable_set.S with Definition A:= term
+    with Definition eq_A_dec := eq_term_dec.
+ +
+
+ +
+

Well-formedness of terms, according to the arity.

+ +
+
+  Fixpoint well_formed (t:term) : Prop :=
+    match t with
+    | Var _ ⇒ True
+    | Term f l ⇒
+      let well_formed_list :=
+          (fix well_formed_list (l:list term) : Prop :=
+             match l with
+             | nil ⇒ True
+             | h :: tl ⇒ well_formed h ∧ well_formed_list tl
+             end) in
+      well_formed_list l ∧
+      (match arity f with
+       | Free n ⇒ length l = n
+       | _ ⇒ length l = 2
+       end)
+    end.
+ +
+  Axiom well_formed_unfold :
+    ∀ t, well_formed t →
+              match t with
+              | Var _ ⇒ True
+              | Term f l ⇒
+                (∀ u, In u l → well_formed u) ∧
+                (match arity f with
+                 | Free n ⇒ length l = n
+                 | _ ⇒ length l = 2
+                 end)
+              end.
+ +
+  Axiom well_formed_fold :
+    ∀ t,
+      match t with
+      | Var _ ⇒ True
+      | Term f l ⇒
+        (∀ u, In u l → well_formed u) ∧
+        (match arity f with
+         | Free n ⇒ length l = n
+         | _ ⇒ length l = 2
+         end)
+      end → well_formed t.
+ +
+  Fixpoint well_formed_list (l : list term) : Prop :=
+    match l with
+    | nil ⇒ True
+    | h :: tl ⇒ well_formed h ∧ well_formed_list tl
+    end.
+ +
+
+ +
+

Substitutions.

+ +
+
+  Definition substitution := list (variable × term).
+ +
+  Fixpoint apply_subst (sigma : substitution) (t : term) {struct t} : term :=
+    match t with
+    | Var v ⇒
+      match find eq_variable_dec v sigma with
+      | None ⇒ t
+      | Some v_sigma ⇒ v_sigma
+      end
+    | Term f l ⇒ Term f (map (apply_subst sigma) l)
+    end.
+ +
+  Axiom empty_subst_is_id : ∀ t, apply_subst nil t = t.
+  Axiom empty_subst_is_id_list : ∀ l, map (apply_subst nil) l = l.
+ +
+
+ +
+Composition of substitutions. +
+
+  Definition map_subst (f : variable → term → term) sigma :=
+    map (fun x ⇒
+           match (x : variable × term) with
+           | (v, v_sigma) ⇒ (v, f v v_sigma)
+           end)
+        sigma.
+ +
+  Definition subst_comp sigma1 sigma2 :=
+    (map_subst (fun _ t ⇒ apply_subst sigma1 t) sigma2)
+      ++
+      (map_subst (fun v t ⇒
+                    match find eq_variable_dec v sigma2 with
+                    | None ⇒ t
+                    | Some v_sigma2 ⇒ apply_subst sigma1 v_sigma2
+                    end)
+                 sigma1).
+ +
+  Axiom subst_comp_is_subst_comp_aux1 :
+    ∀ v sigma f,
+      find eq_variable_dec v (map_subst f sigma) =
+      match find eq_variable_dec v sigma with
+      | None ⇒ None
+      | Some t ⇒ Some (f v t)
+      end.
+ +
+  Axiom subst_comp_is_subst_comp :
+    ∀ sigma1 sigma2 t,
+      apply_subst (subst_comp sigma1 sigma2) t =
+      apply_subst sigma1 (apply_subst sigma2 t).
+ +
+
+ +
+Well-formed substitutions. +
+
+  Definition well_formed_subst sigma :=
+    ∀ v, match find eq_variable_dec v sigma with
+              | None ⇒ True
+              | Some t ⇒ well_formed t
+              end.
+ +
+  Axiom well_formed_apply_subst :
+    ∀ sigma, well_formed_subst sigma →
+                  ∀ t, well_formed t → well_formed (apply_subst sigma t).
+ +
+
+ +
+

Positions in a term.

+ +
+
+  Fixpoint is_a_pos (t : term) (p : list nat) {struct p}: bool :=
+    match p with
+    | nil ⇒ true
+    | i :: q ⇒
+      match t with
+      | Var _ ⇒ false
+      | Term _ l ⇒
+        match nth_error l i with
+        | Some ti ⇒ is_a_pos ti q
+        | None ⇒ false
+        end
+      end
+    end.
+ +
+  Fixpoint subterm_at_pos (t : term) (p : list nat) {struct p}: option term :=
+    match p with
+    | nil ⇒ Some t
+    | i :: q ⇒
+      match t with
+      | Var _ ⇒ None
+      | Term _ l ⇒
+        match nth_error l i with
+        | Some ti ⇒ subterm_at_pos ti q
+        | None ⇒ None
+        end
+      end
+    end.
+ +
+  Axiom size_subterm_at_pos :
+    ∀ t i p, match subterm_at_pos t (i :: p) with
+                  | Some u ⇒ size u < size t
+                  | None ⇒ True
+                  end.
+ +
+  Axiom is_a_pos_exists_subtem :
+    ∀ t p, is_a_pos t p = true → ∃ u, subterm_at_pos t p = Some u.
+ +
+  Fixpoint replace_at_pos (t u : term) (p : list nat) {struct p} : term :=
+    match p with
+    | nil ⇒ u
+    | i :: q ⇒
+      match t with
+      | Var _ ⇒ t
+      | Term f l ⇒
+        let replace_at_pos_list :=
+            (fix replace_at_pos_list j (l : list term) {struct l}: list term :=
+               match l with
+               | nil ⇒ nil
+               | h :: tl ⇒
+                 match j with
+                 | O ⇒ (replace_at_pos h u q) :: tl
+                 | S k ⇒ h :: (replace_at_pos_list k tl)
+                 end
+               end) in
+        Term f (replace_at_pos_list i l)
+      end
+    end.
+ +
+  Fixpoint replace_at_pos_list (l : list term) (u : term) (i : nat) (p : list nat)
+           {struct l}: list term :=
+    match l with
+    | nil ⇒ nil
+    | h :: tl ⇒
+      match i with
+      | O ⇒ (replace_at_pos h u p) :: tl
+      | S j ⇒ h :: (replace_at_pos_list tl u j p)
+      end
+    end.
+ +
+  Axiom replace_at_pos_unfold :
+    ∀ f l u i q,
+      replace_at_pos (Term f l) u (i :: q) = Term f (replace_at_pos_list l u i q).
+ +
+  Axiom replace_at_pos_is_replace_at_pos1 :
+    ∀ t u p, is_a_pos t p = true →
+                  subterm_at_pos (replace_at_pos t u p) p = Some u.
+ +
+  Axiom replace_at_pos_is_replace_at_pos2 :
+    ∀ t p u, subterm_at_pos t p = Some u → replace_at_pos t u p = t.
+ +
+  Axiom subterm_at_pos_apply_subst_apply_subst_subterm_at_pos :
+    ∀ t p sigma,
+      match subterm_at_pos t p with
+      | Some u ⇒
+        subterm_at_pos (apply_subst sigma t) p = Some (apply_subst sigma u)
+      | None ⇒ True
+      end.
+ +
+  Axiom replace_at_pos_list_replace_at_pos_in_subterm :
+    ∀ l1 ui l2 u i p, length l1 = i →
+                           replace_at_pos_list (l1 ++ ui :: l2) u i p =
+                           l1 ++ (replace_at_pos ui u p) :: l2.
+ +
+End Term.
+ +
+
+ +
+

A functor building a Term Module from a Signature and a set of Variables.

+ +
+
+ +
+Module Make (F1 : Signature) (X1 : Variables) <:
+  Term with Module F := F1 with Module X := X1.
+ +
+  Module F := F1.
+  Module X := X1.
+ +
+  Definition symbol := F.symb.
+  Definition variable := X.var.
+ +
+  Import F.
+  Import X.
+ +
+  Module DecVar <: decidable_set.S.
+    Definition A := variable.
+ +
+    Lemma eq_A_dec : ∀ x y : A, { x = y } + { x ≠ y }.
+ +
+  End DecVar.
+ +
+  Module VSet <: list_set.S with Definition DS.A := variable :=
+    list_set.Make (DecVar).
+ +
+
+ +
+Definition of terms. +Arity is not taken into account, and terms may be ill-formed. +
+
+  Inductive term : Set :=
+  | Var : variable → term
+  | Term : symbol → list term → term.
+ +
+  Definition direct_subterm t1 t2 : Prop :=
+    match t2 with
+    | Var _ ⇒ False
+    | Term _ l ⇒ In t1 l
+    end.
+ +
+
+ +
+Definition and a few properties of the size of a term. +
+
+  Fixpoint size (t:term) : nat :=
+    match t with
+    | Var v ⇒ 1
+    | Term f l ⇒ 1 + fold_left (fun size_acc e ⇒ size_acc + size e) l 0
+    end.
+ +
+  Lemma size_unfold :
+    ∀ t,
+      size t = match t with
+               | Var _ ⇒ 1
+               | Term f l ⇒ 1 + list_size size l
+               end.
+ +
+  Lemma size_ge_one : ∀ t, 1 ≤ size t.
+ +
+  Lemma size_direct_subterm :
+    ∀ t1 t2, direct_subterm t1 t2 → size t1 < size t2.
+ +
+
+ +
+

Recursion on terms.

+ +
+
+  Section Recursion.
+    Variable P : term → Type.
+    Variable Pl : list term → Type.
+ +
+    Definition term_rec2 : (∀ n t, size t ≤ n → P t) → ∀ t, P t.
+ +
+    Definition term_rec3 :
+      (∀ v, P (Var v)) → (∀ f l, (∀ t, In t l → P t) → P (Term f l)) → ∀ t, P t.
+ +
+    Definition term_rec4 :
+      (∀ v, P (Var v)) → (∀ f l, Pl l → P (Term f l)) →
+      (∀ l, (∀ t, In t l → P t) → Pl l) → ∀ t, P t.
+  End Recursion.
+ +
+
+ +
+

Double recursion on terms.

+ +
+
+  Section DoubleRecursion.
+    Variable P2 : term → term → Type.
+    Variable Pl2 : list term → list term → Type.
+ +
+    Definition term_rec7 :
+      (∀ v1 t2, P2 (Var v1) t2) →
+      (∀ t1 v2, P2 t1 (Var v2)) →
+      (∀ f1 f2 l1 l2, Pl2 l1 l2 → P2 (Term f1 l1) (Term f2 l2)) →
+      (∀ l1 l2, (∀ t1 t2, In t1 l1 → In t2 l2 → P2 t1 t2) → Pl2 l1 l2) →
+      ∀ t1 t2, P2 t1 t2.
+ +
+    Definition term_rec8 :
+      (∀ v1 t2, P2 (Var v1) t2) →
+      (∀ t1 v2, P2 t1 (Var v2)) →
+      (∀ f1 f2 l1 l2, Pl2 l1 l2 → P2 (Term f1 l1) (Term f2 l2)) →
+      (∀ l1 l2, (∀ t1 t2, In t1 l1 → In t2 l2 → P2 t1 t2) → Pl2 l1 l2) →
+      ∀ l1 l2, Pl2 l1 l2.
+  End DoubleRecursion.
+ +
+
+ +
+

Equality on terms is decidable.

+ +
+
+  Theorem eq_term_dec :
+    ∀ t1 t2:term, {t1 = t2} + {t1 ≠ t2}.
+ +
+  Module Term_eq_dec : decidable_set.S with Definition A:= term
+    with Definition eq_A_dec := eq_term_dec.
+    Definition A := term.
+    Definition eq_A_dec := eq_term_dec.
+  End Term_eq_dec.
+ +
+
+ +
+

Well-formedness of terms, according to the arity.

+ +
+
+  Fixpoint well_formed (t:term) : Prop :=
+    match t with
+    | Var _ ⇒ True
+    | Term f l ⇒
+      let well_formed_list :=
+          (fix well_formed_list (l:list term) : Prop :=
+             match l with
+             | nil ⇒ True
+             | h :: tl ⇒ well_formed h ∧ well_formed_list tl
+             end) in
+      well_formed_list l ∧
+      (match arity f with
+       | Free n ⇒ length l = n
+       | _ ⇒ length l = 2
+       end)
+    end.
+ +
+  Lemma well_formed_unfold :
+    ∀ t, well_formed t →
+              match t with
+              | Var _ ⇒ True
+              | Term f l ⇒
+                (∀ u, In u l → well_formed u) ∧
+                (match arity f with
+                 | Free n ⇒ length l = n
+                 | _ ⇒ length l = 2
+                 end)
+              end.
+ +
+  Lemma well_formed_fold :
+    ∀ t,
+      match t with
+      | Var _ ⇒ True
+      | Term f l ⇒
+        (∀ u, In u l → well_formed u) ∧
+        (match arity f with
+         | Free n ⇒ length l = n
+         | _ ⇒ length l = 2
+         end)
+      end → well_formed t.
+ +
+  Fixpoint well_formed_list (l : list term) : Prop :=
+    match l with
+    | nil ⇒ True
+    | h :: tl ⇒ well_formed h ∧ well_formed_list tl
+    end.
+ +
+
+ +
+

Substitutions.

+ +
+
+  Definition substitution := list (variable × term).
+ +
+  Fixpoint apply_subst (sigma : substitution) (t : term) {struct t} : term :=
+    match t with
+    | Var v ⇒
+      match find eq_variable_dec v sigma with
+      | None ⇒ t
+      | Some v_sigma ⇒ v_sigma
+      end
+    | Term f l ⇒ Term f (map (apply_subst sigma) l)
+    end.
+ +
+  Lemma empty_subst_is_id : ∀ t, apply_subst nil t = t.
+ +
+  Lemma empty_subst_is_id_list : ∀ l, map (apply_subst nil) l = l.
+ +
+
+ +
+Composition of substitutions. +
+
+  Definition map_subst (f : variable → term → term) sigma :=
+    map (fun x ⇒
+           match (x : variable × term) with
+           | (v, v_sigma) ⇒ (v, f v v_sigma)
+           end)
+        sigma.
+ +
+  Definition subst_comp sigma1 sigma2 :=
+    (map_subst (fun _ t ⇒ apply_subst sigma1 t) sigma2)
+      ++
+      (map_subst (fun v t ⇒
+                    match find eq_variable_dec v sigma2 with
+                    | None ⇒ t
+                    | Some v_sigma2 ⇒ apply_subst sigma1 v_sigma2
+                    end)
+                 sigma1).
+ +
+  Lemma subst_comp_is_subst_comp_aux1 :
+    ∀ v sigma f,
+      find eq_variable_dec v (map_subst f sigma) =
+      match find eq_variable_dec v sigma with
+      | None ⇒ None
+      | Some t ⇒ Some (f v t)
+      end.
+ +
+  Lemma subst_comp_is_subst_comp_aux2 :
+    ∀ v sigma1 sigma2,
+      find (B:= term) eq_variable_dec v (sigma1 ++ sigma2) =
+      match find eq_variable_dec v sigma1 with
+      | Some _ ⇒ find eq_variable_dec v sigma1
+      | None ⇒ find eq_variable_dec v sigma2
+      end.
+ +
+  Theorem subst_comp_is_subst_comp :
+    ∀ sigma1 sigma2 t,
+      apply_subst (subst_comp sigma1 sigma2) t =
+      apply_subst sigma1 (apply_subst sigma2 t).
+ +
+
+ +
+Well-formed substitutions. +
+
+  Definition well_formed_subst sigma :=
+    ∀ v, match find eq_variable_dec v sigma with
+              | None ⇒ True
+              | Some t ⇒ well_formed t
+              end.
+ +
+  Theorem well_formed_apply_subst :
+    ∀ sigma, well_formed_subst sigma →
+                  ∀ t, well_formed t → well_formed (apply_subst sigma t).
+ +
+
+ +
+

Positions in a term.

+ +
+
+  Fixpoint is_a_pos (t : term) (p : list nat) {struct p}: bool :=
+    match p with
+    | nil ⇒ true
+    | i :: q ⇒
+      match t with
+      | Var _ ⇒ false
+      | Term _ l ⇒
+        match nth_error l i with
+        | Some ti ⇒ is_a_pos ti q
+        | None ⇒ false
+        end
+      end
+    end.
+ +
+  Fixpoint subterm_at_pos (t : term) (p : list nat) {struct p}: option term :=
+    match p with
+    | nil ⇒ Some t
+    | i :: q ⇒
+      match t with
+      | Var _ ⇒ None
+      | Term _ l ⇒
+        match nth_error l i with
+        | Some ti ⇒ subterm_at_pos ti q
+        | None ⇒ None
+        end
+      end
+    end.
+ +
+  Lemma size_subterm_at_pos :
+    ∀ t i p, match subterm_at_pos t (i :: p) with
+                  | Some u ⇒ size u < size t
+                  | None ⇒ True
+                  end.
+ +
+  Lemma is_a_pos_exists_subtem :
+    ∀ t p, is_a_pos t p = true → ∃ u, subterm_at_pos t p = Some u.
+ +
+  Fixpoint replace_at_pos (t u : term) (p : list nat) {struct p} : term :=
+    match p with
+    | nil ⇒ u
+    | i :: q ⇒
+      match t with
+      | Var _ ⇒ t
+      | Term f l ⇒
+        let replace_at_pos_list :=
+            (fix replace_at_pos_list j (l : list term) {struct l}: list term :=
+               match l with
+               | nil ⇒ nil
+               | h :: tl ⇒
+                 match j with
+                 | O ⇒ (replace_at_pos h u q) :: tl
+                 | S k ⇒ h :: (replace_at_pos_list k tl)
+                 end
+               end) in
+        Term f (replace_at_pos_list i l)
+      end
+    end.
+ +
+  Fixpoint replace_at_pos_list (l : list term) (u : term) (i : nat) (p : list nat)
+           {struct l}: list term :=
+    match l with
+    | nil ⇒ nil
+    | h :: tl ⇒
+      match i with
+      | O ⇒ (replace_at_pos h u p) :: tl
+      | S j ⇒ h :: (replace_at_pos_list tl u j p)
+      end
+    end.
+ +
+  Lemma replace_at_pos_unfold :
+    ∀ f l u i q,
+      replace_at_pos (Term f l) u (i :: q) = Term f (replace_at_pos_list l u i q).
+ +
+  Lemma replace_at_pos_is_replace_at_pos1 :
+    ∀ t u p, is_a_pos t p = true →
+                  subterm_at_pos (replace_at_pos t u p) p = Some u.
+ +
+  Lemma replace_at_pos_is_replace_at_pos2 :
+    ∀ t p u, subterm_at_pos t p = Some u → replace_at_pos t u p = t.
+ +
+  Lemma subterm_at_pos_apply_subst_apply_subst_subterm_at_pos :
+    ∀ t p sigma,
+      match subterm_at_pos t p with
+      | Some u ⇒
+        subterm_at_pos (apply_subst sigma t) p = Some (apply_subst sigma u)
+      | None ⇒ True
+      end.
+ +
+  Lemma replace_at_pos_list_replace_at_pos_in_subterm :
+    ∀ l1 ui l2 u i p, length l1 = i →
+                           replace_at_pos_list (l1 ++ ui :: l2) u i p =
+                           l1 ++ (replace_at_pos ui u p) :: l2.
+ +
+End Make.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.F_3.html b/theories/html/hydras.solutions_exercises.F_3.html new file mode 100644 index 00000000..de8c8d5c --- /dev/null +++ b/theories/html/hydras.solutions_exercises.F_3.html @@ -0,0 +1,200 @@ + + + + + +hydras.solutions_exercises.F_3 + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.F_3

+ +
+From Coq Require Import Arith.
+From hydras Require Import Iterates F_alpha E0.
+From Coq Require Import ArithRing Lia Max.
+Import Exp2.
+From hydras Require Import Compat815.
+Open Scope nat_scope.
+ +
+Lemma LF3 : dominates_from 2 (F_ 3) (fun n ⇒ iterate exp2 n n).
+ +
+
+ +
+

Proof that F_alpha (S n) > exp2 (F alpha n) for alpha >= 3 and

+ + n >= 2 (by induction over alpha) +
+
+ +
+Section S1.
+ +
+ +
+  Let P alpha := ∀ n, 2 ≤ n → exp2 (F_ alpha n) ≤ F_ alpha (S n).
+ +
+  Remark F_3_eqn : ∀ n, F_ 3 n = iterate (F_ 2) (S n) n.
+ +
+
+ +
+

Base case

+ +
+
+ +
+  Lemma P_3 : P 3.
+
+ +
+deprecated, not replaced yet +
+
+ +
+
+ +
+Successor case +
+
+ +
+  Section Successor.
+ +
+    Variable alpha: E0.
+    Hypothesis H_alpha : P alpha.
+ +
+    Remark R1: ∀ n, 2 ≤ n →
+                         ∀ p, n < p →
+                                   exp2 (F_ alpha n) ≤ F_ alpha p.
+ +
+    Section S2.
+      Variable n : nat.
+      Hypothesis Hn : 2 ≤ n.
+ +
+      Remark R3 :
+        F_ (E0_succ alpha) (S n)= F_ alpha (iterate (F_ alpha) (S n) (S n)).
+ +
+      Remark R3' : F_ (E0_succ alpha) n = iterate (F_ alpha) (S n) n.
+ +
+      Remark R4 : F_ (E0_succ alpha) n < iterate (F_ alpha) (S n) (S n).
+ +
+      Lemma L2 : exp2 (F_ (E0_succ alpha) n) ≤ (F_ (E0_succ alpha) (S n)).
+ +
+    End S2.
+ +
+    Lemma L3 : P (E0_succ alpha).
+ +
+  End Successor.
+ +
+
+ +
+

Limit case

+ +
+
+ +
+  Section Limit.
+    Variable lambda : E0.
+    Hypothesis Hlambda : E0limit lambda.
+    Hypothesis IHlambda :
+      ∀ alpha, E0fin 3 o≤ alpha → alpha o< lambda → P alpha.
+ +
+    Remark R5: ∀ beta, 3 o≤ beta → beta o< lambda →
+                            ∀ n, 2 ≤ n →
+                                      ∀ p, n < p →
+                                                exp2 (F_ beta n) ≤ F_ beta p.
+ +
+    Section S3.
+      Variable n: nat.
+      Hypothesis Hn : 2 ≤ n.
+ +
+      Lemma L04 : ∀ beta:T1,
+          T1limit beta →
+          ∀ n, leq lt (\F (S n)) (Canon.canon beta (S n)).
+ +
+      Lemma L04' : ∀ beta, E0limit beta →
+                                ∀ n, (S n) o≤
+                                          (Canon.Canon beta (S n)).
+ +
+      Lemma L4 : exp2 (F_ lambda n) ≤ F_ lambda (S n).
+ +
+    End S3.
+  End Limit.
+ +
+  Lemma L alpha : 3 o≤ alpha → P alpha.
+ +
+End S1.
+ +
+Theorem F_alpha_Sn alpha : 3 o≤ alpha →
+                   ∀ n, 2 ≤ n → exp2 (F_ alpha n) ≤ F_ alpha (S n).
+ +
+
+ +
+What happens if alpha is less than 3, or n less than 2 ? +
+
+ +
+Goal F_ 2 3 = 63.
+ +
+Goal F_ 2 2 = 23.
+ +
+Goal F_ 3 1 = 2047.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.FibonacciPR.html b/theories/html/hydras.solutions_exercises.FibonacciPR.html new file mode 100644 index 00000000..63d8956b --- /dev/null +++ b/theories/html/hydras.solutions_exercises.FibonacciPR.html @@ -0,0 +1,207 @@ + + + + + +hydras.solutions_exercises.FibonacciPR + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.FibonacciPR

+ +
+From hydras Require Import primRec cPair extEqualNat.
+ +
+
+ +
+The famous Fibonacci function +
+
+ +
+ +
+Fixpoint fib (n:nat) : nat :=
+  match n with
+  | 0 ⇒ 1
+  | 1 ⇒ 1
+  | S ((S p) as q) ⇒ fib q + fib p
+  end.
+ +
+ +
+Section Proof_of_FibIsPR.
+ +
+
+ +
+To do : Some parts of this proof may be made more generic +
+ + let us consider another definition of fib, as an application of + nat_rec + +
+
+ +
+  Let fib_step (p: nat × nat) := (fst p + snd p, fst p).
+ +
+  Let fib_iter n p:= (nat_rec (fun _ ⇒ (nat×nat)%type)
+                              p
+                              (fun _ p ⇒ fib_step p)
+                              n).
+  Definition fib_alt n := snd (fib_iter n (1,1)).
+ +
+ +
+
+ +
+The theory of primitive functions deals only with functions + of type naryFunc n. + +
+ + So, let us define a variant of fib_alt + +
+ + +
+
+ +
+  Import LispAbbreviations.
+  Check cPair.
+  Print car.
+  Check car.
+  Check cdr.
+  Search cPair isPR.
+  Search car isPR.
+  Search car cdr cPair.
+ +
+  Let fib_step_cPair p := cPair (car p + cdr p)
+                                (car p).
+ +
+  Let fib_iter_cPair n p := nat_rec (fun _ ⇒ nat)
+                                    p
+                                    (fun _ p ⇒ fib_step_cPair p)
+                                    n.
+ +
+  Definition fibPR n := cdr (fib_iter_cPair n (cPair 1 1)).
+ +
+
+ +
+Let us prove that fibPR is PR +
+
+ +
+  #[export] Instance fibPRIsPR: isPR 1 fibPR.
+ +
+
+ +
+Ok, but we must prove that fibPR is extensionaly equal to fib +
+ + let us consider the following relation +
+
+ +
+  Definition inv (p: nat×nat) (c: nat) := c = cPair (fst p) (snd p).
+ +
+  Lemma inv_Pi : ∀ p c, inv p c → snd p = cPairPi2 c.
+ +
+  Lemma L0: inv (1,1) (cPair 1 1).
+ +
+  Lemma LS : ∀ p c, inv p c → inv (fib_step p) (fib_step_cPair c).
+ +
+  Lemma L1 : ∀ p c,
+      inv p c → ∀ n,
+        inv (fib_iter n p)
+            (fib_iter_cPair n c).
+ +
+  Lemma L2 : extEqual 1 fib_alt fibPR.
+ +
+  #[export] Instance fib_altIsPR : isPR 1 fib_alt.
+ +
+
+ +
+It remains to prove that fib_alt is equivalent to the "classical" fib +
+
+ +
+  Lemma fib_OK0 : ∀ n,
+      fib_iter n (1,1) = (fib (S n), fib n).
+ +
+  Lemma fib_alt_Ok : extEqual 1 fib fib_alt.
+ +
+  #[export] Instance fibIsPR : isPR 1 fib.
+ +
+End Proof_of_FibIsPR.
+ +
+ +
+ +
+
+ +
+Too long ! + +
+ +Time Compute fibPR 3. + +
+
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.Limit_Infinity.html b/theories/html/hydras.solutions_exercises.Limit_Infinity.html new file mode 100644 index 00000000..31f13093 --- /dev/null +++ b/theories/html/hydras.solutions_exercises.Limit_Infinity.html @@ -0,0 +1,107 @@ + + + + + +hydras.solutions_exercises.Limit_Infinity + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.Limit_Infinity

+ +
+
+ +
+Prove that, for any ordinal 0 < alpha < epsilon0, alpha is a limit + if and only if forall ordinal beta < alpha, there exists an infinite + number of ordinals betawee beta and alpha + +
+
+ +
+From hydras Require Import Epsilon0 T1.
+Open Scope t1_scope.
+From Coq Require Import Ensembles Image Compare_dec.
+ +
+Definition Infinite {A: Type} (E: Ensemble A) :=
+  ∃ s: nat → A, injective s ∧ ∀ i, In E (s i).
+ +
+Section On_alpha.
+ +
+  Variable alpha : T1.
+  Hypothesis Halpha : nf alpha.
+  Hypothesis HnonZero : alpha ≠ zero.
+ +
+  Section S1.
+    Hypothesis H : T1limit alpha.
+ +
+    Variable beta : T1.
+    Hypothesis Hbeta : beta t1< alpha.
+ +
+    Definition s (i:nat) := beta + S i.
+ +
+    Lemma L1 (i: nat) : beta t1< s i t1< alpha.
+ +
+
+ +
+Shows that s is infinite +
+
+ +
+    Lemma L2 : ∀ i, s i t1< s (S i).
+ +
+    Lemma L3: injective s.
+ +
+  End S1.
+ +
+  Section S2.
+    Hypothesis H : ∀ beta, beta t1< alpha →
+                                 ∃ gamma, beta t1< gamma t1< alpha.
+ +
+    Lemma L4 : T1limit alpha.
+ +
+  End S2.
+ +
+  Theorem Limit_Infinity : T1limit alpha ↔ (∀ beta,
+                                                beta t1< alpha → Infinite (fun gamma ⇒ beta t1< gamma t1< alpha)).
+ +
+End On_alpha.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.MinPR.html b/theories/html/hydras.solutions_exercises.MinPR.html new file mode 100644 index 00000000..20ec2f3f --- /dev/null +++ b/theories/html/hydras.solutions_exercises.MinPR.html @@ -0,0 +1,65 @@ + + + + + +hydras.solutions_exercises.MinPR + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.MinPR

+ +
+
+ +
+Prove that Stdlib's function min is primitive recursive +
+
+ +
+From hydras Require Import primRec extEqualNat.
+From Coq Require Import Compare_dec Lia ArithRing.
+Import PeanoNat.Nat.
+ +
+Section Proof_of_MinIsPR.
+ +
+  Let min_alt (a b: nat) : nat :=
+    (charFunction 2 leBool a b) × a +
+    (charFunction 2 ltBool b a) × b.
+ +
+  Lemma min_alt_correct : extEqual 2 min_alt Nat.min.
+ +
+  #[local] Instance minPR_PR : isPR 2 min_alt.
+ +
+  #[export] Instance minIsPR : isPR 2 Nat.min.
+ +
+End Proof_of_MinIsPR.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.MinPR2.html b/theories/html/hydras.solutions_exercises.MinPR2.html new file mode 100644 index 00000000..da281986 --- /dev/null +++ b/theories/html/hydras.solutions_exercises.MinPR2.html @@ -0,0 +1,88 @@ + + + + + +hydras.solutions_exercises.MinPR2 + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.MinPR2

+ +
+
+ +
+Prove that Stdlib's function Min is primitive recursive this is a variant of the exercise MinPR +
+
+ +
+From hydras Require Import primRec extEqualNat.
+From Coq Require Import ArithRing Lia Compare_dec.
+ +
+
+ +
+Define an n-ary if-then-else +
+
+ +
+Fixpoint naryIf (n:nat) :
+  naryRel n → naryFunc n → naryFunc n → naryFunc n
+  :=
+    match n return (naryRel n → naryFunc n → naryFunc n → naryFunc n) with
+      0 ⇒ (fun b x y ⇒ if b then x else y)
+    | S m ⇒ fun (p': naryRel (S m)) (g h: naryFunc (S m)) ⇒
+               fun x ⇒ naryIf m (p' x) (g x) (h x)
+    end.
+ +
+#[export] Instance If2IsPR (p: naryRel 2)(f g : naryFunc 2):
+  isPRrel 2 p → isPR 2 f → isPR 2 g →
+  isPR 2 (naryIf 2 p f g).
+ +
+Section Proof_of_MinIsPR.
+ +
+  Let minPR : naryFunc 2 :=
+    naryIf 2 leBool
+           (fun x _ ⇒ x)
+           (fun _ y ⇒ y).
+ +
+  Lemma minPR_correct : extEqual 2 minPR PeanoNat.Nat.min.
+ +
+  #[local] Instance minPR_PR : isPR 2 minPR.
+ +
+  #[export] Instance minIsPR : isPR 2 min.
+ +
+End Proof_of_MinIsPR.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.MorePRExamples.html b/theories/html/hydras.solutions_exercises.MorePRExamples.html new file mode 100644 index 00000000..c91dbf59 --- /dev/null +++ b/theories/html/hydras.solutions_exercises.MorePRExamples.html @@ -0,0 +1,123 @@ + + + + + +hydras.solutions_exercises.MorePRExamples + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.MorePRExamples

+ +
+Require Import primRec Arith ArithRing List.
+Import extEqualNat.
+ +
+
+ +
+

Solution to an exercise

+ +
+
+ +
+Definition double n := n × 2.
+ +
+Lemma doubleIsPR : isPR 1 double.
+ +
+ +
+Fixpoint exp n p :=
+  match p with
+    0 ⇒ 1
+  | S m ⇒ exp n m × n
+  end.
+ +
+ +
+Definition exp_alt := fun a b ⇒ nat_rec (fun _ ⇒ nat)
+                                     1
+                                     (fun _ y ⇒ y × a)
+                                     b.
+ +
+Remark exp_alt_ok : extEqual 2 exp_alt exp.
+ +
+#[local] Instance exp_alt_PR : isPR 2 exp_alt.
+ +
+#[export] Instance expIsPR : isPR 2 exp.
+ +
+ +
+Fixpoint tower2 n :=
+  match n with
+    0 ⇒ 1
+  | S p ⇒ exp 2 (tower2 p)
+  end.
+ +
+ +
+Definition tower2_alt h : nat := nat_rec (fun n ⇒ nat)
+                                1
+                                (fun _ y ⇒ exp 2 y)
+                                h.
+ +
+Remark tower2_alt_ok : extEqual 1 tower2_alt tower2.
+ +
+#[local] Instance tower2_alt_PR : isPR 1 tower2_alt.
+ +
+#[export] Instance tower2IsPR : isPR 1 tower2.
+ +
+Fixpoint fact n :=
+  match n with 0 ⇒ 1
+          | S p ⇒ n × fact p
+  end.
+ +
+Definition fact_alt
+  : nat → nat :=
+  fun a ⇒ nat_rec _ 1 (fun x y ⇒ S x × y) a.
+ +
+Remark fact_alt_ok : extEqual 1 fact_alt fact.
+ +
+#[local] Instance fact_altIsPR : isPR 1 fact_alt.
+ +
+#[export] Instance factIsPR : isPR 1 fact.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.MultisetWf.html b/theories/html/hydras.solutions_exercises.MultisetWf.html new file mode 100644 index 00000000..856c49ee --- /dev/null +++ b/theories/html/hydras.solutions_exercises.MultisetWf.html @@ -0,0 +1,190 @@ + + + + + +hydras.solutions_exercises.MultisetWf + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.MultisetWf

+ +
+Set Implicit Arguments.
+Require Import Relations List Sorted Arith.
+From hydras Require Import Restriction.
+From Coq Require Import Lia.
+ +
+Definition t (A:Type) := list (A × nat).
+ +
+ +
+Section A_given.
+ +
+  Variable A: Type.
+  Variable LtA : relation A.
+ +
+ +
+  Inductive lexpower: relation (t A) :=
+    lex1: ∀ a n l, lexpower nil ((a,n)::l)
+  | lex2: ∀ a n p l l', n < p → lexpower ((a,n)::l) ((a,p)::l')
+  | lex3: ∀ a b n p l l', LtA a b → lexpower ((a,n)::l) ((b,p)::l')
+  | lex4: ∀ a n l l', lexpower l l' → lexpower ((a,n)::l) ((a,n)::l').
+ +
+End A_given.
+ +
+Section Counter_Example.
+  Let R := lexpower lt.
+  Hypothesis Hwf : well_founded R.
+ +
+  Definition seq (n:nat) := repeat (0,0) n ++ ((2,0)::nil).
+ +
+  Lemma decr_seq : ∀ n, R (seq (S n)) (seq n).
+ +
+  Lemma not_acc : ∀ a b, R a b → ¬ Acc R a → ¬ Acc R b.
+ +
+  Let is_in_seq l := ∃ i, l = seq i.
+ +
+  Lemma is_in_seq_not_Acc : ∀ x, is_in_seq x → ¬ Acc R x.
+ +
+  Lemma contrad: False.
+ +
+End Counter_Example.
+ +
+
+ +
+Lists in normal form +
+
+ +
+Definition lexnf {A: Type}(ltA : relation A) (l: t A)
+  := LocallySorted (Basics.flip ltA) (map fst l).
+ +
+Definition lexlt {A}(ltA : relation A) :=
+  restrict (lexnf ltA) (lexpower ltA).
+ +
+Section ProofOfLexwf.
+  Variables (A: Type)
+            (ltA : relation A).
+  Hypothesis HwfA : well_founded ltA.
+ +
+  #[local] Notation NF := (lexnf ltA).
+  #[local] Notation LT := (lexlt ltA).
+ +
+  Theorem lexwf:
+    ∀ l, NF l → Acc LT l.
+ +
+  Lemma NF_inv1 : ∀ a n l, NF ((a,n)::l) → NF l.
+ +
+  Lemma NF_inv2 : ∀ a n b p l, NF((a,n)::(b,p)::l) → ltA b a.
+ +
+  Lemma LT_inv : ∀ a n l l',
+      LT l' ((a,n)::l) →
+      l' = nil ∨
+      (∃ b p l'', l'= ((b,p)::l'') ∧ ltA b a) ∨
+      (∃ l'', l'=(a,n)::l'' ∧ LT l'' l) ∨
+      (∃ p l'', l'= ((a,p)::l'') ∧ p < n).
+ +
+  Let Accs (a:A) := ∀ n l, NF ((a,n)::l) →
+                                Acc LT ((a,n)::l).
+ +
+  Lemma Acc_nil : Acc LT nil.
+ +
+  Lemma Accs_all: ∀ a:A, Accs a.
+ +
+let us prepare an induction on l +
+
+ +
+  Lemma NF_Acc : ∀ l: t A, NF l → Acc LT l.
+ +
+End ProofOfLexwf.
+ +
+Theorem lexwf {A}( ltA : relation A) :
+  well_founded ltA →
+  ∀ l, lexnf ltA l → Acc (lexlt ltA) l. +
+ +
+Example Ex1 : lexpower lt ((2,7)::nil) ((3,0):: nil).
+ +
+Example Ex2 : lexpower lt ((2,7)::(1,0)::(0,33)::nil) ((2,7)::(1,6)::nil).
+ +
+Example Ex3 : lexnf lt ((2,7)::(1,0)::(0,33)::nil).
+ +
+ +
+Section Impossibility1.
+  Variable m : t nat → nat.
+  Hypothesis mDecr : ∀ l l': t nat, lexlt lt l l' → m l < m l'.
+ +
+  Definition iota (n:nat) := (0, n)::nil.
+  Let x := m ((1,0)::nil).
+  Let y := m (iota x).
+ +
+  Fact F1 : y < x.
+  Fact F2 : x ≤ y.
+ +
+  Lemma impossible_nat : False.
+ +
+End Impossibility1.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.OnCodeList.html b/theories/html/hydras.solutions_exercises.OnCodeList.html new file mode 100644 index 00000000..bca6224b --- /dev/null +++ b/theories/html/hydras.solutions_exercises.OnCodeList.html @@ -0,0 +1,50 @@ + + + + + +hydras.solutions_exercises.OnCodeList + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.OnCodeList

+ +
+From hydras.Ackermann Require Import primRec cPair.
+From Coq Require Import Arith.
+From Equations Require Import Equations.
+ +
+Equations members (a:nat): list nat by wf a:=
+  members 0 := List.nil;
+  members (S z) := cPairPi1 z:: members (cPairPi2 z).
+ +
+Lemma membersOk n : n = codeList (members n).
+ +
+Lemma membersOk' l : l = members (codeList l).
+ +
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.T1_ltNotWf.html b/theories/html/hydras.solutions_exercises.T1_ltNotWf.html new file mode 100644 index 00000000..c678f332 --- /dev/null +++ b/theories/html/hydras.solutions_exercises.T1_ltNotWf.html @@ -0,0 +1,93 @@ + + + + + +hydras.solutions_exercises.T1_ltNotWf + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.T1_ltNotWf

+ +
+
+ +
+The order lt on T is not well-founded +
+
+ +
+From hydras Require Import T1.
+From Coq Require Import Relations.
+ +
+Section Proof_of_lt_not_wf.
+ +
+  Hypothesis lt_wf : well_founded lt.
+ +
+  Fixpoint s (i:nat) : T1 :=
+    match i with
+      0 ⇒ phi0 2
+    | S j ⇒ cons 1 0 (s j)
+    end.
+ +
+  Lemma s_decr : ∀ i, lt (s (S i)) (s i).
+ +
+ +
+  Section seq_intro.
+    Variable A: Type.
+    Variable seq : nat → A.
+    Variable R : A → A → Prop.
+    Hypothesis Rwf : well_founded R.
+    Let is_in_seq (x:A) := ∃ i : nat, x = seq i.
+ +
+    Lemma not_acc : ∀ a b:A, R a b → ¬ Acc R a → ¬ Acc R b.
+ +
+    Lemma acc_imp : ∀ a b:A, R a b → Acc R b → Acc R a.
+ +
+    Lemma not_decreasing_aux : ¬ (∀ n:nat, R (seq (S n)) (seq n)).
+ +
+  End seq_intro.
+ +
+  Theorem not_decreasing (A:Type)(R: relation A) (Hwf : well_founded R) :
+    ¬ (∃ seq : nat → A, (∀ i:nat, R (seq (S i)) (seq i))).
+ +
+  Lemma contrad : False.
+ +
+End Proof_of_lt_not_wf.
+ +
+Lemma lt_not_wf : ¬ well_founded lt.
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.ge_omega_iff.html b/theories/html/hydras.solutions_exercises.ge_omega_iff.html new file mode 100644 index 00000000..36c8e98d --- /dev/null +++ b/theories/html/hydras.solutions_exercises.ge_omega_iff.html @@ -0,0 +1,43 @@ + + + + + +hydras.solutions_exercises.ge_omega_iff + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.ge_omega_iff

+ +
+From hydras Require Import T1 E0.
+From Coq Require Import Lia.
+ +
+Open Scope E0_scope.
+ +
+Lemma ge_omega_iff (alpha : E0):
+  E0_omega o≤ alpha ↔ (∀ i:nat, i + alpha = alpha).
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.is_F_monotonous.html b/theories/html/hydras.solutions_exercises.is_F_monotonous.html new file mode 100644 index 00000000..d4ebc210 --- /dev/null +++ b/theories/html/hydras.solutions_exercises.is_F_monotonous.html @@ -0,0 +1,73 @@ + + + + + +hydras.solutions_exercises.is_F_monotonous + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.is_F_monotonous

+ +
+From hydras Require Import Iterates F_alpha E0.
+From Coq Require Import ArithRing Lia Max.
+Import Exp2 Canon.
+From Coq Require Import Mult.
+Open Scope nat_scope.
+ +
+Section S1.
+  Hypothesis H : ∀ (alpha beta:E0), alpha o≤ beta →
+                                         ∀ i, 2 ≤ i →
+                                                    F_ alpha i ≤ F_ beta i.
+ +
+ +
+  Remark R1 : 3 o≤ E0_omega.
+ +
+  Remark R2 : 2 ≤ 2.
+ +
+ Let instance_H := H (E0fin 3) E0_omega R1 _ R2.
+ +
+  Remark R3 : F_ E0_omega 2 = F_ 2 2.
+ +
+ +
+  Remark R4 : F_ 3 2 = F_ 2 (F_ 2 (F_ 2 2)).
+ +
+  Remark R6 : ∀ i, i < F_ 2 i.
+ +
+  Lemma Fake_thm : False.
+ +
+End S1.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.isqrt.html b/theories/html/hydras.solutions_exercises.isqrt.html new file mode 100644 index 00000000..49da54e6 --- /dev/null +++ b/theories/html/hydras.solutions_exercises.isqrt.html @@ -0,0 +1,109 @@ + + + + + +hydras.solutions_exercises.isqrt + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.isqrt

+ +
+ +
+From hydras Require Import primRec extEqualNat ssrnat_extracts.
+From Coq Require Import Min ArithRing Lia Compare_dec Arith Lia.
+ +
+
+ +
+Returns smallest value of x less or equal than b such that (P b x). + Otherwise returns b +
+
+Check boundedSearch.
+Search boundedSearch.
+ +
+Lemma boundedSearch3 :
+  ∀ (P : naryRel 2) (b : nat), boundedSearch P b ≤ b.
+ +
+ +
+Lemma boundedSearch4 :
+  ∀ (P : naryRel 2) (b : nat),
+    P b b = true →
+    P b (boundedSearch P b) = true.
+ +
+Definition isqrt_spec n r := r × r ≤ n < r.+1 × r.+1.
+ +
+Section sqrtIsPR.
+ +
+  Let P (n r: nat) := Nat.ltb n (S r × S r).
+  Definition isqrt := boundedSearch P.
+ +
+  Section Proof_isqrt.
+    Variable n: nat.
+ +
+    Remark R00 : P n (isqrt n) = true.
+ +
+    Lemma R01 : n < (isqrt n).+1 × (isqrt n).+1.
+ +
+    Lemma R02 : isqrt n × isqrt n ≤ n.
+ +
+  End Proof_isqrt.
+ +
+   Lemma sqrt_correct (n: nat) : isqrt_spec n (isqrt n).
+ +
+ #[export] Instance issqrtIsPR : isPR 1 isqrt.
+ +
+End sqrtIsPR.
+ +
+ +
+ +
+
+ +
+Extra work : + Define a faster implementation of sqrt_spec, and prove your function is + extensionnaly equal to isqrt (hence PR!) + +
+
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.lt_succ_le.html b/theories/html/hydras.solutions_exercises.lt_succ_le.html new file mode 100644 index 00000000..33b7ac72 --- /dev/null +++ b/theories/html/hydras.solutions_exercises.lt_succ_le.html @@ -0,0 +1,77 @@ + + + + + +hydras.solutions_exercises.lt_succ_le + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.lt_succ_le

+ +
+From hydras Require Import OrdinalNotations.ON_Generic.
+From Coq Require Import Relations.
+ +
+Section Proofs_of_lt_succ_le.
+ +
+Context (A:Type)
+        (lt : relation A)
+        (cmp : Compare A)
+        (On : ON lt cmp).
+ +
+Section Proofs.
+  Variables alpha beta : A.
+ +
+
+ +
+beta is a successor of alpha +
+
+ +
+  Hypothesis Halphabeta : Successor beta alpha.
+ +
+  Section S1.
+  Variable gamma: A.
+  Hypothesis HGammaBeta : lt gamma beta.
+ +
+  Lemma L1: leq lt gamma alpha.
+ +
+  End S1.
+ +
+End Proofs.
+ +
+End Proofs_of_lt_succ_le.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.predSuccUnicity.html b/theories/html/hydras.solutions_exercises.predSuccUnicity.html new file mode 100644 index 00000000..be9a008e --- /dev/null +++ b/theories/html/hydras.solutions_exercises.predSuccUnicity.html @@ -0,0 +1,109 @@ + + + + + +hydras.solutions_exercises.predSuccUnicity + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.predSuccUnicity

+ +
+From hydras Require Import OrdinalNotations.ON_Generic.
+From Coq Require Import Relations.
+ +
+Section Proofs_of_unicity.
+ +
+Context (A:Type)
+        (lt le : relation A)
+        (cmp : Compare A)
+        (On : ON lt cmp).
+ +
+Section Proofs.
+  Variables alpha beta : A.
+ +
+
+ +
+beta is a successor of alpha +
+
+ +
+  Hypothesis Halphabeta : Successor beta alpha.
+ +
+  Section S1.
+  Variable gamma: A.
+  Hypothesis Halphagamma : Successor gamma alpha.
+ +
+  Lemma L1 : gamma = beta.
+ +
+  End S1.
+ +
+  Section S2.
+    Variable gamma: A.
+    Hypothesis Hgammaalpha : Successor beta gamma.
+ +
+    Lemma L2 : gamma = alpha.
+ +
+  End S2.
+ +
+End Proofs.
+ +
+
+ +
+Please remind that Successor beta alpha must be read as + "beta is a successor of alpha" +
+
+ +
+Lemma Successor_unicity (alpha beta gamma : A):
+  Successor beta alpha →
+  Successor gamma alpha →
+  gamma = beta.
+ +
+Lemma Predecessor_unicity (alpha beta gamma : A):
+  Successor beta alpha →
+  Successor beta gamma →
+  gamma = alpha.
+ +
+End Proofs_of_unicity.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/hydras.solutions_exercises.schutte_cnf_counter_example.html b/theories/html/hydras.solutions_exercises.schutte_cnf_counter_example.html new file mode 100644 index 00000000..63c26f62 --- /dev/null +++ b/theories/html/hydras.solutions_exercises.schutte_cnf_counter_example.html @@ -0,0 +1,65 @@ + + + + + +hydras.solutions_exercises.schutte_cnf_counter_example + + + + +
+ + + +
+ +

Library hydras.solutions_exercises.schutte_cnf_counter_example

+ +
+From hydras Require Import Schutte AP CNF.
+From Coq Require Import List.
+Open Scope schutte_scope.
+ +
+Section Cter_example.
+ +
+  Hypothesis cnf_lt_epsilon0_iff :
+    ∀ l alpha,
+      is_cnf_of alpha l →
+      (alpha < epsilon0 ↔ Forall (fun beta ⇒ beta < alpha) l).
+ +
+  Let alpha := phi0 (succ epsilon0).
+  Let l := succ epsilon0 :: nil.
+ +
+  Remark R1 : epsilon0 < alpha.
+ +
+  Remark R2 : is_cnf_of alpha l.
+ +
+  Remark R3 : Forall (fun beta ⇒ beta < alpha) l.
+ +
+  Lemma counter_ex : False.
+ +
+End Cter_example.
+ +
+
+
+ + + +
+ + + \ No newline at end of file diff --git a/theories/html/index.html b/theories/html/index.html new file mode 100644 index 00000000..5a926a68 --- /dev/null +++ b/theories/html/index.html @@ -0,0 +1,58432 @@ + + + + + +Index + + + + +
+ + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Global IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(28510 entries)
Notation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(182 entries)
Binder IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(20431 entries)
Module IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(94 entries)
Variable IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(764 entries)
Library IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(191 entries)
Lemma IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(3916 entries)
Constructor IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(281 entries)
Axiom IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(52 entries)
Inductive IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(119 entries)
Projection IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(110 entries)
Section IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(284 entries)
Instance IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(366 entries)
Abbreviation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(182 entries)
Definition IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(1486 entries)
Record IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(52 entries)
+
+

Global Index

+

A

+A [definition, in hydras.OrdinalNotations.ON_Omega2]
+Abelian_Monoid [record, in additions.Monoid_def]
+Abelian_EMonoid [record, in additions.Monoid_def]
+Abstract_Properties.alpha [variable, in hydras.Epsilon0.Hprime]
+Abstract_Properties [section, in hydras.Epsilon0.Hprime]
+absurd1 [lemma, in hydras.Ackermann.LNN]
+absurd1 [lemma, in hydras.Ackermann.LNT]
+absurd1 [lemma, in hydras.Ackermann.folLogic]
+AB_given.rel_to_fun.Rf [variable, in hydras.Schutte.PartialFun]
+AB_given.rel_to_fun [section, in hydras.Schutte.PartialFun]
+AB_given.f_given.Rf [variable, in hydras.Schutte.PartialFun]
+AB_given.f_given.f [variable, in hydras.Schutte.PartialFun]
+AB_given.f_given [section, in hydras.Schutte.PartialFun]
+AB_given.DB [variable, in hydras.Schutte.PartialFun]
+AB_given.DA [variable, in hydras.Schutte.PartialFun]
+AB_given.Hb [variable, in hydras.Schutte.PartialFun]
+AB_given.Ha [variable, in hydras.Schutte.PartialFun]
+AB_given.B [variable, in hydras.Schutte.PartialFun]
+AB_given.A [variable, in hydras.Schutte.PartialFun]
+AB_given [section, in hydras.Schutte.PartialFun]
+ab1:18 [binder, in hydras.MoreAck.Ack]
+ab2:19 [binder, in hydras.MoreAck.Ack]
+ab:34 [binder, in hydras.MoreAck.Ack]
+Accs_all [lemma, in hydras.solutions_exercises.MultisetWf]
+acc_from_to_round_plus [lemma, in hydras.Hydra.O2H]
+acc_from [definition, in gaia_hydras.GPaths]
+Acc_nil [lemma, in hydras.solutions_exercises.MultisetWf]
+Acc_inverse_image [lemma, in additions.Wf_transparent]
+Acc_lemma [lemma, in additions.Wf_transparent]
+acc_from_LT [lemma, in hydras.Epsilon0.Paths]
+acc_from_trans [lemma, in hydras.Epsilon0.Paths]
+acc_from [definition, in hydras.Epsilon0.Paths]
+acc_trans [lemma, in hydras.rpo.closure]
+acc_rec [lemma, in gaia_hydras.nfwfgaia]
+acc_imp [lemma, in hydras.solutions_exercises.T1_ltNotWf]
+acc1:10 [binder, in additions.fib]
+acc1:5 [binder, in additions.fib]
+acc2:11 [binder, in additions.fib]
+acc2:6 [binder, in additions.fib]
+acc:108 [binder, in hydras.MoreAck.PrimRecExamples]
+acc:113 [binder, in hydras.rpo.list_set]
+acc:118 [binder, in hydras.rpo.list_set]
+acc:126 [binder, in hydras.rpo.list_set]
+acc:133 [binder, in hydras.rpo.list_set]
+acc:140 [binder, in hydras.rpo.list_set]
+acc:144 [binder, in hydras.rpo.list_set]
+acc:148 [binder, in hydras.rpo.list_set]
+acc:16 [binder, in additions.Compatibility]
+acc:18 [binder, in additions.fib]
+acc:20 [binder, in additions.Compatibility]
+acc:232 [binder, in hydras.rpo.list_set]
+acc:235 [binder, in hydras.rpo.list_set]
+acc:238 [binder, in hydras.rpo.list_set]
+acc:242 [binder, in hydras.rpo.list_set]
+acc:71 [binder, in additions.fib]
+acc:78 [binder, in additions.fib]
+acc:87 [binder, in hydras.rpo.list_set]
+Ack [definition, in hydras.MoreAck.Ack]
+Ack [definition, in hydras.MoreAck.Ack]
+Ack [library]
+Ackermann [module, in gaia_hydras.nfwfgaia]
+Ackermann.addC_CE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.add_normal [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.add_le4 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.add_le3 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.add_inj [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.add_le2 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.add_le1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.add_to_cons [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.add_fin_omega [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.add_int [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.add1Nfin [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.all_zeroE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.all_zero [definition, in gaia_hydras.nfwfgaia]
+Ackermann.ap_pr2CE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.ap_pr4 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.ap_pr3 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.ap_pr2 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.ap_pr1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.ap_limit [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.ap_pr0 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.CompareT3Eq [constructor, in gaia_hydras.nfwfgaia]
+Ackermann.CompareT3Gt [constructor, in gaia_hydras.nfwfgaia]
+Ackermann.CompareT3Lt [constructor, in gaia_hydras.nfwfgaia]
+Ackermann.compare_T3 [inductive, in gaia_hydras.nfwfgaia]
+Ackermann.conc1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.cons [constructor, in gaia_hydras.nfwfgaia]
+Ackermann.epsilon0 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.fincP [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.finite_ltP [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.fooCE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.le_succ_succE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.le_succ_succ [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit_of [definition, in gaia_hydras.nfwfgaia]
+Ackermann.limit_pr [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit_pr1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit10 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit11 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit12a [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit12b1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit12b2 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit12b3 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit12b4 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit12b5 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit12_hyp [definition, in gaia_hydras.nfwfgaia]
+Ackermann.limit2 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit3 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit4 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit5 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit6 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit7 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit8 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.limit9 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.lt_succ_le_4 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.lt_succ_le_3 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.lt_succ_le_2 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.lt_succ_le_1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.lt_succ_succE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.lt_succ_succ [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.lt_omega2 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.lt_omega1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.lt_epsilon0 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.lt_not_wf [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.lt_psi [definition, in gaia_hydras.nfwfgaia]
+Ackermann.lt_rec [definition, in gaia_hydras.nfwfgaia]
+Ackermann.lt_psi_rec [definition, in gaia_hydras.nfwfgaia]
+Ackermann.minus_le [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.minus_lt [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_add [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_sub [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_pred [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_succ [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_finite [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_omega [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_split [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_Wf [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_consE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_cons_cons [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_int [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_psi [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.nf_0 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.normal [definition, in gaia_hydras.nfwfgaia]
+Ackermann.normal_compose [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.normal_limit [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.normal_id [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.omega [definition, in gaia_hydras.nfwfgaia]
+Ackermann.omega_minus_one [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.omega_least_inf2 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.omega_least_inf1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.one [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phia [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phiE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.phiE_3 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.phiE_2 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.phiE_1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.phiL [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.phi_rec [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi_rec_psi [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi0 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi1 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi10 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi11 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi12a [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi12b2 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi12b3 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi12b4 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi12b5 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi3 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi3v [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.phi4 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi5 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi6 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi7 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi8 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.phi9 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.pred_succ [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.pred_lt [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.pred_le [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.psi_succ_pr1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.psi_succ [definition, in gaia_hydras.nfwfgaia]
+Ackermann.psi_lt2 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.psi_lt1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.size [definition, in gaia_hydras.nfwfgaia]
+Ackermann.size_prop [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.size_prop1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.size_psi [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.size_d [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.size_c [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.size_b [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.size_a [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.split_add [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.split_le [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.split_pred [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.split_succ [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.split_is_succ [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.split_limit [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.split_finite [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sub_pr1rCE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sub_pr1r [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sub_nz [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sub_pr1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sub_pr [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sub_le1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sub_1bCE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sub_1aCE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sub_int [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sub1a [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sub1b [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sub1Nfin [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_psiCE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_prCE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_psi2 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_psi1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_is_add_one [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_psi_lt2 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_psi_lt [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_psi [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_nz [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_inj [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_p1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_pred [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.succ_lt [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sup_Oalpha_succ [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sup_Oalpha_limit [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sup_Oalpha_zero [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sup_unique [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.sup_of [definition, in gaia_hydras.nfwfgaia]
+Ackermann.Tf [abbreviation, in gaia_hydras.nfwfgaia]
+Ackermann.TT1T3_inj [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T1succ_nat [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T1T3_surj [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T1T3_inc [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T1T3_lt_epsilon0 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T1_T3 [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3 [inductive, in gaia_hydras.nfwfgaia]
+Ackermann.T3add [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3addA [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3addn0 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3addS [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3add_eq0 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3add0n [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3ap [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3bad [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3eq [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3eqE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3eqP [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3eq_add2l [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3eq_le [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3finite [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3finite_succ [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3finite1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3finite2 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3GeqNotGtn [constructor, in gaia_hydras.nfwfgaia]
+Ackermann.T3GeqNotLtn [constructor, in gaia_hydras.nfwfgaia]
+Ackermann.T3ge1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3GtnNotLeq [constructor, in gaia_hydras.nfwfgaia]
+Ackermann.T3gt1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3is_succ [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3lcp0_pr [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3le [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3leNgt [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lenn [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3len0 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3leP [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3leq_xor_gtn [inductive, in gaia_hydras.nfwfgaia]
+Ackermann.T3le_add2r [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3le_add2l [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3le_psi [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3le_total [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3le_anti [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3le_trans [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3le_lt_trans [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3le_consE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3le_eqVlt [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3le0n [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3limit [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3lta [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3ltE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3ltgtP [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3ltNge [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3ltnn [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3LtnNotGeq [constructor, in gaia_hydras.nfwfgaia]
+Ackermann.T3ltn_xor_geq [inductive, in gaia_hydras.nfwfgaia]
+Ackermann.T3ltn0 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3ltP [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3ltW [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_add2r [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_add2l [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_psi_a [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_psi_c [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_psi_b [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_psi_bc [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_le_trans [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_trans [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_psi' [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_trichotomy [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_anti [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_neAle [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_ne' [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_ne [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_consE [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_psi [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt0n [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3lt1 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3nat [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3nf [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3nf_finite [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3pred [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3split [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3sub [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3subnn [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3subn0 [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3sub0n [lemma, in gaia_hydras.nfwfgaia]
+Ackermann.T3succ [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3_eqType [definition, in gaia_hydras.nfwfgaia]
+Ackermann.T3_eqMixin [definition, in gaia_hydras.nfwfgaia]
+Ackermann.zero [constructor, in gaia_hydras.nfwfgaia]
+_ - _ (ak_scope) [notation, in gaia_hydras.nfwfgaia]
+_ + _ (ak_scope) [notation, in gaia_hydras.nfwfgaia]
+_ > _ (ak_scope) [notation, in gaia_hydras.nfwfgaia]
+_ >= _ (ak_scope) [notation, in gaia_hydras.nfwfgaia]
+_ <= _ (ak_scope) [notation, in gaia_hydras.nfwfgaia]
+_ < _ (ak_scope) [notation, in gaia_hydras.nfwfgaia]
+\F _ (ak_scope) [notation, in gaia_hydras.nfwfgaia]
+[ _ , _ , _ ] (ak_scope) [notation, in gaia_hydras.nfwfgaia]
+_ < _ (cantor_scope) [notation, in gaia_hydras.nfwfgaia]
+Ackm:13 [binder, in hydras.MoreAck.Ack]
+AckNotPR [library]
+Ackn_IsPR [instance, in hydras.MoreAck.AckNotPR]
+AckSn_as_PRiterate [lemma, in hydras.MoreAck.AckNotPR]
+AckSn_as_iterate [lemma, in hydras.MoreAck.AckNotPR]
+Ack_not_PR [lemma, in hydras.MoreAck.AckNotPR]
+Ack_iterate_rw [lemma, in hydras.Epsilon0.F_omega]
+Ack_strict_mono_l [lemma, in hydras.MoreAck.Ack]
+Ack_Sn_plus [lemma, in hydras.MoreAck.Ack]
+Ack_mono_r [lemma, in hydras.MoreAck.Ack]
+Ack_mono_l [lemma, in hydras.MoreAck.Ack]
+Ack_strict_mono [lemma, in hydras.MoreAck.Ack]
+Ack_properties [lemma, in hydras.MoreAck.Ack]
+Ack_m_mono_weak [lemma, in hydras.MoreAck.Ack]
+Ack_Properties.Induc_step.Hm [variable, in hydras.MoreAck.Ack]
+Ack_Properties.Induc_step.m [variable, in hydras.MoreAck.Ack]
+Ack_Properties.Induc_step [section, in hydras.MoreAck.Ack]
+Ack_Properties.P [variable, in hydras.MoreAck.Ack]
+Ack_Properties [section, in hydras.MoreAck.Ack]
+Ack_4_n [lemma, in hydras.MoreAck.Ack]
+Ack_3_n [lemma, in hydras.MoreAck.Ack]
+Ack_2_n [lemma, in hydras.MoreAck.Ack]
+Ack_1_n [lemma, in hydras.MoreAck.Ack]
+Ack_S_S [lemma, in hydras.MoreAck.Ack]
+Ack_S_0 [lemma, in hydras.MoreAck.Ack]
+Ack_0 [lemma, in hydras.MoreAck.Ack]
+Adaptation [section, in additions.Compatibility]
+Add [constructor, in additions.Trace_exercise]
+addExists [definition, in Goedel.PRrepresentable]
+addForalls [definition, in Goedel.PRrepresentable]
+Addition [library]
+Addition_Chains [library]
+addPairwiseEquals [lemma, in hydras.Ackermann.folLogic3]
+add_r_0 [lemma, in hydras.Hydra.Hydra_Lemmas]
+add_r [definition, in hydras.Hydra.Hydra_Definitions]
+add_head_r_plus [definition, in hydras.Hydra.Hydra_Definitions]
+add_head_r [definition, in hydras.Hydra.Hydra_Definitions]
+Aeq:64 [binder, in additions.Monoid_def]
+Alive [definition, in hydras.Hydra.Hydra_Definitions]
+Alive_standard [lemma, in hydras.Hydra.Hydra_Theorems]
+Alive_free [lemma, in hydras.Hydra.Hydra_Theorems]
+all_cases [lemma, in hydras.Hydra.O2H]
+all_ord_acc [lemma, in hydras.Schutte.Schutte_basics]
+Almost_done [lemma, in hydras.Hydra.BigBattle]
+alpha_0_eq [definition, in hydras.Epsilon0.T1]
+alpha_0 [definition, in hydras.Epsilon0.T1]
+alpha_pos [lemma, in hydras.Hydra.O2H]
+alpha_lt_beta [lemma, in hydras.Schutte.AP]
+alpha_plus_zero [lemma, in hydras.Epsilon0.E0]
+alpha_sup [lemma, in hydras.Schutte.Ordering_Functions]
+alpha_A [lemma, in hydras.Schutte.Ordering_Functions]
+alpha_plus_sup [lemma, in hydras.Schutte.Addition]
+alpha_plus_zero [lemma, in hydras.Schutte.Addition]
+alpha':137 [binder, in hydras.Gamma0.Gamma0]
+alpha':193 [binder, in hydras.Epsilon0.T1]
+alpha':322 [binder, in hydras.Epsilon0.T1]
+alpha':462 [binder, in hydras.Gamma0.Gamma0]
+alpha':505 [binder, in hydras.Gamma0.Gamma0]
+alpha':509 [binder, in hydras.Gamma0.Gamma0]
+alpha':548 [binder, in hydras.Gamma0.Gamma0]
+alpha':551 [binder, in hydras.Gamma0.Gamma0]
+alpha':92 [binder, in hydras.Epsilon0.T1]
+alpha1 [definition, in hydras.OrdinalNotations.ON_Finite]
+alpha1:28 [binder, in hydras.Gamma0.T2]
+alpha1:36 [binder, in hydras.Gamma0.T2]
+alpha1:43 [binder, in hydras.Gamma0.T2]
+alpha1:51 [binder, in hydras.Gamma0.T2]
+alpha1:58 [binder, in hydras.Gamma0.T2]
+alpha1:64 [binder, in hydras.Gamma0.T2]
+alpha1:665 [binder, in hydras.Gamma0.Gamma0]
+alpha1:674 [binder, in hydras.Gamma0.Gamma0]
+alpha2:29 [binder, in hydras.Gamma0.T2]
+alpha2:44 [binder, in hydras.Gamma0.T2]
+alpha2:52 [binder, in hydras.Gamma0.T2]
+alpha2:667 [binder, in hydras.Gamma0.Gamma0]
+alpha2:676 [binder, in hydras.Gamma0.Gamma0]
+alpha:1 [binder, in hydras.solutions_exercises.is_F_monotonous]
+alpha:1 [binder, in gaia_hydras.GF_alpha]
+alpha:1 [binder, in gaia_hydras.GHprime]
+alpha:1 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:1 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:1 [binder, in hydras.Schutte.Critical]
+alpha:1 [binder, in hydras.Schutte.AP]
+alpha:1 [binder, in hydras.Epsilon0.L_alpha]
+alpha:1 [binder, in hydras.Hydra.Hydra_Theorems]
+alpha:1 [binder, in hydras.Epsilon0.Canon]
+alpha:1 [binder, in hydras.solutions_exercises.ge_omega_iff]
+alpha:1 [binder, in hydras.Epsilon0.Hprime]
+alpha:1 [binder, in gaia_hydras.GL_alpha]
+alpha:1 [binder, in gaia_hydras.T2Bridge]
+alpha:1 [binder, in hydras.Schutte.Addition]
+alpha:1 [binder, in gaia_hydras.GHessenberg]
+alpha:10 [binder, in hydras.Schutte.Correctness_E0]
+alpha:10 [binder, in gaia_hydras.GF_alpha]
+alpha:10 [binder, in gaia_hydras.GHprime]
+alpha:10 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:10 [binder, in hydras.Schutte.Critical]
+alpha:10 [binder, in hydras.Schutte.Addition]
+alpha:100 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:100 [binder, in hydras.Schutte.Schutte_basics]
+alpha:100 [binder, in hydras.Epsilon0.E0]
+alpha:100 [binder, in hydras.Epsilon0.Paths]
+alpha:101 [binder, in hydras.Hydra.O2H]
+alpha:101 [binder, in hydras.Prelude.Comparable]
+alpha:101 [binder, in hydras.Schutte.Schutte_basics]
+alpha:102 [binder, in hydras.Epsilon0.T1]
+alpha:102 [binder, in hydras.Schutte.Schutte_basics]
+alpha:102 [binder, in hydras.OrdinalNotations.ON_Generic]
+alpha:102 [binder, in hydras.Epsilon0.E0]
+alpha:103 [binder, in hydras.Prelude.Comparable]
+alpha:103 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:103 [binder, in hydras.Epsilon0.Paths]
+alpha:104 [binder, in hydras.Hydra.O2H]
+alpha:104 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:104 [binder, in hydras.Epsilon0.Canon]
+alpha:104 [binder, in hydras.Schutte.Schutte_basics]
+alpha:104 [binder, in hydras.Epsilon0.E0]
+alpha:105 [binder, in hydras.Schutte.Schutte_basics]
+alpha:105 [binder, in hydras.Epsilon0.E0]
+alpha:106 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:106 [binder, in hydras.Epsilon0.Canon]
+alpha:107 [binder, in hydras.Schutte.Schutte_basics]
+alpha:107 [binder, in hydras.Epsilon0.E0]
+alpha:108 [binder, in hydras.Schutte.Schutte_basics]
+alpha:108 [binder, in hydras.Epsilon0.E0]
+alpha:108 [binder, in hydras.Epsilon0.Paths]
+alpha:109 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:109 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:109 [binder, in hydras.Epsilon0.Canon]
+alpha:109 [binder, in hydras.Schutte.Schutte_basics]
+alpha:109 [binder, in hydras.Epsilon0.Paths]
+alpha:11 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:11 [binder, in hydras.Gamma0.T2]
+alpha:11 [binder, in hydras.Schutte.CNF]
+alpha:11 [binder, in hydras.Schutte.Critical]
+alpha:11 [binder, in hydras.Schutte.AP]
+alpha:11 [binder, in hydras.OrdinalNotations.ON_mult]
+alpha:11 [binder, in hydras.OrdinalNotations.ON_Finite]
+alpha:11 [binder, in hydras.Epsilon0.E0]
+alpha:11 [binder, in hydras.Schutte.Addition]
+alpha:110 [binder, in hydras.Epsilon0.T1]
+alpha:110 [binder, in hydras.Hydra.O2H]
+alpha:110 [binder, in hydras.Epsilon0.E0]
+alpha:111 [binder, in hydras.Epsilon0.Canon]
+alpha:111 [binder, in hydras.Schutte.Schutte_basics]
+alpha:111 [binder, in hydras.Epsilon0.E0]
+alpha:112 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:113 [binder, in hydras.Epsilon0.Canon]
+alpha:113 [binder, in hydras.Schutte.Schutte_basics]
+alpha:113 [binder, in hydras.Epsilon0.E0]
+alpha:113 [binder, in hydras.Epsilon0.Paths]
+alpha:114 [binder, in hydras.Epsilon0.F_alpha]
+alpha:114 [binder, in hydras.Epsilon0.Canon]
+alpha:115 [binder, in hydras.Schutte.Schutte_basics]
+alpha:115 [binder, in hydras.OrdinalNotations.ON_Generic]
+alpha:115 [binder, in hydras.Epsilon0.E0]
+alpha:117 [binder, in hydras.Schutte.Schutte_basics]
+alpha:117 [binder, in hydras.Epsilon0.E0]
+alpha:117 [binder, in hydras.Epsilon0.Hprime]
+alpha:118 [binder, in hydras.Epsilon0.E0]
+alpha:118 [binder, in hydras.Epsilon0.Paths]
+alpha:119 [binder, in hydras.Epsilon0.T1]
+alpha:119 [binder, in hydras.Epsilon0.F_alpha]
+alpha:119 [binder, in hydras.Schutte.Schutte_basics]
+alpha:12 [binder, in hydras.Epsilon0.T1]
+alpha:12 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:12 [binder, in hydras.Hydra.O2H]
+alpha:12 [binder, in gaia_hydras.GHprime]
+alpha:12 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:12 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+alpha:12 [binder, in hydras.Epsilon0.E0]
+alpha:12 [binder, in hydras.Hydra.Epsilon0_Needed_Generic]
+alpha:12 [binder, in hydras.Schutte.Addition]
+alpha:120 [binder, in hydras.Schutte.Schutte_basics]
+alpha:120 [binder, in hydras.OrdinalNotations.ON_Generic]
+alpha:120 [binder, in hydras.Epsilon0.E0]
+alpha:121 [binder, in hydras.Epsilon0.T1]
+alpha:121 [binder, in hydras.Schutte.Schutte_basics]
+alpha:123 [binder, in hydras.Epsilon0.Paths]
+alpha:124 [binder, in hydras.Epsilon0.T1]
+alpha:124 [binder, in hydras.Epsilon0.Hprime]
+alpha:124 [binder, in hydras.Gamma0.Gamma0]
+alpha:125 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:127 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:127 [binder, in hydras.Epsilon0.Paths]
+alpha:127 [binder, in hydras.Gamma0.Gamma0]
+alpha:128 [binder, in hydras.Epsilon0.Hprime]
+alpha:129 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:13 [binder, in hydras.solutions_exercises.predSuccUnicity]
+alpha:13 [binder, in gaia_hydras.GF_alpha]
+alpha:13 [binder, in gaia_hydras.GHprime]
+alpha:13 [binder, in hydras.OrdinalNotations.ON_O]
+alpha:13 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:13 [binder, in hydras.Schutte.CNF]
+alpha:13 [binder, in hydras.OrdinalNotations.ON_mult]
+alpha:13 [binder, in hydras.OrdinalNotations.ON_plus]
+alpha:13 [binder, in gaia_hydras.GL_alpha]
+alpha:13 [binder, in hydras.Gamma0.Gamma0]
+alpha:130 [binder, in hydras.Gamma0.Gamma0]
+alpha:131 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:132 [binder, in hydras.OrdinalNotations.ON_Generic]
+alpha:132 [binder, in hydras.Epsilon0.Paths]
+alpha:133 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:133 [binder, in hydras.Gamma0.Gamma0]
+alpha:135 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:136 [binder, in hydras.Schutte.Ordering_Functions]
+alpha:137 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:137 [binder, in hydras.Schutte.Ordering_Functions]
+alpha:139 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:139 [binder, in hydras.Epsilon0.Paths]
+alpha:14 [binder, in hydras.Schutte.Correctness_E0]
+alpha:14 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:14 [binder, in hydras.Schutte.Critical]
+alpha:14 [binder, in hydras.Schutte.AP]
+alpha:14 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+alpha:14 [binder, in hydras.OrdinalNotations.ON_Finite]
+alpha:14 [binder, in hydras.Epsilon0.E0]
+alpha:14 [binder, in hydras.Epsilon0.Paths]
+alpha:14 [binder, in hydras.Gamma0.Gamma0]
+alpha:141 [binder, in hydras.Epsilon0.T1]
+alpha:141 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:145 [binder, in hydras.OrdinalNotations.ON_Generic]
+alpha:145 [binder, in hydras.Epsilon0.Paths]
+alpha:146 [binder, in hydras.Epsilon0.T1]
+alpha:146 [binder, in hydras.Gamma0.Gamma0]
+alpha:147 [binder, in hydras.Schutte.Schutte_basics]
+alpha:147 [binder, in hydras.Gamma0.Gamma0]
+alpha:149 [binder, in hydras.Schutte.Schutte_basics]
+alpha:15 [binder, in hydras.Epsilon0.T1]
+alpha:15 [binder, in hydras.Gamma0.T2]
+alpha:15 [binder, in hydras.Hydra.O2H]
+alpha:15 [binder, in gaia_hydras.GF_alpha]
+alpha:15 [binder, in hydras.Schutte.CNF]
+alpha:15 [binder, in hydras.Schutte.Critical]
+alpha:15 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+alpha:15 [binder, in hydras.OrdinalNotations.ON_mult]
+alpha:15 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+alpha:15 [binder, in hydras.OrdinalNotations.ON_plus]
+alpha:15 [binder, in gaia_hydras.GHydra]
+alpha:15 [binder, in hydras.solutions_exercises.F_3]
+alpha:15 [binder, in gaia_hydras.GL_alpha]
+alpha:15 [binder, in hydras.Gamma0.Gamma0]
+alpha:15 [binder, in hydras.Schutte.Addition]
+alpha:150 [binder, in hydras.Epsilon0.T1]
+alpha:151 [binder, in hydras.Schutte.Schutte_basics]
+alpha:151 [binder, in hydras.Epsilon0.Paths]
+alpha:151 [binder, in hydras.Gamma0.Gamma0]
+alpha:153 [binder, in hydras.Epsilon0.Paths]
+alpha:153 [binder, in hydras.Gamma0.Gamma0]
+alpha:154 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:155 [binder, in hydras.Epsilon0.T1]
+alpha:156 [binder, in hydras.Epsilon0.Paths]
+alpha:157 [binder, in hydras.Epsilon0.T1]
+alpha:157 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:157 [binder, in hydras.Epsilon0.Paths]
+alpha:157 [binder, in hydras.Gamma0.Gamma0]
+alpha:158 [binder, in hydras.Epsilon0.F_alpha]
+alpha:159 [binder, in hydras.Epsilon0.Paths]
+alpha:16 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:16 [binder, in hydras.solutions_exercises.predSuccUnicity]
+alpha:16 [binder, in gaia_hydras.GF_alpha]
+alpha:16 [binder, in hydras.OrdinalNotations.ON_O]
+alpha:16 [binder, in hydras.Epsilon0.Paths]
+alpha:16 [binder, in hydras.Gamma0.Gamma0]
+alpha:16 [binder, in hydras.Schutte.Addition]
+alpha:160 [binder, in hydras.Epsilon0.F_alpha]
+alpha:161 [binder, in hydras.Epsilon0.Paths]
+alpha:162 [binder, in hydras.Epsilon0.F_alpha]
+alpha:164 [binder, in hydras.Epsilon0.Paths]
+alpha:165 [binder, in hydras.Epsilon0.F_alpha]
+alpha:167 [binder, in hydras.Epsilon0.F_alpha]
+alpha:168 [binder, in hydras.Schutte.Ordering_Functions]
+alpha:169 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:169 [binder, in hydras.Epsilon0.Paths]
+alpha:169 [binder, in hydras.Schutte.Ordering_Functions]
+alpha:17 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:17 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+alpha:17 [binder, in hydras.OrdinalNotations.ON_plus]
+alpha:17 [binder, in gaia_hydras.GHydra]
+alpha:17 [binder, in hydras.OrdinalNotations.ON_Finite]
+alpha:17 [binder, in gaia_hydras.GL_alpha]
+alpha:17 [binder, in hydras.Gamma0.Gamma0]
+alpha:17 [binder, in hydras.Schutte.Addition]
+alpha:170 [binder, in hydras.Epsilon0.F_alpha]
+alpha:170 [binder, in gaia_hydras.T1Bridge]
+alpha:172 [binder, in gaia_hydras.T1Bridge]
+alpha:174 [binder, in hydras.Epsilon0.F_alpha]
+alpha:174 [binder, in hydras.Epsilon0.Paths]
+alpha:174 [binder, in hydras.Gamma0.Gamma0]
+alpha:175 [binder, in hydras.Schutte.Schutte_basics]
+alpha:175 [binder, in hydras.Gamma0.Gamma0]
+alpha:176 [binder, in hydras.Epsilon0.F_alpha]
+alpha:176 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:176 [binder, in hydras.Gamma0.Gamma0]
+alpha:177 [binder, in hydras.Schutte.Schutte_basics]
+alpha:177 [binder, in hydras.Epsilon0.Paths]
+alpha:178 [binder, in hydras.Epsilon0.F_alpha]
+alpha:178 [binder, in hydras.Schutte.Schutte_basics]
+alpha:179 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:18 [binder, in hydras.Schutte.Correctness_E0]
+alpha:18 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:18 [binder, in hydras.Hydra.O2H]
+alpha:18 [binder, in gaia_hydras.GF_alpha]
+alpha:18 [binder, in hydras.Schutte.Schutte_basics]
+alpha:180 [binder, in hydras.Epsilon0.F_alpha]
+alpha:180 [binder, in hydras.Gamma0.Gamma0]
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+alpha:516 [binder, in hydras.Epsilon0.T1]
+alpha:516 [binder, in hydras.Epsilon0.Paths]
+alpha:517 [binder, in hydras.Gamma0.Gamma0]
+alpha:52 [binder, in hydras.Epsilon0.F_alpha]
+alpha:52 [binder, in hydras.Schutte.Critical]
+alpha:52 [binder, in hydras.Epsilon0.Canon]
+alpha:52 [binder, in hydras.Epsilon0.E0]
+alpha:523 [binder, in hydras.Epsilon0.Paths]
+alpha:524 [binder, in hydras.Epsilon0.T1]
+alpha:525 [binder, in hydras.Epsilon0.T1]
+alpha:526 [binder, in hydras.Epsilon0.Paths]
+alpha:529 [binder, in hydras.Epsilon0.Paths]
+alpha:53 [binder, in hydras.Schutte.Critical]
+alpha:53 [binder, in hydras.Schutte.AP]
+alpha:53 [binder, in hydras.Schutte.Addition]
+alpha:532 [binder, in hydras.Epsilon0.Paths]
+alpha:536 [binder, in hydras.Gamma0.Gamma0]
+alpha:537 [binder, in hydras.Gamma0.Gamma0]
+alpha:54 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:54 [binder, in hydras.Schutte.CNF]
+alpha:54 [binder, in hydras.Epsilon0.Canon]
+alpha:54 [binder, in hydras.Epsilon0.E0]
+alpha:541 [binder, in hydras.Epsilon0.T1]
+alpha:542 [binder, in hydras.Epsilon0.T1]
+alpha:544 [binder, in hydras.Epsilon0.T1]
+alpha:547 [binder, in hydras.Epsilon0.T1]
+alpha:547 [binder, in hydras.Gamma0.Gamma0]
+alpha:548 [binder, in hydras.Epsilon0.Paths]
+alpha:549 [binder, in hydras.Epsilon0.T1]
+alpha:55 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:55 [binder, in hydras.Schutte.AP]
+alpha:55 [binder, in hydras.Epsilon0.Hprime]
+alpha:55 [binder, in hydras.Epsilon0.Paths]
+alpha:55 [binder, in hydras.Schutte.Addition]
+alpha:550 [binder, in hydras.Gamma0.Gamma0]
+alpha:551 [binder, in hydras.Epsilon0.T1]
+alpha:552 [binder, in hydras.Epsilon0.Paths]
+alpha:553 [binder, in hydras.Gamma0.Gamma0]
+alpha:556 [binder, in hydras.Epsilon0.Paths]
+alpha:558 [binder, in hydras.Epsilon0.T1]
+alpha:56 [binder, in hydras.Epsilon0.F_alpha]
+alpha:56 [binder, in hydras.Schutte.CNF]
+alpha:56 [binder, in hydras.Epsilon0.Canon]
+alpha:56 [binder, in hydras.Epsilon0.E0]
+alpha:56 [binder, in hydras.Schutte.Addition]
+alpha:561 [binder, in hydras.Epsilon0.Paths]
+alpha:562 [binder, in hydras.Gamma0.Gamma0]
+alpha:564 [binder, in hydras.Epsilon0.Paths]
+alpha:568 [binder, in hydras.Epsilon0.T1]
+alpha:569 [binder, in hydras.Epsilon0.Paths]
+alpha:57 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:57 [binder, in hydras.Schutte.AP]
+alpha:57 [binder, in hydras.Epsilon0.L_alpha]
+alpha:57 [binder, in hydras.Epsilon0.E0]
+alpha:573 [binder, in hydras.Epsilon0.Paths]
+alpha:575 [binder, in hydras.Gamma0.Gamma0]
+alpha:576 [binder, in hydras.Epsilon0.Paths]
+alpha:576 [binder, in hydras.Gamma0.Gamma0]
+alpha:58 [binder, in hydras.Epsilon0.F_alpha]
+alpha:58 [binder, in hydras.Schutte.CNF]
+alpha:58 [binder, in hydras.Epsilon0.L_alpha]
+alpha:58 [binder, in hydras.Epsilon0.Paths]
+alpha:581 [binder, in hydras.Gamma0.Gamma0]
+alpha:582 [binder, in hydras.Epsilon0.T1]
+alpha:582 [binder, in hydras.Epsilon0.Paths]
+alpha:585 [binder, in hydras.Epsilon0.T1]
+alpha:586 [binder, in hydras.Epsilon0.Paths]
+alpha:589 [binder, in hydras.Epsilon0.Paths]
+alpha:59 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:59 [binder, in hydras.Schutte.AP]
+alpha:59 [binder, in hydras.Epsilon0.L_alpha]
+alpha:59 [binder, in hydras.Epsilon0.E0]
+alpha:59 [binder, in hydras.Schutte.Addition]
+alpha:6 [binder, in hydras.Schutte.Correctness_E0]
+alpha:6 [binder, in gaia_hydras.GHprime]
+alpha:6 [binder, in hydras.OrdinalNotations.ON_Finite]
+alpha:6 [binder, in hydras.Epsilon0.E0]
+alpha:6 [binder, in gaia_hydras.GL_alpha]
+alpha:6 [binder, in hydras.Schutte.Addition]
+alpha:60 [binder, in hydras.Schutte.AP]
+alpha:60 [binder, in hydras.Epsilon0.E0]
+alpha:60 [binder, in hydras.Epsilon0.Hprime]
+alpha:600 [binder, in hydras.Gamma0.Gamma0]
+alpha:602 [binder, in hydras.Gamma0.Gamma0]
+alpha:603 [binder, in hydras.Gamma0.Gamma0]
+alpha:61 [binder, in hydras.Hydra.O2H]
+alpha:61 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:61 [binder, in hydras.Schutte.CNF]
+alpha:61 [binder, in hydras.Schutte.Schutte_basics]
+alpha:62 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:62 [binder, in hydras.Epsilon0.E0]
+alpha:62 [binder, in hydras.Epsilon0.Paths]
+alpha:621 [binder, in hydras.Gamma0.Gamma0]
+alpha:623 [binder, in hydras.Gamma0.Gamma0]
+alpha:625 [binder, in hydras.Gamma0.Gamma0]
+alpha:629 [binder, in hydras.Gamma0.Gamma0]
+alpha:63 [binder, in hydras.Schutte.Correctness_E0]
+alpha:63 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:63 [binder, in hydras.Epsilon0.F_alpha]
+alpha:63 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:63 [binder, in hydras.Schutte.AP]
+alpha:633 [binder, in hydras.Epsilon0.T1]
+alpha:635 [binder, in hydras.Epsilon0.T1]
+alpha:638 [binder, in hydras.Epsilon0.T1]
+alpha:64 [binder, in hydras.Epsilon0.F_alpha]
+alpha:64 [binder, in hydras.Schutte.AP]
+alpha:64 [binder, in hydras.Schutte.Addition]
+alpha:640 [binder, in hydras.Epsilon0.T1]
+alpha:640 [binder, in hydras.Gamma0.Gamma0]
+alpha:642 [binder, in hydras.Epsilon0.T1]
+alpha:644 [binder, in hydras.Gamma0.Gamma0]
+alpha:645 [binder, in hydras.Gamma0.Gamma0]
+alpha:646 [binder, in hydras.Epsilon0.T1]
+alpha:646 [binder, in hydras.Gamma0.Gamma0]
+alpha:647 [binder, in hydras.Epsilon0.T1]
+alpha:649 [binder, in hydras.Epsilon0.T1]
+alpha:65 [binder, in hydras.Schutte.Correctness_E0]
+alpha:65 [binder, in hydras.Epsilon0.E0]
+alpha:65 [binder, in hydras.Epsilon0.Hprime]
+alpha:650 [binder, in hydras.Epsilon0.T1]
+alpha:652 [binder, in hydras.Epsilon0.T1]
+alpha:654 [binder, in hydras.Epsilon0.T1]
+alpha:655 [binder, in hydras.Epsilon0.T1]
+alpha:656 [binder, in hydras.Epsilon0.T1]
+alpha:657 [binder, in hydras.Epsilon0.T1]
+alpha:66 [binder, in hydras.Schutte.AP]
+alpha:66 [binder, in hydras.OrdinalNotations.ON_Generic]
+alpha:66 [binder, in hydras.Epsilon0.E0]
+alpha:660 [binder, in hydras.Epsilon0.T1]
+alpha:661 [binder, in hydras.Epsilon0.T1]
+alpha:664 [binder, in hydras.Epsilon0.T1]
+alpha:667 [binder, in hydras.Epsilon0.T1]
+alpha:669 [binder, in hydras.Epsilon0.T1]
+alpha:67 [binder, in hydras.Schutte.Correctness_E0]
+alpha:67 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:67 [binder, in hydras.Epsilon0.E0]
+alpha:67 [binder, in hydras.Schutte.Addition]
+alpha:671 [binder, in hydras.Epsilon0.T1]
+alpha:673 [binder, in hydras.Epsilon0.T1]
+alpha:677 [binder, in hydras.Epsilon0.T1]
+alpha:68 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:68 [binder, in hydras.Epsilon0.E0]
+alpha:68 [binder, in hydras.Epsilon0.Hprime]
+alpha:681 [binder, in hydras.Epsilon0.T1]
+alpha:683 [binder, in hydras.Epsilon0.T1]
+alpha:688 [binder, in hydras.Epsilon0.T1]
+alpha:69 [binder, in hydras.Epsilon0.Epsilon0rpo]
+alpha:69 [binder, in hydras.Schutte.AP]
+alpha:69 [binder, in hydras.Epsilon0.E0]
+alpha:692 [binder, in hydras.Epsilon0.T1]
+alpha:694 [binder, in hydras.Gamma0.Gamma0]
+alpha:7 [binder, in hydras.Schutte.Correctness_E0]
+alpha:7 [binder, in gaia_hydras.T1Choice]
+alpha:7 [binder, in hydras.Hydra.O2H]
+alpha:7 [binder, in gaia_hydras.GF_alpha]
+alpha:7 [binder, in hydras.Schutte.CNF]
+alpha:7 [binder, in hydras.Schutte.AP]
+alpha:7 [binder, in hydras.Hydra.Battle_length]
+alpha:7 [binder, in hydras.Epsilon0.E0]
+alpha:7 [binder, in hydras.Epsilon0.Paths]
+alpha:7 [binder, in gaia_hydras.GHessenberg]
+alpha:70 [binder, in hydras.Epsilon0.T1]
+alpha:70 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:70 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:70 [binder, in hydras.Schutte.AP]
+alpha:70 [binder, in hydras.Epsilon0.Paths]
+alpha:700 [binder, in hydras.Epsilon0.T1]
+alpha:700 [binder, in hydras.Gamma0.Gamma0]
+alpha:702 [binder, in hydras.Gamma0.Gamma0]
+alpha:704 [binder, in hydras.Epsilon0.T1]
+alpha:704 [binder, in hydras.Gamma0.Gamma0]
+alpha:705 [binder, in hydras.Epsilon0.T1]
+alpha:709 [binder, in hydras.Epsilon0.T1]
+alpha:71 [binder, in hydras.Schutte.Correctness_E0]
+alpha:71 [binder, in hydras.Epsilon0.E0]
+alpha:710 [binder, in hydras.Gamma0.Gamma0]
+alpha:713 [binder, in hydras.Epsilon0.T1]
+alpha:713 [binder, in hydras.Gamma0.Gamma0]
+alpha:715 [binder, in hydras.Gamma0.Gamma0]
+alpha:716 [binder, in hydras.Epsilon0.T1]
+alpha:719 [binder, in hydras.Epsilon0.T1]
+alpha:72 [binder, in hydras.Schutte.AP]
+alpha:72 [binder, in hydras.Epsilon0.Paths]
+alpha:72 [binder, in hydras.Schutte.Addition]
+alpha:73 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:73 [binder, in hydras.Schutte.AP]
+alpha:73 [binder, in hydras.Epsilon0.E0]
+alpha:74 [binder, in hydras.Schutte.Correctness_E0]
+alpha:74 [binder, in hydras.Schutte.AP]
+alpha:74 [binder, in hydras.Epsilon0.Paths]
+alpha:740 [binder, in hydras.Epsilon0.T1]
+alpha:75 [binder, in hydras.Epsilon0.E0]
+alpha:76 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:76 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:77 [binder, in hydras.Schutte.Correctness_E0]
+alpha:77 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:77 [binder, in hydras.Epsilon0.Canon]
+alpha:77 [binder, in hydras.Epsilon0.E0]
+alpha:78 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:78 [binder, in hydras.Epsilon0.E0]
+alpha:8 [binder, in hydras.Hydra.O2H]
+alpha:8 [binder, in gaia_hydras.GF_alpha]
+alpha:8 [binder, in hydras.Schutte.AP]
+alpha:8 [binder, in gaia_hydras.ON_gfinite]
+alpha:8 [binder, in hydras.Epsilon0.Canon]
+alpha:8 [binder, in hydras.Schutte.Schutte_basics]
+alpha:8 [binder, in hydras.OrdinalNotations.ON_Finite]
+alpha:8 [binder, in hydras.Epsilon0.E0]
+alpha:8 [binder, in gaia_hydras.GL_alpha]
+alpha:80 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:80 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:80 [binder, in hydras.Epsilon0.Canon]
+alpha:80 [binder, in hydras.Epsilon0.E0]
+alpha:81 [binder, in hydras.Schutte.Critical]
+alpha:82 [binder, in hydras.Schutte.Critical]
+alpha:82 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:82 [binder, in hydras.Epsilon0.Paths]
+alpha:83 [binder, in hydras.Schutte.Critical]
+alpha:83 [binder, in hydras.Epsilon0.Canon]
+alpha:83 [binder, in hydras.Epsilon0.E0]
+alpha:83 [binder, in hydras.Schutte.Ordering_Functions]
+alpha:84 [binder, in hydras.Hydra.O2H]
+alpha:84 [binder, in hydras.Schutte.Critical]
+alpha:84 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:84 [binder, in hydras.Epsilon0.Paths]
+alpha:85 [binder, in hydras.Schutte.Critical]
+alpha:85 [binder, in hydras.Epsilon0.Canon]
+alpha:85 [binder, in hydras.Schutte.Schutte_basics]
+alpha:86 [binder, in hydras.Gamma0.T2]
+alpha:86 [binder, in hydras.Epsilon0.E0]
+alpha:86 [binder, in hydras.Epsilon0.Paths]
+alpha:87 [binder, in hydras.Epsilon0.F_alpha]
+alpha:87 [binder, in hydras.Schutte.Schutte_basics]
+alpha:88 [binder, in hydras.Hydra.O2H]
+alpha:88 [binder, in hydras.Epsilon0.F_alpha]
+alpha:89 [binder, in hydras.Epsilon0.F_alpha]
+alpha:89 [binder, in hydras.Epsilon0.E0]
+alpha:89 [binder, in hydras.Epsilon0.Paths]
+alpha:9 [binder, in hydras.Epsilon0.T1]
+alpha:9 [binder, in hydras.OrdinalNotations.ON_Omega2]
+alpha:9 [binder, in gaia_hydras.GF_alpha]
+alpha:9 [binder, in hydras.OrdinalNotations.ON_O]
+alpha:9 [binder, in hydras.Schutte.CNF]
+alpha:9 [binder, in hydras.Epsilon0.Large_Sets]
+alpha:9 [binder, in hydras.Schutte.Critical]
+alpha:9 [binder, in hydras.OrdinalNotations.ON_mult]
+alpha:9 [binder, in hydras.OrdinalNotations.ON_plus]
+alpha:9 [binder, in hydras.Schutte.Schutte_basics]
+alpha:9 [binder, in hydras.Epsilon0.E0]
+alpha:9 [binder, in hydras.Schutte.Addition]
+alpha:90 [binder, in hydras.OrdinalNotations.ON_Generic]
+alpha:91 [binder, in hydras.Epsilon0.T1]
+alpha:91 [binder, in hydras.Epsilon0.F_alpha]
+alpha:92 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:92 [binder, in hydras.Epsilon0.Canon]
+alpha:92 [binder, in hydras.Epsilon0.E0]
+alpha:92 [binder, in hydras.Epsilon0.Hprime]
+alpha:93 [binder, in hydras.Epsilon0.F_alpha]
+alpha:94 [binder, in hydras.Epsilon0.F_alpha]
+alpha:94 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:94 [binder, in hydras.Schutte.Schutte_basics]
+alpha:94 [binder, in hydras.Epsilon0.Paths]
+alpha:95 [binder, in hydras.Epsilon0.F_alpha]
+alpha:95 [binder, in hydras.Epsilon0.E0]
+alpha:95 [binder, in hydras.Epsilon0.Hprime]
+alpha:96 [binder, in hydras.Gamma0.T2]
+alpha:96 [binder, in hydras.Epsilon0.Hessenberg]
+alpha:96 [binder, in hydras.Epsilon0.Paths]
+alpha:96 [binder, in hydras.Gamma0.Gamma0]
+alpha:97 [binder, in hydras.Epsilon0.T1]
+alpha:97 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:97 [binder, in hydras.Epsilon0.Paths]
+alpha:98 [binder, in hydras.OrdinalNotations.OmegaOmega]
+alpha:98 [binder, in hydras.Epsilon0.Canon]
+alpha:98 [binder, in hydras.Epsilon0.E0]
+alpha:99 [binder, in hydras.Gamma0.T2]
+alpha:99 [binder, in hydras.Prelude.Comparable]
+Alt [module, in hydras.Epsilon0.Epsilon0rpo]
+Alt [module, in hydras.MoreAck.Ack]
+Alt [module, in additions.Demo]
+Alt [module, in hydras.MoreAck.PrimRecExamples]
+Alt [module, in hydras.Hydra.Hydra_Definitions]
+alt_double_ok [lemma, in additions.Demo]
+Alt.Ack [definition, in hydras.MoreAck.Ack]
+Alt.addIsPR [instance, in hydras.MoreAck.PrimRecExamples]
+Alt.addIsPR' [instance, in hydras.MoreAck.PrimRecExamples]
+Alt.add' [definition, in hydras.MoreAck.PrimRecExamples]
+Alt.add'_ok [lemma, in hydras.MoreAck.PrimRecExamples]
+Alt.compose_01 [lemma, in hydras.MoreAck.PrimRecExamples]
+Alt.const0_NIsPR [instance, in hydras.MoreAck.PrimRecExamples]
+Alt.double [definition, in additions.Demo]
+Alt.Eps0_rpo [module, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_alg [module, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.prec_transitive [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.prec_antisym [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.prec_dec [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.status [definition, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.Mul [constructor, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.Lex [constructor, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.status_type [inductive, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.prec [definition, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.A [definition, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec [module, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.arity [definition, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.Free [constructor, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.C [constructor, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.AC [constructor, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.arity_type [inductive, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.eq_symbol_dec [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.symb [definition, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.ord_cons [constructor, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.ord_zero [constructor, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.nat_S [constructor, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.nat_0 [constructor, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.symb0 [inductive, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig [module, in hydras.Epsilon0.Epsilon0rpo]
+Alt.hnode [constructor, in hydras.Hydra.Hydra_Definitions]
+Alt.Hydra [inductive, in hydras.Hydra.Hydra_Definitions]
+Alt.lt_inc_rpo_0 [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.lt_subterm2 [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.lt_subterm1 [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.nat_2_term_mono [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.nat_lt_cons [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.nat_2_term [definition, in hydras.Epsilon0.Epsilon0rpo]
+Alt.nf_Acc [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.nf_Wf [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.pi2_5IsPR [instance, in hydras.MoreAck.PrimRecExamples]
+Alt.predIsPR [instance, in hydras.MoreAck.PrimRecExamples]
+Alt.rpo_trans [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.R_inc_rpo [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.R1 [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.R2 [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.R3 [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.R4 [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.succIsPR [instance, in hydras.MoreAck.PrimRecExamples]
+Alt.transfinite_recursor [definition, in hydras.Epsilon0.Epsilon0rpo]
+Alt.transfinite_recursor_lt [definition, in hydras.Epsilon0.Epsilon0rpo]
+Alt.T1_size3 [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.T1_size2 [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.T1_size1 [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.T1_size [definition, in hydras.Epsilon0.Epsilon0rpo]
+Alt.T1_2_term [definition, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Vars [module, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Vars.empty_set [inductive, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Vars.eq_variable_dec [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.Vars.var [definition, in hydras.Epsilon0.Epsilon0rpo]
+Alt.well_founded.R [variable, in hydras.Epsilon0.Epsilon0rpo]
+Alt.well_founded [section, in hydras.Epsilon0.Epsilon0rpo]
+Alt.well_founded_rpo [lemma, in hydras.Epsilon0.Epsilon0rpo]
+Alt.xpred [definition, in hydras.MoreAck.PrimRecExamples]
+Alt.zeroIsPR [instance, in hydras.MoreAck.PrimRecExamples]
+Alt2 [module, in hydras.MoreAck.Ack]
+Alt2.Ack [definition, in hydras.MoreAck.Ack]
+Alt2.test1 [definition, in hydras.MoreAck.Ack]
+Alt2.test2 [definition, in hydras.MoreAck.Ack]
+Alt3 [module, in hydras.MoreAck.Ack]
+Alt3.ack [definition, in hydras.MoreAck.Ack]
+AM [library]
+AM_power_Ok [lemma, in additions.AM]
+AM_power [definition, in additions.AM]
+AM:105 [binder, in additions.Pow]
+AM:96 [binder, in additions.Pow_variant]
+andE1 [lemma, in hydras.Ackermann.LNN]
+andE1 [lemma, in hydras.Ackermann.LNT]
+andE1 [lemma, in hydras.Ackermann.folLogic]
+andE2 [lemma, in hydras.Ackermann.LNN]
+andE2 [lemma, in hydras.Ackermann.LNT]
+andE2 [lemma, in hydras.Ackermann.folLogic]
+andH [definition, in hydras.Ackermann.fol]
+andI [lemma, in hydras.Ackermann.LNN]
+andI [lemma, in hydras.Ackermann.LNT]
+andI [lemma, in hydras.Ackermann.folLogic]
+andRel [definition, in hydras.Ackermann.primRec]
+andRelPR [lemma, in hydras.Ackermann.primRec]
+andSym [lemma, in hydras.Ackermann.LNN]
+andSym [lemma, in hydras.Ackermann.LNT]
+andSym [lemma, in hydras.Ackermann.folLogic]
+anonymous0:360 [binder, in additions.Euclidean_Chains]
+anonymous0:366 [binder, in additions.Euclidean_Chains]
+anonymous0:372 [binder, in additions.Euclidean_Chains]
+anonymous0:379 [binder, in additions.Euclidean_Chains]
+anonymous:359 [binder, in additions.Euclidean_Chains]
+anonymous:365 [binder, in additions.Euclidean_Chains]
+anonymous:371 [binder, in additions.Euclidean_Chains]
+anonymous:378 [binder, in additions.Euclidean_Chains]
+anonymous:386 [binder, in additions.Euclidean_Chains]
+anonymous:392 [binder, in additions.Euclidean_Chains]
+anonymous:399 [binder, in additions.Euclidean_Chains]
+answer [inductive, in hydras.Epsilon0.Large_Sets_Examples]
+any_chain_parametric [definition, in additions.Addition_Chains]
+ap [inductive, in hydras.Epsilon0.T1]
+ap [definition, in hydras.OrdinalNotations.ON_Omega2]
+ap [inductive, in hydras.Gamma0.T2]
+AP [definition, in hydras.Schutte.AP]
+AP [library]
+apply [constructor, in hydras.Ackermann.fol]
+approx [definition, in hydras.Epsilon0.Canon]
+approx_ok [lemma, in hydras.Epsilon0.Canon]
+ap_plusR [lemma, in hydras.Epsilon0.T1]
+ap_plus [lemma, in hydras.Epsilon0.T1]
+ap_intro [constructor, in hydras.Epsilon0.T1]
+ap_cases [lemma, in hydras.OrdinalNotations.ON_Omega2]
+ap_intro [constructor, in hydras.Gamma0.T2]
+AP_plus_AP [lemma, in hydras.Schutte.AP]
+AP_to_phi0 [lemma, in hydras.Schutte.AP]
+AP_phi0 [lemma, in hydras.Schutte.AP]
+AP_o_segment [lemma, in hydras.Schutte.AP]
+AP_closed [lemma, in hydras.Schutte.AP]
+AP_sup [lemma, in hydras.Schutte.AP]
+AP_closed.denM [variable, in hydras.Schutte.AP]
+AP_closed.inhM [variable, in hydras.Schutte.AP]
+AP_closed.OM [variable, in hydras.Schutte.AP]
+AP_closed.M [variable, in hydras.Schutte.AP]
+AP_closed [section, in hydras.Schutte.AP]
+AP_unbounded [lemma, in hydras.Schutte.AP]
+AP_unbounded_0 [lemma, in hydras.Schutte.AP]
+AP_Unbounded.ksi_fixed.n [variable, in hydras.Schutte.AP]
+AP_Unbounded.ksi_fixed.lt_ksi [variable, in hydras.Schutte.AP]
+AP_Unbounded.ksi_fixed.ksi [variable, in hydras.Schutte.AP]
+AP_Unbounded.ksi_fixed [section, in hydras.Schutte.AP]
+AP_Unbounded.beta [variable, in hydras.Schutte.AP]
+AP_Unbounded.seq [variable, in hydras.Schutte.AP]
+AP_Unbounded.alpha [variable, in hydras.Schutte.AP]
+AP_Unbounded [section, in hydras.Schutte.AP]
+AP_mult_fin_r_closed [lemma, in hydras.Schutte.AP]
+AP_mult_Sn_closed [lemma, in hydras.Schutte.AP]
+AP_plus_closed [lemma, in hydras.Schutte.AP]
+AP_finite_eq_one [lemma, in hydras.Schutte.AP]
+AP_omega [lemma, in hydras.Schutte.AP]
+AP_one [lemma, in hydras.Schutte.AP]
+ap_ref [lemma, in gaia_hydras.T1Bridge]
+Arith_lemmas [section, in hydras.Prelude.More_Arith]
+arityF [projection, in hydras.Ackermann.fol]
+arityR [projection, in hydras.Ackermann.fol]
+assoc [projection, in hydras.Prelude.STDPP_compat]
+Assoc [record, in hydras.Prelude.STDPP_compat]
+assoc [constructor, in hydras.Prelude.STDPP_compat]
+Assoc [inductive, in hydras.Prelude.STDPP_compat]
+assoc:105 [binder, in additions.fib]
+assoc:112 [binder, in additions.fib]
+assoc:123 [binder, in additions.fib]
+atomic [constructor, in hydras.Ackermann.fol]
+aux:10 [binder, in additions.Trace_exercise]
+AXM [constructor, in hydras.Ackermann.folProof]
+Axm [lemma, in hydras.Ackermann.LNN]
+Axm [lemma, in hydras.Ackermann.LNT]
+Axm [lemma, in hydras.Ackermann.folLogic]
+AxmEq4 [definition, in hydras.Ackermann.folProof]
+AxmEq5 [definition, in hydras.Ackermann.folProof]
+Axm:120 [binder, in hydras.Ackermann.LNN2LNT]
+axm:126 [binder, in hydras.Ackermann.LNN2LNT]
+Axm:128 [binder, in hydras.Ackermann.LNN2LNT]
+Axm:141 [binder, in hydras.Ackermann.LNN2LNT]
+Axm:145 [binder, in hydras.Ackermann.LNN2LNT]
+Axm:15 [binder, in Goedel.goedel2]
+Axm:19 [binder, in Goedel.goedel2]
+Axm:44 [binder, in Goedel.rosserPA]
+Axm:47 [binder, in Goedel.rosserPA]
+axp_scheme_length1 [lemma, in additions.Addition_Chains]
+axp_correct [lemma, in additions.Addition_Chains]
+axp_scheme [definition, in additions.Addition_Chains]
+AX1 [axiom, in hydras.Schutte.Schutte_basics]
+AX2 [axiom, in hydras.Schutte.Schutte_basics]
+AX3 [axiom, in hydras.Schutte.Schutte_basics]
+ax:252 [binder, in additions.Addition_Chains]
+A_def.m [variable, in hydras.OrdinalNotations.ON_Omega2]
+A_def [section, in hydras.OrdinalNotations.ON_Omega2]
+A_full [lemma, in hydras.Schutte.Critical]
+A_Cr [lemma, in hydras.Schutte.Critical]
+A_ [definition, in hydras.Schutte.Critical]
+A_given.LtA [variable, in hydras.solutions_exercises.MultisetWf]
+A_given.A [variable, in hydras.solutions_exercises.MultisetWf]
+A_given [section, in hydras.solutions_exercises.MultisetWf]
+A_node [constructor, in additions.Addition_Chains]
+A_closed [lemma, in hydras.Schutte.Ordering_Functions]
+A_denum [lemma, in hydras.Schutte.Ordering_Functions]
+a'':44 [binder, in hydras.Hydra.BigBattle]
+a':1022 [binder, in gaia_hydras.nfwfgaia]
+a':11 [binder, in hydras.Prelude.Simple_LexProd]
+a':1150 [binder, in gaia_hydras.nfwfgaia]
+a':116 [binder, in hydras.Epsilon0.T1]
+a':1297 [binder, in gaia_hydras.nfwfgaia]
+a':130 [binder, in gaia_hydras.nfwfgaia]
+a':1301 [binder, in gaia_hydras.nfwfgaia]
+a':1305 [binder, in gaia_hydras.nfwfgaia]
+a':1309 [binder, in gaia_hydras.nfwfgaia]
+a':1335 [binder, in gaia_hydras.nfwfgaia]
+a':1381 [binder, in gaia_hydras.nfwfgaia]
+a':1391 [binder, in gaia_hydras.nfwfgaia]
+a':1419 [binder, in gaia_hydras.nfwfgaia]
+a':1425 [binder, in gaia_hydras.nfwfgaia]
+a':1433 [binder, in gaia_hydras.nfwfgaia]
+a':1497 [binder, in gaia_hydras.nfwfgaia]
+a':1505 [binder, in gaia_hydras.nfwfgaia]
+a':153 [binder, in hydras.rpo.more_list]
+a':1548 [binder, in gaia_hydras.nfwfgaia]
+a':1582 [binder, in gaia_hydras.nfwfgaia]
+a':1606 [binder, in gaia_hydras.nfwfgaia]
+a':1615 [binder, in gaia_hydras.nfwfgaia]
+a':1624 [binder, in gaia_hydras.nfwfgaia]
+a':1632 [binder, in gaia_hydras.nfwfgaia]
+a':164 [binder, in hydras.Epsilon0.T1]
+a':1688 [binder, in gaia_hydras.nfwfgaia]
+a':169 [binder, in hydras.rpo.rpo]
+a':1694 [binder, in gaia_hydras.nfwfgaia]
+a':173 [binder, in gaia_hydras.nfwfgaia]
+a':175 [binder, in hydras.Epsilon0.T1]
+a':18 [binder, in hydras.Epsilon0.Hessenberg]
+a':189 [binder, in hydras.rpo.rpo]
+a':191 [binder, in hydras.rpo.rpo]
+a':2063 [binder, in gaia_hydras.nfwfgaia]
+a':2077 [binder, in gaia_hydras.nfwfgaia]
+a':2082 [binder, in gaia_hydras.nfwfgaia]
+a':22 [binder, in hydras.Schutte.Correctness_E0]
+a':222 [binder, in hydras.Epsilon0.T1]
+a':224 [binder, in gaia_hydras.nfwfgaia]
+a':228 [binder, in hydras.Epsilon0.T1]
+a':23 [binder, in hydras.Hydra.BigBattle]
+a':230 [binder, in gaia_hydras.nfwfgaia]
+a':237 [binder, in hydras.rpo.rpo]
+a':24 [binder, in hydras.Epsilon0.Hessenberg]
+a':24 [binder, in additions.Monoid_instances]
+a':240 [binder, in gaia_hydras.nfwfgaia]
+a':250 [binder, in hydras.rpo.rpo]
+a':255 [binder, in hydras.rpo.rpo]
+a':256 [binder, in hydras.Epsilon0.T1]
+a':257 [binder, in gaia_hydras.nfwfgaia]
+a':300 [binder, in hydras.Gamma0.Gamma0]
+a':307 [binder, in hydras.rpo.rpo]
+a':308 [binder, in hydras.rpo.rpo]
+a':32 [binder, in hydras.Hydra.BigBattle]
+a':33 [binder, in hydras.Epsilon0.T1]
+a':33 [binder, in hydras.Gamma0.Gamma0]
+a':34 [binder, in hydras.Schutte.PartialFun]
+a':346 [binder, in hydras.Epsilon0.T1]
+a':35 [binder, in hydras.Epsilon0.Hessenberg]
+a':358 [binder, in hydras.Epsilon0.T1]
+a':364 [binder, in hydras.Epsilon0.T1]
+a':378 [binder, in hydras.Epsilon0.T1]
+a':388 [binder, in hydras.Epsilon0.T1]
+a':40 [binder, in hydras.Hydra.BigBattle]
+a':41 [binder, in hydras.Epsilon0.Hessenberg]
+a':428 [binder, in hydras.Epsilon0.T1]
+a':429 [binder, in gaia_hydras.nfwfgaia]
+a':439 [binder, in hydras.Epsilon0.T1]
+a':44 [binder, in hydras.Prelude.DecPreOrder_Instances]
+a':44 [binder, in hydras.Schutte.PartialFun]
+a':48 [binder, in hydras.Prelude.DecPreOrder_Instances]
+a':49 [binder, in hydras.Epsilon0.Epsilon0rpo]
+a':51 [binder, in hydras.Epsilon0.Hessenberg]
+a':52 [binder, in hydras.Prelude.DecPreOrder_Instances]
+a':53 [binder, in hydras.Epsilon0.T1]
+a':53 [binder, in hydras.Epsilon0.Epsilon0rpo]
+a':534 [binder, in hydras.Epsilon0.T1]
+a':538 [binder, in hydras.Ackermann.primRec]
+a':56 [binder, in hydras.Prelude.DecPreOrder_Instances]
+a':56 [binder, in hydras.Hydra.BigBattle]
+a':57 [binder, in hydras.Prelude.Sort_spec]
+a':60 [binder, in hydras.Prelude.DecPreOrder_Instances]
+a':67 [binder, in hydras.Epsilon0.T1]
+a':69 [binder, in gaia_hydras.T1Bridge]
+a':725 [binder, in hydras.Epsilon0.T1]
+a':74 [binder, in hydras.Epsilon0.T1]
+a':74 [binder, in hydras.rpo.rpo]
+a':75 [binder, in gaia_hydras.T1Bridge]
+a':789 [binder, in gaia_hydras.nfwfgaia]
+a':81 [binder, in hydras.Gamma0.T2]
+a':846 [binder, in gaia_hydras.nfwfgaia]
+a':869 [binder, in gaia_hydras.nfwfgaia]
+a':896 [binder, in gaia_hydras.nfwfgaia]
+a':900 [binder, in gaia_hydras.nfwfgaia]
+a':906 [binder, in gaia_hydras.nfwfgaia]
+a':914 [binder, in gaia_hydras.nfwfgaia]
+a':942 [binder, in gaia_hydras.nfwfgaia]
+a':990 [binder, in gaia_hydras.nfwfgaia]
+a0:17 [binder, in hydras.Schutte.Well_Orders]
+a0:191 [binder, in hydras.Ackermann.cPair]
+a0:194 [binder, in hydras.Ackermann.cPair]
+a0:198 [binder, in hydras.Ackermann.cPair]
+a0:221 [binder, in hydras.Ackermann.cPair]
+a0:225 [binder, in hydras.Ackermann.cPair]
+a0:229 [binder, in hydras.Ackermann.cPair]
+a0:233 [binder, in hydras.Ackermann.cPair]
+a0:236 [binder, in hydras.Ackermann.cPair]
+a0:238 [binder, in hydras.Ackermann.cPair]
+a0:240 [binder, in hydras.Ackermann.cPair]
+a0:242 [binder, in hydras.Ackermann.cPair]
+a0:301 [binder, in hydras.Ackermann.folProp]
+a0:308 [binder, in hydras.Ackermann.folProp]
+a0:317 [binder, in hydras.Ackermann.folProp]
+a0:324 [binder, in hydras.Ackermann.folProp]
+a0:333 [binder, in hydras.Ackermann.folProp]
+a0:340 [binder, in hydras.Ackermann.folProp]
+a0:349 [binder, in hydras.Ackermann.folProp]
+a0:356 [binder, in hydras.Ackermann.folProp]
+a0:44 [binder, in hydras.Ackermann.LNN2LNT]
+a0:531 [binder, in hydras.Ackermann.primRec]
+a0:533 [binder, in hydras.Ackermann.primRec]
+A1_A2 [lemma, in hydras.Schutte.Ordering_Functions]
+a1:105 [binder, in hydras.rpo.more_list]
+a1:113 [binder, in hydras.rpo.more_list]
+A1:114 [binder, in hydras.Schutte.Ordering_Functions]
+a1:122 [binder, in hydras.Prelude.Merge_Sort]
+a1:130 [binder, in hydras.Prelude.Merge_Sort]
+a1:1360 [binder, in hydras.Ackermann.codeSubFormula]
+a1:137 [binder, in hydras.rpo.more_list]
+a1:146 [binder, in hydras.rpo.more_list]
+a1:156 [binder, in hydras.rpo.more_list]
+a1:165 [binder, in hydras.rpo.more_list]
+a1:172 [binder, in hydras.rpo.more_list]
+a1:180 [binder, in hydras.rpo.more_list]
+a1:188 [binder, in hydras.rpo.more_list]
+a1:192 [binder, in hydras.Ackermann.cPair]
+a1:195 [binder, in hydras.Ackermann.cPair]
+a1:199 [binder, in hydras.Ackermann.cPair]
+a1:2 [binder, in hydras.rpo.decidable_set]
+a1:20 [binder, in hydras.Epsilon0.Epsilon0rpo]
+a1:203 [binder, in hydras.rpo.more_list]
+a1:212 [binder, in hydras.rpo.more_list]
+a1:222 [binder, in hydras.rpo.more_list]
+a1:240 [binder, in hydras.rpo.more_list]
+a1:249 [binder, in hydras.rpo.more_list]
+a1:261 [binder, in hydras.rpo.more_list]
+a1:261 [binder, in hydras.Gamma0.Gamma0]
+a1:27 [binder, in hydras.rpo.rpo]
+a1:3 [binder, in hydras.rpo.rpo]
+a1:35 [binder, in hydras.Gamma0.Gamma0]
+a1:43 [binder, in hydras.Gamma0.Gamma0]
+a1:51 [binder, in hydras.Gamma0.Gamma0]
+a1:59 [binder, in hydras.Gamma0.Gamma0]
+a1:67 [binder, in hydras.Gamma0.Gamma0]
+a1:75 [binder, in hydras.Gamma0.Gamma0]
+a1:79 [binder, in hydras.rpo.more_list]
+a1:81 [binder, in hydras.rpo.list_permut]
+a1:82 [binder, in hydras.Gamma0.Gamma0]
+a1:89 [binder, in hydras.rpo.more_list]
+a1:97 [binder, in hydras.rpo.more_list]
+A2_A1 [lemma, in hydras.Schutte.Ordering_Functions]
+a2:106 [binder, in hydras.rpo.more_list]
+a2:114 [binder, in hydras.rpo.more_list]
+A2:115 [binder, in hydras.Schutte.Ordering_Functions]
+a2:124 [binder, in hydras.Prelude.Merge_Sort]
+a2:132 [binder, in hydras.Prelude.Merge_Sort]
+a2:1361 [binder, in hydras.Ackermann.codeSubFormula]
+a2:138 [binder, in hydras.rpo.more_list]
+a2:147 [binder, in hydras.rpo.more_list]
+a2:157 [binder, in hydras.rpo.more_list]
+a2:166 [binder, in hydras.rpo.more_list]
+a2:173 [binder, in hydras.rpo.more_list]
+a2:181 [binder, in hydras.rpo.more_list]
+a2:189 [binder, in hydras.rpo.more_list]
+a2:196 [binder, in hydras.Ackermann.cPair]
+a2:200 [binder, in hydras.Ackermann.cPair]
+a2:204 [binder, in hydras.rpo.more_list]
+a2:21 [binder, in hydras.Epsilon0.Epsilon0rpo]
+a2:213 [binder, in hydras.rpo.more_list]
+a2:223 [binder, in hydras.rpo.more_list]
+a2:241 [binder, in hydras.rpo.more_list]
+a2:250 [binder, in hydras.rpo.more_list]
+a2:262 [binder, in hydras.rpo.more_list]
+a2:262 [binder, in hydras.Gamma0.Gamma0]
+a2:28 [binder, in hydras.rpo.rpo]
+a2:3 [binder, in hydras.rpo.decidable_set]
+a2:36 [binder, in hydras.Gamma0.Gamma0]
+a2:4 [binder, in hydras.rpo.rpo]
+a2:44 [binder, in hydras.Gamma0.Gamma0]
+a2:52 [binder, in hydras.Gamma0.Gamma0]
+a2:60 [binder, in hydras.Gamma0.Gamma0]
+a2:68 [binder, in hydras.Gamma0.Gamma0]
+a2:76 [binder, in hydras.Gamma0.Gamma0]
+a2:80 [binder, in hydras.rpo.more_list]
+a2:83 [binder, in hydras.rpo.list_permut]
+a2:90 [binder, in hydras.rpo.more_list]
+a2:98 [binder, in hydras.rpo.more_list]
+a:1 [binder, in Goedel.PRrepresentable]
+a:1 [binder, in gaia_hydras.T1Choice]
+A:1 [binder, in hydras.Prelude.DecPreOrder_Instances]
+A:1 [binder, in hydras.Prelude.Iterates]
+A:1 [binder, in hydras.solutions_exercises.predSuccUnicity]
+a:1 [binder, in hydras.Hydra.O2H]
+a:1 [binder, in additions.Compatibility]
+A:1 [binder, in hydras.Prelude.Restriction]
+A:1 [binder, in hydras.solutions_exercises.Limit_Infinity]
+A:1 [binder, in hydras.rpo.more_list]
+A:1 [binder, in hydras.Prelude.Comparable]
+A:1 [binder, in hydras.OrdinalNotations.ON_O]
+A:1 [binder, in additions.Pow_variant]
+a:1 [binder, in Goedel.fixPoint]
+A:1 [binder, in additions.Pow]
+A:1 [binder, in hydras.Prelude.Sort_spec]
+A:1 [binder, in hydras.Prelude.DecPreOrder]
+a:1 [binder, in gaia_hydras.GCanon]
+A:1 [binder, in hydras.Schutte.MoreEpsilonIota]
+a:1 [binder, in hydras.solutions_exercises.OnCodeList]
+A:1 [binder, in hydras.solutions_exercises.MultisetWf]
+A:1 [binder, in additions.Wf_transparent]
+a:1 [binder, in gaia_hydras.T1Bridge]
+A:1 [binder, in hydras.Prelude.MoreOrders]
+A:1 [binder, in additions.Monoid_def]
+A:1 [binder, in additions.Trace_exercise]
+A:1 [binder, in hydras.solutions_exercises.lt_succ_le]
+a:1 [binder, in hydras.Ackermann.prLogic]
+A:1 [binder, in hydras.OrdinalNotations.ON_mult]
+A:1 [binder, in additions.Euclidean_Chains]
+A:1 [binder, in hydras.OrdinalNotations.ON_plus]
+A:1 [binder, in hydras.Prelude.STDPP_compat]
+a:1 [binder, in hydras.solutions_exercises.MinPR]
+a:1 [binder, in hydras.Ackermann.NNtheory]
+a:1 [binder, in gaia_hydras.GLarge_Sets]
+A:1 [binder, in hydras.OrdinalNotations.ON_Generic]
+A:1 [binder, in gaia_hydras.onType]
+A:1 [binder, in additions.Addition_Chains]
+a:1 [binder, in gaia_hydras.GaiaToHydra]
+A:1 [binder, in hydras.Schutte.Ordering_Functions]
+a:1 [binder, in hydras.Gamma0.Gamma0]
+A:1 [binder, in hydras.rpo.closure]
+A:1 [binder, in hydras.rpo.rpo]
+A:1 [binder, in hydras.Prelude.MoreVectors]
+a:10 [binder, in gaia_hydras.T1Choice]
+A:10 [binder, in additions.Pow_variant]
+a:10 [binder, in Goedel.fixPoint]
+A:10 [binder, in additions.Pow]
+A:10 [binder, in hydras.Prelude.DecPreOrder]
+a:10 [binder, in gaia_hydras.GCanon]
+a:10 [binder, in hydras.Hydra.Hydra_Termination]
+a:10 [binder, in hydras.Ackermann.prLogic]
+a:10 [binder, in hydras.Schutte.Ordering_Functions]
+A:10 [binder, in hydras.rpo.closure]
+a:10 [binder, in hydras.Prelude.Simple_LexProd]
+A:10 [binder, in hydras.Ackermann.expressible]
+a:100 [binder, in hydras.rpo.more_list]
+a:100 [binder, in hydras.Ackermann.cPair]
+a:100 [binder, in gaia_hydras.nfwfgaia]
+a:1000 [binder, in gaia_hydras.nfwfgaia]
+a:1003 [binder, in gaia_hydras.nfwfgaia]
+a:1006 [binder, in gaia_hydras.nfwfgaia]
+a:1008 [binder, in gaia_hydras.nfwfgaia]
+A:101 [binder, in additions.fib]
+A:101 [binder, in additions.Addition_Chains]
+a:1012 [binder, in gaia_hydras.nfwfgaia]
+a:1014 [binder, in gaia_hydras.nfwfgaia]
+a:1019 [binder, in gaia_hydras.nfwfgaia]
+a:102 [binder, in hydras.Prelude.Merge_Sort]
+a:102 [binder, in gaia_hydras.T1Bridge]
+a:102 [binder, in hydras.Ackermann.cPair]
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+a:1598 [binder, in gaia_hydras.nfwfgaia]
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+a:43 [binder, in gaia_hydras.T1Bridge]
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+a:44 [binder, in hydras.Prelude.Comparable]
+a:44 [binder, in hydras.Epsilon0.Hessenberg]
+A:44 [binder, in hydras.Prelude.DecPreOrder]
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+a:669 [binder, in hydras.Gamma0.Gamma0]
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+a:675 [binder, in hydras.Ackermann.primRec]
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+a:694 [binder, in hydras.Ackermann.primRec]
+a:695 [binder, in gaia_hydras.nfwfgaia]
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+a:7 [binder, in Goedel.fixPoint]
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+a:7 [binder, in gaia_hydras.T1Bridge]
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+a:7 [binder, in gaia_hydras.GLarge_Sets]
+A:7 [binder, in hydras.OrdinalNotations.ON_Generic]
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+A:71 [binder, in additions.Addition_Chains]
+A:71 [binder, in hydras.Schutte.Ordering_Functions]
+A:71 [binder, in Goedel.rosser]
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+a:756 [binder, in gaia_hydras.nfwfgaia]
+a:758 [binder, in gaia_hydras.nfwfgaia]
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+A:76 [binder, in hydras.Prelude.DecPreOrder]
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+A:76 [binder, in hydras.Schutte.Ordering_Functions]
+a:760 [binder, in hydras.Ackermann.checkPrf]
+a:760 [binder, in gaia_hydras.nfwfgaia]
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+a:77 [binder, in hydras.Ackermann.subAll]
+a:77 [binder, in hydras.Epsilon0.Epsilon0rpo]
+a:77 [binder, in hydras.Epsilon0.Hessenberg]
+A:77 [binder, in Goedel.codeSysPrf]
+a:77 [binder, in hydras.Schutte.Ordering_Functions]
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+a:779 [binder, in gaia_hydras.nfwfgaia]
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+a:788 [binder, in hydras.Ackermann.checkPrf]
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+A:8 [binder, in gaia_hydras.GLarge_Sets]
+a:8 [binder, in hydras.rpo.dickson]
+a:8 [binder, in hydras.MoreAck.FolExamples]
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+a:80 [binder, in gaia_hydras.GPaths]
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+a:807 [binder, in hydras.Ackermann.checkPrf]
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+a:81 [binder, in hydras.Ackermann.cPair]
+a:81 [binder, in hydras.Ackermann.fol]
+a:81 [binder, in hydras.Schutte.Schutte_basics]
+A:81 [binder, in additions.fib]
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+A:81 [binder, in hydras.Schutte.Ordering_Functions]
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+a:819 [binder, in hydras.Ackermann.codeSubFormula]
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+a:826 [binder, in hydras.Ackermann.codeSubFormula]
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+a:84 [binder, in gaia_hydras.T1Bridge]
+A:84 [binder, in Goedel.codeSysPrf]
+A:84 [binder, in hydras.Prelude.MoreVectors]
+a:842 [binder, in hydras.Ackermann.codeSubFormula]
+a:842 [binder, in gaia_hydras.nfwfgaia]
+a:845 [binder, in hydras.Ackermann.codeSubFormula]
+a:848 [binder, in hydras.Ackermann.codeSubFormula]
+a:849 [binder, in hydras.Ackermann.checkPrf]
+A:85 [binder, in Goedel.PRrepresentable]
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+a:85 [binder, in hydras.Ackermann.folLogic3]
+a:851 [binder, in hydras.Ackermann.checkPrf]
+a:852 [binder, in hydras.Ackermann.codeSubFormula]
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+a:855 [binder, in hydras.Ackermann.checkPrf]
+a:856 [binder, in hydras.Ackermann.codeSubFormula]
+a:86 [binder, in hydras.Hydra.O2H]
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+a:87 [binder, in hydras.Ackermann.cPair]
+a:87 [binder, in additions.Addition_Chains]
+A:88 [binder, in hydras.rpo.more_list]
+a:88 [binder, in hydras.Prelude.Comparable]
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+a:88 [binder, in gaia_hydras.T1Bridge]
+A:88 [binder, in additions.Euclidean_Chains]
+A:88 [binder, in hydras.Schutte.PartialFun]
+a:88 [binder, in hydras.Schutte.Schutte_basics]
+A:88 [binder, in Goedel.codeSysPrf]
+A:88 [binder, in additions.Addition_Chains]
+a:88 [binder, in hydras.MoreAck.PrimRecExamples]
+A:88 [binder, in hydras.Prelude.MoreVectors]
+a:885 [binder, in gaia_hydras.nfwfgaia]
+A:89 [binder, in Goedel.PRrepresentable]
+a:89 [binder, in hydras.Prelude.Comparable]
+a:89 [binder, in gaia_hydras.T1Bridge]
+a:89 [binder, in hydras.Ackermann.cPair]
+a:894 [binder, in gaia_hydras.nfwfgaia]
+a:898 [binder, in gaia_hydras.nfwfgaia]
+a:9 [binder, in gaia_hydras.T1Choice]
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+a:9 [binder, in additions.FirstSteps]
+a:9 [binder, in hydras.Prelude.Comparable]
+a:9 [binder, in gaia_hydras.GPaths]
+a:9 [binder, in hydras.Ackermann.PAtheory]
+a:9 [binder, in hydras.solutions_exercises.MultisetWf]
+a:9 [binder, in hydras.OrdinalNotations.OmegaOmega]
+a:9 [binder, in gaia_hydras.T1Bridge]
+a:9 [binder, in hydras.Ackermann.prLogic]
+a:9 [binder, in hydras.Ackermann.NNtheory]
+a:9 [binder, in gaia_hydras.GLarge_Sets]
+a:9 [binder, in hydras.Prelude.MoreDecidable]
+a:9 [binder, in gaia_hydras.T2Bridge]
+a:9 [binder, in hydras.Gamma0.Gamma0]
+A:9 [binder, in hydras.Prelude.MoreVectors]
+a:90 [binder, in hydras.Gamma0.T2]
+a:90 [binder, in hydras.Epsilon0.Hessenberg]
+a:90 [binder, in gaia_hydras.T1Bridge]
+a:90 [binder, in hydras.Schutte.Schutte_basics]
+a:90 [binder, in additions.fib]
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+a:91 [binder, in hydras.Prelude.DecPreOrder]
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+a:91 [binder, in hydras.Ackermann.cPair]
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+a:91 [binder, in hydras.Schutte.Ordering_Functions]
+a:910 [binder, in gaia_hydras.nfwfgaia]
+A:92 [binder, in Goedel.PRrepresentable]
+a:92 [binder, in hydras.Epsilon0.Hessenberg]
+a:92 [binder, in gaia_hydras.T1Bridge]
+a:92 [binder, in hydras.Schutte.Schutte_basics]
+a:92 [binder, in hydras.Prelude.MoreVectors]
+a:923 [binder, in gaia_hydras.nfwfgaia]
+a:925 [binder, in gaia_hydras.nfwfgaia]
+a:927 [binder, in gaia_hydras.nfwfgaia]
+a:929 [binder, in gaia_hydras.nfwfgaia]
+A:93 [binder, in additions.AM]
+a:93 [binder, in hydras.Prelude.Comparable]
+A:93 [binder, in hydras.Prelude.DecPreOrder]
+a:93 [binder, in hydras.Prelude.Merge_Sort]
+a:93 [binder, in gaia_hydras.T1Bridge]
+a:93 [binder, in hydras.Ackermann.cPair]
+A:93 [binder, in hydras.OrdinalNotations.ON_Generic]
+A:93 [binder, in Goedel.codeSysPrf]
+A:93 [binder, in additions.Addition_Chains]
+a:931 [binder, in gaia_hydras.nfwfgaia]
+a:938 [binder, in gaia_hydras.nfwfgaia]
+a:94 [binder, in hydras.Epsilon0.Hessenberg]
+a:94 [binder, in hydras.Ackermann.folLogic3]
+a:944 [binder, in gaia_hydras.nfwfgaia]
+a:95 [binder, in gaia_hydras.T1Bridge]
+a:95 [binder, in hydras.Ackermann.cPair]
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+a:956 [binder, in gaia_hydras.nfwfgaia]
+a:958 [binder, in gaia_hydras.nfwfgaia]
+A:96 [binder, in Goedel.PRrepresentable]
+A:96 [binder, in hydras.rpo.more_list]
+a:96 [binder, in hydras.Prelude.DecPreOrder]
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+a:962 [binder, in gaia_hydras.nfwfgaia]
+a:964 [binder, in gaia_hydras.nfwfgaia]
+a:97 [binder, in hydras.Prelude.Comparable]
+a:97 [binder, in hydras.Epsilon0.Hessenberg]
+a:97 [binder, in gaia_hydras.T1Bridge]
+A:98 [binder, in additions.AM]
+A:98 [binder, in hydras.Hydra.Hydra_Lemmas]
+a:988 [binder, in gaia_hydras.nfwfgaia]
+A:99 [binder, in additions.Euclidean_Chains]
+a:99 [binder, in hydras.Schutte.Schutte_basics]
+a:994 [binder, in gaia_hydras.nfwfgaia]
+

B

+Bad [module, in additions.FirstSteps]
+Bad [module, in hydras.Hydra.Hydra_Examples]
+Bad [module, in additions.Euclidean_Chains]
+Bad [module, in additions.Monoid_instances]
+bad [definition, in hydras.OrdinalNotations.ON_Finite]
+BadExample [section, in hydras.MoreAck.BadSubst]
+BadExample.F [variable, in hydras.MoreAck.BadSubst]
+BadExample.F1 [variable, in hydras.MoreAck.BadSubst]
+BadSubst [module, in hydras.MoreAck.BadSubst]
+BadSubst [library]
+BadSubstF2 [module, in hydras.MoreAck.BadSubst]
+BadSubstF2.substF [definition, in hydras.MoreAck.BadSubst]
+BadSubst.substF [definition, in hydras.MoreAck.BadSubst]
+bad_term [definition, in hydras.Epsilon0.T1]
+Bad.correct_exp_too_strong [lemma, in additions.FirstSteps]
+Bad.correct_expt_function [definition, in additions.FirstSteps]
+Bad.CounterExample [section, in additions.FirstSteps]
+Bad.CounterExample.mul [variable, in additions.FirstSteps]
+Bad.CounterExample.one [variable, in additions.FirstSteps]
+Bad.fact [definition, in additions.Monoid_instances]
+Bad.Fchain_correct [definition, in additions.Euclidean_Chains]
+Bad.F3_correct [lemma, in additions.Euclidean_Chains]
+Bad.height_lt_size [lemma, in hydras.Hydra.Hydra_Examples]
+Bad.int63_from_nat [definition, in additions.Monoid_instances]
+Bad.mul_not_associative [lemma, in additions.FirstSteps]
+Bad.one_not_neutral [lemma, in additions.FirstSteps]
+Bad2 [module, in additions.Euclidean_Chains]
+Bad2.Fcompose_correct [lemma, in additions.Euclidean_Chains]
+Bad3 [module, in additions.Euclidean_Chains]
+Bad3.Fchain_proper_bad_prf [projection, in additions.Euclidean_Chains]
+Bad3.Fchain_proper [record, in additions.Euclidean_Chains]
+Bad3.Fchain_proper_bad_prf [constructor, in additions.Euclidean_Chains]
+Bad3.Fchain_proper [inductive, in additions.Euclidean_Chains]
+Bad3.Fcompose_proper [instance, in additions.Euclidean_Chains]
+Bad4 [module, in additions.Euclidean_Chains]
+Bad4.Fplus [definition, in additions.Euclidean_Chains]
+Bad4.F23 [definition, in additions.Euclidean_Chains]
+Bad4.F23_ok [lemma, in additions.Euclidean_Chains]
+bare_syntax [section, in hydras.MoreAck.LNN_Examples]
+battle [abbreviation, in hydras.Hydra.Hydra_Definitions]
+Battle [record, in hydras.Hydra.Hydra_Definitions]
+battle_length_notPR.m [variable, in hydras.Hydra.Hydra_Theorems]
+battle_length_notPR.h [variable, in hydras.Hydra.Hydra_Theorems]
+battle_length_notPR.alpha [variable, in hydras.Hydra.Hydra_Theorems]
+battle_length_notPR [section, in hydras.Hydra.Hydra_Theorems]
+battle_length_std_Hardy [lemma, in hydras.Hydra.Hydra_Theorems]
+battle_length_std [lemma, in hydras.Hydra.Battle_length]
+Battle_length.l [variable, in hydras.Hydra.Battle_length]
+Battle_length.Hk [variable, in hydras.Hydra.Battle_length]
+Battle_length.k [variable, in hydras.Hydra.Battle_length]
+Battle_length.Halpha [variable, in hydras.Hydra.Battle_length]
+Battle_length.alpha [variable, in hydras.Hydra.Battle_length]
+Battle_length [section, in hydras.Hydra.Battle_length]
+battle_example [lemma, in hydras.Hydra.KP_example]
+battle_length [definition, in hydras.Hydra.Hydra_Definitions]
+battle_ok [projection, in hydras.Hydra.Hydra_Definitions]
+battle_rel [projection, in hydras.Hydra.Hydra_Definitions]
+Battle_length [library]
+Bbeta_denum [lemma, in hydras.Schutte.Ordering_Functions]
+beta [definition, in Goedel.PRrepresentable]
+betaEquiv [lemma, in Goedel.PRrepresentable]
+betaFormula [definition, in Goedel.PRrepresentable]
+betaRepresentable [lemma, in Goedel.PRrepresentable]
+betaThm2 [lemma, in Goedel.PRrepresentable]
+betaThm3 [lemma, in Goedel.PRrepresentable]
+betaThm4 [lemma, in Goedel.PRrepresentable]
+betaThm5 [lemma, in Goedel.PRrepresentable]
+beta_def [lemma, in Goedel.PRrepresentable]
+beta_h [definition, in hydras.Hydra.Epsilon0_Needed_Generic]
+beta':101 [binder, in hydras.Epsilon0.T1]
+beta':105 [binder, in hydras.Epsilon0.T1]
+beta':119 [binder, in hydras.Schutte.Ordering_Functions]
+beta':138 [binder, in hydras.Gamma0.Gamma0]
+beta':195 [binder, in hydras.Epsilon0.T1]
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+beta':326 [binder, in hydras.Epsilon0.T1]
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+beta:5 [binder, in hydras.Epsilon0.E0]
+beta:50 [binder, in hydras.Schutte.AP]
+beta:50 [binder, in hydras.Epsilon0.L_alpha]
+beta:500 [binder, in hydras.Gamma0.Gamma0]
+beta:503 [binder, in hydras.Epsilon0.Paths]
+beta:504 [binder, in hydras.Gamma0.Gamma0]
+beta:507 [binder, in hydras.Epsilon0.Paths]
+beta:508 [binder, in hydras.Gamma0.Gamma0]
+beta:51 [binder, in hydras.Epsilon0.E0]
+beta:512 [binder, in hydras.Epsilon0.Paths]
+beta:512 [binder, in hydras.Gamma0.Gamma0]
+beta:515 [binder, in hydras.Gamma0.Gamma0]
+beta:518 [binder, in hydras.Epsilon0.Paths]
+beta:52 [binder, in hydras.Hydra.O2H]
+beta:52 [binder, in hydras.Schutte.AP]
+beta:52 [binder, in hydras.Epsilon0.Hprime]
+beta:52 [binder, in hydras.Schutte.Ordering_Functions]
+beta:525 [binder, in hydras.Epsilon0.Paths]
+beta:53 [binder, in hydras.Epsilon0.E0]
+beta:54 [binder, in hydras.Schutte.AP]
+beta:549 [binder, in hydras.Epsilon0.Paths]
+beta:55 [binder, in hydras.Epsilon0.E0]
+beta:553 [binder, in hydras.Epsilon0.Paths]
+beta:554 [binder, in hydras.Gamma0.Gamma0]
+beta:557 [binder, in hydras.Epsilon0.Paths]
+beta:56 [binder, in hydras.Epsilon0.Hessenberg]
+beta:56 [binder, in hydras.Schutte.AP]
+beta:56 [binder, in hydras.Epsilon0.Paths]
+beta:560 [binder, in hydras.Epsilon0.T1]
+beta:562 [binder, in hydras.Epsilon0.Paths]
+beta:565 [binder, in hydras.Epsilon0.Paths]
+beta:57 [binder, in hydras.Hydra.O2H]
+beta:57 [binder, in hydras.Schutte.Addition]
+beta:570 [binder, in hydras.Epsilon0.T1]
+beta:570 [binder, in hydras.Epsilon0.Paths]
+beta:577 [binder, in hydras.Epsilon0.Paths]
+beta:577 [binder, in hydras.Gamma0.Gamma0]
+beta:58 [binder, in hydras.Epsilon0.Hessenberg]
+beta:58 [binder, in hydras.Epsilon0.Canon]
+beta:58 [binder, in hydras.Epsilon0.E0]
+beta:582 [binder, in hydras.Gamma0.Gamma0]
+beta:583 [binder, in hydras.Epsilon0.T1]
+beta:583 [binder, in hydras.Epsilon0.Paths]
+beta:586 [binder, in hydras.Epsilon0.T1]
+beta:587 [binder, in hydras.Epsilon0.Paths]
+beta:589 [binder, in hydras.Epsilon0.T1]
+beta:59 [binder, in hydras.Epsilon0.Paths]
+beta:590 [binder, in hydras.Epsilon0.Paths]
+beta:6 [binder, in hydras.Schutte.CNF]
+beta:6 [binder, in gaia_hydras.ON_gfinite]
+beta:60 [binder, in hydras.Schutte.Addition]
+beta:601 [binder, in hydras.Gamma0.Gamma0]
+beta:61 [binder, in hydras.Schutte.AP]
+beta:62 [binder, in hydras.Epsilon0.Hessenberg]
+beta:62 [binder, in hydras.Schutte.CNF]
+beta:62 [binder, in hydras.Epsilon0.Canon]
+beta:622 [binder, in hydras.Gamma0.Gamma0]
+beta:624 [binder, in hydras.Gamma0.Gamma0]
+beta:626 [binder, in hydras.Gamma0.Gamma0]
+beta:63 [binder, in hydras.Epsilon0.Paths]
+beta:630 [binder, in hydras.Gamma0.Gamma0]
+beta:632 [binder, in hydras.Epsilon0.T1]
+beta:634 [binder, in hydras.Epsilon0.T1]
+beta:636 [binder, in hydras.Epsilon0.T1]
+beta:64 [binder, in hydras.Schutte.Correctness_E0]
+beta:64 [binder, in hydras.Epsilon0.Hessenberg]
+beta:641 [binder, in hydras.Gamma0.Gamma0]
+beta:644 [binder, in hydras.Epsilon0.T1]
+beta:645 [binder, in hydras.Epsilon0.T1]
+beta:648 [binder, in hydras.Epsilon0.T1]
+beta:65 [binder, in hydras.Schutte.AP]
+beta:65 [binder, in hydras.Schutte.Addition]
+beta:651 [binder, in hydras.Epsilon0.T1]
+beta:653 [binder, in hydras.Epsilon0.T1]
+beta:659 [binder, in hydras.Epsilon0.T1]
+beta:66 [binder, in hydras.Epsilon0.Paths]
+beta:661 [binder, in hydras.Gamma0.Gamma0]
+beta:662 [binder, in hydras.Gamma0.Gamma0]
+beta:663 [binder, in hydras.Epsilon0.T1]
+beta:666 [binder, in hydras.Epsilon0.T1]
+beta:668 [binder, in hydras.Epsilon0.T1]
+beta:67 [binder, in hydras.Schutte.AP]
+beta:670 [binder, in hydras.Epsilon0.T1]
+beta:672 [binder, in hydras.Epsilon0.T1]
+beta:675 [binder, in hydras.Epsilon0.T1]
+beta:679 [binder, in hydras.Epsilon0.T1]
+beta:679 [binder, in hydras.Gamma0.Gamma0]
+beta:682 [binder, in hydras.Epsilon0.T1]
+beta:682 [binder, in hydras.Gamma0.Gamma0]
+beta:684 [binder, in hydras.Epsilon0.T1]
+beta:687 [binder, in hydras.Gamma0.Gamma0]
+beta:69 [binder, in hydras.Epsilon0.Hessenberg]
+beta:7 [binder, in hydras.Schutte.Critical]
+beta:7 [binder, in hydras.Epsilon0.Large_Sets_Examples]
+beta:70 [binder, in hydras.Epsilon0.E0]
+beta:701 [binder, in hydras.Gamma0.Gamma0]
+beta:703 [binder, in hydras.Gamma0.Gamma0]
+beta:705 [binder, in hydras.Gamma0.Gamma0]
+beta:706 [binder, in hydras.Epsilon0.T1]
+beta:71 [binder, in hydras.OrdinalNotations.ON_Omega2]
+beta:71 [binder, in hydras.Epsilon0.F_alpha]
+beta:71 [binder, in hydras.Epsilon0.Hessenberg]
+beta:711 [binder, in hydras.Gamma0.Gamma0]
+beta:712 [binder, in hydras.Epsilon0.T1]
+beta:714 [binder, in hydras.Gamma0.Gamma0]
+beta:715 [binder, in hydras.Epsilon0.T1]
+beta:716 [binder, in hydras.Gamma0.Gamma0]
+beta:718 [binder, in hydras.Epsilon0.T1]
+beta:72 [binder, in hydras.Epsilon0.E0]
+beta:721 [binder, in hydras.Epsilon0.T1]
+beta:73 [binder, in hydras.Epsilon0.Canon]
+beta:74 [binder, in hydras.Epsilon0.E0]
+beta:75 [binder, in hydras.OrdinalNotations.ON_Generic]
+beta:76 [binder, in hydras.Epsilon0.F_alpha]
+beta:76 [binder, in hydras.Epsilon0.E0]
+beta:76 [binder, in hydras.Epsilon0.Paths]
+beta:78 [binder, in hydras.Epsilon0.Paths]
+beta:79 [binder, in hydras.OrdinalNotations.OmegaOmega]
+beta:79 [binder, in hydras.Epsilon0.E0]
+beta:8 [binder, in hydras.Schutte.Correctness_E0]
+beta:8 [binder, in hydras.Epsilon0.Paths]
+beta:8 [binder, in gaia_hydras.GHessenberg]
+beta:80 [binder, in hydras.Hydra.O2H]
+beta:81 [binder, in hydras.Epsilon0.F_alpha]
+beta:81 [binder, in hydras.Epsilon0.Hessenberg]
+beta:81 [binder, in hydras.OrdinalNotations.OmegaOmega]
+beta:81 [binder, in hydras.Epsilon0.E0]
+beta:83 [binder, in hydras.OrdinalNotations.OmegaOmega]
+beta:84 [binder, in hydras.Epsilon0.E0]
+beta:85 [binder, in hydras.Epsilon0.F_alpha]
+beta:85 [binder, in hydras.OrdinalNotations.OmegaOmega]
+beta:86 [binder, in hydras.Epsilon0.Canon]
+beta:86 [binder, in hydras.Schutte.Schutte_basics]
+beta:87 [binder, in hydras.Epsilon0.E0]
+beta:88 [binder, in hydras.Epsilon0.Paths]
+beta:89 [binder, in hydras.Hydra.O2H]
+beta:9 [binder, in hydras.Schutte.AP]
+beta:9 [binder, in gaia_hydras.ON_gfinite]
+beta:9 [binder, in hydras.OrdinalNotations.ON_Finite]
+beta:90 [binder, in hydras.Epsilon0.E0]
+beta:90 [binder, in hydras.Epsilon0.Paths]
+beta:93 [binder, in hydras.OrdinalNotations.OmegaOmega]
+beta:93 [binder, in hydras.Epsilon0.Canon]
+beta:93 [binder, in hydras.Epsilon0.E0]
+beta:93 [binder, in hydras.Epsilon0.Hprime]
+beta:95 [binder, in hydras.Epsilon0.T1]
+beta:95 [binder, in hydras.OrdinalNotations.OmegaOmega]
+beta:95 [binder, in hydras.Schutte.Schutte_basics]
+beta:96 [binder, in hydras.Epsilon0.F_alpha]
+beta:97 [binder, in hydras.Gamma0.T2]
+beta:99 [binder, in hydras.OrdinalNotations.OmegaOmega]
+beta:99 [binder, in hydras.Epsilon0.E0]
+beta:99 [binder, in hydras.Epsilon0.Paths]
+bigarg [definition, in additions.fib]
+BigBattle [library]
+bigO [definition, in hydras.OrdinalNotations.ON_Generic]
+Bigstep [lemma, in hydras.Hydra.BigBattle]
+big_to_small [lemma, in hydras.Hydra.Epsilon0_Needed_Free]
+big_h [definition, in hydras.Hydra.Epsilon0_Needed_Generic]
+big_to_small [lemma, in hydras.Hydra.Omega2_Small]
+big_to_small [lemma, in hydras.Hydra.Omega_Small]
+big0 [definition, in hydras.Schutte.Schutte_basics]
+BinaryStrat [library]
+binary_power_mult [definition, in additions.FirstSteps]
+binary_power_mult_ok [lemma, in additions.Pow_variant]
+binary_power_mult [definition, in additions.Pow_variant]
+binary_power_mult_ok [lemma, in additions.Pow]
+binary_power_mult [definition, in additions.Pow]
+binary_power_mult [definition, in additions.fib]
+binary_chain_length [lemma, in additions.Addition_Chains]
+binary_generator_not_optimal [lemma, in additions.Addition_Chains]
+binary_generator_correct [lemma, in additions.Addition_Chains]
+binary_correct [lemma, in additions.Addition_Chains]
+binary_power_proof.E_eq [variable, in additions.Addition_Chains]
+binary_power_proof.E_one [variable, in additions.Addition_Chains]
+binary_power_proof.E_op [variable, in additions.Addition_Chains]
+binary_power_proof.A [variable, in additions.Addition_Chains]
+binary_power_proof [section, in additions.Addition_Chains]
+binary_chain [definition, in additions.Addition_Chains]
+Binary_strat [instance, in additions.BinaryStrat]
+bin_pow_scheme_length1 [lemma, in additions.Addition_Chains]
+bin_pow_scheme [definition, in additions.Addition_Chains]
+bits [definition, in additions.fib]
+bitsP [lemma, in additions.fib]
+bits_cat [lemma, in additions.fib]
+bodd [definition, in hydras.Ackermann.primRec]
+boddIsPR [instance, in hydras.Ackermann.primRec]
+bool_eq_iff [lemma, in hydras.Epsilon0.T1]
+bool_and_binop [instance, in additions.Monoid_def]
+bool_decide_eq_false [lemma, in hydras.Prelude.STDPP_compat]
+bool_decide_eq_true [lemma, in hydras.Prelude.STDPP_compat]
+bool_decide [definition, in hydras.Prelude.STDPP_compat]
+boundComputation [definition, in hydras.Ackermann.codeSubFormula]
+boundComputationIsPR [instance, in hydras.Ackermann.codeSubFormula]
+boundComputationMonotone [lemma, in hydras.Ackermann.codeSubFormula]
+Bounded [section, in hydras.Hydra.Epsilon0_Needed_Generic]
+boundedCheck [lemma, in hydras.Prelude.MoreDecidable]
+boundedLT [lemma, in hydras.Ackermann.NNtheory]
+boundedSearch [definition, in hydras.Ackermann.primRec]
+boundedSearchHelp [definition, in hydras.Ackermann.primRec]
+boundedSearch1 [lemma, in hydras.Ackermann.primRec]
+boundedSearch2 [lemma, in hydras.Ackermann.primRec]
+boundedSearch3 [lemma, in hydras.solutions_exercises.isqrt]
+boundedSearch4 [lemma, in hydras.solutions_exercises.isqrt]
+BoundedVariant [record, in hydras.Hydra.Hydra_Definitions]
+bounded_by [definition, in hydras.Prelude.MoreLists]
+bounded_transitionS [definition, in gaia_hydras.GPaths]
+bounded_transitionS [definition, in hydras.Epsilon0.Paths]
+Bounded.m_decrease [variable, in hydras.Hydra.Epsilon0_Needed_Generic]
+boundMakeTrace [lemma, in hydras.Ackermann.codeSubFormula]
+boundSearchIsPR [instance, in hydras.Ackermann.primRec]
+boundSubFormulaHelp [lemma, in hydras.Ackermann.codeSubFormula]
+boundSubFormulaHelp1 [lemma, in hydras.Ackermann.codeSubFormula]
+boundSubFormulaHelp2 [lemma, in hydras.Ackermann.codeSubFormula]
+boundSubTerm [lemma, in hydras.Ackermann.codeSubFormula]
+boundSubTerms [lemma, in hydras.Ackermann.codeSubFormula]
+box [definition, in Goedel.goedel2]
+building_ordering_function_by_induction.B [variable, in hydras.Schutte.Ordering_Functions]
+building_ordering_function_by_induction [section, in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1.g_1 [variable, in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1.beta_fixed.beta_B [variable, in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1.beta_fixed.beta [variable, in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1.beta_fixed [section, in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1.H_B [variable, in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1.B [variable, in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1 [section, in hydras.Schutte.Ordering_Functions]
+Build_OF [constructor, in hydras.Schutte.Ordering_Functions]
+but_last_iota_from' [lemma, in hydras.Prelude.MoreLists]
+but_last_app [lemma, in hydras.Prelude.MoreLists]
+but_last_shift' [lemma, in hydras.Prelude.MoreLists]
+but_last_interval [lemma, in hydras.Prelude.MoreLists]
+but_last_iota_from [lemma, in hydras.Prelude.MoreLists]
+but_last [definition, in hydras.Prelude.MoreLists]
+bVar [instance, in gaia_hydras.GHydra]
+BVar:28 [binder, in gaia_hydras.GHydra]
+B_termination [definition, in hydras.Hydra.Hydra_Definitions]
+b'':45 [binder, in hydras.Hydra.BigBattle]
+b':1023 [binder, in gaia_hydras.nfwfgaia]
+b':1151 [binder, in gaia_hydras.nfwfgaia]
+b':118 [binder, in hydras.Epsilon0.T1]
+b':1280 [binder, in gaia_hydras.nfwfgaia]
+b':1285 [binder, in gaia_hydras.nfwfgaia]
+b':1288 [binder, in gaia_hydras.nfwfgaia]
+b':1291 [binder, in gaia_hydras.nfwfgaia]
+b':1294 [binder, in gaia_hydras.nfwfgaia]
+b':1298 [binder, in gaia_hydras.nfwfgaia]
+b':13 [binder, in hydras.Prelude.Simple_LexProd]
+b':1302 [binder, in gaia_hydras.nfwfgaia]
+b':1306 [binder, in gaia_hydras.nfwfgaia]
+b':1310 [binder, in gaia_hydras.nfwfgaia]
+b':1316 [binder, in gaia_hydras.nfwfgaia]
+b':132 [binder, in gaia_hydras.nfwfgaia]
+b':1336 [binder, in gaia_hydras.nfwfgaia]
+b':1382 [binder, in gaia_hydras.nfwfgaia]
+b':1392 [binder, in gaia_hydras.nfwfgaia]
+b':1420 [binder, in gaia_hydras.nfwfgaia]
+b':1426 [binder, in gaia_hydras.nfwfgaia]
+b':1434 [binder, in gaia_hydras.nfwfgaia]
+b':1498 [binder, in gaia_hydras.nfwfgaia]
+b':1506 [binder, in gaia_hydras.nfwfgaia]
+b':1549 [binder, in gaia_hydras.nfwfgaia]
+b':1583 [binder, in gaia_hydras.nfwfgaia]
+b':16 [binder, in hydras.Prelude.Simple_LexProd]
+b':1607 [binder, in gaia_hydras.nfwfgaia]
+b':1616 [binder, in gaia_hydras.nfwfgaia]
+b':1618 [binder, in gaia_hydras.nfwfgaia]
+b':1625 [binder, in gaia_hydras.nfwfgaia]
+b':1627 [binder, in gaia_hydras.nfwfgaia]
+b':1633 [binder, in gaia_hydras.nfwfgaia]
+b':1689 [binder, in gaia_hydras.nfwfgaia]
+b':1695 [binder, in gaia_hydras.nfwfgaia]
+b':175 [binder, in gaia_hydras.nfwfgaia]
+b':177 [binder, in hydras.Epsilon0.T1]
+b':185 [binder, in hydras.Epsilon0.T1]
+b':20 [binder, in hydras.Epsilon0.Hessenberg]
+b':2064 [binder, in gaia_hydras.nfwfgaia]
+b':2078 [binder, in gaia_hydras.nfwfgaia]
+b':2083 [binder, in gaia_hydras.nfwfgaia]
+b':224 [binder, in hydras.Epsilon0.T1]
+b':226 [binder, in gaia_hydras.nfwfgaia]
+b':232 [binder, in gaia_hydras.nfwfgaia]
+b':233 [binder, in hydras.Epsilon0.T1]
+b':24 [binder, in hydras.Hydra.BigBattle]
+b':247 [binder, in hydras.Epsilon0.T1]
+b':25 [binder, in additions.Monoid_instances]
+b':259 [binder, in gaia_hydras.nfwfgaia]
+b':26 [binder, in hydras.Schutte.Correctness_E0]
+b':26 [binder, in hydras.Epsilon0.Hessenberg]
+b':286 [binder, in hydras.Epsilon0.T1]
+b':295 [binder, in gaia_hydras.nfwfgaia]
+b':30 [binder, in hydras.Schutte.PartialFun]
+b':302 [binder, in hydras.Gamma0.Gamma0]
+b':31 [binder, in hydras.Schutte.Correctness_E0]
+b':31 [binder, in hydras.Epsilon0.Hessenberg]
+b':33 [binder, in hydras.Hydra.BigBattle]
+b':34 [binder, in hydras.Gamma0.Gamma0]
+b':35 [binder, in hydras.Epsilon0.T1]
+b':360 [binder, in hydras.Epsilon0.T1]
+b':365 [binder, in hydras.Epsilon0.T1]
+b':37 [binder, in hydras.Epsilon0.Hessenberg]
+b':379 [binder, in hydras.Epsilon0.T1]
+b':384 [binder, in hydras.Epsilon0.T1]
+b':389 [binder, in hydras.Epsilon0.T1]
+b':41 [binder, in hydras.Hydra.BigBattle]
+b':426 [binder, in gaia_hydras.nfwfgaia]
+b':43 [binder, in hydras.Epsilon0.Hessenberg]
+b':430 [binder, in hydras.Epsilon0.T1]
+b':431 [binder, in gaia_hydras.nfwfgaia]
+b':435 [binder, in hydras.Epsilon0.T1]
+b':441 [binder, in hydras.Epsilon0.T1]
+b':46 [binder, in hydras.Prelude.DecPreOrder_Instances]
+b':491 [binder, in hydras.Gamma0.Gamma0]
+b':50 [binder, in hydras.Prelude.DecPreOrder_Instances]
+b':51 [binder, in hydras.Epsilon0.Epsilon0rpo]
+b':53 [binder, in hydras.Epsilon0.Hessenberg]
+b':536 [binder, in hydras.Epsilon0.T1]
+b':539 [binder, in hydras.Ackermann.primRec]
+b':54 [binder, in hydras.Epsilon0.T1]
+b':54 [binder, in hydras.Prelude.DecPreOrder_Instances]
+b':55 [binder, in gaia_hydras.T1Bridge]
+b':57 [binder, in hydras.Epsilon0.Epsilon0rpo]
+b':57 [binder, in hydras.Hydra.BigBattle]
+b':58 [binder, in hydras.Prelude.DecPreOrder_Instances]
+b':58 [binder, in hydras.Prelude.Sort_spec]
+b':59 [binder, in hydras.OrdinalNotations.OmegaOmega]
+b':60 [binder, in gaia_hydras.T1Bridge]
+b':62 [binder, in hydras.Prelude.DecPreOrder_Instances]
+b':65 [binder, in gaia_hydras.T1Bridge]
+b':69 [binder, in hydras.Epsilon0.T1]
+b':71 [binder, in gaia_hydras.T1Bridge]
+b':727 [binder, in hydras.Epsilon0.T1]
+b':76 [binder, in hydras.Epsilon0.T1]
+b':77 [binder, in gaia_hydras.T1Bridge]
+b':791 [binder, in gaia_hydras.nfwfgaia]
+b':81 [binder, in hydras.Epsilon0.T1]
+b':82 [binder, in hydras.Gamma0.T2]
+b':847 [binder, in gaia_hydras.nfwfgaia]
+b':86 [binder, in hydras.Epsilon0.T1]
+b':870 [binder, in gaia_hydras.nfwfgaia]
+b':897 [binder, in gaia_hydras.nfwfgaia]
+b':9 [binder, in hydras.Schutte.PartialFun]
+b':90 [binder, in hydras.Epsilon0.T1]
+b':901 [binder, in gaia_hydras.nfwfgaia]
+b':907 [binder, in gaia_hydras.nfwfgaia]
+b':915 [binder, in gaia_hydras.nfwfgaia]
+b':943 [binder, in gaia_hydras.nfwfgaia]
+b':991 [binder, in gaia_hydras.nfwfgaia]
+b0:193 [binder, in hydras.Ackermann.cPair]
+b0:197 [binder, in hydras.Ackermann.cPair]
+b0:201 [binder, in hydras.Ackermann.cPair]
+b0:239 [binder, in hydras.Ackermann.cPair]
+b0:241 [binder, in hydras.Ackermann.cPair]
+b0:302 [binder, in hydras.Ackermann.folProp]
+b0:309 [binder, in hydras.Ackermann.folProp]
+b0:318 [binder, in hydras.Ackermann.folProp]
+b0:325 [binder, in hydras.Ackermann.folProp]
+b0:334 [binder, in hydras.Ackermann.folProp]
+b0:341 [binder, in hydras.Ackermann.folProp]
+b0:350 [binder, in hydras.Ackermann.folProp]
+b0:357 [binder, in hydras.Ackermann.folProp]
+b0:45 [binder, in hydras.Ackermann.LNN2LNT]
+b0:718 [binder, in hydras.Ackermann.primRec]
+b0:722 [binder, in hydras.Ackermann.primRec]
+b0:724 [binder, in hydras.Ackermann.primRec]
+b0:727 [binder, in hydras.Ackermann.primRec]
+b0:730 [binder, in hydras.Ackermann.primRec]
+b0:733 [binder, in hydras.Ackermann.primRec]
+b0:736 [binder, in hydras.Ackermann.primRec]
+b0:739 [binder, in hydras.Ackermann.primRec]
+b0:742 [binder, in hydras.Ackermann.primRec]
+b0:745 [binder, in hydras.Ackermann.primRec]
+b0:748 [binder, in hydras.Ackermann.primRec]
+b0:751 [binder, in hydras.Ackermann.primRec]
+b0:754 [binder, in hydras.Ackermann.primRec]
+b0:756 [binder, in hydras.Ackermann.primRec]
+b0:758 [binder, in hydras.Ackermann.primRec]
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+b:157 [binder, in hydras.Ackermann.codePA]
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+b:1599 [binder, in gaia_hydras.nfwfgaia]
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+B:16 [binder, in hydras.MoreAck.FolExamples]
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+b:31 [binder, in hydras.OrdinalNotations.OmegaOmega]
+B:31 [binder, in Goedel.rosserPA]
+b:31 [binder, in hydras.Schutte.PartialFun]
+b:31 [binder, in hydras.Ackermann.NN]
+b:31 [binder, in gaia_hydras.GHessenberg]
+b:311 [binder, in hydras.Epsilon0.T1]
+b:311 [binder, in Goedel.PRrepresentable]
+b:312 [binder, in hydras.Ackermann.primRec]
+b:316 [binder, in hydras.Ackermann.primRec]
+b:319 [binder, in Goedel.PRrepresentable]
+B:319 [binder, in hydras.Ackermann.checkPrf]
+b:319 [binder, in hydras.Ackermann.fol]
+b:32 [binder, in hydras.Epsilon0.T1]
+b:32 [binder, in hydras.Ackermann.codeSubTerm]
+B:32 [binder, in hydras.rpo.more_list]
+b:32 [binder, in hydras.Ackermann.folLogic3]
+b:320 [binder, in hydras.Ackermann.primRec]
+b:323 [binder, in hydras.Ackermann.fol]
+b:325 [binder, in hydras.Ackermann.primRec]
+b:326 [binder, in hydras.Ackermann.fol]
+B:328 [binder, in Goedel.PRrepresentable]
+b:329 [binder, in hydras.Ackermann.primRec]
+B:33 [binder, in hydras.Ackermann.folProof]
+b:33 [binder, in hydras.Ackermann.codeSubTerm]
+b:33 [binder, in hydras.Prelude.Comparable]
+b:33 [binder, in hydras.OrdinalNotations.OmegaOmega]
+B:33 [binder, in hydras.OrdinalNotations.ON_Generic]
+B:33 [binder, in hydras.Schutte.Ordering_Functions]
+b:331 [binder, in hydras.Ackermann.fol]
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+b:333 [binder, in hydras.Ackermann.primRec]
+b:337 [binder, in hydras.Ackermann.primRec]
+B:338 [binder, in Goedel.PRrepresentable]
+b:339 [binder, in hydras.Ackermann.cPair]
+b:34 [binder, in hydras.Prelude.More_Arith]
+b:34 [binder, in hydras.Ackermann.codeSubTerm]
+b:34 [binder, in hydras.Epsilon0.Hessenberg]
+B:34 [binder, in Goedel.rosserPA]
+b:34 [binder, in hydras.Ackermann.cPair]
+b:34 [binder, in gaia_hydras.GHessenberg]
+b:341 [binder, in hydras.Ackermann.primRec]
+b:341 [binder, in hydras.Ackermann.cPair]
+B:342 [binder, in hydras.Ackermann.subAll]
+B:347 [binder, in hydras.Ackermann.subAll]
+b:347 [binder, in hydras.Ackermann.cPair]
+b:348 [binder, in hydras.Epsilon0.T1]
+b:35 [binder, in hydras.Ackermann.codeSubTerm]
+b:35 [binder, in gaia_hydras.GCanon]
+b:35 [binder, in hydras.OrdinalNotations.OmegaOmega]
+b:35 [binder, in hydras.Schutte.PartialFun]
+b:35 [binder, in hydras.Schutte.Well_Orders]
+b:35 [binder, in hydras.Prelude.MoreVectors]
+b:350 [binder, in hydras.Ackermann.cPair]
+b:352 [binder, in Goedel.PRrepresentable]
+B:352 [binder, in hydras.Ackermann.subAll]
+b:353 [binder, in hydras.Ackermann.cPair]
+b:354 [binder, in gaia_hydras.nfwfgaia]
+b:356 [binder, in Goedel.PRrepresentable]
+b:356 [binder, in hydras.Ackermann.cPair]
+b:356 [binder, in gaia_hydras.nfwfgaia]
+b:357 [binder, in hydras.Epsilon0.T1]
+b:358 [binder, in hydras.Ackermann.cPair]
+b:358 [binder, in gaia_hydras.nfwfgaia]
+b:36 [binder, in additions.Compatibility]
+b:36 [binder, in hydras.Prelude.Comparable]
+b:36 [binder, in hydras.Ackermann.folProp]
+b:36 [binder, in hydras.Ackermann.PA]
+b:36 [binder, in hydras.Schutte.Schutte_basics]
+B:36 [binder, in hydras.MoreAck.FolExamples]
+b:360 [binder, in Goedel.PRrepresentable]
+b:360 [binder, in hydras.Ackermann.cPair]
+b:360 [binder, in gaia_hydras.nfwfgaia]
+b:362 [binder, in hydras.Epsilon0.T1]
+b:363 [binder, in hydras.Ackermann.cPair]
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+b:364 [binder, in gaia_hydras.nfwfgaia]
+b:365 [binder, in hydras.Ackermann.fol]
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+b:369 [binder, in hydras.Ackermann.cPair]
+b:369 [binder, in hydras.Ackermann.fol]
+b:37 [binder, in gaia_hydras.GPaths]
+b:37 [binder, in hydras.Hydra.BigBattle]
+b:37 [binder, in gaia_hydras.GHessenberg]
+b:372 [binder, in hydras.Ackermann.fol]
+b:376 [binder, in hydras.Epsilon0.T1]
+b:38 [binder, in hydras.OrdinalNotations.OmegaOmega]
+b:38 [binder, in hydras.Ackermann.cPair]
+b:38 [binder, in hydras.Ackermann.wellFormed]
+b:38 [binder, in gaia_hydras.onType]
+b:38 [binder, in hydras.Schutte.Well_Orders]
+b:383 [binder, in hydras.Epsilon0.T1]
+b:386 [binder, in Goedel.PRrepresentable]
+b:387 [binder, in hydras.Epsilon0.T1]
+b:39 [binder, in hydras.Prelude.Comparable]
+b:39 [binder, in hydras.Ackermann.NN]
+b:39 [binder, in hydras.Schutte.Ordering_Functions]
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+b:392 [binder, in hydras.Epsilon0.T1]
+b:394 [binder, in Goedel.PRrepresentable]
+b:394 [binder, in hydras.Ackermann.checkPrf]
+b:395 [binder, in gaia_hydras.nfwfgaia]
+b:397 [binder, in gaia_hydras.nfwfgaia]
+b:398 [binder, in Goedel.PRrepresentable]
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+b:4 [binder, in hydras.Ackermann.folLogic2]
+b:4 [binder, in hydras.Ackermann.prLogic]
+b:4 [binder, in hydras.Ackermann.NNtheory]
+b:4 [binder, in hydras.Schutte.Well_Orders]
+b:4 [binder, in hydras.Ackermann.NN]
+b:4 [binder, in hydras.solutions_exercises.isqrt]
+B:40 [binder, in hydras.Ackermann.folProof]
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+b:40 [binder, in hydras.Ackermann.cPair]
+b:40 [binder, in hydras.Ackermann.wellFormed]
+b:40 [binder, in gaia_hydras.onType]
+b:40 [binder, in hydras.Schutte.Ordering_Functions]
+b:400 [binder, in gaia_hydras.nfwfgaia]
+b:401 [binder, in Goedel.PRrepresentable]
+b:403 [binder, in Goedel.PRrepresentable]
+b:404 [binder, in hydras.Epsilon0.T1]
+b:406 [binder, in Goedel.PRrepresentable]
+b:409 [binder, in hydras.Ackermann.fol]
+b:409 [binder, in gaia_hydras.nfwfgaia]
+b:41 [binder, in hydras.Epsilon0.Epsilon0rpo]
+b:41 [binder, in hydras.Prelude.Comparable]
+b:41 [binder, in hydras.Schutte.PartialFun]
+b:410 [binder, in Goedel.PRrepresentable]
+b:412 [binder, in gaia_hydras.nfwfgaia]
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+b:415 [binder, in hydras.Ackermann.primRec]
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+b:416 [binder, in gaia_hydras.nfwfgaia]
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+b:419 [binder, in hydras.Ackermann.primRec]
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+b:42 [binder, in hydras.Schutte.Schutte_basics]
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+b:424 [binder, in gaia_hydras.nfwfgaia]
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+b:428 [binder, in gaia_hydras.nfwfgaia]
+b:429 [binder, in hydras.Ackermann.fol]
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+b:43 [binder, in additions.More_on_positive]
+B:43 [binder, in hydras.Schutte.Schutte_basics]
+b:430 [binder, in hydras.Ackermann.primRec]
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+b:432 [binder, in hydras.Ackermann.fol]
+b:433 [binder, in hydras.Epsilon0.T1]
+b:433 [binder, in gaia_hydras.nfwfgaia]
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+b:437 [binder, in hydras.Ackermann.primRec]
+b:438 [binder, in hydras.Epsilon0.T1]
+b:438 [binder, in gaia_hydras.nfwfgaia]
+b:44 [binder, in hydras.Epsilon0.T1]
+b:44 [binder, in hydras.Epsilon0.Epsilon0rpo]
+b:44 [binder, in hydras.rpo.more_list]
+b:44 [binder, in gaia_hydras.T1Bridge]
+b:44 [binder, in hydras.rpo.dickson]
+b:442 [binder, in gaia_hydras.nfwfgaia]
+b:443 [binder, in hydras.Epsilon0.T1]
+b:444 [binder, in hydras.Ackermann.primRec]
+b:444 [binder, in gaia_hydras.nfwfgaia]
+b:445 [binder, in hydras.Ackermann.fol]
+b:446 [binder, in hydras.Ackermann.checkPrf]
+b:446 [binder, in hydras.Ackermann.primRec]
+b:446 [binder, in gaia_hydras.nfwfgaia]
+b:447 [binder, in hydras.Epsilon0.T1]
+b:448 [binder, in gaia_hydras.nfwfgaia]
+b:449 [binder, in hydras.Epsilon0.T1]
+b:45 [binder, in Goedel.PRrepresentable]
+b:45 [binder, in hydras.Prelude.DecPreOrder_Instances]
+b:45 [binder, in hydras.Prelude.Comparable]
+b:45 [binder, in hydras.Epsilon0.Hessenberg]
+b:45 [binder, in hydras.Ackermann.PA]
+b:45 [binder, in hydras.Prelude.MoreOrders]
+b:45 [binder, in hydras.Schutte.PartialFun]
+b:450 [binder, in hydras.Ackermann.checkPrf]
+b:451 [binder, in hydras.Ackermann.primRec]
+b:451 [binder, in gaia_hydras.nfwfgaia]
+b:452 [binder, in hydras.Epsilon0.T1]
+b:453 [binder, in gaia_hydras.nfwfgaia]
+B:454 [binder, in hydras.Ackermann.checkPrf]
+b:455 [binder, in hydras.Epsilon0.T1]
+b:457 [binder, in hydras.Ackermann.primRec]
+b:458 [binder, in gaia_hydras.nfwfgaia]
+b:459 [binder, in hydras.Epsilon0.T1]
+b:459 [binder, in hydras.Ackermann.primRec]
+b:46 [binder, in hydras.solutions_exercises.MultisetWf]
+b:46 [binder, in gaia_hydras.T1Bridge]
+b:46 [binder, in hydras.Ackermann.cPair]
+B:46 [binder, in hydras.Schutte.Ordering_Functions]
+B:46 [binder, in hydras.Ackermann.expressible]
+b:460 [binder, in gaia_hydras.nfwfgaia]
+b:462 [binder, in hydras.Epsilon0.T1]
+b:463 [binder, in gaia_hydras.nfwfgaia]
+b:464 [binder, in hydras.Ackermann.primRec]
+b:465 [binder, in hydras.Epsilon0.T1]
+b:465 [binder, in gaia_hydras.nfwfgaia]
+b:467 [binder, in gaia_hydras.nfwfgaia]
+b:469 [binder, in hydras.Epsilon0.T1]
+b:469 [binder, in gaia_hydras.nfwfgaia]
+b:47 [binder, in hydras.Epsilon0.T1]
+b:47 [binder, in hydras.Epsilon0.Epsilon0rpo]
+b:47 [binder, in hydras.Prelude.Comparable]
+B:47 [binder, in hydras.Prelude.Sort_spec]
+b:47 [binder, in additions.More_on_positive]
+B:47 [binder, in hydras.Schutte.Schutte_basics]
+b:471 [binder, in hydras.Epsilon0.T1]
+b:474 [binder, in gaia_hydras.nfwfgaia]
+b:477 [binder, in hydras.Epsilon0.T1]
+b:478 [binder, in hydras.Ackermann.primRec]
+b:479 [binder, in hydras.Epsilon0.T1]
+b:479 [binder, in hydras.Ackermann.primRec]
+b:48 [binder, in Goedel.PRrepresentable]
+b:48 [binder, in hydras.Prelude.More_Arith]
+b:48 [binder, in hydras.Ackermann.codeSubTerm]
+b:48 [binder, in hydras.Prelude.DecPreOrder]
+b:483 [binder, in hydras.Gamma0.Gamma0]
+b:484 [binder, in hydras.Ackermann.primRec]
+b:487 [binder, in hydras.Ackermann.checkPrf]
+b:49 [binder, in hydras.Epsilon0.T1]
+b:49 [binder, in hydras.Prelude.DecPreOrder_Instances]
+b:49 [binder, in hydras.Ackermann.codeSubTerm]
+b:49 [binder, in hydras.Prelude.Comparable]
+b:49 [binder, in hydras.Ackermann.code]
+b:49 [binder, in hydras.Hydra.BigBattle]
+b:490 [binder, in hydras.Ackermann.primRec]
+b:490 [binder, in hydras.Gamma0.Gamma0]
+b:491 [binder, in hydras.Ackermann.checkPrf]
+b:491 [binder, in gaia_hydras.nfwfgaia]
+b:493 [binder, in gaia_hydras.nfwfgaia]
+b:495 [binder, in hydras.Ackermann.primRec]
+b:495 [binder, in gaia_hydras.nfwfgaia]
+b:497 [binder, in hydras.Epsilon0.T1]
+b:5 [binder, in hydras.Prelude.Restriction]
+b:5 [binder, in hydras.Prelude.DecPreOrder]
+b:5 [binder, in hydras.Ackermann.cPair]
+B:5 [binder, in hydras.OrdinalNotations.ON_mult]
+B:5 [binder, in hydras.OrdinalNotations.ON_plus]
+b:50 [binder, in hydras.Ackermann.codeSubTerm]
+b:50 [binder, in hydras.Epsilon0.Hessenberg]
+b:50 [binder, in gaia_hydras.T1Bridge]
+b:50 [binder, in hydras.Ackermann.cPair]
+b:50 [binder, in hydras.Schutte.Well_Orders]
+B:50 [binder, in hydras.Schutte.Ordering_Functions]
+b:501 [binder, in hydras.Epsilon0.T1]
+b:503 [binder, in hydras.Ackermann.checkPrf]
+b:504 [binder, in hydras.Epsilon0.T1]
+b:504 [binder, in hydras.Ackermann.primRec]
+b:506 [binder, in hydras.Ackermann.primRec]
+b:507 [binder, in hydras.Ackermann.checkPrf]
+b:508 [binder, in hydras.Ackermann.primRec]
+b:51 [binder, in hydras.Epsilon0.T1]
+b:51 [binder, in hydras.Ackermann.codeSubTerm]
+b:51 [binder, in additions.More_on_positive]
+b:51 [binder, in hydras.OrdinalNotations.OmegaOmega]
+B:51 [binder, in hydras.Ackermann.code]
+b:51 [binder, in hydras.Schutte.PartialFun]
+b:51 [binder, in hydras.Ackermann.codeList]
+b:51 [binder, in hydras.rpo.dickson]
+b:510 [binder, in hydras.Epsilon0.T1]
+b:510 [binder, in hydras.Ackermann.primRec]
+b:511 [binder, in hydras.Ackermann.checkPrf]
+b:512 [binder, in hydras.Ackermann.primRec]
+b:514 [binder, in hydras.Ackermann.primRec]
+b:515 [binder, in hydras.Ackermann.checkPrf]
+b:516 [binder, in hydras.Ackermann.primRec]
+b:518 [binder, in hydras.Ackermann.primRec]
+b:519 [binder, in hydras.Epsilon0.T1]
+b:519 [binder, in hydras.Gamma0.Gamma0]
+b:52 [binder, in hydras.Prelude.More_Arith]
+b:52 [binder, in hydras.Prelude.Comparable]
+B:52 [binder, in hydras.OrdinalNotations.ON_Generic]
+b:52 [binder, in hydras.Ackermann.codeList]
+b:520 [binder, in hydras.Ackermann.primRec]
+b:522 [binder, in hydras.Epsilon0.T1]
+b:522 [binder, in hydras.Ackermann.primRec]
+b:524 [binder, in hydras.Ackermann.primRec]
+b:525 [binder, in hydras.Gamma0.Gamma0]
+b:526 [binder, in hydras.Ackermann.primRec]
+b:528 [binder, in hydras.Ackermann.primRec]
+b:528 [binder, in hydras.Gamma0.Gamma0]
+b:53 [binder, in hydras.Prelude.DecPreOrder_Instances]
+b:53 [binder, in hydras.Prelude.More_Arith]
+b:53 [binder, in hydras.Prelude.DecPreOrder]
+b:53 [binder, in hydras.solutions_exercises.MultisetWf]
+b:53 [binder, in gaia_hydras.T1Bridge]
+b:53 [binder, in hydras.Hydra.BigBattle]
+b:530 [binder, in hydras.Epsilon0.T1]
+b:530 [binder, in hydras.Ackermann.primRec]
+b:531 [binder, in hydras.Gamma0.Gamma0]
+b:533 [binder, in hydras.Epsilon0.T1]
+b:534 [binder, in hydras.Gamma0.Gamma0]
+b:537 [binder, in hydras.Ackermann.primRec]
+b:54 [binder, in additions.More_on_positive]
+b:54 [binder, in gaia_hydras.GCanon]
+b:54 [binder, in hydras.OrdinalNotations.OmegaOmega]
+b:54 [binder, in hydras.Ackermann.cPair]
+B:54 [binder, in hydras.Prelude.MoreVectors]
+b:540 [binder, in hydras.Epsilon0.T1]
+b:549 [binder, in hydras.Ackermann.primRec]
+b:55 [binder, in hydras.Ackermann.LNN2LNT]
+b:55 [binder, in hydras.Prelude.Comparable]
+b:55 [binder, in gaia_hydras.GPaths]
+b:55 [binder, in hydras.Prelude.Sort_spec]
+b:55 [binder, in hydras.Ackermann.code]
+B:55 [binder, in hydras.Schutte.Ordering_Functions]
+b:551 [binder, in hydras.Ackermann.primRec]
+b:553 [binder, in hydras.Ackermann.primRec]
+b:555 [binder, in hydras.Ackermann.primRec]
+b:557 [binder, in hydras.Epsilon0.T1]
+b:56 [binder, in hydras.Epsilon0.Epsilon0rpo]
+b:564 [binder, in hydras.Epsilon0.T1]
+b:564 [binder, in hydras.Gamma0.Gamma0]
+b:565 [binder, in gaia_hydras.nfwfgaia]
+b:566 [binder, in hydras.Gamma0.Gamma0]
+b:567 [binder, in gaia_hydras.nfwfgaia]
+b:57 [binder, in hydras.Prelude.DecPreOrder_Instances]
+b:57 [binder, in hydras.Prelude.More_Arith]
+b:57 [binder, in hydras.Ackermann.LNN2LNT]
+b:57 [binder, in hydras.Ackermann.code]
+b:57 [binder, in hydras.Schutte.Ordering_Functions]
+b:570 [binder, in hydras.Gamma0.Gamma0]
+b:572 [binder, in hydras.Epsilon0.T1]
+b:572 [binder, in hydras.Gamma0.Gamma0]
+b:576 [binder, in hydras.Epsilon0.T1]
+b:58 [binder, in hydras.Prelude.Comparable]
+b:58 [binder, in hydras.Prelude.DecPreOrder]
+b:58 [binder, in additions.More_on_positive]
+b:58 [binder, in gaia_hydras.T1Bridge]
+b:58 [binder, in hydras.Prelude.MoreVectors]
+b:580 [binder, in hydras.Epsilon0.T1]
+b:584 [binder, in hydras.Ackermann.primRec]
+b:586 [binder, in hydras.Ackermann.primRec]
+b:586 [binder, in hydras.Gamma0.Gamma0]
+b:588 [binder, in hydras.Ackermann.primRec]
+b:588 [binder, in hydras.Gamma0.Gamma0]
+b:59 [binder, in hydras.Ackermann.LNN2LNT]
+b:59 [binder, in gaia_hydras.GPaths]
+b:59 [binder, in hydras.Schutte.Schutte_basics]
+b:590 [binder, in hydras.Ackermann.primRec]
+b:592 [binder, in hydras.Ackermann.primRec]
+b:593 [binder, in hydras.Gamma0.Gamma0]
+b:594 [binder, in hydras.Ackermann.primRec]
+b:596 [binder, in hydras.Ackermann.primRec]
+b:597 [binder, in hydras.Gamma0.Gamma0]
+b:599 [binder, in hydras.Epsilon0.T1]
+b:6 [binder, in hydras.Ackermann.folProof]
+b:6 [binder, in gaia_hydras.GPaths]
+b:6 [binder, in hydras.Prelude.Sort_spec]
+b:6 [binder, in hydras.Ackermann.prLogic]
+b:6 [binder, in hydras.Ackermann.NNtheory]
+b:6 [binder, in hydras.Ackermann.ListExt]
+b:6 [binder, in hydras.Ackermann.folLogic3]
+b:6 [binder, in hydras.Gamma0.Gamma0]
+b:60 [binder, in hydras.Prelude.Comparable]
+b:60 [binder, in hydras.Schutte.Schutte_basics]
+B:60 [binder, in hydras.Schutte.Ordering_Functions]
+b:603 [binder, in hydras.Epsilon0.T1]
+b:604 [binder, in gaia_hydras.nfwfgaia]
+b:605 [binder, in hydras.Gamma0.Gamma0]
+b:61 [binder, in hydras.Schutte.Correctness_E0]
+b:61 [binder, in hydras.Prelude.DecPreOrder_Instances]
+B:61 [binder, in hydras.Prelude.Sort_spec]
+b:61 [binder, in hydras.Hydra.BigBattle]
+b:611 [binder, in gaia_hydras.nfwfgaia]
+b:615 [binder, in gaia_hydras.nfwfgaia]
+b:62 [binder, in hydras.Prelude.Comparable]
+b:62 [binder, in hydras.Schutte.Ordering_Functions]
+b:620 [binder, in hydras.Epsilon0.T1]
+b:628 [binder, in hydras.Epsilon0.T1]
+b:628 [binder, in gaia_hydras.nfwfgaia]
+b:63 [binder, in hydras.Prelude.DecPreOrder]
+b:63 [binder, in gaia_hydras.T1Bridge]
+b:63 [binder, in hydras.Ackermann.cPair]
+b:63 [binder, in hydras.Schutte.Schutte_basics]
+b:630 [binder, in gaia_hydras.nfwfgaia]
+b:634 [binder, in hydras.Gamma0.Gamma0]
+b:638 [binder, in hydras.Gamma0.Gamma0]
+b:64 [binder, in hydras.Prelude.Comparable]
+b:64 [binder, in hydras.Ackermann.cPair]
+b:64 [binder, in hydras.Hydra.BigBattle]
+b:64 [binder, in hydras.Schutte.PartialFun]
+b:64 [binder, in hydras.OrdinalNotations.ON_Generic]
+b:65 [binder, in gaia_hydras.GPaths]
+b:65 [binder, in hydras.OrdinalNotations.OmegaOmega]
+b:65 [binder, in hydras.Ackermann.code]
+b:65 [binder, in hydras.Schutte.Schutte_basics]
+B:65 [binder, in hydras.Schutte.Ordering_Functions]
+B:654 [binder, in hydras.Ackermann.checkPrf]
+b:656 [binder, in hydras.Ackermann.checkPrf]
+b:658 [binder, in hydras.Ackermann.checkPrf]
+b:66 [binder, in hydras.Epsilon0.T1]
+B:66 [binder, in Goedel.PRrepresentable]
+b:66 [binder, in hydras.Prelude.Comparable]
+b:66 [binder, in hydras.Ackermann.cPair]
+B:66 [binder, in hydras.Schutte.PartialFun]
+b:660 [binder, in hydras.Ackermann.checkPrf]
+b:662 [binder, in hydras.Ackermann.checkPrf]
+b:664 [binder, in hydras.Ackermann.checkPrf]
+b:666 [binder, in hydras.Ackermann.checkPrf]
+b:668 [binder, in hydras.Ackermann.checkPrf]
+b:67 [binder, in hydras.Ackermann.code]
+b:67 [binder, in hydras.Schutte.Schutte_basics]
+B:67 [binder, in hydras.Hydra.Hydra_Lemmas]
+B:67 [binder, in hydras.Ackermann.expressible]
+b:670 [binder, in hydras.Ackermann.checkPrf]
+b:670 [binder, in hydras.Gamma0.Gamma0]
+b:672 [binder, in hydras.Ackermann.checkPrf]
+b:672 [binder, in hydras.Gamma0.Gamma0]
+b:672 [binder, in gaia_hydras.nfwfgaia]
+b:674 [binder, in hydras.Ackermann.checkPrf]
+b:676 [binder, in hydras.Ackermann.checkPrf]
+b:676 [binder, in hydras.Ackermann.primRec]
+b:678 [binder, in hydras.Ackermann.checkPrf]
+b:68 [binder, in hydras.Prelude.Comparable]
+b:68 [binder, in hydras.Prelude.DecPreOrder]
+b:68 [binder, in gaia_hydras.T1Bridge]
+b:68 [binder, in hydras.Ackermann.cPair]
+b:68 [binder, in hydras.Hydra.BigBattle]
+B:68 [binder, in hydras.Schutte.Ordering_Functions]
+b:680 [binder, in hydras.Ackermann.primRec]
+b:683 [binder, in hydras.Ackermann.primRec]
+b:685 [binder, in hydras.Ackermann.primRec]
+b:687 [binder, in hydras.Ackermann.primRec]
+b:689 [binder, in hydras.Ackermann.primRec]
+b:69 [binder, in gaia_hydras.GPaths]
+b:69 [binder, in hydras.Ackermann.code]
+b:69 [binder, in hydras.Hydra.BigBattle]
+b:690 [binder, in hydras.Gamma0.Gamma0]
+b:692 [binder, in hydras.Ackermann.primRec]
+b:695 [binder, in hydras.Ackermann.primRec]
+b:696 [binder, in gaia_hydras.nfwfgaia]
+b:698 [binder, in hydras.Ackermann.primRec]
+b:698 [binder, in gaia_hydras.nfwfgaia]
+b:7 [binder, in hydras.OrdinalNotations.ON_O]
+b:7 [binder, in hydras.solutions_exercises.MorePRExamples]
+B:7 [binder, in hydras.Schutte.Ordering_Functions]
+B:70 [binder, in Goedel.PRrepresentable]
+b:70 [binder, in hydras.Prelude.Comparable]
+b:70 [binder, in hydras.OrdinalNotations.OmegaOmega]
+b:70 [binder, in hydras.Ackermann.cPair]
+b:70 [binder, in hydras.Schutte.Schutte_basics]
+b:70 [binder, in hydras.Ackermann.folLogic3]
+b:70 [binder, in hydras.rpo.dickson]
+b:70 [binder, in hydras.Schutte.Ordering_Functions]
+b:703 [binder, in hydras.Ackermann.primRec]
+b:708 [binder, in hydras.Epsilon0.T1]
+b:709 [binder, in hydras.Ackermann.primRec]
+b:71 [binder, in hydras.Ackermann.code]
+b:712 [binder, in hydras.Ackermann.primRec]
+b:716 [binder, in hydras.Ackermann.primRec]
+b:72 [binder, in gaia_hydras.GPaths]
+b:72 [binder, in hydras.Ackermann.cPair]
+b:72 [binder, in hydras.Ackermann.folLogic3]
+B:72 [binder, in hydras.Schutte.Ordering_Functions]
+b:72 [binder, in hydras.Prelude.MoreVectors]
+b:720 [binder, in hydras.Ackermann.primRec]
+b:721 [binder, in gaia_hydras.nfwfgaia]
+b:724 [binder, in hydras.Epsilon0.T1]
+b:728 [binder, in gaia_hydras.nfwfgaia]
+b:73 [binder, in hydras.Epsilon0.T1]
+b:73 [binder, in hydras.Prelude.Comparable]
+b:73 [binder, in hydras.Ackermann.code]
+B:73 [binder, in hydras.Schutte.PartialFun]
+b:73 [binder, in hydras.Schutte.Schutte_basics]
+B:73 [binder, in hydras.Schutte.Ordering_Functions]
+b:73 [binder, in hydras.rpo.rpo]
+b:730 [binder, in gaia_hydras.nfwfgaia]
+b:731 [binder, in hydras.Epsilon0.T1]
+b:732 [binder, in gaia_hydras.nfwfgaia]
+b:734 [binder, in gaia_hydras.nfwfgaia]
+b:736 [binder, in hydras.Epsilon0.T1]
+b:738 [binder, in gaia_hydras.nfwfgaia]
+b:74 [binder, in hydras.Epsilon0.Hessenberg]
+b:74 [binder, in hydras.Prelude.DecPreOrder]
+b:74 [binder, in gaia_hydras.T1Bridge]
+b:74 [binder, in hydras.Ackermann.cPair]
+b:74 [binder, in hydras.Schutte.Ordering_Functions]
+B:74 [binder, in hydras.Prelude.MoreVectors]
+b:740 [binder, in gaia_hydras.nfwfgaia]
+b:75 [binder, in hydras.Prelude.Comparable]
+b:75 [binder, in hydras.Ackermann.code]
+b:757 [binder, in hydras.Ackermann.checkPrf]
+b:76 [binder, in hydras.Gamma0.T2]
+b:76 [binder, in gaia_hydras.GPaths]
+b:76 [binder, in hydras.Ackermann.cPair]
+b:76 [binder, in hydras.Schutte.Schutte_basics]
+b:76 [binder, in hydras.OrdinalNotations.ON_Generic]
+b:761 [binder, in hydras.Ackermann.checkPrf]
+b:761 [binder, in gaia_hydras.nfwfgaia]
+b:765 [binder, in gaia_hydras.nfwfgaia]
+b:77 [binder, in hydras.Prelude.Comparable]
+b:777 [binder, in gaia_hydras.nfwfgaia]
+b:78 [binder, in hydras.Epsilon0.Hessenberg]
+b:78 [binder, in hydras.Ackermann.cPair]
+b:78 [binder, in hydras.Schutte.Ordering_Functions]
+b:781 [binder, in gaia_hydras.nfwfgaia]
+b:785 [binder, in gaia_hydras.nfwfgaia]
+b:787 [binder, in hydras.Ackermann.checkPrf]
+b:788 [binder, in hydras.Ackermann.primRec]
+b:788 [binder, in gaia_hydras.nfwfgaia]
+b:789 [binder, in hydras.Ackermann.checkPrf]
+b:79 [binder, in hydras.Gamma0.T2]
+b:79 [binder, in hydras.Prelude.Comparable]
+b:79 [binder, in hydras.OrdinalNotations.ON_Generic]
+b:791 [binder, in hydras.Ackermann.primRec]
+b:793 [binder, in gaia_hydras.nfwfgaia]
+b:794 [binder, in hydras.Ackermann.primRec]
+b:798 [binder, in hydras.Ackermann.primRec]
+b:799 [binder, in hydras.Ackermann.primRec]
+b:8 [binder, in hydras.Gamma0.T2]
+b:8 [binder, in hydras.Ackermann.folLogic2]
+b:8 [binder, in gaia_hydras.T1Bridge]
+b:8 [binder, in hydras.Ackermann.NNtheory]
+b:8 [binder, in gaia_hydras.T2Bridge]
+b:8 [binder, in hydras.MoreAck.PrimRecExamples]
+b:80 [binder, in hydras.Epsilon0.T1]
+b:80 [binder, in hydras.Prelude.DecPreOrder]
+b:80 [binder, in hydras.Ackermann.codeSubFormula]
+b:80 [binder, in gaia_hydras.T1Bridge]
+b:80 [binder, in hydras.Schutte.Schutte_basics]
+b:80 [binder, in hydras.Ackermann.folLogic3]
+b:80 [binder, in hydras.rpo.list_permut]
+b:800 [binder, in hydras.Ackermann.primRec]
+b:804 [binder, in hydras.Ackermann.primRec]
+b:805 [binder, in hydras.Ackermann.primRec]
+b:806 [binder, in hydras.Ackermann.primRec]
+b:809 [binder, in hydras.Ackermann.primRec]
+b:81 [binder, in hydras.Prelude.Comparable]
+b:81 [binder, in gaia_hydras.GPaths]
+b:82 [binder, in hydras.Ackermann.cPair]
+b:82 [binder, in hydras.Schutte.Schutte_basics]
+b:82 [binder, in hydras.Ackermann.folLogic3]
+B:82 [binder, in hydras.Schutte.Ordering_Functions]
+b:83 [binder, in hydras.Prelude.Comparable]
+b:83 [binder, in gaia_hydras.T1Bridge]
+B:83 [binder, in hydras.Schutte.PartialFun]
+b:838 [binder, in hydras.Ackermann.codeSubFormula]
+b:84 [binder, in hydras.Ackermann.cPair]
+b:84 [binder, in hydras.Schutte.Schutte_basics]
+b:841 [binder, in hydras.Ackermann.codeSubFormula]
+b:843 [binder, in gaia_hydras.nfwfgaia]
+b:849 [binder, in hydras.Ackermann.codeSubFormula]
+b:85 [binder, in hydras.Epsilon0.T1]
+b:85 [binder, in hydras.Prelude.Comparable]
+b:85 [binder, in gaia_hydras.T1Bridge]
+B:85 [binder, in hydras.OrdinalNotations.ON_Generic]
+b:850 [binder, in hydras.Ackermann.checkPrf]
+b:852 [binder, in hydras.Ackermann.checkPrf]
+b:853 [binder, in hydras.Ackermann.codeSubFormula]
+b:853 [binder, in gaia_hydras.nfwfgaia]
+b:854 [binder, in hydras.Ackermann.checkPrf]
+b:855 [binder, in hydras.Ackermann.codeSubFormula]
+b:856 [binder, in hydras.Ackermann.checkPrf]
+b:857 [binder, in hydras.Ackermann.codeSubFormula]
+b:86 [binder, in hydras.Ackermann.folLogic3]
+b:861 [binder, in gaia_hydras.nfwfgaia]
+b:866 [binder, in gaia_hydras.nfwfgaia]
+b:869 [binder, in hydras.Ackermann.codeSubFormula]
+b:87 [binder, in hydras.Prelude.Comparable]
+b:87 [binder, in hydras.Epsilon0.Hessenberg]
+b:87 [binder, in hydras.OrdinalNotations.OmegaOmega]
+b:87 [binder, in gaia_hydras.T1Bridge]
+b:87 [binder, in gaia_hydras.nfwfgaia]
+b:872 [binder, in hydras.Ackermann.codeSubFormula]
+b:886 [binder, in gaia_hydras.nfwfgaia]
+b:89 [binder, in hydras.Epsilon0.T1]
+b:89 [binder, in hydras.Epsilon0.Hessenberg]
+B:89 [binder, in hydras.Schutte.PartialFun]
+b:89 [binder, in hydras.Schutte.Schutte_basics]
+b:89 [binder, in hydras.MoreAck.PrimRecExamples]
+b:893 [binder, in hydras.Ackermann.codeSubFormula]
+b:895 [binder, in gaia_hydras.nfwfgaia]
+b:896 [binder, in hydras.Ackermann.codeSubFormula]
+b:899 [binder, in gaia_hydras.nfwfgaia]
+B:9 [binder, in hydras.Prelude.DecPreOrder_Instances]
+b:9 [binder, in hydras.Ackermann.extEqualNat]
+b:9 [binder, in hydras.rpo.dickson]
+b:9 [binder, in hydras.MoreAck.FolExamples]
+b:90 [binder, in hydras.Prelude.Comparable]
+b:903 [binder, in gaia_hydras.nfwfgaia]
+b:909 [binder, in hydras.Ackermann.codeSubFormula]
+b:91 [binder, in hydras.Gamma0.T2]
+b:91 [binder, in hydras.Epsilon0.Hessenberg]
+b:91 [binder, in hydras.Schutte.Schutte_basics]
+b:911 [binder, in gaia_hydras.nfwfgaia]
+b:912 [binder, in hydras.Ackermann.codeSubFormula]
+b:92 [binder, in hydras.Prelude.Comparable]
+b:92 [binder, in hydras.Prelude.DecPreOrder]
+b:92 [binder, in hydras.OrdinalNotations.ON_Generic]
+b:924 [binder, in gaia_hydras.nfwfgaia]
+b:926 [binder, in gaia_hydras.nfwfgaia]
+b:928 [binder, in gaia_hydras.nfwfgaia]
+B:93 [binder, in Goedel.PRrepresentable]
+b:93 [binder, in hydras.Epsilon0.Hessenberg]
+b:93 [binder, in hydras.Schutte.Schutte_basics]
+b:930 [binder, in gaia_hydras.nfwfgaia]
+b:932 [binder, in gaia_hydras.nfwfgaia]
+b:936 [binder, in hydras.Ackermann.codeSubFormula]
+b:939 [binder, in hydras.Ackermann.codeSubFormula]
+b:939 [binder, in gaia_hydras.nfwfgaia]
+b:94 [binder, in hydras.Prelude.Comparable]
+b:94 [binder, in hydras.Prelude.Merge_Sort]
+b:94 [binder, in gaia_hydras.T1Bridge]
+b:945 [binder, in gaia_hydras.nfwfgaia]
+b:95 [binder, in hydras.Epsilon0.Hessenberg]
+b:95 [binder, in hydras.Ackermann.folLogic3]
+b:954 [binder, in gaia_hydras.nfwfgaia]
+b:957 [binder, in gaia_hydras.nfwfgaia]
+b:959 [binder, in gaia_hydras.nfwfgaia]
+b:96 [binder, in gaia_hydras.T1Bridge]
+b:961 [binder, in gaia_hydras.nfwfgaia]
+b:963 [binder, in gaia_hydras.nfwfgaia]
+b:965 [binder, in gaia_hydras.nfwfgaia]
+b:97 [binder, in hydras.Prelude.DecPreOrder]
+B:97 [binder, in hydras.OrdinalNotations.ON_Generic]
+b:98 [binder, in hydras.Prelude.Comparable]
+b:98 [binder, in gaia_hydras.T1Bridge]
+b:989 [binder, in gaia_hydras.nfwfgaia]
+b:99 [binder, in hydras.Epsilon0.Hessenberg]
+b:993 [binder, in gaia_hydras.nfwfgaia]
+b:999 [binder, in gaia_hydras.nfwfgaia]
+

C

+callIsPR [instance, in hydras.Ackermann.cPair]
+call_lt_wf [lemma, in hydras.Epsilon0.F_alpha]
+call_lt [definition, in hydras.Epsilon0.F_alpha]
+canon [definition, in hydras.OrdinalNotations.ON_Omega2]
+canon [definition, in gaia_hydras.GCanon]
+Canon [definition, in hydras.Epsilon0.Canon]
+canon [definition, in hydras.Epsilon0.Canon]
+Canon [library]
+canonS [abbreviation, in gaia_hydras.GCanon]
+CanonSSn [lemma, in hydras.Epsilon0.Canon]
+canonSSn [abbreviation, in hydras.Epsilon0.Canon]
+canonS_rel_rounds [lemma, in hydras.Hydra.O2H]
+canonS_iota [lemma, in hydras.Hydra.O2H]
+canonS_iota_i [lemma, in hydras.Hydra.O2H]
+canonS_iota_final [lemma, in hydras.Hydra.O2H]
+canonS_iota_6 [lemma, in hydras.Hydra.O2H]
+canonS_iota_5 [lemma, in hydras.Hydra.O2H]
+canonS_iota_4 [lemma, in hydras.Hydra.O2H]
+canonS_ocons_succE [lemma, in gaia_hydras.GCanon]
+canonS_zero_inv [lemma, in gaia_hydras.GCanon]
+canonS_phi0_succE [lemma, in gaia_hydras.GCanon]
+canonS_cons_not_zero [lemma, in gaia_hydras.GCanon]
+canonS_lt [lemma, in gaia_hydras.GCanon]
+canonS_LE [lemma, in gaia_hydras.GCanon]
+CanonS_phi0_Succ [lemma, in hydras.Epsilon0.Canon]
+CanonS_lt [lemma, in hydras.Epsilon0.Canon]
+CanonS_phi0_lim [lemma, in hydras.Epsilon0.Canon]
+canonS_LE [lemma, in hydras.Epsilon0.Canon]
+canonS_limit_mono [lemma, in hydras.Epsilon0.Canon]
+canonS_limit_lub [lemma, in hydras.Epsilon0.Canon]
+canonS_limit_strong [lemma, in hydras.Epsilon0.Canon]
+canonS_cons_not_zero [lemma, in hydras.Epsilon0.Canon]
+canonS_LT [lemma, in hydras.Epsilon0.Canon]
+canonS_zero_inv [lemma, in hydras.Epsilon0.Canon]
+canonS_cons_succ_eqn2 [lemma, in hydras.Epsilon0.Canon]
+canonS_phi0_succ_eqn [lemma, in hydras.Epsilon0.Canon]
+canonS_succ [lemma, in hydras.Epsilon0.Canon]
+canonS_lim2 [lemma, in hydras.Epsilon0.Canon]
+canonS_lim1 [lemma, in hydras.Epsilon0.Canon]
+CanonS_Phi0_Succ_eqn [lemma, in hydras.Epsilon0.Paths]
+CanonS_plus_1 [lemma, in hydras.Epsilon0.Paths]
+Canon_plus [definition, in gaia_hydras.GF_alpha]
+Canon_plus [definition, in gaia_hydras.GPaths]
+canon_limit_of [lemma, in gaia_hydras.GCanon]
+canon_limit_mono [lemma, in gaia_hydras.GCanon]
+canon_limit_strong [lemma, in gaia_hydras.GCanon]
+canon_lim3 [lemma, in gaia_hydras.GCanon]
+canon_lim2 [lemma, in gaia_hydras.GCanon]
+canon_tail [lemma, in gaia_hydras.GCanon]
+canon_lim1 [lemma, in gaia_hydras.GCanon]
+canon_SSn_zero [lemma, in gaia_hydras.GCanon]
+canon_lt [lemma, in gaia_hydras.GCanon]
+canon_succ [lemma, in gaia_hydras.GCanon]
+canon_not_null [lemma, in hydras.Epsilon0.Large_Sets]
+Canon_of_limit_not_null [lemma, in hydras.Epsilon0.Canon]
+Canon_lt [lemma, in hydras.Epsilon0.Canon]
+Canon_Omega [lemma, in hydras.Epsilon0.Canon]
+Canon_Succ [lemma, in hydras.Epsilon0.Canon]
+canon_limit_mono [lemma, in hydras.Epsilon0.Canon]
+canon_limit_strong [lemma, in hydras.Epsilon0.Canon]
+canon_lt [lemma, in hydras.Epsilon0.Canon]
+canon_LT [lemma, in hydras.Epsilon0.Canon]
+canon_SSn_zero [lemma, in hydras.Epsilon0.Canon]
+canon_succ [lemma, in hydras.Epsilon0.Canon]
+canon_lim3 [lemma, in hydras.Epsilon0.Canon]
+canon_lim2 [lemma, in hydras.Epsilon0.Canon]
+canon_lim1 [lemma, in hydras.Epsilon0.Canon]
+canon_tail [lemma, in hydras.Epsilon0.Canon]
+canon_zero [lemma, in hydras.Epsilon0.Canon]
+Canon_examples [section, in hydras.Epsilon0.Canon]
+Canon_plus_first_step_lim [lemma, in hydras.Epsilon0.Paths]
+Canon_plus_first_step [lemma, in hydras.Epsilon0.Paths]
+Canon_mono1 [lemma, in hydras.Epsilon0.Paths]
+Canon_plus_inv [lemma, in hydras.Epsilon0.Paths]
+Canon_plus [definition, in hydras.Epsilon0.Paths]
+canon0_lt [lemma, in gaia_hydras.GCanon]
+canon0_phi0_succE [lemma, in gaia_hydras.GCanon]
+canon0_LT [lemma, in hydras.Epsilon0.Canon]
+canon0_cons_succ_eqn2 [lemma, in hydras.Epsilon0.Canon]
+canon0_phi0_succ_eqn [lemma, in hydras.Epsilon0.Canon]
+canon0_succ [lemma, in hydras.Epsilon0.Canon]
+canon0_lim2 [lemma, in hydras.Epsilon0.Canon]
+CantorOrdinal [module, in gaia_hydras.nfwfgaia]
+CantorOrdinal.addC_CE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.AddLocalNotation [section, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_simpl3 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_normal [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_le4 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_le3 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_inj [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_le2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_le1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_simpl2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_simpl1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_to_cons [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_fin_omega [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_int [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.add1Nfin [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.ap_pr4 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.ap_pr3 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.ap_pr2CE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.ap_pr2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.ap_pr1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.ap_pr0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.cantorCE2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.cantorCE3 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.cantor_CE1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.cantor_unique [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.cantor_exists [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.CompareT1Eq [constructor, in gaia_hydras.nfwfgaia]
+CantorOrdinal.CompareT1Gt [constructor, in gaia_hydras.nfwfgaia]
+CantorOrdinal.CompareT1Lt [constructor, in gaia_hydras.nfwfgaia]
+CantorOrdinal.compare_T1 [inductive, in gaia_hydras.nfwfgaia]
+CantorOrdinal.cons [constructor, in gaia_hydras.nfwfgaia]
+CantorOrdinal.div_by_omega_pr [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expF_eq1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expF_eq0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expF_1n [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expF_mul [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expF_add [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expO_add [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expO_eq1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expO_eq0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expO_1n [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expO_n0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expO_mul1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expx_nat [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expx_pnat [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expx0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expx1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.expx1CE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_prod [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_sum [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_FO3 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_FO2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_FO1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_FO [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_consCE1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_int [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_eq1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_eq0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_O [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_F [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp0nz [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp00 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp1x [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp2omega [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.ex_middle_pick [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.ex_middle [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.fincP [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.fooCE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.head_lt_cons [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.le_succ_succE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.le_succ_succ [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_prop [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_fct [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_lub [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_unique [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_of [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_CE3 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_CE2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_CE1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_unique2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_unique1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_v2 [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_v1 [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_pr [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_pr1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit3 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.LT [abbreviation, in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ_le_4 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ_le_3 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ_le_2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ_le_1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ_succE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ2CE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ_succ [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_not_wf [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.minus_le [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.minus_lt [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.min_exists [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mulA [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mulnf0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_omega_pr3 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_omega2_pr1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_omega_pr1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_sum_omega [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_omega_limit [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_int_limit [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_fin_omega [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_succ [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_distr [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_phi0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_int [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_na [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_exp [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_expF [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_expO [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_rev_prod [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_rev_sum [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_rev_unique [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_rev [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_revCE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_div_by_omega [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_mul [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_toNF [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_omega_plus_n [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_sub [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_add [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_pred [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_succ [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_log [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_phi0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_finite [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_omega [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_split [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_Wf' [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_Wf [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.normal [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.normal_compose [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.normal_limit [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.normal_id [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.omega_plus_n [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.omega_tower_unbounded [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.omega_tower_nf [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.omega_tower [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.omega_minus_one [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.one [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi0 [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi0_log [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi0_lt1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi0_le [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi0_lt [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi1 [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi2 [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi3 [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.plus_int_Ox [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.pow_mon3 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.pow_mon2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.pow_mon1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.pow_omega [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.pred_succ_CE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.pred_succ [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.pred_lt [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.pred_le [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_add [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_limit1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_le [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_pred [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_succ [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_finite [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_is_succ [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_limit [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_pr1rCE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_nzCE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_nz [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_pr1r [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_pr1CE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_pr1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_pr [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_le1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_int [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_1bCE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_1aCE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub1a [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub1b [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub1Nfin [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_is_add_one [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_injCE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_inj [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_p1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_predCE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_pred [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_lt [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sup [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sup_Oalpha_limit [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sup_Oalpha_succ [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sup_Oalpha_zero [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.sup_unique [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.tail_lt_cons [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.Tf [abbreviation, in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_pred [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_succ [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_ex2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_ex1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_mon [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_nf [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_nz [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1 [inductive, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1add [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1addA [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1addn0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1addS [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1add_eq0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1add0n [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ap [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ap_phi0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1bad [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1div_by_omega [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1eq [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1eqE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1eqP [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1eq_mul2l [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1eq_add2l [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1eq_le [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1exp [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1finite [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1finite_succ [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1finite1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1finite2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1finite2CE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1F_inj [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1GeqNotGtn [constructor, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1GeqNotLtn [constructor, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ge1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1GtnNotLeq [constructor, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1is_succ [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1leNgt [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lenn [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1len0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1leP [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1leq_xor_gtn [inductive, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_pmull [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_mul [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_mulCE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_pmulrl [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_pmulrCE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_pmulr [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_mul2r [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_mul2l [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_add2r [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_add2l [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_cons_le [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_trans [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_lt_trans [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_total [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_consE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_eqVlt [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le0n [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1limit [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1log [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1log_exp2 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1log_exp1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1log_exp0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1log_prod [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1log_phi0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltgtP [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltNge [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltnn [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1LtnNotGeq [constructor, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltn_xor_geq [inductive, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltn0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltP [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltW [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_mul2r [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_mul2l [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_add2r [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_add2l [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_cons_le [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_le_trans [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_trans [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_anti [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_trichotomy [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_neAle [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_ne' [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_ne [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt0n [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1muln0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1muln1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1muln1_CE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul_omega [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul_eq1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul_eq0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul0n [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul1n [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul1nCE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nat [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nat_inc [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nfCE [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_rect [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_finite [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_finite1 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_consb [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_consa [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_cons_cons [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_cons0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1omega [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1phi0_zero' [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1phi0_zero [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1pred [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1split [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1sub [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1subnn [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1subn0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1sub0 [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1succ [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1succ_nat [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1transfinite_induction_Q [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1transfinite_induction [lemma, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1_eqType [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1_eqMixin [definition, in gaia_hydras.nfwfgaia]
+CantorOrdinal.zero [constructor, in gaia_hydras.nfwfgaia]
+_ ^ _ (cantor_scope) [notation, in gaia_hydras.nfwfgaia]
+_ * _ (cantor_scope) [notation, in gaia_hydras.nfwfgaia]
+_ - _ (cantor_scope) [notation, in gaia_hydras.nfwfgaia]
+_ + _ (cantor_scope) [notation, in gaia_hydras.nfwfgaia]
+_ > _ (cantor_scope) [notation, in gaia_hydras.nfwfgaia]
+_ >= _ (cantor_scope) [notation, in gaia_hydras.nfwfgaia]
+_ <= _ (cantor_scope) [notation, in gaia_hydras.nfwfgaia]
+_ < _ (cantor_scope) [notation, in gaia_hydras.nfwfgaia]
+\F _ (cantor_scope) [notation, in gaia_hydras.nfwfgaia]
+Cantor_normal_form [lemma, in hydras.Epsilon0.T1]
+cases_for_mult [lemma, in hydras.Epsilon0.T1]
+case_Eq [lemma, in hydras.Schutte.Correctness_E0]
+case_gt [lemma, in hydras.Schutte.Correctness_E0]
+case_lt [lemma, in hydras.Schutte.Correctness_E0]
+cf:11 [binder, in hydras.Ackermann.code]
+cf:14 [binder, in hydras.Ackermann.code]
+cf:17 [binder, in hydras.Ackermann.code]
+cf:2 [binder, in hydras.Ackermann.codeFreeVar]
+cf:2 [binder, in hydras.Ackermann.codeSubTerm]
+cf:2 [binder, in hydras.Ackermann.checkPrf]
+cf:2 [binder, in hydras.Ackermann.codeSubFormula]
+cf:2 [binder, in hydras.Ackermann.wellFormed]
+cf:2 [binder, in Goedel.codeSysPrf]
+cf:2 [binder, in hydras.Ackermann.codePA]
+chain [definition, in additions.Addition_Chains]
+chain_gen_OK [lemma, in additions.AM]
+chain_apply [definition, in additions.AM]
+chain_trace [definition, in additions.Trace_exercise]
+chain_gen_OK [lemma, in additions.Euclidean_Chains]
+chain_type [definition, in additions.Euclidean_Chains]
+chain_generator [definition, in additions.Addition_Chains]
+chain_correct [definition, in additions.Addition_Chains]
+chain_correct_nat [definition, in additions.Addition_Chains]
+chain_apply [definition, in additions.Addition_Chains]
+chain_execute [definition, in additions.Addition_Chains]
+chain_length [definition, in additions.Addition_Chains]
+charFunction [definition, in hydras.Ackermann.extEqualNat]
+checkPrf [definition, in hydras.Ackermann.checkPrf]
+checkPrf [library]
+checkPrfAXM [definition, in hydras.Ackermann.checkPrf]
+checkPrfAXMIsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfCorrect1 [lemma, in hydras.Ackermann.checkPrf]
+checkPrfCorrect2 [lemma, in hydras.Ackermann.checkPrf]
+checkPrfCP [definition, in hydras.Ackermann.checkPrf]
+checkPrfCPIsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfEQnIsPR [lemma, in hydras.Ackermann.checkPrf]
+checkPrfEQ1 [definition, in hydras.Ackermann.checkPrf]
+checkPrfEQ1IsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfEQ2 [definition, in hydras.Ackermann.checkPrf]
+checkPrfEQ2IsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfEQ3 [definition, in hydras.Ackermann.checkPrf]
+checkPrfEQ3IsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfEQ4 [definition, in hydras.Ackermann.checkPrf]
+checkPrfEQ4IsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfEQ5 [definition, in hydras.Ackermann.checkPrf]
+checkPrfEQ5IsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfFA1 [definition, in hydras.Ackermann.checkPrf]
+checkPrfFA1IsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfFA2 [definition, in hydras.Ackermann.checkPrf]
+checkPrfFA2IsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfFA3 [definition, in hydras.Ackermann.checkPrf]
+checkPrfFA3IsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfGEN [definition, in hydras.Ackermann.checkPrf]
+checkPrfGENIsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfHelp [definition, in hydras.Ackermann.checkPrf]
+checkPrfHelpIsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfIMP1 [definition, in hydras.Ackermann.checkPrf]
+checkPrfIMP1IsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfIMP2 [definition, in hydras.Ackermann.checkPrf]
+checkPrfIMP2IsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfIsPR [instance, in hydras.Ackermann.checkPrf]
+checkPrfMP [definition, in hydras.Ackermann.checkPrf]
+checkPrfMPIsPR [instance, in hydras.Ackermann.checkPrf]
+checkSubFormulaTrace [definition, in hydras.Ackermann.codeSubFormula]
+checkTraceCorrect [lemma, in hydras.Ackermann.codeSubFormula]
+checkTraceIsPR [instance, in hydras.Ackermann.codeSubFormula]
+Check_Proof.Prf [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.wellFormedFormula [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.wellFormedTerm [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.Formula [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.Terms [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.Term [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.codeRInj [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.codeFInj [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.codeArityRIsCorrect2 [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.codeArityRIsCorrect1 [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.codeArityFIsCorrect2 [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.codeArityFIsCorrect1 [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.codeArityR [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.codeArityF [variable, in hydras.Ackermann.checkPrf]
+Check_Proof.L [variable, in hydras.Ackermann.checkPrf]
+Check_Proof [section, in hydras.Ackermann.checkPrf]
+ch:345 [binder, in additions.Euclidean_Chains]
+ch:346 [binder, in additions.Euclidean_Chains]
+ch:349 [binder, in additions.Euclidean_Chains]
+ch:350 [binder, in additions.Euclidean_Chains]
+classic_battle [definition, in hydras.Hydra.Hydra_Lemmas]
+classification [lemma, in hydras.Schutte.Schutte_basics]
+clog2 [definition, in additions.Addition_Chains]
+close [definition, in hydras.Ackermann.folProp]
+close [section, in hydras.Ackermann.codePA]
+closed [definition, in hydras.Ackermann.folProp]
+Closed [definition, in hydras.Schutte.Schutte_basics]
+closedNatToTerm [lemma, in hydras.Ackermann.LNN]
+closedNatToTerm [lemma, in hydras.Ackermann.LNT]
+closedNN [lemma, in hydras.Ackermann.NN]
+closedNN1 [lemma, in hydras.Ackermann.NN]
+closedPA [lemma, in hydras.Ackermann.PA]
+closedPA1 [lemma, in hydras.Ackermann.PA]
+ClosedSystem [definition, in hydras.Ackermann.folProp]
+closedT [lemma, in hydras.Ackermann.expressible]
+Closed_Cr [lemma, in hydras.Schutte.Critical]
+closeFrom [definition, in hydras.Ackermann.subAll]
+closeList [definition, in hydras.Ackermann.folProp]
+close.codeFormula [variable, in hydras.Ackermann.codePA]
+close.Formula [variable, in hydras.Ackermann.codePA]
+close.L [variable, in hydras.Ackermann.codePA]
+closure [library]
+cl:19 [binder, in hydras.Ackermann.code]
+cL:4 [binder, in hydras.Ackermann.codeFreeVar]
+cL:4 [binder, in hydras.Ackermann.codeSubTerm]
+cL:4 [binder, in hydras.Ackermann.checkPrf]
+cL:4 [binder, in hydras.Ackermann.codeSubFormula]
+cL:4 [binder, in hydras.Ackermann.wellFormed]
+cL:4 [binder, in Goedel.codeSysPrf]
+cL:4 [binder, in hydras.Ackermann.codePA]
+cmpA:3 [binder, in hydras.OrdinalNotations.ON_mult]
+cmpA:3 [binder, in hydras.OrdinalNotations.ON_plus]
+cmpA:4 [binder, in hydras.OrdinalNotations.ON_O]
+cmpB:7 [binder, in hydras.OrdinalNotations.ON_mult]
+cmpB:7 [binder, in hydras.OrdinalNotations.ON_plus]
+cmp:13 [binder, in hydras.OrdinalNotations.ON_Generic]
+cmp:14 [binder, in hydras.Prelude.Comparable]
+cmp:3 [binder, in hydras.solutions_exercises.lt_succ_le]
+cmp:3 [binder, in hydras.OrdinalNotations.ON_Generic]
+cmp:4 [binder, in hydras.solutions_exercises.predSuccUnicity]
+cmp:6 [binder, in hydras.Prelude.Comparable]
+cmp:9 [binder, in hydras.OrdinalNotations.ON_Generic]
+cm:106 [binder, in additions.fib]
+cm:113 [binder, in additions.fib]
+cm:124 [binder, in additions.fib]
+cnf [projection, in gaia_hydras.T1Bridge]
+cnf [projection, in hydras.Epsilon0.E0]
+CNF [library]
+cnf_of_epsilon0 [lemma, in hydras.Schutte.CNF]
+cnf_lt_epsilon0 [lemma, in hydras.Schutte.CNF]
+cnf_exists_unique [lemma, in hydras.Schutte.CNF]
+cnf_unicity [lemma, in hydras.Schutte.CNF]
+cnf_exists [lemma, in hydras.Schutte.CNF]
+cnf_plus [lemma, in hydras.Schutte.CNF]
+cnf_plus2 [lemma, in hydras.Schutte.CNF]
+cnf_plus1 [lemma, in hydras.Schutte.CNF]
+cnf_eq [lemma, in hydras.Schutte.CNF]
+cnf_head_eq [lemma, in hydras.Schutte.CNF]
+cnf_of_ap [lemma, in hydras.Schutte.CNF]
+cnf_t [definition, in hydras.Schutte.CNF]
+cnf_Cons [lemma, in hydras.Epsilon0.E0]
+cnf_Omega_term [lemma, in hydras.Epsilon0.E0]
+cnf_Succ [lemma, in hydras.Epsilon0.E0]
+cnf_phi0 [lemma, in hydras.Epsilon0.E0]
+cnf_rw [lemma, in hydras.Epsilon0.E0]
+cnf_ok [projection, in hydras.Epsilon0.E0]
+code [definition, in additions.AM]
+code [library]
+codeAnd [definition, in hydras.Ackermann.code]
+codeAndCorrect [lemma, in hydras.Ackermann.code]
+codeAndIsPR [instance, in hydras.Ackermann.checkPrf]
+codeApp [definition, in hydras.Ackermann.codeList]
+codeAppCorrect [lemma, in hydras.Ackermann.codeList]
+codeAppIsPR [instance, in hydras.Ackermann.codeList]
+codeArityFIsPR:6 [binder, in hydras.Ackermann.wellFormed]
+codeArityFIsPR:7 [binder, in hydras.Ackermann.checkPrf]
+codeArityFIsPR:7 [binder, in Goedel.codeSysPrf]
+codeArityLNNFIsCorrect1 [lemma, in hydras.Ackermann.Languages]
+codeArityLNNFIsCorrect2 [lemma, in hydras.Ackermann.Languages]
+codeArityLNNR [definition, in hydras.Ackermann.Languages]
+codeArityLNNRIsCorrect1 [lemma, in hydras.Ackermann.Languages]
+codeArityLNNRIsCorrect2 [lemma, in hydras.Ackermann.Languages]
+codeArityLNNRIsPR [instance, in hydras.Ackermann.Languages]
+codeArityLNTF [definition, in hydras.Ackermann.Languages]
+codeArityLNTFIsCorrect1 [lemma, in hydras.Ackermann.Languages]
+codeArityLNTFIsCorrect2 [lemma, in hydras.Ackermann.Languages]
+codeArityLNTFIsPR [instance, in hydras.Ackermann.Languages]
+codeArityLNTR [definition, in hydras.Ackermann.Languages]
+codeArityLNTRIsCorrect1 [lemma, in hydras.Ackermann.Languages]
+codeArityLNTRIsCorrect2 [lemma, in hydras.Ackermann.Languages]
+codeArityLNTRIsPR [instance, in hydras.Ackermann.Languages]
+codeArityRIsPR:13 [binder, in hydras.Ackermann.checkPrf]
+codeArityRIsPR:13 [binder, in Goedel.codeSysPrf]
+codeArityRIsPR:181 [binder, in hydras.Ackermann.wellFormed]
+codeAxmEqHelp [definition, in hydras.Ackermann.checkPrf]
+codeAxmEqHelpIsPR [instance, in hydras.Ackermann.checkPrf]
+codeClose [definition, in hydras.Ackermann.codePA]
+codeCloseCorrect [lemma, in hydras.Ackermann.codePA]
+codeCloseIsPR [instance, in hydras.Ackermann.codePA]
+codeCloseList [definition, in hydras.Ackermann.codePA]
+codeCloseListCorrect [lemma, in hydras.Ackermann.codePA]
+codeCloseListIsPR [instance, in hydras.Ackermann.codePA]
+codeF [definition, in hydras.Ackermann.code]
+codeFInj [projection, in hydras.Ackermann.code]
+codeFNN [definition, in Goedel.goedel1]
+codeForall [definition, in hydras.Ackermann.code]
+codeForallCorrect [lemma, in hydras.Ackermann.code]
+codeForallIsPR [lemma, in hydras.Ackermann.prLogic]
+codeFormula [definition, in hydras.Ackermann.code]
+codeFormula [definition, in Goedel.rosserPA]
+codeFormula [definition, in Goedel.rosser]
+codeFormulaInj [lemma, in hydras.Ackermann.code]
+codeFreeVar [library]
+codeFreeVarFormula [definition, in hydras.Ackermann.codeFreeVar]
+codeFreeVarFormulaCorrect [lemma, in hydras.Ackermann.codeFreeVar]
+codeFreeVarFormulaIsPR [instance, in hydras.Ackermann.codeFreeVar]
+codeFreeVarListFormula [definition, in hydras.Ackermann.codeFreeVar]
+codeFreeVarListFormulaCorrect [lemma, in hydras.Ackermann.codeFreeVar]
+codeFreeVarListFormulaIsPR [instance, in hydras.Ackermann.codeFreeVar]
+codeFreeVarTerm [definition, in hydras.Ackermann.codeFreeVar]
+codeFreeVarTermCorrect [lemma, in hydras.Ackermann.codeFreeVar]
+codeFreeVarTermIsPR [lemma, in hydras.Ackermann.codeFreeVar]
+codeFreeVarTerms [definition, in hydras.Ackermann.codeFreeVar]
+codeFreeVarTermsCorrect [lemma, in hydras.Ackermann.codeFreeVar]
+codeFreeVarTermsIsPR [instance, in hydras.Ackermann.codeFreeVar]
+codeFreeVarTermTerms [definition, in hydras.Ackermann.codeFreeVar]
+codeFreeVarTermTermsIsPR [instance, in hydras.Ackermann.codeFreeVar]
+codeIff [definition, in hydras.Ackermann.code]
+codeIffCorrect [lemma, in hydras.Ackermann.code]
+codeIffIsPR [instance, in hydras.Ackermann.checkPrf]
+codeImp [definition, in hydras.Ackermann.code]
+codeImpCorrect [lemma, in hydras.Ackermann.code]
+codeImpIsPR [instance, in hydras.Ackermann.prLogic]
+codeIn [definition, in hydras.Ackermann.codeList]
+codeInCorrect [lemma, in hydras.Ackermann.codeList]
+codeInductionSchema [definition, in hydras.Ackermann.codePA]
+codeInductionSchemaCorrect1 [lemma, in hydras.Ackermann.codePA]
+codeInductionSchemaCorrect2 [lemma, in hydras.Ackermann.codePA]
+codeInductionSchemaCorrect3 [lemma, in hydras.Ackermann.codePA]
+codeInductionSchemaIsPR [instance, in hydras.Ackermann.codePA]
+codeInIsPR [instance, in hydras.Ackermann.codeList]
+codeLength [definition, in hydras.Ackermann.codeList]
+codeLengthCorrect [lemma, in hydras.Ackermann.codeList]
+codeLengthIsPR [instance, in hydras.Ackermann.codeList]
+codeList [definition, in hydras.Ackermann.cPair]
+codeList [library]
+codeListInj [lemma, in hydras.Ackermann.cPair]
+codeListRemove [definition, in hydras.Ackermann.codeList]
+codeListRemoveCorrect [lemma, in hydras.Ackermann.codeList]
+codeListRemoveIsPR [instance, in hydras.Ackermann.codeList]
+codeLNNRelation [definition, in hydras.Ackermann.Languages]
+codeLNNRelationInj [lemma, in hydras.Ackermann.Languages]
+codeLNTFunction [definition, in hydras.Ackermann.Languages]
+codeLNTFunctionInj [lemma, in hydras.Ackermann.Languages]
+codeLNTRelation [definition, in hydras.Ackermann.Languages]
+codeLNTRelationInj [lemma, in hydras.Ackermann.Languages]
+codeNatToTerm [definition, in hydras.Ackermann.codeNatToTerm]
+codeNatToTerm [library]
+codeNatToTermCorrectLNN [lemma, in hydras.Ackermann.codeNatToTerm]
+codeNatToTermCorrectLNT [lemma, in hydras.Ackermann.codeNatToTerm]
+codeNatToTermIsPR [instance, in hydras.Ackermann.codeNatToTerm]
+codeNewVar [definition, in hydras.Ackermann.codeSubFormula]
+codeNewVarCorrect [lemma, in hydras.Ackermann.codeSubFormula]
+codeNewVarIsPR [instance, in hydras.Ackermann.codeSubFormula]
+codeNoDup [definition, in hydras.Ackermann.codeList]
+codeNoDupCorrect [lemma, in hydras.Ackermann.codeList]
+codeNoDupIsPR [instance, in hydras.Ackermann.codeList]
+codeNot [definition, in hydras.Ackermann.code]
+codeNotCorrect [lemma, in hydras.Ackermann.code]
+codeNotIsPR [instance, in hydras.Ackermann.prLogic]
+codeNth [definition, in hydras.Ackermann.cPair]
+codeNthCorrect [lemma, in hydras.Ackermann.cPair]
+codeNthIsPR [instance, in hydras.Ackermann.cPair]
+codeNVarsCorrect [lemma, in hydras.Ackermann.checkPrf]
+codeNVars1 [definition, in hydras.Ackermann.checkPrf]
+codeNVars1IsPR [instance, in hydras.Ackermann.checkPrf]
+codeNVars2 [definition, in hydras.Ackermann.checkPrf]
+codeNVars2IsPR [instance, in hydras.Ackermann.checkPrf]
+codeOpen [definition, in hydras.Ackermann.codePA]
+codeOpenCorrect [lemma, in hydras.Ackermann.codePA]
+codeOpenIsPR [instance, in hydras.Ackermann.codePA]
+codeOr [definition, in hydras.Ackermann.code]
+codeOrCorrect [lemma, in hydras.Ackermann.code]
+codeOrIsPR [instance, in hydras.Ackermann.checkPrf]
+codePA [definition, in hydras.Ackermann.codePA]
+codePA [library]
+codePAcorrect1 [lemma, in hydras.Ackermann.codePA]
+codePAcorrect2 [lemma, in hydras.Ackermann.codePA]
+codePAcorrect3 [lemma, in hydras.Ackermann.codePA]
+codePAIsPR [instance, in hydras.Ackermann.codePA]
+codePrf [definition, in hydras.Ackermann.code]
+codePrf [definition, in Goedel.rosserPA]
+codePrf [definition, in Goedel.rosser]
+codePrfInjAxm [lemma, in hydras.Ackermann.code]
+codeR [definition, in hydras.Ackermann.code]
+codeRInj [projection, in hydras.Ackermann.code]
+codeSubFormula [definition, in hydras.Ackermann.codeSubFormula]
+codeSubFormula [library]
+codeSubFormulaCorrect [lemma, in hydras.Ackermann.codeSubFormula]
+codeSubFormulaIsPR [instance, in hydras.Ackermann.codeSubFormula]
+codeSubTerm [definition, in hydras.Ackermann.codeSubTerm]
+codeSubTerm [library]
+codeSubTermCorrect [lemma, in hydras.Ackermann.codeSubTerm]
+codeSubTermIsPR [instance, in hydras.Ackermann.codeSubTerm]
+codeSubTerms [definition, in hydras.Ackermann.codeSubTerm]
+codeSubTermsCorrect [lemma, in hydras.Ackermann.codeSubTerm]
+codeSubTermsIsPR [lemma, in hydras.Ackermann.codeSubTerm]
+codeSubTermTerms [definition, in hydras.Ackermann.codeSubTerm]
+codeSubTermTermsIsPR [instance, in hydras.Ackermann.codeSubTerm]
+codeSysPf [definition, in Goedel.codeSysPrf]
+codeSysPf [definition, in Goedel.goedel1]
+codeSysPf [definition, in Goedel.goedel2]
+codeSysPfCorrect [lemma, in Goedel.codeSysPrf]
+codeSysPfCorrect [definition, in Goedel.goedel1]
+codeSysPfCorrect [definition, in Goedel.goedel2]
+codeSysPrf [definition, in Goedel.rosserPA]
+codeSysPrf [definition, in Goedel.codeSysPrf]
+codeSysPrf [definition, in Goedel.goedel1]
+codeSysPrf [definition, in Goedel.rosser]
+codeSysPrf [library]
+codeSysPrfCorrect1 [definition, in Goedel.rosserPA]
+codeSysPrfCorrect1 [lemma, in Goedel.codeSysPrf]
+codeSysPrfCorrect1 [definition, in Goedel.rosser]
+codeSysPrfCorrect2 [definition, in Goedel.rosserPA]
+codeSysPrfCorrect2 [lemma, in Goedel.codeSysPrf]
+codeSysPrfCorrect2 [definition, in Goedel.goedel1]
+codeSysPrfCorrect2 [definition, in Goedel.rosser]
+codeSysPrfCorrect3 [definition, in Goedel.rosserPA]
+codeSysPrfCorrect3 [lemma, in Goedel.codeSysPrf]
+codeSysPrfCorrect3 [definition, in Goedel.goedel1]
+codeSysPrfCorrect3 [definition, in Goedel.rosser]
+codeSysPrfNCorrect1 [definition, in Goedel.rosserPA]
+codeSysPrfNCorrect1 [lemma, in Goedel.codeSysPrf]
+codeSysPrfNCorrect1 [definition, in Goedel.rosser]
+codeSysPrfNCorrect2 [definition, in Goedel.rosserPA]
+codeSysPrfNCorrect2 [lemma, in Goedel.codeSysPrf]
+codeSysPrfNCorrect2 [definition, in Goedel.rosser]
+codeSysPrfNCorrect3 [definition, in Goedel.rosserPA]
+codeSysPrfNCorrect3 [lemma, in Goedel.codeSysPrf]
+codeSysPrfNCorrect3 [definition, in Goedel.rosser]
+codeSysPrfNot [definition, in Goedel.rosserPA]
+codeSysPrfNot [definition, in Goedel.codeSysPrf]
+codeSysPrfNot [definition, in Goedel.rosser]
+codeTerm [definition, in hydras.Ackermann.code]
+codeTermFreeVar [lemma, in hydras.Ackermann.codeSubFormula]
+codeTermInj [lemma, in hydras.Ackermann.code]
+codeTerms [definition, in hydras.Ackermann.code]
+codeTermsInj [lemma, in hydras.Ackermann.code]
+Code_Free_Vars.Terms [variable, in hydras.Ackermann.codeFreeVar]
+Code_Free_Vars.Term [variable, in hydras.Ackermann.codeFreeVar]
+Code_Free_Vars.System [variable, in hydras.Ackermann.codeFreeVar]
+Code_Free_Vars.Formulas [variable, in hydras.Ackermann.codeFreeVar]
+Code_Free_Vars.Formula [variable, in hydras.Ackermann.codeFreeVar]
+Code_Free_Vars.L [variable, in hydras.Ackermann.codeFreeVar]
+Code_Free_Vars [section, in hydras.Ackermann.codeFreeVar]
+Code_Substitute_Term.Terms [variable, in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Term.Term [variable, in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Term.System [variable, in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Term.Formulas [variable, in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Term.Formula [variable, in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Term.L [variable, in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Term [section, in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Formula.codeTerm [variable, in hydras.Ackermann.codeSubFormula]
+Code_Substitute_Formula.codeFormula [variable, in hydras.Ackermann.codeSubFormula]
+Code_Substitute_Formula.L [variable, in hydras.Ackermann.codeSubFormula]
+Code_Substitute_Formula [section, in hydras.Ackermann.codeSubFormula]
+Code_Term_Formula_Proof.codeTermFormDef [section, in hydras.Ackermann.code]
+Code_Term_Formula_Proof.SysPrf [variable, in hydras.Ackermann.code]
+Code_Term_Formula_Proof.Prf [variable, in hydras.Ackermann.code]
+Code_Term_Formula_Proof.Terms [variable, in hydras.Ackermann.code]
+Code_Term_Formula_Proof.Term [variable, in hydras.Ackermann.code]
+Code_Term_Formula_Proof.System [variable, in hydras.Ackermann.code]
+Code_Term_Formula_Proof.Formulas [variable, in hydras.Ackermann.code]
+Code_Term_Formula_Proof.Formula [variable, in hydras.Ackermann.code]
+Code_Term_Formula_Proof.LcodeDef.cR [variable, in hydras.Ackermann.code]
+Code_Term_Formula_Proof.LcodeDef.cF [variable, in hydras.Ackermann.code]
+Code_Term_Formula_Proof.LcodeDef [section, in hydras.Ackermann.code]
+Code_Term_Formula_Proof.L [variable, in hydras.Ackermann.code]
+Code_Term_Formula_Proof [section, in hydras.Ackermann.code]
+code_nat_list.drop [variable, in hydras.Ackermann.cPair]
+code_nat_list [section, in hydras.Ackermann.cPair]
+code_SysPrf.LNN.expressU2 [variable, in Goedel.codeSysPrf]
+code_SysPrf.LNN.expressU1 [variable, in Goedel.codeSysPrf]
+code_SysPrf.LNN.freeVarfU [variable, in Goedel.codeSysPrf]
+code_SysPrf.LNN.v0 [variable, in Goedel.codeSysPrf]
+code_SysPrf.LNN.fU [variable, in Goedel.codeSysPrf]
+code_SysPrf.LNN.U [variable, in Goedel.codeSysPrf]
+code_SysPrf.LNN.TextendsNN [variable, in Goedel.codeSysPrf]
+code_SysPrf.LNN.T [variable, in Goedel.codeSysPrf]
+code_SysPrf.LNN [section, in Goedel.codeSysPrf]
+code_SysPrf.codeArityRIsCorrect2 [variable, in Goedel.codeSysPrf]
+code_SysPrf.codeArityRIsCorrect1 [variable, in Goedel.codeSysPrf]
+code_SysPrf.codeArityFIsCorrect2 [variable, in Goedel.codeSysPrf]
+code_SysPrf.codeArityFIsCorrect1 [variable, in Goedel.codeSysPrf]
+code_SysPrf.codeArityR [variable, in Goedel.codeSysPrf]
+code_SysPrf.codeArityF [variable, in Goedel.codeSysPrf]
+code_SysPrf.L [variable, in Goedel.codeSysPrf]
+code_SysPrf [section, in Goedel.codeSysPrf]
+Code_PA.codeFormulaInj [variable, in hydras.Ackermann.codePA]
+Code_PA.codeFormula [variable, in hydras.Ackermann.codePA]
+Code_PA.codeTerm [variable, in hydras.Ackermann.codePA]
+Code_PA [section, in hydras.Ackermann.codePA]
+coeff_lt [lemma, in hydras.Epsilon0.T1]
+coeff_lt [lemma, in hydras.Schutte.Correctness_E0]
+comp [instance, in hydras.OrdinalNotations.ON_Finite]
+Comparable [section, in hydras.Prelude.Comparable]
+Comparable [record, in hydras.Prelude.Comparable]
+Comparable [library]
+comparable_comp_spec [projection, in hydras.Prelude.Comparable]
+comparable_sto [projection, in hydras.Prelude.Comparable]
+Comparable0:15 [binder, in hydras.Prelude.Comparable]
+compare [projection, in hydras.Prelude.Comparable]
+Compare [record, in hydras.Prelude.Comparable]
+compare [constructor, in hydras.Prelude.Comparable]
+Compare [inductive, in hydras.Prelude.Comparable]
+compareA:108 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareA:123 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareA:135 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareA:148 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareA:161 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareA:173 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareA:186 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareA:200 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareA:31 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareA:50 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareA:69 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareA:83 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareA:95 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareB:112 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareB:127 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareB:139 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareB:152 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareB:165 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareB:177 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareB:190 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareB:204 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareB:35 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareB:54 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareB:87 [binder, in hydras.OrdinalNotations.ON_Generic]
+compareB:99 [binder, in hydras.OrdinalNotations.ON_Generic]
+compare_reflectR [lemma, in hydras.Epsilon0.T1]
+compare_of_phi0 [lemma, in hydras.Epsilon0.T1]
+compare_fin_rw [lemma, in hydras.Epsilon0.T1]
+compare_lt_iff [lemma, in hydras.Epsilon0.T1]
+compare_lt_impl [lemma, in hydras.Epsilon0.T1]
+compare_eq_iff [lemma, in hydras.Epsilon0.T1]
+compare_refl [lemma, in hydras.Epsilon0.T1]
+compare_correct [lemma, in hydras.Epsilon0.T1]
+compare_reflect [lemma, in hydras.Epsilon0.T1]
+compare_rev [lemma, in hydras.Epsilon0.T1]
+compare_cons [lemma, in hydras.Epsilon0.T1]
+compare_T1 [instance, in hydras.Epsilon0.T1]
+compare_correct [lemma, in hydras.OrdinalNotations.ON_Omega2]
+compare_reflect [lemma, in hydras.OrdinalNotations.ON_Omega2]
+compare_omega2 [instance, in hydras.OrdinalNotations.ON_Omega2]
+compare_reflect [lemma, in hydras.Prelude.Comparable]
+compare_trans [lemma, in hydras.Prelude.Comparable]
+compare_ge_iff [lemma, in hydras.Prelude.Comparable]
+compare_le_iff [lemma, in hydras.Prelude.Comparable]
+compare_le_iff_refl [lemma, in hydras.Prelude.Comparable]
+compare_gt_not_lt [lemma, in hydras.Prelude.Comparable]
+compare_lt_not_gt [lemma, in hydras.Prelude.Comparable]
+compare_gt_trans [lemma, in hydras.Prelude.Comparable]
+compare_gt_irrefl [lemma, in hydras.Prelude.Comparable]
+compare_gt_iff [lemma, in hydras.Prelude.Comparable]
+compare_eq_trans [lemma, in hydras.Prelude.Comparable]
+compare_refl [lemma, in hydras.Prelude.Comparable]
+compare_eq_iff [lemma, in hydras.Prelude.Comparable]
+compare_lt_irrefl [lemma, in hydras.Prelude.Comparable]
+compare_lt_trans [lemma, in hydras.Prelude.Comparable]
+compare_lt_iff [lemma, in hydras.Prelude.Comparable]
+compare_reflect [lemma, in hydras.OrdinalNotations.ON_O]
+compare_correct [lemma, in hydras.OrdinalNotations.ON_O]
+compare_O [instance, in hydras.OrdinalNotations.ON_O]
+compare_nat_nat [instance, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+compare_h2g [lemma, in gaia_hydras.T1Bridge]
+compare_g2h [lemma, in gaia_hydras.T1Bridge]
+compare_ref [lemma, in gaia_hydras.T1Bridge]
+compare_correct [lemma, in hydras.OrdinalNotations.ON_mult]
+compare_reflect [lemma, in hydras.OrdinalNotations.ON_mult]
+compare_mult [instance, in hydras.OrdinalNotations.ON_mult]
+compare_correct [lemma, in hydras.OrdinalNotations.ON_plus]
+compare_reflect [lemma, in hydras.OrdinalNotations.ON_plus]
+compare_plus [instance, in hydras.OrdinalNotations.ON_plus]
+compare_reflect [lemma, in hydras.OrdinalNotations.ON_Finite]
+compare_correct [lemma, in hydras.OrdinalNotations.ON_Finite]
+compare_fin [instance, in hydras.OrdinalNotations.ON_Finite]
+compare_correct [lemma, in hydras.Epsilon0.E0]
+compare_E0 [instance, in hydras.Epsilon0.E0]
+compare_nat [instance, in hydras.OrdinalNotations.ON_Omega]
+compare_rw_gt [lemma, in hydras.Gamma0.Gamma0]
+compare_rw_eq [lemma, in hydras.Gamma0.Gamma0]
+compare_rw_lt [lemma, in hydras.Gamma0.Gamma0]
+compare_Gt [lemma, in hydras.Gamma0.Gamma0]
+compare_Eq [lemma, in hydras.Gamma0.Gamma0]
+compare_Lt [lemma, in hydras.Gamma0.Gamma0]
+compare_correct [lemma, in hydras.Gamma0.Gamma0]
+compare_reflect [lemma, in hydras.Gamma0.Gamma0]
+compare_T2 [instance, in hydras.Gamma0.Gamma0]
+compare:2 [binder, in additions.Trace_exercise]
+comparison_eq_dec [instance, in hydras.Prelude.DecPreOrder]
+Compatibility [library]
+Compat815 [module, in hydras.Prelude.Compat815]
+Compat815 [library]
+Compat815.ind_0_1_SS [definition, in hydras.Prelude.Compat815]
+Compat815.le_not_lt [lemma, in hydras.Prelude.Compat815]
+Compat815.le_plus_r [lemma, in hydras.Prelude.Compat815]
+Compat815.le_lt_n_Sm [lemma, in hydras.Prelude.Compat815]
+Compat815.le_lt_or_eq [lemma, in hydras.Prelude.Compat815]
+Compat815.le_n_0_eq [lemma, in hydras.Prelude.Compat815]
+Compat815.lt_not_le [lemma, in hydras.Prelude.Compat815]
+Compat815.lt_n_S [lemma, in hydras.Prelude.Compat815]
+Compat815.lt_S_n [lemma, in hydras.Prelude.Compat815]
+Compat815.lt_n_Sm_le [lemma, in hydras.Prelude.Compat815]
+Compat815.mult_O_le [lemma, in hydras.Prelude.Compat815]
+Compat815.n_SSn [lemma, in hydras.Prelude.Compat815]
+Compat815.n_SSSn [lemma, in hydras.Prelude.Compat815]
+Compat815.plus_Snm_nSm [lemma, in hydras.Prelude.Compat815]
+composeFunc [constructor, in hydras.Ackermann.primRec]
+composeSigmaFormula [definition, in Goedel.PRrepresentable]
+composeSigmaRepresentable [lemma, in Goedel.PRrepresentable]
+compose1_NIsPR [instance, in hydras.Ackermann.primRec]
+compose1_3IsPR [instance, in hydras.Ackermann.primRec]
+compose1_2IsPR [instance, in hydras.Ackermann.primRec]
+compose1_1IsPR [instance, in hydras.Ackermann.primRec]
+compose2 [definition, in hydras.Ackermann.primRec]
+compose2Examples [section, in hydras.MoreAck.PrimRecExamples]
+compose2Examples.f [variable, in hydras.MoreAck.PrimRecExamples]
+compose2Examples.f' [variable, in hydras.MoreAck.PrimRecExamples]
+compose2Examples.g [variable, in hydras.MoreAck.PrimRecExamples]
+compose2Examples.g' [variable, in hydras.MoreAck.PrimRecExamples]
+compose2Examples.h [variable, in hydras.MoreAck.PrimRecExamples]
+compose2IsPR [instance, in hydras.Ackermann.primRec]
+compose2_4IsPR [instance, in hydras.Ackermann.primRec]
+compose2_3IsPR [instance, in hydras.Ackermann.primRec]
+compose2_2IsPR [instance, in hydras.Ackermann.primRec]
+compose2_1IsPR [instance, in hydras.Ackermann.primRec]
+compose2_3 [lemma, in hydras.MoreAck.PrimRecExamples]
+compose2_2 [lemma, in hydras.MoreAck.PrimRecExamples]
+compose2_1 [lemma, in hydras.MoreAck.PrimRecExamples]
+compose2_0 [lemma, in hydras.MoreAck.PrimRecExamples]
+compose3_3IsPR [instance, in hydras.Ackermann.primRec]
+compose3_2IsPR [instance, in hydras.Ackermann.primRec]
+compose3_1IsPR [instance, in hydras.Ackermann.primRec]
+compose4_3IsPR [instance, in hydras.Ackermann.primRec]
+compose4_2IsPR [instance, in hydras.Ackermann.primRec]
+Composition [section, in hydras.MoreAck.PrimRecExamples]
+CompositionProofs [section, in additions.AM]
+CompositionProofs.App [section, in additions.AM]
+CompositionProofs.App.cn [variable, in additions.AM]
+CompositionProofs.App.cp [variable, in additions.AM]
+CompositionProofs.App.Hn [variable, in additions.AM]
+CompositionProofs.App.Hp [variable, in additions.AM]
+CompositionProofs.App.n [variable, in additions.AM]
+CompositionProofs.App.p [variable, in additions.AM]
+CompositionProofs.FFK [section, in additions.AM]
+CompositionProofs.FFK.cp [variable, in additions.AM]
+CompositionProofs.FFK.cq [variable, in additions.AM]
+CompositionProofs.FFK.Hp [variable, in additions.AM]
+CompositionProofs.FFK.Hq [variable, in additions.AM]
+CompositionProofs.FFK.p [variable, in additions.AM]
+CompositionProofs.FFK.q [variable, in additions.AM]
+CompositionProofs.FK [section, in additions.AM]
+CompositionProofs.FK.cn [variable, in additions.AM]
+CompositionProofs.FK.Hn [variable, in additions.AM]
+CompositionProofs.FK.n [variable, in additions.AM]
+CompositionProofs.KFK [section, in additions.AM]
+CompositionProofs.KFK.Hpr [variable, in additions.AM]
+CompositionProofs.KFK.Hq [variable, in additions.AM]
+CompositionProofs.KFK.kpr [variable, in additions.AM]
+CompositionProofs.KFK.mq [variable, in additions.AM]
+CompositionProofs.KFK.p [variable, in additions.AM]
+CompositionProofs.KFK.q [variable, in additions.AM]
+CompositionProofs.KFK.r [variable, in additions.AM]
+Composition.f [variable, in hydras.MoreAck.PrimRecExamples]
+Composition.g [variable, in hydras.MoreAck.PrimRecExamples]
+Composition.h [variable, in hydras.MoreAck.PrimRecExamples]
+Composition.k [variable, in hydras.MoreAck.PrimRecExamples]
+Composition.t [variable, in hydras.MoreAck.PrimRecExamples]
+Composition.u [variable, in hydras.MoreAck.PrimRecExamples]
+Composition.v [variable, in hydras.MoreAck.PrimRecExamples]
+Composition.x [variable, in hydras.MoreAck.PrimRecExamples]
+Composition.y [variable, in hydras.MoreAck.PrimRecExamples]
+Composition.z [variable, in hydras.MoreAck.PrimRecExamples]
+computation [inductive, in additions.Addition_Chains]
+computation_equiv [definition, in additions.Euclidean_Chains]
+computation_eval_rw [lemma, in additions.Addition_Chains]
+computation_eval [definition, in additions.Addition_Chains]
+computation_execute [definition, in additions.Addition_Chains]
+computation_length [definition, in additions.Addition_Chains]
+computeEvalStrongRecHelp [lemma, in hydras.Ackermann.cPair]
+comp_res:96 [binder, in hydras.Prelude.Comparable]
+comp_equiv_equivalence [instance, in additions.Euclidean_Chains]
+Comp_equiv [instance, in additions.Euclidean_Chains]
+comp128 [definition, in additions.Addition_Chains]
+Con [definition, in Goedel.goedel2]
+config [definition, in additions.AM]
+config_equiv [definition, in additions.AM]
+cons [constructor, in hydras.Epsilon0.T1]
+Cons [instance, in hydras.Epsilon0.E0]
+Consistent [definition, in hydras.Ackermann.folProof]
+constant_to_standard_path [lemma, in gaia_hydras.GPaths]
+constant_to_standard_path [lemma, in hydras.Epsilon0.Paths]
+constant_to_standard [lemma, in hydras.Epsilon0.Paths]
+constant_to_standard_0 [lemma, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.delta [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.m [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.gamma [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.l [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.l_def [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.P [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.t [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.Hpa [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.Hpos [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.Halpha [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.n [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.beta [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.alpha [variable, in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof [section, in hydras.Epsilon0.Paths]
+consTerms [lemma, in hydras.Ackermann.fol]
+const_path [definition, in gaia_hydras.GPaths]
+const_pathS_LT' [lemma, in hydras.Epsilon0.Paths]
+const_pathS_eps_zero [lemma, in hydras.Epsilon0.Paths]
+const_pathS_first_step [lemma, in hydras.Epsilon0.Paths]
+const_pathS_eps_LE_2 [lemma, in hydras.Epsilon0.Paths]
+const_pathS_eps_trans [lemma, in hydras.Epsilon0.Paths]
+const_pathS_trans [lemma, in hydras.Epsilon0.Paths]
+const_pathS_inv_strong [lemma, in hydras.Epsilon0.Paths]
+const_pathS_inv [lemma, in hydras.Epsilon0.Paths]
+const_pathS_LE [lemma, in hydras.Epsilon0.Paths]
+const_path_LT [lemma, in hydras.Epsilon0.Paths]
+const_pathS_LT [lemma, in hydras.Epsilon0.Paths]
+const_pathS_zero [lemma, in hydras.Epsilon0.Paths]
+const_pathS_nf [lemma, in hydras.Epsilon0.Paths]
+const_pathS_repeatR [lemma, in hydras.Epsilon0.Paths]
+const_pathS_repeat [lemma, in hydras.Epsilon0.Paths]
+const_pathS_eps [definition, in hydras.Epsilon0.Paths]
+const_pathSE [lemma, in hydras.Epsilon0.Paths]
+const_path [definition, in hydras.Epsilon0.Paths]
+const_pathS [definition, in hydras.Epsilon0.Paths]
+const0_NIsPR [instance, in hydras.Ackermann.primRec]
+const1_NIsPR [instance, in hydras.Ackermann.primRec]
+cons_def [lemma, in hydras.Epsilon0.T1]
+cons_nf [lemma, in hydras.Epsilon0.T1]
+cons_succ [constructor, in hydras.Gamma0.T2]
+cons_standard_path [lemma, in hydras.Epsilon0.Paths]
+cons_standard_pathS [lemma, in hydras.Epsilon0.Paths]
+cons_lt_epsilon0 [lemma, in hydras.Gamma0.Gamma0]
+cons_rw [lemma, in hydras.Gamma0.Gamma0]
+continuous [definition, in hydras.Schutte.Schutte_basics]
+contrad [lemma, in hydras.solutions_exercises.MultisetWf]
+contrad [lemma, in hydras.solutions_exercises.T1_ltNotWf]
+contradiction [lemma, in hydras.Ackermann.LNN]
+contradiction [lemma, in hydras.Ackermann.LNT]
+contradiction [lemma, in hydras.Ackermann.folLogic]
+Contradiction [lemma, in hydras.Hydra.Omega_Small]
+Correctness [section, in hydras.Ackermann.fol]
+correctness_statement [definition, in additions.AM]
+correctness_statement [definition, in additions.Euclidean_Chains]
+Correctness_E0 [library]
+Correctness.L [variable, in hydras.Ackermann.fol]
+Correctness.P [variable, in hydras.Ackermann.fol]
+Correctness.Q [variable, in hydras.Ackermann.fol]
+Correctness.R [variable, in hydras.Ackermann.fol]
+correct_app [lemma, in additions.AM]
+correct_generator [definition, in additions.Addition_Chains]
+Cor12 [lemma, in gaia_hydras.GPaths]
+Cor12 [lemma, in hydras.Epsilon0.Paths]
+Cor12_E0 [lemma, in hydras.Epsilon0.Paths]
+Cor12_3 [lemma, in hydras.Epsilon0.Paths]
+Cor12_2 [lemma, in hydras.Epsilon0.Paths]
+Cor12_1 [lemma, in hydras.Epsilon0.Paths]
+Countable [section, in hydras.Schutte.Countable]
+Countable [library]
+countable_members [lemma, in hydras.Schutte.Schutte_basics]
+countable_not_Unbounded [lemma, in hydras.Schutte.Schutte_basics]
+countable_segment_proper [lemma, in hydras.Schutte.Ordering_Functions]
+countable_image [lemma, in hydras.Schutte.Countable]
+countable_bij_funR [lemma, in hydras.Schutte.Countable]
+countable_bij_fun [lemma, in hydras.Schutte.Countable]
+countable_surj [lemma, in hydras.Schutte.Countable]
+Countable.Countable_bijection.g_bij [variable, in hydras.Schutte.Countable]
+Countable.Countable_bijection.g [variable, in hydras.Schutte.Countable]
+Countable.Countable_bijection.B [variable, in hydras.Schutte.Countable]
+Countable.Countable_bijection.A [variable, in hydras.Schutte.Countable]
+Countable.Countable_bijection.V [variable, in hydras.Schutte.Countable]
+Countable.Countable_bijection [section, in hydras.Schutte.Countable]
+Countable.Countable_seq_range [section, in hydras.Schutte.Countable]
+Countable.Definitions [section, in hydras.Schutte.Countable]
+Countable.Definitions.A [variable, in hydras.Schutte.Countable]
+Countable.Definitions.U [variable, in hydras.Schutte.Countable]
+Countable.U [variable, in hydras.Schutte.Countable]
+Counter_Example.is_in_seq [variable, in hydras.solutions_exercises.MultisetWf]
+Counter_Example.Hwf [variable, in hydras.solutions_exercises.MultisetWf]
+Counter_Example.R [variable, in hydras.solutions_exercises.MultisetWf]
+Counter_Example [section, in hydras.solutions_exercises.MultisetWf]
+counter_ex [lemma, in hydras.solutions_exercises.schutte_cnf_counter_example]
+count:9 [binder, in additions.AM]
+CP [constructor, in hydras.Ackermann.folProof]
+cPair [definition, in hydras.Ackermann.cPair]
+cPair [library]
+cPairInjHelp [lemma, in hydras.Ackermann.cPair]
+cPairInj1 [lemma, in hydras.Ackermann.cPair]
+cPairInj2 [lemma, in hydras.Ackermann.cPair]
+cPairIsPR [instance, in hydras.Ackermann.cPair]
+cPairLemma1 [lemma, in hydras.Ackermann.cPair]
+cPairLe1 [lemma, in hydras.Ackermann.cPair]
+cPairLe1A [lemma, in hydras.Ackermann.cPair]
+cPairLe2 [lemma, in hydras.Ackermann.cPair]
+cPairLe2A [lemma, in hydras.Ackermann.cPair]
+cPairLe3 [lemma, in hydras.Ackermann.cPair]
+cPairLt1 [lemma, in hydras.Ackermann.cPair]
+cPairLt2 [lemma, in hydras.Ackermann.cPair]
+cPairPi1 [definition, in hydras.Ackermann.cPair]
+cPairPi1IsPR [instance, in hydras.Ackermann.cPair]
+cPairPi2 [definition, in hydras.Ackermann.cPair]
+cPairPi2IsPR [instance, in hydras.Ackermann.cPair]
+cPairProjections [lemma, in hydras.Ackermann.cPair]
+cPairProjectionsHelp [lemma, in hydras.Ackermann.cPair]
+cPairProjections1 [lemma, in hydras.Ackermann.cPair]
+cPairProjections2 [lemma, in hydras.Ackermann.cPair]
+CPair_Order [section, in hydras.Ackermann.cPair]
+CPair_projections.searchXY [variable, in hydras.Ackermann.cPair]
+CPair_projections [section, in hydras.Ackermann.cPair]
+CPair_Injectivity [section, in hydras.Ackermann.cPair]
+cpower [definition, in additions.Addition_Chains]
+cpower_pos [definition, in additions.Addition_Chains]
+cp1 [lemma, in hydras.Ackermann.LNN]
+cp1 [lemma, in hydras.Ackermann.LNT]
+cp1 [lemma, in hydras.Ackermann.folLogic]
+cp2 [lemma, in hydras.Ackermann.LNN]
+cp2 [lemma, in hydras.Ackermann.LNT]
+cp2 [lemma, in hydras.Ackermann.folLogic]
+Cr [definition, in hydras.Schutte.Critical]
+Critical [library]
+Cr_1_iff [lemma, in hydras.Schutte.Critical]
+Cr_incl [lemma, in hydras.Schutte.Critical]
+Cr_lt [lemma, in hydras.Schutte.Critical]
+Cr_pos_iff [lemma, in hydras.Schutte.Critical]
+Cr_pos_inv [lemma, in hydras.Schutte.Critical]
+Cr_zero_AP [lemma, in hydras.Schutte.Critical]
+Cr_zero_inv [lemma, in hydras.Schutte.Critical]
+Cr_pos [lemma, in hydras.Schutte.Critical]
+Cr_zero [lemma, in hydras.Schutte.Critical]
+Cr_inv [lemma, in hydras.Schutte.Critical]
+Cr_equation [lemma, in hydras.Schutte.Critical]
+Cr_extensional [lemma, in hydras.Schutte.Critical]
+Cr_fun [definition, in hydras.Schutte.Critical]
+cr:12 [binder, in hydras.Ackermann.code]
+cr:15 [binder, in hydras.Ackermann.code]
+cr:18 [binder, in hydras.Ackermann.code]
+cr:3 [binder, in hydras.Ackermann.codeFreeVar]
+cr:3 [binder, in hydras.Ackermann.codeSubTerm]
+cr:3 [binder, in hydras.Ackermann.checkPrf]
+cr:3 [binder, in hydras.Ackermann.codeSubFormula]
+cr:3 [binder, in hydras.Ackermann.wellFormed]
+cr:3 [binder, in Goedel.codeSysPrf]
+cr:3 [binder, in hydras.Ackermann.codePA]
+Cr:5 [binder, in hydras.Schutte.Critical]
+cst_correct [lemma, in hydras.MoreAck.PrimRecExamples]
+cst0_correct [lemma, in hydras.MoreAck.PrimRecExamples]
+Cter_example.l [variable, in hydras.solutions_exercises.schutte_cnf_counter_example]
+Cter_example.alpha [variable, in hydras.solutions_exercises.schutte_cnf_counter_example]
+Cter_example.cnf_lt_epsilon0_iff [variable, in hydras.solutions_exercises.schutte_cnf_counter_example]
+Cter_example [section, in hydras.solutions_exercises.schutte_cnf_counter_example]
+cTriple [definition, in hydras.Ackermann.cPair]
+cTripleIsPR [instance, in hydras.Ackermann.cPair]
+cTriplePi1 [definition, in hydras.Ackermann.cPair]
+cTriplePi1IsPR [instance, in hydras.Ackermann.cPair]
+cTriplePi2 [definition, in hydras.Ackermann.cPair]
+cTriplePi2IsPR [instance, in hydras.Ackermann.cPair]
+cTriplePi3 [definition, in hydras.Ackermann.cPair]
+cTriplePi3IsPR [instance, in hydras.Ackermann.cPair]
+cTripleProj [lemma, in hydras.Ackermann.cPair]
+cTripleProj1 [lemma, in hydras.Ackermann.cPair]
+cTripleProj2 [lemma, in hydras.Ackermann.cPair]
+cTripleProj3 [lemma, in hydras.Ackermann.cPair]
+ctx:317 [binder, in hydras.rpo.rpo]
+ctx:89 [binder, in hydras.rpo.rpo]
+c'':46 [binder, in hydras.Hydra.BigBattle]
+c':1025 [binder, in gaia_hydras.nfwfgaia]
+c':1153 [binder, in gaia_hydras.nfwfgaia]
+c':1337 [binder, in gaia_hydras.nfwfgaia]
+c':1383 [binder, in gaia_hydras.nfwfgaia]
+c':1393 [binder, in gaia_hydras.nfwfgaia]
+c':1421 [binder, in gaia_hydras.nfwfgaia]
+c':1427 [binder, in gaia_hydras.nfwfgaia]
+c':1435 [binder, in gaia_hydras.nfwfgaia]
+c':1499 [binder, in gaia_hydras.nfwfgaia]
+c':1507 [binder, in gaia_hydras.nfwfgaia]
+c':1550 [binder, in gaia_hydras.nfwfgaia]
+c':1584 [binder, in gaia_hydras.nfwfgaia]
+c':1608 [binder, in gaia_hydras.nfwfgaia]
+c':1617 [binder, in gaia_hydras.nfwfgaia]
+c':1619 [binder, in gaia_hydras.nfwfgaia]
+c':1620 [binder, in gaia_hydras.nfwfgaia]
+c':1626 [binder, in gaia_hydras.nfwfgaia]
+c':1628 [binder, in gaia_hydras.nfwfgaia]
+c':1634 [binder, in gaia_hydras.nfwfgaia]
+c':1690 [binder, in gaia_hydras.nfwfgaia]
+c':1696 [binder, in gaia_hydras.nfwfgaia]
+c':2 [binder, in hydras.Epsilon0.F_alpha]
+c':20 [binder, in additions.AM]
+c':2065 [binder, in gaia_hydras.nfwfgaia]
+c':2079 [binder, in gaia_hydras.nfwfgaia]
+c':2084 [binder, in gaia_hydras.nfwfgaia]
+c':25 [binder, in hydras.Hydra.BigBattle]
+c':26 [binder, in additions.Monoid_instances]
+c':28 [binder, in hydras.Gamma0.Gamma0]
+c':303 [binder, in hydras.Gamma0.Gamma0]
+c':34 [binder, in hydras.Hydra.BigBattle]
+c':42 [binder, in hydras.Hydra.BigBattle]
+c':46 [binder, in additions.Euclidean_Chains]
+c':541 [binder, in hydras.Ackermann.primRec]
+c':58 [binder, in additions.AM]
+c':58 [binder, in hydras.Hydra.BigBattle]
+c':584 [binder, in hydras.Gamma0.Gamma0]
+c':81 [binder, in additions.Addition_Chains]
+c':84 [binder, in hydras.Gamma0.T2]
+c':872 [binder, in gaia_hydras.nfwfgaia]
+c':909 [binder, in gaia_hydras.nfwfgaia]
+c':917 [binder, in gaia_hydras.nfwfgaia]
+c':949 [binder, in gaia_hydras.nfwfgaia]
+c00 [projection, in additions.Monoid_instances]
+c01 [projection, in additions.Monoid_instances]
+C1 [definition, in additions.Addition_Chains]
+c10 [projection, in additions.Monoid_instances]
+c11 [projection, in additions.Monoid_instances]
+c153 [definition, in additions.Fib2]
+c1:1173 [binder, in gaia_hydras.nfwfgaia]
+c1:1364 [binder, in hydras.Ackermann.codeSubFormula]
+c1:1817 [binder, in gaia_hydras.nfwfgaia]
+c1:470 [binder, in gaia_hydras.nfwfgaia]
+c1:598 [binder, in gaia_hydras.nfwfgaia]
+c1:73 [binder, in hydras.Gamma0.Gamma0]
+c1:80 [binder, in hydras.Gamma0.Gamma0]
+c1:85 [binder, in hydras.Gamma0.Gamma0]
+c2:1174 [binder, in gaia_hydras.nfwfgaia]
+c2:1365 [binder, in hydras.Ackermann.codeSubFormula]
+c2:1818 [binder, in gaia_hydras.nfwfgaia]
+c2:471 [binder, in gaia_hydras.nfwfgaia]
+c2:599 [binder, in gaia_hydras.nfwfgaia]
+c2:74 [binder, in hydras.Gamma0.Gamma0]
+c2:81 [binder, in hydras.Gamma0.Gamma0]
+c2:86 [binder, in hydras.Gamma0.Gamma0]
+C3 [definition, in additions.Addition_Chains]
+C31_7 [definition, in additions.AM]
+c3:1175 [binder, in gaia_hydras.nfwfgaia]
+c3:1819 [binder, in gaia_hydras.nfwfgaia]
+c3:472 [binder, in gaia_hydras.nfwfgaia]
+C7 [definition, in additions.Addition_Chains]
+C7_ok' [definition, in additions.Addition_Chains]
+C7_ok [definition, in additions.Addition_Chains]
+C87 [definition, in additions.Addition_Chains]
+C87_ok' [lemma, in additions.Addition_Chains]
+C87_ok [definition, in additions.Addition_Chains]
+C87' [definition, in additions.Euclidean_Chains]
+c:1 [binder, in hydras.Epsilon0.F_alpha]
+c:1001 [binder, in gaia_hydras.nfwfgaia]
+c:1004 [binder, in gaia_hydras.nfwfgaia]
+c:1007 [binder, in gaia_hydras.nfwfgaia]
+c:101 [binder, in hydras.Epsilon0.Hessenberg]
+c:101 [binder, in hydras.Ackermann.primRec]
+c:1011 [binder, in gaia_hydras.nfwfgaia]
+c:1032 [binder, in gaia_hydras.nfwfgaia]
+c:1037 [binder, in gaia_hydras.nfwfgaia]
+c:104 [binder, in hydras.MoreAck.PrimRecExamples]
+c:105 [binder, in hydras.Epsilon0.Hessenberg]
+c:1052 [binder, in gaia_hydras.nfwfgaia]
+c:108 [binder, in hydras.Epsilon0.Hessenberg]
+c:113 [binder, in hydras.Gamma0.T2]
+c:113 [binder, in additions.AM]
+c:1147 [binder, in gaia_hydras.nfwfgaia]
+c:115 [binder, in hydras.Ackermann.cPair]
+c:1168 [binder, in gaia_hydras.nfwfgaia]
+c:117 [binder, in hydras.Epsilon0.Hessenberg]
+c:12 [binder, in hydras.Gamma0.Gamma0]
+c:1207 [binder, in gaia_hydras.nfwfgaia]
+c:121 [binder, in hydras.Epsilon0.F_alpha]
+c:1210 [binder, in gaia_hydras.nfwfgaia]
+C:123 [binder, in hydras.rpo.more_list]
+c:1254 [binder, in gaia_hydras.nfwfgaia]
+c:1257 [binder, in gaia_hydras.nfwfgaia]
+c:1260 [binder, in gaia_hydras.nfwfgaia]
+c:1263 [binder, in gaia_hydras.nfwfgaia]
+c:128 [binder, in hydras.Epsilon0.T1]
+c:13 [binder, in additions.AM]
+c:13 [binder, in hydras.Ackermann.code]
+c:1322 [binder, in gaia_hydras.nfwfgaia]
+c:133 [binder, in hydras.Epsilon0.Hessenberg]
+c:1332 [binder, in gaia_hydras.nfwfgaia]
+c:1347 [binder, in gaia_hydras.nfwfgaia]
+c:1352 [binder, in gaia_hydras.nfwfgaia]
+c:1357 [binder, in gaia_hydras.nfwfgaia]
+c:1362 [binder, in gaia_hydras.nfwfgaia]
+c:1367 [binder, in gaia_hydras.nfwfgaia]
+C:1369 [binder, in hydras.Ackermann.codeSubFormula]
+c:1372 [binder, in gaia_hydras.nfwfgaia]
+c:1378 [binder, in gaia_hydras.nfwfgaia]
+c:139 [binder, in hydras.Ackermann.primRec]
+c:1390 [binder, in gaia_hydras.nfwfgaia]
+c:14 [binder, in hydras.Prelude.MoreDecidable]
+c:14 [binder, in hydras.rpo.closure]
+C:1407 [binder, in hydras.Ackermann.codeSubFormula]
+c:1418 [binder, in gaia_hydras.nfwfgaia]
+c:142 [binder, in hydras.Ackermann.primRec]
+c:1424 [binder, in gaia_hydras.nfwfgaia]
+c:143 [binder, in hydras.Gamma0.T2]
+c:1430 [binder, in gaia_hydras.nfwfgaia]
+c:144 [binder, in hydras.Ackermann.LNT]
+c:144 [binder, in hydras.Gamma0.Gamma0]
+c:145 [binder, in hydras.Ackermann.primRec]
+c:145 [binder, in additions.fib]
+c:147 [binder, in hydras.Ackermann.LNN]
+c:148 [binder, in hydras.Ackermann.LNT]
+c:148 [binder, in hydras.Ackermann.primRec]
+c:149 [binder, in hydras.Gamma0.T2]
+c:1494 [binder, in gaia_hydras.nfwfgaia]
+c:15 [binder, in hydras.solutions_exercises.FibonacciPR]
+c:15 [binder, in hydras.Hydra.BigBattle]
+c:1504 [binder, in gaia_hydras.nfwfgaia]
+c:151 [binder, in hydras.Ackermann.LNN]
+C:151 [binder, in hydras.Ackermann.fol]
+c:1510 [binder, in gaia_hydras.nfwfgaia]
+c:152 [binder, in hydras.Ackermann.primRec]
+c:1522 [binder, in gaia_hydras.nfwfgaia]
+c:1525 [binder, in gaia_hydras.nfwfgaia]
+c:1528 [binder, in gaia_hydras.nfwfgaia]
+c:153 [binder, in hydras.Ackermann.LNT]
+c:1533 [binder, in gaia_hydras.nfwfgaia]
+c:1538 [binder, in gaia_hydras.nfwfgaia]
+c:1542 [binder, in gaia_hydras.nfwfgaia]
+c:1545 [binder, in gaia_hydras.nfwfgaia]
+c:1553 [binder, in gaia_hydras.nfwfgaia]
+c:156 [binder, in hydras.Ackermann.LNN]
+c:156 [binder, in hydras.Ackermann.primRec]
+c:1560 [binder, in gaia_hydras.nfwfgaia]
+c:1563 [binder, in gaia_hydras.nfwfgaia]
+c:1566 [binder, in gaia_hydras.nfwfgaia]
+c:157 [binder, in additions.AM]
+c:1570 [binder, in gaia_hydras.nfwfgaia]
+c:1576 [binder, in gaia_hydras.nfwfgaia]
+c:1580 [binder, in gaia_hydras.nfwfgaia]
+c:1589 [binder, in gaia_hydras.nfwfgaia]
+c:16 [binder, in hydras.Ackermann.extEqualNat]
+c:16 [binder, in hydras.Ackermann.code]
+c:16 [binder, in additions.Euclidean_Chains]
+c:16 [binder, in hydras.Ackermann.folLogic3]
+c:16 [binder, in hydras.MoreAck.PrimRecExamples]
+c:160 [binder, in hydras.Ackermann.primRec]
+c:1603 [binder, in gaia_hydras.nfwfgaia]
+c:1614 [binder, in gaia_hydras.nfwfgaia]
+c:1623 [binder, in gaia_hydras.nfwfgaia]
+c:1631 [binder, in gaia_hydras.nfwfgaia]
+c:1637 [binder, in gaia_hydras.nfwfgaia]
+c:164 [binder, in hydras.Ackermann.LNN]
+c:164 [binder, in hydras.Gamma0.Gamma0]
+c:1647 [binder, in gaia_hydras.nfwfgaia]
+c:1652 [binder, in gaia_hydras.nfwfgaia]
+c:1662 [binder, in gaia_hydras.nfwfgaia]
+c:1673 [binder, in gaia_hydras.nfwfgaia]
+c:1676 [binder, in gaia_hydras.nfwfgaia]
+c:168 [binder, in hydras.Gamma0.Gamma0]
+c:1685 [binder, in gaia_hydras.nfwfgaia]
+c:1693 [binder, in gaia_hydras.nfwfgaia]
+c:17 [binder, in hydras.solutions_exercises.FibonacciPR]
+c:172 [binder, in hydras.Gamma0.Gamma0]
+c:1747 [binder, in gaia_hydras.nfwfgaia]
+c:1751 [binder, in gaia_hydras.nfwfgaia]
+c:1754 [binder, in gaia_hydras.nfwfgaia]
+c:179 [binder, in additions.Addition_Chains]
+c:1793 [binder, in gaia_hydras.nfwfgaia]
+c:180 [binder, in additions.Addition_Chains]
+c:181 [binder, in additions.Addition_Chains]
+c:1812 [binder, in gaia_hydras.nfwfgaia]
+c:182 [binder, in hydras.Ackermann.primRec]
+c:185 [binder, in hydras.Ackermann.primRec]
+c:1858 [binder, in gaia_hydras.nfwfgaia]
+c:187 [binder, in additions.Addition_Chains]
+c:1879 [binder, in gaia_hydras.nfwfgaia]
+c:188 [binder, in hydras.Ackermann.primRec]
+c:1888 [binder, in gaia_hydras.nfwfgaia]
+c:19 [binder, in additions.AM]
+c:19 [binder, in hydras.solutions_exercises.FibonacciPR]
+c:190 [binder, in hydras.Ackermann.cPair]
+c:193 [binder, in hydras.Ackermann.primRec]
+c:196 [binder, in hydras.Epsilon0.T1]
+c:196 [binder, in hydras.Ackermann.primRec]
+c:199 [binder, in hydras.Ackermann.primRec]
+c:20 [binder, in hydras.Prelude.MoreDecidable]
+c:2008 [binder, in gaia_hydras.nfwfgaia]
+c:2023 [binder, in gaia_hydras.nfwfgaia]
+c:2027 [binder, in gaia_hydras.nfwfgaia]
+c:2032 [binder, in gaia_hydras.nfwfgaia]
+c:2039 [binder, in gaia_hydras.nfwfgaia]
+c:204 [binder, in hydras.Ackermann.primRec]
+c:2046 [binder, in gaia_hydras.nfwfgaia]
+c:2049 [binder, in gaia_hydras.nfwfgaia]
+c:205 [binder, in additions.Addition_Chains]
+c:2055 [binder, in gaia_hydras.nfwfgaia]
+c:2060 [binder, in gaia_hydras.nfwfgaia]
+c:207 [binder, in hydras.Ackermann.primRec]
+c:207 [binder, in additions.Addition_Chains]
+c:2072 [binder, in gaia_hydras.nfwfgaia]
+c:2075 [binder, in gaia_hydras.nfwfgaia]
+c:208 [binder, in additions.Addition_Chains]
+c:208 [binder, in gaia_hydras.nfwfgaia]
+c:21 [binder, in hydras.solutions_exercises.FibonacciPR]
+c:210 [binder, in hydras.Ackermann.primRec]
+c:211 [binder, in gaia_hydras.nfwfgaia]
+c:213 [binder, in additions.Addition_Chains]
+c:214 [binder, in additions.Addition_Chains]
+c:214 [binder, in gaia_hydras.nfwfgaia]
+c:215 [binder, in hydras.Ackermann.primRec]
+c:215 [binder, in hydras.Ackermann.cPair]
+c:216 [binder, in additions.Addition_Chains]
+c:217 [binder, in gaia_hydras.nfwfgaia]
+c:218 [binder, in hydras.Ackermann.primRec]
+c:218 [binder, in additions.Addition_Chains]
+c:22 [binder, in hydras.Hydra.BigBattle]
+c:22 [binder, in additions.Monoid_instances]
+c:220 [binder, in hydras.Gamma0.Gamma0]
+c:221 [binder, in hydras.Ackermann.primRec]
+c:224 [binder, in hydras.Ackermann.primRec]
+c:226 [binder, in hydras.Gamma0.Gamma0]
+c:227 [binder, in hydras.Ackermann.primRec]
+c:23 [binder, in additions.Trace_exercise]
+c:230 [binder, in hydras.Gamma0.Gamma0]
+c:231 [binder, in hydras.Ackermann.primRec]
+c:234 [binder, in hydras.Ackermann.primRec]
+c:234 [binder, in hydras.Gamma0.Gamma0]
+c:237 [binder, in hydras.Ackermann.primRec]
+c:24 [binder, in hydras.Prelude.Comparable]
+c:24 [binder, in additions.Trace_exercise]
+c:240 [binder, in hydras.Ackermann.primRec]
+c:243 [binder, in hydras.Ackermann.primRec]
+c:247 [binder, in hydras.Ackermann.primRec]
+c:25 [binder, in gaia_hydras.GHessenberg]
+c:250 [binder, in hydras.Ackermann.primRec]
+c:253 [binder, in hydras.Ackermann.primRec]
+c:256 [binder, in hydras.Ackermann.primRec]
+c:259 [binder, in hydras.Ackermann.primRec]
+c:26 [binder, in additions.Fib2]
+c:26 [binder, in hydras.Prelude.More_Arith]
+C:26 [binder, in hydras.Ackermann.fol]
+c:26 [binder, in hydras.Gamma0.Gamma0]
+c:263 [binder, in hydras.Ackermann.primRec]
+C:267 [binder, in hydras.Ackermann.checkPrf]
+c:267 [binder, in hydras.Ackermann.primRec]
+c:270 [binder, in hydras.Gamma0.Gamma0]
+c:271 [binder, in hydras.Ackermann.primRec]
+c:275 [binder, in hydras.Ackermann.primRec]
+c:279 [binder, in hydras.Ackermann.primRec]
+c:28 [binder, in additions.Fib2]
+c:284 [binder, in hydras.Ackermann.primRec]
+c:286 [binder, in hydras.Gamma0.Gamma0]
+c:288 [binder, in hydras.Ackermann.primRec]
+c:288 [binder, in additions.Addition_Chains]
+c:290 [binder, in hydras.Gamma0.Gamma0]
+c:292 [binder, in hydras.Ackermann.primRec]
+c:294 [binder, in hydras.Gamma0.Gamma0]
+c:297 [binder, in hydras.Ackermann.primRec]
+c:297 [binder, in additions.Addition_Chains]
+c:298 [binder, in hydras.Gamma0.Gamma0]
+c:3 [binder, in additions.AM]
+C:3 [binder, in hydras.rpo.more_list]
+c:3 [binder, in hydras.Epsilon0.F_alpha]
+c:3 [binder, in hydras.Prelude.MoreDecidable]
+c:301 [binder, in hydras.Ackermann.primRec]
+c:305 [binder, in hydras.Ackermann.primRec]
+c:305 [binder, in gaia_hydras.nfwfgaia]
+c:308 [binder, in Goedel.PRrepresentable]
+c:308 [binder, in gaia_hydras.nfwfgaia]
+c:309 [binder, in hydras.Ackermann.primRec]
+c:309 [binder, in hydras.Gamma0.Gamma0]
+C:31 [binder, in hydras.Ackermann.folProof]
+c:31 [binder, in hydras.Prelude.Comparable]
+c:31 [binder, in hydras.Hydra.BigBattle]
+c:312 [binder, in hydras.Epsilon0.T1]
+c:313 [binder, in hydras.Ackermann.primRec]
+c:317 [binder, in hydras.Ackermann.primRec]
+c:32 [binder, in hydras.Ackermann.extEqualNat]
+c:32 [binder, in hydras.Gamma0.Gamma0]
+c:32 [binder, in gaia_hydras.GHessenberg]
+c:321 [binder, in hydras.Ackermann.primRec]
+c:326 [binder, in hydras.Ackermann.primRec]
+c:33 [binder, in hydras.Ackermann.folLogic3]
+c:330 [binder, in hydras.Ackermann.primRec]
+c:334 [binder, in hydras.Ackermann.primRec]
+c:338 [binder, in hydras.Ackermann.primRec]
+c:34 [binder, in additions.AM]
+c:340 [binder, in Goedel.PRrepresentable]
+c:342 [binder, in Goedel.PRrepresentable]
+c:342 [binder, in hydras.Ackermann.primRec]
+c:342 [binder, in hydras.Ackermann.cPair]
+c:344 [binder, in Goedel.PRrepresentable]
+c:345 [binder, in Goedel.PRrepresentable]
+c:348 [binder, in hydras.Ackermann.cPair]
+c:35 [binder, in hydras.Ackermann.cPair]
+c:35 [binder, in gaia_hydras.GHessenberg]
+c:351 [binder, in hydras.Ackermann.cPair]
+c:352 [binder, in additions.Euclidean_Chains]
+c:353 [binder, in Goedel.PRrepresentable]
+c:354 [binder, in hydras.Ackermann.cPair]
+c:357 [binder, in Goedel.PRrepresentable]
+c:357 [binder, in hydras.Ackermann.cPair]
+c:359 [binder, in hydras.Ackermann.cPair]
+c:36 [binder, in hydras.Prelude.MoreVectors]
+c:361 [binder, in Goedel.PRrepresentable]
+c:361 [binder, in hydras.Ackermann.cPair]
+c:364 [binder, in hydras.Ackermann.cPair]
+c:365 [binder, in Goedel.PRrepresentable]
+c:367 [binder, in hydras.Ackermann.cPair]
+c:37 [binder, in hydras.Prelude.Comparable]
+c:370 [binder, in hydras.Ackermann.cPair]
+c:38 [binder, in hydras.Hydra.BigBattle]
+c:38 [binder, in gaia_hydras.GHessenberg]
+c:387 [binder, in Goedel.PRrepresentable]
+c:39 [binder, in hydras.OrdinalNotations.OmegaOmega]
+c:39 [binder, in hydras.Schutte.Well_Orders]
+c:391 [binder, in Goedel.PRrepresentable]
+c:395 [binder, in Goedel.PRrepresentable]
+c:399 [binder, in Goedel.PRrepresentable]
+c:4 [binder, in additions.AM]
+c:4 [binder, in hydras.Gamma0.Gamma0]
+c:40 [binder, in additions.Addition_Chains]
+c:401 [binder, in gaia_hydras.nfwfgaia]
+c:407 [binder, in Goedel.PRrepresentable]
+c:407 [binder, in gaia_hydras.nfwfgaia]
+c:410 [binder, in gaia_hydras.nfwfgaia]
+c:411 [binder, in Goedel.PRrepresentable]
+c:414 [binder, in hydras.Epsilon0.T1]
+c:414 [binder, in gaia_hydras.nfwfgaia]
+c:415 [binder, in Goedel.PRrepresentable]
+c:419 [binder, in Goedel.PRrepresentable]
+c:43 [binder, in hydras.Ackermann.cPair]
+c:43 [binder, in additions.Addition_Chains]
+c:432 [binder, in hydras.Gamma0.Gamma0]
+c:438 [binder, in hydras.Ackermann.primRec]
+c:444 [binder, in hydras.Epsilon0.T1]
+c:445 [binder, in hydras.Epsilon0.T1]
+c:45 [binder, in additions.Euclidean_Chains]
+c:450 [binder, in hydras.Epsilon0.T1]
+c:460 [binder, in hydras.Epsilon0.T1]
+c:461 [binder, in gaia_hydras.nfwfgaia]
+c:463 [binder, in hydras.Epsilon0.T1]
+c:47 [binder, in hydras.Ackermann.cPair]
+c:480 [binder, in hydras.Gamma0.Gamma0]
+c:49 [binder, in hydras.Prelude.More_Arith]
+c:50 [binder, in hydras.Prelude.Comparable]
+c:50 [binder, in hydras.Hydra.BigBattle]
+c:505 [binder, in hydras.Epsilon0.T1]
+c:51 [binder, in gaia_hydras.T1Bridge]
+c:51 [binder, in hydras.Ackermann.cPair]
+c:51 [binder, in additions.Addition_Chains]
+c:52 [binder, in additions.Addition_Chains]
+c:526 [binder, in hydras.Gamma0.Gamma0]
+c:529 [binder, in hydras.Gamma0.Gamma0]
+c:53 [binder, in hydras.Prelude.Comparable]
+c:532 [binder, in hydras.Gamma0.Gamma0]
+c:54 [binder, in hydras.Hydra.BigBattle]
+c:540 [binder, in hydras.Ackermann.primRec]
+c:55 [binder, in hydras.Ackermann.cPair]
+c:558 [binder, in hydras.Gamma0.Gamma0]
+c:56 [binder, in hydras.Prelude.Comparable]
+c:56 [binder, in additions.Addition_Chains]
+c:565 [binder, in hydras.Epsilon0.T1]
+c:567 [binder, in hydras.Gamma0.Gamma0]
+c:57 [binder, in additions.AM]
+c:573 [binder, in hydras.Epsilon0.T1]
+c:574 [binder, in hydras.Gamma0.Gamma0]
+c:577 [binder, in hydras.Epsilon0.T1]
+c:578 [binder, in hydras.Gamma0.Gamma0]
+c:580 [binder, in hydras.Gamma0.Gamma0]
+c:581 [binder, in hydras.Epsilon0.T1]
+c:583 [binder, in hydras.Gamma0.Gamma0]
+c:59 [binder, in hydras.Hydra.BigBattle]
+c:591 [binder, in hydras.Gamma0.Gamma0]
+c:594 [binder, in hydras.Gamma0.Gamma0]
+c:599 [binder, in hydras.Gamma0.Gamma0]
+c:604 [binder, in hydras.Epsilon0.T1]
+c:607 [binder, in hydras.Epsilon0.T1]
+c:611 [binder, in hydras.Epsilon0.T1]
+c:614 [binder, in hydras.Epsilon0.T1]
+c:614 [binder, in hydras.Ackermann.primRec]
+c:617 [binder, in hydras.Epsilon0.T1]
+c:621 [binder, in hydras.Epsilon0.T1]
+c:622 [binder, in hydras.Ackermann.primRec]
+c:623 [binder, in hydras.Epsilon0.T1]
+c:626 [binder, in hydras.Ackermann.primRec]
+c:629 [binder, in hydras.Epsilon0.T1]
+c:636 [binder, in hydras.Gamma0.Gamma0]
+c:641 [binder, in hydras.Ackermann.primRec]
+c:646 [binder, in hydras.Ackermann.primRec]
+c:654 [binder, in hydras.Ackermann.primRec]
+c:673 [binder, in hydras.Gamma0.Gamma0]
+c:677 [binder, in hydras.Ackermann.primRec]
+c:68 [binder, in hydras.Schutte.Schutte_basics]
+c:68 [binder, in additions.Addition_Chains]
+c:681 [binder, in hydras.Ackermann.primRec]
+c:684 [binder, in hydras.Ackermann.primRec]
+c:686 [binder, in hydras.Ackermann.primRec]
+c:688 [binder, in hydras.Ackermann.primRec]
+c:69 [binder, in hydras.Prelude.DecPreOrder]
+c:690 [binder, in hydras.Ackermann.primRec]
+c:692 [binder, in hydras.Gamma0.Gamma0]
+c:693 [binder, in hydras.Ackermann.primRec]
+c:696 [binder, in hydras.Ackermann.primRec]
+c:70 [binder, in additions.Addition_Chains]
+c:71 [binder, in hydras.Gamma0.T2]
+c:71 [binder, in additions.AM]
+c:71 [binder, in hydras.Schutte.Schutte_basics]
+c:717 [binder, in hydras.Ackermann.primRec]
+c:721 [binder, in hydras.Ackermann.primRec]
+c:722 [binder, in gaia_hydras.nfwfgaia]
+c:726 [binder, in hydras.Ackermann.primRec]
+c:729 [binder, in hydras.Ackermann.primRec]
+c:732 [binder, in hydras.Ackermann.primRec]
+c:733 [binder, in hydras.Epsilon0.T1]
+c:735 [binder, in hydras.Ackermann.primRec]
+c:737 [binder, in hydras.Epsilon0.T1]
+c:738 [binder, in hydras.Ackermann.primRec]
+c:74 [binder, in hydras.Schutte.Schutte_basics]
+c:741 [binder, in hydras.Ackermann.primRec]
+c:744 [binder, in hydras.Ackermann.primRec]
+c:747 [binder, in hydras.Ackermann.primRec]
+c:75 [binder, in hydras.Prelude.DecPreOrder]
+c:750 [binder, in hydras.Ackermann.primRec]
+c:753 [binder, in hydras.Ackermann.primRec]
+c:759 [binder, in hydras.Ackermann.primRec]
+c:761 [binder, in hydras.Ackermann.primRec]
+c:763 [binder, in hydras.Ackermann.primRec]
+c:765 [binder, in hydras.Ackermann.primRec]
+c:767 [binder, in hydras.Ackermann.primRec]
+c:769 [binder, in hydras.Ackermann.primRec]
+c:77 [binder, in hydras.Schutte.Schutte_basics]
+c:771 [binder, in hydras.Ackermann.primRec]
+c:773 [binder, in hydras.Ackermann.primRec]
+c:775 [binder, in hydras.Ackermann.primRec]
+c:777 [binder, in hydras.Ackermann.primRec]
+c:779 [binder, in hydras.Ackermann.primRec]
+c:78 [binder, in additions.Addition_Chains]
+c:781 [binder, in hydras.Ackermann.primRec]
+c:784 [binder, in hydras.Ackermann.primRec]
+c:787 [binder, in hydras.Ackermann.primRec]
+c:79 [binder, in hydras.Epsilon0.Hessenberg]
+c:795 [binder, in hydras.Ackermann.primRec]
+c:8 [binder, in hydras.Prelude.MoreDecidable]
+c:8 [binder, in hydras.Gamma0.Gamma0]
+c:80 [binder, in additions.Addition_Chains]
+c:801 [binder, in hydras.Ackermann.primRec]
+c:81 [binder, in additions.AM]
+c:850 [binder, in hydras.Ackermann.codeSubFormula]
+c:854 [binder, in gaia_hydras.nfwfgaia]
+c:863 [binder, in gaia_hydras.nfwfgaia]
+c:868 [binder, in gaia_hydras.nfwfgaia]
+c:88 [binder, in hydras.OrdinalNotations.OmegaOmega]
+c:9 [binder, in hydras.MoreAck.PrimRecExamples]
+c:90 [binder, in hydras.MoreAck.PrimRecExamples]
+c:905 [binder, in gaia_hydras.nfwfgaia]
+c:91 [binder, in additions.AM]
+c:913 [binder, in gaia_hydras.nfwfgaia]
+c:92 [binder, in additions.AM]
+c:93 [binder, in hydras.Gamma0.T2]
+c:940 [binder, in gaia_hydras.nfwfgaia]
+c:947 [binder, in gaia_hydras.nfwfgaia]
+c:95 [binder, in hydras.Prelude.Comparable]
+c:995 [binder, in gaia_hydras.nfwfgaia]
+

D

+daughters:7 [binder, in hydras.Hydra.Hydra_Definitions]
+DA:36 [binder, in hydras.Schutte.Countable]
+DA:37 [binder, in hydras.Prelude.DecPreOrder_Instances]
+DB:38 [binder, in hydras.Prelude.DecPreOrder_Instances]
+DecidableSet [definition, in Goedel.rosser]
+decidable_set [library]
+decide [projection, in hydras.Prelude.STDPP_compat]
+decide [constructor, in hydras.Prelude.STDPP_compat]
+decideAxioms [lemma, in Goedel.rosserPA]
+decideAxioms [lemma, in Goedel.rosser]
+decide_hltE [lemma, in gaia_hydras.T1Bridge]
+decide_rel [projection, in hydras.Prelude.STDPP_compat]
+decide_rel [constructor, in hydras.Prelude.STDPP_compat]
+Decision [record, in hydras.Prelude.STDPP_compat]
+Decision [inductive, in hydras.Prelude.STDPP_compat]
+decomp [lemma, in hydras.Prelude.MoreVectors]
+decompose [lemma, in hydras.OrdinalNotations.ON_Omega2]
+decompos2 [lemma, in hydras.Prelude.MoreVectors]
+decomp1 [lemma, in hydras.Prelude.MoreVectors]
+decomp2 [lemma, in hydras.Prelude.MoreVectors]
+DecPreOrder [library]
+DecPreOrder_Instances [library]
+decr [projection, in hydras.Prelude.WfVariant]
+decr_seq [lemma, in hydras.solutions_exercises.MultisetWf]
+dec:20 [binder, in hydras.Prelude.STDPP_compat]
+dec:33 [binder, in hydras.Prelude.Sort_spec]
+dec:42 [binder, in hydras.Prelude.Sort_spec]
+dec:66 [binder, in hydras.Prelude.Sort_spec]
+dec:77 [binder, in hydras.Prelude.Sort_spec]
+dec:90 [binder, in hydras.Prelude.DecPreOrder]
+Deduction [library]
+DeductionTheorem [lemma, in hydras.Ackermann.Deduction]
+Deduction_Theorem.list_incl [variable, in hydras.Ackermann.Deduction]
+Deduction_Theorem.SysPrf [variable, in hydras.Ackermann.Deduction]
+Deduction_Theorem.Prf [variable, in hydras.Ackermann.Deduction]
+Deduction_Theorem.Terms [variable, in hydras.Ackermann.Deduction]
+Deduction_Theorem.Term [variable, in hydras.Ackermann.Deduction]
+Deduction_Theorem.System [variable, in hydras.Ackermann.Deduction]
+Deduction_Theorem.Formulas [variable, in hydras.Ackermann.Deduction]
+Deduction_Theorem.Formula [variable, in hydras.Ackermann.Deduction]
+Deduction_Theorem.L [variable, in hydras.Ackermann.Deduction]
+Deduction_Theorem [section, in hydras.Ackermann.Deduction]
+Definitions [section, in additions.FirstSteps]
+Definitions [section, in hydras.OrdinalNotations.ON_Generic]
+Definitions [section, in hydras.Prelude.Simple_LexProd]
+Definitions.A [variable, in additions.FirstSteps]
+Definitions.A [variable, in hydras.Prelude.Simple_LexProd]
+Definitions.B [variable, in hydras.Prelude.Simple_LexProd]
+Definitions.ltA [variable, in hydras.Prelude.Simple_LexProd]
+Definitions.ltB [variable, in hydras.Prelude.Simple_LexProd]
+Definitions.mult [variable, in additions.FirstSteps]
+Definitions.one [variable, in additions.FirstSteps]
+Definitions.pair2sig [variable, in hydras.Prelude.Simple_LexProd]
+Definitions.wfA [variable, in hydras.Prelude.Simple_LexProd]
+Definitions.wfB [variable, in hydras.Prelude.Simple_LexProd]
+_ ^ _ [notation, in additions.FirstSteps]
+_ * _ [notation, in additions.FirstSteps]
+1 [notation, in additions.FirstSteps]
+Defs [section, in hydras.Prelude.MoreOrders]
+Defs [section, in hydras.OrdinalNotations.ON_mult]
+Defs [section, in hydras.OrdinalNotations.ON_plus]
+Defs.A [variable, in hydras.Prelude.MoreOrders]
+Defs.lt [variable, in hydras.Prelude.MoreOrders]
+_ <= _ [notation, in hydras.Prelude.MoreOrders]
+_ < _ [notation, in hydras.Prelude.MoreOrders]
+delta:593 [binder, in hydras.Epsilon0.T1]
+delta:62 [binder, in hydras.Schutte.Addition]
+Demo [module, in additions.Monoid_def]
+Demo [library]
+Demo_power [library]
+Demo.nat_mult_op [instance, in additions.Monoid_def]
+den_U [lemma, in hydras.Schutte.Ordering_Functions]
+depth [definition, in hydras.Ackermann.fol]
+depthForall [lemma, in hydras.Ackermann.fol]
+depthImp1 [lemma, in hydras.Ackermann.fol]
+depthImp2 [lemma, in hydras.Ackermann.fol]
+depthNot [lemma, in hydras.Ackermann.fol]
+depth_rec_demo.a [variable, in hydras.MoreAck.FolExamples]
+depth_rec_demo.P [variable, in hydras.MoreAck.FolExamples]
+depth_rec_demo.L [variable, in hydras.MoreAck.FolExamples]
+depth_rec_demo [section, in hydras.MoreAck.FolExamples]
+dest:102 [binder, in hydras.Hydra.Hydra_Definitions]
+dicho [definition, in additions.Dichotomy]
+Dichotomy [library]
+Dicho_strat [instance, in additions.Dichotomy]
+dicho_gt [lemma, in additions.Dichotomy]
+dicho_lt [lemma, in additions.Dichotomy]
+dicho_aux_gt [lemma, in additions.Dichotomy]
+dicho_aux_lt [lemma, in additions.Dichotomy]
+dicho_aux_le_xIXI [lemma, in additions.Dichotomy]
+dicho_aux_le_xIXO [lemma, in additions.Dichotomy]
+dicho_aux_le_xOXI [lemma, in additions.Dichotomy]
+dicho_aux_le_xOXO [lemma, in additions.Dichotomy]
+dicho_aux_xIxI [lemma, in additions.Dichotomy]
+dicho_aux_xIxO [lemma, in additions.Dichotomy]
+dicho_aux_xOxI [lemma, in additions.Dichotomy]
+dicho_aux_xOxO [lemma, in additions.Dichotomy]
+dickson [library]
+diffP [lemma, in gaia_hydras.GPrelude]
+Direct_proof.nf_Acc [lemma, in hydras.Epsilon0.T1]
+Direct_proof.Acc_implies_Acc_strong [lemma, in hydras.Epsilon0.T1]
+Direct_proof.Acc_strong_stronger [lemma, in hydras.Epsilon0.T1]
+Direct_proof.acc_impl [lemma, in hydras.Epsilon0.T1]
+Direct_proof.well_foundedness_proof.Acc_strong [variable, in hydras.Epsilon0.T1]
+Direct_proof.wf_LT [lemma, in hydras.Epsilon0.T1]
+Direct_proof.well_foundedness_proof.First_attempt [section, in hydras.Epsilon0.T1]
+Direct_proof.Acc_zero [lemma, in hydras.Epsilon0.T1]
+Direct_proof.well_foundedness_proof [section, in hydras.Epsilon0.T1]
+Direct_proof [module, in hydras.Epsilon0.T1]
+disp:27 [binder, in gaia_hydras.onType]
+div_eucl_unique [lemma, in hydras.Prelude.More_Arith]
+div_not_qlt [lemma, in hydras.Prelude.More_Arith]
+div_gamma_pos [lemma, in additions.Strategies]
+div2IsPR [instance, in hydras.Ackermann.cPair]
+div2_even_plus [lemma, in hydras.Prelude.More_Arith]
+div2_incr [lemma, in hydras.Prelude.More_Arith]
+div2_of_Even [lemma, in hydras.Prelude.More_Arith]
+div2_double_is_id [lemma, in hydras.Prelude.More_Arith]
+dominates [definition, in hydras.Prelude.Iterates]
+dominates [definition, in gaia_hydras.T1Bridge]
+dominates_iterate [lemma, in hydras.Prelude.Iterates]
+dominates_from_le [lemma, in hydras.Prelude.Iterates]
+dominates_trans_strong [lemma, in hydras.Prelude.Iterates]
+dominates_trans [lemma, in hydras.Prelude.Iterates]
+dominates_from_trans [lemma, in hydras.Prelude.Iterates]
+dominates_strong [definition, in hydras.Prelude.Iterates]
+dominates_from [definition, in hydras.Prelude.Iterates]
+dominates_strong [definition, in gaia_hydras.T1Bridge]
+dominates_from [definition, in gaia_hydras.T1Bridge]
+dom_AckNotPR [lemma, in hydras.MoreAck.AckNotPR]
+dom_AckNotPR.Hf [variable, in hydras.MoreAck.AckNotPR]
+dom_AckNotPR.f [variable, in hydras.MoreAck.AckNotPR]
+dom_AckNotPR [section, in hydras.MoreAck.AckNotPR]
+dom:152 [binder, in hydras.rpo.more_list]
+Done [lemma, in hydras.Hydra.BigBattle]
+double [definition, in hydras.solutions_exercises.MorePRExamples]
+double [definition, in hydras.MoreAck.PrimRecExamples]
+doubleIsPR [lemma, in hydras.solutions_exercises.MorePRExamples]
+doubleIsPR [instance, in hydras.MoreAck.PrimRecExamples]
+doubleS [definition, in hydras.Hydra.BigBattle]
+doubleS_law [lemma, in hydras.Hydra.BigBattle]
+double_is_even [lemma, in hydras.Prelude.More_Arith]
+double_inj [lemma, in hydras.Prelude.More_Arith]
+double_plus [lemma, in hydras.Prelude.More_Arith]
+double_S [lemma, in hydras.Prelude.More_Arith]
+DS_iota.Proof_case_6.Hn [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_6.n [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_6.gamma_pos [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_6.Hgamma [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_6.Hbeta [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_6.gamma [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_6.beta [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_6 [section, in hydras.Hydra.O2H]
+DS_iota.Proof_case_5.Hn [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_5.n [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_5.Hgamma [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_5.gamma [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_5 [section, in hydras.Hydra.O2H]
+DS_iota.Proof_case_4.Hn [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_4.n [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_4.Hlim [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_4.Hlambda [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_4.lambda [variable, in hydras.Hydra.O2H]
+DS_iota.Proof_case_4 [section, in hydras.Hydra.O2H]
+DS_iota_3 [lemma, in hydras.Hydra.O2H]
+DS_iota_2 [lemma, in hydras.Hydra.O2H]
+DS_iota_1 [lemma, in hydras.Hydra.O2H]
+DS_iota.Hrec [variable, in hydras.Hydra.O2H]
+DS_iota.nonzero [variable, in hydras.Hydra.O2H]
+DS_iota.Halpha [variable, in hydras.Hydra.O2H]
+DS_iota.i [variable, in hydras.Hydra.O2H]
+DS_iota.alpha [variable, in hydras.Hydra.O2H]
+DS_iota [section, in hydras.Hydra.O2H]
+d':1205 [binder, in gaia_hydras.nfwfgaia]
+d':1339 [binder, in gaia_hydras.nfwfgaia]
+d':1385 [binder, in gaia_hydras.nfwfgaia]
+d':1437 [binder, in gaia_hydras.nfwfgaia]
+d':1501 [binder, in gaia_hydras.nfwfgaia]
+d':1557 [binder, in gaia_hydras.nfwfgaia]
+d':1586 [binder, in gaia_hydras.nfwfgaia]
+d':1610 [binder, in gaia_hydras.nfwfgaia]
+d':1692 [binder, in gaia_hydras.nfwfgaia]
+d':2044 [binder, in gaia_hydras.nfwfgaia]
+d':2067 [binder, in gaia_hydras.nfwfgaia]
+d':27 [binder, in additions.Monoid_instances]
+d':849 [binder, in gaia_hydras.nfwfgaia]
+d:10 [binder, in hydras.MoreAck.PrimRecExamples]
+d:116 [binder, in hydras.Ackermann.cPair]
+d:118 [binder, in hydras.Epsilon0.Hessenberg]
+d:13 [binder, in hydras.Prelude.MoreDecidable]
+d:1308 [binder, in hydras.Ackermann.codeSubFormula]
+d:1334 [binder, in gaia_hydras.nfwfgaia]
+d:1349 [binder, in gaia_hydras.nfwfgaia]
+d:1354 [binder, in gaia_hydras.nfwfgaia]
+d:1359 [binder, in gaia_hydras.nfwfgaia]
+d:1364 [binder, in gaia_hydras.nfwfgaia]
+d:1369 [binder, in gaia_hydras.nfwfgaia]
+d:1374 [binder, in gaia_hydras.nfwfgaia]
+d:1380 [binder, in gaia_hydras.nfwfgaia]
+d:1432 [binder, in gaia_hydras.nfwfgaia]
+d:146 [binder, in additions.fib]
+d:149 [binder, in hydras.Ackermann.LNT]
+d:149 [binder, in hydras.Ackermann.primRec]
+d:1496 [binder, in gaia_hydras.nfwfgaia]
+d:152 [binder, in hydras.Ackermann.LNN]
+d:153 [binder, in hydras.Ackermann.primRec]
+d:1535 [binder, in gaia_hydras.nfwfgaia]
+d:154 [binder, in hydras.Ackermann.LNT]
+d:1546 [binder, in gaia_hydras.nfwfgaia]
+d:1555 [binder, in gaia_hydras.nfwfgaia]
+d:157 [binder, in hydras.Ackermann.LNN]
+d:157 [binder, in hydras.Ackermann.primRec]
+d:1591 [binder, in gaia_hydras.nfwfgaia]
+d:1605 [binder, in gaia_hydras.nfwfgaia]
+d:161 [binder, in hydras.Ackermann.primRec]
+d:165 [binder, in hydras.Ackermann.LNN]
+d:1654 [binder, in gaia_hydras.nfwfgaia]
+d:1678 [binder, in gaia_hydras.nfwfgaia]
+d:1687 [binder, in gaia_hydras.nfwfgaia]
+d:17 [binder, in hydras.MoreAck.PrimRecExamples]
+d:1795 [binder, in gaia_hydras.nfwfgaia]
+d:19 [binder, in hydras.Prelude.MoreDecidable]
+D:2 [binder, in hydras.Schutte.Lub]
+d:2029 [binder, in gaia_hydras.nfwfgaia]
+d:23 [binder, in additions.Monoid_instances]
+d:264 [binder, in hydras.Ackermann.primRec]
+d:268 [binder, in hydras.Ackermann.primRec]
+d:272 [binder, in hydras.Ackermann.primRec]
+d:276 [binder, in hydras.Ackermann.primRec]
+d:280 [binder, in hydras.Ackermann.primRec]
+d:285 [binder, in hydras.Ackermann.primRec]
+d:289 [binder, in hydras.Ackermann.primRec]
+d:293 [binder, in hydras.Ackermann.primRec]
+d:298 [binder, in hydras.Ackermann.primRec]
+d:302 [binder, in hydras.Ackermann.primRec]
+d:306 [binder, in hydras.Ackermann.primRec]
+d:310 [binder, in hydras.Ackermann.primRec]
+d:314 [binder, in hydras.Ackermann.primRec]
+d:318 [binder, in hydras.Ackermann.primRec]
+d:322 [binder, in hydras.Ackermann.primRec]
+d:327 [binder, in hydras.Ackermann.primRec]
+d:331 [binder, in hydras.Ackermann.primRec]
+d:335 [binder, in hydras.Ackermann.primRec]
+d:335 [binder, in additions.Euclidean_Chains]
+d:339 [binder, in hydras.Ackermann.primRec]
+d:34 [binder, in hydras.Ackermann.folLogic3]
+d:343 [binder, in hydras.Ackermann.primRec]
+d:36 [binder, in hydras.Ackermann.cPair]
+d:385 [binder, in additions.Euclidean_Chains]
+d:391 [binder, in additions.Euclidean_Chains]
+d:398 [binder, in additions.Euclidean_Chains]
+d:4 [binder, in hydras.Prelude.MoreDecidable]
+d:44 [binder, in hydras.Ackermann.cPair]
+d:48 [binder, in hydras.Ackermann.cPair]
+d:5 [binder, in hydras.Prelude.MoreDecidable]
+d:52 [binder, in hydras.Ackermann.cPair]
+d:56 [binder, in hydras.Ackermann.cPair]
+d:567 [binder, in hydras.Epsilon0.T1]
+d:595 [binder, in hydras.Gamma0.Gamma0]
+d:609 [binder, in hydras.Epsilon0.T1]
+d:625 [binder, in hydras.Epsilon0.T1]
+d:631 [binder, in hydras.Epsilon0.T1]
+d:655 [binder, in hydras.Ackermann.primRec]
+D:8 [binder, in hydras.Schutte.Lub]
+d:845 [binder, in gaia_hydras.nfwfgaia]
+d:851 [binder, in hydras.Ackermann.codeSubFormula]
+d:91 [binder, in hydras.MoreAck.PrimRecExamples]
+

E

+Elements [definition, in hydras.Epsilon0.T1]
+embedding [lemma, in hydras.Schutte.Correctness_E0]
+EMonoid [record, in additions.Monoid_def]
+empty_interval [lemma, in hydras.Prelude.MoreLists]
+empty_ordering [lemma, in hydras.Schutte.Ordering_Functions]
+Eone_right [projection, in additions.Monoid_def]
+Eone_left [projection, in additions.Monoid_def]
+Eop_proper [instance, in additions.Pow_variant]
+Eop_proper [instance, in additions.Pow]
+Eop_comm [projection, in additions.Monoid_def]
+Eop_assoc [projection, in additions.Monoid_def]
+Eop_proper [projection, in additions.Monoid_def]
+epsilon [definition, in hydras.Gamma0.T2]
+epsilon_0 [definition, in hydras.Epsilon0.T1]
+epsilon_equiv [lemma, in hydras.Schutte.MoreEpsilonIota]
+epsilon_ind [lemma, in hydras.Schutte.MoreEpsilonIota]
+epsilon_fxp [lemma, in hydras.Gamma0.Gamma0]
+epsilon0 [abbreviation, in hydras.Gamma0.T2]
+epsilon0 [definition, in hydras.Schutte.AP]
+Epsilon0 [instance, in gaia_hydras.T1Bridge]
+Epsilon0 [instance, in hydras.Epsilon0.E0]
+Epsilon0 [library]
+Epsilon0rpo [library]
+epsilon0_dec [instance, in hydras.Epsilon0.T1]
+epsilon0_pre_order [instance, in hydras.Epsilon0.T1]
+Epsilon0_correct [instance, in hydras.Schutte.Correctness_E0]
+epsilon0_Cr1 [lemma, in hydras.Schutte.Critical]
+epsilon0_lfp [lemma, in hydras.Schutte.AP]
+epsilon0_AP [lemma, in hydras.Schutte.AP]
+epsilon0_fxp [lemma, in hydras.Schutte.AP]
+epsilon0_as_lub [lemma, in hydras.Gamma0.Gamma0]
+epsilon0_fxp [lemma, in hydras.Gamma0.Gamma0]
+Epsilon0_Needed_Std [library]
+Epsilon0_Needed_Free [library]
+Epsilon0_Needed_Generic [library]
+eqA:107 [binder, in hydras.rpo.more_list]
+eqA:115 [binder, in hydras.rpo.more_list]
+eqA:139 [binder, in hydras.rpo.more_list]
+eqA:148 [binder, in hydras.rpo.more_list]
+eqA:158 [binder, in hydras.rpo.more_list]
+eqA:167 [binder, in hydras.rpo.more_list]
+eqA:174 [binder, in hydras.rpo.more_list]
+eqA:182 [binder, in hydras.rpo.more_list]
+eqA:190 [binder, in hydras.rpo.more_list]
+eqA:205 [binder, in hydras.rpo.more_list]
+eqA:214 [binder, in hydras.rpo.more_list]
+eqA:224 [binder, in hydras.rpo.more_list]
+eqA:242 [binder, in hydras.rpo.more_list]
+eqA:251 [binder, in hydras.rpo.more_list]
+eqA:263 [binder, in hydras.rpo.more_list]
+eqA:81 [binder, in hydras.rpo.more_list]
+eqA:91 [binder, in hydras.rpo.more_list]
+eqA:99 [binder, in hydras.rpo.more_list]
+EqDecision [abbreviation, in hydras.Prelude.STDPP_compat]
+eqDepth [lemma, in hydras.Ackermann.fol]
+eqIsPR [instance, in hydras.Ackermann.primRec]
+eqLT [lemma, in hydras.Ackermann.LNN]
+eqn_simpl2 [lemma, in gaia_hydras.nfwfgaia]
+eqn_simpl1 [lemma, in gaia_hydras.nfwfgaia]
+eqPlus [lemma, in hydras.Ackermann.LNN]
+eqPlus [lemma, in hydras.Ackermann.LNT]
+eqRefl [lemma, in hydras.Ackermann.LNN]
+eqRefl [lemma, in hydras.Ackermann.LNT]
+eqRefl [lemma, in hydras.Ackermann.folLogic3]
+eqSucc [lemma, in hydras.Ackermann.LNN]
+eqSucc [lemma, in hydras.Ackermann.LNT]
+eqSym [lemma, in hydras.Ackermann.LNN]
+eqSym [lemma, in hydras.Ackermann.LNT]
+eqSym [lemma, in hydras.Ackermann.folLogic3]
+eqTimes [lemma, in hydras.Ackermann.LNN]
+eqTimes [lemma, in hydras.Ackermann.LNT]
+eqTrans [lemma, in hydras.Ackermann.LNN]
+eqTrans [lemma, in hydras.Ackermann.LNT]
+eqTrans [lemma, in hydras.Ackermann.folLogic3]
+equal [constructor, in hydras.Ackermann.fol]
+equalFunction [lemma, in hydras.Ackermann.folLogic3]
+Equality_Logic_Rules.termsMap [variable, in hydras.Ackermann.folLogic3]
+Equality_Logic_Rules.SysPrf [variable, in hydras.Ackermann.folLogic3]
+Equality_Logic_Rules.Prf [variable, in hydras.Ackermann.folLogic3]
+Equality_Logic_Rules.L [variable, in hydras.Ackermann.folLogic3]
+Equality_Logic_Rules [section, in hydras.Ackermann.folLogic3]
+equalRelation [lemma, in hydras.Ackermann.folLogic3]
+Equations_for_addition.case3.Hac [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.case3 [section, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.case2.Hac [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.case2 [section, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.case1.Hac [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.case1 [section, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.beta [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.alpha [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.Hnfc [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.Hnfa [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.p [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.n [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.d [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.c [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.b [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition.a [variable, in hydras.Schutte.Correctness_E0]
+Equations_for_addition [section, in hydras.Schutte.Correctness_E0]
+equiv [definition, in hydras.Prelude.Sort_spec]
+equiv [projection, in additions.Monoid_def]
+Equiv [record, in additions.Monoid_def]
+equiv [constructor, in additions.Monoid_def]
+Equiv [inductive, in additions.Monoid_def]
+Equivalence [section, in additions.Compatibility]
+Equivalence.A [variable, in additions.Compatibility]
+Equivalence.is_power_of [variable, in additions.Compatibility]
+Equivalence.one [variable, in additions.Compatibility]
+Equivalence.op [variable, in additions.Compatibility]
+Equivalence.pos_iter_M [variable, in additions.Compatibility]
+equiv_equiv [instance, in hydras.Prelude.Sort_spec]
+equiv_equiv [instance, in hydras.Prelude.DecPreOrder]
+equiv_not_lt [lemma, in hydras.Prelude.DecPreOrder]
+Equiv_Trans [instance, in additions.Monoid_def]
+Equiv_Sym [instance, in additions.Monoid_def]
+Equiv_Refl [instance, in additions.Monoid_def]
+Equiv_Equiv [instance, in additions.Monoid_def]
+equiv:44 [binder, in additions.Euclidean_Chains]
+equiv:49 [binder, in additions.Euclidean_Chains]
+equiv:52 [binder, in additions.Euclidean_Chains]
+equ:101 [binder, in additions.AM]
+equ:106 [binder, in additions.AM]
+equ:111 [binder, in additions.AM]
+equ:121 [binder, in additions.AM]
+equ:126 [binder, in additions.AM]
+equ:38 [binder, in additions.AM]
+equ:44 [binder, in additions.AM]
+equ:55 [binder, in additions.AM]
+equ:62 [binder, in additions.AM]
+equ:75 [binder, in additions.AM]
+equ:85 [binder, in additions.AM]
+equ:96 [binder, in additions.AM]
+eqv:104 [binder, in additions.Addition_Chains]
+eqv:112 [binder, in additions.Addition_Chains]
+eqv:118 [binder, in additions.Addition_Chains]
+eqv:125 [binder, in additions.Addition_Chains]
+eqv:132 [binder, in additions.Addition_Chains]
+eqv:162 [binder, in additions.Addition_Chains]
+eqv:169 [binder, in additions.Addition_Chains]
+eqv:175 [binder, in additions.Addition_Chains]
+eq_succ_lt [lemma, in hydras.Epsilon0.T1]
+eq_succ_LT [lemma, in hydras.Epsilon0.T1]
+eq_dec [lemma, in hydras.OrdinalNotations.ON_Omega2]
+eq_dec [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+eq_equiv [instance, in additions.Monoid_def]
+Eq_equiv [projection, in additions.Monoid_def]
+eq_le [lemma, in hydras.Schutte.Schutte_basics]
+eq_A_dec:19 [binder, in hydras.rpo.rpo]
+eq_A_dec:14 [binder, in hydras.rpo.rpo]
+eq_A_dec:5 [binder, in hydras.rpo.rpo]
+EQ1 [constructor, in hydras.Ackermann.folProof]
+EQ2 [constructor, in hydras.Ackermann.folProof]
+EQ3 [constructor, in hydras.Ackermann.folProof]
+EQ4 [constructor, in hydras.Ackermann.folProof]
+EQ5 [constructor, in hydras.Ackermann.folProof]
+essai [section, in hydras.Epsilon0.T1]
+essai.n [variable, in hydras.Epsilon0.T1]
+eta:80 [binder, in hydras.Schutte.Critical]
+Euclidean_Chains [library]
+Euc1 [lemma, in hydras.Prelude.More_Arith]
+eval [definition, in hydras.Schutte.CNF]
+evalComposeFunc [definition, in hydras.Ackermann.primRec]
+evalConstFunc [definition, in hydras.Ackermann.primRec]
+evalList [section, in hydras.MoreAck.AckNotPR]
+evalList [definition, in hydras.Ackermann.primRec]
+evalListComp [lemma, in hydras.MoreAck.AckNotPR]
+evalListCompose2 [lemma, in hydras.MoreAck.AckNotPR]
+evalListPrimrec_S [lemma, in hydras.MoreAck.AckNotPR]
+evalListPrimrec_0 [lemma, in hydras.MoreAck.AckNotPR]
+evalList_Const [lemma, in hydras.MoreAck.AckNotPR]
+evalOneParamList [definition, in hydras.Ackermann.primRec]
+evalPrimRec [definition, in hydras.Ackermann.primRec]
+evalPrimRecFunc [definition, in hydras.Ackermann.primRec]
+evalPrimRecParam [lemma, in hydras.Ackermann.primRec]
+evalPrimRecs [definition, in hydras.Ackermann.primRec]
+evalProjFunc [definition, in hydras.Ackermann.primRec]
+evalProjFuncInd [lemma, in hydras.Ackermann.primRec]
+evalStrongRec [definition, in hydras.Ackermann.cPair]
+evalStrongRecHelp [definition, in hydras.Ackermann.cPair]
+evalStrongRecHelpParam [lemma, in hydras.Ackermann.cPair]
+evalStrongRecHelp1 [lemma, in hydras.Ackermann.cPair]
+evalStrongRecHelp2 [lemma, in hydras.Ackermann.cPair]
+evalStrongRecIsPR [instance, in hydras.Ackermann.cPair]
+eval_pp [definition, in hydras.Epsilon0.T1]
+even_prod [lemma, in hydras.Prelude.More_Arith]
+every_battle_terminates [lemma, in hydras.Hydra.Hydra_Termination]
+every_battle_terminates [lemma, in gaia_hydras.GHydra]
+Ex [definition, in hydras.Epsilon0.Large_Sets]
+exact_log2_spec [lemma, in additions.More_on_positive]
+exact_log2xOx0 [lemma, in additions.More_on_positive]
+exact_log2 [definition, in additions.More_on_positive]
+Examples [section, in hydras.Ackermann.LNT]
+Examples [section, in hydras.MoreAck.LNN_Examples]
+Examples [module, in hydras.Hydra.Hydra_Examples]
+Examples [module, in hydras.Ackermann.primRec]
+Examples [module, in additions.Euclidean_Chains]
+Examples.big_correct''' [lemma, in additions.Euclidean_Chains]
+Examples.big_correct' [lemma, in additions.Euclidean_Chains]
+Examples.big_correct [lemma, in additions.Euclidean_Chains]
+Examples.big_chain [definition, in additions.Euclidean_Chains]
+Examples.binary_int31_power [definition, in additions.Euclidean_Chains]
+Examples.C87' [definition, in additions.Euclidean_Chains]
+Examples.exact_sqrt [definition, in hydras.Ackermann.primRec]
+Examples.exp56789 [definition, in additions.Euclidean_Chains]
+Examples.Exx [definition, in hydras.Hydra.Hydra_Examples]
+Examples.ex_2 [definition, in hydras.Hydra.Hydra_Examples]
+Examples.ex1 [definition, in hydras.Hydra.Hydra_Examples]
+Examples.ex4 [definition, in hydras.Hydra.Hydra_Examples]
+Examples.Ex5 [definition, in hydras.Hydra.Hydra_Examples]
+Examples.f [variable, in hydras.Ackermann.LNT]
+Examples.fast_int63_power [definition, in additions.Euclidean_Chains]
+Examples.f1 [variable, in hydras.MoreAck.LNN_Examples]
+Examples.f2 [variable, in hydras.MoreAck.LNN_Examples]
+Examples.f2' [variable, in hydras.MoreAck.LNN_Examples]
+Examples.f3 [variable, in hydras.MoreAck.LNN_Examples]
+Examples.f4 [variable, in hydras.MoreAck.LNN_Examples]
+Examples.height_not_strictly_decreasing [definition, in hydras.Hydra.Hydra_Examples]
+Examples.hsize_bigger [definition, in hydras.Hydra.Hydra_Examples]
+Examples.Hy [definition, in hydras.Hydra.Hydra_Examples]
+Examples.Hy_Hy''' [definition, in hydras.Hydra.Hydra_Examples]
+Examples.Hy_3 [definition, in hydras.Hydra.Hydra_Examples]
+Examples.Hy_2 [definition, in hydras.Hydra.Hydra_Examples]
+Examples.Hy_1 [definition, in hydras.Hydra.Hydra_Examples]
+Examples.Hy' [definition, in hydras.Hydra.Hydra_Examples]
+Examples.Hy'H'' [definition, in hydras.Hydra.Hydra_Examples]
+Examples.Hy'' [definition, in hydras.Hydra.Hydra_Examples]
+Examples.Hy''_Hy''' [definition, in hydras.Hydra.Hydra_Examples]
+Examples.Hy''' [definition, in hydras.Hydra.Hydra_Examples]
+Examples.M [definition, in additions.Euclidean_Chains]
+Examples.RM [lemma, in additions.Euclidean_Chains]
+Examples.R2_example [definition, in hydras.Hydra.Hydra_Examples]
+Examples.slow_int31_power [definition, in additions.Euclidean_Chains]
+Examples.sqrtHelp [definition, in hydras.Ackermann.primRec]
+Examples.t1 [variable, in hydras.MoreAck.LNN_Examples]
+Examples.Unnamed_thm [definition, in additions.Euclidean_Chains]
+Example_3PlusOmega [library]
+exec [definition, in additions.AM]
+exec_Proper [instance, in additions.AM]
+exec_equiv [lemma, in additions.AM]
+exec_app [lemma, in additions.AM]
+exH' [abbreviation, in hydras.Ackermann.LNT]
+existE [lemma, in hydras.Ackermann.LNN]
+existE [lemma, in hydras.Ackermann.LNT]
+existE [lemma, in hydras.Ackermann.folLogic]
+existH [definition, in hydras.Ackermann.fol]
+existI [lemma, in hydras.Ackermann.LNN]
+existI [lemma, in hydras.Ackermann.LNT]
+existI [lemma, in hydras.Ackermann.folLogic]
+existSimp [lemma, in hydras.Ackermann.LNN]
+existSimp [lemma, in hydras.Ackermann.LNT]
+existSimp [lemma, in hydras.Ackermann.folLogic]
+existSys [lemma, in hydras.Ackermann.LNN]
+existSys [lemma, in hydras.Ackermann.LNT]
+existSys [lemma, in hydras.Ackermann.folLogic]
+exists_map12_without_repetition [lemma, in hydras.rpo.more_list]
+exists_map_without_repetition [lemma, in hydras.rpo.more_list]
+ExNotIncl [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+exp [definition, in hydras.Epsilon0.T1]
+exp [definition, in hydras.solutions_exercises.MorePRExamples]
+expIsPR [instance, in hydras.solutions_exercises.MorePRExamples]
+exponents [definition, in additions.Trace_exercise]
+exponents_lt_eval [lemma, in hydras.Schutte.CNF]
+exponents_le [definition, in hydras.Schutte.CNF]
+exponents_lt [definition, in hydras.Schutte.CNF]
+exponent_pos_of_nat [lemma, in additions.Addition_Chains]
+exponent_pos2nat [lemma, in additions.Addition_Chains]
+exponent_nat_neq_0 [lemma, in additions.Addition_Chains]
+Expressible [definition, in hydras.Ackermann.expressible]
+expressible [library]
+expressibleAlternate [lemma, in hydras.Ackermann.expressible]
+expressibleExamples [library]
+ExpressibleHelp [definition, in hydras.Ackermann.expressible]
+expressT'1 [lemma, in Goedel.rosserPA]
+expressT'1 [definition, in Goedel.goedel2]
+expressT'2 [lemma, in Goedel.rosserPA]
+expressT'2 [definition, in Goedel.goedel2]
+exp_F_eq [lemma, in hydras.Epsilon0.T1]
+exp_fin_omega [lemma, in hydras.Epsilon0.T1]
+exp_F [definition, in hydras.Epsilon0.T1]
+exp_alt_PR [instance, in hydras.solutions_exercises.MorePRExamples]
+exp_alt_ok [lemma, in hydras.solutions_exercises.MorePRExamples]
+exp_alt [definition, in hydras.solutions_exercises.MorePRExamples]
+exp2 [definition, in additions.Addition_Chains]
+exp2 [definition, in hydras.Prelude.Exp2]
+Exp2 [library]
+exp2S [lemma, in hydras.Prelude.Exp2]
+exp2_as_iterate [lemma, in hydras.Prelude.Iterates]
+exp2_mono_weak [lemma, in hydras.Prelude.Iterates]
+exp2_mono [lemma, in hydras.Prelude.Iterates]
+exp2_ge_S [lemma, in hydras.Prelude.Iterates]
+exp2_k_mult_pos [lemma, in hydras.Epsilon0.Large_Sets]
+exp2_mono1 [lemma, in hydras.Hydra.BigBattle]
+exp2_Plus [lemma, in additions.Addition_Chains]
+exp2_gt_id [lemma, in hydras.Prelude.Exp2]
+exp2_not_zero [lemma, in hydras.Prelude.Exp2]
+exp2_positive [lemma, in hydras.Prelude.Exp2]
+exp:746 [binder, in hydras.Epsilon0.T1]
+exp:751 [binder, in hydras.Epsilon0.T1]
+Exs [section, in hydras.MoreAck.PrimRecExamples]
+Exs.f [variable, in hydras.MoreAck.PrimRecExamples]
+Exs.fdiv2 [variable, in hydras.MoreAck.PrimRecExamples]
+Exs.ffib [variable, in hydras.MoreAck.PrimRecExamples]
+extendsNN [lemma, in Goedel.rosserPA]
+extendsNN [definition, in Goedel.goedel2]
+extEqual [definition, in hydras.Ackermann.extEqualNat]
+extEqualCompose [lemma, in hydras.Ackermann.primRec]
+extEqualCompose2 [lemma, in hydras.Ackermann.primRec]
+extEqualNat [library]
+extEqualOneParamList [lemma, in hydras.Ackermann.primRec]
+extEqualPrimRec [lemma, in hydras.Ackermann.primRec]
+extEqualRefl [lemma, in hydras.Ackermann.extEqualNat]
+extEqualSym [lemma, in hydras.Ackermann.extEqualNat]
+extEqualTrans [lemma, in hydras.Ackermann.extEqualNat]
+extEqualVector [definition, in hydras.Ackermann.primRec]
+extEqualVectorGeneral [definition, in hydras.Ackermann.primRec]
+extEqualVectorRefl [lemma, in hydras.Ackermann.primRec]
+extEqual_ex1 [definition, in hydras.MoreAck.PrimRecExamples]
+ex_pos [definition, in gaia_hydras.T1Choice]
+ex_path4 [definition, in gaia_hydras.GPaths]
+ex_path3 [definition, in gaia_hydras.GPaths]
+ex_path2 [definition, in gaia_hydras.GPaths]
+ex_path1 [definition, in gaia_hydras.GPaths]
+ex_pathS2 [definition, in hydras.Epsilon0.Large_Sets_Examples]
+ex_pathS1 [definition, in hydras.Epsilon0.Large_Sets_Examples]
+ex_bool [definition, in additions.Monoid_def]
+ex_string [definition, in additions.Monoid_def]
+ex_path4 [definition, in hydras.Epsilon0.Paths]
+ex_path3 [definition, in hydras.Epsilon0.Paths]
+ex_path2 [definition, in hydras.Epsilon0.Paths]
+ex_path1 [definition, in hydras.Epsilon0.Paths]
+ex0 [definition, in gaia_hydras.GCanon]
+Ex1 [definition, in hydras.Epsilon0.T1]
+Ex1 [definition, in hydras.OrdinalNotations.ON_Omega2]
+ex1 [definition, in hydras.OrdinalNotations.ON_Omega2]
+Ex1 [definition, in hydras.Gamma0.T2]
+ex1 [definition, in gaia_hydras.GCanon]
+Ex1 [definition, in hydras.Schutte.MoreEpsilonIota]
+ex1 [definition, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Ex1 [definition, in hydras.solutions_exercises.MultisetWf]
+Ex1 [definition, in hydras.Epsilon0.Large_Sets_Examples]
+Ex1 [definition, in hydras.OrdinalNotations.ON_Finite]
+Ex1 [definition, in gaia_hydras.GaiaToHydra]
+Ex1 [definition, in hydras.MoreAck.PrimRecExamples]
+Ex1 [definition, in hydras.Prelude.Simple_LexProd]
+Ex10 [definition, in hydras.Gamma0.Gamma0]
+Ex2 [definition, in hydras.Epsilon0.T1]
+Ex2 [definition, in hydras.Gamma0.T2]
+Ex2 [definition, in hydras.MoreAck.AckNotPR]
+ex2 [definition, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Ex2 [definition, in hydras.solutions_exercises.MultisetWf]
+Ex2 [definition, in hydras.Epsilon0.Large_Sets_Examples]
+Ex2 [definition, in hydras.MoreAck.PrimRecExamples]
+Ex2 [definition, in hydras.Prelude.Simple_LexProd]
+Ex3 [definition, in hydras.Epsilon0.T1]
+Ex3 [definition, in hydras.Gamma0.T2]
+ex3 [definition, in hydras.Epsilon0.Large_Sets]
+Ex3 [definition, in hydras.solutions_exercises.MultisetWf]
+Ex3 [definition, in hydras.MoreAck.PrimRecExamples]
+Ex4 [definition, in hydras.Epsilon0.T1]
+Ex4 [definition, in hydras.Gamma0.T2]
+Ex4 [definition, in hydras.MoreAck.AckNotPR]
+Ex4 [definition, in hydras.MoreAck.PrimRecExamples]
+Ex42 [definition, in hydras.Schutte.Schutte]
+Ex42 [definition, in hydras.OrdinalNotations.OmegaOmega]
+Ex42 [definition, in hydras.Epsilon0.E0]
+Ex5 [definition, in hydras.Epsilon0.T1]
+Ex5 [definition, in hydras.Gamma0.T2]
+Ex6 [definition, in hydras.Gamma0.T2]
+Ex6 [definition, in hydras.Gamma0.Gamma0]
+Ex7 [definition, in hydras.Gamma0.T2]
+Ex7 [definition, in hydras.Gamma0.Gamma0]
+Ex8 [definition, in hydras.Gamma0.T2]
+Ex8 [definition, in hydras.Gamma0.Gamma0]
+Ex9 [definition, in hydras.Gamma0.Gamma0]
+E_eq:240 [binder, in additions.AM]
+E_one:239 [binder, in additions.AM]
+E_op:238 [binder, in additions.AM]
+E_eq:233 [binder, in additions.AM]
+E_one:232 [binder, in additions.AM]
+E_op:231 [binder, in additions.AM]
+E_eq:51 [binder, in additions.Pow_variant]
+E_one:50 [binder, in additions.Pow_variant]
+E_op:49 [binder, in additions.Pow_variant]
+E_eq:42 [binder, in additions.Pow_variant]
+E_one:41 [binder, in additions.Pow_variant]
+E_op:40 [binder, in additions.Pow_variant]
+E_eq:32 [binder, in additions.Pow_variant]
+E_one:31 [binder, in additions.Pow_variant]
+E_op:30 [binder, in additions.Pow_variant]
+E_eq:26 [binder, in additions.Pow_variant]
+E_one:25 [binder, in additions.Pow_variant]
+E_op:24 [binder, in additions.Pow_variant]
+E_eq:19 [binder, in additions.Pow_variant]
+E_one:18 [binder, in additions.Pow_variant]
+E_op:17 [binder, in additions.Pow_variant]
+E_eq:13 [binder, in additions.Pow_variant]
+E_one:12 [binder, in additions.Pow_variant]
+E_op:11 [binder, in additions.Pow_variant]
+E_eq:4 [binder, in additions.Pow_variant]
+E_one:3 [binder, in additions.Pow_variant]
+E_op:2 [binder, in additions.Pow_variant]
+E_eq:59 [binder, in additions.Pow]
+E_op:58 [binder, in additions.Pow]
+E_eq:51 [binder, in additions.Pow]
+E_one:50 [binder, in additions.Pow]
+E_op:49 [binder, in additions.Pow]
+E_eq:42 [binder, in additions.Pow]
+E_one:41 [binder, in additions.Pow]
+E_op:40 [binder, in additions.Pow]
+E_eq:32 [binder, in additions.Pow]
+E_one:31 [binder, in additions.Pow]
+E_op:30 [binder, in additions.Pow]
+E_eq:26 [binder, in additions.Pow]
+E_one:25 [binder, in additions.Pow]
+E_op:24 [binder, in additions.Pow]
+E_eq:19 [binder, in additions.Pow]
+E_one:18 [binder, in additions.Pow]
+E_op:17 [binder, in additions.Pow]
+E_eq:13 [binder, in additions.Pow]
+E_one:12 [binder, in additions.Pow]
+E_op:11 [binder, in additions.Pow]
+E_eq:4 [binder, in additions.Pow]
+E_one:3 [binder, in additions.Pow]
+E_op:2 [binder, in additions.Pow]
+E_eq:54 [binder, in additions.Monoid_def]
+E_one:53 [binder, in additions.Monoid_def]
+E_op:52 [binder, in additions.Monoid_def]
+E_eq:49 [binder, in additions.Monoid_def]
+E_one:48 [binder, in additions.Monoid_def]
+E_op:47 [binder, in additions.Monoid_def]
+E_eq:44 [binder, in additions.Monoid_def]
+E_one:43 [binder, in additions.Monoid_def]
+E_op:42 [binder, in additions.Monoid_def]
+E_eq:39 [binder, in additions.Monoid_def]
+E_one:38 [binder, in additions.Monoid_def]
+E_op:37 [binder, in additions.Monoid_def]
+E_eq:24 [binder, in additions.Monoid_def]
+E_one:23 [binder, in additions.Monoid_def]
+E_op:22 [binder, in additions.Monoid_def]
+E_equiv:197 [binder, in additions.Euclidean_Chains]
+E_one:196 [binder, in additions.Euclidean_Chains]
+E_equiv:185 [binder, in additions.Euclidean_Chains]
+E_one:184 [binder, in additions.Euclidean_Chains]
+E_equiv:171 [binder, in additions.Euclidean_Chains]
+E_one:170 [binder, in additions.Euclidean_Chains]
+E_equiv:165 [binder, in additions.Euclidean_Chains]
+E_one:164 [binder, in additions.Euclidean_Chains]
+E_equiv:113 [binder, in additions.Euclidean_Chains]
+E_one:112 [binder, in additions.Euclidean_Chains]
+E_equiv:102 [binder, in additions.Euclidean_Chains]
+E_one:101 [binder, in additions.Euclidean_Chains]
+E_equiv:91 [binder, in additions.Euclidean_Chains]
+E_one:90 [binder, in additions.Euclidean_Chains]
+E_equiv:71 [binder, in additions.Euclidean_Chains]
+E_one:70 [binder, in additions.Euclidean_Chains]
+E_equiv:64 [binder, in additions.Euclidean_Chains]
+E_one:63 [binder, in additions.Euclidean_Chains]
+E_equiv:56 [binder, in additions.Euclidean_Chains]
+E_one:55 [binder, in additions.Euclidean_Chains]
+E_equiv:38 [binder, in additions.Euclidean_Chains]
+E_one:37 [binder, in additions.Euclidean_Chains]
+E_eq:234 [binder, in additions.Addition_Chains]
+E_one:233 [binder, in additions.Addition_Chains]
+E_op:232 [binder, in additions.Addition_Chains]
+E_eq:226 [binder, in additions.Addition_Chains]
+E_one:225 [binder, in additions.Addition_Chains]
+E_op:224 [binder, in additions.Addition_Chains]
+E_eq:74 [binder, in additions.Addition_Chains]
+E_one:73 [binder, in additions.Addition_Chains]
+E_op:72 [binder, in additions.Addition_Chains]
+E_eq:66 [binder, in additions.Addition_Chains]
+E_one:65 [binder, in additions.Addition_Chains]
+E_op:64 [binder, in additions.Addition_Chains]
+E_eq:60 [binder, in additions.Addition_Chains]
+E_one:59 [binder, in additions.Addition_Chains]
+E_op:58 [binder, in additions.Addition_Chains]
+E_eq:49 [binder, in additions.Addition_Chains]
+E_one:48 [binder, in additions.Addition_Chains]
+E_op:47 [binder, in additions.Addition_Chains]
+e':33 [binder, in hydras.rpo.list_set]
+e':36 [binder, in hydras.rpo.list_set]
+E0 [record, in gaia_hydras.T1Bridge]
+E0 [record, in hydras.Epsilon0.E0]
+E0 [library]
+E0add [instance, in hydras.Epsilon0.E0]
+E0Canon [definition, in gaia_hydras.GCanon]
+E0compare [instance, in gaia_hydras.T1Bridge]
+E0compare_correct [lemma, in gaia_hydras.T1Bridge]
+E0eqb [definition, in gaia_hydras.T1Bridge]
+E0fin [definition, in gaia_hydras.T1Bridge]
+E0fin [instance, in hydras.Epsilon0.E0]
+E0finS [instance, in hydras.Epsilon0.E0]
+E0fin_cnf [lemma, in gaia_hydras.T1Bridge]
+E0g2h_omegaE [lemma, in gaia_hydras.T1Bridge]
+E0g2h_plusE [lemma, in gaia_hydras.T1Bridge]
+E0g2h_mulE [lemma, in gaia_hydras.T1Bridge]
+E0g2h_phi0 [lemma, in gaia_hydras.T1Bridge]
+E0g2h_Fin [lemma, in gaia_hydras.T1Bridge]
+E0is_succE [lemma, in gaia_hydras.T1Bridge]
+E0is_succ_succ [lemma, in gaia_hydras.T1Bridge]
+E0is_succ [definition, in gaia_hydras.T1Bridge]
+E0is_succ [definition, in hydras.Epsilon0.E0]
+E0le [definition, in gaia_hydras.T1Bridge]
+E0le [definition, in hydras.Epsilon0.E0]
+E0limit [definition, in gaia_hydras.T1Bridge]
+E0limit [definition, in hydras.Epsilon0.E0]
+E0lt [definition, in gaia_hydras.T1Bridge]
+E0lt [definition, in hydras.Epsilon0.E0]
+E0lt_wf [lemma, in hydras.Epsilon0.E0]
+E0mul [definition, in gaia_hydras.T1Bridge]
+E0mul [instance, in hydras.Epsilon0.E0]
+E0one [instance, in hydras.Epsilon0.E0]
+E0plus [definition, in gaia_hydras.T1Bridge]
+e0Sub [definition, in gaia_hydras.T1Bridge]
+E0succb [abbreviation, in hydras.Epsilon0.E0]
+E0zero [definition, in gaia_hydras.T1Bridge]
+E0zero [instance, in hydras.Epsilon0.E0]
+E0_plus_correct [lemma, in hydras.Schutte.Correctness_E0]
+E0_canon_lt [lemma, in gaia_hydras.GCanon]
+E0_comp [instance, in gaia_hydras.T1Bridge]
+E0_sto [instance, in gaia_hydras.T1Bridge]
+E0_pred_succK [lemma, in gaia_hydras.T1Bridge]
+E0_diffE [lemma, in gaia_hydras.T1Bridge]
+E0_eqE [lemma, in gaia_hydras.T1Bridge]
+E0_g2h_h2gK [lemma, in gaia_hydras.T1Bridge]
+E0_h2g_g2hK [lemma, in gaia_hydras.T1Bridge]
+E0_h2g_nf [lemma, in gaia_hydras.T1Bridge]
+E0_g2h [definition, in gaia_hydras.T1Bridge]
+E0_h2g [definition, in gaia_hydras.T1Bridge]
+E0_phi0 [definition, in gaia_hydras.T1Bridge]
+E0_omega [definition, in gaia_hydras.T1Bridge]
+E0_pred [definition, in gaia_hydras.T1Bridge]
+E0_succ [definition, in gaia_hydras.T1Bridge]
+E0_eqtype [definition, in gaia_hydras.T1Bridge]
+E0_eq_mixin [definition, in gaia_hydras.T1Bridge]
+E0_lt_ge [lemma, in hydras.Epsilon0.E0]
+E0_lt_eq_lt [lemma, in hydras.Epsilon0.E0]
+E0_not_Lt_zero [lemma, in hydras.Epsilon0.E0]
+E0_Lt_Succ_inv [lemma, in hydras.Epsilon0.E0]
+E0_Lt_irrefl [lemma, in hydras.Epsilon0.E0]
+E0_comp [instance, in hydras.Epsilon0.E0]
+E0_pred_Lt [lemma, in hydras.Epsilon0.E0]
+E0_pred_of_Succ [lemma, in hydras.Epsilon0.E0]
+E0_sto [instance, in hydras.Epsilon0.E0]
+E0_pred [definition, in hydras.Epsilon0.E0]
+E0_eq_dec [lemma, in hydras.Epsilon0.E0]
+E0_eq_iff [lemma, in hydras.Epsilon0.E0]
+E0_eq_intro [lemma, in hydras.Epsilon0.E0]
+E0_phi0 [instance, in hydras.Epsilon0.E0]
+E0_succ [instance, in hydras.Epsilon0.E0]
+E0_omega [instance, in hydras.Epsilon0.E0]
+E1 [definition, in hydras.Epsilon0.T1]
+e1 [definition, in hydras.OrdinalNotations.ON_Omega2]
+e1:103 [binder, in additions.fib]
+e1:110 [binder, in additions.fib]
+e1:179 [binder, in hydras.rpo.list_set]
+E2 [definition, in hydras.Epsilon0.T1]
+e2 [definition, in hydras.OrdinalNotations.ON_Omega2]
+e2:104 [binder, in additions.fib]
+e2:111 [binder, in additions.fib]
+e2:180 [binder, in hydras.rpo.list_set]
+e:10 [binder, in hydras.rpo.list_set]
+e:100 [binder, in hydras.rpo.list_set]
+e:103 [binder, in hydras.rpo.list_set]
+e:106 [binder, in hydras.rpo.list_set]
+e:109 [binder, in hydras.rpo.list_set]
+e:11 [binder, in hydras.MoreAck.PrimRecExamples]
+e:112 [binder, in hydras.rpo.list_set]
+e:119 [binder, in additions.fib]
+e:121 [binder, in hydras.rpo.list_set]
+E:126 [binder, in hydras.Schutte.Ordering_Functions]
+e:129 [binder, in hydras.rpo.list_set]
+e:132 [binder, in hydras.rpo.list_set]
+e:136 [binder, in hydras.rpo.list_set]
+e:139 [binder, in hydras.rpo.list_set]
+e:143 [binder, in hydras.rpo.list_set]
+e:147 [binder, in hydras.rpo.list_set]
+e:151 [binder, in hydras.rpo.list_set]
+e:154 [binder, in hydras.rpo.list_set]
+e:157 [binder, in hydras.rpo.list_set]
+e:170 [binder, in hydras.rpo.list_set]
+e:199 [binder, in hydras.rpo.list_set]
+E:2 [binder, in hydras.Prelude.Restriction]
+E:2 [binder, in hydras.solutions_exercises.Limit_Infinity]
+e:204 [binder, in hydras.rpo.term]
+e:21 [binder, in hydras.rpo.term]
+e:22 [binder, in hydras.rpo.list_set]
+e:24 [binder, in hydras.rpo.list_set]
+e:241 [binder, in hydras.rpo.list_set]
+e:245 [binder, in hydras.rpo.list_set]
+e:252 [binder, in hydras.rpo.list_set]
+e:26 [binder, in hydras.rpo.list_set]
+e:28 [binder, in hydras.rpo.list_set]
+e:29 [binder, in hydras.rpo.list_permut]
+e:30 [binder, in hydras.rpo.list_set]
+e:32 [binder, in hydras.rpo.list_set]
+e:32 [binder, in hydras.rpo.list_permut]
+e:33 [binder, in hydras.rpo.list_permut]
+e:35 [binder, in hydras.rpo.list_set]
+e:39 [binder, in hydras.rpo.list_set]
+e:46 [binder, in hydras.rpo.list_set]
+e:54 [binder, in hydras.Prelude.More_Arith]
+e:55 [binder, in hydras.rpo.list_permut]
+e:57 [binder, in hydras.rpo.list_set]
+e:61 [binder, in hydras.rpo.list_set]
+e:65 [binder, in hydras.rpo.list_set]
+e:69 [binder, in hydras.rpo.list_set]
+e:73 [binder, in hydras.rpo.list_set]
+e:743 [binder, in hydras.Epsilon0.T1]
+e:75 [binder, in hydras.rpo.list_set]
+e:754 [binder, in hydras.Epsilon0.T1]
+e:78 [binder, in hydras.rpo.list_set]
+e:81 [binder, in hydras.rpo.list_set]
+e:84 [binder, in hydras.rpo.list_set]
+e:85 [binder, in hydras.rpo.list_set]
+e:92 [binder, in hydras.MoreAck.PrimRecExamples]
+e:97 [binder, in hydras.rpo.list_set]
+

F

+f [definition, in hydras.OrdinalNotations.Example_3PlusOmega]
+F [abbreviation, in hydras.Schutte.Schutte_basics]
+F [lemma, in gaia_hydras.onType]
+f [axiom, in hydras.Prelude.LibHyps_Experiments]
+F [definition, in Goedel.goedel2]
+fact [definition, in hydras.solutions_exercises.MorePRExamples]
+factIsPR [instance, in hydras.solutions_exercises.MorePRExamples]
+fact_altIsPR [instance, in hydras.solutions_exercises.MorePRExamples]
+fact_alt_ok [lemma, in hydras.solutions_exercises.MorePRExamples]
+fact_alt [definition, in hydras.solutions_exercises.MorePRExamples]
+fact_correct [lemma, in hydras.MoreAck.PrimRecExamples]
+Fake_thm [lemma, in hydras.solutions_exercises.is_F_monotonous]
+Fapply [definition, in additions.Euclidean_Chains]
+fastexp [definition, in additions.fib]
+fastexp2 [definition, in additions.fib]
+fastexp3 [definition, in additions.fib]
+fastexp3P [lemma, in additions.fib]
+fastexp4 [definition, in additions.fib]
+FA1 [constructor, in hydras.Ackermann.folProof]
+FA2 [constructor, in hydras.Ackermann.folProof]
+FA3 [constructor, in hydras.Ackermann.folProof]
+fA:181 [binder, in hydras.OrdinalNotations.ON_Generic]
+fbij [lemma, in hydras.Schutte.Ordering_Functions]
+fB:182 [binder, in hydras.OrdinalNotations.ON_Generic]
+Fchain [definition, in additions.Euclidean_Chains]
+Fchain_correct [definition, in additions.AM]
+Fchain_proper_prf [projection, in additions.Euclidean_Chains]
+Fchain_proper [record, in additions.Euclidean_Chains]
+Fchain_proper_prf [constructor, in additions.Euclidean_Chains]
+Fchain_proper [inductive, in additions.Euclidean_Chains]
+Fchain_correct [definition, in additions.Euclidean_Chains]
+Fchain_correct_nat [definition, in additions.Euclidean_Chains]
+Fcompose [definition, in additions.Euclidean_Chains]
+Fcompose_proper [instance, in additions.Euclidean_Chains]
+Fcompose_correct [lemma, in additions.Euclidean_Chains]
+Fcompose_correct_nat [lemma, in additions.Euclidean_Chains]
+fc1:120 [binder, in additions.Euclidean_Chains]
+fc1:129 [binder, in additions.Euclidean_Chains]
+fc1:133 [binder, in additions.Euclidean_Chains]
+fc2:121 [binder, in additions.Euclidean_Chains]
+fc2:130 [binder, in additions.Euclidean_Chains]
+fc2:134 [binder, in additions.Euclidean_Chains]
+fc:108 [binder, in additions.Euclidean_Chains]
+fc:34 [binder, in additions.Euclidean_Chains]
+fc:79 [binder, in additions.Euclidean_Chains]
+fc:86 [binder, in additions.Euclidean_Chains]
+Fexp2 [definition, in additions.Euclidean_Chains]
+Fexp2_proper [instance, in additions.Euclidean_Chains]
+Fexp2_correct [lemma, in additions.Euclidean_Chains]
+Fexp2_nat_correct [lemma, in additions.Euclidean_Chains]
+Fexp2_nat_proper [instance, in additions.Euclidean_Chains]
+Fexp2_of_nat [definition, in additions.Euclidean_Chains]
+FF [lemma, in hydras.Epsilon0.F_omega]
+FFK [definition, in additions.AM]
+FFK [definition, in additions.Euclidean_Chains]
+FFK_correct [lemma, in additions.AM]
+FFK_proper [instance, in additions.Euclidean_Chains]
+FFK_correct [lemma, in additions.Euclidean_Chains]
+FFK_correct_nat [lemma, in additions.Euclidean_Chains]
+fib [definition, in additions.Fib2]
+fib [definition, in hydras.solutions_exercises.FibonacciPR]
+fib [definition, in additions.fib]
+fib [library]
+fibIsPR [instance, in hydras.solutions_exercises.FibonacciPR]
+fibm [definition, in additions.fib]
+fibmP [lemma, in additions.fib]
+fibonacci [definition, in additions.FirstSteps]
+fibonacci [definition, in additions.Naive]
+FibonacciPR [library]
+fibPR [definition, in hydras.solutions_exercises.FibonacciPR]
+fibPRIsPR [instance, in hydras.solutions_exercises.FibonacciPR]
+fibt [definition, in additions.fib]
+fibtP [lemma, in additions.fib]
+fibt_aux [lemma, in additions.fib]
+fibZ2P [lemma, in additions.fib]
+fibZ3P [lemma, in additions.fib]
+fib_with_chain [definition, in additions.Fib2]
+fib_eucl [definition, in additions.Fib2]
+fib_pos [definition, in additions.Fib2]
+fib_mul2_OK [lemma, in additions.Fib2]
+fib_mul2_OK_0 [lemma, in additions.Fib2]
+fib_mul2 [definition, in additions.Fib2]
+fib_SSn [lemma, in additions.Fib2]
+fib_ind [lemma, in additions.Fib2]
+fib_alt_Ok [lemma, in hydras.solutions_exercises.FibonacciPR]
+fib_OK0 [lemma, in hydras.solutions_exercises.FibonacciPR]
+fib_altIsPR [instance, in hydras.solutions_exercises.FibonacciPR]
+fib_alt [definition, in hydras.solutions_exercises.FibonacciPR]
+Fib2 [library]
+filter001IsPR [instance, in hydras.Ackermann.primRec]
+filter0011IsPR [instance, in hydras.Ackermann.primRec]
+filter01IsPR [instance, in hydras.Ackermann.primRec]
+filter010IsPR [instance, in hydras.Ackermann.primRec]
+filter011IsPR [instance, in hydras.Ackermann.primRec]
+filter10IsPR [instance, in hydras.Ackermann.primRec]
+filter100IsPR [instance, in hydras.Ackermann.primRec]
+filter1000IsPR [instance, in hydras.Ackermann.primRec]
+filter101IsPR [instance, in hydras.Ackermann.primRec]
+filter1011IsPR [instance, in hydras.Ackermann.primRec]
+filter110IsPR [instance, in hydras.Ackermann.primRec]
+filter1100IsPR [instance, in hydras.Ackermann.primRec]
+fin [abbreviation, in hydras.Epsilon0.T1]
+fin [definition, in hydras.OrdinalNotations.ON_Omega2]
+fin [definition, in hydras.Gamma0.T2]
+fin [definition, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Fin [abbreviation, in hydras.Epsilon0.E0]
+find [definition, in hydras.rpo.more_list]
+find_not_mem [lemma, in hydras.rpo.more_list]
+finite [definition, in hydras.Schutte.Schutte_basics]
+finite_ltR [lemma, in hydras.Epsilon0.T1]
+finite_lt [lemma, in hydras.Epsilon0.T1]
+finite_succ [constructor, in hydras.Gamma0.T2]
+Finite_ref [lemma, in gaia_hydras.T1Bridge]
+finite_not_limit [lemma, in hydras.Schutte.Schutte_basics]
+finite_lt_omega [lemma, in hydras.Schutte.Schutte_basics]
+finite_inj [lemma, in hydras.Schutte.Schutte_basics]
+finite_mono [lemma, in hydras.Schutte.Schutte_basics]
+finite_lt_inv [lemma, in hydras.Schutte.Schutte_basics]
+finite_is_finite [lemma, in hydras.Gamma0.Gamma0]
+finite_plus_infinite [lemma, in hydras.Schutte.Addition]
+FinOrd [instance, in hydras.OrdinalNotations.ON_Finite]
+finord_ON [instance, in gaia_hydras.ON_gfinite]
+finord_lt_wf [lemma, in gaia_hydras.ON_gfinite]
+finord_comp [instance, in gaia_hydras.ON_gfinite]
+finord_sto [instance, in gaia_hydras.ON_gfinite]
+finord_compare_correct [lemma, in gaia_hydras.ON_gfinite]
+finord_compare [instance, in gaia_hydras.ON_gfinite]
+finord_lt [definition, in gaia_hydras.ON_gfinite]
+FinOrd_Omega [instance, in hydras.OrdinalNotations.ON_Omega]
+FinS_Succ_eq [lemma, in hydras.Epsilon0.E0]
+FinS_eq [lemma, in hydras.Epsilon0.E0]
+fin_lt_omega [lemma, in hydras.Gamma0.Gamma0]
+fin:90 [binder, in additions.Addition_Chains]
+FirstIncompletenessA [lemma, in Goedel.goedel1]
+FirstIncompletenessA [lemma, in Goedel.goedel2]
+FirstIncompletenessB [lemma, in Goedel.goedel1]
+FirstSteps [library]
+first_toggle [definition, in hydras.Prelude.First_toggle]
+First_Order_Logic.Formula_Depth_Induction [section, in hydras.Ackermann.fol]
+First_Order_Logic.Formula_Decidability.consTermsHelp [variable, in hydras.Ackermann.fol]
+First_Order_Logic.Formula_Decidability.nilTermsHelp [variable, in hydras.Ackermann.fol]
+First_Order_Logic.Formula_Decidability.language_eqdec [variable, in hydras.Ackermann.fol]
+First_Order_Logic.Formula_Decidability [section, in hydras.Ackermann.fol]
+First_Order_Logic.L [variable, in hydras.Ackermann.fol]
+First_Order_Logic [section, in hydras.Ackermann.fol]
+First_toggle [library]
+Fix [section, in hydras.Prelude.Restriction]
+fixPoint [library]
+FixpointDef [module, in gaia_hydras.GHessenberg]
+FixpointDef.oplus [definition, in gaia_hydras.GHessenberg]
+FixPointLNN [lemma, in Goedel.fixPoint]
+FixPointLNT [lemma, in Goedel.fixPoint]
+FixR [definition, in hydras.Prelude.Restriction]
+FixR_F_eq [lemma, in hydras.Prelude.Restriction]
+FixR_F [definition, in hydras.Prelude.Restriction]
+Fix.A [variable, in hydras.Prelude.Restriction]
+Fix.E [variable, in hydras.Prelude.Restriction]
+Fix.F [variable, in hydras.Prelude.Restriction]
+Fix.Hwf [variable, in hydras.Prelude.Restriction]
+Fix.P [variable, in hydras.Prelude.Restriction]
+Fix.R [variable, in hydras.Prelude.Restriction]
+Fix.Rwf [variable, in hydras.Prelude.Restriction]
+FK [definition, in additions.AM]
+FK [definition, in additions.Euclidean_Chains]
+Fkont [definition, in additions.Euclidean_Chains]
+Fkont_equiv [definition, in additions.Euclidean_Chains]
+Fkont_proper_prf [projection, in additions.Euclidean_Chains]
+Fkont_proper [record, in additions.Euclidean_Chains]
+Fkont_proper_prf [constructor, in additions.Euclidean_Chains]
+Fkont_proper [inductive, in additions.Euclidean_Chains]
+FK_correct [lemma, in additions.AM]
+FK_proper [instance, in additions.Euclidean_Chains]
+FK_correct [lemma, in additions.Euclidean_Chains]
+flatten [definition, in additions.Addition_Chains]
+flatten [lemma, in hydras.Epsilon0.Paths]
+flatten_valid_2 [lemma, in additions.Addition_Chains]
+flatten_valid [lemma, in additions.Addition_Chains]
+flatten_aux_valid [lemma, in additions.Addition_Chains]
+flatten_aux [definition, in additions.Addition_Chains]
+fn1:47 [binder, in additions.fib]
+fn2:48 [binder, in additions.fib]
+fn3:49 [binder, in additions.fib]
+fn:33 [binder, in additions.AM]
+fn:443 [binder, in hydras.Ackermann.primRec]
+fn:456 [binder, in hydras.Ackermann.primRec]
+fn:477 [binder, in hydras.Ackermann.primRec]
+FO [constructor, in hydras.Prelude.Fuel]
+fol [library]
+fold_left2 [definition, in hydras.rpo.more_list]
+FolExamples [library]
+folLogic [library]
+folLogic2 [library]
+folLogic3 [library]
+FolNotations [module, in hydras.Ackermann.fol]
+FolNotations.exH' [abbreviation, in hydras.Ackermann.fol]
+_ <->' _ (fol_scope) [notation, in hydras.Ackermann.fol]
+_ <->'' _ (fol_scope) [notation, in hydras.Ackermann.fol]
+_ /\' _ (fol_scope) [notation, in hydras.Ackermann.fol]
+_ \/' _ (fol_scope) [notation, in hydras.Ackermann.fol]
+_ <> _ (fol_scope) [notation, in hydras.Ackermann.fol]
+_ = _ (fol_scope) [notation, in hydras.Ackermann.fol]
+allH _ .. _ , _ (fol_scope) [notation, in hydras.Ackermann.fol]
+exH _ .. _ , _ (fol_scope) [notation, in hydras.Ackermann.fol]
+v# _ (fol_scope) [notation, in hydras.Ackermann.fol]
+_ <-> _ (fol_scope) [notation, in hydras.Ackermann.fol]
+~ _ (fol_scope) [notation, in hydras.Ackermann.fol]
+_ -> _ (fol_scope) [notation, in hydras.Ackermann.fol]
+_ /\ _ (fol_scope) [notation, in hydras.Ackermann.fol]
+_ \/ _ (fol_scope) [notation, in hydras.Ackermann.fol]
+_ = _ (fol_scope) [notation, in hydras.Ackermann.fol]
+folProof [library]
+folProp [library]
+folReplace [library]
+Fol_Properties.Substitution.Substitution_Properties [section, in hydras.Ackermann.folProp]
+Fol_Properties.Substitution.Extensions [section, in hydras.Ackermann.folProp]
+Fol_Properties.Substitution [section, in hydras.Ackermann.folProp]
+Fol_Properties.Free_Variables [section, in hydras.Ackermann.folProp]
+Fol_Properties.lt_depth [variable, in hydras.Ackermann.folProp]
+Fol_Properties.L [variable, in hydras.Ackermann.folProp]
+Fol_Properties [section, in hydras.Ackermann.folProp]
+foo [definition, in hydras.Prelude.Fuel]
+foo:167 [binder, in hydras.Schutte.Ordering_Functions]
+foo:42 [binder, in hydras.Ackermann.fol]
+foo:45 [binder, in hydras.Ackermann.fol]
+forallE [lemma, in hydras.Ackermann.LNN]
+forallE [lemma, in hydras.Ackermann.LNT]
+forallE [lemma, in hydras.Ackermann.folLogic]
+forallH [constructor, in hydras.Ackermann.fol]
+forallI [lemma, in hydras.Ackermann.LNN]
+forallI [lemma, in hydras.Ackermann.LNT]
+forallI [lemma, in hydras.Ackermann.folLogic]
+forallSimp [lemma, in hydras.Ackermann.LNN]
+forallSimp [lemma, in hydras.Ackermann.LNT]
+forallSimp [lemma, in hydras.Ackermann.folLogic]
+forall_weak [lemma, in hydras.Prelude.Sort_spec]
+Forall_forall [lemma, in hydras.Prelude.MoreVectors]
+Forall_and [lemma, in hydras.Prelude.MoreVectors]
+Forall_inv [lemma, in hydras.Prelude.MoreVectors]
+Forall2R [inductive, in hydras.Prelude.MoreLists]
+Forall2R_iff [lemma, in hydras.Prelude.MoreLists]
+Forall2R_cons [lemma, in hydras.Prelude.MoreLists]
+Forall2R_last [constructor, in hydras.Prelude.MoreLists]
+Forall2R_nil [constructor, in hydras.Prelude.MoreLists]
+Forall2_indR [lemma, in hydras.Prelude.MoreLists]
+Forall2_RR [lemma, in hydras.Prelude.MoreLists]
+Forall2_R [lemma, in hydras.Prelude.MoreLists]
+Forall2_right_induction [section, in hydras.Prelude.MoreLists]
+Forall2_inv [lemma, in hydras.Prelude.MoreVectors]
+Formula [abbreviation, in hydras.Ackermann.subAll]
+Formula [definition, in hydras.Ackermann.LNN]
+Formula [definition, in hydras.Ackermann.LNT]
+Formula [abbreviation, in hydras.Ackermann.folProp]
+Formula [abbreviation, in hydras.Ackermann.codeSubFormula]
+Formula [inductive, in hydras.Ackermann.fol]
+Formula [abbreviation, in hydras.Ackermann.folLogic3]
+Formulas [abbreviation, in hydras.Ackermann.subAll]
+Formulas [definition, in hydras.Ackermann.LNN]
+Formulas [definition, in hydras.Ackermann.LNT]
+Formulas [abbreviation, in hydras.Ackermann.folProp]
+Formulas [abbreviation, in hydras.Ackermann.codeSubFormula]
+Formulas [definition, in hydras.Ackermann.fol]
+Formulas [abbreviation, in hydras.Ackermann.folLogic3]
+FormulasToFormula [definition, in Goedel.PRrepresentable]
+FormulasToFuncs [definition, in Goedel.PRrepresentable]
+Formula_depth_ind2 [lemma, in hydras.Ackermann.fol]
+Formula_depth_ind [definition, in hydras.Ackermann.fol]
+Formula_depth_rec2_forall [lemma, in hydras.Ackermann.fol]
+Formula_depth_rec2_not [lemma, in hydras.Ackermann.fol]
+Formula_depth_rec2_imp [lemma, in hydras.Ackermann.fol]
+Formula_depth_rec2rec_nice [lemma, in hydras.Ackermann.fol]
+Formula_depth_rec2 [definition, in hydras.Ackermann.fol]
+Formula_depth_rec2rec [definition, in hydras.Ackermann.fol]
+Formula_depth_rec_indep [lemma, in hydras.Ackermann.fol]
+Formula_depth_rec [definition, in hydras.Ackermann.fol]
+Formula_depth_rec_rec [definition, in hydras.Ackermann.fol]
+formula_eqdec [lemma, in hydras.Ackermann.fol]
+fp:233 [binder, in additions.Euclidean_Chains]
+fp:302 [binder, in additions.Euclidean_Chains]
+fp:318 [binder, in additions.Euclidean_Chains]
+fp:320 [binder, in additions.Euclidean_Chains]
+Fp:323 [binder, in additions.Euclidean_Chains]
+Fp:328 [binder, in additions.Euclidean_Chains]
+fq:223 [binder, in additions.Euclidean_Chains]
+fq:232 [binder, in additions.Euclidean_Chains]
+fq:234 [binder, in additions.Euclidean_Chains]
+fq:294 [binder, in additions.Euclidean_Chains]
+fq:299 [binder, in additions.Euclidean_Chains]
+fq:303 [binder, in additions.Euclidean_Chains]
+fq:319 [binder, in additions.Euclidean_Chains]
+fq:321 [binder, in additions.Euclidean_Chains]
+frec:113 [binder, in hydras.Ackermann.folProp]
+frec:74 [binder, in hydras.Ackermann.codeSubFormula]
+frec:76 [binder, in hydras.Ackermann.folProp]
+frec:98 [binder, in hydras.Ackermann.folProp]
+free [instance, in hydras.Hydra.Hydra_Definitions]
+freeVarAddExists1 [lemma, in Goedel.PRrepresentable]
+freeVarAddExists2 [lemma, in Goedel.PRrepresentable]
+freeVarAddForalls1 [lemma, in Goedel.PRrepresentable]
+freeVarAddForalls2 [lemma, in Goedel.PRrepresentable]
+freeVarClosed [lemma, in hydras.Ackermann.folProp]
+freeVarClosedList1 [lemma, in hydras.Ackermann.folProp]
+freeVarClosedList2 [lemma, in hydras.Ackermann.folProp]
+freeVarCodeSysPf [lemma, in Goedel.codeSysPrf]
+freeVarCodeSysPrf [lemma, in Goedel.codeSysPrf]
+freeVarCodeSysPrfN [definition, in Goedel.rosserPA]
+freeVarCodeSysPrfN [lemma, in Goedel.codeSysPrf]
+freeVarCodeSysPrfN [definition, in Goedel.rosser]
+freeVarF [definition, in hydras.Ackermann.folProp]
+freeVarFormula [abbreviation, in hydras.Ackermann.folProp]
+freeVarG [lemma, in Goedel.goedel1]
+freeVarInterpFormula [lemma, in hydras.Ackermann.model]
+freeVarInterpRel [lemma, in hydras.Ackermann.model]
+freeVarInterpTerm [lemma, in hydras.Ackermann.model]
+freeVarListFormula [definition, in hydras.Ackermann.folProp]
+freeVarListFormulaApp [lemma, in hydras.Ackermann.folProp]
+freeVarLT [lemma, in hydras.Ackermann.LNN]
+freeVarMap [definition, in hydras.Ackermann.subAll]
+freeVarMap_ext [lemma, in hydras.Ackermann.subAll]
+freeVarMap1 [lemma, in hydras.Ackermann.subAll]
+freeVarNNHelp [lemma, in hydras.Ackermann.model]
+freeVarPlus [lemma, in hydras.Ackermann.LNN]
+freeVarPlus [lemma, in hydras.Ackermann.LNT]
+freeVarPrimRecPiFormulaHelp1 [lemma, in Goedel.PRrepresentable]
+freeVarPrimRecSigmaFormulaHelp1 [lemma, in Goedel.PRrepresentable]
+freeVarRepT' [lemma, in Goedel.rosserPA]
+freeVarRepT' [definition, in Goedel.goedel2]
+freeVarSubAllFormula1 [lemma, in hydras.Ackermann.subAll]
+freeVarSubAllFormula2 [lemma, in hydras.Ackermann.subAll]
+freeVarSubAllTerms1 [lemma, in hydras.Ackermann.subAll]
+freeVarSubAllTerms2 [lemma, in hydras.Ackermann.subAll]
+freeVarSubAllTerm1 [lemma, in hydras.Ackermann.subAll]
+freeVarSubAllTerm2 [lemma, in hydras.Ackermann.subAll]
+freeVarSubFormula1 [lemma, in hydras.Ackermann.subProp]
+freeVarSubFormula2 [lemma, in hydras.Ackermann.subProp]
+freeVarSubFormula3 [lemma, in hydras.Ackermann.subProp]
+freeVarSubFormula4 [lemma, in hydras.Ackermann.subProp]
+freeVarSubTerms1 [lemma, in hydras.Ackermann.subProp]
+freeVarSubTerms2 [lemma, in hydras.Ackermann.subProp]
+freeVarSubTerms3 [lemma, in hydras.Ackermann.subProp]
+freeVarSubTerms4 [lemma, in hydras.Ackermann.subProp]
+freeVarSubTerm1 [lemma, in hydras.Ackermann.subProp]
+freeVarSubTerm2 [lemma, in hydras.Ackermann.subProp]
+freeVarSubTerm3 [lemma, in hydras.Ackermann.subProp]
+freeVarSubTerm4 [lemma, in hydras.Ackermann.subProp]
+freeVarSucc [lemma, in hydras.Ackermann.LNN]
+freeVarSucc [lemma, in hydras.Ackermann.LNT]
+freeVarT [definition, in hydras.Ackermann.folProp]
+freeVarTApply [lemma, in hydras.Ackermann.folProp]
+freeVarTerm [abbreviation, in hydras.Ackermann.folProp]
+freeVarTerms [abbreviation, in hydras.Ackermann.folProp]
+freeVarTimes [lemma, in hydras.Ackermann.LNN]
+freeVarTimes [lemma, in hydras.Ackermann.LNT]
+freeVarTs [definition, in hydras.Ackermann.folProp]
+freeVarZero [lemma, in hydras.Ackermann.LNN]
+freeVarZero [lemma, in hydras.Ackermann.LNT]
+Free_Variables [section, in hydras.Ackermann.LNN]
+Free_Variables [section, in hydras.Ackermann.LNT]
+from:16 [binder, in gaia_hydras.GPaths]
+from:34 [binder, in gaia_hydras.GPaths]
+from:44 [binder, in gaia_hydras.GPaths]
+from:47 [binder, in gaia_hydras.GPaths]
+from:64 [binder, in hydras.Prelude.MoreVectors]
+from:70 [binder, in hydras.Prelude.Sort_spec]
+FS [abbreviation, in hydras.Epsilon0.T1]
+FS [abbreviation, in hydras.Gamma0.T2]
+FS [constructor, in hydras.Prelude.Fuel]
+fstar_S [lemma, in hydras.Epsilon0.F_alpha]
+Fstar_S [lemma, in hydras.Epsilon0.F_alpha]
+FS_rw [lemma, in hydras.Epsilon0.T1]
+fs:19 [binder, in hydras.MoreAck.AckNotPR]
+fs:448 [binder, in Goedel.PRrepresentable]
+fs:459 [binder, in Goedel.PRrepresentable]
+fs:467 [binder, in Goedel.PRrepresentable]
+fs:77 [binder, in hydras.Ackermann.primRec]
+fuel [inductive, in hydras.Prelude.Fuel]
+Fuel [library]
+fuel:747 [binder, in hydras.Epsilon0.T1]
+fuel:87 [binder, in hydras.Epsilon0.Canon]
+fuel:94 [binder, in hydras.Epsilon0.Canon]
+func [projection, in hydras.Ackermann.model]
+Functions [projection, in hydras.Ackermann.fol]
+fun_le_trans [lemma, in hydras.Prelude.Iterates]
+fun_le [definition, in hydras.Prelude.Iterates]
+fun_le [definition, in gaia_hydras.T1Bridge]
+fun_bijection_is_ZL_bijection [lemma, in hydras.Schutte.PartialFun]
+fun_bijection_codomain [lemma, in hydras.Schutte.PartialFun]
+fun_restr [definition, in hydras.Schutte.PartialFun]
+fun_bij_i [constructor, in hydras.Schutte.PartialFun]
+fun_bijection [inductive, in hydras.Schutte.PartialFun]
+fun_inj_i [constructor, in hydras.Schutte.PartialFun]
+fun_injection [inductive, in hydras.Schutte.PartialFun]
+fun_inj [definition, in hydras.Schutte.PartialFun]
+fun_onto [definition, in hydras.Schutte.PartialFun]
+fun_codomain [definition, in hydras.Schutte.PartialFun]
+fun_equiv [definition, in hydras.Schutte.Ordering_Functions]
+fun2PR [definition, in hydras.Ackermann.primRec]
+fusion [definition, in additions.Trace_exercise]
+F_alpha_not_PR [lemma, in gaia_hydras.GF_alpha]
+F_alpha_not_PR_E0 [lemma, in gaia_hydras.GF_alpha]
+F_restricted_mono_l [lemma, in gaia_hydras.GF_alpha]
+F_limE [lemma, in gaia_hydras.GF_alpha]
+F_succE [lemma, in gaia_hydras.GF_alpha]
+F_alpha_0_eq [lemma, in gaia_hydras.GF_alpha]
+F_mono_l [lemma, in gaia_hydras.GF_alpha]
+F_zeroE [lemma, in gaia_hydras.GF_alpha]
+F_alpha_positive [lemma, in gaia_hydras.GF_alpha]
+F_alpha_Succ_le [lemma, in gaia_hydras.GF_alpha]
+F_alpha_dom [lemma, in gaia_hydras.GF_alpha]
+F_alpha_mono [lemma, in gaia_hydras.GF_alpha]
+F_alpha_gt [lemma, in gaia_hydras.GF_alpha]
+F_ [definition, in gaia_hydras.GF_alpha]
+f_succ_eqn [lemma, in hydras.Epsilon0.F_alpha]
+f_lim_eqn [lemma, in hydras.Epsilon0.F_alpha]
+f_zero_eqn [lemma, in hydras.Epsilon0.F_alpha]
+f_star_iterate [lemma, in hydras.Epsilon0.F_alpha]
+f_star_Succ [lemma, in hydras.Epsilon0.F_alpha]
+f_eq2 [lemma, in hydras.Epsilon0.F_alpha]
+f_star_zero_eqn [lemma, in hydras.Epsilon0.F_alpha]
+f_ [definition, in hydras.Epsilon0.F_alpha]
+f_star [definition, in hydras.Epsilon0.F_alpha]
+F_mono_l [lemma, in hydras.Epsilon0.F_alpha]
+F_mono_l_0 [lemma, in hydras.Epsilon0.F_alpha]
+F_monotony_l.case_lt.Hd [variable, in hydras.Epsilon0.F_alpha]
+F_monotony_l.case_lt.Hlt [variable, in hydras.Epsilon0.F_alpha]
+F_monotony_l.case_lt.n [variable, in hydras.Epsilon0.F_alpha]
+F_monotony_l.case_lt [section, in hydras.Epsilon0.F_alpha]
+F_monotony_l.case_eq.Heq [variable, in hydras.Epsilon0.F_alpha]
+F_monotony_l.case_eq [section, in hydras.Epsilon0.F_alpha]
+F_monotony_l.H'_beta_alpha [variable, in hydras.Epsilon0.F_alpha]
+F_monotony_l.beta [variable, in hydras.Epsilon0.F_alpha]
+F_monotony_l.alpha [variable, in hydras.Epsilon0.F_alpha]
+F_monotony_l [section, in hydras.Epsilon0.F_alpha]
+F_alpha_ge_S [abbreviation, in hydras.Epsilon0.F_alpha]
+F_restricted_mono_l [lemma, in hydras.Epsilon0.F_alpha]
+F_alpha_dom [lemma, in hydras.Epsilon0.F_alpha]
+F_alpha_Succ_le [lemma, in hydras.Epsilon0.F_alpha]
+F_alpha_positive [lemma, in hydras.Epsilon0.F_alpha]
+F_alpha_gt [lemma, in hydras.Epsilon0.F_alpha]
+F_alpha_mono [lemma, in hydras.Epsilon0.F_alpha]
+F_One_Zero_ge [lemma, in hydras.Epsilon0.F_alpha]
+F_One_Zero_dom [lemma, in hydras.Epsilon0.F_alpha]
+F_alpha_0_eq [lemma, in hydras.Epsilon0.F_alpha]
+F_succ_eqn [lemma, in hydras.Epsilon0.F_alpha]
+F_lim_eqn [lemma, in hydras.Epsilon0.F_alpha]
+F_zero_eqn [lemma, in hydras.Epsilon0.F_alpha]
+F_star_iterate [lemma, in hydras.Epsilon0.F_alpha]
+F_star_Succ [lemma, in hydras.Epsilon0.F_alpha]
+F_eq2 [lemma, in hydras.Epsilon0.F_alpha]
+F_star_zero_eqn [lemma, in hydras.Epsilon0.F_alpha]
+F_ [definition, in hydras.Epsilon0.F_alpha]
+F_star [definition, in hydras.Epsilon0.F_alpha]
+f_ok_inv [lemma, in hydras.Epsilon0.Large_Sets]
+F_alpha_notPR_inv [lemma, in hydras.Epsilon0.F_omega]
+F_alpha_PR_inv [lemma, in hydras.Epsilon0.F_omega]
+F_n_PR [instance, in hydras.Epsilon0.F_omega]
+F_0_isPR [instance, in hydras.Epsilon0.F_omega]
+F_alpha_not_PR [lemma, in hydras.Epsilon0.F_omega]
+F_alpha_notPR.H [variable, in hydras.Epsilon0.F_omega]
+F_alpha_notPR.case_lt.Halpha [variable, in hydras.Epsilon0.F_omega]
+F_alpha_notPR.case_lt [section, in hydras.Epsilon0.F_omega]
+F_alpha_notPR.alpha [variable, in hydras.Epsilon0.F_omega]
+F_alpha_notPR [section, in hydras.Epsilon0.F_omega]
+F_omega_not_PR [lemma, in hydras.Epsilon0.F_omega]
+F_omega_PR:13 [binder, in hydras.Epsilon0.F_omega]
+F_omega_notPR [section, in hydras.Epsilon0.F_omega]
+F_vs_Ack [lemma, in hydras.Epsilon0.F_omega]
+F_iterate_rw [lemma, in hydras.Epsilon0.F_omega]
+f_Sv:55 [binder, in hydras.Ackermann.PA]
+f_0:54 [binder, in hydras.Ackermann.PA]
+f_Sv:4 [binder, in hydras.Ackermann.PA]
+f_0:3 [binder, in hydras.Ackermann.PA]
+F_incl_ij [instance, in hydras.OrdinalNotations.ON_Finite]
+F_alpha_Sn [lemma, in hydras.solutions_exercises.F_3]
+F_3_eqn [lemma, in hydras.solutions_exercises.F_3]
+f_minoration [lemma, in hydras.Epsilon0.Hprime]
+f_sup_commutes [lemma, in hydras.Schutte.Ordering_Functions]
+F_not_lim [lemma, in hydras.Gamma0.Gamma0]
+f_l:286 [binder, in hydras.rpo.rpo]
+f_l:283 [binder, in hydras.rpo.rpo]
+f_l:274 [binder, in hydras.rpo.rpo]
+F_3 [library]
+F_alpha [library]
+F_omega [library]
+f':19 [binder, in gaia_hydras.T1Bridge]
+f':22 [binder, in gaia_hydras.T1Bridge]
+f':33 [binder, in gaia_hydras.T1Bridge]
+f':348 [binder, in hydras.Ackermann.primRec]
+f':36 [binder, in gaia_hydras.T1Bridge]
+f0 [definition, in hydras.MoreAck.LNN_Examples]
+F0 [lemma, in hydras.Epsilon0.Hprime]
+f0:1239 [binder, in hydras.Ackermann.codeSubFormula]
+f0:1242 [binder, in hydras.Ackermann.codeSubFormula]
+f0:1296 [binder, in hydras.Ackermann.codeSubFormula]
+f0:1299 [binder, in hydras.Ackermann.codeSubFormula]
+f0:13 [binder, in hydras.Ackermann.PA]
+f0:148 [binder, in hydras.Ackermann.codePA]
+f0:151 [binder, in hydras.Ackermann.codePA]
+f0:158 [binder, in hydras.Ackermann.codePA]
+f0:160 [binder, in hydras.Ackermann.codePA]
+f0:208 [binder, in hydras.Ackermann.fol]
+f0:231 [binder, in hydras.Ackermann.fol]
+f0:249 [binder, in hydras.Ackermann.fol]
+f0:299 [binder, in hydras.Ackermann.fol]
+f0:345 [binder, in hydras.Ackermann.fol]
+f0:389 [binder, in hydras.Ackermann.fol]
+f0:41 [binder, in hydras.Ackermann.Languages]
+f0:42 [binder, in hydras.Ackermann.Languages]
+f0:43 [binder, in hydras.Ackermann.Languages]
+f0:44 [binder, in hydras.Ackermann.Languages]
+f0:441 [binder, in hydras.Ackermann.fol]
+f0:45 [binder, in hydras.Ackermann.Languages]
+f0:46 [binder, in hydras.Ackermann.Languages]
+f0:62 [binder, in hydras.Ackermann.folLogic3]
+F1 [definition, in additions.AM]
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+F1 [definition, in additions.Euclidean_Chains]
+F1 [lemma, in hydras.Epsilon0.Hprime]
+f1 [definition, in hydras.MoreAck.FolExamples]
+F1 [lemma, in hydras.Hydra.KP_example]
+f1rec:67 [binder, in hydras.Ackermann.codeSubFormula]
+F1_correct [lemma, in additions.AM]
+F1_proper [instance, in additions.Euclidean_Chains]
+F1_correct [lemma, in additions.Euclidean_Chains]
+F1_neutral_r [lemma, in additions.Euclidean_Chains]
+F1_neutral_l [lemma, in additions.Euclidean_Chains]
+F1_simpl [lemma, in hydras.Epsilon0.Hprime]
+F144 [definition, in additions.Euclidean_Chains]
+F197887 [definition, in additions.AM]
+f1:106 [binder, in hydras.Ackermann.primRec]
+f1:109 [binder, in hydras.Ackermann.primRec]
+f1:110 [binder, in hydras.Ackermann.folReplace]
+f1:12 [binder, in hydras.Ackermann.folReplace]
+f1:140 [binder, in additions.Euclidean_Chains]
+f1:144 [binder, in hydras.Ackermann.codeSubFormula]
+f1:152 [binder, in hydras.Ackermann.folProp]
+f1:17 [binder, in hydras.Ackermann.folReplace]
+f1:19 [binder, in additions.Euclidean_Chains]
+f1:2 [binder, in hydras.rpo.term]
+f1:2 [binder, in hydras.Ackermann.folReplace]
+f1:20 [binder, in hydras.Ackermann.folReplace]
+f1:200 [binder, in hydras.Ackermann.codeSubFormula]
+f1:203 [binder, in hydras.Ackermann.fol]
+f1:208 [binder, in hydras.Ackermann.codeSubFormula]
+f1:226 [binder, in hydras.Ackermann.fol]
+f1:23 [binder, in hydras.Ackermann.folReplace]
+f1:232 [binder, in hydras.rpo.term]
+f1:244 [binder, in hydras.Ackermann.fol]
+f1:244 [binder, in hydras.Gamma0.Gamma0]
+f1:246 [binder, in hydras.rpo.term]
+f1:26 [binder, in hydras.Ackermann.folReplace]
+f1:294 [binder, in hydras.Ackermann.fol]
+f1:3 [binder, in hydras.Epsilon0.Epsilon0rpo]
+f1:30 [binder, in hydras.Ackermann.folReplace]
+f1:34 [binder, in hydras.Ackermann.folReplace]
+f1:340 [binder, in hydras.Ackermann.fol]
+f1:351 [binder, in hydras.Ackermann.primRec]
+f1:365 [binder, in hydras.Ackermann.primRec]
+f1:37 [binder, in hydras.Ackermann.subAll]
+f1:371 [binder, in hydras.Ackermann.primRec]
+f1:383 [binder, in hydras.Ackermann.primRec]
+f1:384 [binder, in hydras.Ackermann.fol]
+f1:389 [binder, in hydras.Ackermann.primRec]
+f1:39 [binder, in hydras.Ackermann.folReplace]
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+f1:396 [binder, in hydras.Ackermann.folProp]
+f1:396 [binder, in hydras.Ackermann.primRec]
+f1:403 [binder, in hydras.Ackermann.primRec]
+f1:405 [binder, in hydras.Ackermann.folProp]
+f1:409 [binder, in hydras.Ackermann.folProp]
+f1:44 [binder, in hydras.Ackermann.folReplace]
+f1:49 [binder, in hydras.Ackermann.folReplace]
+f1:54 [binder, in hydras.Ackermann.folReplace]
+f1:55 [binder, in hydras.rpo.term]
+f1:59 [binder, in hydras.Ackermann.folReplace]
+f1:64 [binder, in hydras.Ackermann.folReplace]
+f1:66 [binder, in hydras.Ackermann.codeSubFormula]
+f1:68 [binder, in hydras.Ackermann.folReplace]
+f1:7 [binder, in hydras.Ackermann.folReplace]
+f1:70 [binder, in hydras.rpo.term]
+f1:72 [binder, in hydras.Ackermann.folReplace]
+f1:77 [binder, in hydras.Ackermann.folReplace]
+f1:80 [binder, in additions.Euclidean_Chains]
+f1:82 [binder, in hydras.Ackermann.folReplace]
+f1:87 [binder, in hydras.Ackermann.folReplace]
+f1:97 [binder, in additions.Euclidean_Chains]
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+F2 [definition, in additions.Euclidean_Chains]
+F2 [lemma, in hydras.Epsilon0.Hprime]
+f2 [definition, in hydras.MoreAck.FolExamples]
+F2 [lemma, in hydras.Hydra.KP_example]
+F2C [definition, in additions.Euclidean_Chains]
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+F2q [definition, in additions.AM]
+F2q_correct [lemma, in additions.AM]
+F2q_correct_1 [lemma, in additions.AM]
+F2q_correct_0 [lemma, in additions.AM]
+F2q_of_nat [definition, in additions.AM]
+f2rec:69 [binder, in hydras.Ackermann.codeSubFormula]
+F2_proper [instance, in additions.Euclidean_Chains]
+F2_correct [lemma, in additions.Euclidean_Chains]
+f2:102 [binder, in hydras.Ackermann.codeSubFormula]
+f2:107 [binder, in hydras.Ackermann.primRec]
+f2:110 [binder, in hydras.Ackermann.primRec]
+f2:111 [binder, in hydras.Ackermann.folReplace]
+f2:13 [binder, in hydras.Ackermann.folReplace]
+f2:1370 [binder, in hydras.Ackermann.codeSubFormula]
+f2:1373 [binder, in hydras.Ackermann.codeSubFormula]
+f2:141 [binder, in additions.Euclidean_Chains]
+f2:145 [binder, in hydras.Ackermann.codeSubFormula]
+f2:153 [binder, in hydras.Ackermann.folProp]
+f2:18 [binder, in hydras.Ackermann.folReplace]
+f2:20 [binder, in additions.Euclidean_Chains]
+f2:201 [binder, in hydras.Ackermann.codeSubFormula]
+f2:206 [binder, in hydras.Ackermann.fol]
+f2:209 [binder, in hydras.Ackermann.codeSubFormula]
+f2:21 [binder, in hydras.Ackermann.folReplace]
+f2:229 [binder, in hydras.Ackermann.fol]
+f2:233 [binder, in hydras.rpo.term]
+f2:24 [binder, in hydras.Ackermann.folReplace]
+f2:245 [binder, in hydras.Gamma0.Gamma0]
+f2:247 [binder, in hydras.rpo.term]
+f2:247 [binder, in hydras.Ackermann.fol]
+f2:27 [binder, in hydras.Ackermann.folReplace]
+f2:297 [binder, in hydras.Ackermann.fol]
+f2:3 [binder, in hydras.rpo.term]
+f2:3 [binder, in hydras.Ackermann.folReplace]
+f2:31 [binder, in hydras.Ackermann.folReplace]
+f2:343 [binder, in hydras.Ackermann.fol]
+f2:35 [binder, in hydras.Ackermann.folReplace]
+f2:352 [binder, in hydras.Ackermann.primRec]
+f2:366 [binder, in hydras.Ackermann.primRec]
+f2:372 [binder, in hydras.Ackermann.primRec]
+f2:38 [binder, in hydras.Ackermann.subAll]
+f2:384 [binder, in hydras.Ackermann.primRec]
+f2:387 [binder, in hydras.Ackermann.fol]
+f2:390 [binder, in hydras.Ackermann.primRec]
+f2:393 [binder, in hydras.Ackermann.folProp]
+f2:397 [binder, in hydras.Ackermann.folProp]
+f2:397 [binder, in hydras.Ackermann.primRec]
+f2:4 [binder, in hydras.Epsilon0.Epsilon0rpo]
+f2:40 [binder, in hydras.Ackermann.folReplace]
+f2:404 [binder, in hydras.Ackermann.primRec]
+f2:406 [binder, in hydras.Ackermann.folProp]
+f2:410 [binder, in hydras.Ackermann.folProp]
+f2:45 [binder, in hydras.Ackermann.folReplace]
+f2:50 [binder, in hydras.Ackermann.folReplace]
+f2:55 [binder, in hydras.Ackermann.folReplace]
+f2:56 [binder, in hydras.rpo.term]
+f2:60 [binder, in hydras.Ackermann.folReplace]
+f2:65 [binder, in hydras.Ackermann.folReplace]
+f2:68 [binder, in hydras.Ackermann.codeSubFormula]
+f2:69 [binder, in hydras.Ackermann.folReplace]
+f2:71 [binder, in hydras.rpo.term]
+f2:73 [binder, in hydras.Ackermann.folReplace]
+f2:78 [binder, in hydras.Ackermann.folReplace]
+f2:8 [binder, in hydras.Ackermann.folReplace]
+f2:81 [binder, in additions.Euclidean_Chains]
+f2:83 [binder, in hydras.Ackermann.folReplace]
+f2:88 [binder, in hydras.Ackermann.folReplace]
+f2:93 [binder, in hydras.Ackermann.codeSubFormula]
+f2:98 [binder, in additions.Euclidean_Chains]
+F3 [definition, in additions.AM]
+F3 [definition, in additions.Euclidean_Chains]
+F3 [lemma, in hydras.Epsilon0.Hprime]
+f3 [definition, in hydras.MoreAck.FolExamples]
+F3_correct [lemma, in additions.AM]
+F3_proper [instance, in additions.Euclidean_Chains]
+F3_correct [lemma, in additions.Euclidean_Chains]
+f3:14 [binder, in hydras.Ackermann.folReplace]
+f3:146 [binder, in hydras.Ackermann.codeSubFormula]
+f3:202 [binder, in hydras.Ackermann.codeSubFormula]
+f3:209 [binder, in hydras.Ackermann.fol]
+f3:210 [binder, in hydras.Ackermann.codeSubFormula]
+f3:232 [binder, in hydras.Ackermann.fol]
+f3:250 [binder, in hydras.Ackermann.fol]
+f3:300 [binder, in hydras.Ackermann.fol]
+f3:346 [binder, in hydras.Ackermann.fol]
+f3:353 [binder, in hydras.Ackermann.primRec]
+f3:36 [binder, in hydras.Ackermann.folReplace]
+f3:367 [binder, in hydras.Ackermann.primRec]
+f3:373 [binder, in hydras.Ackermann.primRec]
+f3:390 [binder, in hydras.Ackermann.fol]
+f3:391 [binder, in hydras.Ackermann.primRec]
+f3:4 [binder, in hydras.Ackermann.folReplace]
+f3:405 [binder, in hydras.Ackermann.primRec]
+f3:41 [binder, in hydras.Ackermann.folReplace]
+f3:411 [binder, in hydras.Ackermann.folProp]
+f3:46 [binder, in hydras.Ackermann.folReplace]
+f3:51 [binder, in hydras.Ackermann.folReplace]
+f3:56 [binder, in hydras.Ackermann.folReplace]
+f3:61 [binder, in hydras.Ackermann.folReplace]
+f3:74 [binder, in hydras.Ackermann.folReplace]
+f3:79 [binder, in hydras.Ackermann.folReplace]
+f3:84 [binder, in hydras.Ackermann.folReplace]
+f3:89 [binder, in hydras.Ackermann.folReplace]
+f3:9 [binder, in hydras.Ackermann.folReplace]
+f4:10 [binder, in hydras.Ackermann.folReplace]
+f4:147 [binder, in hydras.Ackermann.codeSubFormula]
+f4:15 [binder, in hydras.Ackermann.folReplace]
+f4:203 [binder, in hydras.Ackermann.codeSubFormula]
+f4:211 [binder, in hydras.Ackermann.codeSubFormula]
+f4:211 [binder, in hydras.Ackermann.fol]
+f4:234 [binder, in hydras.Ackermann.fol]
+f4:261 [binder, in hydras.Ackermann.fol]
+f4:311 [binder, in hydras.Ackermann.fol]
+f4:357 [binder, in hydras.Ackermann.fol]
+f4:37 [binder, in hydras.Ackermann.folReplace]
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+f4:401 [binder, in hydras.Ackermann.fol]
+f4:42 [binder, in hydras.Ackermann.folReplace]
+f4:47 [binder, in hydras.Ackermann.folReplace]
+f4:5 [binder, in hydras.Ackermann.folReplace]
+f4:52 [binder, in hydras.Ackermann.folReplace]
+f4:57 [binder, in hydras.Ackermann.folReplace]
+f4:62 [binder, in hydras.Ackermann.folReplace]
+f4:75 [binder, in hydras.Ackermann.folReplace]
+f4:80 [binder, in hydras.Ackermann.folReplace]
+f4:85 [binder, in hydras.Ackermann.folReplace]
+f4:90 [binder, in hydras.Ackermann.folReplace]
+F5 [lemma, in hydras.Epsilon0.F_alpha]
+f5:148 [binder, in hydras.Ackermann.codeSubFormula]
+f5:204 [binder, in hydras.Ackermann.codeSubFormula]
+f5:212 [binder, in hydras.Ackermann.codeSubFormula]
+f5:215 [binder, in hydras.Ackermann.fol]
+f5:238 [binder, in hydras.Ackermann.fol]
+f5:270 [binder, in hydras.Ackermann.fol]
+f5:320 [binder, in hydras.Ackermann.fol]
+f5:366 [binder, in hydras.Ackermann.fol]
+f5:410 [binder, in hydras.Ackermann.fol]
+f5:91 [binder, in hydras.Ackermann.folReplace]
+F6 [lemma, in hydras.Epsilon0.F_alpha]
+f6:92 [binder, in hydras.Ackermann.folReplace]
+F7 [lemma, in hydras.Epsilon0.F_alpha]
+F8 [lemma, in hydras.Epsilon0.F_alpha]
+F87 [definition, in additions.Euclidean_Chains]
+F9 [lemma, in hydras.Epsilon0.F_alpha]
+F9 [definition, in additions.Euclidean_Chains]
+F9_correct [lemma, in additions.Euclidean_Chains]
+f:1 [binder, in hydras.Ackermann.PAconsistent]
+f:1 [binder, in hydras.Ackermann.NN2PA]
+f:1 [binder, in hydras.MoreAck.AckNotPR]
+f:1 [binder, in hydras.MoreAck.Iterate_compat]
+f:1 [binder, in hydras.Ackermann.PA]
+f:10 [binder, in hydras.Ackermann.folProof]
+f:10 [binder, in hydras.Prelude.DecPreOrder_Instances]
+f:10 [binder, in hydras.MoreAck.AckNotPR]
+f:10 [binder, in hydras.Epsilon0.Large_Sets]
+f:10 [binder, in Goedel.rosserPA]
+f:10 [binder, in hydras.Ackermann.wellFormed]
+f:10 [binder, in gaia_hydras.GLarge_Sets]
+f:10 [binder, in hydras.Ackermann.wConsistent]
+f:100 [binder, in hydras.Ackermann.LNN2LNT]
+f:100 [binder, in hydras.Ackermann.folLogic]
+f:100 [binder, in hydras.Ackermann.folLogic3]
+f:1000 [binder, in hydras.Ackermann.codeSubFormula]
+f:1001 [binder, in hydras.Ackermann.codeSubFormula]
+f:1002 [binder, in hydras.Ackermann.codeSubFormula]
+f:1003 [binder, in hydras.Ackermann.codeSubFormula]
+f:1006 [binder, in hydras.Ackermann.codeSubFormula]
+f:1009 [binder, in hydras.Ackermann.codeSubFormula]
+f:101 [binder, in hydras.Ackermann.LNT]
+f:101 [binder, in hydras.Ackermann.codePA]
+f:1012 [binder, in hydras.Ackermann.codeSubFormula]
+f:1015 [binder, in hydras.Ackermann.codeSubFormula]
+f:1018 [binder, in hydras.Ackermann.codeSubFormula]
+f:102 [binder, in hydras.Ackermann.LNN]
+f:102 [binder, in hydras.Ackermann.folReplace]
+f:102 [binder, in hydras.Ackermann.codePA]
+f:1021 [binder, in hydras.Ackermann.codeSubFormula]
+f:1022 [binder, in hydras.Ackermann.codeSubFormula]
+f:1023 [binder, in hydras.Ackermann.codeSubFormula]
+f:1024 [binder, in hydras.Ackermann.codeSubFormula]
+f:1025 [binder, in hydras.Ackermann.codeSubFormula]
+f:1026 [binder, in hydras.Ackermann.codeSubFormula]
+f:1027 [binder, in hydras.Ackermann.codeSubFormula]
+f:1029 [binder, in hydras.Ackermann.codeSubFormula]
+f:103 [binder, in hydras.Ackermann.folProp]
+f:103 [binder, in hydras.Ackermann.folLogic]
+f:103 [binder, in hydras.OrdinalNotations.ON_Generic]
+f:103 [binder, in gaia_hydras.nfwfgaia]
+f:103 [binder, in hydras.Ackermann.codePA]
+f:1031 [binder, in hydras.Ackermann.codeSubFormula]
+f:1033 [binder, in hydras.Ackermann.codeSubFormula]
+f:1035 [binder, in hydras.Ackermann.codeSubFormula]
+f:1037 [binder, in hydras.Ackermann.codeSubFormula]
+f:1039 [binder, in hydras.Ackermann.codeSubFormula]
+f:104 [binder, in hydras.Prelude.Iterates]
+f:104 [binder, in hydras.Ackermann.LNN]
+f:104 [binder, in hydras.Ackermann.LNT]
+f:104 [binder, in hydras.rpo.rpo]
+f:1041 [binder, in hydras.Ackermann.codeSubFormula]
+f:1043 [binder, in hydras.Ackermann.codeSubFormula]
+f:1045 [binder, in hydras.Ackermann.codeSubFormula]
+f:1047 [binder, in hydras.Ackermann.codeSubFormula]
+f:105 [binder, in hydras.Ackermann.codePA]
+f:1050 [binder, in hydras.Ackermann.codeSubFormula]
+f:1053 [binder, in hydras.Ackermann.codeSubFormula]
+f:1056 [binder, in hydras.Ackermann.codeSubFormula]
+f:1059 [binder, in hydras.Ackermann.codeSubFormula]
+f:106 [binder, in hydras.Ackermann.folLogic]
+f:1062 [binder, in hydras.Ackermann.codeSubFormula]
+f:1065 [binder, in hydras.Ackermann.codeSubFormula]
+f:1068 [binder, in hydras.Ackermann.codeSubFormula]
+f:107 [binder, in hydras.Ackermann.LNN]
+f:107 [binder, in hydras.Ackermann.LNT]
+f:107 [binder, in hydras.Ackermann.folReplace]
+f:107 [binder, in hydras.Ackermann.codePA]
+f:1071 [binder, in hydras.Ackermann.codeSubFormula]
+f:1073 [binder, in hydras.Ackermann.codeSubFormula]
+f:1075 [binder, in hydras.Ackermann.codeSubFormula]
+f:1077 [binder, in hydras.Ackermann.codeSubFormula]
+f:1079 [binder, in hydras.Ackermann.codeSubFormula]
+f:108 [binder, in hydras.Prelude.Iterates]
+f:108 [binder, in hydras.Ackermann.folLogic]
+f:1081 [binder, in hydras.Ackermann.codeSubFormula]
+f:1083 [binder, in hydras.Ackermann.codeSubFormula]
+f:1084 [binder, in hydras.Ackermann.codeSubFormula]
+f:1085 [binder, in hydras.Ackermann.codeSubFormula]
+f:1086 [binder, in hydras.Ackermann.codeSubFormula]
+f:1087 [binder, in hydras.Ackermann.codeSubFormula]
+f:1088 [binder, in hydras.Ackermann.codeSubFormula]
+f:1089 [binder, in hydras.Ackermann.codeSubFormula]
+f:109 [binder, in Goedel.rosserPA]
+f:109 [binder, in hydras.Ackermann.fol]
+f:109 [binder, in hydras.Ackermann.codePA]
+f:1090 [binder, in hydras.Ackermann.codeSubFormula]
+f:1091 [binder, in hydras.Ackermann.codeSubFormula]
+f:1093 [binder, in hydras.Ackermann.codeSubFormula]
+f:1095 [binder, in hydras.Ackermann.codeSubFormula]
+f:1097 [binder, in hydras.Ackermann.codeSubFormula]
+f:1099 [binder, in hydras.Ackermann.codeSubFormula]
+f:11 [binder, in hydras.Prelude.Iterates]
+f:11 [binder, in hydras.Ackermann.checkPrf]
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+

G

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+g [axiom, in hydras.Prelude.LibHyps_Experiments]
+G [definition, in Goedel.goedel1]
+g [definition, in hydras.Schutte.Ordering_Functions]
+G [definition, in Goedel.goedel2]
+GaiaToHydra [library]
+Gamma [section, in additions.AM]
+Gamma [section, in additions.Euclidean_Chains]
+gammaA:295 [binder, in additions.Addition_Chains]
+gammaB:296 [binder, in additions.Addition_Chains]
+gamma_gt [projection, in additions.Strategies]
+gamma_lt [projection, in additions.Strategies]
+gamma_ [definition, in hydras.Schutte.Critical]
+gamma_nat:204 [binder, in additions.Addition_Chains]
+gamma_positive [lemma, in hydras.Epsilon0.Paths]
+gamma':140 [binder, in hydras.Gamma0.Gamma0]
+gamma':161 [binder, in hydras.Gamma0.Gamma0]
+Gamma.All_OK.c [variable, in additions.AM]
+Gamma.All_OK.n [variable, in additions.AM]
+Gamma.All_OK [section, in additions.AM]
+Gamma.gamma [variable, in additions.AM]
+Gamma.gamma [variable, in additions.Euclidean_Chains]
+Gamma0 [definition, in hydras.Schutte.Critical]
+Gamma0 [module, in gaia_hydras.nfwfgaia]
+Gamma0 [library]
+Gamma0_rpo [module, in hydras.Gamma0.Gamma0]
+Gamma0_alg [module, in hydras.Gamma0.Gamma0]
+Gamma0_prec.prec_transitive [lemma, in hydras.Gamma0.Gamma0]
+Gamma0_prec.prec_antisym [lemma, in hydras.Gamma0.Gamma0]
+Gamma0_prec.prec_dec [lemma, in hydras.Gamma0.Gamma0]
+Gamma0_prec.status [definition, in hydras.Gamma0.Gamma0]
+Gamma0_prec.Mul [constructor, in hydras.Gamma0.Gamma0]
+Gamma0_prec.Lex [constructor, in hydras.Gamma0.Gamma0]
+Gamma0_prec.status_type [inductive, in hydras.Gamma0.Gamma0]
+Gamma0_prec.prec [definition, in hydras.Gamma0.Gamma0]
+Gamma0_prec.A [definition, in hydras.Gamma0.Gamma0]
+Gamma0_prec [module, in hydras.Gamma0.Gamma0]
+Gamma0_sig.arity [definition, in hydras.Gamma0.Gamma0]
+Gamma0_sig.Free [constructor, in hydras.Gamma0.Gamma0]
+Gamma0_sig.C [constructor, in hydras.Gamma0.Gamma0]
+Gamma0_sig.AC [constructor, in hydras.Gamma0.Gamma0]
+Gamma0_sig.arity_type [inductive, in hydras.Gamma0.Gamma0]
+Gamma0_sig.eq_symbol_dec [lemma, in hydras.Gamma0.Gamma0]
+Gamma0_sig.symb [definition, in hydras.Gamma0.Gamma0]
+Gamma0_sig.ord_cons [constructor, in hydras.Gamma0.Gamma0]
+Gamma0_sig.ord_psi [constructor, in hydras.Gamma0.Gamma0]
+Gamma0_sig.ord_zero [constructor, in hydras.Gamma0.Gamma0]
+Gamma0_sig.nat_S [constructor, in hydras.Gamma0.Gamma0]
+Gamma0_sig.nat_0 [constructor, in hydras.Gamma0.Gamma0]
+Gamma0_sig.symb0 [inductive, in hydras.Gamma0.Gamma0]
+Gamma0_sig [module, in hydras.Gamma0.Gamma0]
+Gamma0.add_le4 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.add_le3 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.add_inj [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.add_le2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.add_le1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.add_to_cons [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.add_fin_omega [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.add_int [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.add1Nfin [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.ap_pr4 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.ap_pr3 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.ap_pr2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.ap_pr1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.ap_limit [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.ap_pr0 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.CompareT2Eq [constructor, in gaia_hydras.nfwfgaia]
+Gamma0.CompareT2Gt [constructor, in gaia_hydras.nfwfgaia]
+Gamma0.CompareT2Lt [constructor, in gaia_hydras.nfwfgaia]
+Gamma0.compare_T2 [inductive, in gaia_hydras.nfwfgaia]
+Gamma0.cons [constructor, in gaia_hydras.nfwfgaia]
+Gamma0.epsilon0 [definition, in gaia_hydras.nfwfgaia]
+Gamma0.le_succ_succE [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.le_succ_succ [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.limit_pr [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.limit_pr1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.lt_succ_le_4 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.lt_succ_le_3 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.lt_succ_le_2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.lt_succ_le_1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.lt_succ_succE [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.lt_succ_succ [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.lt_tail [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.lt_psi [definition, in gaia_hydras.nfwfgaia]
+Gamma0.lt_rec [definition, in gaia_hydras.nfwfgaia]
+Gamma0.minus_le [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.minus_lt [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.nf_phi [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.nf_add [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.nf_sub [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.nf_pred [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.nf_succ [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.nf_split [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.nf_finite [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.nf_one [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.nf_omega [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.nf_psi [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.no_critical [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.omega [definition, in gaia_hydras.nfwfgaia]
+Gamma0.omega_minus_one [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.omega_least_inf2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.omega_least_inf1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.omega_lt_epsilon0 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.one [definition, in gaia_hydras.nfwfgaia]
+Gamma0.phi [definition, in gaia_hydras.nfwfgaia]
+Gamma0.phi_inv3 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_inv2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_inv0 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_ltE [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_eqE [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_inj1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_inj [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_mono_c [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_mono_b [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_mono_a [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_inv1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_ab_le2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_ab_le1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_spec4c [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_spec4b [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_spec4a [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_spec3 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_spec2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_spec1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_principalR [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_cases [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_succ [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_fix2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_fix1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_le3 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_le2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_le1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.phi_ap [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.pred_succ [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.pred_lt [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.pred_le [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.psi [definition, in gaia_hydras.nfwfgaia]
+Gamma0.psi_phi1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.psi_phi [definition, in gaia_hydras.nfwfgaia]
+Gamma0.psi_phi_aux [definition, in gaia_hydras.nfwfgaia]
+Gamma0.psi_lt2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.psi_lt1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.size [definition, in gaia_hydras.nfwfgaia]
+Gamma0.size_prop [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.size_prop1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.split_add [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.split_le [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.split_pred [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.split_succ [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.split_is_succ [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.split_limit [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.split_finite [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.sub_pr1r [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.sub_nz [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.sub_pr1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.sub_pr [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.sub_le1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.sub_int [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.sub1a [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.sub1b [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.sub1Nfin [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.succ_is_add_one [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.succ_psi_lt2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.succ_psi_lt [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.succ_psi [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.succ_nz [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.succ_inj [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.succ_p1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.succ_pred [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.succ_lt [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T1succ_nat [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T1T2 [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T1T2range1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T1T2range2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T1T2_inc [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T1T2_inj [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2 [inductive, in gaia_hydras.nfwfgaia]
+Gamma0.T2add [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2addA [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2addn0 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2add_eq0 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2add0n [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2ap [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2eq [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2eqE [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2eqP [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2eq_add2l [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2eq_le [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2finite [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2finite_succ [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2finite1 [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2finite2 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2GeqNotGtn [constructor, in gaia_hydras.nfwfgaia]
+Gamma0.T2GeqNotLtn [constructor, in gaia_hydras.nfwfgaia]
+Gamma0.T2ge1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2GtnNotLeq [constructor, in gaia_hydras.nfwfgaia]
+Gamma0.T2gt1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2is_succ [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2le [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2leNgt [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lenn [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2len0 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2leP [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2leq_xor_gtn [inductive, in gaia_hydras.nfwfgaia]
+Gamma0.T2le_add2r [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2le_add2l [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2le_psi_b [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2le_psi1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2le_trans [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2le_lt_trans [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2le_total [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2le_eqVlt [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2le_consE [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2le0n [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2limit [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2lta [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2ltE [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2ltgtP [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2ltNge [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2ltnn [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2LtnNotGeq [constructor, in gaia_hydras.nfwfgaia]
+Gamma0.T2ltn_xor_geq [inductive, in gaia_hydras.nfwfgaia]
+Gamma0.T2ltn0 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2ltP [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2ltW [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_add2r [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_add2l [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_psi_a [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_psi_b [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_le_trans [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_trans [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_psi_aux [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_trichotomy [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_anti [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_neAle [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_ne' [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_ne [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_consE [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_psi [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt0n [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2lt1 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2nat [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2nat_inc [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2nf [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2nf_finite [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2nf_consE [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2nf_cons_cons [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2pred [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2split [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2sub [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2subnn [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2subn0 [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2sub0n [lemma, in gaia_hydras.nfwfgaia]
+Gamma0.T2succ [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2_pr2 [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2_pr1 [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2_eqType [definition, in gaia_hydras.nfwfgaia]
+Gamma0.T2_eqMixin [definition, in gaia_hydras.nfwfgaia]
+Gamma0.zero [constructor, in gaia_hydras.nfwfgaia]
+_ - _ (g0_scope) [notation, in gaia_hydras.nfwfgaia]
+_ + _ (g0_scope) [notation, in gaia_hydras.nfwfgaia]
+_ > _ (g0_scope) [notation, in gaia_hydras.nfwfgaia]
+_ >= _ (g0_scope) [notation, in gaia_hydras.nfwfgaia]
+_ <= _ (g0_scope) [notation, in gaia_hydras.nfwfgaia]
+_ < _ (g0_scope) [notation, in gaia_hydras.nfwfgaia]
+\F _ (g0_scope) [notation, in gaia_hydras.nfwfgaia]
+[ _ , _ ] (g0_scope) [notation, in gaia_hydras.nfwfgaia]
+gamma0:544 [binder, in hydras.Gamma0.Gamma0]
+gamma1 [definition, in hydras.OrdinalNotations.ON_Finite]
+gamma1:34 [binder, in hydras.Gamma0.T2]
+gamma1:41 [binder, in hydras.Gamma0.T2]
+gamma1:49 [binder, in hydras.Gamma0.T2]
+gamma1:56 [binder, in hydras.Gamma0.T2]
+gamma1:62 [binder, in hydras.Gamma0.T2]
+gamma1:67 [binder, in hydras.Gamma0.T2]
+gamma2:35 [binder, in hydras.Gamma0.T2]
+gamma2:42 [binder, in hydras.Gamma0.T2]
+gamma2:50 [binder, in hydras.Gamma0.T2]
+gamma2:57 [binder, in hydras.Gamma0.T2]
+gamma2:63 [binder, in hydras.Gamma0.T2]
+gamma2:68 [binder, in hydras.Gamma0.T2]
+gamma:1 [binder, in additions.Strategies]
+gamma:10 [binder, in hydras.Schutte.AP]
+gamma:102 [binder, in hydras.Gamma0.T2]
+gamma:111 [binder, in hydras.Epsilon0.Hessenberg]
+gamma:112 [binder, in hydras.Epsilon0.T1]
+gamma:114 [binder, in hydras.Epsilon0.Hessenberg]
+gamma:117 [binder, in hydras.Epsilon0.Paths]
+gamma:12 [binder, in hydras.Schutte.Critical]
+gamma:122 [binder, in hydras.Epsilon0.E0]
+gamma:122 [binder, in hydras.Epsilon0.Paths]
+gamma:125 [binder, in hydras.Epsilon0.Paths]
+gamma:126 [binder, in hydras.Gamma0.Gamma0]
+gamma:129 [binder, in hydras.Gamma0.Gamma0]
+gamma:13 [binder, in hydras.Schutte.Correctness_E0]
+gamma:131 [binder, in hydras.Epsilon0.Paths]
+gamma:132 [binder, in hydras.Gamma0.Gamma0]
+gamma:136 [binder, in hydras.Gamma0.Gamma0]
+gamma:138 [binder, in hydras.Epsilon0.Paths]
+gamma:144 [binder, in hydras.Epsilon0.Paths]
+gamma:15 [binder, in hydras.solutions_exercises.predSuccUnicity]
+gamma:150 [binder, in hydras.Gamma0.Gamma0]
+gamma:156 [binder, in hydras.Gamma0.Gamma0]
+gamma:160 [binder, in hydras.Gamma0.Gamma0]
+gamma:162 [binder, in hydras.Gamma0.Gamma0]
+gamma:167 [binder, in hydras.Epsilon0.Paths]
+gamma:17 [binder, in hydras.Schutte.Correctness_E0]
+gamma:172 [binder, in hydras.Epsilon0.Paths]
+gamma:179 [binder, in hydras.Gamma0.Gamma0]
+gamma:18 [binder, in hydras.solutions_exercises.predSuccUnicity]
+gamma:18 [binder, in hydras.Epsilon0.Paths]
+gamma:19 [binder, in hydras.solutions_exercises.Limit_Infinity]
+gamma:20 [binder, in hydras.Gamma0.Gamma0]
+gamma:203 [binder, in additions.Addition_Chains]
+gamma:216 [binder, in hydras.Epsilon0.Paths]
+gamma:22 [binder, in hydras.solutions_exercises.Limit_Infinity]
+gamma:220 [binder, in hydras.Epsilon0.Paths]
+gamma:23 [binder, in additions.Fib2]
+gamma:25 [binder, in gaia_hydras.GCanon]
+gamma:250 [binder, in hydras.Epsilon0.Paths]
+gamma:26 [binder, in hydras.Schutte.Addition]
+gamma:27 [binder, in hydras.Gamma0.T2]
+gamma:298 [binder, in hydras.Epsilon0.Paths]
+gamma:302 [binder, in hydras.Epsilon0.Paths]
+gamma:309 [binder, in hydras.Epsilon0.Paths]
+gamma:31 [binder, in hydras.Schutte.Addition]
+gamma:315 [binder, in hydras.Epsilon0.T1]
+gamma:33 [binder, in hydras.Epsilon0.Paths]
+gamma:339 [binder, in hydras.Epsilon0.T1]
+gamma:34 [binder, in hydras.Schutte.Correctness_E0]
+gamma:34 [binder, in hydras.Epsilon0.Canon]
+gamma:34 [binder, in hydras.Schutte.Addition]
+gamma:342 [binder, in hydras.Epsilon0.T1]
+gamma:35 [binder, in hydras.Epsilon0.Canon]
+gamma:36 [binder, in hydras.Schutte.Correctness_E0]
+gamma:38 [binder, in hydras.Schutte.Correctness_E0]
+gamma:38 [binder, in hydras.Epsilon0.Canon]
+gamma:39 [binder, in gaia_hydras.GCanon]
+gamma:396 [binder, in hydras.Epsilon0.T1]
+gamma:40 [binder, in hydras.Schutte.Correctness_E0]
+gamma:40 [binder, in hydras.Epsilon0.Canon]
+gamma:44 [binder, in hydras.Schutte.Addition]
+gamma:468 [binder, in hydras.Epsilon0.Paths]
+gamma:47 [binder, in hydras.Schutte.Addition]
+gamma:48 [binder, in hydras.Schutte.Addition]
+gamma:498 [binder, in hydras.Gamma0.Gamma0]
+gamma:50 [binder, in hydras.Schutte.CNF]
+gamma:50 [binder, in hydras.Schutte.Addition]
+gamma:509 [binder, in hydras.Epsilon0.Paths]
+gamma:51 [binder, in hydras.Schutte.Addition]
+gamma:513 [binder, in hydras.Epsilon0.Paths]
+gamma:513 [binder, in hydras.Gamma0.Gamma0]
+gamma:516 [binder, in hydras.Gamma0.Gamma0]
+gamma:519 [binder, in hydras.Epsilon0.Paths]
+gamma:52 [binder, in hydras.Schutte.Addition]
+gamma:54 [binder, in hydras.Epsilon0.Hessenberg]
+gamma:542 [binder, in hydras.Epsilon0.Paths]
+gamma:545 [binder, in hydras.Gamma0.Gamma0]
+gamma:58 [binder, in hydras.Hydra.O2H]
+gamma:58 [binder, in hydras.Schutte.Addition]
+gamma:581 [binder, in hydras.Epsilon0.Paths]
+gamma:584 [binder, in hydras.Epsilon0.T1]
+gamma:59 [binder, in hydras.Epsilon0.Hessenberg]
+gamma:590 [binder, in hydras.Epsilon0.T1]
+gamma:60 [binder, in hydras.Hydra.O2H]
+gamma:60 [binder, in hydras.Epsilon0.Hessenberg]
+gamma:61 [binder, in hydras.Schutte.Addition]
+gamma:627 [binder, in hydras.Gamma0.Gamma0]
+gamma:63 [binder, in hydras.Hydra.O2H]
+gamma:631 [binder, in hydras.Gamma0.Gamma0]
+gamma:637 [binder, in hydras.Epsilon0.T1]
+gamma:643 [binder, in hydras.Gamma0.Gamma0]
+gamma:660 [binder, in hydras.Gamma0.Gamma0]
+gamma:663 [binder, in hydras.Gamma0.Gamma0]
+gamma:67 [binder, in hydras.Epsilon0.Hessenberg]
+gamma:676 [binder, in hydras.Epsilon0.T1]
+gamma:68 [binder, in hydras.Schutte.AP]
+gamma:693 [binder, in hydras.Gamma0.Gamma0]
+gamma:7 [binder, in additions.Strategies]
+gamma:72 [binder, in hydras.OrdinalNotations.ON_Omega2]
+gamma:74 [binder, in hydras.Schutte.Critical]
+gamma:82 [binder, in hydras.Hydra.O2H]
+gamma:82 [binder, in hydras.Epsilon0.Hessenberg]
+gamma:82 [binder, in hydras.Epsilon0.E0]
+gamma:85 [binder, in hydras.Epsilon0.E0]
+gamma:91 [binder, in hydras.Epsilon0.Canon]
+gamma:96 [binder, in hydras.Epsilon0.E0]
+gamma:97 [binder, in hydras.Epsilon0.Canon]
+GCanon [library]
+gcanon_limit_v2 [lemma, in gaia_hydras.GCanon]
+gcanon_zero [lemma, in gaia_hydras.GCanon]
+gcons [constructor, in hydras.Gamma0.T2]
+gcons_lt_e0 [constructor, in hydras.Gamma0.T2]
+gcons_nf [constructor, in hydras.Gamma0.T2]
+ge [definition, in hydras.Schutte.Schutte_basics]
+GEN [constructor, in hydras.Ackermann.folProof]
+General_Relations.elagage.to_nat_elagage.R [variable, in hydras.Schutte.GRelations]
+General_Relations.elagage.to_nat_elagage [section, in hydras.Schutte.GRelations]
+General_Relations.elagage [section, in hydras.Schutte.GRelations]
+General_Relations.surjection2injection.R_surj [variable, in hydras.Schutte.GRelations]
+General_Relations.surjection2injection.R [variable, in hydras.Schutte.GRelations]
+General_Relations.surjection2injection [section, in hydras.Schutte.GRelations]
+General_Relations.injection2surjection.R_inj [variable, in hydras.Schutte.GRelations]
+General_Relations.injection2surjection.R [variable, in hydras.Schutte.GRelations]
+General_Relations.injection2surjection [section, in hydras.Schutte.GRelations]
+General_Relations.DB [variable, in hydras.Schutte.GRelations]
+General_Relations.DA [variable, in hydras.Schutte.GRelations]
+General_Relations.B [variable, in hydras.Schutte.GRelations]
+General_Relations.A [variable, in hydras.Schutte.GRelations]
+General_Relations.Definitions.R [variable, in hydras.Schutte.GRelations]
+General_Relations.Definitions.DB [variable, in hydras.Schutte.GRelations]
+General_Relations.Definitions.DA [variable, in hydras.Schutte.GRelations]
+General_Relations.Definitions.B [variable, in hydras.Schutte.GRelations]
+General_Relations.Definitions.A [variable, in hydras.Schutte.GRelations]
+General_Relations.Definitions [section, in hydras.Schutte.GRelations]
+General_Relations [section, in hydras.Schutte.GRelations]
+Generic [section, in hydras.Prelude.Merge_Sort]
+Generic.A [variable, in hydras.Prelude.Merge_Sort]
+Generic.Merging [section, in hydras.Prelude.Merge_Sort]
+Generic.Merging.Correctness [section, in hydras.Prelude.Merge_Sort]
+Generic.Merging.Correctness.merge_sort.split_permutation [variable, in hydras.Prelude.Merge_Sort]
+Generic.Merging.Correctness.merge_sort.split_decr [variable, in hydras.Prelude.Merge_Sort]
+Generic.Merging.Correctness.merge_sort.split [variable, in hydras.Prelude.Merge_Sort]
+Generic.Merging.Correctness.merge_sort [section, in hydras.Prelude.Merge_Sort]
+Generic.Merging.le [variable, in hydras.Prelude.Merge_Sort]
+_ <= _ [notation, in hydras.Prelude.Merge_Sort]
+Generic.Splitting [section, in hydras.Prelude.Merge_Sort]
+gen_K [constructor, in additions.Euclidean_Chains]
+gen_F [constructor, in additions.Euclidean_Chains]
+gen:219 [binder, in additions.Addition_Chains]
+get_decomposition [definition, in hydras.Epsilon0.T1]
+get_predecessor [lemma, in hydras.Prelude.More_Arith]
+ge_omega_iff [lemma, in hydras.solutions_exercises.ge_omega_iff]
+ge_zero [lemma, in hydras.Schutte.Addition]
+ge_o_segment [lemma, in hydras.Schutte.Addition]
+ge_omega_iff [library]
+gE0le_iff [lemma, in gaia_hydras.T1Bridge]
+gE0lt_wf [lemma, in gaia_hydras.T1Bridge]
+gE0lt_iff [lemma, in gaia_hydras.T1Bridge]
+gE0_eq_intro [lemma, in gaia_hydras.T1Bridge]
+GF_alpha [library]
+GHessenberg [library]
+GHprime [library]
+GHydra [library]
+GLarge_Sets [library]
+GL_alpha [library]
+gnaw [definition, in gaia_hydras.GPaths]
+Gnaw [definition, in hydras.Epsilon0.Large_Sets]
+gnaw [definition, in hydras.Epsilon0.Paths]
+gnawS [definition, in gaia_hydras.GPaths]
+GnawS [definition, in hydras.Epsilon0.Large_Sets]
+gnawS [definition, in hydras.Epsilon0.Paths]
+GnawS_omega [lemma, in hydras.Epsilon0.Large_Sets]
+GnawS_Gnaw [lemma, in hydras.Epsilon0.Large_Sets]
+gnawS_path_toS [lemma, in hydras.Epsilon0.Paths]
+gnawS_cut2 [lemma, in hydras.Epsilon0.Paths]
+gnawS_cut1 [lemma, in hydras.Epsilon0.Paths]
+gnawS_SSn [lemma, in hydras.Epsilon0.Paths]
+gnawS_tail [lemma, in hydras.Epsilon0.Paths]
+gnawS_succ_eqn2 [lemma, in hydras.Epsilon0.Paths]
+gnawS_succ_eqn1 [lemma, in hydras.Epsilon0.Paths]
+gnawS_lim2 [lemma, in hydras.Epsilon0.Paths]
+gnawS_lim1 [lemma, in hydras.Epsilon0.Paths]
+gnaws_rw [lemma, in hydras.Epsilon0.Paths]
+gnawS_app [lemma, in hydras.Epsilon0.Paths]
+gnawS_to_path_toS [lemma, in hydras.Epsilon0.Paths]
+gnawS_nf [lemma, in hydras.Epsilon0.Paths]
+gnawS_zero [lemma, in hydras.Epsilon0.Paths]
+gnawS_gnaw [lemma, in hydras.Epsilon0.Paths]
+gnaw_omega_1 [lemma, in hydras.Epsilon0.Large_Sets]
+gnaw_omega_n_SSn [lemma, in hydras.Epsilon0.Large_Sets]
+Gnaw_omega [lemma, in hydras.Epsilon0.Large_Sets]
+Gnaw_GnawS [lemma, in hydras.Epsilon0.Large_Sets]
+gnaw_last_step [lemma, in hydras.Epsilon0.Large_Sets]
+gnaw_n_R [lemma, in hydras.Epsilon0.Large_Sets]
+gnaw_finite [lemma, in hydras.Epsilon0.Large_Sets]
+gnaw_finite_1 [lemma, in hydras.Epsilon0.Large_Sets]
+gnaw_finite_1_iota [lemma, in hydras.Epsilon0.Large_Sets]
+gnaw_path_to [lemma, in hydras.Epsilon0.Paths]
+gnaw_app [lemma, in hydras.Epsilon0.Paths]
+gnaw_to_path_to [lemma, in hydras.Epsilon0.Paths]
+gnaw_rw [lemma, in hydras.Epsilon0.Paths]
+gnaw_succ [lemma, in hydras.Epsilon0.Paths]
+gnaw_zero [lemma, in hydras.Epsilon0.Paths]
+gnaw_nf [lemma, in hydras.Epsilon0.Paths]
+gnaw_gnawS [lemma, in hydras.Epsilon0.Paths]
+Goedel'sIncompleteness1st [lemma, in Goedel.goedel1]
+Goedel's_1st_Incompleteness.expressT2 [variable, in Goedel.goedel1]
+Goedel's_1st_Incompleteness.expressT1 [variable, in Goedel.goedel1]
+Goedel's_1st_Incompleteness.freeVarRepT [variable, in Goedel.goedel1]
+Goedel's_1st_Incompleteness.v0 [variable, in Goedel.goedel1]
+Goedel's_1st_Incompleteness.repT [variable, in Goedel.goedel1]
+Goedel's_1st_Incompleteness.extendsNN [variable, in Goedel.goedel1]
+Goedel's_1st_Incompleteness.T [variable, in Goedel.goedel1]
+Goedel's_1st_Incompleteness [section, in Goedel.goedel1]
+Goedel's_2nd_Incompleteness.HBL3 [variable, in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.HBL2 [variable, in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.Goedel1PA [section, in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.expressT2 [variable, in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.expressT1 [variable, in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.freeVarRepT [variable, in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.v0 [variable, in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.repT [variable, in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.extendsPA [variable, in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.T [variable, in Goedel.goedel2]
+Goedel's_2nd_Incompleteness [section, in Goedel.goedel2]
+goedel1 [library]
+goedel2 [library]
+go:159 [binder, in hydras.Hydra.Hydra_Lemmas]
+GPaths [library]
+GPrelude [library]
+gP:26 [binder, in gaia_hydras.T1Bridge]
+grec:79 [binder, in hydras.Ackermann.folProp]
+GRelation [definition, in hydras.Schutte.GRelations]
+GRelations [library]
+gR:29 [binder, in gaia_hydras.T1Bridge]
+GS [lemma, in Goedel.goedel2]
+gstrict_lub [definition, in gaia_hydras.T1Bridge]
+gs:37 [binder, in hydras.MoreAck.AckNotPR]
+gtail [definition, in hydras.Gamma0.T2]
+gtIsPR [instance, in hydras.Ackermann.primRec]
+g_bij:91 [binder, in hydras.Schutte.PartialFun]
+g_bij:85 [binder, in hydras.Schutte.PartialFun]
+g_1_of [lemma, in hydras.Schutte.Ordering_Functions]
+g_1_bij [lemma, in hydras.Schutte.Ordering_Functions]
+g_bij [lemma, in hydras.Schutte.Ordering_Functions]
+g_mono [lemma, in hydras.Schutte.Ordering_Functions]
+g_lemma [lemma, in hydras.Schutte.Ordering_Functions]
+g_def [lemma, in hydras.Schutte.Ordering_Functions]
+g_beta':130 [binder, in hydras.Schutte.Ordering_Functions]
+g_beta:129 [binder, in hydras.Schutte.Ordering_Functions]
+g_unic [lemma, in hydras.Schutte.Ordering_Functions]
+g_beta:128 [binder, in hydras.Schutte.Ordering_Functions]
+g_def1 [lemma, in hydras.Schutte.Ordering_Functions]
+g_l':275 [binder, in hydras.rpo.rpo]
+G0 [module, in hydras.Gamma0.Gamma0]
+G0.compare_correct [lemma, in hydras.Gamma0.Gamma0]
+G0.compare_G0 [instance, in hydras.Gamma0.Gamma0]
+G0.Ex42 [definition, in hydras.Gamma0.Gamma0]
+G0.Finite [instance, in hydras.Gamma0.Gamma0]
+G0.Gamma0 [instance, in hydras.Gamma0.Gamma0]
+G0.Gamma0_comp [instance, in hydras.Gamma0.Gamma0]
+G0.G0 [record, in hydras.Gamma0.Gamma0]
+G0.le [definition, in hydras.Gamma0.Gamma0]
+G0.lt [definition, in hydras.Gamma0.Gamma0]
+G0.LT [definition, in hydras.Gamma0.Gamma0]
+G0.lt_wf [lemma, in hydras.Gamma0.Gamma0]
+G0.lt_sto [instance, in hydras.Gamma0.Gamma0]
+G0.lt_LT_incl [lemma, in hydras.Gamma0.Gamma0]
+G0.Lt_wf [lemma, in hydras.Gamma0.Gamma0]
+G0.mkg0 [constructor, in hydras.Gamma0.Gamma0]
+G0.nfb_proof_unicity [lemma, in hydras.Gamma0.Gamma0]
+G0.nfb_equiv [lemma, in hydras.Gamma0.Gamma0]
+G0.nfb_a [lemma, in hydras.Gamma0.Gamma0]
+G0.omega [abbreviation, in hydras.Gamma0.Gamma0]
+G0.Omega [instance, in hydras.Gamma0.Gamma0]
+G0.phi [abbreviation, in hydras.Gamma0.Gamma0]
+G0.Phi [instance, in hydras.Gamma0.Gamma0]
+G0.phi0 [abbreviation, in hydras.Gamma0.Gamma0]
+G0.Plus [instance, in hydras.Gamma0.Gamma0]
+G0.vnf [projection, in hydras.Gamma0.Gamma0]
+G0.vnf_ok [projection, in hydras.Gamma0.Gamma0]
+G0.zero [instance, in hydras.Gamma0.Gamma0]
+G0.zero_nfb [lemma, in hydras.Gamma0.Gamma0]
+omega^ (g0_scope) [notation, in hydras.Gamma0.Gamma0]
+_ + _ (g0_scope) [notation, in hydras.Gamma0.Gamma0]
+g0:22 [binder, in hydras.Ackermann.Deduction]
+g0:23 [binder, in hydras.Ackermann.Deduction]
+g0:64 [binder, in hydras.MoreAck.AckNotPR]
+g0:65 [binder, in hydras.MoreAck.AckNotPR]
+g1:111 [binder, in hydras.Ackermann.primRec]
+g1:114 [binder, in hydras.Ackermann.primRec]
+g2h [definition, in gaia_hydras.T1Bridge]
+g2h [definition, in gaia_hydras.T2Bridge]
+g2h_canon [lemma, in gaia_hydras.GCanon]
+g2h_seq [definition, in gaia_hydras.T1Bridge]
+g2h_E0zero [lemma, in gaia_hydras.T1Bridge]
+g2h_E0_succ [lemma, in gaia_hydras.T1Bridge]
+g2h_succ [lemma, in gaia_hydras.T1Bridge]
+g2h_plusE [lemma, in gaia_hydras.T1Bridge]
+g2h_multE [lemma, in gaia_hydras.T1Bridge]
+g2h_zero [lemma, in gaia_hydras.T1Bridge]
+g2h_cons [lemma, in gaia_hydras.T1Bridge]
+g2h_phi0 [lemma, in gaia_hydras.T1Bridge]
+g2h_diffE [lemma, in gaia_hydras.T1Bridge]
+g2h_eqE [lemma, in gaia_hydras.T1Bridge]
+g2h_h2gK [lemma, in gaia_hydras.T1Bridge]
+g2h_eqE [lemma, in gaia_hydras.T2Bridge]
+g2h_h2gK [lemma, in gaia_hydras.T2Bridge]
+g2:112 [binder, in hydras.Ackermann.primRec]
+g2:115 [binder, in hydras.Ackermann.primRec]
+g:10 [binder, in hydras.Ackermann.Deduction]
+g:102 [binder, in hydras.Ackermann.LNT]
+g:103 [binder, in hydras.Ackermann.folReplace]
+g:104 [binder, in hydras.OrdinalNotations.ON_Generic]
+g:105 [binder, in hydras.Ackermann.LNN]
+g:105 [binder, in hydras.Ackermann.LNT]
+g:108 [binder, in hydras.Ackermann.LNN]
+g:108 [binder, in hydras.Ackermann.LNT]
+g:108 [binder, in hydras.Ackermann.folReplace]
+g:109 [binder, in hydras.Ackermann.folLogic]
+g:110 [binder, in hydras.Ackermann.fol]
+g:111 [binder, in hydras.Ackermann.LNN]
+g:113 [binder, in hydras.Ackermann.folLogic]
+g:116 [binder, in hydras.Ackermann.folLogic]
+g:117 [binder, in hydras.Ackermann.fol]
+g:117 [binder, in hydras.OrdinalNotations.ON_Generic]
+g:119 [binder, in hydras.Ackermann.LNT]
+g:12 [binder, in hydras.Ackermann.primRec]
+g:12 [binder, in hydras.Ackermann.Deduction]
+g:12 [binder, in hydras.Ackermann.folLogic]
+g:121 [binder, in hydras.Ackermann.folLogic]
+g:122 [binder, in hydras.Ackermann.LNN2LNT]
+g:122 [binder, in hydras.Ackermann.LNN]
+g:123 [binder, in hydras.Ackermann.LNT]
+g:124 [binder, in hydras.Ackermann.folLogic]
+g:126 [binder, in hydras.Ackermann.LNN]
+g:126 [binder, in hydras.Ackermann.LNT]
+g:127 [binder, in hydras.Ackermann.LNN2LNT]
+g:129 [binder, in hydras.Ackermann.LNN]
+g:13 [binder, in hydras.solutions_exercises.MinPR2]
+g:13 [binder, in Goedel.rosser]
+g:130 [binder, in hydras.Ackermann.LNN2LNT]
+g:131 [binder, in hydras.Ackermann.LNT]
+g:132 [binder, in hydras.Ackermann.LNN2LNT]
+g:133 [binder, in Goedel.PRrepresentable]
+g:133 [binder, in hydras.Ackermann.LNN2LNT]
+g:134 [binder, in hydras.Ackermann.LNN2LNT]
+g:134 [binder, in hydras.Ackermann.LNN]
+g:134 [binder, in hydras.Ackermann.LNT]
+g:134 [binder, in hydras.Ackermann.subProp]
+g:135 [binder, in hydras.Ackermann.LNN2LNT]
+g:137 [binder, in hydras.Ackermann.LNN]
+g:14 [binder, in hydras.Epsilon0.Epsilon0rpo]
+g:14 [binder, in Goedel.rosser]
+G:14 [binder, in hydras.MoreAck.FolExamples]
+g:141 [binder, in hydras.Ackermann.codePA]
+g:141 [binder, in hydras.rpo.rpo]
+g:142 [binder, in hydras.Ackermann.LNN2LNT]
+g:143 [binder, in hydras.OrdinalNotations.ON_Generic]
+g:144 [binder, in Goedel.PRrepresentable]
+g:146 [binder, in hydras.Ackermann.LNN2LNT]
+g:149 [binder, in hydras.Ackermann.codeSubFormula]
+g:15 [binder, in hydras.Ackermann.Deduction]
+g:15 [binder, in hydras.Ackermann.folLogic]
+g:152 [binder, in hydras.Prelude.Iterates]
+g:155 [binder, in Goedel.PRrepresentable]
+g:156 [binder, in hydras.OrdinalNotations.ON_Generic]
+g:16 [binder, in Goedel.rosserPA]
+g:16 [binder, in Goedel.goedel2]
+g:161 [binder, in hydras.Prelude.Iterates]
+g:162 [binder, in hydras.Ackermann.primRec]
+g:167 [binder, in hydras.Ackermann.model]
+g:168 [binder, in hydras.Ackermann.model]
+g:168 [binder, in hydras.OrdinalNotations.ON_Generic]
+g:169 [binder, in hydras.Prelude.Iterates]
+g:169 [binder, in hydras.Ackermann.model]
+g:169 [binder, in hydras.Ackermann.codePA]
+g:17 [binder, in Goedel.rosserPA]
+g:170 [binder, in hydras.Ackermann.primRec]
+g:172 [binder, in hydras.Ackermann.codePA]
+g:174 [binder, in hydras.Prelude.Iterates]
+g:178 [binder, in hydras.Ackermann.codeSubTerm]
+g:178 [binder, in gaia_hydras.T1Bridge]
+g:178 [binder, in hydras.Ackermann.primRec]
+g:18 [binder, in Goedel.rosserPA]
+g:18 [binder, in hydras.Ackermann.folLogic]
+g:18 [binder, in Goedel.goedel1]
+g:180 [binder, in hydras.OrdinalNotations.ON_Generic]
+g:181 [binder, in hydras.Ackermann.codeSubTerm]
+g:181 [binder, in gaia_hydras.T1Bridge]
+g:184 [binder, in gaia_hydras.T1Bridge]
+g:185 [binder, in hydras.Prelude.Iterates]
+g:186 [binder, in hydras.Prelude.Iterates]
+g:188 [binder, in hydras.Prelude.Iterates]
+g:188 [binder, in gaia_hydras.T1Bridge]
+g:189 [binder, in hydras.Ackermann.primRec]
+g:19 [binder, in hydras.Ackermann.LNT]
+g:19 [binder, in hydras.Schutte.Critical]
+g:19 [binder, in Goedel.rosserPA]
+g:19 [binder, in Goedel.goedel1]
+g:19 [binder, in hydras.Ackermann.Languages]
+g:192 [binder, in hydras.Prelude.Iterates]
+g:193 [binder, in hydras.Prelude.Iterates]
+g:193 [binder, in hydras.OrdinalNotations.ON_Generic]
+g:195 [binder, in hydras.Prelude.Iterates]
+g:199 [binder, in hydras.Prelude.Iterates]
+g:20 [binder, in hydras.Ackermann.checkPrf]
+g:20 [binder, in Goedel.goedel1]
+g:20 [binder, in Goedel.goedel2]
+g:200 [binder, in hydras.Ackermann.primRec]
+g:201 [binder, in hydras.Prelude.Iterates]
+g:204 [binder, in hydras.Prelude.Iterates]
+g:205 [binder, in hydras.Ackermann.codeSubFormula]
+g:21 [binder, in Goedel.goedel1]
+g:2108 [binder, in gaia_hydras.nfwfgaia]
+g:211 [binder, in hydras.Ackermann.primRec]
+g:213 [binder, in hydras.Ackermann.codeSubFormula]
+g:216 [binder, in hydras.Prelude.Iterates]
+g:217 [binder, in hydras.Prelude.Iterates]
+g:218 [binder, in hydras.Prelude.Iterates]
+g:219 [binder, in hydras.Prelude.Iterates]
+g:22 [binder, in hydras.Ackermann.LNN]
+g:22 [binder, in hydras.Ackermann.LNT]
+g:22 [binder, in Goedel.goedel1]
+g:220 [binder, in hydras.Prelude.Iterates]
+g:221 [binder, in additions.Addition_Chains]
+g:223 [binder, in hydras.Prelude.Iterates]
+g:227 [binder, in hydras.Prelude.Iterates]
+g:228 [binder, in hydras.Ackermann.primRec]
+g:229 [binder, in additions.Addition_Chains]
+g:23 [binder, in Goedel.goedel1]
+g:234 [binder, in hydras.Ackermann.wellFormed]
+g:237 [binder, in hydras.Ackermann.wellFormed]
+g:238 [binder, in additions.Addition_Chains]
+g:24 [binder, in Goedel.rosserPA]
+g:244 [binder, in hydras.Ackermann.primRec]
+g:25 [binder, in hydras.Ackermann.LNN]
+g:25 [binder, in hydras.Ackermann.LNT]
+g:25 [binder, in Goedel.rosserPA]
+g:25 [binder, in hydras.Ackermann.folLogic]
+g:25 [binder, in Goedel.rosser]
+g:25 [binder, in Goedel.goedel2]
+g:252 [binder, in hydras.Ackermann.fol]
+g:255 [binder, in hydras.Gamma0.Gamma0]
+g:260 [binder, in hydras.Ackermann.primRec]
+g:264 [binder, in hydras.Ackermann.cPair]
+g:278 [binder, in hydras.rpo.rpo]
+g:279 [binder, in hydras.rpo.rpo]
+g:28 [binder, in hydras.Ackermann.LNN]
+g:28 [binder, in hydras.Ackermann.LNT]
+g:28 [binder, in hydras.Ackermann.folLogic]
+g:281 [binder, in hydras.Ackermann.primRec]
+g:29 [binder, in hydras.Prelude.Iterates]
+g:294 [binder, in hydras.Ackermann.primRec]
+g:302 [binder, in hydras.Ackermann.fol]
+g:31 [binder, in hydras.Ackermann.LNN]
+g:31 [binder, in hydras.Ackermann.folLogic]
+g:31 [binder, in hydras.Schutte.Ordering_Functions]
+g:31 [binder, in hydras.MoreAck.FolExamples]
+g:320 [binder, in hydras.rpo.rpo]
+g:321 [binder, in hydras.rpo.rpo]
+g:323 [binder, in hydras.Ackermann.primRec]
+g:327 [binder, in Goedel.PRrepresentable]
+g:33 [binder, in hydras.Prelude.Iterates]
+g:332 [binder, in Goedel.PRrepresentable]
+g:337 [binder, in Goedel.PRrepresentable]
+g:34 [binder, in hydras.Ackermann.folLogic]
+g:34 [binder, in Goedel.codeSysPrf]
+g:345 [binder, in hydras.Ackermann.primRec]
+g:348 [binder, in hydras.Ackermann.fol]
+g:349 [binder, in hydras.Ackermann.primRec]
+g:35 [binder, in hydras.Prelude.Iterates]
+g:354 [binder, in hydras.Ackermann.primRec]
+g:357 [binder, in hydras.Ackermann.primRec]
+g:36 [binder, in Goedel.rosserPA]
+g:361 [binder, in hydras.Ackermann.primRec]
+g:368 [binder, in hydras.Ackermann.primRec]
+g:37 [binder, in hydras.Ackermann.LNT]
+g:375 [binder, in hydras.Ackermann.primRec]
+g:379 [binder, in hydras.Ackermann.primRec]
+g:38 [binder, in hydras.Prelude.Iterates]
+g:385 [binder, in hydras.Ackermann.primRec]
+g:39 [binder, in hydras.MoreAck.AckNotPR]
+g:39 [binder, in Goedel.rosserPA]
+g:39 [binder, in hydras.Ackermann.folLogic]
+g:392 [binder, in hydras.Ackermann.primRec]
+g:392 [binder, in hydras.Ackermann.fol]
+g:398 [binder, in hydras.Ackermann.primRec]
+g:40 [binder, in hydras.Ackermann.LNN]
+g:40 [binder, in hydras.Ackermann.LNT]
+g:406 [binder, in hydras.Ackermann.primRec]
+g:41 [binder, in hydras.Epsilon0.Large_Sets]
+g:42 [binder, in hydras.Ackermann.code]
+g:42 [binder, in Goedel.rosserPA]
+g:423 [binder, in hydras.Ackermann.primRec]
+g:428 [binder, in hydras.Ackermann.primRec]
+g:43 [binder, in hydras.Ackermann.LNN]
+g:43 [binder, in hydras.Ackermann.LNT]
+g:43 [binder, in hydras.MoreAck.AckNotPR]
+g:43 [binder, in hydras.Ackermann.folLogic]
+g:435 [binder, in hydras.Ackermann.primRec]
+g:44 [binder, in hydras.Prelude.Iterates]
+g:45 [binder, in hydras.MoreAck.PrimRecExamples]
+g:46 [binder, in hydras.Ackermann.LNN]
+g:46 [binder, in hydras.Ackermann.LNT]
+g:46 [binder, in Goedel.rosserPA]
+g:46 [binder, in hydras.rpo.rpo]
+g:47 [binder, in hydras.MoreAck.AckNotPR]
+g:47 [binder, in hydras.Ackermann.folLogic]
+g:48 [binder, in hydras.Ackermann.folProof]
+g:48 [binder, in hydras.MoreAck.PrimRecExamples]
+g:49 [binder, in hydras.Ackermann.LNN]
+g:49 [binder, in hydras.Ackermann.LNT]
+g:49 [binder, in Goedel.rosserPA]
+g:49 [binder, in hydras.Ackermann.expressible]
+g:50 [binder, in hydras.Epsilon0.Large_Sets]
+g:50 [binder, in Goedel.rosserPA]
+g:50 [binder, in hydras.Ackermann.folLogic]
+g:51 [binder, in hydras.MoreAck.AckNotPR]
+g:51 [binder, in Goedel.rosserPA]
+g:51 [binder, in hydras.MoreAck.PrimRecExamples]
+g:52 [binder, in hydras.Ackermann.LNN]
+g:52 [binder, in Goedel.rosserPA]
+g:53 [binder, in hydras.Ackermann.LNT]
+g:53 [binder, in Goedel.rosserPA]
+g:53 [binder, in hydras.Ackermann.folLogic]
+g:53 [binder, in gaia_hydras.nfwfgaia]
+g:53 [binder, in hydras.MoreAck.PrimRecExamples]
+g:54 [binder, in Goedel.rosserPA]
+g:55 [binder, in Goedel.rosserPA]
+g:56 [binder, in hydras.Ackermann.LNN]
+g:56 [binder, in hydras.Ackermann.folLogic]
+g:56 [binder, in hydras.MoreAck.PrimRecExamples]
+g:563 [binder, in hydras.Ackermann.checkPrf]
+g:565 [binder, in hydras.Ackermann.checkPrf]
+g:57 [binder, in hydras.Ackermann.LNT]
+g:57 [binder, in hydras.OrdinalNotations.ON_Generic]
+g:58 [binder, in hydras.Prelude.Iterates]
+g:59 [binder, in hydras.Ackermann.folLogic]
+g:59 [binder, in Goedel.codeSysPrf]
+g:597 [binder, in gaia_hydras.nfwfgaia]
+g:6 [binder, in hydras.rpo.more_list]
+g:6 [binder, in hydras.Ackermann.code]
+g:6 [binder, in hydras.Ackermann.PA]
+g:60 [binder, in hydras.Ackermann.LNN]
+g:60 [binder, in hydras.Ackermann.LNT]
+g:60 [binder, in hydras.MoreAck.PrimRecExamples]
+g:62 [binder, in hydras.Ackermann.primRec]
+g:62 [binder, in hydras.Ackermann.folLogic]
+g:63 [binder, in hydras.Prelude.Iterates]
+g:63 [binder, in hydras.Ackermann.LNN]
+g:63 [binder, in hydras.Ackermann.LNT]
+g:63 [binder, in hydras.Ackermann.fol]
+g:64 [binder, in hydras.Ackermann.primRec]
+g:645 [binder, in hydras.Ackermann.primRec]
+g:65 [binder, in hydras.MoreAck.PrimRecExamples]
+g:651 [binder, in hydras.Ackermann.primRec]
+g:656 [binder, in hydras.Ackermann.primRec]
+g:66 [binder, in hydras.Prelude.Iterates]
+g:66 [binder, in hydras.Ackermann.LNN]
+g:66 [binder, in hydras.Ackermann.LNT]
+g:66 [binder, in hydras.MoreAck.AckNotPR]
+g:663 [binder, in hydras.Ackermann.primRec]
+g:668 [binder, in hydras.Ackermann.primRec]
+g:67 [binder, in hydras.MoreAck.AckNotPR]
+g:67 [binder, in hydras.MoreAck.PrimRecExamples]
+g:68 [binder, in hydras.Ackermann.model]
+g:68 [binder, in hydras.Ackermann.primRec]
+g:69 [binder, in hydras.Ackermann.LNN]
+g:69 [binder, in hydras.Ackermann.LNT]
+g:70 [binder, in hydras.Ackermann.fol]
+g:71 [binder, in hydras.MoreAck.PrimRecExamples]
+g:72 [binder, in hydras.Ackermann.LNN]
+g:72 [binder, in hydras.Ackermann.LNT]
+g:74 [binder, in hydras.MoreAck.PrimRecExamples]
+g:75 [binder, in hydras.Ackermann.LNN]
+g:77 [binder, in hydras.Ackermann.folProp]
+g:781 [binder, in hydras.Ackermann.codeSubFormula]
+g:784 [binder, in hydras.Ackermann.codeSubFormula]
+g:79 [binder, in hydras.Ackermann.folLogic]
+g:79 [binder, in hydras.MoreAck.PrimRecExamples]
+g:8 [binder, in hydras.solutions_exercises.MinPR2]
+g:812 [binder, in hydras.Ackermann.primRec]
+g:816 [binder, in hydras.Ackermann.primRec]
+g:82 [binder, in hydras.Prelude.Iterates]
+g:820 [binder, in hydras.Ackermann.primRec]
+g:823 [binder, in hydras.Ackermann.primRec]
+g:83 [binder, in hydras.MoreAck.PrimRecExamples]
+g:84 [binder, in hydras.Schutte.PartialFun]
+g:86 [binder, in hydras.Ackermann.folProp]
+g:86 [binder, in hydras.Ackermann.folLogic]
+g:86 [binder, in Goedel.codeSysPrf]
+g:89 [binder, in hydras.Ackermann.LNT]
+g:89 [binder, in Goedel.codeSysPrf]
+g:89 [binder, in hydras.Epsilon0.Hprime]
+g:9 [binder, in hydras.Ackermann.primRec]
+g:9 [binder, in hydras.Ackermann.folLogic]
+g:90 [binder, in hydras.Schutte.PartialFun]
+g:92 [binder, in hydras.Ackermann.LNN]
+g:92 [binder, in hydras.Ackermann.folLogic]
+g:94 [binder, in hydras.Ackermann.codeSubFormula]
+g:95 [binder, in hydras.Ackermann.folLogic]
+g:96 [binder, in hydras.Ackermann.LNT]
+g:96 [binder, in hydras.MoreAck.AckNotPR]
+g:98 [binder, in hydras.Ackermann.folLogic]
+g:98 [binder, in hydras.Ackermann.folReplace]
+g:99 [binder, in hydras.Ackermann.LNN]
+

H

+h [axiom, in hydras.Prelude.LibHyps_Experiments]
+HAB:175 [binder, in hydras.Epsilon0.Large_Sets]
+HAB:22 [binder, in gaia_hydras.GLarge_Sets]
+HAck:88 [binder, in hydras.MoreAck.AckNotPR]
+half [definition, in additions.BinaryStrat]
+Halpha:14 [binder, in gaia_hydras.GHprime]
+Halpha:170 [binder, in hydras.Epsilon0.Large_Sets]
+Halpha:401 [binder, in hydras.Epsilon0.Paths]
+Halpha:662 [binder, in hydras.Epsilon0.T1]
+Halpha:665 [binder, in hydras.Epsilon0.T1]
+Halpha:710 [binder, in hydras.Epsilon0.T1]
+Ha:17 [binder, in gaia_hydras.GLarge_Sets]
+HA:173 [binder, in hydras.Epsilon0.Large_Sets]
+HA:20 [binder, in gaia_hydras.GLarge_Sets]
+Ha:390 [binder, in hydras.Epsilon0.Paths]
+Ha:50 [binder, in gaia_hydras.GCanon]
+Ha:9 [binder, in gaia_hydras.GCanon]
+Ha:98 [binder, in hydras.Epsilon0.Hessenberg]
+HBL1 [lemma, in Goedel.goedel2]
+hbounded_transitionS [abbreviation, in gaia_hydras.GPaths]
+Hb:100 [binder, in hydras.Epsilon0.Hessenberg]
+HB:174 [binder, in hydras.Epsilon0.Large_Sets]
+HB:21 [binder, in gaia_hydras.GLarge_Sets]
+Hb:391 [binder, in hydras.Epsilon0.Paths]
+hcanon [abbreviation, in gaia_hydras.GCanon]
+hCanon_plus [abbreviation, in gaia_hydras.GPaths]
+hcnf [abbreviation, in gaia_hydras.T1Bridge]
+hcons [abbreviation, in gaia_hydras.T2Bridge]
+hcons [constructor, in hydras.Hydra.Hydra_Definitions]
+hconst_path [abbreviation, in gaia_hydras.GPaths]
+hconst_pathS [abbreviation, in gaia_hydras.GPaths]
+hcons_mult_S2 [lemma, in hydras.Hydra.Hydra_Lemmas]
+hcons_mult_S1 [lemma, in hydras.Hydra.Hydra_Lemmas]
+hcons_mult_S0 [lemma, in hydras.Hydra.Hydra_Lemmas]
+hcons_mult_comm [lemma, in hydras.Hydra.Hydra_Lemmas]
+hcons_mult_app [lemma, in hydras.Hydra.Hydra_Lemmas]
+hcons_mult [definition, in hydras.Hydra.Hydra_Definitions]
+Hc:102 [binder, in hydras.Epsilon0.Hessenberg]
+head [abbreviation, in hydras.Hydra.Hydra_Definitions]
+headbits [lemma, in additions.fib]
+head_LT_cons [lemma, in hydras.Epsilon0.T1]
+head_lt_cons [lemma, in hydras.Epsilon0.T1]
+head_lt [lemma, in hydras.Epsilon0.T1]
+head_lt [lemma, in hydras.Schutte.Correctness_E0]
+head_no_round [lemma, in hydras.Hydra.Hydra_Lemmas]
+head_no_round_n [lemma, in hydras.Hydra.Hydra_Lemmas]
+height [definition, in hydras.Hydra.Hydra_Definitions]
+height_lt_size [lemma, in hydras.Hydra.Hydra_Examples]
+height_bad [lemma, in hydras.Hydra.Omega_Small]
+height_var [instance, in hydras.Hydra.Omega_Small]
+height:17 [binder, in hydras.Epsilon0.T1]
+Hessenberg [library]
+hE0 [abbreviation, in gaia_hydras.T1Bridge]
+hE0fin [abbreviation, in gaia_hydras.T1Bridge]
+hE0is_succ [abbreviation, in gaia_hydras.T1Bridge]
+hE0le [abbreviation, in gaia_hydras.T1Bridge]
+hE0limit [abbreviation, in gaia_hydras.T1Bridge]
+hE0lt [abbreviation, in gaia_hydras.T1Bridge]
+hE0omega [abbreviation, in gaia_hydras.T1Bridge]
+hE0phi0 [abbreviation, in gaia_hydras.T1Bridge]
+hE0zero [abbreviation, in gaia_hydras.T1Bridge]
+HFLim [lemma, in hydras.Epsilon0.F_alpha]
+HFsucc [lemma, in hydras.Epsilon0.F_alpha]
+hF_ [abbreviation, in gaia_hydras.GF_alpha]
+Hf':121 [binder, in hydras.Prelude.Iterates]
+Hf':155 [binder, in hydras.Prelude.Iterates]
+Hf':94 [binder, in hydras.Prelude.Iterates]
+HF0 [lemma, in hydras.Epsilon0.F_alpha]
+Hf:120 [binder, in hydras.Prelude.Iterates]
+Hf:154 [binder, in hydras.Prelude.Iterates]
+Hf:47 [binder, in hydras.Prelude.Iterates]
+Hf:50 [binder, in hydras.Prelude.Iterates]
+Hf:74 [binder, in hydras.Schutte.PartialFun]
+Hf:75 [binder, in hydras.MoreAck.AckNotPR]
+Hf:79 [binder, in hydras.MoreAck.AckNotPR]
+Hf:84 [binder, in hydras.MoreAck.AckNotPR]
+Hf:88 [binder, in hydras.Prelude.Iterates]
+Hf:93 [binder, in hydras.Prelude.Iterates]
+Hgamma:159 [binder, in additions.AM]
+Hgamma:24 [binder, in additions.Fib2]
+Hgamma:354 [binder, in additions.Euclidean_Chains]
+Hgamma:8 [binder, in additions.Strategies]
+hgnaw [abbreviation, in gaia_hydras.GPaths]
+hgnawS [abbreviation, in gaia_hydras.GPaths]
+Hgt:163 [binder, in hydras.Prelude.Iterates]
+hh_2 [constructor, in hydras.Hydra.O2H]
+hh_1 [constructor, in hydras.Hydra.O2H]
+Hij:64 [binder, in gaia_hydras.GCanon]
+Hincl:170 [binder, in hydras.Prelude.MoreLists]
+hinit [definition, in hydras.Hydra.BigBattle]
+Hk:74 [binder, in additions.Euclidean_Chains]
+Hlambda:51 [binder, in gaia_hydras.GCanon]
+hlarge [abbreviation, in gaia_hydras.GLarge_Sets]
+hlargeS [abbreviation, in gaia_hydras.GLarge_Sets]
+Hlim:45 [binder, in gaia_hydras.GCanon]
+Hlim:52 [binder, in gaia_hydras.GCanon]
+Hlim:63 [binder, in gaia_hydras.GCanon]
+Hlim:67 [binder, in gaia_hydras.GCanon]
+hlt_iff [lemma, in gaia_hydras.T1Bridge]
+hL_spec [abbreviation, in gaia_hydras.GLarge_Sets]
+hmlarge [abbreviation, in gaia_hydras.GLarge_Sets]
+hmlargeS [abbreviation, in gaia_hydras.GLarge_Sets]
+Hmono:200 [binder, in hydras.Schutte.Schutte_basics]
+Hmu [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+hmultA [lemma, in gaia_hydras.GaiaToHydra]
+hmult_dist [lemma, in gaia_hydras.GaiaToHydra]
+Hmu:2 [binder, in hydras.Hydra.Epsilon0_Needed_Free]
+Hmu:3 [binder, in hydras.Hydra.Epsilon0_Needed_Generic]
+Hm':165 [binder, in hydras.Prelude.Iterates]
+Hm:164 [binder, in hydras.Prelude.Iterates]
+hnf_g2h [lemma, in gaia_hydras.T1Bridge]
+Hnf:3 [binder, in gaia_hydras.GF_alpha]
+Hnf:34 [binder, in gaia_hydras.GF_alpha]
+Hnf:40 [binder, in gaia_hydras.GCanon]
+Hnf:44 [binder, in gaia_hydras.GCanon]
+Hnf:62 [binder, in gaia_hydras.GCanon]
+Hnf:66 [binder, in gaia_hydras.GCanon]
+hnil [constructor, in hydras.Hydra.Hydra_Definitions]
+hoplus [abbreviation, in gaia_hydras.GHessenberg]
+hpath [abbreviation, in gaia_hydras.GPaths]
+hpathS [abbreviation, in gaia_hydras.GPaths]
+hpath_to [abbreviation, in gaia_hydras.GPaths]
+Hprime [library]
+hP:25 [binder, in gaia_hydras.T1Bridge]
+Hrecb:719 [binder, in hydras.Ackermann.primRec]
+Hrecb:723 [binder, in hydras.Ackermann.primRec]
+Hrecb:725 [binder, in hydras.Ackermann.primRec]
+Hrecb:728 [binder, in hydras.Ackermann.primRec]
+Hrecb:731 [binder, in hydras.Ackermann.primRec]
+Hrecb:734 [binder, in hydras.Ackermann.primRec]
+Hrecb:737 [binder, in hydras.Ackermann.primRec]
+Hrecb:740 [binder, in hydras.Ackermann.primRec]
+Hrecb:743 [binder, in hydras.Ackermann.primRec]
+Hrecb:746 [binder, in hydras.Ackermann.primRec]
+Hrecb:749 [binder, in hydras.Ackermann.primRec]
+Hrecb:752 [binder, in hydras.Ackermann.primRec]
+Hrecb:755 [binder, in hydras.Ackermann.primRec]
+Hrecb:757 [binder, in hydras.Ackermann.primRec]
+Hrecb:783 [binder, in hydras.Ackermann.primRec]
+Hrecb:786 [binder, in hydras.Ackermann.primRec]
+Hrecb:790 [binder, in hydras.Ackermann.primRec]
+Hrecb:793 [binder, in hydras.Ackermann.primRec]
+Hrecb:797 [binder, in hydras.Ackermann.primRec]
+Hrecb:803 [binder, in hydras.Ackermann.primRec]
+Hrecb:808 [binder, in hydras.Ackermann.primRec]
+Hrecb:811 [binder, in hydras.Ackermann.primRec]
+Hrecn:128 [binder, in hydras.Ackermann.cPair]
+Hrecn:144 [binder, in hydras.Ackermann.cPair]
+Hrecn:147 [binder, in hydras.Ackermann.cPair]
+Hrecn:151 [binder, in hydras.Ackermann.cPair]
+Hrecn:155 [binder, in hydras.Ackermann.cPair]
+Hrecn:157 [binder, in hydras.Ackermann.cPair]
+Hrecn:160 [binder, in hydras.Ackermann.cPair]
+Hrecn:178 [binder, in hydras.Ackermann.model]
+Hrecn:185 [binder, in hydras.Ackermann.model]
+Hrecn:204 [binder, in hydras.Ackermann.model]
+Hrecn:211 [binder, in hydras.Ackermann.model]
+Hrecn:64 [binder, in hydras.Ackermann.folLogic3]
+Hrecs:10 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:108 [binder, in hydras.Ackermann.codeList]
+Hrecs:11 [binder, in hydras.Ackermann.codeList]
+Hrecs:113 [binder, in hydras.Ackermann.codeList]
+Hrecs:116 [binder, in hydras.Ackermann.codeList]
+Hrecs:12 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:121 [binder, in hydras.Ackermann.codeList]
+Hrecs:124 [binder, in hydras.Ackermann.codeList]
+Hrecs:127 [binder, in hydras.Ackermann.codeList]
+Hrecs:13 [binder, in hydras.Ackermann.codeList]
+Hrecs:130 [binder, in hydras.Ackermann.codeList]
+Hrecs:133 [binder, in hydras.Ackermann.codeList]
+Hrecs:136 [binder, in hydras.Ackermann.codeList]
+Hrecs:139 [binder, in hydras.Ackermann.codeList]
+Hrecs:14 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:142 [binder, in hydras.Ackermann.codeList]
+Hrecs:145 [binder, in hydras.Ackermann.codeList]
+Hrecs:147 [binder, in hydras.Ackermann.codeList]
+Hrecs:149 [binder, in hydras.Ackermann.codeList]
+Hrecs:15 [binder, in hydras.Ackermann.codeList]
+Hrecs:151 [binder, in hydras.Ackermann.codeList]
+Hrecs:153 [binder, in hydras.Ackermann.codeList]
+Hrecs:155 [binder, in hydras.Ackermann.codeList]
+Hrecs:16 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:165 [binder, in hydras.Ackermann.codeList]
+Hrecs:168 [binder, in hydras.Ackermann.codeList]
+Hrecs:17 [binder, in hydras.Ackermann.codeList]
+Hrecs:171 [binder, in hydras.Ackermann.codeList]
+Hrecs:174 [binder, in hydras.Ackermann.codeList]
+Hrecs:177 [binder, in hydras.Ackermann.codeList]
+Hrecs:18 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:180 [binder, in hydras.Ackermann.codeList]
+Hrecs:19 [binder, in hydras.Ackermann.wellFormed]
+Hrecs:19 [binder, in hydras.Ackermann.codeList]
+Hrecs:197 [binder, in hydras.Ackermann.codeList]
+Hrecs:199 [binder, in hydras.Ackermann.codeList]
+Hrecs:2 [binder, in hydras.Ackermann.codeList]
+Hrecs:20 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:201 [binder, in hydras.Ackermann.codeList]
+Hrecs:203 [binder, in hydras.Ackermann.codeList]
+Hrecs:205 [binder, in hydras.Ackermann.codeList]
+Hrecs:207 [binder, in hydras.Ackermann.codeList]
+Hrecs:21 [binder, in hydras.Ackermann.wellFormed]
+Hrecs:21 [binder, in hydras.Ackermann.codeList]
+Hrecs:213 [binder, in hydras.Ackermann.codeList]
+Hrecs:218 [binder, in hydras.Ackermann.codeList]
+Hrecs:22 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:221 [binder, in hydras.Ackermann.codeList]
+Hrecs:226 [binder, in hydras.Ackermann.codeList]
+Hrecs:229 [binder, in hydras.Ackermann.codeList]
+Hrecs:23 [binder, in hydras.Ackermann.codeList]
+Hrecs:232 [binder, in hydras.Ackermann.codeList]
+Hrecs:235 [binder, in hydras.Ackermann.codeList]
+Hrecs:238 [binder, in hydras.Ackermann.codeList]
+Hrecs:24 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:241 [binder, in hydras.Ackermann.codeList]
+Hrecs:244 [binder, in hydras.Ackermann.codeList]
+Hrecs:247 [binder, in hydras.Ackermann.codeList]
+Hrecs:25 [binder, in hydras.Ackermann.codeList]
+Hrecs:250 [binder, in hydras.Ackermann.codeList]
+Hrecs:253 [binder, in hydras.Ackermann.codeList]
+Hrecs:256 [binder, in hydras.Ackermann.codeList]
+Hrecs:259 [binder, in hydras.Ackermann.codeList]
+Hrecs:26 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:27 [binder, in hydras.Ackermann.codeList]
+Hrecs:276 [binder, in hydras.Ackermann.codeList]
+Hrecs:278 [binder, in hydras.Ackermann.codeList]
+Hrecs:28 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:280 [binder, in hydras.Ackermann.codeList]
+Hrecs:282 [binder, in hydras.Ackermann.codeList]
+Hrecs:284 [binder, in hydras.Ackermann.codeList]
+Hrecs:286 [binder, in hydras.Ackermann.codeList]
+Hrecs:29 [binder, in hydras.Ackermann.codeList]
+Hrecs:30 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:31 [binder, in hydras.Ackermann.codeList]
+Hrecs:32 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:34 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:36 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:41 [binder, in hydras.Ackermann.codeList]
+Hrecs:42 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:44 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:46 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:46 [binder, in hydras.Ackermann.codeList]
+Hrecs:48 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:49 [binder, in hydras.Ackermann.codeList]
+Hrecs:5 [binder, in hydras.Ackermann.codeList]
+Hrecs:54 [binder, in hydras.Ackermann.codeList]
+Hrecs:57 [binder, in hydras.Ackermann.codeList]
+Hrecs:60 [binder, in hydras.Ackermann.codeList]
+Hrecs:63 [binder, in hydras.Ackermann.codeList]
+Hrecs:66 [binder, in hydras.Ackermann.codeList]
+Hrecs:69 [binder, in hydras.Ackermann.codeList]
+Hrecs:7 [binder, in hydras.Ackermann.codeSubFormula]
+Hrecs:7 [binder, in hydras.Ackermann.codeList]
+Hrecs:72 [binder, in hydras.Ackermann.codeList]
+Hrecs:74 [binder, in hydras.Ackermann.codeList]
+Hrecs:76 [binder, in hydras.Ackermann.codeList]
+Hrecs:78 [binder, in hydras.Ackermann.codeList]
+Hrecs:80 [binder, in hydras.Ackermann.codeList]
+Hrecs:82 [binder, in hydras.Ackermann.codeList]
+Hrecs:84 [binder, in hydras.Ackermann.codeList]
+Hrecs:86 [binder, in hydras.Ackermann.codeList]
+Hrecs:9 [binder, in hydras.Ackermann.codeList]
+Hrecs:90 [binder, in hydras.Ackermann.codeList]
+Hrecs:92 [binder, in hydras.Ackermann.codeList]
+Hrecs:94 [binder, in hydras.Ackermann.codeList]
+Hrecs:96 [binder, in hydras.Ackermann.codeList]
+hR:28 [binder, in gaia_hydras.T1Bridge]
+hsize [definition, in hydras.Hydra.Hydra_Definitions]
+hstandard_gnaw [abbreviation, in gaia_hydras.GPaths]
+hs_eq_dec [lemma, in hydras.Hydra.Hydra_Definitions]
+hs':140 [binder, in hydras.Hydra.Hydra_Lemmas]
+hs':142 [binder, in hydras.Hydra.Hydra_Lemmas]
+hs2l [definition, in hydras.Hydra.Hydra_Definitions]
+hs2lK [lemma, in hydras.Hydra.Hydra_Definitions]
+hs:139 [binder, in hydras.Hydra.Hydra_Lemmas]
+hs:141 [binder, in hydras.Hydra.Hydra_Lemmas]
+hs:8 [binder, in gaia_hydras.GHydra]
+hs:9 [binder, in gaia_hydras.GHydra]
+htransition [abbreviation, in gaia_hydras.GPaths]
+Htrans:18 [binder, in hydras.Prelude.Sort_spec]
+Htrans:21 [binder, in hydras.Prelude.Sort_spec]
+Htrans:25 [binder, in hydras.Prelude.Sort_spec]
+hT1 [abbreviation, in gaia_hydras.T1Bridge]
+hT2 [abbreviation, in gaia_hydras.T2Bridge]
+hVar [instance, in gaia_hydras.GHydra]
+HVariant [instance, in hydras.Hydra.Hydra_Termination]
+Hvariant [record, in hydras.Hydra.Hydra_Definitions]
+HVariant_0 [instance, in hydras.Hydra.Hydra_Termination]
+Hvariant_Termination [lemma, in hydras.Hydra.Hydra_Lemmas]
+Hvar:2 [binder, in hydras.Hydra.Omega2_Small]
+Hvar:2 [binder, in hydras.Hydra.Omega_Small]
+Hwf:130 [binder, in hydras.Hydra.Hydra_Lemmas]
+Hwf:21 [binder, in hydras.solutions_exercises.T1_ltNotWf]
+HX:142 [binder, in hydras.Schutte.Schutte_basics]
+HX:145 [binder, in hydras.Schutte.Schutte_basics]
+Hx:27 [binder, in hydras.Prelude.Restriction]
+Hx:31 [binder, in hydras.Prelude.Restriction]
+Hx:38 [binder, in hydras.Prelude.Restriction]
+HX:61 [binder, in hydras.Schutte.Well_Orders]
+hyd [abbreviation, in hydras.Hydra.BigBattle]
+Hydra [inductive, in hydras.Hydra.Hydra_Definitions]
+Hydrae [inductive, in hydras.Hydra.Hydra_Definitions]
+Hydrae_rect2 [definition, in hydras.Hydra.Hydra_Definitions]
+HydraGaia_Examples [library]
+Hydra_rect2 [definition, in hydras.Hydra.Hydra_Definitions]
+Hydra_Termination [library]
+Hydra_Extraction [library]
+Hydra_Lemmas [library]
+Hydra_Definitions [library]
+Hydra_Examples [library]
+Hydra_Theorems [library]
+hyd_mult [definition, in hydras.Hydra.Hydra_Definitions]
+hyd1 [abbreviation, in hydras.Hydra.Hydra_Definitions]
+hyd2 [abbreviation, in hydras.Hydra.Hydra_Definitions]
+hyd3 [abbreviation, in hydras.Hydra.Hydra_Definitions]
+hyd4 [abbreviation, in hydras.Hydra.KP_example]
+hyper_exp2_S [lemma, in hydras.Prelude.Iterates]
+hyper_exp2 [definition, in hydras.Prelude.Iterates]
+Hypos [section, in hydras.Prelude.First_toggle]
+Hypos.Hn [variable, in hydras.Prelude.First_toggle]
+Hypos.Hnp [variable, in hydras.Prelude.First_toggle]
+Hypos.Hp [variable, in hydras.Prelude.First_toggle]
+Hypos.R [variable, in hydras.Prelude.First_toggle]
+Hypos.search_toggle [variable, in hydras.Prelude.First_toggle]
+Hypos.spec [variable, in hydras.Prelude.First_toggle]
+Hyp1:20 [binder, in hydras.Ackermann.folProof]
+Hyp2:21 [binder, in hydras.Ackermann.folProof]
+hyp:220 [binder, in hydras.Ackermann.fol]
+hyp:221 [binder, in hydras.Ackermann.fol]
+hyp:222 [binder, in hydras.Ackermann.fol]
+Hyp:24 [binder, in hydras.Ackermann.folProof]
+Hyp:46 [binder, in hydras.Ackermann.folProof]
+Hyp:48 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+hy_app_l_nil [lemma, in hydras.Hydra.O2H]
+hy_app_assoc [lemma, in hydras.Hydra.Hydra_Lemmas]
+hy_app [definition, in hydras.Hydra.Hydra_Definitions]
+Hy'':5 [binder, in hydras.Hydra.Hydra_Examples]
+Hy:28 [binder, in hydras.Prelude.Restriction]
+Hy:4 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+Hy:5 [binder, in hydras.Hydra.Epsilon0_Needed_Free]
+Hy:6 [binder, in hydras.Hydra.Epsilon0_Needed_Generic]
+hzero [abbreviation, in gaia_hydras.T2Bridge]
+h_eq_dec [lemma, in hydras.Hydra.Hydra_Definitions]
+h_forall [definition, in hydras.Hydra.Hydra_Definitions]
+H'_F [lemma, in gaia_hydras.GF_alpha]
+H'_omega_cube_min [lemma, in gaia_hydras.GHprime]
+H'_alpha_gt [lemma, in gaia_hydras.GHprime]
+H'_alpha_mono [lemma, in gaia_hydras.GHprime]
+H'_dom [lemma, in gaia_hydras.GHprime]
+H'_omega_double [lemma, in gaia_hydras.GHprime]
+H'_omega [lemma, in gaia_hydras.GHprime]
+H'_eq3 [lemma, in gaia_hydras.GHprime]
+H'_eq2 [lemma, in gaia_hydras.GHprime]
+H'_eq1 [lemma, in gaia_hydras.GHprime]
+H'_ [definition, in gaia_hydras.GHprime]
+H'_F [lemma, in hydras.Epsilon0.F_alpha]
+H'_F.IHalpha [variable, in hydras.Epsilon0.F_alpha]
+H'_F.alpha [variable, in hydras.Epsilon0.F_alpha]
+H'_F.P [variable, in hydras.Epsilon0.F_alpha]
+H'_F [section, in hydras.Epsilon0.F_alpha]
+H'_L_ [lemma, in hydras.Epsilon0.L_alpha]
+H'_dom [lemma, in hydras.Epsilon0.Hprime]
+H'_mono_l [lemma, in hydras.Epsilon0.Hprime]
+H'_mono_l_1 [lemma, in hydras.Epsilon0.Hprime]
+H'_mono_l_0 [lemma, in hydras.Epsilon0.Hprime]
+H'_alpha_mono_weak [lemma, in hydras.Epsilon0.Hprime]
+H'_alpha_ge_id [lemma, in hydras.Epsilon0.Hprime]
+H'_restricted_mono_l [lemma, in hydras.Epsilon0.Hprime]
+H'_alpha_dom [lemma, in hydras.Epsilon0.Hprime]
+H'_alpha_Succ_le [lemma, in hydras.Epsilon0.Hprime]
+H'_alpha_gt [lemma, in hydras.Epsilon0.Hprime]
+H'_alpha_mono [lemma, in hydras.Epsilon0.Hprime]
+H'_non_mono1 [lemma, in hydras.Epsilon0.Hprime]
+H'_omega_cube_min [lemma, in hydras.Epsilon0.Hprime]
+H'_omega_sqr_min [lemma, in hydras.Epsilon0.Hprime]
+H'_Phi0_omega_exact_formula [lemma, in hydras.Epsilon0.Hprime]
+H'_Phi0_omega [lemma, in hydras.Epsilon0.Hprime]
+H'_omega_cube_3.N [variable, in hydras.Epsilon0.Hprime]
+H'_omega_cube_3.f [variable, in hydras.Epsilon0.Hprime]
+H'_omega_cube_3 [section, in hydras.Epsilon0.Hprime]
+H'_omega_cube [lemma, in hydras.Epsilon0.Hprime]
+H'_Phi0_Si [lemma, in hydras.Epsilon0.Hprime]
+H'_Phi0_succ [lemma, in hydras.Epsilon0.Hprime]
+H'_Phi0_succ_0 [lemma, in hydras.Epsilon0.Hprime]
+H'_Phi0_succ_1 [lemma, in hydras.Epsilon0.Hprime]
+H'_succ_fun [definition, in hydras.Epsilon0.Hprime]
+H'_Omega_term [lemma, in hydras.Epsilon0.Hprime]
+H'_Fin_iterate [lemma, in hydras.Epsilon0.Hprime]
+H'_Omega_term_0 [lemma, in hydras.Epsilon0.Hprime]
+H'_Omega_term_1 [lemma, in hydras.Epsilon0.Hprime]
+H'_cons [lemma, in hydras.Epsilon0.Hprime]
+H'_cons.i [variable, in hydras.Epsilon0.Hprime]
+H'_cons.alpha [variable, in hydras.Epsilon0.Hprime]
+H'_cons [section, in hydras.Epsilon0.Hprime]
+H'_omega_sqr [lemma, in hydras.Epsilon0.Hprime]
+H'_omega_i [lemma, in hydras.Epsilon0.Hprime]
+H'_omega_4 [lemma, in hydras.Epsilon0.Hprime]
+H'_omega_3 [lemma, in hydras.Epsilon0.Hprime]
+H'_omega_double [lemma, in hydras.Epsilon0.Hprime]
+H'_Plus_Fin [lemma, in hydras.Epsilon0.Hprime]
+H'_omega [lemma, in hydras.Epsilon0.Hprime]
+H'_Fin [lemma, in hydras.Epsilon0.Hprime]
+H'_eq2 [lemma, in hydras.Epsilon0.Hprime]
+H'_eq3 [lemma, in hydras.Epsilon0.Hprime]
+H'_eq2_0 [lemma, in hydras.Epsilon0.Hprime]
+H'_eq1 [lemma, in hydras.Epsilon0.Hprime]
+H'_ [definition, in hydras.Epsilon0.Hprime]
+H'_L_ [lemma, in gaia_hydras.GL_alpha]
+h'':100 [binder, in hydras.Hydra.Hydra_Definitions]
+h'':135 [binder, in hydras.Hydra.Hydra_Definitions]
+h'':83 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':100 [binder, in hydras.Hydra.O2H]
+h':101 [binder, in hydras.Hydra.Hydra_Definitions]
+h':107 [binder, in hydras.Hydra.Hydra_Lemmas]
+H':108 [binder, in hydras.OrdinalNotations.OmegaOmega]
+h':110 [binder, in hydras.Hydra.Hydra_Definitions]
+h':113 [binder, in hydras.Hydra.Hydra_Definitions]
+h':115 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':116 [binder, in hydras.Hydra.Hydra_Definitions]
+h':118 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':12 [binder, in gaia_hydras.GHydra]
+h':121 [binder, in hydras.Hydra.Hydra_Definitions]
+h':122 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':124 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':124 [binder, in hydras.Hydra.Hydra_Definitions]
+h':13 [binder, in hydras.Hydra.Hydra_Definitions]
+h':130 [binder, in hydras.Hydra.Hydra_Definitions]
+h':132 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':134 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':134 [binder, in hydras.Hydra.Hydra_Definitions]
+h':138 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':14 [binder, in gaia_hydras.GHydra]
+h':141 [binder, in hydras.Hydra.Hydra_Definitions]
+h':152 [binder, in hydras.Hydra.Hydra_Definitions]
+h':167 [binder, in hydras.Hydra.Hydra_Definitions]
+h':178 [binder, in hydras.Prelude.Iterates]
+h':19 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':2 [binder, in hydras.Hydra.Hydra_Examples]
+h':23 [binder, in hydras.Hydra.Hydra_Termination]
+h':25 [binder, in hydras.Hydra.Hydra_Termination]
+H':28 [binder, in hydras.Epsilon0.E0]
+h':3 [binder, in hydras.Hydra.Hydra_Examples]
+h':35 [binder, in hydras.Hydra.Hydra_Termination]
+h':36 [binder, in hydras.Hydra.O2H]
+h':4 [binder, in hydras.Hydra.Hydra_Examples]
+h':44 [binder, in hydras.Hydra.Hydra_Termination]
+h':46 [binder, in hydras.Hydra.Hydra_Termination]
+H':489 [binder, in hydras.Epsilon0.T1]
+h':49 [binder, in hydras.Hydra.Hydra_Termination]
+H':493 [binder, in hydras.Epsilon0.T1]
+h':51 [binder, in hydras.Hydra.O2H]
+h':51 [binder, in hydras.Hydra.Hydra_Termination]
+H':696 [binder, in hydras.Gamma0.Gamma0]
+h':7 [binder, in hydras.Hydra.Epsilon0_Needed_Free]
+h':71 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':72 [binder, in hydras.Hydra.Hydra_Definitions]
+h':76 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':80 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':82 [binder, in hydras.Hydra.Hydra_Lemmas]
+h':86 [binder, in hydras.Hydra.Hydra_Definitions]
+h':9 [binder, in hydras.Hydra.Epsilon0_Needed_Generic]
+h':90 [binder, in hydras.Hydra.O2H]
+h':93 [binder, in hydras.Hydra.O2H]
+h':93 [binder, in hydras.Hydra.Hydra_Definitions]
+h':95 [binder, in hydras.Hydra.Hydra_Definitions]
+h':98 [binder, in hydras.Hydra.Hydra_Definitions]
+h0 [definition, in hydras.Hydra.KP_example]
+h0_h1 [lemma, in hydras.Hydra.KP_example]
+H0:21 [binder, in hydras.Epsilon0.F_omega]
+H0:294 [binder, in hydras.rpo.rpo]
+H0:296 [binder, in hydras.rpo.rpo]
+H0:298 [binder, in hydras.rpo.rpo]
+h0:35 [binder, in hydras.Prelude.Restriction]
+h0:41 [binder, in hydras.Prelude.Restriction]
+H0:47 [binder, in hydras.Ackermann.LNN2LNT]
+h0:6 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+H0:7 [binder, in hydras.Prelude.Compat815]
+H0:73 [binder, in hydras.Ackermann.LNN2LNT]
+h0:73 [binder, in hydras.Hydra.Hydra_Lemmas]
+H0:76 [binder, in hydras.Ackermann.LNN2LNT]
+H0:79 [binder, in hydras.Ackermann.LNN2LNT]
+H0:82 [binder, in hydras.Ackermann.LNN2LNT]
+H0:85 [binder, in hydras.Ackermann.LNN2LNT]
+H0:88 [binder, in hydras.Ackermann.LNN2LNT]
+h0:9 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+H0:91 [binder, in hydras.Ackermann.LNN2LNT]
+H0:94 [binder, in hydras.Ackermann.LNN2LNT]
+h1 [abbreviation, in hydras.Hydra.BigBattle]
+h1 [definition, in hydras.Hydra.KP_example]
+h1_h2 [lemma, in hydras.Hydra.KP_example]
+h1:116 [binder, in hydras.Ackermann.primRec]
+H1:310 [binder, in hydras.rpo.rpo]
+H1:312 [binder, in hydras.rpo.rpo]
+H1:32 [binder, in hydras.MoreAck.FolExamples]
+h1:36 [binder, in hydras.Hydra.Hydra_Lemmas]
+H1:371 [binder, in hydras.Epsilon0.T1]
+H1:373 [binder, in hydras.Epsilon0.T1]
+h1:41 [binder, in hydras.Hydra.Hydra_Lemmas]
+H1:48 [binder, in hydras.Ackermann.LNN2LNT]
+H1:60 [binder, in hydras.Epsilon0.T1]
+H1:62 [binder, in hydras.Epsilon0.T1]
+H1:74 [binder, in hydras.Ackermann.LNN2LNT]
+H1:77 [binder, in hydras.Ackermann.LNN2LNT]
+H1:8 [binder, in hydras.Prelude.Compat815]
+H1:80 [binder, in hydras.Ackermann.LNN2LNT]
+H1:83 [binder, in hydras.Ackermann.LNN2LNT]
+H1:86 [binder, in hydras.Ackermann.LNN2LNT]
+H1:89 [binder, in hydras.Ackermann.LNN2LNT]
+H1:92 [binder, in hydras.Ackermann.LNN2LNT]
+H1:95 [binder, in hydras.Ackermann.LNN2LNT]
+h2 [abbreviation, in hydras.Hydra.BigBattle]
+h2 [definition, in hydras.Hydra.KP_example]
+h2g [definition, in gaia_hydras.T1Bridge]
+h2g [definition, in gaia_hydras.T2Bridge]
+h2g_seq [definition, in gaia_hydras.T1Bridge]
+h2g_zero [lemma, in gaia_hydras.T1Bridge]
+h2g_cons [lemma, in gaia_hydras.T1Bridge]
+h2g_diffE [lemma, in gaia_hydras.T1Bridge]
+h2g_eqE [lemma, in gaia_hydras.T1Bridge]
+h2g_g2hK [lemma, in gaia_hydras.T1Bridge]
+h2g_eqE [lemma, in gaia_hydras.T2Bridge]
+h2g_g2hK [lemma, in gaia_hydras.T2Bridge]
+h2_h3 [lemma, in hydras.Hydra.KP_example]
+H2:10 [binder, in hydras.Prelude.Compat815]
+h2:117 [binder, in hydras.Ackermann.primRec]
+H2:33 [binder, in hydras.MoreAck.FolExamples]
+h2:37 [binder, in hydras.Hydra.Hydra_Lemmas]
+H2:374 [binder, in hydras.Epsilon0.T1]
+h2:42 [binder, in hydras.Hydra.Hydra_Lemmas]
+H2:63 [binder, in hydras.Epsilon0.T1]
+h3 [abbreviation, in hydras.Hydra.BigBattle]
+h3 [definition, in hydras.Hydra.KP_example]
+h:1 [binder, in hydras.Hydra.Hydra_Examples]
+h:1 [binder, in hydras.Hydra.Hydra_Termination]
+h:1 [binder, in gaia_hydras.GHydra]
+h:1 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:10 [binder, in hydras.Ackermann.primRec]
+h:10 [binder, in gaia_hydras.GHydra]
+h:10 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:105 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:106 [binder, in hydras.Hydra.Hydra_Definitions]
+H:107 [binder, in hydras.OrdinalNotations.OmegaOmega]
+h:109 [binder, in hydras.Hydra.Hydra_Definitions]
+h:11 [binder, in hydras.Hydra.Hydra_Termination]
+h:11 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+h:11 [binder, in gaia_hydras.GHydra]
+h:11 [binder, in hydras.Hydra.Epsilon0_Needed_Generic]
+H:11 [binder, in hydras.Schutte.Countable]
+h:110 [binder, in hydras.Ackermann.folLogic]
+h:112 [binder, in hydras.Hydra.Hydra_Definitions]
+h:114 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:114 [binder, in hydras.Hydra.Hydra_Definitions]
+h:117 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:12 [binder, in hydras.solutions_exercises.MorePRExamples]
+h:12 [binder, in hydras.Hydra.Hydra_Definitions]
+h:120 [binder, in hydras.Ackermann.LNT]
+h:120 [binder, in additions.fib]
+h:120 [binder, in hydras.Hydra.Hydra_Definitions]
+h:121 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:123 [binder, in hydras.Ackermann.LNN]
+h:123 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:123 [binder, in hydras.Hydra.Hydra_Definitions]
+h:125 [binder, in hydras.Ackermann.folLogic]
+h:129 [binder, in hydras.Hydra.Hydra_Definitions]
+h:13 [binder, in hydras.Ackermann.primRec]
+h:13 [binder, in gaia_hydras.GHydra]
+H:13 [binder, in hydras.MoreAck.PrimRecExamples]
+h:132 [binder, in hydras.Hydra.Hydra_Definitions]
+h:133 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:135 [binder, in hydras.Ackermann.LNT]
+h:135 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:137 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:138 [binder, in hydras.Ackermann.LNN]
+h:138 [binder, in hydras.Hydra.Hydra_Definitions]
+H:14 [binder, in hydras.MoreAck.BadSubst]
+H:14 [binder, in hydras.Prelude.Sort_spec]
+h:142 [binder, in hydras.Hydra.Hydra_Definitions]
+h:144 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:146 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:149 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:15 [binder, in hydras.Hydra.Hydra_Termination]
+h:151 [binder, in hydras.Hydra.Hydra_Definitions]
+h:152 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:155 [binder, in hydras.Hydra.Hydra_Lemmas]
+H:157 [binder, in hydras.Prelude.Iterates]
+H:16 [binder, in hydras.Prelude.First_toggle]
+h:162 [binder, in hydras.Hydra.Hydra_Definitions]
+H:163 [binder, in hydras.Ackermann.primRec]
+h:166 [binder, in hydras.Hydra.Hydra_Definitions]
+H:171 [binder, in hydras.Ackermann.primRec]
+H:172 [binder, in hydras.Ackermann.folProp]
+h:175 [binder, in hydras.Prelude.Iterates]
+H:179 [binder, in hydras.Ackermann.folProp]
+H:179 [binder, in hydras.Ackermann.primRec]
+H:18 [binder, in hydras.Epsilon0.F_omega]
+h:18 [binder, in hydras.Hydra.Hydra_Lemmas]
+H:188 [binder, in hydras.Ackermann.folProp]
+H:190 [binder, in hydras.Ackermann.primRec]
+H:195 [binder, in hydras.Ackermann.folProp]
+H:201 [binder, in hydras.Ackermann.primRec]
+H:201 [binder, in hydras.Schutte.Schutte_basics]
+H:204 [binder, in hydras.Ackermann.folProp]
+h:205 [binder, in hydras.Prelude.Iterates]
+H:21 [binder, in hydras.Schutte.MoreEpsilonIota]
+H:211 [binder, in hydras.Ackermann.folProp]
+H:212 [binder, in hydras.Ackermann.primRec]
+h:22 [binder, in hydras.Hydra.Hydra_Termination]
+H:22 [binder, in hydras.Prelude.STDPP_compat]
+H:220 [binder, in hydras.Ackermann.folProp]
+H:227 [binder, in hydras.Ackermann.folProp]
+H:24 [binder, in hydras.Prelude.Restriction]
+h:24 [binder, in hydras.Schutte.Critical]
+H:24 [binder, in hydras.Schutte.MoreEpsilonIota]
+h:24 [binder, in hydras.Hydra.Hydra_Termination]
+H:24 [binder, in hydras.Prelude.STDPP_compat]
+h:24 [binder, in hydras.Hydra.Hydra_Definitions]
+H:25 [binder, in hydras.Epsilon0.E0]
+H:251 [binder, in hydras.Ackermann.folProp]
+H:258 [binder, in hydras.Ackermann.folProp]
+h:26 [binder, in gaia_hydras.GHydra]
+H:265 [binder, in hydras.Ackermann.cPair]
+H:267 [binder, in hydras.Ackermann.folProp]
+H:27 [binder, in hydras.Epsilon0.E0]
+H:274 [binder, in hydras.Ackermann.folProp]
+h:28 [binder, in hydras.Hydra.O2H]
+H:28 [binder, in hydras.Schutte.Critical]
+H:29 [binder, in hydras.Prelude.Restriction]
+h:30 [binder, in hydras.Hydra.Hydra_Definitions]
+H:300 [binder, in hydras.Ackermann.folProp]
+H:307 [binder, in hydras.Ackermann.folProp]
+H:316 [binder, in hydras.Ackermann.folProp]
+h:32 [binder, in hydras.Hydra.O2H]
+H:323 [binder, in hydras.Ackermann.folProp]
+h:329 [binder, in Goedel.PRrepresentable]
+H:33 [binder, in hydras.MoreAck.AckNotPR]
+H:33 [binder, in hydras.Schutte.Critical]
+H:332 [binder, in hydras.Ackermann.folProp]
+h:334 [binder, in Goedel.PRrepresentable]
+h:339 [binder, in Goedel.PRrepresentable]
+H:339 [binder, in hydras.Ackermann.folProp]
+h:34 [binder, in hydras.Hydra.Hydra_Termination]
+H:34 [binder, in hydras.MoreAck.FolExamples]
+H:348 [binder, in hydras.Ackermann.folProp]
+h:35 [binder, in hydras.Hydra.O2H]
+H:355 [binder, in hydras.Ackermann.folProp]
+h:36 [binder, in hydras.Prelude.Restriction]
+h:36 [binder, in hydras.Hydra.Hydra_Definitions]
+h:362 [binder, in hydras.Ackermann.primRec]
+H:364 [binder, in hydras.Ackermann.folProp]
+H:369 [binder, in hydras.Epsilon0.T1]
+H:37 [binder, in hydras.Schutte.Ordering_Functions]
+H:370 [binder, in hydras.Epsilon0.T1]
+H:371 [binder, in hydras.Ackermann.folProp]
+H:372 [binder, in hydras.Epsilon0.T1]
+h:38 [binder, in hydras.MoreAck.AckNotPR]
+H:38 [binder, in hydras.Schutte.Critical]
+H:38 [binder, in hydras.Hydra.Hydra_Termination]
+H:380 [binder, in hydras.Ackermann.folProp]
+H:385 [binder, in hydras.Epsilon0.Paths]
+H:387 [binder, in hydras.Ackermann.folProp]
+h:40 [binder, in hydras.Hydra.O2H]
+h:40 [binder, in hydras.Ackermann.folLogic]
+H:41 [binder, in hydras.Hydra.Hydra_Termination]
+h:41 [binder, in hydras.Hydra.Hydra_Definitions]
+h:42 [binder, in hydras.Prelude.Restriction]
+H:42 [binder, in hydras.Schutte.Critical]
+H:423 [binder, in hydras.Epsilon0.T1]
+h:43 [binder, in hydras.Hydra.Hydra_Termination]
+H:44 [binder, in hydras.Schutte.Correctness_E0]
+h:44 [binder, in hydras.Ackermann.folLogic]
+h:45 [binder, in hydras.Prelude.Iterates]
+h:45 [binder, in hydras.Hydra.O2H]
+H:45 [binder, in hydras.Prelude.Restriction]
+h:45 [binder, in hydras.Hydra.Hydra_Termination]
+H:46 [binder, in hydras.Ackermann.LNN2LNT]
+h:47 [binder, in hydras.Hydra.Hydra_Termination]
+h:48 [binder, in hydras.Hydra.Hydra_Termination]
+H:487 [binder, in hydras.Epsilon0.T1]
+H:491 [binder, in hydras.Epsilon0.T1]
+H:5 [binder, in hydras.Hydra.Hydra_Theorems]
+h:5 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+h:5 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:50 [binder, in hydras.Ackermann.LNT]
+H:50 [binder, in hydras.Schutte.Critical]
+h:50 [binder, in hydras.Hydra.Hydra_Termination]
+h:50 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:53 [binder, in hydras.Ackermann.LNN]
+H:54 [binder, in hydras.Prelude.Iterates]
+h:54 [binder, in hydras.Ackermann.LNT]
+h:54 [binder, in hydras.Hydra.Hydra_Definitions]
+H:55 [binder, in hydras.MoreAck.AckNotPR]
+h:55 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:57 [binder, in hydras.Ackermann.LNN]
+H:58 [binder, in hydras.Epsilon0.T1]
+H:59 [binder, in hydras.Epsilon0.T1]
+h:59 [binder, in hydras.Prelude.Iterates]
+h:6 [binder, in hydras.Hydra.Epsilon0_Needed_Free]
+h:6 [binder, in hydras.Hydra.Hydra_Examples]
+h:60 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:60 [binder, in hydras.Hydra.Hydra_Definitions]
+H:61 [binder, in hydras.Epsilon0.T1]
+h:63 [binder, in gaia_hydras.nfwfgaia]
+h:64 [binder, in hydras.Prelude.Iterates]
+H:64 [binder, in hydras.Prelude.MoreLists]
+h:65 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:657 [binder, in hydras.Ackermann.primRec]
+h:66 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:67 [binder, in hydras.Prelude.Iterates]
+H:68 [binder, in hydras.Schutte.Correctness_E0]
+h:69 [binder, in hydras.Ackermann.primRec]
+h:69 [binder, in hydras.Hydra.Hydra_Lemmas]
+H:695 [binder, in hydras.Gamma0.Gamma0]
+h:7 [binder, in hydras.Hydra.Hydra_Examples]
+h:7 [binder, in hydras.Hydra.Hydra_Termination]
+h:7 [binder, in gaia_hydras.GHydra]
+h:71 [binder, in hydras.Hydra.Hydra_Definitions]
+H:711 [binder, in hydras.Epsilon0.T1]
+H:72 [binder, in hydras.Schutte.Correctness_E0]
+H:72 [binder, in hydras.Ackermann.LNN2LNT]
+h:73 [binder, in hydras.Hydra.Hydra_Definitions]
+H:75 [binder, in hydras.Schutte.Correctness_E0]
+h:75 [binder, in hydras.Hydra.Hydra_Lemmas]
+H:78 [binder, in hydras.Ackermann.LNN2LNT]
+h:78 [binder, in hydras.Hydra.Hydra_Lemmas]
+H:8 [binder, in hydras.Schutte.Critical]
+h:8 [binder, in hydras.Hydra.Hydra_Examples]
+h:8 [binder, in hydras.Hydra.Epsilon0_Needed_Generic]
+H:8 [binder, in hydras.Schutte.Countable]
+h:81 [binder, in hydras.Hydra.Hydra_Lemmas]
+H:84 [binder, in hydras.Prelude.Iterates]
+H:84 [binder, in hydras.Ackermann.LNN2LNT]
+h:85 [binder, in hydras.Hydra.Hydra_Definitions]
+h:88 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:88 [binder, in hydras.Hydra.Hydra_Definitions]
+h:9 [binder, in hydras.solutions_exercises.MinPR2]
+h:9 [binder, in hydras.Hydra.Hydra_Examples]
+H:90 [binder, in hydras.Ackermann.LNN2LNT]
+h:91 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:92 [binder, in hydras.Hydra.O2H]
+h:92 [binder, in hydras.Hydra.Hydra_Definitions]
+h:93 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:94 [binder, in hydras.Hydra.Hydra_Definitions]
+h:96 [binder, in hydras.Hydra.O2H]
+h:96 [binder, in hydras.Hydra.Hydra_Lemmas]
+h:97 [binder, in hydras.Hydra.Hydra_Definitions]
+h:98 [binder, in hydras.Hydra.O2H]
+h:99 [binder, in hydras.Hydra.Hydra_Definitions]
+

I

+idIsPR [instance, in hydras.Ackermann.primRec]
+idn:126 [binder, in additions.fib]
+id_le_f_alpha [lemma, in hydras.Epsilon0.F_alpha]
+Id2 [definition, in additions.Monoid_instances]
+id:118 [binder, in additions.fib]
+iffExist [lemma, in hydras.Ackermann.folReplace]
+iffE1 [lemma, in hydras.Ackermann.LNN]
+iffE1 [lemma, in hydras.Ackermann.LNT]
+iffE1 [lemma, in hydras.Ackermann.folLogic]
+iffE2 [lemma, in hydras.Ackermann.LNN]
+iffE2 [lemma, in hydras.Ackermann.LNT]
+iffE2 [lemma, in hydras.Ackermann.folLogic]
+iffH [definition, in hydras.Ackermann.fol]
+iffI [lemma, in hydras.Ackermann.LNN]
+iffI [lemma, in hydras.Ackermann.LNT]
+iffI [lemma, in hydras.Ackermann.folLogic]
+iffRefl [lemma, in hydras.Ackermann.LNN]
+iffRefl [lemma, in hydras.Ackermann.LNT]
+iffRefl [lemma, in hydras.Ackermann.folLogic]
+iffSym [lemma, in hydras.Ackermann.LNN]
+iffSym [lemma, in hydras.Ackermann.LNT]
+iffSym [lemma, in hydras.Ackermann.folLogic]
+iffTrans [lemma, in hydras.Ackermann.LNN]
+iffTrans [lemma, in hydras.Ackermann.LNT]
+iffTrans [lemma, in hydras.Ackermann.folLogic]
+ifThenElseH [definition, in hydras.Ackermann.fol]
+if_simpl [lemma, in gaia_hydras.nfwfgaia]
+If2IsPR [instance, in hydras.solutions_exercises.MinPR2]
+ignoreParams [definition, in hydras.Ackermann.primRec]
+ignoreParamsIsPR [instance, in hydras.Ackermann.primRec]
+Ignore_Params [section, in hydras.Ackermann.primRec]
+IHn:102 [binder, in hydras.MoreAck.AckNotPR]
+image [definition, in hydras.Schutte.PartialFun]
+image_as_Im [lemma, in hydras.Schutte.PartialFun]
+image_B_g_seg [lemma, in hydras.Schutte.Ordering_Functions]
+impE [lemma, in hydras.Ackermann.LNN]
+impE [lemma, in hydras.Ackermann.LNT]
+impE [lemma, in hydras.Ackermann.folLogic]
+impForall [lemma, in hydras.Ackermann.folReplace]
+impH [constructor, in hydras.Ackermann.fol]
+impI [lemma, in hydras.Ackermann.LNN]
+impI [lemma, in hydras.Ackermann.LNT]
+impI [lemma, in hydras.Ackermann.folLogic]
+ImpossibilityProof [section, in gaia_hydras.GHydra]
+ImpossibilityProof.m [variable, in gaia_hydras.GHydra]
+ImpossibilityProof.mh [variable, in gaia_hydras.GHydra]
+ImpossibilityProof.mu [variable, in gaia_hydras.GHydra]
+ImpossibilityProof.nfMu [variable, in gaia_hydras.GHydra]
+Impossibility_free [lemma, in hydras.Hydra.Epsilon0_Needed_Free]
+Impossibility_Proof.small_h [variable, in hydras.Hydra.Epsilon0_Needed_Free]
+Impossibility_Proof.big_h [variable, in hydras.Hydra.Epsilon0_Needed_Free]
+Impossibility_Proof [section, in hydras.Hydra.Epsilon0_Needed_Free]
+Impossibility_Proof [section, in hydras.MoreAck.AckNotPR]
+Impossibility_std [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+Impossibility_Proof.small_h [variable, in hydras.Hydra.Epsilon0_Needed_Std]
+Impossibility_Proof.big_h [variable, in hydras.Hydra.Epsilon0_Needed_Std]
+Impossibility_Proof [section, in hydras.Hydra.Epsilon0_Needed_Std]
+Impossibility_std [lemma, in gaia_hydras.GHydra]
+Impossibility_free [lemma, in gaia_hydras.GHydra]
+Impossibility_Proof.small_h [variable, in hydras.Hydra.Omega2_Small]
+Impossibility_Proof.iota [variable, in hydras.Hydra.Omega2_Small]
+Impossibility_Proof.big_h [variable, in hydras.Hydra.Omega2_Small]
+Impossibility_Proof.m [variable, in hydras.Hydra.Omega2_Small]
+Impossibility_Proof [section, in hydras.Hydra.Omega2_Small]
+Impossibility_Proof.small_h [variable, in hydras.Hydra.Omega_Small]
+Impossibility_Proof.big_h [variable, in hydras.Hydra.Omega_Small]
+Impossibility_Proof.iota [variable, in hydras.Hydra.Omega_Small]
+Impossibility_Proof.m [variable, in hydras.Hydra.Omega_Small]
+Impossibility_Proof [section, in hydras.Hydra.Omega_Small]
+Impossibility1 [section, in hydras.solutions_exercises.MultisetWf]
+Impossibility1.m [variable, in hydras.solutions_exercises.MultisetWf]
+Impossibility1.mDecr [variable, in hydras.solutions_exercises.MultisetWf]
+Impossibility1.x [variable, in hydras.solutions_exercises.MultisetWf]
+Impossibility1.y [variable, in hydras.solutions_exercises.MultisetWf]
+Impossible [lemma, in hydras.Hydra.Omega2_Small]
+impossible_nat [lemma, in hydras.solutions_exercises.MultisetWf]
+impRefl [lemma, in hydras.Ackermann.LNN]
+impRefl [lemma, in hydras.Ackermann.LNT]
+impRefl [lemma, in hydras.Ackermann.folLogic]
+impTrans [lemma, in hydras.Ackermann.LNN]
+impTrans [lemma, in hydras.Ackermann.LNT]
+impTrans [lemma, in hydras.Ackermann.folLogic]
+IMP1 [constructor, in hydras.Ackermann.folProof]
+IMP2 [constructor, in hydras.Ackermann.folProof]
+im_U_f [lemma, in hydras.Schutte.Ordering_Functions]
+Incl [instance, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Included_s [definition, in hydras.Prelude.DecPreOrder_Instances]
+Inclusion_ij.b [variable, in hydras.OrdinalNotations.ON_Finite]
+Inclusion_ij.Hij [variable, in hydras.OrdinalNotations.ON_Finite]
+Inclusion_ij.j [variable, in hydras.OrdinalNotations.ON_Finite]
+Inclusion_ij.i [variable, in hydras.OrdinalNotations.ON_Finite]
+Inclusion_ij [section, in hydras.OrdinalNotations.ON_Finite]
+incl_decomp [lemma, in hydras.Prelude.MoreLists]
+incl_inv [lemma, in hydras.Prelude.MoreLists]
+Incompleteness [lemma, in Goedel.rosser]
+Inconsistent [definition, in hydras.Ackermann.folProof]
+Inc_U_A [lemma, in hydras.Schutte.Ordering_Functions]
+independent [definition, in hydras.Ackermann.folProof]
+indIsPR [instance, in hydras.Ackermann.primRec]
+induct [lemma, in hydras.Ackermann.PA]
+InductionSchema [definition, in hydras.Ackermann.PA]
+inductive_step.Hn [variable, in hydras.Epsilon0.F_omega]
+inductive_step.n [variable, in hydras.Epsilon0.F_omega]
+inductive_step [section, in hydras.Epsilon0.F_omega]
+induct_step [lemma, in hydras.Schutte.Critical]
+ind1ParamIsPR [instance, in hydras.Ackermann.primRec]
+ind2ParamIsPR [instance, in hydras.Ackermann.primRec]
+Infinite [definition, in hydras.solutions_exercises.Limit_Infinity]
+info [inductive, in additions.Trace_exercise]
+InH [record, in hydras.Schutte.MoreEpsilonIota]
+InH [inductive, in hydras.Schutte.MoreEpsilonIota]
+inhNat [instance, in hydras.Schutte.MoreEpsilonIota]
+InHWit [projection, in hydras.Schutte.MoreEpsilonIota]
+InHWit [constructor, in hydras.Schutte.MoreEpsilonIota]
+inhX:45 [binder, in hydras.Schutte.Well_Orders]
+Inh_ord [lemma, in hydras.Schutte.Schutte_basics]
+Inh_Ord_Ord [instance, in hydras.Schutte.Schutte_basics]
+Inh_OSets [instance, in hydras.Schutte.Schutte_basics]
+InH_Ord [instance, in hydras.Schutte.Schutte_basics]
+inh_Ord [axiom, in hydras.Schutte.Schutte_basics]
+inh_U [lemma, in hydras.Schutte.Ordering_Functions]
+inh:2 [binder, in hydras.Schutte.MoreEpsilonIota]
+inh:53 [binder, in hydras.Schutte.Well_Orders]
+inh:58 [binder, in hydras.Schutte.Well_Orders]
+Init [constructor, in additions.Trace_exercise]
+inject [definition, in hydras.Schutte.Correctness_E0]
+inject_lt_epsilon0_ex_unique [lemma, in hydras.Schutte.Correctness_E0]
+inject_lt_epsilon0_ex [lemma, in hydras.Schutte.Correctness_E0]
+inject_lt_epsilon0_ex_cnf [lemma, in hydras.Schutte.Correctness_E0]
+inject_mult_fin_r [lemma, in hydras.Schutte.Correctness_E0]
+inject_plus [lemma, in hydras.Schutte.Correctness_E0]
+inject_rw [lemma, in hydras.Schutte.Correctness_E0]
+inject_lt_epsilon0 [lemma, in hydras.Schutte.Correctness_E0]
+inject_monoR [lemma, in hydras.Schutte.Correctness_E0]
+inject_injective [lemma, in hydras.Schutte.Correctness_E0]
+inject_mono [lemma, in hydras.Schutte.Correctness_E0]
+inject_mono_0 [lemma, in hydras.Schutte.Correctness_E0]
+inject_of_phi0 [lemma, in hydras.Schutte.Correctness_E0]
+inject_of_omega [lemma, in hydras.Schutte.Correctness_E0]
+inject_of_finite [lemma, in hydras.Schutte.Correctness_E0]
+inject_of_zero [lemma, in hydras.Schutte.Correctness_E0]
+inject_of_finite_pos [lemma, in hydras.Schutte.Correctness_E0]
+input:61 [binder, in hydras.Ackermann.codeSubFormula]
+instr [inductive, in additions.AM]
+interpFormula [definition, in hydras.Ackermann.model]
+interpRels [definition, in hydras.Ackermann.model]
+interpTerm [definition, in hydras.Ackermann.model]
+interpTerms [definition, in hydras.Ackermann.model]
+interpTermsVector [definition, in hydras.Ackermann.model]
+interval [definition, in hydras.Prelude.MoreLists]
+interval_lt_not_In [lemma, in hydras.Prelude.MoreLists]
+interval_sorted_ge [lemma, in hydras.Prelude.MoreLists]
+interval_unroll [lemma, in hydras.Prelude.MoreLists]
+interval_app [lemma, in hydras.Prelude.MoreLists]
+interval_singleton [lemma, in hydras.Prelude.MoreLists]
+interval_not_empty_iff [lemma, in hydras.Prelude.MoreLists]
+interval_not_empty [lemma, in hydras.Prelude.MoreLists]
+interval_length [lemma, in hydras.Prelude.MoreLists]
+interval_def [lemma, in gaia_hydras.GPaths]
+Int63mult [instance, in additions.Monoid_instances]
+int63_mult_op [instance, in additions.Monoid_instances]
+inv [definition, in hydras.solutions_exercises.FibonacciPR]
+invar:18 [binder, in hydras.Prelude.First_toggle]
+Inverse_fun [instance, in hydras.Prelude.DecPreOrder_Instances]
+Inverse_Image_transp.Rof [variable, in additions.Wf_transparent]
+Inverse_Image_transp.f [variable, in additions.Wf_transparent]
+Inverse_Image_transp.R [variable, in additions.Wf_transparent]
+Inverse_Image_transp.B [variable, in additions.Wf_transparent]
+Inverse_Image_transp.A [variable, in additions.Wf_transparent]
+Inverse_Image_transp [section, in additions.Wf_transparent]
+inversion_of_bijection.f_b [variable, in hydras.Schutte.PartialFun]
+inversion_of_bijection.inv_spec [variable, in hydras.Schutte.PartialFun]
+inversion_of_bijection.f [variable, in hydras.Schutte.PartialFun]
+inversion_of_bijection.DB [variable, in hydras.Schutte.PartialFun]
+inversion_of_bijection.DA [variable, in hydras.Schutte.PartialFun]
+inversion_of_bijection.inhA [variable, in hydras.Schutte.PartialFun]
+inversion_of_bijection.B [variable, in hydras.Schutte.PartialFun]
+inversion_of_bijection.A [variable, in hydras.Schutte.PartialFun]
+inversion_of_bijection [section, in hydras.Schutte.PartialFun]
+inv_Pi [lemma, in hydras.solutions_exercises.FibonacciPR]
+inv_fun_bij [lemma, in hydras.Schutte.PartialFun]
+inv_composeR [lemma, in hydras.Schutte.PartialFun]
+inv_compose [lemma, in hydras.Schutte.PartialFun]
+inv_fun [definition, in hydras.Schutte.PartialFun]
+inv_trans [lemma, in hydras.rpo.closure]
+In_betaFormula_subst_2_1 [lemma, in Goedel.PRrepresentable]
+In_betaFormula_subst_2 [lemma, in Goedel.PRrepresentable]
+In_betaFormula [lemma, in Goedel.PRrepresentable]
+In_betaFormula_subst_1 [lemma, in Goedel.PRrepresentable]
+In_betaFormula_subst_1_2 [lemma, in Goedel.PRrepresentable]
+In_betaFormula_subst_1_2_0 [lemma, in Goedel.PRrepresentable]
+In_sorted_ge_inv [lemma, in hydras.Prelude.MoreLists]
+in_remove [lemma, in hydras.rpo.more_list]
+in_map_in [lemma, in hydras.rpo.more_list]
+in_in_map [lemma, in hydras.rpo.more_list]
+In_freeVarSys [definition, in hydras.Ackermann.folProp]
+In_freeVarListFormulaE [lemma, in hydras.Ackermann.folProp]
+In_freeVarListFormula [lemma, in hydras.Ackermann.folProp]
+In_list_remove2 [abbreviation, in hydras.Ackermann.ListExt]
+in_remove_neq [lemma, in hydras.Ackermann.ListExt]
+In_add2 [lemma, in hydras.MoreAck.FolExamples]
+In_add1 [lemma, in hydras.MoreAck.FolExamples]
+In_cases [lemma, in hydras.Prelude.MoreVectors]
+iota [definition, in hydras.Hydra.O2H]
+iota [definition, in hydras.solutions_exercises.MultisetWf]
+iotas [definition, in hydras.Hydra.O2H]
+iotas_succ [lemma, in hydras.Hydra.O2H]
+iota_phi0 [lemma, in hydras.Hydra.O2H]
+iota_of_succ [lemma, in hydras.Hydra.O2H]
+iota_tail [lemma, in hydras.Hydra.O2H]
+iota_rw4 [lemma, in hydras.Hydra.O2H]
+iota_rw3 [lemma, in hydras.Hydra.O2H]
+iota_rw2 [lemma, in hydras.Hydra.O2H]
+iota_rw1 [lemma, in hydras.Hydra.O2H]
+iota_succ_round [lemma, in hydras.Hydra.O2H]
+iota_succ_round_n [lemma, in hydras.Hydra.O2H]
+iota_succ_R1 [lemma, in hydras.Hydra.O2H]
+iota_iotas [lemma, in hydras.Hydra.O2H]
+iota_from_lt_not_In [lemma, in hydras.Prelude.MoreLists]
+iota_from_sorted_ge [lemma, in hydras.Prelude.MoreLists]
+iota_from_unroll [lemma, in hydras.Prelude.MoreLists]
+iota_from_plus [lemma, in hydras.Prelude.MoreLists]
+iota_from_app [lemma, in hydras.Prelude.MoreLists]
+iota_from [definition, in hydras.Prelude.MoreLists]
+iota_adapt [lemma, in gaia_hydras.GPaths]
+iota_ind [lemma, in hydras.Schutte.MoreEpsilonIota]
+iota_eq [lemma, in hydras.Schutte.MoreEpsilonIota]
+iota_spec_1 [lemma, in hydras.Schutte.MoreEpsilonIota]
+iota_fun_ind [lemma, in hydras.Schutte.PartialFun]
+iota_fun_e [lemma, in hydras.Schutte.PartialFun]
+iota_fun [definition, in hydras.Schutte.PartialFun]
+iota_demo.Bad.le_bottom_zero [lemma, in hydras.Schutte.Schutte_basics]
+iota_demo.Bad.bottom_eq [lemma, in hydras.Schutte.Schutte_basics]
+iota_demo.Bad.le_zero_bottom [lemma, in hydras.Schutte.Schutte_basics]
+iota_demo.Bad.bottom [definition, in hydras.Schutte.Schutte_basics]
+iota_demo.Bad [module, in hydras.Schutte.Schutte_basics]
+iota_demo.zero_le [lemma, in hydras.Schutte.Schutte_basics]
+iota_demo.zero [definition, in hydras.Schutte.Schutte_basics]
+iota_demo.zero [definition, in hydras.Schutte.Schutte_basics]
+iota_demo.R [lemma, in hydras.Schutte.Schutte_basics]
+iota_demo [module, in hydras.Schutte.Schutte_basics]
+iota_mono [lemma, in hydras.OrdinalNotations.ON_Finite]
+iota_compare_commute [lemma, in hydras.OrdinalNotations.ON_Finite]
+iota_ij [definition, in hydras.OrdinalNotations.ON_Finite]
+iota:101 [binder, in hydras.OrdinalNotations.ON_Generic]
+iota:114 [binder, in hydras.OrdinalNotations.ON_Generic]
+iota:129 [binder, in hydras.OrdinalNotations.ON_Generic]
+iota:141 [binder, in hydras.OrdinalNotations.ON_Generic]
+iota:154 [binder, in hydras.OrdinalNotations.ON_Generic]
+iota:38 [binder, in hydras.OrdinalNotations.ON_Generic]
+iota:71 [binder, in hydras.OrdinalNotations.ON_Generic]
+iota:89 [binder, in hydras.OrdinalNotations.ON_Generic]
+Iso_same_op [definition, in hydras.OrdinalNotations.ON_Generic]
+Iso_same_fun [definition, in hydras.OrdinalNotations.ON_Generic]
+Iso_same_cst [definition, in hydras.OrdinalNotations.ON_Generic]
+iso_inv2 [projection, in hydras.OrdinalNotations.ON_Generic]
+iso_inv1 [projection, in hydras.OrdinalNotations.ON_Generic]
+iso_compare [projection, in hydras.OrdinalNotations.ON_Generic]
+isPR [record, in hydras.Ackermann.primRec]
+isPR [inductive, in hydras.Ackermann.primRec]
+isPRextEqual [instance, in hydras.Ackermann.primRec]
+isPRrel [record, in hydras.Ackermann.primRec]
+isPRrel [inductive, in hydras.Ackermann.primRec]
+isPRTrans [instance, in hydras.Ackermann.primRec]
+iSPR_Ack_Sn [instance, in hydras.MoreAck.AckNotPR]
+isqrt [definition, in hydras.solutions_exercises.isqrt]
+isqrt [library]
+isqrt_spec [definition, in hydras.solutions_exercises.isqrt]
+issqrtIsPR [instance, in hydras.solutions_exercises.isqrt]
+is_limit_cons [constructor, in hydras.Gamma0.T2]
+is_limit_0 [constructor, in hydras.Gamma0.T2]
+is_limit [inductive, in hydras.Gamma0.T2]
+is_successor [inductive, in hydras.Gamma0.T2]
+is_finite [inductive, in hydras.Gamma0.T2]
+is_lub [definition, in hydras.Schutte.Lub]
+is_cnf_of [definition, in hydras.Schutte.CNF]
+is_limit_phi0 [lemma, in hydras.Schutte.AP]
+is_in_seq_not_Acc [lemma, in hydras.solutions_exercises.MultisetWf]
+is_pr_rel [projection, in hydras.Ackermann.primRec]
+is_pr_rel [constructor, in hydras.Ackermann.primRec]
+is_pr [projection, in hydras.Ackermann.primRec]
+is_pr [constructor, in hydras.Ackermann.primRec]
+is_limit_sup_members [lemma, in hydras.Schutte.Schutte_basics]
+is_limit_omega [lemma, in hydras.Schutte.Schutte_basics]
+is_limit [definition, in hydras.Schutte.Schutte_basics]
+is_succ [definition, in hydras.Schutte.Schutte_basics]
+is_finite [definition, in hydras.Schutte.Schutte_basics]
+is_successor_of [definition, in gaia_hydras.onType]
+is_limit_ab [lemma, in hydras.Gamma0.Gamma0]
+is_limit_intro [lemma, in hydras.Gamma0.Gamma0]
+is_limit_cons_inv [lemma, in hydras.Gamma0.Gamma0]
+is_limit_not_succ [lemma, in hydras.Gamma0.Gamma0]
+is_finite_finite [lemma, in hydras.Gamma0.Gamma0]
+is_F_monotonous [library]
+iterate [definition, in hydras.Prelude.Iterates]
+iterate [definition, in hydras.Ackermann.primRec]
+iterateIsPR [instance, in hydras.Ackermann.primRec]
+Iterates [library]
+iterate_ge_diag' [lemma, in hydras.Prelude.Iterates]
+iterate_gt_diag' [lemma, in hydras.Prelude.Iterates]
+iterate_ge_from [lemma, in hydras.Prelude.Iterates]
+iterate_ext2 [lemma, in hydras.Prelude.Iterates]
+iterate_dom_prop [lemma, in hydras.Prelude.Iterates]
+iterate_mono_1 [lemma, in hydras.Prelude.Iterates]
+iterate_ge'' [lemma, in hydras.Prelude.Iterates]
+iterate_ge' [lemma, in hydras.Prelude.Iterates]
+iterate_Sge [lemma, in hydras.Prelude.Iterates]
+iterate_ge [lemma, in hydras.Prelude.Iterates]
+iterate_mono [lemma, in hydras.Prelude.Iterates]
+iterate_mono2 [lemma, in hydras.Prelude.Iterates]
+iterate_le_np_le [lemma, in hydras.Prelude.Iterates]
+iterate_le_n_Sn [lemma, in hydras.Prelude.Iterates]
+iterate_lt_from [lemma, in hydras.Prelude.Iterates]
+iterate_lt [lemma, in hydras.Prelude.Iterates]
+iterate_le [lemma, in hydras.Prelude.Iterates]
+iterate_ext [lemma, in hydras.Prelude.Iterates]
+iterate_rw [lemma, in hydras.Prelude.Iterates]
+iterate_S_eqn2 [lemma, in hydras.Prelude.Iterates]
+iterate_S_eqn [lemma, in hydras.Prelude.Iterates]
+iterate_compat3 [lemma, in hydras.Prelude.Iterates]
+iterate_compat2 [lemma, in hydras.Prelude.Iterates]
+iterate_compat [lemma, in hydras.Prelude.Iterates]
+iterate_comm [lemma, in hydras.Prelude.Iterates]
+iterate_nat_rec [lemma, in hydras.MoreAck.AckNotPR]
+iterate_extEqual [lemma, in hydras.MoreAck.Iterate_compat]
+iterate_compat [lemma, in hydras.MoreAck.Iterate_compat]
+Iterate_compat [library]
+iterate2_mono2 [lemma, in hydras.Prelude.Iterates]
+iterate2_mono3 [lemma, in hydras.Prelude.Iterates]
+iterate2_mono_weak [lemma, in hydras.Prelude.Iterates]
+iterate2_mono [lemma, in hydras.Prelude.Iterates]
+iter_combine [lemma, in additions.fib]
+iter_comm [lemma, in additions.fib]
+iter_mul [lemma, in additions.fib]
+I_i_wf [lemma, in gaia_hydras.onType]
+i':179 [binder, in hydras.Prelude.Iterates]
+i1 [definition, in hydras.OrdinalNotations.ON_Finite]
+i2 [definition, in hydras.OrdinalNotations.ON_Finite]
+i:1 [binder, in hydras.OrdinalNotations.ON_Omega2]
+i:1 [binder, in hydras.Prelude.MoreLists]
+i:1 [binder, in gaia_hydras.GPaths]
+i:1 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:1 [binder, in hydras.OrdinalNotations.ON_Omega]
+i:1 [binder, in hydras.Epsilon0.Paths]
+i:10 [binder, in hydras.Prelude.First_toggle]
+i:10 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:10 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+i:10 [binder, in hydras.Hydra.BigBattle]
+i:10 [binder, in hydras.Hydra.Omega2_Small]
+i:10 [binder, in gaia_hydras.GL_alpha]
+i:101 [binder, in hydras.Prelude.Iterates]
+i:102 [binder, in hydras.Prelude.MoreLists]
+i:102 [binder, in hydras.OrdinalNotations.OmegaOmega]
+i:103 [binder, in hydras.Epsilon0.Canon]
+i:104 [binder, in hydras.Epsilon0.F_alpha]
+i:104 [binder, in hydras.Hydra.Hydra_Lemmas]
+i:104 [binder, in gaia_hydras.nfwfgaia]
+i:105 [binder, in hydras.Prelude.MoreLists]
+i:106 [binder, in gaia_hydras.nfwfgaia]
+i:108 [binder, in hydras.Epsilon0.T1]
+i:108 [binder, in hydras.Epsilon0.F_alpha]
+i:108 [binder, in hydras.Epsilon0.Canon]
+i:109 [binder, in hydras.Epsilon0.F_alpha]
+i:109 [binder, in hydras.Epsilon0.Large_Sets]
+i:11 [binder, in hydras.Hydra.O2H]
+i:11 [binder, in hydras.solutions_exercises.Limit_Infinity]
+i:11 [binder, in gaia_hydras.GPaths]
+i:11 [binder, in hydras.Epsilon0.Canon]
+i:11 [binder, in hydras.solutions_exercises.T1_ltNotWf]
+i:110 [binder, in hydras.Epsilon0.F_alpha]
+i:110 [binder, in hydras.Epsilon0.Canon]
+i:111 [binder, in hydras.Epsilon0.F_alpha]
+i:111 [binder, in hydras.Epsilon0.Large_Sets]
+i:112 [binder, in hydras.Epsilon0.Large_Sets]
+i:112 [binder, in hydras.Epsilon0.Canon]
+i:112 [binder, in hydras.Epsilon0.E0]
+i:113 [binder, in hydras.Epsilon0.F_alpha]
+i:113 [binder, in hydras.Epsilon0.Large_Sets]
+i:114 [binder, in hydras.Epsilon0.Large_Sets]
+i:115 [binder, in hydras.Epsilon0.Large_Sets]
+i:115 [binder, in hydras.Epsilon0.Canon]
+i:116 [binder, in hydras.Epsilon0.Large_Sets]
+i:116 [binder, in hydras.Hydra.Hydra_Lemmas]
+i:119 [binder, in hydras.Hydra.Hydra_Lemmas]
+i:119 [binder, in hydras.Hydra.Hydra_Definitions]
+i:12 [binder, in gaia_hydras.GF_alpha]
+i:12 [binder, in hydras.solutions_exercises.Limit_Infinity]
+i:12 [binder, in hydras.Epsilon0.Canon]
+i:12 [binder, in hydras.Hydra.Omega2_Small]
+i:122 [binder, in hydras.Epsilon0.F_alpha]
+i:123 [binder, in hydras.Schutte.Schutte_basics]
+i:125 [binder, in hydras.Hydra.Hydra_Lemmas]
+i:126 [binder, in hydras.Schutte.Schutte_basics]
+i:127 [binder, in additions.fib]
+i:128 [binder, in hydras.Schutte.Schutte_basics]
+i:128 [binder, in additions.fib]
+i:128 [binder, in hydras.Hydra.Hydra_Definitions]
+i:129 [binder, in additions.fib]
+i:13 [binder, in hydras.solutions_exercises.Limit_Infinity]
+i:13 [binder, in hydras.Hydra.Epsilon0_Needed_Generic]
+i:13 [binder, in hydras.Epsilon0.Paths]
+i:130 [binder, in additions.fib]
+i:131 [binder, in additions.fib]
+i:131 [binder, in hydras.Hydra.Hydra_Definitions]
+i:132 [binder, in hydras.Prelude.Iterates]
+i:132 [binder, in additions.fib]
+i:133 [binder, in additions.fib]
+i:14 [binder, in hydras.Hydra.O2H]
+i:14 [binder, in hydras.solutions_exercises.Limit_Infinity]
+i:14 [binder, in gaia_hydras.GCanon]
+i:14 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+i:14 [binder, in hydras.Epsilon0.Canon]
+i:14 [binder, in hydras.Hydra.Hydra_Lemmas]
+i:14 [binder, in hydras.Hydra.Epsilon0_Needed_Generic]
+i:14 [binder, in hydras.Hydra.Omega2_Small]
+i:14 [binder, in gaia_hydras.GL_alpha]
+i:140 [binder, in gaia_hydras.T1Bridge]
+i:142 [binder, in hydras.rpo.term]
+i:142 [binder, in hydras.Prelude.Iterates]
+i:143 [binder, in hydras.Hydra.Hydra_Definitions]
+i:15 [binder, in hydras.Epsilon0.Large_Sets]
+i:15 [binder, in gaia_hydras.GLarge_Sets]
+i:15 [binder, in hydras.Epsilon0.Paths]
+i:150 [binder, in additions.fib]
+i:150 [binder, in hydras.Hydra.Hydra_Definitions]
+i:153 [binder, in hydras.Prelude.MoreLists]
+i:156 [binder, in hydras.Prelude.MoreLists]
+i:157 [binder, in gaia_hydras.T1Bridge]
+i:158 [binder, in hydras.Prelude.Iterates]
+i:159 [binder, in hydras.Prelude.MoreLists]
+i:159 [binder, in hydras.Epsilon0.F_alpha]
+i:159 [binder, in gaia_hydras.T1Bridge]
+i:16 [binder, in hydras.Hydra.O2H]
+i:16 [binder, in hydras.solutions_exercises.Limit_Infinity]
+i:16 [binder, in gaia_hydras.GCanon]
+i:16 [binder, in hydras.Epsilon0.Canon]
+i:16 [binder, in gaia_hydras.GL_alpha]
+i:161 [binder, in hydras.Epsilon0.F_alpha]
+i:162 [binder, in hydras.Prelude.Iterates]
+i:163 [binder, in hydras.rpo.term]
+i:1638 [binder, in gaia_hydras.nfwfgaia]
+i:164 [binder, in hydras.Epsilon0.F_alpha]
+i:1641 [binder, in gaia_hydras.nfwfgaia]
+i:1644 [binder, in gaia_hydras.nfwfgaia]
+i:165 [binder, in hydras.Hydra.Hydra_Definitions]
+i:166 [binder, in hydras.Epsilon0.F_alpha]
+i:167 [binder, in hydras.Prelude.Iterates]
+i:169 [binder, in hydras.Epsilon0.F_alpha]
+i:17 [binder, in gaia_hydras.GF_alpha]
+i:17 [binder, in additions.Compatibility]
+i:17 [binder, in hydras.Prelude.First_toggle]
+i:17 [binder, in hydras.Epsilon0.Large_Sets]
+i:17 [binder, in hydras.Epsilon0.E0]
+i:171 [binder, in hydras.rpo.term]
+i:172 [binder, in hydras.Epsilon0.F_alpha]
+i:173 [binder, in hydras.Epsilon0.F_alpha]
+i:174 [binder, in hydras.Prelude.MoreLists]
+i:175 [binder, in hydras.Epsilon0.F_alpha]
+i:176 [binder, in hydras.Prelude.Iterates]
+i:176 [binder, in hydras.Epsilon0.Paths]
+i:177 [binder, in hydras.Prelude.MoreLists]
+i:177 [binder, in hydras.Epsilon0.F_alpha]
+i:178 [binder, in hydras.Epsilon0.Large_Sets]
+i:179 [binder, in hydras.Epsilon0.F_alpha]
+i:179 [binder, in hydras.Epsilon0.Paths]
+i:18 [binder, in gaia_hydras.GCanon]
+i:18 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+i:18 [binder, in hydras.Epsilon0.E0]
+i:181 [binder, in hydras.Prelude.MoreLists]
+i:181 [binder, in hydras.Schutte.Schutte_basics]
+i:184 [binder, in hydras.Prelude.MoreLists]
+i:184 [binder, in hydras.Schutte.Schutte_basics]
+i:187 [binder, in hydras.Schutte.Schutte_basics]
+i:187 [binder, in hydras.Epsilon0.Paths]
+i:189 [binder, in hydras.Epsilon0.Large_Sets]
+i:189 [binder, in hydras.Epsilon0.Paths]
+i:19 [binder, in hydras.Hydra.O2H]
+i:19 [binder, in gaia_hydras.GF_alpha]
+i:19 [binder, in hydras.Epsilon0.Large_Sets]
+i:19 [binder, in hydras.Hydra.BigBattle]
+i:190 [binder, in hydras.rpo.term]
+i:190 [binder, in hydras.Schutte.Schutte_basics]
+i:191 [binder, in hydras.Epsilon0.Large_Sets]
+i:193 [binder, in hydras.Schutte.Schutte_basics]
+i:199 [binder, in hydras.Schutte.Schutte_basics]
+i:199 [binder, in additions.Addition_Chains]
+i:2 [binder, in gaia_hydras.GHprime]
+i:2 [binder, in gaia_hydras.GCanon]
+i:2 [binder, in hydras.Epsilon0.L_alpha]
+i:2 [binder, in hydras.Epsilon0.Canon]
+i:2 [binder, in hydras.OrdinalNotations.ON_Finite]
+i:2 [binder, in gaia_hydras.GPrelude]
+i:2 [binder, in hydras.solutions_exercises.ge_omega_iff]
+i:2 [binder, in hydras.Epsilon0.Hprime]
+i:2 [binder, in gaia_hydras.GL_alpha]
+i:2 [binder, in hydras.solutions_exercises.T1_ltNotWf]
+i:20 [binder, in hydras.OrdinalNotations.ON_Omega2]
+i:20 [binder, in gaia_hydras.GF_alpha]
+i:20 [binder, in gaia_hydras.GPaths]
+i:20 [binder, in hydras.Prelude.MoreOrders]
+i:202 [binder, in hydras.Epsilon0.Large_Sets]
+i:202 [binder, in hydras.Schutte.Schutte_basics]
+i:204 [binder, in hydras.Schutte.Schutte_basics]
+i:204 [binder, in hydras.Epsilon0.Paths]
+i:205 [binder, in hydras.Schutte.Schutte_basics]
+i:206 [binder, in hydras.Schutte.Schutte_basics]
+i:207 [binder, in hydras.Epsilon0.Paths]
+i:208 [binder, in hydras.Schutte.Schutte_basics]
+i:21 [binder, in hydras.Hydra.O2H]
+i:21 [binder, in gaia_hydras.GF_alpha]
+i:21 [binder, in additions.More_on_positive]
+i:21 [binder, in hydras.Epsilon0.Large_Sets]
+i:21 [binder, in hydras.Schutte.AP]
+i:21 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+i:21 [binder, in hydras.Epsilon0.Canon]
+i:210 [binder, in hydras.Epsilon0.Paths]
+i:214 [binder, in hydras.Epsilon0.Paths]
+i:217 [binder, in hydras.Epsilon0.Paths]
+i:218 [binder, in hydras.Prelude.MoreLists]
+i:22 [binder, in gaia_hydras.GF_alpha]
+i:22 [binder, in gaia_hydras.GCanon]
+i:22 [binder, in hydras.Schutte.AP]
+i:22 [binder, in hydras.Prelude.MoreOrders]
+i:221 [binder, in hydras.Epsilon0.Paths]
+i:226 [binder, in hydras.Epsilon0.Paths]
+i:23 [binder, in gaia_hydras.GPaths]
+i:23 [binder, in hydras.solutions_exercises.T1_ltNotWf]
+i:230 [binder, in hydras.Schutte.Schutte_basics]
+i:233 [binder, in hydras.Prelude.Iterates]
+i:237 [binder, in hydras.Prelude.Iterates]
+i:24 [binder, in hydras.OrdinalNotations.ON_Omega2]
+i:24 [binder, in hydras.Hydra.O2H]
+i:24 [binder, in gaia_hydras.GCanon]
+i:24 [binder, in hydras.Schutte.AP]
+i:24 [binder, in hydras.Epsilon0.Canon]
+i:24 [binder, in additions.fib]
+i:240 [binder, in hydras.Prelude.Iterates]
+i:241 [binder, in gaia_hydras.nfwfgaia]
+i:244 [binder, in hydras.Prelude.Iterates]
+i:244 [binder, in gaia_hydras.nfwfgaia]
+i:247 [binder, in gaia_hydras.nfwfgaia]
+i:248 [binder, in hydras.Prelude.Iterates]
+i:25 [binder, in hydras.Schutte.Schutte_basics]
+i:25 [binder, in additions.fib]
+i:26 [binder, in gaia_hydras.GCanon]
+i:26 [binder, in hydras.Schutte.MoreEpsilonIota]
+i:26 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+i:26 [binder, in additions.fib]
+i:267 [binder, in hydras.Epsilon0.Paths]
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+i:27 [binder, in hydras.Prelude.MoreOrders]
+i:271 [binder, in hydras.Epsilon0.Paths]
+i:28 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:28 [binder, in hydras.Epsilon0.F_omega]
+i:28 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+i:28 [binder, in hydras.Hydra.BigBattle]
+i:28 [binder, in hydras.Epsilon0.Canon]
+i:28 [binder, in hydras.Epsilon0.Paths]
+i:280 [binder, in hydras.Epsilon0.Paths]
+i:283 [binder, in hydras.Epsilon0.Paths]
+i:286 [binder, in hydras.Epsilon0.Paths]
+i:29 [binder, in hydras.Prelude.MoreOrders]
+i:29 [binder, in additions.fib]
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+i:299 [binder, in hydras.Epsilon0.Paths]
+i:3 [binder, in hydras.solutions_exercises.is_F_monotonous]
+i:3 [binder, in gaia_hydras.GHprime]
+i:3 [binder, in gaia_hydras.GCanon]
+i:3 [binder, in hydras.Schutte.AP]
+i:3 [binder, in gaia_hydras.GL_alpha]
+i:3 [binder, in hydras.Hydra.Omega_Small]
+i:30 [binder, in gaia_hydras.GCanon]
+i:30 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:30 [binder, in hydras.solutions_exercises.MultisetWf]
+i:30 [binder, in hydras.Epsilon0.Canon]
+i:30 [binder, in hydras.Epsilon0.Paths]
+i:31 [binder, in gaia_hydras.GCanon]
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+i:318 [binder, in hydras.Epsilon0.Paths]
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+i:337 [binder, in hydras.Epsilon0.Paths]
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+i:349 [binder, in hydras.rpo.term]
+i:349 [binder, in hydras.Epsilon0.Paths]
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+i:36 [binder, in hydras.Epsilon0.Canon]
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+i:367 [binder, in hydras.rpo.term]
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+i:37 [binder, in hydras.Epsilon0.L_alpha]
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+i:37 [binder, in hydras.Epsilon0.Hprime]
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+i:372 [binder, in hydras.rpo.term]
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+i:38 [binder, in hydras.Epsilon0.Hprime]
+i:382 [binder, in hydras.Epsilon0.Paths]
+i:4 [binder, in hydras.solutions_exercises.Limit_Infinity]
+i:4 [binder, in hydras.Epsilon0.F_alpha]
+i:4 [binder, in gaia_hydras.GCanon]
+i:4 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:4 [binder, in hydras.Prelude.Exp2]
+i:4 [binder, in hydras.Epsilon0.Paths]
+i:4 [binder, in hydras.Hydra.Omega_Small]
+i:40 [binder, in hydras.OrdinalNotations.ON_Omega2]
+i:40 [binder, in hydras.Schutte.Addition]
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+i:406 [binder, in hydras.Epsilon0.Paths]
+i:41 [binder, in hydras.Epsilon0.F_alpha]
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+i:41 [binder, in hydras.Epsilon0.Canon]
+i:412 [binder, in hydras.Epsilon0.Paths]
+i:418 [binder, in hydras.Epsilon0.Paths]
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+i:424 [binder, in hydras.Epsilon0.T1]
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+i:430 [binder, in hydras.Epsilon0.Paths]
+i:432 [binder, in hydras.Epsilon0.Paths]
+i:436 [binder, in hydras.Epsilon0.Paths]
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+i:441 [binder, in hydras.Epsilon0.Paths]
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+i:447 [binder, in hydras.Epsilon0.Paths]
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+i:45 [binder, in hydras.OrdinalNotations.OmegaOmega]
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+i:45 [binder, in additions.fib]
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+i:45 [binder, in gaia_hydras.nfwfgaia]
+i:451 [binder, in hydras.Epsilon0.Paths]
+i:453 [binder, in hydras.Epsilon0.Paths]
+i:457 [binder, in hydras.Epsilon0.Paths]
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+i:46 [binder, in gaia_hydras.GPaths]
+i:46 [binder, in gaia_hydras.GCanon]
+i:46 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:46 [binder, in hydras.Epsilon0.Canon]
+i:46 [binder, in gaia_hydras.nfwfgaia]
+i:461 [binder, in hydras.Epsilon0.Paths]
+i:469 [binder, in hydras.Epsilon0.Paths]
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+i:47 [binder, in hydras.Hydra.BigBattle]
+i:47 [binder, in hydras.Epsilon0.Hprime]
+i:47 [binder, in hydras.MoreAck.FolExamples]
+i:471 [binder, in hydras.Epsilon0.Paths]
+i:476 [binder, in hydras.Epsilon0.Paths]
+i:478 [binder, in hydras.Epsilon0.Paths]
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+i:48 [binder, in hydras.Epsilon0.Paths]
+i:480 [binder, in hydras.Epsilon0.Paths]
+i:485 [binder, in hydras.Epsilon0.Paths]
+i:489 [binder, in hydras.Epsilon0.Paths]
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+i:49 [binder, in hydras.Epsilon0.Canon]
+i:49 [binder, in hydras.MoreAck.FolExamples]
+i:492 [binder, in hydras.Epsilon0.Paths]
+i:496 [binder, in hydras.Epsilon0.Paths]
+i:5 [binder, in hydras.OrdinalNotations.ON_Omega2]
+i:5 [binder, in hydras.solutions_exercises.is_F_monotonous]
+i:5 [binder, in hydras.Prelude.MoreLists]
+i:5 [binder, in gaia_hydras.GHprime]
+i:5 [binder, in hydras.Epsilon0.Large_Sets_Examples]
+i:5 [binder, in hydras.solutions_exercises.F_3]
+i:5 [binder, in hydras.Prelude.Exp2]
+i:5 [binder, in gaia_hydras.GL_alpha]
+i:5 [binder, in hydras.Hydra.Omega_Small]
+i:5 [binder, in hydras.solutions_exercises.T1_ltNotWf]
+i:50 [binder, in gaia_hydras.GPaths]
+i:50 [binder, in additions.More_on_positive]
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+i:504 [binder, in hydras.Epsilon0.Paths]
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+i:51 [binder, in hydras.Epsilon0.F_alpha]
+i:51 [binder, in hydras.Hydra.BigBattle]
+i:51 [binder, in hydras.Epsilon0.Canon]
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+i:51 [binder, in hydras.Hydra.Hydra_Lemmas]
+i:51 [binder, in hydras.Hydra.Hydra_Definitions]
+i:514 [binder, in hydras.Epsilon0.Paths]
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+i:520 [binder, in hydras.Epsilon0.Paths]
+i:522 [binder, in hydras.Epsilon0.Paths]
+i:527 [binder, in hydras.Epsilon0.Paths]
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+i:530 [binder, in hydras.Epsilon0.Paths]
+i:533 [binder, in hydras.Epsilon0.Paths]
+i:54 [binder, in hydras.Epsilon0.F_alpha]
+i:54 [binder, in hydras.Epsilon0.L_alpha]
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+i:544 [binder, in hydras.Epsilon0.Paths]
+i:545 [binder, in hydras.Epsilon0.Paths]
+i:546 [binder, in hydras.Epsilon0.Paths]
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+i:55 [binder, in hydras.Epsilon0.F_alpha]
+i:55 [binder, in gaia_hydras.GCanon]
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+i:55 [binder, in hydras.Epsilon0.Canon]
+i:55 [binder, in hydras.Hydra.Hydra_Definitions]
+i:554 [binder, in hydras.Epsilon0.Paths]
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+i:56 [binder, in hydras.Hydra.Hydra_Lemmas]
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+i:560 [binder, in hydras.Epsilon0.Paths]
+i:567 [binder, in hydras.Epsilon0.Paths]
+i:57 [binder, in hydras.Prelude.MoreLists]
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+i:57 [binder, in additions.More_on_positive]
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+i:571 [binder, in hydras.Epsilon0.Paths]
+i:574 [binder, in hydras.Epsilon0.Paths]
+i:579 [binder, in hydras.Epsilon0.Paths]
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+i:585 [binder, in hydras.Epsilon0.Paths]
+i:588 [binder, in hydras.Epsilon0.Paths]
+i:59 [binder, in hydras.Prelude.MoreLists]
+i:59 [binder, in hydras.Epsilon0.F_alpha]
+i:59 [binder, in hydras.Epsilon0.Canon]
+i:59 [binder, in additions.fib]
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+i:6 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:6 [binder, in hydras.solutions_exercises.F_3]
+i:6 [binder, in hydras.Hydra.Omega2_Small]
+i:60 [binder, in hydras.Prelude.Iterates]
+i:60 [binder, in hydras.Epsilon0.F_alpha]
+i:60 [binder, in additions.More_on_positive]
+i:60 [binder, in gaia_hydras.GCanon]
+i:60 [binder, in hydras.Epsilon0.L_alpha]
+i:61 [binder, in hydras.Epsilon0.F_alpha]
+i:61 [binder, in hydras.Epsilon0.E0]
+i:61 [binder, in hydras.Hydra.Hydra_Lemmas]
+i:61 [binder, in hydras.Epsilon0.Hprime]
+i:61 [binder, in hydras.Epsilon0.Paths]
+i:62 [binder, in hydras.Epsilon0.F_alpha]
+i:62 [binder, in gaia_hydras.GPaths]
+i:62 [binder, in hydras.Hydra.BigBattle]
+i:63 [binder, in hydras.Epsilon0.Canon]
+i:63 [binder, in hydras.Epsilon0.E0]
+i:64 [binder, in hydras.Epsilon0.Canon]
+i:64 [binder, in hydras.Epsilon0.E0]
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+i:64 [binder, in hydras.Epsilon0.Paths]
+i:65 [binder, in hydras.OrdinalNotations.ON_Omega2]
+i:65 [binder, in hydras.Hydra.BigBattle]
+i:65 [binder, in hydras.Epsilon0.Canon]
+i:66 [binder, in gaia_hydras.GPaths]
+i:66 [binder, in hydras.Hydra.BigBattle]
+i:66 [binder, in hydras.Epsilon0.Canon]
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+i:67 [binder, in hydras.Epsilon0.Canon]
+i:68 [binder, in hydras.OrdinalNotations.ON_Omega2]
+i:68 [binder, in hydras.MoreAck.AckNotPR]
+i:68 [binder, in hydras.Epsilon0.Canon]
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+i:68 [binder, in hydras.MoreAck.PrimRecExamples]
+i:687 [binder, in hydras.Epsilon0.T1]
+i:689 [binder, in hydras.Epsilon0.T1]
+i:69 [binder, in gaia_hydras.GCanon]
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+i:69 [binder, in hydras.Epsilon0.Canon]
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+i:693 [binder, in hydras.Epsilon0.T1]
+i:696 [binder, in hydras.Epsilon0.T1]
+i:7 [binder, in gaia_hydras.GHprime]
+i:7 [binder, in gaia_hydras.GPaths]
+i:7 [binder, in gaia_hydras.GCanon]
+i:7 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+i:7 [binder, in hydras.Epsilon0.Canon]
+i:7 [binder, in hydras.Hydra.Epsilon0_Needed_Generic]
+i:7 [binder, in gaia_hydras.GL_alpha]
+i:70 [binder, in hydras.MoreAck.AckNotPR]
+i:70 [binder, in hydras.Epsilon0.Large_Sets]
+i:70 [binder, in hydras.Hydra.BigBattle]
+i:70 [binder, in hydras.Epsilon0.Canon]
+i:70 [binder, in hydras.Epsilon0.Hprime]
+i:702 [binder, in hydras.Epsilon0.T1]
+i:703 [binder, in hydras.Epsilon0.T1]
+i:71 [binder, in gaia_hydras.GCanon]
+i:71 [binder, in hydras.Epsilon0.Large_Sets]
+i:71 [binder, in hydras.Schutte.AP]
+i:71 [binder, in hydras.Epsilon0.Canon]
+i:71 [binder, in hydras.Epsilon0.Paths]
+i:717 [binder, in hydras.Epsilon0.T1]
+i:73 [binder, in gaia_hydras.GPaths]
+i:73 [binder, in hydras.Epsilon0.Large_Sets]
+i:73 [binder, in hydras.Epsilon0.Paths]
+i:74 [binder, in hydras.Epsilon0.Large_Sets]
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+i:78 [binder, in hydras.Epsilon0.Canon]
+i:79 [binder, in gaia_hydras.GPaths]
+i:79 [binder, in hydras.Epsilon0.Large_Sets]
+i:8 [binder, in gaia_hydras.GPaths]
+i:8 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
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+i:81 [binder, in hydras.Epsilon0.Large_Sets]
+i:81 [binder, in hydras.Epsilon0.Canon]
+i:81 [binder, in hydras.Epsilon0.Paths]
+i:82 [binder, in hydras.Prelude.MoreLists]
+i:82 [binder, in hydras.Epsilon0.Large_Sets]
+i:83 [binder, in hydras.Hydra.O2H]
+i:83 [binder, in hydras.Epsilon0.Paths]
+i:84 [binder, in hydras.Prelude.MoreLists]
+i:84 [binder, in hydras.Epsilon0.Large_Sets]
+i:84 [binder, in hydras.MoreAck.PrimRecExamples]
+i:85 [binder, in hydras.Gamma0.T2]
+i:85 [binder, in hydras.Hydra.O2H]
+i:85 [binder, in hydras.Hydra.Hydra_Lemmas]
+i:85 [binder, in hydras.Epsilon0.Paths]
+i:86 [binder, in hydras.Epsilon0.Hprime]
+i:87 [binder, in hydras.Epsilon0.Large_Sets]
+i:87 [binder, in hydras.Hydra.Hydra_Lemmas]
+i:88 [binder, in hydras.Prelude.MoreLists]
+i:88 [binder, in hydras.Epsilon0.Canon]
+i:88 [binder, in hydras.Epsilon0.E0]
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+i:89 [binder, in hydras.Prelude.MoreLists]
+i:89 [binder, in hydras.Epsilon0.Large_Sets]
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+i:9 [binder, in hydras.Epsilon0.Paths]
+i:9 [binder, in gaia_hydras.GL_alpha]
+i:90 [binder, in hydras.Epsilon0.Large_Sets]
+i:91 [binder, in hydras.Epsilon0.Large_Sets]
+i:91 [binder, in hydras.Epsilon0.Paths]
+i:92 [binder, in hydras.Prelude.MoreLists]
+i:92 [binder, in hydras.Hydra.Hydra_Lemmas]
+i:94 [binder, in hydras.Prelude.MoreLists]
+i:94 [binder, in hydras.Epsilon0.E0]
+i:95 [binder, in hydras.Prelude.Iterates]
+i:95 [binder, in hydras.Hydra.O2H]
+i:95 [binder, in hydras.Epsilon0.Canon]
+i:95 [binder, in hydras.Hydra.Hydra_Lemmas]
+i:96 [binder, in hydras.OrdinalNotations.OmegaOmega]
+i:97 [binder, in hydras.Hydra.O2H]
+i:97 [binder, in hydras.Epsilon0.E0]
+i:98 [binder, in hydras.Epsilon0.F_alpha]
+i:99 [binder, in hydras.Epsilon0.Canon]
+

J

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+j:106 [binder, in hydras.Prelude.MoreLists]
+j:106 [binder, in hydras.Hydra.Hydra_Lemmas]
+j:109 [binder, in hydras.Epsilon0.T1]
+j:11 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+j:11 [binder, in hydras.Hydra.Omega2_Small]
+j:12 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+j:120 [binder, in hydras.Hydra.Hydra_Lemmas]
+j:124 [binder, in hydras.Prelude.Iterates]
+j:125 [binder, in hydras.Schutte.Schutte_basics]
+j:126 [binder, in hydras.Hydra.Hydra_Lemmas]
+j:127 [binder, in hydras.Prelude.Iterates]
+j:127 [binder, in hydras.Schutte.Schutte_basics]
+j:129 [binder, in hydras.Schutte.Schutte_basics]
+j:13 [binder, in hydras.Hydra.Omega2_Small]
+j:130 [binder, in hydras.Prelude.Iterates]
+j:133 [binder, in hydras.Hydra.Hydra_Definitions]
+j:15 [binder, in hydras.solutions_exercises.Limit_Infinity]
+j:15 [binder, in hydras.Hydra.Omega2_Small]
+j:151 [binder, in additions.fib]
+j:154 [binder, in hydras.Prelude.MoreLists]
+j:155 [binder, in hydras.rpo.term]
+j:157 [binder, in hydras.Prelude.MoreLists]
+j:16 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+j:160 [binder, in hydras.Prelude.MoreLists]
+j:168 [binder, in hydras.Prelude.Iterates]
+j:17 [binder, in hydras.solutions_exercises.Limit_Infinity]
+j:182 [binder, in hydras.Schutte.Schutte_basics]
+j:183 [binder, in hydras.Prelude.MoreLists]
+j:185 [binder, in hydras.Prelude.MoreLists]
+j:194 [binder, in hydras.Schutte.Schutte_basics]
+j:200 [binder, in additions.Addition_Chains]
+j:203 [binder, in hydras.Schutte.Schutte_basics]
+j:21 [binder, in hydras.OrdinalNotations.ON_Omega2]
+j:23 [binder, in hydras.Schutte.AP]
+j:238 [binder, in hydras.Prelude.Iterates]
+j:243 [binder, in hydras.Prelude.Iterates]
+j:247 [binder, in hydras.Prelude.Iterates]
+j:25 [binder, in hydras.OrdinalNotations.ON_Omega2]
+j:25 [binder, in gaia_hydras.GPaths]
+j:25 [binder, in hydras.Schutte.AP]
+j:251 [binder, in hydras.Prelude.Iterates]
+j:29 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+j:3 [binder, in gaia_hydras.GPrelude]
+j:31 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+j:333 [binder, in hydras.rpo.term]
+j:338 [binder, in hydras.Epsilon0.Paths]
+j:34 [binder, in hydras.Prelude.MoreLists]
+j:36 [binder, in hydras.Prelude.MoreLists]
+j:39 [binder, in hydras.Hydra.BigBattle]
+j:404 [binder, in hydras.Epsilon0.Paths]
+j:408 [binder, in hydras.Epsilon0.Paths]
+j:41 [binder, in hydras.Schutte.Addition]
+j:414 [binder, in hydras.Epsilon0.Paths]
+j:416 [binder, in hydras.Epsilon0.Paths]
+j:42 [binder, in hydras.OrdinalNotations.ON_Omega2]
+j:420 [binder, in hydras.Epsilon0.Paths]
+j:424 [binder, in hydras.Epsilon0.Paths]
+j:428 [binder, in hydras.Epsilon0.Paths]
+j:43 [binder, in additions.fib]
+j:434 [binder, in hydras.Epsilon0.Paths]
+j:438 [binder, in hydras.Epsilon0.Paths]
+j:449 [binder, in hydras.Epsilon0.Paths]
+j:455 [binder, in hydras.Epsilon0.Paths]
+j:458 [binder, in hydras.Epsilon0.Paths]
+j:46 [binder, in additions.fib]
+j:463 [binder, in hydras.Epsilon0.Paths]
+j:465 [binder, in hydras.Epsilon0.Paths]
+j:47 [binder, in hydras.Prelude.MoreLists]
+j:481 [binder, in hydras.Epsilon0.Paths]
+j:49 [binder, in hydras.OrdinalNotations.OmegaOmega]
+j:493 [binder, in hydras.Epsilon0.Paths]
+j:497 [binder, in hydras.Epsilon0.Paths]
+j:5 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+j:50 [binder, in hydras.Prelude.MoreLists]
+j:501 [binder, in hydras.Epsilon0.Paths]
+j:506 [binder, in hydras.Epsilon0.Paths]
+j:515 [binder, in hydras.Epsilon0.Paths]
+j:52 [binder, in additions.fib]
+j:521 [binder, in hydras.Epsilon0.Paths]
+j:524 [binder, in hydras.Epsilon0.Paths]
+j:531 [binder, in hydras.Epsilon0.Paths]
+j:54 [binder, in hydras.Prelude.MoreLists]
+j:54 [binder, in gaia_hydras.GPaths]
+j:55 [binder, in hydras.Hydra.BigBattle]
+j:55 [binder, in additions.fib]
+j:555 [binder, in hydras.Epsilon0.Paths]
+j:559 [binder, in hydras.Epsilon0.Paths]
+j:568 [binder, in hydras.Epsilon0.Paths]
+j:57 [binder, in hydras.OrdinalNotations.OmegaOmega]
+j:575 [binder, in hydras.Epsilon0.Paths]
+j:58 [binder, in hydras.Prelude.MoreLists]
+j:6 [binder, in hydras.OrdinalNotations.ON_Omega2]
+j:6 [binder, in hydras.Prelude.MoreLists]
+j:6 [binder, in hydras.Epsilon0.Large_Sets_Examples]
+j:60 [binder, in hydras.Prelude.MoreLists]
+j:60 [binder, in additions.fib]
+j:61 [binder, in hydras.Prelude.Iterates]
+j:61 [binder, in additions.More_on_positive]
+j:61 [binder, in gaia_hydras.GCanon]
+j:63 [binder, in gaia_hydras.GPaths]
+j:65 [binder, in hydras.Epsilon0.Paths]
+j:66 [binder, in hydras.OrdinalNotations.ON_Omega2]
+j:69 [binder, in hydras.OrdinalNotations.ON_Omega2]
+j:7 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+j:7 [binder, in hydras.Hydra.Omega2_Small]
+j:70 [binder, in hydras.Hydra.Hydra_Lemmas]
+j:74 [binder, in gaia_hydras.GPaths]
+j:75 [binder, in hydras.Epsilon0.Paths]
+j:77 [binder, in hydras.Epsilon0.Paths]
+j:78 [binder, in gaia_hydras.GPaths]
+j:79 [binder, in hydras.Epsilon0.Canon]
+j:79 [binder, in hydras.Hydra.Hydra_Lemmas]
+j:8 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+j:82 [binder, in hydras.Epsilon0.Canon]
+j:83 [binder, in hydras.Prelude.MoreLists]
+j:85 [binder, in hydras.Prelude.MoreLists]
+j:87 [binder, in hydras.Epsilon0.Paths]
+j:9 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+j:9 [binder, in hydras.Hydra.Omega2_Small]
+j:90 [binder, in hydras.Prelude.Iterates]
+j:90 [binder, in hydras.Prelude.MoreLists]
+j:90 [binder, in hydras.Hydra.Hydra_Lemmas]
+j:92 [binder, in hydras.Epsilon0.Paths]
+j:94 [binder, in hydras.Hydra.Hydra_Lemmas]
+j:95 [binder, in hydras.Prelude.MoreLists]
+j:96 [binder, in hydras.Prelude.Iterates]
+j:96 [binder, in hydras.Epsilon0.Canon]
+j:97 [binder, in hydras.Hydra.Hydra_Lemmas]
+j:99 [binder, in hydras.Hydra.O2H]
+

K

+kbr:222 [binder, in additions.Euclidean_Chains]
+kbr:231 [binder, in additions.Euclidean_Chains]
+kbr:293 [binder, in additions.Euclidean_Chains]
+kbr:298 [binder, in additions.Euclidean_Chains]
+Kchain [definition, in additions.Euclidean_Chains]
+Kchain_correct [definition, in additions.AM]
+Kchain_correct_conv [lemma, in additions.Euclidean_Chains]
+Kchain_proper_prf [projection, in additions.Euclidean_Chains]
+Kchain_proper [record, in additions.Euclidean_Chains]
+Kchain_proper_prf [constructor, in additions.Euclidean_Chains]
+Kchain_proper [inductive, in additions.Euclidean_Chains]
+Kchain_correct [definition, in additions.Euclidean_Chains]
+Kchain_correct_nat [definition, in additions.Euclidean_Chains]
+kc:181 [binder, in additions.Euclidean_Chains]
+kc:191 [binder, in additions.Euclidean_Chains]
+kc:192 [binder, in additions.Euclidean_Chains]
+kc:204 [binder, in additions.Euclidean_Chains]
+kc:218 [binder, in additions.Euclidean_Chains]
+kc:221 [binder, in additions.Euclidean_Chains]
+KFF [definition, in additions.AM]
+KFF [definition, in additions.Euclidean_Chains]
+KFF_correct [lemma, in additions.AM]
+KFF_correct [lemma, in additions.Euclidean_Chains]
+KFF_proper [instance, in additions.Euclidean_Chains]
+KFF_correct_nat [lemma, in additions.Euclidean_Chains]
+KFK [definition, in additions.AM]
+KFK [definition, in additions.Euclidean_Chains]
+KFK_correct [lemma, in additions.AM]
+KFK_correct [lemma, in additions.Euclidean_Chains]
+KFK_proper [lemma, in additions.Euclidean_Chains]
+KFK_correct_nat [lemma, in additions.Euclidean_Chains]
+KFK_proof.Hq_prop [variable, in additions.Euclidean_Chains]
+KFK_proof.Hbr_prop [variable, in additions.Euclidean_Chains]
+KFK_proof.Hq [variable, in additions.Euclidean_Chains]
+KFK_proof.Hbr [variable, in additions.Euclidean_Chains]
+KFK_proof.fq [variable, in additions.Euclidean_Chains]
+KFK_proof.kbr [variable, in additions.Euclidean_Chains]
+KFK_proof.r [variable, in additions.Euclidean_Chains]
+KFK_proof.q [variable, in additions.Euclidean_Chains]
+KFK_proof.b [variable, in additions.Euclidean_Chains]
+KFK_proof [section, in additions.Euclidean_Chains]
+khi:170 [binder, in hydras.Schutte.Ordering_Functions]
+khi:171 [binder, in hydras.Schutte.Ordering_Functions]
+khi:172 [binder, in hydras.Schutte.Ordering_Functions]
+khi:173 [binder, in hydras.Schutte.Ordering_Functions]
+khi:49 [binder, in hydras.Schutte.Addition]
+khi:92 [binder, in hydras.Schutte.Ordering_Functions]
+khi:93 [binder, in hydras.Schutte.Ordering_Functions]
+Kkont [definition, in additions.Euclidean_Chains]
+Kkont_equiv [definition, in additions.Euclidean_Chains]
+Kkont_proper [definition, in additions.Euclidean_Chains]
+knp:207 [binder, in additions.Euclidean_Chains]
+knp:211 [binder, in additions.Euclidean_Chains]
+kont_type [definition, in additions.Euclidean_Chains]
+KP [section, in hydras.Hydra.KP_example]
+kpr:27 [binder, in additions.AM]
+kpr:31 [binder, in additions.AM]
+KP_5_iii [lemma, in hydras.Epsilon0.Paths]
+KP_arrowS_zero [lemma, in hydras.Epsilon0.Paths]
+KP_arrowS [definition, in hydras.Epsilon0.Paths]
+KP_example [library]
+KP.h' [variable, in hydras.Hydra.KP_example]
+ksi_plus_beta_eq [lemma, in hydras.Schutte.AP]
+ksi_plus_beta [lemma, in hydras.Schutte.AP]
+ksi_plus_seq_n' [lemma, in hydras.Schutte.AP]
+ksi_plus_seq_n [lemma, in hydras.Schutte.AP]
+ksi:58 [binder, in hydras.Schutte.AP]
+KS_thm_2_4 [lemma, in gaia_hydras.GPaths]
+KS_thm_2_4_E0 [lemma, in hydras.Epsilon0.Paths]
+KS_thm_2_4 [lemma, in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma5 [lemma, in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma4 [lemma, in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma3 [lemma, in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma3_0 [lemma, in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma2 [lemma, in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma1' [lemma, in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma1 [lemma, in hydras.Epsilon0.Paths]
+k':105 [binder, in additions.Euclidean_Chains]
+k':116 [binder, in additions.Euclidean_Chains]
+k':174 [binder, in additions.Euclidean_Chains]
+k':200 [binder, in additions.Euclidean_Chains]
+k10_7:330 [binder, in additions.Euclidean_Chains]
+k17_7 [definition, in additions.Euclidean_Chains]
+K2F [definition, in additions.Euclidean_Chains]
+K2F_proper [instance, in additions.Euclidean_Chains]
+K2F_correct [lemma, in additions.Euclidean_Chains]
+K2F_correct_nat [lemma, in additions.Euclidean_Chains]
+k3_1_proper [lemma, in additions.Euclidean_Chains]
+k3_1_correct [lemma, in additions.Euclidean_Chains]
+k3_1 [definition, in additions.Euclidean_Chains]
+k7_3:329 [binder, in additions.Euclidean_Chains]
+k7_3_correct [lemma, in additions.Euclidean_Chains]
+k7_3_proper [instance, in additions.Euclidean_Chains]
+k7_3 [definition, in additions.Euclidean_Chains]
+K99_24 [definition, in additions.AM]
+k:10 [binder, in hydras.Hydra.Hydra_Theorems]
+k:102 [binder, in hydras.Epsilon0.Canon]
+k:104 [binder, in additions.Euclidean_Chains]
+k:107 [binder, in hydras.Prelude.MoreLists]
+k:107 [binder, in hydras.Epsilon0.Canon]
+k:11 [binder, in hydras.Epsilon0.Large_Sets]
+k:11 [binder, in gaia_hydras.GL_alpha]
+k:110 [binder, in hydras.Epsilon0.Large_Sets]
+k:115 [binder, in additions.Euclidean_Chains]
+k:117 [binder, in hydras.Epsilon0.Large_Sets]
+k:118 [binder, in hydras.Epsilon0.Large_Sets]
+k:12 [binder, in additions.Euclidean_Chains]
+k:12 [binder, in gaia_hydras.GLarge_Sets]
+k:12 [binder, in gaia_hydras.GL_alpha]
+k:122 [binder, in hydras.Epsilon0.Large_Sets]
+k:122 [binder, in hydras.Epsilon0.Hprime]
+k:133 [binder, in hydras.Prelude.Iterates]
+k:137 [binder, in hydras.Hydra.Hydra_Definitions]
+k:14 [binder, in hydras.Epsilon0.Large_Sets]
+k:1402 [binder, in gaia_hydras.nfwfgaia]
+k:143 [binder, in hydras.Prelude.Iterates]
+k:143 [binder, in additions.Euclidean_Chains]
+k:15 [binder, in additions.fib]
+k:15 [binder, in hydras.rpo.dickson]
+k:151 [binder, in additions.Euclidean_Chains]
+k:153 [binder, in hydras.Prelude.Iterates]
+k:156 [binder, in additions.Euclidean_Chains]
+k:16 [binder, in hydras.Epsilon0.T1]
+k:16 [binder, in gaia_hydras.GHprime]
+k:16 [binder, in gaia_hydras.GHydra]
+k:166 [binder, in hydras.Prelude.Iterates]
+k:167 [binder, in additions.Euclidean_Chains]
+k:17 [binder, in hydras.Epsilon0.F_omega]
+k:17 [binder, in hydras.MoreAck.Ack]
+k:173 [binder, in additions.Euclidean_Chains]
+k:18 [binder, in gaia_hydras.GHydra]
+k:187 [binder, in hydras.Prelude.Iterates]
+k:187 [binder, in additions.Euclidean_Chains]
+k:1889 [binder, in gaia_hydras.nfwfgaia]
+k:194 [binder, in hydras.Prelude.Iterates]
+k:199 [binder, in additions.Euclidean_Chains]
+k:2 [binder, in hydras.Hydra.Hydra_Theorems]
+k:20 [binder, in gaia_hydras.GL_alpha]
+k:2013 [binder, in gaia_hydras.nfwfgaia]
+k:207 [binder, in hydras.Prelude.Iterates]
+k:209 [binder, in additions.Euclidean_Chains]
+k:21 [binder, in gaia_hydras.GHydra]
+k:210 [binder, in hydras.Prelude.Iterates]
+k:211 [binder, in hydras.Prelude.Iterates]
+k:212 [binder, in hydras.Prelude.Iterates]
+k:213 [binder, in hydras.Prelude.Iterates]
+k:22 [binder, in hydras.OrdinalNotations.ON_Omega2]
+k:22 [binder, in additions.Euclidean_Chains]
+k:22 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+k:225 [binder, in additions.Euclidean_Chains]
+k:226 [binder, in hydras.Prelude.Iterates]
+k:23 [binder, in hydras.Prelude.More_Arith]
+k:23 [binder, in gaia_hydras.GL_alpha]
+k:234 [binder, in hydras.Prelude.Iterates]
+k:236 [binder, in additions.Euclidean_Chains]
+k:242 [binder, in additions.Euclidean_Chains]
+k:25 [binder, in hydras.Epsilon0.Large_Sets]
+k:252 [binder, in hydras.Prelude.Iterates]
+k:26 [binder, in hydras.OrdinalNotations.ON_Omega2]
+k:26 [binder, in gaia_hydras.GL_alpha]
+k:260 [binder, in hydras.rpo.rpo]
+k:32 [binder, in hydras.MoreAck.AckNotPR]
+k:38 [binder, in hydras.Epsilon0.Large_Sets]
+k:39 [binder, in hydras.Epsilon0.L_alpha]
+k:39 [binder, in hydras.Epsilon0.Hprime]
+k:4 [binder, in additions.Euclidean_Chains]
+k:40 [binder, in hydras.Epsilon0.L_alpha]
+k:40 [binder, in additions.Euclidean_Chains]
+k:40 [binder, in hydras.Epsilon0.Hprime]
+k:40 [binder, in hydras.MoreAck.PrimRecExamples]
+k:410 [binder, in hydras.Epsilon0.Paths]
+k:43 [binder, in hydras.Hydra.BigBattle]
+k:43 [binder, in hydras.Epsilon0.Hprime]
+k:44 [binder, in hydras.Epsilon0.Large_Sets]
+k:44 [binder, in hydras.Epsilon0.Hprime]
+k:45 [binder, in hydras.Epsilon0.Hprime]
+k:46 [binder, in hydras.Epsilon0.L_alpha]
+k:46 [binder, in hydras.Epsilon0.Hprime]
+k:467 [binder, in hydras.Epsilon0.Paths]
+k:47 [binder, in hydras.Epsilon0.Large_Sets]
+k:48 [binder, in hydras.Epsilon0.Hprime]
+k:49 [binder, in hydras.Epsilon0.Hprime]
+k:508 [binder, in hydras.Epsilon0.Paths]
+k:51 [binder, in hydras.Epsilon0.Large_Sets]
+k:51 [binder, in hydras.Epsilon0.L_alpha]
+k:52 [binder, in hydras.Epsilon0.L_alpha]
+k:53 [binder, in hydras.Epsilon0.L_alpha]
+k:53 [binder, in hydras.Epsilon0.Hprime]
+k:54 [binder, in hydras.Epsilon0.Hprime]
+k:54 [binder, in gaia_hydras.nfwfgaia]
+k:541 [binder, in hydras.Epsilon0.Paths]
+k:55 [binder, in hydras.Epsilon0.L_alpha]
+k:57 [binder, in hydras.Epsilon0.Hprime]
+k:578 [binder, in hydras.Epsilon0.Paths]
+k:58 [binder, in hydras.Epsilon0.Large_Sets]
+k:58 [binder, in additions.Euclidean_Chains]
+k:58 [binder, in gaia_hydras.nfwfgaia]
+k:59 [binder, in hydras.Epsilon0.Hprime]
+k:6 [binder, in hydras.Hydra.Hydra_Theorems]
+k:6 [binder, in hydras.Hydra.Battle_length]
+k:60 [binder, in gaia_hydras.nfwfgaia]
+k:62 [binder, in hydras.Epsilon0.Hprime]
+k:63 [binder, in hydras.Epsilon0.Large_Sets]
+k:64 [binder, in hydras.Epsilon0.Hprime]
+k:65 [binder, in hydras.Epsilon0.Large_Sets]
+k:66 [binder, in hydras.Epsilon0.Large_Sets]
+k:66 [binder, in hydras.Epsilon0.Hprime]
+k:67 [binder, in hydras.Epsilon0.Hprime]
+k:68 [binder, in hydras.Epsilon0.Large_Sets]
+k:69 [binder, in hydras.Epsilon0.Hprime]
+k:7 [binder, in hydras.OrdinalNotations.ON_Omega2]
+k:7 [binder, in hydras.Hydra.Hydra_Theorems]
+k:7 [binder, in additions.Addition_Chains]
+k:71 [binder, in hydras.Epsilon0.Hprime]
+k:72 [binder, in hydras.Epsilon0.Large_Sets]
+k:72 [binder, in hydras.Hydra.Hydra_Lemmas]
+k:72 [binder, in hydras.Epsilon0.Hprime]
+k:73 [binder, in additions.Euclidean_Chains]
+k:73 [binder, in hydras.Epsilon0.Hprime]
+k:74 [binder, in hydras.Epsilon0.Hprime]
+k:75 [binder, in hydras.Epsilon0.Large_Sets]
+k:76 [binder, in hydras.Epsilon0.Large_Sets]
+k:77 [binder, in hydras.Epsilon0.Large_Sets]
+k:77 [binder, in hydras.MoreAck.Ack]
+k:79 [binder, in hydras.Epsilon0.Hprime]
+k:8 [binder, in gaia_hydras.GHprime]
+k:8 [binder, in hydras.Epsilon0.Large_Sets]
+k:8 [binder, in hydras.Hydra.Hydra_Theorems]
+k:8 [binder, in hydras.Hydra.Battle_length]
+k:80 [binder, in hydras.Epsilon0.Hprime]
+k:81 [binder, in hydras.Prelude.MoreLists]
+k:83 [binder, in hydras.Epsilon0.Large_Sets]
+k:83 [binder, in hydras.Epsilon0.Hprime]
+k:84 [binder, in hydras.Epsilon0.Hprime]
+k:85 [binder, in hydras.Epsilon0.Large_Sets]
+k:88 [binder, in hydras.Epsilon0.Hprime]
+k:880 [binder, in gaia_hydras.nfwfgaia]
+k:9 [binder, in gaia_hydras.GHprime]
+k:9 [binder, in hydras.Hydra.Hydra_Theorems]
+k:9 [binder, in additions.Euclidean_Chains]
+k:90 [binder, in hydras.Epsilon0.Hprime]
+k:91 [binder, in hydras.Prelude.MoreLists]
+k:91 [binder, in hydras.Epsilon0.Hprime]
+k:92 [binder, in hydras.Epsilon0.Large_Sets]
+k:93 [binder, in hydras.MoreAck.AckNotPR]
+k:93 [binder, in hydras.Epsilon0.Large_Sets]
+k:93 [binder, in additions.Euclidean_Chains]
+k:94 [binder, in hydras.Epsilon0.Hprime]
+k:95 [binder, in hydras.MoreAck.AckNotPR]
+k:95 [binder, in hydras.Epsilon0.Large_Sets]
+k:96 [binder, in hydras.Epsilon0.Large_Sets]
+k:99 [binder, in hydras.Prelude.Iterates]
+

L

+L [lemma, in hydras.Epsilon0.F_omega]
+L [lemma, in hydras.solutions_exercises.F_3]
+L [lemma, in additions.Addition_Chains]
+lambda:13 [binder, in hydras.Epsilon0.Canon]
+lambda:15 [binder, in hydras.Epsilon0.Canon]
+lambda:156 [binder, in gaia_hydras.T1Bridge]
+lambda:161 [binder, in gaia_hydras.T1Bridge]
+lambda:18 [binder, in hydras.Epsilon0.Canon]
+lambda:20 [binder, in hydras.Epsilon0.Canon]
+lambda:209 [binder, in hydras.Epsilon0.Paths]
+lambda:213 [binder, in hydras.Epsilon0.Paths]
+lambda:23 [binder, in gaia_hydras.GCanon]
+lambda:23 [binder, in hydras.Epsilon0.Canon]
+lambda:27 [binder, in hydras.Epsilon0.Canon]
+lambda:29 [binder, in hydras.Schutte.Schutte_basics]
+lambda:32 [binder, in gaia_hydras.GCanon]
+lambda:336 [binder, in hydras.Epsilon0.Paths]
+lambda:43 [binder, in gaia_hydras.GCanon]
+lambda:49 [binder, in gaia_hydras.GCanon]
+lambda:53 [binder, in gaia_hydras.GCanon]
+lambda:566 [binder, in hydras.Epsilon0.Paths]
+lambda:58 [binder, in gaia_hydras.GCanon]
+lambda:59 [binder, in gaia_hydras.GCanon]
+lambda:60 [binder, in hydras.Epsilon0.Canon]
+lambda:61 [binder, in gaia_hydras.GPaths]
+lambda:61 [binder, in hydras.Epsilon0.Canon]
+lambda:62 [binder, in hydras.Hydra.O2H]
+lambda:65 [binder, in gaia_hydras.GCanon]
+lambda:686 [binder, in hydras.Epsilon0.T1]
+lambda:72 [binder, in hydras.Epsilon0.Canon]
+lambda:75 [binder, in hydras.Epsilon0.Canon]
+Language [record, in hydras.Ackermann.fol]
+language [constructor, in hydras.Ackermann.fol]
+Languages [library]
+language_decidable [definition, in hydras.Ackermann.fol]
+Large [definition, in hydras.Epsilon0.Large_Sets]
+large [definition, in hydras.Epsilon0.Large_Sets]
+large [definition, in gaia_hydras.GLarge_Sets]
+Largeb [definition, in hydras.Epsilon0.Large_Sets]
+largeb [definition, in hydras.Epsilon0.Large_Sets]
+largeb_n_R [lemma, in hydras.Epsilon0.Large_Sets]
+largeb_n [lemma, in hydras.Epsilon0.Large_Sets]
+largeb_finite [lemma, in hydras.Epsilon0.Large_Sets]
+Largeb_Sb [lemma, in hydras.Epsilon0.Large_Sets]
+largeb_Sb [lemma, in hydras.Epsilon0.Large_Sets]
+LargeS [definition, in hydras.Epsilon0.Large_Sets]
+largeS [definition, in hydras.Epsilon0.Large_Sets]
+largeS [definition, in gaia_hydras.GLarge_Sets]
+LargeSb [definition, in hydras.Epsilon0.Large_Sets]
+largeSb [definition, in hydras.Epsilon0.Large_Sets]
+LargeSb_b [lemma, in hydras.Epsilon0.Large_Sets]
+largeSb_b [lemma, in hydras.Epsilon0.Large_Sets]
+largeS_iff [lemma, in hydras.Epsilon0.Large_Sets]
+large_omega_iff [lemma, in hydras.Epsilon0.Large_Sets]
+large_omega_2 [lemma, in hydras.Epsilon0.Large_Sets]
+large_omega_1 [lemma, in hydras.Epsilon0.Large_Sets]
+large_n_iff [lemma, in hydras.Epsilon0.Large_Sets]
+large_iff [lemma, in hydras.Epsilon0.Large_Sets]
+large_set_check [definition, in hydras.Epsilon0.Large_Sets_Examples]
+Large_Sets [library]
+Large_Sets_Examples [library]
+la:116 [binder, in hydras.rpo.more_list]
+la:7 [binder, in hydras.rpo.dickson]
+Lcode [record, in hydras.Ackermann.code]
+LcodeLNN [instance, in hydras.Ackermann.codeNatToTerm]
+LcodeLNT [instance, in hydras.Ackermann.codeNatToTerm]
+lc:165 [binder, in hydras.rpo.rpo]
+lc:233 [binder, in hydras.rpo.rpo]
+lc:242 [binder, in hydras.rpo.rpo]
+lc:70 [binder, in hydras.rpo.rpo]
+LE [definition, in hydras.Epsilon0.T1]
+le [abbreviation, in hydras.Epsilon0.T1]
+le [definition, in hydras.Gamma0.T2]
+le [abbreviation, in hydras.Prelude.Comparable]
+le [definition, in hydras.OrdinalNotations.ON_O]
+LE [abbreviation, in gaia_hydras.T1Bridge]
+le [definition, in hydras.OrdinalNotations.ON_mult]
+le [definition, in hydras.OrdinalNotations.ON_plus]
+le [definition, in hydras.Schutte.Schutte_basics]
+Le [definition, in hydras.Schutte.Well_Orders]
+Least [definition, in hydras.Prelude.MoreOrders]
+Least_zero [lemma, in hydras.OrdinalNotations.ON_Omega2]
+least_AP [lemma, in hydras.Schutte.AP]
+Least_is_0 [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Least_not_Limit [lemma, in hydras.Prelude.MoreOrders]
+Least_not_Succ [lemma, in hydras.Prelude.MoreOrders]
+least_member_of_eq [lemma, in hydras.Schutte.Well_Orders]
+least_member_ex_unique [lemma, in hydras.Schutte.Well_Orders]
+least_member_unicity [lemma, in hydras.Schutte.Well_Orders]
+least_member_glb [lemma, in hydras.Schutte.Well_Orders]
+least_member_lower_bound [lemma, in hydras.Schutte.Well_Orders]
+least_fixpoint [definition, in hydras.Schutte.Well_Orders]
+least_member [definition, in hydras.Schutte.Well_Orders]
+leA:48 [binder, in hydras.Prelude.Sort_spec]
+leA:62 [binder, in hydras.Prelude.Sort_spec]
+leBool [definition, in hydras.Ackermann.primRec]
+leb:107 [binder, in hydras.Prelude.Merge_Sort]
+leB:11 [binder, in hydras.Prelude.DecPreOrder_Instances]
+leb:177 [binder, in hydras.Prelude.Merge_Sort]
+leB:18 [binder, in hydras.Prelude.DecPreOrder_Instances]
+leb:183 [binder, in hydras.Prelude.Merge_Sort]
+leb:185 [binder, in hydras.Prelude.Merge_Sort]
+leb:188 [binder, in hydras.Prelude.Merge_Sort]
+leb:193 [binder, in hydras.Prelude.Merge_Sort]
+leb:195 [binder, in hydras.Prelude.Merge_Sort]
+leB:4 [binder, in hydras.Prelude.DecPreOrder_Instances]
+leB:49 [binder, in hydras.Prelude.Sort_spec]
+leB:63 [binder, in hydras.Prelude.Sort_spec]
+leIsPR [instance, in hydras.Ackermann.primRec]
+lemmas_on_length [section, in hydras.Gamma0.Gamma0]
+Lemma_4_5_2 [lemma, in hydras.Epsilon0.Large_Sets]
+Lemma_4_5_1 [lemma, in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_2 [lemma, in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.H0 [variable, in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.H [variable, in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.Hs [variable, in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.s [variable, in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.n [variable, in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.Hbeta [variable, in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.Halpha [variable, in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.beta [variable, in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.alpha [variable, in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof [section, in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_0 [lemma, in hydras.Epsilon0.Paths]
+Lemma_4_4_1 [lemma, in hydras.Epsilon0.Paths]
+Lemma_4_3 [lemma, in hydras.Epsilon0.Paths]
+Lemma_4_3_0 [lemma, in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.H4 [variable, in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.H1 [variable, in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.H0 [variable, in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.n2 [variable, in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.n1 [variable, in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.n0 [variable, in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.nf2 [variable, in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.nf1 [variable, in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.H00 [variable, in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.beta [variable, in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.alpha [variable, in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof [section, in hydras.Epsilon0.Paths]
+Lemma2_6_1 [lemma, in gaia_hydras.GPaths]
+Lemma2_6_1_E0 [lemma, in hydras.Epsilon0.Paths]
+Lemma2_6_1 [lemma, in hydras.Epsilon0.Paths]
+Lemma4_4 [lemma, in hydras.Epsilon0.Large_Sets]
+Lemma5 [lemma, in hydras.Schutte.Critical]
+Lemma5_7 [lemma, in hydras.Schutte.Critical]
+Lemma5_2 [lemma, in hydras.Schutte.Critical]
+Lemma5_1 [lemma, in hydras.Schutte.Critical]
+Lemma5_04 [lemma, in hydras.Schutte.Critical]
+Lemma5_03 [lemma, in hydras.Schutte.Critical]
+Lemma5_02 [lemma, in hydras.Schutte.Critical]
+Lemma5_01 [lemma, in hydras.Schutte.Critical]
+Lemma5_0 [lemma, in hydras.Schutte.Critical]
+lengthTerms [lemma, in hydras.Ackermann.wellFormed]
+length_abnc [lemma, in hydras.Gamma0.T2]
+length_ab [lemma, in hydras.Gamma0.T2]
+length_psi [lemma, in hydras.Gamma0.T2]
+length_n [lemma, in hydras.Gamma0.T2]
+length_c [lemma, in hydras.Gamma0.T2]
+length_b [lemma, in hydras.Gamma0.T2]
+length_a [lemma, in hydras.Gamma0.T2]
+length_map [lemma, in hydras.rpo.more_list]
+leq [definition, in hydras.Prelude.MoreOrders]
+leq_trans [instance, in hydras.Prelude.MoreOrders]
+lex [definition, in hydras.rpo.rpo]
+LExamples [section, in hydras.Ackermann.fol]
+LExamples.ex1 [variable, in hydras.Ackermann.fol]
+LExamples.ex2 [variable, in hydras.Ackermann.fol]
+LExamples.ex3 [variable, in hydras.Ackermann.fol]
+LExamples.L [variable, in hydras.Ackermann.fol]
+LExamples.P [variable, in hydras.Ackermann.fol]
+LExamples.Q [variable, in hydras.Ackermann.fol]
+lexico [inductive, in hydras.Prelude.Simple_LexProd]
+lexlt [definition, in hydras.solutions_exercises.MultisetWf]
+lexnf [definition, in hydras.solutions_exercises.MultisetWf]
+lexpower [inductive, in hydras.solutions_exercises.MultisetWf]
+lexprod [section, in hydras.Prelude.DecPreOrder_Instances]
+lexprod.A [variable, in hydras.Prelude.DecPreOrder_Instances]
+lexprod.B [variable, in hydras.Prelude.DecPreOrder_Instances]
+lexprod.leA [variable, in hydras.Prelude.DecPreOrder_Instances]
+lexprod.leB [variable, in hydras.Prelude.DecPreOrder_Instances]
+_ <lex _ [notation, in hydras.Prelude.DecPreOrder_Instances]
+_ <=lex _ [notation, in hydras.Prelude.DecPreOrder_Instances]
+_ =B= _ [notation, in hydras.Prelude.DecPreOrder_Instances]
+_ =A= _ [notation, in hydras.Prelude.DecPreOrder_Instances]
+_ <B _ [notation, in hydras.Prelude.DecPreOrder_Instances]
+_ <=B _ [notation, in hydras.Prelude.DecPreOrder_Instances]
+_ <A _ [notation, in hydras.Prelude.DecPreOrder_Instances]
+_ <=A _ [notation, in hydras.Prelude.DecPreOrder_Instances]
+lexwf [lemma, in hydras.solutions_exercises.MultisetWf]
+lexwf [lemma, in hydras.solutions_exercises.MultisetWf]
+lex_prod_dec [instance, in hydras.Prelude.DecPreOrder_Instances]
+lex_strict_intro_left [lemma, in hydras.Prelude.DecPreOrder_Instances]
+lex_strict_intro_right [lemma, in hydras.Prelude.DecPreOrder_Instances]
+lex_inv_right [lemma, in hydras.Prelude.DecPreOrder_Instances]
+lex_inv_left [lemma, in hydras.Prelude.DecPreOrder_Instances]
+lex_of_equiv [lemma, in hydras.Prelude.DecPreOrder_Instances]
+lex_prod [inductive, in hydras.Prelude.DecPreOrder_Instances]
+Lex_nat_wf [instance, in hydras.MoreAck.Ack]
+lex_nat_wf [lemma, in hydras.MoreAck.Ack]
+lex_nat [definition, in hydras.MoreAck.Ack]
+lex_2 [constructor, in hydras.Prelude.Simple_LexProd]
+lex_1 [constructor, in hydras.Prelude.Simple_LexProd]
+lex_trans [lemma, in hydras.rpo.rpo]
+lex1 [constructor, in hydras.Prelude.DecPreOrder_Instances]
+lex1 [constructor, in hydras.solutions_exercises.MultisetWf]
+lex2 [constructor, in hydras.Prelude.DecPreOrder_Instances]
+lex2 [constructor, in hydras.solutions_exercises.MultisetWf]
+lex3 [constructor, in hydras.solutions_exercises.MultisetWf]
+lex4 [constructor, in hydras.solutions_exercises.MultisetWf]
+LE_LT_eq_dec [lemma, in hydras.Epsilon0.T1]
+LE_r [lemma, in hydras.Epsilon0.T1]
+le_lt_LT [lemma, in hydras.Epsilon0.T1]
+LE_phi0 [lemma, in hydras.Epsilon0.T1]
+LE_LT_trans [lemma, in hydras.Epsilon0.T1]
+LE_antisym [lemma, in hydras.Epsilon0.T1]
+LE_trans [lemma, in hydras.Epsilon0.T1]
+LE_refl [lemma, in hydras.Epsilon0.T1]
+LE_zero [lemma, in hydras.Epsilon0.T1]
+LE_le [lemma, in hydras.Epsilon0.T1]
+LE_nf_r [lemma, in hydras.Epsilon0.T1]
+LE_nf_l [lemma, in hydras.Epsilon0.T1]
+le_phi0 [lemma, in hydras.Epsilon0.T1]
+le_tail [lemma, in hydras.Epsilon0.T1]
+le_zero_inv [lemma, in hydras.Epsilon0.T1]
+le_inv [lemma, in hydras.Epsilon0.T1]
+le_eq_lt_dec [lemma, in hydras.Epsilon0.T1]
+le_0 [lemma, in hydras.OrdinalNotations.ON_Omega2]
+le_intror [lemma, in hydras.OrdinalNotations.ON_Omega2]
+le_min_bd [lemma, in hydras.Prelude.Comparable]
+le_min_a [lemma, in hydras.Prelude.Comparable]
+le_max_b [lemma, in hydras.Prelude.Comparable]
+le_max_a [lemma, in hydras.Prelude.Comparable]
+le_not_gt [lemma, in hydras.Prelude.Comparable]
+le_lt_trans [lemma, in hydras.Prelude.Comparable]
+le_trans [lemma, in hydras.Prelude.Comparable]
+le_refl [lemma, in hydras.Prelude.Comparable]
+le_lt_equiv_dec [lemma, in hydras.Prelude.DecPreOrder]
+le_lt_weak [lemma, in hydras.Prelude.DecPreOrder]
+le_lt_trans [lemma, in hydras.Prelude.DecPreOrder]
+le_not_gt [lemma, in hydras.Prelude.DecPreOrder]
+le_phi0 [lemma, in hydras.Schutte.AP]
+le_lt_trans [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+le_0 [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+le_intror [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+le_introl [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+le_S_Ack [lemma, in hydras.MoreAck.Ack]
+le_dec:118 [binder, in hydras.Prelude.Merge_Sort]
+le_total:117 [binder, in hydras.Prelude.Merge_Sort]
+LE_ref [lemma, in gaia_hydras.T1Bridge]
+le_ref [lemma, in gaia_hydras.T1Bridge]
+le_lt_eq [lemma, in hydras.Prelude.MoreOrders]
+le_alpha_zero [lemma, in hydras.Schutte.Schutte_basics]
+le_sup_members [lemma, in hydras.Schutte.Schutte_basics]
+le_disj [lemma, in hydras.Schutte.Schutte_basics]
+le_not_gt [lemma, in hydras.Schutte.Schutte_basics]
+le_eq_or_lt [lemma, in hydras.Schutte.Schutte_basics]
+le_antisym [lemma, in hydras.Schutte.Schutte_basics]
+le_trans [lemma, in hydras.Schutte.Schutte_basics]
+le_lt_trans [lemma, in hydras.Schutte.Schutte_basics]
+le_refl [lemma, in hydras.Schutte.Schutte_basics]
+Le_refl [lemma, in hydras.Epsilon0.E0]
+Le_trans [lemma, in hydras.Epsilon0.E0]
+le_lt_eq_dec [lemma, in hydras.Epsilon0.E0]
+Le_iff [lemma, in hydras.Epsilon0.E0]
+Le_Lt_trans [lemma, in hydras.Schutte.Well_Orders]
+Le_trans [instance, in hydras.Schutte.Well_Orders]
+Le_antisym [lemma, in hydras.Schutte.Well_Orders]
+Le_refl [lemma, in hydras.Schutte.Well_Orders]
+le_plus_l [lemma, in hydras.Gamma0.Gamma0]
+le_plus_r [lemma, in hydras.Gamma0.Gamma0]
+le_b_phi_ab [lemma, in hydras.Gamma0.Gamma0]
+le_one_cons [lemma, in hydras.Gamma0.Gamma0]
+le_cons_tail [lemma, in hydras.Gamma0.Gamma0]
+le_trans [lemma, in hydras.Gamma0.Gamma0]
+le_lt_trans [lemma, in hydras.Gamma0.Gamma0]
+le_inv_nc [lemma, in hydras.Gamma0.Gamma0]
+le_psi_term_le [lemma, in hydras.Gamma0.Gamma0]
+le_zero_alpha [lemma, in hydras.Gamma0.Gamma0]
+le_a_mult_Sn_a [lemma, in hydras.Schutte.Addition]
+le_plus_r [lemma, in hydras.Schutte.Addition]
+le_plus_l [lemma, in hydras.Schutte.Addition]
+le:16 [binder, in hydras.Prelude.DecPreOrder]
+le:21 [binder, in hydras.Prelude.DecPreOrder]
+le:26 [binder, in hydras.Prelude.DecPreOrder]
+le:3 [binder, in hydras.solutions_exercises.predSuccUnicity]
+Le:3 [binder, in hydras.OrdinalNotations.ON_O]
+le:31 [binder, in hydras.Prelude.Sort_spec]
+le:36 [binder, in hydras.Prelude.DecPreOrder]
+le:40 [binder, in hydras.Prelude.Sort_spec]
+le:40 [binder, in hydras.Prelude.DecPreOrder]
+le:45 [binder, in hydras.Prelude.DecPreOrder]
+le:50 [binder, in hydras.Prelude.DecPreOrder]
+le:55 [binder, in hydras.Prelude.DecPreOrder]
+le:60 [binder, in hydras.Prelude.DecPreOrder]
+le:65 [binder, in hydras.Prelude.DecPreOrder]
+le:71 [binder, in hydras.Prelude.DecPreOrder]
+le:75 [binder, in hydras.Prelude.Sort_spec]
+le:77 [binder, in hydras.Prelude.DecPreOrder]
+le:82 [binder, in hydras.Prelude.DecPreOrder]
+le:85 [binder, in hydras.Prelude.DecPreOrder]
+le:88 [binder, in hydras.Prelude.DecPreOrder]
+le:94 [binder, in hydras.Prelude.DecPreOrder]
+LF1 [lemma, in gaia_hydras.GF_alpha]
+LF1 [lemma, in hydras.Epsilon0.F_alpha]
+LF2 [lemma, in gaia_hydras.GF_alpha]
+LF2 [lemma, in hydras.Epsilon0.F_alpha]
+LF2_0 [lemma, in hydras.Epsilon0.F_alpha]
+LF2' [lemma, in gaia_hydras.GF_alpha]
+LF2' [lemma, in hydras.Epsilon0.F_alpha]
+LF3 [lemma, in hydras.solutions_exercises.F_3]
+LF3_2 [lemma, in gaia_hydras.GF_alpha]
+LF3_2 [lemma, in hydras.Epsilon0.F_alpha]
+lg:163 [binder, in hydras.rpo.rpo]
+lg:231 [binder, in hydras.rpo.rpo]
+lg:68 [binder, in hydras.rpo.rpo]
+lheight [definition, in hydras.Hydra.Hydra_Definitions]
+lhsize [definition, in hydras.Hydra.Hydra_Definitions]
+LibHyps_Experiments [library]
+liftCloseFrom [lemma, in hydras.Ackermann.subAll]
+lim [section, in hydras.Epsilon0.Large_Sets]
+Limit [definition, in hydras.Prelude.MoreOrders]
+limitb [abbreviation, in hydras.Epsilon0.T1]
+limitb [definition, in hydras.OrdinalNotations.ON_Omega2]
+limitb [definition, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Limitb [abbreviation, in hydras.Epsilon0.E0]
+limitb_limit [lemma, in hydras.OrdinalNotations.ON_Omega2]
+LimitNotSucc [lemma, in hydras.Prelude.Comparable]
+limit_is_lub [lemma, in hydras.OrdinalNotations.ON_Omega2]
+limit_is_lub_0 [lemma, in hydras.OrdinalNotations.ON_Omega2]
+limit_no_R1 [lemma, in hydras.Hydra.O2H]
+limit_no_head [lemma, in hydras.Hydra.O2H]
+Limit_Infinity [lemma, in hydras.solutions_exercises.Limit_Infinity]
+limit_canonS_not_zero [lemma, in gaia_hydras.GCanon]
+limit_iff [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+limit_is_omega [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Limit_gt_Zero [lemma, in hydras.Epsilon0.E0]
+Limit_not_Zero [lemma, in hydras.Epsilon0.E0]
+limit_of [definition, in gaia_hydras.onType]
+limit_v2 [definition, in gaia_hydras.onType]
+limit_rounds [lemma, in hydras.Hydra.Omega2_Small]
+limit_rounds_0 [lemma, in hydras.Hydra.Omega2_Small]
+limit_step [constructor, in hydras.Hydra.Omega2_Small]
+limit_plus_fin_ok [lemma, in hydras.Gamma0.Gamma0]
+limit_plus_fin_inv0 [lemma, in hydras.Gamma0.Gamma0]
+limit_plus_fin_lim [lemma, in hydras.Gamma0.Gamma0]
+limit_plus_fin_plus [lemma, in hydras.Gamma0.Gamma0]
+limit_plus_F_cons [constructor, in hydras.Gamma0.Gamma0]
+limit_plus_F_0 [constructor, in hydras.Gamma0.Gamma0]
+limit_plus_fin [inductive, in hydras.Gamma0.Gamma0]
+Limit_Infinity [library]
+lim.f [variable, in hydras.Epsilon0.Large_Sets]
+lim.H [variable, in hydras.Epsilon0.Large_Sets]
+lim.Hlim [variable, in hydras.Epsilon0.Large_Sets]
+lim.Hnf [variable, in hydras.Epsilon0.Large_Sets]
+lim.lambda [variable, in hydras.Epsilon0.Large_Sets]
+LispAbbreviations [module, in hydras.Ackermann.cPair]
+LispAbbreviations.caar [abbreviation, in hydras.Ackermann.cPair]
+LispAbbreviations.caddr [abbreviation, in hydras.Ackermann.cPair]
+LispAbbreviations.cadr [abbreviation, in hydras.Ackermann.cPair]
+LispAbbreviations.car [abbreviation, in hydras.Ackermann.cPair]
+LispAbbreviations.cddddr [abbreviation, in hydras.Ackermann.cPair]
+LispAbbreviations.cdddr [abbreviation, in hydras.Ackermann.cPair]
+LispAbbreviations.cddr [abbreviation, in hydras.Ackermann.cPair]
+LispAbbreviations.cdr [abbreviation, in hydras.Ackermann.cPair]
+ListExt [library]
+listValues [definition, in hydras.Ackermann.cPair]
+List_length [instance, in hydras.Prelude.DecPreOrder_Instances]
+list_rec3 [definition, in hydras.rpo.more_list]
+list_rec2 [definition, in hydras.rpo.more_list]
+list_size_size_eq [lemma, in hydras.rpo.more_list]
+list_size_fold [lemma, in hydras.rpo.more_list]
+list_size_app [lemma, in hydras.rpo.more_list]
+list_size_tl_compat [lemma, in hydras.rpo.more_list]
+list_size [definition, in hydras.rpo.more_list]
+list_app_length [lemma, in hydras.rpo.more_list]
+List_Remove.Aeq_dec [variable, in hydras.Ackermann.ListExt]
+List_Remove.A [variable, in hydras.Ackermann.ListExt]
+List_Remove [section, in hydras.Ackermann.ListExt]
+list_permut [library]
+list_set [library]
+LL [lemma, in hydras.Epsilon0.F_alpha]
+LNN [definition, in hydras.Ackermann.Languages]
+LNN [library]
+LNNArityF [definition, in hydras.Ackermann.Languages]
+LNNArityR [definition, in hydras.Ackermann.Languages]
+LNNRelation [inductive, in hydras.Ackermann.Languages]
+LNN_eqdec [lemma, in hydras.Ackermann.LNN]
+LNN_FixPoint.codeFormula [variable, in Goedel.fixPoint]
+LNN_FixPoint [section, in Goedel.fixPoint]
+LNN_Examples [library]
+LNN2LNT [library]
+LNN2LNT_subFormula [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_freeVarF2 [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_freeVarF1 [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_freeVarF [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_exist [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_iff [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_and [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_or [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_formula [definition, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_subTerms [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_subTerm [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_freeVarTs [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_freeVarT [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_natToTerm [lemma, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_terms [definition, in hydras.Ackermann.LNN2LNT]
+LNN2LNT_term [definition, in hydras.Ackermann.LNN2LNT]
+LNotPR [lemma, in hydras.Hydra.Hydra_Theorems]
+LNT [module, in hydras.Ackermann.LNT]
+LNT [definition, in hydras.Ackermann.Languages]
+LNT [library]
+LNTFunction [inductive, in hydras.Ackermann.Languages]
+LNTFunctionArity [definition, in hydras.Ackermann.Languages]
+LNTRelation [inductive, in hydras.Ackermann.Languages]
+LNTRelationR [definition, in hydras.Ackermann.Languages]
+LNT_eqdec [lemma, in hydras.Ackermann.LNT]
+LNT_FixPoint.codeFormula [variable, in Goedel.fixPoint]
+LNT_FixPoint [section, in Goedel.fixPoint]
+LNT.Plus [definition, in hydras.Ackermann.LNT]
+LNT.Succ [definition, in hydras.Ackermann.LNT]
+LNT.Times [definition, in hydras.Ackermann.LNT]
+LNT.Zero [definition, in hydras.Ackermann.LNT]
+LNT2LNN_subFormula [lemma, in hydras.Ackermann.LNN2LNT]
+LNT2LNN_freeVarF [lemma, in hydras.Ackermann.LNN2LNT]
+LNT2LNN_freeVarT [lemma, in hydras.Ackermann.LNN2LNT]
+LNT2LNN_subTerm [lemma, in hydras.Ackermann.LNN2LNT]
+LNT2LNN_formula [definition, in hydras.Ackermann.LNN2LNT]
+LNT2LNN_natToTerm [lemma, in hydras.Ackermann.LNN2LNT]
+LNT2LNN_terms [definition, in hydras.Ackermann.LNN2LNT]
+LNT2LNN_term [definition, in hydras.Ackermann.LNN2LNT]
+LNT2LNT_formula [lemma, in hydras.Ackermann.LNN2LNT]
+LNT2LNT_term [lemma, in hydras.Ackermann.LNN2LNT]
+LO [module, in hydras.OrdinalNotations.OmegaOmega]
+LocallySorted_inv_In [lemma, in hydras.Prelude.Sort_spec]
+LocallySorted_trans [lemma, in hydras.Prelude.Sort_spec]
+LocallySorted_cons' [lemma, in hydras.Prelude.Sort_spec]
+LocallySorted_cons [lemma, in hydras.Prelude.Sort_spec]
+Logic [section, in hydras.Ackermann.LNN]
+Logic [section, in hydras.Ackermann.LNT]
+Logic_Rules.Other_Rules [section, in hydras.Ackermann.folLogic]
+Logic_Rules.Not_Rules [section, in hydras.Ackermann.folLogic]
+Logic_Rules.SysPrf [variable, in hydras.Ackermann.folLogic]
+Logic_Rules.Prf [variable, in hydras.Ackermann.folLogic]
+Logic_Rules.Terms [variable, in hydras.Ackermann.folLogic]
+Logic_Rules.Term [variable, in hydras.Ackermann.folLogic]
+Logic_Rules.System [variable, in hydras.Ackermann.folLogic]
+Logic_Rules.Formulas [variable, in hydras.Ackermann.folLogic]
+Logic_Rules.Formula [variable, in hydras.Ackermann.folLogic]
+Logic_Rules.L [variable, in hydras.Ackermann.folLogic]
+Logic_Rules [section, in hydras.Ackermann.folLogic]
+LO.ap [inductive, in hydras.OrdinalNotations.OmegaOmega]
+LO.ap_intro [constructor, in hydras.OrdinalNotations.OmegaOmega]
+LO.compare_correct [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.compare_reflect [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.compare_rev [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.compare_ref [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.compare_oo [instance, in hydras.OrdinalNotations.OmegaOmega]
+LO.cons [abbreviation, in hydras.OrdinalNotations.OmegaOmega]
+LO.cons_nf [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.eq_ref [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.fin [definition, in hydras.OrdinalNotations.OmegaOmega]
+LO.fin_nf [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.FS [abbreviation, in hydras.OrdinalNotations.OmegaOmega]
+LO.limitb [definition, in hydras.OrdinalNotations.OmegaOmega]
+LO.limitb_ref [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.lo_comparable [instance, in hydras.OrdinalNotations.OmegaOmega]
+LO.lo_strorder [instance, in hydras.OrdinalNotations.OmegaOmega]
+LO.lt [definition, in hydras.OrdinalNotations.OmegaOmega]
+LO.lt_trans [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.lt_irrefl [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.lt_ref [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.mult [definition, in hydras.OrdinalNotations.OmegaOmega]
+LO.mult_plus_distr_l [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.mult_nf [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.mult_ref [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.nf [definition, in hydras.OrdinalNotations.OmegaOmega]
+LO.nf_ref [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.nf_inv3 [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.nf_inv2 [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.nf_b [definition, in hydras.OrdinalNotations.OmegaOmega]
+LO.omega [abbreviation, in hydras.OrdinalNotations.OmegaOmega]
+LO.phi0 [abbreviation, in hydras.OrdinalNotations.OmegaOmega]
+LO.phi0_nf [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.phi0_ref [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.plus [definition, in hydras.OrdinalNotations.OmegaOmega]
+LO.plus_assoc [instance, in hydras.OrdinalNotations.OmegaOmega]
+LO.plus_nf [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.plus_ref [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.refine [definition, in hydras.OrdinalNotations.OmegaOmega]
+LO.single_nf [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.succ [definition, in hydras.OrdinalNotations.OmegaOmega]
+LO.succb [definition, in hydras.OrdinalNotations.OmegaOmega]
+LO.succb_ref [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.succ_nf [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.succ_ref [lemma, in hydras.OrdinalNotations.OmegaOmega]
+LO.t [definition, in hydras.OrdinalNotations.OmegaOmega]
+LO.zero [definition, in hydras.OrdinalNotations.OmegaOmega]
+LO.zero_nf [lemma, in hydras.OrdinalNotations.OmegaOmega]
+_ * _ (lo_scope) [notation, in hydras.OrdinalNotations.OmegaOmega]
+_ + _ (lo_scope) [notation, in hydras.OrdinalNotations.OmegaOmega]
+LS [lemma, in hydras.solutions_exercises.FibonacciPR]
+LS [lemma, in hydras.Hydra.BigBattle]
+ls1:246 [binder, in hydras.rpo.rpo]
+ls1:251 [binder, in hydras.rpo.rpo]
+ls2:247 [binder, in hydras.rpo.rpo]
+ls2:252 [binder, in hydras.rpo.rpo]
+ls:164 [binder, in hydras.rpo.rpo]
+ls:232 [binder, in hydras.rpo.rpo]
+ls:241 [binder, in hydras.rpo.rpo]
+ls:69 [binder, in hydras.rpo.rpo]
+LT [definition, in hydras.Epsilon0.T1]
+lt [definition, in hydras.Epsilon0.T1]
+lt [inductive, in hydras.Gamma0.T2]
+lt [definition, in hydras.OrdinalNotations.ON_O]
+lt [definition, in hydras.Prelude.DecPreOrder]
+LT [abbreviation, in hydras.solutions_exercises.MultisetWf]
+LT [abbreviation, in gaia_hydras.T1Bridge]
+lt [definition, in hydras.OrdinalNotations.ON_mult]
+lt [definition, in hydras.OrdinalNotations.ON_plus]
+lt [axiom, in hydras.Schutte.Schutte_basics]
+lt [definition, in hydras.OrdinalNotations.ON_Finite]
+ltA:107 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltA:122 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltA:134 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltA:147 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltA:160 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltA:172 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltA:185 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltA:199 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltA:2 [binder, in hydras.OrdinalNotations.ON_mult]
+ltA:2 [binder, in hydras.OrdinalNotations.ON_plus]
+ltA:30 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltA:33 [binder, in hydras.solutions_exercises.MultisetWf]
+ltA:36 [binder, in hydras.solutions_exercises.MultisetWf]
+ltA:49 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltA:68 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltA:69 [binder, in hydras.solutions_exercises.MultisetWf]
+ltA:82 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltA:94 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltBool [definition, in hydras.Ackermann.primRec]
+ltBoolFalse [lemma, in hydras.Ackermann.primRec]
+ltBoolTrue [lemma, in hydras.Ackermann.primRec]
+Ltb_ij [lemma, in hydras.OrdinalNotations.ON_Finite]
+ltB:111 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltB:126 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltB:138 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltB:151 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltB:164 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltB:176 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltB:189 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltB:203 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltB:34 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltB:53 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltB:6 [binder, in hydras.OrdinalNotations.ON_mult]
+ltB:6 [binder, in hydras.OrdinalNotations.ON_plus]
+ltB:86 [binder, in hydras.OrdinalNotations.ON_Generic]
+ltB:98 [binder, in hydras.OrdinalNotations.ON_Generic]
+LTFormula [definition, in hydras.Ackermann.LNN2LNT]
+ltIsPR [instance, in hydras.Ackermann.primRec]
+ltn_add_ll [lemma, in gaia_hydras.nfwfgaia]
+ltn_add_el [lemma, in gaia_hydras.nfwfgaia]
+ltn_add_le [lemma, in gaia_hydras.nfwfgaia]
+ltn_simpl2 [lemma, in gaia_hydras.nfwfgaia]
+ltn_simpl1 [lemma, in gaia_hydras.nfwfgaia]
+lt_succ_le_2' [lemma, in hydras.Epsilon0.T1]
+LT_succ_LT_eq_dec [lemma, in hydras.Epsilon0.T1]
+lt_one [lemma, in hydras.Epsilon0.T1]
+lt_cons_phi0_inv [lemma, in hydras.Epsilon0.T1]
+LT_eq_LT_dec [lemma, in hydras.Epsilon0.T1]
+LT_succ [lemma, in hydras.Epsilon0.T1]
+LT_add [lemma, in hydras.Epsilon0.T1]
+lt_plus_r [lemma, in hydras.Epsilon0.T1]
+lt_plus_l [lemma, in hydras.Epsilon0.T1]
+LT_of_finite [lemma, in hydras.Epsilon0.T1]
+lt_omega_inv [lemma, in hydras.Epsilon0.T1]
+LT_one [lemma, in hydras.Epsilon0.T1]
+LT_succ_LE_2 [lemma, in hydras.Epsilon0.T1]
+LT_succ_LE [lemma, in hydras.Epsilon0.T1]
+lt_succ_le [lemma, in hydras.Epsilon0.T1]
+lt_succ_le_2 [lemma, in hydras.Epsilon0.T1]
+LT_succ_LE_R [lemma, in hydras.Epsilon0.T1]
+lt_succ_le_R [lemma, in hydras.Epsilon0.T1]
+LT_cons_0 [lemma, in hydras.Epsilon0.T1]
+LT_inv_strong [lemma, in hydras.Epsilon0.T1]
+LT_right [constructor, in hydras.Epsilon0.T1]
+LT_middle [constructor, in hydras.Epsilon0.T1]
+LT_left [constructor, in hydras.Epsilon0.T1]
+LT_cases [inductive, in hydras.Epsilon0.T1]
+LT_inv [lemma, in hydras.Epsilon0.T1]
+LT_St [instance, in hydras.Epsilon0.T1]
+LT_LE_trans [lemma, in hydras.Epsilon0.T1]
+LT_irrefl [lemma, in hydras.Epsilon0.T1]
+LT_trans [lemma, in hydras.Epsilon0.T1]
+LT_lt [lemma, in hydras.Epsilon0.T1]
+LT_nf_r [lemma, in hydras.Epsilon0.T1]
+LT_nf_l [lemma, in hydras.Epsilon0.T1]
+lt_a_phi0_b_iff [lemma, in hydras.Epsilon0.T1]
+lt_a_phi0_b_def [lemma, in hydras.Epsilon0.T1]
+lt_a_phi0_b_phi0R [lemma, in hydras.Epsilon0.T1]
+lt_a_phi0_b_phi0 [lemma, in hydras.Epsilon0.T1]
+lt_a_phi0_b_intro [lemma, in hydras.Epsilon0.T1]
+lt_a_phi0_b_inv1 [lemma, in hydras.Epsilon0.T1]
+lt_le_dec [definition, in hydras.Epsilon0.T1]
+lt_eq_lt_dec [definition, in hydras.Epsilon0.T1]
+lt_a_phi0_b_inv [lemma, in hydras.Epsilon0.T1]
+lt_a_phi0_a [lemma, in hydras.Epsilon0.T1]
+lt_succ [lemma, in hydras.Epsilon0.T1]
+lt_a_phi0_b_c [constructor, in hydras.Epsilon0.T1]
+lt_a_phi0_b_z [constructor, in hydras.Epsilon0.T1]
+lt_a_phi0_b [inductive, in hydras.Epsilon0.T1]
+lt_dec [instance, in hydras.Epsilon0.T1]
+lt_inv_head [lemma, in hydras.Epsilon0.T1]
+lt_trans [lemma, in hydras.Epsilon0.T1]
+lt_fin_iff [lemma, in hydras.Epsilon0.T1]
+lt_inv_tail [lemma, in hydras.Epsilon0.T1]
+lt_inv_coeff_dec [lemma, in hydras.Epsilon0.T1]
+lt_inv_coeff [lemma, in hydras.Epsilon0.T1]
+lt_inv [lemma, in hydras.Epsilon0.T1]
+lt_irrefl [lemma, in hydras.Epsilon0.T1]
+lt_inv_strong [lemma, in hydras.Epsilon0.T1]
+lt_right [constructor, in hydras.Epsilon0.T1]
+lt_middle [constructor, in hydras.Epsilon0.T1]
+lt_left [constructor, in hydras.Epsilon0.T1]
+lt_cases [inductive, in hydras.Epsilon0.T1]
+lt_omega [lemma, in hydras.OrdinalNotations.ON_Omega2]
+lt_eq_lt_dec [lemma, in hydras.OrdinalNotations.ON_Omega2]
+lt_succ [lemma, in hydras.OrdinalNotations.ON_Omega2]
+lt_succ_le [lemma, in hydras.OrdinalNotations.ON_Omega2]
+lt_epsilon0 [inductive, in hydras.Gamma0.T2]
+lt_7 [constructor, in hydras.Gamma0.T2]
+lt_6 [constructor, in hydras.Gamma0.T2]
+lt_5 [constructor, in hydras.Gamma0.T2]
+lt_4 [constructor, in hydras.Gamma0.T2]
+lt_3 [constructor, in hydras.Gamma0.T2]
+lt_2 [constructor, in hydras.Gamma0.T2]
+lt_1 [constructor, in hydras.Gamma0.T2]
+lt_lt_Sn [lemma, in hydras.Prelude.More_Arith]
+LT_to_round_plus [lemma, in hydras.Hydra.O2H]
+Lt_n_F_Zero_n [lemma, in hydras.Epsilon0.F_alpha]
+lt_eq_lt_dec [definition, in hydras.Prelude.Comparable]
+lt_eq_lt [lemma, in hydras.Prelude.Comparable]
+lt_incl_le [lemma, in hydras.Prelude.Comparable]
+lt_le_trans [lemma, in hydras.Prelude.Comparable]
+lt_not_ge [lemma, in hydras.Prelude.Comparable]
+lt_not_gt [lemma, in hydras.Prelude.Comparable]
+lt_irrefl [abbreviation, in hydras.Prelude.Comparable]
+lt_trans [abbreviation, in hydras.Prelude.Comparable]
+lt_wf [lemma, in hydras.OrdinalNotations.ON_O]
+LT_to_standard_path [lemma, in gaia_hydras.GPaths]
+LT_path_to [lemma, in gaia_hydras.GPaths]
+lt_a_phi0_b_oplus [lemma, in hydras.Epsilon0.Hessenberg]
+lt_transitive [instance, in hydras.Prelude.DecPreOrder]
+lt_le_trans [lemma, in hydras.Prelude.DecPreOrder]
+lt_not_equiv [lemma, in hydras.Prelude.DecPreOrder]
+lt_not_ge [lemma, in hydras.Prelude.DecPreOrder]
+lt_irreflexive [lemma, in hydras.Prelude.DecPreOrder]
+lt_S_2i [lemma, in additions.More_on_positive]
+lt_phi0 [lemma, in hydras.Schutte.AP]
+lt_beta_exists [lemma, in hydras.Schutte.AP]
+lt_omega [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+lt_succ [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+lt_succ_le [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+lt_le_trans [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+LT_inv [lemma, in hydras.solutions_exercises.MultisetWf]
+lt_wf_double_ind [lemma, in hydras.MoreAck.Ack]
+lt_wf_ind [lemma, in hydras.MoreAck.Ack]
+LT_ref [lemma, in gaia_hydras.T1Bridge]
+lt_ref [lemma, in gaia_hydras.T1Bridge]
+lt_eq_lt_dec [lemma, in hydras.OrdinalNotations.ON_mult]
+lt_wf [lemma, in hydras.OrdinalNotations.ON_mult]
+lt_strorder [instance, in hydras.OrdinalNotations.ON_mult]
+LT_to_standard_battle [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+lt_eq_lt_dec [lemma, in hydras.OrdinalNotations.ON_plus]
+lt_wf [lemma, in hydras.OrdinalNotations.ON_plus]
+lt_strorder [instance, in hydras.OrdinalNotations.ON_plus]
+lt_depth [definition, in hydras.Ackermann.fol]
+lt_omega_finite [lemma, in hydras.Schutte.Schutte_basics]
+lt_omega_limit_lt_exists_lt [lemma, in hydras.Schutte.Schutte_basics]
+lt_omega_limit [lemma, in hydras.Schutte.Schutte_basics]
+lt_sup_exists_lt [lemma, in hydras.Schutte.Schutte_basics]
+lt_sup_exists_leq [lemma, in hydras.Schutte.Schutte_basics]
+lt_succ_lt [lemma, in hydras.Schutte.Schutte_basics]
+lt_succ_le_2 [lemma, in hydras.Schutte.Schutte_basics]
+lt_succ_le [lemma, in hydras.Schutte.Schutte_basics]
+lt_succ [lemma, in hydras.Schutte.Schutte_basics]
+lt_or_ge [lemma, in hydras.Schutte.Schutte_basics]
+lt_irrefl [lemma, in hydras.Schutte.Schutte_basics]
+lt_le_trans [lemma, in hydras.Schutte.Schutte_basics]
+lt_trans [lemma, in hydras.Schutte.Schutte_basics]
+lt_le [lemma, in hydras.Schutte.Schutte_basics]
+lt_wf [lemma, in hydras.OrdinalNotations.ON_Finite]
+lt_omega_inv [lemma, in hydras.Epsilon0.E0]
+Lt_Le_incl [lemma, in hydras.Epsilon0.E0]
+Lt_trans [lemma, in hydras.Epsilon0.E0]
+lt_Succ_le_2 [lemma, in hydras.Epsilon0.E0]
+lt_Succ_inv [lemma, in hydras.Epsilon0.E0]
+Lt_Succ [lemma, in hydras.Epsilon0.E0]
+Lt_Succ_Le [lemma, in hydras.Epsilon0.E0]
+Lt_not_Gt [lemma, in hydras.Schutte.Well_Orders]
+Lt_Le_trans [lemma, in hydras.Schutte.Well_Orders]
+Lt_connect [lemma, in hydras.Schutte.Well_Orders]
+Lt_irreflexive [projection, in hydras.Schutte.Well_Orders]
+Lt_trans [projection, in hydras.Schutte.Well_Orders]
+LT_to_standard_path [lemma, in hydras.Epsilon0.Paths]
+LT_acc_from [lemma, in hydras.Epsilon0.Paths]
+LT_path_to [lemma, in hydras.Epsilon0.Paths]
+LT_path_toS [lemma, in hydras.Epsilon0.Paths]
+lt_not_gt [lemma, in hydras.Gamma0.Gamma0]
+lt_epsilon0_succ [lemma, in hydras.Gamma0.Gamma0]
+lt_epsilon0_trans [lemma, in hydras.Gamma0.Gamma0]
+lt_epsilon0_okR [lemma, in hydras.Gamma0.Gamma0]
+lt_epsilon0_ok [lemma, in hydras.Gamma0.Gamma0]
+lt_a_phi_ab [lemma, in hydras.Gamma0.Gamma0]
+lt_inc_rpo_0 [lemma, in hydras.Gamma0.Gamma0]
+lt_rpo_cons_cons [lemma, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.H [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.nf2 [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.nf1 [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.Hrec [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.Hsize [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.n2 [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.n1 [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.c2 [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.b2 [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.a2 [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.c1 [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.b1 [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.a1 [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo.s [variable, in hydras.Gamma0.Gamma0]
+lt_incl_rpo [section, in hydras.Gamma0.Gamma0]
+lt_subterm1 [lemma, in hydras.Gamma0.Gamma0]
+lt_succ_le [lemma, in hydras.Gamma0.Gamma0]
+lt_succ [lemma, in hydras.Gamma0.Gamma0]
+lt_compatR [lemma, in hydras.Gamma0.Gamma0]
+lt_compat [lemma, in hydras.Gamma0.Gamma0]
+lt_omega_is_fin [lemma, in hydras.Gamma0.Gamma0]
+lt_omega_inv [lemma, in hydras.Gamma0.Gamma0]
+lt_cons_omega_inv [lemma, in hydras.Gamma0.Gamma0]
+lt_one_inv [lemma, in hydras.Gamma0.Gamma0]
+lt_tail [lemma, in hydras.Gamma0.Gamma0]
+lt_tail0 [lemma, in hydras.Gamma0.Gamma0]
+lt_alpha_cons [lemma, in hydras.Gamma0.Gamma0]
+lt_alpha_psi [lemma, in hydras.Gamma0.Gamma0]
+lt_beta_cons [lemma, in hydras.Gamma0.Gamma0]
+lt_beta_psi [lemma, in hydras.Gamma0.Gamma0]
+lt_le_trans [lemma, in hydras.Gamma0.Gamma0]
+lt_trans [lemma, in hydras.Gamma0.Gamma0]
+lt_trans0 [lemma, in hydras.Gamma0.Gamma0]
+lt_ge_dec [definition, in hydras.Gamma0.Gamma0]
+lt_eq_lt_dec [definition, in hydras.Gamma0.Gamma0]
+lt_than_psi [lemma, in hydras.Gamma0.Gamma0]
+lt_irr [lemma, in hydras.Gamma0.Gamma0]
+lt_wf [lemma, in gaia_hydras.nfwfgaia]
+LT_ [constructor, in hydras.Ackermann.Languages]
+lt_not_wf [lemma, in hydras.solutions_exercises.T1_ltNotWf]
+lt_succ_le [library]
+LT1 [lemma, in hydras.Epsilon0.T1]
+LT2 [lemma, in hydras.Epsilon0.T1]
+LT3 [lemma, in hydras.Epsilon0.T1]
+LT4 [lemma, in hydras.Epsilon0.T1]
+Lt:109 [binder, in hydras.Hydra.Hydra_Lemmas]
+lt:12 [binder, in hydras.OrdinalNotations.ON_Generic]
+lt:129 [binder, in hydras.Hydra.Hydra_Lemmas]
+lt:13 [binder, in hydras.Prelude.Comparable]
+Lt:145 [binder, in hydras.Hydra.Hydra_Definitions]
+Lt:155 [binder, in hydras.Hydra.Hydra_Definitions]
+lt:2 [binder, in hydras.solutions_exercises.predSuccUnicity]
+Lt:2 [binder, in hydras.OrdinalNotations.ON_O]
+lt:2 [binder, in hydras.Prelude.MoreOrders]
+lt:2 [binder, in hydras.solutions_exercises.lt_succ_le]
+lt:2 [binder, in hydras.OrdinalNotations.ON_Generic]
+lt:23 [binder, in hydras.Prelude.MoreOrders]
+lt:3 [binder, in hydras.Schutte.Lub]
+lt:43 [binder, in hydras.Prelude.MoreOrders]
+lt:47 [binder, in hydras.Prelude.MoreOrders]
+lt:5 [binder, in hydras.Prelude.Comparable]
+Lt:52 [binder, in hydras.Schutte.Well_Orders]
+Lt:57 [binder, in hydras.Schutte.Well_Orders]
+lt:8 [binder, in hydras.OrdinalNotations.ON_Generic]
+lt:9 [binder, in hydras.Schutte.Lub]
+Lt:99 [binder, in hydras.Hydra.Hydra_Lemmas]
+lub [definition, in hydras.Gamma0.Gamma0]
+Lub [library]
+lub_mono [lemma, in hydras.Gamma0.Gamma0]
+lub_unicity [lemma, in hydras.Gamma0.Gamma0]
+Lvar [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+L_3_plus_omega [definition, in hydras.OrdinalNotations.ON_Omega2]
+L_omega_cube_3_eq [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_cube_eqn [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_cube_ok [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_cube [definition, in hydras.Epsilon0.Large_Sets]
+L_omega_square_times_ok [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_square_times [definition, in hydras.Epsilon0.Large_Sets]
+L_phi0_mult_Sle [lemma, in hydras.Epsilon0.Large_Sets]
+L_phi0_mult_smono [lemma, in hydras.Epsilon0.Large_Sets]
+L_phi0_mult_ok [lemma, in hydras.Epsilon0.Large_Sets]
+L_phi0_mult [definition, in hydras.Epsilon0.Large_Sets]
+L_omega_square_ok [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_square_smono [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_square_Sle [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_square_eqn [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_square_eqn1 [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_square [definition, in hydras.Epsilon0.Large_Sets]
+L_omega_mult_eqn [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_mult_ok [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_mult_Sle [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_mult [definition, in hydras.Epsilon0.Large_Sets]
+L_omega_ok [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_smono [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega_Sle [lemma, in hydras.Epsilon0.Large_Sets]
+L_omega [definition, in hydras.Epsilon0.Large_Sets]
+L_fin_ok [lemma, in hydras.Epsilon0.Large_Sets]
+L_S_succ_rw [lemma, in hydras.Epsilon0.Large_Sets]
+L_fin_smono [lemma, in hydras.Epsilon0.Large_Sets]
+L_finS_S_le [lemma, in hydras.Epsilon0.Large_Sets]
+L_fin [definition, in hydras.Epsilon0.Large_Sets]
+L_lim_ok [lemma, in hydras.Epsilon0.Large_Sets]
+L_lim [definition, in hydras.Epsilon0.Large_Sets]
+L_succ_ok [lemma, in hydras.Epsilon0.Large_Sets]
+L_succ_Sle [lemma, in hydras.Epsilon0.Large_Sets]
+L_succ_mono [lemma, in hydras.Epsilon0.Large_Sets]
+L_succ [definition, in hydras.Epsilon0.Large_Sets]
+L_spec_unicity [lemma, in hydras.Epsilon0.Large_Sets]
+L_pos_inv [lemma, in hydras.Epsilon0.Large_Sets]
+L_zero_inv [lemma, in hydras.Epsilon0.Large_Sets]
+L_spec_compat [lemma, in hydras.Epsilon0.Large_Sets]
+L_spec_inv2 [lemma, in hydras.Epsilon0.Large_Sets]
+L_test [definition, in hydras.Epsilon0.Large_Sets]
+L_spec1 [constructor, in hydras.Epsilon0.Large_Sets]
+L_spec0 [constructor, in hydras.Epsilon0.Large_Sets]
+L_spec [inductive, in hydras.Epsilon0.Large_Sets]
+L_correct [lemma, in hydras.Epsilon0.L_alpha]
+L_ok [lemma, in hydras.Epsilon0.L_alpha]
+L_ok_lim [lemma, in hydras.Epsilon0.L_alpha]
+L_ok_succ [lemma, in hydras.Epsilon0.L_alpha]
+L_ok0 [lemma, in hydras.Epsilon0.L_alpha]
+L_correct_proof.P [variable, in hydras.Epsilon0.L_alpha]
+L_correct_proof [section, in hydras.Epsilon0.L_alpha]
+L_succ_ok [lemma, in hydras.Epsilon0.L_alpha]
+L_ge_S [lemma, in hydras.Epsilon0.L_alpha]
+L_ge_id [lemma, in hydras.Epsilon0.L_alpha]
+L_omega [lemma, in hydras.Epsilon0.L_alpha]
+L_finite [lemma, in hydras.Epsilon0.L_alpha]
+L_lim_eqn [lemma, in hydras.Epsilon0.L_alpha]
+L_succ_eqn [lemma, in hydras.Epsilon0.L_alpha]
+L_eq2 [lemma, in hydras.Epsilon0.L_alpha]
+L_zero_eqn [lemma, in hydras.Epsilon0.L_alpha]
+L_ [definition, in hydras.Epsilon0.L_alpha]
+L_3_plus_omega [instance, in hydras.OrdinalNotations.Example_3PlusOmega]
+l_std_ok [lemma, in hydras.Hydra.Battle_length]
+l_std [definition, in hydras.Hydra.Battle_length]
+L_0_3 [lemma, in hydras.Hydra.BigBattle]
+L_2_3 [lemma, in hydras.Hydra.BigBattle]
+L_0_2 [lemma, in hydras.Hydra.BigBattle]
+l_std_ok [lemma, in gaia_hydras.GHydra]
+l_stdE [lemma, in gaia_hydras.GHydra]
+l_std [definition, in gaia_hydras.GHydra]
+L_fin_ok [lemma, in gaia_hydras.GLarge_Sets]
+L_spec [definition, in gaia_hydras.GLarge_Sets]
+L_3_plus_omega [definition, in hydras.Epsilon0.E0]
+L_correct [lemma, in gaia_hydras.GL_alpha]
+L_pos_inv [lemma, in gaia_hydras.GL_alpha]
+L_spec1 [lemma, in gaia_hydras.GL_alpha]
+L_spec0 [lemma, in gaia_hydras.GL_alpha]
+L_spec [definition, in gaia_hydras.GL_alpha]
+L_ge_S [lemma, in gaia_hydras.GL_alpha]
+L_ge_id [lemma, in gaia_hydras.GL_alpha]
+L_omega [lemma, in gaia_hydras.GL_alpha]
+L_finite [lemma, in gaia_hydras.GL_alpha]
+L_limE [lemma, in gaia_hydras.GL_alpha]
+L_succE [lemma, in gaia_hydras.GL_alpha]
+L_eq2 [lemma, in gaia_hydras.GL_alpha]
+L_zeroE [lemma, in gaia_hydras.GL_alpha]
+L_ [definition, in gaia_hydras.GL_alpha]
+L_3_plus_omega [definition, in hydras.Schutte.Addition]
+L_alpha [library]
+l'':55 [binder, in hydras.solutions_exercises.MultisetWf]
+l'':56 [binder, in hydras.solutions_exercises.MultisetWf]
+l'':58 [binder, in hydras.solutions_exercises.MultisetWf]
+l'1:118 [binder, in hydras.Prelude.MoreLists]
+l'1:150 [binder, in hydras.Prelude.MoreLists]
+l':105 [binder, in hydras.Prelude.Merge_Sort]
+l':114 [binder, in hydras.Prelude.MoreLists]
+l':123 [binder, in hydras.Prelude.MoreLists]
+l':13 [binder, in hydras.solutions_exercises.MultisetWf]
+l':130 [binder, in hydras.Prelude.MoreLists]
+l':135 [binder, in hydras.Prelude.MoreLists]
+l':139 [binder, in hydras.Schutte.Schutte_basics]
+l':14 [binder, in gaia_hydras.GLarge_Sets]
+l':140 [binder, in hydras.Prelude.MoreLists]
+l':143 [binder, in hydras.rpo.rpo]
+l':146 [binder, in hydras.Prelude.MoreLists]
+l':147 [binder, in hydras.rpo.rpo]
+l':15 [binder, in hydras.Hydra.Hydra_Definitions]
+l':151 [binder, in hydras.rpo.rpo]
+l':158 [binder, in hydras.rpo.rpo]
+l':161 [binder, in hydras.rpo.rpo]
+l':167 [binder, in hydras.rpo.rpo]
+l':171 [binder, in hydras.rpo.rpo]
+l':19 [binder, in hydras.solutions_exercises.MultisetWf]
+l':192 [binder, in hydras.rpo.more_list]
+l':20 [binder, in additions.Trace_exercise]
+l':218 [binder, in hydras.rpo.rpo]
+l':222 [binder, in hydras.rpo.rpo]
+l':224 [binder, in hydras.rpo.rpo]
+l':23 [binder, in hydras.solutions_exercises.MultisetWf]
+l':235 [binder, in hydras.rpo.rpo]
+l':244 [binder, in hydras.rpo.rpo]
+l':267 [binder, in hydras.rpo.rpo]
+l':27 [binder, in hydras.Epsilon0.Large_Sets]
+l':30 [binder, in hydras.Prelude.DecPreOrder_Instances]
+l':34 [binder, in hydras.Prelude.Merge_Sort]
+l':37 [binder, in hydras.Prelude.Sort_spec]
+l':41 [binder, in hydras.Prelude.Merge_Sort]
+l':43 [binder, in hydras.Prelude.Merge_Sort]
+l':46 [binder, in hydras.Prelude.Merge_Sort]
+l':48 [binder, in hydras.Hydra.O2H]
+l':48 [binder, in hydras.rpo.rpo]
+l':52 [binder, in hydras.solutions_exercises.MultisetWf]
+l':52 [binder, in hydras.Prelude.Merge_Sort]
+l':52 [binder, in hydras.rpo.rpo]
+l':56 [binder, in hydras.rpo.rpo]
+l':57 [binder, in hydras.Schutte.CNF]
+l':614 [binder, in hydras.Gamma0.Gamma0]
+l':618 [binder, in hydras.Gamma0.Gamma0]
+l':63 [binder, in hydras.rpo.rpo]
+l':66 [binder, in hydras.rpo.rpo]
+l':699 [binder, in hydras.Epsilon0.T1]
+l':72 [binder, in hydras.rpo.rpo]
+l':73 [binder, in hydras.solutions_exercises.MultisetWf]
+l':8 [binder, in hydras.Prelude.Sort_spec]
+l':89 [binder, in hydras.Ackermann.primRec]
+l':93 [binder, in hydras.Ackermann.primRec]
+l':97 [binder, in hydras.Prelude.Merge_Sort]
+l':99 [binder, in hydras.Prelude.Merge_Sort]
+L0 [lemma, in hydras.Epsilon0.T1]
+L0 [lemma, in additions.AM]
+L0 [lemma, in hydras.solutions_exercises.FibonacciPR]
+L0_95 [lemma, in hydras.Hydra.BigBattle]
+L00 [lemma, in hydras.Epsilon0.F_omega]
+L02 [lemma, in hydras.Epsilon0.F_omega]
+L03 [lemma, in hydras.Epsilon0.T1]
+L04 [lemma, in hydras.solutions_exercises.F_3]
+L04' [lemma, in hydras.solutions_exercises.F_3]
+L05 [lemma, in hydras.Epsilon0.T1]
+L06 [lemma, in hydras.Hydra.Battle_length]
+l0:152 [binder, in hydras.Prelude.MoreLists]
+l0:296 [binder, in hydras.Ackermann.codeFreeVar]
+l0:298 [binder, in hydras.Ackermann.codeFreeVar]
+l0:298 [binder, in hydras.Ackermann.codeList]
+l0:300 [binder, in hydras.Ackermann.codeList]
+L1 [lemma, in hydras.Epsilon0.T1]
+L1 [lemma, in hydras.solutions_exercises.predSuccUnicity]
+L1 [lemma, in hydras.MoreAck.expressibleExamples]
+L1 [lemma, in hydras.solutions_exercises.Limit_Infinity]
+L1 [lemma, in hydras.solutions_exercises.FibonacciPR]
+L1 [lemma, in hydras.solutions_exercises.lt_succ_le]
+L1' [lemma, in gaia_hydras.T1Bridge]
+L10 [lemma, in hydras.Epsilon0.T1]
+L10 [lemma, in hydras.Hydra.BigBattle]
+L11 [lemma, in hydras.Epsilon0.T1]
+L12 [lemma, in hydras.Epsilon0.T1]
+L13 [lemma, in hydras.Epsilon0.T1]
+L14 [lemma, in hydras.Epsilon0.T1]
+l1:100 [binder, in hydras.Prelude.Merge_Sort]
+l1:101 [binder, in hydras.rpo.list_set]
+l1:101 [binder, in hydras.rpo.list_permut]
+l1:104 [binder, in hydras.Ackermann.primRec]
+l1:107 [binder, in hydras.rpo.list_set]
+l1:108 [binder, in hydras.Prelude.Merge_Sort]
+l1:11 [binder, in hydras.rpo.list_permut]
+l1:110 [binder, in hydras.rpo.more_list]
+l1:114 [binder, in hydras.rpo.list_set]
+l1:117 [binder, in hydras.Prelude.MoreLists]
+l1:119 [binder, in hydras.rpo.list_set]
+l1:12 [binder, in hydras.Prelude.Merge_Sort]
+l1:122 [binder, in hydras.rpo.list_set]
+l1:123 [binder, in hydras.Prelude.Merge_Sort]
+l1:126 [binder, in hydras.rpo.more_list]
+l1:127 [binder, in hydras.rpo.list_set]
+l1:13 [binder, in hydras.rpo.list_permut]
+l1:131 [binder, in hydras.Prelude.Merge_Sort]
+l1:134 [binder, in hydras.rpo.list_set]
+l1:138 [binder, in hydras.Prelude.Merge_Sort]
+l1:141 [binder, in additions.fib]
+l1:141 [binder, in hydras.rpo.list_set]
+l1:145 [binder, in hydras.rpo.list_set]
+l1:146 [binder, in hydras.Prelude.Merge_Sort]
+l1:149 [binder, in hydras.Prelude.MoreLists]
+l1:149 [binder, in hydras.rpo.list_set]
+l1:157 [binder, in hydras.Prelude.Merge_Sort]
+l1:161 [binder, in hydras.Prelude.Merge_Sort]
+l1:165 [binder, in hydras.Prelude.Merge_Sort]
+l1:17 [binder, in hydras.Prelude.Merge_Sort]
+l1:170 [binder, in hydras.Prelude.Merge_Sort]
+l1:178 [binder, in hydras.Prelude.MoreLists]
+l1:18 [binder, in hydras.rpo.more_list]
+l1:18 [binder, in hydras.rpo.dickson]
+l1:181 [binder, in hydras.Prelude.Merge_Sort]
+l1:184 [binder, in hydras.rpo.more_list]
+l1:185 [binder, in hydras.Ackermann.fol]
+l1:186 [binder, in hydras.rpo.term]
+l1:191 [binder, in hydras.Prelude.Merge_Sort]
+l1:191 [binder, in hydras.Ackermann.fol]
+l1:195 [binder, in hydras.Prelude.MoreLists]
+l1:197 [binder, in hydras.Prelude.MoreLists]
+l1:197 [binder, in hydras.rpo.more_list]
+l1:198 [binder, in hydras.Prelude.Merge_Sort]
+l1:199 [binder, in hydras.rpo.more_list]
+l1:203 [binder, in hydras.Prelude.MoreLists]
+l1:21 [binder, in hydras.rpo.list_permut]
+l1:212 [binder, in hydras.Prelude.MoreLists]
+l1:219 [binder, in hydras.Prelude.MoreLists]
+l1:22 [binder, in hydras.rpo.dickson]
+l1:23 [binder, in hydras.rpo.list_permut]
+l1:233 [binder, in hydras.rpo.list_set]
+l1:234 [binder, in hydras.rpo.term]
+l1:236 [binder, in hydras.rpo.term]
+l1:236 [binder, in hydras.rpo.list_set]
+l1:239 [binder, in hydras.rpo.list_set]
+l1:243 [binder, in hydras.rpo.list_set]
+l1:248 [binder, in hydras.rpo.term]
+l1:250 [binder, in hydras.rpo.term]
+l1:254 [binder, in hydras.rpo.term]
+l1:257 [binder, in hydras.rpo.list_set]
+l1:26 [binder, in hydras.rpo.dickson]
+l1:27 [binder, in hydras.rpo.list_permut]
+l1:3 [binder, in additions.Trace_exercise]
+l1:30 [binder, in hydras.rpo.dickson]
+l1:30 [binder, in hydras.rpo.list_permut]
+l1:324 [binder, in hydras.rpo.rpo]
+l1:327 [binder, in hydras.rpo.rpo]
+l1:34 [binder, in hydras.rpo.dickson]
+l1:34 [binder, in hydras.rpo.list_permut]
+l1:368 [binder, in hydras.rpo.term]
+l1:37 [binder, in hydras.rpo.list_permut]
+l1:38 [binder, in hydras.Prelude.Merge_Sort]
+l1:39 [binder, in hydras.rpo.dickson]
+l1:4 [binder, in hydras.rpo.dickson]
+l1:4 [binder, in hydras.rpo.list_permut]
+l1:43 [binder, in hydras.Ackermann.codeList]
+l1:45 [binder, in hydras.rpo.list_permut]
+l1:48 [binder, in hydras.rpo.more_list]
+l1:48 [binder, in hydras.rpo.list_permut]
+l1:49 [binder, in hydras.Prelude.Merge_Sort]
+l1:5 [binder, in hydras.Prelude.Merge_Sort]
+l1:51 [binder, in hydras.rpo.list_permut]
+l1:53 [binder, in hydras.rpo.list_permut]
+l1:56 [binder, in hydras.Prelude.Merge_Sort]
+l1:56 [binder, in additions.fib]
+l1:56 [binder, in hydras.rpo.list_permut]
+l1:57 [binder, in hydras.rpo.term]
+l1:59 [binder, in hydras.rpo.term]
+l1:59 [binder, in hydras.rpo.list_permut]
+l1:61 [binder, in additions.fib]
+l1:62 [binder, in hydras.rpo.list_permut]
+l1:67 [binder, in hydras.rpo.list_permut]
+l1:69 [binder, in hydras.rpo.more_list]
+l1:70 [binder, in hydras.rpo.list_permut]
+l1:72 [binder, in hydras.rpo.term]
+l1:74 [binder, in hydras.rpo.term]
+l1:76 [binder, in hydras.rpo.list_permut]
+l1:78 [binder, in hydras.rpo.term]
+l1:79 [binder, in hydras.rpo.list_set]
+l1:82 [binder, in hydras.rpo.list_set]
+l1:85 [binder, in hydras.rpo.list_permut]
+l1:86 [binder, in hydras.rpo.more_list]
+l1:9 [binder, in hydras.rpo.more_list]
+l1:91 [binder, in hydras.Prelude.Merge_Sort]
+l1:91 [binder, in hydras.rpo.list_set]
+l1:93 [binder, in hydras.rpo.list_permut]
+l1:94 [binder, in hydras.rpo.more_list]
+l1:95 [binder, in hydras.rpo.list_set]
+l1:99 [binder, in hydras.Ackermann.primRec]
+l1:99 [binder, in additions.fib]
+L2 [lemma, in hydras.Epsilon0.T1]
+L2 [lemma, in hydras.solutions_exercises.predSuccUnicity]
+L2 [lemma, in hydras.MoreAck.expressibleExamples]
+L2 [lemma, in hydras.solutions_exercises.Limit_Infinity]
+L2 [lemma, in hydras.Epsilon0.F_omega]
+L2 [lemma, in hydras.solutions_exercises.FibonacciPR]
+L2 [lemma, in hydras.solutions_exercises.F_3]
+L2 [lemma, in additions.Addition_Chains]
+l2hs [definition, in hydras.Hydra.Hydra_Definitions]
+l2hsK [lemma, in hydras.Hydra.Hydra_Definitions]
+L2_95_S [lemma, in hydras.Hydra.BigBattle]
+L2_95 [lemma, in hydras.Hydra.BigBattle]
+L2_6_2 [lemma, in hydras.Epsilon0.Paths]
+l2':253 [binder, in hydras.rpo.list_set]
+l2':254 [binder, in hydras.rpo.list_set]
+L22 [lemma, in hydras.Hydra.BigBattle]
+l2:10 [binder, in hydras.rpo.more_list]
+l2:100 [binder, in hydras.Ackermann.primRec]
+l2:100 [binder, in additions.fib]
+l2:101 [binder, in hydras.Prelude.Merge_Sort]
+l2:102 [binder, in hydras.rpo.list_set]
+l2:102 [binder, in hydras.rpo.list_permut]
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+l:36 [binder, in hydras.Ackermann.subAll]
+l:36 [binder, in hydras.Ackermann.primRec]
+l:36 [binder, in hydras.rpo.dickson]
+l:360 [binder, in hydras.rpo.term]
+l:364 [binder, in hydras.rpo.term]
+l:366 [binder, in hydras.rpo.term]
+l:37 [binder, in hydras.Ackermann.Languages]
+l:37 [binder, in hydras.Ackermann.codePA]
+l:38 [binder, in hydras.rpo.term]
+l:38 [binder, in hydras.rpo.more_list]
+l:39 [binder, in hydras.Ackermann.Languages]
+l:4 [binder, in hydras.rpo.more_list]
+l:4 [binder, in hydras.Prelude.Sort_spec]
+l:4 [binder, in hydras.Schutte.CNF]
+l:40 [binder, in hydras.Ackermann.subAll]
+l:40 [binder, in hydras.solutions_exercises.MultisetWf]
+l:40 [binder, in hydras.Prelude.Merge_Sort]
+l:40 [binder, in hydras.Hydra.Hydra_Lemmas]
+l:40 [binder, in hydras.Ackermann.codePA]
+l:41 [binder, in hydras.rpo.list_set]
+l:411 [binder, in hydras.Epsilon0.Paths]
+l:42 [binder, in hydras.Prelude.Merge_Sort]
+l:42 [binder, in hydras.Ackermann.codePA]
+l:42 [binder, in hydras.rpo.rpo]
+l:43 [binder, in hydras.Prelude.Sort_spec]
+l:43 [binder, in hydras.solutions_exercises.MultisetWf]
+l:43 [binder, in hydras.MoreAck.FolExamples]
+l:44 [binder, in hydras.rpo.term]
+l:44 [binder, in hydras.rpo.list_permut]
+l:44 [binder, in hydras.Ackermann.codePA]
+l:440 [binder, in hydras.Epsilon0.Paths]
+l:444 [binder, in hydras.Epsilon0.Paths]
+l:45 [binder, in hydras.rpo.term]
+l:45 [binder, in hydras.rpo.more_list]
+l:45 [binder, in hydras.Prelude.Merge_Sort]
+l:45 [binder, in hydras.Ackermann.primRec]
+l:45 [binder, in hydras.Hydra.Hydra_Lemmas]
+l:46 [binder, in hydras.Ackermann.codePA]
+l:47 [binder, in hydras.Hydra.O2H]
+l:47 [binder, in hydras.rpo.list_set]
+l:47 [binder, in hydras.rpo.list_permut]
+l:47 [binder, in hydras.rpo.rpo]
+l:477 [binder, in hydras.Epsilon0.Paths]
+l:48 [binder, in hydras.solutions_exercises.MultisetWf]
+l:48 [binder, in hydras.Ackermann.codePA]
+l:484 [binder, in hydras.Epsilon0.Paths]
+l:488 [binder, in hydras.Epsilon0.Paths]
+l:49 [binder, in gaia_hydras.GPaths]
+l:49 [binder, in hydras.Hydra.Hydra_Lemmas]
+l:5 [binder, in hydras.Ackermann.codeSubFormula]
+l:5 [binder, in hydras.Ackermann.codePA]
+l:50 [binder, in additions.fib]
+l:50 [binder, in hydras.rpo.list_set]
+l:50 [binder, in hydras.Epsilon0.Paths]
+l:50 [binder, in hydras.rpo.list_permut]
+l:50 [binder, in hydras.Ackermann.codePA]
+l:51 [binder, in hydras.solutions_exercises.MultisetWf]
+l:51 [binder, in hydras.Prelude.Merge_Sort]
+l:51 [binder, in hydras.rpo.rpo]
+l:52 [binder, in hydras.Prelude.MoreLists]
+l:52 [binder, in hydras.rpo.more_list]
+l:52 [binder, in gaia_hydras.GPaths]
+l:52 [binder, in hydras.Schutte.CNF]
+l:52 [binder, in hydras.Ackermann.codePA]
+l:53 [binder, in hydras.Schutte.CNF]
+l:53 [binder, in additions.fib]
+l:54 [binder, in hydras.Ackermann.codePA]
+l:547 [binder, in hydras.Epsilon0.Paths]
+l:55 [binder, in hydras.Schutte.CNF]
+l:55 [binder, in hydras.Ackermann.primRec]
+l:55 [binder, in hydras.rpo.rpo]
+l:551 [binder, in hydras.Epsilon0.Paths]
+l:56 [binder, in hydras.Prelude.MoreLists]
+l:56 [binder, in hydras.Prelude.Sort_spec]
+l:56 [binder, in hydras.rpo.list_set]
+l:56 [binder, in hydras.Ackermann.codePA]
+l:58 [binder, in additions.fib]
+l:58 [binder, in hydras.rpo.list_permut]
+l:58 [binder, in hydras.Ackermann.codePA]
+l:59 [binder, in hydras.rpo.more_list]
+l:59 [binder, in hydras.Schutte.CNF]
+l:6 [binder, in hydras.rpo.dickson]
+l:60 [binder, in hydras.Schutte.CNF]
+l:60 [binder, in hydras.Ackermann.codePA]
+l:61 [binder, in hydras.Prelude.MoreLists]
+l:61 [binder, in hydras.solutions_exercises.MultisetWf]
+l:61 [binder, in hydras.rpo.list_permut]
+l:61 [binder, in hydras.Ackermann.codePA]
+l:613 [binder, in hydras.Gamma0.Gamma0]
+l:617 [binder, in hydras.Gamma0.Gamma0]
+l:62 [binder, in hydras.Prelude.MoreLists]
+l:62 [binder, in hydras.Schutte.Well_Orders]
+l:62 [binder, in hydras.Ackermann.codePA]
+l:62 [binder, in hydras.rpo.rpo]
+l:63 [binder, in hydras.Prelude.MoreLists]
+l:63 [binder, in hydras.Ackermann.codePA]
+l:64 [binder, in hydras.solutions_exercises.MultisetWf]
+l:64 [binder, in hydras.rpo.list_set]
+l:65 [binder, in hydras.rpo.rpo]
+l:66 [binder, in hydras.rpo.more_list]
+l:66 [binder, in hydras.solutions_exercises.MultisetWf]
+l:67 [binder, in hydras.Prelude.MoreLists]
+l:67 [binder, in hydras.rpo.more_list]
+l:67 [binder, in hydras.Prelude.Sort_spec]
+l:67 [binder, in hydras.solutions_exercises.MultisetWf]
+l:68 [binder, in hydras.Ackermann.folProp]
+l:69 [binder, in hydras.Schutte.Correctness_E0]
+l:69 [binder, in hydras.Prelude.Sort_spec]
+l:69 [binder, in hydras.Ackermann.folProp]
+l:691 [binder, in hydras.Epsilon0.T1]
+l:695 [binder, in hydras.Epsilon0.T1]
+l:698 [binder, in hydras.Epsilon0.T1]
+L:7 [binder, in hydras.MoreAck.BadSubst]
+l:7 [binder, in hydras.Prelude.Sort_spec]
+l:7 [binder, in hydras.solutions_exercises.OnCodeList]
+l:7 [binder, in hydras.Prelude.Merge_Sort]
+l:7 [binder, in hydras.Ackermann.ListExt]
+l:70 [binder, in hydras.solutions_exercises.MultisetWf]
+l:70 [binder, in hydras.rpo.list_set]
+l:71 [binder, in hydras.rpo.rpo]
+l:72 [binder, in hydras.solutions_exercises.MultisetWf]
+l:73 [binder, in hydras.Prelude.MoreLists]
+l:73 [binder, in hydras.Ackermann.folProp]
+l:74 [binder, in hydras.rpo.list_set]
+l:75 [binder, in hydras.Prelude.MoreLists]
+l:76 [binder, in hydras.rpo.more_list]
+l:76 [binder, in hydras.rpo.list_set]
+l:77 [binder, in hydras.rpo.more_list]
+l:77 [binder, in additions.fib]
+l:77 [binder, in hydras.rpo.list_set]
+l:78 [binder, in hydras.Prelude.Sort_spec]
+l:794 [binder, in hydras.Ackermann.checkPrf]
+l:8 [binder, in hydras.Schutte.CNF]
+l:8 [binder, in hydras.Ackermann.codeSubFormula]
+l:8 [binder, in hydras.solutions_exercises.MultisetWf]
+l:8 [binder, in hydras.Prelude.Merge_Sort]
+l:8 [binder, in hydras.rpo.list_permut]
+l:8 [binder, in hydras.Ackermann.codePA]
+l:80 [binder, in hydras.Prelude.Merge_Sort]
+l:800 [binder, in hydras.Ackermann.checkPrf]
+l:805 [binder, in hydras.Ackermann.checkPrf]
+l:809 [binder, in hydras.Ackermann.checkPrf]
+l:82 [binder, in hydras.rpo.more_list]
+l:82 [binder, in hydras.Prelude.Merge_Sort]
+l:83 [binder, in hydras.Prelude.Merge_Sort]
+l:835 [binder, in hydras.Ackermann.checkPrf]
+l:84 [binder, in hydras.Hydra.Hydra_Lemmas]
+l:847 [binder, in hydras.Ackermann.checkPrf]
+l:85 [binder, in hydras.Prelude.Merge_Sort]
+l:86 [binder, in hydras.Prelude.Merge_Sort]
+l:86 [binder, in hydras.Hydra.Hydra_Lemmas]
+l:86 [binder, in hydras.rpo.list_set]
+l:860 [binder, in hydras.Ackermann.checkPrf]
+l:864 [binder, in gaia_hydras.nfwfgaia]
+l:865 [binder, in hydras.Ackermann.checkPrf]
+l:87 [binder, in hydras.rpo.term]
+l:87 [binder, in hydras.Ackermann.primRec]
+l:87 [binder, in hydras.Gamma0.Gamma0]
+l:873 [binder, in gaia_hydras.nfwfgaia]
+l:878 [binder, in hydras.Ackermann.checkPrf]
+l:88 [binder, in hydras.rpo.list_set]
+l:881 [binder, in hydras.Ackermann.checkPrf]
+l:9 [binder, in hydras.Hydra.O2H]
+l:9 [binder, in hydras.Prelude.First_toggle]
+l:92 [binder, in hydras.Ackermann.primRec]
+l:93 [binder, in hydras.Prelude.MoreLists]
+l:93 [binder, in hydras.rpo.more_list]
+l:94 [binder, in hydras.Hydra.O2H]
+l:95 [binder, in hydras.Prelude.Merge_Sort]
+l:96 [binder, in hydras.Prelude.Merge_Sort]
+l:96 [binder, in hydras.Ackermann.primRec]
+l:98 [binder, in hydras.Prelude.Merge_Sort]
+l:99 [binder, in hydras.rpo.term]
+

M

+m [definition, in hydras.Hydra.Hydra_Termination]
+M [definition, in hydras.Hydra.BigBattle]
+m [definition, in gaia_hydras.GHydra]
+majorAnyPR [lemma, in hydras.MoreAck.AckNotPR]
+majorized [definition, in hydras.MoreAck.AckNotPR]
+majorizedPR [definition, in hydras.MoreAck.AckNotPR]
+majorizedS [definition, in hydras.MoreAck.AckNotPR]
+majorizedSPR [definition, in hydras.MoreAck.AckNotPR]
+majorProjection [lemma, in hydras.MoreAck.AckNotPR]
+majorPR1 [lemma, in hydras.MoreAck.AckNotPR]
+majorPR2 [lemma, in hydras.MoreAck.AckNotPR]
+majorPR2_strict [lemma, in hydras.MoreAck.AckNotPR]
+majorSucc [lemma, in hydras.MoreAck.AckNotPR]
+majorZero [lemma, in hydras.MoreAck.AckNotPR]
+Make [module, in hydras.rpo.term]
+Make [module, in hydras.rpo.list_set]
+Make [module, in hydras.rpo.dickson]
+Make [module, in hydras.rpo.list_permut]
+Make [module, in hydras.rpo.rpo]
+makeTrace [definition, in hydras.Ackermann.codeSubFormula]
+makeTraceCorrect [lemma, in hydras.Ackermann.codeSubFormula]
+makeTraceForall [definition, in hydras.Ackermann.codeSubFormula]
+makeTraceForallNice [lemma, in hydras.Ackermann.codeSubFormula]
+makeTraceImp [definition, in hydras.Ackermann.codeSubFormula]
+makeTraceImpNice [lemma, in hydras.Ackermann.codeSubFormula]
+makeTraceNot [definition, in hydras.Ackermann.codeSubFormula]
+makeTraceNotNice [lemma, in hydras.Ackermann.codeSubFormula]
+makeTrace1 [lemma, in hydras.Ackermann.codeSubFormula]
+makeTrace2 [lemma, in hydras.Ackermann.codeSubFormula]
+make_chain [definition, in additions.AM]
+make_chain_correct [lemma, in additions.Euclidean_Chains]
+make_chain [definition, in additions.Euclidean_Chains]
+Make.acc_build [lemma, in hydras.rpo.rpo]
+Make.acc_lex_drop_proof [lemma, in hydras.rpo.rpo]
+Make.ac_syntactic [lemma, in hydras.rpo.list_permut]
+Make.ac_syntactic_aux [lemma, in hydras.rpo.list_permut]
+Make.add [definition, in hydras.rpo.list_set]
+Make.add_comm [lemma, in hydras.rpo.list_set]
+Make.add_without_red [definition, in hydras.rpo.list_set]
+Make.add_12 [lemma, in hydras.rpo.list_set]
+Make.add_2 [lemma, in hydras.rpo.list_set]
+Make.add_1 [lemma, in hydras.rpo.list_set]
+Make.add_prf [lemma, in hydras.rpo.list_set]
+Make.apply_subst [definition, in hydras.rpo.term]
+Make.build_list_of_SN_terms [definition, in hydras.rpo.rpo]
+Make.cardinal [definition, in hydras.rpo.list_set]
+Make.cardinal_eq_set [lemma, in hydras.rpo.list_set]
+Make.cardinal_union [lemma, in hydras.rpo.list_set]
+Make.cardinal_union_inter_12 [lemma, in hydras.rpo.list_set]
+Make.cardinal_union_2 [lemma, in hydras.rpo.list_set]
+Make.cardinal_union_1 [lemma, in hydras.rpo.list_set]
+Make.cardinal_subset [lemma, in hydras.rpo.list_set]
+Make.cons_permut_in [lemma, in hydras.rpo.list_permut]
+Make.context_multiset_extension_step_app2 [lemma, in hydras.rpo.dickson]
+Make.context_trans_clos_multiset_extension_step_app1 [lemma, in hydras.rpo.dickson]
+Make.context_multiset_extension_step_app1 [lemma, in hydras.rpo.dickson]
+Make.context_list_permut_app2 [lemma, in hydras.rpo.list_permut]
+Make.context_list_permut_app1 [lemma, in hydras.rpo.list_permut]
+Make.context_list_permut_cons [lemma, in hydras.rpo.list_permut]
+Make.DecVar [module, in hydras.rpo.term]
+Make.DecVar.A [definition, in hydras.rpo.term]
+Make.DecVar.eq_A_dec [lemma, in hydras.rpo.term]
+Make.dickson [lemma, in hydras.rpo.dickson]
+Make.dickson_aux3 [lemma, in hydras.rpo.dickson]
+Make.dickson_aux2 [lemma, in hydras.rpo.dickson]
+Make.dickson_aux1 [lemma, in hydras.rpo.dickson]
+Make.direct_subterm [definition, in hydras.rpo.term]
+Make.DoubleRecursion [section, in hydras.rpo.term]
+Make.DoubleRecursion.Pl2 [variable, in hydras.rpo.term]
+Make.DoubleRecursion.P2 [variable, in hydras.rpo.term]
+Make.DS [module, in hydras.rpo.list_set]
+Make.DS [module, in hydras.rpo.dickson]
+Make.DS [module, in hydras.rpo.list_permut]
+Make.elt [definition, in hydras.rpo.list_set]
+Make.elt [definition, in hydras.rpo.list_permut]
+Make.empty [definition, in hydras.rpo.list_set]
+Make.empty_subst_is_id_list [lemma, in hydras.rpo.term]
+Make.empty_subst_is_id [lemma, in hydras.rpo.term]
+Make.Eq [constructor, in hydras.rpo.rpo]
+Make.eq_term_dec [lemma, in hydras.rpo.term]
+Make.eq_set_list_permut_support [lemma, in hydras.rpo.list_set]
+Make.eq_set_trans [lemma, in hydras.rpo.list_set]
+Make.eq_set_sym [lemma, in hydras.rpo.list_set]
+Make.eq_set_refl [lemma, in hydras.rpo.list_set]
+Make.eq_set_dec [lemma, in hydras.rpo.list_set]
+Make.eq_set [definition, in hydras.rpo.list_set]
+Make.eq_elt_dec [definition, in hydras.rpo.list_set]
+Make.eq_elt_dec [definition, in hydras.rpo.list_permut]
+Make.F [module, in hydras.rpo.term]
+Make.filter [definition, in hydras.rpo.list_set]
+Make.filter_union [lemma, in hydras.rpo.list_set]
+Make.filter_2 [lemma, in hydras.rpo.list_set]
+Make.filter_2_list [lemma, in hydras.rpo.list_set]
+Make.filter_1 [lemma, in hydras.rpo.list_set]
+Make.filter_1_list [lemma, in hydras.rpo.list_set]
+Make.filter_aux [definition, in hydras.rpo.list_set]
+Make.included_remove_red [lemma, in hydras.rpo.list_set]
+Make.included_filter_aux [lemma, in hydras.rpo.list_set]
+Make.inter [definition, in hydras.rpo.list_set]
+Make.inter_12 [lemma, in hydras.rpo.list_set]
+Make.inter_12_aux [lemma, in hydras.rpo.list_set]
+Make.inter_2 [lemma, in hydras.rpo.list_set]
+Make.inter_2_aux [lemma, in hydras.rpo.list_set]
+Make.inter_1 [lemma, in hydras.rpo.list_set]
+Make.inter_1_aux [lemma, in hydras.rpo.list_set]
+Make.in_permut_in [lemma, in hydras.rpo.list_permut]
+Make.in_mult_S [lemma, in hydras.rpo.list_permut]
+Make.in_sn_sn [lemma, in hydras.rpo.rpo]
+Make.is_a_pos_exists_subtem [lemma, in hydras.rpo.term]
+Make.is_a_pos [definition, in hydras.rpo.term]
+Make.is_a_set [projection, in hydras.rpo.list_set]
+Make.lex1 [lemma, in hydras.rpo.rpo]
+Make.lex1_bis [lemma, in hydras.rpo.rpo]
+Make.lex2 [lemma, in hydras.rpo.rpo]
+Make.lex3 [lemma, in hydras.rpo.rpo]
+Make.list_permut_acc [lemma, in hydras.rpo.dickson]
+Make.list_permut_multiset_extension_step_2 [lemma, in hydras.rpo.dickson]
+Make.list_permut_multiset_extension_step_1 [lemma, in hydras.rpo.dickson]
+Make.list_permut_dec [lemma, in hydras.rpo.list_permut]
+Make.list_permut_length_2 [lemma, in hydras.rpo.list_permut]
+Make.list_permut_length_1 [lemma, in hydras.rpo.list_permut]
+Make.list_permut_map [lemma, in hydras.rpo.list_permut]
+Make.list_permut_size [lemma, in hydras.rpo.list_permut]
+Make.list_permut_length [lemma, in hydras.rpo.list_permut]
+Make.list_permut_remove_hd [lemma, in hydras.rpo.list_permut]
+Make.list_permut_app_app [lemma, in hydras.rpo.list_permut]
+Make.list_permut_add_cons_inside [lemma, in hydras.rpo.list_permut]
+Make.list_permut_add_inside [lemma, in hydras.rpo.list_permut]
+Make.list_permut_nil [lemma, in hydras.rpo.list_permut]
+Make.list_permut_trans [lemma, in hydras.rpo.list_permut]
+Make.list_permut_sym [lemma, in hydras.rpo.list_permut]
+Make.list_permut_refl [lemma, in hydras.rpo.list_permut]
+Make.list_permut [definition, in hydras.rpo.list_permut]
+Make.list_to_multiset [definition, in hydras.rpo.list_permut]
+Make.list_permut_map_acc [lemma, in hydras.rpo.rpo]
+Make.List_mul_rest_step [constructor, in hydras.rpo.rpo]
+Make.List_mul_rest [constructor, in hydras.rpo.rpo]
+Make.List_eq_rest [constructor, in hydras.rpo.rpo]
+Make.List_gt_rest [constructor, in hydras.rpo.rpo]
+Make.List_mul [constructor, in hydras.rpo.rpo]
+Make.List_eq [constructor, in hydras.rpo.rpo]
+Make.List_gt [constructor, in hydras.rpo.rpo]
+Make.LP [module, in hydras.rpo.list_set]
+Make.LP [module, in hydras.rpo.dickson]
+Make.LP [module, in hydras.rpo.rpo]
+Make.Lt [constructor, in hydras.rpo.rpo]
+Make.make_set [definition, in hydras.rpo.list_set]
+Make.map_subst [definition, in hydras.rpo.term]
+Make.mem [definition, in hydras.rpo.list_set]
+Make.mem_dec [lemma, in hydras.rpo.list_set]
+Make.mk_set [constructor, in hydras.rpo.list_set]
+Make.mk_sn [constructor, in hydras.rpo.rpo]
+Make.multiplicity_app [lemma, in hydras.rpo.list_permut]
+Make.multiset_closure [lemma, in hydras.rpo.dickson]
+Make.multiset_extension_step [inductive, in hydras.rpo.dickson]
+Make.out_mult_O [lemma, in hydras.rpo.list_permut]
+Make.o_size3_trans [lemma, in hydras.rpo.rpo]
+Make.o_size3 [definition, in hydras.rpo.rpo]
+Make.o_size2 [definition, in hydras.rpo.rpo]
+Make.o_size [definition, in hydras.rpo.rpo]
+Make.P [module, in hydras.rpo.rpo]
+Make.projection_list_of_SN_terms [lemma, in hydras.rpo.rpo]
+Make.Recursion [section, in hydras.rpo.term]
+Make.Recursion.P [variable, in hydras.rpo.term]
+Make.Recursion.Pl [variable, in hydras.rpo.term]
+Make.remove_not_common [definition, in hydras.rpo.list_set]
+Make.remove_red_included [lemma, in hydras.rpo.list_set]
+Make.remove_red [definition, in hydras.rpo.list_set]
+Make.remove_context_list_permut_app2 [lemma, in hydras.rpo.list_permut]
+Make.remove_context_list_permut_cons [lemma, in hydras.rpo.list_permut]
+Make.replace_at_pos_list_replace_at_pos_in_subterm [lemma, in hydras.rpo.term]
+Make.replace_at_pos_is_replace_at_pos2 [lemma, in hydras.rpo.term]
+Make.replace_at_pos_is_replace_at_pos1 [lemma, in hydras.rpo.term]
+Make.replace_at_pos_unfold [lemma, in hydras.rpo.term]
+Make.replace_at_pos_list [definition, in hydras.rpo.term]
+Make.replace_at_pos [definition, in hydras.rpo.term]
+Make.rmv_case [constructor, in hydras.rpo.dickson]
+Make.rpo [inductive, in hydras.rpo.rpo]
+Make.rpo_add_context [lemma, in hydras.rpo.rpo]
+Make.rpo_subst [lemma, in hydras.rpo.rpo]
+Make.rpo_term [definition, in hydras.rpo.rpo]
+Make.rpo_mul_trans_clos [lemma, in hydras.rpo.rpo]
+Make.rpo_mul_rest_step [inductive, in hydras.rpo.rpo]
+Make.rpo_mul_rest [inductive, in hydras.rpo.rpo]
+Make.rpo_lex_rest_same_length [lemma, in hydras.rpo.rpo]
+Make.rpo_lex_rest [inductive, in hydras.rpo.rpo]
+Make.rpo_rest [definition, in hydras.rpo.rpo]
+Make.rpo_trans [lemma, in hydras.rpo.rpo]
+Make.rpo_closure [lemma, in hydras.rpo.rpo]
+Make.rpo_subterm [lemma, in hydras.rpo.rpo]
+Make.rpo_lex_same_length [lemma, in hydras.rpo.rpo]
+Make.rpo_mul [inductive, in hydras.rpo.rpo]
+Make.rpo_lex [inductive, in hydras.rpo.rpo]
+Make.rpo_eq [inductive, in hydras.rpo.rpo]
+Make.singleton [definition, in hydras.rpo.list_set]
+Make.size [definition, in hydras.rpo.term]
+Make.size_subterm_at_pos [lemma, in hydras.rpo.term]
+Make.size_direct_subterm [lemma, in hydras.rpo.term]
+Make.size_ge_one [lemma, in hydras.rpo.term]
+Make.size_unfold [lemma, in hydras.rpo.term]
+Make.size2 [definition, in hydras.rpo.rpo]
+Make.size3 [definition, in hydras.rpo.rpo]
+Make.sn [projection, in hydras.rpo.rpo]
+Make.SN_term [record, in hydras.rpo.rpo]
+Make.subset [definition, in hydras.rpo.list_set]
+Make.subset_cardinal_not_eq_not_eq_set [lemma, in hydras.rpo.list_set]
+Make.subset_subset_union [lemma, in hydras.rpo.list_set]
+Make.subset_compat [lemma, in hydras.rpo.list_set]
+Make.subset_compat_2 [lemma, in hydras.rpo.list_set]
+Make.subset_compat_1 [lemma, in hydras.rpo.list_set]
+Make.subset_filter [lemma, in hydras.rpo.list_set]
+Make.subset_inter_2 [lemma, in hydras.rpo.list_set]
+Make.subset_inter_1 [lemma, in hydras.rpo.list_set]
+Make.subset_union_2 [lemma, in hydras.rpo.list_set]
+Make.subset_union_1 [lemma, in hydras.rpo.list_set]
+Make.subset_dec [lemma, in hydras.rpo.list_set]
+Make.substitution [definition, in hydras.rpo.term]
+Make.subst_comp_is_subst_comp [lemma, in hydras.rpo.term]
+Make.subst_comp_is_subst_comp_aux2 [lemma, in hydras.rpo.term]
+Make.subst_comp_is_subst_comp_aux1 [lemma, in hydras.rpo.term]
+Make.subst_comp [definition, in hydras.rpo.term]
+Make.Subterm [constructor, in hydras.rpo.rpo]
+Make.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos [lemma, in hydras.rpo.term]
+Make.subterm_at_pos [definition, in hydras.rpo.term]
+Make.support [projection, in hydras.rpo.list_set]
+Make.symbol [definition, in hydras.rpo.term]
+Make.t [record, in hydras.rpo.list_set]
+Make.T [module, in hydras.rpo.rpo]
+Make.Term [constructor, in hydras.rpo.term]
+Make.term [inductive, in hydras.rpo.term]
+Make.Term_eq_dec.eq_A_dec [definition, in hydras.rpo.term]
+Make.Term_eq_dec.A [definition, in hydras.rpo.term]
+Make.Term_eq_dec [module, in hydras.rpo.term]
+Make.term_rec8 [definition, in hydras.rpo.term]
+Make.term_rec7 [definition, in hydras.rpo.term]
+Make.term_rec4 [definition, in hydras.rpo.term]
+Make.term_rec3 [definition, in hydras.rpo.term]
+Make.term_rec2 [definition, in hydras.rpo.term]
+Make.Top_eq_mul [constructor, in hydras.rpo.rpo]
+Make.Top_eq_lex [constructor, in hydras.rpo.rpo]
+Make.Top_gt [constructor, in hydras.rpo.rpo]
+Make.tt [projection, in hydras.rpo.rpo]
+Make.two_cases [lemma, in hydras.rpo.dickson]
+Make.two_cases_rpo [lemma, in hydras.rpo.rpo]
+Make.union [definition, in hydras.rpo.list_set]
+Make.union_compat_eq_set [lemma, in hydras.rpo.list_set]
+Make.union_compat_subset_2 [lemma, in hydras.rpo.list_set]
+Make.union_compat_subset_1 [lemma, in hydras.rpo.list_set]
+Make.union_assoc [lemma, in hydras.rpo.list_set]
+Make.union_comm [lemma, in hydras.rpo.list_set]
+Make.union_empty_2 [lemma, in hydras.rpo.list_set]
+Make.union_empty_1 [lemma, in hydras.rpo.list_set]
+Make.union_12 [lemma, in hydras.rpo.list_set]
+Make.union_12_aux [lemma, in hydras.rpo.list_set]
+Make.union_2 [lemma, in hydras.rpo.list_set]
+Make.union_2_aux [lemma, in hydras.rpo.list_set]
+Make.union_1 [lemma, in hydras.rpo.list_set]
+Make.union_1_aux [lemma, in hydras.rpo.list_set]
+Make.Var [constructor, in hydras.rpo.term]
+Make.variable [definition, in hydras.rpo.term]
+Make.VSet [module, in hydras.rpo.term]
+Make.well_formed_apply_subst [lemma, in hydras.rpo.term]
+Make.well_formed_subst [definition, in hydras.rpo.term]
+Make.well_formed_list [definition, in hydras.rpo.term]
+Make.well_formed_fold [lemma, in hydras.rpo.term]
+Make.well_formed_unfold [lemma, in hydras.rpo.term]
+Make.well_formed [definition, in hydras.rpo.term]
+Make.wf_rpo [lemma, in hydras.rpo.rpo]
+Make.wf_rpo_term [lemma, in hydras.rpo.rpo]
+Make.wf_on_mul_rest [lemma, in hydras.rpo.rpo]
+Make.wf_on_lex_rest [lemma, in hydras.rpo.rpo]
+Make.wf_on_rest [lemma, in hydras.rpo.rpo]
+Make.wf_size3 [lemma, in hydras.rpo.rpo]
+Make.wf_size2 [lemma, in hydras.rpo.rpo]
+Make.wf_size [lemma, in hydras.rpo.rpo]
+Make.without_red_permut [lemma, in hydras.rpo.list_set]
+Make.without_red_remove_not_common [lemma, in hydras.rpo.list_set]
+Make.without_red_remove_not_common_aux [lemma, in hydras.rpo.list_set]
+Make.without_red_add_without_red [lemma, in hydras.rpo.list_set]
+Make.without_red_singleton [lemma, in hydras.rpo.list_set]
+Make.without_red_nil [lemma, in hydras.rpo.list_set]
+Make.without_red_add [lemma, in hydras.rpo.list_set]
+Make.without_red_remove [lemma, in hydras.rpo.list_set]
+Make.without_red_remove_red [lemma, in hydras.rpo.list_set]
+Make.without_red_filter_aux [lemma, in hydras.rpo.list_set]
+Make.without_red [definition, in hydras.rpo.list_set]
+Make.X [module, in hydras.rpo.term]
+mapListLemma [lemma, in hydras.Ackermann.codeSubFormula]
+map_without_repetition [definition, in hydras.rpo.more_list]
+map_app [lemma, in hydras.rpo.more_list]
+map_map [lemma, in hydras.rpo.more_list]
+map12_without_repetition [definition, in hydras.rpo.more_list]
+map:296 [binder, in hydras.Ackermann.subAll]
+map:340 [binder, in hydras.Ackermann.subAll]
+map:345 [binder, in hydras.Ackermann.subAll]
+map:350 [binder, in hydras.Ackermann.subAll]
+map:373 [binder, in hydras.Ackermann.subAll]
+mark [definition, in hydras.Prelude.Sort_spec]
+marked [inductive, in hydras.Prelude.Sort_spec]
+marked0 [constructor, in hydras.Prelude.Sort_spec]
+marked1 [constructor, in hydras.Prelude.Sort_spec]
+match2 [definition, in hydras.Prelude.MoreVectors]
+max [definition, in hydras.Prelude.Comparable]
+maxApp [lemma, in hydras.Ackermann.codeSubFormula]
+maximal_critical [definition, in hydras.Schutte.Critical]
+maxIsPR [instance, in hydras.Ackermann.primRec]
+maxLemma [lemma, in hydras.Ackermann.codeSubFormula]
+maxLemma2 [lemma, in hydras.Ackermann.codeSubFormula]
+maxLemma3 [lemma, in hydras.Ackermann.codeSubFormula]
+maxSubTerm [lemma, in hydras.Ackermann.codeSubFormula]
+maxSubTerms [lemma, in hydras.Ackermann.codeSubFormula]
+maxVarFreeVar [lemma, in hydras.Ackermann.codeSubFormula]
+max_nf [lemma, in hydras.Epsilon0.T1]
+max_le_regL [lemma, in hydras.Prelude.More_Arith]
+max_le_regR [lemma, in hydras.Prelude.More_Arith]
+max_le_plus [lemma, in hydras.Prelude.More_Arith]
+max_assoc [instance, in hydras.Prelude.Comparable]
+max_refl [lemma, in hydras.Prelude.Comparable]
+max_ge_b [lemma, in hydras.Prelude.Comparable]
+max_ge_a [lemma, in hydras.Prelude.Comparable]
+max_comm [lemma, in hydras.Prelude.Comparable]
+max_dec [lemma, in hydras.Prelude.Comparable]
+max_v_tl [lemma, in hydras.Prelude.MoreVectors]
+max_v_ge [lemma, in hydras.Prelude.MoreVectors]
+max_v_lub [lemma, in hydras.Prelude.MoreVectors]
+max_v_2 [lemma, in hydras.Prelude.MoreVectors]
+max_v [definition, in hydras.Prelude.MoreVectors]
+measure_lt [definition, in hydras.OrdinalNotations.ON_Generic]
+mem [definition, in hydras.Ackermann.fol]
+members [definition, in hydras.solutions_exercises.OnCodeList]
+members [definition, in hydras.Schutte.Schutte_basics]
+membersOk [lemma, in hydras.solutions_exercises.OnCodeList]
+membersOk' [lemma, in hydras.solutions_exercises.OnCodeList]
+members_omega [lemma, in hydras.Schutte.Schutte_basics]
+members_eq [lemma, in hydras.Schutte.Schutte_basics]
+members_proper [lemma, in hydras.Schutte.Ordering_Functions]
+mem_head_mult_inv [lemma, in hydras.Hydra.O2H]
+mem_head [inductive, in hydras.Hydra.O2H]
+merge [definition, in hydras.Prelude.Merge_Sort]
+merge_sort_correct [lemma, in hydras.Prelude.Merge_Sort]
+merge_permutation [lemma, in hydras.Prelude.Merge_Sort]
+merge_LocallySorted [lemma, in hydras.Prelude.Merge_Sort]
+merge_Forall [lemma, in hydras.Prelude.Merge_Sort]
+merge_equation [lemma, in hydras.Prelude.Merge_Sort]
+merge_rec [definition, in hydras.Prelude.Merge_Sort]
+merge_ind [definition, in hydras.Prelude.Merge_Sort]
+merge_rect [lemma, in hydras.Prelude.Merge_Sort]
+merge_aux:115 [binder, in hydras.Prelude.Merge_Sort]
+Merge_Sort [library]
+min [definition, in hydras.Prelude.Comparable]
+minimize [definition, in Goedel.PRrepresentable]
+minimize1 [lemma, in Goedel.PRrepresentable]
+minIsPR [instance, in hydras.solutions_exercises.MinPR2]
+minIsPR [instance, in hydras.solutions_exercises.MinPR]
+minoration [lemma, in hydras.Hydra.BigBattle]
+minoration_3 [lemma, in hydras.Hydra.BigBattle]
+minoration_2 [lemma, in hydras.Hydra.BigBattle]
+minoration_1 [lemma, in hydras.Hydra.BigBattle]
+minoration_0 [lemma, in hydras.Hydra.BigBattle]
+MinPR [library]
+minPR_PR [instance, in hydras.solutions_exercises.MinPR2]
+minPR_correct [lemma, in hydras.solutions_exercises.MinPR2]
+minPR_PR [instance, in hydras.solutions_exercises.MinPR]
+MinPR2 [library]
+minus [definition, in hydras.Epsilon0.T1]
+minusIndIsPR [instance, in hydras.Ackermann.primRec]
+minusIsPR [instance, in hydras.Ackermann.primRec]
+minus_le [lemma, in hydras.Epsilon0.T1]
+minus_a_a [lemma, in hydras.Epsilon0.T1]
+minus_lt [lemma, in hydras.Epsilon0.T1]
+minus_semi_assoc [lemma, in hydras.Prelude.More_Arith]
+minus_exists [lemma, in hydras.Schutte.Addition]
+min_assoc [instance, in hydras.Prelude.Comparable]
+min_refl [lemma, in hydras.Prelude.Comparable]
+min_le_b [lemma, in hydras.Prelude.Comparable]
+min_le_ad [lemma, in hydras.Prelude.Comparable]
+min_dec [lemma, in hydras.Prelude.Comparable]
+min_comm [lemma, in hydras.Prelude.Comparable]
+min_max_iff [lemma, in hydras.Prelude.Comparable]
+min_alt_correct [lemma, in hydras.solutions_exercises.MinPR]
+misc [library]
+mIsPR [instance, in hydras.Hydra.Hydra_Theorems]
+mkE0 [constructor, in gaia_hydras.T1Bridge]
+mkord [constructor, in hydras.Epsilon0.E0]
+mkP [constructor, in hydras.Epsilon0.F_alpha]
+mkP [constructor, in hydras.Epsilon0.Hprime]
+mkQ [constructor, in hydras.Epsilon0.F_alpha]
+mks [constructor, in hydras.Hydra.BigBattle]
+mlarge [definition, in hydras.Epsilon0.Large_Sets]
+mlarge [definition, in gaia_hydras.GLarge_Sets]
+mlargeS [definition, in hydras.Epsilon0.Large_Sets]
+mlargeS [definition, in gaia_hydras.GLarge_Sets]
+mlargeS_iff [lemma, in hydras.Epsilon0.Large_Sets]
+mlarge_omega [lemma, in hydras.Epsilon0.Large_Sets]
+mlarge_FS [lemma, in hydras.Epsilon0.Large_Sets]
+mlarge_iff [lemma, in hydras.Epsilon0.Large_Sets]
+mlarge_unshift [lemma, in hydras.Epsilon0.Large_Sets]
+mlarge_unicity [lemma, in hydras.Epsilon0.Large_Sets]
+mlarge_unicity [lemma, in gaia_hydras.GLarge_Sets]
+ml_1 [lemma, in hydras.Gamma0.Gamma0]
+ml_psi [lemma, in hydras.Gamma0.Gamma0]
+Model [record, in hydras.Ackermann.model]
+model [constructor, in hydras.Ackermann.model]
+model [library]
+ModelConsistent [lemma, in hydras.Ackermann.model]
+Model_Theory.Consistent_Theory.T [variable, in hydras.Ackermann.model]
+Model_Theory.Consistent_Theory [section, in hydras.Ackermann.model]
+Model_Theory.M [variable, in hydras.Ackermann.model]
+Model_Theory.L [variable, in hydras.Ackermann.model]
+Model_Theory [section, in hydras.Ackermann.model]
+mod_Equiv [instance, in additions.Monoid_instances]
+mod_op [instance, in additions.Monoid_instances]
+mod_equiv [instance, in additions.Monoid_instances]
+mod_eq [definition, in additions.Monoid_instances]
+Monoid [record, in additions.Monoid_def]
+Monoid_EMonoid [instance, in additions.Monoid_def]
+Monoid_Exp [inductive, in additions.Addition_Chains]
+Monoid_instances [library]
+Monoid_def [library]
+mono_weak [lemma, in hydras.Prelude.Iterates]
+mono_injective [lemma, in hydras.Prelude.Iterates]
+mono_le [lemma, in hydras.Prelude.Iterates]
+mono_F_Zero [lemma, in hydras.Epsilon0.F_alpha]
+mono_seq_weak2 [lemma, in hydras.Schutte.AP]
+mono_seq2 [lemma, in hydras.Schutte.AP]
+mono_seq [lemma, in hydras.Schutte.AP]
+MoreDecidable [library]
+MoreEpsilonIota [library]
+MoreExamples [module, in hydras.MoreAck.PrimRecExamples]
+MoreExamples.cst [definition, in hydras.MoreAck.PrimRecExamples]
+MoreExamples.cst0 [definition, in hydras.MoreAck.PrimRecExamples]
+MoreExamples.fact [definition, in hydras.MoreAck.PrimRecExamples]
+MoreExamples.mult [definition, in hydras.MoreAck.PrimRecExamples]
+MoreExamples.plus [definition, in hydras.MoreAck.PrimRecExamples]
+MoreLibHyps [library]
+MoreLists [library]
+MoreOrders [library]
+MoreOrderType [section, in gaia_hydras.onType]
+MoreOrderType.disp [variable, in gaia_hydras.onType]
+MoreOrderType.Succ_no_limit.Hlim [variable, in gaia_hydras.onType]
+MoreOrderType.Succ_no_limit.Hsucc [variable, in gaia_hydras.onType]
+MoreOrderType.Succ_no_limit.s [variable, in gaia_hydras.onType]
+MoreOrderType.Succ_no_limit.y [variable, in gaia_hydras.onType]
+MoreOrderType.Succ_no_limit.x [variable, in gaia_hydras.onType]
+MoreOrderType.Succ_no_limit [section, in gaia_hydras.onType]
+MoreOrderType.T [variable, in gaia_hydras.onType]
+MorePRExamples [library]
+MoreVectors [library]
+More_Logic_Rules.SysPrf [variable, in hydras.Ackermann.folLogic2]
+More_Logic_Rules.Prf [variable, in hydras.Ackermann.folLogic2]
+More_Logic_Rules.Terms [variable, in hydras.Ackermann.folLogic2]
+More_Logic_Rules.Term [variable, in hydras.Ackermann.folLogic2]
+More_Logic_Rules.System [variable, in hydras.Ackermann.folLogic2]
+More_Logic_Rules.Formulas [variable, in hydras.Ackermann.folLogic2]
+More_Logic_Rules.Formula [variable, in hydras.Ackermann.folLogic2]
+More_Logic_Rules.L [variable, in hydras.Ackermann.folLogic2]
+More_Logic_Rules [section, in hydras.Ackermann.folLogic2]
+More_Arith [library]
+more_list [library]
+More_on_positive [library]
+moser_lepper [definition, in hydras.Gamma0.Gamma0]
+MP [constructor, in hydras.Ackermann.folProof]
+MPSys [lemma, in hydras.MoreAck.FolExamples]
+mp:29 [binder, in additions.AM]
+mq:28 [binder, in additions.AM]
+mq:30 [binder, in additions.AM]
+mq:32 [binder, in additions.AM]
+ms [definition, in hydras.Hydra.Hydra_Termination]
+ms [definition, in gaia_hydras.GHydra]
+ms_eqn3 [lemma, in hydras.Hydra.Hydra_Termination]
+ms_nf [lemma, in hydras.Hydra.Hydra_Termination]
+ms_eqn2 [lemma, in hydras.Hydra.Hydra_Termination]
+MUL [constructor, in additions.AM]
+mult [abbreviation, in hydras.Epsilon0.T1]
+Mult [abbreviation, in hydras.Epsilon0.E0]
+Mult [constructor, in additions.Addition_Chains]
+multE4 [lemma, in gaia_hydras.T1Bridge]
+multE5 [lemma, in gaia_hydras.T1Bridge]
+multIndIsPR [instance, in hydras.Ackermann.primRec]
+MultisetWf [library]
+multIsPR [instance, in hydras.Ackermann.primRec]
+multLemma1 [lemma, in hydras.Ackermann.wellFormed]
+multLemma2 [lemma, in hydras.Ackermann.wellFormed]
+mults_squares [definition, in additions.AM]
+mults:5 [binder, in additions.AM]
+mult_plus_distr_l [lemma, in hydras.Epsilon0.T1]
+mult_mono [lemma, in hydras.Epsilon0.T1]
+mult_nf [lemma, in hydras.Epsilon0.T1]
+mult_Sn_add [lemma, in hydras.Epsilon0.T1]
+mult_0_a [lemma, in hydras.Epsilon0.T1]
+mult_nf_fin [lemma, in hydras.Epsilon0.T1]
+mult_a_1 [lemma, in hydras.Epsilon0.T1]
+mult_1_a [lemma, in hydras.Epsilon0.T1]
+mult_a_0 [lemma, in hydras.Epsilon0.T1]
+mult_compat [lemma, in hydras.Epsilon0.T1]
+mult_fin_omega [lemma, in hydras.Epsilon0.T1]
+mult_Sn_dist [lemma, in hydras.Schutte.Correctness_E0]
+mult_fin_l [definition, in hydras.OrdinalNotations.ON_Omega2]
+mult_fin_r [definition, in hydras.OrdinalNotations.ON_Omega2]
+mult_lt_lt [lemma, in hydras.Prelude.More_Arith]
+mult_ref [lemma, in gaia_hydras.T1Bridge]
+mult_ref0 [lemma, in gaia_hydras.T1Bridge]
+mult_op [projection, in additions.Monoid_def]
+Mult_op [record, in additions.Monoid_def]
+mult_op [constructor, in additions.Monoid_def]
+Mult_op [inductive, in additions.Monoid_def]
+mult_comp [instance, in hydras.OrdinalNotations.ON_mult]
+mult_mod_associative [lemma, in additions.Monoid_instances]
+mult_mod_proper [instance, in additions.Monoid_instances]
+mult_mod [definition, in additions.Monoid_instances]
+mult_plus_distr_l [lemma, in hydras.Epsilon0.E0]
+mult_fin_rw [lemma, in hydras.Epsilon0.E0]
+Mult_i [instance, in hydras.Epsilon0.E0]
+mult_Sn_mono3 [lemma, in hydras.Schutte.Addition]
+mult_Sn_mono2 [lemma, in hydras.Schutte.Addition]
+mult_fin_r_mono [lemma, in hydras.Schutte.Addition]
+mult_fin_r_one [lemma, in hydras.Schutte.Addition]
+mult_fin_r [definition, in hydras.Schutte.Addition]
+mult_Sn [definition, in hydras.Schutte.Addition]
+mult_correct [lemma, in hydras.MoreAck.PrimRecExamples]
+mult_eqn1 [lemma, in hydras.MoreAck.PrimRecExamples]
+mult:35 [binder, in additions.FirstSteps]
+mult:41 [binder, in additions.FirstSteps]
+Mul_node [constructor, in additions.Addition_Chains]
+Mul2 [instance, in additions.Fib2]
+mul2 [definition, in additions.Fib2]
+mul:75 [binder, in additions.fib]
+mul:82 [binder, in additions.fib]
+mul:88 [binder, in additions.fib]
+mul:94 [binder, in additions.fib]
+mu_beta_h [lemma, in hydras.Hydra.Epsilon0_Needed_Generic]
+mu_positive [lemma, in hydras.Hydra.Epsilon0_Needed_Generic]
+mu:1 [binder, in hydras.Hydra.Epsilon0_Needed_Free]
+mu:1 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+mu:160 [binder, in hydras.Hydra.Hydra_Definitions]
+mu:2 [binder, in hydras.Hydra.Epsilon0_Needed_Generic]
+mu:29 [binder, in gaia_hydras.GHydra]
+mu:32 [binder, in gaia_hydras.GHydra]
+mVariant [lemma, in gaia_hydras.GHydra]
+my_pow_m2lmul [lemma, in additions.fib]
+my_pow_m3lmul [lemma, in additions.fib]
+my_powP [lemma, in additions.fib]
+my_pow [definition, in additions.fib]
+m_lt [lemma, in hydras.Hydra.Epsilon0_Needed_Free]
+m_variant_LT [lemma, in hydras.Hydra.Epsilon0_Needed_Free]
+m_ge [lemma, in hydras.Hydra.Epsilon0_Needed_Free]
+M_given.Power_of_op [section, in additions.Pow_variant]
+M_given.About_power [section, in additions.Pow_variant]
+M_given.E_eq [variable, in additions.Pow_variant]
+M_given.E_one [variable, in additions.Pow_variant]
+M_given.E_op [variable, in additions.Pow_variant]
+M_given.A [variable, in additions.Pow_variant]
+M_given [section, in additions.Pow_variant]
+M_given.Power_of_op [section, in additions.Pow]
+M_given.E_one [variable, in additions.Pow]
+M_given.A [variable, in additions.Pow]
+M_given [section, in additions.Pow]
+m_dominates_Ack [lemma, in hydras.Hydra.Hydra_Theorems]
+m_dominates_Ack_from_3 [lemma, in hydras.Hydra.Hydra_Theorems]
+m_ge_Ack [lemma, in hydras.Hydra.Hydra_Theorems]
+m_ge_F_omega [lemma, in hydras.Hydra.Hydra_Theorems]
+m_eqn [lemma, in hydras.Hydra.Hydra_Theorems]
+m_ms [lemma, in hydras.Hydra.Hydra_Termination]
+m_nf [lemma, in hydras.Hydra.Hydra_Termination]
+m_lt [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+m_decrease [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+m_ge [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+m_neq_0 [lemma, in additions.Monoid_instances]
+m_nf [lemma, in gaia_hydras.GHydra]
+m_ref [lemma, in gaia_hydras.GHydra]
+m_strict_mono [lemma, in hydras.Hydra.Hydra_Lemmas]
+m_ge_generic [lemma, in hydras.Hydra.Epsilon0_Needed_Generic]
+m_ge_0 [lemma, in hydras.Hydra.Epsilon0_Needed_Generic]
+m_ge [lemma, in hydras.Hydra.Omega2_Small]
+m_lt [lemma, in hydras.Hydra.Omega2_Small]
+m_big_h_not_null [lemma, in hydras.Hydra.Omega2_Small]
+m_def [lemma, in hydras.Epsilon0.Paths]
+m_ge [lemma, in hydras.Hydra.Omega_Small]
+m_lt [lemma, in hydras.Hydra.Omega_Small]
+m_bounded [projection, in hydras.Hydra.Hydra_Definitions]
+M'':37 [binder, in additions.Naive]
+m':19 [binder, in additions.Monoid_instances]
+M':29 [binder, in additions.Naive]
+M':32 [binder, in additions.FirstSteps]
+M':36 [binder, in additions.Naive]
+m':47 [binder, in hydras.rpo.dickson]
+m':88 [binder, in hydras.Ackermann.primRec]
+m0:60 [binder, in hydras.Ackermann.folLogic3]
+m1:12 [binder, in gaia_hydras.nfwfgaia]
+m1:14 [binder, in hydras.Ackermann.subAll]
+m1:152 [binder, in additions.fib]
+m1:16 [binder, in gaia_hydras.nfwfgaia]
+m1:19 [binder, in hydras.Ackermann.subAll]
+m1:20 [binder, in gaia_hydras.nfwfgaia]
+m1:208 [binder, in hydras.Ackermann.subAll]
+m1:213 [binder, in hydras.Ackermann.subAll]
+m1:218 [binder, in hydras.Ackermann.subAll]
+m1:223 [binder, in hydras.Ackermann.subAll]
+m1:228 [binder, in hydras.Ackermann.subAll]
+m1:24 [binder, in hydras.Ackermann.subAll]
+m1:29 [binder, in hydras.Ackermann.subAll]
+m1:51 [binder, in hydras.Ackermann.subAll]
+m1:659 [binder, in gaia_hydras.nfwfgaia]
+m1:663 [binder, in gaia_hydras.nfwfgaia]
+M2 [module, in additions.FirstSteps]
+M2 [module, in additions.Naive]
+M2 [record, in additions.Monoid_instances]
+m2lfib [definition, in additions.fib]
+m2lfibP [lemma, in additions.fib]
+m2lmul [definition, in additions.fib]
+m2lmulP [lemma, in additions.fib]
+m2lmx [definition, in additions.fib]
+m2lpow [definition, in additions.fib]
+M2N [definition, in additions.Monoid_instances]
+M2_Monoid [instance, in additions.Monoid_instances]
+M2_op [instance, in additions.Monoid_instances]
+M2_eq_intros [lemma, in additions.Monoid_instances]
+M2_mult [definition, in additions.Monoid_instances]
+M2_def.rt [variable, in additions.Monoid_instances]
+_ * _ [notation, in additions.Monoid_instances]
+_ + _ [notation, in additions.Monoid_instances]
+1 [notation, in additions.Monoid_instances]
+0 [notation, in additions.Monoid_instances]
+M2_def.mult [variable, in additions.Monoid_instances]
+M2_def.plus [variable, in additions.Monoid_instances]
+M2_def.one [variable, in additions.Monoid_instances]
+M2_def.zero [variable, in additions.Monoid_instances]
+M2_def.A [variable, in additions.Monoid_instances]
+M2_def [section, in additions.Monoid_instances]
+M2.c00 [projection, in additions.FirstSteps]
+M2.c00 [projection, in additions.Naive]
+M2.c01 [projection, in additions.FirstSteps]
+M2.c01 [projection, in additions.Naive]
+M2.c10 [projection, in additions.FirstSteps]
+M2.c10 [projection, in additions.Naive]
+M2.c11 [projection, in additions.FirstSteps]
+M2.c11 [projection, in additions.Naive]
+M2.Definitions [section, in additions.Naive]
+M2.Definitions.A [variable, in additions.Naive]
+M2.Definitions.mult [variable, in additions.Naive]
+M2.Definitions.one [variable, in additions.Naive]
+M2.Definitions.plus [variable, in additions.Naive]
+M2.Definitions.rt [variable, in additions.Naive]
+M2.Definitions.zero [variable, in additions.Naive]
+_ ** _ [notation, in additions.Naive]
+_ * _ [notation, in additions.Naive]
+_ + _ [notation, in additions.Naive]
+0 [notation, in additions.Naive]
+1 [notation, in additions.Naive]
+M2.Id2 [definition, in additions.FirstSteps]
+M2.Id2 [definition, in additions.Naive]
+M2.Id2_neutral [lemma, in additions.Naive]
+M2.mat [constructor, in additions.FirstSteps]
+M2.mat [constructor, in additions.Naive]
+M2.M2_mult [definition, in additions.FirstSteps]
+_ * _ [notation, in additions.FirstSteps]
+_ + _ [notation, in additions.FirstSteps]
+1 [notation, in additions.FirstSteps]
+0 [notation, in additions.FirstSteps]
+M2.M2_Definitions.rt [variable, in additions.FirstSteps]
+M2.M2_Definitions.mult [variable, in additions.FirstSteps]
+M2.M2_Definitions.plus [variable, in additions.FirstSteps]
+M2.M2_Definitions.one [variable, in additions.FirstSteps]
+M2.M2_Definitions.zero [variable, in additions.FirstSteps]
+M2.M2_Definitions.A [variable, in additions.FirstSteps]
+M2.M2_Definitions [section, in additions.FirstSteps]
+M2.M2_mult_assoc [lemma, in additions.Naive]
+M2.M2_mult [definition, in additions.Naive]
+M2.power [definition, in additions.Naive]
+M2.power_of_plus [lemma, in additions.Naive]
+M2.t [record, in additions.FirstSteps]
+M2.t [record, in additions.Naive]
+m2:13 [binder, in gaia_hydras.nfwfgaia]
+m2:15 [binder, in hydras.Ackermann.subAll]
+m2:153 [binder, in additions.fib]
+m2:17 [binder, in gaia_hydras.nfwfgaia]
+m2:20 [binder, in hydras.Ackermann.subAll]
+m2:209 [binder, in hydras.Ackermann.subAll]
+m2:21 [binder, in gaia_hydras.nfwfgaia]
+m2:214 [binder, in hydras.Ackermann.subAll]
+m2:219 [binder, in hydras.Ackermann.subAll]
+m2:224 [binder, in hydras.Ackermann.subAll]
+m2:229 [binder, in hydras.Ackermann.subAll]
+m2:25 [binder, in hydras.Ackermann.subAll]
+m2:30 [binder, in hydras.Ackermann.subAll]
+m2:52 [binder, in hydras.Ackermann.subAll]
+m2:660 [binder, in gaia_hydras.nfwfgaia]
+m2:664 [binder, in gaia_hydras.nfwfgaia]
+m3lfib [definition, in additions.fib]
+m3lfibP [lemma, in additions.fib]
+m3lid [definition, in additions.fib]
+m3lmul [definition, in additions.fib]
+m3lmulP [lemma, in additions.fib]
+m3lmx [definition, in additions.fib]
+m3lpow [definition, in additions.fib]
+m4lfib [definition, in additions.fib]
+m4lmul [definition, in additions.fib]
+m4lmx [definition, in additions.fib]
+m4lval [definition, in additions.fib]
+M7_3 [definition, in additions.AM]
+M:1 [binder, in hydras.Schutte.Lub]
+m:1 [binder, in hydras.MoreAck.Ack]
+m:102 [binder, in Goedel.PRrepresentable]
+M:102 [binder, in additions.AM]
+m:102 [binder, in hydras.Hydra.Hydra_Lemmas]
+m:103 [binder, in hydras.Ackermann.primRec]
+M:103 [binder, in additions.Euclidean_Chains]
+M:105 [binder, in additions.Addition_Chains]
+m:107 [binder, in Goedel.PRrepresentable]
+M:107 [binder, in additions.AM]
+m:109 [binder, in hydras.Ackermann.subAll]
+m:11 [binder, in hydras.Prelude.More_Arith]
+M:11 [binder, in additions.Compatibility]
+M:112 [binder, in additions.AM]
+m:112 [binder, in hydras.Hydra.Hydra_Lemmas]
+m:113 [binder, in Goedel.PRrepresentable]
+M:113 [binder, in additions.Addition_Chains]
+m:1134 [binder, in gaia_hydras.nfwfgaia]
+M:114 [binder, in additions.Euclidean_Chains]
+m:1141 [binder, in gaia_hydras.nfwfgaia]
+m:115 [binder, in hydras.Ackermann.subAll]
+m:115 [binder, in additions.fib]
+m:1156 [binder, in gaia_hydras.nfwfgaia]
+m:117 [binder, in Goedel.PRrepresentable]
+m:1177 [binder, in gaia_hydras.nfwfgaia]
+m:1180 [binder, in gaia_hydras.nfwfgaia]
+m:1183 [binder, in gaia_hydras.nfwfgaia]
+m:1186 [binder, in gaia_hydras.nfwfgaia]
+m:1189 [binder, in gaia_hydras.nfwfgaia]
+m:119 [binder, in Goedel.PRrepresentable]
+M:119 [binder, in hydras.Epsilon0.Large_Sets]
+M:119 [binder, in additions.Addition_Chains]
+m:12 [binder, in hydras.Ackermann.subAll]
+m:12 [binder, in hydras.rpo.dickson]
+M:122 [binder, in additions.AM]
+m:1227 [binder, in hydras.Ackermann.codeSubFormula]
+m:123 [binder, in Goedel.PRrepresentable]
+m:1230 [binder, in hydras.Ackermann.codeSubFormula]
+m:1246 [binder, in hydras.Ackermann.codeSubFormula]
+m:1249 [binder, in hydras.Ackermann.codeSubFormula]
+m:125 [binder, in hydras.Ackermann.cPair]
+M:126 [binder, in additions.Addition_Chains]
+M:127 [binder, in additions.AM]
+m:127 [binder, in hydras.Ackermann.cPair]
+m:130 [binder, in Goedel.PRrepresentable]
+m:1303 [binder, in hydras.Ackermann.codeSubFormula]
+m:1306 [binder, in hydras.Ackermann.codeSubFormula]
+m:131 [binder, in hydras.Hydra.Hydra_Lemmas]
+M:133 [binder, in additions.Addition_Chains]
+m:135 [binder, in Goedel.PRrepresentable]
+m:135 [binder, in hydras.Ackermann.codeSubFormula]
+m:136 [binder, in additions.fib]
+m:1383 [binder, in hydras.Ackermann.codeSubFormula]
+m:1386 [binder, in hydras.Ackermann.codeSubFormula]
+m:139 [binder, in additions.fib]
+M:14 [binder, in additions.Pow_variant]
+M:14 [binder, in additions.Pow]
+m:14 [binder, in hydras.MoreAck.Ack]
+m:141 [binder, in Goedel.PRrepresentable]
+m:144 [binder, in hydras.Ackermann.codePA]
+m:145 [binder, in hydras.Ackermann.codePA]
+m:146 [binder, in Goedel.PRrepresentable]
+m:1472 [binder, in gaia_hydras.nfwfgaia]
+m:1474 [binder, in gaia_hydras.nfwfgaia]
+m:1478 [binder, in gaia_hydras.nfwfgaia]
+m:148 [binder, in hydras.Hydra.Hydra_Definitions]
+m:1482 [binder, in gaia_hydras.nfwfgaia]
+m:1488 [binder, in gaia_hydras.nfwfgaia]
+m:1490 [binder, in gaia_hydras.nfwfgaia]
+m:15 [binder, in hydras.Ackermann.NNtheory]
+m:15 [binder, in hydras.Schutte.Countable]
+m:150 [binder, in hydras.Ackermann.cPair]
+m:152 [binder, in Goedel.PRrepresentable]
+m:1529 [binder, in gaia_hydras.nfwfgaia]
+m:153 [binder, in hydras.Ackermann.codePA]
+m:154 [binder, in hydras.Ackermann.cPair]
+m:154 [binder, in additions.fib]
+m:156 [binder, in hydras.Ackermann.cPair]
+m:156 [binder, in hydras.Ackermann.codePA]
+m:157 [binder, in Goedel.PRrepresentable]
+m:158 [binder, in hydras.Ackermann.cPair]
+m:158 [binder, in additions.fib]
+m:158 [binder, in hydras.Hydra.Hydra_Definitions]
+m:159 [binder, in hydras.Ackermann.cPair]
+m:16 [binder, in hydras.Ackermann.subAll]
+m:16 [binder, in hydras.Ackermann.primRec]
+m:16 [binder, in hydras.OrdinalNotations.ON_Generic]
+m:16 [binder, in gaia_hydras.onType]
+m:160 [binder, in additions.fib]
+m:161 [binder, in hydras.Ackermann.cPair]
+m:163 [binder, in hydras.Prelude.MoreLists]
+M:163 [binder, in additions.Addition_Chains]
+M:166 [binder, in additions.Euclidean_Chains]
+m:17 [binder, in hydras.Ackermann.codePA]
+M:170 [binder, in additions.Addition_Chains]
+M:172 [binder, in additions.Euclidean_Chains]
+M:176 [binder, in additions.Addition_Chains]
+m:1776 [binder, in gaia_hydras.nfwfgaia]
+m:1785 [binder, in gaia_hydras.nfwfgaia]
+m:18 [binder, in hydras.Prelude.More_Arith]
+m:18 [binder, in hydras.MoreAck.AckNotPR]
+m:18 [binder, in hydras.Ackermann.primRec]
+m:18 [binder, in additions.Monoid_instances]
+m:1800 [binder, in gaia_hydras.nfwfgaia]
+m:1821 [binder, in gaia_hydras.nfwfgaia]
+m:1824 [binder, in gaia_hydras.nfwfgaia]
+m:1827 [binder, in gaia_hydras.nfwfgaia]
+m:183 [binder, in hydras.Ackermann.fol]
+m:1830 [binder, in gaia_hydras.nfwfgaia]
+m:1833 [binder, in gaia_hydras.nfwfgaia]
+m:184 [binder, in gaia_hydras.nfwfgaia]
+m:1848 [binder, in gaia_hydras.nfwfgaia]
+m:1855 [binder, in gaia_hydras.nfwfgaia]
+M:186 [binder, in additions.Euclidean_Chains]
+m:186 [binder, in gaia_hydras.nfwfgaia]
+m:1875 [binder, in gaia_hydras.nfwfgaia]
+m:188 [binder, in gaia_hydras.nfwfgaia]
+m:1881 [binder, in gaia_hydras.nfwfgaia]
+m:189 [binder, in hydras.Ackermann.fol]
+m:1898 [binder, in gaia_hydras.nfwfgaia]
+m:19 [binder, in hydras.Prelude.Compat815]
+m:19 [binder, in hydras.Ackermann.model]
+m:19 [binder, in hydras.Schutte.Countable]
+m:1907 [binder, in gaia_hydras.nfwfgaia]
+m:1916 [binder, in gaia_hydras.nfwfgaia]
+M:192 [binder, in additions.Addition_Chains]
+m:192 [binder, in gaia_hydras.nfwfgaia]
+m:1937 [binder, in gaia_hydras.nfwfgaia]
+m:1942 [binder, in gaia_hydras.nfwfgaia]
+m:1947 [binder, in gaia_hydras.nfwfgaia]
+m:196 [binder, in hydras.Ackermann.wellFormed]
+m:196 [binder, in gaia_hydras.nfwfgaia]
+M:198 [binder, in additions.Euclidean_Chains]
+m:1982 [binder, in gaia_hydras.nfwfgaia]
+m:1987 [binder, in gaia_hydras.nfwfgaia]
+m:199 [binder, in hydras.Ackermann.wellFormed]
+m:1993 [binder, in gaia_hydras.nfwfgaia]
+m:1998 [binder, in gaia_hydras.nfwfgaia]
+m:2 [binder, in hydras.Ackermann.NN2PA]
+m:2 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+m:20 [binder, in hydras.Prelude.More_Arith]
+M:20 [binder, in additions.Pow_variant]
+M:20 [binder, in additions.Pow]
+m:20 [binder, in hydras.OrdinalNotations.ON_Generic]
+m:20 [binder, in hydras.Ackermann.codePA]
+m:2004 [binder, in gaia_hydras.nfwfgaia]
+m:202 [binder, in gaia_hydras.nfwfgaia]
+m:204 [binder, in gaia_hydras.nfwfgaia]
+m:208 [binder, in hydras.Ackermann.cPair]
+m:21 [binder, in hydras.Ackermann.subAll]
+m:21 [binder, in hydras.Prelude.Compat815]
+m:22 [binder, in hydras.Ackermann.primRec]
+M:227 [binder, in additions.Addition_Chains]
+m:23 [binder, in hydras.Prelude.Compat815]
+M:234 [binder, in additions.AM]
+M:235 [binder, in additions.Addition_Chains]
+m:237 [binder, in Goedel.PRrepresentable]
+m:240 [binder, in Goedel.PRrepresentable]
+M:241 [binder, in additions.AM]
+m:243 [binder, in Goedel.PRrepresentable]
+m:246 [binder, in Goedel.PRrepresentable]
+m:246 [binder, in hydras.Ackermann.cPair]
+m:25 [binder, in hydras.Prelude.Compat815]
+m:25 [binder, in hydras.MoreAck.AckNotPR]
+m:25 [binder, in hydras.MoreAck.Ack]
+m:25 [binder, in hydras.Ackermann.primRec]
+m:257 [binder, in hydras.Ackermann.subAll]
+m:257 [binder, in hydras.rpo.rpo]
+m:26 [binder, in hydras.Ackermann.subAll]
+m:26 [binder, in hydras.Ackermann.primRec]
+M:269 [binder, in additions.Addition_Chains]
+m:269 [binder, in hydras.rpo.rpo]
+m:27 [binder, in hydras.Prelude.Compat815]
+M:27 [binder, in additions.Pow_variant]
+M:27 [binder, in additions.Pow]
+m:271 [binder, in hydras.rpo.rpo]
+m:272 [binder, in hydras.rpo.rpo]
+m:273 [binder, in hydras.rpo.rpo]
+m:28 [binder, in hydras.MoreAck.Ack]
+m:28 [binder, in hydras.Ackermann.primRec]
+M:28 [binder, in additions.Naive]
+m:285 [binder, in hydras.Ackermann.subAll]
+m:29 [binder, in hydras.Prelude.Compat815]
+m:290 [binder, in hydras.Ackermann.subAll]
+m:295 [binder, in hydras.Ackermann.subAll]
+m:3 [binder, in hydras.Prelude.Compat815]
+m:3 [binder, in hydras.Hydra.Epsilon0_Needed_Free]
+m:3 [binder, in hydras.Ackermann.PAtheory]
+m:30 [binder, in hydras.Ackermann.model]
+m:30 [binder, in hydras.Schutte.AP]
+M:30 [binder, in additions.Naive]
+m:30 [binder, in gaia_hydras.GHydra]
+m:31 [binder, in hydras.Ackermann.subAll]
+M:31 [binder, in additions.FirstSteps]
+m:31 [binder, in hydras.Prelude.Compat815]
+m:315 [binder, in hydras.Ackermann.subAll]
+m:32 [binder, in hydras.Ackermann.primRec]
+M:33 [binder, in additions.Pow_variant]
+M:33 [binder, in additions.Pow]
+m:33 [binder, in hydras.Schutte.AP]
+m:33 [binder, in hydras.MoreAck.Ack]
+m:33 [binder, in gaia_hydras.GHydra]
+M:34 [binder, in additions.Naive]
+m:35 [binder, in hydras.Ackermann.subAll]
+m:35 [binder, in hydras.Prelude.Compat815]
+m:35 [binder, in hydras.Ackermann.primRec]
+M:35 [binder, in additions.Naive]
+m:35 [binder, in hydras.Ackermann.folLogic3]
+m:36 [binder, in hydras.MoreAck.AckNotPR]
+m:37 [binder, in additions.Addition_Chains]
+M:38 [binder, in additions.Naive]
+m:385 [binder, in gaia_hydras.nfwfgaia]
+m:387 [binder, in gaia_hydras.nfwfgaia]
+m:39 [binder, in hydras.Ackermann.subAll]
+M:39 [binder, in additions.AM]
+M:39 [binder, in additions.Euclidean_Chains]
+m:4 [binder, in hydras.Hydra.Epsilon0_Needed_Generic]
+M:40 [binder, in additions.Monoid_def]
+m:40 [binder, in additions.fib]
+m:41 [binder, in hydras.Ackermann.subAll]
+M:43 [binder, in additions.Pow_variant]
+M:43 [binder, in additions.Pow]
+m:43 [binder, in hydras.Ackermann.primRec]
+m:43 [binder, in hydras.Ackermann.wellFormed]
+m:439 [binder, in gaia_hydras.nfwfgaia]
+M:44 [binder, in hydras.Schutte.Schutte_basics]
+m:447 [binder, in Goedel.PRrepresentable]
+m:45 [binder, in hydras.Ackermann.subAll]
+M:45 [binder, in additions.AM]
+M:45 [binder, in additions.Monoid_def]
+m:45 [binder, in hydras.Ackermann.wellFormed]
+m:45 [binder, in hydras.rpo.dickson]
+m:458 [binder, in Goedel.PRrepresentable]
+m:46 [binder, in hydras.MoreAck.Ack]
+m:46 [binder, in hydras.rpo.dickson]
+m:461 [binder, in Goedel.PRrepresentable]
+m:466 [binder, in Goedel.PRrepresentable]
+m:469 [binder, in Goedel.PRrepresentable]
+m:47 [binder, in hydras.MoreAck.Ack]
+m:476 [binder, in gaia_hydras.nfwfgaia]
+m:479 [binder, in gaia_hydras.nfwfgaia]
+m:48 [binder, in hydras.Ackermann.wellFormed]
+m:482 [binder, in gaia_hydras.nfwfgaia]
+m:485 [binder, in gaia_hydras.nfwfgaia]
+m:488 [binder, in gaia_hydras.nfwfgaia]
+m:49 [binder, in hydras.rpo.dickson]
+m:5 [binder, in hydras.Prelude.Compat815]
+M:5 [binder, in additions.Pow_variant]
+M:5 [binder, in additions.Pow]
+m:5 [binder, in hydras.Ackermann.PAtheory]
+m:50 [binder, in Goedel.PRrepresentable]
+M:50 [binder, in additions.Monoid_def]
+m:50 [binder, in hydras.Ackermann.primRec]
+m:50 [binder, in hydras.Ackermann.wellFormed]
+M:50 [binder, in additions.Addition_Chains]
+M:51 [binder, in hydras.Schutte.Well_Orders]
+M:52 [binder, in additions.Pow_variant]
+M:52 [binder, in additions.Pow]
+m:52 [binder, in hydras.rpo.dickson]
+m:53 [binder, in hydras.Ackermann.subAll]
+m:53 [binder, in hydras.MoreAck.Ack]
+m:53 [binder, in hydras.Ackermann.primRec]
+m:53 [binder, in hydras.Ackermann.wellFormed]
+m:53 [binder, in hydras.rpo.dickson]
+m:539 [binder, in hydras.Ackermann.checkPrf]
+m:54 [binder, in hydras.MoreAck.AckNotPR]
+m:54 [binder, in hydras.Ackermann.primRec]
+m:54 [binder, in hydras.rpo.dickson]
+m:542 [binder, in hydras.Ackermann.checkPrf]
+m:545 [binder, in gaia_hydras.nfwfgaia]
+m:546 [binder, in hydras.Ackermann.checkPrf]
+m:549 [binder, in hydras.Ackermann.checkPrf]
+m:549 [binder, in gaia_hydras.nfwfgaia]
+M:55 [binder, in additions.Monoid_def]
+m:551 [binder, in hydras.Ackermann.checkPrf]
+m:553 [binder, in hydras.Ackermann.checkPrf]
+m:555 [binder, in hydras.Ackermann.checkPrf]
+m:557 [binder, in hydras.Ackermann.checkPrf]
+m:559 [binder, in hydras.Ackermann.checkPrf]
+m:56 [binder, in Goedel.PRrepresentable]
+M:56 [binder, in additions.AM]
+M:56 [binder, in hydras.Schutte.Well_Orders]
+m:56 [binder, in gaia_hydras.nfwfgaia]
+m:561 [binder, in hydras.Ackermann.checkPrf]
+m:562 [binder, in hydras.Ackermann.checkPrf]
+m:569 [binder, in hydras.Ackermann.checkPrf]
+M:57 [binder, in additions.Euclidean_Chains]
+m:57 [binder, in hydras.rpo.dickson]
+m:570 [binder, in hydras.Ackermann.checkPrf]
+m:571 [binder, in hydras.Ackermann.checkPrf]
+m:572 [binder, in hydras.Ackermann.checkPrf]
+m:573 [binder, in hydras.Ackermann.checkPrf]
+m:574 [binder, in hydras.Ackermann.checkPrf]
+m:574 [binder, in gaia_hydras.nfwfgaia]
+m:575 [binder, in hydras.Ackermann.checkPrf]
+m:575 [binder, in gaia_hydras.nfwfgaia]
+m:576 [binder, in hydras.Ackermann.checkPrf]
+m:577 [binder, in hydras.Ackermann.checkPrf]
+m:578 [binder, in hydras.Ackermann.checkPrf]
+m:579 [binder, in hydras.Ackermann.checkPrf]
+m:580 [binder, in hydras.Ackermann.checkPrf]
+m:581 [binder, in hydras.Ackermann.checkPrf]
+m:582 [binder, in hydras.Ackermann.checkPrf]
+m:583 [binder, in hydras.Ackermann.checkPrf]
+m:584 [binder, in hydras.Ackermann.checkPrf]
+m:585 [binder, in hydras.Ackermann.checkPrf]
+m:586 [binder, in hydras.Ackermann.checkPrf]
+m:587 [binder, in hydras.Ackermann.checkPrf]
+m:588 [binder, in hydras.Ackermann.checkPrf]
+m:59 [binder, in hydras.Ackermann.subAll]
+m:59 [binder, in hydras.MoreAck.AckNotPR]
+m:59 [binder, in hydras.rpo.dickson]
+m:590 [binder, in hydras.Ackermann.checkPrf]
+m:592 [binder, in hydras.Ackermann.checkPrf]
+m:594 [binder, in hydras.Ackermann.checkPrf]
+m:596 [binder, in hydras.Ackermann.checkPrf]
+m:598 [binder, in hydras.Ackermann.checkPrf]
+m:598 [binder, in hydras.Ackermann.primRec]
+m:6 [binder, in hydras.MoreAck.Ack]
+m:6 [binder, in hydras.Ackermann.primRec]
+m:60 [binder, in Goedel.PRrepresentable]
+M:60 [binder, in additions.Pow_variant]
+M:60 [binder, in additions.Pow]
+m:60 [binder, in hydras.MoreAck.Ack]
+M:60 [binder, in additions.Monoid_def]
+m:600 [binder, in hydras.Ackermann.checkPrf]
+m:602 [binder, in hydras.Ackermann.checkPrf]
+m:603 [binder, in hydras.Ackermann.primRec]
+m:604 [binder, in hydras.Ackermann.checkPrf]
+m:606 [binder, in hydras.Ackermann.checkPrf]
+m:606 [binder, in hydras.Ackermann.primRec]
+m:608 [binder, in hydras.Ackermann.checkPrf]
+m:609 [binder, in hydras.Ackermann.checkPrf]
+m:609 [binder, in hydras.Ackermann.primRec]
+m:609 [binder, in gaia_hydras.nfwfgaia]
+m:61 [binder, in hydras.MoreAck.Ack]
+M:61 [binder, in additions.Addition_Chains]
+m:61 [binder, in hydras.rpo.dickson]
+m:610 [binder, in hydras.Ackermann.checkPrf]
+m:611 [binder, in hydras.Ackermann.checkPrf]
+m:612 [binder, in hydras.Ackermann.checkPrf]
+m:612 [binder, in hydras.Ackermann.primRec]
+m:613 [binder, in hydras.Ackermann.checkPrf]
+m:614 [binder, in hydras.Ackermann.checkPrf]
+m:615 [binder, in hydras.Ackermann.checkPrf]
+m:616 [binder, in hydras.Ackermann.checkPrf]
+m:617 [binder, in hydras.Ackermann.checkPrf]
+m:619 [binder, in hydras.Ackermann.checkPrf]
+m:62 [binder, in hydras.MoreAck.AckNotPR]
+m:62 [binder, in hydras.MoreAck.Ack]
+m:62 [binder, in hydras.Ackermann.wellFormed]
+m:620 [binder, in hydras.Ackermann.primRec]
+m:621 [binder, in hydras.Ackermann.checkPrf]
+m:623 [binder, in hydras.Ackermann.checkPrf]
+m:624 [binder, in hydras.Ackermann.primRec]
+m:625 [binder, in hydras.Ackermann.checkPrf]
+m:627 [binder, in hydras.Ackermann.checkPrf]
+m:629 [binder, in hydras.Ackermann.checkPrf]
+m:629 [binder, in hydras.Ackermann.primRec]
+M:63 [binder, in additions.AM]
+m:63 [binder, in hydras.MoreAck.Ack]
+m:63 [binder, in additions.fib]
+m:631 [binder, in hydras.Ackermann.checkPrf]
+m:632 [binder, in hydras.Ackermann.primRec]
+m:633 [binder, in hydras.Ackermann.checkPrf]
+m:635 [binder, in hydras.Ackermann.checkPrf]
+m:637 [binder, in hydras.Ackermann.checkPrf]
+m:637 [binder, in hydras.Ackermann.primRec]
+m:638 [binder, in hydras.Ackermann.checkPrf]
+m:639 [binder, in hydras.Ackermann.checkPrf]
+m:64 [binder, in Goedel.PRrepresentable]
+m:640 [binder, in hydras.Ackermann.checkPrf]
+m:641 [binder, in hydras.Ackermann.checkPrf]
+m:642 [binder, in hydras.Ackermann.checkPrf]
+m:643 [binder, in hydras.Ackermann.checkPrf]
+m:643 [binder, in hydras.Ackermann.primRec]
+m:644 [binder, in hydras.Ackermann.checkPrf]
+m:645 [binder, in hydras.Ackermann.checkPrf]
+m:646 [binder, in hydras.Ackermann.checkPrf]
+m:647 [binder, in hydras.Ackermann.checkPrf]
+m:648 [binder, in hydras.Ackermann.checkPrf]
+m:648 [binder, in hydras.Ackermann.primRec]
+M:65 [binder, in additions.Monoid_def]
+M:65 [binder, in additions.Euclidean_Chains]
+m:651 [binder, in gaia_hydras.nfwfgaia]
+m:66 [binder, in hydras.MoreAck.Ack]
+M:67 [binder, in additions.Addition_Chains]
+m:68 [binder, in Goedel.PRrepresentable]
+m:68 [binder, in gaia_hydras.nfwfgaia]
+m:69 [binder, in hydras.MoreAck.Ack]
+m:69 [binder, in additions.fib]
+M:7 [binder, in hydras.Schutte.Lub]
+m:70 [binder, in hydras.Ackermann.subAll]
+m:704 [binder, in gaia_hydras.nfwfgaia]
+m:71 [binder, in hydras.MoreAck.Ack]
+m:714 [binder, in gaia_hydras.nfwfgaia]
+m:72 [binder, in Goedel.PRrepresentable]
+M:72 [binder, in additions.Euclidean_Chains]
+m:73 [binder, in hydras.MoreAck.Ack]
+M:73 [binder, in additions.Monoid_def]
+m:74 [binder, in hydras.Ackermann.subAll]
+m:75 [binder, in hydras.MoreAck.Ack]
+M:75 [binder, in additions.Addition_Chains]
+m:751 [binder, in gaia_hydras.nfwfgaia]
+m:754 [binder, in gaia_hydras.nfwfgaia]
+M:76 [binder, in additions.AM]
+m:76 [binder, in hydras.Ackermann.primRec]
+m:76 [binder, in additions.fib]
+m:77 [binder, in Goedel.PRrepresentable]
+m:78 [binder, in hydras.MoreAck.Ack]
+m:79 [binder, in hydras.Prelude.Sort_spec]
+m:8 [binder, in hydras.Ackermann.subAll]
+m:8 [binder, in hydras.Prelude.More_Arith]
+m:8 [binder, in hydras.Ackermann.primRec]
+m:807 [binder, in gaia_hydras.nfwfgaia]
+m:809 [binder, in gaia_hydras.nfwfgaia]
+m:82 [binder, in hydras.MoreAck.Ack]
+m:83 [binder, in Goedel.PRrepresentable]
+m:83 [binder, in hydras.Ackermann.subAll]
+m:83 [binder, in additions.fib]
+m:837 [binder, in hydras.Ackermann.codeSubFormula]
+m:840 [binder, in hydras.Ackermann.codeSubFormula]
+m:845 [binder, in hydras.Ackermann.checkPrf]
+m:85 [binder, in hydras.MoreAck.Ack]
+m:85 [binder, in additions.fib]
+m:857 [binder, in hydras.Ackermann.checkPrf]
+M:86 [binder, in additions.AM]
+m:86 [binder, in hydras.Ackermann.primRec]
+m:862 [binder, in hydras.Ackermann.checkPrf]
+m:868 [binder, in hydras.Ackermann.codeSubFormula]
+m:87 [binder, in Goedel.PRrepresentable]
+m:87 [binder, in hydras.Ackermann.subAll]
+m:871 [binder, in hydras.Ackermann.codeSubFormula]
+m:892 [binder, in hydras.Ackermann.codeSubFormula]
+m:895 [binder, in hydras.Ackermann.codeSubFormula]
+m:9 [binder, in hydras.Ackermann.subAll]
+m:908 [binder, in hydras.Ackermann.codeSubFormula]
+m:91 [binder, in Goedel.PRrepresentable]
+m:91 [binder, in hydras.Ackermann.primRec]
+m:911 [binder, in hydras.Ackermann.codeSubFormula]
+M:92 [binder, in additions.Euclidean_Chains]
+m:93 [binder, in hydras.Ackermann.subAll]
+m:935 [binder, in hydras.Ackermann.codeSubFormula]
+m:938 [binder, in hydras.Ackermann.codeSubFormula]
+m:95 [binder, in Goedel.PRrepresentable]
+m:95 [binder, in hydras.Ackermann.primRec]
+m:966 [binder, in gaia_hydras.nfwfgaia]
+m:968 [binder, in gaia_hydras.nfwfgaia]
+m:97 [binder, in hydras.Ackermann.subAll]
+M:97 [binder, in additions.AM]
+m:970 [binder, in gaia_hydras.nfwfgaia]
+m:974 [binder, in gaia_hydras.nfwfgaia]
+m:978 [binder, in gaia_hydras.nfwfgaia]
+m:98 [binder, in hydras.Ackermann.primRec]
+m:984 [binder, in gaia_hydras.nfwfgaia]
+m:986 [binder, in gaia_hydras.nfwfgaia]
+

N

+N [definition, in hydras.Hydra.BigBattle]
+Naive [library]
+nAnd [lemma, in hydras.Ackermann.LNN]
+nAnd [lemma, in hydras.Ackermann.LNT]
+nAnd [lemma, in hydras.Ackermann.folLogic]
+naryFunc [definition, in hydras.Ackermann.extEqualNat]
+naryFunc [definition, in hydras.Ackermann.model]
+naryIf [definition, in hydras.solutions_exercises.MinPR2]
+naryRel [definition, in hydras.Ackermann.extEqualNat]
+naryRel [definition, in hydras.Ackermann.model]
+natLE [lemma, in hydras.Ackermann.NNtheory]
+natLT [lemma, in hydras.Ackermann.NNtheory]
+natModel [definition, in hydras.Ackermann.PAconsistent]
+Natmult [instance, in additions.Monoid_instances]
+natNE [lemma, in hydras.Ackermann.NNtheory]
+Natplus [instance, in additions.Monoid_instances]
+natPlus [lemma, in hydras.Ackermann.NNtheory]
+natTimes [lemma, in hydras.Ackermann.NNtheory]
+natToTerm [definition, in hydras.Ackermann.LNN]
+natToTerm [definition, in hydras.Ackermann.LNT]
+natToTermLNN [definition, in hydras.Ackermann.codeNatToTerm]
+natToTermLNT [definition, in hydras.Ackermann.codeNatToTerm]
+Nat_le_TO [instance, in hydras.Prelude.DecPreOrder_Instances]
+Nat_le_dec [instance, in hydras.Prelude.DecPreOrder_Instances]
+nat_double_or_s_double [lemma, in hydras.Prelude.More_Arith]
+nat_power_ok [lemma, in additions.Compatibility]
+nat_eqb_false [lemma, in hydras.Ackermann.primRec]
+nat_plus_op [instance, in additions.Monoid_instances]
+nat_mult_op [instance, in additions.Monoid_instances]
+nat_2_term_mono [lemma, in hydras.Gamma0.Gamma0]
+nat_lt_psi [lemma, in hydras.Gamma0.Gamma0]
+nat_lt_cons [lemma, in hydras.Gamma0.Gamma0]
+nat_2_term [definition, in hydras.Gamma0.Gamma0]
+NA:4 [binder, in hydras.OrdinalNotations.ON_mult]
+NA:4 [binder, in hydras.OrdinalNotations.ON_plus]
+nbterms [definition, in hydras.Gamma0.T2]
+nb_occ_app [lemma, in hydras.rpo.more_list]
+nb_occ [definition, in hydras.rpo.more_list]
+NB:8 [binder, in hydras.OrdinalNotations.ON_mult]
+NB:8 [binder, in hydras.OrdinalNotations.ON_plus]
+neqIsPR [instance, in hydras.Ackermann.primRec]
+nested_Ack_bound [lemma, in hydras.MoreAck.Ack]
+neutral_r [lemma, in additions.Fib2]
+neutral_l [lemma, in additions.Fib2]
+NewNotations [library]
+newVar [definition, in hydras.Ackermann.folProp]
+newVar1 [lemma, in hydras.Ackermann.folProp]
+newVar2 [lemma, in hydras.Ackermann.folProp]
+nExist [lemma, in hydras.Ackermann.LNN]
+nExist [lemma, in hydras.Ackermann.LNT]
+nExist [lemma, in hydras.Ackermann.folLogic]
+next [definition, in hydras.Hydra.BigBattle]
+next_fib [lemma, in additions.Fib2]
+next_step [definition, in hydras.Hydra.Hydra_Lemmas]
+next_round [definition, in hydras.Hydra.Hydra_Lemmas]
+next_round_dec [definition, in hydras.Hydra.Hydra_Lemmas]
+nf [definition, in hydras.Epsilon0.T1]
+nf [inductive, in hydras.Gamma0.T2]
+NF [abbreviation, in hydras.solutions_exercises.MultisetWf]
+nfb [definition, in hydras.Gamma0.Gamma0]
+nForall [lemma, in hydras.Ackermann.LNN]
+nForall [lemma, in hydras.Ackermann.LNT]
+nForall [lemma, in hydras.Ackermann.folLogic]
+nfwfgaia [library]
+nf_LT_right [lemma, in hydras.Epsilon0.T1]
+nf_exp_F [lemma, in hydras.Epsilon0.T1]
+nf_LT_iff [lemma, in hydras.Epsilon0.T1]
+nf_omega_omega [lemma, in hydras.Epsilon0.T1]
+nf_phi0 [lemma, in hydras.Epsilon0.T1]
+nf_omega [lemma, in hydras.Epsilon0.T1]
+nf_Wf [lemma, in hydras.Epsilon0.T1]
+nf_Acc [definition, in hydras.Epsilon0.T1]
+nf_tail_lt [lemma, in hydras.Epsilon0.T1]
+nf_cons_LT [lemma, in hydras.Epsilon0.T1]
+nf_rect [definition, in hydras.Epsilon0.T1]
+nf_omega_tower [lemma, in hydras.Epsilon0.T1]
+nf_coeff_irrelevance [lemma, in hydras.Epsilon0.T1]
+nf_intro' [lemma, in hydras.Epsilon0.T1]
+nf_intro [lemma, in hydras.Epsilon0.T1]
+nf_tail_lt_nf [lemma, in hydras.Epsilon0.T1]
+nf_of_finite [lemma, in hydras.Epsilon0.T1]
+nf_fin [lemma, in hydras.Epsilon0.T1]
+nf_FS [lemma, in hydras.Epsilon0.T1]
+nf_b_cons_eq [lemma, in hydras.Epsilon0.T1]
+nf_cons_iff [lemma, in hydras.Epsilon0.T1]
+nf_cons_inv [lemma, in hydras.Epsilon0.T1]
+nf_inv3 [lemma, in hydras.Epsilon0.T1]
+nf_inv2 [lemma, in hydras.Epsilon0.T1]
+nf_inv1 [lemma, in hydras.Epsilon0.T1]
+nf_b [definition, in hydras.Epsilon0.T1]
+nf_epsilon [lemma, in hydras.Gamma0.T2]
+nf_epsilon0 [lemma, in hydras.Gamma0.T2]
+nf_omega [lemma, in hydras.Gamma0.T2]
+nf_fin [lemma, in hydras.Gamma0.T2]
+nf_bounded [lemma, in hydras.Schutte.CNF]
+NF_Acc [lemma, in hydras.solutions_exercises.MultisetWf]
+NF_inv2 [lemma, in hydras.solutions_exercises.MultisetWf]
+NF_inv1 [lemma, in hydras.solutions_exercises.MultisetWf]
+nf_ref [lemma, in gaia_hydras.T1Bridge]
+nf_canon [lemma, in hydras.Epsilon0.Canon]
+nf_proof_unicity [lemma, in hydras.Epsilon0.E0]
+nf_m [lemma, in hydras.Hydra.Epsilon0_Needed_Generic]
+nf_intro [lemma, in hydras.Gamma0.Gamma0]
+nf_coeff_irrelevance [lemma, in hydras.Gamma0.Gamma0]
+nf_Wf [lemma, in hydras.Gamma0.Gamma0]
+nf_c2 [lemma, in hydras.Gamma0.Gamma0]
+nf_c1 [lemma, in hydras.Gamma0.Gamma0]
+nf_b2 [lemma, in hydras.Gamma0.Gamma0]
+nf_b1 [lemma, in hydras.Gamma0.Gamma0]
+nf_a2 [lemma, in hydras.Gamma0.Gamma0]
+nf_a1 [lemma, in hydras.Gamma0.Gamma0]
+nf_subterm [lemma, in hydras.Gamma0.Gamma0]
+nf_fin_inv [lemma, in hydras.Gamma0.Gamma0]
+nf_inv_tail [lemma, in hydras.Gamma0.Gamma0]
+nf_c [lemma, in hydras.Gamma0.Gamma0]
+nf_b [lemma, in hydras.Gamma0.Gamma0]
+nf_a [lemma, in hydras.Gamma0.Gamma0]
+nh [projection, in hydras.Hydra.BigBattle]
+nilTerms [lemma, in hydras.Ackermann.fol]
+nImp [lemma, in hydras.Ackermann.LNN]
+nImp [lemma, in hydras.Ackermann.LNT]
+nImp [lemma, in hydras.Ackermann.folLogic]
+Nmodulo [section, in additions.Monoid_instances]
+Nmodulo.m [variable, in additions.Monoid_instances]
+Nmodulo.m_gt_1 [variable, in additions.Monoid_instances]
+Nmod_Monoid [instance, in additions.Monoid_instances]
+NMult [instance, in additions.Monoid_instances]
+NN [definition, in hydras.Ackermann.NN]
+NN [section, in hydras.Ackermann.NN]
+NN [library]
+nnE [lemma, in hydras.Ackermann.LNN]
+nnE [lemma, in hydras.Ackermann.LNT]
+nnE [lemma, in hydras.Ackermann.folLogic]
+nnHelp [definition, in hydras.Ackermann.model]
+nnI [lemma, in hydras.Ackermann.LNN]
+nnI [lemma, in hydras.Ackermann.LNT]
+nnI [lemma, in hydras.Ackermann.folLogic]
+NNnotations [module, in hydras.Ackermann.LNN]
+NNnotations.LT [definition, in hydras.Ackermann.LNN]
+NNnotations.Plus [definition, in hydras.Ackermann.LNN]
+NNnotations.Succ [definition, in hydras.Ackermann.LNN]
+NNnotations.S_ [abbreviation, in hydras.Ackermann.LNN]
+NNnotations.Times [definition, in hydras.Ackermann.LNN]
+NNnotations.Zero [definition, in hydras.Ackermann.LNN]
+fol:_ * _ [notation, in hydras.Ackermann.LNN]
+fol:_ + _ [notation, in hydras.Ackermann.LNN]
+_ < _ (fol_scope) [notation, in hydras.Ackermann.LNN]
+_ * _ (fol_scope) [notation, in hydras.Ackermann.LNN]
+_ + _ (fol_scope) [notation, in hydras.Ackermann.LNN]
+nnPlusNotNeeded [lemma, in hydras.Ackermann.NNtheory]
+NNtheory [library]
+nnTranslate [definition, in hydras.Ackermann.model]
+nn1 [lemma, in hydras.Ackermann.NN]
+NN1 [definition, in hydras.Ackermann.NN]
+nn2 [lemma, in hydras.Ackermann.NN]
+NN2 [definition, in hydras.Ackermann.NN]
+NN2PA [lemma, in hydras.Ackermann.NN2PA]
+NN2PA [library]
+nn3 [lemma, in hydras.Ackermann.NN]
+NN3 [definition, in hydras.Ackermann.NN]
+nn4 [lemma, in hydras.Ackermann.NN]
+NN4 [definition, in hydras.Ackermann.NN]
+nn5 [lemma, in hydras.Ackermann.NN]
+NN5 [definition, in hydras.Ackermann.NN]
+nn6 [lemma, in hydras.Ackermann.NN]
+NN6 [definition, in hydras.Ackermann.NN]
+nn7 [lemma, in hydras.Ackermann.NN]
+NN7 [definition, in hydras.Ackermann.NN]
+NN72PA [lemma, in hydras.Ackermann.PAtheory]
+nn8 [lemma, in hydras.Ackermann.NN]
+NN8 [definition, in hydras.Ackermann.NN]
+NN82PA [lemma, in hydras.Ackermann.PAtheory]
+nn9 [lemma, in hydras.Ackermann.NN]
+NN9 [definition, in hydras.Ackermann.NN]
+NN92PA [lemma, in hydras.Ackermann.PAtheory]
+node [constructor, in hydras.Hydra.Hydra_Definitions]
+noMiddle [lemma, in hydras.Ackermann.LNN]
+noMiddle [lemma, in hydras.Ackermann.LNT]
+noMiddle [lemma, in hydras.Ackermann.folLogic]
+none_nb_occ_O [lemma, in hydras.rpo.more_list]
+non_commutativity_of_plus [definition, in hydras.OrdinalNotations.ON_Omega2]
+Non_denum [lemma, in hydras.Schutte.Schutte_basics]
+non_optimality_proof.binary_opt [variable, in additions.Addition_Chains]
+non_optimality_proof [section, in additions.Addition_Chains]
+nOr [lemma, in hydras.Ackermann.LNN]
+nOr [lemma, in hydras.Ackermann.LNT]
+nOr [lemma, in hydras.Ackermann.folLogic]
+normal [definition, in hydras.Schutte.Ordering_Functions]
+normal_phi0 [lemma, in hydras.Schutte.AP]
+normal_plus_alpha [lemma, in hydras.Schutte.Addition]
+notBoundedForall [lemma, in Goedel.PRrepresentable]
+notCon2wNotCon [lemma, in hydras.Ackermann.wConsistent]
+notH [constructor, in hydras.Ackermann.fol]
+NotIncl [section, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+notInFreeVarSys [lemma, in hydras.Ackermann.folProp]
+notRel [definition, in hydras.Ackermann.primRec]
+notRelPR [lemma, in hydras.Ackermann.primRec]
+notZero [definition, in hydras.Ackermann.primRec]
+notZeroIsPR [instance, in hydras.Ackermann.primRec]
+not_LT_zero [lemma, in hydras.Epsilon0.T1]
+not_zero_lt [lemma, in hydras.Epsilon0.T1]
+not_zero_gt_0 [lemma, in hydras.Epsilon0.T1]
+not_lt_zero [lemma, in hydras.Epsilon0.T1]
+not_double_is_s_double [lemma, in hydras.Prelude.More_Arith]
+not_le_gt [lemma, in hydras.Prelude.DecPreOrder]
+not_le_ge [lemma, in hydras.Prelude.DecPreOrder]
+not_AP_inv2 [lemma, in hydras.Schutte.CNF]
+not_AP_inv_0 [lemma, in hydras.Schutte.CNF]
+not_acc [lemma, in hydras.solutions_exercises.MultisetWf]
+not_countable_unbounded [lemma, in hydras.Schutte.Schutte_basics]
+Not_Unbounded_countable [lemma, in hydras.Schutte.Schutte_basics]
+Not_Unbounded_bounded [lemma, in hydras.Schutte.Schutte_basics]
+not_is_succ_limit [lemma, in hydras.Schutte.Schutte_basics]
+not_is_limit_succ [lemma, in hydras.Schutte.Schutte_basics]
+not_is_limit_zero [lemma, in hydras.Schutte.Schutte_basics]
+not_is_succ_zero [lemma, in hydras.Schutte.Schutte_basics]
+not_lt_zero [lemma, in hydras.Schutte.Schutte_basics]
+not_gt_le [lemma, in hydras.Schutte.Schutte_basics]
+not_lt_zero [lemma, in hydras.Gamma0.Gamma0]
+not_decreasing [lemma, in gaia_hydras.nfwfgaia]
+not_decreasing [lemma, in hydras.solutions_exercises.T1_ltNotWf]
+not_decreasing_aux [lemma, in hydras.solutions_exercises.T1_ltNotWf]
+not_acc [lemma, in hydras.solutions_exercises.T1_ltNotWf]
+NPlus [instance, in additions.Monoid_instances]
+Npos_power_compat [lemma, in additions.Compatibility]
+Npos_gt_0 [lemma, in additions.More_on_positive]
+Npos_diff_zero [lemma, in additions.More_on_positive]
+nth_error_ok_in [lemma, in hydras.rpo.more_list]
+nth_error_map [lemma, in hydras.rpo.more_list]
+nVars [definition, in hydras.Ackermann.folProof]
+nv:118 [binder, in hydras.Ackermann.folProp]
+nv:13 [binder, in hydras.MoreAck.BadSubst]
+nv:2 [binder, in hydras.Ackermann.PAtheory]
+nv:283 [binder, in hydras.Ackermann.folProp]
+nv:30 [binder, in Goedel.codeSysPrf]
+nv:404 [binder, in hydras.Ackermann.folProp]
+nv:432 [binder, in hydras.Ackermann.folProp]
+nv:437 [binder, in hydras.Ackermann.folProp]
+nv:48 [binder, in hydras.Ackermann.subAll]
+nv:65 [binder, in hydras.Ackermann.codeSubFormula]
+N_size_gt [lemma, in additions.Compatibility]
+N_pow_compat [lemma, in additions.Compatibility]
+N_pow_power [lemma, in additions.Compatibility]
+N_pow [definition, in additions.Compatibility]
+N_bpow [definition, in additions.FirstSteps]
+N_bpow_commute [lemma, in additions.Pow_variant]
+N_bpow_ok_R [lemma, in additions.Pow_variant]
+N_bpow_ok [lemma, in additions.Pow_variant]
+N_bpow [definition, in additions.Pow_variant]
+N_bpow_commute [lemma, in additions.Pow]
+N_bpow_ok_R [lemma, in additions.Pow]
+N_bpow_ok [lemma, in additions.Pow]
+N_bpow [definition, in additions.Pow]
+N_le_mul_pos [lemma, in additions.More_on_positive]
+N_pos_div_eucl_q0 [lemma, in additions.More_on_positive]
+N_pos_div_eucl_rest [lemma, in additions.More_on_positive]
+N_pos_div_eucl_divides [lemma, in additions.More_on_positive]
+N_pos_N2pos [lemma, in additions.More_on_positive]
+N_le_1_pos [lemma, in additions.More_on_positive]
+N_0_le_n [lemma, in additions.More_on_positive]
+N_mod.power [definition, in additions.Naive]
+N_mod.mult_mod [definition, in additions.Naive]
+N_mod.m_fixed.m [variable, in additions.Naive]
+N_mod.m_fixed [section, in additions.Naive]
+N_mod [module, in additions.Naive]
+N_plus_op [instance, in additions.Monoid_instances]
+N_mult_op [instance, in additions.Monoid_instances]
+N_simpl [lemma, in hydras.Epsilon0.Hprime]
+n'':11 [binder, in gaia_hydras.nfwfgaia]
+n'':8 [binder, in gaia_hydras.nfwfgaia]
+n':10 [binder, in gaia_hydras.nfwfgaia]
+n':1024 [binder, in gaia_hydras.nfwfgaia]
+n':1152 [binder, in gaia_hydras.nfwfgaia]
+n':117 [binder, in hydras.Epsilon0.T1]
+n':1204 [binder, in gaia_hydras.nfwfgaia]
+n':131 [binder, in gaia_hydras.nfwfgaia]
+n':1338 [binder, in gaia_hydras.nfwfgaia]
+n':1384 [binder, in gaia_hydras.nfwfgaia]
+n':139 [binder, in hydras.Gamma0.Gamma0]
+n':14 [binder, in hydras.rpo.dickson]
+n':1436 [binder, in gaia_hydras.nfwfgaia]
+n':1500 [binder, in gaia_hydras.nfwfgaia]
+n':1556 [binder, in gaia_hydras.nfwfgaia]
+n':1585 [binder, in gaia_hydras.nfwfgaia]
+n':1609 [binder, in gaia_hydras.nfwfgaia]
+n':165 [binder, in hydras.Epsilon0.T1]
+n':1691 [binder, in gaia_hydras.nfwfgaia]
+n':174 [binder, in gaia_hydras.nfwfgaia]
+n':176 [binder, in hydras.Epsilon0.T1]
+n':19 [binder, in hydras.Epsilon0.Hessenberg]
+n':194 [binder, in hydras.Epsilon0.T1]
+n':2043 [binder, in gaia_hydras.nfwfgaia]
+n':2066 [binder, in gaia_hydras.nfwfgaia]
+n':223 [binder, in hydras.Epsilon0.T1]
+n':225 [binder, in gaia_hydras.nfwfgaia]
+n':231 [binder, in gaia_hydras.nfwfgaia]
+n':24 [binder, in hydras.Schutte.Correctness_E0]
+n':25 [binder, in hydras.Epsilon0.Hessenberg]
+n':258 [binder, in gaia_hydras.nfwfgaia]
+n':259 [binder, in hydras.rpo.rpo]
+n':27 [binder, in hydras.Gamma0.Gamma0]
+n':285 [binder, in hydras.Epsilon0.T1]
+n':29 [binder, in hydras.Schutte.Correctness_E0]
+n':294 [binder, in gaia_hydras.nfwfgaia]
+n':3 [binder, in gaia_hydras.nfwfgaia]
+n':30 [binder, in hydras.Epsilon0.Hessenberg]
+n':301 [binder, in hydras.Gamma0.Gamma0]
+n':305 [binder, in hydras.Gamma0.Gamma0]
+n':324 [binder, in hydras.Epsilon0.T1]
+n':329 [binder, in hydras.Epsilon0.T1]
+n':34 [binder, in hydras.Epsilon0.T1]
+n':347 [binder, in hydras.Epsilon0.T1]
+n':359 [binder, in hydras.Epsilon0.T1]
+n':36 [binder, in hydras.Epsilon0.Hessenberg]
+n':366 [binder, in hydras.Epsilon0.T1]
+n':380 [binder, in hydras.Epsilon0.T1]
+n':42 [binder, in hydras.Epsilon0.Hessenberg]
+n':425 [binder, in gaia_hydras.nfwfgaia]
+n':429 [binder, in hydras.Epsilon0.T1]
+n':430 [binder, in gaia_hydras.nfwfgaia]
+n':434 [binder, in hydras.Epsilon0.T1]
+n':440 [binder, in hydras.Epsilon0.T1]
+n':5 [binder, in gaia_hydras.nfwfgaia]
+n':50 [binder, in hydras.Epsilon0.Epsilon0rpo]
+n':50 [binder, in hydras.OrdinalNotations.OmegaOmega]
+n':52 [binder, in hydras.Epsilon0.Hessenberg]
+n':535 [binder, in hydras.Epsilon0.T1]
+n':54 [binder, in gaia_hydras.T1Bridge]
+n':55 [binder, in hydras.Epsilon0.T1]
+n':55 [binder, in hydras.Epsilon0.Epsilon0rpo]
+n':58 [binder, in hydras.OrdinalNotations.OmegaOmega]
+n':59 [binder, in hydras.Epsilon0.Epsilon0rpo]
+n':59 [binder, in gaia_hydras.T1Bridge]
+n':590 [binder, in hydras.Gamma0.Gamma0]
+n':64 [binder, in gaia_hydras.T1Bridge]
+n':68 [binder, in hydras.Epsilon0.T1]
+n':7 [binder, in gaia_hydras.nfwfgaia]
+n':70 [binder, in gaia_hydras.T1Bridge]
+n':726 [binder, in hydras.Epsilon0.T1]
+n':75 [binder, in hydras.Epsilon0.T1]
+n':76 [binder, in gaia_hydras.T1Bridge]
+n':79 [binder, in hydras.Epsilon0.T1]
+n':790 [binder, in gaia_hydras.nfwfgaia]
+n':83 [binder, in hydras.Gamma0.T2]
+n':84 [binder, in hydras.Epsilon0.T1]
+n':848 [binder, in gaia_hydras.nfwfgaia]
+n':871 [binder, in gaia_hydras.nfwfgaia]
+n':908 [binder, in gaia_hydras.nfwfgaia]
+n':916 [binder, in gaia_hydras.nfwfgaia]
+n':94 [binder, in hydras.Epsilon0.T1]
+n':948 [binder, in gaia_hydras.nfwfgaia]
+n':99 [binder, in hydras.Epsilon0.T1]
+n0:18 [binder, in hydras.Ackermann.wellFormed]
+n0:194 [binder, in hydras.Ackermann.fol]
+n0:195 [binder, in hydras.Ackermann.fol]
+n0:196 [binder, in hydras.Ackermann.fol]
+n0:197 [binder, in hydras.Ackermann.fol]
+n0:20 [binder, in hydras.Ackermann.wellFormed]
+n0:231 [binder, in hydras.Ackermann.subAll]
+n0:232 [binder, in hydras.Ackermann.subAll]
+n0:241 [binder, in hydras.Ackermann.subAll]
+n0:243 [binder, in hydras.Ackermann.subAll]
+n0:31 [binder, in hydras.Schutte.AP]
+n0:32 [binder, in hydras.Schutte.AP]
+n0:379 [binder, in hydras.Epsilon0.Paths]
+n0:650 [binder, in hydras.Ackermann.primRec]
+n1 [projection, in hydras.Hydra.BigBattle]
+n1:105 [binder, in Goedel.rosserPA]
+n1:107 [binder, in hydras.Epsilon0.T1]
+n1:107 [binder, in Goedel.rosserPA]
+n1:122 [binder, in additions.Euclidean_Chains]
+n1:131 [binder, in additions.Euclidean_Chains]
+n1:14 [binder, in gaia_hydras.nfwfgaia]
+n1:15 [binder, in hydras.Prelude.Compat815]
+n1:18 [binder, in gaia_hydras.nfwfgaia]
+n1:198 [binder, in hydras.Ackermann.fol]
+n1:199 [binder, in hydras.Ackermann.fol]
+n1:22 [binder, in gaia_hydras.nfwfgaia]
+n1:223 [binder, in Goedel.PRrepresentable]
+n1:230 [binder, in Goedel.PRrepresentable]
+n1:318 [binder, in hydras.Gamma0.Gamma0]
+n1:32 [binder, in hydras.Gamma0.T2]
+n1:324 [binder, in hydras.Gamma0.Gamma0]
+n1:340 [binder, in hydras.Gamma0.Gamma0]
+n1:345 [binder, in hydras.Gamma0.Gamma0]
+n1:362 [binder, in hydras.Gamma0.Gamma0]
+n1:368 [binder, in hydras.Gamma0.Gamma0]
+n1:38 [binder, in Goedel.rosser]
+n1:380 [binder, in hydras.Epsilon0.Paths]
+n1:384 [binder, in hydras.Gamma0.Gamma0]
+n1:389 [binder, in hydras.Gamma0.Gamma0]
+n1:39 [binder, in hydras.Gamma0.T2]
+n1:39 [binder, in hydras.Gamma0.Gamma0]
+n1:400 [binder, in hydras.Gamma0.Gamma0]
+n1:410 [binder, in hydras.Gamma0.Gamma0]
+n1:43 [binder, in Goedel.rosser]
+n1:45 [binder, in Goedel.rosser]
+n1:47 [binder, in hydras.Gamma0.T2]
+n1:47 [binder, in hydras.Gamma0.Gamma0]
+n1:50 [binder, in Goedel.rosser]
+n1:54 [binder, in hydras.Gamma0.T2]
+n1:55 [binder, in hydras.Gamma0.Gamma0]
+n1:60 [binder, in hydras.Gamma0.T2]
+n1:63 [binder, in hydras.Gamma0.Gamma0]
+n1:66 [binder, in hydras.Gamma0.T2]
+n1:661 [binder, in gaia_hydras.nfwfgaia]
+n1:665 [binder, in gaia_hydras.nfwfgaia]
+n1:68 [binder, in Goedel.rosserPA]
+n1:71 [binder, in hydras.Gamma0.Gamma0]
+n1:73 [binder, in Goedel.rosserPA]
+n1:75 [binder, in Goedel.rosserPA]
+n1:78 [binder, in hydras.Gamma0.Gamma0]
+n1:80 [binder, in Goedel.rosserPA]
+n1:82 [binder, in additions.Euclidean_Chains]
+n1:84 [binder, in hydras.Gamma0.Gamma0]
+n1:94 [binder, in Goedel.rosserPA]
+n1:99 [binder, in Goedel.rosserPA]
+n2 [projection, in hydras.Hydra.BigBattle]
+N2pos [definition, in additions.More_on_positive]
+N2pos_lt_switch2 [lemma, in additions.More_on_positive]
+N2pos_lt_switch [lemma, in additions.More_on_positive]
+N2pos_pos [lemma, in additions.More_on_positive]
+n2t [abbreviation, in hydras.Ackermann.LNN]
+n2:123 [binder, in additions.Euclidean_Chains]
+n2:132 [binder, in additions.Euclidean_Chains]
+n2:15 [binder, in gaia_hydras.nfwfgaia]
+n2:19 [binder, in gaia_hydras.nfwfgaia]
+n2:23 [binder, in gaia_hydras.nfwfgaia]
+n2:325 [binder, in hydras.Gamma0.Gamma0]
+n2:33 [binder, in hydras.Gamma0.T2]
+n2:332 [binder, in hydras.Gamma0.Gamma0]
+n2:346 [binder, in hydras.Gamma0.Gamma0]
+n2:352 [binder, in hydras.Gamma0.Gamma0]
+n2:369 [binder, in hydras.Gamma0.Gamma0]
+n2:376 [binder, in hydras.Gamma0.Gamma0]
+n2:381 [binder, in hydras.Epsilon0.Paths]
+n2:390 [binder, in hydras.Gamma0.Gamma0]
+n2:396 [binder, in hydras.Gamma0.Gamma0]
+n2:40 [binder, in hydras.Gamma0.T2]
+n2:40 [binder, in hydras.Gamma0.Gamma0]
+n2:401 [binder, in hydras.Gamma0.Gamma0]
+n2:406 [binder, in hydras.Gamma0.Gamma0]
+n2:48 [binder, in hydras.Gamma0.T2]
+n2:48 [binder, in hydras.Gamma0.Gamma0]
+n2:55 [binder, in hydras.Gamma0.T2]
+n2:56 [binder, in hydras.Gamma0.Gamma0]
+n2:61 [binder, in hydras.Gamma0.T2]
+n2:64 [binder, in hydras.Gamma0.Gamma0]
+n2:662 [binder, in gaia_hydras.nfwfgaia]
+n2:666 [binder, in gaia_hydras.nfwfgaia]
+n2:72 [binder, in hydras.Gamma0.Gamma0]
+n2:79 [binder, in hydras.Gamma0.Gamma0]
+n2:83 [binder, in additions.Euclidean_Chains]
+n2:871 [binder, in hydras.Ackermann.checkPrf]
+n2:874 [binder, in hydras.Ackermann.checkPrf]
+n3:17 [binder, in hydras.Prelude.Compat815]
+n3:48 [binder, in Goedel.rosser]
+n3:53 [binder, in Goedel.rosser]
+n3:78 [binder, in Goedel.rosserPA]
+n3:83 [binder, in Goedel.rosserPA]
+n:1 [binder, in hydras.solutions_exercises.MinPR2]
+n:1 [binder, in additions.Fib2]
+n:1 [binder, in hydras.Prelude.More_Arith]
+n:1 [binder, in hydras.Ackermann.extEqualNat]
+n:1 [binder, in hydras.MoreAck.expressibleExamples]
+n:1 [binder, in hydras.Prelude.Compat815]
+n:1 [binder, in hydras.Prelude.Fuel]
+n:1 [binder, in hydras.Epsilon0.F_omega]
+n:1 [binder, in hydras.OrdinalNotations.OmegaOmega]
+n:1 [binder, in hydras.solutions_exercises.MorePRExamples]
+n:1 [binder, in hydras.solutions_exercises.FibonacciPR]
+n:1 [binder, in hydras.Ackermann.cPair]
+n:1 [binder, in gaia_hydras.ON_gfinite]
+n:1 [binder, in additions.Monoid_instances]
+n:1 [binder, in hydras.OrdinalNotations.ON_Finite]
+n:1 [binder, in additions.fib]
+n:1 [binder, in hydras.Ackermann.codeList]
+n:1 [binder, in hydras.Prelude.LibHyps_Experiments]
+n:1 [binder, in hydras.solutions_exercises.F_3]
+n:1 [binder, in hydras.Ackermann.codeNatToTerm]
+n:1 [binder, in hydras.Prelude.Exp2]
+n:1 [binder, in additions.Demo]
+n:1 [binder, in hydras.MoreAck.PrimRecExamples]
+n:10 [binder, in hydras.Prelude.More_Arith]
+n:10 [binder, in hydras.Ackermann.codeFreeVar]
+n:10 [binder, in hydras.MoreAck.expressibleExamples]
+n:10 [binder, in hydras.Ackermann.checkPrf]
+n:10 [binder, in hydras.solutions_exercises.MultisetWf]
+n:10 [binder, in hydras.Prelude.Fuel]
+n:10 [binder, in hydras.Epsilon0.F_omega]
+n:10 [binder, in hydras.MoreAck.Ack]
+n:10 [binder, in hydras.solutions_exercises.FibonacciPR]
+n:10 [binder, in gaia_hydras.ON_gfinite]
+n:10 [binder, in hydras.OrdinalNotations.ON_Finite]
+n:10 [binder, in gaia_hydras.onType]
+n:10 [binder, in hydras.Ackermann.codeList]
+n:10 [binder, in Goedel.codeSysPrf]
+n:10 [binder, in hydras.solutions_exercises.isqrt]
+n:100 [binder, in Goedel.PRrepresentable]
+n:100 [binder, in hydras.Ackermann.subAll]
+n:100 [binder, in hydras.Prelude.Iterates]
+n:100 [binder, in hydras.Prelude.MoreLists]
+n:100 [binder, in hydras.Ackermann.fol]
+n:100 [binder, in hydras.Ackermann.codeList]
+n:100 [binder, in hydras.MoreAck.PrimRecExamples]
+n:101 [binder, in hydras.Gamma0.T2]
+n:101 [binder, in hydras.Ackermann.subAll]
+n:101 [binder, in hydras.Ackermann.folProp]
+n:101 [binder, in Goedel.rosserPA]
+n:101 [binder, in hydras.Ackermann.fol]
+n:101 [binder, in hydras.Epsilon0.Canon]
+n:101 [binder, in hydras.Ackermann.codeList]
+n:101 [binder, in hydras.MoreAck.PrimRecExamples]
+n:1010 [binder, in gaia_hydras.nfwfgaia]
+n:102 [binder, in hydras.Ackermann.primRec]
+n:102 [binder, in hydras.Ackermann.fol]
+n:102 [binder, in hydras.Ackermann.codeList]
+n:1021 [binder, in gaia_hydras.nfwfgaia]
+n:1028 [binder, in gaia_hydras.nfwfgaia]
+n:103 [binder, in hydras.Epsilon0.T1]
+n:103 [binder, in hydras.Ackermann.subAll]
+n:103 [binder, in Goedel.rosserPA]
+n:103 [binder, in hydras.Ackermann.codeList]
+n:103 [binder, in hydras.Epsilon0.Hprime]
+n:103 [binder, in hydras.Prelude.MoreVectors]
+n:1031 [binder, in gaia_hydras.nfwfgaia]
+n:1033 [binder, in gaia_hydras.nfwfgaia]
+n:1036 [binder, in gaia_hydras.nfwfgaia]
+n:104 [binder, in hydras.Ackermann.subAll]
+n:104 [binder, in hydras.Ackermann.LNN2LNT]
+n:104 [binder, in hydras.Ackermann.codeList]
+n:105 [binder, in hydras.Ackermann.subProp]
+n:105 [binder, in hydras.Ackermann.fol]
+n:105 [binder, in hydras.Epsilon0.Canon]
+n:105 [binder, in hydras.Hydra.Hydra_Definitions]
+n:1051 [binder, in gaia_hydras.nfwfgaia]
+n:106 [binder, in hydras.Epsilon0.T1]
+n:106 [binder, in Goedel.PRrepresentable]
+n:106 [binder, in hydras.Ackermann.subAll]
+n:106 [binder, in hydras.Prelude.Iterates]
+n:106 [binder, in hydras.Ackermann.LNN2LNT]
+n:106 [binder, in additions.Pow]
+n:106 [binder, in hydras.Ackermann.fol]
+n:106 [binder, in hydras.Prelude.MoreVectors]
+n:107 [binder, in hydras.Ackermann.subAll]
+n:107 [binder, in hydras.MoreAck.AckNotPR]
+n:107 [binder, in hydras.Ackermann.subProp]
+n:107 [binder, in additions.fib]
+n:107 [binder, in hydras.Ackermann.codeList]
+n:107 [binder, in hydras.MoreAck.PrimRecExamples]
+n:107 [binder, in hydras.Hydra.Hydra_Definitions]
+n:1073 [binder, in gaia_hydras.nfwfgaia]
+n:108 [binder, in hydras.Ackermann.primRec]
+n:109 [binder, in hydras.Ackermann.LNN2LNT]
+n:109 [binder, in hydras.Ackermann.folProp]
+n:11 [binder, in hydras.Ackermann.folProof]
+n:11 [binder, in gaia_hydras.GF_alpha]
+n:11 [binder, in hydras.Prelude.Compat815]
+n:11 [binder, in hydras.Ackermann.LNN2LNT]
+n:11 [binder, in gaia_hydras.GCanon]
+n:11 [binder, in hydras.Ackermann.subProp]
+n:11 [binder, in hydras.Ackermann.codeSubFormula]
+n:11 [binder, in hydras.Hydra.Hydra_Theorems]
+n:11 [binder, in hydras.Ackermann.primRec]
+n:11 [binder, in gaia_hydras.ON_gfinite]
+n:11 [binder, in hydras.Gamma0.Gamma0]
+n:110 [binder, in hydras.Prelude.Iterates]
+n:110 [binder, in hydras.Epsilon0.Hprime]
+n:111 [binder, in hydras.Ackermann.LNN2LNT]
+n:112 [binder, in Goedel.PRrepresentable]
+n:112 [binder, in hydras.Gamma0.T2]
+n:112 [binder, in hydras.Ackermann.subAll]
+n:112 [binder, in hydras.Epsilon0.F_alpha]
+n:112 [binder, in hydras.Ackermann.codeList]
+n:112 [binder, in hydras.Epsilon0.Hprime]
+n:113 [binder, in gaia_hydras.T1Bridge]
+n:113 [binder, in hydras.Ackermann.primRec]
+n:1133 [binder, in gaia_hydras.nfwfgaia]
+n:114 [binder, in hydras.Epsilon0.T1]
+n:114 [binder, in hydras.Ackermann.subProp]
+n:114 [binder, in additions.fib]
+n:114 [binder, in hydras.Epsilon0.Hprime]
+n:1140 [binder, in gaia_hydras.nfwfgaia]
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+n:1534 [binder, in gaia_hydras.nfwfgaia]
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+n:19 [binder, in hydras.Epsilon0.Canon]
+n:19 [binder, in hydras.Ackermann.NNtheory]
+n:19 [binder, in hydras.Gamma0.Gamma0]
+n:19 [binder, in hydras.Ackermann.codePA]
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+n:1919 [binder, in gaia_hydras.nfwfgaia]
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+n:20 [binder, in gaia_hydras.GCanon]
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+n:22 [binder, in hydras.MoreAck.Ack]
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+n:23 [binder, in hydras.Prelude.Iterates]
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+n:23 [binder, in hydras.Prelude.Fuel]
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+n:23 [binder, in hydras.OrdinalNotations.ON_Finite]
+n:23 [binder, in hydras.Epsilon0.E0]
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+n:233 [binder, in hydras.Gamma0.Gamma0]
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+n:238 [binder, in hydras.Epsilon0.Paths]
+n:238 [binder, in gaia_hydras.nfwfgaia]
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+n:24 [binder, in hydras.Ackermann.extEqualNat]
+n:24 [binder, in gaia_hydras.GF_alpha]
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+n:24 [binder, in hydras.Ackermann.LNN2LNT]
+n:24 [binder, in hydras.MoreAck.AckNotPR]
+n:24 [binder, in hydras.Prelude.Fuel]
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+n:24 [binder, in hydras.Ackermann.codeList]
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+n:24 [binder, in hydras.Prelude.MoreVectors]
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+n:25 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+n:25 [binder, in hydras.Epsilon0.Canon]
+n:25 [binder, in hydras.OrdinalNotations.ON_Finite]
+n:25 [binder, in gaia_hydras.onType]
+n:25 [binder, in hydras.solutions_exercises.F_3]
+n:25 [binder, in hydras.Gamma0.Gamma0]
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+n:258 [binder, in hydras.rpo.rpo]
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+n:26 [binder, in hydras.Prelude.Iterates]
+n:26 [binder, in hydras.Ackermann.extEqualNat]
+n:26 [binder, in hydras.Hydra.O2H]
+n:26 [binder, in hydras.Epsilon0.Epsilon0rpo]
+n:26 [binder, in hydras.Prelude.Compat815]
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+n:26 [binder, in hydras.solutions_exercises.MultisetWf]
+n:26 [binder, in hydras.Prelude.Fuel]
+n:26 [binder, in hydras.Hydra.Hydra_Termination]
+n:26 [binder, in hydras.Ackermann.codeList]
+n:26 [binder, in hydras.MoreAck.FolExamples]
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+n:28 [binder, in hydras.Ackermann.codeSubTerm]
+n:28 [binder, in hydras.Epsilon0.Hessenberg]
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+n:28 [binder, in hydras.Schutte.AP]
+n:28 [binder, in hydras.Prelude.Fuel]
+n:28 [binder, in hydras.Ackermann.codeList]
+n:28 [binder, in hydras.solutions_exercises.F_3]
+n:28 [binder, in hydras.Hydra.Hydra_Lemmas]
+n:28 [binder, in hydras.MoreAck.PrimRecExamples]
+n:28 [binder, in hydras.Ackermann.expressible]
+n:28 [binder, in hydras.Prelude.MoreVectors]
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+n:289 [binder, in hydras.Epsilon0.Paths]
+n:289 [binder, in hydras.Gamma0.Gamma0]
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+n:29 [binder, in gaia_hydras.GF_alpha]
+n:29 [binder, in additions.Compatibility]
+n:29 [binder, in hydras.Ackermann.LNN2LNT]
+n:29 [binder, in hydras.Ackermann.subProp]
+n:29 [binder, in hydras.Schutte.GRelations]
+n:29 [binder, in hydras.Ackermann.codeSubFormula]
+n:29 [binder, in hydras.Schutte.AP]
+n:29 [binder, in hydras.Prelude.Fuel]
+n:29 [binder, in hydras.Ackermann.code]
+n:29 [binder, in hydras.Hydra.Hydra_Termination]
+n:29 [binder, in additions.Euclidean_Chains]
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+n:29 [binder, in Goedel.rosser]
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+n:290 [binder, in hydras.Ackermann.cPair]
+n:290 [binder, in hydras.Ackermann.codeList]
+n:290 [binder, in gaia_hydras.nfwfgaia]
+n:291 [binder, in hydras.Ackermann.codeList]
+n:292 [binder, in hydras.Ackermann.cPair]
+n:292 [binder, in hydras.Ackermann.codeList]
+n:292 [binder, in hydras.Epsilon0.Paths]
+n:292 [binder, in gaia_hydras.nfwfgaia]
+n:293 [binder, in hydras.Ackermann.codeList]
+n:293 [binder, in hydras.Gamma0.Gamma0]
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+n:294 [binder, in hydras.Ackermann.codeList]
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+n:296 [binder, in Goedel.PRrepresentable]
+n:296 [binder, in hydras.Ackermann.cPair]
+n:297 [binder, in hydras.Gamma0.Gamma0]
+n:298 [binder, in hydras.Ackermann.cPair]
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+n:3 [binder, in hydras.Prelude.Iterates]
+n:3 [binder, in hydras.MoreAck.expressibleExamples]
+n:3 [binder, in hydras.Ackermann.model]
+n:3 [binder, in Goedel.fixPoint]
+n:3 [binder, in hydras.Ackermann.folProp]
+n:3 [binder, in additions.More_on_positive]
+n:3 [binder, in hydras.Epsilon0.F_omega]
+n:3 [binder, in hydras.OrdinalNotations.ON_Finite]
+n:3 [binder, in hydras.solutions_exercises.F_3]
+n:3 [binder, in hydras.Gamma0.Gamma0]
+n:3 [binder, in hydras.Schutte.Addition]
+n:3 [binder, in gaia_hydras.GHessenberg]
+n:30 [binder, in hydras.Prelude.Compat815]
+n:30 [binder, in hydras.MoreAck.AckNotPR]
+n:30 [binder, in hydras.Ackermann.codeList]
+n:30 [binder, in gaia_hydras.nfwfgaia]
+n:30 [binder, in hydras.Ackermann.Languages]
+n:30 [binder, in hydras.Prelude.MoreVectors]
+n:300 [binder, in hydras.Ackermann.cPair]
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+n:302 [binder, in hydras.Ackermann.cPair]
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+n:40 [binder, in hydras.Ackermann.model]
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+n:40 [binder, in hydras.Prelude.Fuel]
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+n:43 [binder, in hydras.Prelude.Fuel]
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+n:43 [binder, in gaia_hydras.nfwfgaia]
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+n:44 [binder, in hydras.Prelude.Fuel]
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+n:46 [binder, in hydras.Hydra.Hydra_Lemmas]
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+n:48 [binder, in hydras.MoreAck.AckNotPR]
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+n:49 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+n:49 [binder, in hydras.Prelude.Fuel]
+n:49 [binder, in hydras.MoreAck.Ack]
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+n:49 [binder, in hydras.Ackermann.Languages]
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+n:496 [binder, in gaia_hydras.nfwfgaia]
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+n:499 [binder, in gaia_hydras.nfwfgaia]
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+n:5 [binder, in hydras.MoreAck.expressibleExamples]
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+n:5 [binder, in hydras.Prelude.Fuel]
+n:5 [binder, in hydras.solutions_exercises.FibonacciPR]
+n:5 [binder, in hydras.Ackermann.primRec]
+n:5 [binder, in hydras.Ackermann.codeNatToTerm]
+n:5 [binder, in hydras.Ackermann.wConsistent]
+n:5 [binder, in hydras.solutions_exercises.isqrt]
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+n:50 [binder, in hydras.Epsilon0.F_alpha]
+n:50 [binder, in hydras.Ackermann.folProp]
+n:50 [binder, in hydras.Ackermann.codeSubFormula]
+n:50 [binder, in hydras.solutions_exercises.MultisetWf]
+n:50 [binder, in hydras.MoreAck.Ack]
+n:50 [binder, in gaia_hydras.nfwfgaia]
+n:50 [binder, in hydras.Prelude.MoreVectors]
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+n:506 [binder, in hydras.Epsilon0.T1]
+n:508 [binder, in hydras.Epsilon0.T1]
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+n:51 [binder, in hydras.Ackermann.codeSubFormula]
+n:51 [binder, in hydras.MoreAck.Ack]
+n:51 [binder, in hydras.Ackermann.wellFormed]
+n:51 [binder, in gaia_hydras.nfwfgaia]
+n:51 [binder, in hydras.Ackermann.expressible]
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+n:511 [binder, in hydras.Epsilon0.Paths]
+n:516 [binder, in gaia_hydras.nfwfgaia]
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+n:517 [binder, in hydras.Epsilon0.Paths]
+n:518 [binder, in hydras.Ackermann.checkPrf]
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+n:52 [binder, in hydras.Epsilon0.T1]
+n:52 [binder, in hydras.OrdinalNotations.ON_Omega2]
+n:52 [binder, in hydras.Ackermann.model]
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+n:52 [binder, in hydras.Ackermann.codeSubFormula]
+n:52 [binder, in hydras.MoreAck.Ack]
+n:52 [binder, in gaia_hydras.T1Bridge]
+n:521 [binder, in gaia_hydras.nfwfgaia]
+n:523 [binder, in hydras.Epsilon0.T1]
+n:523 [binder, in hydras.Gamma0.Gamma0]
+n:523 [binder, in gaia_hydras.nfwfgaia]
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+n:53 [binder, in hydras.rpo.more_list]
+n:53 [binder, in hydras.Epsilon0.F_alpha]
+n:53 [binder, in hydras.MoreAck.AckNotPR]
+n:53 [binder, in hydras.Ackermann.codeSubFormula]
+n:53 [binder, in hydras.OrdinalNotations.OmegaOmega]
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+n:53 [binder, in hydras.Epsilon0.Paths]
+n:530 [binder, in gaia_hydras.nfwfgaia]
+n:531 [binder, in gaia_hydras.nfwfgaia]
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+n:535 [binder, in hydras.Ackermann.primRec]
+n:535 [binder, in hydras.Gamma0.Gamma0]
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+n:539 [binder, in hydras.Epsilon0.T1]
+n:54 [binder, in hydras.Epsilon0.Epsilon0rpo]
+n:54 [binder, in additions.Pow_variant]
+n:54 [binder, in additions.Pow]
+n:54 [binder, in hydras.Ackermann.codeSubFormula]
+n:54 [binder, in hydras.MoreAck.Ack]
+n:54 [binder, in hydras.Ackermann.folLogic3]
+n:54 [binder, in hydras.Hydra.Hydra_Lemmas]
+n:54 [binder, in hydras.Schutte.Addition]
+n:542 [binder, in hydras.Ackermann.primRec]
+n:543 [binder, in hydras.Epsilon0.T1]
+n:543 [binder, in hydras.Gamma0.Gamma0]
+n:543 [binder, in gaia_hydras.nfwfgaia]
+n:544 [binder, in gaia_hydras.nfwfgaia]
+n:545 [binder, in hydras.Epsilon0.T1]
+n:545 [binder, in hydras.Ackermann.primRec]
+n:548 [binder, in gaia_hydras.nfwfgaia]
+n:55 [binder, in Goedel.PRrepresentable]
+n:55 [binder, in hydras.Prelude.More_Arith]
+n:55 [binder, in hydras.Prelude.Iterates]
+n:55 [binder, in hydras.Ackermann.codeSubFormula]
+n:55 [binder, in hydras.Ackermann.folLogic3]
+n:55 [binder, in Goedel.rosser]
+n:55 [binder, in gaia_hydras.nfwfgaia]
+n:550 [binder, in hydras.Epsilon0.T1]
+n:550 [binder, in hydras.Epsilon0.Paths]
+n:552 [binder, in hydras.Epsilon0.T1]
+n:554 [binder, in hydras.Epsilon0.T1]
+n:556 [binder, in hydras.Epsilon0.T1]
+n:556 [binder, in hydras.Ackermann.primRec]
+n:556 [binder, in gaia_hydras.nfwfgaia]
+n:557 [binder, in hydras.Gamma0.Gamma0]
+n:558 [binder, in hydras.Epsilon0.Paths]
+n:559 [binder, in hydras.Epsilon0.T1]
+n:559 [binder, in hydras.Ackermann.primRec]
+n:559 [binder, in gaia_hydras.nfwfgaia]
+n:56 [binder, in hydras.Ackermann.model]
+n:56 [binder, in hydras.MoreAck.AckNotPR]
+n:56 [binder, in hydras.Ackermann.codeSubFormula]
+n:56 [binder, in hydras.OrdinalNotations.OmegaOmega]
+n:56 [binder, in hydras.Ackermann.fol]
+n:56 [binder, in hydras.Ackermann.codeList]
+n:56 [binder, in hydras.Prelude.MoreVectors]
+n:561 [binder, in hydras.Gamma0.Gamma0]
+n:561 [binder, in gaia_hydras.nfwfgaia]
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+n:563 [binder, in hydras.Epsilon0.Paths]
+n:563 [binder, in gaia_hydras.nfwfgaia]
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+n:568 [binder, in hydras.Gamma0.Gamma0]
+n:569 [binder, in hydras.Epsilon0.T1]
+n:57 [binder, in hydras.Ackermann.subProp]
+n:57 [binder, in hydras.MoreAck.Ack]
+n:57 [binder, in gaia_hydras.T1Bridge]
+n:57 [binder, in hydras.Epsilon0.Canon]
+n:57 [binder, in hydras.Ackermann.folLogic3]
+n:57 [binder, in Goedel.rosser]
+n:57 [binder, in gaia_hydras.nfwfgaia]
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+n:573 [binder, in hydras.Gamma0.Gamma0]
+n:574 [binder, in hydras.Epsilon0.T1]
+n:574 [binder, in hydras.Ackermann.primRec]
+n:578 [binder, in hydras.Epsilon0.T1]
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+n:58 [binder, in hydras.Epsilon0.Epsilon0rpo]
+n:58 [binder, in hydras.MoreAck.AckNotPR]
+n:58 [binder, in hydras.Ackermann.primRec]
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+n:589 [binder, in hydras.Gamma0.Gamma0]
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+n:59 [binder, in Goedel.rosserPA]
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+n:59 [binder, in hydras.Hydra.Hydra_Lemmas]
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+n:598 [binder, in hydras.Gamma0.Gamma0]
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+n:60 [binder, in hydras.Ackermann.code]
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+n:659 [binder, in hydras.Gamma0.Gamma0]
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+n:664 [binder, in hydras.Gamma0.Gamma0]
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+n:668 [binder, in gaia_hydras.nfwfgaia]
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+n:67 [binder, in hydras.Ackermann.primRec]
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+n:67 [binder, in hydras.Hydra.Hydra_Definitions]
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+n:682 [binder, in hydras.Ackermann.primRec]
+n:683 [binder, in hydras.Gamma0.Gamma0]
+n:683 [binder, in gaia_hydras.nfwfgaia]
+n:688 [binder, in hydras.Gamma0.Gamma0]
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+n:7 [binder, in hydras.MoreAck.expressibleExamples]
+n:7 [binder, in additions.Pow_variant]
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+n:7 [binder, in additions.More_on_positive]
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+n:7 [binder, in hydras.MoreAck.Ack]
+n:7 [binder, in hydras.Ackermann.primRec]
+n:7 [binder, in gaia_hydras.ON_gfinite]
+n:7 [binder, in hydras.OrdinalNotations.ON_Finite]
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+n:7 [binder, in hydras.Gamma0.Gamma0]
+n:7 [binder, in hydras.Schutte.Addition]
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+n:70 [binder, in gaia_hydras.GPaths]
+n:70 [binder, in additions.Pow]
+n:70 [binder, in hydras.Ackermann.folProp]
+n:70 [binder, in hydras.MoreAck.Ack]
+n:70 [binder, in Goedel.rosserPA]
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+n:702 [binder, in gaia_hydras.nfwfgaia]
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+n:71 [binder, in hydras.Ackermann.codeList]
+n:71 [binder, in gaia_hydras.nfwfgaia]
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+n:72 [binder, in hydras.MoreAck.Ack]
+n:72 [binder, in hydras.Hydra.BigBattle]
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+n:723 [binder, in hydras.Epsilon0.T1]
+n:724 [binder, in gaia_hydras.nfwfgaia]
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+n:74 [binder, in hydras.MoreAck.Ack]
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+n:75 [binder, in hydras.Ackermann.primRec]
+n:75 [binder, in hydras.Hydra.BigBattle]
+n:75 [binder, in hydras.Ackermann.codeList]
+n:75 [binder, in hydras.Epsilon0.Hprime]
+n:75 [binder, in gaia_hydras.nfwfgaia]
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+n:76 [binder, in Goedel.codeSysPrf]
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+n:764 [binder, in gaia_hydras.nfwfgaia]
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+n:769 [binder, in gaia_hydras.nfwfgaia]
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+n:784 [binder, in gaia_hydras.nfwfgaia]
+n:787 [binder, in gaia_hydras.nfwfgaia]
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+n:795 [binder, in gaia_hydras.nfwfgaia]
+n:796 [binder, in hydras.Ackermann.checkPrf]
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+n:8 [binder, in hydras.Hydra.BigBattle]
+n:8 [binder, in additions.Naive]
+n:8 [binder, in hydras.Ackermann.codeList]
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+n:806 [binder, in hydras.Ackermann.checkPrf]
+n:806 [binder, in gaia_hydras.nfwfgaia]
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+n:81 [binder, in hydras.Ackermann.LNN2LNT]
+n:81 [binder, in additions.Pow_variant]
+n:81 [binder, in hydras.Ackermann.model]
+n:81 [binder, in hydras.Ackermann.codeList]
+n:81 [binder, in hydras.Epsilon0.Hprime]
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+n:818 [binder, in hydras.Ackermann.primRec]
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+n:82 [binder, in hydras.Ackermann.folProp]
+n:82 [binder, in gaia_hydras.T1Bridge]
+n:82 [binder, in gaia_hydras.nfwfgaia]
+n:821 [binder, in hydras.Ackermann.primRec]
+n:825 [binder, in hydras.Ackermann.codeSubFormula]
+n:829 [binder, in hydras.Ackermann.codeSubFormula]
+n:83 [binder, in hydras.Epsilon0.T1]
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+n:83 [binder, in additions.Pow]
+n:83 [binder, in hydras.Ackermann.folProp]
+n:83 [binder, in hydras.MoreAck.Ack]
+n:83 [binder, in hydras.Ackermann.codeList]
+n:83 [binder, in gaia_hydras.nfwfgaia]
+n:834 [binder, in gaia_hydras.nfwfgaia]
+n:836 [binder, in hydras.Ackermann.codeSubFormula]
+n:839 [binder, in hydras.Ackermann.codeSubFormula]
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+n:84 [binder, in hydras.MoreAck.Ack]
+n:84 [binder, in hydras.Epsilon0.Canon]
+n:844 [binder, in hydras.Ackermann.checkPrf]
+n:844 [binder, in gaia_hydras.nfwfgaia]
+n:85 [binder, in hydras.Ackermann.subAll]
+n:85 [binder, in hydras.Prelude.Iterates]
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+n:85 [binder, in additions.Pow]
+n:85 [binder, in hydras.MoreAck.AckNotPR]
+n:85 [binder, in hydras.Ackermann.subProp]
+n:85 [binder, in Goedel.rosserPA]
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+n:85 [binder, in hydras.Ackermann.codeList]
+n:85 [binder, in gaia_hydras.nfwfgaia]
+n:858 [binder, in hydras.Ackermann.checkPrf]
+n:86 [binder, in Goedel.PRrepresentable]
+n:86 [binder, in hydras.Prelude.MoreLists]
+n:86 [binder, in hydras.Epsilon0.F_alpha]
+n:86 [binder, in hydras.Prelude.MoreVectors]
+n:862 [binder, in gaia_hydras.nfwfgaia]
+n:863 [binder, in hydras.Ackermann.checkPrf]
+n:867 [binder, in hydras.Ackermann.codeSubFormula]
+n:867 [binder, in gaia_hydras.nfwfgaia]
+n:87 [binder, in hydras.Hydra.O2H]
+n:87 [binder, in hydras.Ackermann.LNN2LNT]
+n:87 [binder, in additions.Pow_variant]
+n:87 [binder, in Goedel.rosserPA]
+n:87 [binder, in hydras.Ackermann.codeList]
+n:870 [binder, in hydras.Ackermann.codeSubFormula]
+n:873 [binder, in hydras.Ackermann.codeSubFormula]
+n:877 [binder, in hydras.Ackermann.codeSubFormula]
+n:88 [binder, in hydras.Epsilon0.T1]
+n:88 [binder, in hydras.Ackermann.codeList]
+n:88 [binder, in gaia_hydras.nfwfgaia]
+n:881 [binder, in hydras.Ackermann.codeSubFormula]
+n:89 [binder, in hydras.Gamma0.T2]
+n:89 [binder, in hydras.Ackermann.subAll]
+n:89 [binder, in hydras.Ackermann.codeList]
+n:89 [binder, in gaia_hydras.nfwfgaia]
+n:891 [binder, in hydras.Ackermann.codeSubFormula]
+n:891 [binder, in gaia_hydras.nfwfgaia]
+n:894 [binder, in hydras.Ackermann.codeSubFormula]
+n:9 [binder, in hydras.Ackermann.folProof]
+n:9 [binder, in hydras.Schutte.Correctness_E0]
+n:9 [binder, in hydras.Gamma0.T2]
+n:9 [binder, in hydras.Prelude.More_Arith]
+n:9 [binder, in hydras.Prelude.Iterates]
+n:9 [binder, in hydras.Prelude.Compat815]
+n:9 [binder, in hydras.Ackermann.LNN2LNT]
+n:9 [binder, in Goedel.fixPoint]
+n:9 [binder, in hydras.Epsilon0.Hessenberg]
+n:9 [binder, in additions.More_on_positive]
+n:9 [binder, in hydras.MoreAck.AckNotPR]
+n:9 [binder, in hydras.Ackermann.codeSubFormula]
+n:9 [binder, in hydras.Prelude.Fuel]
+n:9 [binder, in hydras.solutions_exercises.MorePRExamples]
+n:9 [binder, in hydras.Hydra.Hydra_Termination]
+n:9 [binder, in hydras.Hydra.BigBattle]
+n:9 [binder, in hydras.Ackermann.wellFormed]
+n:9 [binder, in hydras.Epsilon0.Canon]
+n:9 [binder, in additions.fib]
+n:9 [binder, in hydras.solutions_exercises.F_3]
+n:9 [binder, in gaia_hydras.nfwfgaia]
+n:9 [binder, in hydras.Schutte.Countable]
+n:90 [binder, in Goedel.PRrepresentable]
+n:90 [binder, in hydras.Epsilon0.F_alpha]
+n:90 [binder, in additions.Pow]
+n:90 [binder, in hydras.MoreAck.AckNotPR]
+n:90 [binder, in hydras.Ackermann.primRec]
+n:90 [binder, in hydras.Ackermann.folLogic3]
+n:90 [binder, in hydras.Prelude.MoreVectors]
+n:904 [binder, in gaia_hydras.nfwfgaia]
+n:907 [binder, in hydras.Ackermann.codeSubFormula]
+n:91 [binder, in Goedel.rosserPA]
+n:91 [binder, in hydras.Epsilon0.E0]
+n:91 [binder, in hydras.Ackermann.codeList]
+n:91 [binder, in hydras.Hydra.Hydra_Definitions]
+n:910 [binder, in hydras.Ackermann.codeSubFormula]
+n:912 [binder, in gaia_hydras.nfwfgaia]
+n:913 [binder, in hydras.Ackermann.codeSubFormula]
+n:918 [binder, in hydras.Ackermann.codeSubFormula]
+n:92 [binder, in hydras.Gamma0.T2]
+n:92 [binder, in hydras.Epsilon0.F_alpha]
+n:92 [binder, in hydras.MoreAck.AckNotPR]
+n:92 [binder, in hydras.Ackermann.fol]
+n:92 [binder, in hydras.Ackermann.folLogic3]
+n:92 [binder, in Goedel.codeSysPrf]
+n:923 [binder, in hydras.Ackermann.codeSubFormula]
+n:93 [binder, in hydras.Epsilon0.T1]
+n:93 [binder, in hydras.Ackermann.LNN2LNT]
+n:93 [binder, in additions.Pow]
+n:93 [binder, in hydras.Ackermann.codeList]
+n:93 [binder, in hydras.Prelude.MoreVectors]
+n:934 [binder, in hydras.Ackermann.codeSubFormula]
+n:936 [binder, in gaia_hydras.nfwfgaia]
+n:937 [binder, in hydras.Ackermann.codeSubFormula]
+n:94 [binder, in Goedel.PRrepresentable]
+n:94 [binder, in hydras.MoreAck.AckNotPR]
+n:94 [binder, in hydras.Ackermann.primRec]
+n:941 [binder, in gaia_hydras.nfwfgaia]
+n:946 [binder, in gaia_hydras.nfwfgaia]
+n:95 [binder, in hydras.Ackermann.subAll]
+n:95 [binder, in hydras.Ackermann.fol]
+n:95 [binder, in hydras.Ackermann.codeList]
+n:95 [binder, in hydras.MoreAck.PrimRecExamples]
+n:950 [binder, in gaia_hydras.nfwfgaia]
+n:952 [binder, in gaia_hydras.nfwfgaia]
+n:96 [binder, in additions.Pow]
+n:96 [binder, in Goedel.rosserPA]
+n:96 [binder, in hydras.Ackermann.folLogic3]
+n:96 [binder, in hydras.Hydra.Hydra_Definitions]
+n:967 [binder, in gaia_hydras.nfwfgaia]
+n:969 [binder, in gaia_hydras.nfwfgaia]
+n:97 [binder, in hydras.Prelude.MoreLists]
+n:97 [binder, in hydras.Epsilon0.F_alpha]
+n:97 [binder, in additions.Pow_variant]
+n:97 [binder, in hydras.MoreAck.AckNotPR]
+n:97 [binder, in hydras.Ackermann.subProp]
+n:97 [binder, in hydras.Ackermann.primRec]
+n:97 [binder, in hydras.Ackermann.codeList]
+n:97 [binder, in gaia_hydras.nfwfgaia]
+n:971 [binder, in gaia_hydras.nfwfgaia]
+n:975 [binder, in gaia_hydras.nfwfgaia]
+n:979 [binder, in gaia_hydras.nfwfgaia]
+n:98 [binder, in hydras.Epsilon0.T1]
+n:98 [binder, in hydras.Gamma0.T2]
+n:98 [binder, in hydras.Ackermann.codeList]
+n:98 [binder, in hydras.Epsilon0.Hprime]
+n:985 [binder, in gaia_hydras.nfwfgaia]
+n:987 [binder, in gaia_hydras.nfwfgaia]
+n:99 [binder, in hydras.Epsilon0.F_alpha]
+n:99 [binder, in hydras.Ackermann.subProp]
+n:99 [binder, in hydras.Ackermann.codeList]
+n:99 [binder, in hydras.Prelude.MoreVectors]
+

O

+OA_given [section, in hydras.OrdinalNotations.ON_O]
+OA:109 [binder, in hydras.OrdinalNotations.ON_Generic]
+OA:124 [binder, in hydras.OrdinalNotations.ON_Generic]
+OA:136 [binder, in hydras.OrdinalNotations.ON_Generic]
+OA:149 [binder, in hydras.OrdinalNotations.ON_Generic]
+OA:162 [binder, in hydras.OrdinalNotations.ON_Generic]
+OA:174 [binder, in hydras.OrdinalNotations.ON_Generic]
+OA:187 [binder, in hydras.OrdinalNotations.ON_Generic]
+OA:201 [binder, in hydras.OrdinalNotations.ON_Generic]
+OA:32 [binder, in hydras.OrdinalNotations.ON_Generic]
+OA:5 [binder, in hydras.OrdinalNotations.ON_O]
+OA:51 [binder, in hydras.OrdinalNotations.ON_Generic]
+OA:70 [binder, in hydras.OrdinalNotations.ON_Generic]
+OA:84 [binder, in hydras.OrdinalNotations.ON_Generic]
+OA:96 [binder, in hydras.OrdinalNotations.ON_Generic]
+OB:100 [binder, in hydras.OrdinalNotations.ON_Generic]
+OB:113 [binder, in hydras.OrdinalNotations.ON_Generic]
+OB:128 [binder, in hydras.OrdinalNotations.ON_Generic]
+OB:140 [binder, in hydras.OrdinalNotations.ON_Generic]
+OB:153 [binder, in hydras.OrdinalNotations.ON_Generic]
+OB:166 [binder, in hydras.OrdinalNotations.ON_Generic]
+OB:178 [binder, in hydras.OrdinalNotations.ON_Generic]
+OB:191 [binder, in hydras.OrdinalNotations.ON_Generic]
+OB:205 [binder, in hydras.OrdinalNotations.ON_Generic]
+OB:36 [binder, in hydras.OrdinalNotations.ON_Generic]
+OB:55 [binder, in hydras.OrdinalNotations.ON_Generic]
+OB:88 [binder, in hydras.OrdinalNotations.ON_Generic]
+of_image [lemma, in hydras.Schutte.Ordering_Functions]
+of_beta' [lemma, in hydras.Schutte.Ordering_Functions]
+OF_mono [projection, in hydras.Schutte.Ordering_Functions]
+OF_onto [projection, in hydras.Schutte.Ordering_Functions]
+OF_total [projection, in hydras.Schutte.Ordering_Functions]
+OF_segment [projection, in hydras.Schutte.Ordering_Functions]
+of_u [lemma, in hydras.Schutte.Addition]
+of_g [lemma, in hydras.Schutte.Addition]
+OK [definition, in additions.AM]
+Ok [constructor, in hydras.Epsilon0.Large_Sets_Examples]
+OK [definition, in additions.Euclidean_Chains]
+OK87 [lemma, in additions.Euclidean_Chains]
+Olt [instance, in hydras.Epsilon0.F_alpha]
+Olt [instance, in hydras.Epsilon0.L_alpha]
+Olt [instance, in hydras.Epsilon0.Hprime]
+ol'':40 [binder, in hydras.Schutte.CNF]
+ol'':45 [binder, in hydras.Schutte.CNF]
+ol':30 [binder, in hydras.Schutte.CNF]
+ol':34 [binder, in hydras.Schutte.CNF]
+ol':37 [binder, in hydras.Schutte.CNF]
+ol':39 [binder, in hydras.Schutte.CNF]
+ol':43 [binder, in hydras.Schutte.CNF]
+ol:29 [binder, in hydras.Schutte.CNF]
+ol:33 [binder, in hydras.Schutte.CNF]
+ol:35 [binder, in hydras.Schutte.CNF]
+ol:38 [binder, in hydras.Schutte.CNF]
+ol:41 [binder, in hydras.Schutte.CNF]
+omega [abbreviation, in hydras.Epsilon0.T1]
+omega [abbreviation, in hydras.OrdinalNotations.ON_Omega2]
+omega [abbreviation, in hydras.Gamma0.T2]
+omega [abbreviation, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+omega [abbreviation, in hydras.Schutte.Schutte_basics]
+Omega [instance, in hydras.OrdinalNotations.ON_Omega]
+omeganf [lemma, in gaia_hydras.T1Bridge]
+OmegaOmega [library]
+omega_limit [lemma, in hydras.Epsilon0.T1]
+omega_omega [definition, in hydras.Epsilon0.T1]
+omega_term_def [lemma, in hydras.Epsilon0.T1]
+omega_term_plus_rw [lemma, in hydras.Epsilon0.T1]
+omega_tower [definition, in hydras.Epsilon0.T1]
+omega_term [definition, in hydras.Epsilon0.T1]
+omega_ap [lemma, in hydras.OrdinalNotations.ON_Omega2]
+Omega_limit_limitb [lemma, in hydras.OrdinalNotations.ON_Omega2]
+omega_is_limit [lemma, in hydras.OrdinalNotations.ON_Omega2]
+omega_ex2 [definition, in hydras.Epsilon0.Large_Sets]
+omega_ex1 [definition, in hydras.Epsilon0.Large_Sets]
+omega_square_thrice_eqn [definition, in hydras.Epsilon0.Large_Sets]
+omega_mult_mlarge_0 [lemma, in hydras.Epsilon0.Large_Sets]
+omega_tower_mono [lemma, in hydras.Schutte.AP]
+omega_eqn [lemma, in hydras.Schutte.AP]
+omega_second_AP [lemma, in hydras.Schutte.AP]
+omega_tower [definition, in hydras.Schutte.AP]
+Omega_as_lub [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+omega_not_succ [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+omega_is_limit [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Omega_plus_Omega [instance, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+omega_omega_1_3 [definition, in hydras.Epsilon0.Large_Sets_Examples]
+omega_omega_1_4 [definition, in hydras.Epsilon0.Large_Sets_Examples]
+omega_ref [lemma, in gaia_hydras.T1Bridge]
+Omega_limit_Limit [lemma, in hydras.Prelude.MoreOrders]
+Omega_limit_not_Succ [lemma, in hydras.Prelude.MoreOrders]
+Omega_limit_type [definition, in hydras.Prelude.MoreOrders]
+Omega_limit [definition, in hydras.Prelude.MoreOrders]
+omega_limit_least_gt [lemma, in hydras.Schutte.Schutte_basics]
+omega_limit_least [lemma, in hydras.Schutte.Schutte_basics]
+omega_limit [definition, in hydras.Schutte.Schutte_basics]
+Omega_term_plus [lemma, in hydras.Epsilon0.E0]
+Omega_2 [instance, in hydras.Epsilon0.E0]
+Omega_term [instance, in hydras.Epsilon0.E0]
+omega_lt_wf [lemma, in gaia_hydras.onType]
+omega_ok [instance, in hydras.OrdinalNotations.ON_Omega]
+Omega_comp [instance, in hydras.OrdinalNotations.ON_Omega]
+omega_lt_epsilon [lemma, in hydras.Gamma0.Gamma0]
+omega_lt_epsilon0 [lemma, in hydras.Gamma0.Gamma0]
+Omega_Small [library]
+Omega2 [instance, in hydras.OrdinalNotations.ON_Omega2]
+Omega2_Small [library]
+om12 [definition, in gaia_hydras.onType]
+om67 [definition, in gaia_hydras.onType]
+ON [record, in hydras.OrdinalNotations.ON_Generic]
+OnCodeList [library]
+ONDef [module, in gaia_hydras.onType]
+ONDef.Exports [module, in gaia_hydras.onType]
+ONDef.Exports.Lemmas [section, in gaia_hydras.onType]
+ONDef.Exports.Lemmas.disp [variable, in gaia_hydras.onType]
+ONDef.Exports.Lemmas.U [variable, in gaia_hydras.onType]
+ONDef.Exports.ON [abbreviation, in gaia_hydras.onType]
+ONDef.Exports.on [abbreviation, in gaia_hydras.onType]
+ONDef.Exports.ONMixin [abbreviation, in gaia_hydras.onType]
+ONDef.Exports.tricho [definition, in gaia_hydras.onType]
+ONDef.Exports.trichoP [lemma, in gaia_hydras.onType]
+ONDef.Exports.wf [lemma, in gaia_hydras.onType]
+ONDef.Mixin [constructor, in gaia_hydras.onType]
+ONDef.mixin_of [record, in gaia_hydras.onType]
+ONDef.on_struct [definition, in gaia_hydras.onType]
+ONDef.Pack [constructor, in gaia_hydras.onType]
+ONDef.Packing [section, in gaia_hydras.onType]
+ONDef.Packing.cT [variable, in gaia_hydras.onType]
+ONDef.Packing.disp [variable, in gaia_hydras.onType]
+ONDef.pack_type [record, in gaia_hydras.onType]
+ONDef.type [projection, in gaia_hydras.onType]
+one [abbreviation, in hydras.Epsilon0.T1]
+one [abbreviation, in hydras.Gamma0.T2]
+ONEpsilon0Def [section, in gaia_hydras.onType]
+one_ref [lemma, in gaia_hydras.T1Bridge]
+one_right [projection, in additions.Monoid_def]
+one_left [projection, in additions.Monoid_def]
+one_step [inductive, in hydras.Hydra.BigBattle]
+one_mod_neutral_r [lemma, in additions.Monoid_instances]
+one_mod_neutral_l [lemma, in additions.Monoid_instances]
+One_node [constructor, in additions.Addition_Chains]
+one_plus_infinite [lemma, in hydras.Schutte.Addition]
+one_plus_omega [lemma, in hydras.Schutte.Addition]
+one:100 [binder, in additions.AM]
+one:103 [binder, in additions.Addition_Chains]
+one:105 [binder, in additions.AM]
+one:110 [binder, in additions.AM]
+one:111 [binder, in additions.Addition_Chains]
+one:117 [binder, in additions.Addition_Chains]
+one:120 [binder, in additions.AM]
+one:124 [binder, in additions.Addition_Chains]
+one:125 [binder, in additions.AM]
+one:131 [binder, in additions.Addition_Chains]
+one:161 [binder, in additions.Addition_Chains]
+one:168 [binder, in additions.Addition_Chains]
+one:174 [binder, in additions.Addition_Chains]
+one:36 [binder, in additions.FirstSteps]
+one:37 [binder, in additions.AM]
+one:42 [binder, in additions.FirstSteps]
+one:43 [binder, in additions.AM]
+one:54 [binder, in additions.AM]
+one:59 [binder, in additions.Monoid_def]
+one:61 [binder, in additions.AM]
+one:63 [binder, in additions.Monoid_def]
+one:74 [binder, in additions.AM]
+one:8 [binder, in additions.Monoid_def]
+one:84 [binder, in additions.AM]
+one:95 [binder, in additions.AM]
+onFiniteDef [section, in gaia_hydras.onType]
+onFiniteDef.i [variable, in gaia_hydras.onType]
+onFiniteMixin [definition, in gaia_hydras.onType]
+onFiniteType [definition, in gaia_hydras.onType]
+onOmegaDef [section, in gaia_hydras.onType]
+onOmegaMixin [definition, in gaia_hydras.onType]
+onOmegaType [definition, in gaia_hydras.onType]
+onType [library]
+on_length [section, in hydras.Gamma0.T2]
+On_alpha.S2.H [variable, in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.S2 [section, in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.S1.Hbeta [variable, in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.S1.beta [variable, in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.S1.H [variable, in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.S1 [section, in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.HnonZero [variable, in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.Halpha [variable, in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.alpha [variable, in hydras.solutions_exercises.Limit_Infinity]
+On_alpha [section, in hydras.solutions_exercises.Limit_Infinity]
+ON_O [instance, in hydras.OrdinalNotations.ON_O]
+ON_O_comp [instance, in hydras.OrdinalNotations.ON_O]
+On_Iota.unique_P [variable, in hydras.Schutte.MoreEpsilonIota]
+On_Iota.inhA [variable, in hydras.Schutte.MoreEpsilonIota]
+On_Iota.P [variable, in hydras.Schutte.MoreEpsilonIota]
+On_Iota.A [variable, in hydras.Schutte.MoreEpsilonIota]
+On_Iota [section, in hydras.Schutte.MoreEpsilonIota]
+ON_mult [instance, in hydras.OrdinalNotations.ON_mult]
+ON_plus [instance, in hydras.OrdinalNotations.ON_plus]
+ON_op_ok [definition, in hydras.OrdinalNotations.ON_Generic]
+ON_fun_ok [definition, in hydras.OrdinalNotations.ON_Generic]
+ON_cst_ok [definition, in hydras.OrdinalNotations.ON_Generic]
+On_compare_spec [projection, in hydras.OrdinalNotations.ON_Generic]
+ON_correct_onto [projection, in hydras.OrdinalNotations.ON_Generic]
+ON_correct_inj [projection, in hydras.OrdinalNotations.ON_Generic]
+ON_correct [record, in hydras.OrdinalNotations.ON_Generic]
+ON_Iso [record, in hydras.OrdinalNotations.ON_Generic]
+ON_le [definition, in hydras.OrdinalNotations.ON_Generic]
+ON_lt [definition, in hydras.OrdinalNotations.ON_Generic]
+ON_compare [definition, in hydras.OrdinalNotations.ON_Generic]
+ON_t [definition, in hydras.OrdinalNotations.ON_Generic]
+ON_wf [projection, in hydras.OrdinalNotations.ON_Generic]
+ON_comp [projection, in hydras.OrdinalNotations.ON_Generic]
+ON_Generic [library]
+ON_mult [library]
+ON_Finite [library]
+ON_Omega_plus_omega [library]
+ON_Omega [library]
+ON_Omega2 [library]
+ON_plus [library]
+ON_gfinite [library]
+ON_O [library]
+on:10 [binder, in hydras.OrdinalNotations.ON_Generic]
+on:14 [binder, in hydras.OrdinalNotations.ON_Generic]
+On:4 [binder, in hydras.solutions_exercises.lt_succ_le]
+On:5 [binder, in hydras.solutions_exercises.predSuccUnicity]
+OO [module, in hydras.OrdinalNotations.OmegaOmega]
+OO.compare_OO [instance, in hydras.OrdinalNotations.OmegaOmega]
+OO.data [projection, in hydras.OrdinalNotations.OmegaOmega]
+OO.data_ok [projection, in hydras.OrdinalNotations.OmegaOmega]
+OO.embed [instance, in hydras.OrdinalNotations.OmegaOmega]
+OO.Fin [instance, in hydras.OrdinalNotations.OmegaOmega]
+OO.le [definition, in hydras.OrdinalNotations.OmegaOmega]
+OO.lt [definition, in hydras.OrdinalNotations.OmegaOmega]
+OO.lt_wf [lemma, in hydras.OrdinalNotations.OmegaOmega]
+OO.lt_embed [lemma, in hydras.OrdinalNotations.OmegaOmega]
+OO.mkord [constructor, in hydras.OrdinalNotations.OmegaOmega]
+OO.mult [instance, in hydras.OrdinalNotations.OmegaOmega]
+OO.nf_proof_unicity [lemma, in hydras.OrdinalNotations.OmegaOmega]
+OO.omega [abbreviation, in hydras.OrdinalNotations.OmegaOmega]
+OO.ON_OO [instance, in hydras.OrdinalNotations.OmegaOmega]
+OO.OO [record, in hydras.OrdinalNotations.OmegaOmega]
+OO.OO_comp [instance, in hydras.OrdinalNotations.OmegaOmega]
+OO.OO_eq_intro [lemma, in hydras.OrdinalNotations.OmegaOmega]
+OO.oo_str [instance, in hydras.OrdinalNotations.OmegaOmega]
+OO.phi0 [instance, in hydras.OrdinalNotations.OmegaOmega]
+OO.plus [instance, in hydras.OrdinalNotations.OmegaOmega]
+OO.Succ [instance, in hydras.OrdinalNotations.OmegaOmega]
+OO.Zero [instance, in hydras.OrdinalNotations.OmegaOmega]
+OO._Omega [instance, in hydras.OrdinalNotations.OmegaOmega]
+_ * _ (OO_scope) [notation, in hydras.OrdinalNotations.OmegaOmega]
+omega^ (OO_scope) [notation, in hydras.OrdinalNotations.OmegaOmega]
+_ * _ (OO_scope) [notation, in hydras.OrdinalNotations.OmegaOmega]
+_ + _ (OO_scope) [notation, in hydras.OrdinalNotations.OmegaOmega]
+OpaqueWf [module, in hydras.Prelude.Fuel]
+OpaqueWf.lt_wf [lemma, in hydras.Prelude.Fuel]
+opA:194 [binder, in hydras.OrdinalNotations.ON_Generic]
+opB:195 [binder, in hydras.OrdinalNotations.ON_Generic]
+open [definition, in hydras.Ackermann.PA]
+openClosed [lemma, in hydras.Ackermann.folLogic]
+oplus [definition, in hydras.Epsilon0.Hessenberg]
+Oplus [instance, in hydras.Epsilon0.E0]
+oplus [definition, in gaia_hydras.GHessenberg]
+oplusA [lemma, in gaia_hydras.GHessenberg]
+oplusa0 [lemma, in gaia_hydras.GHessenberg]
+oplusC [lemma, in gaia_hydras.GHessenberg]
+oplusE [lemma, in gaia_hydras.GHessenberg]
+oplus_lt_phi0 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_of_phi0 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_of_phi0_0 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_strict_mono_bi [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_strict_mono_LT_r [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_strict_mono_LT_l [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_strict_mono_l [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_strict_mono_r [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_strict_mono_0 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_le2 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_le1 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_lt2 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_lt1 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_lt_0 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_assoc [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_assoc_0 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_lt_rw2 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_comm [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_comm_0 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_nf [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_nf_0 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_bounded_phi0 [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_cons_cons [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_eqn [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_compare_Gt [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_gt_rw [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_eq_rw [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_lt_rw [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_compare_Lt [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_0_l [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_0_r [lemma, in hydras.Epsilon0.Hessenberg]
+oplus_aux:8 [binder, in hydras.Epsilon0.Hessenberg]
+oPlus_rw [lemma, in hydras.Epsilon0.E0]
+Oplus_assoc [instance, in hydras.Epsilon0.E0]
+oplus_lt_phi0 [lemma, in gaia_hydras.GHessenberg]
+oplus_strict_mono_r [lemma, in gaia_hydras.GHessenberg]
+oplus_strict_mono_l [lemma, in gaia_hydras.GHessenberg]
+oplus_lt2 [lemma, in gaia_hydras.GHessenberg]
+oplus_lt1 [lemma, in gaia_hydras.GHessenberg]
+oplus_nf [lemma, in gaia_hydras.GHessenberg]
+oplus_aux:14 [binder, in gaia_hydras.GHessenberg]
+oplus0b [lemma, in gaia_hydras.GHessenberg]
+optimal [definition, in additions.Addition_Chains]
+optimal_upper_bound [lemma, in additions.Addition_Chains]
+optimal_generator [definition, in additions.Addition_Chains]
+op_comm [projection, in additions.Monoid_def]
+op_assoc [projection, in additions.Monoid_def]
+op:100 [binder, in additions.Euclidean_Chains]
+op:102 [binder, in additions.fib]
+op:102 [binder, in additions.Addition_Chains]
+op:104 [binder, in additions.AM]
+op:109 [binder, in additions.AM]
+op:109 [binder, in additions.fib]
+op:110 [binder, in additions.Addition_Chains]
+op:111 [binder, in additions.Euclidean_Chains]
+op:116 [binder, in additions.Addition_Chains]
+op:117 [binder, in additions.fib]
+op:119 [binder, in additions.AM]
+op:123 [binder, in additions.Addition_Chains]
+op:124 [binder, in additions.AM]
+op:130 [binder, in additions.Addition_Chains]
+op:160 [binder, in additions.Addition_Chains]
+op:163 [binder, in additions.Euclidean_Chains]
+op:167 [binder, in additions.Addition_Chains]
+op:169 [binder, in additions.Euclidean_Chains]
+op:173 [binder, in additions.Addition_Chains]
+op:183 [binder, in additions.Euclidean_Chains]
+op:195 [binder, in additions.Euclidean_Chains]
+op:36 [binder, in additions.AM]
+op:36 [binder, in additions.Euclidean_Chains]
+op:42 [binder, in additions.AM]
+op:42 [binder, in additions.Addition_Chains]
+op:43 [binder, in additions.Euclidean_Chains]
+op:48 [binder, in additions.Euclidean_Chains]
+op:5 [binder, in additions.Monoid_def]
+op:51 [binder, in additions.Euclidean_Chains]
+op:53 [binder, in additions.AM]
+op:54 [binder, in additions.Euclidean_Chains]
+op:54 [binder, in additions.Addition_Chains]
+op:58 [binder, in additions.Monoid_def]
+op:60 [binder, in additions.AM]
+op:62 [binder, in additions.Monoid_def]
+op:62 [binder, in additions.Euclidean_Chains]
+op:69 [binder, in additions.Euclidean_Chains]
+op:7 [binder, in additions.Monoid_def]
+op:73 [binder, in additions.AM]
+op:83 [binder, in additions.AM]
+op:89 [binder, in additions.Euclidean_Chains]
+op:94 [binder, in additions.AM]
+op:99 [binder, in additions.AM]
+Ord [axiom, in hydras.Schutte.Schutte_basics]
+ord [definition, in hydras.Schutte.Ordering_Functions]
+ordering_function_least_least [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_unbounded_unbounded [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_function_ex [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_segment_ex_unique [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_segments_of_B [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_function_seg_unicity [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.O2 [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.O1 [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.f2 [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.f1 [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.A2 [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.A1 [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.B [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity [section, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.O2 [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.O1 [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.f2 [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.f1 [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.A2 [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.A1 [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.B [variable, in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1 [section, in hydras.Schutte.Ordering_Functions]
+ordering_le [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_function_seg [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_function_mono_weakR [lemma, in hydras.Schutte.Ordering_Functions]
+Ordering_bijection [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_function_monoR [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_function_mono_weak [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_function_mono [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_function_In [lemma, in hydras.Schutte.Ordering_Functions]
+ordering_segment [definition, in hydras.Schutte.Ordering_Functions]
+ordering_function [record, in hydras.Schutte.Ordering_Functions]
+Ordering_Functions [library]
+ordinal [abbreviation, in hydras.Schutte.Schutte_basics]
+ordinal_ok [lemma, in hydras.Schutte.Schutte_basics]
+ordinal_segment [lemma, in hydras.Schutte.Ordering_Functions]
+OrdNotations [library]
+ord_eq [lemma, in hydras.Schutte.Ordering_Functions]
+ord_ok [lemma, in hydras.Schutte.Ordering_Functions]
+orE [lemma, in hydras.Ackermann.LNN]
+orE [lemma, in hydras.Ackermann.LNT]
+orE [lemma, in hydras.Ackermann.folLogic]
+orH [definition, in hydras.Ackermann.fol]
+orI1 [lemma, in hydras.Ackermann.LNN]
+orI1 [lemma, in hydras.Ackermann.LNT]
+orI1 [lemma, in hydras.Ackermann.folLogic]
+orI2 [lemma, in hydras.Ackermann.LNN]
+orI2 [lemma, in hydras.Ackermann.LNT]
+orI2 [lemma, in hydras.Ackermann.folLogic]
+orRel [definition, in hydras.Ackermann.primRec]
+orRelPR [lemma, in hydras.Ackermann.primRec]
+orSym [lemma, in hydras.Ackermann.LNN]
+orSym [lemma, in hydras.Ackermann.LNT]
+orSym [lemma, in hydras.Ackermann.folLogic]
+orSys [lemma, in hydras.Ackermann.LNN]
+orSys [lemma, in hydras.Ackermann.LNT]
+orSys [lemma, in hydras.Ackermann.folLogic]
+output:62 [binder, in hydras.Ackermann.codeSubFormula]
+o_length [definition, in hydras.rpo.more_list]
+o_finite_mult_mono [lemma, in hydras.Epsilon0.Hessenberg]
+o_finite_mult_lt_phi0_1 [lemma, in hydras.Epsilon0.Hessenberg]
+o_finite_mult_rw [lemma, in hydras.Epsilon0.Hessenberg]
+o_finite_mult_nf [lemma, in hydras.Epsilon0.Hessenberg]
+o_finite_mult [definition, in hydras.Epsilon0.Hessenberg]
+o_finite_mult_S_rw [lemma, in hydras.Hydra.Hydra_Termination]
+o_36_of_42 [definition, in gaia_hydras.ON_gfinite]
+o_33_of_42 [definition, in gaia_hydras.ON_gfinite]
+o_finite_mult [definition, in gaia_hydras.GHessenberg]
+o':423 [binder, in hydras.Gamma0.Gamma0]
+o':434 [binder, in hydras.Gamma0.Gamma0]
+o':437 [binder, in hydras.Gamma0.Gamma0]
+o':61 [binder, in hydras.Epsilon0.Epsilon0rpo]
+o':64 [binder, in hydras.Epsilon0.Epsilon0rpo]
+o1:15 [binder, in hydras.rpo.rpo]
+o1:20 [binder, in hydras.rpo.rpo]
+o1:6 [binder, in hydras.rpo.rpo]
+O2H [library]
+o2iota [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+o2iota_1 [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+o2iota_0 [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+o2:16 [binder, in hydras.rpo.rpo]
+o2:21 [binder, in hydras.rpo.rpo]
+o2:7 [binder, in hydras.rpo.rpo]
+O3O [definition, in hydras.OrdinalNotations.Example_3PlusOmega]
+o:10 [binder, in hydras.Hydra.O2H]
+o:124 [binder, in hydras.Schutte.Schutte_basics]
+o:13 [binder, in hydras.Hydra.O2H]
+o:131 [binder, in hydras.Schutte.Ordering_Functions]
+o:18 [binder, in hydras.Schutte.Addition]
+o:19 [binder, in hydras.Schutte.Addition]
+o:280 [binder, in hydras.Gamma0.Gamma0]
+o:36 [binder, in hydras.Epsilon0.Epsilon0rpo]
+o:424 [binder, in hydras.Gamma0.Gamma0]
+o:435 [binder, in hydras.Gamma0.Gamma0]
+o:436 [binder, in hydras.Gamma0.Gamma0]
+o:537 [binder, in hydras.Epsilon0.T1]
+o:619 [binder, in hydras.Gamma0.Gamma0]
+o:62 [binder, in hydras.Epsilon0.Epsilon0rpo]
+o:628 [binder, in hydras.Gamma0.Gamma0]
+o:63 [binder, in hydras.Epsilon0.Epsilon0rpo]
+o:63 [binder, in hydras.Schutte.CNF]
+o:64 [binder, in hydras.Schutte.CNF]
+o:75 [binder, in hydras.Schutte.Ordering_Functions]
+

P

+P [record, in hydras.Epsilon0.F_alpha]
+P [axiom, in hydras.Prelude.LibHyps_Experiments]
+P [record, in hydras.Epsilon0.Hprime]
+PA [definition, in hydras.Prelude.DecPreOrder_Instances]
+PA [projection, in hydras.Epsilon0.F_alpha]
+PA [definition, in hydras.Ackermann.PA]
+PA [section, in hydras.Ackermann.PA]
+PA [projection, in hydras.Epsilon0.Hprime]
+PA [library]
+PAboundedLT [lemma, in hydras.Ackermann.NN2PA]
+PAconsistent [lemma, in hydras.Ackermann.PAconsistent]
+PAconsistent [library]
+PAdec [lemma, in hydras.Ackermann.PA]
+PAIncomplete [lemma, in Goedel.rosserPA]
+PairwiseEqual [definition, in hydras.Ackermann.folLogic3]
+pair:134 [binder, in Goedel.PRrepresentable]
+pair:145 [binder, in Goedel.PRrepresentable]
+pair:156 [binder, in Goedel.PRrepresentable]
+pair:239 [binder, in Goedel.PRrepresentable]
+pair:245 [binder, in Goedel.PRrepresentable]
+pair:460 [binder, in Goedel.PRrepresentable]
+pair:468 [binder, in Goedel.PRrepresentable]
+paPlusSym [lemma, in hydras.Ackermann.PAtheory]
+parametric [definition, in additions.Addition_Chains]
+param_correctness_for_free [lemma, in additions.Addition_Chains]
+param_correctness [lemma, in additions.Addition_Chains]
+param_correctness_nat [lemma, in additions.Addition_Chains]
+param_correctness_aux [lemma, in additions.Addition_Chains]
+PartialFun [library]
+path [abbreviation, in gaia_hydras.GPaths]
+path [definition, in hydras.Epsilon0.Paths]
+PAtheory [library]
+pathS [definition, in gaia_hydras.GPaths]
+pathS [definition, in hydras.Epsilon0.Paths]
+Paths [library]
+path_to_round_plus [lemma, in hydras.Hydra.O2H]
+path_toS_round_plus [lemma, in hydras.Hydra.O2H]
+path_toS_trace [lemma, in hydras.Hydra.O2H]
+path_to_LT [lemma, in gaia_hydras.GPaths]
+path_to_iff1 [lemma, in gaia_hydras.GPaths]
+path_to_inv1 [lemma, in gaia_hydras.GPaths]
+path_tob [definition, in gaia_hydras.GPaths]
+path_to [definition, in gaia_hydras.GPaths]
+path_to_omega_mult [lemma, in hydras.Epsilon0.Large_Sets]
+path_to_standard_equiv [lemma, in hydras.Epsilon0.Paths]
+path_toS_standardS_equiv [lemma, in hydras.Epsilon0.Paths]
+path_to_S_standard_pathS [lemma, in hydras.Epsilon0.Paths]
+path_to_S_iota_from [lemma, in hydras.Epsilon0.Paths]
+path_to_LT [lemma, in hydras.Epsilon0.Paths]
+path_toS_LT [lemma, in hydras.Epsilon0.Paths]
+path_to_gnaw [lemma, in hydras.Epsilon0.Paths]
+path_to_mult [lemma, in hydras.Epsilon0.Paths]
+path_toS_mult [lemma, in hydras.Epsilon0.Paths]
+path_to_tail [lemma, in hydras.Epsilon0.Paths]
+path_toS_tail [lemma, in hydras.Epsilon0.Paths]
+path_to_not_In_zero [lemma, in hydras.Epsilon0.Paths]
+path_to_zero [lemma, in hydras.Epsilon0.Paths]
+path_toS_zero [lemma, in hydras.Epsilon0.Paths]
+path_toS_zero_inv [lemma, in hydras.Epsilon0.Paths]
+path_toS_zero_one [lemma, in hydras.Epsilon0.Paths]
+path_toS_zero_but_last [lemma, in hydras.Epsilon0.Paths]
+path_to_appR [lemma, in hydras.Epsilon0.Paths]
+path_toS_appR [lemma, in hydras.Epsilon0.Paths]
+path_to_decompose [lemma, in hydras.Epsilon0.Paths]
+path_toS_decompose [lemma, in hydras.Epsilon0.Paths]
+path_to_app [lemma, in hydras.Epsilon0.Paths]
+path_toS_app [lemma, in hydras.Epsilon0.Paths]
+path_toS_gnawS [lemma, in hydras.Epsilon0.Paths]
+path_acc_from [lemma, in hydras.Epsilon0.Paths]
+path_toS_nf [lemma, in hydras.Epsilon0.Paths]
+path_to_path_toS_iff [lemma, in hydras.Epsilon0.Paths]
+path_to_path_toS [lemma, in hydras.Epsilon0.Paths]
+path_toS_path_to [lemma, in hydras.Epsilon0.Paths]
+path_to_interval_inv_le [lemma, in hydras.Epsilon0.Paths]
+path_toS_cons [constructor, in hydras.Epsilon0.Paths]
+path_toS_1 [constructor, in hydras.Epsilon0.Paths]
+path_toS [inductive, in hydras.Epsilon0.Paths]
+path_to_not_nil [lemma, in hydras.Epsilon0.Paths]
+path_to_cons [constructor, in hydras.Epsilon0.Paths]
+path_to_1 [constructor, in hydras.Epsilon0.Paths]
+path_to [inductive, in hydras.Epsilon0.Paths]
+paZeroOrSucc [lemma, in hydras.Ackermann.PAtheory]
+PA_Succ [lemma, in hydras.Epsilon0.Hprime]
+PA_Zero [lemma, in hydras.Epsilon0.Hprime]
+pa1 [lemma, in hydras.Ackermann.PA]
+PA1 [definition, in hydras.Ackermann.PA]
+pa2 [lemma, in hydras.Ackermann.PA]
+PA2 [definition, in hydras.Ackermann.PA]
+pa3 [lemma, in hydras.Ackermann.PA]
+PA3 [definition, in hydras.Ackermann.PA]
+pa4 [lemma, in hydras.Ackermann.PA]
+PA4 [definition, in hydras.Ackermann.PA]
+pa5 [lemma, in hydras.Ackermann.PA]
+PA5 [definition, in hydras.Ackermann.PA]
+pa6 [lemma, in hydras.Ackermann.PA]
+PA6 [definition, in hydras.Ackermann.PA]
+PA7 [definition, in hydras.Ackermann.PA]
+pA:50 [binder, in hydras.Prelude.Sort_spec]
+PA:64 [binder, in hydras.Prelude.Sort_spec]
+PB [definition, in hydras.Prelude.DecPreOrder_Instances]
+PB [projection, in hydras.Epsilon0.F_alpha]
+PB [projection, in hydras.Epsilon0.Hprime]
+PB:5 [binder, in hydras.Prelude.DecPreOrder_Instances]
+pB:51 [binder, in hydras.Prelude.Sort_spec]
+PB:65 [binder, in hydras.Prelude.Sort_spec]
+PC [projection, in hydras.Epsilon0.F_alpha]
+PC [projection, in hydras.Epsilon0.Hprime]
+PC_Zero [lemma, in hydras.Epsilon0.Hprime]
+PD [projection, in hydras.Epsilon0.F_alpha]
+PD [projection, in hydras.Epsilon0.Hprime]
+Pdec:3 [binder, in hydras.Prelude.First_toggle]
+PD_Zero [lemma, in hydras.Epsilon0.Hprime]
+PE [projection, in hydras.Epsilon0.F_alpha]
+PE [projection, in hydras.Epsilon0.Hprime]
+Permut [module, in hydras.rpo.list_permut]
+Permut.DS [module, in hydras.rpo.list_permut]
+Permut.elt [definition, in hydras.rpo.list_permut]
+Permut.eq_elt_dec [definition, in hydras.rpo.list_permut]
+Permut.list_permut [definition, in hydras.rpo.list_permut]
+Permut.list_to_multiset [definition, in hydras.rpo.list_permut]
+PF87 [lemma, in additions.Euclidean_Chains]
+pf:39 [binder, in hydras.MoreAck.FolExamples]
+pf:40 [binder, in hydras.MoreAck.FolExamples]
+phi [definition, in hydras.Schutte.Critical]
+phi [definition, in hydras.Gamma0.Gamma0]
+phi_mono_RR [lemma, in hydras.Gamma0.Gamma0]
+phi_psi [lemma, in hydras.Gamma0.Gamma0]
+phi_to_psi_6 [lemma, in hydras.Gamma0.Gamma0]
+phi_to_psi_5 [lemma, in hydras.Gamma0.Gamma0]
+phi_to_psi_4 [lemma, in hydras.Gamma0.Gamma0]
+phi_to_psi_3 [lemma, in hydras.Gamma0.Gamma0]
+phi_to_psi_2 [lemma, in hydras.Gamma0.Gamma0]
+phi_to_psi_1 [lemma, in hydras.Gamma0.Gamma0]
+phi_to_psi.alpha [variable, in hydras.Gamma0.Gamma0]
+phi_to_psi [section, in hydras.Gamma0.Gamma0]
+phi_inj_r [lemma, in hydras.Gamma0.Gamma0]
+phi_mono_weak_r [lemma, in hydras.Gamma0.Gamma0]
+phi_mono_r [lemma, in hydras.Gamma0.Gamma0]
+phi_of_psi_plus_fin [lemma, in hydras.Gamma0.Gamma0]
+phi_alpha_zero_gt_alpha [lemma, in hydras.Gamma0.Gamma0]
+phi_principalR [lemma, in hydras.Gamma0.Gamma0]
+phi_spec1 [lemma, in hydras.Gamma0.Gamma0]
+phi_le_ge [lemma, in hydras.Gamma0.Gamma0]
+phi_le [lemma, in hydras.Gamma0.Gamma0]
+phi_fix [lemma, in hydras.Gamma0.Gamma0]
+phi_of_any_cons [lemma, in hydras.Gamma0.Gamma0]
+phi_nf [lemma, in hydras.Gamma0.Gamma0]
+phi_cases [lemma, in hydras.Gamma0.Gamma0]
+phi_cases' [lemma, in hydras.Gamma0.Gamma0]
+phi_cases_aux [lemma, in hydras.Gamma0.Gamma0]
+phi_of_psi_succ [lemma, in hydras.Gamma0.Gamma0]
+phi_alpha_zero [lemma, in hydras.Gamma0.Gamma0]
+phi_principal [lemma, in hydras.Gamma0.Gamma0]
+phi_to_psi [lemma, in hydras.Gamma0.Gamma0]
+phi_of_psi [lemma, in hydras.Gamma0.Gamma0]
+phi0 [abbreviation, in hydras.Epsilon0.T1]
+phi0 [abbreviation, in hydras.Schutte.AP]
+phi0_eq [lemma, in hydras.Epsilon0.T1]
+phi0_eq_bad [lemma, in hydras.Epsilon0.T1]
+phi0_plus_mult [lemma, in hydras.Epsilon0.T1]
+phi0_mono [lemma, in hydras.Epsilon0.T1]
+phi0_mono_strict_LT [lemma, in hydras.Epsilon0.T1]
+phi0_mono_strict [lemma, in hydras.Epsilon0.T1]
+phi0_ltR [lemma, in hydras.Epsilon0.T1]
+phi0_lt [lemma, in hydras.Epsilon0.T1]
+phi0_mult_plus_lt_phi0R [lemma, in hydras.Schutte.Correctness_E0]
+phi0_mult_plus_lt_phi0 [lemma, in hydras.Schutte.Correctness_E0]
+phi0_mult_lt_phi0 [lemma, in hydras.Schutte.Correctness_E0]
+phi0_mult.f_ok [variable, in hydras.Epsilon0.Large_Sets]
+phi0_mult.f_Sle [variable, in hydras.Epsilon0.Large_Sets]
+phi0_mult.f_mono [variable, in hydras.Epsilon0.Large_Sets]
+phi0_mult.Halpha [variable, in hydras.Epsilon0.Large_Sets]
+phi0_mult.f [variable, in hydras.Epsilon0.Large_Sets]
+phi0_mult.alpha [variable, in hydras.Epsilon0.Large_Sets]
+phi0_mult [section, in hydras.Epsilon0.Large_Sets]
+phi0_well_named [lemma, in hydras.Schutte.Critical]
+phi0_lt_epsilon0_R [lemma, in hydras.Schutte.AP]
+phi0_lt_epsilon0 [lemma, in hydras.Schutte.AP]
+phi0_of_limit [lemma, in hydras.Schutte.AP]
+phi0_sup [lemma, in hydras.Schutte.AP]
+phi0_alpha_phi0_beta [lemma, in hydras.Schutte.AP]
+phi0_positive [lemma, in hydras.Schutte.AP]
+phi0_inj [lemma, in hydras.Schutte.AP]
+phi0_mono_R_weak [lemma, in hydras.Schutte.AP]
+phi0_mono_R [lemma, in hydras.Schutte.AP]
+phi0_mono_weak [lemma, in hydras.Schutte.AP]
+phi0_mono [lemma, in hydras.Schutte.AP]
+phi0_zero [lemma, in hydras.Schutte.AP]
+phi0_elim [lemma, in hydras.Schutte.AP]
+phi0_ordering [lemma, in hydras.Schutte.AP]
+phi0_ref [lemma, in gaia_hydras.T1Bridge]
+phi0_mono [lemma, in hydras.Epsilon0.E0]
+Phi0_def [lemma, in hydras.Epsilon0.Hprime]
+phi:198 [binder, in hydras.Prelude.Iterates]
+phi:214 [binder, in hydras.Prelude.Iterates]
+phi:29 [binder, in hydras.MoreAck.FolExamples]
+phi:69 [binder, in hydras.MoreAck.PrimRecExamples]
+phi:76 [binder, in hydras.MoreAck.PrimRecExamples]
+phi:85 [binder, in hydras.MoreAck.PrimRecExamples]
+pi1_4IsPR [instance, in hydras.Ackermann.primRec]
+pi1_3IsPR [instance, in hydras.Ackermann.primRec]
+pi1_2IsPR [instance, in hydras.Ackermann.primRec]
+pi2_4IsPR [instance, in hydras.Ackermann.primRec]
+pi2_3IsPR [instance, in hydras.Ackermann.primRec]
+pi2_2IsPR [instance, in hydras.Ackermann.primRec]
+pi3_4IsPR [instance, in hydras.Ackermann.primRec]
+pi3_3IsPR [instance, in hydras.Ackermann.primRec]
+pi4_4IsPR [instance, in hydras.Ackermann.primRec]
+plus [abbreviation, in hydras.Epsilon0.T1]
+plus [definition, in hydras.OrdinalNotations.ON_Omega2]
+Plus [abbreviation, in hydras.Epsilon0.E0]
+plus [definition, in hydras.Gamma0.Gamma0]
+plus [definition, in hydras.Schutte.Addition]
+plusIndIsPR [instance, in hydras.Ackermann.primRec]
+plusIsPR [instance, in hydras.Ackermann.primRec]
+plus_cons_cons_eqn [lemma, in hydras.Epsilon0.T1]
+plus_smono_LT_r [lemma, in hydras.Epsilon0.T1]
+plus_compat [lemma, in hydras.Epsilon0.T1]
+plus_left_absorb [lemma, in hydras.Epsilon0.T1]
+plus_is_zero [lemma, in hydras.Epsilon0.T1]
+plus_nf [lemma, in hydras.Epsilon0.T1]
+plus_nf0 [lemma, in hydras.Epsilon0.T1]
+plus_cons_cons_rw3 [lemma, in hydras.Epsilon0.T1]
+plus_cons_cons_rw2 [lemma, in hydras.Epsilon0.T1]
+plus_cons_cons_rw1 [lemma, in hydras.Epsilon0.T1]
+plus_zero_r [lemma, in hydras.Epsilon0.T1]
+plus_zero [lemma, in hydras.Epsilon0.T1]
+plus_alpha_mult_phi0 [lemma, in hydras.Schutte.Correctness_E0]
+plus_assoc [instance, in hydras.OrdinalNotations.ON_Omega2]
+plus_compat [lemma, in hydras.OrdinalNotations.ON_Omega2]
+plus_2 [lemma, in hydras.Prelude.More_Arith]
+plus_lt_phi0 [lemma, in hydras.Schutte.AP]
+plus_ref [lemma, in gaia_hydras.T1Bridge]
+plus_comp [instance, in hydras.OrdinalNotations.ON_plus]
+plus_assoc [instance, in hydras.Epsilon0.E0]
+Plus_rw [lemma, in hydras.Epsilon0.E0]
+plus_mono_l_weak [lemma, in hydras.Gamma0.Gamma0]
+plus_mono_r [lemma, in hydras.Gamma0.Gamma0]
+plus_nf [lemma, in hydras.Gamma0.Gamma0]
+plus_alpha_0 [lemma, in hydras.Gamma0.Gamma0]
+Plus_ [constructor, in hydras.Ackermann.Languages]
+plus_mono_bi [lemma, in hydras.Schutte.Addition]
+plus_mono_weak_l [lemma, in hydras.Schutte.Addition]
+plus_assoc [instance, in hydras.Schutte.Addition]
+plus_assoc' [lemma, in hydras.Schutte.Addition]
+plus_assoc3 [lemma, in hydras.Schutte.Addition]
+plus_assoc2 [lemma, in hydras.Schutte.Addition]
+plus_assoc1 [lemma, in hydras.Schutte.Addition]
+plus_FF [lemma, in hydras.Schutte.Addition]
+plus_limit [lemma, in hydras.Schutte.Addition]
+plus_reg_r [lemma, in hydras.Schutte.Addition]
+plus_mono_r_weak [lemma, in hydras.Schutte.Addition]
+plus_of_succ [lemma, in hydras.Schutte.Addition]
+plus_mono_r [lemma, in hydras.Schutte.Addition]
+plus_elim [lemma, in hydras.Schutte.Addition]
+plus_ordering [lemma, in hydras.Schutte.Addition]
+plus_correct [lemma, in hydras.MoreAck.PrimRecExamples]
+PMult [instance, in additions.Monoid_instances]
+positive_4step_ind [lemma, in additions.More_on_positive]
+Pos_to_nat_diff_0 [lemma, in additions.AM]
+Pos_pow_compat [lemma, in additions.Compatibility]
+Pos_pow_power [lemma, in additions.Compatibility]
+Pos_iter_ok [lemma, in additions.Compatibility]
+Pos_iter_op_ok [lemma, in additions.Compatibility]
+Pos_iter_op_ok_0 [lemma, in additions.Compatibility]
+Pos_pow [definition, in additions.Compatibility]
+Pos_bpow [definition, in additions.FirstSteps]
+Pos_bpow_ok_R [lemma, in additions.Pow_variant]
+Pos_bpow_ok [lemma, in additions.Pow_variant]
+Pos_bpow [definition, in additions.Pow_variant]
+Pos_bpow_of_bpow [lemma, in additions.Pow]
+Pos_bpow_of_plus [lemma, in additions.Pow]
+Pos_bpow_ok_R [lemma, in additions.Pow]
+Pos_bpow_proper [instance, in additions.Pow]
+Pos_bpow_ok [lemma, in additions.Pow]
+Pos_bpow [definition, in additions.Pow]
+pos_div_eucl_quotient_lt [lemma, in additions.More_on_positive]
+pos_div_eucl_quotient_pos [lemma, in additions.More_on_positive]
+pos_gt_3 [lemma, in additions.More_on_positive]
+pos_lt_wf [lemma, in additions.More_on_positive]
+pos_lt_mul [lemma, in additions.More_on_positive]
+pos_le_mul [lemma, in additions.More_on_positive]
+Pos_to_nat_neq_0 [lemma, in additions.More_on_positive]
+pos_eq_dec [definition, in additions.More_on_positive]
+Pos2Nat_morph [lemma, in additions.Compatibility]
+Pos2Nat_le_n_pn [lemma, in additions.More_on_positive]
+Pos2Nat_le_1_p [lemma, in additions.More_on_positive]
+pos2N_inj_add [lemma, in additions.More_on_positive]
+pos2N_inj_lt [lemma, in additions.More_on_positive]
+Pow [library]
+power [abbreviation, in hydras.Prelude.More_Arith]
+power [definition, in additions.FirstSteps]
+power [definition, in additions.Pow_variant]
+power [definition, in additions.Pow]
+power_of_1 [lemma, in hydras.Prelude.More_Arith]
+power_t [definition, in additions.FirstSteps]
+power_of_mult [lemma, in additions.Pow_variant]
+power_of_square [lemma, in additions.Pow_variant]
+power_of_power [lemma, in additions.Pow_variant]
+power_commute_with_x [lemma, in additions.Pow_variant]
+power_commute [lemma, in additions.Pow_variant]
+power_of_plus [lemma, in additions.Pow_variant]
+power_proper [instance, in additions.Pow_variant]
+power_eq3 [lemma, in additions.Pow_variant]
+power_eq2 [lemma, in additions.Pow_variant]
+power_eq1 [lemma, in additions.Pow_variant]
+power_of_mult [lemma, in additions.Pow]
+power_of_square [lemma, in additions.Pow]
+power_of_power_comm [lemma, in additions.Pow]
+power_of_power [lemma, in additions.Pow]
+power_commute_with_x [lemma, in additions.Pow]
+power_commute [lemma, in additions.Pow]
+power_of_plus [lemma, in additions.Pow]
+power_proper [instance, in additions.Pow]
+power_eq3 [lemma, in additions.Pow]
+power_eq2 [lemma, in additions.Pow]
+power_eq1 [lemma, in additions.Pow]
+power_R_is_a_refinement [lemma, in additions.Addition_Chains]
+power_R_1 [lemma, in additions.Addition_Chains]
+power_R_Mult [lemma, in additions.Addition_Chains]
+power_R [definition, in additions.Addition_Chains]
+pow_17 [definition, in additions.Pow_variant]
+pow_17 [definition, in additions.Pow]
+Pow_variant [library]
+pow3 [definition, in hydras.Ackermann.codeSubFormula]
+pow3IsPR [instance, in hydras.Ackermann.codeSubFormula]
+pow3Min [lemma, in hydras.Ackermann.codeSubFormula]
+pow3Monotone [lemma, in hydras.Ackermann.codeSubFormula]
+PO_lex_prod [instance, in hydras.Prelude.DecPreOrder_Instances]
+PO1 [lemma, in additions.Euclidean_Chains]
+PO2 [lemma, in additions.Euclidean_Chains]
+PO3 [lemma, in additions.Euclidean_Chains]
+PO4 [lemma, in additions.Euclidean_Chains]
+PO6 [lemma, in additions.Euclidean_Chains]
+PO8 [lemma, in additions.Euclidean_Chains]
+PO9 [lemma, in additions.Euclidean_Chains]
+pp [definition, in hydras.Epsilon0.T1]
+ppE0 [definition, in gaia_hydras.T1Bridge]
+ppT1 [inductive, in hydras.Epsilon0.T1]
+pp_size [definition, in hydras.Epsilon0.T1]
+PP_omega [constructor, in hydras.Epsilon0.T1]
+PP_exp [constructor, in hydras.Epsilon0.T1]
+PP_mult [constructor, in hydras.Epsilon0.T1]
+PP_add [constructor, in hydras.Epsilon0.T1]
+PP_fin [constructor, in hydras.Epsilon0.T1]
+pp0 [definition, in hydras.Epsilon0.T1]
+pq:68 [binder, in hydras.rpo.dickson]
+PRcompose1 [definition, in hydras.MoreAck.PrimRecExamples]
+PRcons [constructor, in hydras.Ackermann.primRec]
+Precedence [module, in hydras.rpo.rpo]
+Precedence.A [axiom, in hydras.rpo.rpo]
+Precedence.Lex [constructor, in hydras.rpo.rpo]
+Precedence.Mul [constructor, in hydras.rpo.rpo]
+Precedence.prec [axiom, in hydras.rpo.rpo]
+Precedence.prec_transitive [axiom, in hydras.rpo.rpo]
+Precedence.prec_antisym [axiom, in hydras.rpo.rpo]
+Precedence.prec_dec [axiom, in hydras.rpo.rpo]
+Precedence.status [axiom, in hydras.rpo.rpo]
+Precedence.status_type [inductive, in hydras.rpo.rpo]
+pred [definition, in hydras.Epsilon0.T1]
+pred [definition, in hydras.Gamma0.T2]
+Predecessor_unicity [lemma, in hydras.solutions_exercises.predSuccUnicity]
+predIsPR [instance, in hydras.Ackermann.primRec]
+predSuccUnicity [library]
+pred_nf [lemma, in hydras.Epsilon0.T1]
+pred_LT [lemma, in hydras.Epsilon0.T1]
+pred_of_limitR [lemma, in hydras.Epsilon0.T1]
+pred_of_limit [lemma, in hydras.Epsilon0.T1]
+pred_of_succ [lemma, in hydras.Epsilon0.T1]
+pred_of_power [lemma, in hydras.Prelude.More_Arith]
+pred_of_succ [lemma, in hydras.Gamma0.Gamma0]
+pred_of_limit [lemma, in hydras.Gamma0.Gamma0]
+pred_of_cons' [lemma, in hydras.Gamma0.Gamma0]
+pred_of_cons [lemma, in hydras.Gamma0.Gamma0]
+preorder_equiv [definition, in hydras.Prelude.DecPreOrder]
+preserveValue [lemma, in hydras.Ackermann.model]
+PReval [abbreviation, in hydras.Ackermann.primRec]
+PRevalN [abbreviation, in hydras.Ackermann.primRec]
+Prf [inductive, in hydras.Ackermann.folProof]
+prf:11 [binder, in hydras.Ackermann.Deduction]
+prf:121 [binder, in hydras.Ackermann.LNN2LNT]
+prf:129 [binder, in hydras.Ackermann.LNN2LNT]
+prf:45 [binder, in hydras.Ackermann.code]
+prf:47 [binder, in hydras.Ackermann.folProof]
+Primitive_Recursive_Representable [section, in Goedel.PRrepresentable]
+Primitive_recursion.f [variable, in hydras.MoreAck.PrimRecExamples]
+Primitive_recursion.h [variable, in hydras.MoreAck.PrimRecExamples]
+Primitive_recursion.g [variable, in hydras.MoreAck.PrimRecExamples]
+Primitive_recursion.z [variable, in hydras.MoreAck.PrimRecExamples]
+Primitive_recursion.y [variable, in hydras.MoreAck.PrimRecExamples]
+Primitive_recursion.x [variable, in hydras.MoreAck.PrimRecExamples]
+Primitive_recursion [section, in hydras.MoreAck.PrimRecExamples]
+PrimRec [inductive, in hydras.Ackermann.primRec]
+primRec [library]
+PrimRecExamples [library]
+primRecFormula [definition, in Goedel.PRrepresentable]
+primRecFunc [constructor, in hydras.Ackermann.primRec]
+primRecPiFormulaHelp [definition, in Goedel.PRrepresentable]
+primRecRepresentable [lemma, in Goedel.PRrepresentable]
+primRecRepresentable1 [lemma, in Goedel.PRrepresentable]
+PrimRecs [inductive, in hydras.Ackermann.primRec]
+primRecsFormula [definition, in Goedel.PRrepresentable]
+primRecSigmaFormula [definition, in Goedel.PRrepresentable]
+primRecSigmaFormulaHelp [definition, in Goedel.PRrepresentable]
+primRecSigmaRepresentable [lemma, in Goedel.PRrepresentable]
+PrimRecs_PrimRec_ind [definition, in hydras.Ackermann.primRec]
+PrimRecs_PrimRec_rec [definition, in hydras.Ackermann.primRec]
+PrimRec_PrimRecs_ind [definition, in hydras.Ackermann.primRec]
+PrimRec_PrimRecs_rec [definition, in hydras.Ackermann.primRec]
+PrimRec_2_S [lemma, in hydras.MoreAck.PrimRecExamples]
+PrimRec_2_0 [lemma, in hydras.MoreAck.PrimRecExamples]
+PrimRec_1_S [lemma, in hydras.MoreAck.PrimRecExamples]
+PrimRec_1_0 [lemma, in hydras.MoreAck.PrimRecExamples]
+PrimRec_0_S [lemma, in hydras.MoreAck.PrimRecExamples]
+PrimRec_0_0 [lemma, in hydras.MoreAck.PrimRecExamples]
+prLogic [library]
+PRnil [constructor, in hydras.Ackermann.primRec]
+PRNotations [module, in hydras.Ackermann.primRec]
+PRNotations.pi1_3 [abbreviation, in hydras.Ackermann.primRec]
+PRNotations.pi1_2 [abbreviation, in hydras.Ackermann.primRec]
+PRNotations.pi1_1 [abbreviation, in hydras.Ackermann.primRec]
+PRNotations.pi2_3 [abbreviation, in hydras.Ackermann.primRec]
+PRNotations.pi2_2 [abbreviation, in hydras.Ackermann.primRec]
+PRNotations.pi3_3 [abbreviation, in hydras.Ackermann.primRec]
+PRNotations.PRcomp [abbreviation, in hydras.Ackermann.primRec]
+PRNotations.PRrec [abbreviation, in hydras.Ackermann.primRec]
+[ _ ; _ ; .. ; _ ] (pr_scope) [notation, in hydras.Ackermann.primRec]
+[ _ ] (pr_scope) [notation, in hydras.Ackermann.primRec]
+_ :: _ (pr_scope) [notation, in hydras.Ackermann.primRec]
+progressive [definition, in hydras.Schutte.Schutte_basics]
+Progressive_Acc [lemma, in hydras.Schutte.Schutte_basics]
+projectionList [definition, in hydras.Ackermann.primRec]
+projectionListApplyParam [lemma, in hydras.Ackermann.primRec]
+projectionListId [lemma, in hydras.Ackermann.primRec]
+projectionListInd [lemma, in hydras.Ackermann.primRec]
+projectionListPR [definition, in hydras.Ackermann.primRec]
+projFormula [definition, in Goedel.PRrepresentable]
+projFunc [constructor, in hydras.Ackermann.primRec]
+projRepresentable [lemma, in Goedel.PRrepresentable]
+proj_le_max [lemma, in hydras.MoreAck.AckNotPR]
+ProofH [section, in hydras.Ackermann.folProof]
+ProofH.Example [section, in hydras.Ackermann.folProof]
+ProofH.Formula [variable, in hydras.Ackermann.folProof]
+ProofH.Formulas [variable, in hydras.Ackermann.folProof]
+ProofH.L [variable, in hydras.Ackermann.folProof]
+ProofH.System [variable, in hydras.Ackermann.folProof]
+ProofH.Term [variable, in hydras.Ackermann.folProof]
+ProofH.Terms [variable, in hydras.Ackermann.folProof]
+ProofOfLexwf [section, in hydras.solutions_exercises.MultisetWf]
+ProofOfLexwf.A [variable, in hydras.solutions_exercises.MultisetWf]
+ProofOfLexwf.Accs [variable, in hydras.solutions_exercises.MultisetWf]
+ProofOfLexwf.HwfA [variable, in hydras.solutions_exercises.MultisetWf]
+ProofOfLexwf.ltA [variable, in hydras.solutions_exercises.MultisetWf]
+Proofs_of_unicity.Proofs.S2.Hgammaalpha [variable, in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.S2.gamma [variable, in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.S2 [section, in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.S1.Halphagamma [variable, in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.S1.gamma [variable, in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.S1 [section, in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.Halphabeta [variable, in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.beta [variable, in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.alpha [variable, in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs [section, in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity [section, in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_lt_succ_le.Proofs.S1.HGammaBeta [variable, in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le.Proofs.S1.gamma [variable, in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le.Proofs.S1 [section, in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le.Proofs.Halphabeta [variable, in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le.Proofs.beta [variable, in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le.Proofs.alpha [variable, in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le.Proofs [section, in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le [section, in hydras.solutions_exercises.lt_succ_le]
+Proof_of_dist.P [variable, in hydras.Epsilon0.T1]
+Proof_of_dist [section, in hydras.Epsilon0.T1]
+Proof_of_mult_nf.Induction.IHbeta [variable, in hydras.Epsilon0.T1]
+Proof_of_mult_nf.Induction.Hbeta [variable, in hydras.Epsilon0.T1]
+Proof_of_mult_nf.Induction.beta [variable, in hydras.Epsilon0.T1]
+Proof_of_mult_nf.Induction [section, in hydras.Epsilon0.T1]
+Proof_of_mult_nf.P [variable, in hydras.Epsilon0.T1]
+Proof_of_mult_nf.Halpha [variable, in hydras.Epsilon0.T1]
+Proof_of_mult_nf.alpha [variable, in hydras.Epsilon0.T1]
+Proof_of_mult_nf [section, in hydras.Epsilon0.T1]
+Proof_of_MinIsPR.minPR [variable, in hydras.solutions_exercises.MinPR2]
+Proof_of_MinIsPR [section, in hydras.solutions_exercises.MinPR2]
+Proof_of_oplus_lt1.H0 [variable, in hydras.Epsilon0.Hessenberg]
+Proof_of_oplus_lt1.n [variable, in hydras.Epsilon0.Hessenberg]
+Proof_of_oplus_lt1.a2 [variable, in hydras.Epsilon0.Hessenberg]
+Proof_of_oplus_lt1.a1 [variable, in hydras.Epsilon0.Hessenberg]
+Proof_of_oplus_lt1 [section, in hydras.Epsilon0.Hessenberg]
+Proof_of_oplus_assoc [section, in hydras.Epsilon0.Hessenberg]
+Proof_of_oplus_comm [section, in hydras.Epsilon0.Hessenberg]
+Proof_of_plus_nf [section, in hydras.Epsilon0.Hessenberg]
+Proof_of_Ackn_PR.S_step.n [variable, in hydras.MoreAck.AckNotPR]
+Proof_of_Ackn_PR.S_step [section, in hydras.MoreAck.AckNotPR]
+Proof_of_Ackn_PR [section, in hydras.MoreAck.AckNotPR]
+Proof_of_4_5_2.HlargeA [variable, in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.HAB1 [variable, in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.HB [variable, in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.HA [variable, in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.Halpha [variable, in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.alpha [variable, in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.B2 [variable, in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.B1 [variable, in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.A [variable, in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2 [section, in hydras.Epsilon0.Large_Sets]
+Proof_of_Lemma5.Alpha_positive.closedness.IM [variable, in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.closedness.CM [variable, in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.closedness.HM [variable, in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.closedness.M [variable, in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.closedness [section, in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.Proof_unbounded.gamma [variable, in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.Proof_unbounded.beta [variable, in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.Proof_unbounded [section, in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.IHalpha [variable, in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.alpha_pos [variable, in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.alpha [variable, in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive [section, in hydras.Schutte.Critical]
+Proof_of_Lemma5.P [variable, in hydras.Schutte.Critical]
+Proof_of_Lemma5 [section, in hydras.Schutte.Critical]
+Proof_of_nested_Ack_bound [section, in hydras.MoreAck.Ack]
+Proof_of_mult_ref [section, in gaia_hydras.T1Bridge]
+Proof_of_FibIsPR.fib_iter_cPair [variable, in hydras.solutions_exercises.FibonacciPR]
+Proof_of_FibIsPR.fib_step_cPair [variable, in hydras.solutions_exercises.FibonacciPR]
+Proof_of_FibIsPR.fib_iter [variable, in hydras.solutions_exercises.FibonacciPR]
+Proof_of_FibIsPR.fib_step [variable, in hydras.solutions_exercises.FibonacciPR]
+Proof_of_FibIsPR [section, in hydras.solutions_exercises.FibonacciPR]
+Proof_of_MinIsPR.min_alt [variable, in hydras.solutions_exercises.MinPR]
+Proof_of_MinIsPR [section, in hydras.solutions_exercises.MinPR]
+Proof_of_H'_mono_l.Limit_case.Hbeta [variable, in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.Limit_case [section, in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.Succ_case.Hgamma [variable, in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.Succ_case.gamma [variable, in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.Succ_case [section, in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.H_alpha_beta [variable, in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.beta [variable, in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.alpha [variable, in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l [section, in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.alpha_limit.Hlim [variable, in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.alpha_limit [section, in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.alpha_Succ.alpha_def [variable, in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.alpha_Succ.beta [variable, in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.alpha_Succ [section, in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.Halpha [variable, in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.alpha [variable, in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction [section, in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties [section, in hydras.Epsilon0.Hprime]
+proof_of_associativity.g_alpha_beta [variable, in hydras.Schutte.Addition]
+proof_of_associativity.f_alpha_beta [variable, in hydras.Schutte.Addition]
+proof_of_associativity.beta [variable, in hydras.Schutte.Addition]
+proof_of_associativity.alpha [variable, in hydras.Schutte.Addition]
+proof_of_associativity [section, in hydras.Schutte.Addition]
+Proof_of_lt_not_wf.seq_intro.is_in_seq [variable, in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf.seq_intro.Rwf [variable, in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf.seq_intro.R [variable, in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf.seq_intro.seq [variable, in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf.seq_intro.A [variable, in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf.seq_intro [section, in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf.lt_wf [variable, in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf [section, in hydras.solutions_exercises.T1_ltNotWf]
+proof:202 [binder, in hydras.rpo.rpo]
+proof:206 [binder, in hydras.rpo.rpo]
+Properties [section, in hydras.Epsilon0.F_alpha]
+Properties_of_f_alpha.The_induction [section, in hydras.Epsilon0.F_alpha]
+Properties_of_f_alpha [section, in hydras.Epsilon0.F_alpha]
+Properties.The_induction.alpha_limit.Hlim [variable, in hydras.Epsilon0.F_alpha]
+Properties.The_induction.alpha_limit [section, in hydras.Epsilon0.F_alpha]
+Properties.The_induction.alpha_Succ.alpha_def [variable, in hydras.Epsilon0.F_alpha]
+Properties.The_induction.alpha_Succ.beta [variable, in hydras.Epsilon0.F_alpha]
+Properties.The_induction.alpha_Succ [section, in hydras.Epsilon0.F_alpha]
+Properties.The_induction.Halpha [variable, in hydras.Epsilon0.F_alpha]
+Properties.The_induction.alpha [variable, in hydras.Epsilon0.F_alpha]
+Properties.The_induction [section, in hydras.Epsilon0.F_alpha]
+proper_statement [definition, in additions.Euclidean_Chains]
+Proper_A [lemma, in hydras.Schutte.Ordering_Functions]
+proper_of_proper [lemma, in hydras.Schutte.Ordering_Functions]
+proper_members [lemma, in hydras.Schutte.Ordering_Functions]
+proper_segment_of [definition, in hydras.Schutte.Ordering_Functions]
+proper_segment [definition, in hydras.Schutte.Ordering_Functions]
+Proposition_2_3a [lemma, in hydras.Epsilon0.Paths]
+prop_map12_without_repetition [lemma, in hydras.rpo.more_list]
+prop_map_without_repetition [lemma, in hydras.rpo.more_list]
+PrO_Included [instance, in hydras.Prelude.DecPreOrder_Instances]
+PRrepresentable [library]
+pr:120 [binder, in Goedel.PRrepresentable]
+pr:633 [binder, in hydras.Ackermann.primRec]
+pr:638 [binder, in hydras.Ackermann.primRec]
+psi [definition, in hydras.Gamma0.T2]
+psi_eq [lemma, in hydras.Gamma0.T2]
+psi_term [definition, in hydras.Gamma0.T2]
+psi_principal [lemma, in hydras.Gamma0.Gamma0]
+psi_lt_epsilon0 [lemma, in hydras.Gamma0.Gamma0]
+psi_lt_head [lemma, in hydras.Gamma0.Gamma0]
+psi_le_cons [lemma, in hydras.Gamma0.Gamma0]
+psi:215 [binder, in hydras.Prelude.Iterates]
+ptwise_le [definition, in hydras.Prelude.MoreLists]
+PUSH [constructor, in additions.AM]
+PZero [lemma, in hydras.Epsilon0.F_alpha]
+PZero [lemma, in hydras.Epsilon0.Hprime]
+P_mult_op [instance, in additions.Monoid_instances]
+P_3 [lemma, in hydras.solutions_exercises.F_3]
+P_alpha [lemma, in hydras.Epsilon0.Hprime]
+P_alpha_0 [lemma, in hydras.Epsilon0.Hprime]
+P_dec:190 [binder, in hydras.rpo.list_set]
+P_dec:67 [binder, in hydras.rpo.list_set]
+P_dec:63 [binder, in hydras.rpo.list_set]
+P_dec:59 [binder, in hydras.rpo.list_set]
+P_dec:55 [binder, in hydras.rpo.list_set]
+P_dec:52 [binder, in hydras.rpo.list_set]
+P_dec:49 [binder, in hydras.rpo.list_set]
+P_dec:45 [binder, in hydras.rpo.list_set]
+P_dec:40 [binder, in hydras.rpo.list_set]
+P_dec [instance, in hydras.Epsilon0.Paths]
+P_well_founded_induction_type [lemma, in gaia_hydras.nfwfgaia]
+p':2 [binder, in additions.More_on_positive]
+p':22 [binder, in hydras.Prelude.More_Arith]
+p':40 [binder, in hydras.Prelude.DecPreOrder_Instances]
+p':66 [binder, in hydras.rpo.dickson]
+p':7 [binder, in hydras.solutions_exercises.MinPR2]
+P0 [lemma, in hydras.MoreAck.Ack]
+p0:395 [binder, in additions.Euclidean_Chains]
+p0:402 [binder, in additions.Euclidean_Chains]
+P0:46 [binder, in hydras.Prelude.DecPreOrder]
+p0:838 [binder, in hydras.Ackermann.checkPrf]
+p0:841 [binder, in hydras.Ackermann.checkPrf]
+P0:89 [binder, in hydras.Prelude.DecPreOrder]
+P0:95 [binder, in hydras.Prelude.DecPreOrder]
+P1_dec:195 [binder, in hydras.rpo.list_set]
+p1:109 [binder, in hydras.Ackermann.codeList]
+p1:114 [binder, in hydras.Ackermann.codeList]
+p1:117 [binder, in hydras.Ackermann.codeList]
+p1:122 [binder, in hydras.Ackermann.codeList]
+p1:125 [binder, in hydras.Ackermann.codeList]
+p1:128 [binder, in hydras.Ackermann.codeList]
+p1:1309 [binder, in hydras.Ackermann.codeSubFormula]
+p1:131 [binder, in hydras.Ackermann.codeList]
+p1:1313 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1315 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1318 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1320 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1323 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1326 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1329 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1332 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1335 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1338 [binder, in hydras.Ackermann.codeSubFormula]
+p1:134 [binder, in hydras.Ackermann.codeList]
+p1:1348 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1350 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1352 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1354 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1356 [binder, in hydras.Ackermann.codeSubFormula]
+p1:1358 [binder, in hydras.Ackermann.codeSubFormula]
+p1:137 [binder, in hydras.Ackermann.codeList]
+p1:140 [binder, in hydras.Ackermann.codeList]
+p1:143 [binder, in hydras.Ackermann.codeList]
+p1:166 [binder, in hydras.Ackermann.codeList]
+p1:169 [binder, in hydras.Ackermann.codeList]
+p1:172 [binder, in hydras.Ackermann.codeList]
+p1:175 [binder, in hydras.Ackermann.codeList]
+p1:178 [binder, in hydras.Ackermann.codeList]
+p1:181 [binder, in hydras.Ackermann.codeList]
+p1:183 [binder, in hydras.Ackermann.codeList]
+p1:185 [binder, in hydras.Ackermann.codeList]
+p1:187 [binder, in hydras.Ackermann.codeList]
+p1:189 [binder, in hydras.Ackermann.codeList]
+p1:191 [binder, in hydras.Ackermann.codeList]
+p1:193 [binder, in hydras.Ackermann.codeList]
+P1:193 [binder, in hydras.rpo.list_set]
+p1:214 [binder, in hydras.Ackermann.codeList]
+p1:219 [binder, in hydras.Ackermann.codeList]
+p1:222 [binder, in hydras.Ackermann.codeList]
+p1:227 [binder, in hydras.Ackermann.codeList]
+p1:230 [binder, in hydras.Ackermann.codeList]
+p1:233 [binder, in hydras.Ackermann.codeList]
+p1:236 [binder, in hydras.Ackermann.codeList]
+p1:239 [binder, in hydras.Ackermann.codeList]
+p1:242 [binder, in hydras.Ackermann.codeList]
+p1:245 [binder, in hydras.Ackermann.codeList]
+p1:248 [binder, in hydras.Ackermann.codeList]
+p1:251 [binder, in hydras.Ackermann.codeList]
+p1:254 [binder, in hydras.Ackermann.codeList]
+p1:257 [binder, in hydras.Ackermann.codeList]
+p1:260 [binder, in hydras.Ackermann.codeList]
+p1:262 [binder, in hydras.Ackermann.codeList]
+p1:264 [binder, in hydras.Ackermann.codeList]
+p1:266 [binder, in hydras.Ackermann.codeList]
+p1:268 [binder, in hydras.Ackermann.codeList]
+p1:270 [binder, in hydras.Ackermann.codeList]
+p1:272 [binder, in hydras.Ackermann.codeList]
+p1:33 [binder, in hydras.Ackermann.primRec]
+p1:42 [binder, in hydras.Ackermann.codeList]
+p1:47 [binder, in hydras.Ackermann.codeList]
+p1:50 [binder, in hydras.Ackermann.codeList]
+p1:55 [binder, in hydras.Ackermann.codeList]
+p1:58 [binder, in hydras.Ackermann.codeList]
+p1:61 [binder, in hydras.Ackermann.codeList]
+p1:610 [binder, in hydras.Ackermann.primRec]
+p1:615 [binder, in hydras.Ackermann.primRec]
+p1:64 [binder, in hydras.Ackermann.codeList]
+p1:67 [binder, in hydras.Ackermann.codeList]
+p1:70 [binder, in hydras.Ackermann.codeList]
+P2_dec:196 [binder, in hydras.rpo.list_set]
+p2:1310 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1314 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1316 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1319 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1321 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1324 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1327 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1330 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1333 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1336 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1339 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1349 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1351 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1353 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1355 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1357 [binder, in hydras.Ackermann.codeSubFormula]
+p2:1359 [binder, in hydras.Ackermann.codeSubFormula]
+P2:194 [binder, in hydras.rpo.list_set]
+p2:34 [binder, in hydras.Ackermann.primRec]
+p2:611 [binder, in hydras.Ackermann.primRec]
+p2:616 [binder, in hydras.Ackermann.primRec]
+P2:79 [binder, in hydras.Prelude.MoreVectors]
+P87 [definition, in additions.Addition_Chains]
+P:1 [binder, in hydras.Prelude.First_toggle]
+p:1 [binder, in additions.More_on_positive]
+P:1 [binder, in hydras.Prelude.MoreDecidable]
+p:1 [binder, in additions.BinaryStrat]
+p:1 [binder, in additions.Dichotomy]
+P:1 [binder, in hydras.solutions_exercises.isqrt]
+p:1 [binder, in gaia_hydras.nfwfgaia]
+p:10 [binder, in additions.FirstSteps]
+p:10 [binder, in additions.More_on_positive]
+p:10 [binder, in hydras.solutions_exercises.F_3]
+p:101 [binder, in hydras.Ackermann.checkPrf]
+p:103 [binder, in hydras.Ackermann.checkPrf]
+p:105 [binder, in hydras.Ackermann.checkPrf]
+p:106 [binder, in hydras.Ackermann.checkPrf]
+p:107 [binder, in hydras.Ackermann.checkPrf]
+p:108 [binder, in hydras.Ackermann.checkPrf]
+p:109 [binder, in hydras.Ackermann.checkPrf]
+p:11 [binder, in hydras.solutions_exercises.MinPR2]
+p:11 [binder, in hydras.MoreAck.expressibleExamples]
+p:11 [binder, in additions.More_on_positive]
+p:11 [binder, in hydras.solutions_exercises.MultisetWf]
+p:11 [binder, in hydras.Epsilon0.F_omega]
+p:11 [binder, in hydras.solutions_exercises.FibonacciPR]
+P:11 [binder, in hydras.Prelude.MoreDecidable]
+p:110 [binder, in hydras.Ackermann.checkPrf]
+p:111 [binder, in hydras.Prelude.Iterates]
+p:111 [binder, in hydras.Ackermann.checkPrf]
+p:111 [binder, in hydras.Ackermann.fol]
+p:112 [binder, in hydras.Ackermann.checkPrf]
+p:113 [binder, in hydras.Ackermann.checkPrf]
+p:114 [binder, in hydras.Ackermann.folProp]
+p:115 [binder, in hydras.Ackermann.checkPrf]
+p:117 [binder, in hydras.Ackermann.checkPrf]
+p:1176 [binder, in gaia_hydras.nfwfgaia]
+p:1179 [binder, in gaia_hydras.nfwfgaia]
+p:118 [binder, in hydras.Ackermann.fol]
+p:1182 [binder, in gaia_hydras.nfwfgaia]
+p:1185 [binder, in gaia_hydras.nfwfgaia]
+p:1188 [binder, in gaia_hydras.nfwfgaia]
+p:119 [binder, in hydras.Ackermann.checkPrf]
+P:119 [binder, in hydras.Prelude.Merge_Sort]
+p:12 [binder, in additions.Fib2]
+p:12 [binder, in additions.More_on_positive]
+p:12 [binder, in hydras.solutions_exercises.FibonacciPR]
+p:121 [binder, in hydras.Ackermann.checkPrf]
+P:121 [binder, in hydras.Epsilon0.Large_Sets]
+p:121 [binder, in hydras.Ackermann.primRec]
+p:123 [binder, in hydras.Ackermann.checkPrf]
+P:124 [binder, in hydras.Epsilon0.Large_Sets]
+p:125 [binder, in hydras.Ackermann.checkPrf]
+p:125 [binder, in hydras.Ackermann.primRec]
+p:127 [binder, in hydras.Ackermann.checkPrf]
+p:127 [binder, in hydras.Epsilon0.Hprime]
+p:129 [binder, in hydras.Ackermann.checkPrf]
+p:13 [binder, in additions.Fib2]
+P:13 [binder, in hydras.Prelude.Sort_spec]
+p:13 [binder, in additions.More_on_positive]
+p:13 [binder, in additions.Trace_exercise]
+P:13 [binder, in hydras.Schutte.Addition]
+p:13 [binder, in hydras.Schutte.Countable]
+p:130 [binder, in hydras.Ackermann.checkPrf]
+p:131 [binder, in hydras.Ackermann.checkPrf]
+p:131 [binder, in hydras.Epsilon0.Hprime]
+p:132 [binder, in hydras.rpo.term]
+p:132 [binder, in hydras.Ackermann.checkPrf]
+p:133 [binder, in hydras.Ackermann.checkPrf]
+p:134 [binder, in additions.fib]
+p:1340 [binder, in gaia_hydras.nfwfgaia]
+p:135 [binder, in hydras.Ackermann.checkPrf]
+p:137 [binder, in hydras.rpo.term]
+p:137 [binder, in hydras.Ackermann.checkPrf]
+p:137 [binder, in additions.Euclidean_Chains]
+p:137 [binder, in additions.fib]
+p:138 [binder, in hydras.Ackermann.checkPrf]
+p:138 [binder, in additions.Euclidean_Chains]
+p:138 [binder, in hydras.Epsilon0.Hprime]
+p:139 [binder, in hydras.Gamma0.T2]
+p:139 [binder, in hydras.Ackermann.checkPrf]
+p:14 [binder, in hydras.Prelude.More_Arith]
+p:14 [binder, in additions.FirstSteps]
+p:14 [binder, in hydras.solutions_exercises.FibonacciPR]
+P:14 [binder, in hydras.Schutte.PartialFun]
+p:140 [binder, in hydras.Ackermann.checkPrf]
+p:140 [binder, in additions.fib]
+p:140 [binder, in hydras.Epsilon0.Hprime]
+p:1408 [binder, in hydras.Ackermann.codeSubFormula]
+p:141 [binder, in hydras.Ackermann.checkPrf]
+p:1413 [binder, in hydras.Ackermann.codeSubFormula]
+p:1415 [binder, in hydras.Ackermann.codeSubFormula]
+P:142 [binder, in hydras.Prelude.Merge_Sort]
+p:1420 [binder, in hydras.Ackermann.codeSubFormula]
+p:1425 [binder, in hydras.Ackermann.codeSubFormula]
+p:143 [binder, in hydras.rpo.term]
+p:143 [binder, in hydras.Ackermann.checkPrf]
+P:143 [binder, in hydras.Prelude.Merge_Sort]
+p:143 [binder, in hydras.Epsilon0.Hprime]
+P:144 [binder, in hydras.Prelude.MoreLists]
+p:145 [binder, in hydras.Ackermann.checkPrf]
+p:145 [binder, in hydras.Epsilon0.Hprime]
+p:1458 [binder, in gaia_hydras.nfwfgaia]
+p:146 [binder, in hydras.rpo.term]
+p:147 [binder, in hydras.Ackermann.checkPrf]
+p:149 [binder, in hydras.Ackermann.checkPrf]
+p:15 [binder, in additions.More_on_positive]
+p:151 [binder, in hydras.rpo.term]
+p:151 [binder, in hydras.Ackermann.checkPrf]
+P:152 [binder, in hydras.Ackermann.fol]
+p:153 [binder, in hydras.Ackermann.checkPrf]
+p:155 [binder, in hydras.Ackermann.checkPrf]
+p:155 [binder, in additions.fib]
+p:156 [binder, in additions.fib]
+p:1569 [binder, in hydras.Ackermann.codeSubFormula]
+p:157 [binder, in hydras.Ackermann.checkPrf]
+P:157 [binder, in hydras.Ackermann.fol]
+p:157 [binder, in additions.fib]
+p:1571 [binder, in hydras.Ackermann.codeSubFormula]
+p:1573 [binder, in hydras.Ackermann.codeSubFormula]
+p:1575 [binder, in hydras.Ackermann.codeSubFormula]
+p:1577 [binder, in hydras.Ackermann.codeSubFormula]
+p:1579 [binder, in hydras.Ackermann.codeSubFormula]
+p:1581 [binder, in hydras.Ackermann.codeSubFormula]
+p:1583 [binder, in hydras.Ackermann.codeSubFormula]
+p:1585 [binder, in hydras.Ackermann.codeSubFormula]
+p:1587 [binder, in hydras.Ackermann.codeSubFormula]
+p:1589 [binder, in hydras.Ackermann.codeSubFormula]
+p:159 [binder, in hydras.Ackermann.checkPrf]
+p:1591 [binder, in hydras.Ackermann.codeSubFormula]
+p:16 [binder, in hydras.solutions_exercises.FibonacciPR]
+p:16 [binder, in hydras.Hydra.Omega2_Small]
+p:161 [binder, in hydras.Ackermann.checkPrf]
+p:163 [binder, in hydras.Ackermann.checkPrf]
+P:163 [binder, in hydras.Ackermann.fol]
+p:164 [binder, in hydras.rpo.term]
+p:164 [binder, in hydras.Ackermann.checkPrf]
+p:165 [binder, in hydras.Ackermann.checkPrf]
+p:166 [binder, in hydras.Ackermann.checkPrf]
+p:167 [binder, in hydras.Ackermann.checkPrf]
+p:168 [binder, in hydras.Ackermann.checkPrf]
+p:169 [binder, in hydras.Ackermann.checkPrf]
+p:169 [binder, in gaia_hydras.nfwfgaia]
+p:17 [binder, in hydras.Prelude.MoreLists]
+p:17 [binder, in additions.More_on_positive]
+p:17 [binder, in hydras.solutions_exercises.MultisetWf]
+p:17 [binder, in additions.fib]
+P:17 [binder, in hydras.Prelude.MoreDecidable]
+p:17 [binder, in hydras.Schutte.Countable]
+p:170 [binder, in hydras.Ackermann.checkPrf]
+p:171 [binder, in hydras.Ackermann.checkPrf]
+p:172 [binder, in hydras.Ackermann.checkPrf]
+p:172 [binder, in hydras.Ackermann.fol]
+p:173 [binder, in hydras.Ackermann.checkPrf]
+p:174 [binder, in hydras.Ackermann.checkPrf]
+p:175 [binder, in hydras.Ackermann.checkPrf]
+p:175 [binder, in hydras.rpo.rpo]
+p:176 [binder, in hydras.rpo.term]
+p:176 [binder, in hydras.Ackermann.checkPrf]
+p:176 [binder, in gaia_hydras.T1Bridge]
+p:177 [binder, in hydras.Ackermann.checkPrf]
+p:178 [binder, in hydras.Ackermann.checkPrf]
+p:178 [binder, in hydras.rpo.rpo]
+p:179 [binder, in hydras.rpo.term]
+p:179 [binder, in hydras.Ackermann.checkPrf]
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+p:78 [binder, in hydras.Epsilon0.Hprime]
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+p:781 [binder, in hydras.Ackermann.checkPrf]
+p:782 [binder, in hydras.Ackermann.checkPrf]
+p:783 [binder, in hydras.Ackermann.checkPrf]
+p:784 [binder, in hydras.Ackermann.checkPrf]
+p:785 [binder, in hydras.Ackermann.checkPrf]
+p:79 [binder, in hydras.Ackermann.checkPrf]
+p:79 [binder, in additions.Addition_Chains]
+p:790 [binder, in hydras.Ackermann.checkPrf]
+p:797 [binder, in hydras.Ackermann.checkPrf]
+p:8 [binder, in gaia_hydras.T1Choice]
+p:8 [binder, in hydras.OrdinalNotations.ON_Omega2]
+p:8 [binder, in additions.Fib2]
+p:8 [binder, in additions.More_on_positive]
+P:8 [binder, in hydras.Schutte.MoreEpsilonIota]
+P:8 [binder, in hydras.Hydra.Hydra_Definitions]
+p:80 [binder, in additions.AM]
+p:80 [binder, in hydras.Prelude.MoreLists]
+p:80 [binder, in hydras.Ackermann.folProp]
+p:803 [binder, in hydras.Ackermann.checkPrf]
+p:81 [binder, in hydras.Ackermann.checkPrf]
+p:81 [binder, in additions.Pow]
+p:81 [binder, in hydras.MoreAck.Ack]
+p:813 [binder, in hydras.Ackermann.checkPrf]
+p:815 [binder, in hydras.Ackermann.checkPrf]
+p:817 [binder, in hydras.Ackermann.checkPrf]
+p:818 [binder, in hydras.Ackermann.checkPrf]
+p:819 [binder, in hydras.Ackermann.checkPrf]
+p:82 [binder, in hydras.Ackermann.checkPrf]
+p:82 [binder, in hydras.Ackermann.codeSubFormula]
+p:82 [binder, in hydras.Epsilon0.Hprime]
+p:820 [binder, in hydras.Ackermann.checkPrf]
+p:822 [binder, in hydras.Ackermann.checkPrf]
+p:824 [binder, in hydras.Ackermann.checkPrf]
+p:826 [binder, in hydras.Ackermann.checkPrf]
+p:828 [binder, in hydras.Ackermann.checkPrf]
+p:83 [binder, in hydras.Ackermann.checkPrf]
+P:83 [binder, in hydras.Prelude.DecPreOrder]
+p:830 [binder, in hydras.Ackermann.checkPrf]
+p:832 [binder, in hydras.Ackermann.checkPrf]
+p:834 [binder, in hydras.Ackermann.checkPrf]
+p:837 [binder, in hydras.Ackermann.checkPrf]
+p:84 [binder, in hydras.Ackermann.checkPrf]
+p:84 [binder, in additions.fib]
+p:848 [binder, in hydras.Ackermann.checkPrf]
+p:85 [binder, in hydras.Ackermann.checkPrf]
+p:85 [binder, in hydras.Ackermann.codeSubFormula]
+p:85 [binder, in Goedel.codeSysPrf]
+P:85 [binder, in hydras.Prelude.MoreVectors]
+p:850 [binder, in gaia_hydras.nfwfgaia]
+p:86 [binder, in hydras.Ackermann.checkPrf]
+P:86 [binder, in hydras.Prelude.DecPreOrder]
+p:86 [binder, in hydras.Epsilon0.Large_Sets]
+p:86 [binder, in hydras.MoreAck.Ack]
+p:86 [binder, in additions.fib]
+p:861 [binder, in hydras.Ackermann.checkPrf]
+p:866 [binder, in hydras.Ackermann.checkPrf]
+p:867 [binder, in hydras.Ackermann.checkPrf]
+p:869 [binder, in hydras.Ackermann.checkPrf]
+p:87 [binder, in hydras.Prelude.MoreLists]
+p:87 [binder, in hydras.Ackermann.checkPrf]
+p:87 [binder, in additions.Pow]
+p:872 [binder, in hydras.Ackermann.checkPrf]
+p:875 [binder, in hydras.Ackermann.checkPrf]
+p:879 [binder, in hydras.Ackermann.checkPrf]
+p:88 [binder, in additions.Pow_variant]
+p:88 [binder, in hydras.Ackermann.checkPrf]
+p:88 [binder, in hydras.rpo.rpo]
+p:882 [binder, in hydras.Ackermann.checkPrf]
+p:89 [binder, in hydras.Ackermann.checkPrf]
+P:89 [binder, in hydras.Prelude.MoreVectors]
+p:9 [binder, in hydras.MoreAck.expressibleExamples]
+p:9 [binder, in additions.Strategies]
+p:9 [binder, in hydras.Epsilon0.F_omega]
+p:9 [binder, in hydras.solutions_exercises.FibonacciPR]
+p:9 [binder, in additions.Naive]
+P:9 [binder, in hydras.Prelude.STDPP_compat]
+p:9 [binder, in additions.Dichotomy]
+p:90 [binder, in hydras.Ackermann.checkPrf]
+p:90 [binder, in hydras.Ackermann.codeSubFormula]
+p:90 [binder, in Goedel.codeSysPrf]
+p:91 [binder, in hydras.Ackermann.checkPrf]
+p:91 [binder, in additions.Pow]
+p:91 [binder, in additions.fib]
+p:92 [binder, in hydras.Ackermann.checkPrf]
+p:93 [binder, in hydras.Ackermann.checkPrf]
+p:937 [binder, in gaia_hydras.nfwfgaia]
+p:94 [binder, in hydras.Ackermann.checkPrf]
+p:94 [binder, in additions.Pow]
+p:94 [binder, in Goedel.codeSysPrf]
+p:95 [binder, in hydras.Ackermann.checkPrf]
+p:951 [binder, in gaia_hydras.nfwfgaia]
+p:953 [binder, in gaia_hydras.nfwfgaia]
+p:96 [binder, in hydras.Prelude.MoreLists]
+p:96 [binder, in hydras.Ackermann.checkPrf]
+p:96 [binder, in additions.fib]
+p:97 [binder, in hydras.Ackermann.checkPrf]
+p:97 [binder, in additions.Pow]
+p:99 [binder, in hydras.Prelude.MoreLists]
+p:99 [binder, in hydras.Ackermann.checkPrf]
+p:99 [binder, in hydras.Ackermann.folProp]
+

Q

+Q [record, in hydras.Epsilon0.F_alpha]
+QA [projection, in hydras.Epsilon0.F_alpha]
+QA0 [lemma, in hydras.Epsilon0.F_alpha]
+QD [projection, in hydras.Epsilon0.F_alpha]
+QD0 [lemma, in hydras.Epsilon0.F_alpha]
+QE [projection, in hydras.Epsilon0.F_alpha]
+q':11 [binder, in hydras.Prelude.First_toggle]
+q':30 [binder, in hydras.Prelude.More_Arith]
+q':36 [binder, in hydras.Prelude.More_Arith]
+q':59 [binder, in hydras.Prelude.More_Arith]
+q':67 [binder, in hydras.rpo.dickson]
+q:10 [binder, in additions.Strategies]
+q:100 [binder, in hydras.Ackermann.codeSubFormula]
+q:101 [binder, in hydras.Prelude.MoreLists]
+q:101 [binder, in hydras.Ackermann.codeSubFormula]
+q:105 [binder, in hydras.Ackermann.codeSubFormula]
+q:106 [binder, in hydras.Ackermann.codeSubFormula]
+q:107 [binder, in hydras.Ackermann.folProp]
+q:108 [binder, in hydras.Ackermann.folProp]
+q:113 [binder, in hydras.Ackermann.codeSubFormula]
+q:114 [binder, in hydras.Ackermann.fol]
+Q:114 [binder, in gaia_hydras.nfwfgaia]
+q:115 [binder, in hydras.Ackermann.codeSubFormula]
+q:12 [binder, in hydras.Prelude.First_toggle]
+q:12 [binder, in hydras.MoreAck.AckNotPR]
+q:121 [binder, in hydras.Ackermann.fol]
+q:126 [binder, in hydras.Ackermann.fol]
+q:13 [binder, in hydras.Prelude.First_toggle]
+q:130 [binder, in hydras.Ackermann.folProp]
+q:131 [binder, in hydras.Ackermann.fol]
+q:132 [binder, in hydras.Ackermann.folProp]
+q:14 [binder, in hydras.Prelude.First_toggle]
+q:14 [binder, in additions.More_on_positive]
+q:14 [binder, in additions.Trace_exercise]
+q:15 [binder, in hydras.Prelude.More_Arith]
+q:15 [binder, in hydras.Prelude.First_toggle]
+Q:15 [binder, in hydras.Schutte.MoreEpsilonIota]
+Q:15 [binder, in hydras.Schutte.PartialFun]
+q:16 [binder, in additions.More_on_positive]
+Q:162 [binder, in hydras.Ackermann.fol]
+q:17 [binder, in hydras.Hydra.Omega2_Small]
+q:172 [binder, in hydras.rpo.term]
+q:173 [binder, in hydras.Ackermann.fol]
+q:174 [binder, in hydras.Ackermann.fol]
+q:176 [binder, in hydras.Ackermann.fol]
+q:18 [binder, in additions.More_on_positive]
+q:187 [binder, in hydras.Ackermann.fol]
+q:19 [binder, in hydras.Prelude.MoreLists]
+q:193 [binder, in hydras.Ackermann.fol]
+q:21 [binder, in Goedel.rosser]
+q:22 [binder, in hydras.Prelude.Simple_LexProd]
+q:23 [binder, in additions.AM]
+q:24 [binder, in Goedel.rosser]
+q:24 [binder, in hydras.Prelude.Simple_LexProd]
+Q:240 [binder, in hydras.Ackermann.fol]
+q:255 [binder, in hydras.Ackermann.fol]
+q:258 [binder, in hydras.Ackermann.fol]
+q:259 [binder, in hydras.Ackermann.fol]
+q:264 [binder, in hydras.Ackermann.subAll]
+q:265 [binder, in hydras.Ackermann.subAll]
+q:265 [binder, in hydras.Ackermann.fol]
+q:266 [binder, in hydras.Ackermann.fol]
+q:27 [binder, in hydras.Schutte.GRelations]
+q:277 [binder, in hydras.Ackermann.fol]
+q:279 [binder, in hydras.Ackermann.fol]
+Q:280 [binder, in gaia_hydras.nfwfgaia]
+q:286 [binder, in hydras.Ackermann.fol]
+q:287 [binder, in hydras.Ackermann.fol]
+q:29 [binder, in hydras.Prelude.More_Arith]
+Q:290 [binder, in hydras.Ackermann.fol]
+q:291 [binder, in additions.Euclidean_Chains]
+q:296 [binder, in additions.Euclidean_Chains]
+q:3 [binder, in hydras.Prelude.LibHyps_Experiments]
+q:3 [binder, in hydras.MoreAck.PrimRecExamples]
+q:30 [binder, in hydras.MoreAck.Ack]
+q:301 [binder, in additions.Euclidean_Chains]
+q:305 [binder, in hydras.Ackermann.fol]
+q:308 [binder, in hydras.Ackermann.fol]
+q:309 [binder, in hydras.Ackermann.fol]
+q:31 [binder, in hydras.Schutte.GRelations]
+q:315 [binder, in hydras.Ackermann.fol]
+q:316 [binder, in hydras.Ackermann.fol]
+q:317 [binder, in additions.Euclidean_Chains]
+q:32 [binder, in Goedel.rosserPA]
+q:327 [binder, in hydras.Ackermann.fol]
+q:329 [binder, in hydras.Ackermann.fol]
+q:332 [binder, in hydras.Ackermann.fol]
+Q:336 [binder, in hydras.Ackermann.fol]
+q:34 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+q:35 [binder, in hydras.Prelude.More_Arith]
+q:35 [binder, in Goedel.rosserPA]
+q:350 [binder, in hydras.rpo.term]
+q:351 [binder, in hydras.Ackermann.fol]
+q:354 [binder, in hydras.Ackermann.fol]
+q:355 [binder, in hydras.Ackermann.fol]
+q:361 [binder, in additions.Euclidean_Chains]
+q:361 [binder, in hydras.Ackermann.fol]
+q:362 [binder, in hydras.Ackermann.fol]
+q:367 [binder, in additions.Euclidean_Chains]
+q:37 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+q:373 [binder, in additions.Euclidean_Chains]
+q:373 [binder, in hydras.Ackermann.fol]
+q:375 [binder, in hydras.Ackermann.fol]
+q:377 [binder, in hydras.Ackermann.fol]
+q:380 [binder, in additions.Euclidean_Chains]
+Q:380 [binder, in hydras.Ackermann.fol]
+q:387 [binder, in additions.Euclidean_Chains]
+q:393 [binder, in additions.Euclidean_Chains]
+q:395 [binder, in hydras.Ackermann.fol]
+q:398 [binder, in hydras.Ackermann.fol]
+q:399 [binder, in hydras.Ackermann.fol]
+Q:4 [binder, in hydras.Schutte.MoreEpsilonIota]
+q:400 [binder, in additions.Euclidean_Chains]
+q:405 [binder, in hydras.Ackermann.fol]
+q:406 [binder, in hydras.Ackermann.fol]
+q:417 [binder, in hydras.Ackermann.fol]
+q:419 [binder, in hydras.Ackermann.fol]
+q:422 [binder, in hydras.Ackermann.fol]
+q:425 [binder, in hydras.Ackermann.fol]
+q:43 [binder, in hydras.Prelude.More_Arith]
+q:44 [binder, in additions.More_on_positive]
+Q:447 [binder, in hydras.Gamma0.Gamma0]
+q:46 [binder, in hydras.Prelude.More_Arith]
+q:48 [binder, in additions.More_on_positive]
+q:49 [binder, in additions.FirstSteps]
+Q:504 [binder, in gaia_hydras.nfwfgaia]
+q:52 [binder, in additions.More_on_positive]
+q:53 [binder, in hydras.Ackermann.code]
+q:55 [binder, in additions.More_on_positive]
+q:58 [binder, in hydras.Prelude.More_Arith]
+q:6 [binder, in hydras.Prelude.LibHyps_Experiments]
+q:61 [binder, in hydras.Ackermann.folLogic3]
+Q:616 [binder, in hydras.Gamma0.Gamma0]
+q:62 [binder, in additions.More_on_positive]
+q:65 [binder, in hydras.rpo.dickson]
+q:664 [binder, in hydras.Ackermann.primRec]
+q:67 [binder, in hydras.Ackermann.fol]
+q:701 [binder, in hydras.Ackermann.primRec]
+q:707 [binder, in hydras.Ackermann.primRec]
+q:74 [binder, in hydras.Ackermann.fol]
+q:79 [binder, in hydras.Ackermann.fol]
+q:84 [binder, in hydras.Ackermann.fol]
+q:873 [binder, in hydras.Ackermann.checkPrf]
+q:876 [binder, in hydras.Ackermann.checkPrf]
+q:9 [binder, in additions.Fib2]
+q:90 [binder, in hydras.Ackermann.folProp]
+q:94 [binder, in hydras.Ackermann.folProp]
+q:95 [binder, in hydras.Ackermann.folProp]
+q:97 [binder, in hydras.Ackermann.codeSubFormula]
+q:98 [binder, in hydras.Prelude.MoreLists]
+

R

+RAlim [lemma, in hydras.Epsilon0.F_alpha]
+RAlim [lemma, in hydras.Epsilon0.Hprime]
+RB [lemma, in hydras.Epsilon0.F_alpha]
+RB [lemma, in hydras.Epsilon0.Hprime]
+RBlim [lemma, in hydras.Epsilon0.F_alpha]
+RBlim [lemma, in hydras.Epsilon0.Hprime]
+RC [lemma, in hydras.Epsilon0.F_alpha]
+RC [lemma, in hydras.Epsilon0.Hprime]
+RClim [lemma, in hydras.Epsilon0.F_alpha]
+RClim [lemma, in hydras.Epsilon0.Hprime]
+RD [lemma, in hydras.Epsilon0.F_alpha]
+RD [lemma, in hydras.Epsilon0.Hprime]
+RDB:19 [binder, in hydras.Prelude.DecPreOrder_Instances]
+RDlim [lemma, in hydras.Epsilon0.F_alpha]
+RDlim [lemma, in hydras.Epsilon0.Hprime]
+RE [lemma, in hydras.Epsilon0.F_alpha]
+RE [lemma, in hydras.Epsilon0.Hprime]
+reachable [definition, in hydras.Hydra.BigBattle]
+reachable_S [lemma, in hydras.Hydra.BigBattle]
+reassoc [definition, in hydras.Epsilon0.T1]
+rebindExist [lemma, in hydras.Ackermann.folLogic2]
+rebindForall [lemma, in hydras.Ackermann.folLogic2]
+recf:16 [binder, in hydras.Ackermann.PA]
+recs:100 [binder, in hydras.Ackermann.checkPrf]
+recs:100 [binder, in hydras.Ackermann.codePA]
+recs:1004 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1007 [binder, in hydras.Ackermann.codeSubFormula]
+recs:101 [binder, in hydras.Ackermann.codeFreeVar]
+recs:101 [binder, in hydras.Ackermann.codeSubTerm]
+recs:1010 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1013 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1016 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1019 [binder, in hydras.Ackermann.codeSubFormula]
+recs:102 [binder, in hydras.Ackermann.checkPrf]
+recs:103 [binder, in hydras.Ackermann.codeSubTerm]
+recs:103 [binder, in hydras.Ackermann.wellFormed]
+recs:104 [binder, in hydras.Ackermann.checkPrf]
+recs:104 [binder, in hydras.Ackermann.codePA]
+recs:1048 [binder, in hydras.Ackermann.codeSubFormula]
+recs:105 [binder, in hydras.Ackermann.codeSubTerm]
+recs:105 [binder, in hydras.Ackermann.wellFormed]
+recs:1051 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1054 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1057 [binder, in hydras.Ackermann.codeSubFormula]
+recs:106 [binder, in hydras.Ackermann.codePA]
+recs:1060 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1063 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1066 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1069 [binder, in hydras.Ackermann.codeSubFormula]
+recs:107 [binder, in hydras.Ackermann.codeSubTerm]
+recs:107 [binder, in hydras.Ackermann.wellFormed]
+recs:1072 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1074 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1076 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1078 [binder, in hydras.Ackermann.codeSubFormula]
+recs:108 [binder, in hydras.Ackermann.codePA]
+recs:1080 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1082 [binder, in hydras.Ackermann.codeSubFormula]
+recs:109 [binder, in hydras.Ackermann.codeSubTerm]
+recs:109 [binder, in hydras.Ackermann.wellFormed]
+recs:1092 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1094 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1096 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1098 [binder, in hydras.Ackermann.codeSubFormula]
+recs:11 [binder, in hydras.Ackermann.codePA]
+recs:110 [binder, in hydras.Ackermann.codePA]
+recs:1100 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1102 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1104 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1106 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1108 [binder, in hydras.Ackermann.codeSubFormula]
+recs:111 [binder, in hydras.Ackermann.codeFreeVar]
+recs:111 [binder, in hydras.Ackermann.codeSubTerm]
+recs:111 [binder, in hydras.Ackermann.wellFormed]
+recs:1110 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1118 [binder, in hydras.Ackermann.codeSubFormula]
+recs:112 [binder, in hydras.Ackermann.codePA]
+recs:1120 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1122 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1124 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1126 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1128 [binder, in hydras.Ackermann.codeSubFormula]
+recs:113 [binder, in hydras.Ackermann.codeFreeVar]
+recs:113 [binder, in hydras.Ackermann.codeSubTerm]
+recs:113 [binder, in hydras.Ackermann.wellFormed]
+recs:1130 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1132 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1134 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1136 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1138 [binder, in hydras.Ackermann.codeSubFormula]
+recs:114 [binder, in hydras.Ackermann.checkPrf]
+recs:114 [binder, in hydras.Ackermann.codePA]
+recs:1140 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1142 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1144 [binder, in hydras.Ackermann.codeSubFormula]
+recs:115 [binder, in hydras.Ackermann.codeFreeVar]
+recs:115 [binder, in hydras.Ackermann.codeSubTerm]
+recs:1154 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1156 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1158 [binder, in hydras.Ackermann.codeSubFormula]
+recs:116 [binder, in hydras.Ackermann.checkPrf]
+recs:1160 [binder, in hydras.Ackermann.codeSubFormula]
+recs:117 [binder, in hydras.Ackermann.codeFreeVar]
+recs:118 [binder, in hydras.Ackermann.checkPrf]
+recs:119 [binder, in hydras.Ackermann.codeFreeVar]
+recs:120 [binder, in hydras.Ackermann.checkPrf]
+recs:120 [binder, in hydras.Ackermann.codePA]
+recs:121 [binder, in hydras.Ackermann.codeFreeVar]
+recs:121 [binder, in hydras.Ackermann.wellFormed]
+recs:122 [binder, in hydras.Ackermann.checkPrf]
+recs:122 [binder, in hydras.Ackermann.codePA]
+recs:1221 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1224 [binder, in hydras.Ackermann.codeSubFormula]
+recs:123 [binder, in hydras.Ackermann.wellFormed]
+recs:124 [binder, in hydras.Ackermann.checkPrf]
+recs:124 [binder, in hydras.Ackermann.codePA]
+recs:1240 [binder, in hydras.Ackermann.codeSubFormula]
+recs:1243 [binder, in hydras.Ackermann.codeSubFormula]
+recs:125 [binder, in hydras.Ackermann.wellFormed]
+recs:126 [binder, in hydras.Ackermann.checkPrf]
+recs:126 [binder, in hydras.Ackermann.codeSubFormula]
+recs:126 [binder, in hydras.Ackermann.codePA]
+recs:127 [binder, in hydras.Ackermann.wellFormed]
+recs:128 [binder, in hydras.Ackermann.checkPrf]
+recs:128 [binder, in hydras.Ackermann.codeSubFormula]
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+rec:88 [binder, in Goedel.rosserPA]
+rec:9 [binder, in hydras.Ackermann.codeNatToTerm]
+rec:95 [binder, in Goedel.rosserPA]
+reduceAddExists [lemma, in Goedel.PRrepresentable]
+reduceAddExistsOneWay [lemma, in Goedel.PRrepresentable]
+reduceAddForalls [lemma, in Goedel.PRrepresentable]
+reduceAnd [lemma, in hydras.Ackermann.folReplace]
+reduceCloseList [lemma, in hydras.Ackermann.folReplace]
+reduceExist [lemma, in hydras.Ackermann.folReplace]
+reduceForall [lemma, in hydras.Ackermann.folReplace]
+reduceIff [lemma, in hydras.Ackermann.folReplace]
+reduceIfThenElse [lemma, in hydras.Ackermann.folReplace]
+reduceImp [lemma, in hydras.Ackermann.folReplace]
+reduceNot [lemma, in hydras.Ackermann.folReplace]
+reduceOr [lemma, in hydras.Ackermann.folReplace]
+reduceSub [lemma, in hydras.Ackermann.folReplace]
+reduceSubAll [lemma, in hydras.Ackermann.subAll]
+reduce_lt_plus [lemma, in hydras.Epsilon0.T1]
+reduce_assoc_list [lemma, in hydras.rpo.more_list]
+reduce1stCompose [lemma, in hydras.Ackermann.primRec]
+reduce2ndCompose [lemma, in hydras.Ackermann.primRec]
+Refinement_proof.E_eq [variable, in additions.Addition_Chains]
+Refinement_proof.E_one [variable, in additions.Addition_Chains]
+Refinement_proof.E_op [variable, in additions.Addition_Chains]
+Refinement_proof.A [variable, in additions.Addition_Chains]
+Refinement_proof [section, in additions.Addition_Chains]
+refinesPred [definition, in gaia_hydras.T1Bridge]
+refinesRel [definition, in gaia_hydras.T1Bridge]
+refines0 [definition, in gaia_hydras.T1Bridge]
+refines1 [definition, in gaia_hydras.T1Bridge]
+refines1_R [lemma, in gaia_hydras.T1Bridge]
+refines2 [definition, in gaia_hydras.T1Bridge]
+refines2_R [lemma, in gaia_hydras.T1Bridge]
+reflection [abbreviation, in Goedel.goedel1]
+rel [projection, in hydras.Ackermann.model]
+Relations [projection, in hydras.Ackermann.fol]
+RelDecision [record, in hydras.Prelude.STDPP_compat]
+RelDecision [inductive, in hydras.Prelude.STDPP_compat]
+RelDecision_Inverse_fun [instance, in hydras.Prelude.DecPreOrder_Instances]
+RElim [lemma, in hydras.Epsilon0.F_alpha]
+RElim [lemma, in hydras.Epsilon0.Hprime]
+rel_surj_i [constructor, in hydras.Schutte.GRelations]
+rel_surjection [inductive, in hydras.Schutte.GRelations]
+rel_inj_i [constructor, in hydras.Schutte.GRelations]
+rel_injection [inductive, in hydras.Schutte.GRelations]
+rel_inv [definition, in hydras.Schutte.PartialFun]
+rel_bij_i [constructor, in hydras.Schutte.PartialFun]
+rel_bijection [inductive, in hydras.Schutte.PartialFun]
+rel_inj_i [constructor, in hydras.Schutte.PartialFun]
+rel_injection [inductive, in hydras.Schutte.PartialFun]
+rel_inj [definition, in hydras.Schutte.PartialFun]
+rel_onto [definition, in hydras.Schutte.PartialFun]
+rel_functional [definition, in hydras.Schutte.PartialFun]
+rel_codomain [definition, in hydras.Schutte.PartialFun]
+rel_domain [definition, in hydras.Schutte.PartialFun]
+rel_enumerates [definition, in hydras.Schutte.Countable]
+rel_numbers [definition, in hydras.Schutte.Countable]
+rem [abbreviation, in Goedel.PRrepresentable]
+Remaining [constructor, in hydras.Epsilon0.Large_Sets_Examples]
+remove [definition, in hydras.rpo.more_list]
+remove_first_sumand [lemma, in hydras.Epsilon0.T1]
+remove_list [definition, in hydras.rpo.more_list]
+remove_heads_r_free [lemma, in hydras.Hydra.Hydra_Lemmas]
+remove_heads_r [lemma, in hydras.Hydra.Hydra_Lemmas]
+Rem0 [lemma, in hydras.Epsilon0.Paths]
+Rem01 [lemma, in hydras.Epsilon0.Paths]
+Rem02 [lemma, in hydras.Epsilon0.Paths]
+Rem03 [lemma, in hydras.Epsilon0.Paths]
+Rem04 [lemma, in hydras.Epsilon0.Paths]
+Rem05 [lemma, in hydras.Epsilon0.Paths]
+Rem06 [lemma, in hydras.Epsilon0.Paths]
+Rem08 [lemma, in hydras.Epsilon0.Paths]
+Rem09 [lemma, in hydras.Epsilon0.Paths]
+rem1 [lemma, in hydras.Hydra.O2H]
+Rem1 [lemma, in hydras.MoreAck.Ack]
+Rem1 [lemma, in hydras.Epsilon0.Paths]
+Rem10 [lemma, in hydras.Epsilon0.Paths]
+Rem11 [lemma, in hydras.Epsilon0.Paths]
+rem2 [lemma, in hydras.Hydra.O2H]
+Rem2 [lemma, in hydras.MoreAck.Ack]
+Rem2 [lemma, in hydras.Epsilon0.Paths]
+rem3 [lemma, in hydras.Hydra.O2H]
+Rem3 [lemma, in hydras.MoreAck.Ack]
+Rem3 [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+Rem4 [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+rem6 [lemma, in hydras.Hydra.O2H]
+rem61 [lemma, in hydras.Hydra.O2H]
+rem62 [lemma, in hydras.Hydra.O2H]
+rep [definition, in hydras.OrdinalNotations.ON_Generic]
+replaceCompose2 [lemma, in hydras.Ackermann.primRec]
+ReplaceFormulaTerm [definition, in hydras.Ackermann.codeSubFormula]
+ReplaceFormulaTermIsPR [instance, in hydras.Ackermann.codeSubFormula]
+ReplaceFormulaTermMonotone [lemma, in hydras.Ackermann.codeSubFormula]
+ReplaceFormulaTermSub [lemma, in hydras.Ackermann.codeSubFormula]
+Replacement [section, in hydras.Ackermann.folReplace]
+Replacement.Formula [variable, in hydras.Ackermann.folReplace]
+Replacement.Formulas [variable, in hydras.Ackermann.folReplace]
+Replacement.L [variable, in hydras.Ackermann.folReplace]
+Replacement.SysPrf [variable, in hydras.Ackermann.folReplace]
+Replacement.System [variable, in hydras.Ackermann.folReplace]
+Replacement.Term [variable, in hydras.Ackermann.folReplace]
+Replacement.Terms [variable, in hydras.Ackermann.folReplace]
+ReplaceTermsTerm [definition, in hydras.Ackermann.codeSubFormula]
+ReplaceTermsTermIsPR [instance, in hydras.Ackermann.codeSubFormula]
+ReplaceTermsTermMonotone [lemma, in hydras.Ackermann.codeSubFormula]
+ReplaceTermsTermSub [lemma, in hydras.Ackermann.codeSubFormula]
+ReplaceTermTerm [definition, in hydras.Ackermann.codeSubFormula]
+ReplaceTermTermIsPR [instance, in hydras.Ackermann.codeSubFormula]
+ReplaceTermTermMonotone [lemma, in hydras.Ackermann.codeSubFormula]
+ReplaceTermTermsTerm [definition, in hydras.Ackermann.codeSubFormula]
+ReplaceTermTermsTermIsPR [instance, in hydras.Ackermann.codeSubFormula]
+ReplaceTermTermsTermMonotone [lemma, in hydras.Ackermann.codeSubFormula]
+ReplaceTermTermSub [lemma, in hydras.Ackermann.codeSubFormula]
+replace_at_pos_list:338 [binder, in hydras.rpo.term]
+replace_at_pos_list:160 [binder, in hydras.rpo.term]
+Representable [definition, in Goedel.PRrepresentable]
+Representable [definition, in hydras.Ackermann.expressible]
+RepresentableAlternate [definition, in Goedel.PRrepresentable]
+RepresentableAlternate [lemma, in hydras.Ackermann.expressible]
+RepresentableExpressible [section, in hydras.Ackermann.expressible]
+RepresentableExpressible.closedT1 [variable, in hydras.Ackermann.expressible]
+RepresentableExpressible.nn1 [variable, in hydras.Ackermann.expressible]
+RepresentableExpressible.T [variable, in hydras.Ackermann.expressible]
+RepresentableHalfHelp [lemma, in hydras.Ackermann.expressible]
+RepresentableHalf1 [definition, in hydras.Ackermann.expressible]
+RepresentableHalf1Alternate [lemma, in hydras.Ackermann.expressible]
+RepresentableHalf2 [definition, in hydras.Ackermann.expressible]
+RepresentableHalf2Alternate [lemma, in hydras.Ackermann.expressible]
+RepresentableHelp [definition, in Goedel.PRrepresentable]
+RepresentableHelp [definition, in hydras.Ackermann.expressible]
+RepresentablesHelp [definition, in Goedel.PRrepresentable]
+Representable_ext [definition, in Goedel.PRrepresentable]
+Representable_ext [lemma, in hydras.Ackermann.expressible]
+Representable2Expressible [lemma, in hydras.Ackermann.expressible]
+RepresentsInSelf [definition, in Goedel.rosser]
+rep:66 [binder, in Goedel.rosser]
+restrict [definition, in hydras.Prelude.Restriction]
+restrict [definition, in gaia_hydras.onType]
+restrict [lemma, in hydras.Schutte.Ordering_Functions]
+restrict [definition, in gaia_hydras.nfwfgaia]
+restricted_recursion.R [variable, in hydras.Prelude.Restriction]
+restricted_recursion.E [variable, in hydras.Prelude.Restriction]
+restricted_recursion.A [variable, in hydras.Prelude.Restriction]
+restricted_recursion [section, in hydras.Prelude.Restriction]
+restricted_recursion.R [variable, in gaia_hydras.nfwfgaia]
+restricted_recursion.P [variable, in gaia_hydras.nfwfgaia]
+restricted_recursion.A [variable, in gaia_hydras.nfwfgaia]
+restricted_recursion [section, in gaia_hydras.nfwfgaia]
+Restriction [library]
+restriction_fwd [lemma, in hydras.Prelude.Restriction]
+restrict_build [definition, in hydras.Prelude.Restriction]
+rest:3 [binder, in hydras.Epsilon0.Large_Sets_Examples]
+rest:63 [binder, in hydras.Ackermann.codeSubFormula]
+result_equiv_equiv [instance, in additions.AM]
+result_equiv_success [constructor, in additions.AM]
+result_equiv_fail [constructor, in additions.AM]
+result_equiv [inductive, in additions.AM]
+Return [constructor, in additions.Addition_Chains]
+Return_proper [instance, in additions.Euclidean_Chains]
+rosser [library]
+rosserPA [library]
+Rosser'sIncompleteness [lemma, in Goedel.rosserPA]
+Rosser'sIncompleteness [lemma, in Goedel.rosser]
+Rosser's_Incompleteness.expressT2 [variable, in Goedel.rosserPA]
+Rosser's_Incompleteness.expressT1 [variable, in Goedel.rosserPA]
+Rosser's_Incompleteness.freeVarRepT [variable, in Goedel.rosserPA]
+Rosser's_Incompleteness.v0 [variable, in Goedel.rosserPA]
+Rosser's_Incompleteness.repT [variable, in Goedel.rosserPA]
+Rosser's_Incompleteness.extendsPA [variable, in Goedel.rosserPA]
+Rosser's_Incompleteness.T [variable, in Goedel.rosserPA]
+Rosser's_Incompleteness [section, in Goedel.rosserPA]
+Rosser's_Incompleteness.expressT2 [variable, in Goedel.rosser]
+Rosser's_Incompleteness.expressT1 [variable, in Goedel.rosser]
+Rosser's_Incompleteness.freeVarRepT [variable, in Goedel.rosser]
+Rosser's_Incompleteness.v0 [variable, in Goedel.rosser]
+Rosser's_Incompleteness.repT [variable, in Goedel.rosser]
+Rosser's_Incompleteness.extendsNN [variable, in Goedel.rosser]
+Rosser's_Incompleteness.T [variable, in Goedel.rosser]
+Rosser's_Incompleteness [section, in Goedel.rosser]
+round [projection, in hydras.Hydra.BigBattle]
+round [definition, in hydras.Hydra.Hydra_Definitions]
+rounds [inductive, in hydras.Hydra.Hydra_Definitions]
+rounds_free_equiv2 [lemma, in hydras.Hydra.Hydra_Lemmas]
+rounds_free_equiv1 [lemma, in hydras.Hydra.Hydra_Lemmas]
+rounds_trans [lemma, in hydras.Hydra.Hydra_Lemmas]
+rounds_n [constructor, in hydras.Hydra.Hydra_Definitions]
+rounds_1 [constructor, in hydras.Hydra.Hydra_Definitions]
+round_decr [lemma, in hydras.Hydra.Hydra_Termination]
+round_n_inv [lemma, in hydras.Hydra.Hydra_Lemmas]
+round_n_remove_h [lemma, in hydras.Hydra.Hydra_Lemmas]
+round_plus_trans [lemma, in hydras.Hydra.Hydra_Lemmas]
+round_S [lemma, in hydras.Hydra.Omega_Small]
+round_t [definition, in hydras.Hydra.Hydra_Definitions]
+round_spec [definition, in hydras.Hydra.Hydra_Definitions]
+round_star_intro [lemma, in hydras.Hydra.Hydra_Definitions]
+round_star [definition, in hydras.Hydra.Hydra_Definitions]
+round_plus [definition, in hydras.Hydra.Hydra_Definitions]
+round_n [definition, in hydras.Hydra.Hydra_Definitions]
+RP [lemma, in hydras.Epsilon0.F_alpha]
+RP [lemma, in hydras.Epsilon0.Hprime]
+RPO [module, in hydras.rpo.rpo]
+rpo [library]
+rpo_7_1 [lemma, in hydras.Gamma0.Gamma0]
+rpo_6_4 [lemma, in hydras.Gamma0.Gamma0]
+rpo_6_1 [lemma, in hydras.Gamma0.Gamma0]
+rpo_5_4 [lemma, in hydras.Gamma0.Gamma0]
+rpo_5_1 [lemma, in hydras.Gamma0.Gamma0]
+rpo_5_3 [lemma, in hydras.Gamma0.Gamma0]
+rpo_5_2 [lemma, in hydras.Gamma0.Gamma0]
+rpo_4_4 [lemma, in hydras.Gamma0.Gamma0]
+rpo_4_1 [lemma, in hydras.Gamma0.Gamma0]
+rpo_4_3 [lemma, in hydras.Gamma0.Gamma0]
+rpo_4_2 [lemma, in hydras.Gamma0.Gamma0]
+rpo_3_4 [lemma, in hydras.Gamma0.Gamma0]
+rpo_3_1 [lemma, in hydras.Gamma0.Gamma0]
+rpo_3_3 [lemma, in hydras.Gamma0.Gamma0]
+rpo_3_2 [lemma, in hydras.Gamma0.Gamma0]
+rpo_2_4 [lemma, in hydras.Gamma0.Gamma0]
+rpo_2_1 [lemma, in hydras.Gamma0.Gamma0]
+rpo_2_3 [lemma, in hydras.Gamma0.Gamma0]
+rpo_2_2 [lemma, in hydras.Gamma0.Gamma0]
+rpo_trans [lemma, in hydras.Gamma0.Gamma0]
+RPO.Eq [constructor, in hydras.rpo.rpo]
+RPO.List_mul [constructor, in hydras.rpo.rpo]
+RPO.List_eq [constructor, in hydras.rpo.rpo]
+RPO.List_gt [constructor, in hydras.rpo.rpo]
+RPO.LP [module, in hydras.rpo.rpo]
+RPO.Lt [constructor, in hydras.rpo.rpo]
+RPO.P [module, in hydras.rpo.rpo]
+RPO.rpo [inductive, in hydras.rpo.rpo]
+RPO.rpo_add_context [axiom, in hydras.rpo.rpo]
+RPO.rpo_subst [axiom, in hydras.rpo.rpo]
+RPO.rpo_trans [axiom, in hydras.rpo.rpo]
+RPO.rpo_closure [axiom, in hydras.rpo.rpo]
+RPO.rpo_mul [inductive, in hydras.rpo.rpo]
+RPO.rpo_lex [inductive, in hydras.rpo.rpo]
+RPO.rpo_eq [inductive, in hydras.rpo.rpo]
+RPO.Subterm [constructor, in hydras.rpo.rpo]
+RPO.T [module, in hydras.rpo.rpo]
+RPO.Top_eq_mul [constructor, in hydras.rpo.rpo]
+RPO.Top_eq_lex [constructor, in hydras.rpo.rpo]
+RPO.Top_gt [constructor, in hydras.rpo.rpo]
+RPO.wf_rpo [axiom, in hydras.rpo.rpo]
+rt:9 [binder, in hydras.Prelude.DecPreOrder]
+Rwf [lemma, in hydras.Prelude.First_toggle]
+R_given.R [variable, in hydras.Prelude.Sort_spec]
+R_given.A [variable, in hydras.Prelude.Sort_spec]
+R_given [section, in hydras.Prelude.Sort_spec]
+R_nat_elaguee_domain [lemma, in hydras.Schutte.GRelations]
+R_nat_elaguee_fun [lemma, in hydras.Schutte.GRelations]
+R_nat_elaguee [definition, in hydras.Schutte.GRelations]
+R_inv_inj [lemma, in hydras.Schutte.GRelations]
+R_inv [definition, in hydras.Schutte.GRelations]
+R_inv_surj [lemma, in hydras.Schutte.GRelations]
+R_pred_Sn [lemma, in hydras.Gamma0.Gamma0]
+R_inc_rpo [lemma, in hydras.Gamma0.Gamma0]
+r':32 [binder, in hydras.Prelude.More_Arith]
+r':38 [binder, in hydras.Prelude.More_Arith]
+R':547 [binder, in hydras.Ackermann.primRec]
+R':561 [binder, in hydras.Ackermann.primRec]
+r':61 [binder, in hydras.Prelude.More_Arith]
+r':90 [binder, in hydras.Hydra.Hydra_Definitions]
+R0 [lemma, in hydras.MoreAck.Ack]
+R0 [lemma, in hydras.Hydra.Battle_length]
+R0 [lemma, in gaia_hydras.onType]
+R0 [lemma, in hydras.Epsilon0.Hprime]
+R00 [lemma, in hydras.Epsilon0.F_omega]
+R00 [lemma, in hydras.solutions_exercises.isqrt]
+R01 [instance, in hydras.Epsilon0.F_omega]
+R01 [lemma, in hydras.solutions_exercises.isqrt]
+R02 [instance, in hydras.Epsilon0.F_omega]
+R02 [lemma, in hydras.solutions_exercises.isqrt]
+R03 [instance, in hydras.Epsilon0.F_omega]
+R1 [lemma, in hydras.solutions_exercises.is_F_monotonous]
+R1 [lemma, in hydras.Epsilon0.F_alpha]
+R1 [lemma, in hydras.MoreAck.AckNotPR]
+R1 [lemma, in hydras.Epsilon0.Large_Sets]
+R1 [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+R1 [lemma, in hydras.MoreAck.Ack]
+R1 [lemma, in hydras.solutions_exercises.schutte_cnf_counter_example]
+R1 [lemma, in gaia_hydras.onType]
+R1 [lemma, in hydras.solutions_exercises.F_3]
+R1 [lemma, in hydras.Epsilon0.Hprime]
+R1 [definition, in hydras.Schutte.Ordering_Functions]
+R1 [lemma, in hydras.Gamma0.Gamma0]
+R1 [lemma, in hydras.MoreAck.FolExamples]
+R1 [inductive, in hydras.Hydra.Hydra_Definitions]
+R1_mem_head [lemma, in hydras.Hydra.O2H]
+R1_R2 [lemma, in hydras.Hydra.O2H]
+R1_hcons_mult [lemma, in hydras.Hydra.O2H]
+R1_hcons [lemma, in hydras.Hydra.O2H]
+R1_decr [lemma, in hydras.Hydra.Hydra_Termination]
+R1_decr_0 [lemma, in hydras.Hydra.Hydra_Termination]
+R1_remove_r_i [lemma, in hydras.Hydra.Hydra_Lemmas]
+R1_app [lemma, in hydras.Hydra.Hydra_Lemmas]
+R1_iff [lemma, in hydras.Hydra.Hydra_Lemmas]
+R1_aux [lemma, in hydras.Schutte.Ordering_Functions]
+R1_intro [constructor, in hydras.Hydra.Hydra_Definitions]
+R12 [lemma, in hydras.Epsilon0.Paths]
+R13 [lemma, in hydras.Epsilon0.Paths]
+R14 [lemma, in hydras.Epsilon0.Paths]
+R15 [lemma, in hydras.Epsilon0.Paths]
+R16 [lemma, in hydras.Epsilon0.Paths]
+R17 [lemma, in hydras.Epsilon0.Paths]
+R18 [lemma, in hydras.Epsilon0.Paths]
+R19 [lemma, in hydras.Epsilon0.Paths]
+R1:2 [binder, in additions.Wf_transparent]
+r1:41 [binder, in hydras.Gamma0.Gamma0]
+r1:49 [binder, in hydras.Gamma0.Gamma0]
+r1:57 [binder, in hydras.Gamma0.Gamma0]
+r1:65 [binder, in hydras.Gamma0.Gamma0]
+R2 [lemma, in hydras.solutions_exercises.is_F_monotonous]
+R2 [instance, in hydras.MoreAck.AckNotPR]
+R2 [lemma, in hydras.Epsilon0.Large_Sets]
+R2 [lemma, in hydras.MoreAck.Ack]
+R2 [lemma, in hydras.solutions_exercises.schutte_cnf_counter_example]
+R2 [lemma, in hydras.Hydra.Battle_length]
+R2 [lemma, in hydras.Epsilon0.Hprime]
+R2 [lemma, in hydras.Schutte.Ordering_Functions]
+R2 [lemma, in hydras.Gamma0.Gamma0]
+R2 [inductive, in hydras.Hydra.Hydra_Definitions]
+r2i [definition, in hydras.Schutte.PartialFun]
+R2_decr [lemma, in hydras.Hydra.Hydra_Termination]
+R2_decr_0 [lemma, in hydras.Hydra.Hydra_Termination]
+R2_app [lemma, in hydras.Hydra.Hydra_Lemmas]
+R2_ind2 [definition, in hydras.Hydra.Hydra_Lemmas]
+R2_iff [lemma, in hydras.Hydra.Hydra_Lemmas]
+R2_intro_2 [constructor, in hydras.Hydra.Hydra_Definitions]
+R2_intro [constructor, in hydras.Hydra.Hydra_Definitions]
+R20 [lemma, in hydras.Epsilon0.Paths]
+R21 [lemma, in hydras.Epsilon0.Paths]
+R22 [lemma, in hydras.Epsilon0.Paths]
+R23 [lemma, in hydras.Epsilon0.Paths]
+R24 [lemma, in hydras.Epsilon0.Paths]
+R25 [lemma, in hydras.Epsilon0.Paths]
+R26 [lemma, in hydras.Epsilon0.Paths]
+R27 [lemma, in hydras.Epsilon0.Paths]
+R28 [lemma, in hydras.Epsilon0.Paths]
+R29 [lemma, in hydras.Epsilon0.Paths]
+R2:3 [binder, in additions.Wf_transparent]
+r2:42 [binder, in hydras.Gamma0.Gamma0]
+r2:50 [binder, in hydras.Gamma0.Gamma0]
+r2:58 [binder, in hydras.Gamma0.Gamma0]
+r2:66 [binder, in hydras.Gamma0.Gamma0]
+R3 [lemma, in hydras.solutions_exercises.is_F_monotonous]
+R3 [lemma, in hydras.MoreAck.Ack]
+R3 [lemma, in hydras.solutions_exercises.schutte_cnf_counter_example]
+R3 [lemma, in hydras.Hydra.Battle_length]
+R3 [lemma, in hydras.solutions_exercises.F_3]
+R3 [lemma, in hydras.Epsilon0.Hprime]
+R3 [lemma, in hydras.Schutte.Ordering_Functions]
+R3 [lemma, in hydras.Gamma0.Gamma0]
+R3' [lemma, in hydras.solutions_exercises.F_3]
+R30 [lemma, in hydras.Epsilon0.Paths]
+R31 [lemma, in hydras.Epsilon0.Paths]
+R31_0 [lemma, in hydras.Epsilon0.Paths]
+R4 [lemma, in hydras.solutions_exercises.is_F_monotonous]
+R4 [lemma, in hydras.Hydra.Battle_length]
+R4 [lemma, in hydras.solutions_exercises.F_3]
+R4 [lemma, in hydras.Epsilon0.Hprime]
+R4 [lemma, in hydras.Schutte.Ordering_Functions]
+R4 [lemma, in hydras.Gamma0.Gamma0]
+R4_3_4 [lemma, in hydras.Epsilon0.Paths]
+R4_3_3 [lemma, in hydras.Epsilon0.Paths]
+R4_3_2 [lemma, in hydras.Epsilon0.Paths]
+R4_3_1 [lemma, in hydras.Epsilon0.Paths]
+R4' [lemma, in hydras.Schutte.Ordering_Functions]
+R4'' [lemma, in hydras.Schutte.Ordering_Functions]
+R42 [lemma, in hydras.Schutte.Ordering_Functions]
+R5 [lemma, in hydras.Epsilon0.F_omega]
+R5 [lemma, in hydras.MoreAck.Ack]
+R5 [lemma, in hydras.Hydra.Battle_length]
+R5 [lemma, in hydras.solutions_exercises.F_3]
+R5 [lemma, in hydras.Epsilon0.Hprime]
+R5 [lemma, in hydras.Schutte.Ordering_Functions]
+R5 [lemma, in hydras.Gamma0.Gamma0]
+R6 [lemma, in hydras.solutions_exercises.is_F_monotonous]
+R6 [lemma, in hydras.Hydra.Battle_length]
+R6 [lemma, in hydras.Schutte.Ordering_Functions]
+R7 [lemma, in hydras.Schutte.Ordering_Functions]
+R:1 [binder, in hydras.rpo.dickson]
+r:10 [binder, in hydras.Ackermann.PA]
+R:10 [binder, in hydras.Schutte.PartialFun]
+R:10 [binder, in hydras.rpo.dickson]
+r:11 [binder, in additions.Strategies]
+R:11 [binder, in hydras.Prelude.DecPreOrder]
+R:11 [binder, in hydras.rpo.closure]
+R:110 [binder, in hydras.Prelude.MoreLists]
+r:114 [binder, in hydras.Ackermann.codeSubFormula]
+R:121 [binder, in hydras.Prelude.MoreLists]
+R:127 [binder, in hydras.Ackermann.primRec]
+R:128 [binder, in hydras.Prelude.MoreLists]
+r:131 [binder, in hydras.Ackermann.folProp]
+R:133 [binder, in hydras.Prelude.MoreLists]
+r:137 [binder, in hydras.Gamma0.T2]
+R:138 [binder, in hydras.Prelude.MoreLists]
+r:14 [binder, in hydras.Ackermann.model]
+r:14 [binder, in hydras.Ackermann.checkPrf]
+R:14 [binder, in hydras.Prelude.STDPP_compat]
+r:14 [binder, in Goedel.codeSysPrf]
+R:143 [binder, in hydras.Prelude.MoreLists]
+r:148 [binder, in hydras.Ackermann.folProp]
+r:15 [binder, in hydras.Ackermann.fol]
+R:16 [binder, in hydras.Schutte.PartialFun]
+R:16 [binder, in hydras.rpo.closure]
+R:16 [binder, in hydras.Ackermann.Languages]
+R:166 [binder, in hydras.Prelude.MoreLists]
+r:17 [binder, in hydras.Ackermann.checkPrf]
+r:17 [binder, in Goedel.codeSysPrf]
+R:17 [binder, in hydras.rpo.dickson]
+r:177 [binder, in hydras.Ackermann.folProp]
+r:182 [binder, in hydras.Ackermann.wellFormed]
+R:184 [binder, in additions.Addition_Chains]
+r:185 [binder, in hydras.Ackermann.wellFormed]
+R:19 [binder, in hydras.rpo.closure]
+r:193 [binder, in hydras.Ackermann.folProp]
+R:2 [binder, in hydras.Prelude.DecPreOrder]
+R:2 [binder, in hydras.Prelude.STDPP_compat]
+R:2 [binder, in hydras.rpo.closure]
+R:20 [binder, in hydras.Ackermann.extEqualNat]
+R:20 [binder, in hydras.Ackermann.Languages]
+R:20 [binder, in hydras.solutions_exercises.T1_ltNotWf]
+r:204 [binder, in hydras.Ackermann.fol]
+r:209 [binder, in hydras.Ackermann.folProp]
+R:21 [binder, in hydras.rpo.dickson]
+R:21 [binder, in hydras.rpo.closure]
+R:22 [binder, in hydras.Ackermann.extEqualNat]
+R:22 [binder, in hydras.Ackermann.checkPrf]
+r:22 [binder, in hydras.Ackermann.Languages]
+r:225 [binder, in hydras.Ackermann.folProp]
+r:227 [binder, in hydras.Ackermann.fol]
+r:23 [binder, in hydras.Ackermann.Languages]
+r:245 [binder, in hydras.Ackermann.fol]
+R:25 [binder, in hydras.rpo.dickson]
+r:25 [binder, in hydras.Ackermann.Languages]
+r:256 [binder, in hydras.Ackermann.folProp]
+r:27 [binder, in additions.Fib2]
+r:272 [binder, in hydras.Ackermann.folProp]
+r:278 [binder, in hydras.Ackermann.fol]
+r:284 [binder, in hydras.Ackermann.subAll]
+r:289 [binder, in hydras.Ackermann.subAll]
+R:29 [binder, in hydras.rpo.dickson]
+r:292 [binder, in additions.Euclidean_Chains]
+r:295 [binder, in hydras.Ackermann.fol]
+r:297 [binder, in additions.Euclidean_Chains]
+R:3 [binder, in hydras.Prelude.Restriction]
+r:3 [binder, in gaia_hydras.onType]
+R:3 [binder, in hydras.Schutte.Countable]
+r:305 [binder, in hydras.Ackermann.folProp]
+r:31 [binder, in hydras.Prelude.More_Arith]
+r:31 [binder, in hydras.Ackermann.model]
+R:31 [binder, in hydras.Prelude.DecPreOrder]
+r:319 [binder, in hydras.Ackermann.subAll]
+r:321 [binder, in hydras.Ackermann.subAll]
+r:321 [binder, in hydras.Ackermann.folProp]
+r:328 [binder, in hydras.Ackermann.fol]
+r:33 [binder, in hydras.Schutte.GRelations]
+R:33 [binder, in hydras.rpo.dickson]
+r:337 [binder, in hydras.Ackermann.folProp]
+r:34 [binder, in hydras.Schutte.GRelations]
+r:341 [binder, in hydras.Ackermann.fol]
+r:35 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+r:353 [binder, in hydras.Ackermann.folProp]
+r:362 [binder, in additions.Euclidean_Chains]
+r:368 [binder, in additions.Euclidean_Chains]
+r:369 [binder, in hydras.Ackermann.folProp]
+r:37 [binder, in hydras.Prelude.More_Arith]
+r:37 [binder, in hydras.Ackermann.model]
+R:37 [binder, in hydras.rpo.dickson]
+r:374 [binder, in additions.Euclidean_Chains]
+r:374 [binder, in hydras.Ackermann.fol]
+r:38 [binder, in hydras.Ackermann.model]
+r:38 [binder, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+R:38 [binder, in hydras.rpo.dickson]
+r:381 [binder, in additions.Euclidean_Chains]
+r:385 [binder, in hydras.Ackermann.folProp]
+r:385 [binder, in hydras.Ackermann.fol]
+r:388 [binder, in additions.Euclidean_Chains]
+r:39 [binder, in hydras.Prelude.Restriction]
+r:394 [binder, in additions.Euclidean_Chains]
+R:4 [binder, in hydras.Schutte.Countable]
+r:401 [binder, in additions.Euclidean_Chains]
+R:402 [binder, in hydras.Epsilon0.T1]
+R:41 [binder, in hydras.rpo.dickson]
+r:418 [binder, in hydras.Ackermann.fol]
+R:42 [binder, in hydras.Ackermann.folProof]
+R:42 [binder, in hydras.rpo.dickson]
+r:438 [binder, in hydras.Ackermann.fol]
+r:45 [binder, in additions.More_on_positive]
+R:48 [binder, in hydras.rpo.dickson]
+r:49 [binder, in additions.More_on_positive]
+R:5 [binder, in hydras.Schutte.Countable]
+R:51 [binder, in hydras.Ackermann.Languages]
+R:52 [binder, in hydras.Ackermann.Languages]
+r:54 [binder, in hydras.Ackermann.Languages]
+R:54 [binder, in hydras.Ackermann.expressible]
+R:543 [binder, in hydras.Ackermann.primRec]
+R:546 [binder, in hydras.Ackermann.primRec]
+R:55 [binder, in hydras.rpo.dickson]
+r:55 [binder, in hydras.Ackermann.Languages]
+R:557 [binder, in hydras.Ackermann.primRec]
+r:56 [binder, in additions.More_on_positive]
+r:56 [binder, in hydras.Ackermann.Languages]
+R:560 [binder, in hydras.Ackermann.primRec]
+R:563 [binder, in hydras.Ackermann.primRec]
+R:565 [binder, in hydras.Ackermann.primRec]
+r:57 [binder, in hydras.Ackermann.Languages]
+R:57 [binder, in hydras.Ackermann.expressible]
+r:58 [binder, in hydras.Ackermann.Languages]
+r:59 [binder, in additions.More_on_positive]
+r:59 [binder, in hydras.Ackermann.Languages]
+r:6 [binder, in hydras.solutions_exercises.isqrt]
+r:60 [binder, in hydras.Prelude.More_Arith]
+r:60 [binder, in hydras.Ackermann.Languages]
+r:61 [binder, in hydras.Ackermann.Languages]
+R:61 [binder, in hydras.Ackermann.expressible]
+R:62 [binder, in hydras.rpo.dickson]
+R:63 [binder, in hydras.rpo.dickson]
+r:63 [binder, in hydras.Ackermann.Languages]
+r:64 [binder, in hydras.Ackermann.model]
+R:65 [binder, in hydras.Ackermann.expressible]
+r:652 [binder, in hydras.Ackermann.checkPrf]
+r:66 [binder, in hydras.Ackermann.folLogic3]
+r:67 [binder, in additions.fib]
+r:68 [binder, in additions.fib]
+R:7 [binder, in hydras.Ackermann.folProof]
+r:7 [binder, in hydras.Ackermann.PAconsistent]
+R:7 [binder, in hydras.Schutte.PartialFun]
+R:70 [binder, in hydras.Ackermann.expressible]
+R:8 [binder, in hydras.Prelude.DecPreOrder]
+R:8 [binder, in hydras.Ackermann.code]
+r:8 [binder, in hydras.solutions_exercises.isqrt]
+r:85 [binder, in hydras.Ackermann.model]
+r:88 [binder, in hydras.Ackermann.codeSubFormula]
+r:89 [binder, in hydras.Hydra.Hydra_Definitions]
+

S

+s [definition, in hydras.solutions_exercises.Limit_Infinity]
+S [module, in hydras.rpo.decidable_set]
+S [module, in hydras.rpo.list_set]
+s [definition, in hydras.solutions_exercises.T1_ltNotWf]
+SA1 [lemma, in hydras.Schutte.Ordering_Functions]
+SA2 [lemma, in hydras.Schutte.Ordering_Functions]
+SA:17 [binder, in hydras.Prelude.Simple_LexProd]
+SA:19 [binder, in hydras.Prelude.Simple_LexProd]
+SB:18 [binder, in hydras.Prelude.Simple_LexProd]
+SB:20 [binder, in hydras.Prelude.Simple_LexProd]
+Schutte [library]
+Schutte_basics [library]
+schutte_cnf_counter_example [library]
+searchProof [lemma, in Goedel.rosserPA]
+searchProof [lemma, in Goedel.rosser]
+searchXYIsPR [instance, in hydras.Ackermann.cPair]
+SecondIncompletness [lemma, in Goedel.goedel2]
+segment [definition, in hydras.Schutte.Ordering_Functions]
+segment_lt_closed [lemma, in hydras.Schutte.Ordering_Functions]
+segment_the_ordering_segment [lemma, in hydras.Schutte.Ordering_Functions]
+segment_unbounded [lemma, in hydras.Schutte.Ordering_Functions]
+segment_lt [lemma, in hydras.Schutte.Ordering_Functions]
+self_lt_free [lemma, in hydras.Hydra.Epsilon0_Needed_Free]
+self_lt_standard [lemma, in hydras.Hydra.Epsilon0_Needed_Std]
+Semantics [section, in additions.AM]
+Semantics.A [variable, in additions.AM]
+Semantics.mul [variable, in additions.AM]
+Semantics.one [variable, in additions.AM]
+Semibundled [module, in hydras.Prelude.DecPreOrder]
+Semibundled.dec_dec [projection, in hydras.Prelude.DecPreOrder]
+Semibundled.TotalDec [record, in hydras.Prelude.DecPreOrder]
+Semibundled.TotalDecPreOrder [record, in hydras.Prelude.DecPreOrder]
+Semibundled.total_dec_total [projection, in hydras.Prelude.DecPreOrder]
+Semibundled.total_dec_pre_order [projection, in hydras.Prelude.DecPreOrder]
+Semibundled.total_dec [projection, in hydras.Prelude.DecPreOrder]
+Sentence [definition, in hydras.Ackermann.LNN]
+Sentence [definition, in hydras.Ackermann.LNT]
+Sentence [definition, in hydras.Ackermann.folProp]
+seq [definition, in hydras.solutions_exercises.MultisetWf]
+Sequences [section, in gaia_hydras.nfwfgaia]
+Sequences.A [variable, in gaia_hydras.nfwfgaia]
+Sequences.R [variable, in gaia_hydras.nfwfgaia]
+Sequences.W [variable, in gaia_hydras.nfwfgaia]
+seq_mono_inj [lemma, in hydras.Schutte.Schutte_basics]
+seq_mono_intro [lemma, in hydras.Schutte.Schutte_basics]
+seq_mono [definition, in hydras.Schutte.Schutte_basics]
+seq_range_countable [lemma, in hydras.Schutte.Countable]
+seq_range [definition, in hydras.Schutte.Countable]
+seq:22 [binder, in hydras.solutions_exercises.T1_ltNotWf]
+set_eq [definition, in hydras.Schutte.Schutte_basics]
+shift [definition, in hydras.Prelude.MoreLists]
+shift_unshift [lemma, in hydras.Prelude.MoreLists]
+shift_no_zero [lemma, in hydras.Prelude.MoreLists]
+shift_interval [lemma, in hydras.Prelude.MoreLists]
+shift_iota_from [lemma, in hydras.Prelude.MoreLists]
+SigA:279 [binder, in Goedel.PRrepresentable]
+SigA:282 [binder, in Goedel.PRrepresentable]
+SigA:297 [binder, in Goedel.PRrepresentable]
+SigB:280 [binder, in Goedel.PRrepresentable]
+SigB:283 [binder, in Goedel.PRrepresentable]
+SigB:298 [binder, in Goedel.PRrepresentable]
+sigma1:113 [binder, in hydras.rpo.term]
+sigma1:122 [binder, in hydras.rpo.term]
+sigma1:291 [binder, in hydras.rpo.term]
+sigma1:300 [binder, in hydras.rpo.term]
+sigma1:302 [binder, in hydras.rpo.term]
+sigma2:114 [binder, in hydras.rpo.term]
+sigma2:123 [binder, in hydras.rpo.term]
+sigma2:292 [binder, in hydras.rpo.term]
+sigma2:301 [binder, in hydras.rpo.term]
+sigma2:303 [binder, in hydras.rpo.term]
+sigma:102 [binder, in hydras.rpo.term]
+sigma:111 [binder, in hydras.rpo.term]
+sigma:119 [binder, in hydras.rpo.term]
+sigma:126 [binder, in hydras.rpo.term]
+sigma:128 [binder, in hydras.rpo.term]
+sigma:184 [binder, in hydras.rpo.term]
+sigma:280 [binder, in hydras.rpo.term]
+sigma:289 [binder, in hydras.rpo.term]
+sigma:297 [binder, in hydras.rpo.term]
+sigma:307 [binder, in hydras.rpo.term]
+sigma:309 [binder, in hydras.rpo.term]
+sigma:315 [binder, in hydras.rpo.rpo]
+sigma:363 [binder, in hydras.rpo.term]
+sigma:86 [binder, in hydras.rpo.rpo]
+Signature [module, in hydras.rpo.term]
+signature [inductive, in additions.Euclidean_Chains]
+signature_measure [definition, in additions.Euclidean_Chains]
+signature_exponent [definition, in additions.Euclidean_Chains]
+Signature.AC [constructor, in hydras.rpo.term]
+Signature.arity [axiom, in hydras.rpo.term]
+Signature.arity_type [inductive, in hydras.rpo.term]
+Signature.C [constructor, in hydras.rpo.term]
+Signature.eq_symbol_dec [axiom, in hydras.rpo.term]
+Signature.Free [constructor, in hydras.rpo.term]
+Signature.symb [axiom, in hydras.rpo.term]
+sig_eq_intro [lemma, in hydras.OrdinalNotations.ON_Finite]
+simple_last_app1 [lemma, in hydras.Prelude.MoreLists]
+simple_last_app [lemma, in hydras.Prelude.MoreLists]
+simple_last_correct [lemma, in hydras.Prelude.MoreLists]
+simple_last [definition, in hydras.Prelude.MoreLists]
+Simple_LexProd [library]
+single_nf [lemma, in hydras.Epsilon0.T1]
+single_nf [constructor, in hydras.Gamma0.T2]
+size_acc:203 [binder, in hydras.rpo.term]
+size_acc:20 [binder, in hydras.rpo.term]
+size_acc:54 [binder, in hydras.rpo.more_list]
+size1:57 [binder, in hydras.rpo.more_list]
+size2:58 [binder, in hydras.rpo.more_list]
+size:37 [binder, in hydras.rpo.more_list]
+size:42 [binder, in hydras.rpo.more_list]
+size:47 [binder, in hydras.rpo.more_list]
+size:51 [binder, in hydras.rpo.more_list]
+size:69 [binder, in hydras.rpo.list_permut]
+size:73 [binder, in hydras.rpo.more_list]
+size:74 [binder, in hydras.rpo.list_permut]
+slowexp [definition, in additions.fib]
+smallestExists [lemma, in hydras.Prelude.MoreDecidable]
+small_h [definition, in hydras.Hydra.Epsilon0_Needed_Generic]
+small_lemma [lemma, in hydras.Epsilon0.Paths]
+SmNotPR [lemma, in hydras.Hydra.Hydra_Theorems]
+smono_Sle [lemma, in hydras.Prelude.Iterates]
+sn_tt:288 [binder, in hydras.rpo.rpo]
+sn_tt:285 [binder, in hydras.rpo.rpo]
+sn_tt:282 [binder, in hydras.rpo.rpo]
+some [definition, in hydras.Schutte.MoreEpsilonIota]
+some_pos' [definition, in gaia_hydras.T1Choice]
+some_pos [definition, in gaia_hydras.T1Choice]
+some_nb_occ_Sn [lemma, in hydras.rpo.more_list]
+some_pos [definition, in hydras.Schutte.MoreEpsilonIota]
+sort [definition, in hydras.Prelude.Sort_spec]
+sorted [definition, in hydras.Schutte.CNF]
+sorted_ge_suffix [lemma, in hydras.Prelude.MoreLists]
+sorted_max_2 [lemma, in hydras.Prelude.MoreLists]
+sorted_cut [lemma, in hydras.Prelude.MoreLists]
+sorted_max_1 [lemma, in hydras.Prelude.MoreLists]
+sorted_not_in_tail [lemma, in hydras.Prelude.MoreLists]
+sorted_In [lemma, in hydras.Prelude.MoreLists]
+sorted_ge_prefix [lemma, in hydras.Prelude.MoreLists]
+sorted_ge_iff [lemma, in hydras.Prelude.MoreLists]
+sorted_ge_iff0 [lemma, in hydras.Prelude.MoreLists]
+Sorted_mono [lemma, in hydras.Prelude.MoreLists]
+sorted_head [lemma, in hydras.Prelude.MoreLists]
+sorted_tail' [lemma, in hydras.Prelude.MoreLists]
+sorted_tail [lemma, in hydras.Prelude.MoreLists]
+sorted_le [lemma, in hydras.Prelude.MoreLists]
+sorted_inv_gt [lemma, in hydras.Prelude.MoreLists]
+sorted_ge_not_In [lemma, in hydras.Prelude.MoreLists]
+sorted_ge_trans [lemma, in hydras.Prelude.MoreLists]
+sorted_ge_Forall [lemma, in hydras.Prelude.MoreLists]
+sorted_ge_cons [constructor, in hydras.Prelude.MoreLists]
+sorted_ge_one [constructor, in hydras.Prelude.MoreLists]
+sorted_ge_nil [constructor, in hydras.Prelude.MoreLists]
+sorted_ge [inductive, in hydras.Prelude.MoreLists]
+sorted_lt [lemma, in hydras.Schutte.CNF]
+sorted_lt_lt_2 [lemma, in hydras.Schutte.CNF]
+sorted_lt_lt [lemma, in hydras.Schutte.CNF]
+sorted_tail [lemma, in hydras.Schutte.CNF]
+sort_spec [definition, in hydras.Prelude.Sort_spec]
+sort_correct [definition, in hydras.Prelude.Sort_spec]
+sort_rel [definition, in hydras.Prelude.Sort_spec]
+sort_fun_t [definition, in hydras.Prelude.Sort_spec]
+Sort_spec [library]
+split_list_app_cons [lemma, in hydras.rpo.more_list]
+split_list [definition, in hydras.rpo.more_list]
+split_permutation [lemma, in hydras.Prelude.Merge_Sort]
+split_decr [lemma, in hydras.Prelude.Merge_Sort]
+split'_permutation [lemma, in hydras.Prelude.Merge_Sort]
+split'_aux_eq [lemma, in hydras.Prelude.Merge_Sort]
+split'_decr [lemma, in hydras.Prelude.Merge_Sort]
+split'_aux_length_fst [lemma, in hydras.Prelude.Merge_Sort]
+split'_aux_length_preserve [lemma, in hydras.Prelude.Merge_Sort]
+sp_mergesort_OK [lemma, in hydras.Prelude.Merge_Sort]
+sp_merge_sort [definition, in hydras.Prelude.Merge_Sort]
+SQR [constructor, in additions.AM]
+sqrtIsPR [section, in hydras.solutions_exercises.isqrt]
+sqrtIsPR.P [variable, in hydras.solutions_exercises.isqrt]
+sqrtIsPR.Proof_isqrt.n [variable, in hydras.solutions_exercises.isqrt]
+sqrtIsPR.Proof_isqrt [section, in hydras.solutions_exercises.isqrt]
+sqrt_correct [lemma, in hydras.solutions_exercises.isqrt]
+sqr_eqn [lemma, in additions.Pow_variant]
+sqr_eqn [lemma, in additions.Pow]
+squares:6 [binder, in additions.AM]
+ssrnat_extracts [library]
+ss:113 [binder, in hydras.Ackermann.fol]
+ss:120 [binder, in hydras.Ackermann.fol]
+ss:125 [binder, in hydras.Ackermann.fol]
+ss:130 [binder, in hydras.Ackermann.fol]
+ss:28 [binder, in hydras.Ackermann.folLogic3]
+ss:30 [binder, in hydras.Ackermann.folLogic3]
+ss:31 [binder, in hydras.Ackermann.code]
+ss:34 [binder, in hydras.Ackermann.code]
+ss:37 [binder, in hydras.Ackermann.code]
+ss:37 [binder, in hydras.Ackermann.folLogic3]
+ss:4 [binder, in hydras.Ackermann.folProp]
+ss:44 [binder, in hydras.Ackermann.folLogic3]
+ss:51 [binder, in hydras.Ackermann.folProp]
+ss:51 [binder, in hydras.Ackermann.folLogic3]
+ss:59 [binder, in hydras.Ackermann.folLogic3]
+ss:66 [binder, in hydras.Ackermann.fol]
+ss:68 [binder, in hydras.Ackermann.folLogic3]
+ss:73 [binder, in hydras.Ackermann.fol]
+ss:78 [binder, in hydras.Ackermann.fol]
+ss:78 [binder, in hydras.Ackermann.folLogic3]
+ss:83 [binder, in hydras.Ackermann.fol]
+ss:94 [binder, in hydras.Ackermann.fol]
+ss:97 [binder, in hydras.Ackermann.fol]
+stable [definition, in hydras.Prelude.Sort_spec]
+stable_test [definition, in hydras.Prelude.Sort_spec]
+stable_mergesort_OK [lemma, in hydras.Prelude.Merge_Sort]
+stable_merge_sort [definition, in hydras.Prelude.Merge_Sort]
+stack [definition, in additions.AM]
+Stack_equiv_equiv [instance, in additions.AM]
+Stack_equiv_refl [instance, in additions.AM]
+stack_equivn [constructor, in additions.AM]
+stack_equiv0 [constructor, in additions.AM]
+stack_equiv [inductive, in additions.AM]
+standard [instance, in hydras.Hydra.Hydra_Definitions]
+standard_gnaw_iota_from [lemma, in gaia_hydras.GPaths]
+standard_gnaw [definition, in gaia_hydras.GPaths]
+standard_path [definition, in gaia_hydras.GPaths]
+standard_battle_head [lemma, in hydras.Hydra.Hydra_Lemmas]
+standard_incl_free [lemma, in hydras.Hydra.Hydra_Lemmas]
+standard_gnaw_2_zero [lemma, in hydras.Epsilon0.Paths]
+standard_path_to_zero [lemma, in hydras.Epsilon0.Paths]
+standard_gnaw_to_zero [lemma, in hydras.Epsilon0.Paths]
+standard_pathS_app [lemma, in hydras.Epsilon0.Paths]
+standard_path_equiv_2 [lemma, in hydras.Epsilon0.Paths]
+standard_path_equiv_1 [lemma, in hydras.Epsilon0.Paths]
+standard_gnaw_nf [lemma, in hydras.Epsilon0.Paths]
+standard_gnaw_to_path [lemma, in hydras.Epsilon0.Paths]
+standard_gnaw_S_zero [lemma, in hydras.Epsilon0.Paths]
+standard_path_compose [lemma, in hydras.Epsilon0.Paths]
+standard_path_lt2 [lemma, in hydras.Epsilon0.Paths]
+standard_path_zero [lemma, in hydras.Epsilon0.Paths]
+standard_path_to_nf [lemma, in hydras.Epsilon0.Paths]
+standard_path_LE [lemma, in hydras.Epsilon0.Paths]
+standard_gnaw_plus [lemma, in hydras.Epsilon0.Paths]
+standard_gnaw_iota_from [lemma, in hydras.Epsilon0.Paths]
+standard_gnaw_zero [lemma, in hydras.Epsilon0.Paths]
+standard_path_path_to [lemma, in hydras.Epsilon0.Paths]
+standard_pathS_path_toS [lemma, in hydras.Epsilon0.Paths]
+standard_path_origin [lemma, in hydras.Epsilon0.Paths]
+standard_path_toS_zero [lemma, in hydras.Epsilon0.Paths]
+standard_path_to_le_inv [lemma, in hydras.Epsilon0.Paths]
+standard_path_toS_le_inv [lemma, in hydras.Epsilon0.Paths]
+standard_path_unshift [lemma, in hydras.Epsilon0.Paths]
+standard_path_unshift_0 [lemma, in hydras.Epsilon0.Paths]
+standard_path_shift [lemma, in hydras.Epsilon0.Paths]
+standard_path [definition, in hydras.Epsilon0.Paths]
+standard_path_to [inductive, in hydras.Epsilon0.Paths]
+standard_pathS [definition, in hydras.Epsilon0.Paths]
+standard_path_toS [inductive, in hydras.Epsilon0.Paths]
+standard_gnaw [definition, in hydras.Epsilon0.Paths]
+state [record, in hydras.Hydra.BigBattle]
+STDPP_compat [library]
+stdS_S [constructor, in hydras.Epsilon0.Paths]
+stdS_1 [constructor, in hydras.Epsilon0.Paths]
+std_S [constructor, in hydras.Epsilon0.Paths]
+std_1 [constructor, in hydras.Epsilon0.Paths]
+step [section, in hydras.Epsilon0.F_omega]
+step [inductive, in hydras.Hydra.Omega2_Small]
+steps [inductive, in hydras.Hydra.BigBattle]
+steps_rounds [lemma, in hydras.Hydra.BigBattle]
+steps_S [constructor, in hydras.Hydra.BigBattle]
+steps1 [constructor, in hydras.Hydra.BigBattle]
+step_rounds [lemma, in hydras.Hydra.BigBattle]
+step_to_battle [lemma, in hydras.Hydra.Omega2_Small]
+step.F [variable, in hydras.Epsilon0.F_omega]
+step.Hn [variable, in hydras.Epsilon0.F_omega]
+step.n [variable, in hydras.Epsilon0.F_omega]
+step1 [constructor, in hydras.Hydra.BigBattle]
+step2 [constructor, in hydras.Hydra.BigBattle]
+step3 [constructor, in hydras.Hydra.BigBattle]
+sto [instance, in hydras.OrdinalNotations.ON_O]
+sto [instance, in hydras.OrdinalNotations.ON_Finite]
+sto:12 [binder, in hydras.Prelude.MoreOrders]
+sto:17 [binder, in hydras.Prelude.MoreOrders]
+sto:24 [binder, in hydras.Prelude.MoreOrders]
+sto:30 [binder, in hydras.Prelude.MoreOrders]
+sto:34 [binder, in hydras.Prelude.MoreOrders]
+sto:37 [binder, in hydras.Prelude.MoreOrders]
+sto:40 [binder, in hydras.Prelude.MoreOrders]
+sto:48 [binder, in hydras.Prelude.MoreOrders]
+sto:5 [binder, in hydras.Prelude.MoreOrders]
+sto:8 [binder, in hydras.Prelude.MoreOrders]
+Strategies [library]
+Strategy [record, in additions.Strategies]
+strict_lub_T1limit [lemma, in hydras.Epsilon0.T1]
+strict_lub_unique [lemma, in hydras.Epsilon0.T1]
+strict_lub_maj [definition, in hydras.Epsilon0.T1]
+strict_lub_lub [definition, in hydras.Epsilon0.T1]
+strict_lub [definition, in hydras.Epsilon0.T1]
+strict_mono_iterate_S [lemma, in hydras.Prelude.Iterates]
+strict_mono [definition, in hydras.Prelude.Iterates]
+strict_mono [definition, in gaia_hydras.T1Bridge]
+strict_lub_ref [lemma, in gaia_hydras.T1Bridge]
+Strict_lex [instance, in hydras.Prelude.Simple_LexProd]
+string_op [instance, in additions.Monoid_def]
+strongly_critical [definition, in hydras.Schutte.Critical]
+Strong_Recursion [section, in hydras.Ackermann.cPair]
+St:110 [binder, in hydras.Hydra.Hydra_Lemmas]
+subAddExistsNice [lemma, in Goedel.PRrepresentable]
+subAddForallsNice [lemma, in Goedel.PRrepresentable]
+subAll [library]
+subAllCloseFrom [lemma, in hydras.Ackermann.subAll]
+subAllCloseFrom1 [lemma, in hydras.Ackermann.subAll]
+subAllFormula [definition, in hydras.Ackermann.subAll]
+subAllFormulaId [lemma, in hydras.Ackermann.subAll]
+subAllFormula_ext [lemma, in hydras.Ackermann.subAll]
+subAllnVars1 [lemma, in hydras.Ackermann.folLogic3]
+subAllnVars2 [lemma, in hydras.Ackermann.folLogic3]
+subAllSubAllFormula [lemma, in hydras.Ackermann.subAll]
+subAllSubAllTerm [lemma, in hydras.Ackermann.subAll]
+subAllSubAllTerms [lemma, in hydras.Ackermann.subAll]
+subAllSubFormula [lemma, in hydras.Ackermann.subAll]
+subAllTerm [definition, in hydras.Ackermann.subAll]
+subAllTermId [lemma, in hydras.Ackermann.subAll]
+subAllTerms [definition, in hydras.Ackermann.subAll]
+subAllTermsId [lemma, in hydras.Ackermann.subAll]
+subAllTerms_ext [lemma, in hydras.Ackermann.subAll]
+subAllTerm_ext [lemma, in hydras.Ackermann.subAll]
+SubAllVars [section, in hydras.Ackermann.subAll]
+SubAllVars.L [variable, in hydras.Ackermann.subAll]
+SubAllVars.subAllCloseFrom [section, in hydras.Ackermann.subAll]
+subFormulaAnd [lemma, in hydras.Ackermann.folProp]
+subFormulaDepth [lemma, in hydras.Ackermann.folProp]
+subFormulaEqual [lemma, in hydras.Ackermann.folProp]
+subFormulaExch [lemma, in hydras.Ackermann.subProp]
+subFormulaExist [lemma, in hydras.Ackermann.folProp]
+subFormulaExist2 [lemma, in hydras.Ackermann.folProp]
+subFormulaForall [lemma, in hydras.Ackermann.folProp]
+subFormulaForall2 [lemma, in hydras.Ackermann.folProp]
+subFormulaId [lemma, in hydras.Ackermann.folProp]
+subFormulaIff [lemma, in hydras.Ackermann.folProp]
+subFormulaIfThenElse [lemma, in hydras.Ackermann.folProp]
+subFormulaImp [lemma, in hydras.Ackermann.folProp]
+subFormulaMinimize [lemma, in Goedel.PRrepresentable]
+subFormulaNil [lemma, in hydras.Ackermann.subProp]
+subFormulaNot [lemma, in hydras.Ackermann.folProp]
+subFormulaNTE [lemma, in hydras.Ackermann.subProp]
+subFormulaNTEHelp [lemma, in hydras.Ackermann.subProp]
+subFormulaOr [lemma, in hydras.Ackermann.folProp]
+subFormulaRelation [lemma, in hydras.Ackermann.folProp]
+subFormulaTrans [lemma, in hydras.Ackermann.subProp]
+subInterpFormula [lemma, in hydras.Ackermann.model]
+subInterpFormula1 [lemma, in hydras.Ackermann.model]
+subInterpFormula2 [lemma, in hydras.Ackermann.model]
+subInterpRel [lemma, in hydras.Ackermann.model]
+subInterpTerm [lemma, in hydras.Ackermann.model]
+subNNHelp [lemma, in hydras.Ackermann.model]
+SubON [record, in hydras.OrdinalNotations.ON_Generic]
+SubON_least [lemma, in hydras.OrdinalNotations.ON_Generic]
+SubON_limit [lemma, in hydras.OrdinalNotations.ON_Generic]
+SubON_successor [lemma, in hydras.OrdinalNotations.ON_Generic]
+SubON_inj [lemma, in hydras.OrdinalNotations.ON_Generic]
+SubON_mono [lemma, in hydras.OrdinalNotations.ON_Generic]
+SubON_properties [section, in hydras.OrdinalNotations.ON_Generic]
+SubON_same_op [definition, in hydras.OrdinalNotations.ON_Generic]
+SubON_same_fun [definition, in hydras.OrdinalNotations.ON_Generic]
+SubON_same_cst [definition, in hydras.OrdinalNotations.ON_Generic]
+SubON_onto [projection, in hydras.OrdinalNotations.ON_Generic]
+SubON_incl [projection, in hydras.OrdinalNotations.ON_Generic]
+SubON_compare [projection, in hydras.OrdinalNotations.ON_Generic]
+subProp [library]
+subStar [definition, in Goedel.fixPoint]
+subStarIsPR [instance, in Goedel.fixPoint]
+substExHC [lemma, in hydras.Ackermann.folProp]
+substF [definition, in hydras.Ackermann.folProp]
+substF2 [abbreviation, in hydras.Ackermann.folProp]
+substF3 [abbreviation, in hydras.Ackermann.folProp]
+substF4 [abbreviation, in hydras.Ackermann.folProp]
+substF5 [abbreviation, in hydras.Ackermann.folProp]
+substF6 [abbreviation, in hydras.Ackermann.folProp]
+substF7 [abbreviation, in hydras.Ackermann.folProp]
+substF8 [abbreviation, in hydras.Ackermann.folProp]
+substF9 [abbreviation, in hydras.Ackermann.folProp]
+substituteFormula [abbreviation, in hydras.Ackermann.folProp]
+substituteFormulaForall [definition, in hydras.Ackermann.folProp]
+substituteFormulaForallNice [lemma, in hydras.Ackermann.folProp]
+substituteFormulaHelp [definition, in hydras.Ackermann.folProp]
+substituteFormulaImp [definition, in hydras.Ackermann.folProp]
+substituteFormulaImpNice [lemma, in hydras.Ackermann.folProp]
+substituteFormulaNot [definition, in hydras.Ackermann.folProp]
+substituteFormulaNotNice [lemma, in hydras.Ackermann.folProp]
+substituteTerm [abbreviation, in hydras.Ackermann.folProp]
+substituteTerms [abbreviation, in hydras.Ackermann.folProp]
+Substitution_Properties.SysPrf [variable, in hydras.Ackermann.subProp]
+Substitution_Properties.Terms [variable, in hydras.Ackermann.subProp]
+Substitution_Properties.Term [variable, in hydras.Ackermann.subProp]
+Substitution_Properties.System [variable, in hydras.Ackermann.subProp]
+Substitution_Properties.Formulas [variable, in hydras.Ackermann.subProp]
+Substitution_Properties.Formula [variable, in hydras.Ackermann.subProp]
+Substitution_Properties.L [variable, in hydras.Ackermann.subProp]
+Substitution_Properties [section, in hydras.Ackermann.subProp]
+substT [definition, in hydras.Ackermann.folProp]
+substTs [definition, in hydras.Ackermann.folProp]
+subSubAllFormula [lemma, in hydras.Ackermann.subAll]
+subSubAllTerm [lemma, in hydras.Ackermann.subAll]
+subSubAllTerms [lemma, in hydras.Ackermann.subAll]
+subSubFormula [lemma, in hydras.Ackermann.folLogic2]
+subSubTerm [lemma, in hydras.Ackermann.folLogic2]
+subSubTerms [lemma, in hydras.Ackermann.folLogic2]
+subterm [inductive, in hydras.Gamma0.Gamma0]
+subTermApply [lemma, in hydras.Ackermann.folProp]
+subTermExch [lemma, in hydras.Ackermann.subProp]
+subTermId [lemma, in hydras.Ackermann.folProp]
+subTermNil [lemma, in hydras.Ackermann.subProp]
+subTermsExch [lemma, in hydras.Ackermann.subProp]
+subTermsId [lemma, in hydras.Ackermann.folProp]
+subTermsNil [lemma, in hydras.Ackermann.subProp]
+subTermsTrans [lemma, in hydras.Ackermann.subProp]
+subTermTrans [lemma, in hydras.Ackermann.subProp]
+subTermVar1 [lemma, in hydras.Ackermann.folProp]
+subTermVar2 [lemma, in hydras.Ackermann.folProp]
+subterm_lt [lemma, in hydras.Gamma0.Gamma0]
+subterm_trans [constructor, in hydras.Gamma0.Gamma0]
+subterm_c [constructor, in hydras.Gamma0.Gamma0]
+subterm_b [constructor, in hydras.Gamma0.Gamma0]
+subterm_a [constructor, in hydras.Gamma0.Gamma0]
+subToSubAll [lemma, in hydras.Ackermann.subAll]
+subWithEquals [lemma, in hydras.Ackermann.folLogic3]
+subWithEqualsTerm [lemma, in hydras.Ackermann.folLogic3]
+subWithEqualsTerms [lemma, in hydras.Ackermann.folLogic3]
+succ [definition, in hydras.Epsilon0.T1]
+succ [definition, in hydras.OrdinalNotations.ON_Omega2]
+succ [definition, in hydras.Gamma0.T2]
+succ [section, in hydras.Epsilon0.Large_Sets]
+succ [definition, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+succ [definition, in hydras.Schutte.Schutte_basics]
+Succ [abbreviation, in hydras.Epsilon0.E0]
+succb [abbreviation, in hydras.Epsilon0.T1]
+succb [definition, in hydras.OrdinalNotations.ON_Omega2]
+succb [definition, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+succb_correct [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Succb_Succ [lemma, in hydras.Epsilon0.E0]
+succb_not_lim [lemma, in hydras.Gamma0.Gamma0]
+Successor [definition, in hydras.Prelude.MoreOrders]
+Successor_succ [lemma, in hydras.OrdinalNotations.ON_Omega2]
+Successor_not [lemma, in hydras.OrdinalNotations.ON_Omega2]
+Successor_inv [lemma, in hydras.OrdinalNotations.ON_Omega2]
+Successor_unicity [lemma, in hydras.solutions_exercises.predSuccUnicity]
+Successor_succ [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Successor_inv4 [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Successor_inv3 [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Successor_inv2 [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Successor_inv1 [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+succFormula [definition, in Goedel.PRrepresentable]
+succFunc [constructor, in hydras.Ackermann.primRec]
+succIsPR [instance, in hydras.Ackermann.primRec]
+succRepresentable [lemma, in Goedel.PRrepresentable]
+succ_rw1 [lemma, in hydras.Epsilon0.T1]
+succ_cons' [lemma, in hydras.Epsilon0.T1]
+succ_cons [lemma, in hydras.Epsilon0.T1]
+succ_lt_limit [lemma, in hydras.Epsilon0.T1]
+succ_not_limit [lemma, in hydras.Epsilon0.T1]
+succ_compat [lemma, in hydras.Epsilon0.T1]
+succ_compatS [lemma, in hydras.Epsilon0.T1]
+succ_injective [lemma, in hydras.Epsilon0.T1]
+succ_monomorphism [lemma, in hydras.Epsilon0.T1]
+succ_strict_monoR [lemma, in hydras.Epsilon0.T1]
+succ_mono [lemma, in hydras.Epsilon0.T1]
+succ_strict_mono_LT [lemma, in hydras.Epsilon0.T1]
+succ_strict_mono [lemma, in hydras.Epsilon0.T1]
+succ_of_plus_finite [lemma, in hydras.Epsilon0.T1]
+succ_is_plus_one [lemma, in hydras.Epsilon0.T1]
+succ_nf [lemma, in hydras.Epsilon0.T1]
+succ_lt_a_phi0_b [lemma, in hydras.Epsilon0.T1]
+succ_is_succ [lemma, in hydras.Epsilon0.T1]
+succ_not_zero [lemma, in hydras.Epsilon0.T1]
+succ_is_plus_1 [lemma, in hydras.OrdinalNotations.ON_Omega2]
+succ_ok [lemma, in hydras.OrdinalNotations.ON_Omega2]
+succ_finite [constructor, in hydras.Gamma0.T2]
+succ_correct [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+succ_ref [lemma, in gaia_hydras.T1Bridge]
+succ_zero_diff [lemma, in hydras.Schutte.Schutte_basics]
+succ_injection [lemma, in hydras.Schutte.Schutte_basics]
+succ_monoR [lemma, in hydras.Schutte.Schutte_basics]
+succ_mono [lemma, in hydras.Schutte.Schutte_basics]
+succ_ok [lemma, in hydras.Schutte.Schutte_basics]
+succ_spec [definition, in hydras.Schutte.Schutte_basics]
+succ_correct [lemma, in hydras.Epsilon0.E0]
+Succ_of_cons [lemma, in hydras.Epsilon0.E0]
+Succ_rw [lemma, in hydras.Epsilon0.E0]
+Succ_lt_T1limit [lemma, in hydras.Epsilon0.E0]
+Succ_not_T1limit [lemma, in hydras.Epsilon0.E0]
+Succ_not_Zero [lemma, in hydras.Epsilon0.E0]
+Succ_Succb [lemma, in hydras.Epsilon0.E0]
+Succ_inj [lemma, in hydras.Epsilon0.E0]
+succ_rounds [lemma, in hydras.Hydra.Omega2_Small]
+succ_step [constructor, in hydras.Hydra.Omega2_Small]
+succ_limit_dec [lemma, in hydras.Gamma0.Gamma0]
+succ_nf [lemma, in hydras.Gamma0.Gamma0]
+succ_as_plus [lemma, in hydras.Gamma0.Gamma0]
+succ_of_cons [lemma, in hydras.Gamma0.Gamma0]
+succ_lt_le [lemma, in hydras.Gamma0.Gamma0]
+Succ_ [constructor, in hydras.Ackermann.Languages]
+succ_is_plus_1 [lemma, in hydras.Schutte.Addition]
+succ.beta [variable, in hydras.Epsilon0.Large_Sets]
+succ.f [variable, in hydras.Epsilon0.Large_Sets]
+succ.f_ok [variable, in hydras.Epsilon0.Large_Sets]
+succ.f_Sle [variable, in hydras.Epsilon0.Large_Sets]
+succ.f_mono [variable, in hydras.Epsilon0.Large_Sets]
+succ.Hbeta [variable, in hydras.Epsilon0.Large_Sets]
+sumToN [definition, in hydras.Ackermann.cPair]
+sumToNIsPR [instance, in hydras.Ackermann.cPair]
+sumToN1 [lemma, in hydras.Ackermann.cPair]
+sumToN2 [lemma, in hydras.Ackermann.cPair]
+sup [definition, in hydras.Schutte.Schutte_basics]
+SuperbigStep [lemma, in hydras.Hydra.BigBattle]
+supM_gt0 [lemma, in hydras.Schutte.AP]
+sup_members_disj [lemma, in hydras.Schutte.Schutte_basics]
+sup_members_not_succ [lemma, in hydras.Schutte.Schutte_basics]
+sup_members_succ [lemma, in hydras.Schutte.Schutte_basics]
+sup_eq_intro [lemma, in hydras.Schutte.Schutte_basics]
+sup_mono [lemma, in hydras.Schutte.Schutte_basics]
+sup_of_leq [lemma, in hydras.Schutte.Schutte_basics]
+sup_least_upper_bound [lemma, in hydras.Schutte.Schutte_basics]
+sup_upper_bound [lemma, in hydras.Schutte.Schutte_basics]
+sup_ok1 [lemma, in hydras.Schutte.Schutte_basics]
+sup_spec_unicity [lemma, in hydras.Schutte.Schutte_basics]
+sup_unicity [lemma, in hydras.Schutte.Schutte_basics]
+sup_exists [lemma, in hydras.Schutte.Schutte_basics]
+sup_spec [definition, in hydras.Schutte.Schutte_basics]
+sup_M_in_B [lemma, in hydras.Schutte.Ordering_Functions]
+Su:208 [binder, in hydras.OrdinalNotations.ON_Generic]
+SWAP [constructor, in additions.AM]
+swapIsPR [instance, in hydras.Ackermann.primRec]
+switchIsPR [instance, in hydras.Ackermann.primRec]
+switchPR [definition, in hydras.Ackermann.primRec]
+switch5IsPR [lemma, in hydras.Ackermann.codeSubFormula]
+sysExtend [lemma, in hydras.Ackermann.LNN]
+sysExtend [lemma, in hydras.Ackermann.LNT]
+sysExtend [lemma, in hydras.Ackermann.folLogic]
+SysPrf [definition, in hydras.Ackermann.folProof]
+SysPrf [abbreviation, in hydras.Ackermann.subAll]
+SysPrf [definition, in hydras.Ackermann.LNN]
+SysPrf [definition, in hydras.Ackermann.LNT]
+SysPrf_rephrase [lemma, in hydras.Ackermann.Deduction]
+System [abbreviation, in hydras.Ackermann.subAll]
+System [definition, in hydras.Ackermann.LNN]
+System [definition, in hydras.Ackermann.LNT]
+System [abbreviation, in hydras.Ackermann.folProp]
+System [abbreviation, in hydras.Ackermann.codeSubFormula]
+System [definition, in hydras.Ackermann.fol]
+System [abbreviation, in hydras.Ackermann.folLogic3]
+sysWeaken [lemma, in hydras.Ackermann.LNN]
+sysWeaken [lemma, in hydras.Ackermann.LNT]
+sysWeaken [lemma, in hydras.Ackermann.folLogic]
+S_pred_rw [lemma, in hydras.Prelude.Iterates]
+S_ [abbreviation, in hydras.Ackermann.LNT]
+s_decr [lemma, in hydras.solutions_exercises.T1_ltNotWf]
+s':117 [binder, in additions.AM]
+s':13 [binder, in hydras.Hydra.Hydra_Lemmas]
+s':144 [binder, in hydras.rpo.rpo]
+s':148 [binder, in hydras.rpo.rpo]
+s':16 [binder, in hydras.Hydra.Hydra_Lemmas]
+s':19 [binder, in hydras.Hydra.Hydra_Termination]
+s':199 [binder, in hydras.Epsilon0.Paths]
+s':202 [binder, in hydras.Epsilon0.Paths]
+s':21 [binder, in hydras.Hydra.Hydra_Termination]
+s':21 [binder, in hydras.Hydra.Hydra_Lemmas]
+s':28 [binder, in hydras.Hydra.Hydra_Termination]
+s':289 [binder, in hydras.rpo.rpo]
+s':290 [binder, in hydras.rpo.rpo]
+s':291 [binder, in hydras.rpo.rpo]
+s':292 [binder, in hydras.rpo.rpo]
+s':299 [binder, in hydras.rpo.rpo]
+s':300 [binder, in hydras.rpo.rpo]
+s':31 [binder, in hydras.Hydra.Hydra_Termination]
+s':37 [binder, in hydras.Hydra.Hydra_Termination]
+s':40 [binder, in hydras.Hydra.Hydra_Termination]
+s':43 [binder, in hydras.Hydra.O2H]
+s':49 [binder, in hydras.rpo.rpo]
+s':51 [binder, in additions.AM]
+s':53 [binder, in hydras.Hydra.Hydra_Lemmas]
+s':53 [binder, in hydras.rpo.rpo]
+s':58 [binder, in hydras.Hydra.Hydra_Lemmas]
+s':62 [binder, in hydras.Hydra.Hydra_Definitions]
+s':63 [binder, in hydras.Hydra.Hydra_Lemmas]
+s':66 [binder, in hydras.Hydra.Hydra_Definitions]
+s':69 [binder, in additions.AM]
+s':75 [binder, in hydras.Hydra.Hydra_Definitions]
+s':8 [binder, in hydras.Hydra.Hydra_Lemmas]
+s':82 [binder, in hydras.Hydra.Hydra_Definitions]
+s':84 [binder, in hydras.Hydra.Hydra_Definitions]
+S.A [axiom, in hydras.rpo.decidable_set]
+S.cardinal [definition, in hydras.rpo.list_set]
+S.cardinal_subset [axiom, in hydras.rpo.list_set]
+S.DS [module, in hydras.rpo.list_set]
+S.elt [definition, in hydras.rpo.list_set]
+S.eq_A_dec [axiom, in hydras.rpo.decidable_set]
+S.eq_elt_dec [definition, in hydras.rpo.list_set]
+S.is_a_set [projection, in hydras.rpo.list_set]
+S.mk_set [constructor, in hydras.rpo.list_set]
+S.subset [definition, in hydras.rpo.list_set]
+S.support [projection, in hydras.rpo.list_set]
+S.t [record, in hydras.rpo.list_set]
+S.without_red [definition, in hydras.rpo.list_set]
+S0 [inductive, in hydras.Hydra.Hydra_Definitions]
+S0_mem_head [lemma, in hydras.Hydra.O2H]
+S0_decr [lemma, in hydras.Hydra.Hydra_Termination]
+S0_decr_0 [lemma, in hydras.Hydra.Hydra_Termination]
+S0_remove_r_i [lemma, in hydras.Hydra.Hydra_Lemmas]
+S0_remove_r [lemma, in hydras.Hydra.Hydra_Lemmas]
+S0_app [lemma, in hydras.Hydra.Hydra_Lemmas]
+S0_rest [constructor, in hydras.Hydra.Hydra_Definitions]
+S0_first [constructor, in hydras.Hydra.Hydra_Definitions]
+s0:1241 [binder, in hydras.Ackermann.codeSubFormula]
+s0:1244 [binder, in hydras.Ackermann.codeSubFormula]
+s0:1298 [binder, in hydras.Ackermann.codeSubFormula]
+s0:1301 [binder, in hydras.Ackermann.codeSubFormula]
+s0:1372 [binder, in hydras.Ackermann.codeSubFormula]
+s0:1375 [binder, in hydras.Ackermann.codeSubFormula]
+s0:21 [binder, in hydras.Ackermann.codeSubTerm]
+s0:25 [binder, in hydras.Ackermann.codeSubTerm]
+s0:43 [binder, in hydras.Ackermann.codeSubTerm]
+s0:47 [binder, in hydras.Ackermann.codeSubTerm]
+s0:863 [binder, in hydras.Ackermann.codeSubFormula]
+s0:866 [binder, in hydras.Ackermann.codeSubFormula]
+s0:887 [binder, in hydras.Ackermann.codeSubFormula]
+s0:890 [binder, in hydras.Ackermann.codeSubFormula]
+s0:903 [binder, in hydras.Ackermann.codeSubFormula]
+s0:906 [binder, in hydras.Ackermann.codeSubFormula]
+s0:930 [binder, in hydras.Ackermann.codeSubFormula]
+s0:933 [binder, in hydras.Ackermann.codeSubFormula]
+S1 [section, in hydras.solutions_exercises.is_F_monotonous]
+S1 [section, in hydras.solutions_exercises.F_3]
+S1 [section, in additions.Addition_Chains]
+S1 [inductive, in hydras.Hydra.Hydra_Definitions]
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+S1_decr_0 [lemma, in hydras.Hydra.Hydra_Termination]
+S1_app [lemma, in hydras.Hydra.Hydra_Lemmas]
+S1_next [constructor, in hydras.Hydra.Hydra_Definitions]
+S1_first [constructor, in hydras.Hydra.Hydra_Definitions]
+s1':201 [binder, in hydras.rpo.list_set]
+s1':207 [binder, in hydras.rpo.list_set]
+s1':217 [binder, in hydras.rpo.list_set]
+S1.H [variable, in hydras.solutions_exercises.is_F_monotonous]
+S1.instance_H [variable, in hydras.solutions_exercises.is_F_monotonous]
+S1.Limit [section, in hydras.solutions_exercises.F_3]
+S1.Limit.Hlambda [variable, in hydras.solutions_exercises.F_3]
+S1.Limit.IHlambda [variable, in hydras.solutions_exercises.F_3]
+S1.Limit.lambda [variable, in hydras.solutions_exercises.F_3]
+S1.Limit.S3 [section, in hydras.solutions_exercises.F_3]
+S1.Limit.S3.Hn [variable, in hydras.solutions_exercises.F_3]
+S1.Limit.S3.n [variable, in hydras.solutions_exercises.F_3]
+S1.P [variable, in hydras.solutions_exercises.F_3]
+S1.Successor [section, in hydras.solutions_exercises.F_3]
+S1.Successor.alpha [variable, in hydras.solutions_exercises.F_3]
+S1.Successor.H_alpha [variable, in hydras.solutions_exercises.F_3]
+S1.Successor.S2 [section, in hydras.solutions_exercises.F_3]
+S1.Successor.S2.Hn [variable, in hydras.solutions_exercises.F_3]
+S1.Successor.S2.n [variable, in hydras.solutions_exercises.F_3]
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+s2':209 [binder, in hydras.rpo.list_set]
+s2':219 [binder, in hydras.rpo.list_set]
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+s:95 [binder, in hydras.Epsilon0.Paths]
+s:95 [binder, in hydras.rpo.rpo]
+s:951 [binder, in hydras.Ackermann.codeSubFormula]
+s:954 [binder, in hydras.Ackermann.codeSubFormula]
+s:957 [binder, in hydras.Ackermann.codeSubFormula]
+s:96 [binder, in hydras.Ackermann.subProp]
+s:96 [binder, in hydras.Ackermann.folReplace]
+s:960 [binder, in hydras.Ackermann.codeSubFormula]
+s:963 [binder, in hydras.Ackermann.codeSubFormula]
+s:966 [binder, in hydras.Ackermann.codeSubFormula]
+s:969 [binder, in hydras.Ackermann.codeSubFormula]
+s:97 [binder, in hydras.rpo.rpo]
+s:972 [binder, in hydras.Ackermann.codeSubFormula]
+s:975 [binder, in hydras.Ackermann.codeSubFormula]
+s:978 [binder, in hydras.Ackermann.codeSubFormula]
+s:98 [binder, in Goedel.PRrepresentable]
+s:98 [binder, in hydras.Epsilon0.Paths]
+s:983 [binder, in hydras.Ackermann.codeSubFormula]
+s:986 [binder, in hydras.Ackermann.codeSubFormula]
+s:989 [binder, in hydras.Ackermann.codeSubFormula]
+s:99 [binder, in hydras.Ackermann.subAll]
+s:99 [binder, in hydras.rpo.rpo]
+s:992 [binder, in hydras.Ackermann.codeSubFormula]
+s:995 [binder, in hydras.Ackermann.codeSubFormula]
+s:998 [binder, in hydras.Ackermann.codeSubFormula]
+

T

+t [definition, in hydras.OrdinalNotations.ON_Omega2]
+t [definition, in hydras.OrdinalNotations.ON_O]
+t [definition, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+t [definition, in hydras.solutions_exercises.MultisetWf]
+t [definition, in hydras.OrdinalNotations.ON_mult]
+t [definition, in hydras.OrdinalNotations.ON_plus]
+t [definition, in hydras.OrdinalNotations.ON_Finite]
+tail_LT_cons [lemma, in hydras.Epsilon0.T1]
+tail_lt_cons [lemma, in hydras.Epsilon0.T1]
+tail_lt [lemma, in hydras.Epsilon0.T1]
+ta1:310 [binder, in hydras.Gamma0.Gamma0]
+ta1:314 [binder, in hydras.Gamma0.Gamma0]
+ta1:320 [binder, in hydras.Gamma0.Gamma0]
+ta1:328 [binder, in hydras.Gamma0.Gamma0]
+ta1:334 [binder, in hydras.Gamma0.Gamma0]
+ta1:337 [binder, in hydras.Gamma0.Gamma0]
+ta1:342 [binder, in hydras.Gamma0.Gamma0]
+ta1:349 [binder, in hydras.Gamma0.Gamma0]
+ta1:354 [binder, in hydras.Gamma0.Gamma0]
+ta1:358 [binder, in hydras.Gamma0.Gamma0]
+ta1:364 [binder, in hydras.Gamma0.Gamma0]
+ta1:372 [binder, in hydras.Gamma0.Gamma0]
+ta1:378 [binder, in hydras.Gamma0.Gamma0]
+ta1:381 [binder, in hydras.Gamma0.Gamma0]
+ta1:386 [binder, in hydras.Gamma0.Gamma0]
+ta1:393 [binder, in hydras.Gamma0.Gamma0]
+ta1:398 [binder, in hydras.Gamma0.Gamma0]
+ta1:404 [binder, in hydras.Gamma0.Gamma0]
+ta1:408 [binder, in hydras.Gamma0.Gamma0]
+ta2:311 [binder, in hydras.Gamma0.Gamma0]
+ta2:315 [binder, in hydras.Gamma0.Gamma0]
+ta2:321 [binder, in hydras.Gamma0.Gamma0]
+ta2:329 [binder, in hydras.Gamma0.Gamma0]
+ta2:355 [binder, in hydras.Gamma0.Gamma0]
+ta2:359 [binder, in hydras.Gamma0.Gamma0]
+ta2:365 [binder, in hydras.Gamma0.Gamma0]
+ta2:373 [binder, in hydras.Gamma0.Gamma0]
+ta2:379 [binder, in hydras.Gamma0.Gamma0]
+ta2:382 [binder, in hydras.Gamma0.Gamma0]
+ta2:387 [binder, in hydras.Gamma0.Gamma0]
+ta2:394 [binder, in hydras.Gamma0.Gamma0]
+TA:35 [binder, in hydras.Prelude.DecPreOrder_Instances]
+tb1:312 [binder, in hydras.Gamma0.Gamma0]
+tb1:316 [binder, in hydras.Gamma0.Gamma0]
+tb1:322 [binder, in hydras.Gamma0.Gamma0]
+tb1:330 [binder, in hydras.Gamma0.Gamma0]
+tb1:335 [binder, in hydras.Gamma0.Gamma0]
+tb1:338 [binder, in hydras.Gamma0.Gamma0]
+tb1:343 [binder, in hydras.Gamma0.Gamma0]
+tb1:350 [binder, in hydras.Gamma0.Gamma0]
+tb1:356 [binder, in hydras.Gamma0.Gamma0]
+tb1:360 [binder, in hydras.Gamma0.Gamma0]
+tb1:366 [binder, in hydras.Gamma0.Gamma0]
+tb1:374 [binder, in hydras.Gamma0.Gamma0]
+tb1:380 [binder, in hydras.Gamma0.Gamma0]
+tb1:383 [binder, in hydras.Gamma0.Gamma0]
+tb1:388 [binder, in hydras.Gamma0.Gamma0]
+tb1:395 [binder, in hydras.Gamma0.Gamma0]
+tb1:399 [binder, in hydras.Gamma0.Gamma0]
+tb1:405 [binder, in hydras.Gamma0.Gamma0]
+tb1:409 [binder, in hydras.Gamma0.Gamma0]
+tb2:313 [binder, in hydras.Gamma0.Gamma0]
+tb2:317 [binder, in hydras.Gamma0.Gamma0]
+tb2:323 [binder, in hydras.Gamma0.Gamma0]
+tb2:331 [binder, in hydras.Gamma0.Gamma0]
+tb2:336 [binder, in hydras.Gamma0.Gamma0]
+tb2:339 [binder, in hydras.Gamma0.Gamma0]
+tb2:344 [binder, in hydras.Gamma0.Gamma0]
+tb2:351 [binder, in hydras.Gamma0.Gamma0]
+tb2:357 [binder, in hydras.Gamma0.Gamma0]
+tb2:361 [binder, in hydras.Gamma0.Gamma0]
+tb2:367 [binder, in hydras.Gamma0.Gamma0]
+tb2:375 [binder, in hydras.Gamma0.Gamma0]
+TB:12 [binder, in hydras.Prelude.DecPreOrder_Instances]
+TB:36 [binder, in hydras.Prelude.DecPreOrder_Instances]
+Tcons [constructor, in hydras.Ackermann.fol]
+tc1:319 [binder, in hydras.Gamma0.Gamma0]
+tc1:326 [binder, in hydras.Gamma0.Gamma0]
+tc1:341 [binder, in hydras.Gamma0.Gamma0]
+tc1:347 [binder, in hydras.Gamma0.Gamma0]
+tc1:363 [binder, in hydras.Gamma0.Gamma0]
+tc1:370 [binder, in hydras.Gamma0.Gamma0]
+tc1:385 [binder, in hydras.Gamma0.Gamma0]
+tc1:391 [binder, in hydras.Gamma0.Gamma0]
+tc1:402 [binder, in hydras.Gamma0.Gamma0]
+tc1:411 [binder, in hydras.Gamma0.Gamma0]
+tc2:327 [binder, in hydras.Gamma0.Gamma0]
+tc2:333 [binder, in hydras.Gamma0.Gamma0]
+tc2:348 [binder, in hydras.Gamma0.Gamma0]
+tc2:353 [binder, in hydras.Gamma0.Gamma0]
+tc2:371 [binder, in hydras.Gamma0.Gamma0]
+tc2:377 [binder, in hydras.Gamma0.Gamma0]
+tc2:392 [binder, in hydras.Gamma0.Gamma0]
+tc2:397 [binder, in hydras.Gamma0.Gamma0]
+tc2:403 [binder, in hydras.Gamma0.Gamma0]
+tc2:407 [binder, in hydras.Gamma0.Gamma0]
+tc2:412 [binder, in hydras.Gamma0.Gamma0]
+ten [definition, in hydras.Epsilon0.T1]
+Term [module, in hydras.rpo.term]
+Term [abbreviation, in hydras.Ackermann.subAll]
+Term [definition, in hydras.Ackermann.LNN]
+Term [definition, in hydras.Ackermann.LNT]
+Term [abbreviation, in hydras.Ackermann.folProp]
+Term [abbreviation, in hydras.Ackermann.codeSubFormula]
+Term [inductive, in hydras.Ackermann.fol]
+Term [abbreviation, in hydras.Ackermann.folLogic3]
+term [library]
+terminates [definition, in hydras.Prelude.WfVariant]
+Termination [definition, in hydras.Hydra.Hydra_Definitions]
+Termination_strong [lemma, in hydras.Hydra.Hydra_Lemmas]
+Terms [abbreviation, in hydras.Ackermann.subAll]
+Terms [definition, in hydras.Ackermann.LNN]
+Terms [definition, in hydras.Ackermann.LNT]
+Terms [abbreviation, in hydras.Ackermann.folProp]
+Terms [abbreviation, in hydras.Ackermann.codeSubFormula]
+Terms [inductive, in hydras.Ackermann.fol]
+Terms [abbreviation, in hydras.Ackermann.folLogic3]
+terms_eqdec [lemma, in hydras.Ackermann.fol]
+Terms_Term_rec_full [definition, in hydras.Ackermann.fol]
+Terms_Term_rec [definition, in hydras.Ackermann.fol]
+Terms_Term_ind [definition, in hydras.Ackermann.fol]
+term_eqdec [lemma, in hydras.Ackermann.fol]
+Term_Terms_rec_full [definition, in hydras.Ackermann.fol]
+Term_Terms_rec [definition, in hydras.Ackermann.fol]
+Term_Terms_ind [definition, in hydras.Ackermann.fol]
+Term.apply_subst [definition, in hydras.rpo.term]
+Term.direct_subterm [definition, in hydras.rpo.term]
+Term.DoubleRecursion [section, in hydras.rpo.term]
+Term.DoubleRecursion.Pl2 [variable, in hydras.rpo.term]
+Term.DoubleRecursion.P2 [variable, in hydras.rpo.term]
+Term.empty_subst_is_id_list [axiom, in hydras.rpo.term]
+Term.empty_subst_is_id [axiom, in hydras.rpo.term]
+Term.eq_term_dec [axiom, in hydras.rpo.term]
+Term.F [module, in hydras.rpo.term]
+Term.is_a_pos_exists_subtem [axiom, in hydras.rpo.term]
+Term.is_a_pos [definition, in hydras.rpo.term]
+Term.map_subst [definition, in hydras.rpo.term]
+Term.Recursion [section, in hydras.rpo.term]
+Term.Recursion.P [variable, in hydras.rpo.term]
+Term.Recursion.Pl [variable, in hydras.rpo.term]
+Term.replace_at_pos_list_replace_at_pos_in_subterm [axiom, in hydras.rpo.term]
+Term.replace_at_pos_is_replace_at_pos2 [axiom, in hydras.rpo.term]
+Term.replace_at_pos_is_replace_at_pos1 [axiom, in hydras.rpo.term]
+Term.replace_at_pos_unfold [axiom, in hydras.rpo.term]
+Term.replace_at_pos_list [definition, in hydras.rpo.term]
+Term.replace_at_pos [definition, in hydras.rpo.term]
+Term.size [definition, in hydras.rpo.term]
+Term.size_subterm_at_pos [axiom, in hydras.rpo.term]
+Term.size_direct_subterm [axiom, in hydras.rpo.term]
+Term.size_ge_one [axiom, in hydras.rpo.term]
+Term.size_unfold [axiom, in hydras.rpo.term]
+Term.substitution [definition, in hydras.rpo.term]
+Term.subst_comp_is_subst_comp [axiom, in hydras.rpo.term]
+Term.subst_comp_is_subst_comp_aux1 [axiom, in hydras.rpo.term]
+Term.subst_comp [definition, in hydras.rpo.term]
+Term.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos [axiom, in hydras.rpo.term]
+Term.subterm_at_pos [definition, in hydras.rpo.term]
+Term.symbol [definition, in hydras.rpo.term]
+Term.Term [constructor, in hydras.rpo.term]
+Term.term [inductive, in hydras.rpo.term]
+Term.Term_eq_dec [module, in hydras.rpo.term]
+Term.term_rec8 [axiom, in hydras.rpo.term]
+Term.term_rec7 [axiom, in hydras.rpo.term]
+Term.term_rec4 [axiom, in hydras.rpo.term]
+Term.term_rec3 [axiom, in hydras.rpo.term]
+Term.term_rec2 [axiom, in hydras.rpo.term]
+Term.Var [constructor, in hydras.rpo.term]
+Term.variable [definition, in hydras.rpo.term]
+Term.well_formed_apply_subst [axiom, in hydras.rpo.term]
+Term.well_formed_subst [definition, in hydras.rpo.term]
+Term.well_formed_list [definition, in hydras.rpo.term]
+Term.well_formed_fold [axiom, in hydras.rpo.term]
+Term.well_formed_unfold [axiom, in hydras.rpo.term]
+Term.well_formed [definition, in hydras.rpo.term]
+Term.X [module, in hydras.rpo.term]
+test [definition, in hydras.Hydra.BigBattle]
+the [definition, in hydras.Schutte.MoreEpsilonIota]
+Theorem_4_5 [lemma, in hydras.Epsilon0.Large_Sets]
+Theorem_4_5 [lemma, in gaia_hydras.GLarge_Sets]
+the_least_ok [lemma, in hydras.Schutte.Schutte_basics]
+the_least_unicity [lemma, in hydras.Schutte.Well_Orders]
+the_least [definition, in hydras.Schutte.Well_Orders]
+the_context.About_WO [section, in hydras.Schutte.Well_Orders]
+the_context.Lt [variable, in hydras.Schutte.Well_Orders]
+the_context.M [variable, in hydras.Schutte.Well_Orders]
+the_context [section, in hydras.Schutte.Well_Orders]
+the_exponent [definition, in additions.Addition_Chains]
+the_exponent_nat [definition, in additions.Addition_Chains]
+the_ordering_segment [definition, in hydras.Schutte.Ordering_Functions]
+TH_packed [lemma, in hydras.Epsilon0.F_alpha]
+th_In [lemma, in hydras.Schutte.Ordering_Functions]
+TH_13_6R [lemma, in hydras.Schutte.Ordering_Functions]
+TH_13_6 [lemma, in hydras.Schutte.Ordering_Functions]
+Th_13_5_2 [lemma, in hydras.Schutte.Ordering_Functions]
+Th_13_5_1 [lemma, in hydras.Schutte.Ordering_Functions]
+th_14_6 [lemma, in hydras.Gamma0.Gamma0]
+th_14_5 [lemma, in hydras.Gamma0.Gamma0]
+Th13_8_1 [lemma, in hydras.Schutte.Critical]
+Th13_8 [lemma, in hydras.Schutte.Critical]
+Th13_5.verso.U_fixed.alpha_ [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.verso.U_fixed.U_inc_A [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.verso.U_fixed.U_den [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.verso.U_fixed.U_non_empty [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.verso.U_fixed.U [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.verso.U_fixed [section, in hydras.Schutte.Ordering_Functions]
+Th13_5.verso.B_closed [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.verso [section, in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.M_fixed.U [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.M_fixed.den [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.M_fixed.ne [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.M_fixed.inc [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.M_fixed.M [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.M_fixed [section, in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.f_cont [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.recto [section, in hydras.Schutte.Ordering_Functions]
+Th13_5.f_ord [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.f [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.B [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5.A [variable, in hydras.Schutte.Ordering_Functions]
+Th13_5 [section, in hydras.Schutte.Ordering_Functions]
+Times_ [constructor, in hydras.Ackermann.Languages]
+tj:174 [binder, in hydras.rpo.rpo]
+tj:177 [binder, in hydras.rpo.rpo]
+tj:180 [binder, in hydras.rpo.rpo]
+Tnil [constructor, in hydras.Ackermann.fol]
+Too_far [constructor, in hydras.Epsilon0.Large_Sets_Examples]
+Total [record, in hydras.Prelude.DecPreOrder]
+Total [inductive, in hydras.Prelude.DecPreOrder]
+totalness [projection, in hydras.Prelude.DecPreOrder]
+totalness [constructor, in hydras.Prelude.DecPreOrder]
+TotalPreOrder [record, in hydras.Prelude.DecPreOrder]
+Total_Inverse_fun [instance, in hydras.Prelude.DecPreOrder_Instances]
+total_pre_order_total [projection, in hydras.Prelude.DecPreOrder]
+total_pre_order_pre [projection, in hydras.Prelude.DecPreOrder]
+Total_Reflexive [instance, in hydras.Prelude.DecPreOrder]
+tower2 [definition, in hydras.solutions_exercises.MorePRExamples]
+tower2IsPR [instance, in hydras.solutions_exercises.MorePRExamples]
+tower2_alt_PR [instance, in hydras.solutions_exercises.MorePRExamples]
+tower2_alt_ok [lemma, in hydras.solutions_exercises.MorePRExamples]
+tower2_alt [definition, in hydras.solutions_exercises.MorePRExamples]
+Toy [module, in hydras.MoreAck.FolExamples]
+Toy.A [abbreviation, in hydras.MoreAck.FolExamples]
+Toy.a [abbreviation, in hydras.MoreAck.FolExamples]
+Toy.Adds [definition, in hydras.MoreAck.FolExamples]
+Toy.arityF [definition, in hydras.MoreAck.FolExamples]
+Toy.arityR [definition, in hydras.MoreAck.FolExamples]
+Toy.a_ [constructor, in hydras.MoreAck.FolExamples]
+Toy.A_ [constructor, in hydras.MoreAck.FolExamples]
+Toy.B [abbreviation, in hydras.MoreAck.FolExamples]
+Toy.b [abbreviation, in hydras.MoreAck.FolExamples]
+Toy.b_ [constructor, in hydras.MoreAck.FolExamples]
+Toy.B_ [constructor, in hydras.MoreAck.FolExamples]
+Toy.C [abbreviation, in hydras.MoreAck.FolExamples]
+Toy.C_ [constructor, in hydras.MoreAck.FolExamples]
+Toy.ded1 [lemma, in hydras.MoreAck.FolExamples]
+Toy.ded2 [lemma, in hydras.MoreAck.FolExamples]
+Toy.ded3 [lemma, in hydras.MoreAck.FolExamples]
+Toy.drinkers_thm [lemma, in hydras.MoreAck.FolExamples]
+Toy.Drinkers_theorem.f [variable, in hydras.MoreAck.FolExamples]
+Toy.Drinkers_theorem [section, in hydras.MoreAck.FolExamples]
+Toy.D0 [lemma, in hydras.MoreAck.FolExamples]
+Toy.D01 [lemma, in hydras.MoreAck.FolExamples]
+Toy.eq_refl [lemma, in hydras.MoreAck.FolExamples]
+Toy.f [abbreviation, in hydras.MoreAck.FolExamples]
+Toy.Fun [inductive, in hydras.MoreAck.FolExamples]
+Toy.f_ [constructor, in hydras.MoreAck.FolExamples]
+Toy.F1 [definition, in hydras.MoreAck.FolExamples]
+Toy.F2 [definition, in hydras.MoreAck.FolExamples]
+Toy.F3 [definition, in hydras.MoreAck.FolExamples]
+Toy.F4 [definition, in hydras.MoreAck.FolExamples]
+Toy.F5 [definition, in hydras.MoreAck.FolExamples]
+Toy.F6 [definition, in hydras.MoreAck.FolExamples]
+Toy.g [abbreviation, in hydras.MoreAck.FolExamples]
+Toy.g_ [constructor, in hydras.MoreAck.FolExamples]
+Toy.h [abbreviation, in hydras.MoreAck.FolExamples]
+Toy.h_ [constructor, in hydras.MoreAck.FolExamples]
+Toy.L [definition, in hydras.MoreAck.FolExamples]
+Toy.Ldec [definition, in hydras.MoreAck.FolExamples]
+Toy.MP' [lemma, in hydras.MoreAck.FolExamples]
+Toy.OnSubstF [section, in hydras.MoreAck.FolExamples]
+Toy.OnSubstF.F [variable, in hydras.MoreAck.FolExamples]
+Toy.P [abbreviation, in hydras.MoreAck.FolExamples]
+Toy.peirce [lemma, in hydras.MoreAck.FolExamples]
+Toy.Peirce [definition, in hydras.MoreAck.FolExamples]
+Toy.PeirceProof [section, in hydras.MoreAck.FolExamples]
+Toy.PrfContrex7 [definition, in hydras.MoreAck.FolExamples]
+Toy.PrfContrex9 [definition, in hydras.MoreAck.FolExamples]
+Toy.PrfEx1 [definition, in hydras.MoreAck.FolExamples]
+Toy.PrfEx10 [definition, in hydras.MoreAck.FolExamples]
+Toy.PrfEx11 [definition, in hydras.MoreAck.FolExamples]
+Toy.PrfEx2 [lemma, in hydras.MoreAck.FolExamples]
+Toy.PrfEx2' [definition, in hydras.MoreAck.FolExamples]
+Toy.PrfEx3 [definition, in hydras.MoreAck.FolExamples]
+Toy.PrfEx4 [definition, in hydras.MoreAck.FolExamples]
+Toy.PrfEx5 [definition, in hydras.MoreAck.FolExamples]
+Toy.PrfEx6 [definition, in hydras.MoreAck.FolExamples]
+Toy.PrfEx8 [definition, in hydras.MoreAck.FolExamples]
+Toy.PrfEx9 [definition, in hydras.MoreAck.FolExamples]
+Toy.PrimedSymbols [section, in hydras.MoreAck.FolExamples]
+Toy.ProofOfEx3 [section, in hydras.MoreAck.FolExamples]
+Toy.P_ [constructor, in hydras.MoreAck.FolExamples]
+Toy.Q [abbreviation, in hydras.MoreAck.FolExamples]
+Toy.Q_ [constructor, in hydras.MoreAck.FolExamples]
+Toy.R [abbreviation, in hydras.MoreAck.FolExamples]
+Toy.Rel [inductive, in hydras.MoreAck.FolExamples]
+Toy.R_ [constructor, in hydras.MoreAck.FolExamples]
+Toy.SysPrfEx2 [definition, in hydras.MoreAck.FolExamples]
+Toy.t0 [definition, in hydras.MoreAck.FolExamples]
+Toy.t1 [definition, in hydras.MoreAck.FolExamples]
+Toy.t2 [definition, in hydras.MoreAck.FolExamples]
+Toy.t3 [definition, in hydras.MoreAck.FolExamples]
+Toy.t4 [definition, in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm8 [definition, in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm7 [definition, in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm6 [definition, in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm5 [definition, in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm4 [definition, in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm3 [definition, in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm2 [definition, in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm1 [definition, in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm0 [definition, in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm [definition, in hydras.MoreAck.FolExamples]
+To_lex_prod [instance, in hydras.Prelude.DecPreOrder_Instances]
+to:14 [binder, in gaia_hydras.GPaths]
+to:36 [binder, in gaia_hydras.GPaths]
+to:42 [binder, in gaia_hydras.GPaths]
+to:45 [binder, in gaia_hydras.GPaths]
+Tprf2T'prf [lemma, in Goedel.rosserPA]
+Tprf2T'prf [definition, in Goedel.goedel2]
+trace [definition, in hydras.Hydra.Hydra_Definitions]
+trace_to_std [lemma, in hydras.Hydra.O2H]
+trace_to_std_0 [lemma, in hydras.Hydra.O2H]
+trace_to_round_plus [lemma, in hydras.Hydra.O2H]
+trace_mult [definition, in additions.Trace_exercise]
+trace_compare [definition, in additions.Trace_exercise]
+trace_toS [constructor, in hydras.Hydra.Hydra_Definitions]
+trace_to1 [constructor, in hydras.Hydra.Hydra_Definitions]
+trace_to [inductive, in hydras.Hydra.Hydra_Definitions]
+Trace_exercise [library]
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+trace:662 [binder, in hydras.Ackermann.codeSubFormula]
+trace:663 [binder, in hydras.Ackermann.codeSubFormula]
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+trace:666 [binder, in hydras.Ackermann.codeSubFormula]
+trace:667 [binder, in hydras.Ackermann.codeSubFormula]
+trace:668 [binder, in hydras.Ackermann.codeSubFormula]
+trace:669 [binder, in hydras.Ackermann.codeSubFormula]
+trace:670 [binder, in hydras.Ackermann.codeSubFormula]
+trace:671 [binder, in hydras.Ackermann.codeSubFormula]
+trace:672 [binder, in hydras.Ackermann.codeSubFormula]
+trace:673 [binder, in hydras.Ackermann.codeSubFormula]
+trace:674 [binder, in hydras.Ackermann.codeSubFormula]
+trace:675 [binder, in hydras.Ackermann.codeSubFormula]
+trace:676 [binder, in hydras.Ackermann.codeSubFormula]
+trace:677 [binder, in hydras.Ackermann.codeSubFormula]
+trace:678 [binder, in hydras.Ackermann.codeSubFormula]
+trace:679 [binder, in hydras.Ackermann.codeSubFormula]
+trace:680 [binder, in hydras.Ackermann.codeSubFormula]
+trace:681 [binder, in hydras.Ackermann.codeSubFormula]
+trace:682 [binder, in hydras.Ackermann.codeSubFormula]
+trace:683 [binder, in hydras.Ackermann.codeSubFormula]
+trace:684 [binder, in hydras.Ackermann.codeSubFormula]
+trace:685 [binder, in hydras.Ackermann.codeSubFormula]
+trace:686 [binder, in hydras.Ackermann.codeSubFormula]
+trace:687 [binder, in hydras.Ackermann.codeSubFormula]
+trace:688 [binder, in hydras.Ackermann.codeSubFormula]
+trace:689 [binder, in hydras.Ackermann.codeSubFormula]
+trace:690 [binder, in hydras.Ackermann.codeSubFormula]
+trace:691 [binder, in hydras.Ackermann.codeSubFormula]
+transfinite_recursor [definition, in hydras.Epsilon0.T1]
+transfinite_recursor_lt [definition, in hydras.Epsilon0.T1]
+transfinite_induction [lemma, in hydras.Schutte.Schutte_basics]
+transfinite_induction_Q [definition, in hydras.Gamma0.Gamma0]
+transfinite_induction [definition, in hydras.Gamma0.Gamma0]
+transition [definition, in gaia_hydras.GPaths]
+transition [definition, in hydras.Epsilon0.Paths]
+transitionb [definition, in gaia_hydras.GPaths]
+transitionP [lemma, in gaia_hydras.GPaths]
+transition_S [definition, in hydras.Epsilon0.Paths]
+translateLT [definition, in hydras.Ackermann.LNN2LNT]
+translateLT1 [lemma, in hydras.Ackermann.LNN2LNT]
+translatePrf [lemma, in hydras.Ackermann.LNN2LNT]
+translateProof [lemma, in hydras.Ackermann.LNN2LNT]
+Translate_Proof.AxiomsOK [variable, in hydras.Ackermann.LNN2LNT]
+Translate_Proof.V [variable, in hydras.Ackermann.LNN2LNT]
+Translate_Proof.U [variable, in hydras.Ackermann.LNN2LNT]
+Translate_Proof [section, in hydras.Ackermann.LNN2LNT]
+trans_aux [lemma, in hydras.Gamma0.Gamma0]
+trans_proof.induc [variable, in hydras.Gamma0.Gamma0]
+trans_proof.H23 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.H12 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.n3 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.n2 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.n1 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.c3 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.b3 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.a3 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.c2 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.b2 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.a2 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.c1 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.b1 [variable, in hydras.Gamma0.Gamma0]
+trans_proof.a1 [variable, in hydras.Gamma0.Gamma0]
+trans_proof [section, in hydras.Gamma0.Gamma0]
+trans_clos_is_trans [lemma, in hydras.rpo.closure]
+trans_clos [inductive, in hydras.rpo.closure]
+Trans_lex [instance, in hydras.Prelude.Simple_LexProd]
+TreeT1K [lemma, in gaia_hydras.T1Choice]
+Tree2T1 [definition, in gaia_hydras.T1Choice]
+trichotomy [lemma, in hydras.Schutte.Schutte_basics]
+tricho_aux [lemma, in hydras.Gamma0.Gamma0]
+tricho_lt_7 [lemma, in hydras.Gamma0.Gamma0]
+tricho_lt_5 [lemma, in hydras.Gamma0.Gamma0]
+tricho_lt_4' [lemma, in hydras.Gamma0.Gamma0]
+tricho_lt_4 [lemma, in hydras.Gamma0.Gamma0]
+tricho_lt_3 [lemma, in hydras.Gamma0.Gamma0]
+tricho_lt_2' [lemma, in hydras.Gamma0.Gamma0]
+tricho_lt_2 [lemma, in hydras.Gamma0.Gamma0]
+triple:184 [binder, in hydras.rpo.rpo]
+triple:186 [binder, in hydras.rpo.rpo]
+Tr:100 [binder, in hydras.Hydra.Hydra_Lemmas]
+ts:10 [binder, in hydras.Ackermann.folProp]
+ts:100 [binder, in hydras.Ackermann.subProp]
+ts:102 [binder, in hydras.Ackermann.subAll]
+ts:105 [binder, in hydras.Ackermann.subAll]
+ts:105 [binder, in hydras.Ackermann.LNN2LNT]
+ts:106 [binder, in hydras.Ackermann.subProp]
+ts:107 [binder, in hydras.Ackermann.LNN2LNT]
+ts:108 [binder, in hydras.Ackermann.subAll]
+ts:108 [binder, in hydras.Ackermann.subProp]
+ts:11 [binder, in hydras.Ackermann.codeFreeVar]
+ts:11 [binder, in hydras.Ackermann.PA]
+ts:110 [binder, in hydras.Ackermann.LNN2LNT]
+ts:112 [binder, in hydras.Ackermann.LNN2LNT]
+ts:112 [binder, in hydras.Ackermann.fol]
+ts:115 [binder, in hydras.Ackermann.subProp]
+ts:117 [binder, in hydras.Ackermann.subProp]
+ts:119 [binder, in hydras.Ackermann.subProp]
+ts:119 [binder, in hydras.Ackermann.fol]
+ts:12 [binder, in hydras.Ackermann.LNN2LNT]
+ts:12 [binder, in hydras.Ackermann.subProp]
+ts:123 [binder, in hydras.Ackermann.subProp]
+ts:124 [binder, in hydras.Ackermann.fol]
+ts:1253 [binder, in hydras.Ackermann.codeSubFormula]
+ts:1255 [binder, in hydras.Ackermann.codeSubFormula]
+ts:1261 [binder, in hydras.Ackermann.codeSubFormula]
+ts:1263 [binder, in hydras.Ackermann.codeSubFormula]
+ts:1265 [binder, in hydras.Ackermann.codeSubFormula]
+ts:128 [binder, in hydras.Ackermann.subProp]
+ts:129 [binder, in hydras.Ackermann.fol]
+ts:13 [binder, in hydras.Ackermann.codeFreeVar]
+ts:14 [binder, in hydras.Ackermann.LNN2LNT]
+ts:149 [binder, in hydras.Ackermann.folProp]
+ts:15 [binder, in hydras.Ackermann.wellFormed]
+ts:16 [binder, in hydras.Ackermann.folLogic2]
+ts:16 [binder, in hydras.Ackermann.LNN2LNT]
+ts:161 [binder, in hydras.Ackermann.model]
+ts:17 [binder, in hydras.Ackermann.subProp]
+ts:17 [binder, in hydras.Ackermann.wellFormed]
+ts:18 [binder, in hydras.Ackermann.subAll]
+ts:18 [binder, in hydras.Ackermann.folLogic2]
+ts:180 [binder, in hydras.Ackermann.subAll]
+ts:183 [binder, in hydras.Ackermann.subAll]
+ts:186 [binder, in hydras.Ackermann.subAll]
+ts:20 [binder, in hydras.Ackermann.folLogic2]
+ts:21 [binder, in hydras.Ackermann.LNN2LNT]
+ts:212 [binder, in hydras.Ackermann.subAll]
+ts:217 [binder, in hydras.Ackermann.subAll]
+ts:22 [binder, in hydras.Ackermann.model]
+ts:22 [binder, in hydras.Ackermann.code]
+ts:222 [binder, in hydras.Ackermann.subAll]
+ts:23 [binder, in hydras.Ackermann.subAll]
+ts:23 [binder, in hydras.Ackermann.codeFreeVar]
+ts:23 [binder, in hydras.Ackermann.LNN2LNT]
+ts:25 [binder, in hydras.Ackermann.LNN2LNT]
+ts:27 [binder, in hydras.Ackermann.codeSubTerm]
+ts:27 [binder, in hydras.Ackermann.folLogic3]
+ts:28 [binder, in hydras.Ackermann.subAll]
+ts:28 [binder, in hydras.Ackermann.wellFormed]
+ts:29 [binder, in hydras.Ackermann.codeSubTerm]
+ts:29 [binder, in hydras.Ackermann.folLogic3]
+ts:3 [binder, in hydras.Ackermann.LNN2LNT]
+ts:30 [binder, in hydras.Ackermann.LNN2LNT]
+ts:30 [binder, in hydras.Ackermann.subProp]
+ts:30 [binder, in hydras.Ackermann.code]
+ts:30 [binder, in hydras.Ackermann.wellFormed]
+ts:32 [binder, in hydras.Ackermann.wellFormed]
+ts:33 [binder, in hydras.Ackermann.model]
+ts:33 [binder, in hydras.Ackermann.code]
+ts:35 [binder, in hydras.Ackermann.subProp]
+ts:36 [binder, in hydras.Ackermann.code]
+ts:36 [binder, in hydras.Ackermann.folLogic3]
+ts:37 [binder, in hydras.Ackermann.codeSubTerm]
+ts:39 [binder, in hydras.Ackermann.LNN2LNT]
+ts:4 [binder, in hydras.Ackermann.subAll]
+ts:40 [binder, in hydras.Ackermann.subProp]
+ts:41 [binder, in hydras.Ackermann.LNN2LNT]
+ts:42 [binder, in hydras.Ackermann.LNN2LNT]
+ts:420 [binder, in hydras.Ackermann.folProp]
+ts:422 [binder, in hydras.Ackermann.folProp]
+ts:424 [binder, in hydras.Ackermann.folProp]
+ts:43 [binder, in hydras.Ackermann.folLogic3]
+ts:44 [binder, in hydras.Ackermann.wellFormed]
+ts:49 [binder, in hydras.Ackermann.wellFormed]
+ts:50 [binder, in hydras.Ackermann.folLogic3]
+ts:53 [binder, in hydras.Ackermann.LNN2LNT]
+ts:53 [binder, in hydras.Ackermann.model]
+ts:53 [binder, in hydras.Ackermann.subProp]
+ts:54 [binder, in hydras.Ackermann.wellFormed]
+ts:57 [binder, in hydras.Ackermann.model]
+ts:58 [binder, in hydras.Ackermann.subProp]
+ts:58 [binder, in hydras.Ackermann.folLogic3]
+ts:63 [binder, in hydras.Ackermann.subAll]
+ts:63 [binder, in hydras.Ackermann.model]
+ts:63 [binder, in hydras.Ackermann.subProp]
+ts:63 [binder, in hydras.Ackermann.wellFormed]
+ts:65 [binder, in hydras.Ackermann.folProp]
+ts:65 [binder, in hydras.Ackermann.fol]
+ts:66 [binder, in hydras.Ackermann.subAll]
+ts:67 [binder, in hydras.Ackermann.folLogic3]
+ts:69 [binder, in hydras.Ackermann.subAll]
+ts:7 [binder, in hydras.Ackermann.subProp]
+ts:72 [binder, in hydras.Ackermann.fol]
+ts:75 [binder, in hydras.Ackermann.model]
+ts:76 [binder, in hydras.Ackermann.subProp]
+ts:77 [binder, in hydras.Ackermann.fol]
+ts:77 [binder, in hydras.Ackermann.folLogic3]
+ts:78 [binder, in hydras.Ackermann.subAll]
+ts:78 [binder, in hydras.Ackermann.model]
+ts:80 [binder, in hydras.Ackermann.subAll]
+ts:81 [binder, in hydras.Ackermann.subProp]
+ts:82 [binder, in hydras.Ackermann.subAll]
+ts:82 [binder, in hydras.Ackermann.model]
+ts:82 [binder, in hydras.Ackermann.fol]
+ts:86 [binder, in hydras.Ackermann.subProp]
+ts:874 [binder, in hydras.Ackermann.codeSubFormula]
+ts:878 [binder, in hydras.Ackermann.codeSubFormula]
+ts:882 [binder, in hydras.Ackermann.codeSubFormula]
+ts:91 [binder, in hydras.Ackermann.folLogic3]
+ts:914 [binder, in hydras.Ackermann.codeSubFormula]
+ts:919 [binder, in hydras.Ackermann.codeSubFormula]
+ts:924 [binder, in hydras.Ackermann.codeSubFormula]
+ts:93 [binder, in hydras.Ackermann.fol]
+ts:93 [binder, in hydras.Ackermann.folLogic3]
+ts:96 [binder, in hydras.Ackermann.fol]
+ts:97 [binder, in hydras.Ackermann.folLogic3]
+ts:98 [binder, in hydras.Ackermann.subProp]
+Two [instance, in hydras.Epsilon0.E0]
+two [definition, in additions.BinaryStrat]
+Two_strat [instance, in additions.BinaryStrat]
+type:64 [binder, in hydras.Ackermann.codeSubFormula]
+type:792 [binder, in hydras.Ackermann.checkPrf]
+type:840 [binder, in hydras.Ackermann.checkPrf]
+type:843 [binder, in hydras.Ackermann.checkPrf]
+t_trans [constructor, in hydras.rpo.closure]
+t_step [constructor, in hydras.rpo.closure]
+t_0_nil [lemma, in hydras.Prelude.MoreVectors]
+T' [definition, in Goedel.rosserPA]
+T' [definition, in Goedel.goedel2]
+T'prf2Tprf [lemma, in Goedel.rosserPA]
+t'':115 [binder, in hydras.Gamma0.Gamma0]
+t':114 [binder, in hydras.Gamma0.Gamma0]
+t':16 [binder, in additions.Trace_exercise]
+t':165 [binder, in additions.Addition_Chains]
+t':178 [binder, in additions.Addition_Chains]
+t':322 [binder, in hydras.rpo.rpo]
+t':323 [binder, in hydras.rpo.rpo]
+t':70 [binder, in hydras.Gamma0.T2]
+t':86 [binder, in additions.Addition_Chains]
+t':89 [binder, in hydras.Gamma0.Gamma0]
+t':91 [binder, in hydras.Gamma0.Gamma0]
+t':93 [binder, in hydras.Gamma0.Gamma0]
+t0_empty [lemma, in hydras.OrdinalNotations.ON_Finite]
+t0:171 [binder, in hydras.Ackermann.folProp]
+t0:18 [binder, in hydras.Ackermann.codeFreeVar]
+t0:18 [binder, in hydras.Ackermann.codeSubTerm]
+t0:187 [binder, in hydras.Ackermann.folProp]
+t0:20 [binder, in hydras.Ackermann.codeFreeVar]
+t0:202 [binder, in hydras.Ackermann.fol]
+t0:203 [binder, in hydras.Ackermann.folProp]
+t0:219 [binder, in hydras.Ackermann.folProp]
+t0:22 [binder, in hydras.Ackermann.codeSubTerm]
+t0:225 [binder, in hydras.Ackermann.fol]
+t0:24 [binder, in hydras.Ackermann.codeFreeVar]
+t0:243 [binder, in hydras.Ackermann.fol]
+t0:250 [binder, in hydras.Ackermann.folProp]
+t0:26 [binder, in hydras.Ackermann.codeFreeVar]
+t0:266 [binder, in hydras.Ackermann.folProp]
+t0:293 [binder, in hydras.Ackermann.fol]
+t0:299 [binder, in hydras.Ackermann.folProp]
+t0:315 [binder, in hydras.Ackermann.folProp]
+t0:331 [binder, in hydras.Ackermann.folProp]
+t0:339 [binder, in hydras.Ackermann.fol]
+t0:347 [binder, in hydras.Ackermann.folProp]
+t0:363 [binder, in hydras.Ackermann.folProp]
+t0:379 [binder, in hydras.Ackermann.folProp]
+t0:383 [binder, in hydras.Ackermann.fol]
+t0:40 [binder, in hydras.Ackermann.codeSubTerm]
+t0:437 [binder, in hydras.Ackermann.fol]
+t0:44 [binder, in hydras.Ackermann.codeSubTerm]
+t0:84 [binder, in hydras.Ackermann.codeSubFormula]
+t0:861 [binder, in hydras.Ackermann.codeSubFormula]
+t0:864 [binder, in hydras.Ackermann.codeSubFormula]
+t0:885 [binder, in hydras.Ackermann.codeSubFormula]
+t0:888 [binder, in hydras.Ackermann.codeSubFormula]
+t0:9 [binder, in hydras.Ackermann.PA]
+t0:901 [binder, in hydras.Ackermann.codeSubFormula]
+t0:904 [binder, in hydras.Ackermann.codeSubFormula]
+t0:928 [binder, in hydras.Ackermann.codeSubFormula]
+t0:931 [binder, in hydras.Ackermann.codeSubFormula]
+T1 [inductive, in hydras.Epsilon0.T1]
+T1 [abbreviation, in gaia_hydras.T1Bridge]
+T1 [library]
+T1add [definition, in hydras.Epsilon0.T1]
+T1addA [lemma, in hydras.Epsilon0.T1]
+T1addAssoc [instance, in hydras.Epsilon0.T1]
+T1add_not_monotonous_l [lemma, in hydras.Epsilon0.T1]
+T1Bridge [library]
+T1Choice [definition, in gaia_hydras.T1Choice]
+T1Choice [library]
+T1compare [instance, in gaia_hydras.T1Bridge]
+T1compare_correct [lemma, in gaia_hydras.T1Bridge]
+T1eqE [lemma, in gaia_hydras.T1Bridge]
+T1eq_h2g [lemma, in gaia_hydras.T1Bridge]
+T1eq_refl [lemma, in gaia_hydras.T1Bridge]
+T1F_ [definition, in gaia_hydras.GF_alpha]
+T1is_succ_iff [lemma, in hydras.Epsilon0.T1]
+T1is_succ_def [lemma, in hydras.Epsilon0.T1]
+T1is_succ [definition, in hydras.Epsilon0.T1]
+T1is_succ_ref [lemma, in gaia_hydras.T1Bridge]
+T1leOrderMixin [definition, in gaia_hydras.T1Choice]
+T1le_asym [lemma, in gaia_hydras.T1Choice]
+T1le_iff [lemma, in gaia_hydras.T1Bridge]
+T1limit [definition, in hydras.Epsilon0.T1]
+T1limit_succ_tail [lemma, in hydras.Epsilon0.T1]
+T1limit_not_zero [lemma, in hydras.Epsilon0.T1]
+T1limit_succ [lemma, in hydras.Epsilon0.T1]
+T1limit_cases [lemma, in hydras.Epsilon0.T1]
+T1limit_ref [lemma, in gaia_hydras.T1Bridge]
+T1limit_canonS_not_zero [lemma, in hydras.Epsilon0.Canon]
+T1limit_plus [lemma, in hydras.Epsilon0.E0]
+T1limit_phi0 [lemma, in hydras.Epsilon0.E0]
+T1limit_Omega_term [lemma, in hydras.Epsilon0.E0]
+T1ltE [lemma, in gaia_hydras.T1Choice]
+T1lt_iff [lemma, in gaia_hydras.T1Bridge]
+T1max [definition, in gaia_hydras.T1Choice]
+T1maxE [lemma, in gaia_hydras.T1Choice]
+T1min [definition, in gaia_hydras.T1Choice]
+T1minE [lemma, in gaia_hydras.T1Choice]
+T1mixin [definition, in gaia_hydras.T1Choice]
+T1mul [definition, in hydras.Epsilon0.T1]
+T1mulE4 [lemma, in gaia_hydras.T1Bridge]
+T1mulE5 [lemma, in gaia_hydras.T1Bridge]
+T1mul_not_monotonous [lemma, in hydras.Epsilon0.T1]
+T1mul_cons_cons_E [lemma, in gaia_hydras.T1Bridge]
+T1mul_a0E [lemma, in gaia_hydras.T1Bridge]
+T1nat [definition, in hydras.Epsilon0.T1]
+T1nf_canon [lemma, in gaia_hydras.GCanon]
+T1omega [abbreviation, in hydras.Epsilon0.T1]
+T1orderType [definition, in gaia_hydras.T1Choice]
+T1pp [definition, in gaia_hydras.T1Bridge]
+T1toH [definition, in gaia_hydras.GHydra]
+T1_wf [lemma, in hydras.Epsilon0.T1]
+T1_eq_dec [definition, in hydras.Epsilon0.T1]
+t1_strorder [instance, in hydras.Epsilon0.T1]
+T1_le_Mixin [definition, in gaia_hydras.T1Choice]
+T1_to_T2 [definition, in hydras.Gamma0.T2]
+t1_0 [definition, in hydras.MoreAck.LNN_Examples]
+T1_to_T2_monoR [lemma, in hydras.Gamma0.Gamma0]
+T1_to_T2_mono [lemma, in hydras.Gamma0.Gamma0]
+T1_to_T2_R [definition, in hydras.Gamma0.Gamma0]
+T1_to_T2_lt [lemma, in hydras.Gamma0.Gamma0]
+T1_to_T2_inj [lemma, in hydras.Gamma0.Gamma0]
+t1_0 [definition, in hydras.MoreAck.FolExamples]
+T1_ltNotWf [library]
+T12Tree [definition, in gaia_hydras.T1Choice]
+T12Tree_inj [lemma, in gaia_hydras.T1Choice]
+t1:106 [binder, in hydras.rpo.rpo]
+t1:113 [binder, in hydras.rpo.rpo]
+t1:118 [binder, in hydras.Gamma0.Gamma0]
+t1:121 [binder, in hydras.Gamma0.Gamma0]
+t1:14 [binder, in hydras.rpo.term]
+t1:14 [binder, in hydras.Ackermann.codeFreeVar]
+t1:14 [binder, in hydras.rpo.list_set]
+t1:144 [binder, in hydras.Ackermann.folProp]
+t1:16 [binder, in hydras.Ackermann.codeFreeVar]
+t1:197 [binder, in hydras.rpo.term]
+t1:206 [binder, in hydras.Gamma0.Gamma0]
+t1:208 [binder, in hydras.rpo.term]
+t1:230 [binder, in hydras.rpo.term]
+t1:236 [binder, in hydras.Gamma0.Gamma0]
+t1:238 [binder, in hydras.rpo.term]
+t1:240 [binder, in hydras.rpo.term]
+t1:244 [binder, in hydras.rpo.term]
+t1:252 [binder, in hydras.rpo.term]
+t1:256 [binder, in hydras.rpo.term]
+t1:27 [binder, in hydras.rpo.term]
+t1:275 [binder, in hydras.Gamma0.Gamma0]
+t1:31 [binder, in hydras.Epsilon0.Epsilon0rpo]
+t1:36 [binder, in Goedel.PRrepresentable]
+t1:40 [binder, in Goedel.PRrepresentable]
+t1:445 [binder, in hydras.Epsilon0.Paths]
+t1:49 [binder, in Goedel.codeSysPrf]
+t1:53 [binder, in hydras.rpo.term]
+t1:53 [binder, in Goedel.codeSysPrf]
+t1:6 [binder, in hydras.rpo.list_permut]
+t1:61 [binder, in hydras.rpo.term]
+t1:63 [binder, in hydras.rpo.term]
+t1:65 [binder, in Goedel.codeSysPrf]
+t1:68 [binder, in hydras.rpo.term]
+t1:684 [binder, in hydras.Gamma0.Gamma0]
+t1:69 [binder, in Goedel.codeSysPrf]
+t1:76 [binder, in hydras.rpo.term]
+t1:81 [binder, in hydras.rpo.term]
+t1:94 [binder, in hydras.Gamma0.Gamma0]
+T2 [inductive, in hydras.Gamma0.T2]
+T2 [abbreviation, in gaia_hydras.T2Bridge]
+T2 [library]
+T2Bridge [library]
+t2_length_aux [definition, in hydras.Gamma0.T2]
+t2_length [definition, in hydras.Gamma0.T2]
+T2_size_psi [lemma, in hydras.Gamma0.Gamma0]
+T2_size4 [lemma, in hydras.Gamma0.Gamma0]
+T2_size3 [lemma, in hydras.Gamma0.Gamma0]
+T2_size2 [lemma, in hydras.Gamma0.Gamma0]
+T2_size1 [lemma, in hydras.Gamma0.Gamma0]
+T2_size [definition, in hydras.Gamma0.Gamma0]
+T2_2_term [definition, in hydras.Gamma0.Gamma0]
+t2:108 [binder, in hydras.rpo.rpo]
+t2:115 [binder, in hydras.rpo.rpo]
+t2:119 [binder, in hydras.Gamma0.Gamma0]
+t2:122 [binder, in hydras.Gamma0.Gamma0]
+t2:145 [binder, in hydras.Ackermann.folProp]
+t2:15 [binder, in hydras.rpo.term]
+t2:15 [binder, in hydras.rpo.list_set]
+t2:198 [binder, in hydras.rpo.term]
+t2:207 [binder, in hydras.Gamma0.Gamma0]
+t2:209 [binder, in hydras.rpo.term]
+t2:229 [binder, in hydras.rpo.term]
+t2:237 [binder, in hydras.Gamma0.Gamma0]
+t2:239 [binder, in hydras.rpo.term]
+t2:241 [binder, in hydras.rpo.term]
+t2:243 [binder, in hydras.rpo.term]
+t2:253 [binder, in hydras.rpo.term]
+t2:257 [binder, in hydras.rpo.term]
+t2:261 [binder, in hydras.rpo.term]
+t2:263 [binder, in hydras.rpo.term]
+t2:276 [binder, in hydras.Gamma0.Gamma0]
+t2:28 [binder, in hydras.rpo.term]
+t2:32 [binder, in hydras.Epsilon0.Epsilon0rpo]
+t2:37 [binder, in Goedel.PRrepresentable]
+t2:41 [binder, in Goedel.PRrepresentable]
+t2:446 [binder, in hydras.Epsilon0.Paths]
+t2:50 [binder, in Goedel.codeSysPrf]
+t2:52 [binder, in hydras.rpo.term]
+t2:54 [binder, in Goedel.codeSysPrf]
+t2:62 [binder, in hydras.rpo.term]
+t2:64 [binder, in hydras.rpo.term]
+t2:66 [binder, in Goedel.codeSysPrf]
+t2:67 [binder, in hydras.rpo.term]
+t2:685 [binder, in hydras.Gamma0.Gamma0]
+t2:7 [binder, in hydras.rpo.list_permut]
+t2:70 [binder, in Goedel.codeSysPrf]
+t2:77 [binder, in hydras.rpo.term]
+t2:82 [binder, in hydras.rpo.term]
+t2:95 [binder, in hydras.Gamma0.Gamma0]
+t3:120 [binder, in hydras.Gamma0.Gamma0]
+t3:123 [binder, in hydras.Gamma0.Gamma0]
+t:1 [binder, in hydras.Schutte.Correctness_E0]
+t:1 [binder, in hydras.Ackermann.LNN2LNT]
+t:1 [binder, in hydras.Ackermann.PAtheory]
+T:1 [binder, in gaia_hydras.GPrelude]
+T:1 [binder, in hydras.Ackermann.wConsistent]
+t:10 [binder, in hydras.MoreAck.BadSubst]
+t:10 [binder, in hydras.Ackermann.folLogic2]
+t:10 [binder, in hydras.Ackermann.LNN2LNT]
+T:10 [binder, in hydras.Ackermann.folLogic]
+t:100 [binder, in hydras.Ackermann.codeFreeVar]
+t:100 [binder, in hydras.Ackermann.codeSubTerm]
+T:100 [binder, in hydras.Ackermann.LNT]
+t:100 [binder, in hydras.Ackermann.wellFormed]
+t:100 [binder, in additions.Addition_Chains]
+t:101 [binder, in hydras.Ackermann.LNN2LNT]
+T:101 [binder, in hydras.Ackermann.LNN]
+t:101 [binder, in hydras.Ackermann.subProp]
+t:101 [binder, in hydras.Ackermann.wellFormed]
+t:102 [binder, in hydras.Ackermann.codeFreeVar]
+t:102 [binder, in hydras.Ackermann.codeSubTerm]
+t:102 [binder, in hydras.Ackermann.wellFormed]
+T:102 [binder, in hydras.Ackermann.folLogic]
+t:102 [binder, in hydras.rpo.rpo]
+t:103 [binder, in hydras.rpo.term]
+t:103 [binder, in hydras.Ackermann.codeFreeVar]
+T:103 [binder, in hydras.Ackermann.LNN]
+T:103 [binder, in hydras.Ackermann.LNT]
+t:104 [binder, in hydras.Ackermann.codeFreeVar]
+t:104 [binder, in hydras.Ackermann.codeSubTerm]
+t:104 [binder, in hydras.Ackermann.wellFormed]
+T:104 [binder, in hydras.Ackermann.folLogic3]
+T:104 [binder, in hydras.Ackermann.folReplace]
+t:105 [binder, in hydras.Ackermann.codeFreeVar]
+T:105 [binder, in hydras.Ackermann.folLogic]
+t:106 [binder, in hydras.rpo.term]
+t:106 [binder, in hydras.Ackermann.codeFreeVar]
+t:106 [binder, in hydras.Ackermann.codeSubTerm]
+T:106 [binder, in hydras.Ackermann.LNN]
+T:106 [binder, in hydras.Ackermann.LNT]
+t:106 [binder, in hydras.Ackermann.wellFormed]
+t:106 [binder, in additions.Addition_Chains]
+t:107 [binder, in hydras.Ackermann.codeFreeVar]
+T:107 [binder, in hydras.Ackermann.folLogic]
+t:108 [binder, in hydras.Ackermann.codeFreeVar]
+t:108 [binder, in hydras.Ackermann.codeSubTerm]
+t:108 [binder, in hydras.Ackermann.LNN2LNT]
+t:108 [binder, in hydras.Ackermann.wellFormed]
+t:108 [binder, in hydras.Hydra.Hydra_Definitions]
+t:109 [binder, in hydras.Ackermann.codeFreeVar]
+T:109 [binder, in hydras.Ackermann.LNN]
+T:109 [binder, in hydras.Ackermann.LNT]
+t:109 [binder, in hydras.Ackermann.subProp]
+T:11 [binder, in hydras.Ackermann.folReplace]
+t:110 [binder, in hydras.Ackermann.codeFreeVar]
+t:110 [binder, in hydras.Ackermann.codeSubTerm]
+t:110 [binder, in hydras.Ackermann.wellFormed]
+T:111 [binder, in hydras.Ackermann.folLogic]
+t:111 [binder, in hydras.Hydra.Hydra_Definitions]
+t:112 [binder, in hydras.Ackermann.codeFreeVar]
+t:112 [binder, in hydras.Ackermann.codeSubTerm]
+T:112 [binder, in hydras.Ackermann.LNN]
+T:112 [binder, in hydras.Ackermann.LNT]
+t:112 [binder, in hydras.Ackermann.wellFormed]
+T:112 [binder, in hydras.Ackermann.folReplace]
+T:113 [binder, in hydras.Ackermann.subAll]
+t:113 [binder, in hydras.Gamma0.Gamma0]
+t:114 [binder, in hydras.Gamma0.T2]
+t:114 [binder, in hydras.Ackermann.codeFreeVar]
+t:114 [binder, in hydras.Ackermann.codeSubTerm]
+t:114 [binder, in hydras.Ackermann.wellFormed]
+T:114 [binder, in hydras.Ackermann.folLogic]
+t:114 [binder, in additions.Addition_Chains]
+t:115 [binder, in hydras.rpo.term]
+T:115 [binder, in hydras.Ackermann.LNN]
+T:115 [binder, in hydras.Ackermann.LNT]
+t:115 [binder, in hydras.Ackermann.wellFormed]
+t:116 [binder, in hydras.Ackermann.codeFreeVar]
+t:116 [binder, in hydras.Ackermann.codeSubTerm]
+t:116 [binder, in hydras.Ackermann.wellFormed]
+t:116 [binder, in hydras.Epsilon0.Paths]
+t:117 [binder, in hydras.rpo.term]
+t:117 [binder, in hydras.Gamma0.T2]
+t:117 [binder, in hydras.Ackermann.codeSubTerm]
+T:117 [binder, in hydras.Ackermann.LNT]
+t:117 [binder, in hydras.Ackermann.wellFormed]
+T:117 [binder, in hydras.Ackermann.folLogic]
+t:117 [binder, in hydras.rpo.rpo]
+t:118 [binder, in hydras.Gamma0.T2]
+t:118 [binder, in hydras.Ackermann.codeFreeVar]
+t:118 [binder, in hydras.Ackermann.codeSubTerm]
+T:118 [binder, in hydras.Ackermann.LNN]
+t:118 [binder, in hydras.Ackermann.wellFormed]
+t:119 [binder, in hydras.Ackermann.codeSubTerm]
+t:119 [binder, in hydras.Ackermann.wellFormed]
+T:119 [binder, in hydras.Ackermann.folLogic]
+t:12 [binder, in hydras.Ackermann.codeSubTerm]
+T:12 [binder, in hydras.Ackermann.LNT]
+t:12 [binder, in hydras.Ackermann.wellFormed]
+t:12 [binder, in additions.Addition_Chains]
+t:12 [binder, in hydras.MoreAck.FolExamples]
+t:120 [binder, in hydras.Ackermann.codeFreeVar]
+t:120 [binder, in hydras.Ackermann.codeSubTerm]
+T:120 [binder, in hydras.Ackermann.LNN]
+t:120 [binder, in hydras.Ackermann.wellFormed]
+t:120 [binder, in additions.Addition_Chains]
+t:121 [binder, in hydras.Ackermann.codeSubTerm]
+T:121 [binder, in hydras.Ackermann.LNT]
+t:121 [binder, in hydras.Epsilon0.Paths]
+t:122 [binder, in hydras.Ackermann.codeFreeVar]
+t:122 [binder, in hydras.Ackermann.codeSubTerm]
+t:122 [binder, in hydras.Ackermann.wellFormed]
+T:122 [binder, in hydras.Ackermann.folLogic]
+t:123 [binder, in hydras.Ackermann.codeFreeVar]
+t:124 [binder, in hydras.rpo.term]
+t:124 [binder, in hydras.Ackermann.codeFreeVar]
+T:124 [binder, in hydras.Ackermann.LNN]
+T:124 [binder, in hydras.Ackermann.LNT]
+t:124 [binder, in hydras.Ackermann.wellFormed]
+t:125 [binder, in hydras.Ackermann.codeFreeVar]
+t:125 [binder, in hydras.Ackermann.codeSubTerm]
+t:1257 [binder, in hydras.Ackermann.codeSubFormula]
+t:126 [binder, in hydras.Ackermann.codeFreeVar]
+t:126 [binder, in hydras.Ackermann.wellFormed]
+T:126 [binder, in hydras.Ackermann.folLogic]
+t:127 [binder, in hydras.Ackermann.codeFreeVar]
+T:127 [binder, in hydras.Ackermann.LNN]
+T:127 [binder, in hydras.Ackermann.LNT]
+t:127 [binder, in additions.Addition_Chains]
+t:127 [binder, in hydras.rpo.rpo]
+t:128 [binder, in hydras.Ackermann.codeFreeVar]
+t:128 [binder, in hydras.Ackermann.codeSubTerm]
+t:128 [binder, in hydras.Ackermann.wellFormed]
+t:129 [binder, in hydras.rpo.term]
+t:129 [binder, in hydras.Ackermann.codeFreeVar]
+T:129 [binder, in hydras.Ackermann.LNT]
+t:129 [binder, in hydras.Epsilon0.Paths]
+t:13 [binder, in hydras.Ackermann.subAll]
+T:13 [binder, in hydras.Ackermann.folLogic]
+T:13 [binder, in hydras.Ackermann.folLogic3]
+T:130 [binder, in hydras.Ackermann.LNN]
+t:130 [binder, in hydras.Ackermann.wellFormed]
+t:131 [binder, in hydras.rpo.term]
+t:131 [binder, in hydras.Ackermann.codeSubTerm]
+T:132 [binder, in hydras.Ackermann.LNN]
+T:132 [binder, in hydras.Ackermann.LNT]
+t:132 [binder, in hydras.Ackermann.wellFormed]
+t:134 [binder, in hydras.Ackermann.codeSubTerm]
+t:134 [binder, in hydras.Ackermann.wellFormed]
+t:134 [binder, in additions.Addition_Chains]
+T:135 [binder, in hydras.Ackermann.LNN]
+t:135 [binder, in hydras.Epsilon0.Paths]
+t:136 [binder, in hydras.rpo.term]
+T:136 [binder, in hydras.Ackermann.LNT]
+t:136 [binder, in hydras.Ackermann.wellFormed]
+t:137 [binder, in hydras.Ackermann.codeSubTerm]
+T:138 [binder, in hydras.Ackermann.LNT]
+T:138 [binder, in hydras.Ackermann.subProp]
+t:138 [binder, in hydras.Ackermann.wellFormed]
+t:138 [binder, in hydras.rpo.rpo]
+T:139 [binder, in hydras.Ackermann.LNN]
+T:14 [binder, in hydras.Ackermann.LNT]
+T:14 [binder, in hydras.Ackermann.folProp]
+t:14 [binder, in hydras.Ackermann.wellFormed]
+T:14 [binder, in hydras.Ackermann.wConsistent]
+t:140 [binder, in hydras.Ackermann.codeSubTerm]
+t:140 [binder, in hydras.Ackermann.wellFormed]
+t:141 [binder, in hydras.rpo.term]
+T:141 [binder, in hydras.Ackermann.LNN]
+T:141 [binder, in hydras.Ackermann.LNT]
+t:141 [binder, in hydras.Epsilon0.Paths]
+t:142 [binder, in hydras.Ackermann.wellFormed]
+t:143 [binder, in hydras.Ackermann.codeSubTerm]
+T:144 [binder, in hydras.Ackermann.LNN]
+t:144 [binder, in hydras.Ackermann.wellFormed]
+t:145 [binder, in hydras.rpo.term]
+T:145 [binder, in hydras.Ackermann.LNT]
+t:146 [binder, in hydras.Ackermann.codeSubTerm]
+t:146 [binder, in hydras.Ackermann.wellFormed]
+t:147 [binder, in additions.Euclidean_Chains]
+t:147 [binder, in hydras.Epsilon0.Paths]
+t:148 [binder, in hydras.Ackermann.codeSubTerm]
+T:148 [binder, in hydras.Ackermann.LNN]
+t:148 [binder, in hydras.Ackermann.wellFormed]
+t:149 [binder, in hydras.rpo.term]
+t:15 [binder, in hydras.Ackermann.codeSubTerm]
+T:15 [binder, in hydras.Ackermann.LNN]
+t:15 [binder, in additions.Trace_exercise]
+t:15 [binder, in hydras.rpo.list_permut]
+t:150 [binder, in hydras.Ackermann.codeSubTerm]
+T:150 [binder, in hydras.Ackermann.LNT]
+t:150 [binder, in hydras.Ackermann.wellFormed]
+t:151 [binder, in hydras.Hydra.Hydra_Lemmas]
+t:152 [binder, in hydras.Ackermann.codeSubTerm]
+t:152 [binder, in hydras.Ackermann.wellFormed]
+t:152 [binder, in hydras.rpo.rpo]
+T:153 [binder, in hydras.Ackermann.LNN]
+t:153 [binder, in hydras.Ackermann.wellFormed]
+t:154 [binder, in hydras.Ackermann.codeSubTerm]
+t:154 [binder, in hydras.Ackermann.wellFormed]
+t:154 [binder, in hydras.Hydra.Hydra_Lemmas]
+t:154 [binder, in hydras.rpo.rpo]
+T:155 [binder, in hydras.Ackermann.LNT]
+t:155 [binder, in hydras.Ackermann.wellFormed]
+t:156 [binder, in hydras.Ackermann.codeSubTerm]
+t:156 [binder, in hydras.Ackermann.wellFormed]
+t:156 [binder, in hydras.rpo.rpo]
+T:157 [binder, in hydras.Ackermann.subProp]
+t:157 [binder, in hydras.Ackermann.wellFormed]
+t:158 [binder, in hydras.Ackermann.codeSubTerm]
+T:158 [binder, in hydras.Ackermann.LNN]
+t:158 [binder, in hydras.Ackermann.wellFormed]
+t:159 [binder, in hydras.Epsilon0.Large_Sets]
+t:159 [binder, in hydras.Ackermann.wellFormed]
+T:16 [binder, in hydras.Ackermann.folLogic]
+t:16 [binder, in hydras.rpo.list_permut]
+T:16 [binder, in hydras.Ackermann.folReplace]
+t:160 [binder, in hydras.Ackermann.codeSubTerm]
+t:160 [binder, in hydras.Ackermann.wellFormed]
+T:161 [binder, in hydras.Ackermann.LNN]
+T:161 [binder, in hydras.Ackermann.subProp]
+t:162 [binder, in hydras.Ackermann.codeSubTerm]
+t:162 [binder, in hydras.Ackermann.wellFormed]
+t:164 [binder, in hydras.Ackermann.codeSubTerm]
+t:164 [binder, in hydras.Ackermann.wellFormed]
+t:164 [binder, in additions.Addition_Chains]
+t:166 [binder, in hydras.Ackermann.codeSubTerm]
+T:166 [binder, in hydras.Ackermann.subProp]
+t:166 [binder, in hydras.Ackermann.wellFormed]
+t:168 [binder, in hydras.Ackermann.codeSubTerm]
+t:168 [binder, in hydras.Ackermann.wellFormed]
+t:17 [binder, in hydras.rpo.term]
+t:17 [binder, in hydras.Ackermann.LNN2LNT]
+T:17 [binder, in hydras.Ackermann.LNN]
+T:17 [binder, in hydras.Ackermann.LNT]
+t:170 [binder, in hydras.Ackermann.codeSubTerm]
+t:170 [binder, in hydras.Ackermann.folProp]
+t:170 [binder, in hydras.Ackermann.wellFormed]
+t:171 [binder, in additions.Addition_Chains]
+t:172 [binder, in hydras.Ackermann.codeSubTerm]
+t:172 [binder, in hydras.Ackermann.wellFormed]
+t:173 [binder, in hydras.Ackermann.wellFormed]
+t:173 [binder, in hydras.rpo.rpo]
+t:174 [binder, in hydras.rpo.term]
+t:174 [binder, in hydras.Ackermann.codeSubTerm]
+t:174 [binder, in hydras.Ackermann.wellFormed]
+t:175 [binder, in hydras.Ackermann.wellFormed]
+t:176 [binder, in hydras.Ackermann.codeSubTerm]
+t:176 [binder, in hydras.Ackermann.wellFormed]
+t:177 [binder, in hydras.Ackermann.subAll]
+t:177 [binder, in hydras.Ackermann.wellFormed]
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+t:791 [binder, in hydras.Ackermann.codeSubFormula]
+t:793 [binder, in hydras.Ackermann.codeSubFormula]
+t:795 [binder, in hydras.Ackermann.codeSubFormula]
+t:797 [binder, in hydras.Ackermann.codeSubFormula]
+t:799 [binder, in hydras.Ackermann.codeSubFormula]
+t:8 [binder, in hydras.Ackermann.codeFreeVar]
+t:8 [binder, in hydras.MoreAck.AckNotPR]
+t:8 [binder, in hydras.Ackermann.PA]
+T:8 [binder, in hydras.Ackermann.Deduction]
+t:80 [binder, in hydras.Ackermann.codeFreeVar]
+t:80 [binder, in hydras.Ackermann.codeSubTerm]
+T:80 [binder, in hydras.Ackermann.LNT]
+t:80 [binder, in hydras.Ackermann.wellFormed]
+t:80 [binder, in hydras.rpo.rpo]
+t:800 [binder, in hydras.Ackermann.codeSubFormula]
+t:801 [binder, in hydras.Ackermann.codeSubFormula]
+t:802 [binder, in hydras.Ackermann.codeSubFormula]
+t:803 [binder, in hydras.Ackermann.codeSubFormula]
+t:804 [binder, in hydras.Ackermann.codeSubFormula]
+t:805 [binder, in hydras.Ackermann.codeSubFormula]
+t:806 [binder, in hydras.Ackermann.codeSubFormula]
+t:807 [binder, in hydras.Ackermann.codeSubFormula]
+t:809 [binder, in hydras.Ackermann.codeSubFormula]
+T:81 [binder, in hydras.Ackermann.folLogic]
+T:81 [binder, in hydras.Ackermann.folReplace]
+t:811 [binder, in hydras.Ackermann.codeSubFormula]
+t:811 [binder, in gaia_hydras.nfwfgaia]
+t:813 [binder, in hydras.Ackermann.codeSubFormula]
+t:815 [binder, in hydras.Ackermann.codeSubFormula]
+t:817 [binder, in hydras.Ackermann.codeSubFormula]
+t:82 [binder, in hydras.Ackermann.codeFreeVar]
+t:82 [binder, in hydras.Ackermann.LNN]
+t:82 [binder, in hydras.Ackermann.wellFormed]
+t:83 [binder, in hydras.rpo.more_list]
+T:83 [binder, in hydras.Ackermann.LNN]
+T:83 [binder, in hydras.Ackermann.LNT]
+t:83 [binder, in hydras.Ackermann.codeSubFormula]
+t:830 [binder, in hydras.Ackermann.codeSubFormula]
+t:833 [binder, in hydras.Ackermann.codeSubFormula]
+t:84 [binder, in hydras.rpo.term]
+t:84 [binder, in hydras.Ackermann.codeFreeVar]
+t:84 [binder, in hydras.Ackermann.codeSubTerm]
+t:84 [binder, in hydras.Ackermann.wellFormed]
+T:84 [binder, in hydras.Ackermann.folLogic]
+t:84 [binder, in hydras.rpo.rpo]
+t:85 [binder, in additions.Addition_Chains]
+t:858 [binder, in hydras.Ackermann.codeSubFormula]
+t:86 [binder, in hydras.Ackermann.codeFreeVar]
+T:86 [binder, in hydras.Ackermann.LNN]
+t:86 [binder, in hydras.Ackermann.LNT]
+t:86 [binder, in hydras.Ackermann.wellFormed]
+T:86 [binder, in hydras.Ackermann.folReplace]
+T:87 [binder, in hydras.Ackermann.LNT]
+t:87 [binder, in hydras.Ackermann.folLogic3]
+t:88 [binder, in hydras.Ackermann.codeFreeVar]
+t:88 [binder, in hydras.Ackermann.codeSubTerm]
+t:88 [binder, in hydras.Ackermann.wellFormed]
+T:88 [binder, in hydras.Ackermann.folLogic]
+t:88 [binder, in hydras.Gamma0.Gamma0]
+t:89 [binder, in hydras.Ackermann.LNN]
+t:89 [binder, in hydras.Ackermann.codeSubFormula]
+T:89 [binder, in hydras.Ackermann.folLogic3]
+t:89 [binder, in additions.Addition_Chains]
+t:897 [binder, in hydras.Ackermann.codeSubFormula]
+t:9 [binder, in hydras.Ackermann.codeFreeVar]
+t:9 [binder, in hydras.Ackermann.codeSubTerm]
+T:9 [binder, in hydras.Ackermann.wConsistent]
+t:9 [binder, in hydras.rpo.rpo]
+t:90 [binder, in hydras.Ackermann.codeFreeVar]
+T:90 [binder, in hydras.Ackermann.LNN]
+t:90 [binder, in hydras.Ackermann.wellFormed]
+T:90 [binder, in hydras.Ackermann.folLogic]
+t:90 [binder, in hydras.Gamma0.Gamma0]
+t:91 [binder, in hydras.rpo.term]
+t:91 [binder, in hydras.Hydra.O2H]
+T:91 [binder, in hydras.Ackermann.LNT]
+t:91 [binder, in hydras.rpo.rpo]
+t:92 [binder, in hydras.Ackermann.codeFreeVar]
+t:92 [binder, in hydras.Ackermann.codeSubTerm]
+t:92 [binder, in hydras.rpo.more_list]
+t:92 [binder, in hydras.Ackermann.wellFormed]
+t:92 [binder, in hydras.Gamma0.Gamma0]
+t:93 [binder, in hydras.Ackermann.wellFormed]
+T:93 [binder, in hydras.Ackermann.folLogic]
+T:93 [binder, in hydras.Ackermann.folReplace]
+t:94 [binder, in hydras.Ackermann.codeFreeVar]
+t:94 [binder, in hydras.Ackermann.codeSubTerm]
+T:94 [binder, in hydras.Ackermann.LNN]
+T:94 [binder, in hydras.Ackermann.LNT]
+t:94 [binder, in hydras.Ackermann.subProp]
+t:94 [binder, in hydras.Ackermann.wellFormed]
+t:94 [binder, in additions.Addition_Chains]
+T:94 [binder, in hydras.Ackermann.folReplace]
+t:94 [binder, in hydras.MoreAck.PrimRecExamples]
+t:94 [binder, in hydras.rpo.rpo]
+t:95 [binder, in hydras.rpo.term]
+t:95 [binder, in hydras.Ackermann.wellFormed]
+t:96 [binder, in hydras.Ackermann.subAll]
+t:96 [binder, in hydras.Ackermann.codeFreeVar]
+t:96 [binder, in hydras.Ackermann.codeSubTerm]
+t:96 [binder, in hydras.Ackermann.wellFormed]
+T:96 [binder, in hydras.Ackermann.folLogic]
+T:97 [binder, in hydras.Ackermann.LNN]
+t:97 [binder, in hydras.Ackermann.wellFormed]
+t:98 [binder, in hydras.Ackermann.codeFreeVar]
+t:98 [binder, in hydras.Ackermann.codeSubTerm]
+T:98 [binder, in hydras.Ackermann.LNT]
+t:98 [binder, in hydras.Ackermann.wellFormed]
+t:98 [binder, in hydras.rpo.rpo]
+t:99 [binder, in hydras.Ackermann.wellFormed]
+T:99 [binder, in hydras.Ackermann.folLogic]
+T:99 [binder, in hydras.Ackermann.folLogic3]
+T:99 [binder, in hydras.Ackermann.folReplace]
+t:99 [binder, in hydras.MoreAck.PrimRecExamples]
+

U

+U [projection, in hydras.Ackermann.model]
+ui:187 [binder, in hydras.rpo.term]
+ui:325 [binder, in hydras.rpo.rpo]
+ui:328 [binder, in hydras.rpo.rpo]
+ui:369 [binder, in hydras.rpo.term]
+unbounded [lemma, in hydras.Schutte.Schutte_basics]
+Unbounded [definition, in hydras.Schutte.Schutte_basics]
+Unbounded_Cr [lemma, in hydras.Schutte.Critical]
+Unbounded_not_countable [lemma, in hydras.Schutte.Schutte_basics]
+Unbounded_ge [lemma, in hydras.Schutte.Addition]
+undecidable [abbreviation, in hydras.Ackermann.folProof]
+unique_decomposition [lemma, in hydras.Epsilon0.T1]
+unique_decomposition [lemma, in hydras.OrdinalNotations.ON_Omega2]
+Unnamed_thm9 [definition, in gaia_hydras.T1Choice]
+Unnamed_thm8 [definition, in gaia_hydras.T1Choice]
+Unnamed_thm7 [definition, in gaia_hydras.T1Choice]
+Unnamed_thm6 [definition, in gaia_hydras.T1Choice]
+Unnamed_thm5 [definition, in gaia_hydras.T1Choice]
+Unnamed_thm5 [definition, in gaia_hydras.T1Choice]
+Unnamed_thm4 [definition, in gaia_hydras.T1Choice]
+Unnamed_thm3 [definition, in gaia_hydras.T1Choice]
+Unnamed_thm2 [definition, in gaia_hydras.T1Choice]
+Unnamed_thm1 [definition, in gaia_hydras.T1Choice]
+Unnamed_thm0 [definition, in gaia_hydras.T1Choice]
+Unnamed_thm [definition, in gaia_hydras.T1Choice]
+Unnamed_thm [definition, in hydras.Prelude.More_Arith]
+Unnamed_thm [definition, in hydras.MoreAck.expressibleExamples]
+Unnamed_thm0 [definition, in hydras.MoreAck.LNN_Examples]
+Unnamed_thm [definition, in hydras.MoreAck.LNN_Examples]
+Unnamed_thm [definition, in additions.Monoid_def]
+Unnamed_thm0 [definition, in hydras.Ackermann.fol]
+Unnamed_thm [definition, in hydras.Ackermann.fol]
+Unnamed_thm [definition, in hydras.Epsilon0.Canon]
+Unnamed_thm [definition, in hydras.OrdinalNotations.ON_Finite]
+Unnamed_thm1 [definition, in hydras.Prelude.LibHyps_Experiments]
+Unnamed_thm0 [definition, in hydras.Prelude.LibHyps_Experiments]
+Unnamed_thm [definition, in hydras.Prelude.LibHyps_Experiments]
+Unnamed_thm1 [definition, in hydras.solutions_exercises.F_3]
+Unnamed_thm0 [definition, in hydras.solutions_exercises.F_3]
+Unnamed_thm [definition, in hydras.solutions_exercises.F_3]
+Unnamed_thm [definition, in additions.Addition_Chains]
+Unnamed_thm [definition, in hydras.Epsilon0.Hprime]
+Unnamed_thm [definition, in hydras.MoreAck.FolExamples]
+Unnamed_thm [definition, in hydras.MoreAck.FolExamples]
+Unnamed_thm2 [definition, in hydras.MoreAck.PrimRecExamples]
+Unnamed_thm1 [definition, in hydras.MoreAck.PrimRecExamples]
+Unnamed_thm0 [definition, in hydras.MoreAck.PrimRecExamples]
+Unnamed_thm [definition, in hydras.MoreAck.PrimRecExamples]
+unshift [definition, in hydras.Prelude.MoreLists]
+unshift_pred [lemma, in hydras.Prelude.MoreLists]
+unshift_shift [lemma, in hydras.Prelude.MoreLists]
+unshift_interval_pred [lemma, in hydras.Prelude.MoreLists]
+unshift_interval [lemma, in hydras.Prelude.MoreLists]
+unshift_iota_from [lemma, in hydras.Prelude.MoreLists]
+unshift_not_nil [lemma, in hydras.Prelude.MoreLists]
+unshift_app [lemma, in hydras.Prelude.MoreLists]
+unshift_but_last [lemma, in hydras.Prelude.MoreLists]
+updateValue [definition, in hydras.Ackermann.model]
+upper_bound [definition, in hydras.Schutte.Lub]
+Usual [module, in Goedel.PRrepresentable]
+Usual.beta [definition, in Goedel.PRrepresentable]
+Usual.β [abbreviation, in Goedel.PRrepresentable]
+u':693 [binder, in gaia_hydras.nfwfgaia]
+u1:107 [binder, in hydras.rpo.rpo]
+u1:114 [binder, in hydras.rpo.rpo]
+u1:121 [binder, in hydras.rpo.rpo]
+u1:89 [binder, in hydras.rpo.list_permut]
+u1:97 [binder, in hydras.rpo.list_permut]
+u2:109 [binder, in hydras.rpo.rpo]
+u2:116 [binder, in hydras.rpo.rpo]
+u2:122 [binder, in hydras.rpo.rpo]
+u2:90 [binder, in hydras.rpo.list_permut]
+u2:98 [binder, in hydras.rpo.list_permut]
+u3:91 [binder, in hydras.rpo.list_permut]
+u3:99 [binder, in hydras.rpo.list_permut]
+u4:100 [binder, in hydras.rpo.list_permut]
+u4:92 [binder, in hydras.rpo.list_permut]
+u:1202 [binder, in gaia_hydras.nfwfgaia]
+u:1213 [binder, in gaia_hydras.nfwfgaia]
+u:123 [binder, in hydras.rpo.rpo]
+u:1238 [binder, in gaia_hydras.nfwfgaia]
+u:1242 [binder, in gaia_hydras.nfwfgaia]
+u:1245 [binder, in gaia_hydras.nfwfgaia]
+u:1265 [binder, in gaia_hydras.nfwfgaia]
+u:1281 [binder, in gaia_hydras.nfwfgaia]
+u:130 [binder, in hydras.Epsilon0.Paths]
+U:136 [binder, in hydras.Ackermann.LNN2LNT]
+u:136 [binder, in hydras.Epsilon0.Paths]
+u:137 [binder, in hydras.Epsilon0.Paths]
+u:143 [binder, in hydras.Schutte.Schutte_basics]
+u:143 [binder, in hydras.Epsilon0.Paths]
+u:147 [binder, in hydras.rpo.term]
+U:15 [binder, in hydras.Ackermann.LNT]
+u:150 [binder, in hydras.rpo.term]
+u:150 [binder, in hydras.Epsilon0.Paths]
+u:1512 [binder, in gaia_hydras.nfwfgaia]
+u:1515 [binder, in gaia_hydras.nfwfgaia]
+u:1518 [binder, in gaia_hydras.nfwfgaia]
+u:155 [binder, in hydras.Epsilon0.Paths]
+u:161 [binder, in hydras.Schutte.Ordering_Functions]
+u:162 [binder, in hydras.rpo.term]
+u:170 [binder, in hydras.rpo.term]
+u:175 [binder, in hydras.rpo.term]
+u:1788 [binder, in gaia_hydras.nfwfgaia]
+U:18 [binder, in hydras.Ackermann.LNN]
+u:180 [binder, in hydras.rpo.term]
+u:183 [binder, in hydras.rpo.rpo]
+u:189 [binder, in hydras.rpo.term]
+u:196 [binder, in hydras.rpo.rpo]
+u:2041 [binder, in gaia_hydras.nfwfgaia]
+u:2052 [binder, in gaia_hydras.nfwfgaia]
+u:2080 [binder, in gaia_hydras.nfwfgaia]
+u:2085 [binder, in gaia_hydras.nfwfgaia]
+U:24 [binder, in hydras.Prelude.DecPreOrder_Instances]
+U:25 [binder, in hydras.Prelude.DecPreOrder_Instances]
+u:273 [binder, in hydras.rpo.term]
+u:276 [binder, in hydras.rpo.term]
+u:326 [binder, in hydras.rpo.term]
+u:328 [binder, in hydras.rpo.term]
+U:34 [binder, in hydras.Schutte.Countable]
+u:340 [binder, in hydras.rpo.term]
+u:348 [binder, in hydras.rpo.term]
+u:354 [binder, in hydras.rpo.term]
+u:358 [binder, in hydras.rpo.term]
+u:371 [binder, in hydras.rpo.term]
+u:390 [binder, in gaia_hydras.nfwfgaia]
+u:413 [binder, in gaia_hydras.nfwfgaia]
+U:48 [binder, in hydras.Schutte.Schutte_basics]
+U:5 [binder, in hydras.Ackermann.folLogic]
+U:62 [binder, in hydras.Schutte.AP]
+U:67 [binder, in hydras.Schutte.PartialFun]
+u:679 [binder, in gaia_hydras.nfwfgaia]
+u:686 [binder, in gaia_hydras.nfwfgaia]
+u:688 [binder, in gaia_hydras.nfwfgaia]
+U:69 [binder, in hydras.Schutte.PartialFun]
+u:691 [binder, in gaia_hydras.nfwfgaia]
+u:701 [binder, in gaia_hydras.nfwfgaia]
+u:711 [binder, in gaia_hydras.nfwfgaia]
+u:719 [binder, in gaia_hydras.nfwfgaia]
+u:745 [binder, in gaia_hydras.nfwfgaia]
+u:77 [binder, in hydras.rpo.rpo]
+u:799 [binder, in gaia_hydras.nfwfgaia]
+U:80 [binder, in hydras.Schutte.PartialFun]
+u:81 [binder, in hydras.rpo.rpo]
+u:810 [binder, in gaia_hydras.nfwfgaia]
+u:814 [binder, in gaia_hydras.nfwfgaia]
+U:86 [binder, in hydras.Schutte.PartialFun]
+u:93 [binder, in hydras.rpo.term]
+u:97 [binder, in hydras.rpo.term]
+u:996 [binder, in gaia_hydras.nfwfgaia]
+

V

+value:103 [binder, in hydras.Ackermann.model]
+value:107 [binder, in hydras.Ackermann.model]
+value:111 [binder, in hydras.Ackermann.model]
+value:115 [binder, in hydras.Ackermann.model]
+value:127 [binder, in hydras.Ackermann.model]
+value:131 [binder, in hydras.Ackermann.model]
+value:135 [binder, in hydras.Ackermann.model]
+value:139 [binder, in hydras.Ackermann.model]
+value:142 [binder, in hydras.Ackermann.model]
+value:146 [binder, in hydras.Ackermann.model]
+value:159 [binder, in hydras.Ackermann.model]
+value:165 [binder, in hydras.Ackermann.model]
+value:17 [binder, in hydras.Ackermann.model]
+value:21 [binder, in hydras.Ackermann.model]
+value:222 [binder, in hydras.Ackermann.model]
+value:32 [binder, in hydras.Ackermann.model]
+value:39 [binder, in hydras.Ackermann.model]
+value:43 [binder, in hydras.Ackermann.model]
+value:70 [binder, in hydras.Ackermann.model]
+value:80 [binder, in hydras.Ackermann.model]
+value:86 [binder, in hydras.Ackermann.model]
+var [instance, in hydras.Hydra.Hydra_Termination]
+var [constructor, in hydras.Ackermann.fol]
+varFormula [definition, in hydras.Ackermann.codeSubFormula]
+Variables [module, in hydras.rpo.term]
+Variables.eq_variable_dec [axiom, in hydras.rpo.term]
+Variables.var [axiom, in hydras.rpo.term]
+Variants [section, in hydras.Prelude.WfVariant]
+Variants.E [variable, in hydras.Prelude.WfVariant]
+Variants.lt [variable, in hydras.Prelude.WfVariant]
+Variants.m [variable, in hydras.Prelude.WfVariant]
+Variants.T [variable, in hydras.Prelude.WfVariant]
+Variants.tr [variable, in hydras.Prelude.WfVariant]
+_ < _ [notation, in hydras.Prelude.WfVariant]
+Variant_termination [lemma, in hydras.Prelude.WfVariant]
+Variant_lt_standard [lemma, in hydras.Hydra.Hydra_Theorems]
+Variant_LT_standard [lemma, in hydras.Hydra.Hydra_Theorems]
+Variant_lt_free [lemma, in hydras.Hydra.Hydra_Theorems]
+Variant_LT_free_0 [lemma, in hydras.Hydra.Hydra_Theorems]
+variant_mono_free [lemma, in hydras.Hydra.Hydra_Lemmas]
+variant_decr [projection, in hydras.Hydra.Hydra_Definitions]
+Vars [module, in hydras.Gamma0.Gamma0]
+Vars.empty_set [inductive, in hydras.Gamma0.Gamma0]
+Vars.eq_variable_dec [lemma, in hydras.Gamma0.Gamma0]
+Vars.var [definition, in hydras.Gamma0.Gamma0]
+Var:11 [binder, in hydras.Prelude.WfVariant]
+Var:159 [binder, in hydras.Hydra.Hydra_Definitions]
+Var:27 [binder, in gaia_hydras.GHydra]
+Var:3 [binder, in hydras.Hydra.Epsilon0_Needed_Std]
+Var:31 [binder, in gaia_hydras.GHydra]
+Var:34 [binder, in gaia_hydras.GHydra]
+Var:4 [binder, in hydras.Hydra.Epsilon0_Needed_Free]
+Var:5 [binder, in hydras.Hydra.Epsilon0_Needed_Generic]
+vector_nth [definition, in hydras.Prelude.MoreVectors]
+vector_triple_rect [definition, in hydras.Prelude.MoreVectors]
+vector_double_rect [definition, in hydras.Prelude.MoreVectors]
+vect_from_fun [definition, in hydras.Prelude.MoreVectors]
+Vec2_proj [definition, in hydras.Prelude.MoreVectors]
+vfourth [abbreviation, in hydras.Prelude.MoreVectors]
+vfst [abbreviation, in hydras.Prelude.MoreVectors]
+Vid [definition, in hydras.Prelude.MoreVectors]
+Vid_eq [lemma, in hydras.Prelude.MoreVectors]
+vsnd [abbreviation, in hydras.Prelude.MoreVectors]
+vs:103 [binder, in Goedel.PRrepresentable]
+vs:108 [binder, in Goedel.PRrepresentable]
+vs:114 [binder, in Goedel.PRrepresentable]
+vthird [abbreviation, in hydras.Prelude.MoreVectors]
+v':694 [binder, in gaia_hydras.nfwfgaia]
+v0:107 [binder, in hydras.Ackermann.codeSubFormula]
+v0:1286 [binder, in hydras.Ackermann.codeSubFormula]
+v0:1290 [binder, in hydras.Ackermann.codeSubFormula]
+v0:20 [binder, in hydras.Ackermann.codeSubTerm]
+v0:24 [binder, in hydras.Ackermann.codeSubTerm]
+v0:42 [binder, in hydras.Ackermann.codeSubTerm]
+v0:46 [binder, in hydras.Ackermann.codeSubTerm]
+v1:102 [binder, in hydras.Ackermann.subProp]
+v1:11 [binder, in hydras.Epsilon0.Epsilon0rpo]
+v1:11 [binder, in hydras.Ackermann.folLogic2]
+v1:110 [binder, in hydras.Ackermann.subProp]
+v1:121 [binder, in hydras.Ackermann.subAll]
+v1:122 [binder, in hydras.Ackermann.subAll]
+v1:124 [binder, in hydras.Ackermann.subProp]
+v1:129 [binder, in hydras.Ackermann.subAll]
+v1:129 [binder, in hydras.Ackermann.subProp]
+v1:130 [binder, in hydras.Ackermann.subAll]
+v1:131 [binder, in hydras.Ackermann.subAll]
+v1:132 [binder, in hydras.Ackermann.subAll]
+v1:133 [binder, in hydras.Ackermann.subAll]
+v1:134 [binder, in hydras.Ackermann.subAll]
+v1:135 [binder, in hydras.Ackermann.subAll]
+v1:136 [binder, in hydras.Ackermann.subAll]
+v1:137 [binder, in hydras.Ackermann.subAll]
+v1:138 [binder, in hydras.Ackermann.subAll]
+v1:139 [binder, in hydras.Ackermann.subAll]
+v1:140 [binder, in hydras.Ackermann.subAll]
+v1:141 [binder, in hydras.Ackermann.subAll]
+v1:141 [binder, in hydras.Ackermann.subProp]
+v1:142 [binder, in hydras.Ackermann.subAll]
+v1:143 [binder, in hydras.Ackermann.subAll]
+v1:144 [binder, in hydras.Ackermann.subAll]
+v1:144 [binder, in hydras.Ackermann.subProp]
+v1:145 [binder, in hydras.Ackermann.subAll]
+v1:146 [binder, in hydras.Ackermann.subAll]
+v1:147 [binder, in hydras.Ackermann.subAll]
+v1:148 [binder, in hydras.Ackermann.subAll]
+v1:148 [binder, in hydras.Ackermann.subProp]
+v1:149 [binder, in hydras.Ackermann.subAll]
+v1:150 [binder, in hydras.Ackermann.subAll]
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+v:427 [binder, in hydras.Ackermann.folProp]
+v:43 [binder, in hydras.Ackermann.folProp]
+v:43 [binder, in Goedel.rosserPA]
+v:43 [binder, in Goedel.codeSysPrf]
+v:430 [binder, in hydras.Ackermann.folProp]
+v:435 [binder, in hydras.Ackermann.folProp]
+v:439 [binder, in hydras.Ackermann.folProp]
+v:44 [binder, in Goedel.codeSysPrf]
+v:44 [binder, in hydras.Prelude.MoreVectors]
+v:440 [binder, in Goedel.PRrepresentable]
+v:441 [binder, in Goedel.PRrepresentable]
+v:442 [binder, in Goedel.PRrepresentable]
+v:443 [binder, in Goedel.PRrepresentable]
+v:443 [binder, in hydras.Ackermann.folProp]
+v:443 [binder, in hydras.Ackermann.fol]
+v:45 [binder, in hydras.MoreAck.AckNotPR]
+v:45 [binder, in hydras.Ackermann.subProp]
+v:45 [binder, in Goedel.rosserPA]
+v:45 [binder, in hydras.Ackermann.folLogic3]
+v:455 [binder, in hydras.Ackermann.checkPrf]
+v:46 [binder, in hydras.Ackermann.folLogic3]
+v:462 [binder, in Goedel.PRrepresentable]
+v:464 [binder, in Goedel.PRrepresentable]
+v:470 [binder, in Goedel.PRrepresentable]
+v:472 [binder, in Goedel.PRrepresentable]
+v:48 [binder, in Goedel.rosserPA]
+v:49 [binder, in hydras.Ackermann.subAll]
+v:49 [binder, in hydras.MoreAck.AckNotPR]
+v:49 [binder, in hydras.Ackermann.subProp]
+v:5 [binder, in Goedel.fixPoint]
+v:5 [binder, in Goedel.goedel1]
+v:5 [binder, in Goedel.rosser]
+v:5 [binder, in Goedel.goedel2]
+v:5 [binder, in hydras.Prelude.MoreVectors]
+v:52 [binder, in hydras.Ackermann.folLogic3]
+v:52 [binder, in Goedel.codeSysPrf]
+v:52 [binder, in hydras.Prelude.MoreVectors]
+v:53 [binder, in hydras.Ackermann.PA]
+v:53 [binder, in hydras.Ackermann.folLogic3]
+v:54 [binder, in hydras.Ackermann.subAll]
+v:54 [binder, in hydras.Ackermann.codeSubTerm]
+v:54 [binder, in hydras.Ackermann.subProp]
+v:55 [binder, in hydras.Ackermann.subAll]
+v:56 [binder, in hydras.Ackermann.subAll]
+v:56 [binder, in Goedel.codeSysPrf]
+v:57 [binder, in Goedel.PRrepresentable]
+v:57 [binder, in hydras.Ackermann.subAll]
+v:58 [binder, in hydras.Ackermann.codeSubTerm]
+v:58 [binder, in Goedel.rosserPA]
+v:59 [binder, in hydras.Ackermann.folProp]
+v:59 [binder, in hydras.Ackermann.subProp]
+v:59 [binder, in hydras.Ackermann.codeSubFormula]
+v:59 [binder, in hydras.Prelude.MoreVectors]
+v:6 [binder, in Goedel.rosserPA]
+v:60 [binder, in hydras.Ackermann.subAll]
+v:60 [binder, in hydras.Ackermann.LNN2LNT]
+v:61 [binder, in Goedel.PRrepresentable]
+v:61 [binder, in hydras.Ackermann.folProp]
+v:61 [binder, in Goedel.codeSysPrf]
+v:62 [binder, in hydras.Ackermann.codeSubTerm]
+v:62 [binder, in Goedel.codeSysPrf]
+v:623 [binder, in hydras.Ackermann.primRec]
+v:627 [binder, in hydras.Ackermann.primRec]
+v:63 [binder, in hydras.Ackermann.LNN2LNT]
+v:63 [binder, in Goedel.codeSysPrf]
+v:63 [binder, in hydras.Ackermann.expressible]
+v:64 [binder, in hydras.Ackermann.subProp]
+v:64 [binder, in Goedel.codeSysPrf]
+v:644 [binder, in hydras.Ackermann.primRec]
+v:649 [binder, in hydras.Ackermann.primRec]
+v:65 [binder, in hydras.Ackermann.LNN2LNT]
+v:65 [binder, in hydras.Ackermann.folLogic]
+v:66 [binder, in hydras.Ackermann.codeSubTerm]
+v:66 [binder, in hydras.Ackermann.folProp]
+v:66 [binder, in hydras.Ackermann.folReplace]
+v:67 [binder, in hydras.Ackermann.LNN2LNT]
+v:67 [binder, in Goedel.rosser]
+v:68 [binder, in hydras.Ackermann.subProp]
+v:68 [binder, in hydras.Ackermann.folLogic]
+v:68 [binder, in Goedel.codeSysPrf]
+v:68 [binder, in hydras.Prelude.MoreVectors]
+v:680 [binder, in gaia_hydras.nfwfgaia]
+v:687 [binder, in gaia_hydras.nfwfgaia]
+v:689 [binder, in gaia_hydras.nfwfgaia]
+v:69 [binder, in hydras.MoreAck.AckNotPR]
+v:692 [binder, in gaia_hydras.nfwfgaia]
+v:7 [binder, in hydras.Ackermann.codeSubTerm]
+v:7 [binder, in hydras.Ackermann.PA]
+v:70 [binder, in hydras.Ackermann.codeSubTerm]
+v:70 [binder, in hydras.Ackermann.LNN2LNT]
+V:70 [binder, in hydras.Schutte.PartialFun]
+v:70 [binder, in hydras.Ackermann.folReplace]
+v:70 [binder, in hydras.Prelude.MoreVectors]
+v:703 [binder, in gaia_hydras.nfwfgaia]
+v:71 [binder, in hydras.Ackermann.subAll]
+v:71 [binder, in hydras.MoreAck.AckNotPR]
+v:71 [binder, in hydras.Ackermann.codeSubFormula]
+v:713 [binder, in gaia_hydras.nfwfgaia]
+v:72 [binder, in hydras.Ackermann.model]
+v:72 [binder, in hydras.Ackermann.subProp]
+v:72 [binder, in hydras.Ackermann.folLogic]
+v:72 [binder, in Goedel.codeSysPrf]
+v:73 [binder, in Goedel.codeSysPrf]
+v:74 [binder, in Goedel.PRrepresentable]
+v:74 [binder, in hydras.Ackermann.codeSubTerm]
+v:74 [binder, in Goedel.codeSysPrf]
+v:746 [binder, in gaia_hydras.nfwfgaia]
+v:75 [binder, in hydras.Ackermann.subAll]
+v:75 [binder, in hydras.Ackermann.LNT]
+v:75 [binder, in hydras.Ackermann.folLogic]
+v:76 [binder, in Goedel.PRrepresentable]
+v:76 [binder, in hydras.Ackermann.codeSubFormula]
+v:76 [binder, in hydras.Prelude.MoreVectors]
+v:77 [binder, in hydras.Ackermann.subProp]
+v:78 [binder, in hydras.Ackermann.codeSubTerm]
+v:78 [binder, in hydras.Ackermann.LNN]
+v:78 [binder, in hydras.Ackermann.LNT]
+v:79 [binder, in Goedel.codeSysPrf]
+v:8 [binder, in Goedel.fixPoint]
+v:8 [binder, in hydras.Ackermann.subProp]
+v:8 [binder, in hydras.Prelude.MoreVectors]
+v:80 [binder, in hydras.Ackermann.folLogic]
+v:80 [binder, in Goedel.codeSysPrf]
+v:800 [binder, in gaia_hydras.nfwfgaia]
+v:803 [binder, in gaia_hydras.nfwfgaia]
+v:805 [binder, in gaia_hydras.nfwfgaia]
+v:81 [binder, in hydras.Ackermann.LNN]
+V:81 [binder, in hydras.Schutte.PartialFun]
+v:82 [binder, in hydras.Ackermann.codeSubTerm]
+v:82 [binder, in hydras.Ackermann.LNT]
+v:82 [binder, in hydras.Ackermann.subProp]
+v:82 [binder, in Goedel.codeSysPrf]
+v:82 [binder, in hydras.Prelude.MoreVectors]
+v:83 [binder, in hydras.Ackermann.model]
+v:83 [binder, in hydras.Ackermann.folLogic]
+v:84 [binder, in Goedel.PRrepresentable]
+v:84 [binder, in hydras.Ackermann.subAll]
+v:85 [binder, in hydras.Ackermann.LNN]
+v:85 [binder, in hydras.Ackermann.LNT]
+v:859 [binder, in hydras.Ackermann.codeSubFormula]
+v:86 [binder, in hydras.Ackermann.codeSubTerm]
+v:86 [binder, in hydras.Ackermann.codeSubFormula]
+v:87 [binder, in hydras.Ackermann.subProp]
+V:87 [binder, in hydras.Schutte.PartialFun]
+v:87 [binder, in hydras.Ackermann.folLogic]
+v:87 [binder, in hydras.Prelude.MoreVectors]
+v:875 [binder, in hydras.Ackermann.codeSubFormula]
+v:879 [binder, in hydras.Ackermann.codeSubFormula]
+v:88 [binder, in Goedel.PRrepresentable]
+v:88 [binder, in hydras.Ackermann.subAll]
+v:88 [binder, in hydras.Ackermann.LNN]
+v:88 [binder, in hydras.Ackermann.model]
+v:88 [binder, in hydras.Ackermann.folLogic3]
+v:883 [binder, in hydras.Ackermann.codeSubFormula]
+v:898 [binder, in hydras.Ackermann.codeSubFormula]
+v:9 [binder, in hydras.MoreAck.BadSubst]
+v:90 [binder, in hydras.Ackermann.subAll]
+v:90 [binder, in hydras.Ackermann.codeSubTerm]
+v:90 [binder, in hydras.Ackermann.LNT]
+v:91 [binder, in hydras.Ackermann.subAll]
+v:91 [binder, in hydras.Ackermann.subProp]
+v:91 [binder, in hydras.Ackermann.codeSubFormula]
+v:91 [binder, in hydras.Prelude.MoreVectors]
+v:915 [binder, in hydras.Ackermann.codeSubFormula]
+v:920 [binder, in hydras.Ackermann.codeSubFormula]
+v:925 [binder, in hydras.Ackermann.codeSubFormula]
+v:93 [binder, in hydras.Ackermann.LNN]
+v:93 [binder, in hydras.Ackermann.LNT]
+v:94 [binder, in hydras.Ackermann.subAll]
+v:94 [binder, in hydras.Prelude.MoreVectors]
+v:95 [binder, in hydras.Ackermann.subProp]
+v:95 [binder, in hydras.Ackermann.folReplace]
+v:96 [binder, in hydras.Ackermann.LNN]
+v:97 [binder, in Goedel.PRrepresentable]
+v:97 [binder, in hydras.Ackermann.LNT]
+v:98 [binder, in hydras.Ackermann.subAll]
+v:98 [binder, in hydras.Ackermann.folLogic3]
+v:99 [binder, in Goedel.PRrepresentable]
+v:997 [binder, in gaia_hydras.nfwfgaia]
+

W

+wConsistent [definition, in hydras.Ackermann.wConsistent]
+wConsistent [library]
+wCon2Con [lemma, in hydras.Ackermann.wConsistent]
+wellFormed [library]
+wellFormedFormula [definition, in hydras.Ackermann.wellFormed]
+wellFormedFormulaCorrect1 [lemma, in hydras.Ackermann.wellFormed]
+wellFormedFormulaCorrect2 [lemma, in hydras.Ackermann.wellFormed]
+wellFormedFormulaIsPR [instance, in hydras.Ackermann.wellFormed]
+wellFormedTerm [definition, in hydras.Ackermann.wellFormed]
+wellFormedTermCorrect1 [lemma, in hydras.Ackermann.wellFormed]
+wellFormedTermCorrect2 [lemma, in hydras.Ackermann.wellFormed]
+wellFormedTermIsPR [instance, in hydras.Ackermann.wellFormed]
+wellFormedTerms [definition, in hydras.Ackermann.wellFormed]
+wellFormedTermsCorrect1 [lemma, in hydras.Ackermann.wellFormed]
+wellFormedTermsCorrect2 [lemma, in hydras.Ackermann.wellFormed]
+wellFormedTermsIsPR [instance, in hydras.Ackermann.wellFormed]
+wellFormedTermTerms [definition, in hydras.Ackermann.wellFormed]
+wellFormedTermTermsCorrect2 [lemma, in hydras.Ackermann.wellFormed]
+wellFormedTermTermsIsPR [instance, in hydras.Ackermann.wellFormed]
+well_formed_list:270 [binder, in hydras.rpo.term]
+well_formed_list:90 [binder, in hydras.rpo.term]
+well_founded_restriction_rect [definition, in hydras.Prelude.Restriction]
+well_founded_restriction [definition, in hydras.Prelude.Restriction]
+well_founded_length [lemma, in hydras.rpo.more_list]
+Well_Formed_Term.Well_Formed_Formula.Formula [variable, in hydras.Ackermann.wellFormed]
+Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect2 [variable, in hydras.Ackermann.wellFormed]
+Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect1 [variable, in hydras.Ackermann.wellFormed]
+Well_Formed_Term.Well_Formed_Formula.codeArityR [variable, in hydras.Ackermann.wellFormed]
+Well_Formed_Term.Well_Formed_Formula [section, in hydras.Ackermann.wellFormed]
+Well_Formed_Term.Terms [variable, in hydras.Ackermann.wellFormed]
+Well_Formed_Term.Term [variable, in hydras.Ackermann.wellFormed]
+Well_Formed_Term.codeArityFIsCorrect2 [variable, in hydras.Ackermann.wellFormed]
+Well_Formed_Term.codeArityFIsCorrect1 [variable, in hydras.Ackermann.wellFormed]
+Well_Formed_Term.codeArityF [variable, in hydras.Ackermann.wellFormed]
+Well_Formed_Term.L [variable, in hydras.Ackermann.wellFormed]
+Well_Formed_Term [section, in hydras.Ackermann.wellFormed]
+well_order [projection, in hydras.Schutte.Well_Orders]
+well_founded.R [variable, in hydras.Gamma0.Gamma0]
+well_founded [section, in hydras.Gamma0.Gamma0]
+well_founded_rpo [lemma, in hydras.Gamma0.Gamma0]
+well_founded_P [definition, in gaia_hydras.nfwfgaia]
+Well_Orders [library]
+WF [instance, in hydras.OrdinalNotations.ON_Omega2]
+wf [projection, in hydras.Prelude.WfVariant]
+WF [instance, in hydras.Epsilon0.F_alpha]
+Wfsum [module, in gaia_hydras.nfwfgaia]
+Wfsum.extension [definition, in gaia_hydras.nfwfgaia]
+Wfsum.f [definition, in gaia_hydras.nfwfgaia]
+Wfsum.f_correct [lemma, in gaia_hydras.nfwfgaia]
+Wfsum.f_eqn [lemma, in gaia_hydras.nfwfgaia]
+Wfsum.f_spec_simp [lemma, in gaia_hydras.nfwfgaia]
+Wfsum.f_spec [definition, in gaia_hydras.nfwfgaia]
+Wfsum.f1 [definition, in gaia_hydras.nfwfgaia]
+Wfsum.f2 [definition, in gaia_hydras.nfwfgaia]
+Wfsum.lt_dec [lemma, in gaia_hydras.nfwfgaia]
+Wfsum.psum [definition, in gaia_hydras.nfwfgaia]
+Wfsum.psum_exten [lemma, in gaia_hydras.nfwfgaia]
+WfVariant [record, in hydras.Prelude.WfVariant]
+WfVariant [library]
+wf_inverse_image_transparent [lemma, in additions.Wf_transparent]
+wf_incl_transparent [lemma, in additions.Wf_transparent]
+wf_measure [lemma, in hydras.OrdinalNotations.ON_Generic]
+wf_ltn [lemma, in gaia_hydras.onType]
+wf_trans [lemma, in hydras.rpo.closure]
+Wf_ex3.f_correct [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex3.f_eqn [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex3.f [definition, in gaia_hydras.nfwfgaia]
+Wf_ex3.f2 [definition, in gaia_hydras.nfwfgaia]
+Wf_ex3.f1 [definition, in gaia_hydras.nfwfgaia]
+Wf_ex3.odd_dec [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex3.f0c [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex3.f0b [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex3.f0a [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex3.f_spec_simp2 [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex3.f_spec_simp1 [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex3.f_spec_simp [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex3.f_spec [definition, in gaia_hydras.nfwfgaia]
+Wf_ex3.lte_wf [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex3.lte [definition, in gaia_hydras.nfwfgaia]
+Wf_ex3 [module, in gaia_hydras.nfwfgaia]
+Wf_ex.f_correct [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex.f_eqn [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex.f [definition, in gaia_hydras.nfwfgaia]
+Wf_ex.f2 [definition, in gaia_hydras.nfwfgaia]
+Wf_ex.f1 [definition, in gaia_hydras.nfwfgaia]
+Wf_ex.f0 [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex.f_spec_simp [lemma, in gaia_hydras.nfwfgaia]
+Wf_ex.f_spec [definition, in gaia_hydras.nfwfgaia]
+Wf_ex [module, in gaia_hydras.nfwfgaia]
+wf_lexico [lemma, in hydras.Prelude.Simple_LexProd]
+wf_lex [lemma, in hydras.rpo.rpo]
+Wf_transparent [library]
+Wf:101 [binder, in hydras.Hydra.Hydra_Lemmas]
+Wf:111 [binder, in hydras.Hydra.Hydra_Lemmas]
+Wf:146 [binder, in hydras.Hydra.Hydra_Definitions]
+Wf:156 [binder, in hydras.Hydra.Hydra_Definitions]
+wInconsistent [definition, in hydras.Ackermann.wConsistent]
+with_matrices.R [variable, in additions.fib]
+with_matrices [section, in additions.fib]
+WO [record, in hydras.Schutte.Well_Orders]
+WO_ord [instance, in hydras.Schutte.Schutte_basics]
+WO_nat [instance, in hydras.Schutte.Well_Orders]
+Wo:19 [binder, in hydras.Schutte.Well_Orders]
+WO:54 [binder, in hydras.Schutte.Well_Orders]
+WO:59 [binder, in hydras.Schutte.Well_Orders]
+w:101 [binder, in Goedel.PRrepresentable]
+w:122 [binder, in Goedel.PRrepresentable]
+w:1237 [binder, in hydras.Ackermann.codeSubFormula]
+w:129 [binder, in Goedel.PRrepresentable]
+w:1388 [binder, in hydras.Ackermann.codeSubFormula]
+w:1392 [binder, in hydras.Ackermann.codeSubFormula]
+w:14 [binder, in hydras.Prelude.MoreOrders]
+w:140 [binder, in Goedel.PRrepresentable]
+w:151 [binder, in Goedel.PRrepresentable]
+w:1513 [binder, in gaia_hydras.nfwfgaia]
+w:1516 [binder, in gaia_hydras.nfwfgaia]
+w:1519 [binder, in gaia_hydras.nfwfgaia]
+w:300 [binder, in hydras.Ackermann.subAll]
+w:399 [binder, in hydras.Ackermann.primRec]
+w:407 [binder, in hydras.Ackermann.primRec]
+w:60 [binder, in hydras.Prelude.MoreVectors]
+w:705 [binder, in gaia_hydras.nfwfgaia]
+w:715 [binder, in gaia_hydras.nfwfgaia]
+w:815 [binder, in gaia_hydras.nfwfgaia]
+w:822 [binder, in gaia_hydras.nfwfgaia]
+w:899 [binder, in hydras.Ackermann.codeSubFormula]
+w:916 [binder, in hydras.Ackermann.codeSubFormula]
+w:921 [binder, in hydras.Ackermann.codeSubFormula]
+w:926 [binder, in hydras.Ackermann.codeSubFormula]
+w:998 [binder, in gaia_hydras.nfwfgaia]
+

X

+xb:227 [binder, in additions.Euclidean_Chains]
+xb:238 [binder, in additions.Euclidean_Chains]
+xb:254 [binder, in additions.Euclidean_Chains]
+xb:258 [binder, in additions.Euclidean_Chains]
+xb:262 [binder, in additions.Euclidean_Chains]
+xb:266 [binder, in additions.Euclidean_Chains]
+xb:304 [binder, in additions.Euclidean_Chains]
+xb:306 [binder, in additions.Euclidean_Chains]
+xb:308 [binder, in additions.Euclidean_Chains]
+xb:310 [binder, in additions.Euclidean_Chains]
+xi:13 [binder, in hydras.Schutte.Critical]
+xi:56 [binder, in hydras.Schutte.Critical]
+xi:63 [binder, in hydras.Schutte.Critical]
+xi:64 [binder, in hydras.Schutte.Critical]
+xi:73 [binder, in hydras.Schutte.Critical]
+xi:79 [binder, in hydras.Schutte.Critical]
+xPair [definition, in hydras.Ackermann.cPair]
+xPairDef [lemma, in hydras.Ackermann.cPair]
+xr:228 [binder, in additions.Euclidean_Chains]
+xr:255 [binder, in additions.Euclidean_Chains]
+xr:259 [binder, in additions.Euclidean_Chains]
+xr:263 [binder, in additions.Euclidean_Chains]
+xr:267 [binder, in additions.Euclidean_Chains]
+x':1395 [binder, in gaia_hydras.nfwfgaia]
+x':526 [binder, in gaia_hydras.nfwfgaia]
+x':529 [binder, in gaia_hydras.nfwfgaia]
+x0:123 [binder, in hydras.Ackermann.model]
+x0:125 [binder, in hydras.Ackermann.model]
+x0:200 [binder, in hydras.Ackermann.folProp]
+x0:216 [binder, in hydras.Ackermann.folProp]
+x0:226 [binder, in Goedel.PRrepresentable]
+x0:228 [binder, in Goedel.PRrepresentable]
+x0:233 [binder, in Goedel.PRrepresentable]
+x0:235 [binder, in Goedel.PRrepresentable]
+x0:296 [binder, in hydras.Ackermann.folProp]
+x0:312 [binder, in hydras.Ackermann.folProp]
+x0:329 [binder, in hydras.Ackermann.subAll]
+x0:331 [binder, in hydras.Ackermann.subAll]
+x0:332 [binder, in hydras.Ackermann.subAll]
+x0:334 [binder, in hydras.Ackermann.subAll]
+x0:335 [binder, in hydras.Ackermann.subAll]
+x0:336 [binder, in hydras.Ackermann.subAll]
+x0:337 [binder, in hydras.Ackermann.subAll]
+x0:338 [binder, in hydras.Ackermann.subAll]
+x0:360 [binder, in hydras.Ackermann.folProp]
+x0:376 [binder, in hydras.Ackermann.folProp]
+x0:90 [binder, in hydras.Ackermann.model]
+x0:91 [binder, in hydras.Ackermann.model]
+x0:92 [binder, in hydras.Ackermann.model]
+x0:93 [binder, in hydras.Ackermann.model]
+x0:95 [binder, in hydras.Ackermann.model]
+x0:97 [binder, in hydras.Ackermann.model]
+x10:30 [binder, in additions.Addition_Chains]
+x16:104 [binder, in additions.Pow]
+x16:95 [binder, in additions.Pow_variant]
+x1:328 [binder, in hydras.Ackermann.folProp]
+x1:344 [binder, in hydras.Ackermann.folProp]
+x20:31 [binder, in additions.Addition_Chains]
+x2:101 [binder, in additions.Pow]
+x2:153 [binder, in additions.Euclidean_Chains]
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+y:304 [binder, in hydras.Ackermann.folProp]
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+y:34 [binder, in additions.fib]
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+y:343 [binder, in hydras.Ackermann.folProp]
+y:344 [binder, in gaia_hydras.nfwfgaia]
+y:345 [binder, in hydras.Ackermann.folProp]
+y:346 [binder, in gaia_hydras.nfwfgaia]
+y:348 [binder, in gaia_hydras.nfwfgaia]
+y:35 [binder, in additions.Monoid_instances]
+y:350 [binder, in gaia_hydras.nfwfgaia]
+y:351 [binder, in hydras.Ackermann.folProp]
+y:352 [binder, in hydras.Ackermann.folProp]
+y:352 [binder, in gaia_hydras.nfwfgaia]
+y:358 [binder, in hydras.Ackermann.folProp]
+y:359 [binder, in hydras.Ackermann.folProp]
+y:359 [binder, in hydras.Ackermann.primRec]
+y:36 [binder, in hydras.Prelude.Sort_spec]
+y:36 [binder, in hydras.Epsilon0.F_omega]
+y:36 [binder, in gaia_hydras.nfwfgaia]
+y:361 [binder, in hydras.Ackermann.folProp]
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+y:366 [binder, in gaia_hydras.nfwfgaia]
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+y:368 [binder, in hydras.Ackermann.folProp]
+y:37 [binder, in gaia_hydras.T1Bridge]
+y:37 [binder, in gaia_hydras.nfwfgaia]
+y:370 [binder, in hydras.Ackermann.primRec]
+y:371 [binder, in gaia_hydras.nfwfgaia]
+y:374 [binder, in hydras.Ackermann.folProp]
+y:375 [binder, in hydras.Ackermann.folProp]
+y:377 [binder, in Goedel.PRrepresentable]
+y:377 [binder, in hydras.Ackermann.folProp]
+y:377 [binder, in hydras.Ackermann.primRec]
+y:379 [binder, in Goedel.PRrepresentable]
+Y:38 [binder, in hydras.Schutte.Schutte_basics]
+y:38 [binder, in additions.fib]
+y:381 [binder, in Goedel.PRrepresentable]
+y:381 [binder, in hydras.Ackermann.primRec]
+y:383 [binder, in Goedel.PRrepresentable]
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+y:387 [binder, in hydras.Ackermann.primRec]
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+y:4 [binder, in hydras.Ackermann.LNT]
+y:40 [binder, in hydras.Prelude.Restriction]
+y:40 [binder, in gaia_hydras.nfwfgaia]
+y:401 [binder, in hydras.Ackermann.primRec]
+y:409 [binder, in hydras.Ackermann.primRec]
+y:41 [binder, in hydras.OrdinalNotations.ON_Generic]
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+y:426 [binder, in hydras.Ackermann.primRec]
+y:432 [binder, in hydras.Ackermann.primRec]
+y:439 [binder, in hydras.Gamma0.Gamma0]
+y:440 [binder, in hydras.Ackermann.primRec]
+y:441 [binder, in hydras.Gamma0.Gamma0]
+y:444 [binder, in hydras.Gamma0.Gamma0]
+y:448 [binder, in hydras.Ackermann.primRec]
+y:449 [binder, in hydras.Gamma0.Gamma0]
+y:45 [binder, in hydras.Prelude.Sort_spec]
+y:45 [binder, in hydras.OrdinalNotations.ON_Generic]
+y:453 [binder, in hydras.Ackermann.primRec]
+y:461 [binder, in hydras.Ackermann.primRec]
+y:466 [binder, in hydras.Ackermann.primRec]
+y:470 [binder, in hydras.Ackermann.primRec]
+y:474 [binder, in hydras.Ackermann.primRec]
+y:48 [binder, in gaia_hydras.T1Bridge]
+Y:48 [binder, in hydras.Schutte.Well_Orders]
+y:482 [binder, in hydras.Ackermann.primRec]
+y:487 [binder, in hydras.Ackermann.primRec]
+y:49 [binder, in additions.AM]
+y:492 [binder, in hydras.Ackermann.primRec]
+y:497 [binder, in hydras.Ackermann.primRec]
+y:5 [binder, in gaia_hydras.onType]
+y:5 [binder, in hydras.MoreAck.PrimRecExamples]
+y:501 [binder, in hydras.Ackermann.primRec]
+y:510 [binder, in gaia_hydras.nfwfgaia]
+y:517 [binder, in gaia_hydras.nfwfgaia]
+y:522 [binder, in gaia_hydras.nfwfgaia]
+y:552 [binder, in gaia_hydras.nfwfgaia]
+y:555 [binder, in gaia_hydras.nfwfgaia]
+y:579 [binder, in gaia_hydras.nfwfgaia]
+y:58 [binder, in hydras.MoreAck.PrimRecExamples]
+y:581 [binder, in gaia_hydras.nfwfgaia]
+y:585 [binder, in gaia_hydras.nfwfgaia]
+y:59 [binder, in hydras.Ackermann.cPair]
+y:6 [binder, in hydras.solutions_exercises.MinPR2]
+y:6 [binder, in hydras.Ackermann.PAconsistent]
+y:6 [binder, in additions.Compatibility]
+y:6 [binder, in hydras.MoreAck.AckNotPR]
+y:6 [binder, in additions.Euclidean_Chains]
+y:6 [binder, in hydras.Prelude.STDPP_compat]
+y:6 [binder, in additions.Addition_Chains]
+y:6 [binder, in hydras.rpo.closure]
+y:60 [binder, in hydras.MoreAck.AckNotPR]
+y:60 [binder, in hydras.Schutte.PartialFun]
+y:60 [binder, in hydras.OrdinalNotations.ON_Generic]
+y:609 [binder, in hydras.Gamma0.Gamma0]
+y:61 [binder, in hydras.Ackermann.fol]
+y:610 [binder, in hydras.Gamma0.Gamma0]
+y:613 [binder, in gaia_hydras.nfwfgaia]
+y:62 [binder, in hydras.Schutte.Critical]
+y:62 [binder, in hydras.MoreAck.PrimRecExamples]
+y:622 [binder, in gaia_hydras.nfwfgaia]
+y:624 [binder, in gaia_hydras.nfwfgaia]
+y:63 [binder, in hydras.MoreAck.AckNotPR]
+y:637 [binder, in gaia_hydras.nfwfgaia]
+y:645 [binder, in gaia_hydras.nfwfgaia]
+y:648 [binder, in gaia_hydras.nfwfgaia]
+y:65 [binder, in hydras.Ackermann.cPair]
+y:65 [binder, in gaia_hydras.nfwfgaia]
+y:659 [binder, in hydras.Ackermann.primRec]
+y:66 [binder, in hydras.Epsilon0.Epsilon0rpo]
+y:669 [binder, in gaia_hydras.nfwfgaia]
+y:67 [binder, in hydras.Ackermann.cPair]
+y:673 [binder, in hydras.Ackermann.primRec]
+y:68 [binder, in additions.AM]
+y:68 [binder, in hydras.Epsilon0.Epsilon0rpo]
+y:68 [binder, in additions.Monoid_def]
+y:682 [binder, in gaia_hydras.nfwfgaia]
+y:69 [binder, in hydras.Ackermann.cPair]
+y:7 [binder, in Goedel.PRrepresentable]
+y:7 [binder, in hydras.Prelude.DecPreOrder_Instances]
+y:7 [binder, in hydras.Prelude.More_Arith]
+y:7 [binder, in hydras.Ackermann.LNN]
+y:7 [binder, in hydras.Prelude.MoreOrders]
+y:700 [binder, in gaia_hydras.nfwfgaia]
+y:707 [binder, in hydras.Gamma0.Gamma0]
+y:709 [binder, in hydras.Gamma0.Gamma0]
+y:71 [binder, in hydras.Ackermann.cPair]
+y:710 [binder, in gaia_hydras.nfwfgaia]
+y:72 [binder, in hydras.Epsilon0.Epsilon0rpo]
+y:729 [binder, in hydras.Epsilon0.T1]
+y:73 [binder, in hydras.Ackermann.cPair]
+y:75 [binder, in hydras.Ackermann.folProp]
+y:75 [binder, in hydras.Ackermann.cPair]
+y:76 [binder, in hydras.Epsilon0.Epsilon0rpo]
+y:76 [binder, in additions.Monoid_def]
+y:76 [binder, in hydras.Schutte.PartialFun]
+y:77 [binder, in hydras.Ackermann.cPair]
+y:772 [binder, in gaia_hydras.nfwfgaia]
+y:775 [binder, in gaia_hydras.nfwfgaia]
+y:78 [binder, in hydras.Ackermann.folProp]
+y:79 [binder, in hydras.Ackermann.cPair]
+y:79 [binder, in hydras.Schutte.PartialFun]
+y:8 [binder, in hydras.Ackermann.LNT]
+y:8 [binder, in hydras.solutions_exercises.MorePRExamples]
+y:8 [binder, in hydras.Ackermann.cPair]
+y:8 [binder, in hydras.rpo.closure]
+y:81 [binder, in hydras.Ackermann.folProp]
+y:81 [binder, in hydras.MoreAck.PrimRecExamples]
+y:817 [binder, in gaia_hydras.nfwfgaia]
+y:82 [binder, in hydras.MoreAck.AckNotPR]
+y:820 [binder, in gaia_hydras.nfwfgaia]
+y:824 [binder, in gaia_hydras.nfwfgaia]
+y:827 [binder, in gaia_hydras.nfwfgaia]
+y:832 [binder, in gaia_hydras.nfwfgaia]
+y:838 [binder, in gaia_hydras.nfwfgaia]
+y:84 [binder, in hydras.Ackermann.folProp]
+y:84 [binder, in additions.Euclidean_Chains]
+y:84 [binder, in gaia_hydras.nfwfgaia]
+y:85 [binder, in additions.Euclidean_Chains]
+y:86 [binder, in hydras.Ackermann.cPair]
+y:87 [binder, in hydras.Ackermann.folProp]
+y:87 [binder, in hydras.MoreAck.AckNotPR]
+y:87 [binder, in hydras.MoreAck.PrimRecExamples]
+y:876 [binder, in gaia_hydras.nfwfgaia]
+y:88 [binder, in hydras.Ackermann.cPair]
+y:884 [binder, in gaia_hydras.nfwfgaia]
+y:888 [binder, in gaia_hydras.nfwfgaia]
+y:890 [binder, in gaia_hydras.nfwfgaia]
+y:893 [binder, in gaia_hydras.nfwfgaia]
+y:9 [binder, in hydras.Prelude.WfVariant]
+y:9 [binder, in hydras.Ackermann.LNN]
+y:9 [binder, in additions.Wf_transparent]
+y:9 [binder, in hydras.Prelude.MoreOrders]
+y:90 [binder, in hydras.Ackermann.cPair]
+y:91 [binder, in hydras.Ackermann.folProp]
+y:92 [binder, in hydras.Ackermann.cPair]
+y:92 [binder, in gaia_hydras.nfwfgaia]
+y:93 [binder, in gaia_hydras.nfwfgaia]
+y:94 [binder, in hydras.Ackermann.cPair]
+y:94 [binder, in hydras.Schutte.Ordering_Functions]
+y:95 [binder, in additions.Euclidean_Chains]
+y:95 [binder, in hydras.Schutte.Ordering_Functions]
+y:96 [binder, in hydras.Ackermann.cPair]
+y:96 [binder, in hydras.Schutte.Ordering_Functions]
+y:96 [binder, in gaia_hydras.nfwfgaia]
+y:97 [binder, in hydras.Ackermann.folProp]
+y:97 [binder, in hydras.Ackermann.cPair]
+y:97 [binder, in hydras.Schutte.Ordering_Functions]
+y:97 [binder, in hydras.MoreAck.PrimRecExamples]
+y:98 [binder, in hydras.Ackermann.cPair]
+y:98 [binder, in additions.Addition_Chains]
+y:98 [binder, in hydras.Prelude.MoreVectors]
+y:983 [binder, in gaia_hydras.nfwfgaia]
+y:99 [binder, in additions.Pow_variant]
+y:99 [binder, in hydras.Schutte.Ordering_Functions]
+

Z

+Z [module, in additions.Naive]
+zero [constructor, in hydras.Epsilon0.T1]
+zero [definition, in hydras.OrdinalNotations.ON_Omega2]
+zero [constructor, in hydras.Gamma0.T2]
+zero [definition, in hydras.Schutte.Schutte_basics]
+Zero [abbreviation, in hydras.Epsilon0.E0]
+zeroFormula [definition, in Goedel.PRrepresentable]
+zeroFunc [constructor, in hydras.Ackermann.primRec]
+ZeroLimitSucc_dec [definition, in hydras.OrdinalNotations.ON_Generic]
+zeroRepresentable [lemma, in Goedel.PRrepresentable]
+zero_limit_succ_dec [definition, in hydras.Epsilon0.T1]
+zero_le [lemma, in hydras.Epsilon0.T1]
+zero_nf [lemma, in hydras.Epsilon0.T1]
+zero_lt [lemma, in hydras.Epsilon0.T1]
+zero_lt [lemma, in hydras.Schutte.Correctness_E0]
+zero_limit_succ_dec [definition, in hydras.OrdinalNotations.ON_Omega2]
+Zero_limit_succ_dec [definition, in hydras.OrdinalNotations.ON_Omega2]
+zero_le [lemma, in hydras.OrdinalNotations.ON_Omega2]
+zero_lt_e0 [constructor, in hydras.Gamma0.T2]
+zero_nf [constructor, in hydras.Gamma0.T2]
+zero_finite [constructor, in hydras.Gamma0.T2]
+zero_lt_beta [lemma, in hydras.Schutte.AP]
+Zero_limit_succ_dec [definition, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+zero_ref [lemma, in gaia_hydras.T1Bridge]
+zero_lt_omega [lemma, in hydras.Schutte.Schutte_basics]
+zero_lt_succ [lemma, in hydras.Schutte.Schutte_basics]
+zero_or_positive [lemma, in hydras.Schutte.Schutte_basics]
+zero_or_greater [lemma, in hydras.Schutte.Schutte_basics]
+zero_le [lemma, in hydras.Schutte.Schutte_basics]
+Zero_limit_succ_dec [definition, in hydras.OrdinalNotations.ON_Finite]
+Zero_limit_succ_dec [definition, in hydras.Epsilon0.E0]
+Zero_Limit_Succ_dec [definition, in hydras.Epsilon0.E0]
+Zero_limit_succ_dec [definition, in hydras.OrdinalNotations.ON_Omega]
+zero_not_lim [lemma, in hydras.Gamma0.Gamma0]
+zero_lt_succ [lemma, in hydras.Gamma0.Gamma0]
+Zero_ [constructor, in hydras.Ackermann.Languages]
+zero_plus_alpha [lemma, in hydras.Schutte.Addition]
+Zfibt [definition, in additions.fib]
+ZLS_dec [lemma, in hydras.OrdinalNotations.ON_Omega_plus_omega]
+ZMult [instance, in additions.Monoid_instances]
+ZMult_Abelian [instance, in additions.Monoid_instances]
+ZtoR [definition, in additions.fib]
+ZtoRD [lemma, in additions.fib]
+ZtoRM [lemma, in additions.fib]
+Z_le_TO [instance, in hydras.Prelude.DecPreOrder_Instances]
+Z_le_dec [instance, in hydras.Prelude.DecPreOrder_Instances]
+Z_pow_compat [lemma, in additions.Compatibility]
+Z_pow_compat_pos [lemma, in additions.Compatibility]
+Z_pow [definition, in additions.Compatibility]
+Z_mult_op [instance, in additions.Monoid_instances]
+z':2092 [binder, in gaia_hydras.nfwfgaia]
+z':2102 [binder, in gaia_hydras.nfwfgaia]
+z':580 [binder, in gaia_hydras.nfwfgaia]
+z':590 [binder, in gaia_hydras.nfwfgaia]
+Z.power [definition, in additions.Naive]
+Z.power_of_plus [lemma, in additions.Naive]
+Z.power_S [lemma, in additions.Naive]
+z0:273 [binder, in additions.Euclidean_Chains]
+z0:275 [binder, in additions.Euclidean_Chains]
+z0:279 [binder, in additions.Euclidean_Chains]
+z0:281 [binder, in additions.Euclidean_Chains]
+z1:103 [binder, in hydras.Ackermann.codeSubFormula]
+z1:105 [binder, in hydras.Ackermann.folProp]
+z1:110 [binder, in hydras.Ackermann.codeSubFormula]
+z1:127 [binder, in hydras.Ackermann.folProp]
+z1:169 [binder, in hydras.Ackermann.fol]
+z1:253 [binder, in hydras.Ackermann.fol]
+z1:263 [binder, in hydras.Ackermann.fol]
+z1:274 [binder, in hydras.Ackermann.fol]
+z1:282 [binder, in hydras.Ackermann.fol]
+z1:303 [binder, in hydras.Ackermann.fol]
+z1:313 [binder, in hydras.Ackermann.fol]
+z1:324 [binder, in hydras.Ackermann.fol]
+z1:349 [binder, in hydras.Ackermann.fol]
+z1:359 [binder, in hydras.Ackermann.fol]
+z1:370 [binder, in hydras.Ackermann.fol]
+z1:393 [binder, in hydras.Ackermann.fol]
+z1:403 [binder, in hydras.Ackermann.fol]
+z1:414 [binder, in hydras.Ackermann.fol]
+z1:88 [binder, in hydras.Ackermann.folProp]
+z1:95 [binder, in hydras.Ackermann.codeSubFormula]
+z2:104 [binder, in hydras.Ackermann.codeSubFormula]
+z2:106 [binder, in hydras.Ackermann.folProp]
+z2:111 [binder, in hydras.Ackermann.codeSubFormula]
+z2:128 [binder, in hydras.Ackermann.folProp]
+z2:170 [binder, in hydras.Ackermann.fol]
+z2:254 [binder, in hydras.Ackermann.fol]
+z2:264 [binder, in hydras.Ackermann.fol]
+z2:275 [binder, in hydras.Ackermann.fol]
+z2:283 [binder, in hydras.Ackermann.fol]
+z2:304 [binder, in hydras.Ackermann.fol]
+z2:314 [binder, in hydras.Ackermann.fol]
+z2:325 [binder, in hydras.Ackermann.fol]
+z2:350 [binder, in hydras.Ackermann.fol]
+z2:360 [binder, in hydras.Ackermann.fol]
+z2:371 [binder, in hydras.Ackermann.fol]
+z2:394 [binder, in hydras.Ackermann.fol]
+z2:404 [binder, in hydras.Ackermann.fol]
+z2:415 [binder, in hydras.Ackermann.fol]
+z2:89 [binder, in hydras.Ackermann.folProp]
+z2:96 [binder, in hydras.Ackermann.codeSubFormula]
+z3:256 [binder, in hydras.Ackermann.fol]
+z3:306 [binder, in hydras.Ackermann.fol]
+z3:352 [binder, in hydras.Ackermann.fol]
+z3:396 [binder, in hydras.Ackermann.fol]
+z3:92 [binder, in hydras.Ackermann.folProp]
+z3:98 [binder, in hydras.Ackermann.codeSubFormula]
+z4:257 [binder, in hydras.Ackermann.fol]
+z4:307 [binder, in hydras.Ackermann.fol]
+z4:353 [binder, in hydras.Ackermann.fol]
+z4:397 [binder, in hydras.Ackermann.fol]
+z4:93 [binder, in hydras.Ackermann.folProp]
+z4:99 [binder, in hydras.Ackermann.codeSubFormula]
+z:1 [binder, in hydras.OrdinalNotations.Example_3PlusOmega]
+z:100 [binder, in hydras.Epsilon0.Large_Sets]
+z:101 [binder, in hydras.Epsilon0.Large_Sets]
+z:102 [binder, in hydras.Epsilon0.Large_Sets]
+z:103 [binder, in hydras.Prelude.Iterates]
+z:106 [binder, in hydras.Schutte.Schutte_basics]
+z:11 [binder, in Goedel.PRrepresentable]
+z:11 [binder, in hydras.Prelude.MoreOrders]
+z:11 [binder, in additions.Addition_Chains]
+z:12 [binder, in additions.Monoid_def]
+z:13 [binder, in hydras.Ackermann.Deduction]
+z:133 [binder, in hydras.Schutte.Schutte_basics]
+z:134 [binder, in hydras.Prelude.Iterates]
+z:135 [binder, in hydras.Prelude.Iterates]
+z:135 [binder, in hydras.Schutte.Schutte_basics]
+z:136 [binder, in hydras.Prelude.Iterates]
+z:137 [binder, in hydras.Prelude.Iterates]
+z:138 [binder, in hydras.Prelude.Iterates]
+z:139 [binder, in hydras.Prelude.Iterates]
+z:140 [binder, in hydras.Prelude.Iterates]
+z:144 [binder, in hydras.Prelude.Iterates]
+z:145 [binder, in hydras.Prelude.Iterates]
+z:146 [binder, in hydras.Prelude.Iterates]
+z:146 [binder, in additions.Euclidean_Chains]
+z:147 [binder, in hydras.Prelude.Iterates]
+z:148 [binder, in hydras.Prelude.Iterates]
+z:149 [binder, in hydras.Prelude.Iterates]
+z:15 [binder, in Goedel.PRrepresentable]
+z:150 [binder, in hydras.Prelude.Iterates]
+z:16 [binder, in hydras.Prelude.MoreOrders]
+z:16 [binder, in hydras.Ackermann.Deduction]
+z:170 [binder, in hydras.Ackermann.cPair]
+z:171 [binder, in hydras.Ackermann.cPair]
+z:177 [binder, in additions.Euclidean_Chains]
+z:182 [binder, in hydras.Ackermann.cPair]
+z:183 [binder, in hydras.Ackermann.cPair]
+z:184 [binder, in hydras.Ackermann.cPair]
+z:185 [binder, in hydras.Ackermann.cPair]
+z:186 [binder, in hydras.Ackermann.cPair]
+z:187 [binder, in hydras.Ackermann.cPair]
+z:19 [binder, in Goedel.PRrepresentable]
+z:19 [binder, in gaia_hydras.onType]
+z:1924 [binder, in gaia_hydras.nfwfgaia]
+z:1930 [binder, in gaia_hydras.nfwfgaia]
+z:1940 [binder, in gaia_hydras.nfwfgaia]
+z:1945 [binder, in gaia_hydras.nfwfgaia]
+z:1962 [binder, in gaia_hydras.nfwfgaia]
+z:1973 [binder, in gaia_hydras.nfwfgaia]
+z:1990 [binder, in gaia_hydras.nfwfgaia]
+z:2 [binder, in Goedel.PRrepresentable]
+z:2001 [binder, in gaia_hydras.nfwfgaia]
+z:2087 [binder, in gaia_hydras.nfwfgaia]
+z:2090 [binder, in gaia_hydras.nfwfgaia]
+z:2101 [binder, in gaia_hydras.nfwfgaia]
+z:23 [binder, in Goedel.PRrepresentable]
+z:230 [binder, in additions.Euclidean_Chains]
+z:231 [binder, in hydras.Schutte.Schutte_basics]
+z:257 [binder, in additions.Euclidean_Chains]
+z:26 [binder, in gaia_hydras.onType]
+z:261 [binder, in additions.Euclidean_Chains]
+z:265 [binder, in additions.Euclidean_Chains]
+z:269 [binder, in additions.Euclidean_Chains]
+z:27 [binder, in Goedel.PRrepresentable]
+z:27 [binder, in hydras.Schutte.Well_Orders]
+z:276 [binder, in Goedel.PRrepresentable]
+z:283 [binder, in additions.Euclidean_Chains]
+z:285 [binder, in additions.Euclidean_Chains]
+z:287 [binder, in additions.Euclidean_Chains]
+z:289 [binder, in additions.Euclidean_Chains]
+z:30 [binder, in additions.Monoid_def]
+z:30 [binder, in hydras.Schutte.Well_Orders]
+z:341 [binder, in Goedel.PRrepresentable]
+z:343 [binder, in Goedel.PRrepresentable]
+z:346 [binder, in Goedel.PRrepresentable]
+z:347 [binder, in Goedel.PRrepresentable]
+z:348 [binder, in Goedel.PRrepresentable]
+z:349 [binder, in Goedel.PRrepresentable]
+z:35 [binder, in gaia_hydras.nfwfgaia]
+z:36 [binder, in hydras.Prelude.MoreOrders]
+z:36 [binder, in additions.Monoid_instances]
+z:366 [binder, in Goedel.PRrepresentable]
+z:367 [binder, in Goedel.PRrepresentable]
+z:368 [binder, in Goedel.PRrepresentable]
+z:369 [binder, in Goedel.PRrepresentable]
+z:372 [binder, in Goedel.PRrepresentable]
+z:373 [binder, in Goedel.PRrepresentable]
+z:374 [binder, in Goedel.PRrepresentable]
+z:375 [binder, in Goedel.PRrepresentable]
+z:38 [binder, in gaia_hydras.T1Bridge]
+z:382 [binder, in hydras.Ackermann.primRec]
+z:388 [binder, in hydras.Ackermann.primRec]
+z:395 [binder, in hydras.Ackermann.primRec]
+z:402 [binder, in hydras.Ackermann.primRec]
+z:402 [binder, in gaia_hydras.nfwfgaia]
+z:403 [binder, in gaia_hydras.nfwfgaia]
+z:404 [binder, in gaia_hydras.nfwfgaia]
+z:405 [binder, in gaia_hydras.nfwfgaia]
+z:406 [binder, in gaia_hydras.nfwfgaia]
+z:410 [binder, in hydras.Ackermann.primRec]
+z:417 [binder, in gaia_hydras.nfwfgaia]
+z:418 [binder, in gaia_hydras.nfwfgaia]
+z:419 [binder, in gaia_hydras.nfwfgaia]
+z:420 [binder, in gaia_hydras.nfwfgaia]
+z:421 [binder, in gaia_hydras.nfwfgaia]
+Z:43 [binder, in hydras.Ackermann.code]
+z:43 [binder, in hydras.Ackermann.fol]
+z:454 [binder, in gaia_hydras.nfwfgaia]
+z:455 [binder, in gaia_hydras.nfwfgaia]
+z:456 [binder, in gaia_hydras.nfwfgaia]
+z:46 [binder, in hydras.Ackermann.fol]
+z:49 [binder, in hydras.Ackermann.fol]
+z:5 [binder, in Goedel.PRrepresentable]
+z:511 [binder, in gaia_hydras.nfwfgaia]
+z:512 [binder, in gaia_hydras.nfwfgaia]
+z:513 [binder, in gaia_hydras.nfwfgaia]
+z:52 [binder, in hydras.Ackermann.fol]
+z:57 [binder, in hydras.Schutte.Schutte_basics]
+z:578 [binder, in gaia_hydras.nfwfgaia]
+z:58 [binder, in hydras.Schutte.Schutte_basics]
+z:589 [binder, in gaia_hydras.nfwfgaia]
+z:62 [binder, in gaia_hydras.nfwfgaia]
+z:625 [binder, in gaia_hydras.nfwfgaia]
+z:63 [binder, in hydras.MoreAck.PrimRecExamples]
+z:631 [binder, in gaia_hydras.nfwfgaia]
+z:632 [binder, in gaia_hydras.nfwfgaia]
+z:633 [binder, in gaia_hydras.nfwfgaia]
+z:634 [binder, in gaia_hydras.nfwfgaia]
+z:635 [binder, in gaia_hydras.nfwfgaia]
+z:638 [binder, in gaia_hydras.nfwfgaia]
+z:639 [binder, in gaia_hydras.nfwfgaia]
+z:640 [binder, in gaia_hydras.nfwfgaia]
+z:641 [binder, in gaia_hydras.nfwfgaia]
+z:642 [binder, in gaia_hydras.nfwfgaia]
+z:643 [binder, in gaia_hydras.nfwfgaia]
+z:646 [binder, in gaia_hydras.nfwfgaia]
+z:649 [binder, in gaia_hydras.nfwfgaia]
+z:7 [binder, in hydras.MoreAck.AckNotPR]
+z:7 [binder, in additions.Euclidean_Chains]
+z:7 [binder, in hydras.Prelude.STDPP_compat]
+z:730 [binder, in hydras.Epsilon0.T1]
+z:741 [binder, in gaia_hydras.nfwfgaia]
+z:742 [binder, in gaia_hydras.nfwfgaia]
+z:743 [binder, in gaia_hydras.nfwfgaia]
+z:744 [binder, in gaia_hydras.nfwfgaia]
+z:796 [binder, in gaia_hydras.nfwfgaia]
+z:798 [binder, in gaia_hydras.nfwfgaia]
+z:8 [binder, in Goedel.PRrepresentable]
+z:8 [binder, in additions.Addition_Chains]
+z:801 [binder, in gaia_hydras.nfwfgaia]
+z:804 [binder, in gaia_hydras.nfwfgaia]
+z:808 [binder, in gaia_hydras.nfwfgaia]
+z:812 [binder, in gaia_hydras.nfwfgaia]
+z:818 [binder, in gaia_hydras.nfwfgaia]
+z:821 [binder, in gaia_hydras.nfwfgaia]
+z:825 [binder, in gaia_hydras.nfwfgaia]
+z:828 [binder, in gaia_hydras.nfwfgaia]
+z:833 [binder, in gaia_hydras.nfwfgaia]
+z:86 [binder, in hydras.Ackermann.fol]
+z:88 [binder, in hydras.Ackermann.fol]
+z:9 [binder, in hydras.rpo.closure]
+z:91 [binder, in hydras.Prelude.Iterates]
+z:91 [binder, in gaia_hydras.nfwfgaia]
+z:94 [binder, in hydras.Epsilon0.Large_Sets]
+z:96 [binder, in hydras.Schutte.Schutte_basics]
+z:97 [binder, in hydras.Prelude.Iterates]
+z:97 [binder, in hydras.Epsilon0.Large_Sets]
+z:97 [binder, in hydras.Schutte.Schutte_basics]
+z:98 [binder, in hydras.Epsilon0.Large_Sets]
+z:98 [binder, in hydras.Ackermann.fol]
+z:98 [binder, in hydras.MoreAck.PrimRecExamples]
+z:99 [binder, in hydras.Epsilon0.Large_Sets]
+z:99 [binder, in hydras.Ackermann.fol]
+z:99 [binder, in additions.Addition_Chains]
+

_

+_phi0 [definition, in hydras.Schutte.AP]
+_omega [definition, in hydras.Schutte.Schutte_basics]
+_Omega [abbreviation, in hydras.Epsilon0.E0]
+_f [definition, in hydras.Schutte.Ordering_Functions]
+_A [definition, in hydras.Schutte.Ordering_Functions]
+

other

+fol:_ <> _ [notation, in hydras.Ackermann.fol]
+fol:_ = _ [notation, in hydras.Ackermann.fol]
+fol:_ <-> _ [notation, in hydras.Ackermann.fol]
+fol:_ -> _ [notation, in hydras.Ackermann.fol]
+fol:_ ∧ _ [notation, in hydras.Ackermann.fol]
+fol:_ ∨ _ [notation, in hydras.Ackermann.fol]
+fol:( _ ) [notation, in hydras.Ackermann.fol]
+fol:{ _ } [notation, in hydras.Ackermann.fol]
+fol:~ _ [notation, in hydras.Ackermann.fol]
+fol:∀ _ , _ [notation, in hydras.Ackermann.fol]
+fol:∃ _ , _ [notation, in hydras.Ackermann.fol]
+_ o+ _ (cantor_scope) [notation, in gaia_hydras.GHessenberg]
+_ O+ _ (E0_scope) [notation, in hydras.Epsilon0.E0]
+_ * _ (E0_scope) [notation, in hydras.Epsilon0.E0]
+E0_omega^ (E0_scope) [notation, in hydras.Epsilon0.E0]
+_ + _ (E0_scope) [notation, in hydras.Epsilon0.E0]
+_ o<= _ (E0_scope) [notation, in hydras.Epsilon0.E0]
+_ o< _ (E0_scope) [notation, in hydras.Epsilon0.E0]
+_ ^ _ (M_scope) [notation, in additions.Pow_variant]
+_ ^b _ (M_scope) [notation, in additions.Pow_variant]
+_ ^ _ (M_scope) [notation, in additions.Pow_variant]
+_ ^ _ (M_scope) [notation, in additions.Pow]
+_ ^b _ (M_scope) [notation, in additions.Pow]
+_ ^ _ (M_scope) [notation, in additions.Pow]
+_ * _ (M_scope) [notation, in additions.Monoid_def]
+_ .-4 (nat_scope) [notation, in hydras.Prelude.ssrnat_extracts]
+_ .-3 (nat_scope) [notation, in hydras.Prelude.ssrnat_extracts]
+_ .-2 (nat_scope) [notation, in hydras.Prelude.ssrnat_extracts]
+_ .-1 (nat_scope) [notation, in hydras.Prelude.ssrnat_extracts]
+_ .+5 (nat_scope) [notation, in hydras.Prelude.ssrnat_extracts]
+_ .+4 (nat_scope) [notation, in hydras.Prelude.ssrnat_extracts]
+_ .+3 (nat_scope) [notation, in hydras.Prelude.ssrnat_extracts]
+_ .+2 (nat_scope) [notation, in hydras.Prelude.ssrnat_extracts]
+_ .+1 (nat_scope) [notation, in hydras.Prelude.ssrnat_extracts]
+_ <->' _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ <->'' _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ /\' _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ \/' _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ * _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ + _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+allH _ .. _ , _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+exH _ .. _ , _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+v# _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ <> _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ = _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ <-> _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+~ _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ -> _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ /\ _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ \/ _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ = _ (nt_scope) [notation, in hydras.Ackermann.LNT]
+_ * _ (ON_scope) [notation, in hydras.OrdinalNotations.ON_Omega2]
+_ + _ (ON_scope) [notation, in hydras.OrdinalNotations.ON_Omega2]
+_ o?= _ (ON_scope) [notation, in hydras.OrdinalNotations.ON_Generic]
+_ o<= _ (ON_scope) [notation, in hydras.OrdinalNotations.ON_Generic]
+_ o< _ (ON_scope) [notation, in hydras.OrdinalNotations.ON_Generic]
+_ ^ _ (ppT1_scope) [notation, in hydras.Epsilon0.T1]
+_ * _ (ppT1_scope) [notation, in hydras.Epsilon0.T1]
+_ + _ (ppT1_scope) [notation, in hydras.Epsilon0.T1]
+omega^ (schutte_scope) [notation, in hydras.Schutte.AP]
+|_| _ (schutte_scope) [notation, in hydras.Schutte.Schutte_basics]
+_ <= _ (schutte_scope) [notation, in hydras.Schutte.Schutte_basics]
+_ < _ (schutte_scope) [notation, in hydras.Schutte.Schutte_basics]
+_ * _ (schutte_scope) [notation, in hydras.Schutte.Addition]
+_ + _ (schutte_scope) [notation, in hydras.Schutte.Addition]
+_ == _ (type_scope) [notation, in additions.Monoid_def]
+_ == _ (type_scope) [notation, in additions.Addition_Chains]
+_ t1< _ t1< _ (t1_scope) [notation, in hydras.Epsilon0.T1]
+_ t1<= _ (t1_scope) [notation, in hydras.Epsilon0.T1]
+_ t1< _ (t1_scope) [notation, in hydras.Epsilon0.T1]
+_ ^ _ (t1_scope) [notation, in hydras.Epsilon0.T1]
+_ - _ (t1_scope) [notation, in hydras.Epsilon0.T1]
+_ * _ (t1_scope) [notation, in hydras.Epsilon0.T1]
+_ + _ (t1_scope) [notation, in hydras.Epsilon0.T1]
+\F _ (t1_scope) [notation, in hydras.Epsilon0.T1]
+_ t2<= _ (T2_scope) [notation, in hydras.Gamma0.T2]
+_ t2< _ (T2_scope) [notation, in hydras.Gamma0.T2]
+[ _ , _ ] (T2_scope) [notation, in hydras.Gamma0.T2]
+_ + _ (T2_scope) [notation, in hydras.Gamma0.Gamma0]
+_ <_phi0 _ [notation, in hydras.Epsilon0.T1]
+_ < _ [notation, in gaia_hydras.T1Choice]
+_ <= _ [notation, in gaia_hydras.T1Choice]
+_ '<s' _ [notation, in hydras.Prelude.DecPreOrder_Instances]
+_ <: _ [notation, in hydras.Prelude.DecPreOrder_Instances]
+_ >>s _ [notation, in hydras.Prelude.Iterates]
+_ >> _ [notation, in hydras.Prelude.Iterates]
+_ <<= _ [notation, in hydras.Prelude.Iterates]
+_ o+ _ [notation, in hydras.Epsilon0.Hessenberg]
+_ =x= _ [notation, in hydras.Ackermann.primRec]
+_ <--- _ times _ ; _ [notation, in additions.Addition_Chains]
+_ -*-> _ [notation, in hydras.Hydra.Hydra_Definitions]
+_ -+-> _ [notation, in hydras.Hydra.Hydra_Definitions]
+_ -1-> _ [notation, in hydras.Hydra.Hydra_Definitions]
+f[ _ ]f [notation, in hydras.Ackermann.fol]
+SetAdds _ _ .. _ [notation, in hydras.Ackermann.NewNotations]
+SetEnum _ _ .. _ [notation, in hydras.Ackermann.NewNotations]
+v_apply _ _ [notation, in hydras.MoreAck.AckNotPR]
+α [definition, in gaia_hydras.HydraGaia_Examples]
+β [abbreviation, in Goedel.PRrepresentable]
+β [definition, in gaia_hydras.HydraGaia_Examples]
+βR [abbreviation, in Goedel.PRrepresentable]
+ω [abbreviation, in hydras.Epsilon0.T1]
+

+

Notation Index

+

A

+_ - _ (ak_scope) [in gaia_hydras.nfwfgaia]
+_ + _ (ak_scope) [in gaia_hydras.nfwfgaia]
+_ > _ (ak_scope) [in gaia_hydras.nfwfgaia]
+_ >= _ (ak_scope) [in gaia_hydras.nfwfgaia]
+_ <= _ (ak_scope) [in gaia_hydras.nfwfgaia]
+_ < _ (ak_scope) [in gaia_hydras.nfwfgaia]
+\F _ (ak_scope) [in gaia_hydras.nfwfgaia]
+[ _ , _ , _ ] (ak_scope) [in gaia_hydras.nfwfgaia]
+_ < _ (cantor_scope) [in gaia_hydras.nfwfgaia]
+

C

+_ ^ _ (cantor_scope) [in gaia_hydras.nfwfgaia]
+_ * _ (cantor_scope) [in gaia_hydras.nfwfgaia]
+_ - _ (cantor_scope) [in gaia_hydras.nfwfgaia]
+_ + _ (cantor_scope) [in gaia_hydras.nfwfgaia]
+_ > _ (cantor_scope) [in gaia_hydras.nfwfgaia]
+_ >= _ (cantor_scope) [in gaia_hydras.nfwfgaia]
+_ <= _ (cantor_scope) [in gaia_hydras.nfwfgaia]
+_ < _ (cantor_scope) [in gaia_hydras.nfwfgaia]
+\F _ (cantor_scope) [in gaia_hydras.nfwfgaia]
+

D

+_ ^ _ [in additions.FirstSteps]
+_ * _ [in additions.FirstSteps]
+1 [in additions.FirstSteps]
+_ <= _ [in hydras.Prelude.MoreOrders]
+_ < _ [in hydras.Prelude.MoreOrders]
+

F

+_ <->' _ (fol_scope) [in hydras.Ackermann.fol]
+_ <->'' _ (fol_scope) [in hydras.Ackermann.fol]
+_ /\' _ (fol_scope) [in hydras.Ackermann.fol]
+_ \/' _ (fol_scope) [in hydras.Ackermann.fol]
+_ <> _ (fol_scope) [in hydras.Ackermann.fol]
+_ = _ (fol_scope) [in hydras.Ackermann.fol]
+allH _ .. _ , _ (fol_scope) [in hydras.Ackermann.fol]
+exH _ .. _ , _ (fol_scope) [in hydras.Ackermann.fol]
+v# _ (fol_scope) [in hydras.Ackermann.fol]
+_ <-> _ (fol_scope) [in hydras.Ackermann.fol]
+~ _ (fol_scope) [in hydras.Ackermann.fol]
+_ -> _ (fol_scope) [in hydras.Ackermann.fol]
+_ /\ _ (fol_scope) [in hydras.Ackermann.fol]
+_ \/ _ (fol_scope) [in hydras.Ackermann.fol]
+_ = _ (fol_scope) [in hydras.Ackermann.fol]
+

G

+_ - _ (g0_scope) [in gaia_hydras.nfwfgaia]
+_ + _ (g0_scope) [in gaia_hydras.nfwfgaia]
+_ > _ (g0_scope) [in gaia_hydras.nfwfgaia]
+_ >= _ (g0_scope) [in gaia_hydras.nfwfgaia]
+_ <= _ (g0_scope) [in gaia_hydras.nfwfgaia]
+_ < _ (g0_scope) [in gaia_hydras.nfwfgaia]
+\F _ (g0_scope) [in gaia_hydras.nfwfgaia]
+[ _ , _ ] (g0_scope) [in gaia_hydras.nfwfgaia]
+_ <= _ [in hydras.Prelude.Merge_Sort]
+omega^ (g0_scope) [in hydras.Gamma0.Gamma0]
+_ + _ (g0_scope) [in hydras.Gamma0.Gamma0]
+

L

+_ <lex _ [in hydras.Prelude.DecPreOrder_Instances]
+_ <=lex _ [in hydras.Prelude.DecPreOrder_Instances]
+_ =B= _ [in hydras.Prelude.DecPreOrder_Instances]
+_ =A= _ [in hydras.Prelude.DecPreOrder_Instances]
+_ <B _ [in hydras.Prelude.DecPreOrder_Instances]
+_ <=B _ [in hydras.Prelude.DecPreOrder_Instances]
+_ <A _ [in hydras.Prelude.DecPreOrder_Instances]
+_ <=A _ [in hydras.Prelude.DecPreOrder_Instances]
+_ * _ (lo_scope) [in hydras.OrdinalNotations.OmegaOmega]
+_ + _ (lo_scope) [in hydras.OrdinalNotations.OmegaOmega]
+

M

+_ * _ [in additions.Monoid_instances]
+_ + _ [in additions.Monoid_instances]
+1 [in additions.Monoid_instances]
+0 [in additions.Monoid_instances]
+_ ** _ [in additions.Naive]
+_ * _ [in additions.Naive]
+_ + _ [in additions.Naive]
+0 [in additions.Naive]
+1 [in additions.Naive]
+_ * _ [in additions.FirstSteps]
+_ + _ [in additions.FirstSteps]
+1 [in additions.FirstSteps]
+0 [in additions.FirstSteps]
+

N

+fol:_ * _ [in hydras.Ackermann.LNN]
+fol:_ + _ [in hydras.Ackermann.LNN]
+_ < _ (fol_scope) [in hydras.Ackermann.LNN]
+_ * _ (fol_scope) [in hydras.Ackermann.LNN]
+_ + _ (fol_scope) [in hydras.Ackermann.LNN]
+

O

+_ * _ (OO_scope) [in hydras.OrdinalNotations.OmegaOmega]
+omega^ (OO_scope) [in hydras.OrdinalNotations.OmegaOmega]
+_ * _ (OO_scope) [in hydras.OrdinalNotations.OmegaOmega]
+_ + _ (OO_scope) [in hydras.OrdinalNotations.OmegaOmega]
+

P

+[ _ ; _ ; .. ; _ ] (pr_scope) [in hydras.Ackermann.primRec]
+[ _ ] (pr_scope) [in hydras.Ackermann.primRec]
+_ :: _ (pr_scope) [in hydras.Ackermann.primRec]
+

V

+_ < _ [in hydras.Prelude.WfVariant]
+

other

+fol:_ <> _ [in hydras.Ackermann.fol]
+fol:_ = _ [in hydras.Ackermann.fol]
+fol:_ <-> _ [in hydras.Ackermann.fol]
+fol:_ -> _ [in hydras.Ackermann.fol]
+fol:_ ∧ _ [in hydras.Ackermann.fol]
+fol:_ ∨ _ [in hydras.Ackermann.fol]
+fol:( _ ) [in hydras.Ackermann.fol]
+fol:{ _ } [in hydras.Ackermann.fol]
+fol:~ _ [in hydras.Ackermann.fol]
+fol:∀ _ , _ [in hydras.Ackermann.fol]
+fol:∃ _ , _ [in hydras.Ackermann.fol]
+_ o+ _ (cantor_scope) [in gaia_hydras.GHessenberg]
+_ O+ _ (E0_scope) [in hydras.Epsilon0.E0]
+_ * _ (E0_scope) [in hydras.Epsilon0.E0]
+E0_omega^ (E0_scope) [in hydras.Epsilon0.E0]
+_ + _ (E0_scope) [in hydras.Epsilon0.E0]
+_ o<= _ (E0_scope) [in hydras.Epsilon0.E0]
+_ o< _ (E0_scope) [in hydras.Epsilon0.E0]
+_ ^ _ (M_scope) [in additions.Pow_variant]
+_ ^b _ (M_scope) [in additions.Pow_variant]
+_ ^ _ (M_scope) [in additions.Pow_variant]
+_ ^ _ (M_scope) [in additions.Pow]
+_ ^b _ (M_scope) [in additions.Pow]
+_ ^ _ (M_scope) [in additions.Pow]
+_ * _ (M_scope) [in additions.Monoid_def]
+_ .-4 (nat_scope) [in hydras.Prelude.ssrnat_extracts]
+_ .-3 (nat_scope) [in hydras.Prelude.ssrnat_extracts]
+_ .-2 (nat_scope) [in hydras.Prelude.ssrnat_extracts]
+_ .-1 (nat_scope) [in hydras.Prelude.ssrnat_extracts]
+_ .+5 (nat_scope) [in hydras.Prelude.ssrnat_extracts]
+_ .+4 (nat_scope) [in hydras.Prelude.ssrnat_extracts]
+_ .+3 (nat_scope) [in hydras.Prelude.ssrnat_extracts]
+_ .+2 (nat_scope) [in hydras.Prelude.ssrnat_extracts]
+_ .+1 (nat_scope) [in hydras.Prelude.ssrnat_extracts]
+_ <->' _ (nt_scope) [in hydras.Ackermann.LNT]
+_ <->'' _ (nt_scope) [in hydras.Ackermann.LNT]
+_ /\' _ (nt_scope) [in hydras.Ackermann.LNT]
+_ \/' _ (nt_scope) [in hydras.Ackermann.LNT]
+_ * _ (nt_scope) [in hydras.Ackermann.LNT]
+_ + _ (nt_scope) [in hydras.Ackermann.LNT]
+allH _ .. _ , _ (nt_scope) [in hydras.Ackermann.LNT]
+exH _ .. _ , _ (nt_scope) [in hydras.Ackermann.LNT]
+v# _ (nt_scope) [in hydras.Ackermann.LNT]
+_ <> _ (nt_scope) [in hydras.Ackermann.LNT]
+_ = _ (nt_scope) [in hydras.Ackermann.LNT]
+_ <-> _ (nt_scope) [in hydras.Ackermann.LNT]
+~ _ (nt_scope) [in hydras.Ackermann.LNT]
+_ -> _ (nt_scope) [in hydras.Ackermann.LNT]
+_ /\ _ (nt_scope) [in hydras.Ackermann.LNT]
+_ \/ _ (nt_scope) [in hydras.Ackermann.LNT]
+_ = _ (nt_scope) [in hydras.Ackermann.LNT]
+_ * _ (ON_scope) [in hydras.OrdinalNotations.ON_Omega2]
+_ + _ (ON_scope) [in hydras.OrdinalNotations.ON_Omega2]
+_ o?= _ (ON_scope) [in hydras.OrdinalNotations.ON_Generic]
+_ o<= _ (ON_scope) [in hydras.OrdinalNotations.ON_Generic]
+_ o< _ (ON_scope) [in hydras.OrdinalNotations.ON_Generic]
+_ ^ _ (ppT1_scope) [in hydras.Epsilon0.T1]
+_ * _ (ppT1_scope) [in hydras.Epsilon0.T1]
+_ + _ (ppT1_scope) [in hydras.Epsilon0.T1]
+omega^ (schutte_scope) [in hydras.Schutte.AP]
+|_| _ (schutte_scope) [in hydras.Schutte.Schutte_basics]
+_ <= _ (schutte_scope) [in hydras.Schutte.Schutte_basics]
+_ < _ (schutte_scope) [in hydras.Schutte.Schutte_basics]
+_ * _ (schutte_scope) [in hydras.Schutte.Addition]
+_ + _ (schutte_scope) [in hydras.Schutte.Addition]
+_ == _ (type_scope) [in additions.Monoid_def]
+_ == _ (type_scope) [in additions.Addition_Chains]
+_ t1< _ t1< _ (t1_scope) [in hydras.Epsilon0.T1]
+_ t1<= _ (t1_scope) [in hydras.Epsilon0.T1]
+_ t1< _ (t1_scope) [in hydras.Epsilon0.T1]
+_ ^ _ (t1_scope) [in hydras.Epsilon0.T1]
+_ - _ (t1_scope) [in hydras.Epsilon0.T1]
+_ * _ (t1_scope) [in hydras.Epsilon0.T1]
+_ + _ (t1_scope) [in hydras.Epsilon0.T1]
+\F _ (t1_scope) [in hydras.Epsilon0.T1]
+_ t2<= _ (T2_scope) [in hydras.Gamma0.T2]
+_ t2< _ (T2_scope) [in hydras.Gamma0.T2]
+[ _ , _ ] (T2_scope) [in hydras.Gamma0.T2]
+_ + _ (T2_scope) [in hydras.Gamma0.Gamma0]
+_ <_phi0 _ [in hydras.Epsilon0.T1]
+_ < _ [in gaia_hydras.T1Choice]
+_ <= _ [in gaia_hydras.T1Choice]
+_ '<s' _ [in hydras.Prelude.DecPreOrder_Instances]
+_ <: _ [in hydras.Prelude.DecPreOrder_Instances]
+_ >>s _ [in hydras.Prelude.Iterates]
+_ >> _ [in hydras.Prelude.Iterates]
+_ <<= _ [in hydras.Prelude.Iterates]
+_ o+ _ [in hydras.Epsilon0.Hessenberg]
+_ =x= _ [in hydras.Ackermann.primRec]
+_ <--- _ times _ ; _ [in additions.Addition_Chains]
+_ -*-> _ [in hydras.Hydra.Hydra_Definitions]
+_ -+-> _ [in hydras.Hydra.Hydra_Definitions]
+_ -1-> _ [in hydras.Hydra.Hydra_Definitions]
+f[ _ ]f [in hydras.Ackermann.fol]
+SetAdds _ _ .. _ [in hydras.Ackermann.NewNotations]
+SetEnum _ _ .. _ [in hydras.Ackermann.NewNotations]
+v_apply _ _ [in hydras.MoreAck.AckNotPR]
+

+

Binder Index

+

A

+ab1:18 [in hydras.MoreAck.Ack]
+ab2:19 [in hydras.MoreAck.Ack]
+ab:34 [in hydras.MoreAck.Ack]
+acc1:10 [in additions.fib]
+acc1:5 [in additions.fib]
+acc2:11 [in additions.fib]
+acc2:6 [in additions.fib]
+acc:108 [in hydras.MoreAck.PrimRecExamples]
+acc:113 [in hydras.rpo.list_set]
+acc:118 [in hydras.rpo.list_set]
+acc:126 [in hydras.rpo.list_set]
+acc:133 [in hydras.rpo.list_set]
+acc:140 [in hydras.rpo.list_set]
+acc:144 [in hydras.rpo.list_set]
+acc:148 [in hydras.rpo.list_set]
+acc:16 [in additions.Compatibility]
+acc:18 [in additions.fib]
+acc:20 [in additions.Compatibility]
+acc:232 [in hydras.rpo.list_set]
+acc:235 [in hydras.rpo.list_set]
+acc:238 [in hydras.rpo.list_set]
+acc:242 [in hydras.rpo.list_set]
+acc:71 [in additions.fib]
+acc:78 [in additions.fib]
+acc:87 [in hydras.rpo.list_set]
+Ackm:13 [in hydras.MoreAck.Ack]
+Aeq:64 [in additions.Monoid_def]
+alpha':137 [in hydras.Gamma0.Gamma0]
+alpha':193 [in hydras.Epsilon0.T1]
+alpha':322 [in hydras.Epsilon0.T1]
+alpha':462 [in hydras.Gamma0.Gamma0]
+alpha':505 [in hydras.Gamma0.Gamma0]
+alpha':509 [in hydras.Gamma0.Gamma0]
+alpha':548 [in hydras.Gamma0.Gamma0]
+alpha':551 [in hydras.Gamma0.Gamma0]
+alpha':92 [in hydras.Epsilon0.T1]
+alpha1:28 [in hydras.Gamma0.T2]
+alpha1:36 [in hydras.Gamma0.T2]
+alpha1:43 [in hydras.Gamma0.T2]
+alpha1:51 [in hydras.Gamma0.T2]
+alpha1:58 [in hydras.Gamma0.T2]
+alpha1:64 [in hydras.Gamma0.T2]
+alpha1:665 [in hydras.Gamma0.Gamma0]
+alpha1:674 [in hydras.Gamma0.Gamma0]
+alpha2:29 [in hydras.Gamma0.T2]
+alpha2:44 [in hydras.Gamma0.T2]
+alpha2:52 [in hydras.Gamma0.T2]
+alpha2:667 [in hydras.Gamma0.Gamma0]
+alpha2:676 [in hydras.Gamma0.Gamma0]
+alpha:1 [in hydras.solutions_exercises.is_F_monotonous]
+alpha:1 [in gaia_hydras.GF_alpha]
+alpha:1 [in gaia_hydras.GHprime]
+alpha:1 [in hydras.Epsilon0.Hessenberg]
+alpha:1 [in hydras.Epsilon0.Large_Sets]
+alpha:1 [in hydras.Schutte.Critical]
+alpha:1 [in hydras.Schutte.AP]
+alpha:1 [in hydras.Epsilon0.L_alpha]
+alpha:1 [in hydras.Hydra.Hydra_Theorems]
+alpha:1 [in hydras.Epsilon0.Canon]
+alpha:1 [in hydras.solutions_exercises.ge_omega_iff]
+alpha:1 [in hydras.Epsilon0.Hprime]
+alpha:1 [in gaia_hydras.GL_alpha]
+alpha:1 [in gaia_hydras.T2Bridge]
+alpha:1 [in hydras.Schutte.Addition]
+alpha:1 [in gaia_hydras.GHessenberg]
+alpha:10 [in hydras.Schutte.Correctness_E0]
+alpha:10 [in gaia_hydras.GF_alpha]
+alpha:10 [in gaia_hydras.GHprime]
+alpha:10 [in hydras.Epsilon0.Hessenberg]
+alpha:10 [in hydras.Schutte.Critical]
+alpha:10 [in hydras.Schutte.Addition]
+alpha:100 [in hydras.OrdinalNotations.OmegaOmega]
+alpha:100 [in hydras.Schutte.Schutte_basics]
+alpha:100 [in hydras.Epsilon0.E0]
+alpha:100 [in hydras.Epsilon0.Paths]
+alpha:101 [in hydras.Hydra.O2H]
+alpha:101 [in hydras.Prelude.Comparable]
+alpha:101 [in hydras.Schutte.Schutte_basics]
+alpha:102 [in hydras.Epsilon0.T1]
+alpha:102 [in hydras.Schutte.Schutte_basics]
+alpha:102 [in hydras.OrdinalNotations.ON_Generic]
+alpha:102 [in hydras.Epsilon0.E0]
+alpha:103 [in hydras.Prelude.Comparable]
+alpha:103 [in hydras.OrdinalNotations.OmegaOmega]
+alpha:103 [in hydras.Epsilon0.Paths]
+alpha:104 [in hydras.Hydra.O2H]
+alpha:104 [in hydras.OrdinalNotations.OmegaOmega]
+alpha:104 [in hydras.Epsilon0.Canon]
+alpha:104 [in hydras.Schutte.Schutte_basics]
+alpha:104 [in hydras.Epsilon0.E0]
+alpha:105 [in hydras.Schutte.Schutte_basics]
+alpha:105 [in hydras.Epsilon0.E0]
+alpha:106 [in hydras.OrdinalNotations.OmegaOmega]
+alpha:106 [in hydras.Epsilon0.Canon]
+alpha:107 [in hydras.Schutte.Schutte_basics]
+alpha:107 [in hydras.Epsilon0.E0]
+alpha:108 [in hydras.Schutte.Schutte_basics]
+alpha:108 [in hydras.Epsilon0.E0]
+alpha:108 [in hydras.Epsilon0.Paths]
+alpha:109 [in hydras.Epsilon0.Hessenberg]
+alpha:109 [in hydras.OrdinalNotations.OmegaOmega]
+alpha:109 [in hydras.Epsilon0.Canon]
+alpha:109 [in hydras.Schutte.Schutte_basics]
+alpha:109 [in hydras.Epsilon0.Paths]
+alpha:11 [in hydras.OrdinalNotations.ON_Omega2]
+alpha:11 [in hydras.Gamma0.T2]
+alpha:11 [in hydras.Schutte.CNF]
+alpha:11 [in hydras.Schutte.Critical]
+alpha:11 [in hydras.Schutte.AP]
+alpha:11 [in hydras.OrdinalNotations.ON_mult]
+alpha:11 [in hydras.OrdinalNotations.ON_Finite]
+alpha:11 [in hydras.Epsilon0.E0]
+alpha:11 [in hydras.Schutte.Addition]
+alpha:110 [in hydras.Epsilon0.T1]
+alpha:110 [in hydras.Hydra.O2H]
+alpha:110 [in hydras.Epsilon0.E0]
+alpha:111 [in hydras.Epsilon0.Canon]
+alpha:111 [in hydras.Schutte.Schutte_basics]
+alpha:111 [in hydras.Epsilon0.E0]
+alpha:112 [in hydras.Epsilon0.Hessenberg]
+alpha:113 [in hydras.Epsilon0.Canon]
+alpha:113 [in hydras.Schutte.Schutte_basics]
+alpha:113 [in hydras.Epsilon0.E0]
+alpha:113 [in hydras.Epsilon0.Paths]
+alpha:114 [in hydras.Epsilon0.F_alpha]
+alpha:114 [in hydras.Epsilon0.Canon]
+alpha:115 [in hydras.Schutte.Schutte_basics]
+alpha:115 [in hydras.OrdinalNotations.ON_Generic]
+alpha:115 [in hydras.Epsilon0.E0]
+alpha:117 [in hydras.Schutte.Schutte_basics]
+alpha:117 [in hydras.Epsilon0.E0]
+alpha:117 [in hydras.Epsilon0.Hprime]
+alpha:118 [in hydras.Epsilon0.E0]
+alpha:118 [in hydras.Epsilon0.Paths]
+alpha:119 [in hydras.Epsilon0.T1]
+alpha:119 [in hydras.Epsilon0.F_alpha]
+alpha:119 [in hydras.Schutte.Schutte_basics]
+alpha:12 [in hydras.Epsilon0.T1]
+alpha:12 [in hydras.OrdinalNotations.ON_Omega2]
+alpha:12 [in hydras.Hydra.O2H]
+alpha:12 [in gaia_hydras.GHprime]
+alpha:12 [in hydras.Epsilon0.Large_Sets]
+alpha:12 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+alpha:12 [in hydras.Epsilon0.E0]
+alpha:12 [in hydras.Hydra.Epsilon0_Needed_Generic]
+alpha:12 [in hydras.Schutte.Addition]
+alpha:120 [in hydras.Schutte.Schutte_basics]
+alpha:120 [in hydras.OrdinalNotations.ON_Generic]
+alpha:120 [in hydras.Epsilon0.E0]
+alpha:121 [in hydras.Epsilon0.T1]
+alpha:121 [in hydras.Schutte.Schutte_basics]
+alpha:123 [in hydras.Epsilon0.Paths]
+alpha:124 [in hydras.Epsilon0.T1]
+alpha:124 [in hydras.Epsilon0.Hprime]
+alpha:124 [in hydras.Gamma0.Gamma0]
+alpha:125 [in hydras.Epsilon0.Large_Sets]
+alpha:127 [in hydras.Epsilon0.Large_Sets]
+alpha:127 [in hydras.Epsilon0.Paths]
+alpha:127 [in hydras.Gamma0.Gamma0]
+alpha:128 [in hydras.Epsilon0.Hprime]
+alpha:129 [in hydras.Epsilon0.Large_Sets]
+alpha:13 [in hydras.solutions_exercises.predSuccUnicity]
+alpha:13 [in gaia_hydras.GF_alpha]
+alpha:13 [in gaia_hydras.GHprime]
+alpha:13 [in hydras.OrdinalNotations.ON_O]
+alpha:13 [in hydras.Epsilon0.Hessenberg]
+alpha:13 [in hydras.Schutte.CNF]
+alpha:13 [in hydras.OrdinalNotations.ON_mult]
+alpha:13 [in hydras.OrdinalNotations.ON_plus]
+alpha:13 [in gaia_hydras.GL_alpha]
+alpha:13 [in hydras.Gamma0.Gamma0]
+alpha:130 [in hydras.Gamma0.Gamma0]
+alpha:131 [in hydras.Epsilon0.Large_Sets]
+alpha:132 [in hydras.OrdinalNotations.ON_Generic]
+alpha:132 [in hydras.Epsilon0.Paths]
+alpha:133 [in hydras.Epsilon0.Large_Sets]
+alpha:133 [in hydras.Gamma0.Gamma0]
+alpha:135 [in hydras.Epsilon0.Large_Sets]
+alpha:136 [in hydras.Schutte.Ordering_Functions]
+alpha:137 [in hydras.Epsilon0.Large_Sets]
+alpha:137 [in hydras.Schutte.Ordering_Functions]
+alpha:139 [in hydras.Epsilon0.Large_Sets]
+alpha:139 [in hydras.Epsilon0.Paths]
+alpha:14 [in hydras.Schutte.Correctness_E0]
+alpha:14 [in hydras.OrdinalNotations.ON_Omega2]
+alpha:14 [in hydras.Schutte.Critical]
+alpha:14 [in hydras.Schutte.AP]
+alpha:14 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+alpha:14 [in hydras.OrdinalNotations.ON_Finite]
+alpha:14 [in hydras.Epsilon0.E0]
+alpha:14 [in hydras.Epsilon0.Paths]
+alpha:14 [in hydras.Gamma0.Gamma0]
+alpha:141 [in hydras.Epsilon0.T1]
+alpha:141 [in hydras.Epsilon0.Large_Sets]
+alpha:145 [in hydras.OrdinalNotations.ON_Generic]
+alpha:145 [in hydras.Epsilon0.Paths]
+alpha:146 [in hydras.Epsilon0.T1]
+alpha:146 [in hydras.Gamma0.Gamma0]
+alpha:147 [in hydras.Schutte.Schutte_basics]
+alpha:147 [in hydras.Gamma0.Gamma0]
+alpha:149 [in hydras.Schutte.Schutte_basics]
+alpha:15 [in hydras.Epsilon0.T1]
+alpha:15 [in hydras.Gamma0.T2]
+alpha:15 [in hydras.Hydra.O2H]
+alpha:15 [in gaia_hydras.GF_alpha]
+alpha:15 [in hydras.Schutte.CNF]
+alpha:15 [in hydras.Schutte.Critical]
+alpha:15 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+alpha:15 [in hydras.OrdinalNotations.ON_mult]
+alpha:15 [in hydras.Hydra.Epsilon0_Needed_Std]
+alpha:15 [in hydras.OrdinalNotations.ON_plus]
+alpha:15 [in gaia_hydras.GHydra]
+alpha:15 [in hydras.solutions_exercises.F_3]
+alpha:15 [in gaia_hydras.GL_alpha]
+alpha:15 [in hydras.Gamma0.Gamma0]
+alpha:15 [in hydras.Schutte.Addition]
+alpha:150 [in hydras.Epsilon0.T1]
+alpha:151 [in hydras.Schutte.Schutte_basics]
+alpha:151 [in hydras.Epsilon0.Paths]
+alpha:151 [in hydras.Gamma0.Gamma0]
+alpha:153 [in hydras.Epsilon0.Paths]
+alpha:153 [in hydras.Gamma0.Gamma0]
+alpha:154 [in hydras.Epsilon0.Large_Sets]
+alpha:155 [in hydras.Epsilon0.T1]
+alpha:156 [in hydras.Epsilon0.Paths]
+alpha:157 [in hydras.Epsilon0.T1]
+alpha:157 [in hydras.Epsilon0.Large_Sets]
+alpha:157 [in hydras.Epsilon0.Paths]
+alpha:157 [in hydras.Gamma0.Gamma0]
+alpha:158 [in hydras.Epsilon0.F_alpha]
+alpha:159 [in hydras.Epsilon0.Paths]
+alpha:16 [in hydras.OrdinalNotations.ON_Omega2]
+alpha:16 [in hydras.solutions_exercises.predSuccUnicity]
+alpha:16 [in gaia_hydras.GF_alpha]
+alpha:16 [in hydras.OrdinalNotations.ON_O]
+alpha:16 [in hydras.Epsilon0.Paths]
+alpha:16 [in hydras.Gamma0.Gamma0]
+alpha:16 [in hydras.Schutte.Addition]
+alpha:160 [in hydras.Epsilon0.F_alpha]
+alpha:161 [in hydras.Epsilon0.Paths]
+alpha:162 [in hydras.Epsilon0.F_alpha]
+alpha:164 [in hydras.Epsilon0.Paths]
+alpha:165 [in hydras.Epsilon0.F_alpha]
+alpha:167 [in hydras.Epsilon0.F_alpha]
+alpha:168 [in hydras.Schutte.Ordering_Functions]
+alpha:169 [in hydras.Epsilon0.Large_Sets]
+alpha:169 [in hydras.Epsilon0.Paths]
+alpha:169 [in hydras.Schutte.Ordering_Functions]
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+alpha:17 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+alpha:17 [in hydras.OrdinalNotations.ON_plus]
+alpha:17 [in gaia_hydras.GHydra]
+alpha:17 [in hydras.OrdinalNotations.ON_Finite]
+alpha:17 [in gaia_hydras.GL_alpha]
+alpha:17 [in hydras.Gamma0.Gamma0]
+alpha:17 [in hydras.Schutte.Addition]
+alpha:170 [in hydras.Epsilon0.F_alpha]
+alpha:170 [in gaia_hydras.T1Bridge]
+alpha:172 [in gaia_hydras.T1Bridge]
+alpha:174 [in hydras.Epsilon0.F_alpha]
+alpha:174 [in hydras.Epsilon0.Paths]
+alpha:174 [in hydras.Gamma0.Gamma0]
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+alpha:176 [in hydras.Epsilon0.Large_Sets]
+alpha:176 [in hydras.Gamma0.Gamma0]
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+alpha:178 [in hydras.Schutte.Schutte_basics]
+alpha:179 [in hydras.Epsilon0.Large_Sets]
+alpha:18 [in hydras.Schutte.Correctness_E0]
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+alpha:18 [in hydras.Hydra.O2H]
+alpha:18 [in gaia_hydras.GF_alpha]
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+alpha:180 [in hydras.Gamma0.Gamma0]
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+alpha:186 [in hydras.Epsilon0.Large_Sets]
+alpha:186 [in hydras.Epsilon0.Paths]
+alpha:188 [in hydras.Epsilon0.Large_Sets]
+alpha:188 [in hydras.Schutte.Ordering_Functions]
+alpha:188 [in hydras.Gamma0.Gamma0]
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+alpha:19 [in hydras.Schutte.CNF]
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+alpha:19 [in hydras.OrdinalNotations.ON_Finite]
+alpha:19 [in hydras.Epsilon0.Paths]
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+alpha:197 [in hydras.Schutte.Schutte_basics]
+alpha:198 [in hydras.Gamma0.Gamma0]
+alpha:199 [in hydras.Epsilon0.Large_Sets]
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+alpha:2 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+alpha:2 [in hydras.solutions_exercises.schutte_cnf_counter_example]
+alpha:2 [in gaia_hydras.ON_gfinite]
+alpha:2 [in hydras.solutions_exercises.F_3]
+alpha:2 [in hydras.OrdinalNotations.ON_Omega]
+alpha:2 [in hydras.Epsilon0.Paths]
+alpha:2 [in hydras.Schutte.Ordering_Functions]
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+alpha:20 [in gaia_hydras.GHydra]
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+alpha:200 [in hydras.Gamma0.Gamma0]
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+alpha:247 [in hydras.Epsilon0.Paths]
+alpha:248 [in hydras.Epsilon0.Paths]
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+alpha:257 [in hydras.Epsilon0.Paths]
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+alpha:264 [in hydras.Epsilon0.Paths]
+alpha:265 [in hydras.Epsilon0.Paths]
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+alpha:323 [in hydras.Epsilon0.Paths]
+alpha:325 [in hydras.Epsilon0.Paths]
+alpha:327 [in hydras.Epsilon0.T1]
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+alpha:337 [in hydras.Epsilon0.T1]
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+alpha:34 [in hydras.Epsilon0.Hprime]
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+alpha:343 [in hydras.Epsilon0.T1]
+alpha:343 [in hydras.Epsilon0.Paths]
+alpha:347 [in hydras.Epsilon0.Paths]
+alpha:349 [in hydras.Epsilon0.T1]
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+alpha:357 [in hydras.Epsilon0.Paths]
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+alpha:36 [in hydras.Epsilon0.Hprime]
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+alpha:364 [in hydras.Epsilon0.Paths]
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+alpha:397 [in hydras.Epsilon0.T1]
+alpha:398 [in hydras.Epsilon0.T1]
+alpha:399 [in hydras.Epsilon0.Paths]
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+alpha:4 [in hydras.Hydra.Hydra_Theorems]
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+alpha:4 [in hydras.Epsilon0.E0]
+alpha:4 [in gaia_hydras.GL_alpha]
+alpha:4 [in gaia_hydras.T2Bridge]
+alpha:4 [in gaia_hydras.GHessenberg]
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+alpha:40 [in hydras.Epsilon0.F_alpha]
+alpha:40 [in hydras.Epsilon0.F_omega]
+alpha:40 [in hydras.OrdinalNotations.OmegaOmega]
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+alpha:406 [in hydras.Epsilon0.T1]
+alpha:407 [in hydras.Epsilon0.Paths]
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+alpha:41 [in hydras.Epsilon0.Hprime]
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+alpha:419 [in hydras.Epsilon0.T1]
+alpha:419 [in hydras.Epsilon0.Paths]
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+alpha:42 [in hydras.Schutte.CNF]
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+alpha:443 [in hydras.Epsilon0.Paths]
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+alpha:456 [in hydras.Epsilon0.T1]
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+alpha:474 [in hydras.Epsilon0.T1]
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+alpha:479 [in hydras.Epsilon0.Paths]
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+alpha:486 [in hydras.Epsilon0.T1]
+alpha:486 [in hydras.Epsilon0.Paths]
+alpha:49 [in hydras.Hydra.O2H]
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+alpha:49 [in hydras.Schutte.AP]
+alpha:49 [in hydras.Epsilon0.L_alpha]
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+alpha:490 [in hydras.Epsilon0.Paths]
+alpha:492 [in hydras.Gamma0.Gamma0]
+alpha:494 [in hydras.Epsilon0.Paths]
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+alpha:498 [in hydras.Epsilon0.Paths]
+alpha:499 [in hydras.Gamma0.Gamma0]
+alpha:5 [in hydras.Gamma0.T2]
+alpha:5 [in gaia_hydras.GF_alpha]
+alpha:5 [in hydras.Schutte.CNF]
+alpha:5 [in hydras.Hydra.Battle_length]
+alpha:5 [in gaia_hydras.ON_gfinite]
+alpha:50 [in hydras.Hydra.O2H]
+alpha:50 [in hydras.Epsilon0.Canon]
+alpha:50 [in hydras.Epsilon0.E0]
+alpha:502 [in hydras.Epsilon0.Paths]
+alpha:503 [in hydras.Gamma0.Gamma0]
+alpha:505 [in hydras.Epsilon0.Paths]
+alpha:507 [in hydras.Gamma0.Gamma0]
+alpha:51 [in hydras.Schutte.CNF]
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+alpha:51 [in hydras.Schutte.AP]
+alpha:51 [in hydras.Schutte.Ordering_Functions]
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+a':74 [in hydras.rpo.rpo]
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+a':789 [in gaia_hydras.nfwfgaia]
+a':81 [in hydras.Gamma0.T2]
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+a':869 [in gaia_hydras.nfwfgaia]
+a':896 [in gaia_hydras.nfwfgaia]
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+a':906 [in gaia_hydras.nfwfgaia]
+a':914 [in gaia_hydras.nfwfgaia]
+a':942 [in gaia_hydras.nfwfgaia]
+a':990 [in gaia_hydras.nfwfgaia]
+a0:17 [in hydras.Schutte.Well_Orders]
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+a0:194 [in hydras.Ackermann.cPair]
+a0:198 [in hydras.Ackermann.cPair]
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+a0:238 [in hydras.Ackermann.cPair]
+a0:240 [in hydras.Ackermann.cPair]
+a0:242 [in hydras.Ackermann.cPair]
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+a0:308 [in hydras.Ackermann.folProp]
+a0:317 [in hydras.Ackermann.folProp]
+a0:324 [in hydras.Ackermann.folProp]
+a0:333 [in hydras.Ackermann.folProp]
+a0:340 [in hydras.Ackermann.folProp]
+a0:349 [in hydras.Ackermann.folProp]
+a0:356 [in hydras.Ackermann.folProp]
+a0:44 [in hydras.Ackermann.LNN2LNT]
+a0:531 [in hydras.Ackermann.primRec]
+a0:533 [in hydras.Ackermann.primRec]
+a1:105 [in hydras.rpo.more_list]
+a1:113 [in hydras.rpo.more_list]
+A1:114 [in hydras.Schutte.Ordering_Functions]
+a1:122 [in hydras.Prelude.Merge_Sort]
+a1:130 [in hydras.Prelude.Merge_Sort]
+a1:1360 [in hydras.Ackermann.codeSubFormula]
+a1:137 [in hydras.rpo.more_list]
+a1:146 [in hydras.rpo.more_list]
+a1:156 [in hydras.rpo.more_list]
+a1:165 [in hydras.rpo.more_list]
+a1:172 [in hydras.rpo.more_list]
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+a:1299 [in gaia_hydras.nfwfgaia]
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+a:13 [in hydras.Ackermann.prLogic]
+a:13 [in hydras.Hydra.BigBattle]
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+a:13 [in hydras.Schutte.Well_Orders]
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+a:13 [in hydras.solutions_exercises.isqrt]
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+a:1318 [in gaia_hydras.nfwfgaia]
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+a:1330 [in gaia_hydras.nfwfgaia]
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+a:135 [in hydras.Gamma0.T2]
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+a:1350 [in gaia_hydras.nfwfgaia]
+a:1355 [in gaia_hydras.nfwfgaia]
+a:136 [in hydras.Epsilon0.T1]
+A:136 [in hydras.Prelude.MoreLists]
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+a:1365 [in gaia_hydras.nfwfgaia]
+a:137 [in hydras.Ackermann.LNT]
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+A:137 [in hydras.Ackermann.fol]
+a:1370 [in gaia_hydras.nfwfgaia]
+a:1376 [in gaia_hydras.nfwfgaia]
+a:138 [in hydras.Ackermann.codeSubFormula]
+a:138 [in additions.fib]
+a:1388 [in gaia_hydras.nfwfgaia]
+a:139 [in hydras.Ackermann.LNT]
+a:14 [in Goedel.PRrepresentable]
+a:14 [in hydras.solutions_exercises.MinPR2]
+a:14 [in hydras.Prelude.WfVariant]
+a:14 [in hydras.Ackermann.extEqualNat]
+a:14 [in hydras.Schutte.MoreEpsilonIota]
+a:14 [in hydras.solutions_exercises.MultisetWf]
+a:14 [in hydras.Ackermann.folLogic3]
+A:14 [in additions.Addition_Chains]
+a:14 [in hydras.solutions_exercises.T1_ltNotWf]
+a:14 [in hydras.MoreAck.PrimRecExamples]
+a:14 [in hydras.Prelude.Simple_LexProd]
+A:14 [in hydras.Prelude.MoreVectors]
+a:140 [in hydras.Gamma0.T2]
+a:140 [in hydras.rpo.more_list]
+a:140 [in hydras.Ackermann.LNN]
+a:140 [in hydras.Ackermann.primRec]
+A:140 [in hydras.Ackermann.fol]
+a:1407 [in gaia_hydras.nfwfgaia]
+A:141 [in hydras.Prelude.MoreLists]
+a:141 [in gaia_hydras.T1Bridge]
+a:141 [in hydras.Gamma0.Gamma0]
+a:1416 [in gaia_hydras.nfwfgaia]
+A:142 [in Goedel.PRrepresentable]
+a:142 [in hydras.Ackermann.LNN]
+a:142 [in hydras.Ackermann.LNT]
+a:142 [in gaia_hydras.T1Bridge]
+A:142 [in additions.Euclidean_Chains]
+A:142 [in hydras.Ackermann.fol]
+a:1422 [in gaia_hydras.nfwfgaia]
+a:1428 [in gaia_hydras.nfwfgaia]
+a:143 [in hydras.Ackermann.primRec]
+a:143 [in additions.fib]
+A:143 [in hydras.Schutte.Ordering_Functions]
+a:144 [in hydras.Gamma0.T2]
+A:144 [in hydras.rpo.more_list]
+a:144 [in gaia_hydras.T1Bridge]
+A:144 [in hydras.Ackermann.fol]
+A:144 [in hydras.Hydra.Hydra_Definitions]
+a:1441 [in gaia_hydras.nfwfgaia]
+a:1443 [in gaia_hydras.nfwfgaia]
+a:1445 [in gaia_hydras.nfwfgaia]
+a:1447 [in gaia_hydras.nfwfgaia]
+a:1449 [in gaia_hydras.nfwfgaia]
+a:145 [in hydras.Ackermann.LNN]
+a:146 [in hydras.Gamma0.T2]
+a:146 [in hydras.Ackermann.LNT]
+a:146 [in hydras.Ackermann.primRec]
+A:146 [in hydras.Ackermann.fol]
+A:146 [in hydras.OrdinalNotations.ON_Generic]
+a:1460 [in gaia_hydras.nfwfgaia]
+a:1461 [in gaia_hydras.nfwfgaia]
+a:1463 [in gaia_hydras.nfwfgaia]
+a:1465 [in gaia_hydras.nfwfgaia]
+a:1468 [in gaia_hydras.nfwfgaia]
+A:147 [in hydras.Ackermann.fol]
+A:147 [in hydras.Schutte.Ordering_Functions]
+a:1470 [in gaia_hydras.nfwfgaia]
+A:148 [in additions.Euclidean_Chains]
+a:148 [in additions.fib]
+a:149 [in hydras.rpo.more_list]
+a:149 [in hydras.Ackermann.LNN]
+A:149 [in additions.Euclidean_Chains]
+A:149 [in hydras.Ackermann.fol]
+a:1492 [in gaia_hydras.nfwfgaia]
+A:15 [in hydras.Prelude.DecPreOrder_Instances]
+A:15 [in hydras.rpo.more_list]
+a:15 [in hydras.OrdinalNotations.ON_O]
+a:15 [in hydras.Epsilon0.Hessenberg]
+A:15 [in hydras.Prelude.DecPreOrder]
+a:15 [in gaia_hydras.GCanon]
+a:15 [in hydras.OrdinalNotations.OmegaOmega]
+a:15 [in gaia_hydras.T1Bridge]
+a:15 [in hydras.Schutte.Schutte_basics]
+a:15 [in hydras.Prelude.MoreDecidable]
+a:15 [in hydras.Schutte.Ordering_Functions]
+A:15 [in hydras.rpo.closure]
+A:15 [in hydras.MoreAck.FolExamples]
+a:15 [in gaia_hydras.GHessenberg]
+a:150 [in gaia_hydras.T1Bridge]
+a:150 [in hydras.Ackermann.primRec]
+A:150 [in additions.Euclidean_Chains]
+A:150 [in hydras.Schutte.Ordering_Functions]
+a:1502 [in gaia_hydras.nfwfgaia]
+a:1509 [in gaia_hydras.nfwfgaia]
+a:151 [in hydras.Ackermann.LNT]
+a:152 [in gaia_hydras.nfwfgaia]
+a:1521 [in gaia_hydras.nfwfgaia]
+a:1524 [in gaia_hydras.nfwfgaia]
+a:1527 [in gaia_hydras.nfwfgaia]
+A:153 [in Goedel.PRrepresentable]
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+a:1531 [in gaia_hydras.nfwfgaia]
+a:1536 [in gaia_hydras.nfwfgaia]
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+a:154 [in hydras.Ackermann.LNN]
+a:154 [in hydras.Ackermann.primRec]
+a:154 [in gaia_hydras.nfwfgaia]
+A:154 [in hydras.Hydra.Hydra_Definitions]
+a:1540 [in gaia_hydras.nfwfgaia]
+a:1543 [in gaia_hydras.nfwfgaia]
+A:155 [in additions.Euclidean_Chains]
+a:1551 [in gaia_hydras.nfwfgaia]
+a:1558 [in gaia_hydras.nfwfgaia]
+a:156 [in hydras.Ackermann.LNT]
+a:156 [in gaia_hydras.nfwfgaia]
+a:1561 [in gaia_hydras.nfwfgaia]
+a:1564 [in gaia_hydras.nfwfgaia]
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+a:1574 [in gaia_hydras.nfwfgaia]
+a:1578 [in gaia_hydras.nfwfgaia]
+a:158 [in gaia_hydras.T1Bridge]
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+a:158 [in gaia_hydras.nfwfgaia]
+a:1587 [in gaia_hydras.nfwfgaia]
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+a:1592 [in gaia_hydras.nfwfgaia]
+a:1595 [in gaia_hydras.nfwfgaia]
+a:1598 [in gaia_hydras.nfwfgaia]
+a:16 [in hydras.solutions_exercises.MinPR2]
+a:16 [in additions.Fib2]
+a:16 [in hydras.Prelude.Comparable]
+A:16 [in additions.Pow_variant]
+a:16 [in Goedel.fixPoint]
+A:16 [in additions.Pow]
+a:16 [in hydras.Prelude.Sort_spec]
+A:16 [in hydras.MoreAck.AckNotPR]
+a:16 [in hydras.Schutte.MoreEpsilonIota]
+a:16 [in hydras.OrdinalNotations.OmegaOmega]
+a:16 [in hydras.Hydra.BigBattle]
+A:16 [in hydras.Ackermann.fol]
+a:16 [in hydras.Ackermann.NNtheory]
+a:16 [in gaia_hydras.GLarge_Sets]
+a:16 [in hydras.Schutte.Well_Orders]
+a:16 [in Goedel.rosser]
+A:16 [in hydras.Ackermann.expressible]
+A:16 [in hydras.Prelude.MoreVectors]
+a:160 [in hydras.Epsilon0.T1]
+a:160 [in gaia_hydras.nfwfgaia]
+a:1601 [in gaia_hydras.nfwfgaia]
+a:161 [in hydras.Ackermann.LNT]
+a:161 [in hydras.Ackermann.fol]
+a:1612 [in gaia_hydras.nfwfgaia]
+a:162 [in hydras.Epsilon0.T1]
+a:162 [in hydras.Ackermann.LNN]
+a:162 [in gaia_hydras.T1Bridge]
+A:162 [in additions.Euclidean_Chains]
+a:162 [in hydras.rpo.rpo]
+a:1621 [in gaia_hydras.nfwfgaia]
+a:1629 [in gaia_hydras.nfwfgaia]
+A:163 [in hydras.rpo.more_list]
+a:1635 [in gaia_hydras.nfwfgaia]
+a:164 [in gaia_hydras.T1Bridge]
+a:164 [in hydras.Ackermann.primRec]
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+A:165 [in hydras.Prelude.MoreLists]
+a:165 [in hydras.Gamma0.Gamma0]
+a:1650 [in gaia_hydras.nfwfgaia]
+a:166 [in gaia_hydras.T1Bridge]
+a:166 [in hydras.Ackermann.primRec]
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+a:1660 [in gaia_hydras.nfwfgaia]
+a:167 [in hydras.Epsilon0.T1]
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+a:1671 [in gaia_hydras.nfwfgaia]
+a:1674 [in gaia_hydras.nfwfgaia]
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+a:168 [in hydras.Ackermann.primRec]
+A:168 [in additions.Euclidean_Chains]
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+a:169 [in hydras.OrdinalNotations.ON_Generic]
+a:169 [in hydras.Gamma0.Gamma0]
+A:17 [in hydras.Prelude.Iterates]
+a:17 [in gaia_hydras.GPaths]
+a:17 [in hydras.Ackermann.folProp]
+a:17 [in gaia_hydras.GCanon]
+A:17 [in hydras.Schutte.MoreEpsilonIota]
+a:17 [in hydras.OrdinalNotations.OmegaOmega]
+A:17 [in additions.Euclidean_Chains]
+a:17 [in hydras.Schutte.PartialFun]
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+a:17 [in hydras.solutions_exercises.T1_ltNotWf]
+A:17 [in hydras.rpo.rpo]
+a:170 [in hydras.Epsilon0.T1]
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+a:170 [in hydras.Ackermann.model]
+a:170 [in gaia_hydras.nfwfgaia]
+A:171 [in hydras.Epsilon0.Large_Sets]
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+a:1718 [in gaia_hydras.nfwfgaia]
+a:1719 [in gaia_hydras.nfwfgaia]
+A:172 [in hydras.Prelude.Iterates]
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+a:1720 [in gaia_hydras.nfwfgaia]
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+a:1722 [in gaia_hydras.nfwfgaia]
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+a:1740 [in gaia_hydras.nfwfgaia]
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+a:1752 [in gaia_hydras.nfwfgaia]
+a:176 [in hydras.Ackermann.primRec]
+a:176 [in hydras.Ackermann.cPair]
+A:176 [in hydras.Schutte.Ordering_Functions]
+a:176 [in gaia_hydras.nfwfgaia]
+a:1768 [in gaia_hydras.nfwfgaia]
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+a:1796 [in gaia_hydras.nfwfgaia]
+a:1798 [in gaia_hydras.nfwfgaia]
+a:18 [in Goedel.PRrepresentable]
+a:18 [in hydras.Prelude.Comparable]
+a:18 [in hydras.OrdinalNotations.ON_O]
+a:18 [in hydras.solutions_exercises.MorePRExamples]
+A:18 [in additions.Monoid_def]
+a:18 [in hydras.Hydra.BigBattle]
+A:18 [in hydras.Ackermann.fol]
+A:18 [in gaia_hydras.GLarge_Sets]
+A:18 [in additions.Addition_Chains]
+A:18 [in hydras.Schutte.Ordering_Functions]
+A:18 [in Goedel.rosser]
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+a:18 [in hydras.solutions_exercises.T1_ltNotWf]
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+a:1808 [in gaia_hydras.nfwfgaia]
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+a:1835 [in gaia_hydras.nfwfgaia]
+a:1837 [in gaia_hydras.nfwfgaia]
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+a:1843 [in gaia_hydras.nfwfgaia]
+a:185 [in additions.Addition_Chains]
+a:185 [in hydras.Schutte.Ordering_Functions]
+a:1856 [in gaia_hydras.nfwfgaia]
+a:186 [in hydras.Epsilon0.T1]
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+a:186 [in hydras.Ackermann.model]
+a:186 [in hydras.Ackermann.primRec]
+a:1860 [in gaia_hydras.nfwfgaia]
+a:1865 [in gaia_hydras.nfwfgaia]
+a:1869 [in gaia_hydras.nfwfgaia]
+A:187 [in hydras.Prelude.MoreLists]
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+a:1877 [in gaia_hydras.nfwfgaia]
+a:188 [in additions.Euclidean_Chains]
+a:1886 [in gaia_hydras.nfwfgaia]
+a:189 [in hydras.Ackermann.folProp]
+A:19 [in hydras.Ackermann.folProof]
+a:19 [in Goedel.fixPoint]
+a:19 [in gaia_hydras.GCanon]
+a:19 [in hydras.Ackermann.cPair]
+a:19 [in gaia_hydras.GHydra]
+A:19 [in hydras.solutions_exercises.T1_ltNotWf]
+a:19 [in gaia_hydras.GHessenberg]
+a:19 [in hydras.Ackermann.expressible]
+a:191 [in hydras.Ackermann.primRec]
+a:192 [in hydras.Ackermann.model]
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+a:193 [in additions.Addition_Chains]
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+a:196 [in hydras.Ackermann.model]
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+A:198 [in hydras.OrdinalNotations.ON_Generic]
+a:2 [in hydras.Hydra.O2H]
+a:2 [in gaia_hydras.GF_alpha]
+a:2 [in gaia_hydras.GPaths]
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+a:2 [in hydras.Ackermann.ListExt]
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+a:20 [in hydras.Prelude.Comparable]
+a:20 [in hydras.Prelude.Sort_spec]
+A:20 [in hydras.Prelude.DecPreOrder]
+a:20 [in hydras.Ackermann.folProp]
+A:20 [in hydras.MoreAck.AckNotPR]
+A:20 [in hydras.Schutte.MoreEpsilonIota]
+a:20 [in hydras.solutions_exercises.MultisetWf]
+a:20 [in hydras.Hydra.BigBattle]
+A:20 [in hydras.Ackermann.fol]
+a:20 [in additions.Monoid_instances]
+a:20 [in hydras.Schutte.Well_Orders]
+a:20 [in hydras.Ackermann.NN]
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+A:200 [in hydras.Ackermann.model]
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+a:2021 [in gaia_hydras.nfwfgaia]
+a:2025 [in gaia_hydras.nfwfgaia]
+a:2030 [in gaia_hydras.nfwfgaia]
+a:2037 [in gaia_hydras.nfwfgaia]
+a:2047 [in gaia_hydras.nfwfgaia]
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+a:2050 [in gaia_hydras.nfwfgaia]
+a:2053 [in gaia_hydras.nfwfgaia]
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+a:2070 [in gaia_hydras.nfwfgaia]
+a:2073 [in gaia_hydras.nfwfgaia]
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+a:209 [in hydras.OrdinalNotations.ON_Generic]
+a:21 [in hydras.Schutte.Correctness_E0]
+a:21 [in additions.Fib2]
+A:21 [in hydras.Prelude.MoreLists]
+a:21 [in hydras.OrdinalNotations.ON_O]
+a:21 [in gaia_hydras.GPaths]
+a:21 [in hydras.Epsilon0.Hessenberg]
+A:21 [in additions.Monoid_def]
+a:21 [in hydras.Ackermann.cPair]
+A:21 [in additions.Euclidean_Chains]
+a:21 [in hydras.Prelude.MoreDecidable]
+a:21 [in hydras.Ackermann.NN]
+a:21 [in gaia_hydras.GL_alpha]
+a:21 [in gaia_hydras.GHessenberg]
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+a:210 [in gaia_hydras.nfwfgaia]
+a:2109 [in gaia_hydras.nfwfgaia]
+a:211 [in hydras.Epsilon0.T1]
+a:211 [in hydras.OrdinalNotations.ON_Generic]
+a:212 [in hydras.Epsilon0.T1]
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+a:213 [in hydras.Ackermann.primRec]
+a:213 [in hydras.Ackermann.cPair]
+a:213 [in hydras.Ackermann.fol]
+a:213 [in hydras.OrdinalNotations.ON_Generic]
+a:213 [in gaia_hydras.nfwfgaia]
+a:214 [in hydras.Epsilon0.T1]
+a:215 [in hydras.OrdinalNotations.ON_Generic]
+a:215 [in hydras.Ackermann.codeList]
+a:215 [in hydras.Gamma0.Gamma0]
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+a:216 [in hydras.Ackermann.fol]
+a:216 [in hydras.OrdinalNotations.ON_Generic]
+a:216 [in gaia_hydras.nfwfgaia]
+a:217 [in hydras.Gamma0.Gamma0]
+a:218 [in hydras.Epsilon0.T1]
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+a:218 [in hydras.Ackermann.model]
+a:218 [in gaia_hydras.nfwfgaia]
+a:219 [in hydras.Epsilon0.T1]
+a:219 [in hydras.Ackermann.primRec]
+A:22 [in hydras.Ackermann.folProof]
+a:22 [in hydras.Epsilon0.T1]
+a:22 [in Goedel.PRrepresentable]
+a:22 [in hydras.Prelude.Comparable]
+a:22 [in hydras.OrdinalNotations.ON_O]
+A:22 [in Goedel.fixPoint]
+a:22 [in hydras.OrdinalNotations.OmegaOmega]
+a:22 [in additions.fib]
+A:22 [in hydras.Ackermann.expressible]
+a:22 [in hydras.Prelude.MoreVectors]
+A:220 [in hydras.rpo.more_list]
+a:220 [in hydras.Ackermann.model]
+a:221 [in hydras.Ackermann.folProp]
+a:221 [in gaia_hydras.nfwfgaia]
+a:222 [in hydras.Ackermann.primRec]
+A:223 [in additions.Addition_Chains]
+a:223 [in hydras.Gamma0.Gamma0]
+A:224 [in additions.Euclidean_Chains]
+a:225 [in hydras.Epsilon0.T1]
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+a:226 [in additions.Euclidean_Chains]
+a:227 [in hydras.Gamma0.Gamma0]
+a:227 [in gaia_hydras.nfwfgaia]
+a:228 [in hydras.rpo.more_list]
+a:228 [in hydras.Ackermann.folProp]
+a:228 [in additions.Addition_Chains]
+a:229 [in hydras.Epsilon0.T1]
+a:229 [in hydras.Ackermann.primRec]
+a:23 [in hydras.Ackermann.extEqualNat]
+a:23 [in hydras.rpo.more_list]
+a:23 [in hydras.OrdinalNotations.ON_O]
+A:23 [in additions.Pow_variant]
+A:23 [in additions.Pow]
+a:23 [in hydras.Prelude.Sort_spec]
+a:23 [in hydras.Prelude.DecPreOrder]
+A:23 [in hydras.Schutte.MoreEpsilonIota]
+a:23 [in hydras.Ackermann.cPair]
+A:23 [in hydras.Ackermann.fol]
+a:23 [in hydras.Schutte.Well_Orders]
+a:23 [in hydras.Gamma0.Gamma0]
+a:23 [in hydras.rpo.closure]
+a:23 [in gaia_hydras.GHessenberg]
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+A:230 [in additions.AM]
+a:230 [in hydras.rpo.rpo]
+a:231 [in hydras.rpo.more_list]
+A:231 [in additions.Addition_Chains]
+a:231 [in hydras.Gamma0.Gamma0]
+a:232 [in hydras.Ackermann.primRec]
+A:233 [in hydras.Ackermann.checkPrf]
+a:233 [in gaia_hydras.nfwfgaia]
+a:234 [in hydras.Epsilon0.T1]
+a:234 [in hydras.rpo.more_list]
+a:235 [in hydras.Ackermann.primRec]
+A:235 [in additions.Euclidean_Chains]
+a:235 [in gaia_hydras.nfwfgaia]
+a:236 [in hydras.Ackermann.fol]
+a:236 [in additions.Addition_Chains]
+a:237 [in hydras.Epsilon0.T1]
+A:237 [in additions.AM]
+a:237 [in additions.Euclidean_Chains]
+a:237 [in gaia_hydras.nfwfgaia]
+A:238 [in hydras.rpo.more_list]
+a:238 [in hydras.Ackermann.primRec]
+a:239 [in hydras.Ackermann.fol]
+a:24 [in hydras.Prelude.More_Arith]
+a:24 [in hydras.OrdinalNotations.ON_O]
+a:24 [in gaia_hydras.GPaths]
+a:24 [in hydras.Epsilon0.F_omega]
+a:24 [in hydras.OrdinalNotations.OmegaOmega]
+A:24 [in hydras.Ackermann.fol]
+a:24 [in hydras.Schutte.PartialFun]
+A:24 [in additions.Addition_Chains]
+a:24 [in gaia_hydras.GL_alpha]
+a:24 [in gaia_hydras.nfwfgaia]
+A:240 [in additions.Addition_Chains]
+a:240 [in hydras.rpo.rpo]
+a:241 [in hydras.Ackermann.primRec]
+A:241 [in additions.Euclidean_Chains]
+a:243 [in hydras.Ackermann.cPair]
+a:243 [in additions.Euclidean_Chains]
+a:244 [in additions.Addition_Chains]
+a:245 [in hydras.Ackermann.primRec]
+A:247 [in hydras.rpo.more_list]
+a:247 [in additions.Addition_Chains]
+a:248 [in hydras.Epsilon0.T1]
+a:248 [in hydras.Ackermann.primRec]
+a:249 [in hydras.Ackermann.subAll]
+A:25 [in hydras.Ackermann.folProof]
+a:25 [in hydras.Ackermann.extEqualNat]
+A:25 [in hydras.rpo.more_list]
+a:25 [in hydras.Prelude.Comparable]
+A:25 [in hydras.Prelude.DecPreOrder]
+a:25 [in hydras.Ackermann.PA]
+a:25 [in hydras.rpo.closure]
+a:250 [in additions.Addition_Chains]
+a:251 [in hydras.Ackermann.primRec]
+a:252 [in hydras.Epsilon0.T1]
+a:252 [in hydras.Ackermann.folProp]
+a:253 [in hydras.Epsilon0.T1]
+a:253 [in gaia_hydras.nfwfgaia]
+a:254 [in hydras.Ackermann.primRec]
+A:254 [in additions.Addition_Chains]
+a:255 [in hydras.rpo.more_list]
+a:255 [in gaia_hydras.nfwfgaia]
+a:256 [in hydras.rpo.rpo]
+a:257 [in hydras.Epsilon0.T1]
+a:257 [in hydras.Ackermann.primRec]
+A:259 [in hydras.rpo.more_list]
+a:259 [in hydras.Ackermann.folProp]
+a:26 [in Goedel.PRrepresentable]
+A:26 [in hydras.Prelude.DecPreOrder_Instances]
+A:26 [in hydras.Prelude.MoreLists]
+a:26 [in additions.Compatibility]
+a:26 [in hydras.Prelude.Comparable]
+A:26 [in Goedel.fixPoint]
+a:26 [in hydras.OrdinalNotations.OmegaOmega]
+a:26 [in hydras.Ackermann.PA]
+A:26 [in additions.Euclidean_Chains]
+a:26 [in hydras.Schutte.PartialFun]
+a:26 [in gaia_hydras.GHessenberg]
+a:26 [in hydras.MoreAck.PrimRecExamples]
+A:26 [in hydras.Ackermann.expressible]
+a:260 [in hydras.Epsilon0.T1]
+a:260 [in gaia_hydras.nfwfgaia]
+A:261 [in Goedel.PRrepresentable]
+a:261 [in hydras.Ackermann.primRec]
+a:263 [in hydras.Epsilon0.T1]
+a:263 [in gaia_hydras.nfwfgaia]
+A:264 [in additions.Addition_Chains]
+A:265 [in Goedel.PRrepresentable]
+A:265 [in hydras.Ackermann.checkPrf]
+a:265 [in hydras.Ackermann.primRec]
+a:266 [in hydras.Epsilon0.T1]
+a:266 [in hydras.Ackermann.cPair]
+a:267 [in hydras.rpo.more_list]
+a:268 [in hydras.Ackermann.folProp]
+a:268 [in hydras.Ackermann.cPair]
+a:268 [in hydras.Ackermann.fol]
+a:268 [in hydras.rpo.rpo]
+a:269 [in Goedel.PRrepresentable]
+a:269 [in hydras.Ackermann.primRec]
+A:27 [in hydras.Ackermann.folProof]
+a:27 [in hydras.Schutte.Correctness_E0]
+a:27 [in hydras.Prelude.More_Arith]
+a:27 [in hydras.Ackermann.extEqualNat]
+a:27 [in hydras.Epsilon0.Epsilon0rpo]
+a:27 [in hydras.Epsilon0.Hessenberg]
+a:27 [in gaia_hydras.GCanon]
+a:27 [in hydras.solutions_exercises.MultisetWf]
+a:27 [in Goedel.rosserPA]
+a:27 [in hydras.OrdinalNotations.ON_Generic]
+a:27 [in gaia_hydras.GL_alpha]
+a:27 [in hydras.Ackermann.Languages]
+A:27 [in hydras.Prelude.MoreVectors]
+a:270 [in hydras.Epsilon0.T1]
+a:270 [in hydras.Ackermann.cPair]
+a:270 [in hydras.rpo.rpo]
+a:271 [in additions.Addition_Chains]
+a:272 [in hydras.Epsilon0.T1]
+A:272 [in Goedel.PRrepresentable]
+a:272 [in hydras.Ackermann.cPair]
+a:272 [in hydras.Ackermann.fol]
+a:272 [in hydras.Gamma0.Gamma0]
+a:273 [in hydras.Ackermann.primRec]
+a:274 [in hydras.Epsilon0.T1]
+a:274 [in hydras.Ackermann.cPair]
+a:275 [in hydras.Ackermann.folProp]
+a:276 [in hydras.Epsilon0.T1]
+a:276 [in hydras.Ackermann.cPair]
+a:277 [in hydras.Ackermann.primRec]
+a:277 [in hydras.Ackermann.cPair]
+a:277 [in hydras.Gamma0.Gamma0]
+a:278 [in hydras.Ackermann.cPair]
+a:278 [in gaia_hydras.nfwfgaia]
+a:279 [in hydras.Ackermann.cPair]
+A:28 [in Goedel.PRrepresentable]
+A:28 [in hydras.Prelude.DecPreOrder_Instances]
+a:28 [in additions.Compatibility]
+a:28 [in hydras.Prelude.Comparable]
+a:28 [in gaia_hydras.GPaths]
+a:28 [in hydras.Prelude.DecPreOrder]
+a:28 [in hydras.OrdinalNotations.OmegaOmega]
+a:28 [in hydras.Schutte.PartialFun]
+a:28 [in gaia_hydras.GL_alpha]
+a:28 [in gaia_hydras.GHessenberg]
+a:280 [in hydras.Ackermann.cPair]
+a:280 [in hydras.Ackermann.fol]
+a:281 [in hydras.Ackermann.cPair]
+a:282 [in hydras.Epsilon0.T1]
+a:282 [in hydras.Ackermann.primRec]
+a:283 [in hydras.Gamma0.Gamma0]
+a:283 [in gaia_hydras.nfwfgaia]
+a:284 [in gaia_hydras.nfwfgaia]
+A:285 [in Goedel.PRrepresentable]
+a:286 [in hydras.Ackermann.primRec]
+a:287 [in hydras.Epsilon0.T1]
+a:287 [in hydras.Gamma0.Gamma0]
+A:29 [in hydras.Ackermann.folProof]
+a:29 [in hydras.Prelude.Comparable]
+A:29 [in additions.Pow_variant]
+A:29 [in additions.Pow]
+a:29 [in gaia_hydras.GCanon]
+A:29 [in Goedel.rosserPA]
+a:29 [in hydras.Hydra.BigBattle]
+A:29 [in hydras.OrdinalNotations.ON_Generic]
+a:29 [in hydras.Ackermann.NN]
+a:29 [in hydras.Gamma0.Gamma0]
+a:29 [in hydras.MoreAck.PrimRecExamples]
+a:290 [in hydras.Ackermann.primRec]
+A:291 [in Goedel.PRrepresentable]
+a:291 [in hydras.Gamma0.Gamma0]
+a:291 [in gaia_hydras.nfwfgaia]
+a:295 [in hydras.Ackermann.primRec]
+a:295 [in hydras.Gamma0.Gamma0]
+a:296 [in gaia_hydras.nfwfgaia]
+a:299 [in hydras.Ackermann.primRec]
+a:299 [in hydras.Gamma0.Gamma0]
+a:3 [in additions.Compatibility]
+a:3 [in hydras.Ackermann.folLogic2]
+a:3 [in hydras.OrdinalNotations.OmegaOmega]
+a:3 [in hydras.OrdinalNotations.Example_3PlusOmega]
+a:3 [in hydras.Ackermann.prLogic]
+A:3 [in additions.Euclidean_Chains]
+a:3 [in hydras.Ackermann.NNtheory]
+a:3 [in gaia_hydras.GLarge_Sets]
+a:3 [in hydras.Schutte.Well_Orders]
+a:3 [in hydras.Ackermann.folLogic3]
+a:3 [in hydras.Ackermann.NN]
+a:30 [in hydras.Epsilon0.T1]
+a:30 [in hydras.Ackermann.extEqualNat]
+a:30 [in additions.Compatibility]
+a:30 [in hydras.rpo.more_list]
+a:30 [in gaia_hydras.GPaths]
+A:30 [in hydras.Prelude.Sort_spec]
+A:30 [in hydras.Prelude.DecPreOrder]
+a:30 [in hydras.OrdinalNotations.OmegaOmega]
+a:30 [in hydras.Ackermann.primRec]
+a:30 [in hydras.Ackermann.NN]
+a:30 [in gaia_hydras.GHessenberg]
+a:303 [in hydras.Ackermann.primRec]
+a:306 [in Goedel.PRrepresentable]
+a:306 [in hydras.Gamma0.Gamma0]
+a:307 [in hydras.Ackermann.primRec]
+a:31 [in Goedel.PRrepresentable]
+A:31 [in hydras.rpo.more_list]
+a:31 [in hydras.Ackermann.folLogic3]
+a:310 [in hydras.Epsilon0.T1]
+a:310 [in Goedel.PRrepresentable]
+a:311 [in hydras.Ackermann.primRec]
+a:313 [in Goedel.PRrepresentable]
+a:315 [in hydras.Ackermann.primRec]
+a:316 [in Goedel.PRrepresentable]
+a:318 [in Goedel.PRrepresentable]
+A:318 [in hydras.Ackermann.checkPrf]
+a:318 [in hydras.Ackermann.fol]
+a:319 [in hydras.Ackermann.primRec]
+A:32 [in hydras.Ackermann.folProof]
+A:32 [in Goedel.PRrepresentable]
+a:32 [in additions.Compatibility]
+a:32 [in hydras.Prelude.Restriction]
+a:32 [in hydras.Prelude.Comparable]
+a:32 [in gaia_hydras.GPaths]
+a:32 [in hydras.Epsilon0.Hessenberg]
+A:32 [in hydras.solutions_exercises.MultisetWf]
+a:32 [in hydras.OrdinalNotations.OmegaOmega]
+a:32 [in hydras.Schutte.PartialFun]
+A:32 [in Goedel.codeSysPrf]
+A:32 [in hydras.Schutte.Ordering_Functions]
+a:32 [in hydras.Ackermann.Languages]
+A:32 [in hydras.Ackermann.expressible]
+a:322 [in hydras.Ackermann.fol]
+a:324 [in hydras.Ackermann.primRec]
+A:326 [in Goedel.PRrepresentable]
+a:327 [in gaia_hydras.nfwfgaia]
+a:328 [in hydras.Ackermann.primRec]
+a:328 [in gaia_hydras.nfwfgaia]
+a:33 [in hydras.Prelude.More_Arith]
+a:33 [in hydras.Epsilon0.Epsilon0rpo]
+a:33 [in gaia_hydras.GCanon]
+a:33 [in hydras.Ackermann.cPair]
+a:33 [in hydras.Schutte.PartialFun]
+a:33 [in gaia_hydras.GHessenberg]
+a:330 [in hydras.Ackermann.fol]
+A:331 [in Goedel.PRrepresentable]
+a:332 [in hydras.Ackermann.primRec]
+a:334 [in gaia_hydras.nfwfgaia]
+a:335 [in gaia_hydras.nfwfgaia]
+A:336 [in Goedel.PRrepresentable]
+a:336 [in hydras.Ackermann.primRec]
+a:336 [in gaia_hydras.nfwfgaia]
+a:337 [in gaia_hydras.nfwfgaia]
+a:338 [in hydras.Ackermann.cPair]
+a:338 [in gaia_hydras.nfwfgaia]
+A:339 [in additions.Euclidean_Chains]
+A:34 [in hydras.Ackermann.folProof]
+a:34 [in additions.Compatibility]
+A:34 [in additions.FirstSteps]
+a:34 [in hydras.Prelude.Comparable]
+a:34 [in hydras.OrdinalNotations.OmegaOmega]
+a:34 [in hydras.Ackermann.PA]
+a:34 [in hydras.Schutte.Well_Orders]
+a:34 [in hydras.Schutte.Ordering_Functions]
+A:34 [in hydras.Ackermann.expressible]
+a:34 [in hydras.Prelude.MoreVectors]
+a:340 [in hydras.Ackermann.primRec]
+a:340 [in hydras.Ackermann.cPair]
+A:341 [in hydras.Ackermann.subAll]
+a:344 [in hydras.Epsilon0.T1]
+A:346 [in hydras.Ackermann.subAll]
+a:346 [in hydras.Ackermann.cPair]
+a:349 [in hydras.Ackermann.cPair]
+a:35 [in Goedel.PRrepresentable]
+A:35 [in additions.AM]
+a:35 [in additions.Compatibility]
+a:35 [in hydras.Prelude.Comparable]
+a:35 [in additions.Pow_variant]
+a:35 [in additions.Pow]
+A:35 [in hydras.Prelude.DecPreOrder]
+a:35 [in hydras.Ackermann.folProp]
+A:35 [in hydras.solutions_exercises.MultisetWf]
+a:35 [in hydras.Ackermann.PA]
+A:35 [in additions.Euclidean_Chains]
+a:35 [in hydras.Schutte.Schutte_basics]
+a:35 [in Goedel.codeSysPrf]
+A:35 [in additions.Addition_Chains]
+A:35 [in hydras.MoreAck.FolExamples]
+a:35 [in hydras.Ackermann.expressible]
+a:351 [in Goedel.PRrepresentable]
+A:351 [in hydras.Ackermann.subAll]
+a:352 [in hydras.Ackermann.cPair]
+a:353 [in gaia_hydras.nfwfgaia]
+A:354 [in hydras.Ackermann.checkPrf]
+a:355 [in hydras.Epsilon0.T1]
+a:355 [in Goedel.PRrepresentable]
+a:355 [in hydras.Ackermann.cPair]
+a:355 [in gaia_hydras.nfwfgaia]
+A:357 [in hydras.Ackermann.subAll]
+a:357 [in gaia_hydras.nfwfgaia]
+a:359 [in Goedel.PRrepresentable]
+a:359 [in gaia_hydras.nfwfgaia]
+A:36 [in hydras.rpo.more_list]
+a:36 [in hydras.OrdinalNotations.OmegaOmega]
+A:36 [in additions.Monoid_def]
+a:36 [in hydras.Hydra.BigBattle]
+a:36 [in additions.Addition_Chains]
+a:36 [in hydras.rpo.list_permut]
+A:36 [in hydras.Schutte.Ordering_Functions]
+a:36 [in gaia_hydras.GHessenberg]
+a:361 [in hydras.Epsilon0.T1]
+a:361 [in gaia_hydras.nfwfgaia]
+a:362 [in hydras.Ackermann.cPair]
+a:362 [in gaia_hydras.nfwfgaia]
+a:363 [in Goedel.PRrepresentable]
+a:364 [in hydras.Ackermann.fol]
+a:365 [in hydras.Ackermann.folProp]
+a:365 [in hydras.Ackermann.cPair]
+a:368 [in hydras.Ackermann.cPair]
+a:368 [in hydras.Ackermann.fol]
+A:37 [in hydras.Ackermann.folProof]
+a:37 [in hydras.OrdinalNotations.OmegaOmega]
+a:37 [in hydras.Ackermann.cPair]
+a:37 [in hydras.Ackermann.wellFormed]
+a:37 [in gaia_hydras.onType]
+a:37 [in hydras.Schutte.Well_Orders]
+a:37 [in Goedel.codeSysPrf]
+A:37 [in hydras.Schutte.Addition]
+A:370 [in hydras.Ackermann.subAll]
+a:371 [in hydras.Ackermann.cPair]
+a:372 [in hydras.Ackermann.folProp]
+a:375 [in hydras.Epsilon0.T1]
+a:375 [in gaia_hydras.nfwfgaia]
+a:376 [in hydras.Ackermann.fol]
+a:376 [in gaia_hydras.nfwfgaia]
+a:377 [in gaia_hydras.nfwfgaia]
+a:38 [in additions.Compatibility]
+a:38 [in hydras.Prelude.Comparable]
+a:38 [in hydras.Epsilon0.Hessenberg]
+a:38 [in hydras.Prelude.DecPreOrder]
+a:38 [in hydras.Ackermann.NN]
+a:38 [in hydras.Schutte.Ordering_Functions]
+A:38 [in hydras.MoreAck.FolExamples]
+A:38 [in hydras.Ackermann.expressible]
+a:380 [in gaia_hydras.nfwfgaia]
+a:381 [in hydras.Epsilon0.T1]
+a:381 [in hydras.Ackermann.folProp]
+a:381 [in gaia_hydras.nfwfgaia]
+a:385 [in hydras.Epsilon0.T1]
+a:385 [in Goedel.PRrepresentable]
+a:388 [in hydras.Ackermann.folProp]
+a:389 [in Goedel.PRrepresentable]
+A:39 [in hydras.Ackermann.folProof]
+a:39 [in Goedel.PRrepresentable]
+a:39 [in hydras.Epsilon0.Epsilon0rpo]
+A:39 [in additions.Pow_variant]
+a:39 [in gaia_hydras.GPaths]
+A:39 [in additions.Pow]
+A:39 [in hydras.Prelude.Sort_spec]
+A:39 [in hydras.Prelude.DecPreOrder]
+a:39 [in hydras.Ackermann.cPair]
+a:39 [in hydras.Ackermann.wellFormed]
+a:39 [in hydras.Schutte.Schutte_basics]
+a:39 [in gaia_hydras.onType]
+a:390 [in hydras.Epsilon0.T1]
+a:393 [in Goedel.PRrepresentable]
+a:393 [in hydras.Ackermann.checkPrf]
+a:393 [in gaia_hydras.nfwfgaia]
+a:396 [in gaia_hydras.nfwfgaia]
+a:397 [in Goedel.PRrepresentable]
+a:397 [in hydras.Ackermann.checkPrf]
+a:398 [in gaia_hydras.nfwfgaia]
+a:4 [in hydras.Prelude.Restriction]
+A:4 [in hydras.Prelude.Comparable]
+a:4 [in Goedel.fixPoint]
+a:4 [in hydras.Prelude.DecPreOrder]
+a:4 [in gaia_hydras.T1Bridge]
+A:4 [in additions.Monoid_def]
+a:4 [in additions.Addition_Chains]
+A:4 [in hydras.Schutte.Ordering_Functions]
+A:40 [in additions.FirstSteps]
+a:40 [in hydras.Prelude.Comparable]
+a:40 [in hydras.Schutte.PartialFun]
+a:400 [in Goedel.PRrepresentable]
+A:401 [in hydras.Epsilon0.T1]
+a:402 [in Goedel.PRrepresentable]
+a:403 [in hydras.Epsilon0.T1]
+a:405 [in Goedel.PRrepresentable]
+A:407 [in hydras.Ackermann.checkPrf]
+a:408 [in hydras.Ackermann.fol]
+a:408 [in gaia_hydras.nfwfgaia]
+a:409 [in Goedel.PRrepresentable]
+A:41 [in additions.AM]
+A:41 [in hydras.rpo.more_list]
+a:41 [in hydras.solutions_exercises.MultisetWf]
+a:41 [in gaia_hydras.T1Bridge]
+A:41 [in additions.Monoid_def]
+a:41 [in hydras.Ackermann.cPair]
+a:41 [in additions.Euclidean_Chains]
+a:41 [in hydras.Schutte.Schutte_basics]
+a:41 [in hydras.Schutte.Well_Orders]
+A:41 [in additions.Addition_Chains]
+A:41 [in hydras.MoreAck.FolExamples]
+A:41 [in hydras.Ackermann.expressible]
+A:41 [in hydras.Prelude.MoreVectors]
+a:411 [in gaia_hydras.nfwfgaia]
+a:412 [in hydras.Ackermann.fol]
+a:413 [in Goedel.PRrepresentable]
+a:414 [in hydras.Ackermann.primRec]
+a:415 [in hydras.Epsilon0.T1]
+a:415 [in gaia_hydras.nfwfgaia]
+a:416 [in hydras.Ackermann.primRec]
+a:417 [in Goedel.PRrepresentable]
+a:418 [in hydras.Ackermann.primRec]
+a:42 [in hydras.Epsilon0.Epsilon0rpo]
+a:42 [in hydras.Prelude.Comparable]
+a:42 [in hydras.Prelude.DecPreOrder]
+a:42 [in additions.More_on_positive]
+a:42 [in gaia_hydras.T1Bridge]
+A:42 [in hydras.Prelude.MoreOrders]
+A:42 [in additions.Euclidean_Chains]
+a:42 [in hydras.Schutte.PartialFun]
+a:42 [in hydras.Ackermann.folLogic3]
+a:420 [in Goedel.PRrepresentable]
+a:420 [in hydras.Ackermann.primRec]
+a:421 [in hydras.Epsilon0.T1]
+a:421 [in hydras.Ackermann.fol]
+a:422 [in hydras.Epsilon0.T1]
+a:422 [in gaia_hydras.nfwfgaia]
+a:424 [in Goedel.PRrepresentable]
+a:424 [in hydras.Ackermann.primRec]
+a:425 [in hydras.Epsilon0.T1]
+a:428 [in hydras.Ackermann.fol]
+a:429 [in Goedel.PRrepresentable]
+a:429 [in hydras.Ackermann.primRec]
+a:429 [in hydras.Gamma0.Gamma0]
+a:43 [in hydras.Epsilon0.T1]
+a:43 [in Goedel.PRrepresentable]
+a:43 [in hydras.Prelude.DecPreOrder_Instances]
+a:43 [in hydras.rpo.more_list]
+a:43 [in hydras.Ackermann.LNN2LNT]
+a:43 [in hydras.Prelude.Comparable]
+a:43 [in gaia_hydras.T1Bridge]
+a:43 [in hydras.Ackermann.PA]
+a:43 [in hydras.Schutte.PartialFun]
+a:43 [in hydras.rpo.dickson]
+a:43 [in hydras.rpo.list_permut]
+A:43 [in hydras.Schutte.Ordering_Functions]
+a:430 [in hydras.Ackermann.fol]
+a:431 [in hydras.Epsilon0.T1]
+a:432 [in Goedel.PRrepresentable]
+a:432 [in gaia_hydras.nfwfgaia]
+a:435 [in Goedel.PRrepresentable]
+a:436 [in hydras.Epsilon0.T1]
+a:436 [in hydras.Ackermann.primRec]
+a:437 [in gaia_hydras.nfwfgaia]
+a:438 [in Goedel.PRrepresentable]
+A:438 [in hydras.Ackermann.folProp]
+a:44 [in Goedel.PRrepresentable]
+A:44 [in hydras.Prelude.MoreLists]
+a:44 [in hydras.Prelude.Comparable]
+a:44 [in hydras.Epsilon0.Hessenberg]
+A:44 [in hydras.Prelude.DecPreOrder]
+a:44 [in hydras.solutions_exercises.MultisetWf]
+a:44 [in hydras.Ackermann.PA]
+a:44 [in hydras.Prelude.MoreOrders]
+a:44 [in hydras.Ackermann.primRec]
+a:441 [in gaia_hydras.nfwfgaia]
+a:442 [in hydras.Epsilon0.T1]
+a:443 [in gaia_hydras.nfwfgaia]
+a:444 [in hydras.Ackermann.fol]
+a:445 [in hydras.Ackermann.checkPrf]
+a:445 [in hydras.Ackermann.primRec]
+a:445 [in hydras.Gamma0.Gamma0]
+a:445 [in gaia_hydras.nfwfgaia]
+a:446 [in hydras.Epsilon0.T1]
+a:447 [in gaia_hydras.nfwfgaia]
+a:448 [in hydras.Epsilon0.T1]
+a:449 [in hydras.Ackermann.checkPrf]
+a:449 [in gaia_hydras.nfwfgaia]
+a:45 [in hydras.Epsilon0.Epsilon0rpo]
+a:45 [in gaia_hydras.T1Bridge]
+a:45 [in hydras.Ackermann.cPair]
+a:45 [in Goedel.codeSysPrf]
+A:45 [in hydras.Schutte.Ordering_Functions]
+A:45 [in hydras.Ackermann.expressible]
+a:450 [in hydras.Ackermann.primRec]
+a:450 [in hydras.Gamma0.Gamma0]
+a:450 [in gaia_hydras.nfwfgaia]
+a:451 [in hydras.Epsilon0.T1]
+a:451 [in hydras.Gamma0.Gamma0]
+a:452 [in hydras.Gamma0.Gamma0]
+a:452 [in gaia_hydras.nfwfgaia]
+a:453 [in hydras.Epsilon0.T1]
+A:453 [in hydras.Ackermann.checkPrf]
+a:457 [in hydras.Gamma0.Gamma0]
+a:457 [in gaia_hydras.nfwfgaia]
+a:458 [in hydras.Epsilon0.T1]
+a:458 [in hydras.Ackermann.primRec]
+a:459 [in gaia_hydras.nfwfgaia]
+a:46 [in hydras.Epsilon0.T1]
+A:46 [in hydras.rpo.more_list]
+a:46 [in hydras.Prelude.Comparable]
+A:46 [in hydras.Prelude.Sort_spec]
+a:46 [in additions.More_on_positive]
+A:46 [in hydras.Prelude.MoreOrders]
+A:46 [in additions.Monoid_def]
+a:46 [in hydras.Schutte.PartialFun]
+A:46 [in hydras.Schutte.Schutte_basics]
+a:46 [in hydras.Schutte.Well_Orders]
+A:46 [in additions.Addition_Chains]
+a:461 [in hydras.Epsilon0.T1]
+a:462 [in gaia_hydras.nfwfgaia]
+a:463 [in hydras.Ackermann.primRec]
+a:464 [in hydras.Epsilon0.T1]
+a:464 [in gaia_hydras.nfwfgaia]
+a:466 [in gaia_hydras.nfwfgaia]
+a:467 [in hydras.Gamma0.Gamma0]
+a:468 [in hydras.Epsilon0.T1]
+a:468 [in hydras.Ackermann.primRec]
+a:468 [in gaia_hydras.nfwfgaia]
+a:47 [in Goedel.PRrepresentable]
+a:47 [in hydras.Prelude.DecPreOrder_Instances]
+a:47 [in hydras.Prelude.More_Arith]
+a:47 [in hydras.Prelude.DecPreOrder]
+A:47 [in additions.Euclidean_Chains]
+a:47 [in Goedel.codeSysPrf]
+a:47 [in hydras.Schutte.Ordering_Functions]
+a:470 [in hydras.Epsilon0.T1]
+a:472 [in hydras.Ackermann.primRec]
+a:473 [in gaia_hydras.nfwfgaia]
+a:476 [in hydras.Epsilon0.T1]
+a:478 [in hydras.Epsilon0.T1]
+a:48 [in hydras.Epsilon0.T1]
+a:48 [in hydras.Epsilon0.Epsilon0rpo]
+a:48 [in hydras.Prelude.Comparable]
+A:48 [in additions.Pow_variant]
+A:48 [in additions.Pow]
+a:48 [in hydras.Epsilon0.Hessenberg]
+a:48 [in gaia_hydras.GCanon]
+a:48 [in hydras.Ackermann.code]
+a:48 [in hydras.Hydra.BigBattle]
+A:48 [in hydras.OrdinalNotations.ON_Generic]
+A:48 [in hydras.Prelude.MoreVectors]
+a:480 [in hydras.Ackermann.primRec]
+a:482 [in hydras.Gamma0.Gamma0]
+a:485 [in hydras.Ackermann.primRec]
+a:486 [in hydras.Ackermann.checkPrf]
+a:489 [in hydras.Ackermann.primRec]
+a:489 [in hydras.Gamma0.Gamma0]
+A:49 [in hydras.Prelude.DecPreOrder]
+a:49 [in hydras.solutions_exercises.MultisetWf]
+a:49 [in gaia_hydras.T1Bridge]
+a:49 [in hydras.Ackermann.cPair]
+a:49 [in hydras.Schutte.Well_Orders]
+a:49 [in hydras.Ackermann.folLogic3]
+A:49 [in hydras.Schutte.Ordering_Functions]
+a:490 [in hydras.Ackermann.checkPrf]
+a:490 [in gaia_hydras.nfwfgaia]
+a:492 [in gaia_hydras.nfwfgaia]
+a:494 [in hydras.Ackermann.primRec]
+a:494 [in gaia_hydras.nfwfgaia]
+a:496 [in hydras.Epsilon0.T1]
+a:499 [in hydras.Ackermann.primRec]
+a:5 [in hydras.Ackermann.folProof]
+a:5 [in gaia_hydras.GPaths]
+a:5 [in hydras.Schutte.Lub]
+a:5 [in hydras.Prelude.Sort_spec]
+a:5 [in gaia_hydras.GCanon]
+a:5 [in hydras.Schutte.MoreEpsilonIota]
+a:5 [in hydras.Ackermann.prLogic]
+a:5 [in hydras.Schutte.Schutte_basics]
+a:5 [in hydras.Ackermann.NNtheory]
+a:5 [in gaia_hydras.GLarge_Sets]
+a:5 [in hydras.Ackermann.ListExt]
+a:5 [in hydras.Ackermann.folLogic3]
+a:5 [in hydras.Gamma0.Gamma0]
+a:5 [in hydras.MoreAck.FolExamples]
+a:50 [in hydras.Epsilon0.T1]
+A:50 [in hydras.rpo.more_list]
+A:50 [in hydras.Ackermann.code]
+A:50 [in additions.Euclidean_Chains]
+a:50 [in hydras.rpo.dickson]
+a:50 [in hydras.MoreAck.PrimRecExamples]
+A:50 [in hydras.Ackermann.expressible]
+a:500 [in hydras.Epsilon0.T1]
+a:502 [in hydras.Epsilon0.T1]
+a:502 [in hydras.Ackermann.checkPrf]
+a:503 [in hydras.Ackermann.primRec]
+a:505 [in hydras.Ackermann.primRec]
+a:506 [in hydras.Ackermann.checkPrf]
+a:507 [in hydras.Ackermann.primRec]
+a:509 [in hydras.Epsilon0.T1]
+a:509 [in hydras.Ackermann.primRec]
+a:51 [in hydras.Prelude.DecPreOrder_Instances]
+a:51 [in hydras.Prelude.More_Arith]
+a:51 [in hydras.Prelude.Comparable]
+a:51 [in gaia_hydras.GPaths]
+A:51 [in additions.Monoid_def]
+a:51 [in hydras.Prelude.MoreVectors]
+a:510 [in hydras.Ackermann.checkPrf]
+a:511 [in hydras.Ackermann.primRec]
+a:513 [in hydras.Epsilon0.T1]
+a:513 [in hydras.Ackermann.primRec]
+a:514 [in hydras.Epsilon0.T1]
+a:514 [in hydras.Ackermann.checkPrf]
+a:515 [in hydras.Epsilon0.T1]
+a:515 [in hydras.Ackermann.primRec]
+a:517 [in hydras.Ackermann.primRec]
+a:518 [in hydras.Epsilon0.T1]
+a:518 [in hydras.Gamma0.Gamma0]
+a:519 [in hydras.Ackermann.primRec]
+A:52 [in additions.AM]
+a:52 [in hydras.Epsilon0.Epsilon0rpo]
+a:52 [in hydras.Prelude.DecPreOrder]
+a:52 [in hydras.MoreAck.AckNotPR]
+a:52 [in hydras.Hydra.BigBattle]
+a:52 [in hydras.Schutte.PartialFun]
+a:52 [in hydras.Schutte.Schutte_basics]
+a:520 [in hydras.Epsilon0.T1]
+a:520 [in hydras.Gamma0.Gamma0]
+a:521 [in hydras.Epsilon0.T1]
+a:521 [in hydras.Ackermann.primRec]
+a:523 [in hydras.Ackermann.primRec]
+a:524 [in hydras.Gamma0.Gamma0]
+a:525 [in hydras.Ackermann.primRec]
+a:526 [in hydras.Epsilon0.T1]
+a:527 [in hydras.Ackermann.primRec]
+a:527 [in hydras.Gamma0.Gamma0]
+a:528 [in hydras.Epsilon0.T1]
+a:529 [in hydras.Ackermann.primRec]
+a:53 [in hydras.Ackermann.cPair]
+A:53 [in additions.Euclidean_Chains]
+a:53 [in hydras.Schutte.Schutte_basics]
+A:53 [in additions.Addition_Chains]
+A:53 [in hydras.Prelude.MoreVectors]
+a:530 [in hydras.Gamma0.Gamma0]
+a:531 [in hydras.Epsilon0.T1]
+a:532 [in hydras.Ackermann.primRec]
+a:533 [in hydras.Gamma0.Gamma0]
+a:534 [in hydras.Ackermann.primRec]
+a:536 [in hydras.Ackermann.primRec]
+a:538 [in hydras.Epsilon0.T1]
+a:54 [in hydras.Ackermann.LNN2LNT]
+a:54 [in hydras.Prelude.Comparable]
+a:54 [in hydras.Prelude.Sort_spec]
+A:54 [in hydras.Prelude.DecPreOrder]
+a:54 [in hydras.Ackermann.code]
+a:54 [in hydras.Schutte.Schutte_basics]
+A:54 [in hydras.Schutte.Ordering_Functions]
+a:548 [in hydras.Epsilon0.T1]
+a:548 [in hydras.Ackermann.primRec]
+a:55 [in hydras.Prelude.DecPreOrder_Instances]
+a:55 [in hydras.rpo.more_list]
+a:55 [in hydras.Schutte.Schutte_basics]
+a:55 [in additions.Addition_Chains]
+A:55 [in hydras.Ackermann.expressible]
+a:550 [in hydras.Ackermann.primRec]
+a:552 [in hydras.Ackermann.primRec]
+a:553 [in hydras.Epsilon0.T1]
+a:554 [in hydras.Ackermann.primRec]
+a:555 [in hydras.Epsilon0.T1]
+a:557 [in gaia_hydras.nfwfgaia]
+A:56 [in hydras.rpo.more_list]
+a:56 [in hydras.Ackermann.LNN2LNT]
+a:56 [in gaia_hydras.GCanon]
+a:56 [in hydras.Ackermann.code]
+a:56 [in gaia_hydras.T1Bridge]
+A:56 [in additions.Monoid_def]
+a:56 [in hydras.Schutte.Schutte_basics]
+a:56 [in hydras.rpo.dickson]
+a:56 [in hydras.Schutte.Ordering_Functions]
+a:562 [in hydras.Epsilon0.T1]
+a:562 [in gaia_hydras.nfwfgaia]
+a:563 [in hydras.Gamma0.Gamma0]
+a:564 [in hydras.Ackermann.checkPrf]
+a:564 [in gaia_hydras.nfwfgaia]
+a:565 [in hydras.Gamma0.Gamma0]
+a:566 [in hydras.Ackermann.checkPrf]
+a:567 [in hydras.Ackermann.checkPrf]
+a:568 [in hydras.Ackermann.checkPrf]
+a:569 [in hydras.Gamma0.Gamma0]
+a:569 [in gaia_hydras.nfwfgaia]
+a:57 [in hydras.Prelude.Comparable]
+a:57 [in gaia_hydras.GPaths]
+a:57 [in hydras.Prelude.DecPreOrder]
+A:57 [in additions.Monoid_def]
+a:57 [in hydras.Ackermann.primRec]
+a:57 [in hydras.Ackermann.cPair]
+A:57 [in additions.Addition_Chains]
+a:57 [in hydras.Prelude.MoreVectors]
+a:571 [in hydras.Epsilon0.T1]
+a:571 [in hydras.Gamma0.Gamma0]
+a:575 [in hydras.Epsilon0.T1]
+a:579 [in hydras.Epsilon0.T1]
+a:579 [in hydras.Gamma0.Gamma0]
+A:58 [in Goedel.PRrepresentable]
+a:58 [in hydras.Ackermann.LNN2LNT]
+a:58 [in gaia_hydras.GPaths]
+a:58 [in hydras.Ackermann.code]
+a:58 [in hydras.Ackermann.cPair]
+A:58 [in Goedel.codeSysPrf]
+a:58 [in hydras.rpo.dickson]
+A:58 [in hydras.Ackermann.expressible]
+a:583 [in hydras.Ackermann.primRec]
+a:585 [in hydras.Ackermann.primRec]
+a:585 [in hydras.Gamma0.Gamma0]
+a:587 [in hydras.Ackermann.primRec]
+a:587 [in hydras.Gamma0.Gamma0]
+a:589 [in hydras.Ackermann.primRec]
+a:59 [in hydras.Prelude.DecPreOrder_Instances]
+A:59 [in additions.AM]
+a:59 [in hydras.Prelude.Comparable]
+A:59 [in hydras.Prelude.DecPreOrder]
+a:59 [in hydras.solutions_exercises.MultisetWf]
+a:59 [in hydras.Ackermann.code]
+A:59 [in hydras.Schutte.Ordering_Functions]
+a:59 [in hydras.Ackermann.expressible]
+a:591 [in hydras.Ackermann.primRec]
+a:592 [in hydras.Gamma0.Gamma0]
+a:593 [in hydras.Ackermann.primRec]
+a:595 [in hydras.Ackermann.primRec]
+a:595 [in gaia_hydras.nfwfgaia]
+a:596 [in hydras.Gamma0.Gamma0]
+a:597 [in hydras.Epsilon0.T1]
+a:6 [in hydras.OrdinalNotations.ON_O]
+A:6 [in hydras.Ackermann.model]
+A:6 [in hydras.Schutte.MoreEpsilonIota]
+a:6 [in hydras.solutions_exercises.MultisetWf]
+a:6 [in hydras.solutions_exercises.MorePRExamples]
+A:6 [in additions.Monoid_def]
+a:6 [in hydras.Ackermann.cPair]
+A:6 [in gaia_hydras.GLarge_Sets]
+a:6 [in hydras.Schutte.Well_Orders]
+A:6 [in hydras.Schutte.Ordering_Functions]
+a:6 [in hydras.MoreAck.PrimRecExamples]
+a:60 [in hydras.Schutte.Correctness_E0]
+a:60 [in hydras.rpo.more_list]
+A:60 [in hydras.Prelude.Sort_spec]
+a:60 [in hydras.OrdinalNotations.OmegaOmega]
+a:60 [in hydras.Ackermann.cPair]
+a:60 [in hydras.Hydra.BigBattle]
+a:60 [in hydras.rpo.dickson]
+a:601 [in hydras.Epsilon0.T1]
+a:604 [in hydras.Gamma0.Gamma0]
+a:606 [in hydras.Gamma0.Gamma0]
+a:61 [in hydras.Ackermann.LNN2LNT]
+a:61 [in hydras.Prelude.Comparable]
+a:61 [in hydras.OrdinalNotations.OmegaOmega]
+a:61 [in hydras.Ackermann.code]
+a:61 [in gaia_hydras.T1Bridge]
+A:61 [in additions.Monoid_def]
+a:61 [in hydras.Ackermann.primRec]
+a:61 [in hydras.Ackermann.cPair]
+A:61 [in additions.Euclidean_Chains]
+a:61 [in hydras.Schutte.Ordering_Functions]
+A:61 [in hydras.Prelude.MoreVectors]
+a:610 [in gaia_hydras.nfwfgaia]
+a:614 [in gaia_hydras.nfwfgaia]
+a:618 [in hydras.Epsilon0.T1]
+A:62 [in Goedel.PRrepresentable]
+a:62 [in hydras.Prelude.DecPreOrder]
+a:62 [in hydras.solutions_exercises.MultisetWf]
+a:62 [in hydras.Ackermann.cPair]
+a:62 [in hydras.Schutte.Schutte_basics]
+a:62 [in hydras.OrdinalNotations.ON_Generic]
+a:62 [in additions.Addition_Chains]
+A:62 [in hydras.Ackermann.expressible]
+a:620 [in hydras.Gamma0.Gamma0]
+a:626 [in hydras.Epsilon0.T1]
+a:626 [in gaia_hydras.nfwfgaia]
+a:629 [in gaia_hydras.nfwfgaia]
+A:63 [in hydras.rpo.more_list]
+a:63 [in hydras.Prelude.Comparable]
+a:63 [in hydras.Ackermann.code]
+a:63 [in hydras.Hydra.BigBattle]
+A:63 [in additions.Addition_Chains]
+a:633 [in hydras.Gamma0.Gamma0]
+a:637 [in hydras.Gamma0.Gamma0]
+a:64 [in hydras.Epsilon0.T1]
+a:64 [in hydras.Ackermann.subAll]
+a:64 [in gaia_hydras.GPaths]
+A:64 [in hydras.Prelude.DecPreOrder]
+a:64 [in hydras.OrdinalNotations.OmegaOmega]
+a:64 [in hydras.Ackermann.code]
+a:64 [in hydras.Schutte.Schutte_basics]
+a:64 [in hydras.rpo.list_permut]
+A:64 [in hydras.Schutte.Ordering_Functions]
+a:64 [in hydras.MoreAck.PrimRecExamples]
+A:65 [in Goedel.PRrepresentable]
+a:65 [in hydras.Prelude.Comparable]
+a:65 [in hydras.Epsilon0.Hessenberg]
+a:65 [in hydras.Ackermann.primRec]
+A:65 [in hydras.Schutte.PartialFun]
+A:653 [in hydras.Ackermann.checkPrf]
+a:654 [in gaia_hydras.nfwfgaia]
+a:655 [in hydras.Ackermann.checkPrf]
+a:656 [in gaia_hydras.nfwfgaia]
+a:657 [in hydras.Ackermann.checkPrf]
+a:658 [in gaia_hydras.nfwfgaia]
+a:659 [in hydras.Ackermann.checkPrf]
+a:66 [in hydras.Ackermann.code]
+a:66 [in gaia_hydras.T1Bridge]
+a:66 [in hydras.Schutte.Schutte_basics]
+a:66 [in hydras.MoreAck.PrimRecExamples]
+A:66 [in hydras.Ackermann.expressible]
+a:661 [in hydras.Ackermann.checkPrf]
+a:663 [in hydras.Ackermann.checkPrf]
+a:665 [in hydras.Ackermann.checkPrf]
+a:667 [in hydras.Ackermann.checkPrf]
+a:669 [in hydras.Ackermann.checkPrf]
+a:669 [in hydras.Gamma0.Gamma0]
+a:67 [in hydras.Ackermann.subAll]
+a:67 [in hydras.Prelude.Comparable]
+a:67 [in hydras.Prelude.DecPreOrder]
+a:67 [in hydras.Hydra.BigBattle]
+A:67 [in hydras.OrdinalNotations.ON_Generic]
+A:67 [in hydras.Schutte.Ordering_Functions]
+a:67 [in hydras.rpo.rpo]
+A:67 [in hydras.Prelude.MoreVectors]
+a:670 [in gaia_hydras.nfwfgaia]
+a:671 [in hydras.Ackermann.checkPrf]
+a:671 [in hydras.Gamma0.Gamma0]
+a:673 [in hydras.Ackermann.checkPrf]
+a:675 [in hydras.Ackermann.checkPrf]
+a:675 [in hydras.Ackermann.primRec]
+a:677 [in hydras.Ackermann.checkPrf]
+a:679 [in hydras.Ackermann.primRec]
+A:68 [in hydras.Prelude.Iterates]
+A:68 [in hydras.rpo.more_list]
+a:68 [in gaia_hydras.GPaths]
+A:68 [in hydras.Prelude.Sort_spec]
+a:68 [in gaia_hydras.GCanon]
+A:68 [in hydras.solutions_exercises.MultisetWf]
+a:68 [in hydras.Ackermann.code]
+A:68 [in additions.Euclidean_Chains]
+a:689 [in hydras.Gamma0.Gamma0]
+A:69 [in Goedel.PRrepresentable]
+a:69 [in hydras.Prelude.Comparable]
+a:69 [in hydras.OrdinalNotations.OmegaOmega]
+a:69 [in hydras.Schutte.Schutte_basics]
+a:69 [in hydras.Ackermann.folLogic3]
+a:69 [in hydras.rpo.dickson]
+a:69 [in hydras.Schutte.Ordering_Functions]
+a:69 [in hydras.Schutte.Addition]
+A:69 [in hydras.Prelude.MoreVectors]
+a:691 [in hydras.Ackermann.primRec]
+a:694 [in hydras.Ackermann.primRec]
+a:695 [in gaia_hydras.nfwfgaia]
+a:697 [in gaia_hydras.nfwfgaia]
+a:7 [in hydras.Gamma0.T2]
+A:7 [in hydras.Prelude.Iterates]
+a:7 [in hydras.Ackermann.folLogic2]
+a:7 [in Goedel.fixPoint]
+A:7 [in hydras.Prelude.DecPreOrder]
+a:7 [in hydras.Schutte.MoreEpsilonIota]
+a:7 [in gaia_hydras.T1Bridge]
+a:7 [in hydras.Ackermann.prLogic]
+a:7 [in hydras.Ackermann.NNtheory]
+a:7 [in gaia_hydras.GLarge_Sets]
+A:7 [in hydras.OrdinalNotations.ON_Generic]
+a:7 [in gaia_hydras.T2Bridge]
+a:7 [in hydras.MoreAck.PrimRecExamples]
+A:7 [in hydras.Prelude.MoreVectors]
+A:70 [in hydras.Prelude.DecPreOrder]
+a:70 [in gaia_hydras.GCanon]
+a:70 [in hydras.Ackermann.code]
+a:70 [in hydras.Ackermann.primRec]
+a:707 [in hydras.Epsilon0.T1]
+a:707 [in gaia_hydras.nfwfgaia]
+a:71 [in hydras.Epsilon0.T1]
+A:71 [in hydras.rpo.more_list]
+a:71 [in hydras.Prelude.Comparable]
+a:71 [in gaia_hydras.GPaths]
+a:71 [in hydras.Hydra.BigBattle]
+a:71 [in hydras.Ackermann.folLogic3]
+A:71 [in additions.Addition_Chains]
+A:71 [in hydras.Schutte.Ordering_Functions]
+A:71 [in Goedel.rosser]
+A:71 [in hydras.Ackermann.expressible]
+a:71 [in hydras.Prelude.MoreVectors]
+a:717 [in gaia_hydras.nfwfgaia]
+a:718 [in gaia_hydras.nfwfgaia]
+a:72 [in hydras.Ackermann.subAll]
+A:72 [in additions.AM]
+A:72 [in hydras.Prelude.Iterates]
+A:72 [in hydras.rpo.more_list]
+a:72 [in hydras.Prelude.Comparable]
+a:72 [in hydras.Epsilon0.Hessenberg]
+a:72 [in hydras.Ackermann.code]
+a:72 [in gaia_hydras.T1Bridge]
+A:72 [in hydras.Schutte.PartialFun]
+a:72 [in hydras.Schutte.Schutte_basics]
+a:720 [in gaia_hydras.nfwfgaia]
+a:722 [in hydras.Epsilon0.T1]
+a:723 [in gaia_hydras.nfwfgaia]
+a:727 [in gaia_hydras.nfwfgaia]
+a:729 [in gaia_hydras.nfwfgaia]
+A:73 [in Goedel.PRrepresentable]
+a:73 [in hydras.Epsilon0.Epsilon0rpo]
+a:73 [in hydras.Prelude.DecPreOrder]
+a:73 [in hydras.OrdinalNotations.ON_Generic]
+A:73 [in hydras.Prelude.MoreVectors]
+a:731 [in gaia_hydras.nfwfgaia]
+a:732 [in hydras.Epsilon0.T1]
+a:733 [in gaia_hydras.nfwfgaia]
+a:735 [in hydras.Epsilon0.T1]
+a:737 [in gaia_hydras.nfwfgaia]
+a:739 [in gaia_hydras.nfwfgaia]
+a:74 [in hydras.Prelude.Comparable]
+A:74 [in hydras.Prelude.Sort_spec]
+a:74 [in hydras.Ackermann.code]
+A:74 [in additions.fib]
+a:747 [in gaia_hydras.nfwfgaia]
+a:749 [in gaia_hydras.nfwfgaia]
+a:75 [in hydras.Gamma0.T2]
+a:75 [in gaia_hydras.GPaths]
+a:75 [in additions.Euclidean_Chains]
+a:75 [in hydras.Schutte.Schutte_basics]
+a:752 [in gaia_hydras.nfwfgaia]
+a:756 [in hydras.Ackermann.checkPrf]
+a:756 [in gaia_hydras.nfwfgaia]
+a:758 [in gaia_hydras.nfwfgaia]
+A:76 [in hydras.Prelude.Iterates]
+a:76 [in hydras.Prelude.Comparable]
+a:76 [in additions.Pow_variant]
+A:76 [in hydras.Prelude.DecPreOrder]
+a:76 [in hydras.Ackermann.fol]
+A:76 [in hydras.Schutte.Ordering_Functions]
+a:760 [in hydras.Ackermann.checkPrf]
+a:760 [in gaia_hydras.nfwfgaia]
+a:77 [in hydras.Epsilon0.T1]
+a:77 [in hydras.Ackermann.subAll]
+a:77 [in hydras.Epsilon0.Epsilon0rpo]
+a:77 [in hydras.Epsilon0.Hessenberg]
+A:77 [in Goedel.codeSysPrf]
+a:77 [in hydras.Schutte.Ordering_Functions]
+a:776 [in gaia_hydras.nfwfgaia]
+a:779 [in gaia_hydras.nfwfgaia]
+a:78 [in hydras.Gamma0.T2]
+A:78 [in hydras.rpo.more_list]
+a:78 [in hydras.Prelude.Comparable]
+a:78 [in gaia_hydras.T1Bridge]
+a:78 [in hydras.Schutte.Schutte_basics]
+a:78 [in hydras.OrdinalNotations.ON_Generic]
+A:78 [in hydras.Prelude.MoreVectors]
+a:783 [in gaia_hydras.nfwfgaia]
+a:786 [in hydras.Ackermann.checkPrf]
+a:786 [in gaia_hydras.nfwfgaia]
+a:788 [in hydras.Ackermann.checkPrf]
+a:79 [in hydras.Ackermann.subAll]
+a:79 [in additions.Pow]
+a:79 [in hydras.Prelude.DecPreOrder]
+a:79 [in hydras.Schutte.Schutte_basics]
+a:79 [in hydras.Ackermann.folLogic3]
+a:79 [in hydras.rpo.list_permut]
+A:79 [in hydras.Schutte.Ordering_Functions]
+a:792 [in gaia_hydras.nfwfgaia]
+a:793 [in hydras.Ackermann.checkPrf]
+a:799 [in hydras.Ackermann.checkPrf]
+A:8 [in hydras.Prelude.DecPreOrder_Instances]
+a:8 [in hydras.Ackermann.extEqualNat]
+A:8 [in hydras.rpo.more_list]
+a:8 [in hydras.OrdinalNotations.ON_O]
+a:8 [in gaia_hydras.GCanon]
+a:8 [in hydras.OrdinalNotations.OmegaOmega]
+a:8 [in hydras.Ackermann.prLogic]
+A:8 [in additions.Euclidean_Chains]
+a:8 [in hydras.Schutte.PartialFun]
+A:8 [in gaia_hydras.GLarge_Sets]
+a:8 [in hydras.rpo.dickson]
+a:8 [in hydras.MoreAck.FolExamples]
+A:8 [in hydras.Ackermann.expressible]
+A:80 [in hydras.Prelude.Iterates]
+a:80 [in hydras.Prelude.Comparable]
+a:80 [in gaia_hydras.GPaths]
+a:80 [in hydras.Ackermann.cPair]
+A:80 [in hydras.Prelude.MoreVectors]
+a:807 [in hydras.Ackermann.checkPrf]
+a:81 [in hydras.Ackermann.subAll]
+A:81 [in hydras.Prelude.DecPreOrder]
+a:81 [in gaia_hydras.T1Bridge]
+a:81 [in hydras.Ackermann.cPair]
+a:81 [in hydras.Ackermann.fol]
+a:81 [in hydras.Schutte.Schutte_basics]
+A:81 [in additions.fib]
+A:81 [in hydras.OrdinalNotations.ON_Generic]
+a:81 [in hydras.Ackermann.folLogic3]
+A:81 [in hydras.Schutte.Ordering_Functions]
+a:811 [in hydras.Ackermann.checkPrf]
+a:819 [in hydras.Ackermann.codeSubFormula]
+a:82 [in hydras.Epsilon0.T1]
+A:82 [in additions.AM]
+a:82 [in hydras.Prelude.Comparable]
+A:82 [in hydras.Schutte.PartialFun]
+A:82 [in additions.Addition_Chains]
+a:822 [in hydras.Ackermann.codeSubFormula]
+a:826 [in hydras.Ackermann.codeSubFormula]
+a:83 [in hydras.Ackermann.cPair]
+a:83 [in hydras.Schutte.Schutte_basics]
+a:830 [in gaia_hydras.nfwfgaia]
+a:84 [in hydras.Prelude.Comparable]
+A:84 [in hydras.Prelude.DecPreOrder]
+a:84 [in gaia_hydras.T1Bridge]
+A:84 [in Goedel.codeSysPrf]
+A:84 [in hydras.Prelude.MoreVectors]
+a:842 [in hydras.Ackermann.codeSubFormula]
+a:842 [in gaia_hydras.nfwfgaia]
+a:845 [in hydras.Ackermann.codeSubFormula]
+a:848 [in hydras.Ackermann.codeSubFormula]
+a:849 [in hydras.Ackermann.checkPrf]
+A:85 [in Goedel.PRrepresentable]
+a:85 [in hydras.Ackermann.cPair]
+a:85 [in hydras.Ackermann.folLogic3]
+a:851 [in hydras.Ackermann.checkPrf]
+a:852 [in hydras.Ackermann.codeSubFormula]
+a:852 [in gaia_hydras.nfwfgaia]
+a:853 [in hydras.Ackermann.checkPrf]
+a:854 [in hydras.Ackermann.codeSubFormula]
+a:855 [in hydras.Ackermann.checkPrf]
+a:856 [in hydras.Ackermann.codeSubFormula]
+a:86 [in hydras.Hydra.O2H]
+a:86 [in hydras.Prelude.Comparable]
+a:86 [in hydras.OrdinalNotations.OmegaOmega]
+a:86 [in gaia_hydras.T1Bridge]
+a:86 [in gaia_hydras.nfwfgaia]
+a:860 [in gaia_hydras.nfwfgaia]
+a:865 [in gaia_hydras.nfwfgaia]
+a:87 [in hydras.Epsilon0.T1]
+A:87 [in hydras.Prelude.DecPreOrder]
+a:87 [in hydras.Ackermann.cPair]
+a:87 [in additions.Addition_Chains]
+A:88 [in hydras.rpo.more_list]
+a:88 [in hydras.Prelude.Comparable]
+a:88 [in hydras.Epsilon0.Hessenberg]
+a:88 [in gaia_hydras.T1Bridge]
+A:88 [in additions.Euclidean_Chains]
+A:88 [in hydras.Schutte.PartialFun]
+a:88 [in hydras.Schutte.Schutte_basics]
+A:88 [in Goedel.codeSysPrf]
+A:88 [in additions.Addition_Chains]
+a:88 [in hydras.MoreAck.PrimRecExamples]
+A:88 [in hydras.Prelude.MoreVectors]
+a:885 [in gaia_hydras.nfwfgaia]
+A:89 [in Goedel.PRrepresentable]
+a:89 [in hydras.Prelude.Comparable]
+a:89 [in gaia_hydras.T1Bridge]
+a:89 [in hydras.Ackermann.cPair]
+a:894 [in gaia_hydras.nfwfgaia]
+a:898 [in gaia_hydras.nfwfgaia]
+a:9 [in gaia_hydras.T1Choice]
+a:9 [in hydras.Prelude.Restriction]
+a:9 [in additions.FirstSteps]
+a:9 [in hydras.Prelude.Comparable]
+a:9 [in gaia_hydras.GPaths]
+a:9 [in hydras.Ackermann.PAtheory]
+a:9 [in hydras.solutions_exercises.MultisetWf]
+a:9 [in hydras.OrdinalNotations.OmegaOmega]
+a:9 [in gaia_hydras.T1Bridge]
+a:9 [in hydras.Ackermann.prLogic]
+a:9 [in hydras.Ackermann.NNtheory]
+a:9 [in gaia_hydras.GLarge_Sets]
+a:9 [in hydras.Prelude.MoreDecidable]
+a:9 [in gaia_hydras.T2Bridge]
+a:9 [in hydras.Gamma0.Gamma0]
+A:9 [in hydras.Prelude.MoreVectors]
+a:90 [in hydras.Gamma0.T2]
+a:90 [in hydras.Epsilon0.Hessenberg]
+a:90 [in gaia_hydras.T1Bridge]
+a:90 [in hydras.Schutte.Schutte_basics]
+a:90 [in additions.fib]
+a:902 [in gaia_hydras.nfwfgaia]
+a:91 [in hydras.Prelude.Comparable]
+a:91 [in hydras.Prelude.DecPreOrder]
+a:91 [in gaia_hydras.T1Bridge]
+a:91 [in hydras.Ackermann.cPair]
+a:91 [in hydras.OrdinalNotations.ON_Generic]
+a:91 [in hydras.Schutte.Ordering_Functions]
+a:910 [in gaia_hydras.nfwfgaia]
+A:92 [in Goedel.PRrepresentable]
+a:92 [in hydras.Epsilon0.Hessenberg]
+a:92 [in gaia_hydras.T1Bridge]
+a:92 [in hydras.Schutte.Schutte_basics]
+a:92 [in hydras.Prelude.MoreVectors]
+a:923 [in gaia_hydras.nfwfgaia]
+a:925 [in gaia_hydras.nfwfgaia]
+a:927 [in gaia_hydras.nfwfgaia]
+a:929 [in gaia_hydras.nfwfgaia]
+A:93 [in additions.AM]
+a:93 [in hydras.Prelude.Comparable]
+A:93 [in hydras.Prelude.DecPreOrder]
+a:93 [in hydras.Prelude.Merge_Sort]
+a:93 [in gaia_hydras.T1Bridge]
+a:93 [in hydras.Ackermann.cPair]
+A:93 [in hydras.OrdinalNotations.ON_Generic]
+A:93 [in Goedel.codeSysPrf]
+A:93 [in additions.Addition_Chains]
+a:931 [in gaia_hydras.nfwfgaia]
+a:938 [in gaia_hydras.nfwfgaia]
+a:94 [in hydras.Epsilon0.Hessenberg]
+a:94 [in hydras.Ackermann.folLogic3]
+a:944 [in gaia_hydras.nfwfgaia]
+a:95 [in gaia_hydras.T1Bridge]
+a:95 [in hydras.Ackermann.cPair]
+a:955 [in gaia_hydras.nfwfgaia]
+a:956 [in gaia_hydras.nfwfgaia]
+a:958 [in gaia_hydras.nfwfgaia]
+A:96 [in Goedel.PRrepresentable]
+A:96 [in hydras.rpo.more_list]
+a:96 [in hydras.Prelude.DecPreOrder]
+a:960 [in gaia_hydras.nfwfgaia]
+a:962 [in gaia_hydras.nfwfgaia]
+a:964 [in gaia_hydras.nfwfgaia]
+a:97 [in hydras.Prelude.Comparable]
+a:97 [in hydras.Epsilon0.Hessenberg]
+a:97 [in gaia_hydras.T1Bridge]
+A:98 [in additions.AM]
+A:98 [in hydras.Hydra.Hydra_Lemmas]
+a:988 [in gaia_hydras.nfwfgaia]
+A:99 [in additions.Euclidean_Chains]
+a:99 [in hydras.Schutte.Schutte_basics]
+a:994 [in gaia_hydras.nfwfgaia]
+

B

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+beta':105 [in hydras.Epsilon0.T1]
+beta':119 [in hydras.Schutte.Ordering_Functions]
+beta':138 [in hydras.Gamma0.Gamma0]
+beta':195 [in hydras.Epsilon0.T1]
+beta':314 [in hydras.Epsilon0.Paths]
+beta':326 [in hydras.Epsilon0.T1]
+beta':331 [in hydras.Epsilon0.T1]
+beta':335 [in hydras.Epsilon0.T1]
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+beta1:501 [in hydras.Gamma0.Gamma0]
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+beta1:675 [in hydras.Gamma0.Gamma0]
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+beta2:38 [in hydras.Gamma0.T2]
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+beta2:502 [in hydras.Gamma0.Gamma0]
+beta2:542 [in hydras.Gamma0.Gamma0]
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+beta2:653 [in hydras.Gamma0.Gamma0]
+beta2:656 [in hydras.Gamma0.Gamma0]
+beta2:658 [in hydras.Gamma0.Gamma0]
+beta2:668 [in hydras.Gamma0.Gamma0]
+beta2:677 [in hydras.Gamma0.Gamma0]
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+beta:10 [in hydras.OrdinalNotations.ON_O]
+beta:10 [in hydras.Schutte.CNF]
+beta:10 [in hydras.OrdinalNotations.ON_mult]
+beta:10 [in hydras.OrdinalNotations.ON_plus]
+beta:10 [in hydras.Epsilon0.Canon]
+beta:10 [in hydras.Epsilon0.E0]
+beta:10 [in hydras.Epsilon0.Paths]
+beta:10 [in gaia_hydras.GHessenberg]
+beta:100 [in hydras.Epsilon0.T1]
+beta:100 [in hydras.Gamma0.T2]
+beta:100 [in hydras.Prelude.Comparable]
+beta:100 [in hydras.Epsilon0.Canon]
+beta:101 [in hydras.OrdinalNotations.OmegaOmega]
+beta:101 [in hydras.Epsilon0.E0]
+beta:102 [in hydras.Prelude.Comparable]
+beta:102 [in hydras.Epsilon0.Hprime]
+beta:102 [in hydras.Epsilon0.Paths]
+beta:103 [in hydras.Hydra.O2H]
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+beta:106 [in hydras.Hydra.O2H]
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+beta:106 [in hydras.Epsilon0.Hprime]
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+beta:11 [in gaia_hydras.GHprime]
+beta:110 [in hydras.Epsilon0.Hessenberg]
+beta:110 [in hydras.OrdinalNotations.OmegaOmega]
+beta:110 [in hydras.Schutte.Schutte_basics]
+beta:111 [in hydras.Epsilon0.T1]
+beta:111 [in hydras.Hydra.O2H]
+beta:111 [in hydras.Epsilon0.Hprime]
+beta:111 [in hydras.Epsilon0.Paths]
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+beta:115 [in hydras.Epsilon0.Paths]
+beta:116 [in hydras.Schutte.Schutte_basics]
+beta:116 [in hydras.Epsilon0.E0]
+beta:117 [in hydras.Epsilon0.F_alpha]
+beta:118 [in gaia_hydras.T1Bridge]
+beta:118 [in hydras.Schutte.Schutte_basics]
+beta:118 [in hydras.Schutte.Ordering_Functions]
+beta:119 [in hydras.Epsilon0.E0]
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+beta:12 [in hydras.Schutte.AP]
+beta:12 [in hydras.OrdinalNotations.ON_mult]
+beta:12 [in hydras.OrdinalNotations.ON_Finite]
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+beta:120 [in gaia_hydras.T1Bridge]
+beta:120 [in hydras.Epsilon0.Hprime]
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+beta:121 [in hydras.Schutte.Ordering_Functions]
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+beta:125 [in hydras.Epsilon0.Hprime]
+beta:125 [in hydras.Gamma0.Gamma0]
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+beta:128 [in hydras.Gamma0.Gamma0]
+beta:129 [in hydras.Epsilon0.Hprime]
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+beta:133 [in hydras.Schutte.Ordering_Functions]
+beta:134 [in gaia_hydras.T1Bridge]
+beta:134 [in hydras.Gamma0.Gamma0]
+beta:14 [in hydras.solutions_exercises.predSuccUnicity]
+beta:14 [in gaia_hydras.GF_alpha]
+beta:14 [in hydras.OrdinalNotations.ON_O]
+beta:14 [in hydras.Epsilon0.Hessenberg]
+beta:14 [in hydras.OrdinalNotations.ON_mult]
+beta:14 [in hydras.OrdinalNotations.ON_plus]
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+beta:148 [in hydras.Gamma0.Gamma0]
+beta:15 [in hydras.Schutte.Correctness_E0]
+beta:15 [in hydras.OrdinalNotations.ON_Omega2]
+beta:15 [in hydras.Schutte.AP]
+beta:15 [in hydras.OrdinalNotations.ON_Finite]
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+beta:152 [in hydras.Gamma0.Gamma0]
+beta:154 [in hydras.Gamma0.Gamma0]
+beta:155 [in hydras.Epsilon0.Large_Sets]
+beta:156 [in hydras.Epsilon0.T1]
+beta:158 [in hydras.Gamma0.Gamma0]
+beta:159 [in hydras.Epsilon0.T1]
+beta:16 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+beta:16 [in hydras.OrdinalNotations.ON_mult]
+beta:16 [in hydras.OrdinalNotations.ON_plus]
+beta:16 [in hydras.Epsilon0.E0]
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+beta:163 [in hydras.Epsilon0.Paths]
+beta:165 [in gaia_hydras.T1Bridge]
+beta:166 [in hydras.Epsilon0.Paths]
+beta:167 [in gaia_hydras.T1Bridge]
+beta:17 [in hydras.solutions_exercises.predSuccUnicity]
+beta:17 [in hydras.OrdinalNotations.ON_O]
+beta:17 [in hydras.Schutte.CNF]
+beta:17 [in hydras.Hydra.Epsilon0_Needed_Std]
+beta:17 [in hydras.solutions_exercises.F_3]
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+beta:171 [in hydras.Epsilon0.Paths]
+beta:173 [in gaia_hydras.T1Bridge]
+beta:176 [in hydras.Schutte.Schutte_basics]
+beta:177 [in hydras.Gamma0.Gamma0]
+beta:179 [in hydras.Schutte.Schutte_basics]
+beta:18 [in hydras.solutions_exercises.Limit_Infinity]
+beta:18 [in hydras.OrdinalNotations.ON_plus]
+beta:18 [in hydras.OrdinalNotations.ON_Finite]
+beta:18 [in hydras.Gamma0.Gamma0]
+beta:184 [in hydras.Epsilon0.F_alpha]
+beta:19 [in hydras.OrdinalNotations.ON_Omega2]
+beta:19 [in hydras.Schutte.Schutte_basics]
+beta:191 [in hydras.Gamma0.Gamma0]
+beta:193 [in hydras.Gamma0.Gamma0]
+beta:194 [in hydras.Epsilon0.Paths]
+beta:195 [in hydras.Gamma0.Gamma0]
+beta:197 [in hydras.Epsilon0.Paths]
+beta:197 [in hydras.Gamma0.Gamma0]
+beta:199 [in hydras.Gamma0.Gamma0]
+beta:2 [in hydras.solutions_exercises.is_F_monotonous]
+beta:2 [in hydras.Epsilon0.Hessenberg]
+beta:2 [in hydras.Schutte.Critical]
+beta:2 [in hydras.Schutte.AP]
+beta:2 [in gaia_hydras.GHessenberg]
+beta:20 [in hydras.Schutte.Correctness_E0]
+beta:20 [in hydras.OrdinalNotations.ON_O]
+beta:20 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+beta:20 [in hydras.OrdinalNotations.ON_Finite]
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+beta:203 [in hydras.Gamma0.Gamma0]
+beta:205 [in hydras.Gamma0.Gamma0]
+beta:209 [in hydras.Epsilon0.T1]
+beta:21 [in hydras.Gamma0.T2]
+beta:21 [in hydras.solutions_exercises.Limit_Infinity]
+beta:21 [in hydras.Epsilon0.E0]
+beta:21 [in hydras.Epsilon0.Paths]
+beta:21 [in hydras.Schutte.Addition]
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+beta:217 [in hydras.Schutte.Schutte_basics]
+beta:219 [in hydras.Schutte.Schutte_basics]
+beta:22 [in hydras.Schutte.CNF]
+beta:22 [in hydras.solutions_exercises.F_3]
+beta:22 [in hydras.Gamma0.Gamma0]
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+beta:225 [in hydras.Epsilon0.Paths]
+beta:23 [in hydras.Schutte.Addition]
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+beta:239 [in hydras.Gamma0.Gamma0]
+beta:24 [in hydras.Epsilon0.T1]
+beta:24 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+beta:24 [in hydras.Hydra.Epsilon0_Needed_Std]
+beta:24 [in hydras.Schutte.Schutte_basics]
+beta:24 [in hydras.solutions_exercises.F_3]
+beta:24 [in hydras.Epsilon0.Paths]
+beta:241 [in hydras.Epsilon0.T1]
+beta:241 [in hydras.Gamma0.Gamma0]
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+beta:243 [in hydras.Epsilon0.Paths]
+beta:245 [in hydras.Epsilon0.T1]
+beta:249 [in hydras.Epsilon0.Paths]
+beta:25 [in hydras.Gamma0.T2]
+beta:25 [in hydras.Epsilon0.Paths]
+beta:25 [in hydras.Schutte.Addition]
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+beta:255 [in hydras.Epsilon0.Paths]
+beta:258 [in hydras.Epsilon0.Paths]
+beta:259 [in hydras.Epsilon0.Paths]
+beta:26 [in gaia_hydras.GF_alpha]
+beta:26 [in hydras.OrdinalNotations.ON_Generic]
+beta:26 [in hydras.Schutte.Ordering_Functions]
+beta:262 [in hydras.Epsilon0.Paths]
+beta:266 [in hydras.Epsilon0.Paths]
+beta:269 [in hydras.Epsilon0.Paths]
+beta:27 [in hydras.Hydra.O2H]
+beta:27 [in hydras.Schutte.Critical]
+beta:27 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+beta:274 [in hydras.Epsilon0.Paths]
+beta:277 [in hydras.Epsilon0.Paths]
+beta:28 [in gaia_hydras.GF_alpha]
+beta:28 [in hydras.Schutte.CNF]
+beta:28 [in hydras.Schutte.Addition]
+beta:282 [in hydras.Epsilon0.Paths]
+beta:285 [in hydras.Epsilon0.Paths]
+beta:288 [in hydras.Epsilon0.Paths]
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+b0:341 [in hydras.Ackermann.folProp]
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+b0:357 [in hydras.Ackermann.folProp]
+b0:45 [in hydras.Ackermann.LNN2LNT]
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+b0:776 [in hydras.Ackermann.primRec]
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+b0:782 [in hydras.Ackermann.primRec]
+b0:785 [in hydras.Ackermann.primRec]
+b0:789 [in hydras.Ackermann.primRec]
+b0:792 [in hydras.Ackermann.primRec]
+b0:796 [in hydras.Ackermann.primRec]
+b0:802 [in hydras.Ackermann.primRec]
+b0:807 [in hydras.Ackermann.primRec]
+b0:810 [in hydras.Ackermann.primRec]
+b1:1362 [in hydras.Ackermann.codeSubFormula]
+b1:366 [in hydras.Ackermann.folProp]
+b1:37 [in hydras.Gamma0.Gamma0]
+b1:373 [in hydras.Ackermann.folProp]
+b1:382 [in hydras.Ackermann.folProp]
+b1:389 [in hydras.Ackermann.folProp]
+b1:45 [in hydras.Gamma0.Gamma0]
+b1:458 [in hydras.Gamma0.Gamma0]
+b1:468 [in hydras.Gamma0.Gamma0]
+b1:472 [in hydras.Gamma0.Gamma0]
+b1:474 [in hydras.Gamma0.Gamma0]
+b1:477 [in hydras.Gamma0.Gamma0]
+b1:484 [in hydras.Gamma0.Gamma0]
+b1:486 [in hydras.Gamma0.Gamma0]
+b1:521 [in hydras.Gamma0.Gamma0]
+b1:53 [in hydras.Gamma0.Gamma0]
+b1:555 [in hydras.Gamma0.Gamma0]
+b1:559 [in hydras.Gamma0.Gamma0]
+b1:61 [in hydras.Gamma0.Gamma0]
+b1:69 [in hydras.Gamma0.Gamma0]
+b1:77 [in hydras.Gamma0.Gamma0]
+b1:82 [in hydras.rpo.list_permut]
+b1:83 [in hydras.Gamma0.Gamma0]
+b2:1363 [in hydras.Ackermann.codeSubFormula]
+b2:38 [in hydras.Gamma0.Gamma0]
+b2:459 [in hydras.Gamma0.Gamma0]
+b2:46 [in hydras.Gamma0.Gamma0]
+b2:469 [in hydras.Gamma0.Gamma0]
+b2:473 [in hydras.Gamma0.Gamma0]
+b2:475 [in hydras.Gamma0.Gamma0]
+b2:478 [in hydras.Gamma0.Gamma0]
+b2:485 [in hydras.Gamma0.Gamma0]
+b2:487 [in hydras.Gamma0.Gamma0]
+b2:522 [in hydras.Gamma0.Gamma0]
+b2:54 [in hydras.Gamma0.Gamma0]
+b2:556 [in hydras.Gamma0.Gamma0]
+b2:560 [in hydras.Gamma0.Gamma0]
+b2:62 [in hydras.Gamma0.Gamma0]
+b2:70 [in hydras.Gamma0.Gamma0]
+b2:84 [in hydras.rpo.list_permut]
+B:1 [in hydras.Hydra.Epsilon0_Needed_Generic]
+b:10 [in hydras.Prelude.Comparable]
+b:10 [in gaia_hydras.GPaths]
+b:10 [in hydras.Ackermann.PAtheory]
+b:10 [in gaia_hydras.T1Bridge]
+b:10 [in hydras.Ackermann.NNtheory]
+b:10 [in hydras.Prelude.MoreDecidable]
+b:10 [in gaia_hydras.T2Bridge]
+b:10 [in hydras.Gamma0.Gamma0]
+b:1002 [in gaia_hydras.nfwfgaia]
+b:1005 [in gaia_hydras.nfwfgaia]
+b:1009 [in gaia_hydras.nfwfgaia]
+b:101 [in gaia_hydras.nfwfgaia]
+b:1013 [in gaia_hydras.nfwfgaia]
+b:1015 [in gaia_hydras.nfwfgaia]
+b:102 [in hydras.Ackermann.model]
+b:1020 [in gaia_hydras.nfwfgaia]
+b:1027 [in gaia_hydras.nfwfgaia]
+b:103 [in hydras.Prelude.Merge_Sort]
+b:103 [in gaia_hydras.T1Bridge]
+b:103 [in hydras.Ackermann.folLogic3]
+b:1030 [in gaia_hydras.nfwfgaia]
+b:1035 [in gaia_hydras.nfwfgaia]
+b:104 [in hydras.Epsilon0.Hessenberg]
+b:104 [in hydras.Ackermann.cPair]
+b:105 [in gaia_hydras.T1Bridge]
+b:1050 [in gaia_hydras.nfwfgaia]
+b:106 [in hydras.Ackermann.model]
+b:106 [in hydras.Ackermann.cPair]
+b:107 [in hydras.Epsilon0.Hessenberg]
+b:1078 [in gaia_hydras.nfwfgaia]
+b:108 [in gaia_hydras.T1Bridge]
+b:108 [in hydras.Ackermann.cPair]
+b:109 [in hydras.Hydra.O2H]
+B:109 [in hydras.Prelude.MoreLists]
+b:109 [in hydras.Ackermann.codeSubFormula]
+b:1098 [in gaia_hydras.nfwfgaia]
+b:11 [in gaia_hydras.T1Choice]
+b:110 [in hydras.Ackermann.model]
+b:110 [in gaia_hydras.T1Bridge]
+B:110 [in hydras.OrdinalNotations.ON_Generic]
+b:1100 [in gaia_hydras.nfwfgaia]
+b:1102 [in gaia_hydras.nfwfgaia]
+b:1104 [in gaia_hydras.nfwfgaia]
+b:1107 [in gaia_hydras.nfwfgaia]
+b:111 [in hydras.Gamma0.T2]
+b:111 [in hydras.Ackermann.folProp]
+b:111 [in hydras.Ackermann.cPair]
+b:1110 [in gaia_hydras.nfwfgaia]
+b:1112 [in gaia_hydras.nfwfgaia]
+b:112 [in hydras.Ackermann.codeSubFormula]
+b:112 [in gaia_hydras.nfwfgaia]
+b:1127 [in gaia_hydras.nfwfgaia]
+b:113 [in hydras.Hydra.O2H]
+b:1130 [in gaia_hydras.nfwfgaia]
+b:1132 [in gaia_hydras.nfwfgaia]
+b:114 [in hydras.Ackermann.model]
+b:114 [in hydras.Ackermann.cPair]
+b:1145 [in gaia_hydras.nfwfgaia]
+b:1149 [in gaia_hydras.nfwfgaia]
+b:115 [in hydras.Epsilon0.T1]
+b:1155 [in gaia_hydras.nfwfgaia]
+b:1159 [in gaia_hydras.nfwfgaia]
+b:116 [in hydras.Epsilon0.Hessenberg]
+B:116 [in hydras.Schutte.Ordering_Functions]
+b:1161 [in gaia_hydras.nfwfgaia]
+b:1163 [in gaia_hydras.nfwfgaia]
+b:1165 [in gaia_hydras.nfwfgaia]
+b:1167 [in gaia_hydras.nfwfgaia]
+B:117 [in hydras.Schutte.Ordering_Functions]
+b:1170 [in gaia_hydras.nfwfgaia]
+b:1172 [in gaia_hydras.nfwfgaia]
+b:118 [in hydras.Ackermann.cPair]
+b:118 [in hydras.Ackermann.codeList]
+b:119 [in hydras.Ackermann.codeList]
+b:1192 [in gaia_hydras.nfwfgaia]
+b:1194 [in gaia_hydras.nfwfgaia]
+b:1196 [in gaia_hydras.nfwfgaia]
+B:12 [in hydras.rpo.more_list]
+b:12 [in gaia_hydras.T1Bridge]
+b:12 [in hydras.Ackermann.prLogic]
+b:12 [in hydras.Schutte.PartialFun]
+b:12 [in hydras.Ackermann.NNtheory]
+b:12 [in hydras.Schutte.Ordering_Functions]
+b:12 [in hydras.solutions_exercises.isqrt]
+b:12 [in gaia_hydras.T2Bridge]
+b:12 [in hydras.Prelude.Simple_LexProd]
+B:120 [in hydras.Prelude.MoreLists]
+b:120 [in hydras.Epsilon0.Hessenberg]
+b:120 [in hydras.Ackermann.cPair]
+b:1200 [in gaia_hydras.nfwfgaia]
+b:1209 [in gaia_hydras.nfwfgaia]
+b:1212 [in gaia_hydras.nfwfgaia]
+b:1215 [in gaia_hydras.nfwfgaia]
+b:1219 [in gaia_hydras.nfwfgaia]
+B:122 [in hydras.rpo.more_list]
+b:122 [in hydras.Epsilon0.Hessenberg]
+b:1227 [in gaia_hydras.nfwfgaia]
+b:1228 [in hydras.Ackermann.codeSubFormula]
+b:1231 [in hydras.Ackermann.codeSubFormula]
+b:1232 [in gaia_hydras.nfwfgaia]
+b:1234 [in gaia_hydras.nfwfgaia]
+b:1236 [in gaia_hydras.nfwfgaia]
+b:124 [in hydras.Gamma0.T2]
+b:1241 [in gaia_hydras.nfwfgaia]
+b:1247 [in hydras.Ackermann.codeSubFormula]
+b:1249 [in gaia_hydras.nfwfgaia]
+B:125 [in Goedel.PRrepresentable]
+b:125 [in hydras.Ackermann.folProp]
+B:125 [in hydras.OrdinalNotations.ON_Generic]
+B:125 [in hydras.Hydra.Hydra_Definitions]
+b:1250 [in hydras.Ackermann.codeSubFormula]
+b:1253 [in gaia_hydras.nfwfgaia]
+b:1256 [in gaia_hydras.nfwfgaia]
+b:126 [in hydras.Ackermann.model]
+b:126 [in hydras.Epsilon0.Hessenberg]
+b:1264 [in gaia_hydras.nfwfgaia]
+b:1269 [in gaia_hydras.nfwfgaia]
+B:127 [in hydras.Prelude.MoreLists]
+B:127 [in hydras.Hydra.Hydra_Lemmas]
+b:1272 [in gaia_hydras.nfwfgaia]
+b:1274 [in gaia_hydras.nfwfgaia]
+b:1276 [in gaia_hydras.nfwfgaia]
+b:1279 [in gaia_hydras.nfwfgaia]
+b:128 [in hydras.Gamma0.T2]
+b:1282 [in hydras.Ackermann.codeSubFormula]
+b:1284 [in gaia_hydras.nfwfgaia]
+b:1287 [in gaia_hydras.nfwfgaia]
+b:1288 [in hydras.Ackermann.codeSubFormula]
+b:129 [in hydras.Epsilon0.Hessenberg]
+b:129 [in hydras.Ackermann.folProp]
+b:129 [in hydras.Ackermann.codeSubFormula]
+b:129 [in gaia_hydras.nfwfgaia]
+b:1290 [in gaia_hydras.nfwfgaia]
+b:1293 [in gaia_hydras.nfwfgaia]
+b:1296 [in gaia_hydras.nfwfgaia]
+b:13 [in gaia_hydras.T1Choice]
+b:13 [in hydras.Prelude.WfVariant]
+b:13 [in hydras.Ackermann.extEqualNat]
+b:13 [in gaia_hydras.GPaths]
+b:13 [in hydras.Schutte.PartialFun]
+B:13 [in hydras.Prelude.STDPP_compat]
+b:13 [in hydras.Ackermann.NN]
+b:13 [in hydras.rpo.closure]
+b:13 [in hydras.solutions_exercises.T1_ltNotWf]
+B:13 [in hydras.rpo.rpo]
+b:130 [in hydras.Ackermann.model]
+b:130 [in hydras.Ackermann.codeSubFormula]
+b:1300 [in gaia_hydras.nfwfgaia]
+b:1304 [in hydras.Ackermann.codeSubFormula]
+b:1304 [in gaia_hydras.nfwfgaia]
+b:1307 [in hydras.Ackermann.codeSubFormula]
+b:1308 [in gaia_hydras.nfwfgaia]
+b:131 [in hydras.OrdinalNotations.ON_Generic]
+b:1313 [in gaia_hydras.nfwfgaia]
+b:1315 [in gaia_hydras.nfwfgaia]
+b:1319 [in gaia_hydras.nfwfgaia]
+b:132 [in hydras.Epsilon0.T1]
+B:132 [in Goedel.PRrepresentable]
+b:132 [in hydras.Gamma0.T2]
+B:132 [in hydras.Prelude.MoreLists]
+b:132 [in hydras.Epsilon0.Hessenberg]
+b:1321 [in gaia_hydras.nfwfgaia]
+b:1331 [in gaia_hydras.nfwfgaia]
+b:134 [in hydras.Ackermann.model]
+b:134 [in hydras.Ackermann.primRec]
+b:1346 [in gaia_hydras.nfwfgaia]
+b:1351 [in gaia_hydras.nfwfgaia]
+b:1356 [in gaia_hydras.nfwfgaia]
+b:136 [in hydras.Gamma0.T2]
+B:136 [in hydras.rpo.more_list]
+b:136 [in hydras.Ackermann.primRec]
+B:136 [in hydras.Hydra.Hydra_Definitions]
+b:1361 [in gaia_hydras.nfwfgaia]
+b:1366 [in gaia_hydras.nfwfgaia]
+b:137 [in hydras.Epsilon0.T1]
+B:137 [in hydras.Prelude.MoreLists]
+b:137 [in hydras.Ackermann.codeSubFormula]
+B:137 [in hydras.OrdinalNotations.ON_Generic]
+b:1371 [in gaia_hydras.nfwfgaia]
+b:1377 [in gaia_hydras.nfwfgaia]
+b:138 [in hydras.Ackermann.model]
+b:138 [in hydras.Ackermann.primRec]
+b:1384 [in hydras.Ackermann.codeSubFormula]
+b:1387 [in hydras.Ackermann.codeSubFormula]
+b:1389 [in gaia_hydras.nfwfgaia]
+b:139 [in hydras.Ackermann.codeSubFormula]
+b:14 [in gaia_hydras.T1Bridge]
+b:14 [in hydras.Ackermann.prLogic]
+b:14 [in hydras.Hydra.BigBattle]
+b:14 [in hydras.Ackermann.NNtheory]
+b:14 [in hydras.solutions_exercises.isqrt]
+b:140 [in hydras.Ackermann.LNT]
+B:140 [in hydras.Hydra.Hydra_Definitions]
+b:1402 [in hydras.Ackermann.codeSubFormula]
+b:1403 [in hydras.Ackermann.codeSubFormula]
+b:1408 [in gaia_hydras.nfwfgaia]
+b:141 [in hydras.Gamma0.T2]
+b:141 [in hydras.Ackermann.primRec]
+B:141 [in hydras.Ackermann.fol]
+b:1417 [in gaia_hydras.nfwfgaia]
+B:142 [in hydras.Prelude.MoreLists]
+b:142 [in hydras.Gamma0.Gamma0]
+b:1423 [in gaia_hydras.nfwfgaia]
+b:1429 [in gaia_hydras.nfwfgaia]
+B:143 [in Goedel.PRrepresentable]
+b:143 [in hydras.Ackermann.LNN]
+b:143 [in hydras.Ackermann.LNT]
+b:143 [in gaia_hydras.T1Bridge]
+B:143 [in hydras.Ackermann.fol]
+b:144 [in hydras.Ackermann.primRec]
+b:144 [in additions.fib]
+b:1442 [in gaia_hydras.nfwfgaia]
+b:1444 [in gaia_hydras.nfwfgaia]
+b:1446 [in gaia_hydras.nfwfgaia]
+b:1448 [in gaia_hydras.nfwfgaia]
+b:145 [in hydras.Gamma0.T2]
+B:145 [in hydras.rpo.more_list]
+b:145 [in gaia_hydras.T1Bridge]
+B:145 [in hydras.Ackermann.fol]
+b:1450 [in gaia_hydras.nfwfgaia]
+b:1459 [in gaia_hydras.nfwfgaia]
+b:146 [in hydras.Ackermann.LNN]
+b:1462 [in gaia_hydras.nfwfgaia]
+b:1464 [in gaia_hydras.nfwfgaia]
+b:1466 [in gaia_hydras.nfwfgaia]
+b:1469 [in gaia_hydras.nfwfgaia]
+b:147 [in hydras.Gamma0.T2]
+b:147 [in hydras.Ackermann.LNT]
+b:147 [in hydras.Ackermann.primRec]
+B:147 [in hydras.Hydra.Hydra_Definitions]
+b:1471 [in gaia_hydras.nfwfgaia]
+b:149 [in additions.fib]
+b:1493 [in gaia_hydras.nfwfgaia]
+b:15 [in hydras.solutions_exercises.MinPR2]
+b:15 [in hydras.Prelude.WfVariant]
+b:15 [in hydras.Ackermann.extEqualNat]
+b:15 [in hydras.solutions_exercises.MultisetWf]
+B:15 [in hydras.OrdinalNotations.ON_Generic]
+b:15 [in hydras.Ackermann.folLogic3]
+b:15 [in hydras.solutions_exercises.T1_ltNotWf]
+b:15 [in hydras.MoreAck.PrimRecExamples]
+b:15 [in hydras.Prelude.Simple_LexProd]
+b:150 [in hydras.rpo.more_list]
+b:150 [in hydras.Ackermann.LNN]
+B:150 [in hydras.Ackermann.fol]
+B:150 [in hydras.OrdinalNotations.ON_Generic]
+b:1503 [in gaia_hydras.nfwfgaia]
+b:1508 [in gaia_hydras.nfwfgaia]
+b:151 [in hydras.Ackermann.primRec]
+B:151 [in hydras.Schutte.Ordering_Functions]
+b:152 [in hydras.Ackermann.LNT]
+b:1520 [in gaia_hydras.nfwfgaia]
+b:1523 [in gaia_hydras.nfwfgaia]
+b:1526 [in gaia_hydras.nfwfgaia]
+b:153 [in gaia_hydras.nfwfgaia]
+b:1532 [in gaia_hydras.nfwfgaia]
+b:1537 [in gaia_hydras.nfwfgaia]
+B:154 [in Goedel.PRrepresentable]
+b:154 [in hydras.Ackermann.fol]
+b:154 [in hydras.Ackermann.codePA]
+b:1541 [in gaia_hydras.nfwfgaia]
+b:1544 [in gaia_hydras.nfwfgaia]
+B:155 [in hydras.rpo.more_list]
+b:155 [in hydras.Ackermann.LNN]
+b:155 [in hydras.Ackermann.primRec]
+b:155 [in gaia_hydras.nfwfgaia]
+b:1552 [in gaia_hydras.nfwfgaia]
+b:1559 [in gaia_hydras.nfwfgaia]
+b:156 [in hydras.Ackermann.fol]
+b:1562 [in gaia_hydras.nfwfgaia]
+b:1565 [in gaia_hydras.nfwfgaia]
+b:1569 [in gaia_hydras.nfwfgaia]
+b:157 [in hydras.Ackermann.LNT]
+b:157 [in gaia_hydras.nfwfgaia]
+b:157 [in hydras.Ackermann.codePA]
+B:157 [in hydras.Hydra.Hydra_Definitions]
+b:1575 [in gaia_hydras.nfwfgaia]
+b:1579 [in gaia_hydras.nfwfgaia]
+b:1588 [in gaia_hydras.nfwfgaia]
+b:159 [in hydras.Ackermann.primRec]
+b:159 [in hydras.Ackermann.fol]
+b:159 [in gaia_hydras.nfwfgaia]
+b:1593 [in gaia_hydras.nfwfgaia]
+b:1596 [in gaia_hydras.nfwfgaia]
+b:1599 [in gaia_hydras.nfwfgaia]
+B:16 [in hydras.Prelude.DecPreOrder_Instances]
+B:16 [in hydras.rpo.more_list]
+b:16 [in hydras.Prelude.MoreDecidable]
+b:16 [in hydras.rpo.dickson]
+b:16 [in hydras.Schutte.Ordering_Functions]
+B:16 [in hydras.MoreAck.FolExamples]
+b:16 [in gaia_hydras.GHessenberg]
+b:160 [in hydras.Ackermann.LNN]
+b:1602 [in gaia_hydras.nfwfgaia]
+b:161 [in gaia_hydras.nfwfgaia]
+b:1613 [in gaia_hydras.nfwfgaia]
+b:1622 [in gaia_hydras.nfwfgaia]
+b:163 [in hydras.Ackermann.LNN]
+B:163 [in hydras.OrdinalNotations.ON_Generic]
+b:1630 [in gaia_hydras.nfwfgaia]
+b:1636 [in gaia_hydras.nfwfgaia]
+B:164 [in hydras.rpo.more_list]
+B:164 [in hydras.Hydra.Hydra_Definitions]
+b:165 [in hydras.Ackermann.primRec]
+b:165 [in hydras.Ackermann.fol]
+b:1651 [in gaia_hydras.nfwfgaia]
+b:166 [in hydras.Epsilon0.T1]
+b:166 [in hydras.Gamma0.Gamma0]
+b:1661 [in gaia_hydras.nfwfgaia]
+b:167 [in hydras.Ackermann.primRec]
+b:167 [in hydras.Ackermann.cPair]
+b:1672 [in gaia_hydras.nfwfgaia]
+b:1675 [in gaia_hydras.nfwfgaia]
+b:168 [in hydras.Ackermann.fol]
+b:168 [in hydras.rpo.rpo]
+b:169 [in hydras.Epsilon0.T1]
+b:169 [in gaia_hydras.T1Bridge]
+b:169 [in hydras.Ackermann.primRec]
+b:17 [in hydras.solutions_exercises.MinPR2]
+b:17 [in additions.Fib2]
+b:17 [in hydras.Prelude.Comparable]
+b:17 [in hydras.Prelude.Sort_spec]
+b:17 [in hydras.Epsilon0.Hessenberg]
+b:17 [in hydras.Hydra.BigBattle]
+B:17 [in hydras.Ackermann.fol]
+b:17 [in Goedel.rosser]
+b:170 [in hydras.OrdinalNotations.ON_Generic]
+b:170 [in hydras.Gamma0.Gamma0]
+B:171 [in hydras.rpo.more_list]
+b:171 [in hydras.Ackermann.model]
+b:171 [in hydras.Ackermann.fol]
+b:172 [in hydras.Epsilon0.T1]
+B:172 [in hydras.Epsilon0.Large_Sets]
+b:172 [in gaia_hydras.nfwfgaia]
+b:173 [in hydras.Ackermann.model]
+b:173 [in hydras.Prelude.Merge_Sort]
+b:173 [in hydras.Ackermann.primRec]
+b:1737 [in gaia_hydras.nfwfgaia]
+b:1739 [in gaia_hydras.nfwfgaia]
+b:174 [in hydras.Ackermann.folProp]
+B:174 [in hydras.Schutte.Ordering_Functions]
+b:1741 [in gaia_hydras.nfwfgaia]
+b:1743 [in gaia_hydras.nfwfgaia]
+b:1746 [in gaia_hydras.nfwfgaia]
+b:175 [in hydras.Ackermann.primRec]
+b:175 [in hydras.Ackermann.cPair]
+B:175 [in hydras.OrdinalNotations.ON_Generic]
+b:1750 [in gaia_hydras.nfwfgaia]
+b:1753 [in gaia_hydras.nfwfgaia]
+b:176 [in hydras.Ackermann.model]
+b:1769 [in gaia_hydras.nfwfgaia]
+b:177 [in hydras.rpo.more_list]
+b:177 [in hydras.Ackermann.primRec]
+b:177 [in hydras.Ackermann.cPair]
+B:177 [in hydras.Schutte.Ordering_Functions]
+b:177 [in gaia_hydras.nfwfgaia]
+b:1772 [in gaia_hydras.nfwfgaia]
+b:1774 [in gaia_hydras.nfwfgaia]
+b:178 [in hydras.Ackermann.fol]
+b:178 [in gaia_hydras.nfwfgaia]
+B:179 [in hydras.rpo.more_list]
+b:179 [in hydras.Ackermann.cPair]
+b:1792 [in gaia_hydras.nfwfgaia]
+b:1797 [in gaia_hydras.nfwfgaia]
+b:1799 [in gaia_hydras.nfwfgaia]
+b:18 [in gaia_hydras.GPaths]
+b:18 [in hydras.OrdinalNotations.OmegaOmega]
+b:18 [in hydras.Ackermann.codePA]
+B:18 [in hydras.rpo.rpo]
+b:180 [in hydras.Epsilon0.T1]
+b:180 [in hydras.Ackermann.model]
+B:180 [in hydras.Schutte.Ordering_Functions]
+b:1803 [in gaia_hydras.nfwfgaia]
+b:1805 [in gaia_hydras.nfwfgaia]
+b:1807 [in gaia_hydras.nfwfgaia]
+b:1809 [in gaia_hydras.nfwfgaia]
+b:181 [in hydras.Ackermann.folProp]
+b:181 [in hydras.Ackermann.primRec]
+b:181 [in hydras.Ackermann.cPair]
+b:181 [in gaia_hydras.nfwfgaia]
+b:1811 [in gaia_hydras.nfwfgaia]
+b:1814 [in gaia_hydras.nfwfgaia]
+b:1816 [in gaia_hydras.nfwfgaia]
+B:182 [in hydras.Schutte.Ordering_Functions]
+b:183 [in hydras.Epsilon0.T1]
+b:183 [in hydras.Ackermann.model]
+B:183 [in additions.Addition_Chains]
+b:183 [in gaia_hydras.nfwfgaia]
+b:1836 [in gaia_hydras.nfwfgaia]
+b:1838 [in gaia_hydras.nfwfgaia]
+b:184 [in hydras.Epsilon0.T1]
+b:184 [in hydras.Ackermann.primRec]
+b:184 [in hydras.Ackermann.fol]
+b:1840 [in gaia_hydras.nfwfgaia]
+b:1842 [in gaia_hydras.nfwfgaia]
+b:1844 [in gaia_hydras.nfwfgaia]
+b:1857 [in gaia_hydras.nfwfgaia]
+b:186 [in additions.Addition_Chains]
+b:186 [in hydras.Schutte.Ordering_Functions]
+b:1866 [in gaia_hydras.nfwfgaia]
+B:187 [in hydras.rpo.more_list]
+b:187 [in hydras.Ackermann.model]
+b:187 [in hydras.Ackermann.primRec]
+b:1870 [in gaia_hydras.nfwfgaia]
+b:1878 [in gaia_hydras.nfwfgaia]
+b:188 [in hydras.Epsilon0.T1]
+B:188 [in hydras.OrdinalNotations.ON_Generic]
+b:188 [in hydras.rpo.rpo]
+b:1887 [in gaia_hydras.nfwfgaia]
+b:19 [in hydras.Prelude.Comparable]
+B:19 [in hydras.Ackermann.fol]
+b:19 [in hydras.Schutte.PartialFun]
+B:19 [in gaia_hydras.GLarge_Sets]
+B:19 [in hydras.OrdinalNotations.ON_Generic]
+B:19 [in hydras.Schutte.Ordering_Functions]
+b:190 [in hydras.Ackermann.folProp]
+b:190 [in hydras.Ackermann.fol]
+b:190 [in hydras.rpo.rpo]
+b:192 [in hydras.Ackermann.primRec]
+b:193 [in hydras.Ackermann.model]
+b:195 [in hydras.Ackermann.model]
+b:195 [in hydras.Ackermann.primRec]
+b:197 [in hydras.Ackermann.model]
+b:197 [in hydras.Ackermann.folProp]
+b:198 [in hydras.Ackermann.primRec]
+b:199 [in hydras.Epsilon0.T1]
+b:199 [in hydras.Ackermann.model]
+B:2 [in hydras.Prelude.DecPreOrder_Instances]
+B:2 [in hydras.rpo.more_list]
+b:2 [in hydras.Ackermann.prLogic]
+b:2 [in hydras.solutions_exercises.MinPR]
+b:2 [in hydras.Ackermann.NNtheory]
+b:2 [in hydras.solutions_exercises.isqrt]
+b:2 [in hydras.Gamma0.Gamma0]
+B:2 [in hydras.rpo.rpo]
+b:20 [in hydras.Ackermann.cPair]
+b:20 [in hydras.Schutte.PartialFun]
+B:20 [in Goedel.rosser]
+b:20 [in gaia_hydras.GHessenberg]
+b:2007 [in gaia_hydras.nfwfgaia]
+B:202 [in hydras.rpo.more_list]
+b:202 [in hydras.Ackermann.model]
+B:202 [in hydras.OrdinalNotations.ON_Generic]
+b:2022 [in gaia_hydras.nfwfgaia]
+b:2026 [in gaia_hydras.nfwfgaia]
+b:203 [in hydras.Ackermann.primRec]
+b:2031 [in gaia_hydras.nfwfgaia]
+b:2038 [in gaia_hydras.nfwfgaia]
+b:2048 [in gaia_hydras.nfwfgaia]
+b:2051 [in gaia_hydras.nfwfgaia]
+b:2054 [in gaia_hydras.nfwfgaia]
+b:2059 [in gaia_hydras.nfwfgaia]
+b:206 [in hydras.Ackermann.model]
+b:206 [in hydras.Ackermann.folProp]
+b:206 [in hydras.Ackermann.primRec]
+b:206 [in gaia_hydras.nfwfgaia]
+b:2062 [in gaia_hydras.nfwfgaia]
+b:2071 [in gaia_hydras.nfwfgaia]
+b:2074 [in gaia_hydras.nfwfgaia]
+b:209 [in hydras.Ackermann.model]
+b:209 [in hydras.Ackermann.primRec]
+b:209 [in gaia_hydras.nfwfgaia]
+B:21 [in hydras.rpo.more_list]
+b:21 [in hydras.Prelude.Comparable]
+b:21 [in gaia_hydras.GCanon]
+b:21 [in hydras.Hydra.BigBattle]
+B:21 [in hydras.Ackermann.fol]
+b:21 [in additions.Monoid_instances]
+b:21 [in hydras.Schutte.PartialFun]
+b:21 [in hydras.Schutte.Well_Orders]
+B:21 [in hydras.Schutte.Ordering_Functions]
+b:21 [in hydras.Ackermann.codePA]
+b:210 [in hydras.OrdinalNotations.ON_Generic]
+B:211 [in hydras.rpo.more_list]
+b:212 [in hydras.OrdinalNotations.ON_Generic]
+b:212 [in gaia_hydras.nfwfgaia]
+b:213 [in hydras.Epsilon0.T1]
+b:213 [in hydras.Ackermann.model]
+b:213 [in hydras.Ackermann.folProp]
+b:214 [in hydras.Ackermann.primRec]
+b:214 [in hydras.Ackermann.fol]
+b:214 [in hydras.OrdinalNotations.ON_Generic]
+b:215 [in hydras.Epsilon0.T1]
+b:215 [in gaia_hydras.nfwfgaia]
+b:216 [in hydras.Gamma0.Gamma0]
+b:217 [in hydras.Epsilon0.T1]
+b:217 [in hydras.Ackermann.primRec]
+b:217 [in hydras.Ackermann.fol]
+b:218 [in hydras.Gamma0.Gamma0]
+b:219 [in hydras.rpo.more_list]
+b:219 [in hydras.Ackermann.model]
+b:219 [in hydras.Ackermann.fol]
+b:22 [in additions.Fib2]
+b:22 [in gaia_hydras.GPaths]
+b:22 [in hydras.Ackermann.cPair]
+b:22 [in gaia_hydras.GHydra]
+b:22 [in hydras.Prelude.MoreDecidable]
+b:22 [in hydras.Ackermann.NN]
+b:22 [in gaia_hydras.GHessenberg]
+b:220 [in hydras.Ackermann.primRec]
+b:220 [in gaia_hydras.nfwfgaia]
+b:221 [in hydras.Epsilon0.T1]
+B:221 [in hydras.rpo.more_list]
+b:221 [in hydras.Ackermann.model]
+b:222 [in hydras.Ackermann.folProp]
+b:222 [in hydras.Ackermann.cPair]
+b:223 [in hydras.Ackermann.primRec]
+b:223 [in hydras.Ackermann.codeList]
+b:223 [in gaia_hydras.nfwfgaia]
+b:224 [in hydras.Ackermann.codeList]
+b:224 [in hydras.Gamma0.Gamma0]
+b:226 [in hydras.Ackermann.primRec]
+b:226 [in hydras.Ackermann.cPair]
+b:227 [in hydras.Epsilon0.T1]
+b:228 [in hydras.Gamma0.Gamma0]
+b:229 [in hydras.rpo.more_list]
+b:229 [in hydras.Ackermann.folProp]
+b:229 [in gaia_hydras.nfwfgaia]
+B:23 [in hydras.Ackermann.folProof]
+b:23 [in hydras.Prelude.Comparable]
+b:23 [in hydras.Epsilon0.Hessenberg]
+b:23 [in hydras.OrdinalNotations.OmegaOmega]
+B:23 [in hydras.Schutte.Ordering_Functions]
+B:23 [in Goedel.rosser]
+B:23 [in hydras.Ackermann.expressible]
+b:23 [in hydras.Prelude.MoreVectors]
+b:230 [in hydras.rpo.more_list]
+b:230 [in hydras.Ackermann.primRec]
+b:230 [in hydras.Ackermann.cPair]
+b:232 [in hydras.Epsilon0.T1]
+b:232 [in hydras.Gamma0.Gamma0]
+b:233 [in hydras.rpo.more_list]
+b:233 [in hydras.Ackermann.primRec]
+B:234 [in hydras.Ackermann.checkPrf]
+b:234 [in hydras.Ackermann.cPair]
+b:234 [in gaia_hydras.nfwfgaia]
+b:235 [in hydras.Ackermann.cPair]
+b:236 [in hydras.Epsilon0.T1]
+b:236 [in hydras.rpo.more_list]
+b:236 [in hydras.Ackermann.primRec]
+b:236 [in gaia_hydras.nfwfgaia]
+b:236 [in hydras.rpo.rpo]
+b:237 [in hydras.rpo.more_list]
+b:237 [in hydras.Ackermann.cPair]
+b:237 [in hydras.Ackermann.fol]
+b:239 [in hydras.Epsilon0.T1]
+B:239 [in hydras.rpo.more_list]
+b:239 [in hydras.Ackermann.primRec]
+b:239 [in gaia_hydras.nfwfgaia]
+B:24 [in Goedel.fixPoint]
+b:24 [in hydras.Prelude.DecPreOrder]
+b:24 [in hydras.Ackermann.cPair]
+b:24 [in hydras.Schutte.Well_Orders]
+b:24 [in hydras.Gamma0.Gamma0]
+b:24 [in hydras.rpo.closure]
+b:24 [in gaia_hydras.GHessenberg]
+b:242 [in hydras.Ackermann.primRec]
+b:245 [in hydras.rpo.rpo]
+b:246 [in hydras.Epsilon0.T1]
+b:246 [in hydras.Ackermann.primRec]
+b:247 [in hydras.Ackermann.cPair]
+B:248 [in hydras.rpo.more_list]
+b:248 [in hydras.rpo.rpo]
+b:249 [in hydras.Ackermann.primRec]
+b:249 [in hydras.rpo.rpo]
+b:25 [in hydras.Schutte.Correctness_E0]
+b:25 [in hydras.Prelude.More_Arith]
+b:25 [in hydras.Epsilon0.F_omega]
+b:25 [in hydras.OrdinalNotations.OmegaOmega]
+B:25 [in hydras.Ackermann.fol]
+b:25 [in hydras.Schutte.PartialFun]
+B:25 [in hydras.Schutte.Ordering_Functions]
+b:25 [in gaia_hydras.nfwfgaia]
+b:250 [in hydras.Epsilon0.T1]
+b:252 [in hydras.Ackermann.primRec]
+b:252 [in hydras.Ackermann.cPair]
+b:253 [in hydras.Ackermann.folProp]
+b:253 [in hydras.Ackermann.cPair]
+b:253 [in hydras.rpo.rpo]
+b:254 [in hydras.rpo.rpo]
+b:255 [in hydras.Epsilon0.T1]
+b:255 [in hydras.Ackermann.primRec]
+b:256 [in hydras.rpo.more_list]
+b:257 [in hydras.rpo.more_list]
+b:258 [in hydras.rpo.more_list]
+b:258 [in hydras.Ackermann.primRec]
+b:259 [in hydras.Epsilon0.T1]
+B:26 [in hydras.rpo.more_list]
+b:26 [in gaia_hydras.GPaths]
+b:26 [in hydras.rpo.closure]
+B:260 [in hydras.rpo.more_list]
+b:260 [in hydras.Ackermann.folProp]
+b:260 [in hydras.Ackermann.cPair]
+b:261 [in hydras.rpo.rpo]
+b:262 [in hydras.Epsilon0.T1]
+B:262 [in Goedel.PRrepresentable]
+b:262 [in hydras.Ackermann.primRec]
+b:262 [in gaia_hydras.nfwfgaia]
+b:263 [in hydras.Ackermann.cPair]
+b:265 [in hydras.Epsilon0.T1]
+b:265 [in gaia_hydras.nfwfgaia]
+B:266 [in Goedel.PRrepresentable]
+B:266 [in hydras.Ackermann.checkPrf]
+b:266 [in hydras.Ackermann.primRec]
+b:267 [in hydras.Epsilon0.T1]
+b:267 [in gaia_hydras.nfwfgaia]
+b:268 [in hydras.rpo.more_list]
+b:269 [in hydras.rpo.more_list]
+b:269 [in hydras.Ackermann.folProp]
+b:269 [in hydras.Ackermann.fol]
+b:269 [in gaia_hydras.nfwfgaia]
+B:27 [in hydras.Prelude.DecPreOrder_Instances]
+b:27 [in hydras.Prelude.Comparable]
+b:27 [in hydras.OrdinalNotations.OmegaOmega]
+b:27 [in hydras.Ackermann.PA]
+b:27 [in hydras.Schutte.PartialFun]
+b:27 [in gaia_hydras.GHessenberg]
+b:27 [in hydras.MoreAck.PrimRecExamples]
+B:27 [in hydras.Ackermann.expressible]
+b:270 [in Goedel.PRrepresentable]
+b:270 [in hydras.rpo.more_list]
+b:270 [in hydras.Ackermann.primRec]
+b:271 [in hydras.Epsilon0.T1]
+b:271 [in Goedel.PRrepresentable]
+b:273 [in hydras.Epsilon0.T1]
+B:273 [in Goedel.PRrepresentable]
+b:273 [in hydras.Ackermann.fol]
+b:273 [in hydras.Gamma0.Gamma0]
+b:273 [in gaia_hydras.nfwfgaia]
+b:274 [in hydras.Ackermann.primRec]
+b:274 [in gaia_hydras.nfwfgaia]
+b:275 [in hydras.Epsilon0.T1]
+b:276 [in hydras.Ackermann.folProp]
+b:276 [in hydras.Ackermann.fol]
+b:277 [in hydras.Epsilon0.T1]
+b:278 [in hydras.Ackermann.primRec]
+B:28 [in hydras.Ackermann.folProof]
+b:28 [in hydras.Prelude.More_Arith]
+b:28 [in hydras.Ackermann.extEqualNat]
+b:28 [in hydras.rpo.more_list]
+B:28 [in Goedel.fixPoint]
+b:28 [in hydras.solutions_exercises.MultisetWf]
+b:28 [in Goedel.rosserPA]
+b:281 [in hydras.Ackermann.fol]
+b:283 [in hydras.Ackermann.primRec]
+b:284 [in hydras.Epsilon0.T1]
+b:284 [in hydras.Ackermann.fol]
+b:284 [in hydras.Gamma0.Gamma0]
+B:286 [in Goedel.PRrepresentable]
+b:287 [in hydras.Ackermann.primRec]
+b:288 [in hydras.Gamma0.Gamma0]
+b:29 [in hydras.Epsilon0.Epsilon0rpo]
+b:29 [in hydras.Epsilon0.Hessenberg]
+b:29 [in hydras.Prelude.DecPreOrder]
+b:29 [in hydras.OrdinalNotations.OmegaOmega]
+b:29 [in hydras.Schutte.PartialFun]
+B:29 [in hydras.Schutte.Ordering_Functions]
+b:29 [in gaia_hydras.GHessenberg]
+b:290 [in additions.Euclidean_Chains]
+b:291 [in hydras.Ackermann.primRec]
+B:292 [in Goedel.PRrepresentable]
+b:292 [in hydras.Gamma0.Gamma0]
+b:293 [in gaia_hydras.nfwfgaia]
+b:295 [in additions.Euclidean_Chains]
+b:296 [in hydras.Ackermann.primRec]
+b:296 [in hydras.Gamma0.Gamma0]
+b:3 [in gaia_hydras.GPaths]
+b:3 [in hydras.Ackermann.ListExt]
+B:30 [in hydras.Ackermann.folProof]
+b:30 [in hydras.Schutte.Correctness_E0]
+b:30 [in hydras.Ackermann.codeSubTerm]
+b:30 [in hydras.Prelude.Comparable]
+b:30 [in hydras.Hydra.BigBattle]
+b:30 [in hydras.Gamma0.Gamma0]
+b:30 [in hydras.MoreAck.PrimRecExamples]
+b:300 [in Goedel.PRrepresentable]
+b:300 [in hydras.Ackermann.primRec]
+b:304 [in hydras.Ackermann.primRec]
+b:307 [in Goedel.PRrepresentable]
+b:307 [in hydras.Gamma0.Gamma0]
+b:308 [in hydras.Ackermann.primRec]
+b:31 [in hydras.Ackermann.extEqualNat]
+b:31 [in hydras.Ackermann.codeSubTerm]
+b:31 [in hydras.OrdinalNotations.OmegaOmega]
+B:31 [in Goedel.rosserPA]
+b:31 [in hydras.Schutte.PartialFun]
+b:31 [in hydras.Ackermann.NN]
+b:31 [in gaia_hydras.GHessenberg]
+b:311 [in hydras.Epsilon0.T1]
+b:311 [in Goedel.PRrepresentable]
+b:312 [in hydras.Ackermann.primRec]
+b:316 [in hydras.Ackermann.primRec]
+b:319 [in Goedel.PRrepresentable]
+B:319 [in hydras.Ackermann.checkPrf]
+b:319 [in hydras.Ackermann.fol]
+b:32 [in hydras.Epsilon0.T1]
+b:32 [in hydras.Ackermann.codeSubTerm]
+B:32 [in hydras.rpo.more_list]
+b:32 [in hydras.Ackermann.folLogic3]
+b:320 [in hydras.Ackermann.primRec]
+b:323 [in hydras.Ackermann.fol]
+b:325 [in hydras.Ackermann.primRec]
+b:326 [in hydras.Ackermann.fol]
+B:328 [in Goedel.PRrepresentable]
+b:329 [in hydras.Ackermann.primRec]
+B:33 [in hydras.Ackermann.folProof]
+b:33 [in hydras.Ackermann.codeSubTerm]
+b:33 [in hydras.Prelude.Comparable]
+b:33 [in hydras.OrdinalNotations.OmegaOmega]
+B:33 [in hydras.OrdinalNotations.ON_Generic]
+B:33 [in hydras.Schutte.Ordering_Functions]
+b:331 [in hydras.Ackermann.fol]
+B:333 [in Goedel.PRrepresentable]
+b:333 [in hydras.Ackermann.primRec]
+b:337 [in hydras.Ackermann.primRec]
+B:338 [in Goedel.PRrepresentable]
+b:339 [in hydras.Ackermann.cPair]
+b:34 [in hydras.Prelude.More_Arith]
+b:34 [in hydras.Ackermann.codeSubTerm]
+b:34 [in hydras.Epsilon0.Hessenberg]
+B:34 [in Goedel.rosserPA]
+b:34 [in hydras.Ackermann.cPair]
+b:34 [in gaia_hydras.GHessenberg]
+b:341 [in hydras.Ackermann.primRec]
+b:341 [in hydras.Ackermann.cPair]
+B:342 [in hydras.Ackermann.subAll]
+B:347 [in hydras.Ackermann.subAll]
+b:347 [in hydras.Ackermann.cPair]
+b:348 [in hydras.Epsilon0.T1]
+b:35 [in hydras.Ackermann.codeSubTerm]
+b:35 [in gaia_hydras.GCanon]
+b:35 [in hydras.OrdinalNotations.OmegaOmega]
+b:35 [in hydras.Schutte.PartialFun]
+b:35 [in hydras.Schutte.Well_Orders]
+b:35 [in hydras.Prelude.MoreVectors]
+b:350 [in hydras.Ackermann.cPair]
+b:352 [in Goedel.PRrepresentable]
+B:352 [in hydras.Ackermann.subAll]
+b:353 [in hydras.Ackermann.cPair]
+b:354 [in gaia_hydras.nfwfgaia]
+b:356 [in Goedel.PRrepresentable]
+b:356 [in hydras.Ackermann.cPair]
+b:356 [in gaia_hydras.nfwfgaia]
+b:357 [in hydras.Epsilon0.T1]
+b:358 [in hydras.Ackermann.cPair]
+b:358 [in gaia_hydras.nfwfgaia]
+b:36 [in additions.Compatibility]
+b:36 [in hydras.Prelude.Comparable]
+b:36 [in hydras.Ackermann.folProp]
+b:36 [in hydras.Ackermann.PA]
+b:36 [in hydras.Schutte.Schutte_basics]
+B:36 [in hydras.MoreAck.FolExamples]
+b:360 [in Goedel.PRrepresentable]
+b:360 [in hydras.Ackermann.cPair]
+b:360 [in gaia_hydras.nfwfgaia]
+b:362 [in hydras.Epsilon0.T1]
+b:363 [in hydras.Ackermann.cPair]
+b:364 [in Goedel.PRrepresentable]
+b:364 [in gaia_hydras.nfwfgaia]
+b:365 [in hydras.Ackermann.fol]
+b:366 [in hydras.Ackermann.cPair]
+b:369 [in hydras.Ackermann.cPair]
+b:369 [in hydras.Ackermann.fol]
+b:37 [in gaia_hydras.GPaths]
+b:37 [in hydras.Hydra.BigBattle]
+b:37 [in gaia_hydras.GHessenberg]
+b:372 [in hydras.Ackermann.fol]
+b:376 [in hydras.Epsilon0.T1]
+b:38 [in hydras.OrdinalNotations.OmegaOmega]
+b:38 [in hydras.Ackermann.cPair]
+b:38 [in hydras.Ackermann.wellFormed]
+b:38 [in gaia_hydras.onType]
+b:38 [in hydras.Schutte.Well_Orders]
+b:383 [in hydras.Epsilon0.T1]
+b:386 [in Goedel.PRrepresentable]
+b:387 [in hydras.Epsilon0.T1]
+b:39 [in hydras.Prelude.Comparable]
+b:39 [in hydras.Ackermann.NN]
+b:39 [in hydras.Schutte.Ordering_Functions]
+b:390 [in Goedel.PRrepresentable]
+b:392 [in hydras.Epsilon0.T1]
+b:394 [in Goedel.PRrepresentable]
+b:394 [in hydras.Ackermann.checkPrf]
+b:395 [in gaia_hydras.nfwfgaia]
+b:397 [in gaia_hydras.nfwfgaia]
+b:398 [in Goedel.PRrepresentable]
+b:398 [in hydras.Ackermann.checkPrf]
+b:4 [in hydras.solutions_exercises.MinPR2]
+b:4 [in hydras.Ackermann.folLogic2]
+b:4 [in hydras.Ackermann.prLogic]
+b:4 [in hydras.Ackermann.NNtheory]
+b:4 [in hydras.Schutte.Well_Orders]
+b:4 [in hydras.Ackermann.NN]
+b:4 [in hydras.solutions_exercises.isqrt]
+B:40 [in hydras.Ackermann.folProof]
+b:40 [in hydras.Epsilon0.Hessenberg]
+b:40 [in hydras.Ackermann.cPair]
+b:40 [in hydras.Ackermann.wellFormed]
+b:40 [in gaia_hydras.onType]
+b:40 [in hydras.Schutte.Ordering_Functions]
+b:400 [in gaia_hydras.nfwfgaia]
+b:401 [in Goedel.PRrepresentable]
+b:403 [in Goedel.PRrepresentable]
+b:404 [in hydras.Epsilon0.T1]
+b:406 [in Goedel.PRrepresentable]
+b:409 [in hydras.Ackermann.fol]
+b:409 [in gaia_hydras.nfwfgaia]
+b:41 [in hydras.Epsilon0.Epsilon0rpo]
+b:41 [in hydras.Prelude.Comparable]
+b:41 [in hydras.Schutte.PartialFun]
+b:410 [in Goedel.PRrepresentable]
+b:412 [in gaia_hydras.nfwfgaia]
+b:413 [in hydras.Ackermann.fol]
+b:414 [in Goedel.PRrepresentable]
+b:415 [in hydras.Ackermann.primRec]
+b:416 [in hydras.Ackermann.fol]
+b:416 [in gaia_hydras.nfwfgaia]
+b:417 [in hydras.Epsilon0.T1]
+b:417 [in hydras.Ackermann.primRec]
+b:418 [in Goedel.PRrepresentable]
+b:419 [in hydras.Ackermann.primRec]
+b:42 [in hydras.Ackermann.cPair]
+b:42 [in hydras.Schutte.Schutte_basics]
+b:42 [in hydras.Schutte.Well_Orders]
+b:421 [in Goedel.PRrepresentable]
+b:421 [in hydras.Ackermann.primRec]
+b:424 [in hydras.Ackermann.fol]
+b:424 [in gaia_hydras.nfwfgaia]
+b:425 [in Goedel.PRrepresentable]
+b:427 [in hydras.Epsilon0.T1]
+b:428 [in gaia_hydras.nfwfgaia]
+b:429 [in hydras.Ackermann.fol]
+b:43 [in hydras.Prelude.DecPreOrder]
+b:43 [in additions.More_on_positive]
+B:43 [in hydras.Schutte.Schutte_basics]
+b:430 [in hydras.Ackermann.primRec]
+b:430 [in hydras.Gamma0.Gamma0]
+b:432 [in hydras.Ackermann.fol]
+b:433 [in hydras.Epsilon0.T1]
+b:433 [in gaia_hydras.nfwfgaia]
+b:434 [in hydras.Ackermann.fol]
+b:436 [in gaia_hydras.nfwfgaia]
+b:437 [in hydras.Ackermann.primRec]
+b:438 [in hydras.Epsilon0.T1]
+b:438 [in gaia_hydras.nfwfgaia]
+b:44 [in hydras.Epsilon0.T1]
+b:44 [in hydras.Epsilon0.Epsilon0rpo]
+b:44 [in hydras.rpo.more_list]
+b:44 [in gaia_hydras.T1Bridge]
+b:44 [in hydras.rpo.dickson]
+b:442 [in gaia_hydras.nfwfgaia]
+b:443 [in hydras.Epsilon0.T1]
+b:444 [in hydras.Ackermann.primRec]
+b:444 [in gaia_hydras.nfwfgaia]
+b:445 [in hydras.Ackermann.fol]
+b:446 [in hydras.Ackermann.checkPrf]
+b:446 [in hydras.Ackermann.primRec]
+b:446 [in gaia_hydras.nfwfgaia]
+b:447 [in hydras.Epsilon0.T1]
+b:448 [in gaia_hydras.nfwfgaia]
+b:449 [in hydras.Epsilon0.T1]
+b:45 [in Goedel.PRrepresentable]
+b:45 [in hydras.Prelude.DecPreOrder_Instances]
+b:45 [in hydras.Prelude.Comparable]
+b:45 [in hydras.Epsilon0.Hessenberg]
+b:45 [in hydras.Ackermann.PA]
+b:45 [in hydras.Prelude.MoreOrders]
+b:45 [in hydras.Schutte.PartialFun]
+b:450 [in hydras.Ackermann.checkPrf]
+b:451 [in hydras.Ackermann.primRec]
+b:451 [in gaia_hydras.nfwfgaia]
+b:452 [in hydras.Epsilon0.T1]
+b:453 [in gaia_hydras.nfwfgaia]
+B:454 [in hydras.Ackermann.checkPrf]
+b:455 [in hydras.Epsilon0.T1]
+b:457 [in hydras.Ackermann.primRec]
+b:458 [in gaia_hydras.nfwfgaia]
+b:459 [in hydras.Epsilon0.T1]
+b:459 [in hydras.Ackermann.primRec]
+b:46 [in hydras.solutions_exercises.MultisetWf]
+b:46 [in gaia_hydras.T1Bridge]
+b:46 [in hydras.Ackermann.cPair]
+B:46 [in hydras.Schutte.Ordering_Functions]
+B:46 [in hydras.Ackermann.expressible]
+b:460 [in gaia_hydras.nfwfgaia]
+b:462 [in hydras.Epsilon0.T1]
+b:463 [in gaia_hydras.nfwfgaia]
+b:464 [in hydras.Ackermann.primRec]
+b:465 [in hydras.Epsilon0.T1]
+b:465 [in gaia_hydras.nfwfgaia]
+b:467 [in gaia_hydras.nfwfgaia]
+b:469 [in hydras.Epsilon0.T1]
+b:469 [in gaia_hydras.nfwfgaia]
+b:47 [in hydras.Epsilon0.T1]
+b:47 [in hydras.Epsilon0.Epsilon0rpo]
+b:47 [in hydras.Prelude.Comparable]
+B:47 [in hydras.Prelude.Sort_spec]
+b:47 [in additions.More_on_positive]
+B:47 [in hydras.Schutte.Schutte_basics]
+b:471 [in hydras.Epsilon0.T1]
+b:474 [in gaia_hydras.nfwfgaia]
+b:477 [in hydras.Epsilon0.T1]
+b:478 [in hydras.Ackermann.primRec]
+b:479 [in hydras.Epsilon0.T1]
+b:479 [in hydras.Ackermann.primRec]
+b:48 [in Goedel.PRrepresentable]
+b:48 [in hydras.Prelude.More_Arith]
+b:48 [in hydras.Ackermann.codeSubTerm]
+b:48 [in hydras.Prelude.DecPreOrder]
+b:483 [in hydras.Gamma0.Gamma0]
+b:484 [in hydras.Ackermann.primRec]
+b:487 [in hydras.Ackermann.checkPrf]
+b:49 [in hydras.Epsilon0.T1]
+b:49 [in hydras.Prelude.DecPreOrder_Instances]
+b:49 [in hydras.Ackermann.codeSubTerm]
+b:49 [in hydras.Prelude.Comparable]
+b:49 [in hydras.Ackermann.code]
+b:49 [in hydras.Hydra.BigBattle]
+b:490 [in hydras.Ackermann.primRec]
+b:490 [in hydras.Gamma0.Gamma0]
+b:491 [in hydras.Ackermann.checkPrf]
+b:491 [in gaia_hydras.nfwfgaia]
+b:493 [in gaia_hydras.nfwfgaia]
+b:495 [in hydras.Ackermann.primRec]
+b:495 [in gaia_hydras.nfwfgaia]
+b:497 [in hydras.Epsilon0.T1]
+b:5 [in hydras.Prelude.Restriction]
+b:5 [in hydras.Prelude.DecPreOrder]
+b:5 [in hydras.Ackermann.cPair]
+B:5 [in hydras.OrdinalNotations.ON_mult]
+B:5 [in hydras.OrdinalNotations.ON_plus]
+b:50 [in hydras.Ackermann.codeSubTerm]
+b:50 [in hydras.Epsilon0.Hessenberg]
+b:50 [in gaia_hydras.T1Bridge]
+b:50 [in hydras.Ackermann.cPair]
+b:50 [in hydras.Schutte.Well_Orders]
+B:50 [in hydras.Schutte.Ordering_Functions]
+b:501 [in hydras.Epsilon0.T1]
+b:503 [in hydras.Ackermann.checkPrf]
+b:504 [in hydras.Epsilon0.T1]
+b:504 [in hydras.Ackermann.primRec]
+b:506 [in hydras.Ackermann.primRec]
+b:507 [in hydras.Ackermann.checkPrf]
+b:508 [in hydras.Ackermann.primRec]
+b:51 [in hydras.Epsilon0.T1]
+b:51 [in hydras.Ackermann.codeSubTerm]
+b:51 [in additions.More_on_positive]
+b:51 [in hydras.OrdinalNotations.OmegaOmega]
+B:51 [in hydras.Ackermann.code]
+b:51 [in hydras.Schutte.PartialFun]
+b:51 [in hydras.Ackermann.codeList]
+b:51 [in hydras.rpo.dickson]
+b:510 [in hydras.Epsilon0.T1]
+b:510 [in hydras.Ackermann.primRec]
+b:511 [in hydras.Ackermann.checkPrf]
+b:512 [in hydras.Ackermann.primRec]
+b:514 [in hydras.Ackermann.primRec]
+b:515 [in hydras.Ackermann.checkPrf]
+b:516 [in hydras.Ackermann.primRec]
+b:518 [in hydras.Ackermann.primRec]
+b:519 [in hydras.Epsilon0.T1]
+b:519 [in hydras.Gamma0.Gamma0]
+b:52 [in hydras.Prelude.More_Arith]
+b:52 [in hydras.Prelude.Comparable]
+B:52 [in hydras.OrdinalNotations.ON_Generic]
+b:52 [in hydras.Ackermann.codeList]
+b:520 [in hydras.Ackermann.primRec]
+b:522 [in hydras.Epsilon0.T1]
+b:522 [in hydras.Ackermann.primRec]
+b:524 [in hydras.Ackermann.primRec]
+b:525 [in hydras.Gamma0.Gamma0]
+b:526 [in hydras.Ackermann.primRec]
+b:528 [in hydras.Ackermann.primRec]
+b:528 [in hydras.Gamma0.Gamma0]
+b:53 [in hydras.Prelude.DecPreOrder_Instances]
+b:53 [in hydras.Prelude.More_Arith]
+b:53 [in hydras.Prelude.DecPreOrder]
+b:53 [in hydras.solutions_exercises.MultisetWf]
+b:53 [in gaia_hydras.T1Bridge]
+b:53 [in hydras.Hydra.BigBattle]
+b:530 [in hydras.Epsilon0.T1]
+b:530 [in hydras.Ackermann.primRec]
+b:531 [in hydras.Gamma0.Gamma0]
+b:533 [in hydras.Epsilon0.T1]
+b:534 [in hydras.Gamma0.Gamma0]
+b:537 [in hydras.Ackermann.primRec]
+b:54 [in additions.More_on_positive]
+b:54 [in gaia_hydras.GCanon]
+b:54 [in hydras.OrdinalNotations.OmegaOmega]
+b:54 [in hydras.Ackermann.cPair]
+B:54 [in hydras.Prelude.MoreVectors]
+b:540 [in hydras.Epsilon0.T1]
+b:549 [in hydras.Ackermann.primRec]
+b:55 [in hydras.Ackermann.LNN2LNT]
+b:55 [in hydras.Prelude.Comparable]
+b:55 [in gaia_hydras.GPaths]
+b:55 [in hydras.Prelude.Sort_spec]
+b:55 [in hydras.Ackermann.code]
+B:55 [in hydras.Schutte.Ordering_Functions]
+b:551 [in hydras.Ackermann.primRec]
+b:553 [in hydras.Ackermann.primRec]
+b:555 [in hydras.Ackermann.primRec]
+b:557 [in hydras.Epsilon0.T1]
+b:56 [in hydras.Epsilon0.Epsilon0rpo]
+b:564 [in hydras.Epsilon0.T1]
+b:564 [in hydras.Gamma0.Gamma0]
+b:565 [in gaia_hydras.nfwfgaia]
+b:566 [in hydras.Gamma0.Gamma0]
+b:567 [in gaia_hydras.nfwfgaia]
+b:57 [in hydras.Prelude.DecPreOrder_Instances]
+b:57 [in hydras.Prelude.More_Arith]
+b:57 [in hydras.Ackermann.LNN2LNT]
+b:57 [in hydras.Ackermann.code]
+b:57 [in hydras.Schutte.Ordering_Functions]
+b:570 [in hydras.Gamma0.Gamma0]
+b:572 [in hydras.Epsilon0.T1]
+b:572 [in hydras.Gamma0.Gamma0]
+b:576 [in hydras.Epsilon0.T1]
+b:58 [in hydras.Prelude.Comparable]
+b:58 [in hydras.Prelude.DecPreOrder]
+b:58 [in additions.More_on_positive]
+b:58 [in gaia_hydras.T1Bridge]
+b:58 [in hydras.Prelude.MoreVectors]
+b:580 [in hydras.Epsilon0.T1]
+b:584 [in hydras.Ackermann.primRec]
+b:586 [in hydras.Ackermann.primRec]
+b:586 [in hydras.Gamma0.Gamma0]
+b:588 [in hydras.Ackermann.primRec]
+b:588 [in hydras.Gamma0.Gamma0]
+b:59 [in hydras.Ackermann.LNN2LNT]
+b:59 [in gaia_hydras.GPaths]
+b:59 [in hydras.Schutte.Schutte_basics]
+b:590 [in hydras.Ackermann.primRec]
+b:592 [in hydras.Ackermann.primRec]
+b:593 [in hydras.Gamma0.Gamma0]
+b:594 [in hydras.Ackermann.primRec]
+b:596 [in hydras.Ackermann.primRec]
+b:597 [in hydras.Gamma0.Gamma0]
+b:599 [in hydras.Epsilon0.T1]
+b:6 [in hydras.Ackermann.folProof]
+b:6 [in gaia_hydras.GPaths]
+b:6 [in hydras.Prelude.Sort_spec]
+b:6 [in hydras.Ackermann.prLogic]
+b:6 [in hydras.Ackermann.NNtheory]
+b:6 [in hydras.Ackermann.ListExt]
+b:6 [in hydras.Ackermann.folLogic3]
+b:6 [in hydras.Gamma0.Gamma0]
+b:60 [in hydras.Prelude.Comparable]
+b:60 [in hydras.Schutte.Schutte_basics]
+B:60 [in hydras.Schutte.Ordering_Functions]
+b:603 [in hydras.Epsilon0.T1]
+b:604 [in gaia_hydras.nfwfgaia]
+b:605 [in hydras.Gamma0.Gamma0]
+b:61 [in hydras.Schutte.Correctness_E0]
+b:61 [in hydras.Prelude.DecPreOrder_Instances]
+B:61 [in hydras.Prelude.Sort_spec]
+b:61 [in hydras.Hydra.BigBattle]
+b:611 [in gaia_hydras.nfwfgaia]
+b:615 [in gaia_hydras.nfwfgaia]
+b:62 [in hydras.Prelude.Comparable]
+b:62 [in hydras.Schutte.Ordering_Functions]
+b:620 [in hydras.Epsilon0.T1]
+b:628 [in hydras.Epsilon0.T1]
+b:628 [in gaia_hydras.nfwfgaia]
+b:63 [in hydras.Prelude.DecPreOrder]
+b:63 [in gaia_hydras.T1Bridge]
+b:63 [in hydras.Ackermann.cPair]
+b:63 [in hydras.Schutte.Schutte_basics]
+b:630 [in gaia_hydras.nfwfgaia]
+b:634 [in hydras.Gamma0.Gamma0]
+b:638 [in hydras.Gamma0.Gamma0]
+b:64 [in hydras.Prelude.Comparable]
+b:64 [in hydras.Ackermann.cPair]
+b:64 [in hydras.Hydra.BigBattle]
+b:64 [in hydras.Schutte.PartialFun]
+b:64 [in hydras.OrdinalNotations.ON_Generic]
+b:65 [in gaia_hydras.GPaths]
+b:65 [in hydras.OrdinalNotations.OmegaOmega]
+b:65 [in hydras.Ackermann.code]
+b:65 [in hydras.Schutte.Schutte_basics]
+B:65 [in hydras.Schutte.Ordering_Functions]
+B:654 [in hydras.Ackermann.checkPrf]
+b:656 [in hydras.Ackermann.checkPrf]
+b:658 [in hydras.Ackermann.checkPrf]
+b:66 [in hydras.Epsilon0.T1]
+B:66 [in Goedel.PRrepresentable]
+b:66 [in hydras.Prelude.Comparable]
+b:66 [in hydras.Ackermann.cPair]
+B:66 [in hydras.Schutte.PartialFun]
+b:660 [in hydras.Ackermann.checkPrf]
+b:662 [in hydras.Ackermann.checkPrf]
+b:664 [in hydras.Ackermann.checkPrf]
+b:666 [in hydras.Ackermann.checkPrf]
+b:668 [in hydras.Ackermann.checkPrf]
+b:67 [in hydras.Ackermann.code]
+b:67 [in hydras.Schutte.Schutte_basics]
+B:67 [in hydras.Hydra.Hydra_Lemmas]
+B:67 [in hydras.Ackermann.expressible]
+b:670 [in hydras.Ackermann.checkPrf]
+b:670 [in hydras.Gamma0.Gamma0]
+b:672 [in hydras.Ackermann.checkPrf]
+b:672 [in hydras.Gamma0.Gamma0]
+b:672 [in gaia_hydras.nfwfgaia]
+b:674 [in hydras.Ackermann.checkPrf]
+b:676 [in hydras.Ackermann.checkPrf]
+b:676 [in hydras.Ackermann.primRec]
+b:678 [in hydras.Ackermann.checkPrf]
+b:68 [in hydras.Prelude.Comparable]
+b:68 [in hydras.Prelude.DecPreOrder]
+b:68 [in gaia_hydras.T1Bridge]
+b:68 [in hydras.Ackermann.cPair]
+b:68 [in hydras.Hydra.BigBattle]
+B:68 [in hydras.Schutte.Ordering_Functions]
+b:680 [in hydras.Ackermann.primRec]
+b:683 [in hydras.Ackermann.primRec]
+b:685 [in hydras.Ackermann.primRec]
+b:687 [in hydras.Ackermann.primRec]
+b:689 [in hydras.Ackermann.primRec]
+b:69 [in gaia_hydras.GPaths]
+b:69 [in hydras.Ackermann.code]
+b:69 [in hydras.Hydra.BigBattle]
+b:690 [in hydras.Gamma0.Gamma0]
+b:692 [in hydras.Ackermann.primRec]
+b:695 [in hydras.Ackermann.primRec]
+b:696 [in gaia_hydras.nfwfgaia]
+b:698 [in hydras.Ackermann.primRec]
+b:698 [in gaia_hydras.nfwfgaia]
+b:7 [in hydras.OrdinalNotations.ON_O]
+b:7 [in hydras.solutions_exercises.MorePRExamples]
+B:7 [in hydras.Schutte.Ordering_Functions]
+B:70 [in Goedel.PRrepresentable]
+b:70 [in hydras.Prelude.Comparable]
+b:70 [in hydras.OrdinalNotations.OmegaOmega]
+b:70 [in hydras.Ackermann.cPair]
+b:70 [in hydras.Schutte.Schutte_basics]
+b:70 [in hydras.Ackermann.folLogic3]
+b:70 [in hydras.rpo.dickson]
+b:70 [in hydras.Schutte.Ordering_Functions]
+b:703 [in hydras.Ackermann.primRec]
+b:708 [in hydras.Epsilon0.T1]
+b:709 [in hydras.Ackermann.primRec]
+b:71 [in hydras.Ackermann.code]
+b:712 [in hydras.Ackermann.primRec]
+b:716 [in hydras.Ackermann.primRec]
+b:72 [in gaia_hydras.GPaths]
+b:72 [in hydras.Ackermann.cPair]
+b:72 [in hydras.Ackermann.folLogic3]
+B:72 [in hydras.Schutte.Ordering_Functions]
+b:72 [in hydras.Prelude.MoreVectors]
+b:720 [in hydras.Ackermann.primRec]
+b:721 [in gaia_hydras.nfwfgaia]
+b:724 [in hydras.Epsilon0.T1]
+b:728 [in gaia_hydras.nfwfgaia]
+b:73 [in hydras.Epsilon0.T1]
+b:73 [in hydras.Prelude.Comparable]
+b:73 [in hydras.Ackermann.code]
+B:73 [in hydras.Schutte.PartialFun]
+b:73 [in hydras.Schutte.Schutte_basics]
+B:73 [in hydras.Schutte.Ordering_Functions]
+b:73 [in hydras.rpo.rpo]
+b:730 [in gaia_hydras.nfwfgaia]
+b:731 [in hydras.Epsilon0.T1]
+b:732 [in gaia_hydras.nfwfgaia]
+b:734 [in gaia_hydras.nfwfgaia]
+b:736 [in hydras.Epsilon0.T1]
+b:738 [in gaia_hydras.nfwfgaia]
+b:74 [in hydras.Epsilon0.Hessenberg]
+b:74 [in hydras.Prelude.DecPreOrder]
+b:74 [in gaia_hydras.T1Bridge]
+b:74 [in hydras.Ackermann.cPair]
+b:74 [in hydras.Schutte.Ordering_Functions]
+B:74 [in hydras.Prelude.MoreVectors]
+b:740 [in gaia_hydras.nfwfgaia]
+b:75 [in hydras.Prelude.Comparable]
+b:75 [in hydras.Ackermann.code]
+b:757 [in hydras.Ackermann.checkPrf]
+b:76 [in hydras.Gamma0.T2]
+b:76 [in gaia_hydras.GPaths]
+b:76 [in hydras.Ackermann.cPair]
+b:76 [in hydras.Schutte.Schutte_basics]
+b:76 [in hydras.OrdinalNotations.ON_Generic]
+b:761 [in hydras.Ackermann.checkPrf]
+b:761 [in gaia_hydras.nfwfgaia]
+b:765 [in gaia_hydras.nfwfgaia]
+b:77 [in hydras.Prelude.Comparable]
+b:777 [in gaia_hydras.nfwfgaia]
+b:78 [in hydras.Epsilon0.Hessenberg]
+b:78 [in hydras.Ackermann.cPair]
+b:78 [in hydras.Schutte.Ordering_Functions]
+b:781 [in gaia_hydras.nfwfgaia]
+b:785 [in gaia_hydras.nfwfgaia]
+b:787 [in hydras.Ackermann.checkPrf]
+b:788 [in hydras.Ackermann.primRec]
+b:788 [in gaia_hydras.nfwfgaia]
+b:789 [in hydras.Ackermann.checkPrf]
+b:79 [in hydras.Gamma0.T2]
+b:79 [in hydras.Prelude.Comparable]
+b:79 [in hydras.OrdinalNotations.ON_Generic]
+b:791 [in hydras.Ackermann.primRec]
+b:793 [in gaia_hydras.nfwfgaia]
+b:794 [in hydras.Ackermann.primRec]
+b:798 [in hydras.Ackermann.primRec]
+b:799 [in hydras.Ackermann.primRec]
+b:8 [in hydras.Gamma0.T2]
+b:8 [in hydras.Ackermann.folLogic2]
+b:8 [in gaia_hydras.T1Bridge]
+b:8 [in hydras.Ackermann.NNtheory]
+b:8 [in gaia_hydras.T2Bridge]
+b:8 [in hydras.MoreAck.PrimRecExamples]
+b:80 [in hydras.Epsilon0.T1]
+b:80 [in hydras.Prelude.DecPreOrder]
+b:80 [in hydras.Ackermann.codeSubFormula]
+b:80 [in gaia_hydras.T1Bridge]
+b:80 [in hydras.Schutte.Schutte_basics]
+b:80 [in hydras.Ackermann.folLogic3]
+b:80 [in hydras.rpo.list_permut]
+b:800 [in hydras.Ackermann.primRec]
+b:804 [in hydras.Ackermann.primRec]
+b:805 [in hydras.Ackermann.primRec]
+b:806 [in hydras.Ackermann.primRec]
+b:809 [in hydras.Ackermann.primRec]
+b:81 [in hydras.Prelude.Comparable]
+b:81 [in gaia_hydras.GPaths]
+b:82 [in hydras.Ackermann.cPair]
+b:82 [in hydras.Schutte.Schutte_basics]
+b:82 [in hydras.Ackermann.folLogic3]
+B:82 [in hydras.Schutte.Ordering_Functions]
+b:83 [in hydras.Prelude.Comparable]
+b:83 [in gaia_hydras.T1Bridge]
+B:83 [in hydras.Schutte.PartialFun]
+b:838 [in hydras.Ackermann.codeSubFormula]
+b:84 [in hydras.Ackermann.cPair]
+b:84 [in hydras.Schutte.Schutte_basics]
+b:841 [in hydras.Ackermann.codeSubFormula]
+b:843 [in gaia_hydras.nfwfgaia]
+b:849 [in hydras.Ackermann.codeSubFormula]
+b:85 [in hydras.Epsilon0.T1]
+b:85 [in hydras.Prelude.Comparable]
+b:85 [in gaia_hydras.T1Bridge]
+B:85 [in hydras.OrdinalNotations.ON_Generic]
+b:850 [in hydras.Ackermann.checkPrf]
+b:852 [in hydras.Ackermann.checkPrf]
+b:853 [in hydras.Ackermann.codeSubFormula]
+b:853 [in gaia_hydras.nfwfgaia]
+b:854 [in hydras.Ackermann.checkPrf]
+b:855 [in hydras.Ackermann.codeSubFormula]
+b:856 [in hydras.Ackermann.checkPrf]
+b:857 [in hydras.Ackermann.codeSubFormula]
+b:86 [in hydras.Ackermann.folLogic3]
+b:861 [in gaia_hydras.nfwfgaia]
+b:866 [in gaia_hydras.nfwfgaia]
+b:869 [in hydras.Ackermann.codeSubFormula]
+b:87 [in hydras.Prelude.Comparable]
+b:87 [in hydras.Epsilon0.Hessenberg]
+b:87 [in hydras.OrdinalNotations.OmegaOmega]
+b:87 [in gaia_hydras.T1Bridge]
+b:87 [in gaia_hydras.nfwfgaia]
+b:872 [in hydras.Ackermann.codeSubFormula]
+b:886 [in gaia_hydras.nfwfgaia]
+b:89 [in hydras.Epsilon0.T1]
+b:89 [in hydras.Epsilon0.Hessenberg]
+B:89 [in hydras.Schutte.PartialFun]
+b:89 [in hydras.Schutte.Schutte_basics]
+b:89 [in hydras.MoreAck.PrimRecExamples]
+b:893 [in hydras.Ackermann.codeSubFormula]
+b:895 [in gaia_hydras.nfwfgaia]
+b:896 [in hydras.Ackermann.codeSubFormula]
+b:899 [in gaia_hydras.nfwfgaia]
+B:9 [in hydras.Prelude.DecPreOrder_Instances]
+b:9 [in hydras.Ackermann.extEqualNat]
+b:9 [in hydras.rpo.dickson]
+b:9 [in hydras.MoreAck.FolExamples]
+b:90 [in hydras.Prelude.Comparable]
+b:903 [in gaia_hydras.nfwfgaia]
+b:909 [in hydras.Ackermann.codeSubFormula]
+b:91 [in hydras.Gamma0.T2]
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+b:91 [in hydras.Schutte.Schutte_basics]
+b:911 [in gaia_hydras.nfwfgaia]
+b:912 [in hydras.Ackermann.codeSubFormula]
+b:92 [in hydras.Prelude.Comparable]
+b:92 [in hydras.Prelude.DecPreOrder]
+b:92 [in hydras.OrdinalNotations.ON_Generic]
+b:924 [in gaia_hydras.nfwfgaia]
+b:926 [in gaia_hydras.nfwfgaia]
+b:928 [in gaia_hydras.nfwfgaia]
+B:93 [in Goedel.PRrepresentable]
+b:93 [in hydras.Epsilon0.Hessenberg]
+b:93 [in hydras.Schutte.Schutte_basics]
+b:930 [in gaia_hydras.nfwfgaia]
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+b:939 [in gaia_hydras.nfwfgaia]
+b:94 [in hydras.Prelude.Comparable]
+b:94 [in hydras.Prelude.Merge_Sort]
+b:94 [in gaia_hydras.T1Bridge]
+b:945 [in gaia_hydras.nfwfgaia]
+b:95 [in hydras.Epsilon0.Hessenberg]
+b:95 [in hydras.Ackermann.folLogic3]
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+b:957 [in gaia_hydras.nfwfgaia]
+b:959 [in gaia_hydras.nfwfgaia]
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+b:963 [in gaia_hydras.nfwfgaia]
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+B:97 [in hydras.OrdinalNotations.ON_Generic]
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+b:98 [in gaia_hydras.T1Bridge]
+b:989 [in gaia_hydras.nfwfgaia]
+b:99 [in hydras.Epsilon0.Hessenberg]
+b:993 [in gaia_hydras.nfwfgaia]
+b:999 [in gaia_hydras.nfwfgaia]
+

C

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+c:1004 [in gaia_hydras.nfwfgaia]
+c:1007 [in gaia_hydras.nfwfgaia]
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+c:1011 [in gaia_hydras.nfwfgaia]
+c:1032 [in gaia_hydras.nfwfgaia]
+c:1037 [in gaia_hydras.nfwfgaia]
+c:104 [in hydras.MoreAck.PrimRecExamples]
+c:105 [in hydras.Epsilon0.Hessenberg]
+c:1052 [in gaia_hydras.nfwfgaia]
+c:108 [in hydras.Epsilon0.Hessenberg]
+c:113 [in hydras.Gamma0.T2]
+c:113 [in additions.AM]
+c:1147 [in gaia_hydras.nfwfgaia]
+c:115 [in hydras.Ackermann.cPair]
+c:1168 [in gaia_hydras.nfwfgaia]
+c:117 [in hydras.Epsilon0.Hessenberg]
+c:12 [in hydras.Gamma0.Gamma0]
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+c:121 [in hydras.Epsilon0.F_alpha]
+c:1210 [in gaia_hydras.nfwfgaia]
+C:123 [in hydras.rpo.more_list]
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+c:1257 [in gaia_hydras.nfwfgaia]
+c:1260 [in gaia_hydras.nfwfgaia]
+c:1263 [in gaia_hydras.nfwfgaia]
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+c:133 [in hydras.Epsilon0.Hessenberg]
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+c:1352 [in gaia_hydras.nfwfgaia]
+c:1357 [in gaia_hydras.nfwfgaia]
+c:1362 [in gaia_hydras.nfwfgaia]
+c:1367 [in gaia_hydras.nfwfgaia]
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+c:1378 [in gaia_hydras.nfwfgaia]
+c:139 [in hydras.Ackermann.primRec]
+c:1390 [in gaia_hydras.nfwfgaia]
+c:14 [in hydras.Prelude.MoreDecidable]
+c:14 [in hydras.rpo.closure]
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+c:1418 [in gaia_hydras.nfwfgaia]
+c:142 [in hydras.Ackermann.primRec]
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+c:143 [in hydras.Gamma0.T2]
+c:1430 [in gaia_hydras.nfwfgaia]
+c:144 [in hydras.Ackermann.LNT]
+c:144 [in hydras.Gamma0.Gamma0]
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+c:145 [in additions.fib]
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+c:148 [in hydras.Ackermann.primRec]
+c:149 [in hydras.Gamma0.T2]
+c:1494 [in gaia_hydras.nfwfgaia]
+c:15 [in hydras.solutions_exercises.FibonacciPR]
+c:15 [in hydras.Hydra.BigBattle]
+c:1504 [in gaia_hydras.nfwfgaia]
+c:151 [in hydras.Ackermann.LNN]
+C:151 [in hydras.Ackermann.fol]
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+c:1528 [in gaia_hydras.nfwfgaia]
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+c:1538 [in gaia_hydras.nfwfgaia]
+c:1542 [in gaia_hydras.nfwfgaia]
+c:1545 [in gaia_hydras.nfwfgaia]
+c:1553 [in gaia_hydras.nfwfgaia]
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+c:1566 [in gaia_hydras.nfwfgaia]
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+c:1580 [in gaia_hydras.nfwfgaia]
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+c:1623 [in gaia_hydras.nfwfgaia]
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+c:1693 [in gaia_hydras.nfwfgaia]
+c:17 [in hydras.solutions_exercises.FibonacciPR]
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+c:2049 [in gaia_hydras.nfwfgaia]
+c:205 [in additions.Addition_Chains]
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+c:2075 [in gaia_hydras.nfwfgaia]
+c:208 [in additions.Addition_Chains]
+c:208 [in gaia_hydras.nfwfgaia]
+c:21 [in hydras.solutions_exercises.FibonacciPR]
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+c:211 [in gaia_hydras.nfwfgaia]
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+c:214 [in gaia_hydras.nfwfgaia]
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+c:215 [in hydras.Ackermann.cPair]
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+c:217 [in gaia_hydras.nfwfgaia]
+c:218 [in hydras.Ackermann.primRec]
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+c:22 [in hydras.Hydra.BigBattle]
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+c:220 [in hydras.Gamma0.Gamma0]
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+c:224 [in hydras.Ackermann.primRec]
+c:226 [in hydras.Gamma0.Gamma0]
+c:227 [in hydras.Ackermann.primRec]
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+c:234 [in hydras.Ackermann.primRec]
+c:234 [in hydras.Gamma0.Gamma0]
+c:237 [in hydras.Ackermann.primRec]
+c:24 [in hydras.Prelude.Comparable]
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+c:240 [in hydras.Ackermann.primRec]
+c:243 [in hydras.Ackermann.primRec]
+c:247 [in hydras.Ackermann.primRec]
+c:25 [in gaia_hydras.GHessenberg]
+c:250 [in hydras.Ackermann.primRec]
+c:253 [in hydras.Ackermann.primRec]
+c:256 [in hydras.Ackermann.primRec]
+c:259 [in hydras.Ackermann.primRec]
+c:26 [in additions.Fib2]
+c:26 [in hydras.Prelude.More_Arith]
+C:26 [in hydras.Ackermann.fol]
+c:26 [in hydras.Gamma0.Gamma0]
+c:263 [in hydras.Ackermann.primRec]
+C:267 [in hydras.Ackermann.checkPrf]
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+c:270 [in hydras.Gamma0.Gamma0]
+c:271 [in hydras.Ackermann.primRec]
+c:275 [in hydras.Ackermann.primRec]
+c:279 [in hydras.Ackermann.primRec]
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+c:286 [in hydras.Gamma0.Gamma0]
+c:288 [in hydras.Ackermann.primRec]
+c:288 [in additions.Addition_Chains]
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+c:297 [in hydras.Ackermann.primRec]
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+c:3 [in additions.AM]
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+c:3 [in hydras.Epsilon0.F_alpha]
+c:3 [in hydras.Prelude.MoreDecidable]
+c:301 [in hydras.Ackermann.primRec]
+c:305 [in hydras.Ackermann.primRec]
+c:305 [in gaia_hydras.nfwfgaia]
+c:308 [in Goedel.PRrepresentable]
+c:308 [in gaia_hydras.nfwfgaia]
+c:309 [in hydras.Ackermann.primRec]
+c:309 [in hydras.Gamma0.Gamma0]
+C:31 [in hydras.Ackermann.folProof]
+c:31 [in hydras.Prelude.Comparable]
+c:31 [in hydras.Hydra.BigBattle]
+c:312 [in hydras.Epsilon0.T1]
+c:313 [in hydras.Ackermann.primRec]
+c:317 [in hydras.Ackermann.primRec]
+c:32 [in hydras.Ackermann.extEqualNat]
+c:32 [in hydras.Gamma0.Gamma0]
+c:32 [in gaia_hydras.GHessenberg]
+c:321 [in hydras.Ackermann.primRec]
+c:326 [in hydras.Ackermann.primRec]
+c:33 [in hydras.Ackermann.folLogic3]
+c:330 [in hydras.Ackermann.primRec]
+c:334 [in hydras.Ackermann.primRec]
+c:338 [in hydras.Ackermann.primRec]
+c:34 [in additions.AM]
+c:340 [in Goedel.PRrepresentable]
+c:342 [in Goedel.PRrepresentable]
+c:342 [in hydras.Ackermann.primRec]
+c:342 [in hydras.Ackermann.cPair]
+c:344 [in Goedel.PRrepresentable]
+c:345 [in Goedel.PRrepresentable]
+c:348 [in hydras.Ackermann.cPair]
+c:35 [in hydras.Ackermann.cPair]
+c:35 [in gaia_hydras.GHessenberg]
+c:351 [in hydras.Ackermann.cPair]
+c:352 [in additions.Euclidean_Chains]
+c:353 [in Goedel.PRrepresentable]
+c:354 [in hydras.Ackermann.cPair]
+c:357 [in Goedel.PRrepresentable]
+c:357 [in hydras.Ackermann.cPair]
+c:359 [in hydras.Ackermann.cPair]
+c:36 [in hydras.Prelude.MoreVectors]
+c:361 [in Goedel.PRrepresentable]
+c:361 [in hydras.Ackermann.cPair]
+c:364 [in hydras.Ackermann.cPair]
+c:365 [in Goedel.PRrepresentable]
+c:367 [in hydras.Ackermann.cPair]
+c:37 [in hydras.Prelude.Comparable]
+c:370 [in hydras.Ackermann.cPair]
+c:38 [in hydras.Hydra.BigBattle]
+c:38 [in gaia_hydras.GHessenberg]
+c:387 [in Goedel.PRrepresentable]
+c:39 [in hydras.OrdinalNotations.OmegaOmega]
+c:39 [in hydras.Schutte.Well_Orders]
+c:391 [in Goedel.PRrepresentable]
+c:395 [in Goedel.PRrepresentable]
+c:399 [in Goedel.PRrepresentable]
+c:4 [in additions.AM]
+c:4 [in hydras.Gamma0.Gamma0]
+c:40 [in additions.Addition_Chains]
+c:401 [in gaia_hydras.nfwfgaia]
+c:407 [in Goedel.PRrepresentable]
+c:407 [in gaia_hydras.nfwfgaia]
+c:410 [in gaia_hydras.nfwfgaia]
+c:411 [in Goedel.PRrepresentable]
+c:414 [in hydras.Epsilon0.T1]
+c:414 [in gaia_hydras.nfwfgaia]
+c:415 [in Goedel.PRrepresentable]
+c:419 [in Goedel.PRrepresentable]
+c:43 [in hydras.Ackermann.cPair]
+c:43 [in additions.Addition_Chains]
+c:432 [in hydras.Gamma0.Gamma0]
+c:438 [in hydras.Ackermann.primRec]
+c:444 [in hydras.Epsilon0.T1]
+c:445 [in hydras.Epsilon0.T1]
+c:45 [in additions.Euclidean_Chains]
+c:450 [in hydras.Epsilon0.T1]
+c:460 [in hydras.Epsilon0.T1]
+c:461 [in gaia_hydras.nfwfgaia]
+c:463 [in hydras.Epsilon0.T1]
+c:47 [in hydras.Ackermann.cPair]
+c:480 [in hydras.Gamma0.Gamma0]
+c:49 [in hydras.Prelude.More_Arith]
+c:50 [in hydras.Prelude.Comparable]
+c:50 [in hydras.Hydra.BigBattle]
+c:505 [in hydras.Epsilon0.T1]
+c:51 [in gaia_hydras.T1Bridge]
+c:51 [in hydras.Ackermann.cPair]
+c:51 [in additions.Addition_Chains]
+c:52 [in additions.Addition_Chains]
+c:526 [in hydras.Gamma0.Gamma0]
+c:529 [in hydras.Gamma0.Gamma0]
+c:53 [in hydras.Prelude.Comparable]
+c:532 [in hydras.Gamma0.Gamma0]
+c:54 [in hydras.Hydra.BigBattle]
+c:540 [in hydras.Ackermann.primRec]
+c:55 [in hydras.Ackermann.cPair]
+c:558 [in hydras.Gamma0.Gamma0]
+c:56 [in hydras.Prelude.Comparable]
+c:56 [in additions.Addition_Chains]
+c:565 [in hydras.Epsilon0.T1]
+c:567 [in hydras.Gamma0.Gamma0]
+c:57 [in additions.AM]
+c:573 [in hydras.Epsilon0.T1]
+c:574 [in hydras.Gamma0.Gamma0]
+c:577 [in hydras.Epsilon0.T1]
+c:578 [in hydras.Gamma0.Gamma0]
+c:580 [in hydras.Gamma0.Gamma0]
+c:581 [in hydras.Epsilon0.T1]
+c:583 [in hydras.Gamma0.Gamma0]
+c:59 [in hydras.Hydra.BigBattle]
+c:591 [in hydras.Gamma0.Gamma0]
+c:594 [in hydras.Gamma0.Gamma0]
+c:599 [in hydras.Gamma0.Gamma0]
+c:604 [in hydras.Epsilon0.T1]
+c:607 [in hydras.Epsilon0.T1]
+c:611 [in hydras.Epsilon0.T1]
+c:614 [in hydras.Epsilon0.T1]
+c:614 [in hydras.Ackermann.primRec]
+c:617 [in hydras.Epsilon0.T1]
+c:621 [in hydras.Epsilon0.T1]
+c:622 [in hydras.Ackermann.primRec]
+c:623 [in hydras.Epsilon0.T1]
+c:626 [in hydras.Ackermann.primRec]
+c:629 [in hydras.Epsilon0.T1]
+c:636 [in hydras.Gamma0.Gamma0]
+c:641 [in hydras.Ackermann.primRec]
+c:646 [in hydras.Ackermann.primRec]
+c:654 [in hydras.Ackermann.primRec]
+c:673 [in hydras.Gamma0.Gamma0]
+c:677 [in hydras.Ackermann.primRec]
+c:68 [in hydras.Schutte.Schutte_basics]
+c:68 [in additions.Addition_Chains]
+c:681 [in hydras.Ackermann.primRec]
+c:684 [in hydras.Ackermann.primRec]
+c:686 [in hydras.Ackermann.primRec]
+c:688 [in hydras.Ackermann.primRec]
+c:69 [in hydras.Prelude.DecPreOrder]
+c:690 [in hydras.Ackermann.primRec]
+c:692 [in hydras.Gamma0.Gamma0]
+c:693 [in hydras.Ackermann.primRec]
+c:696 [in hydras.Ackermann.primRec]
+c:70 [in additions.Addition_Chains]
+c:71 [in hydras.Gamma0.T2]
+c:71 [in additions.AM]
+c:71 [in hydras.Schutte.Schutte_basics]
+c:717 [in hydras.Ackermann.primRec]
+c:721 [in hydras.Ackermann.primRec]
+c:722 [in gaia_hydras.nfwfgaia]
+c:726 [in hydras.Ackermann.primRec]
+c:729 [in hydras.Ackermann.primRec]
+c:732 [in hydras.Ackermann.primRec]
+c:733 [in hydras.Epsilon0.T1]
+c:735 [in hydras.Ackermann.primRec]
+c:737 [in hydras.Epsilon0.T1]
+c:738 [in hydras.Ackermann.primRec]
+c:74 [in hydras.Schutte.Schutte_basics]
+c:741 [in hydras.Ackermann.primRec]
+c:744 [in hydras.Ackermann.primRec]
+c:747 [in hydras.Ackermann.primRec]
+c:75 [in hydras.Prelude.DecPreOrder]
+c:750 [in hydras.Ackermann.primRec]
+c:753 [in hydras.Ackermann.primRec]
+c:759 [in hydras.Ackermann.primRec]
+c:761 [in hydras.Ackermann.primRec]
+c:763 [in hydras.Ackermann.primRec]
+c:765 [in hydras.Ackermann.primRec]
+c:767 [in hydras.Ackermann.primRec]
+c:769 [in hydras.Ackermann.primRec]
+c:77 [in hydras.Schutte.Schutte_basics]
+c:771 [in hydras.Ackermann.primRec]
+c:773 [in hydras.Ackermann.primRec]
+c:775 [in hydras.Ackermann.primRec]
+c:777 [in hydras.Ackermann.primRec]
+c:779 [in hydras.Ackermann.primRec]
+c:78 [in additions.Addition_Chains]
+c:781 [in hydras.Ackermann.primRec]
+c:784 [in hydras.Ackermann.primRec]
+c:787 [in hydras.Ackermann.primRec]
+c:79 [in hydras.Epsilon0.Hessenberg]
+c:795 [in hydras.Ackermann.primRec]
+c:8 [in hydras.Prelude.MoreDecidable]
+c:8 [in hydras.Gamma0.Gamma0]
+c:80 [in additions.Addition_Chains]
+c:801 [in hydras.Ackermann.primRec]
+c:81 [in additions.AM]
+c:850 [in hydras.Ackermann.codeSubFormula]
+c:854 [in gaia_hydras.nfwfgaia]
+c:863 [in gaia_hydras.nfwfgaia]
+c:868 [in gaia_hydras.nfwfgaia]
+c:88 [in hydras.OrdinalNotations.OmegaOmega]
+c:9 [in hydras.MoreAck.PrimRecExamples]
+c:90 [in hydras.MoreAck.PrimRecExamples]
+c:905 [in gaia_hydras.nfwfgaia]
+c:91 [in additions.AM]
+c:913 [in gaia_hydras.nfwfgaia]
+c:92 [in additions.AM]
+c:93 [in hydras.Gamma0.T2]
+c:940 [in gaia_hydras.nfwfgaia]
+c:947 [in gaia_hydras.nfwfgaia]
+c:95 [in hydras.Prelude.Comparable]
+c:995 [in gaia_hydras.nfwfgaia]
+

D

+daughters:7 [in hydras.Hydra.Hydra_Definitions]
+DA:36 [in hydras.Schutte.Countable]
+DA:37 [in hydras.Prelude.DecPreOrder_Instances]
+DB:38 [in hydras.Prelude.DecPreOrder_Instances]
+dec:20 [in hydras.Prelude.STDPP_compat]
+dec:33 [in hydras.Prelude.Sort_spec]
+dec:42 [in hydras.Prelude.Sort_spec]
+dec:66 [in hydras.Prelude.Sort_spec]
+dec:77 [in hydras.Prelude.Sort_spec]
+dec:90 [in hydras.Prelude.DecPreOrder]
+delta:593 [in hydras.Epsilon0.T1]
+delta:62 [in hydras.Schutte.Addition]
+dest:102 [in hydras.Hydra.Hydra_Definitions]
+disp:27 [in gaia_hydras.onType]
+dom:152 [in hydras.rpo.more_list]
+d':1205 [in gaia_hydras.nfwfgaia]
+d':1339 [in gaia_hydras.nfwfgaia]
+d':1385 [in gaia_hydras.nfwfgaia]
+d':1437 [in gaia_hydras.nfwfgaia]
+d':1501 [in gaia_hydras.nfwfgaia]
+d':1557 [in gaia_hydras.nfwfgaia]
+d':1586 [in gaia_hydras.nfwfgaia]
+d':1610 [in gaia_hydras.nfwfgaia]
+d':1692 [in gaia_hydras.nfwfgaia]
+d':2044 [in gaia_hydras.nfwfgaia]
+d':2067 [in gaia_hydras.nfwfgaia]
+d':27 [in additions.Monoid_instances]
+d':849 [in gaia_hydras.nfwfgaia]
+d:10 [in hydras.MoreAck.PrimRecExamples]
+d:116 [in hydras.Ackermann.cPair]
+d:118 [in hydras.Epsilon0.Hessenberg]
+d:13 [in hydras.Prelude.MoreDecidable]
+d:1308 [in hydras.Ackermann.codeSubFormula]
+d:1334 [in gaia_hydras.nfwfgaia]
+d:1349 [in gaia_hydras.nfwfgaia]
+d:1354 [in gaia_hydras.nfwfgaia]
+d:1359 [in gaia_hydras.nfwfgaia]
+d:1364 [in gaia_hydras.nfwfgaia]
+d:1369 [in gaia_hydras.nfwfgaia]
+d:1374 [in gaia_hydras.nfwfgaia]
+d:1380 [in gaia_hydras.nfwfgaia]
+d:1432 [in gaia_hydras.nfwfgaia]
+d:146 [in additions.fib]
+d:149 [in hydras.Ackermann.LNT]
+d:149 [in hydras.Ackermann.primRec]
+d:1496 [in gaia_hydras.nfwfgaia]
+d:152 [in hydras.Ackermann.LNN]
+d:153 [in hydras.Ackermann.primRec]
+d:1535 [in gaia_hydras.nfwfgaia]
+d:154 [in hydras.Ackermann.LNT]
+d:1546 [in gaia_hydras.nfwfgaia]
+d:1555 [in gaia_hydras.nfwfgaia]
+d:157 [in hydras.Ackermann.LNN]
+d:157 [in hydras.Ackermann.primRec]
+d:1591 [in gaia_hydras.nfwfgaia]
+d:1605 [in gaia_hydras.nfwfgaia]
+d:161 [in hydras.Ackermann.primRec]
+d:165 [in hydras.Ackermann.LNN]
+d:1654 [in gaia_hydras.nfwfgaia]
+d:1678 [in gaia_hydras.nfwfgaia]
+d:1687 [in gaia_hydras.nfwfgaia]
+d:17 [in hydras.MoreAck.PrimRecExamples]
+d:1795 [in gaia_hydras.nfwfgaia]
+d:19 [in hydras.Prelude.MoreDecidable]
+D:2 [in hydras.Schutte.Lub]
+d:2029 [in gaia_hydras.nfwfgaia]
+d:23 [in additions.Monoid_instances]
+d:264 [in hydras.Ackermann.primRec]
+d:268 [in hydras.Ackermann.primRec]
+d:272 [in hydras.Ackermann.primRec]
+d:276 [in hydras.Ackermann.primRec]
+d:280 [in hydras.Ackermann.primRec]
+d:285 [in hydras.Ackermann.primRec]
+d:289 [in hydras.Ackermann.primRec]
+d:293 [in hydras.Ackermann.primRec]
+d:298 [in hydras.Ackermann.primRec]
+d:302 [in hydras.Ackermann.primRec]
+d:306 [in hydras.Ackermann.primRec]
+d:310 [in hydras.Ackermann.primRec]
+d:314 [in hydras.Ackermann.primRec]
+d:318 [in hydras.Ackermann.primRec]
+d:322 [in hydras.Ackermann.primRec]
+d:327 [in hydras.Ackermann.primRec]
+d:331 [in hydras.Ackermann.primRec]
+d:335 [in hydras.Ackermann.primRec]
+d:335 [in additions.Euclidean_Chains]
+d:339 [in hydras.Ackermann.primRec]
+d:34 [in hydras.Ackermann.folLogic3]
+d:343 [in hydras.Ackermann.primRec]
+d:36 [in hydras.Ackermann.cPair]
+d:385 [in additions.Euclidean_Chains]
+d:391 [in additions.Euclidean_Chains]
+d:398 [in additions.Euclidean_Chains]
+d:4 [in hydras.Prelude.MoreDecidable]
+d:44 [in hydras.Ackermann.cPair]
+d:48 [in hydras.Ackermann.cPair]
+d:5 [in hydras.Prelude.MoreDecidable]
+d:52 [in hydras.Ackermann.cPair]
+d:56 [in hydras.Ackermann.cPair]
+d:567 [in hydras.Epsilon0.T1]
+d:595 [in hydras.Gamma0.Gamma0]
+d:609 [in hydras.Epsilon0.T1]
+d:625 [in hydras.Epsilon0.T1]
+d:631 [in hydras.Epsilon0.T1]
+d:655 [in hydras.Ackermann.primRec]
+D:8 [in hydras.Schutte.Lub]
+d:845 [in gaia_hydras.nfwfgaia]
+d:851 [in hydras.Ackermann.codeSubFormula]
+d:91 [in hydras.MoreAck.PrimRecExamples]
+

E

+eqA:107 [in hydras.rpo.more_list]
+eqA:115 [in hydras.rpo.more_list]
+eqA:139 [in hydras.rpo.more_list]
+eqA:148 [in hydras.rpo.more_list]
+eqA:158 [in hydras.rpo.more_list]
+eqA:167 [in hydras.rpo.more_list]
+eqA:174 [in hydras.rpo.more_list]
+eqA:182 [in hydras.rpo.more_list]
+eqA:190 [in hydras.rpo.more_list]
+eqA:205 [in hydras.rpo.more_list]
+eqA:214 [in hydras.rpo.more_list]
+eqA:224 [in hydras.rpo.more_list]
+eqA:242 [in hydras.rpo.more_list]
+eqA:251 [in hydras.rpo.more_list]
+eqA:263 [in hydras.rpo.more_list]
+eqA:81 [in hydras.rpo.more_list]
+eqA:91 [in hydras.rpo.more_list]
+eqA:99 [in hydras.rpo.more_list]
+equiv:44 [in additions.Euclidean_Chains]
+equiv:49 [in additions.Euclidean_Chains]
+equiv:52 [in additions.Euclidean_Chains]
+equ:101 [in additions.AM]
+equ:106 [in additions.AM]
+equ:111 [in additions.AM]
+equ:121 [in additions.AM]
+equ:126 [in additions.AM]
+equ:38 [in additions.AM]
+equ:44 [in additions.AM]
+equ:55 [in additions.AM]
+equ:62 [in additions.AM]
+equ:75 [in additions.AM]
+equ:85 [in additions.AM]
+equ:96 [in additions.AM]
+eqv:104 [in additions.Addition_Chains]
+eqv:112 [in additions.Addition_Chains]
+eqv:118 [in additions.Addition_Chains]
+eqv:125 [in additions.Addition_Chains]
+eqv:132 [in additions.Addition_Chains]
+eqv:162 [in additions.Addition_Chains]
+eqv:169 [in additions.Addition_Chains]
+eqv:175 [in additions.Addition_Chains]
+eq_A_dec:19 [in hydras.rpo.rpo]
+eq_A_dec:14 [in hydras.rpo.rpo]
+eq_A_dec:5 [in hydras.rpo.rpo]
+eta:80 [in hydras.Schutte.Critical]
+exp:746 [in hydras.Epsilon0.T1]
+exp:751 [in hydras.Epsilon0.T1]
+E_eq:240 [in additions.AM]
+E_one:239 [in additions.AM]
+E_op:238 [in additions.AM]
+E_eq:233 [in additions.AM]
+E_one:232 [in additions.AM]
+E_op:231 [in additions.AM]
+E_eq:51 [in additions.Pow_variant]
+E_one:50 [in additions.Pow_variant]
+E_op:49 [in additions.Pow_variant]
+E_eq:42 [in additions.Pow_variant]
+E_one:41 [in additions.Pow_variant]
+E_op:40 [in additions.Pow_variant]
+E_eq:32 [in additions.Pow_variant]
+E_one:31 [in additions.Pow_variant]
+E_op:30 [in additions.Pow_variant]
+E_eq:26 [in additions.Pow_variant]
+E_one:25 [in additions.Pow_variant]
+E_op:24 [in additions.Pow_variant]
+E_eq:19 [in additions.Pow_variant]
+E_one:18 [in additions.Pow_variant]
+E_op:17 [in additions.Pow_variant]
+E_eq:13 [in additions.Pow_variant]
+E_one:12 [in additions.Pow_variant]
+E_op:11 [in additions.Pow_variant]
+E_eq:4 [in additions.Pow_variant]
+E_one:3 [in additions.Pow_variant]
+E_op:2 [in additions.Pow_variant]
+E_eq:59 [in additions.Pow]
+E_op:58 [in additions.Pow]
+E_eq:51 [in additions.Pow]
+E_one:50 [in additions.Pow]
+E_op:49 [in additions.Pow]
+E_eq:42 [in additions.Pow]
+E_one:41 [in additions.Pow]
+E_op:40 [in additions.Pow]
+E_eq:32 [in additions.Pow]
+E_one:31 [in additions.Pow]
+E_op:30 [in additions.Pow]
+E_eq:26 [in additions.Pow]
+E_one:25 [in additions.Pow]
+E_op:24 [in additions.Pow]
+E_eq:19 [in additions.Pow]
+E_one:18 [in additions.Pow]
+E_op:17 [in additions.Pow]
+E_eq:13 [in additions.Pow]
+E_one:12 [in additions.Pow]
+E_op:11 [in additions.Pow]
+E_eq:4 [in additions.Pow]
+E_one:3 [in additions.Pow]
+E_op:2 [in additions.Pow]
+E_eq:54 [in additions.Monoid_def]
+E_one:53 [in additions.Monoid_def]
+E_op:52 [in additions.Monoid_def]
+E_eq:49 [in additions.Monoid_def]
+E_one:48 [in additions.Monoid_def]
+E_op:47 [in additions.Monoid_def]
+E_eq:44 [in additions.Monoid_def]
+E_one:43 [in additions.Monoid_def]
+E_op:42 [in additions.Monoid_def]
+E_eq:39 [in additions.Monoid_def]
+E_one:38 [in additions.Monoid_def]
+E_op:37 [in additions.Monoid_def]
+E_eq:24 [in additions.Monoid_def]
+E_one:23 [in additions.Monoid_def]
+E_op:22 [in additions.Monoid_def]
+E_equiv:197 [in additions.Euclidean_Chains]
+E_one:196 [in additions.Euclidean_Chains]
+E_equiv:185 [in additions.Euclidean_Chains]
+E_one:184 [in additions.Euclidean_Chains]
+E_equiv:171 [in additions.Euclidean_Chains]
+E_one:170 [in additions.Euclidean_Chains]
+E_equiv:165 [in additions.Euclidean_Chains]
+E_one:164 [in additions.Euclidean_Chains]
+E_equiv:113 [in additions.Euclidean_Chains]
+E_one:112 [in additions.Euclidean_Chains]
+E_equiv:102 [in additions.Euclidean_Chains]
+E_one:101 [in additions.Euclidean_Chains]
+E_equiv:91 [in additions.Euclidean_Chains]
+E_one:90 [in additions.Euclidean_Chains]
+E_equiv:71 [in additions.Euclidean_Chains]
+E_one:70 [in additions.Euclidean_Chains]
+E_equiv:64 [in additions.Euclidean_Chains]
+E_one:63 [in additions.Euclidean_Chains]
+E_equiv:56 [in additions.Euclidean_Chains]
+E_one:55 [in additions.Euclidean_Chains]
+E_equiv:38 [in additions.Euclidean_Chains]
+E_one:37 [in additions.Euclidean_Chains]
+E_eq:234 [in additions.Addition_Chains]
+E_one:233 [in additions.Addition_Chains]
+E_op:232 [in additions.Addition_Chains]
+E_eq:226 [in additions.Addition_Chains]
+E_one:225 [in additions.Addition_Chains]
+E_op:224 [in additions.Addition_Chains]
+E_eq:74 [in additions.Addition_Chains]
+E_one:73 [in additions.Addition_Chains]
+E_op:72 [in additions.Addition_Chains]
+E_eq:66 [in additions.Addition_Chains]
+E_one:65 [in additions.Addition_Chains]
+E_op:64 [in additions.Addition_Chains]
+E_eq:60 [in additions.Addition_Chains]
+E_one:59 [in additions.Addition_Chains]
+E_op:58 [in additions.Addition_Chains]
+E_eq:49 [in additions.Addition_Chains]
+E_one:48 [in additions.Addition_Chains]
+E_op:47 [in additions.Addition_Chains]
+e':33 [in hydras.rpo.list_set]
+e':36 [in hydras.rpo.list_set]
+e1:103 [in additions.fib]
+e1:110 [in additions.fib]
+e1:179 [in hydras.rpo.list_set]
+e2:104 [in additions.fib]
+e2:111 [in additions.fib]
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+e:11 [in hydras.MoreAck.PrimRecExamples]
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+E:126 [in hydras.Schutte.Ordering_Functions]
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+E:2 [in hydras.Prelude.Restriction]
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+e:97 [in hydras.rpo.list_set]
+

F

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+f:1 [in hydras.Ackermann.PAconsistent]
+f:1 [in hydras.Ackermann.NN2PA]
+f:1 [in hydras.MoreAck.AckNotPR]
+f:1 [in hydras.MoreAck.Iterate_compat]
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+f:10 [in hydras.Ackermann.folProof]
+f:10 [in hydras.Prelude.DecPreOrder_Instances]
+f:10 [in hydras.MoreAck.AckNotPR]
+f:10 [in hydras.Epsilon0.Large_Sets]
+f:10 [in Goedel.rosserPA]
+f:10 [in hydras.Ackermann.wellFormed]
+f:10 [in gaia_hydras.GLarge_Sets]
+f:10 [in hydras.Ackermann.wConsistent]
+f:100 [in hydras.Ackermann.LNN2LNT]
+f:100 [in hydras.Ackermann.folLogic]
+f:100 [in hydras.Ackermann.folLogic3]
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+f:1015 [in hydras.Ackermann.codeSubFormula]
+f:1018 [in hydras.Ackermann.codeSubFormula]
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+f:103 [in gaia_hydras.nfwfgaia]
+f:103 [in hydras.Ackermann.codePA]
+f:1031 [in hydras.Ackermann.codeSubFormula]
+f:1033 [in hydras.Ackermann.codeSubFormula]
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+f:104 [in hydras.rpo.rpo]
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+f:12 [in hydras.Ackermann.folProof]
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+f:12 [in hydras.Ackermann.PA]
+f:12 [in Goedel.goedel1]
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+f:12 [in hydras.MoreAck.PrimRecExamples]
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+f:1235 [in hydras.Ackermann.codeSubFormula]
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+f:13 [in hydras.Epsilon0.Large_Sets]
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+f:1396 [in gaia_hydras.nfwfgaia]
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+f:14 [in Goedel.goedel1]
+f:14 [in hydras.Ackermann.Languages]
+f:14 [in hydras.Schutte.Addition]
+f:14 [in Goedel.goedel2]
+f:140 [in hydras.Ackermann.LNN2LNT]
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+f:145 [in hydras.rpo.rpo]
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+f:1457 [in hydras.Ackermann.codeSubFormula]
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+f:1496 [in hydras.Ackermann.codeSubFormula]
+f:1499 [in hydras.Ackermann.codeSubFormula]
+f:15 [in hydras.Ackermann.folProof]
+f:15 [in additions.Euclidean_Chains]
+f:15 [in Goedel.goedel1]
+f:15 [in hydras.Ackermann.codePA]
+f:15 [in hydras.Ackermann.expressible]
+f:150 [in hydras.Ackermann.model]
+f:150 [in hydras.Hydra.Hydra_Lemmas]
+f:1502 [in hydras.Ackermann.codeSubFormula]
+f:1505 [in hydras.Ackermann.codeSubFormula]
+f:1508 [in hydras.Ackermann.codeSubFormula]
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+f:151 [in hydras.Ackermann.codeFreeVar]
+f:1511 [in hydras.Ackermann.codeSubFormula]
+f:1513 [in hydras.Ackermann.codeSubFormula]
+f:1515 [in hydras.Ackermann.codeSubFormula]
+f:1518 [in hydras.Ackermann.codeSubFormula]
+f:1521 [in hydras.Ackermann.codeSubFormula]
+f:1524 [in hydras.Ackermann.codeSubFormula]
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+f:1551 [in hydras.Ackermann.codeSubFormula]
+f:1554 [in hydras.Ackermann.codeSubFormula]
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+f:1560 [in hydras.Ackermann.codeSubFormula]
+f:1563 [in hydras.Ackermann.codeSubFormula]
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+f:167 [in hydras.OrdinalNotations.ON_Generic]
+f:167 [in hydras.Ackermann.codePA]
+f:168 [in hydras.rpo.term]
+f:169 [in hydras.Ackermann.codeFreeVar]
+f:169 [in hydras.Ackermann.cPair]
+f:17 [in hydras.Prelude.DecPreOrder_Instances]
+f:17 [in hydras.rpo.more_list]
+f:17 [in hydras.Ackermann.folLogic]
+f:17 [in hydras.Ackermann.expressible]
+f:170 [in hydras.Prelude.Iterates]
+f:170 [in hydras.Ackermann.codePA]
+f:171 [in hydras.Prelude.Iterates]
+f:173 [in hydras.Prelude.Iterates]
+f:173 [in hydras.Ackermann.codeFreeVar]
+f:173 [in hydras.Ackermann.cPair]
+f:173 [in hydras.Ackermann.codePA]
+f:174 [in gaia_hydras.T1Bridge]
+f:174 [in hydras.Ackermann.codePA]
+f:175 [in hydras.Ackermann.codeFreeVar]
+f:175 [in hydras.Schutte.Ordering_Functions]
+f:175 [in hydras.Ackermann.codePA]
+f:176 [in hydras.Ackermann.codePA]
+f:177 [in hydras.Ackermann.codeFreeVar]
+f:177 [in hydras.Ackermann.codePA]
+f:178 [in hydras.Schutte.Ordering_Functions]
+f:178 [in hydras.Ackermann.codePA]
+f:179 [in hydras.Ackermann.codeFreeVar]
+f:179 [in gaia_hydras.T1Bridge]
+f:179 [in hydras.OrdinalNotations.ON_Generic]
+f:179 [in hydras.Ackermann.codePA]
+f:18 [in hydras.Prelude.Iterates]
+f:18 [in hydras.Ackermann.LNT]
+f:18 [in hydras.Schutte.Critical]
+f:18 [in gaia_hydras.T1Bridge]
+f:18 [in gaia_hydras.GL_alpha]
+f:18 [in hydras.Ackermann.Languages]
+f:18 [in Goedel.goedel2]
+f:180 [in hydras.Ackermann.codePA]
+f:181 [in hydras.Ackermann.codeFreeVar]
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+f:181 [in hydras.Ackermann.codePA]
+f:182 [in gaia_hydras.T1Bridge]
+f:182 [in hydras.Ackermann.codePA]
+f:183 [in hydras.Ackermann.codeFreeVar]
+f:183 [in hydras.Schutte.Ordering_Functions]
+f:183 [in hydras.Ackermann.codePA]
+f:184 [in hydras.Prelude.Iterates]
+f:184 [in hydras.Ackermann.codePA]
+f:1845 [in gaia_hydras.nfwfgaia]
+f:185 [in gaia_hydras.T1Bridge]
+f:185 [in hydras.Ackermann.codePA]
+f:1852 [in gaia_hydras.nfwfgaia]
+f:186 [in hydras.Ackermann.codePA]
+f:1861 [in gaia_hydras.nfwfgaia]
+f:1863 [in gaia_hydras.nfwfgaia]
+f:1867 [in gaia_hydras.nfwfgaia]
+f:187 [in hydras.Ackermann.codeFreeVar]
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+f:187 [in hydras.Ackermann.wellFormed]
+f:187 [in hydras.Ackermann.codePA]
+f:1871 [in gaia_hydras.nfwfgaia]
+f:1872 [in gaia_hydras.nfwfgaia]
+f:1876 [in gaia_hydras.nfwfgaia]
+f:188 [in hydras.Ackermann.codePA]
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+f:189 [in hydras.Ackermann.cPair]
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+f:189 [in hydras.Ackermann.codePA]
+f:19 [in hydras.Epsilon0.Epsilon0rpo]
+f:19 [in hydras.Ackermann.LNN]
+f:19 [in hydras.Ackermann.checkPrf]
+f:19 [in hydras.Ackermann.PA]
+f:19 [in gaia_hydras.GL_alpha]
+f:190 [in hydras.Ackermann.wellFormed]
+f:190 [in hydras.Ackermann.codePA]
+f:191 [in hydras.Prelude.Iterates]
+f:191 [in hydras.Ackermann.codeFreeVar]
+f:191 [in hydras.Ackermann.codePA]
+f:192 [in hydras.Ackermann.wellFormed]
+f:192 [in hydras.OrdinalNotations.ON_Generic]
+f:192 [in hydras.Ackermann.codePA]
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+f:193 [in hydras.Ackermann.codePA]
+f:194 [in hydras.Ackermann.codePA]
+f:195 [in hydras.Ackermann.codeFreeVar]
+f:195 [in hydras.Ackermann.wellFormed]
+f:195 [in hydras.Ackermann.codePA]
+f:1950 [in gaia_hydras.nfwfgaia]
+f:1956 [in gaia_hydras.nfwfgaia]
+f:196 [in hydras.Ackermann.codePA]
+f:1963 [in gaia_hydras.nfwfgaia]
+f:1968 [in gaia_hydras.nfwfgaia]
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+f:1974 [in gaia_hydras.nfwfgaia]
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+f:1980 [in gaia_hydras.nfwfgaia]
+f:1985 [in gaia_hydras.nfwfgaia]
+f:199 [in hydras.Ackermann.codePA]
+f:1991 [in gaia_hydras.nfwfgaia]
+f:1996 [in gaia_hydras.nfwfgaia]
+F:2 [in hydras.MoreAck.BadSubst]
+f:2 [in hydras.Prelude.Iterates]
+f:2 [in hydras.Ackermann.wConsistent]
+f:20 [in hydras.Ackermann.model]
+f:20 [in Goedel.rosserPA]
+f:20 [in hydras.Ackermann.folLogic]
+f:20 [in hydras.Schutte.Ordering_Functions]
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+f:200 [in hydras.Ackermann.codePA]
+f:2002 [in gaia_hydras.nfwfgaia]
+f:2005 [in gaia_hydras.nfwfgaia]
+f:2009 [in gaia_hydras.nfwfgaia]
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+f:204 [in hydras.Ackermann.codePA]
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+f:205 [in hydras.Ackermann.codePA]
+f:206 [in hydras.rpo.more_list]
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+f:53 [in hydras.Schutte.Ordering_Functions]
+f:538 [in hydras.Ackermann.checkPrf]
+f:54 [in hydras.Ackermann.folProof]
+f:54 [in hydras.Ackermann.model]
+f:54 [in hydras.rpo.rpo]
+f:541 [in hydras.Ackermann.checkPrf]
+f:542 [in gaia_hydras.nfwfgaia]
+f:544 [in hydras.Ackermann.checkPrf]
+f:545 [in hydras.Ackermann.checkPrf]
+f:546 [in gaia_hydras.nfwfgaia]
+f:548 [in hydras.Ackermann.checkPrf]
+f:55 [in hydras.Ackermann.LNN]
+f:55 [in hydras.Ackermann.folLogic]
+f:55 [in hydras.MoreAck.PrimRecExamples]
+f:550 [in gaia_hydras.nfwfgaia]
+f:553 [in gaia_hydras.nfwfgaia]
+f:558 [in gaia_hydras.nfwfgaia]
+f:56 [in hydras.Ackermann.LNT]
+f:56 [in hydras.Ackermann.primRec]
+f:56 [in hydras.OrdinalNotations.ON_Generic]
+f:560 [in gaia_hydras.nfwfgaia]
+f:566 [in gaia_hydras.nfwfgaia]
+f:568 [in gaia_hydras.nfwfgaia]
+f:57 [in hydras.Prelude.Iterates]
+f:57 [in Goedel.rosserPA]
+f:57 [in Goedel.codeSysPrf]
+f:576 [in gaia_hydras.nfwfgaia]
+f:58 [in hydras.Ackermann.model]
+f:58 [in hydras.Ackermann.folLogic]
+f:58 [in hydras.Schutte.Ordering_Functions]
+f:582 [in gaia_hydras.nfwfgaia]
+f:587 [in gaia_hydras.nfwfgaia]
+f:59 [in hydras.Ackermann.LNN]
+f:59 [in hydras.Ackermann.LNT]
+f:59 [in hydras.Prelude.Sort_spec]
+f:59 [in hydras.MoreAck.PrimRecExamples]
+f:593 [in gaia_hydras.nfwfgaia]
+f:596 [in gaia_hydras.nfwfgaia]
+f:599 [in hydras.Ackermann.primRec]
+F:6 [in hydras.Ackermann.Deduction]
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+f:61 [in hydras.Ackermann.folLogic]
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+f:621 [in hydras.Ackermann.primRec]
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+f:63 [in hydras.Epsilon0.Hprime]
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+f:65 [in hydras.Ackermann.Languages]
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+f:7 [in hydras.Epsilon0.Large_Sets]
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+f:7 [in Goedel.goedel1]
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+f:999 [in hydras.Ackermann.codeSubFormula]
+

G

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+g:132 [in hydras.Ackermann.LNN2LNT]
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+g:134 [in hydras.Ackermann.LNN]
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+g:142 [in hydras.Ackermann.LNN2LNT]
+g:143 [in hydras.OrdinalNotations.ON_Generic]
+g:144 [in Goedel.PRrepresentable]
+g:146 [in hydras.Ackermann.LNN2LNT]
+g:149 [in hydras.Ackermann.codeSubFormula]
+g:15 [in hydras.Ackermann.Deduction]
+g:15 [in hydras.Ackermann.folLogic]
+g:152 [in hydras.Prelude.Iterates]
+g:155 [in Goedel.PRrepresentable]
+g:156 [in hydras.OrdinalNotations.ON_Generic]
+g:16 [in Goedel.rosserPA]
+g:16 [in Goedel.goedel2]
+g:161 [in hydras.Prelude.Iterates]
+g:162 [in hydras.Ackermann.primRec]
+g:167 [in hydras.Ackermann.model]
+g:168 [in hydras.Ackermann.model]
+g:168 [in hydras.OrdinalNotations.ON_Generic]
+g:169 [in hydras.Prelude.Iterates]
+g:169 [in hydras.Ackermann.model]
+g:169 [in hydras.Ackermann.codePA]
+g:17 [in Goedel.rosserPA]
+g:170 [in hydras.Ackermann.primRec]
+g:172 [in hydras.Ackermann.codePA]
+g:174 [in hydras.Prelude.Iterates]
+g:178 [in hydras.Ackermann.codeSubTerm]
+g:178 [in gaia_hydras.T1Bridge]
+g:178 [in hydras.Ackermann.primRec]
+g:18 [in Goedel.rosserPA]
+g:18 [in hydras.Ackermann.folLogic]
+g:18 [in Goedel.goedel1]
+g:180 [in hydras.OrdinalNotations.ON_Generic]
+g:181 [in hydras.Ackermann.codeSubTerm]
+g:181 [in gaia_hydras.T1Bridge]
+g:184 [in gaia_hydras.T1Bridge]
+g:185 [in hydras.Prelude.Iterates]
+g:186 [in hydras.Prelude.Iterates]
+g:188 [in hydras.Prelude.Iterates]
+g:188 [in gaia_hydras.T1Bridge]
+g:189 [in hydras.Ackermann.primRec]
+g:19 [in hydras.Ackermann.LNT]
+g:19 [in hydras.Schutte.Critical]
+g:19 [in Goedel.rosserPA]
+g:19 [in Goedel.goedel1]
+g:19 [in hydras.Ackermann.Languages]
+g:192 [in hydras.Prelude.Iterates]
+g:193 [in hydras.Prelude.Iterates]
+g:193 [in hydras.OrdinalNotations.ON_Generic]
+g:195 [in hydras.Prelude.Iterates]
+g:199 [in hydras.Prelude.Iterates]
+g:20 [in hydras.Ackermann.checkPrf]
+g:20 [in Goedel.goedel1]
+g:20 [in Goedel.goedel2]
+g:200 [in hydras.Ackermann.primRec]
+g:201 [in hydras.Prelude.Iterates]
+g:204 [in hydras.Prelude.Iterates]
+g:205 [in hydras.Ackermann.codeSubFormula]
+g:21 [in Goedel.goedel1]
+g:2108 [in gaia_hydras.nfwfgaia]
+g:211 [in hydras.Ackermann.primRec]
+g:213 [in hydras.Ackermann.codeSubFormula]
+g:216 [in hydras.Prelude.Iterates]
+g:217 [in hydras.Prelude.Iterates]
+g:218 [in hydras.Prelude.Iterates]
+g:219 [in hydras.Prelude.Iterates]
+g:22 [in hydras.Ackermann.LNN]
+g:22 [in hydras.Ackermann.LNT]
+g:22 [in Goedel.goedel1]
+g:220 [in hydras.Prelude.Iterates]
+g:221 [in additions.Addition_Chains]
+g:223 [in hydras.Prelude.Iterates]
+g:227 [in hydras.Prelude.Iterates]
+g:228 [in hydras.Ackermann.primRec]
+g:229 [in additions.Addition_Chains]
+g:23 [in Goedel.goedel1]
+g:234 [in hydras.Ackermann.wellFormed]
+g:237 [in hydras.Ackermann.wellFormed]
+g:238 [in additions.Addition_Chains]
+g:24 [in Goedel.rosserPA]
+g:244 [in hydras.Ackermann.primRec]
+g:25 [in hydras.Ackermann.LNN]
+g:25 [in hydras.Ackermann.LNT]
+g:25 [in Goedel.rosserPA]
+g:25 [in hydras.Ackermann.folLogic]
+g:25 [in Goedel.rosser]
+g:25 [in Goedel.goedel2]
+g:252 [in hydras.Ackermann.fol]
+g:255 [in hydras.Gamma0.Gamma0]
+g:260 [in hydras.Ackermann.primRec]
+g:264 [in hydras.Ackermann.cPair]
+g:278 [in hydras.rpo.rpo]
+g:279 [in hydras.rpo.rpo]
+g:28 [in hydras.Ackermann.LNN]
+g:28 [in hydras.Ackermann.LNT]
+g:28 [in hydras.Ackermann.folLogic]
+g:281 [in hydras.Ackermann.primRec]
+g:29 [in hydras.Prelude.Iterates]
+g:294 [in hydras.Ackermann.primRec]
+g:302 [in hydras.Ackermann.fol]
+g:31 [in hydras.Ackermann.LNN]
+g:31 [in hydras.Ackermann.folLogic]
+g:31 [in hydras.Schutte.Ordering_Functions]
+g:31 [in hydras.MoreAck.FolExamples]
+g:320 [in hydras.rpo.rpo]
+g:321 [in hydras.rpo.rpo]
+g:323 [in hydras.Ackermann.primRec]
+g:327 [in Goedel.PRrepresentable]
+g:33 [in hydras.Prelude.Iterates]
+g:332 [in Goedel.PRrepresentable]
+g:337 [in Goedel.PRrepresentable]
+g:34 [in hydras.Ackermann.folLogic]
+g:34 [in Goedel.codeSysPrf]
+g:345 [in hydras.Ackermann.primRec]
+g:348 [in hydras.Ackermann.fol]
+g:349 [in hydras.Ackermann.primRec]
+g:35 [in hydras.Prelude.Iterates]
+g:354 [in hydras.Ackermann.primRec]
+g:357 [in hydras.Ackermann.primRec]
+g:36 [in Goedel.rosserPA]
+g:361 [in hydras.Ackermann.primRec]
+g:368 [in hydras.Ackermann.primRec]
+g:37 [in hydras.Ackermann.LNT]
+g:375 [in hydras.Ackermann.primRec]
+g:379 [in hydras.Ackermann.primRec]
+g:38 [in hydras.Prelude.Iterates]
+g:385 [in hydras.Ackermann.primRec]
+g:39 [in hydras.MoreAck.AckNotPR]
+g:39 [in Goedel.rosserPA]
+g:39 [in hydras.Ackermann.folLogic]
+g:392 [in hydras.Ackermann.primRec]
+g:392 [in hydras.Ackermann.fol]
+g:398 [in hydras.Ackermann.primRec]
+g:40 [in hydras.Ackermann.LNN]
+g:40 [in hydras.Ackermann.LNT]
+g:406 [in hydras.Ackermann.primRec]
+g:41 [in hydras.Epsilon0.Large_Sets]
+g:42 [in hydras.Ackermann.code]
+g:42 [in Goedel.rosserPA]
+g:423 [in hydras.Ackermann.primRec]
+g:428 [in hydras.Ackermann.primRec]
+g:43 [in hydras.Ackermann.LNN]
+g:43 [in hydras.Ackermann.LNT]
+g:43 [in hydras.MoreAck.AckNotPR]
+g:43 [in hydras.Ackermann.folLogic]
+g:435 [in hydras.Ackermann.primRec]
+g:44 [in hydras.Prelude.Iterates]
+g:45 [in hydras.MoreAck.PrimRecExamples]
+g:46 [in hydras.Ackermann.LNN]
+g:46 [in hydras.Ackermann.LNT]
+g:46 [in Goedel.rosserPA]
+g:46 [in hydras.rpo.rpo]
+g:47 [in hydras.MoreAck.AckNotPR]
+g:47 [in hydras.Ackermann.folLogic]
+g:48 [in hydras.Ackermann.folProof]
+g:48 [in hydras.MoreAck.PrimRecExamples]
+g:49 [in hydras.Ackermann.LNN]
+g:49 [in hydras.Ackermann.LNT]
+g:49 [in Goedel.rosserPA]
+g:49 [in hydras.Ackermann.expressible]
+g:50 [in hydras.Epsilon0.Large_Sets]
+g:50 [in Goedel.rosserPA]
+g:50 [in hydras.Ackermann.folLogic]
+g:51 [in hydras.MoreAck.AckNotPR]
+g:51 [in Goedel.rosserPA]
+g:51 [in hydras.MoreAck.PrimRecExamples]
+g:52 [in hydras.Ackermann.LNN]
+g:52 [in Goedel.rosserPA]
+g:53 [in hydras.Ackermann.LNT]
+g:53 [in Goedel.rosserPA]
+g:53 [in hydras.Ackermann.folLogic]
+g:53 [in gaia_hydras.nfwfgaia]
+g:53 [in hydras.MoreAck.PrimRecExamples]
+g:54 [in Goedel.rosserPA]
+g:55 [in Goedel.rosserPA]
+g:56 [in hydras.Ackermann.LNN]
+g:56 [in hydras.Ackermann.folLogic]
+g:56 [in hydras.MoreAck.PrimRecExamples]
+g:563 [in hydras.Ackermann.checkPrf]
+g:565 [in hydras.Ackermann.checkPrf]
+g:57 [in hydras.Ackermann.LNT]
+g:57 [in hydras.OrdinalNotations.ON_Generic]
+g:58 [in hydras.Prelude.Iterates]
+g:59 [in hydras.Ackermann.folLogic]
+g:59 [in Goedel.codeSysPrf]
+g:597 [in gaia_hydras.nfwfgaia]
+g:6 [in hydras.rpo.more_list]
+g:6 [in hydras.Ackermann.code]
+g:6 [in hydras.Ackermann.PA]
+g:60 [in hydras.Ackermann.LNN]
+g:60 [in hydras.Ackermann.LNT]
+g:60 [in hydras.MoreAck.PrimRecExamples]
+g:62 [in hydras.Ackermann.primRec]
+g:62 [in hydras.Ackermann.folLogic]
+g:63 [in hydras.Prelude.Iterates]
+g:63 [in hydras.Ackermann.LNN]
+g:63 [in hydras.Ackermann.LNT]
+g:63 [in hydras.Ackermann.fol]
+g:64 [in hydras.Ackermann.primRec]
+g:645 [in hydras.Ackermann.primRec]
+g:65 [in hydras.MoreAck.PrimRecExamples]
+g:651 [in hydras.Ackermann.primRec]
+g:656 [in hydras.Ackermann.primRec]
+g:66 [in hydras.Prelude.Iterates]
+g:66 [in hydras.Ackermann.LNN]
+g:66 [in hydras.Ackermann.LNT]
+g:66 [in hydras.MoreAck.AckNotPR]
+g:663 [in hydras.Ackermann.primRec]
+g:668 [in hydras.Ackermann.primRec]
+g:67 [in hydras.MoreAck.AckNotPR]
+g:67 [in hydras.MoreAck.PrimRecExamples]
+g:68 [in hydras.Ackermann.model]
+g:68 [in hydras.Ackermann.primRec]
+g:69 [in hydras.Ackermann.LNN]
+g:69 [in hydras.Ackermann.LNT]
+g:70 [in hydras.Ackermann.fol]
+g:71 [in hydras.MoreAck.PrimRecExamples]
+g:72 [in hydras.Ackermann.LNN]
+g:72 [in hydras.Ackermann.LNT]
+g:74 [in hydras.MoreAck.PrimRecExamples]
+g:75 [in hydras.Ackermann.LNN]
+g:77 [in hydras.Ackermann.folProp]
+g:781 [in hydras.Ackermann.codeSubFormula]
+g:784 [in hydras.Ackermann.codeSubFormula]
+g:79 [in hydras.Ackermann.folLogic]
+g:79 [in hydras.MoreAck.PrimRecExamples]
+g:8 [in hydras.solutions_exercises.MinPR2]
+g:812 [in hydras.Ackermann.primRec]
+g:816 [in hydras.Ackermann.primRec]
+g:82 [in hydras.Prelude.Iterates]
+g:820 [in hydras.Ackermann.primRec]
+g:823 [in hydras.Ackermann.primRec]
+g:83 [in hydras.MoreAck.PrimRecExamples]
+g:84 [in hydras.Schutte.PartialFun]
+g:86 [in hydras.Ackermann.folProp]
+g:86 [in hydras.Ackermann.folLogic]
+g:86 [in Goedel.codeSysPrf]
+g:89 [in hydras.Ackermann.LNT]
+g:89 [in Goedel.codeSysPrf]
+g:89 [in hydras.Epsilon0.Hprime]
+g:9 [in hydras.Ackermann.primRec]
+g:9 [in hydras.Ackermann.folLogic]
+g:90 [in hydras.Schutte.PartialFun]
+g:92 [in hydras.Ackermann.LNN]
+g:92 [in hydras.Ackermann.folLogic]
+g:94 [in hydras.Ackermann.codeSubFormula]
+g:95 [in hydras.Ackermann.folLogic]
+g:96 [in hydras.Ackermann.LNT]
+g:96 [in hydras.MoreAck.AckNotPR]
+g:98 [in hydras.Ackermann.folLogic]
+g:98 [in hydras.Ackermann.folReplace]
+g:99 [in hydras.Ackermann.LNN]
+

H

+HAB:175 [in hydras.Epsilon0.Large_Sets]
+HAB:22 [in gaia_hydras.GLarge_Sets]
+HAck:88 [in hydras.MoreAck.AckNotPR]
+Halpha:14 [in gaia_hydras.GHprime]
+Halpha:170 [in hydras.Epsilon0.Large_Sets]
+Halpha:401 [in hydras.Epsilon0.Paths]
+Halpha:662 [in hydras.Epsilon0.T1]
+Halpha:665 [in hydras.Epsilon0.T1]
+Halpha:710 [in hydras.Epsilon0.T1]
+Ha:17 [in gaia_hydras.GLarge_Sets]
+HA:173 [in hydras.Epsilon0.Large_Sets]
+HA:20 [in gaia_hydras.GLarge_Sets]
+Ha:390 [in hydras.Epsilon0.Paths]
+Ha:50 [in gaia_hydras.GCanon]
+Ha:9 [in gaia_hydras.GCanon]
+Ha:98 [in hydras.Epsilon0.Hessenberg]
+Hb:100 [in hydras.Epsilon0.Hessenberg]
+HB:174 [in hydras.Epsilon0.Large_Sets]
+HB:21 [in gaia_hydras.GLarge_Sets]
+Hb:391 [in hydras.Epsilon0.Paths]
+Hc:102 [in hydras.Epsilon0.Hessenberg]
+height:17 [in hydras.Epsilon0.T1]
+Hf':121 [in hydras.Prelude.Iterates]
+Hf':155 [in hydras.Prelude.Iterates]
+Hf':94 [in hydras.Prelude.Iterates]
+Hf:120 [in hydras.Prelude.Iterates]
+Hf:154 [in hydras.Prelude.Iterates]
+Hf:47 [in hydras.Prelude.Iterates]
+Hf:50 [in hydras.Prelude.Iterates]
+Hf:74 [in hydras.Schutte.PartialFun]
+Hf:75 [in hydras.MoreAck.AckNotPR]
+Hf:79 [in hydras.MoreAck.AckNotPR]
+Hf:84 [in hydras.MoreAck.AckNotPR]
+Hf:88 [in hydras.Prelude.Iterates]
+Hf:93 [in hydras.Prelude.Iterates]
+Hgamma:159 [in additions.AM]
+Hgamma:24 [in additions.Fib2]
+Hgamma:354 [in additions.Euclidean_Chains]
+Hgamma:8 [in additions.Strategies]
+Hgt:163 [in hydras.Prelude.Iterates]
+Hij:64 [in gaia_hydras.GCanon]
+Hincl:170 [in hydras.Prelude.MoreLists]
+Hk:74 [in additions.Euclidean_Chains]
+Hlambda:51 [in gaia_hydras.GCanon]
+Hlim:45 [in gaia_hydras.GCanon]
+Hlim:52 [in gaia_hydras.GCanon]
+Hlim:63 [in gaia_hydras.GCanon]
+Hlim:67 [in gaia_hydras.GCanon]
+Hmono:200 [in hydras.Schutte.Schutte_basics]
+Hmu:2 [in hydras.Hydra.Epsilon0_Needed_Free]
+Hmu:3 [in hydras.Hydra.Epsilon0_Needed_Generic]
+Hm':165 [in hydras.Prelude.Iterates]
+Hm:164 [in hydras.Prelude.Iterates]
+Hnf:3 [in gaia_hydras.GF_alpha]
+Hnf:34 [in gaia_hydras.GF_alpha]
+Hnf:40 [in gaia_hydras.GCanon]
+Hnf:44 [in gaia_hydras.GCanon]
+Hnf:62 [in gaia_hydras.GCanon]
+Hnf:66 [in gaia_hydras.GCanon]
+hP:25 [in gaia_hydras.T1Bridge]
+Hrecb:719 [in hydras.Ackermann.primRec]
+Hrecb:723 [in hydras.Ackermann.primRec]
+Hrecb:725 [in hydras.Ackermann.primRec]
+Hrecb:728 [in hydras.Ackermann.primRec]
+Hrecb:731 [in hydras.Ackermann.primRec]
+Hrecb:734 [in hydras.Ackermann.primRec]
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+Hrecb:740 [in hydras.Ackermann.primRec]
+Hrecb:743 [in hydras.Ackermann.primRec]
+Hrecb:746 [in hydras.Ackermann.primRec]
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+Hrecb:755 [in hydras.Ackermann.primRec]
+Hrecb:757 [in hydras.Ackermann.primRec]
+Hrecb:783 [in hydras.Ackermann.primRec]
+Hrecb:786 [in hydras.Ackermann.primRec]
+Hrecb:790 [in hydras.Ackermann.primRec]
+Hrecb:793 [in hydras.Ackermann.primRec]
+Hrecb:797 [in hydras.Ackermann.primRec]
+Hrecb:803 [in hydras.Ackermann.primRec]
+Hrecb:808 [in hydras.Ackermann.primRec]
+Hrecb:811 [in hydras.Ackermann.primRec]
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+Hrecn:147 [in hydras.Ackermann.cPair]
+Hrecn:151 [in hydras.Ackermann.cPair]
+Hrecn:155 [in hydras.Ackermann.cPair]
+Hrecn:157 [in hydras.Ackermann.cPair]
+Hrecn:160 [in hydras.Ackermann.cPair]
+Hrecn:178 [in hydras.Ackermann.model]
+Hrecn:185 [in hydras.Ackermann.model]
+Hrecn:204 [in hydras.Ackermann.model]
+Hrecn:211 [in hydras.Ackermann.model]
+Hrecn:64 [in hydras.Ackermann.folLogic3]
+Hrecs:10 [in hydras.Ackermann.codeSubFormula]
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+Hrecs:116 [in hydras.Ackermann.codeList]
+Hrecs:12 [in hydras.Ackermann.codeSubFormula]
+Hrecs:121 [in hydras.Ackermann.codeList]
+Hrecs:124 [in hydras.Ackermann.codeList]
+Hrecs:127 [in hydras.Ackermann.codeList]
+Hrecs:13 [in hydras.Ackermann.codeList]
+Hrecs:130 [in hydras.Ackermann.codeList]
+Hrecs:133 [in hydras.Ackermann.codeList]
+Hrecs:136 [in hydras.Ackermann.codeList]
+Hrecs:139 [in hydras.Ackermann.codeList]
+Hrecs:14 [in hydras.Ackermann.codeSubFormula]
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+Hrecs:145 [in hydras.Ackermann.codeList]
+Hrecs:147 [in hydras.Ackermann.codeList]
+Hrecs:149 [in hydras.Ackermann.codeList]
+Hrecs:15 [in hydras.Ackermann.codeList]
+Hrecs:151 [in hydras.Ackermann.codeList]
+Hrecs:153 [in hydras.Ackermann.codeList]
+Hrecs:155 [in hydras.Ackermann.codeList]
+Hrecs:16 [in hydras.Ackermann.codeSubFormula]
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+Hrecs:168 [in hydras.Ackermann.codeList]
+Hrecs:17 [in hydras.Ackermann.codeList]
+Hrecs:171 [in hydras.Ackermann.codeList]
+Hrecs:174 [in hydras.Ackermann.codeList]
+Hrecs:177 [in hydras.Ackermann.codeList]
+Hrecs:18 [in hydras.Ackermann.codeSubFormula]
+Hrecs:180 [in hydras.Ackermann.codeList]
+Hrecs:19 [in hydras.Ackermann.wellFormed]
+Hrecs:19 [in hydras.Ackermann.codeList]
+Hrecs:197 [in hydras.Ackermann.codeList]
+Hrecs:199 [in hydras.Ackermann.codeList]
+Hrecs:2 [in hydras.Ackermann.codeList]
+Hrecs:20 [in hydras.Ackermann.codeSubFormula]
+Hrecs:201 [in hydras.Ackermann.codeList]
+Hrecs:203 [in hydras.Ackermann.codeList]
+Hrecs:205 [in hydras.Ackermann.codeList]
+Hrecs:207 [in hydras.Ackermann.codeList]
+Hrecs:21 [in hydras.Ackermann.wellFormed]
+Hrecs:21 [in hydras.Ackermann.codeList]
+Hrecs:213 [in hydras.Ackermann.codeList]
+Hrecs:218 [in hydras.Ackermann.codeList]
+Hrecs:22 [in hydras.Ackermann.codeSubFormula]
+Hrecs:221 [in hydras.Ackermann.codeList]
+Hrecs:226 [in hydras.Ackermann.codeList]
+Hrecs:229 [in hydras.Ackermann.codeList]
+Hrecs:23 [in hydras.Ackermann.codeList]
+Hrecs:232 [in hydras.Ackermann.codeList]
+Hrecs:235 [in hydras.Ackermann.codeList]
+Hrecs:238 [in hydras.Ackermann.codeList]
+Hrecs:24 [in hydras.Ackermann.codeSubFormula]
+Hrecs:241 [in hydras.Ackermann.codeList]
+Hrecs:244 [in hydras.Ackermann.codeList]
+Hrecs:247 [in hydras.Ackermann.codeList]
+Hrecs:25 [in hydras.Ackermann.codeList]
+Hrecs:250 [in hydras.Ackermann.codeList]
+Hrecs:253 [in hydras.Ackermann.codeList]
+Hrecs:256 [in hydras.Ackermann.codeList]
+Hrecs:259 [in hydras.Ackermann.codeList]
+Hrecs:26 [in hydras.Ackermann.codeSubFormula]
+Hrecs:27 [in hydras.Ackermann.codeList]
+Hrecs:276 [in hydras.Ackermann.codeList]
+Hrecs:278 [in hydras.Ackermann.codeList]
+Hrecs:28 [in hydras.Ackermann.codeSubFormula]
+Hrecs:280 [in hydras.Ackermann.codeList]
+Hrecs:282 [in hydras.Ackermann.codeList]
+Hrecs:284 [in hydras.Ackermann.codeList]
+Hrecs:286 [in hydras.Ackermann.codeList]
+Hrecs:29 [in hydras.Ackermann.codeList]
+Hrecs:30 [in hydras.Ackermann.codeSubFormula]
+Hrecs:31 [in hydras.Ackermann.codeList]
+Hrecs:32 [in hydras.Ackermann.codeSubFormula]
+Hrecs:34 [in hydras.Ackermann.codeSubFormula]
+Hrecs:36 [in hydras.Ackermann.codeSubFormula]
+Hrecs:41 [in hydras.Ackermann.codeList]
+Hrecs:42 [in hydras.Ackermann.codeSubFormula]
+Hrecs:44 [in hydras.Ackermann.codeSubFormula]
+Hrecs:46 [in hydras.Ackermann.codeSubFormula]
+Hrecs:46 [in hydras.Ackermann.codeList]
+Hrecs:48 [in hydras.Ackermann.codeSubFormula]
+Hrecs:49 [in hydras.Ackermann.codeList]
+Hrecs:5 [in hydras.Ackermann.codeList]
+Hrecs:54 [in hydras.Ackermann.codeList]
+Hrecs:57 [in hydras.Ackermann.codeList]
+Hrecs:60 [in hydras.Ackermann.codeList]
+Hrecs:63 [in hydras.Ackermann.codeList]
+Hrecs:66 [in hydras.Ackermann.codeList]
+Hrecs:69 [in hydras.Ackermann.codeList]
+Hrecs:7 [in hydras.Ackermann.codeSubFormula]
+Hrecs:7 [in hydras.Ackermann.codeList]
+Hrecs:72 [in hydras.Ackermann.codeList]
+Hrecs:74 [in hydras.Ackermann.codeList]
+Hrecs:76 [in hydras.Ackermann.codeList]
+Hrecs:78 [in hydras.Ackermann.codeList]
+Hrecs:80 [in hydras.Ackermann.codeList]
+Hrecs:82 [in hydras.Ackermann.codeList]
+Hrecs:84 [in hydras.Ackermann.codeList]
+Hrecs:86 [in hydras.Ackermann.codeList]
+Hrecs:9 [in hydras.Ackermann.codeList]
+Hrecs:90 [in hydras.Ackermann.codeList]
+Hrecs:92 [in hydras.Ackermann.codeList]
+Hrecs:94 [in hydras.Ackermann.codeList]
+Hrecs:96 [in hydras.Ackermann.codeList]
+hR:28 [in gaia_hydras.T1Bridge]
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+hs':142 [in hydras.Hydra.Hydra_Lemmas]
+hs:139 [in hydras.Hydra.Hydra_Lemmas]
+hs:141 [in hydras.Hydra.Hydra_Lemmas]
+hs:8 [in gaia_hydras.GHydra]
+hs:9 [in gaia_hydras.GHydra]
+Htrans:18 [in hydras.Prelude.Sort_spec]
+Htrans:21 [in hydras.Prelude.Sort_spec]
+Htrans:25 [in hydras.Prelude.Sort_spec]
+Hvar:2 [in hydras.Hydra.Omega2_Small]
+Hvar:2 [in hydras.Hydra.Omega_Small]
+Hwf:130 [in hydras.Hydra.Hydra_Lemmas]
+Hwf:21 [in hydras.solutions_exercises.T1_ltNotWf]
+HX:142 [in hydras.Schutte.Schutte_basics]
+HX:145 [in hydras.Schutte.Schutte_basics]
+Hx:27 [in hydras.Prelude.Restriction]
+Hx:31 [in hydras.Prelude.Restriction]
+Hx:38 [in hydras.Prelude.Restriction]
+HX:61 [in hydras.Schutte.Well_Orders]
+Hyp1:20 [in hydras.Ackermann.folProof]
+Hyp2:21 [in hydras.Ackermann.folProof]
+hyp:220 [in hydras.Ackermann.fol]
+hyp:221 [in hydras.Ackermann.fol]
+hyp:222 [in hydras.Ackermann.fol]
+Hyp:24 [in hydras.Ackermann.folProof]
+Hyp:46 [in hydras.Ackermann.folProof]
+Hyp:48 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
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+Hy:5 [in hydras.Hydra.Epsilon0_Needed_Free]
+Hy:6 [in hydras.Hydra.Epsilon0_Needed_Generic]
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+h'':135 [in hydras.Hydra.Hydra_Definitions]
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+h':100 [in hydras.Hydra.O2H]
+h':101 [in hydras.Hydra.Hydra_Definitions]
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+H':108 [in hydras.OrdinalNotations.OmegaOmega]
+h':110 [in hydras.Hydra.Hydra_Definitions]
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+h':118 [in hydras.Hydra.Hydra_Lemmas]
+h':12 [in gaia_hydras.GHydra]
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+h':124 [in hydras.Hydra.Hydra_Definitions]
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+h':130 [in hydras.Hydra.Hydra_Definitions]
+h':132 [in hydras.Hydra.Hydra_Lemmas]
+h':134 [in hydras.Hydra.Hydra_Lemmas]
+h':134 [in hydras.Hydra.Hydra_Definitions]
+h':138 [in hydras.Hydra.Hydra_Lemmas]
+h':14 [in gaia_hydras.GHydra]
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+h':152 [in hydras.Hydra.Hydra_Definitions]
+h':167 [in hydras.Hydra.Hydra_Definitions]
+h':178 [in hydras.Prelude.Iterates]
+h':19 [in hydras.Hydra.Hydra_Lemmas]
+h':2 [in hydras.Hydra.Hydra_Examples]
+h':23 [in hydras.Hydra.Hydra_Termination]
+h':25 [in hydras.Hydra.Hydra_Termination]
+H':28 [in hydras.Epsilon0.E0]
+h':3 [in hydras.Hydra.Hydra_Examples]
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+h':36 [in hydras.Hydra.O2H]
+h':4 [in hydras.Hydra.Hydra_Examples]
+h':44 [in hydras.Hydra.Hydra_Termination]
+h':46 [in hydras.Hydra.Hydra_Termination]
+H':489 [in hydras.Epsilon0.T1]
+h':49 [in hydras.Hydra.Hydra_Termination]
+H':493 [in hydras.Epsilon0.T1]
+h':51 [in hydras.Hydra.O2H]
+h':51 [in hydras.Hydra.Hydra_Termination]
+H':696 [in hydras.Gamma0.Gamma0]
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+h':72 [in hydras.Hydra.Hydra_Definitions]
+h':76 [in hydras.Hydra.Hydra_Lemmas]
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+h':82 [in hydras.Hydra.Hydra_Lemmas]
+h':86 [in hydras.Hydra.Hydra_Definitions]
+h':9 [in hydras.Hydra.Epsilon0_Needed_Generic]
+h':90 [in hydras.Hydra.O2H]
+h':93 [in hydras.Hydra.O2H]
+h':93 [in hydras.Hydra.Hydra_Definitions]
+h':95 [in hydras.Hydra.Hydra_Definitions]
+h':98 [in hydras.Hydra.Hydra_Definitions]
+H0:21 [in hydras.Epsilon0.F_omega]
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+H0:296 [in hydras.rpo.rpo]
+H0:298 [in hydras.rpo.rpo]
+h0:35 [in hydras.Prelude.Restriction]
+h0:41 [in hydras.Prelude.Restriction]
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+H0:85 [in hydras.Ackermann.LNN2LNT]
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+H:107 [in hydras.OrdinalNotations.OmegaOmega]
+h:109 [in hydras.Hydra.Hydra_Definitions]
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+h:11 [in hydras.Hydra.Epsilon0_Needed_Std]
+h:11 [in gaia_hydras.GHydra]
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+H:11 [in hydras.Schutte.Countable]
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+h:112 [in hydras.Hydra.Hydra_Definitions]
+h:114 [in hydras.Hydra.Hydra_Lemmas]
+h:114 [in hydras.Hydra.Hydra_Definitions]
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+h:12 [in hydras.solutions_exercises.MorePRExamples]
+h:12 [in hydras.Hydra.Hydra_Definitions]
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+h:120 [in hydras.Hydra.Hydra_Definitions]
+h:121 [in hydras.Hydra.Hydra_Lemmas]
+h:123 [in hydras.Ackermann.LNN]
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+h:123 [in hydras.Hydra.Hydra_Definitions]
+h:125 [in hydras.Ackermann.folLogic]
+h:129 [in hydras.Hydra.Hydra_Definitions]
+h:13 [in hydras.Ackermann.primRec]
+h:13 [in gaia_hydras.GHydra]
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+h:151 [in hydras.Hydra.Hydra_Definitions]
+h:152 [in hydras.Hydra.Hydra_Lemmas]
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+H:157 [in hydras.Prelude.Iterates]
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+h:166 [in hydras.Hydra.Hydra_Definitions]
+H:171 [in hydras.Ackermann.primRec]
+H:172 [in hydras.Ackermann.folProp]
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+H:179 [in hydras.Ackermann.folProp]
+H:179 [in hydras.Ackermann.primRec]
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+H:204 [in hydras.Ackermann.folProp]
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+H:212 [in hydras.Ackermann.primRec]
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+H:22 [in hydras.Prelude.STDPP_compat]
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+h:24 [in hydras.Hydra.Hydra_Termination]
+H:24 [in hydras.Prelude.STDPP_compat]
+h:24 [in hydras.Hydra.Hydra_Definitions]
+H:25 [in hydras.Epsilon0.E0]
+H:251 [in hydras.Ackermann.folProp]
+H:258 [in hydras.Ackermann.folProp]
+h:26 [in gaia_hydras.GHydra]
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+H:29 [in hydras.Prelude.Restriction]
+h:30 [in hydras.Hydra.Hydra_Definitions]
+H:300 [in hydras.Ackermann.folProp]
+H:307 [in hydras.Ackermann.folProp]
+H:316 [in hydras.Ackermann.folProp]
+h:32 [in hydras.Hydra.O2H]
+H:323 [in hydras.Ackermann.folProp]
+h:329 [in Goedel.PRrepresentable]
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+H:33 [in hydras.Schutte.Critical]
+H:332 [in hydras.Ackermann.folProp]
+h:334 [in Goedel.PRrepresentable]
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+H:339 [in hydras.Ackermann.folProp]
+h:34 [in hydras.Hydra.Hydra_Termination]
+H:34 [in hydras.MoreAck.FolExamples]
+H:348 [in hydras.Ackermann.folProp]
+h:35 [in hydras.Hydra.O2H]
+H:355 [in hydras.Ackermann.folProp]
+h:36 [in hydras.Prelude.Restriction]
+h:36 [in hydras.Hydra.Hydra_Definitions]
+h:362 [in hydras.Ackermann.primRec]
+H:364 [in hydras.Ackermann.folProp]
+H:369 [in hydras.Epsilon0.T1]
+H:37 [in hydras.Schutte.Ordering_Functions]
+H:370 [in hydras.Epsilon0.T1]
+H:371 [in hydras.Ackermann.folProp]
+H:372 [in hydras.Epsilon0.T1]
+h:38 [in hydras.MoreAck.AckNotPR]
+H:38 [in hydras.Schutte.Critical]
+H:38 [in hydras.Hydra.Hydra_Termination]
+H:380 [in hydras.Ackermann.folProp]
+H:385 [in hydras.Epsilon0.Paths]
+H:387 [in hydras.Ackermann.folProp]
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+h:40 [in hydras.Ackermann.folLogic]
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+h:41 [in hydras.Hydra.Hydra_Definitions]
+h:42 [in hydras.Prelude.Restriction]
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+H:423 [in hydras.Epsilon0.T1]
+h:43 [in hydras.Hydra.Hydra_Termination]
+H:44 [in hydras.Schutte.Correctness_E0]
+h:44 [in hydras.Ackermann.folLogic]
+h:45 [in hydras.Prelude.Iterates]
+h:45 [in hydras.Hydra.O2H]
+H:45 [in hydras.Prelude.Restriction]
+h:45 [in hydras.Hydra.Hydra_Termination]
+H:46 [in hydras.Ackermann.LNN2LNT]
+h:47 [in hydras.Hydra.Hydra_Termination]
+h:48 [in hydras.Hydra.Hydra_Termination]
+H:487 [in hydras.Epsilon0.T1]
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+h:50 [in hydras.Hydra.Hydra_Termination]
+h:50 [in hydras.Hydra.Hydra_Lemmas]
+h:53 [in hydras.Ackermann.LNN]
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+h:54 [in hydras.Hydra.Hydra_Definitions]
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+H:59 [in hydras.Epsilon0.T1]
+h:59 [in hydras.Prelude.Iterates]
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+h:6 [in hydras.Hydra.Hydra_Examples]
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+h:60 [in hydras.Hydra.Hydra_Definitions]
+H:61 [in hydras.Epsilon0.T1]
+h:63 [in gaia_hydras.nfwfgaia]
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+H:64 [in hydras.Prelude.MoreLists]
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+h:67 [in hydras.Prelude.Iterates]
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+h:7 [in gaia_hydras.GHydra]
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+h:78 [in hydras.Hydra.Hydra_Lemmas]
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+h:8 [in hydras.Hydra.Hydra_Examples]
+h:8 [in hydras.Hydra.Epsilon0_Needed_Generic]
+H:8 [in hydras.Schutte.Countable]
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+h:85 [in hydras.Hydra.Hydra_Definitions]
+h:88 [in hydras.Hydra.Hydra_Lemmas]
+h:88 [in hydras.Hydra.Hydra_Definitions]
+h:9 [in hydras.solutions_exercises.MinPR2]
+h:9 [in hydras.Hydra.Hydra_Examples]
+H:90 [in hydras.Ackermann.LNN2LNT]
+h:91 [in hydras.Hydra.Hydra_Lemmas]
+h:92 [in hydras.Hydra.O2H]
+h:92 [in hydras.Hydra.Hydra_Definitions]
+h:93 [in hydras.Hydra.Hydra_Lemmas]
+h:94 [in hydras.Hydra.Hydra_Definitions]
+h:96 [in hydras.Hydra.O2H]
+h:96 [in hydras.Hydra.Hydra_Lemmas]
+h:97 [in hydras.Hydra.Hydra_Definitions]
+h:98 [in hydras.Hydra.O2H]
+h:99 [in hydras.Hydra.Hydra_Definitions]
+

I

+idn:126 [in additions.fib]
+id:118 [in additions.fib]
+IHn:102 [in hydras.MoreAck.AckNotPR]
+inhX:45 [in hydras.Schutte.Well_Orders]
+inh:2 [in hydras.Schutte.MoreEpsilonIota]
+inh:53 [in hydras.Schutte.Well_Orders]
+inh:58 [in hydras.Schutte.Well_Orders]
+input:61 [in hydras.Ackermann.codeSubFormula]
+invar:18 [in hydras.Prelude.First_toggle]
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+iota:114 [in hydras.OrdinalNotations.ON_Generic]
+iota:129 [in hydras.OrdinalNotations.ON_Generic]
+iota:141 [in hydras.OrdinalNotations.ON_Generic]
+iota:154 [in hydras.OrdinalNotations.ON_Generic]
+iota:38 [in hydras.OrdinalNotations.ON_Generic]
+iota:71 [in hydras.OrdinalNotations.ON_Generic]
+iota:89 [in hydras.OrdinalNotations.ON_Generic]
+i':179 [in hydras.Prelude.Iterates]
+i:1 [in hydras.OrdinalNotations.ON_Omega2]
+i:1 [in hydras.Prelude.MoreLists]
+i:1 [in gaia_hydras.GPaths]
+i:1 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:1 [in hydras.OrdinalNotations.ON_Omega]
+i:1 [in hydras.Epsilon0.Paths]
+i:10 [in hydras.Prelude.First_toggle]
+i:10 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:10 [in hydras.Hydra.Epsilon0_Needed_Std]
+i:10 [in hydras.Hydra.BigBattle]
+i:10 [in hydras.Hydra.Omega2_Small]
+i:10 [in gaia_hydras.GL_alpha]
+i:101 [in hydras.Prelude.Iterates]
+i:102 [in hydras.Prelude.MoreLists]
+i:102 [in hydras.OrdinalNotations.OmegaOmega]
+i:103 [in hydras.Epsilon0.Canon]
+i:104 [in hydras.Epsilon0.F_alpha]
+i:104 [in hydras.Hydra.Hydra_Lemmas]
+i:104 [in gaia_hydras.nfwfgaia]
+i:105 [in hydras.Prelude.MoreLists]
+i:106 [in gaia_hydras.nfwfgaia]
+i:108 [in hydras.Epsilon0.T1]
+i:108 [in hydras.Epsilon0.F_alpha]
+i:108 [in hydras.Epsilon0.Canon]
+i:109 [in hydras.Epsilon0.F_alpha]
+i:109 [in hydras.Epsilon0.Large_Sets]
+i:11 [in hydras.Hydra.O2H]
+i:11 [in hydras.solutions_exercises.Limit_Infinity]
+i:11 [in gaia_hydras.GPaths]
+i:11 [in hydras.Epsilon0.Canon]
+i:11 [in hydras.solutions_exercises.T1_ltNotWf]
+i:110 [in hydras.Epsilon0.F_alpha]
+i:110 [in hydras.Epsilon0.Canon]
+i:111 [in hydras.Epsilon0.F_alpha]
+i:111 [in hydras.Epsilon0.Large_Sets]
+i:112 [in hydras.Epsilon0.Large_Sets]
+i:112 [in hydras.Epsilon0.Canon]
+i:112 [in hydras.Epsilon0.E0]
+i:113 [in hydras.Epsilon0.F_alpha]
+i:113 [in hydras.Epsilon0.Large_Sets]
+i:114 [in hydras.Epsilon0.Large_Sets]
+i:115 [in hydras.Epsilon0.Large_Sets]
+i:115 [in hydras.Epsilon0.Canon]
+i:116 [in hydras.Epsilon0.Large_Sets]
+i:116 [in hydras.Hydra.Hydra_Lemmas]
+i:119 [in hydras.Hydra.Hydra_Lemmas]
+i:119 [in hydras.Hydra.Hydra_Definitions]
+i:12 [in gaia_hydras.GF_alpha]
+i:12 [in hydras.solutions_exercises.Limit_Infinity]
+i:12 [in hydras.Epsilon0.Canon]
+i:12 [in hydras.Hydra.Omega2_Small]
+i:122 [in hydras.Epsilon0.F_alpha]
+i:123 [in hydras.Schutte.Schutte_basics]
+i:125 [in hydras.Hydra.Hydra_Lemmas]
+i:126 [in hydras.Schutte.Schutte_basics]
+i:127 [in additions.fib]
+i:128 [in hydras.Schutte.Schutte_basics]
+i:128 [in additions.fib]
+i:128 [in hydras.Hydra.Hydra_Definitions]
+i:129 [in additions.fib]
+i:13 [in hydras.solutions_exercises.Limit_Infinity]
+i:13 [in hydras.Hydra.Epsilon0_Needed_Generic]
+i:13 [in hydras.Epsilon0.Paths]
+i:130 [in additions.fib]
+i:131 [in additions.fib]
+i:131 [in hydras.Hydra.Hydra_Definitions]
+i:132 [in hydras.Prelude.Iterates]
+i:132 [in additions.fib]
+i:133 [in additions.fib]
+i:14 [in hydras.Hydra.O2H]
+i:14 [in hydras.solutions_exercises.Limit_Infinity]
+i:14 [in gaia_hydras.GCanon]
+i:14 [in hydras.Hydra.Epsilon0_Needed_Std]
+i:14 [in hydras.Epsilon0.Canon]
+i:14 [in hydras.Hydra.Hydra_Lemmas]
+i:14 [in hydras.Hydra.Epsilon0_Needed_Generic]
+i:14 [in hydras.Hydra.Omega2_Small]
+i:14 [in gaia_hydras.GL_alpha]
+i:140 [in gaia_hydras.T1Bridge]
+i:142 [in hydras.rpo.term]
+i:142 [in hydras.Prelude.Iterates]
+i:143 [in hydras.Hydra.Hydra_Definitions]
+i:15 [in hydras.Epsilon0.Large_Sets]
+i:15 [in gaia_hydras.GLarge_Sets]
+i:15 [in hydras.Epsilon0.Paths]
+i:150 [in additions.fib]
+i:150 [in hydras.Hydra.Hydra_Definitions]
+i:153 [in hydras.Prelude.MoreLists]
+i:156 [in hydras.Prelude.MoreLists]
+i:157 [in gaia_hydras.T1Bridge]
+i:158 [in hydras.Prelude.Iterates]
+i:159 [in hydras.Prelude.MoreLists]
+i:159 [in hydras.Epsilon0.F_alpha]
+i:159 [in gaia_hydras.T1Bridge]
+i:16 [in hydras.Hydra.O2H]
+i:16 [in hydras.solutions_exercises.Limit_Infinity]
+i:16 [in gaia_hydras.GCanon]
+i:16 [in hydras.Epsilon0.Canon]
+i:16 [in gaia_hydras.GL_alpha]
+i:161 [in hydras.Epsilon0.F_alpha]
+i:162 [in hydras.Prelude.Iterates]
+i:163 [in hydras.rpo.term]
+i:1638 [in gaia_hydras.nfwfgaia]
+i:164 [in hydras.Epsilon0.F_alpha]
+i:1641 [in gaia_hydras.nfwfgaia]
+i:1644 [in gaia_hydras.nfwfgaia]
+i:165 [in hydras.Hydra.Hydra_Definitions]
+i:166 [in hydras.Epsilon0.F_alpha]
+i:167 [in hydras.Prelude.Iterates]
+i:169 [in hydras.Epsilon0.F_alpha]
+i:17 [in gaia_hydras.GF_alpha]
+i:17 [in additions.Compatibility]
+i:17 [in hydras.Prelude.First_toggle]
+i:17 [in hydras.Epsilon0.Large_Sets]
+i:17 [in hydras.Epsilon0.E0]
+i:171 [in hydras.rpo.term]
+i:172 [in hydras.Epsilon0.F_alpha]
+i:173 [in hydras.Epsilon0.F_alpha]
+i:174 [in hydras.Prelude.MoreLists]
+i:175 [in hydras.Epsilon0.F_alpha]
+i:176 [in hydras.Prelude.Iterates]
+i:176 [in hydras.Epsilon0.Paths]
+i:177 [in hydras.Prelude.MoreLists]
+i:177 [in hydras.Epsilon0.F_alpha]
+i:178 [in hydras.Epsilon0.Large_Sets]
+i:179 [in hydras.Epsilon0.F_alpha]
+i:179 [in hydras.Epsilon0.Paths]
+i:18 [in gaia_hydras.GCanon]
+i:18 [in hydras.Hydra.Epsilon0_Needed_Std]
+i:18 [in hydras.Epsilon0.E0]
+i:181 [in hydras.Prelude.MoreLists]
+i:181 [in hydras.Schutte.Schutte_basics]
+i:184 [in hydras.Prelude.MoreLists]
+i:184 [in hydras.Schutte.Schutte_basics]
+i:187 [in hydras.Schutte.Schutte_basics]
+i:187 [in hydras.Epsilon0.Paths]
+i:189 [in hydras.Epsilon0.Large_Sets]
+i:189 [in hydras.Epsilon0.Paths]
+i:19 [in hydras.Hydra.O2H]
+i:19 [in gaia_hydras.GF_alpha]
+i:19 [in hydras.Epsilon0.Large_Sets]
+i:19 [in hydras.Hydra.BigBattle]
+i:190 [in hydras.rpo.term]
+i:190 [in hydras.Schutte.Schutte_basics]
+i:191 [in hydras.Epsilon0.Large_Sets]
+i:193 [in hydras.Schutte.Schutte_basics]
+i:199 [in hydras.Schutte.Schutte_basics]
+i:199 [in additions.Addition_Chains]
+i:2 [in gaia_hydras.GHprime]
+i:2 [in gaia_hydras.GCanon]
+i:2 [in hydras.Epsilon0.L_alpha]
+i:2 [in hydras.Epsilon0.Canon]
+i:2 [in hydras.OrdinalNotations.ON_Finite]
+i:2 [in gaia_hydras.GPrelude]
+i:2 [in hydras.solutions_exercises.ge_omega_iff]
+i:2 [in hydras.Epsilon0.Hprime]
+i:2 [in gaia_hydras.GL_alpha]
+i:2 [in hydras.solutions_exercises.T1_ltNotWf]
+i:20 [in hydras.OrdinalNotations.ON_Omega2]
+i:20 [in gaia_hydras.GF_alpha]
+i:20 [in gaia_hydras.GPaths]
+i:20 [in hydras.Prelude.MoreOrders]
+i:202 [in hydras.Epsilon0.Large_Sets]
+i:202 [in hydras.Schutte.Schutte_basics]
+i:204 [in hydras.Schutte.Schutte_basics]
+i:204 [in hydras.Epsilon0.Paths]
+i:205 [in hydras.Schutte.Schutte_basics]
+i:206 [in hydras.Schutte.Schutte_basics]
+i:207 [in hydras.Epsilon0.Paths]
+i:208 [in hydras.Schutte.Schutte_basics]
+i:21 [in hydras.Hydra.O2H]
+i:21 [in gaia_hydras.GF_alpha]
+i:21 [in additions.More_on_positive]
+i:21 [in hydras.Epsilon0.Large_Sets]
+i:21 [in hydras.Schutte.AP]
+i:21 [in hydras.Hydra.Epsilon0_Needed_Std]
+i:21 [in hydras.Epsilon0.Canon]
+i:210 [in hydras.Epsilon0.Paths]
+i:214 [in hydras.Epsilon0.Paths]
+i:217 [in hydras.Epsilon0.Paths]
+i:218 [in hydras.Prelude.MoreLists]
+i:22 [in gaia_hydras.GF_alpha]
+i:22 [in gaia_hydras.GCanon]
+i:22 [in hydras.Schutte.AP]
+i:22 [in hydras.Prelude.MoreOrders]
+i:221 [in hydras.Epsilon0.Paths]
+i:226 [in hydras.Epsilon0.Paths]
+i:23 [in gaia_hydras.GPaths]
+i:23 [in hydras.solutions_exercises.T1_ltNotWf]
+i:230 [in hydras.Schutte.Schutte_basics]
+i:233 [in hydras.Prelude.Iterates]
+i:237 [in hydras.Prelude.Iterates]
+i:24 [in hydras.OrdinalNotations.ON_Omega2]
+i:24 [in hydras.Hydra.O2H]
+i:24 [in gaia_hydras.GCanon]
+i:24 [in hydras.Schutte.AP]
+i:24 [in hydras.Epsilon0.Canon]
+i:24 [in additions.fib]
+i:240 [in hydras.Prelude.Iterates]
+i:241 [in gaia_hydras.nfwfgaia]
+i:244 [in hydras.Prelude.Iterates]
+i:244 [in gaia_hydras.nfwfgaia]
+i:247 [in gaia_hydras.nfwfgaia]
+i:248 [in hydras.Prelude.Iterates]
+i:25 [in hydras.Schutte.Schutte_basics]
+i:25 [in additions.fib]
+i:26 [in gaia_hydras.GCanon]
+i:26 [in hydras.Schutte.MoreEpsilonIota]
+i:26 [in hydras.Hydra.Epsilon0_Needed_Std]
+i:26 [in additions.fib]
+i:267 [in hydras.Epsilon0.Paths]
+i:27 [in gaia_hydras.GPaths]
+i:27 [in hydras.Prelude.MoreOrders]
+i:271 [in hydras.Epsilon0.Paths]
+i:28 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:28 [in hydras.Epsilon0.F_omega]
+i:28 [in hydras.Hydra.Epsilon0_Needed_Std]
+i:28 [in hydras.Hydra.BigBattle]
+i:28 [in hydras.Epsilon0.Canon]
+i:28 [in hydras.Epsilon0.Paths]
+i:280 [in hydras.Epsilon0.Paths]
+i:283 [in hydras.Epsilon0.Paths]
+i:286 [in hydras.Epsilon0.Paths]
+i:29 [in hydras.Prelude.MoreOrders]
+i:29 [in additions.fib]
+i:29 [in gaia_hydras.GL_alpha]
+i:299 [in hydras.Epsilon0.Paths]
+i:3 [in hydras.solutions_exercises.is_F_monotonous]
+i:3 [in gaia_hydras.GHprime]
+i:3 [in gaia_hydras.GCanon]
+i:3 [in hydras.Schutte.AP]
+i:3 [in gaia_hydras.GL_alpha]
+i:3 [in hydras.Hydra.Omega_Small]
+i:30 [in gaia_hydras.GCanon]
+i:30 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:30 [in hydras.solutions_exercises.MultisetWf]
+i:30 [in hydras.Epsilon0.Canon]
+i:30 [in hydras.Epsilon0.Paths]
+i:31 [in gaia_hydras.GCanon]
+i:31 [in hydras.Epsilon0.L_alpha]
+i:31 [in hydras.Epsilon0.Hprime]
+i:310 [in hydras.Epsilon0.Paths]
+i:315 [in hydras.Epsilon0.Paths]
+i:318 [in hydras.Epsilon0.Paths]
+i:32 [in hydras.Prelude.MoreLists]
+i:322 [in hydras.rpo.term]
+i:324 [in hydras.Epsilon0.Paths]
+i:327 [in hydras.Epsilon0.Paths]
+i:33 [in hydras.Prelude.MoreLists]
+i:33 [in hydras.Epsilon0.L_alpha]
+i:33 [in hydras.Epsilon0.Canon]
+i:33 [in hydras.Epsilon0.Hprime]
+i:331 [in hydras.Epsilon0.Paths]
+i:333 [in hydras.Epsilon0.Paths]
+i:337 [in hydras.Epsilon0.Paths]
+i:341 [in hydras.rpo.term]
+i:341 [in hydras.Epsilon0.Paths]
+i:345 [in hydras.Epsilon0.Paths]
+i:349 [in hydras.rpo.term]
+i:349 [in hydras.Epsilon0.Paths]
+i:35 [in hydras.Prelude.MoreLists]
+i:35 [in hydras.rpo.more_list]
+i:35 [in additions.More_on_positive]
+i:35 [in hydras.Epsilon0.L_alpha]
+i:35 [in hydras.Hydra.BigBattle]
+i:35 [in hydras.Epsilon0.Hprime]
+i:353 [in hydras.Epsilon0.Paths]
+i:358 [in additions.Euclidean_Chains]
+i:36 [in gaia_hydras.GCanon]
+i:36 [in hydras.Epsilon0.Canon]
+i:36 [in hydras.MoreAck.PrimRecExamples]
+i:360 [in hydras.Epsilon0.Paths]
+i:364 [in additions.Euclidean_Chains]
+i:365 [in hydras.rpo.term]
+i:367 [in hydras.rpo.term]
+i:37 [in hydras.OrdinalNotations.ON_Omega2]
+i:37 [in additions.More_on_positive]
+i:37 [in gaia_hydras.GCanon]
+i:37 [in hydras.Epsilon0.L_alpha]
+i:37 [in hydras.Epsilon0.F_omega]
+i:37 [in hydras.Epsilon0.Hprime]
+i:370 [in additions.Euclidean_Chains]
+i:372 [in hydras.rpo.term]
+i:377 [in additions.Euclidean_Chains]
+i:38 [in hydras.Epsilon0.L_alpha]
+i:38 [in hydras.Epsilon0.Hprime]
+i:382 [in hydras.Epsilon0.Paths]
+i:4 [in hydras.solutions_exercises.Limit_Infinity]
+i:4 [in hydras.Epsilon0.F_alpha]
+i:4 [in gaia_hydras.GCanon]
+i:4 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:4 [in hydras.Prelude.Exp2]
+i:4 [in hydras.Epsilon0.Paths]
+i:4 [in hydras.Hydra.Omega_Small]
+i:40 [in hydras.OrdinalNotations.ON_Omega2]
+i:40 [in hydras.Schutte.Addition]
+i:402 [in hydras.Epsilon0.Paths]
+i:406 [in hydras.Epsilon0.Paths]
+i:41 [in hydras.Epsilon0.F_alpha]
+i:41 [in gaia_hydras.GCanon]
+i:41 [in hydras.Epsilon0.Canon]
+i:412 [in hydras.Epsilon0.Paths]
+i:418 [in hydras.Epsilon0.Paths]
+i:42 [in hydras.Epsilon0.L_alpha]
+i:42 [in additions.fib]
+i:42 [in hydras.Epsilon0.Hprime]
+i:422 [in hydras.Epsilon0.Paths]
+i:424 [in hydras.Epsilon0.T1]
+i:426 [in hydras.Epsilon0.Paths]
+i:43 [in hydras.Epsilon0.F_alpha]
+i:43 [in gaia_hydras.GPaths]
+i:430 [in hydras.Epsilon0.Paths]
+i:432 [in hydras.Epsilon0.Paths]
+i:436 [in hydras.Epsilon0.Paths]
+i:44 [in hydras.OrdinalNotations.OmegaOmega]
+i:44 [in gaia_hydras.nfwfgaia]
+i:441 [in hydras.Epsilon0.Paths]
+i:442 [in hydras.Epsilon0.Paths]
+i:447 [in hydras.Epsilon0.Paths]
+i:45 [in hydras.OrdinalNotations.ON_Omega2]
+i:45 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:45 [in hydras.OrdinalNotations.OmegaOmega]
+i:45 [in hydras.Epsilon0.Canon]
+i:45 [in additions.fib]
+i:45 [in hydras.Epsilon0.E0]
+i:45 [in gaia_hydras.nfwfgaia]
+i:451 [in hydras.Epsilon0.Paths]
+i:453 [in hydras.Epsilon0.Paths]
+i:457 [in hydras.Epsilon0.Paths]
+i:46 [in hydras.Epsilon0.F_alpha]
+i:46 [in gaia_hydras.GPaths]
+i:46 [in gaia_hydras.GCanon]
+i:46 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:46 [in hydras.Epsilon0.Canon]
+i:46 [in gaia_hydras.nfwfgaia]
+i:461 [in hydras.Epsilon0.Paths]
+i:469 [in hydras.Epsilon0.Paths]
+i:47 [in hydras.OrdinalNotations.OmegaOmega]
+i:47 [in hydras.Hydra.BigBattle]
+i:47 [in hydras.Epsilon0.Hprime]
+i:47 [in hydras.MoreAck.FolExamples]
+i:471 [in hydras.Epsilon0.Paths]
+i:476 [in hydras.Epsilon0.Paths]
+i:478 [in hydras.Epsilon0.Paths]
+i:48 [in hydras.Prelude.MoreLists]
+i:48 [in hydras.Epsilon0.F_alpha]
+i:48 [in gaia_hydras.GPaths]
+i:48 [in hydras.Epsilon0.Paths]
+i:480 [in hydras.Epsilon0.Paths]
+i:485 [in hydras.Epsilon0.Paths]
+i:489 [in hydras.Epsilon0.Paths]
+i:49 [in hydras.Prelude.MoreLists]
+i:49 [in hydras.Epsilon0.Canon]
+i:49 [in hydras.MoreAck.FolExamples]
+i:492 [in hydras.Epsilon0.Paths]
+i:496 [in hydras.Epsilon0.Paths]
+i:5 [in hydras.OrdinalNotations.ON_Omega2]
+i:5 [in hydras.solutions_exercises.is_F_monotonous]
+i:5 [in hydras.Prelude.MoreLists]
+i:5 [in gaia_hydras.GHprime]
+i:5 [in hydras.Epsilon0.Large_Sets_Examples]
+i:5 [in hydras.solutions_exercises.F_3]
+i:5 [in hydras.Prelude.Exp2]
+i:5 [in gaia_hydras.GL_alpha]
+i:5 [in hydras.Hydra.Omega_Small]
+i:5 [in hydras.solutions_exercises.T1_ltNotWf]
+i:50 [in gaia_hydras.GPaths]
+i:50 [in additions.More_on_positive]
+i:500 [in hydras.Epsilon0.Paths]
+i:504 [in hydras.Epsilon0.Paths]
+i:51 [in hydras.Prelude.MoreLists]
+i:51 [in hydras.Epsilon0.F_alpha]
+i:51 [in hydras.Hydra.BigBattle]
+i:51 [in hydras.Epsilon0.Canon]
+i:51 [in additions.fib]
+i:51 [in hydras.Hydra.Hydra_Lemmas]
+i:51 [in hydras.Hydra.Hydra_Definitions]
+i:514 [in hydras.Epsilon0.Paths]
+i:52 [in hydras.OrdinalNotations.OmegaOmega]
+i:520 [in hydras.Epsilon0.Paths]
+i:522 [in hydras.Epsilon0.Paths]
+i:527 [in hydras.Epsilon0.Paths]
+i:53 [in hydras.Prelude.MoreLists]
+i:53 [in gaia_hydras.GPaths]
+i:53 [in additions.More_on_positive]
+i:53 [in hydras.Epsilon0.Canon]
+i:530 [in hydras.Epsilon0.Paths]
+i:533 [in hydras.Epsilon0.Paths]
+i:54 [in hydras.Epsilon0.F_alpha]
+i:54 [in hydras.Epsilon0.L_alpha]
+i:54 [in additions.fib]
+i:54 [in hydras.Epsilon0.Paths]
+i:544 [in hydras.Epsilon0.Paths]
+i:545 [in hydras.Epsilon0.Paths]
+i:546 [in hydras.Epsilon0.Paths]
+i:55 [in hydras.Prelude.MoreLists]
+i:55 [in hydras.Epsilon0.F_alpha]
+i:55 [in gaia_hydras.GCanon]
+i:55 [in hydras.OrdinalNotations.OmegaOmega]
+i:55 [in hydras.Epsilon0.Canon]
+i:55 [in hydras.Hydra.Hydra_Definitions]
+i:554 [in hydras.Epsilon0.Paths]
+i:56 [in hydras.Epsilon0.L_alpha]
+i:56 [in hydras.Hydra.Hydra_Lemmas]
+i:56 [in hydras.Epsilon0.Hprime]
+i:560 [in hydras.Epsilon0.Paths]
+i:567 [in hydras.Epsilon0.Paths]
+i:57 [in hydras.Prelude.MoreLists]
+i:57 [in hydras.Epsilon0.F_alpha]
+i:57 [in additions.More_on_positive]
+i:57 [in gaia_hydras.GCanon]
+i:57 [in hydras.Epsilon0.Paths]
+i:571 [in hydras.Epsilon0.Paths]
+i:574 [in hydras.Epsilon0.Paths]
+i:579 [in hydras.Epsilon0.Paths]
+i:58 [in hydras.Epsilon0.Hprime]
+i:580 [in hydras.Epsilon0.Paths]
+i:585 [in hydras.Epsilon0.Paths]
+i:588 [in hydras.Epsilon0.Paths]
+i:59 [in hydras.Prelude.MoreLists]
+i:59 [in hydras.Epsilon0.F_alpha]
+i:59 [in hydras.Epsilon0.Canon]
+i:59 [in additions.fib]
+i:6 [in gaia_hydras.GCanon]
+i:6 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:6 [in hydras.solutions_exercises.F_3]
+i:6 [in hydras.Hydra.Omega2_Small]
+i:60 [in hydras.Prelude.Iterates]
+i:60 [in hydras.Epsilon0.F_alpha]
+i:60 [in additions.More_on_positive]
+i:60 [in gaia_hydras.GCanon]
+i:60 [in hydras.Epsilon0.L_alpha]
+i:61 [in hydras.Epsilon0.F_alpha]
+i:61 [in hydras.Epsilon0.E0]
+i:61 [in hydras.Hydra.Hydra_Lemmas]
+i:61 [in hydras.Epsilon0.Hprime]
+i:61 [in hydras.Epsilon0.Paths]
+i:62 [in hydras.Epsilon0.F_alpha]
+i:62 [in gaia_hydras.GPaths]
+i:62 [in hydras.Hydra.BigBattle]
+i:63 [in hydras.Epsilon0.Canon]
+i:63 [in hydras.Epsilon0.E0]
+i:64 [in hydras.Epsilon0.Canon]
+i:64 [in hydras.Epsilon0.E0]
+i:64 [in hydras.Hydra.Hydra_Lemmas]
+i:64 [in hydras.Epsilon0.Paths]
+i:65 [in hydras.OrdinalNotations.ON_Omega2]
+i:65 [in hydras.Hydra.BigBattle]
+i:65 [in hydras.Epsilon0.Canon]
+i:66 [in gaia_hydras.GPaths]
+i:66 [in hydras.Hydra.BigBattle]
+i:66 [in hydras.Epsilon0.Canon]
+i:67 [in hydras.Epsilon0.Large_Sets]
+i:67 [in hydras.Epsilon0.Canon]
+i:68 [in hydras.OrdinalNotations.ON_Omega2]
+i:68 [in hydras.MoreAck.AckNotPR]
+i:68 [in hydras.Epsilon0.Canon]
+i:68 [in hydras.Hydra.Hydra_Lemmas]
+i:68 [in hydras.MoreAck.PrimRecExamples]
+i:687 [in hydras.Epsilon0.T1]
+i:689 [in hydras.Epsilon0.T1]
+i:69 [in gaia_hydras.GCanon]
+i:69 [in hydras.Epsilon0.Large_Sets]
+i:69 [in hydras.Epsilon0.Canon]
+i:69 [in hydras.Epsilon0.Paths]
+i:693 [in hydras.Epsilon0.T1]
+i:696 [in hydras.Epsilon0.T1]
+i:7 [in gaia_hydras.GHprime]
+i:7 [in gaia_hydras.GPaths]
+i:7 [in gaia_hydras.GCanon]
+i:7 [in hydras.Hydra.Epsilon0_Needed_Std]
+i:7 [in hydras.Epsilon0.Canon]
+i:7 [in hydras.Hydra.Epsilon0_Needed_Generic]
+i:7 [in gaia_hydras.GL_alpha]
+i:70 [in hydras.MoreAck.AckNotPR]
+i:70 [in hydras.Epsilon0.Large_Sets]
+i:70 [in hydras.Hydra.BigBattle]
+i:70 [in hydras.Epsilon0.Canon]
+i:70 [in hydras.Epsilon0.Hprime]
+i:702 [in hydras.Epsilon0.T1]
+i:703 [in hydras.Epsilon0.T1]
+i:71 [in gaia_hydras.GCanon]
+i:71 [in hydras.Epsilon0.Large_Sets]
+i:71 [in hydras.Schutte.AP]
+i:71 [in hydras.Epsilon0.Canon]
+i:71 [in hydras.Epsilon0.Paths]
+i:717 [in hydras.Epsilon0.T1]
+i:73 [in gaia_hydras.GPaths]
+i:73 [in hydras.Epsilon0.Large_Sets]
+i:73 [in hydras.Epsilon0.Paths]
+i:74 [in hydras.Epsilon0.Large_Sets]
+i:74 [in hydras.OrdinalNotations.OmegaOmega]
+i:74 [in hydras.Epsilon0.Canon]
+i:74 [in hydras.Hydra.Hydra_Lemmas]
+i:75 [in hydras.OrdinalNotations.OmegaOmega]
+i:75 [in hydras.MoreAck.PrimRecExamples]
+i:76 [in hydras.Epsilon0.Canon]
+i:77 [in hydras.Hydra.Hydra_Lemmas]
+i:78 [in hydras.Epsilon0.Large_Sets]
+i:78 [in hydras.Epsilon0.Canon]
+i:79 [in gaia_hydras.GPaths]
+i:79 [in hydras.Epsilon0.Large_Sets]
+i:8 [in gaia_hydras.GPaths]
+i:8 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+i:8 [in hydras.Hydra.Omega2_Small]
+i:80 [in hydras.Epsilon0.Large_Sets]
+i:81 [in hydras.Epsilon0.Large_Sets]
+i:81 [in hydras.Epsilon0.Canon]
+i:81 [in hydras.Epsilon0.Paths]
+i:82 [in hydras.Prelude.MoreLists]
+i:82 [in hydras.Epsilon0.Large_Sets]
+i:83 [in hydras.Hydra.O2H]
+i:83 [in hydras.Epsilon0.Paths]
+i:84 [in hydras.Prelude.MoreLists]
+i:84 [in hydras.Epsilon0.Large_Sets]
+i:84 [in hydras.MoreAck.PrimRecExamples]
+i:85 [in hydras.Gamma0.T2]
+i:85 [in hydras.Hydra.O2H]
+i:85 [in hydras.Hydra.Hydra_Lemmas]
+i:85 [in hydras.Epsilon0.Paths]
+i:86 [in hydras.Epsilon0.Hprime]
+i:87 [in hydras.Epsilon0.Large_Sets]
+i:87 [in hydras.Hydra.Hydra_Lemmas]
+i:88 [in hydras.Prelude.MoreLists]
+i:88 [in hydras.Epsilon0.Canon]
+i:88 [in hydras.Epsilon0.E0]
+i:89 [in hydras.Prelude.Iterates]
+i:89 [in hydras.Prelude.MoreLists]
+i:89 [in hydras.Epsilon0.Large_Sets]
+i:89 [in hydras.Hydra.Hydra_Lemmas]
+i:9 [in hydras.Prelude.MoreLists]
+i:9 [in hydras.Hydra.Hydra_Lemmas]
+i:9 [in hydras.Epsilon0.Paths]
+i:9 [in gaia_hydras.GL_alpha]
+i:90 [in hydras.Epsilon0.Large_Sets]
+i:91 [in hydras.Epsilon0.Large_Sets]
+i:91 [in hydras.Epsilon0.Paths]
+i:92 [in hydras.Prelude.MoreLists]
+i:92 [in hydras.Hydra.Hydra_Lemmas]
+i:94 [in hydras.Prelude.MoreLists]
+i:94 [in hydras.Epsilon0.E0]
+i:95 [in hydras.Prelude.Iterates]
+i:95 [in hydras.Hydra.O2H]
+i:95 [in hydras.Epsilon0.Canon]
+i:95 [in hydras.Hydra.Hydra_Lemmas]
+i:96 [in hydras.OrdinalNotations.OmegaOmega]
+i:97 [in hydras.Hydra.O2H]
+i:97 [in hydras.Epsilon0.E0]
+i:98 [in hydras.Epsilon0.F_alpha]
+i:99 [in hydras.Epsilon0.Canon]
+

J

+j:102 [in hydras.Prelude.Iterates]
+j:103 [in hydras.Prelude.MoreLists]
+j:106 [in hydras.Prelude.MoreLists]
+j:106 [in hydras.Hydra.Hydra_Lemmas]
+j:109 [in hydras.Epsilon0.T1]
+j:11 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+j:11 [in hydras.Hydra.Omega2_Small]
+j:12 [in hydras.Hydra.Epsilon0_Needed_Std]
+j:120 [in hydras.Hydra.Hydra_Lemmas]
+j:124 [in hydras.Prelude.Iterates]
+j:125 [in hydras.Schutte.Schutte_basics]
+j:126 [in hydras.Hydra.Hydra_Lemmas]
+j:127 [in hydras.Prelude.Iterates]
+j:127 [in hydras.Schutte.Schutte_basics]
+j:129 [in hydras.Schutte.Schutte_basics]
+j:13 [in hydras.Hydra.Omega2_Small]
+j:130 [in hydras.Prelude.Iterates]
+j:133 [in hydras.Hydra.Hydra_Definitions]
+j:15 [in hydras.solutions_exercises.Limit_Infinity]
+j:15 [in hydras.Hydra.Omega2_Small]
+j:151 [in additions.fib]
+j:154 [in hydras.Prelude.MoreLists]
+j:155 [in hydras.rpo.term]
+j:157 [in hydras.Prelude.MoreLists]
+j:16 [in hydras.Hydra.Epsilon0_Needed_Std]
+j:160 [in hydras.Prelude.MoreLists]
+j:168 [in hydras.Prelude.Iterates]
+j:17 [in hydras.solutions_exercises.Limit_Infinity]
+j:182 [in hydras.Schutte.Schutte_basics]
+j:183 [in hydras.Prelude.MoreLists]
+j:185 [in hydras.Prelude.MoreLists]
+j:194 [in hydras.Schutte.Schutte_basics]
+j:200 [in additions.Addition_Chains]
+j:203 [in hydras.Schutte.Schutte_basics]
+j:21 [in hydras.OrdinalNotations.ON_Omega2]
+j:23 [in hydras.Schutte.AP]
+j:238 [in hydras.Prelude.Iterates]
+j:243 [in hydras.Prelude.Iterates]
+j:247 [in hydras.Prelude.Iterates]
+j:25 [in hydras.OrdinalNotations.ON_Omega2]
+j:25 [in gaia_hydras.GPaths]
+j:25 [in hydras.Schutte.AP]
+j:251 [in hydras.Prelude.Iterates]
+j:29 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+j:3 [in gaia_hydras.GPrelude]
+j:31 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+j:333 [in hydras.rpo.term]
+j:338 [in hydras.Epsilon0.Paths]
+j:34 [in hydras.Prelude.MoreLists]
+j:36 [in hydras.Prelude.MoreLists]
+j:39 [in hydras.Hydra.BigBattle]
+j:404 [in hydras.Epsilon0.Paths]
+j:408 [in hydras.Epsilon0.Paths]
+j:41 [in hydras.Schutte.Addition]
+j:414 [in hydras.Epsilon0.Paths]
+j:416 [in hydras.Epsilon0.Paths]
+j:42 [in hydras.OrdinalNotations.ON_Omega2]
+j:420 [in hydras.Epsilon0.Paths]
+j:424 [in hydras.Epsilon0.Paths]
+j:428 [in hydras.Epsilon0.Paths]
+j:43 [in additions.fib]
+j:434 [in hydras.Epsilon0.Paths]
+j:438 [in hydras.Epsilon0.Paths]
+j:449 [in hydras.Epsilon0.Paths]
+j:455 [in hydras.Epsilon0.Paths]
+j:458 [in hydras.Epsilon0.Paths]
+j:46 [in additions.fib]
+j:463 [in hydras.Epsilon0.Paths]
+j:465 [in hydras.Epsilon0.Paths]
+j:47 [in hydras.Prelude.MoreLists]
+j:481 [in hydras.Epsilon0.Paths]
+j:49 [in hydras.OrdinalNotations.OmegaOmega]
+j:493 [in hydras.Epsilon0.Paths]
+j:497 [in hydras.Epsilon0.Paths]
+j:5 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+j:50 [in hydras.Prelude.MoreLists]
+j:501 [in hydras.Epsilon0.Paths]
+j:506 [in hydras.Epsilon0.Paths]
+j:515 [in hydras.Epsilon0.Paths]
+j:52 [in additions.fib]
+j:521 [in hydras.Epsilon0.Paths]
+j:524 [in hydras.Epsilon0.Paths]
+j:531 [in hydras.Epsilon0.Paths]
+j:54 [in hydras.Prelude.MoreLists]
+j:54 [in gaia_hydras.GPaths]
+j:55 [in hydras.Hydra.BigBattle]
+j:55 [in additions.fib]
+j:555 [in hydras.Epsilon0.Paths]
+j:559 [in hydras.Epsilon0.Paths]
+j:568 [in hydras.Epsilon0.Paths]
+j:57 [in hydras.OrdinalNotations.OmegaOmega]
+j:575 [in hydras.Epsilon0.Paths]
+j:58 [in hydras.Prelude.MoreLists]
+j:6 [in hydras.OrdinalNotations.ON_Omega2]
+j:6 [in hydras.Prelude.MoreLists]
+j:6 [in hydras.Epsilon0.Large_Sets_Examples]
+j:60 [in hydras.Prelude.MoreLists]
+j:60 [in additions.fib]
+j:61 [in hydras.Prelude.Iterates]
+j:61 [in additions.More_on_positive]
+j:61 [in gaia_hydras.GCanon]
+j:63 [in gaia_hydras.GPaths]
+j:65 [in hydras.Epsilon0.Paths]
+j:66 [in hydras.OrdinalNotations.ON_Omega2]
+j:69 [in hydras.OrdinalNotations.ON_Omega2]
+j:7 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+j:7 [in hydras.Hydra.Omega2_Small]
+j:70 [in hydras.Hydra.Hydra_Lemmas]
+j:74 [in gaia_hydras.GPaths]
+j:75 [in hydras.Epsilon0.Paths]
+j:77 [in hydras.Epsilon0.Paths]
+j:78 [in gaia_hydras.GPaths]
+j:79 [in hydras.Epsilon0.Canon]
+j:79 [in hydras.Hydra.Hydra_Lemmas]
+j:8 [in hydras.Hydra.Epsilon0_Needed_Std]
+j:82 [in hydras.Epsilon0.Canon]
+j:83 [in hydras.Prelude.MoreLists]
+j:85 [in hydras.Prelude.MoreLists]
+j:87 [in hydras.Epsilon0.Paths]
+j:9 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+j:9 [in hydras.Hydra.Omega2_Small]
+j:90 [in hydras.Prelude.Iterates]
+j:90 [in hydras.Prelude.MoreLists]
+j:90 [in hydras.Hydra.Hydra_Lemmas]
+j:92 [in hydras.Epsilon0.Paths]
+j:94 [in hydras.Hydra.Hydra_Lemmas]
+j:95 [in hydras.Prelude.MoreLists]
+j:96 [in hydras.Prelude.Iterates]
+j:96 [in hydras.Epsilon0.Canon]
+j:97 [in hydras.Hydra.Hydra_Lemmas]
+j:99 [in hydras.Hydra.O2H]
+

K

+kbr:222 [in additions.Euclidean_Chains]
+kbr:231 [in additions.Euclidean_Chains]
+kbr:293 [in additions.Euclidean_Chains]
+kbr:298 [in additions.Euclidean_Chains]
+kc:181 [in additions.Euclidean_Chains]
+kc:191 [in additions.Euclidean_Chains]
+kc:192 [in additions.Euclidean_Chains]
+kc:204 [in additions.Euclidean_Chains]
+kc:218 [in additions.Euclidean_Chains]
+kc:221 [in additions.Euclidean_Chains]
+khi:170 [in hydras.Schutte.Ordering_Functions]
+khi:171 [in hydras.Schutte.Ordering_Functions]
+khi:172 [in hydras.Schutte.Ordering_Functions]
+khi:173 [in hydras.Schutte.Ordering_Functions]
+khi:49 [in hydras.Schutte.Addition]
+khi:92 [in hydras.Schutte.Ordering_Functions]
+khi:93 [in hydras.Schutte.Ordering_Functions]
+knp:207 [in additions.Euclidean_Chains]
+knp:211 [in additions.Euclidean_Chains]
+kpr:27 [in additions.AM]
+kpr:31 [in additions.AM]
+ksi:58 [in hydras.Schutte.AP]
+k':105 [in additions.Euclidean_Chains]
+k':116 [in additions.Euclidean_Chains]
+k':174 [in additions.Euclidean_Chains]
+k':200 [in additions.Euclidean_Chains]
+k10_7:330 [in additions.Euclidean_Chains]
+k7_3:329 [in additions.Euclidean_Chains]
+k:10 [in hydras.Hydra.Hydra_Theorems]
+k:102 [in hydras.Epsilon0.Canon]
+k:104 [in additions.Euclidean_Chains]
+k:107 [in hydras.Prelude.MoreLists]
+k:107 [in hydras.Epsilon0.Canon]
+k:11 [in hydras.Epsilon0.Large_Sets]
+k:11 [in gaia_hydras.GL_alpha]
+k:110 [in hydras.Epsilon0.Large_Sets]
+k:115 [in additions.Euclidean_Chains]
+k:117 [in hydras.Epsilon0.Large_Sets]
+k:118 [in hydras.Epsilon0.Large_Sets]
+k:12 [in additions.Euclidean_Chains]
+k:12 [in gaia_hydras.GLarge_Sets]
+k:12 [in gaia_hydras.GL_alpha]
+k:122 [in hydras.Epsilon0.Large_Sets]
+k:122 [in hydras.Epsilon0.Hprime]
+k:133 [in hydras.Prelude.Iterates]
+k:137 [in hydras.Hydra.Hydra_Definitions]
+k:14 [in hydras.Epsilon0.Large_Sets]
+k:1402 [in gaia_hydras.nfwfgaia]
+k:143 [in hydras.Prelude.Iterates]
+k:143 [in additions.Euclidean_Chains]
+k:15 [in additions.fib]
+k:15 [in hydras.rpo.dickson]
+k:151 [in additions.Euclidean_Chains]
+k:153 [in hydras.Prelude.Iterates]
+k:156 [in additions.Euclidean_Chains]
+k:16 [in hydras.Epsilon0.T1]
+k:16 [in gaia_hydras.GHprime]
+k:16 [in gaia_hydras.GHydra]
+k:166 [in hydras.Prelude.Iterates]
+k:167 [in additions.Euclidean_Chains]
+k:17 [in hydras.Epsilon0.F_omega]
+k:17 [in hydras.MoreAck.Ack]
+k:173 [in additions.Euclidean_Chains]
+k:18 [in gaia_hydras.GHydra]
+k:187 [in hydras.Prelude.Iterates]
+k:187 [in additions.Euclidean_Chains]
+k:1889 [in gaia_hydras.nfwfgaia]
+k:194 [in hydras.Prelude.Iterates]
+k:199 [in additions.Euclidean_Chains]
+k:2 [in hydras.Hydra.Hydra_Theorems]
+k:20 [in gaia_hydras.GL_alpha]
+k:2013 [in gaia_hydras.nfwfgaia]
+k:207 [in hydras.Prelude.Iterates]
+k:209 [in additions.Euclidean_Chains]
+k:21 [in gaia_hydras.GHydra]
+k:210 [in hydras.Prelude.Iterates]
+k:211 [in hydras.Prelude.Iterates]
+k:212 [in hydras.Prelude.Iterates]
+k:213 [in hydras.Prelude.Iterates]
+k:22 [in hydras.OrdinalNotations.ON_Omega2]
+k:22 [in additions.Euclidean_Chains]
+k:22 [in hydras.Hydra.Epsilon0_Needed_Std]
+k:225 [in additions.Euclidean_Chains]
+k:226 [in hydras.Prelude.Iterates]
+k:23 [in hydras.Prelude.More_Arith]
+k:23 [in gaia_hydras.GL_alpha]
+k:234 [in hydras.Prelude.Iterates]
+k:236 [in additions.Euclidean_Chains]
+k:242 [in additions.Euclidean_Chains]
+k:25 [in hydras.Epsilon0.Large_Sets]
+k:252 [in hydras.Prelude.Iterates]
+k:26 [in hydras.OrdinalNotations.ON_Omega2]
+k:26 [in gaia_hydras.GL_alpha]
+k:260 [in hydras.rpo.rpo]
+k:32 [in hydras.MoreAck.AckNotPR]
+k:38 [in hydras.Epsilon0.Large_Sets]
+k:39 [in hydras.Epsilon0.L_alpha]
+k:39 [in hydras.Epsilon0.Hprime]
+k:4 [in additions.Euclidean_Chains]
+k:40 [in hydras.Epsilon0.L_alpha]
+k:40 [in additions.Euclidean_Chains]
+k:40 [in hydras.Epsilon0.Hprime]
+k:40 [in hydras.MoreAck.PrimRecExamples]
+k:410 [in hydras.Epsilon0.Paths]
+k:43 [in hydras.Hydra.BigBattle]
+k:43 [in hydras.Epsilon0.Hprime]
+k:44 [in hydras.Epsilon0.Large_Sets]
+k:44 [in hydras.Epsilon0.Hprime]
+k:45 [in hydras.Epsilon0.Hprime]
+k:46 [in hydras.Epsilon0.L_alpha]
+k:46 [in hydras.Epsilon0.Hprime]
+k:467 [in hydras.Epsilon0.Paths]
+k:47 [in hydras.Epsilon0.Large_Sets]
+k:48 [in hydras.Epsilon0.Hprime]
+k:49 [in hydras.Epsilon0.Hprime]
+k:508 [in hydras.Epsilon0.Paths]
+k:51 [in hydras.Epsilon0.Large_Sets]
+k:51 [in hydras.Epsilon0.L_alpha]
+k:52 [in hydras.Epsilon0.L_alpha]
+k:53 [in hydras.Epsilon0.L_alpha]
+k:53 [in hydras.Epsilon0.Hprime]
+k:54 [in hydras.Epsilon0.Hprime]
+k:54 [in gaia_hydras.nfwfgaia]
+k:541 [in hydras.Epsilon0.Paths]
+k:55 [in hydras.Epsilon0.L_alpha]
+k:57 [in hydras.Epsilon0.Hprime]
+k:578 [in hydras.Epsilon0.Paths]
+k:58 [in hydras.Epsilon0.Large_Sets]
+k:58 [in additions.Euclidean_Chains]
+k:58 [in gaia_hydras.nfwfgaia]
+k:59 [in hydras.Epsilon0.Hprime]
+k:6 [in hydras.Hydra.Hydra_Theorems]
+k:6 [in hydras.Hydra.Battle_length]
+k:60 [in gaia_hydras.nfwfgaia]
+k:62 [in hydras.Epsilon0.Hprime]
+k:63 [in hydras.Epsilon0.Large_Sets]
+k:64 [in hydras.Epsilon0.Hprime]
+k:65 [in hydras.Epsilon0.Large_Sets]
+k:66 [in hydras.Epsilon0.Large_Sets]
+k:66 [in hydras.Epsilon0.Hprime]
+k:67 [in hydras.Epsilon0.Hprime]
+k:68 [in hydras.Epsilon0.Large_Sets]
+k:69 [in hydras.Epsilon0.Hprime]
+k:7 [in hydras.OrdinalNotations.ON_Omega2]
+k:7 [in hydras.Hydra.Hydra_Theorems]
+k:7 [in additions.Addition_Chains]
+k:71 [in hydras.Epsilon0.Hprime]
+k:72 [in hydras.Epsilon0.Large_Sets]
+k:72 [in hydras.Hydra.Hydra_Lemmas]
+k:72 [in hydras.Epsilon0.Hprime]
+k:73 [in additions.Euclidean_Chains]
+k:73 [in hydras.Epsilon0.Hprime]
+k:74 [in hydras.Epsilon0.Hprime]
+k:75 [in hydras.Epsilon0.Large_Sets]
+k:76 [in hydras.Epsilon0.Large_Sets]
+k:77 [in hydras.Epsilon0.Large_Sets]
+k:77 [in hydras.MoreAck.Ack]
+k:79 [in hydras.Epsilon0.Hprime]
+k:8 [in gaia_hydras.GHprime]
+k:8 [in hydras.Epsilon0.Large_Sets]
+k:8 [in hydras.Hydra.Hydra_Theorems]
+k:8 [in hydras.Hydra.Battle_length]
+k:80 [in hydras.Epsilon0.Hprime]
+k:81 [in hydras.Prelude.MoreLists]
+k:83 [in hydras.Epsilon0.Large_Sets]
+k:83 [in hydras.Epsilon0.Hprime]
+k:84 [in hydras.Epsilon0.Hprime]
+k:85 [in hydras.Epsilon0.Large_Sets]
+k:88 [in hydras.Epsilon0.Hprime]
+k:880 [in gaia_hydras.nfwfgaia]
+k:9 [in gaia_hydras.GHprime]
+k:9 [in hydras.Hydra.Hydra_Theorems]
+k:9 [in additions.Euclidean_Chains]
+k:90 [in hydras.Epsilon0.Hprime]
+k:91 [in hydras.Prelude.MoreLists]
+k:91 [in hydras.Epsilon0.Hprime]
+k:92 [in hydras.Epsilon0.Large_Sets]
+k:93 [in hydras.MoreAck.AckNotPR]
+k:93 [in hydras.Epsilon0.Large_Sets]
+k:93 [in additions.Euclidean_Chains]
+k:94 [in hydras.Epsilon0.Hprime]
+k:95 [in hydras.MoreAck.AckNotPR]
+k:95 [in hydras.Epsilon0.Large_Sets]
+k:96 [in hydras.Epsilon0.Large_Sets]
+k:99 [in hydras.Prelude.Iterates]
+

L

+lambda:13 [in hydras.Epsilon0.Canon]
+lambda:15 [in hydras.Epsilon0.Canon]
+lambda:156 [in gaia_hydras.T1Bridge]
+lambda:161 [in gaia_hydras.T1Bridge]
+lambda:18 [in hydras.Epsilon0.Canon]
+lambda:20 [in hydras.Epsilon0.Canon]
+lambda:209 [in hydras.Epsilon0.Paths]
+lambda:213 [in hydras.Epsilon0.Paths]
+lambda:23 [in gaia_hydras.GCanon]
+lambda:23 [in hydras.Epsilon0.Canon]
+lambda:27 [in hydras.Epsilon0.Canon]
+lambda:29 [in hydras.Schutte.Schutte_basics]
+lambda:32 [in gaia_hydras.GCanon]
+lambda:336 [in hydras.Epsilon0.Paths]
+lambda:43 [in gaia_hydras.GCanon]
+lambda:49 [in gaia_hydras.GCanon]
+lambda:53 [in gaia_hydras.GCanon]
+lambda:566 [in hydras.Epsilon0.Paths]
+lambda:58 [in gaia_hydras.GCanon]
+lambda:59 [in gaia_hydras.GCanon]
+lambda:60 [in hydras.Epsilon0.Canon]
+lambda:61 [in gaia_hydras.GPaths]
+lambda:61 [in hydras.Epsilon0.Canon]
+lambda:62 [in hydras.Hydra.O2H]
+lambda:65 [in gaia_hydras.GCanon]
+lambda:686 [in hydras.Epsilon0.T1]
+lambda:72 [in hydras.Epsilon0.Canon]
+lambda:75 [in hydras.Epsilon0.Canon]
+la:116 [in hydras.rpo.more_list]
+la:7 [in hydras.rpo.dickson]
+lc:165 [in hydras.rpo.rpo]
+lc:233 [in hydras.rpo.rpo]
+lc:242 [in hydras.rpo.rpo]
+lc:70 [in hydras.rpo.rpo]
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+leb:185 [in hydras.Prelude.Merge_Sort]
+leb:188 [in hydras.Prelude.Merge_Sort]
+leb:193 [in hydras.Prelude.Merge_Sort]
+leb:195 [in hydras.Prelude.Merge_Sort]
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+le:21 [in hydras.Prelude.DecPreOrder]
+le:26 [in hydras.Prelude.DecPreOrder]
+le:3 [in hydras.solutions_exercises.predSuccUnicity]
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+le:40 [in hydras.Prelude.Sort_spec]
+le:40 [in hydras.Prelude.DecPreOrder]
+le:45 [in hydras.Prelude.DecPreOrder]
+le:50 [in hydras.Prelude.DecPreOrder]
+le:55 [in hydras.Prelude.DecPreOrder]
+le:60 [in hydras.Prelude.DecPreOrder]
+le:65 [in hydras.Prelude.DecPreOrder]
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+le:75 [in hydras.Prelude.Sort_spec]
+le:77 [in hydras.Prelude.DecPreOrder]
+le:82 [in hydras.Prelude.DecPreOrder]
+le:85 [in hydras.Prelude.DecPreOrder]
+le:88 [in hydras.Prelude.DecPreOrder]
+le:94 [in hydras.Prelude.DecPreOrder]
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+ls2:252 [in hydras.rpo.rpo]
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+ls:232 [in hydras.rpo.rpo]
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+ltA:134 [in hydras.OrdinalNotations.ON_Generic]
+ltA:147 [in hydras.OrdinalNotations.ON_Generic]
+ltA:160 [in hydras.OrdinalNotations.ON_Generic]
+ltA:172 [in hydras.OrdinalNotations.ON_Generic]
+ltA:185 [in hydras.OrdinalNotations.ON_Generic]
+ltA:199 [in hydras.OrdinalNotations.ON_Generic]
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+ltA:2 [in hydras.OrdinalNotations.ON_plus]
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+ltA:33 [in hydras.solutions_exercises.MultisetWf]
+ltA:36 [in hydras.solutions_exercises.MultisetWf]
+ltA:49 [in hydras.OrdinalNotations.ON_Generic]
+ltA:68 [in hydras.OrdinalNotations.ON_Generic]
+ltA:69 [in hydras.solutions_exercises.MultisetWf]
+ltA:82 [in hydras.OrdinalNotations.ON_Generic]
+ltA:94 [in hydras.OrdinalNotations.ON_Generic]
+ltB:111 [in hydras.OrdinalNotations.ON_Generic]
+ltB:126 [in hydras.OrdinalNotations.ON_Generic]
+ltB:138 [in hydras.OrdinalNotations.ON_Generic]
+ltB:151 [in hydras.OrdinalNotations.ON_Generic]
+ltB:164 [in hydras.OrdinalNotations.ON_Generic]
+ltB:176 [in hydras.OrdinalNotations.ON_Generic]
+ltB:189 [in hydras.OrdinalNotations.ON_Generic]
+ltB:203 [in hydras.OrdinalNotations.ON_Generic]
+ltB:34 [in hydras.OrdinalNotations.ON_Generic]
+ltB:53 [in hydras.OrdinalNotations.ON_Generic]
+ltB:6 [in hydras.OrdinalNotations.ON_mult]
+ltB:6 [in hydras.OrdinalNotations.ON_plus]
+ltB:86 [in hydras.OrdinalNotations.ON_Generic]
+ltB:98 [in hydras.OrdinalNotations.ON_Generic]
+Lt:109 [in hydras.Hydra.Hydra_Lemmas]
+lt:12 [in hydras.OrdinalNotations.ON_Generic]
+lt:129 [in hydras.Hydra.Hydra_Lemmas]
+lt:13 [in hydras.Prelude.Comparable]
+Lt:145 [in hydras.Hydra.Hydra_Definitions]
+Lt:155 [in hydras.Hydra.Hydra_Definitions]
+lt:2 [in hydras.solutions_exercises.predSuccUnicity]
+Lt:2 [in hydras.OrdinalNotations.ON_O]
+lt:2 [in hydras.Prelude.MoreOrders]
+lt:2 [in hydras.solutions_exercises.lt_succ_le]
+lt:2 [in hydras.OrdinalNotations.ON_Generic]
+lt:23 [in hydras.Prelude.MoreOrders]
+lt:3 [in hydras.Schutte.Lub]
+lt:43 [in hydras.Prelude.MoreOrders]
+lt:47 [in hydras.Prelude.MoreOrders]
+lt:5 [in hydras.Prelude.Comparable]
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+Lt:57 [in hydras.Schutte.Well_Orders]
+lt:8 [in hydras.OrdinalNotations.ON_Generic]
+lt:9 [in hydras.Schutte.Lub]
+Lt:99 [in hydras.Hydra.Hydra_Lemmas]
+l'':55 [in hydras.solutions_exercises.MultisetWf]
+l'':56 [in hydras.solutions_exercises.MultisetWf]
+l'':58 [in hydras.solutions_exercises.MultisetWf]
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+l':130 [in hydras.Prelude.MoreLists]
+l':135 [in hydras.Prelude.MoreLists]
+l':139 [in hydras.Schutte.Schutte_basics]
+l':14 [in gaia_hydras.GLarge_Sets]
+l':140 [in hydras.Prelude.MoreLists]
+l':143 [in hydras.rpo.rpo]
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+l':147 [in hydras.rpo.rpo]
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+l':158 [in hydras.rpo.rpo]
+l':161 [in hydras.rpo.rpo]
+l':167 [in hydras.rpo.rpo]
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+l':235 [in hydras.rpo.rpo]
+l':244 [in hydras.rpo.rpo]
+l':267 [in hydras.rpo.rpo]
+l':27 [in hydras.Epsilon0.Large_Sets]
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+l':34 [in hydras.Prelude.Merge_Sort]
+l':37 [in hydras.Prelude.Sort_spec]
+l':41 [in hydras.Prelude.Merge_Sort]
+l':43 [in hydras.Prelude.Merge_Sort]
+l':46 [in hydras.Prelude.Merge_Sort]
+l':48 [in hydras.Hydra.O2H]
+l':48 [in hydras.rpo.rpo]
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+l':52 [in hydras.Prelude.Merge_Sort]
+l':52 [in hydras.rpo.rpo]
+l':56 [in hydras.rpo.rpo]
+l':57 [in hydras.Schutte.CNF]
+l':614 [in hydras.Gamma0.Gamma0]
+l':618 [in hydras.Gamma0.Gamma0]
+l':63 [in hydras.rpo.rpo]
+l':66 [in hydras.rpo.rpo]
+l':699 [in hydras.Epsilon0.T1]
+l':72 [in hydras.rpo.rpo]
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+l':99 [in hydras.Prelude.Merge_Sort]
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+l0:298 [in hydras.Ackermann.codeList]
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+l1:101 [in hydras.rpo.list_permut]
+l1:104 [in hydras.Ackermann.primRec]
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+l1:108 [in hydras.Prelude.Merge_Sort]
+l1:11 [in hydras.rpo.list_permut]
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+l1:114 [in hydras.rpo.list_set]
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+l1:119 [in hydras.rpo.list_set]
+l1:12 [in hydras.Prelude.Merge_Sort]
+l1:122 [in hydras.rpo.list_set]
+l1:123 [in hydras.Prelude.Merge_Sort]
+l1:126 [in hydras.rpo.more_list]
+l1:127 [in hydras.rpo.list_set]
+l1:13 [in hydras.rpo.list_permut]
+l1:131 [in hydras.Prelude.Merge_Sort]
+l1:134 [in hydras.rpo.list_set]
+l1:138 [in hydras.Prelude.Merge_Sort]
+l1:141 [in additions.fib]
+l1:141 [in hydras.rpo.list_set]
+l1:145 [in hydras.rpo.list_set]
+l1:146 [in hydras.Prelude.Merge_Sort]
+l1:149 [in hydras.Prelude.MoreLists]
+l1:149 [in hydras.rpo.list_set]
+l1:157 [in hydras.Prelude.Merge_Sort]
+l1:161 [in hydras.Prelude.Merge_Sort]
+l1:165 [in hydras.Prelude.Merge_Sort]
+l1:17 [in hydras.Prelude.Merge_Sort]
+l1:170 [in hydras.Prelude.Merge_Sort]
+l1:178 [in hydras.Prelude.MoreLists]
+l1:18 [in hydras.rpo.more_list]
+l1:18 [in hydras.rpo.dickson]
+l1:181 [in hydras.Prelude.Merge_Sort]
+l1:184 [in hydras.rpo.more_list]
+l1:185 [in hydras.Ackermann.fol]
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+l1:191 [in hydras.Ackermann.fol]
+l1:195 [in hydras.Prelude.MoreLists]
+l1:197 [in hydras.Prelude.MoreLists]
+l1:197 [in hydras.rpo.more_list]
+l1:198 [in hydras.Prelude.Merge_Sort]
+l1:199 [in hydras.rpo.more_list]
+l1:203 [in hydras.Prelude.MoreLists]
+l1:21 [in hydras.rpo.list_permut]
+l1:212 [in hydras.Prelude.MoreLists]
+l1:219 [in hydras.Prelude.MoreLists]
+l1:22 [in hydras.rpo.dickson]
+l1:23 [in hydras.rpo.list_permut]
+l1:233 [in hydras.rpo.list_set]
+l1:234 [in hydras.rpo.term]
+l1:236 [in hydras.rpo.term]
+l1:236 [in hydras.rpo.list_set]
+l1:239 [in hydras.rpo.list_set]
+l1:243 [in hydras.rpo.list_set]
+l1:248 [in hydras.rpo.term]
+l1:250 [in hydras.rpo.term]
+l1:254 [in hydras.rpo.term]
+l1:257 [in hydras.rpo.list_set]
+l1:26 [in hydras.rpo.dickson]
+l1:27 [in hydras.rpo.list_permut]
+l1:3 [in additions.Trace_exercise]
+l1:30 [in hydras.rpo.dickson]
+l1:30 [in hydras.rpo.list_permut]
+l1:324 [in hydras.rpo.rpo]
+l1:327 [in hydras.rpo.rpo]
+l1:34 [in hydras.rpo.dickson]
+l1:34 [in hydras.rpo.list_permut]
+l1:368 [in hydras.rpo.term]
+l1:37 [in hydras.rpo.list_permut]
+l1:38 [in hydras.Prelude.Merge_Sort]
+l1:39 [in hydras.rpo.dickson]
+l1:4 [in hydras.rpo.dickson]
+l1:4 [in hydras.rpo.list_permut]
+l1:43 [in hydras.Ackermann.codeList]
+l1:45 [in hydras.rpo.list_permut]
+l1:48 [in hydras.rpo.more_list]
+l1:48 [in hydras.rpo.list_permut]
+l1:49 [in hydras.Prelude.Merge_Sort]
+l1:5 [in hydras.Prelude.Merge_Sort]
+l1:51 [in hydras.rpo.list_permut]
+l1:53 [in hydras.rpo.list_permut]
+l1:56 [in hydras.Prelude.Merge_Sort]
+l1:56 [in additions.fib]
+l1:56 [in hydras.rpo.list_permut]
+l1:57 [in hydras.rpo.term]
+l1:59 [in hydras.rpo.term]
+l1:59 [in hydras.rpo.list_permut]
+l1:61 [in additions.fib]
+l1:62 [in hydras.rpo.list_permut]
+l1:67 [in hydras.rpo.list_permut]
+l1:69 [in hydras.rpo.more_list]
+l1:70 [in hydras.rpo.list_permut]
+l1:72 [in hydras.rpo.term]
+l1:74 [in hydras.rpo.term]
+l1:76 [in hydras.rpo.list_permut]
+l1:78 [in hydras.rpo.term]
+l1:79 [in hydras.rpo.list_set]
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+l1:85 [in hydras.rpo.list_permut]
+l1:86 [in hydras.rpo.more_list]
+l1:9 [in hydras.rpo.more_list]
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+l1:91 [in hydras.rpo.list_set]
+l1:93 [in hydras.rpo.list_permut]
+l1:94 [in hydras.rpo.more_list]
+l1:95 [in hydras.rpo.list_set]
+l1:99 [in hydras.Ackermann.primRec]
+l1:99 [in additions.fib]
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+l2':254 [in hydras.rpo.list_set]
+l2:10 [in hydras.rpo.more_list]
+l2:100 [in hydras.Ackermann.primRec]
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+l2:102 [in hydras.rpo.list_permut]
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+l2:111 [in hydras.Prelude.Merge_Sort]
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+l2:135 [in hydras.rpo.list_set]
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+l2:14 [in hydras.rpo.list_permut]
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+l2:142 [in hydras.rpo.list_set]
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+l2:147 [in hydras.Prelude.Merge_Sort]
+l2:150 [in hydras.rpo.list_set]
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+l2:180 [in hydras.Prelude.MoreLists]
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+l2:185 [in hydras.rpo.more_list]
+l2:186 [in hydras.Ackermann.fol]
+l2:188 [in hydras.rpo.term]
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+l2:19 [in hydras.rpo.dickson]
+l2:192 [in hydras.Prelude.Merge_Sort]
+l2:192 [in hydras.Ackermann.fol]
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+l2:198 [in hydras.Prelude.MoreLists]
+l2:198 [in hydras.rpo.more_list]
+l2:199 [in hydras.Prelude.Merge_Sort]
+l2:200 [in hydras.rpo.more_list]
+l2:215 [in hydras.Prelude.MoreLists]
+l2:22 [in hydras.rpo.list_permut]
+l2:221 [in hydras.Prelude.MoreLists]
+l2:23 [in hydras.rpo.dickson]
+l2:234 [in hydras.rpo.list_set]
+l2:235 [in hydras.rpo.term]
+l2:237 [in hydras.rpo.term]
+l2:237 [in hydras.rpo.list_set]
+l2:24 [in hydras.rpo.list_permut]
+l2:240 [in hydras.rpo.list_set]
+l2:244 [in hydras.rpo.list_set]
+l2:249 [in hydras.rpo.term]
+l2:251 [in hydras.rpo.term]
+l2:255 [in hydras.rpo.term]
+l2:258 [in hydras.rpo.term]
+l2:258 [in hydras.rpo.list_set]
+l2:259 [in hydras.rpo.term]
+l2:27 [in hydras.rpo.dickson]
+l2:28 [in hydras.rpo.list_permut]
+l2:31 [in hydras.rpo.dickson]
+l2:31 [in hydras.rpo.list_permut]
+l2:326 [in hydras.rpo.rpo]
+l2:329 [in hydras.rpo.rpo]
+l2:35 [in hydras.rpo.dickson]
+l2:35 [in hydras.rpo.list_permut]
+l2:370 [in hydras.rpo.term]
+l2:38 [in hydras.rpo.list_permut]
+l2:39 [in hydras.Prelude.Merge_Sort]
+l2:4 [in additions.Trace_exercise]
+l2:40 [in hydras.rpo.dickson]
+l2:44 [in hydras.Ackermann.codeList]
+l2:46 [in hydras.rpo.list_permut]
+l2:49 [in hydras.rpo.more_list]
+l2:49 [in hydras.rpo.list_permut]
+l2:5 [in hydras.rpo.dickson]
+l2:5 [in hydras.rpo.list_permut]
+l2:50 [in hydras.Prelude.Merge_Sort]
+l2:52 [in hydras.rpo.list_permut]
+l2:54 [in hydras.rpo.list_permut]
+l2:57 [in hydras.Prelude.Merge_Sort]
+l2:57 [in additions.fib]
+l2:57 [in hydras.rpo.list_permut]
+l2:58 [in hydras.rpo.term]
+l2:6 [in hydras.Prelude.Merge_Sort]
+l2:6 [in additions.Trace_exercise]
+l2:60 [in hydras.rpo.term]
+l2:60 [in hydras.rpo.list_permut]
+l2:62 [in additions.fib]
+l2:63 [in hydras.rpo.list_permut]
+l2:68 [in hydras.rpo.list_permut]
+l2:70 [in hydras.rpo.more_list]
+l2:71 [in hydras.rpo.list_permut]
+l2:73 [in hydras.rpo.term]
+l2:75 [in hydras.rpo.term]
+l2:77 [in hydras.rpo.list_permut]
+l2:79 [in hydras.rpo.term]
+l2:80 [in hydras.rpo.list_set]
+l2:83 [in hydras.rpo.list_set]
+l2:86 [in hydras.rpo.list_permut]
+l2:87 [in hydras.rpo.more_list]
+l2:92 [in hydras.Prelude.Merge_Sort]
+l2:92 [in hydras.rpo.list_set]
+l2:94 [in hydras.rpo.list_permut]
+l2:95 [in hydras.rpo.more_list]
+l2:96 [in hydras.rpo.list_set]
+l3:199 [in hydras.Prelude.MoreLists]
+l3:25 [in hydras.rpo.list_permut]
+l3:39 [in hydras.rpo.list_permut]
+l3:87 [in hydras.rpo.list_permut]
+l3:95 [in hydras.rpo.list_permut]
+l4:200 [in hydras.Prelude.MoreLists]
+l4:40 [in hydras.rpo.list_permut]
+l4:88 [in hydras.rpo.list_permut]
+l4:96 [in hydras.rpo.list_permut]
+L:1 [in hydras.MoreAck.BadSubst]
+l:1 [in hydras.Schutte.CNF]
+l:1 [in hydras.solutions_exercises.schutte_cnf_counter_example]
+l:1 [in hydras.rpo.list_set]
+l:1 [in hydras.rpo.list_permut]
+l:10 [in hydras.Prelude.MoreLists]
+l:10 [in hydras.Prelude.Sort_spec]
+l:10 [in hydras.Prelude.Merge_Sort]
+l:10 [in hydras.Ackermann.codePA]
+l:101 [in hydras.rpo.more_list]
+l:104 [in hydras.Prelude.MoreLists]
+l:104 [in hydras.Prelude.Merge_Sort]
+l:105 [in hydras.rpo.rpo]
+l:106 [in hydras.Prelude.Merge_Sort]
+l:106 [in hydras.Ackermann.codeList]
+l:108 [in hydras.rpo.term]
+l:109 [in hydras.rpo.more_list]
+l:109 [in hydras.Ackermann.folReplace]
+l:11 [in hydras.Prelude.MoreLists]
+L:11 [in hydras.MoreAck.FolExamples]
+l:111 [in hydras.Ackermann.codeList]
+l:112 [in hydras.rpo.rpo]
+l:113 [in hydras.Prelude.MoreLists]
+l:117 [in hydras.rpo.more_list]
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+

M

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+m2:30 [in hydras.Ackermann.subAll]
+m2:52 [in hydras.Ackermann.subAll]
+m2:660 [in gaia_hydras.nfwfgaia]
+m2:664 [in gaia_hydras.nfwfgaia]
+M:1 [in hydras.Schutte.Lub]
+m:1 [in hydras.MoreAck.Ack]
+m:102 [in Goedel.PRrepresentable]
+M:102 [in additions.AM]
+m:102 [in hydras.Hydra.Hydra_Lemmas]
+m:103 [in hydras.Ackermann.primRec]
+M:103 [in additions.Euclidean_Chains]
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+M:11 [in additions.Compatibility]
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+m:1134 [in gaia_hydras.nfwfgaia]
+M:114 [in additions.Euclidean_Chains]
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+m:115 [in additions.fib]
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+m:1186 [in gaia_hydras.nfwfgaia]
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+M:119 [in hydras.Epsilon0.Large_Sets]
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+m:12 [in hydras.rpo.dickson]
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+m:1306 [in hydras.Ackermann.codeSubFormula]
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+M:14 [in additions.Pow]
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+m:1488 [in gaia_hydras.nfwfgaia]
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+m:15 [in hydras.Schutte.Countable]
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+m:1529 [in gaia_hydras.nfwfgaia]
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+m:159 [in hydras.Ackermann.cPair]
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+m:16 [in hydras.Ackermann.primRec]
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+m:16 [in gaia_hydras.onType]
+m:160 [in additions.fib]
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+m:163 [in hydras.Prelude.MoreLists]
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+m:18 [in hydras.Prelude.More_Arith]
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+m:189 [in hydras.Ackermann.fol]
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+m:20 [in hydras.Ackermann.codePA]
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+m:25 [in hydras.MoreAck.Ack]
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+m:273 [in hydras.rpo.rpo]
+m:28 [in hydras.MoreAck.Ack]
+m:28 [in hydras.Ackermann.primRec]
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+m:3 [in hydras.Ackermann.PAtheory]
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+m:30 [in hydras.Schutte.AP]
+M:30 [in additions.Naive]
+m:30 [in gaia_hydras.GHydra]
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+M:31 [in additions.FirstSteps]
+m:31 [in hydras.Prelude.Compat815]
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+m:33 [in gaia_hydras.GHydra]
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+m:37 [in additions.Addition_Chains]
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+m:39 [in hydras.Ackermann.subAll]
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+m:439 [in gaia_hydras.nfwfgaia]
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+m:45 [in hydras.rpo.dickson]
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+m:46 [in hydras.MoreAck.Ack]
+m:46 [in hydras.rpo.dickson]
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+m:485 [in gaia_hydras.nfwfgaia]
+m:488 [in gaia_hydras.nfwfgaia]
+m:49 [in hydras.rpo.dickson]
+m:5 [in hydras.Prelude.Compat815]
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+m:5 [in hydras.Ackermann.PAtheory]
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+m:50 [in hydras.Ackermann.primRec]
+m:50 [in hydras.Ackermann.wellFormed]
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+m:52 [in hydras.rpo.dickson]
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+m:53 [in hydras.Ackermann.wellFormed]
+m:53 [in hydras.rpo.dickson]
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+m:54 [in hydras.Ackermann.primRec]
+m:54 [in hydras.rpo.dickson]
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+m:546 [in hydras.Ackermann.checkPrf]
+m:549 [in hydras.Ackermann.checkPrf]
+m:549 [in gaia_hydras.nfwfgaia]
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+m:555 [in hydras.Ackermann.checkPrf]
+m:557 [in hydras.Ackermann.checkPrf]
+m:559 [in hydras.Ackermann.checkPrf]
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+m:562 [in hydras.Ackermann.checkPrf]
+m:569 [in hydras.Ackermann.checkPrf]
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+m:571 [in hydras.Ackermann.checkPrf]
+m:572 [in hydras.Ackermann.checkPrf]
+m:573 [in hydras.Ackermann.checkPrf]
+m:574 [in hydras.Ackermann.checkPrf]
+m:574 [in gaia_hydras.nfwfgaia]
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+m:579 [in hydras.Ackermann.checkPrf]
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+m:97 [in hydras.Ackermann.subAll]
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+m:970 [in gaia_hydras.nfwfgaia]
+m:974 [in gaia_hydras.nfwfgaia]
+m:978 [in gaia_hydras.nfwfgaia]
+m:98 [in hydras.Ackermann.primRec]
+m:984 [in gaia_hydras.nfwfgaia]
+m:986 [in gaia_hydras.nfwfgaia]
+

N

+NA:4 [in hydras.OrdinalNotations.ON_mult]
+NA:4 [in hydras.OrdinalNotations.ON_plus]
+NB:8 [in hydras.OrdinalNotations.ON_mult]
+NB:8 [in hydras.OrdinalNotations.ON_plus]
+nv:118 [in hydras.Ackermann.folProp]
+nv:13 [in hydras.MoreAck.BadSubst]
+nv:2 [in hydras.Ackermann.PAtheory]
+nv:283 [in hydras.Ackermann.folProp]
+nv:30 [in Goedel.codeSysPrf]
+nv:404 [in hydras.Ackermann.folProp]
+nv:432 [in hydras.Ackermann.folProp]
+nv:437 [in hydras.Ackermann.folProp]
+nv:48 [in hydras.Ackermann.subAll]
+nv:65 [in hydras.Ackermann.codeSubFormula]
+n'':11 [in gaia_hydras.nfwfgaia]
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+n':1338 [in gaia_hydras.nfwfgaia]
+n':1384 [in gaia_hydras.nfwfgaia]
+n':139 [in hydras.Gamma0.Gamma0]
+n':14 [in hydras.rpo.dickson]
+n':1436 [in gaia_hydras.nfwfgaia]
+n':1500 [in gaia_hydras.nfwfgaia]
+n':1556 [in gaia_hydras.nfwfgaia]
+n':1585 [in gaia_hydras.nfwfgaia]
+n':1609 [in gaia_hydras.nfwfgaia]
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+n':1691 [in gaia_hydras.nfwfgaia]
+n':174 [in gaia_hydras.nfwfgaia]
+n':176 [in hydras.Epsilon0.T1]
+n':19 [in hydras.Epsilon0.Hessenberg]
+n':194 [in hydras.Epsilon0.T1]
+n':2043 [in gaia_hydras.nfwfgaia]
+n':2066 [in gaia_hydras.nfwfgaia]
+n':223 [in hydras.Epsilon0.T1]
+n':225 [in gaia_hydras.nfwfgaia]
+n':231 [in gaia_hydras.nfwfgaia]
+n':24 [in hydras.Schutte.Correctness_E0]
+n':25 [in hydras.Epsilon0.Hessenberg]
+n':258 [in gaia_hydras.nfwfgaia]
+n':259 [in hydras.rpo.rpo]
+n':27 [in hydras.Gamma0.Gamma0]
+n':285 [in hydras.Epsilon0.T1]
+n':29 [in hydras.Schutte.Correctness_E0]
+n':294 [in gaia_hydras.nfwfgaia]
+n':3 [in gaia_hydras.nfwfgaia]
+n':30 [in hydras.Epsilon0.Hessenberg]
+n':301 [in hydras.Gamma0.Gamma0]
+n':305 [in hydras.Gamma0.Gamma0]
+n':324 [in hydras.Epsilon0.T1]
+n':329 [in hydras.Epsilon0.T1]
+n':34 [in hydras.Epsilon0.T1]
+n':347 [in hydras.Epsilon0.T1]
+n':359 [in hydras.Epsilon0.T1]
+n':36 [in hydras.Epsilon0.Hessenberg]
+n':366 [in hydras.Epsilon0.T1]
+n':380 [in hydras.Epsilon0.T1]
+n':42 [in hydras.Epsilon0.Hessenberg]
+n':425 [in gaia_hydras.nfwfgaia]
+n':429 [in hydras.Epsilon0.T1]
+n':430 [in gaia_hydras.nfwfgaia]
+n':434 [in hydras.Epsilon0.T1]
+n':440 [in hydras.Epsilon0.T1]
+n':5 [in gaia_hydras.nfwfgaia]
+n':50 [in hydras.Epsilon0.Epsilon0rpo]
+n':50 [in hydras.OrdinalNotations.OmegaOmega]
+n':52 [in hydras.Epsilon0.Hessenberg]
+n':535 [in hydras.Epsilon0.T1]
+n':54 [in gaia_hydras.T1Bridge]
+n':55 [in hydras.Epsilon0.T1]
+n':55 [in hydras.Epsilon0.Epsilon0rpo]
+n':58 [in hydras.OrdinalNotations.OmegaOmega]
+n':59 [in hydras.Epsilon0.Epsilon0rpo]
+n':59 [in gaia_hydras.T1Bridge]
+n':590 [in hydras.Gamma0.Gamma0]
+n':64 [in gaia_hydras.T1Bridge]
+n':68 [in hydras.Epsilon0.T1]
+n':7 [in gaia_hydras.nfwfgaia]
+n':70 [in gaia_hydras.T1Bridge]
+n':726 [in hydras.Epsilon0.T1]
+n':75 [in hydras.Epsilon0.T1]
+n':76 [in gaia_hydras.T1Bridge]
+n':79 [in hydras.Epsilon0.T1]
+n':790 [in gaia_hydras.nfwfgaia]
+n':83 [in hydras.Gamma0.T2]
+n':84 [in hydras.Epsilon0.T1]
+n':848 [in gaia_hydras.nfwfgaia]
+n':871 [in gaia_hydras.nfwfgaia]
+n':908 [in gaia_hydras.nfwfgaia]
+n':916 [in gaia_hydras.nfwfgaia]
+n':94 [in hydras.Epsilon0.T1]
+n':948 [in gaia_hydras.nfwfgaia]
+n':99 [in hydras.Epsilon0.T1]
+n0:18 [in hydras.Ackermann.wellFormed]
+n0:194 [in hydras.Ackermann.fol]
+n0:195 [in hydras.Ackermann.fol]
+n0:196 [in hydras.Ackermann.fol]
+n0:197 [in hydras.Ackermann.fol]
+n0:20 [in hydras.Ackermann.wellFormed]
+n0:231 [in hydras.Ackermann.subAll]
+n0:232 [in hydras.Ackermann.subAll]
+n0:241 [in hydras.Ackermann.subAll]
+n0:243 [in hydras.Ackermann.subAll]
+n0:31 [in hydras.Schutte.AP]
+n0:32 [in hydras.Schutte.AP]
+n0:379 [in hydras.Epsilon0.Paths]
+n0:650 [in hydras.Ackermann.primRec]
+n1:105 [in Goedel.rosserPA]
+n1:107 [in hydras.Epsilon0.T1]
+n1:107 [in Goedel.rosserPA]
+n1:122 [in additions.Euclidean_Chains]
+n1:131 [in additions.Euclidean_Chains]
+n1:14 [in gaia_hydras.nfwfgaia]
+n1:15 [in hydras.Prelude.Compat815]
+n1:18 [in gaia_hydras.nfwfgaia]
+n1:198 [in hydras.Ackermann.fol]
+n1:199 [in hydras.Ackermann.fol]
+n1:22 [in gaia_hydras.nfwfgaia]
+n1:223 [in Goedel.PRrepresentable]
+n1:230 [in Goedel.PRrepresentable]
+n1:318 [in hydras.Gamma0.Gamma0]
+n1:32 [in hydras.Gamma0.T2]
+n1:324 [in hydras.Gamma0.Gamma0]
+n1:340 [in hydras.Gamma0.Gamma0]
+n1:345 [in hydras.Gamma0.Gamma0]
+n1:362 [in hydras.Gamma0.Gamma0]
+n1:368 [in hydras.Gamma0.Gamma0]
+n1:38 [in Goedel.rosser]
+n1:380 [in hydras.Epsilon0.Paths]
+n1:384 [in hydras.Gamma0.Gamma0]
+n1:389 [in hydras.Gamma0.Gamma0]
+n1:39 [in hydras.Gamma0.T2]
+n1:39 [in hydras.Gamma0.Gamma0]
+n1:400 [in hydras.Gamma0.Gamma0]
+n1:410 [in hydras.Gamma0.Gamma0]
+n1:43 [in Goedel.rosser]
+n1:45 [in Goedel.rosser]
+n1:47 [in hydras.Gamma0.T2]
+n1:47 [in hydras.Gamma0.Gamma0]
+n1:50 [in Goedel.rosser]
+n1:54 [in hydras.Gamma0.T2]
+n1:55 [in hydras.Gamma0.Gamma0]
+n1:60 [in hydras.Gamma0.T2]
+n1:63 [in hydras.Gamma0.Gamma0]
+n1:66 [in hydras.Gamma0.T2]
+n1:661 [in gaia_hydras.nfwfgaia]
+n1:665 [in gaia_hydras.nfwfgaia]
+n1:68 [in Goedel.rosserPA]
+n1:71 [in hydras.Gamma0.Gamma0]
+n1:73 [in Goedel.rosserPA]
+n1:75 [in Goedel.rosserPA]
+n1:78 [in hydras.Gamma0.Gamma0]
+n1:80 [in Goedel.rosserPA]
+n1:82 [in additions.Euclidean_Chains]
+n1:84 [in hydras.Gamma0.Gamma0]
+n1:94 [in Goedel.rosserPA]
+n1:99 [in Goedel.rosserPA]
+n2:123 [in additions.Euclidean_Chains]
+n2:132 [in additions.Euclidean_Chains]
+n2:15 [in gaia_hydras.nfwfgaia]
+n2:19 [in gaia_hydras.nfwfgaia]
+n2:23 [in gaia_hydras.nfwfgaia]
+n2:325 [in hydras.Gamma0.Gamma0]
+n2:33 [in hydras.Gamma0.T2]
+n2:332 [in hydras.Gamma0.Gamma0]
+n2:346 [in hydras.Gamma0.Gamma0]
+n2:352 [in hydras.Gamma0.Gamma0]
+n2:369 [in hydras.Gamma0.Gamma0]
+n2:376 [in hydras.Gamma0.Gamma0]
+n2:381 [in hydras.Epsilon0.Paths]
+n2:390 [in hydras.Gamma0.Gamma0]
+n2:396 [in hydras.Gamma0.Gamma0]
+n2:40 [in hydras.Gamma0.T2]
+n2:40 [in hydras.Gamma0.Gamma0]
+n2:401 [in hydras.Gamma0.Gamma0]
+n2:406 [in hydras.Gamma0.Gamma0]
+n2:48 [in hydras.Gamma0.T2]
+n2:48 [in hydras.Gamma0.Gamma0]
+n2:55 [in hydras.Gamma0.T2]
+n2:56 [in hydras.Gamma0.Gamma0]
+n2:61 [in hydras.Gamma0.T2]
+n2:64 [in hydras.Gamma0.Gamma0]
+n2:662 [in gaia_hydras.nfwfgaia]
+n2:666 [in gaia_hydras.nfwfgaia]
+n2:72 [in hydras.Gamma0.Gamma0]
+n2:79 [in hydras.Gamma0.Gamma0]
+n2:83 [in additions.Euclidean_Chains]
+n2:871 [in hydras.Ackermann.checkPrf]
+n2:874 [in hydras.Ackermann.checkPrf]
+n3:17 [in hydras.Prelude.Compat815]
+n3:48 [in Goedel.rosser]
+n3:53 [in Goedel.rosser]
+n3:78 [in Goedel.rosserPA]
+n3:83 [in Goedel.rosserPA]
+n:1 [in hydras.solutions_exercises.MinPR2]
+n:1 [in additions.Fib2]
+n:1 [in hydras.Prelude.More_Arith]
+n:1 [in hydras.Ackermann.extEqualNat]
+n:1 [in hydras.MoreAck.expressibleExamples]
+n:1 [in hydras.Prelude.Compat815]
+n:1 [in hydras.Prelude.Fuel]
+n:1 [in hydras.Epsilon0.F_omega]
+n:1 [in hydras.OrdinalNotations.OmegaOmega]
+n:1 [in hydras.solutions_exercises.MorePRExamples]
+n:1 [in hydras.solutions_exercises.FibonacciPR]
+n:1 [in hydras.Ackermann.cPair]
+n:1 [in gaia_hydras.ON_gfinite]
+n:1 [in additions.Monoid_instances]
+n:1 [in hydras.OrdinalNotations.ON_Finite]
+n:1 [in additions.fib]
+n:1 [in hydras.Ackermann.codeList]
+n:1 [in hydras.Prelude.LibHyps_Experiments]
+n:1 [in hydras.solutions_exercises.F_3]
+n:1 [in hydras.Ackermann.codeNatToTerm]
+n:1 [in hydras.Prelude.Exp2]
+n:1 [in additions.Demo]
+n:1 [in hydras.MoreAck.PrimRecExamples]
+n:10 [in hydras.Prelude.More_Arith]
+n:10 [in hydras.Ackermann.codeFreeVar]
+n:10 [in hydras.MoreAck.expressibleExamples]
+n:10 [in hydras.Ackermann.checkPrf]
+n:10 [in hydras.solutions_exercises.MultisetWf]
+n:10 [in hydras.Prelude.Fuel]
+n:10 [in hydras.Epsilon0.F_omega]
+n:10 [in hydras.MoreAck.Ack]
+n:10 [in hydras.solutions_exercises.FibonacciPR]
+n:10 [in gaia_hydras.ON_gfinite]
+n:10 [in hydras.OrdinalNotations.ON_Finite]
+n:10 [in gaia_hydras.onType]
+n:10 [in hydras.Ackermann.codeList]
+n:10 [in Goedel.codeSysPrf]
+n:10 [in hydras.solutions_exercises.isqrt]
+n:100 [in Goedel.PRrepresentable]
+n:100 [in hydras.Ackermann.subAll]
+n:100 [in hydras.Prelude.Iterates]
+n:100 [in hydras.Prelude.MoreLists]
+n:100 [in hydras.Ackermann.fol]
+n:100 [in hydras.Ackermann.codeList]
+n:100 [in hydras.MoreAck.PrimRecExamples]
+n:101 [in hydras.Gamma0.T2]
+n:101 [in hydras.Ackermann.subAll]
+n:101 [in hydras.Ackermann.folProp]
+n:101 [in Goedel.rosserPA]
+n:101 [in hydras.Ackermann.fol]
+n:101 [in hydras.Epsilon0.Canon]
+n:101 [in hydras.Ackermann.codeList]
+n:101 [in hydras.MoreAck.PrimRecExamples]
+n:1010 [in gaia_hydras.nfwfgaia]
+n:102 [in hydras.Ackermann.primRec]
+n:102 [in hydras.Ackermann.fol]
+n:102 [in hydras.Ackermann.codeList]
+n:1021 [in gaia_hydras.nfwfgaia]
+n:1028 [in gaia_hydras.nfwfgaia]
+n:103 [in hydras.Epsilon0.T1]
+n:103 [in hydras.Ackermann.subAll]
+n:103 [in Goedel.rosserPA]
+n:103 [in hydras.Ackermann.codeList]
+n:103 [in hydras.Epsilon0.Hprime]
+n:103 [in hydras.Prelude.MoreVectors]
+n:1031 [in gaia_hydras.nfwfgaia]
+n:1033 [in gaia_hydras.nfwfgaia]
+n:1036 [in gaia_hydras.nfwfgaia]
+n:104 [in hydras.Ackermann.subAll]
+n:104 [in hydras.Ackermann.LNN2LNT]
+n:104 [in hydras.Ackermann.codeList]
+n:105 [in hydras.Ackermann.subProp]
+n:105 [in hydras.Ackermann.fol]
+n:105 [in hydras.Epsilon0.Canon]
+n:105 [in hydras.Hydra.Hydra_Definitions]
+n:1051 [in gaia_hydras.nfwfgaia]
+n:106 [in hydras.Epsilon0.T1]
+n:106 [in Goedel.PRrepresentable]
+n:106 [in hydras.Ackermann.subAll]
+n:106 [in hydras.Prelude.Iterates]
+n:106 [in hydras.Ackermann.LNN2LNT]
+n:106 [in additions.Pow]
+n:106 [in hydras.Ackermann.fol]
+n:106 [in hydras.Prelude.MoreVectors]
+n:107 [in hydras.Ackermann.subAll]
+n:107 [in hydras.MoreAck.AckNotPR]
+n:107 [in hydras.Ackermann.subProp]
+n:107 [in additions.fib]
+n:107 [in hydras.Ackermann.codeList]
+n:107 [in hydras.MoreAck.PrimRecExamples]
+n:107 [in hydras.Hydra.Hydra_Definitions]
+n:1073 [in gaia_hydras.nfwfgaia]
+n:108 [in hydras.Ackermann.primRec]
+n:109 [in hydras.Ackermann.LNN2LNT]
+n:109 [in hydras.Ackermann.folProp]
+n:11 [in hydras.Ackermann.folProof]
+n:11 [in gaia_hydras.GF_alpha]
+n:11 [in hydras.Prelude.Compat815]
+n:11 [in hydras.Ackermann.LNN2LNT]
+n:11 [in gaia_hydras.GCanon]
+n:11 [in hydras.Ackermann.subProp]
+n:11 [in hydras.Ackermann.codeSubFormula]
+n:11 [in hydras.Hydra.Hydra_Theorems]
+n:11 [in hydras.Ackermann.primRec]
+n:11 [in gaia_hydras.ON_gfinite]
+n:11 [in hydras.Gamma0.Gamma0]
+n:110 [in hydras.Prelude.Iterates]
+n:110 [in hydras.Epsilon0.Hprime]
+n:111 [in hydras.Ackermann.LNN2LNT]
+n:112 [in Goedel.PRrepresentable]
+n:112 [in hydras.Gamma0.T2]
+n:112 [in hydras.Ackermann.subAll]
+n:112 [in hydras.Epsilon0.F_alpha]
+n:112 [in hydras.Ackermann.codeList]
+n:112 [in hydras.Epsilon0.Hprime]
+n:113 [in gaia_hydras.T1Bridge]
+n:113 [in hydras.Ackermann.primRec]
+n:1133 [in gaia_hydras.nfwfgaia]
+n:114 [in hydras.Epsilon0.T1]
+n:114 [in hydras.Ackermann.subProp]
+n:114 [in additions.fib]
+n:114 [in hydras.Epsilon0.Hprime]
+n:1140 [in gaia_hydras.nfwfgaia]
+n:1142 [in gaia_hydras.nfwfgaia]
+n:1146 [in gaia_hydras.nfwfgaia]
+n:115 [in hydras.Epsilon0.F_alpha]
+n:115 [in hydras.Ackermann.codeList]
+n:115 [in hydras.Hydra.Hydra_Definitions]
+n:1157 [in gaia_hydras.nfwfgaia]
+n:116 [in hydras.Prelude.Iterates]
+n:116 [in hydras.Ackermann.subProp]
+n:116 [in hydras.Epsilon0.Hprime]
+n:117 [in hydras.Gamma0.Gamma0]
+n:1178 [in gaia_hydras.nfwfgaia]
+n:118 [in Goedel.PRrepresentable]
+n:118 [in hydras.Ackermann.subAll]
+n:118 [in hydras.Ackermann.subProp]
+n:118 [in hydras.Ackermann.primRec]
+n:1181 [in gaia_hydras.nfwfgaia]
+n:1184 [in gaia_hydras.nfwfgaia]
+n:1187 [in gaia_hydras.nfwfgaia]
+n:119 [in hydras.Ackermann.subAll]
+n:119 [in hydras.Ackermann.folProp]
+n:119 [in hydras.Epsilon0.Hprime]
+n:1190 [in gaia_hydras.nfwfgaia]
+n:12 [in hydras.Schutte.Correctness_E0]
+n:12 [in Goedel.PRrepresentable]
+n:12 [in hydras.Prelude.More_Arith]
+n:12 [in hydras.Prelude.Iterates]
+n:12 [in hydras.Ackermann.codeFreeVar]
+n:12 [in hydras.Prelude.Compat815]
+n:12 [in Goedel.fixPoint]
+n:12 [in hydras.Epsilon0.F_omega]
+n:12 [in hydras.Hydra.Hydra_Theorems]
+n:12 [in gaia_hydras.ON_gfinite]
+n:12 [in hydras.Ackermann.fol]
+n:12 [in gaia_hydras.onType]
+n:12 [in hydras.Ackermann.codeList]
+n:12 [in hydras.Ackermann.expressible]
+n:12 [in hydras.Prelude.MoreVectors]
+n:120 [in hydras.Ackermann.subAll]
+n:120 [in hydras.Epsilon0.F_alpha]
+N:120 [in hydras.Epsilon0.Large_Sets]
+n:120 [in hydras.Ackermann.codeList]
+n:121 [in Goedel.PRrepresentable]
+n:121 [in hydras.Ackermann.folProp]
+n:121 [in gaia_hydras.T1Bridge]
+n:121 [in hydras.Epsilon0.Hprime]
+n:1215 [in hydras.Ackermann.codeSubFormula]
+n:1219 [in hydras.Ackermann.codeSubFormula]
+n:122 [in hydras.Prelude.Iterates]
+n:122 [in hydras.Ackermann.subProp]
+n:1226 [in hydras.Ackermann.codeSubFormula]
+n:1229 [in hydras.Ackermann.codeSubFormula]
+n:123 [in hydras.Ackermann.subAll]
+N:123 [in hydras.Epsilon0.Large_Sets]
+n:123 [in hydras.Ackermann.primRec]
+n:123 [in hydras.Ackermann.codeList]
+n:124 [in hydras.Ackermann.subAll]
+n:124 [in hydras.Epsilon0.Hessenberg]
+n:1245 [in hydras.Ackermann.codeSubFormula]
+n:1247 [in gaia_hydras.nfwfgaia]
+n:1248 [in hydras.Ackermann.codeSubFormula]
+n:125 [in hydras.Gamma0.T2]
+n:125 [in hydras.Ackermann.subAll]
+n:125 [in hydras.Prelude.Iterates]
+n:1252 [in hydras.Ackermann.codeSubFormula]
+n:1254 [in hydras.Ackermann.codeSubFormula]
+n:126 [in hydras.Ackermann.subAll]
+n:126 [in hydras.Ackermann.primRec]
+n:126 [in hydras.Ackermann.cPair]
+n:126 [in hydras.Ackermann.codeList]
+n:126 [in hydras.Epsilon0.Hprime]
+n:1260 [in hydras.Ackermann.codeSubFormula]
+n:1262 [in hydras.Ackermann.codeSubFormula]
+n:1264 [in hydras.Ackermann.codeSubFormula]
+n:127 [in hydras.Ackermann.subAll]
+n:127 [in hydras.Epsilon0.Hessenberg]
+n:127 [in hydras.Ackermann.subProp]
+n:1277 [in gaia_hydras.nfwfgaia]
+n:128 [in Goedel.PRrepresentable]
+n:128 [in hydras.Ackermann.subAll]
+n:128 [in additions.AM]
+n:128 [in gaia_hydras.nfwfgaia]
+n:129 [in hydras.Gamma0.T2]
+n:129 [in hydras.Prelude.Iterates]
+n:129 [in hydras.Ackermann.codeList]
+n:13 [in hydras.Prelude.More_Arith]
+n:13 [in hydras.Ackermann.LNN2LNT]
+n:13 [in hydras.Ackermann.codeSubFormula]
+n:13 [in hydras.Schutte.AP]
+n:13 [in hydras.Hydra.Hydra_Theorems]
+n:13 [in hydras.solutions_exercises.FibonacciPR]
+n:13 [in gaia_hydras.ON_gfinite]
+n:13 [in hydras.OrdinalNotations.ON_Finite]
+n:13 [in hydras.Epsilon0.E0]
+n:13 [in hydras.Ackermann.wConsistent]
+n:13 [in hydras.rpo.dickson]
+n:130 [in hydras.Epsilon0.Hessenberg]
+n:130 [in hydras.Ackermann.primRec]
+n:130 [in hydras.Ackermann.cPair]
+n:130 [in hydras.Epsilon0.Hprime]
+n:1302 [in hydras.Ackermann.codeSubFormula]
+n:1305 [in hydras.Ackermann.codeSubFormula]
+n:131 [in additions.AM]
+n:131 [in hydras.Ackermann.primRec]
+n:1317 [in gaia_hydras.nfwfgaia]
+n:132 [in additions.AM]
+n:132 [in hydras.rpo.more_list]
+n:132 [in hydras.Ackermann.codeList]
+n:133 [in hydras.Gamma0.T2]
+n:133 [in additions.AM]
+n:1333 [in gaia_hydras.nfwfgaia]
+n:134 [in hydras.Ackermann.codeSubFormula]
+n:134 [in hydras.Ackermann.cPair]
+n:134 [in gaia_hydras.nfwfgaia]
+n:1348 [in gaia_hydras.nfwfgaia]
+n:135 [in additions.Euclidean_Chains]
+n:135 [in hydras.Ackermann.codeList]
+n:135 [in hydras.Gamma0.Gamma0]
+n:1353 [in gaia_hydras.nfwfgaia]
+n:1358 [in gaia_hydras.nfwfgaia]
+n:136 [in additions.Euclidean_Chains]
+n:136 [in hydras.Hydra.Hydra_Lemmas]
+n:1363 [in gaia_hydras.nfwfgaia]
+n:1368 [in gaia_hydras.nfwfgaia]
+n:137 [in hydras.Epsilon0.Hprime]
+n:1373 [in gaia_hydras.nfwfgaia]
+n:1379 [in gaia_hydras.nfwfgaia]
+n:138 [in hydras.Gamma0.T2]
+n:138 [in hydras.Ackermann.codeList]
+n:1382 [in hydras.Ackermann.codeSubFormula]
+n:1385 [in hydras.Ackermann.codeSubFormula]
+n:139 [in Goedel.PRrepresentable]
+n:139 [in hydras.Epsilon0.Hprime]
+n:1390 [in hydras.Ackermann.codeSubFormula]
+n:1394 [in hydras.Ackermann.codeSubFormula]
+n:14 [in hydras.Ackermann.folProof]
+n:14 [in hydras.Gamma0.T2]
+n:14 [in additions.Fib2]
+n:14 [in hydras.Prelude.MoreLists]
+n:14 [in hydras.MoreAck.AckNotPR]
+n:14 [in hydras.Hydra.Hydra_Theorems]
+n:14 [in hydras.solutions_exercises.MorePRExamples]
+n:14 [in hydras.Ackermann.primRec]
+n:14 [in additions.Naive]
+n:14 [in additions.fib]
+n:14 [in hydras.Ackermann.codeList]
+n:140 [in hydras.Ackermann.codePA]
+n:141 [in hydras.Ackermann.codeList]
+n:1413 [in gaia_hydras.nfwfgaia]
+n:142 [in hydras.Gamma0.T2]
+n:142 [in hydras.Ackermann.cPair]
+n:142 [in hydras.Epsilon0.Hprime]
+n:143 [in hydras.Hydra.Hydra_Lemmas]
+n:143 [in hydras.Gamma0.Gamma0]
+n:1431 [in gaia_hydras.nfwfgaia]
+n:144 [in hydras.Ackermann.codeList]
+n:144 [in hydras.Epsilon0.Hprime]
+n:145 [in hydras.Ackermann.cPair]
+n:145 [in hydras.Hydra.Hydra_Lemmas]
+n:1457 [in gaia_hydras.nfwfgaia]
+n:146 [in hydras.Ackermann.codeList]
+n:147 [in hydras.Epsilon0.T1]
+n:1473 [in gaia_hydras.nfwfgaia]
+n:1475 [in gaia_hydras.nfwfgaia]
+n:1479 [in gaia_hydras.nfwfgaia]
+n:148 [in hydras.Gamma0.T2]
+n:148 [in hydras.Ackermann.codeList]
+n:148 [in hydras.Hydra.Hydra_Lemmas]
+n:1483 [in gaia_hydras.nfwfgaia]
+n:1489 [in gaia_hydras.nfwfgaia]
+n:149 [in hydras.Ackermann.cPair]
+n:149 [in hydras.Gamma0.Gamma0]
+n:1491 [in gaia_hydras.nfwfgaia]
+n:1495 [in gaia_hydras.nfwfgaia]
+n:15 [in additions.Fib2]
+n:15 [in gaia_hydras.GHprime]
+n:15 [in hydras.Ackermann.folLogic2]
+n:15 [in hydras.Ackermann.LNN2LNT]
+n:15 [in Goedel.fixPoint]
+n:15 [in hydras.Ackermann.codeSubFormula]
+n:15 [in hydras.Hydra.Hydra_Theorems]
+n:15 [in hydras.Ackermann.PA]
+n:15 [in hydras.solutions_exercises.MorePRExamples]
+n:15 [in hydras.Ackermann.primRec]
+n:15 [in hydras.Epsilon0.E0]
+n:15 [in gaia_hydras.onType]
+n:150 [in Goedel.PRrepresentable]
+n:150 [in hydras.Ackermann.codeSubFormula]
+n:150 [in hydras.Ackermann.codeList]
+n:152 [in gaia_hydras.T1Bridge]
+n:152 [in hydras.Ackermann.codeList]
+n:152 [in hydras.Ackermann.codePA]
+n:153 [in hydras.Epsilon0.Large_Sets]
+n:153 [in hydras.Ackermann.cPair]
+n:153 [in hydras.Hydra.Hydra_Lemmas]
+n:1530 [in gaia_hydras.nfwfgaia]
+n:1534 [in gaia_hydras.nfwfgaia]
+n:154 [in gaia_hydras.T1Bridge]
+n:154 [in hydras.Ackermann.codeList]
+n:1547 [in gaia_hydras.nfwfgaia]
+n:155 [in hydras.Ackermann.fol]
+n:155 [in hydras.Gamma0.Gamma0]
+n:155 [in hydras.Ackermann.codePA]
+n:1554 [in gaia_hydras.nfwfgaia]
+n:156 [in hydras.Ackermann.subAll]
+n:156 [in hydras.Prelude.Iterates]
+n:156 [in hydras.Epsilon0.Large_Sets]
+n:156 [in hydras.Ackermann.codeList]
+n:157 [in hydras.Ackermann.codeList]
+n:1577 [in gaia_hydras.nfwfgaia]
+n:158 [in hydras.Epsilon0.T1]
+n:158 [in hydras.Ackermann.subAll]
+n:158 [in hydras.Ackermann.LNT]
+n:158 [in hydras.Ackermann.codeList]
+n:1581 [in gaia_hydras.nfwfgaia]
+n:159 [in hydras.Prelude.Iterates]
+n:159 [in additions.fib]
+n:159 [in hydras.Ackermann.codeList]
+n:159 [in hydras.Gamma0.Gamma0]
+n:1590 [in gaia_hydras.nfwfgaia]
+n:16 [in hydras.Ackermann.folProof]
+n:16 [in hydras.Schutte.Correctness_E0]
+n:16 [in Goedel.PRrepresentable]
+n:16 [in hydras.Prelude.More_Arith]
+n:16 [in hydras.Ackermann.checkPrf]
+n:16 [in hydras.Epsilon0.Hessenberg]
+n:16 [in hydras.Ackermann.subProp]
+n:16 [in hydras.Epsilon0.Large_Sets]
+n:16 [in hydras.Schutte.AP]
+n:16 [in hydras.solutions_exercises.MultisetWf]
+N:16 [in hydras.Epsilon0.F_omega]
+n:16 [in hydras.Hydra.Hydra_Termination]
+n:16 [in hydras.Ackermann.wellFormed]
+n:16 [in hydras.OrdinalNotations.ON_Finite]
+n:16 [in additions.fib]
+n:16 [in hydras.Ackermann.codeList]
+n:16 [in Goedel.codeSysPrf]
+n:16 [in hydras.Ackermann.codePA]
+n:16 [in hydras.solutions_exercises.T1_ltNotWf]
+n:160 [in hydras.Ackermann.subAll]
+n:160 [in hydras.Ackermann.model]
+n:160 [in hydras.Ackermann.codeList]
+n:1604 [in gaia_hydras.nfwfgaia]
+n:161 [in hydras.Epsilon0.T1]
+n:161 [in additions.fib]
+n:161 [in hydras.Ackermann.codeList]
+n:1611 [in gaia_hydras.nfwfgaia]
+n:162 [in hydras.Ackermann.subAll]
+n:162 [in hydras.Prelude.MoreLists]
+n:162 [in hydras.Ackermann.codeList]
+n:163 [in hydras.Epsilon0.T1]
+n:163 [in hydras.Epsilon0.F_alpha]
+n:163 [in hydras.Ackermann.codeList]
+n:163 [in hydras.Gamma0.Gamma0]
+n:164 [in hydras.Ackermann.cPair]
+n:164 [in hydras.Ackermann.codeList]
+n:1653 [in gaia_hydras.nfwfgaia]
+n:166 [in hydras.Ackermann.LNN]
+n:166 [in hydras.Schutte.Schutte_basics]
+n:167 [in hydras.Ackermann.subAll]
+n:167 [in hydras.Schutte.Schutte_basics]
+n:167 [in hydras.Ackermann.codeList]
+n:167 [in hydras.Gamma0.Gamma0]
+n:1677 [in gaia_hydras.nfwfgaia]
+n:168 [in hydras.Epsilon0.T1]
+n:168 [in hydras.Ackermann.subAll]
+n:168 [in hydras.Epsilon0.F_alpha]
+n:168 [in hydras.Ackermann.cPair]
+n:168 [in hydras.Epsilon0.Paths]
+n:168 [in gaia_hydras.nfwfgaia]
+n:168 [in hydras.Ackermann.codePA]
+n:1680 [in gaia_hydras.nfwfgaia]
+n:1686 [in gaia_hydras.nfwfgaia]
+n:17 [in hydras.Ackermann.subAll]
+n:17 [in hydras.Prelude.More_Arith]
+n:17 [in hydras.Ackermann.extEqualNat]
+n:17 [in hydras.Hydra.O2H]
+n:17 [in hydras.Ackermann.folLogic2]
+n:17 [in hydras.MoreAck.AckNotPR]
+n:17 [in hydras.Ackermann.codeSubFormula]
+n:17 [in hydras.Ackermann.primRec]
+n:17 [in hydras.Epsilon0.Canon]
+n:17 [in hydras.Hydra.Hydra_Lemmas]
+n:17 [in hydras.Prelude.MoreVectors]
+n:170 [in hydras.Ackermann.subAll]
+n:170 [in hydras.Ackermann.codeList]
+n:171 [in hydras.Epsilon0.T1]
+n:171 [in hydras.Epsilon0.F_alpha]
+n:171 [in hydras.Gamma0.Gamma0]
+n:171 [in gaia_hydras.nfwfgaia]
+n:171 [in hydras.Ackermann.codePA]
+n:1713 [in gaia_hydras.nfwfgaia]
+n:1714 [in gaia_hydras.nfwfgaia]
+n:172 [in hydras.Ackermann.subAll]
+n:172 [in hydras.Ackermann.cPair]
+n:173 [in hydras.Prelude.MoreLists]
+n:173 [in hydras.Schutte.Schutte_basics]
+n:173 [in hydras.Ackermann.codeList]
+n:173 [in hydras.Epsilon0.Paths]
+n:173 [in hydras.Gamma0.Gamma0]
+n:174 [in hydras.Epsilon0.T1]
+n:174 [in hydras.Ackermann.subAll]
+n:174 [in hydras.Schutte.Schutte_basics]
+n:175 [in gaia_hydras.T1Bridge]
+n:176 [in hydras.Ackermann.subAll]
+n:176 [in hydras.Prelude.MoreLists]
+n:176 [in hydras.Ackermann.codeList]
+n:177 [in hydras.Ackermann.model]
+n:177 [in gaia_hydras.T1Bridge]
+n:1775 [in gaia_hydras.nfwfgaia]
+n:178 [in hydras.Gamma0.Gamma0]
+n:1784 [in gaia_hydras.nfwfgaia]
+n:1786 [in gaia_hydras.nfwfgaia]
+n:179 [in hydras.Epsilon0.T1]
+n:179 [in hydras.Ackermann.subAll]
+n:179 [in hydras.Prelude.MoreLists]
+n:179 [in additions.Euclidean_Chains]
+n:179 [in hydras.Ackermann.codeList]
+n:1794 [in gaia_hydras.nfwfgaia]
+n:18 [in additions.Fib2]
+n:18 [in additions.FirstSteps]
+n:18 [in hydras.Prelude.Compat815]
+n:18 [in Goedel.fixPoint]
+n:18 [in hydras.Epsilon0.Large_Sets]
+n:18 [in hydras.Schutte.AP]
+n:18 [in hydras.Ackermann.NNtheory]
+n:18 [in hydras.Ackermann.codeList]
+n:18 [in hydras.solutions_exercises.F_3]
+n:1801 [in gaia_hydras.nfwfgaia]
+n:181 [in hydras.Gamma0.Gamma0]
+n:182 [in hydras.Epsilon0.T1]
+n:182 [in hydras.Ackermann.subAll]
+n:182 [in hydras.Prelude.Iterates]
+n:182 [in hydras.Ackermann.fol]
+n:182 [in hydras.Ackermann.codeList]
+n:1822 [in gaia_hydras.nfwfgaia]
+n:1825 [in gaia_hydras.nfwfgaia]
+n:1828 [in gaia_hydras.nfwfgaia]
+n:183 [in gaia_hydras.T1Bridge]
+n:183 [in hydras.Gamma0.Gamma0]
+n:1831 [in gaia_hydras.nfwfgaia]
+n:1834 [in gaia_hydras.nfwfgaia]
+n:184 [in hydras.Ackermann.model]
+n:184 [in hydras.Ackermann.wellFormed]
+n:184 [in hydras.Ackermann.codeList]
+n:1847 [in gaia_hydras.nfwfgaia]
+n:1849 [in gaia_hydras.nfwfgaia]
+n:185 [in hydras.Ackermann.subAll]
+n:185 [in hydras.Epsilon0.F_alpha]
+n:185 [in hydras.Gamma0.Gamma0]
+n:185 [in gaia_hydras.nfwfgaia]
+n:1851 [in gaia_hydras.nfwfgaia]
+n:1853 [in gaia_hydras.nfwfgaia]
+n:1854 [in gaia_hydras.nfwfgaia]
+n:186 [in gaia_hydras.T1Bridge]
+n:186 [in hydras.Ackermann.codeList]
+n:1862 [in gaia_hydras.nfwfgaia]
+n:1864 [in gaia_hydras.nfwfgaia]
+n:1868 [in gaia_hydras.nfwfgaia]
+n:187 [in hydras.Epsilon0.T1]
+n:187 [in hydras.Gamma0.Gamma0]
+n:187 [in gaia_hydras.nfwfgaia]
+n:1874 [in gaia_hydras.nfwfgaia]
+n:188 [in hydras.Ackermann.cPair]
+n:188 [in hydras.Ackermann.fol]
+n:188 [in hydras.Ackermann.codeList]
+n:1880 [in gaia_hydras.nfwfgaia]
+n:1883 [in gaia_hydras.nfwfgaia]
+n:189 [in gaia_hydras.T1Bridge]
+n:189 [in additions.Euclidean_Chains]
+n:189 [in hydras.Gamma0.Gamma0]
+n:189 [in gaia_hydras.nfwfgaia]
+n:1893 [in gaia_hydras.nfwfgaia]
+n:1897 [in gaia_hydras.nfwfgaia]
+n:19 [in hydras.Schutte.Correctness_E0]
+n:19 [in additions.Fib2]
+n:19 [in hydras.Prelude.More_Arith]
+n:19 [in hydras.Prelude.Iterates]
+n:19 [in hydras.Ackermann.folLogic2]
+n:19 [in additions.More_on_positive]
+n:19 [in hydras.Ackermann.codeSubFormula]
+n:19 [in hydras.Epsilon0.F_omega]
+n:19 [in gaia_hydras.ON_gfinite]
+n:19 [in hydras.Epsilon0.Canon]
+n:19 [in hydras.Ackermann.NNtheory]
+n:19 [in hydras.Gamma0.Gamma0]
+n:19 [in hydras.Ackermann.codePA]
+n:19 [in hydras.Prelude.MoreVectors]
+n:190 [in hydras.Ackermann.codeList]
+n:1901 [in gaia_hydras.nfwfgaia]
+n:1906 [in gaia_hydras.nfwfgaia]
+n:1910 [in gaia_hydras.nfwfgaia]
+n:1915 [in gaia_hydras.nfwfgaia]
+n:1919 [in gaia_hydras.nfwfgaia]
+n:192 [in hydras.Ackermann.codeList]
+n:1925 [in gaia_hydras.nfwfgaia]
+n:193 [in gaia_hydras.nfwfgaia]
+n:1931 [in gaia_hydras.nfwfgaia]
+n:1936 [in gaia_hydras.nfwfgaia]
+n:194 [in hydras.Ackermann.wellFormed]
+n:194 [in hydras.Ackermann.codeList]
+n:1941 [in gaia_hydras.nfwfgaia]
+n:1946 [in gaia_hydras.nfwfgaia]
+n:195 [in hydras.Ackermann.codeList]
+n:195 [in additions.Addition_Chains]
+n:1951 [in gaia_hydras.nfwfgaia]
+n:1957 [in gaia_hydras.nfwfgaia]
+n:196 [in hydras.Ackermann.codeList]
+n:1964 [in gaia_hydras.nfwfgaia]
+n:1969 [in gaia_hydras.nfwfgaia]
+n:197 [in hydras.Ackermann.wellFormed]
+n:197 [in gaia_hydras.nfwfgaia]
+n:1975 [in gaia_hydras.nfwfgaia]
+n:198 [in hydras.Epsilon0.T1]
+n:198 [in hydras.Ackermann.codeList]
+n:1981 [in gaia_hydras.nfwfgaia]
+n:1986 [in gaia_hydras.nfwfgaia]
+n:1992 [in gaia_hydras.nfwfgaia]
+n:1997 [in gaia_hydras.nfwfgaia]
+n:2 [in hydras.Ackermann.folProof]
+n:2 [in hydras.Prelude.MoreLists]
+n:2 [in hydras.MoreAck.expressibleExamples]
+n:2 [in additions.Compatibility]
+n:2 [in hydras.Prelude.Compat815]
+n:2 [in hydras.Prelude.First_toggle]
+n:2 [in hydras.Ackermann.LNN2LNT]
+n:2 [in hydras.MoreAck.Iterate_compat]
+n:2 [in hydras.MoreAck.Ack]
+n:2 [in hydras.solutions_exercises.MorePRExamples]
+n:2 [in additions.Naive]
+n:2 [in gaia_hydras.nfwfgaia]
+n:2 [in additions.Demo]
+n:2 [in hydras.Prelude.MoreVectors]
+n:20 [in Goedel.PRrepresentable]
+n:20 [in additions.Fib2]
+n:20 [in hydras.Prelude.Compat815]
+n:20 [in hydras.Ackermann.LNN2LNT]
+n:20 [in gaia_hydras.GCanon]
+n:20 [in hydras.Schutte.GRelations]
+n:20 [in hydras.Epsilon0.Large_Sets]
+n:20 [in hydras.Prelude.Fuel]
+n:20 [in hydras.Ackermann.codeList]
+n:20 [in hydras.Schutte.Countable]
+n:20 [in hydras.Ackermann.expressible]
+n:200 [in hydras.Epsilon0.T1]
+n:200 [in hydras.Ackermann.wellFormed]
+n:200 [in hydras.Ackermann.codeList]
+n:2003 [in gaia_hydras.nfwfgaia]
+n:201 [in hydras.Epsilon0.T1]
+n:201 [in hydras.Epsilon0.Large_Sets]
+n:2019 [in gaia_hydras.nfwfgaia]
+n:202 [in hydras.Ackermann.codeList]
+n:2024 [in gaia_hydras.nfwfgaia]
+n:2028 [in gaia_hydras.nfwfgaia]
+n:203 [in hydras.Ackermann.model]
+n:203 [in hydras.Epsilon0.Large_Sets]
+n:203 [in hydras.Ackermann.cPair]
+n:203 [in gaia_hydras.nfwfgaia]
+n:204 [in hydras.Epsilon0.T1]
+n:204 [in hydras.Ackermann.codeList]
+n:205 [in additions.Euclidean_Chains]
+n:205 [in gaia_hydras.nfwfgaia]
+n:206 [in hydras.Epsilon0.T1]
+n:206 [in hydras.Epsilon0.Large_Sets]
+n:206 [in hydras.Ackermann.codeList]
+n:207 [in hydras.Ackermann.cPair]
+n:208 [in hydras.Epsilon0.Large_Sets]
+n:208 [in hydras.Ackermann.codeList]
+n:209 [in hydras.Epsilon0.Large_Sets]
+n:209 [in hydras.Ackermann.codeList]
+n:209 [in additions.Addition_Chains]
+n:21 [in Goedel.fixPoint]
+N:21 [in hydras.MoreAck.AckNotPR]
+n:21 [in hydras.Ackermann.codeSubFormula]
+n:21 [in hydras.solutions_exercises.MultisetWf]
+n:21 [in hydras.Ackermann.code]
+n:21 [in hydras.Ackermann.primRec]
+n:21 [in gaia_hydras.ON_gfinite]
+n:21 [in hydras.OrdinalNotations.ON_Finite]
+n:21 [in hydras.Schutte.Countable]
+n:210 [in hydras.Ackermann.subAll]
+n:210 [in hydras.Ackermann.model]
+n:210 [in hydras.Epsilon0.Large_Sets]
+n:210 [in hydras.Ackermann.cPair]
+n:211 [in hydras.Ackermann.subAll]
+n:211 [in hydras.Prelude.MoreLists]
+n:211 [in additions.Addition_Chains]
+n:211 [in hydras.Epsilon0.Paths]
+n:212 [in hydras.rpo.term]
+n:212 [in hydras.Epsilon0.Large_Sets]
+n:212 [in hydras.Ackermann.cPair]
+n:212 [in additions.Euclidean_Chains]
+n:212 [in hydras.Ackermann.codeList]
+n:213 [in hydras.Prelude.MoreLists]
+n:214 [in hydras.Epsilon0.Large_Sets]
+n:214 [in hydras.Ackermann.cPair]
+n:215 [in hydras.Ackermann.subAll]
+n:216 [in hydras.Epsilon0.T1]
+n:216 [in hydras.Ackermann.subAll]
+n:216 [in hydras.Epsilon0.Large_Sets]
+n:217 [in hydras.Prelude.MoreLists]
+n:217 [in hydras.Ackermann.codeList]
+n:218 [in hydras.Epsilon0.Paths]
+n:219 [in additions.Euclidean_Chains]
+n:219 [in hydras.Gamma0.Gamma0]
+n:219 [in gaia_hydras.nfwfgaia]
+n:22 [in hydras.Ackermann.subAll]
+n:22 [in hydras.Ackermann.codeFreeVar]
+n:22 [in hydras.Hydra.O2H]
+n:22 [in hydras.Prelude.Compat815]
+n:22 [in hydras.Ackermann.LNN2LNT]
+n:22 [in additions.Pow_variant]
+n:22 [in additions.Pow]
+n:22 [in hydras.Epsilon0.Hessenberg]
+n:22 [in additions.More_on_positive]
+n:22 [in hydras.MoreAck.Ack]
+n:22 [in hydras.solutions_exercises.FibonacciPR]
+n:22 [in hydras.Epsilon0.Canon]
+n:22 [in hydras.OrdinalNotations.ON_Finite]
+n:22 [in hydras.Ackermann.codeList]
+n:22 [in hydras.MoreAck.PrimRecExamples]
+n:220 [in hydras.Epsilon0.T1]
+n:220 [in hydras.Ackermann.subAll]
+n:220 [in hydras.Prelude.MoreLists]
+n:220 [in hydras.Ackermann.codeList]
+n:221 [in hydras.Ackermann.subAll]
+n:222 [in gaia_hydras.nfwfgaia]
+n:224 [in hydras.Epsilon0.Paths]
+n:225 [in hydras.Ackermann.subAll]
+n:225 [in hydras.Ackermann.codeList]
+n:225 [in hydras.Gamma0.Gamma0]
+n:226 [in hydras.Epsilon0.T1]
+n:227 [in additions.AM]
+n:228 [in hydras.Ackermann.codeList]
+n:228 [in gaia_hydras.nfwfgaia]
+n:229 [in hydras.Epsilon0.Paths]
+n:229 [in hydras.Gamma0.Gamma0]
+n:23 [in hydras.Schutte.Correctness_E0]
+n:23 [in hydras.Prelude.Iterates]
+n:23 [in gaia_hydras.GF_alpha]
+n:23 [in hydras.Epsilon0.Epsilon0rpo]
+n:23 [in additions.More_on_positive]
+n:23 [in hydras.Schutte.GRelations]
+n:23 [in hydras.Ackermann.codeSubFormula]
+n:23 [in hydras.Epsilon0.Large_Sets]
+n:23 [in hydras.Prelude.Fuel]
+n:23 [in hydras.solutions_exercises.FibonacciPR]
+n:23 [in gaia_hydras.ON_gfinite]
+n:23 [in hydras.OrdinalNotations.ON_Finite]
+n:23 [in hydras.Epsilon0.E0]
+n:23 [in hydras.solutions_exercises.F_3]
+n:230 [in hydras.Ackermann.subAll]
+n:230 [in hydras.Prelude.Iterates]
+n:230 [in additions.Addition_Chains]
+n:231 [in hydras.Epsilon0.T1]
+n:231 [in hydras.Ackermann.codeList]
+n:232 [in hydras.Prelude.Iterates]
+n:232 [in hydras.Epsilon0.Paths]
+n:232 [in hydras.Ackermann.codePA]
+n:233 [in hydras.Gamma0.Gamma0]
+n:233 [in hydras.Ackermann.codePA]
+n:234 [in hydras.Ackermann.codeList]
+n:235 [in hydras.Epsilon0.T1]
+n:235 [in hydras.Prelude.Iterates]
+n:236 [in additions.AM]
+n:237 [in hydras.Ackermann.codeList]
+n:238 [in hydras.Epsilon0.T1]
+n:238 [in hydras.Epsilon0.Paths]
+n:238 [in gaia_hydras.nfwfgaia]
+n:239 [in hydras.Prelude.Iterates]
+n:24 [in Goedel.PRrepresentable]
+n:24 [in hydras.Ackermann.extEqualNat]
+n:24 [in gaia_hydras.GF_alpha]
+n:24 [in hydras.Prelude.Compat815]
+n:24 [in hydras.Ackermann.LNN2LNT]
+n:24 [in hydras.MoreAck.AckNotPR]
+n:24 [in hydras.Prelude.Fuel]
+n:24 [in hydras.MoreAck.Ack]
+n:24 [in hydras.OrdinalNotations.ON_Finite]
+n:24 [in hydras.Ackermann.codeList]
+n:24 [in hydras.Ackermann.folLogic3]
+n:24 [in hydras.Ackermann.Languages]
+n:24 [in hydras.Ackermann.expressible]
+n:24 [in hydras.Prelude.MoreVectors]
+n:240 [in hydras.Ackermann.codeList]
+n:242 [in hydras.Prelude.Iterates]
+n:243 [in additions.AM]
+n:243 [in hydras.Ackermann.codeList]
+n:245 [in hydras.Ackermann.cPair]
+n:246 [in hydras.Prelude.Iterates]
+n:246 [in hydras.Ackermann.codeList]
+n:249 [in hydras.Epsilon0.T1]
+n:249 [in hydras.Ackermann.codeList]
+n:25 [in additions.Fib2]
+n:25 [in additions.Compatibility]
+n:25 [in additions.More_on_positive]
+n:25 [in hydras.Schutte.GRelations]
+n:25 [in hydras.Ackermann.codeSubFormula]
+n:25 [in hydras.solutions_exercises.MultisetWf]
+n:25 [in hydras.Hydra.Epsilon0_Needed_Std]
+n:25 [in hydras.Epsilon0.Canon]
+n:25 [in hydras.OrdinalNotations.ON_Finite]
+n:25 [in gaia_hydras.onType]
+n:25 [in hydras.solutions_exercises.F_3]
+n:25 [in hydras.Gamma0.Gamma0]
+n:250 [in hydras.Ackermann.subAll]
+n:250 [in hydras.Prelude.Iterates]
+n:251 [in hydras.Epsilon0.T1]
+n:252 [in hydras.Ackermann.codeList]
+n:254 [in hydras.Epsilon0.T1]
+n:254 [in hydras.Ackermann.subAll]
+n:254 [in gaia_hydras.nfwfgaia]
+n:255 [in hydras.Ackermann.codeList]
+n:256 [in gaia_hydras.nfwfgaia]
+n:258 [in hydras.Epsilon0.T1]
+n:258 [in hydras.Ackermann.codeList]
+n:258 [in hydras.rpo.rpo]
+n:26 [in hydras.Gamma0.T2]
+n:26 [in hydras.Prelude.Iterates]
+n:26 [in hydras.Ackermann.extEqualNat]
+n:26 [in hydras.Hydra.O2H]
+n:26 [in hydras.Epsilon0.Epsilon0rpo]
+n:26 [in hydras.Prelude.Compat815]
+n:26 [in hydras.Ackermann.codeSubTerm]
+n:26 [in hydras.solutions_exercises.MultisetWf]
+n:26 [in hydras.Prelude.Fuel]
+n:26 [in hydras.Hydra.Hydra_Termination]
+n:26 [in hydras.Ackermann.codeList]
+n:26 [in hydras.MoreAck.FolExamples]
+n:261 [in hydras.Epsilon0.T1]
+n:261 [in hydras.Ackermann.codeList]
+n:261 [in gaia_hydras.nfwfgaia]
+n:263 [in hydras.Ackermann.codeList]
+n:264 [in hydras.Epsilon0.T1]
+n:264 [in hydras.Gamma0.Gamma0]
+n:264 [in gaia_hydras.nfwfgaia]
+n:265 [in hydras.Ackermann.codeList]
+n:266 [in gaia_hydras.nfwfgaia]
+n:267 [in hydras.Ackermann.codeList]
+n:267 [in hydras.Gamma0.Gamma0]
+n:268 [in hydras.Epsilon0.T1]
+n:268 [in gaia_hydras.nfwfgaia]
+n:269 [in hydras.Ackermann.codeList]
+n:27 [in hydras.Ackermann.subAll]
+n:27 [in additions.Compatibility]
+n:27 [in hydras.MoreAck.AckNotPR]
+n:27 [in hydras.Ackermann.codeSubFormula]
+n:27 [in hydras.MoreAck.Ack]
+n:27 [in hydras.Hydra.Epsilon0_Needed_Std]
+n:27 [in hydras.Ackermann.wellFormed]
+n:27 [in gaia_hydras.nfwfgaia]
+n:270 [in hydras.Epsilon0.Paths]
+n:271 [in hydras.Ackermann.codeList]
+n:271 [in hydras.Gamma0.Gamma0]
+n:272 [in hydras.Epsilon0.Paths]
+n:273 [in hydras.Ackermann.codeList]
+n:274 [in hydras.Ackermann.codeList]
+n:275 [in hydras.Ackermann.codeList]
+n:275 [in hydras.Epsilon0.Paths]
+n:277 [in hydras.Ackermann.codeList]
+n:278 [in hydras.Epsilon0.T1]
+n:278 [in Goedel.PRrepresentable]
+n:278 [in hydras.Epsilon0.Paths]
+n:279 [in hydras.Epsilon0.T1]
+n:279 [in hydras.Ackermann.codeList]
+n:279 [in additions.Addition_Chains]
+n:28 [in hydras.Schutte.Correctness_E0]
+n:28 [in hydras.Prelude.Iterates]
+n:28 [in hydras.Prelude.Compat815]
+n:28 [in hydras.Ackermann.codeSubTerm]
+n:28 [in hydras.Epsilon0.Hessenberg]
+n:28 [in gaia_hydras.GCanon]
+n:28 [in hydras.Schutte.AP]
+n:28 [in hydras.Prelude.Fuel]
+n:28 [in hydras.Ackermann.codeList]
+n:28 [in hydras.solutions_exercises.F_3]
+n:28 [in hydras.Hydra.Hydra_Lemmas]
+n:28 [in hydras.MoreAck.PrimRecExamples]
+n:28 [in hydras.Ackermann.expressible]
+n:28 [in hydras.Prelude.MoreVectors]
+n:281 [in hydras.Epsilon0.T1]
+n:281 [in Goedel.PRrepresentable]
+n:281 [in hydras.Ackermann.codeList]
+n:282 [in hydras.Ackermann.cPair]
+n:282 [in additions.Addition_Chains]
+n:283 [in hydras.Epsilon0.T1]
+n:283 [in hydras.Ackermann.codeList]
+n:284 [in Goedel.PRrepresentable]
+n:284 [in hydras.Ackermann.cPair]
+n:285 [in hydras.Ackermann.codeList]
+n:285 [in hydras.Gamma0.Gamma0]
+n:286 [in hydras.Ackermann.cPair]
+n:287 [in hydras.Ackermann.codeList]
+n:288 [in hydras.Ackermann.cPair]
+n:288 [in hydras.Ackermann.codeList]
+n:289 [in hydras.Ackermann.codeList]
+n:289 [in hydras.Epsilon0.Paths]
+n:289 [in hydras.Gamma0.Gamma0]
+n:29 [in hydras.Ackermann.extEqualNat]
+n:29 [in gaia_hydras.GF_alpha]
+n:29 [in additions.Compatibility]
+n:29 [in hydras.Ackermann.LNN2LNT]
+n:29 [in hydras.Ackermann.subProp]
+n:29 [in hydras.Schutte.GRelations]
+n:29 [in hydras.Ackermann.codeSubFormula]
+n:29 [in hydras.Schutte.AP]
+n:29 [in hydras.Prelude.Fuel]
+n:29 [in hydras.Ackermann.code]
+n:29 [in hydras.Hydra.Hydra_Termination]
+n:29 [in additions.Euclidean_Chains]
+n:29 [in hydras.Ackermann.wellFormed]
+n:29 [in Goedel.rosser]
+n:290 [in Goedel.PRrepresentable]
+n:290 [in hydras.Ackermann.cPair]
+n:290 [in hydras.Ackermann.codeList]
+n:290 [in gaia_hydras.nfwfgaia]
+n:291 [in hydras.Ackermann.codeList]
+n:292 [in hydras.Ackermann.cPair]
+n:292 [in hydras.Ackermann.codeList]
+n:292 [in hydras.Epsilon0.Paths]
+n:292 [in gaia_hydras.nfwfgaia]
+n:293 [in hydras.Ackermann.codeList]
+n:293 [in hydras.Gamma0.Gamma0]
+n:294 [in hydras.Ackermann.subAll]
+n:294 [in hydras.Ackermann.cPair]
+n:294 [in hydras.Ackermann.codeList]
+n:295 [in hydras.Epsilon0.Paths]
+n:296 [in Goedel.PRrepresentable]
+n:296 [in hydras.Ackermann.cPair]
+n:297 [in hydras.Gamma0.Gamma0]
+n:298 [in hydras.Ackermann.cPair]
+n:3 [in hydras.Ackermann.subAll]
+n:3 [in hydras.Prelude.Iterates]
+n:3 [in hydras.MoreAck.expressibleExamples]
+n:3 [in hydras.Ackermann.model]
+n:3 [in Goedel.fixPoint]
+n:3 [in hydras.Ackermann.folProp]
+n:3 [in additions.More_on_positive]
+n:3 [in hydras.Epsilon0.F_omega]
+n:3 [in hydras.OrdinalNotations.ON_Finite]
+n:3 [in hydras.solutions_exercises.F_3]
+n:3 [in hydras.Gamma0.Gamma0]
+n:3 [in hydras.Schutte.Addition]
+n:3 [in gaia_hydras.GHessenberg]
+n:30 [in hydras.Prelude.Compat815]
+n:30 [in hydras.MoreAck.AckNotPR]
+n:30 [in hydras.Ackermann.codeList]
+n:30 [in gaia_hydras.nfwfgaia]
+n:30 [in hydras.Ackermann.Languages]
+n:30 [in hydras.Prelude.MoreVectors]
+n:300 [in hydras.Ackermann.cPair]
+n:302 [in Goedel.PRrepresentable]
+n:302 [in hydras.Ackermann.cPair]
+n:303 [in Goedel.PRrepresentable]
+n:303 [in hydras.Epsilon0.Paths]
+n:304 [in Goedel.PRrepresentable]
+n:304 [in hydras.Ackermann.cPair]
+n:304 [in hydras.Gamma0.Gamma0]
+n:305 [in Goedel.PRrepresentable]
+n:306 [in hydras.Ackermann.cPair]
+n:306 [in hydras.Epsilon0.Paths]
+n:308 [in hydras.Ackermann.cPair]
+n:308 [in hydras.Gamma0.Gamma0]
+n:31 [in hydras.Epsilon0.T1]
+n:31 [in hydras.Hydra.O2H]
+n:31 [in gaia_hydras.GF_alpha]
+n:31 [in additions.Compatibility]
+n:31 [in hydras.Ackermann.codeSubFormula]
+n:31 [in hydras.Ackermann.primRec]
+n:31 [in additions.Naive]
+n:31 [in hydras.Ackermann.wellFormed]
+n:31 [in Goedel.rosser]
+n:31 [in hydras.Gamma0.Gamma0]
+n:31 [in gaia_hydras.nfwfgaia]
+n:310 [in hydras.Ackermann.cPair]
+n:312 [in hydras.Ackermann.cPair]
+n:312 [in hydras.Epsilon0.Paths]
+n:314 [in hydras.Ackermann.subAll]
+n:314 [in hydras.Ackermann.cPair]
+n:316 [in hydras.Ackermann.cPair]
+n:318 [in hydras.Ackermann.cPair]
+n:319 [in hydras.Epsilon0.T1]
+n:32 [in hydras.rpo.term]
+n:32 [in hydras.Prelude.Compat815]
+n:32 [in hydras.Ackermann.code]
+n:32 [in hydras.MoreAck.Ack]
+n:32 [in hydras.Ackermann.fol]
+n:32 [in hydras.Ackermann.codeList]
+n:32 [in hydras.Hydra.Hydra_Lemmas]
+n:32 [in gaia_hydras.nfwfgaia]
+n:320 [in hydras.Ackermann.cPair]
+n:320 [in hydras.Epsilon0.Paths]
+n:321 [in gaia_hydras.nfwfgaia]
+n:322 [in hydras.Ackermann.cPair]
+n:322 [in hydras.Epsilon0.Paths]
+n:323 [in hydras.Epsilon0.T1]
+n:323 [in gaia_hydras.nfwfgaia]
+n:324 [in hydras.Ackermann.cPair]
+n:325 [in Goedel.PRrepresentable]
+n:325 [in gaia_hydras.nfwfgaia]
+n:326 [in hydras.Ackermann.cPair]
+n:326 [in hydras.Epsilon0.Paths]
+n:326 [in gaia_hydras.nfwfgaia]
+n:327 [in hydras.Ackermann.cPair]
+n:328 [in hydras.Epsilon0.T1]
+n:328 [in hydras.Ackermann.cPair]
+n:328 [in hydras.Epsilon0.Paths]
+n:329 [in hydras.Ackermann.cPair]
+n:33 [in additions.Compatibility]
+n:33 [in additions.FirstSteps]
+n:33 [in hydras.Prelude.Compat815]
+n:33 [in hydras.Epsilon0.Hessenberg]
+n:33 [in hydras.Ackermann.codeSubFormula]
+n:33 [in hydras.Hydra.Hydra_Termination]
+n:33 [in additions.Euclidean_Chains]
+n:33 [in hydras.Ackermann.codeList]
+n:330 [in Goedel.PRrepresentable]
+n:330 [in hydras.Ackermann.cPair]
+n:331 [in hydras.Ackermann.cPair]
+n:332 [in hydras.Ackermann.cPair]
+n:333 [in hydras.Epsilon0.T1]
+n:333 [in hydras.Ackermann.cPair]
+n:333 [in additions.Euclidean_Chains]
+n:334 [in hydras.Ackermann.cPair]
+n:335 [in Goedel.PRrepresentable]
+n:336 [in hydras.Ackermann.cPair]
+n:34 [in hydras.Prelude.Iterates]
+n:34 [in hydras.Prelude.Compat815]
+n:34 [in gaia_hydras.GCanon]
+n:34 [in hydras.MoreAck.AckNotPR]
+n:34 [in hydras.Ackermann.subProp]
+n:34 [in hydras.Schutte.AP]
+n:34 [in hydras.Ackermann.fol]
+n:34 [in hydras.Ackermann.codeList]
+n:342 [in hydras.Epsilon0.Paths]
+n:343 [in hydras.Ackermann.cPair]
+n:344 [in hydras.Ackermann.cPair]
+n:345 [in hydras.Epsilon0.T1]
+n:345 [in hydras.Ackermann.cPair]
+n:346 [in hydras.Epsilon0.Paths]
+n:35 [in hydras.Ackermann.codeSubFormula]
+n:35 [in hydras.Ackermann.code]
+n:35 [in hydras.Ackermann.codeList]
+n:35 [in Goedel.rosser]
+n:35 [in hydras.Ackermann.Languages]
+n:350 [in hydras.Epsilon0.T1]
+n:350 [in hydras.Epsilon0.Paths]
+n:353 [in hydras.Epsilon0.T1]
+n:354 [in hydras.Epsilon0.Paths]
+n:355 [in hydras.Epsilon0.Paths]
+n:356 [in hydras.Epsilon0.T1]
+n:359 [in hydras.Epsilon0.Paths]
+n:36 [in hydras.Ackermann.codeSubTerm]
+n:36 [in hydras.Ackermann.LNN2LNT]
+n:36 [in hydras.Ackermann.codeList]
+n:36 [in hydras.Ackermann.expressible]
+n:363 [in hydras.Epsilon0.T1]
+n:363 [in gaia_hydras.nfwfgaia]
+n:364 [in hydras.Ackermann.subAll]
+n:367 [in hydras.Ackermann.subAll]
+n:37 [in hydras.Prelude.Iterates]
+n:37 [in additions.Compatibility]
+n:37 [in hydras.Ackermann.codeSubFormula]
+n:37 [in hydras.Schutte.AP]
+n:37 [in hydras.Ackermann.fol]
+n:37 [in hydras.Epsilon0.Canon]
+n:37 [in hydras.Ackermann.codeList]
+n:37 [in hydras.Hydra.Hydra_Definitions]
+n:37 [in hydras.Prelude.MoreVectors]
+n:370 [in Goedel.PRrepresentable]
+n:371 [in Goedel.PRrepresentable]
+n:376 [in hydras.Ackermann.subAll]
+n:377 [in hydras.Epsilon0.T1]
+n:379 [in hydras.Ackermann.subAll]
+n:38 [in additions.FirstSteps]
+n:38 [in hydras.Ackermann.LNN2LNT]
+n:38 [in gaia_hydras.GCanon]
+n:38 [in hydras.Ackermann.codeSubFormula]
+n:38 [in hydras.Epsilon0.F_omega]
+n:38 [in hydras.Ackermann.codeList]
+n:38 [in hydras.Ackermann.folLogic3]
+n:38 [in hydras.Ackermann.Languages]
+n:382 [in hydras.Epsilon0.T1]
+n:384 [in gaia_hydras.nfwfgaia]
+n:386 [in hydras.Epsilon0.T1]
+n:386 [in gaia_hydras.nfwfgaia]
+n:387 [in hydras.Epsilon0.Paths]
+n:388 [in gaia_hydras.nfwfgaia]
+n:39 [in hydras.Prelude.More_Arith]
+n:39 [in hydras.Epsilon0.Hessenberg]
+n:39 [in hydras.Ackermann.subProp]
+n:39 [in hydras.Ackermann.codeSubFormula]
+n:39 [in gaia_hydras.T1Bridge]
+n:39 [in additions.Naive]
+n:39 [in hydras.Epsilon0.Canon]
+n:39 [in additions.fib]
+n:39 [in hydras.Ackermann.codeList]
+n:39 [in hydras.Ackermann.folLogic3]
+n:39 [in hydras.Ackermann.expressible]
+n:391 [in hydras.Epsilon0.T1]
+n:392 [in hydras.Epsilon0.Paths]
+n:394 [in hydras.Epsilon0.T1]
+n:394 [in hydras.Epsilon0.Paths]
+n:394 [in gaia_hydras.nfwfgaia]
+n:397 [in hydras.Epsilon0.Paths]
+n:399 [in hydras.Epsilon0.T1]
+n:4 [in hydras.Epsilon0.T1]
+n:4 [in hydras.Schutte.Correctness_E0]
+n:4 [in hydras.Prelude.More_Arith]
+n:4 [in hydras.Ackermann.extEqualNat]
+n:4 [in gaia_hydras.GF_alpha]
+n:4 [in hydras.MoreAck.expressibleExamples]
+n:4 [in additions.Compatibility]
+n:4 [in hydras.Prelude.Compat815]
+n:4 [in hydras.Prelude.First_toggle]
+n:4 [in gaia_hydras.GPaths]
+n:4 [in hydras.Ackermann.PAtheory]
+n:4 [in hydras.Prelude.Fuel]
+n:4 [in hydras.Ackermann.cPair]
+n:4 [in gaia_hydras.ON_gfinite]
+n:4 [in additions.Monoid_instances]
+n:4 [in additions.fib]
+n:4 [in hydras.Ackermann.codeList]
+n:4 [in hydras.Prelude.LibHyps_Experiments]
+n:4 [in hydras.solutions_exercises.F_3]
+n:4 [in hydras.Ackermann.codeNatToTerm]
+n:4 [in gaia_hydras.nfwfgaia]
+n:4 [in hydras.Ackermann.expressible]
+n:40 [in hydras.Prelude.Iterates]
+n:40 [in hydras.Epsilon0.Epsilon0rpo]
+n:40 [in hydras.Ackermann.LNN2LNT]
+n:40 [in hydras.Ackermann.model]
+n:40 [in hydras.MoreAck.AckNotPR]
+n:40 [in hydras.Ackermann.codeSubFormula]
+n:40 [in hydras.Prelude.Fuel]
+n:40 [in hydras.Ackermann.fol]
+n:40 [in hydras.Ackermann.codeList]
+n:40 [in hydras.Ackermann.folLogic3]
+n:40 [in Goedel.rosser]
+n:40 [in hydras.Ackermann.Languages]
+n:41 [in hydras.Prelude.More_Arith]
+n:41 [in hydras.Ackermann.codeSubFormula]
+n:41 [in additions.Naive]
+n:41 [in hydras.Ackermann.wellFormed]
+n:41 [in gaia_hydras.onType]
+n:41 [in hydras.Ackermann.folLogic3]
+n:41 [in hydras.rpo.list_permut]
+n:41 [in gaia_hydras.nfwfgaia]
+n:41 [in hydras.MoreAck.PrimRecExamples]
+n:416 [in hydras.Epsilon0.T1]
+n:419 [in hydras.Ackermann.folProp]
+n:42 [in gaia_hydras.GCanon]
+n:42 [in hydras.Epsilon0.Large_Sets]
+n:42 [in hydras.solutions_exercises.MultisetWf]
+n:42 [in hydras.Ackermann.primRec]
+n:42 [in hydras.Hydra.Hydra_Termination]
+n:42 [in hydras.rpo.list_permut]
+n:42 [in hydras.Hydra.Hydra_Definitions]
+n:42 [in hydras.Prelude.MoreVectors]
+n:421 [in hydras.Ackermann.folProp]
+n:423 [in hydras.Ackermann.folProp]
+n:423 [in gaia_hydras.nfwfgaia]
+n:426 [in hydras.Epsilon0.T1]
+n:427 [in gaia_hydras.nfwfgaia]
+n:43 [in hydras.Ackermann.subAll]
+n:43 [in additions.Compatibility]
+n:43 [in hydras.Epsilon0.Epsilon0rpo]
+n:43 [in hydras.Ackermann.codeSubFormula]
+n:43 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+n:43 [in hydras.Prelude.Fuel]
+n:43 [in hydras.Epsilon0.Canon]
+n:43 [in hydras.Epsilon0.E0]
+n:43 [in gaia_hydras.nfwfgaia]
+n:43 [in hydras.MoreAck.PrimRecExamples]
+n:43 [in hydras.Ackermann.expressible]
+n:431 [in hydras.Ackermann.fol]
+n:431 [in hydras.Gamma0.Gamma0]
+n:432 [in hydras.Epsilon0.T1]
+n:433 [in hydras.Ackermann.primRec]
+n:433 [in hydras.Ackermann.fol]
+n:433 [in hydras.Gamma0.Gamma0]
+n:437 [in hydras.Epsilon0.T1]
+n:44 [in hydras.Prelude.More_Arith]
+n:44 [in hydras.Hydra.O2H]
+n:44 [in additions.FirstSteps]
+n:44 [in hydras.MoreAck.AckNotPR]
+n:44 [in hydras.Prelude.Fuel]
+n:44 [in additions.fib]
+n:44 [in hydras.MoreAck.PrimRecExamples]
+n:440 [in gaia_hydras.nfwfgaia]
+n:441 [in hydras.Ackermann.primRec]
+n:442 [in hydras.Ackermann.primRec]
+n:444 [in Goedel.PRrepresentable]
+n:446 [in Goedel.PRrepresentable]
+n:449 [in hydras.Ackermann.primRec]
+n:45 [in hydras.Schutte.Correctness_E0]
+n:45 [in additions.FirstSteps]
+n:45 [in hydras.Epsilon0.F_alpha]
+n:45 [in hydras.Ackermann.codeSubFormula]
+n:45 [in hydras.solutions_exercises.MultisetWf]
+n:45 [in hydras.Ackermann.codeList]
+n:454 [in hydras.Epsilon0.T1]
+n:454 [in hydras.Ackermann.primRec]
+n:455 [in Goedel.PRrepresentable]
+n:455 [in hydras.Ackermann.primRec]
+n:457 [in Goedel.PRrepresentable]
+n:46 [in hydras.Epsilon0.Epsilon0rpo]
+n:46 [in hydras.Prelude.Fuel]
+n:46 [in hydras.OrdinalNotations.OmegaOmega]
+n:46 [in hydras.Ackermann.wellFormed]
+n:46 [in hydras.Hydra.Hydra_Lemmas]
+n:462 [in hydras.Ackermann.primRec]
+n:465 [in Goedel.PRrepresentable]
+n:467 [in hydras.Ackermann.primRec]
+n:47 [in hydras.Schutte.Correctness_E0]
+n:47 [in additions.FirstSteps]
+n:47 [in gaia_hydras.GCanon]
+n:47 [in hydras.Ackermann.codeSubFormula]
+n:47 [in hydras.Ackermann.folLogic3]
+n:47 [in hydras.Ackermann.expressible]
+n:470 [in additions.Euclidean_Chains]
+n:470 [in hydras.Gamma0.Gamma0]
+n:471 [in hydras.Ackermann.primRec]
+n:472 [in hydras.Epsilon0.Paths]
+n:473 [in Goedel.PRrepresentable]
+n:474 [in additions.Euclidean_Chains]
+n:475 [in hydras.Ackermann.primRec]
+n:476 [in hydras.Ackermann.primRec]
+n:476 [in additions.Euclidean_Chains]
+n:476 [in hydras.Gamma0.Gamma0]
+n:477 [in gaia_hydras.nfwfgaia]
+n:478 [in additions.Euclidean_Chains]
+n:479 [in hydras.Gamma0.Gamma0]
+n:48 [in hydras.Prelude.Iterates]
+n:48 [in hydras.MoreAck.AckNotPR]
+n:48 [in hydras.Prelude.Fuel]
+n:48 [in hydras.OrdinalNotations.OmegaOmega]
+n:48 [in hydras.Ackermann.codeList]
+n:48 [in hydras.Ackermann.folLogic3]
+n:48 [in gaia_hydras.nfwfgaia]
+n:480 [in gaia_hydras.nfwfgaia]
+n:483 [in hydras.Ackermann.primRec]
+n:483 [in gaia_hydras.nfwfgaia]
+n:486 [in gaia_hydras.nfwfgaia]
+n:488 [in hydras.Ackermann.primRec]
+n:488 [in hydras.Gamma0.Gamma0]
+n:489 [in gaia_hydras.nfwfgaia]
+n:49 [in hydras.Epsilon0.Hessenberg]
+n:49 [in hydras.Ackermann.codeSubFormula]
+n:49 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+n:49 [in hydras.Prelude.Fuel]
+n:49 [in hydras.MoreAck.Ack]
+n:49 [in hydras.Ackermann.primRec]
+n:49 [in hydras.Ackermann.Languages]
+n:493 [in hydras.Ackermann.primRec]
+n:494 [in hydras.Epsilon0.T1]
+n:495 [in hydras.Epsilon0.T1]
+n:496 [in gaia_hydras.nfwfgaia]
+n:497 [in hydras.Gamma0.Gamma0]
+n:498 [in hydras.Ackermann.primRec]
+n:499 [in gaia_hydras.nfwfgaia]
+n:5 [in hydras.Schutte.Correctness_E0]
+n:5 [in additions.Fib2]
+n:5 [in hydras.Prelude.More_Arith]
+n:5 [in hydras.MoreAck.expressibleExamples]
+n:5 [in hydras.Ackermann.NN2PA]
+n:5 [in additions.FirstSteps]
+n:5 [in hydras.MoreAck.Iterate_compat]
+n:5 [in hydras.Prelude.Fuel]
+n:5 [in hydras.solutions_exercises.FibonacciPR]
+n:5 [in hydras.Ackermann.primRec]
+n:5 [in hydras.Ackermann.codeNatToTerm]
+n:5 [in hydras.Ackermann.wConsistent]
+n:5 [in hydras.solutions_exercises.isqrt]
+n:50 [in additions.FirstSteps]
+n:50 [in hydras.Epsilon0.F_alpha]
+n:50 [in hydras.Ackermann.folProp]
+n:50 [in hydras.Ackermann.codeSubFormula]
+n:50 [in hydras.solutions_exercises.MultisetWf]
+n:50 [in hydras.MoreAck.Ack]
+n:50 [in gaia_hydras.nfwfgaia]
+n:50 [in hydras.Prelude.MoreVectors]
+n:501 [in gaia_hydras.nfwfgaia]
+n:502 [in hydras.Ackermann.primRec]
+n:503 [in hydras.Epsilon0.T1]
+n:506 [in hydras.Epsilon0.T1]
+n:508 [in hydras.Epsilon0.T1]
+n:51 [in Goedel.PRrepresentable]
+n:51 [in hydras.Prelude.Iterates]
+n:51 [in hydras.Ackermann.codeSubFormula]
+n:51 [in hydras.MoreAck.Ack]
+n:51 [in hydras.Ackermann.wellFormed]
+n:51 [in gaia_hydras.nfwfgaia]
+n:51 [in hydras.Ackermann.expressible]
+n:511 [in hydras.Epsilon0.T1]
+n:511 [in hydras.Epsilon0.Paths]
+n:516 [in gaia_hydras.nfwfgaia]
+n:517 [in hydras.Epsilon0.T1]
+n:517 [in hydras.Epsilon0.Paths]
+n:518 [in hydras.Ackermann.checkPrf]
+n:518 [in gaia_hydras.nfwfgaia]
+n:52 [in hydras.Epsilon0.T1]
+n:52 [in hydras.OrdinalNotations.ON_Omega2]
+n:52 [in hydras.Ackermann.model]
+n:52 [in hydras.Ackermann.subProp]
+n:52 [in hydras.Ackermann.codeSubFormula]
+n:52 [in hydras.MoreAck.Ack]
+n:52 [in gaia_hydras.T1Bridge]
+n:521 [in gaia_hydras.nfwfgaia]
+n:523 [in hydras.Epsilon0.T1]
+n:523 [in hydras.Gamma0.Gamma0]
+n:523 [in gaia_hydras.nfwfgaia]
+n:527 [in hydras.Epsilon0.T1]
+n:529 [in hydras.Epsilon0.T1]
+n:53 [in hydras.rpo.more_list]
+n:53 [in hydras.Epsilon0.F_alpha]
+n:53 [in hydras.MoreAck.AckNotPR]
+n:53 [in hydras.Ackermann.codeSubFormula]
+n:53 [in hydras.OrdinalNotations.OmegaOmega]
+n:53 [in hydras.Ackermann.fol]
+n:53 [in hydras.Ackermann.codeList]
+n:53 [in hydras.Epsilon0.Paths]
+n:530 [in gaia_hydras.nfwfgaia]
+n:531 [in gaia_hydras.nfwfgaia]
+n:532 [in hydras.Epsilon0.T1]
+n:535 [in hydras.Ackermann.primRec]
+n:535 [in hydras.Gamma0.Gamma0]
+n:537 [in hydras.Ackermann.checkPrf]
+n:539 [in hydras.Epsilon0.T1]
+n:54 [in hydras.Epsilon0.Epsilon0rpo]
+n:54 [in additions.Pow_variant]
+n:54 [in additions.Pow]
+n:54 [in hydras.Ackermann.codeSubFormula]
+n:54 [in hydras.MoreAck.Ack]
+n:54 [in hydras.Ackermann.folLogic3]
+n:54 [in hydras.Hydra.Hydra_Lemmas]
+n:54 [in hydras.Schutte.Addition]
+n:542 [in hydras.Ackermann.primRec]
+n:543 [in hydras.Epsilon0.T1]
+n:543 [in hydras.Gamma0.Gamma0]
+n:543 [in gaia_hydras.nfwfgaia]
+n:544 [in gaia_hydras.nfwfgaia]
+n:545 [in hydras.Epsilon0.T1]
+n:545 [in hydras.Ackermann.primRec]
+n:548 [in gaia_hydras.nfwfgaia]
+n:55 [in Goedel.PRrepresentable]
+n:55 [in hydras.Prelude.More_Arith]
+n:55 [in hydras.Prelude.Iterates]
+n:55 [in hydras.Ackermann.codeSubFormula]
+n:55 [in hydras.Ackermann.folLogic3]
+n:55 [in Goedel.rosser]
+n:55 [in gaia_hydras.nfwfgaia]
+n:550 [in hydras.Epsilon0.T1]
+n:550 [in hydras.Epsilon0.Paths]
+n:552 [in hydras.Epsilon0.T1]
+n:554 [in hydras.Epsilon0.T1]
+n:556 [in hydras.Epsilon0.T1]
+n:556 [in hydras.Ackermann.primRec]
+n:556 [in gaia_hydras.nfwfgaia]
+n:557 [in hydras.Gamma0.Gamma0]
+n:558 [in hydras.Epsilon0.Paths]
+n:559 [in hydras.Epsilon0.T1]
+n:559 [in hydras.Ackermann.primRec]
+n:559 [in gaia_hydras.nfwfgaia]
+n:56 [in hydras.Ackermann.model]
+n:56 [in hydras.MoreAck.AckNotPR]
+n:56 [in hydras.Ackermann.codeSubFormula]
+n:56 [in hydras.OrdinalNotations.OmegaOmega]
+n:56 [in hydras.Ackermann.fol]
+n:56 [in hydras.Ackermann.codeList]
+n:56 [in hydras.Prelude.MoreVectors]
+n:561 [in hydras.Gamma0.Gamma0]
+n:561 [in gaia_hydras.nfwfgaia]
+n:562 [in hydras.Ackermann.primRec]
+n:563 [in hydras.Epsilon0.T1]
+n:563 [in hydras.Epsilon0.Paths]
+n:563 [in gaia_hydras.nfwfgaia]
+n:564 [in hydras.Ackermann.primRec]
+n:568 [in hydras.Gamma0.Gamma0]
+n:569 [in hydras.Epsilon0.T1]
+n:57 [in hydras.Ackermann.subProp]
+n:57 [in hydras.MoreAck.Ack]
+n:57 [in gaia_hydras.T1Bridge]
+n:57 [in hydras.Epsilon0.Canon]
+n:57 [in hydras.Ackermann.folLogic3]
+n:57 [in Goedel.rosser]
+n:57 [in gaia_hydras.nfwfgaia]
+n:572 [in hydras.Epsilon0.Paths]
+n:573 [in hydras.Gamma0.Gamma0]
+n:574 [in hydras.Epsilon0.T1]
+n:574 [in hydras.Ackermann.primRec]
+n:578 [in hydras.Epsilon0.T1]
+n:58 [in hydras.OrdinalNotations.ON_Omega2]
+n:58 [in hydras.Epsilon0.Epsilon0rpo]
+n:58 [in hydras.MoreAck.AckNotPR]
+n:58 [in hydras.Ackermann.primRec]
+n:58 [in hydras.Ackermann.fol]
+n:584 [in hydras.Epsilon0.Paths]
+n:589 [in hydras.Ackermann.checkPrf]
+n:589 [in hydras.Gamma0.Gamma0]
+n:59 [in Goedel.PRrepresentable]
+n:59 [in hydras.Schutte.Critical]
+n:59 [in hydras.MoreAck.Ack]
+n:59 [in Goedel.rosserPA]
+n:59 [in hydras.Ackermann.wellFormed]
+n:59 [in hydras.Ackermann.codeList]
+n:59 [in hydras.Hydra.Hydra_Lemmas]
+n:595 [in hydras.Epsilon0.T1]
+n:597 [in hydras.Ackermann.primRec]
+n:598 [in hydras.Epsilon0.T1]
+n:598 [in hydras.Gamma0.Gamma0]
+n:6 [in hydras.Epsilon0.T1]
+n:6 [in additions.Fib2]
+n:6 [in gaia_hydras.GF_alpha]
+n:6 [in hydras.MoreAck.expressibleExamples]
+n:6 [in Goedel.fixPoint]
+n:6 [in hydras.Ackermann.PAtheory]
+n:6 [in hydras.Ackermann.subProp]
+n:6 [in hydras.Ackermann.codeSubFormula]
+n:6 [in hydras.solutions_exercises.OnCodeList]
+n:6 [in hydras.Schutte.AP]
+n:6 [in additions.Naive]
+n:6 [in hydras.Ackermann.codeList]
+n:6 [in hydras.Hydra.Hydra_Lemmas]
+n:6 [in hydras.Prelude.Exp2]
+n:6 [in hydras.Epsilon0.Paths]
+n:6 [in hydras.Hydra.Omega_Small]
+n:6 [in gaia_hydras.nfwfgaia]
+n:6 [in hydras.Schutte.Countable]
+n:6 [in hydras.Prelude.MoreVectors]
+n:60 [in hydras.Epsilon0.Epsilon0rpo]
+n:60 [in hydras.solutions_exercises.MultisetWf]
+n:60 [in hydras.Ackermann.code]
+n:60 [in hydras.Ackermann.expressible]
+n:602 [in hydras.Epsilon0.T1]
+n:602 [in hydras.Ackermann.primRec]
+n:603 [in gaia_hydras.nfwfgaia]
+n:605 [in hydras.Ackermann.primRec]
+n:606 [in hydras.Epsilon0.T1]
+n:608 [in hydras.Ackermann.primRec]
+n:608 [in gaia_hydras.nfwfgaia]
+n:61 [in hydras.Ackermann.subAll]
+n:61 [in hydras.rpo.more_list]
+n:61 [in hydras.MoreAck.AckNotPR]
+n:61 [in Goedel.rosserPA]
+n:61 [in hydras.Ackermann.wellFormed]
+n:61 [in Goedel.rosser]
+n:61 [in gaia_hydras.nfwfgaia]
+n:610 [in hydras.Epsilon0.T1]
+n:613 [in hydras.Epsilon0.T1]
+n:613 [in hydras.Ackermann.primRec]
+n:616 [in hydras.Epsilon0.T1]
+n:617 [in hydras.Ackermann.primRec]
+n:618 [in hydras.Ackermann.checkPrf]
+n:619 [in hydras.Epsilon0.T1]
+n:62 [in hydras.Schutte.Correctness_E0]
+n:62 [in hydras.Ackermann.subAll]
+n:62 [in hydras.rpo.more_list]
+n:62 [in additions.Pow_variant]
+n:62 [in hydras.Ackermann.model]
+n:62 [in additions.Pow]
+n:62 [in hydras.Ackermann.subProp]
+n:62 [in hydras.Prelude.Fuel]
+n:62 [in hydras.Ackermann.code]
+n:62 [in gaia_hydras.T1Bridge]
+n:62 [in hydras.Ackermann.codeList]
+n:62 [in hydras.Ackermann.Languages]
+n:622 [in hydras.Epsilon0.T1]
+n:627 [in hydras.Epsilon0.T1]
+n:627 [in gaia_hydras.nfwfgaia]
+n:628 [in hydras.Ackermann.primRec]
+n:63 [in Goedel.PRrepresentable]
+n:63 [in hydras.solutions_exercises.MultisetWf]
+n:63 [in hydras.Ackermann.folLogic3]
+n:63 [in Goedel.rosser]
+n:63 [in hydras.Schutte.Addition]
+n:63 [in hydras.Prelude.MoreVectors]
+n:631 [in hydras.Ackermann.primRec]
+n:632 [in hydras.Gamma0.Gamma0]
+n:635 [in hydras.Gamma0.Gamma0]
+n:636 [in hydras.Ackermann.primRec]
+n:639 [in hydras.Epsilon0.T1]
+n:639 [in hydras.Gamma0.Gamma0]
+n:64 [in hydras.OrdinalNotations.ON_Omega2]
+n:64 [in hydras.MoreAck.Ack]
+n:64 [in hydras.Ackermann.Languages]
+n:64 [in hydras.Ackermann.expressible]
+n:641 [in hydras.Epsilon0.T1]
+n:642 [in hydras.Ackermann.primRec]
+n:642 [in hydras.Gamma0.Gamma0]
+n:643 [in hydras.Epsilon0.T1]
+n:647 [in hydras.Ackermann.primRec]
+n:647 [in hydras.Gamma0.Gamma0]
+n:649 [in hydras.Ackermann.checkPrf]
+n:65 [in hydras.Epsilon0.T1]
+n:65 [in hydras.Ackermann.subAll]
+n:65 [in hydras.rpo.more_list]
+n:65 [in additions.Pow_variant]
+n:65 [in additions.Pow]
+n:65 [in hydras.Schutte.Critical]
+n:65 [in hydras.solutions_exercises.MultisetWf]
+n:65 [in Goedel.rosserPA]
+n:65 [in hydras.Ackermann.codeList]
+n:65 [in hydras.rpo.list_permut]
+n:651 [in hydras.Gamma0.Gamma0]
+n:652 [in gaia_hydras.nfwfgaia]
+n:653 [in hydras.Ackermann.primRec]
+n:654 [in hydras.Gamma0.Gamma0]
+n:658 [in hydras.Epsilon0.T1]
+n:659 [in hydras.Gamma0.Gamma0]
+n:66 [in hydras.Schutte.Correctness_E0]
+n:66 [in hydras.Prelude.MoreLists]
+n:66 [in hydras.Epsilon0.Hessenberg]
+n:66 [in hydras.Schutte.Critical]
+n:66 [in additions.Euclidean_Chains]
+n:66 [in hydras.rpo.list_permut]
+n:66 [in hydras.Schutte.Addition]
+n:660 [in hydras.Ackermann.primRec]
+n:664 [in hydras.Gamma0.Gamma0]
+n:667 [in hydras.Ackermann.primRec]
+n:667 [in gaia_hydras.nfwfgaia]
+n:668 [in gaia_hydras.nfwfgaia]
+n:67 [in Goedel.PRrepresentable]
+n:67 [in hydras.Epsilon0.F_alpha]
+n:67 [in gaia_hydras.GPaths]
+n:67 [in hydras.MoreAck.Ack]
+n:67 [in gaia_hydras.T1Bridge]
+n:67 [in hydras.Ackermann.primRec]
+n:67 [in gaia_hydras.nfwfgaia]
+n:67 [in hydras.Hydra.Hydra_Definitions]
+n:671 [in hydras.Ackermann.primRec]
+n:671 [in gaia_hydras.nfwfgaia]
+n:674 [in hydras.Epsilon0.T1]
+n:678 [in hydras.Epsilon0.T1]
+n:678 [in hydras.Ackermann.primRec]
+n:68 [in hydras.Ackermann.subAll]
+n:68 [in additions.Pow_variant]
+n:68 [in additions.Pow]
+n:68 [in hydras.Ackermann.codeList]
+n:68 [in hydras.Schutte.Addition]
+n:680 [in hydras.Gamma0.Gamma0]
+n:682 [in hydras.Ackermann.primRec]
+n:683 [in hydras.Gamma0.Gamma0]
+n:683 [in gaia_hydras.nfwfgaia]
+n:688 [in hydras.Gamma0.Gamma0]
+n:69 [in additions.Addition_Chains]
+n:69 [in gaia_hydras.nfwfgaia]
+n:69 [in hydras.Ackermann.expressible]
+n:691 [in hydras.Gamma0.Gamma0]
+n:7 [in additions.Fib2]
+n:7 [in hydras.Ackermann.extEqualNat]
+n:7 [in hydras.Prelude.MoreLists]
+n:7 [in hydras.MoreAck.expressibleExamples]
+n:7 [in additions.Pow_variant]
+n:7 [in hydras.Ackermann.model]
+n:7 [in additions.Pow]
+n:7 [in additions.More_on_positive]
+n:7 [in hydras.solutions_exercises.MultisetWf]
+n:7 [in hydras.Prelude.Fuel]
+n:7 [in hydras.MoreAck.Ack]
+n:7 [in hydras.Ackermann.primRec]
+n:7 [in gaia_hydras.ON_gfinite]
+n:7 [in hydras.OrdinalNotations.ON_Finite]
+n:7 [in hydras.Prelude.Exp2]
+n:7 [in hydras.solutions_exercises.isqrt]
+n:7 [in hydras.Gamma0.Gamma0]
+n:7 [in hydras.Schutte.Addition]
+n:70 [in hydras.Prelude.Iterates]
+n:70 [in additions.Pow_variant]
+n:70 [in gaia_hydras.GPaths]
+n:70 [in additions.Pow]
+n:70 [in hydras.Ackermann.folProp]
+n:70 [in hydras.MoreAck.Ack]
+n:70 [in Goedel.rosserPA]
+n:70 [in hydras.Schutte.Addition]
+n:702 [in gaia_hydras.nfwfgaia]
+n:704 [in hydras.Ackermann.primRec]
+n:706 [in hydras.Ackermann.primRec]
+n:708 [in gaia_hydras.nfwfgaia]
+n:71 [in Goedel.PRrepresentable]
+n:71 [in hydras.Prelude.MoreLists]
+n:71 [in hydras.Ackermann.folProp]
+n:71 [in hydras.Schutte.Critical]
+n:71 [in hydras.Ackermann.codeList]
+n:71 [in gaia_hydras.nfwfgaia]
+n:712 [in hydras.Gamma0.Gamma0]
+n:712 [in gaia_hydras.nfwfgaia]
+n:714 [in hydras.Epsilon0.T1]
+n:72 [in hydras.Epsilon0.T1]
+n:72 [in hydras.Epsilon0.F_alpha]
+n:72 [in hydras.Ackermann.folProp]
+n:72 [in hydras.Schutte.Critical]
+n:72 [in hydras.MoreAck.Ack]
+n:72 [in hydras.Hydra.BigBattle]
+n:72 [in hydras.rpo.list_permut]
+n:720 [in hydras.Epsilon0.T1]
+n:723 [in hydras.Epsilon0.T1]
+n:724 [in gaia_hydras.nfwfgaia]
+n:73 [in additions.Pow]
+n:73 [in hydras.Epsilon0.Hessenberg]
+n:73 [in gaia_hydras.T1Bridge]
+n:73 [in hydras.Ackermann.primRec]
+n:73 [in hydras.Hydra.BigBattle]
+n:73 [in hydras.Ackermann.codeList]
+n:73 [in hydras.rpo.list_permut]
+n:73 [in hydras.Schutte.Addition]
+n:736 [in gaia_hydras.nfwfgaia]
+n:74 [in hydras.Prelude.Iterates]
+n:74 [in hydras.Prelude.MoreLists]
+n:74 [in hydras.Epsilon0.F_alpha]
+n:74 [in additions.Pow_variant]
+n:74 [in hydras.Ackermann.model]
+n:74 [in hydras.MoreAck.Ack]
+n:74 [in gaia_hydras.nfwfgaia]
+n:748 [in gaia_hydras.nfwfgaia]
+n:75 [in hydras.rpo.more_list]
+n:75 [in hydras.Ackermann.LNN2LNT]
+n:75 [in hydras.Ackermann.subProp]
+n:75 [in hydras.solutions_exercises.MultisetWf]
+n:75 [in hydras.Ackermann.primRec]
+n:75 [in hydras.Hydra.BigBattle]
+n:75 [in hydras.Ackermann.codeList]
+n:75 [in hydras.Epsilon0.Hprime]
+n:75 [in gaia_hydras.nfwfgaia]
+n:750 [in gaia_hydras.nfwfgaia]
+n:753 [in gaia_hydras.nfwfgaia]
+n:755 [in gaia_hydras.nfwfgaia]
+n:756 [in hydras.Epsilon0.T1]
+n:757 [in gaia_hydras.nfwfgaia]
+n:759 [in gaia_hydras.nfwfgaia]
+n:76 [in hydras.Ackermann.subAll]
+n:76 [in hydras.Prelude.MoreLists]
+n:76 [in hydras.MoreAck.AckNotPR]
+n:76 [in hydras.MoreAck.Ack]
+n:76 [in Goedel.codeSysPrf]
+n:76 [in gaia_hydras.nfwfgaia]
+n:76 [in hydras.Hydra.Hydra_Definitions]
+n:764 [in gaia_hydras.nfwfgaia]
+n:767 [in gaia_hydras.nfwfgaia]
+n:769 [in gaia_hydras.nfwfgaia]
+n:77 [in hydras.Gamma0.T2]
+n:77 [in gaia_hydras.GPaths]
+n:77 [in hydras.Ackermann.model]
+n:77 [in additions.Pow]
+n:77 [in hydras.Ackermann.codeList]
+n:77 [in hydras.Epsilon0.Hprime]
+n:78 [in hydras.Epsilon0.T1]
+n:78 [in Goedel.PRrepresentable]
+n:78 [in hydras.Prelude.Iterates]
+n:78 [in hydras.Ackermann.codeSubFormula]
+n:78 [in gaia_hydras.nfwfgaia]
+n:780 [in gaia_hydras.nfwfgaia]
+n:784 [in gaia_hydras.nfwfgaia]
+n:787 [in gaia_hydras.nfwfgaia]
+n:79 [in additions.AM]
+n:79 [in hydras.Prelude.MoreLists]
+n:79 [in hydras.MoreAck.Ack]
+n:79 [in gaia_hydras.T1Bridge]
+n:79 [in hydras.Ackermann.codeList]
+n:79 [in gaia_hydras.nfwfgaia]
+n:795 [in gaia_hydras.nfwfgaia]
+n:796 [in hydras.Ackermann.checkPrf]
+n:8 [in hydras.Epsilon0.T1]
+n:8 [in hydras.MoreAck.expressibleExamples]
+n:8 [in hydras.solutions_exercises.FibonacciPR]
+n:8 [in hydras.Hydra.BigBattle]
+n:8 [in additions.Naive]
+n:8 [in hydras.Ackermann.codeList]
+n:80 [in hydras.Gamma0.T2]
+n:80 [in hydras.Epsilon0.F_alpha]
+n:80 [in hydras.MoreAck.AckNotPR]
+n:80 [in hydras.Ackermann.subProp]
+n:80 [in hydras.MoreAck.Ack]
+n:802 [in hydras.Ackermann.checkPrf]
+n:802 [in gaia_hydras.nfwfgaia]
+n:806 [in hydras.Ackermann.checkPrf]
+n:806 [in gaia_hydras.nfwfgaia]
+n:81 [in hydras.Hydra.O2H]
+n:81 [in hydras.Ackermann.LNN2LNT]
+n:81 [in additions.Pow_variant]
+n:81 [in hydras.Ackermann.model]
+n:81 [in hydras.Ackermann.codeList]
+n:81 [in hydras.Epsilon0.Hprime]
+n:81 [in gaia_hydras.nfwfgaia]
+n:81 [in hydras.Prelude.MoreVectors]
+n:810 [in hydras.Ackermann.checkPrf]
+n:817 [in hydras.Ackermann.primRec]
+n:818 [in hydras.Ackermann.primRec]
+n:82 [in Goedel.PRrepresentable]
+n:82 [in hydras.Epsilon0.F_alpha]
+n:82 [in hydras.Ackermann.folProp]
+n:82 [in gaia_hydras.T1Bridge]
+n:82 [in gaia_hydras.nfwfgaia]
+n:821 [in hydras.Ackermann.primRec]
+n:825 [in hydras.Ackermann.codeSubFormula]
+n:829 [in hydras.Ackermann.codeSubFormula]
+n:83 [in hydras.Epsilon0.T1]
+n:83 [in additions.Pow_variant]
+n:83 [in additions.Pow]
+n:83 [in hydras.Ackermann.folProp]
+n:83 [in hydras.MoreAck.Ack]
+n:83 [in hydras.Ackermann.codeList]
+n:83 [in gaia_hydras.nfwfgaia]
+n:834 [in gaia_hydras.nfwfgaia]
+n:836 [in hydras.Ackermann.codeSubFormula]
+n:839 [in hydras.Ackermann.codeSubFormula]
+n:84 [in hydras.Epsilon0.F_alpha]
+n:84 [in hydras.MoreAck.Ack]
+n:84 [in hydras.Epsilon0.Canon]
+n:844 [in hydras.Ackermann.checkPrf]
+n:844 [in gaia_hydras.nfwfgaia]
+n:85 [in hydras.Ackermann.subAll]
+n:85 [in hydras.Prelude.Iterates]
+n:85 [in additions.Pow_variant]
+n:85 [in additions.Pow]
+n:85 [in hydras.MoreAck.AckNotPR]
+n:85 [in hydras.Ackermann.subProp]
+n:85 [in Goedel.rosserPA]
+n:85 [in hydras.Ackermann.primRec]
+n:85 [in hydras.Ackermann.codeList]
+n:85 [in gaia_hydras.nfwfgaia]
+n:858 [in hydras.Ackermann.checkPrf]
+n:86 [in Goedel.PRrepresentable]
+n:86 [in hydras.Prelude.MoreLists]
+n:86 [in hydras.Epsilon0.F_alpha]
+n:86 [in hydras.Prelude.MoreVectors]
+n:862 [in gaia_hydras.nfwfgaia]
+n:863 [in hydras.Ackermann.checkPrf]
+n:867 [in hydras.Ackermann.codeSubFormula]
+n:867 [in gaia_hydras.nfwfgaia]
+n:87 [in hydras.Hydra.O2H]
+n:87 [in hydras.Ackermann.LNN2LNT]
+n:87 [in additions.Pow_variant]
+n:87 [in Goedel.rosserPA]
+n:87 [in hydras.Ackermann.codeList]
+n:870 [in hydras.Ackermann.codeSubFormula]
+n:873 [in hydras.Ackermann.codeSubFormula]
+n:877 [in hydras.Ackermann.codeSubFormula]
+n:88 [in hydras.Epsilon0.T1]
+n:88 [in hydras.Ackermann.codeList]
+n:88 [in gaia_hydras.nfwfgaia]
+n:881 [in hydras.Ackermann.codeSubFormula]
+n:89 [in hydras.Gamma0.T2]
+n:89 [in hydras.Ackermann.subAll]
+n:89 [in hydras.Ackermann.codeList]
+n:89 [in gaia_hydras.nfwfgaia]
+n:891 [in hydras.Ackermann.codeSubFormula]
+n:891 [in gaia_hydras.nfwfgaia]
+n:894 [in hydras.Ackermann.codeSubFormula]
+n:9 [in hydras.Ackermann.folProof]
+n:9 [in hydras.Schutte.Correctness_E0]
+n:9 [in hydras.Gamma0.T2]
+n:9 [in hydras.Prelude.More_Arith]
+n:9 [in hydras.Prelude.Iterates]
+n:9 [in hydras.Prelude.Compat815]
+n:9 [in hydras.Ackermann.LNN2LNT]
+n:9 [in Goedel.fixPoint]
+n:9 [in hydras.Epsilon0.Hessenberg]
+n:9 [in additions.More_on_positive]
+n:9 [in hydras.MoreAck.AckNotPR]
+n:9 [in hydras.Ackermann.codeSubFormula]
+n:9 [in hydras.Prelude.Fuel]
+n:9 [in hydras.solutions_exercises.MorePRExamples]
+n:9 [in hydras.Hydra.Hydra_Termination]
+n:9 [in hydras.Hydra.BigBattle]
+n:9 [in hydras.Ackermann.wellFormed]
+n:9 [in hydras.Epsilon0.Canon]
+n:9 [in additions.fib]
+n:9 [in hydras.solutions_exercises.F_3]
+n:9 [in gaia_hydras.nfwfgaia]
+n:9 [in hydras.Schutte.Countable]
+n:90 [in Goedel.PRrepresentable]
+n:90 [in hydras.Epsilon0.F_alpha]
+n:90 [in additions.Pow]
+n:90 [in hydras.MoreAck.AckNotPR]
+n:90 [in hydras.Ackermann.primRec]
+n:90 [in hydras.Ackermann.folLogic3]
+n:90 [in hydras.Prelude.MoreVectors]
+n:904 [in gaia_hydras.nfwfgaia]
+n:907 [in hydras.Ackermann.codeSubFormula]
+n:91 [in Goedel.rosserPA]
+n:91 [in hydras.Epsilon0.E0]
+n:91 [in hydras.Ackermann.codeList]
+n:91 [in hydras.Hydra.Hydra_Definitions]
+n:910 [in hydras.Ackermann.codeSubFormula]
+n:912 [in gaia_hydras.nfwfgaia]
+n:913 [in hydras.Ackermann.codeSubFormula]
+n:918 [in hydras.Ackermann.codeSubFormula]
+n:92 [in hydras.Gamma0.T2]
+n:92 [in hydras.Epsilon0.F_alpha]
+n:92 [in hydras.MoreAck.AckNotPR]
+n:92 [in hydras.Ackermann.fol]
+n:92 [in hydras.Ackermann.folLogic3]
+n:92 [in Goedel.codeSysPrf]
+n:923 [in hydras.Ackermann.codeSubFormula]
+n:93 [in hydras.Epsilon0.T1]
+n:93 [in hydras.Ackermann.LNN2LNT]
+n:93 [in additions.Pow]
+n:93 [in hydras.Ackermann.codeList]
+n:93 [in hydras.Prelude.MoreVectors]
+n:934 [in hydras.Ackermann.codeSubFormula]
+n:936 [in gaia_hydras.nfwfgaia]
+n:937 [in hydras.Ackermann.codeSubFormula]
+n:94 [in Goedel.PRrepresentable]
+n:94 [in hydras.MoreAck.AckNotPR]
+n:94 [in hydras.Ackermann.primRec]
+n:941 [in gaia_hydras.nfwfgaia]
+n:946 [in gaia_hydras.nfwfgaia]
+n:95 [in hydras.Ackermann.subAll]
+n:95 [in hydras.Ackermann.fol]
+n:95 [in hydras.Ackermann.codeList]
+n:95 [in hydras.MoreAck.PrimRecExamples]
+n:950 [in gaia_hydras.nfwfgaia]
+n:952 [in gaia_hydras.nfwfgaia]
+n:96 [in additions.Pow]
+n:96 [in Goedel.rosserPA]
+n:96 [in hydras.Ackermann.folLogic3]
+n:96 [in hydras.Hydra.Hydra_Definitions]
+n:967 [in gaia_hydras.nfwfgaia]
+n:969 [in gaia_hydras.nfwfgaia]
+n:97 [in hydras.Prelude.MoreLists]
+n:97 [in hydras.Epsilon0.F_alpha]
+n:97 [in additions.Pow_variant]
+n:97 [in hydras.MoreAck.AckNotPR]
+n:97 [in hydras.Ackermann.subProp]
+n:97 [in hydras.Ackermann.primRec]
+n:97 [in hydras.Ackermann.codeList]
+n:97 [in gaia_hydras.nfwfgaia]
+n:971 [in gaia_hydras.nfwfgaia]
+n:975 [in gaia_hydras.nfwfgaia]
+n:979 [in gaia_hydras.nfwfgaia]
+n:98 [in hydras.Epsilon0.T1]
+n:98 [in hydras.Gamma0.T2]
+n:98 [in hydras.Ackermann.codeList]
+n:98 [in hydras.Epsilon0.Hprime]
+n:985 [in gaia_hydras.nfwfgaia]
+n:987 [in gaia_hydras.nfwfgaia]
+n:99 [in hydras.Epsilon0.F_alpha]
+n:99 [in hydras.Ackermann.subProp]
+n:99 [in hydras.Ackermann.codeList]
+n:99 [in hydras.Prelude.MoreVectors]
+

O

+OA:109 [in hydras.OrdinalNotations.ON_Generic]
+OA:124 [in hydras.OrdinalNotations.ON_Generic]
+OA:136 [in hydras.OrdinalNotations.ON_Generic]
+OA:149 [in hydras.OrdinalNotations.ON_Generic]
+OA:162 [in hydras.OrdinalNotations.ON_Generic]
+OA:174 [in hydras.OrdinalNotations.ON_Generic]
+OA:187 [in hydras.OrdinalNotations.ON_Generic]
+OA:201 [in hydras.OrdinalNotations.ON_Generic]
+OA:32 [in hydras.OrdinalNotations.ON_Generic]
+OA:5 [in hydras.OrdinalNotations.ON_O]
+OA:51 [in hydras.OrdinalNotations.ON_Generic]
+OA:70 [in hydras.OrdinalNotations.ON_Generic]
+OA:84 [in hydras.OrdinalNotations.ON_Generic]
+OA:96 [in hydras.OrdinalNotations.ON_Generic]
+OB:100 [in hydras.OrdinalNotations.ON_Generic]
+OB:113 [in hydras.OrdinalNotations.ON_Generic]
+OB:128 [in hydras.OrdinalNotations.ON_Generic]
+OB:140 [in hydras.OrdinalNotations.ON_Generic]
+OB:153 [in hydras.OrdinalNotations.ON_Generic]
+OB:166 [in hydras.OrdinalNotations.ON_Generic]
+OB:178 [in hydras.OrdinalNotations.ON_Generic]
+OB:191 [in hydras.OrdinalNotations.ON_Generic]
+OB:205 [in hydras.OrdinalNotations.ON_Generic]
+OB:36 [in hydras.OrdinalNotations.ON_Generic]
+OB:55 [in hydras.OrdinalNotations.ON_Generic]
+OB:88 [in hydras.OrdinalNotations.ON_Generic]
+ol'':40 [in hydras.Schutte.CNF]
+ol'':45 [in hydras.Schutte.CNF]
+ol':30 [in hydras.Schutte.CNF]
+ol':34 [in hydras.Schutte.CNF]
+ol':37 [in hydras.Schutte.CNF]
+ol':39 [in hydras.Schutte.CNF]
+ol':43 [in hydras.Schutte.CNF]
+ol:29 [in hydras.Schutte.CNF]
+ol:33 [in hydras.Schutte.CNF]
+ol:35 [in hydras.Schutte.CNF]
+ol:38 [in hydras.Schutte.CNF]
+ol:41 [in hydras.Schutte.CNF]
+one:100 [in additions.AM]
+one:103 [in additions.Addition_Chains]
+one:105 [in additions.AM]
+one:110 [in additions.AM]
+one:111 [in additions.Addition_Chains]
+one:117 [in additions.Addition_Chains]
+one:120 [in additions.AM]
+one:124 [in additions.Addition_Chains]
+one:125 [in additions.AM]
+one:131 [in additions.Addition_Chains]
+one:161 [in additions.Addition_Chains]
+one:168 [in additions.Addition_Chains]
+one:174 [in additions.Addition_Chains]
+one:36 [in additions.FirstSteps]
+one:37 [in additions.AM]
+one:42 [in additions.FirstSteps]
+one:43 [in additions.AM]
+one:54 [in additions.AM]
+one:59 [in additions.Monoid_def]
+one:61 [in additions.AM]
+one:63 [in additions.Monoid_def]
+one:74 [in additions.AM]
+one:8 [in additions.Monoid_def]
+one:84 [in additions.AM]
+one:95 [in additions.AM]
+on:10 [in hydras.OrdinalNotations.ON_Generic]
+on:14 [in hydras.OrdinalNotations.ON_Generic]
+On:4 [in hydras.solutions_exercises.lt_succ_le]
+On:5 [in hydras.solutions_exercises.predSuccUnicity]
+opA:194 [in hydras.OrdinalNotations.ON_Generic]
+opB:195 [in hydras.OrdinalNotations.ON_Generic]
+oplus_aux:8 [in hydras.Epsilon0.Hessenberg]
+oplus_aux:14 [in gaia_hydras.GHessenberg]
+op:100 [in additions.Euclidean_Chains]
+op:102 [in additions.fib]
+op:102 [in additions.Addition_Chains]
+op:104 [in additions.AM]
+op:109 [in additions.AM]
+op:109 [in additions.fib]
+op:110 [in additions.Addition_Chains]
+op:111 [in additions.Euclidean_Chains]
+op:116 [in additions.Addition_Chains]
+op:117 [in additions.fib]
+op:119 [in additions.AM]
+op:123 [in additions.Addition_Chains]
+op:124 [in additions.AM]
+op:130 [in additions.Addition_Chains]
+op:160 [in additions.Addition_Chains]
+op:163 [in additions.Euclidean_Chains]
+op:167 [in additions.Addition_Chains]
+op:169 [in additions.Euclidean_Chains]
+op:173 [in additions.Addition_Chains]
+op:183 [in additions.Euclidean_Chains]
+op:195 [in additions.Euclidean_Chains]
+op:36 [in additions.AM]
+op:36 [in additions.Euclidean_Chains]
+op:42 [in additions.AM]
+op:42 [in additions.Addition_Chains]
+op:43 [in additions.Euclidean_Chains]
+op:48 [in additions.Euclidean_Chains]
+op:5 [in additions.Monoid_def]
+op:51 [in additions.Euclidean_Chains]
+op:53 [in additions.AM]
+op:54 [in additions.Euclidean_Chains]
+op:54 [in additions.Addition_Chains]
+op:58 [in additions.Monoid_def]
+op:60 [in additions.AM]
+op:62 [in additions.Monoid_def]
+op:62 [in additions.Euclidean_Chains]
+op:69 [in additions.Euclidean_Chains]
+op:7 [in additions.Monoid_def]
+op:73 [in additions.AM]
+op:83 [in additions.AM]
+op:89 [in additions.Euclidean_Chains]
+op:94 [in additions.AM]
+op:99 [in additions.AM]
+output:62 [in hydras.Ackermann.codeSubFormula]
+o':423 [in hydras.Gamma0.Gamma0]
+o':434 [in hydras.Gamma0.Gamma0]
+o':437 [in hydras.Gamma0.Gamma0]
+o':61 [in hydras.Epsilon0.Epsilon0rpo]
+o':64 [in hydras.Epsilon0.Epsilon0rpo]
+o1:15 [in hydras.rpo.rpo]
+o1:20 [in hydras.rpo.rpo]
+o1:6 [in hydras.rpo.rpo]
+o2:16 [in hydras.rpo.rpo]
+o2:21 [in hydras.rpo.rpo]
+o2:7 [in hydras.rpo.rpo]
+o:10 [in hydras.Hydra.O2H]
+o:124 [in hydras.Schutte.Schutte_basics]
+o:13 [in hydras.Hydra.O2H]
+o:131 [in hydras.Schutte.Ordering_Functions]
+o:18 [in hydras.Schutte.Addition]
+o:19 [in hydras.Schutte.Addition]
+o:280 [in hydras.Gamma0.Gamma0]
+o:36 [in hydras.Epsilon0.Epsilon0rpo]
+o:424 [in hydras.Gamma0.Gamma0]
+o:435 [in hydras.Gamma0.Gamma0]
+o:436 [in hydras.Gamma0.Gamma0]
+o:537 [in hydras.Epsilon0.T1]
+o:619 [in hydras.Gamma0.Gamma0]
+o:62 [in hydras.Epsilon0.Epsilon0rpo]
+o:628 [in hydras.Gamma0.Gamma0]
+o:63 [in hydras.Epsilon0.Epsilon0rpo]
+o:63 [in hydras.Schutte.CNF]
+o:64 [in hydras.Schutte.CNF]
+o:75 [in hydras.Schutte.Ordering_Functions]
+

P

+pair:134 [in Goedel.PRrepresentable]
+pair:145 [in Goedel.PRrepresentable]
+pair:156 [in Goedel.PRrepresentable]
+pair:239 [in Goedel.PRrepresentable]
+pair:245 [in Goedel.PRrepresentable]
+pair:460 [in Goedel.PRrepresentable]
+pair:468 [in Goedel.PRrepresentable]
+pA:50 [in hydras.Prelude.Sort_spec]
+PA:64 [in hydras.Prelude.Sort_spec]
+PB:5 [in hydras.Prelude.DecPreOrder_Instances]
+pB:51 [in hydras.Prelude.Sort_spec]
+PB:65 [in hydras.Prelude.Sort_spec]
+Pdec:3 [in hydras.Prelude.First_toggle]
+pf:39 [in hydras.MoreAck.FolExamples]
+pf:40 [in hydras.MoreAck.FolExamples]
+phi:198 [in hydras.Prelude.Iterates]
+phi:214 [in hydras.Prelude.Iterates]
+phi:29 [in hydras.MoreAck.FolExamples]
+phi:69 [in hydras.MoreAck.PrimRecExamples]
+phi:76 [in hydras.MoreAck.PrimRecExamples]
+phi:85 [in hydras.MoreAck.PrimRecExamples]
+pq:68 [in hydras.rpo.dickson]
+prf:11 [in hydras.Ackermann.Deduction]
+prf:121 [in hydras.Ackermann.LNN2LNT]
+prf:129 [in hydras.Ackermann.LNN2LNT]
+prf:45 [in hydras.Ackermann.code]
+prf:47 [in hydras.Ackermann.folProof]
+proof:202 [in hydras.rpo.rpo]
+proof:206 [in hydras.rpo.rpo]
+pr:120 [in Goedel.PRrepresentable]
+pr:633 [in hydras.Ackermann.primRec]
+pr:638 [in hydras.Ackermann.primRec]
+psi:215 [in hydras.Prelude.Iterates]
+P_dec:190 [in hydras.rpo.list_set]
+P_dec:67 [in hydras.rpo.list_set]
+P_dec:63 [in hydras.rpo.list_set]
+P_dec:59 [in hydras.rpo.list_set]
+P_dec:55 [in hydras.rpo.list_set]
+P_dec:52 [in hydras.rpo.list_set]
+P_dec:49 [in hydras.rpo.list_set]
+P_dec:45 [in hydras.rpo.list_set]
+P_dec:40 [in hydras.rpo.list_set]
+p':2 [in additions.More_on_positive]
+p':22 [in hydras.Prelude.More_Arith]
+p':40 [in hydras.Prelude.DecPreOrder_Instances]
+p':66 [in hydras.rpo.dickson]
+p':7 [in hydras.solutions_exercises.MinPR2]
+p0:395 [in additions.Euclidean_Chains]
+p0:402 [in additions.Euclidean_Chains]
+P0:46 [in hydras.Prelude.DecPreOrder]
+p0:838 [in hydras.Ackermann.checkPrf]
+p0:841 [in hydras.Ackermann.checkPrf]
+P0:89 [in hydras.Prelude.DecPreOrder]
+P0:95 [in hydras.Prelude.DecPreOrder]
+P1_dec:195 [in hydras.rpo.list_set]
+p1:109 [in hydras.Ackermann.codeList]
+p1:114 [in hydras.Ackermann.codeList]
+p1:117 [in hydras.Ackermann.codeList]
+p1:122 [in hydras.Ackermann.codeList]
+p1:125 [in hydras.Ackermann.codeList]
+p1:128 [in hydras.Ackermann.codeList]
+p1:1309 [in hydras.Ackermann.codeSubFormula]
+p1:131 [in hydras.Ackermann.codeList]
+p1:1313 [in hydras.Ackermann.codeSubFormula]
+p1:1315 [in hydras.Ackermann.codeSubFormula]
+p1:1318 [in hydras.Ackermann.codeSubFormula]
+p1:1320 [in hydras.Ackermann.codeSubFormula]
+p1:1323 [in hydras.Ackermann.codeSubFormula]
+p1:1326 [in hydras.Ackermann.codeSubFormula]
+p1:1329 [in hydras.Ackermann.codeSubFormula]
+p1:1332 [in hydras.Ackermann.codeSubFormula]
+p1:1335 [in hydras.Ackermann.codeSubFormula]
+p1:1338 [in hydras.Ackermann.codeSubFormula]
+p1:134 [in hydras.Ackermann.codeList]
+p1:1348 [in hydras.Ackermann.codeSubFormula]
+p1:1350 [in hydras.Ackermann.codeSubFormula]
+p1:1352 [in hydras.Ackermann.codeSubFormula]
+p1:1354 [in hydras.Ackermann.codeSubFormula]
+p1:1356 [in hydras.Ackermann.codeSubFormula]
+p1:1358 [in hydras.Ackermann.codeSubFormula]
+p1:137 [in hydras.Ackermann.codeList]
+p1:140 [in hydras.Ackermann.codeList]
+p1:143 [in hydras.Ackermann.codeList]
+p1:166 [in hydras.Ackermann.codeList]
+p1:169 [in hydras.Ackermann.codeList]
+p1:172 [in hydras.Ackermann.codeList]
+p1:175 [in hydras.Ackermann.codeList]
+p1:178 [in hydras.Ackermann.codeList]
+p1:181 [in hydras.Ackermann.codeList]
+p1:183 [in hydras.Ackermann.codeList]
+p1:185 [in hydras.Ackermann.codeList]
+p1:187 [in hydras.Ackermann.codeList]
+p1:189 [in hydras.Ackermann.codeList]
+p1:191 [in hydras.Ackermann.codeList]
+p1:193 [in hydras.Ackermann.codeList]
+P1:193 [in hydras.rpo.list_set]
+p1:214 [in hydras.Ackermann.codeList]
+p1:219 [in hydras.Ackermann.codeList]
+p1:222 [in hydras.Ackermann.codeList]
+p1:227 [in hydras.Ackermann.codeList]
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+p:714 [in hydras.Ackermann.checkPrf]
+P:714 [in hydras.Ackermann.primRec]
+p:715 [in hydras.Ackermann.checkPrf]
+P:715 [in hydras.Ackermann.primRec]
+p:716 [in hydras.Ackermann.checkPrf]
+p:716 [in gaia_hydras.nfwfgaia]
+p:717 [in hydras.Ackermann.checkPrf]
+p:718 [in hydras.Ackermann.checkPrf]
+p:719 [in hydras.Ackermann.checkPrf]
+p:72 [in hydras.Prelude.MoreLists]
+p:72 [in hydras.Ackermann.checkPrf]
+P:72 [in hydras.Prelude.DecPreOrder]
+p:720 [in hydras.Ackermann.checkPrf]
+p:721 [in hydras.Ackermann.checkPrf]
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+p:732 [in hydras.Ackermann.checkPrf]
+p:733 [in hydras.Ackermann.checkPrf]
+p:734 [in hydras.Epsilon0.T1]
+p:734 [in hydras.Ackermann.checkPrf]
+p:735 [in hydras.Ackermann.checkPrf]
+p:736 [in hydras.Ackermann.checkPrf]
+p:737 [in hydras.Ackermann.checkPrf]
+p:738 [in hydras.Ackermann.checkPrf]
+p:739 [in hydras.Ackermann.checkPrf]
+P:74 [in hydras.Epsilon0.Epsilon0rpo]
+P:74 [in hydras.rpo.more_list]
+p:74 [in additions.Pow]
+p:74 [in hydras.Schutte.Addition]
+p:740 [in hydras.Ackermann.checkPrf]
+p:741 [in hydras.Ackermann.checkPrf]
+p:742 [in hydras.Ackermann.checkPrf]
+p:743 [in hydras.Ackermann.checkPrf]
+p:744 [in hydras.Ackermann.checkPrf]
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+p:748 [in hydras.Ackermann.checkPrf]
+p:749 [in hydras.Ackermann.checkPrf]
+p:75 [in hydras.OrdinalNotations.ON_Omega2]
+p:75 [in additions.Pow_variant]
+p:75 [in hydras.Ackermann.checkPrf]
+p:75 [in hydras.Ackermann.codeSubFormula]
+p:750 [in hydras.Ackermann.checkPrf]
+p:751 [in hydras.Ackermann.checkPrf]
+p:752 [in hydras.Ackermann.checkPrf]
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+p:755 [in hydras.Ackermann.checkPrf]
+p:758 [in hydras.Ackermann.checkPrf]
+p:759 [in hydras.Ackermann.checkPrf]
+P:76 [in hydras.Prelude.Sort_spec]
+p:76 [in additions.Euclidean_Chains]
+p:76 [in hydras.Hydra.BigBattle]
+p:76 [in hydras.Epsilon0.Hprime]
+p:762 [in hydras.Ackermann.checkPrf]
+p:763 [in hydras.Ackermann.checkPrf]
+p:764 [in hydras.Ackermann.checkPrf]
+p:765 [in hydras.Ackermann.checkPrf]
+p:766 [in hydras.Ackermann.checkPrf]
+p:767 [in hydras.Ackermann.checkPrf]
+p:768 [in hydras.Ackermann.checkPrf]
+p:769 [in hydras.Ackermann.checkPrf]
+p:77 [in hydras.Prelude.MoreLists]
+p:77 [in hydras.Ackermann.checkPrf]
+p:77 [in additions.Addition_Chains]
+p:770 [in hydras.Ackermann.checkPrf]
+p:771 [in hydras.Ackermann.checkPrf]
+p:772 [in hydras.Ackermann.checkPrf]
+p:773 [in hydras.Ackermann.checkPrf]
+p:774 [in hydras.Ackermann.checkPrf]
+p:775 [in hydras.Ackermann.checkPrf]
+p:776 [in hydras.Ackermann.checkPrf]
+p:777 [in hydras.Ackermann.checkPrf]
+p:778 [in hydras.Ackermann.checkPrf]
+p:779 [in hydras.Ackermann.checkPrf]
+p:78 [in additions.Pow_variant]
+p:78 [in additions.Pow]
+P:78 [in hydras.Prelude.DecPreOrder]
+p:78 [in additions.Euclidean_Chains]
+p:78 [in hydras.Schutte.PartialFun]
+p:78 [in Goedel.codeSysPrf]
+p:78 [in hydras.Epsilon0.Hprime]
+p:780 [in hydras.Ackermann.checkPrf]
+p:781 [in hydras.Ackermann.checkPrf]
+p:782 [in hydras.Ackermann.checkPrf]
+p:783 [in hydras.Ackermann.checkPrf]
+p:784 [in hydras.Ackermann.checkPrf]
+p:785 [in hydras.Ackermann.checkPrf]
+p:79 [in hydras.Ackermann.checkPrf]
+p:79 [in additions.Addition_Chains]
+p:790 [in hydras.Ackermann.checkPrf]
+p:797 [in hydras.Ackermann.checkPrf]
+p:8 [in gaia_hydras.T1Choice]
+p:8 [in hydras.OrdinalNotations.ON_Omega2]
+p:8 [in additions.Fib2]
+p:8 [in additions.More_on_positive]
+P:8 [in hydras.Schutte.MoreEpsilonIota]
+P:8 [in hydras.Hydra.Hydra_Definitions]
+p:80 [in additions.AM]
+p:80 [in hydras.Prelude.MoreLists]
+p:80 [in hydras.Ackermann.folProp]
+p:803 [in hydras.Ackermann.checkPrf]
+p:81 [in hydras.Ackermann.checkPrf]
+p:81 [in additions.Pow]
+p:81 [in hydras.MoreAck.Ack]
+p:813 [in hydras.Ackermann.checkPrf]
+p:815 [in hydras.Ackermann.checkPrf]
+p:817 [in hydras.Ackermann.checkPrf]
+p:818 [in hydras.Ackermann.checkPrf]
+p:819 [in hydras.Ackermann.checkPrf]
+p:82 [in hydras.Ackermann.checkPrf]
+p:82 [in hydras.Ackermann.codeSubFormula]
+p:82 [in hydras.Epsilon0.Hprime]
+p:820 [in hydras.Ackermann.checkPrf]
+p:822 [in hydras.Ackermann.checkPrf]
+p:824 [in hydras.Ackermann.checkPrf]
+p:826 [in hydras.Ackermann.checkPrf]
+p:828 [in hydras.Ackermann.checkPrf]
+p:83 [in hydras.Ackermann.checkPrf]
+P:83 [in hydras.Prelude.DecPreOrder]
+p:830 [in hydras.Ackermann.checkPrf]
+p:832 [in hydras.Ackermann.checkPrf]
+p:834 [in hydras.Ackermann.checkPrf]
+p:837 [in hydras.Ackermann.checkPrf]
+p:84 [in hydras.Ackermann.checkPrf]
+p:84 [in additions.fib]
+p:848 [in hydras.Ackermann.checkPrf]
+p:85 [in hydras.Ackermann.checkPrf]
+p:85 [in hydras.Ackermann.codeSubFormula]
+p:85 [in Goedel.codeSysPrf]
+P:85 [in hydras.Prelude.MoreVectors]
+p:850 [in gaia_hydras.nfwfgaia]
+p:86 [in hydras.Ackermann.checkPrf]
+P:86 [in hydras.Prelude.DecPreOrder]
+p:86 [in hydras.Epsilon0.Large_Sets]
+p:86 [in hydras.MoreAck.Ack]
+p:86 [in additions.fib]
+p:861 [in hydras.Ackermann.checkPrf]
+p:866 [in hydras.Ackermann.checkPrf]
+p:867 [in hydras.Ackermann.checkPrf]
+p:869 [in hydras.Ackermann.checkPrf]
+p:87 [in hydras.Prelude.MoreLists]
+p:87 [in hydras.Ackermann.checkPrf]
+p:87 [in additions.Pow]
+p:872 [in hydras.Ackermann.checkPrf]
+p:875 [in hydras.Ackermann.checkPrf]
+p:879 [in hydras.Ackermann.checkPrf]
+p:88 [in additions.Pow_variant]
+p:88 [in hydras.Ackermann.checkPrf]
+p:88 [in hydras.rpo.rpo]
+p:882 [in hydras.Ackermann.checkPrf]
+p:89 [in hydras.Ackermann.checkPrf]
+P:89 [in hydras.Prelude.MoreVectors]
+p:9 [in hydras.MoreAck.expressibleExamples]
+p:9 [in additions.Strategies]
+p:9 [in hydras.Epsilon0.F_omega]
+p:9 [in hydras.solutions_exercises.FibonacciPR]
+p:9 [in additions.Naive]
+P:9 [in hydras.Prelude.STDPP_compat]
+p:9 [in additions.Dichotomy]
+p:90 [in hydras.Ackermann.checkPrf]
+p:90 [in hydras.Ackermann.codeSubFormula]
+p:90 [in Goedel.codeSysPrf]
+p:91 [in hydras.Ackermann.checkPrf]
+p:91 [in additions.Pow]
+p:91 [in additions.fib]
+p:92 [in hydras.Ackermann.checkPrf]
+p:93 [in hydras.Ackermann.checkPrf]
+p:937 [in gaia_hydras.nfwfgaia]
+p:94 [in hydras.Ackermann.checkPrf]
+p:94 [in additions.Pow]
+p:94 [in Goedel.codeSysPrf]
+p:95 [in hydras.Ackermann.checkPrf]
+p:951 [in gaia_hydras.nfwfgaia]
+p:953 [in gaia_hydras.nfwfgaia]
+p:96 [in hydras.Prelude.MoreLists]
+p:96 [in hydras.Ackermann.checkPrf]
+p:96 [in additions.fib]
+p:97 [in hydras.Ackermann.checkPrf]
+p:97 [in additions.Pow]
+p:99 [in hydras.Prelude.MoreLists]
+p:99 [in hydras.Ackermann.checkPrf]
+p:99 [in hydras.Ackermann.folProp]
+

Q

+q':11 [in hydras.Prelude.First_toggle]
+q':30 [in hydras.Prelude.More_Arith]
+q':36 [in hydras.Prelude.More_Arith]
+q':59 [in hydras.Prelude.More_Arith]
+q':67 [in hydras.rpo.dickson]
+q:10 [in additions.Strategies]
+q:100 [in hydras.Ackermann.codeSubFormula]
+q:101 [in hydras.Prelude.MoreLists]
+q:101 [in hydras.Ackermann.codeSubFormula]
+q:105 [in hydras.Ackermann.codeSubFormula]
+q:106 [in hydras.Ackermann.codeSubFormula]
+q:107 [in hydras.Ackermann.folProp]
+q:108 [in hydras.Ackermann.folProp]
+q:113 [in hydras.Ackermann.codeSubFormula]
+q:114 [in hydras.Ackermann.fol]
+Q:114 [in gaia_hydras.nfwfgaia]
+q:115 [in hydras.Ackermann.codeSubFormula]
+q:12 [in hydras.Prelude.First_toggle]
+q:12 [in hydras.MoreAck.AckNotPR]
+q:121 [in hydras.Ackermann.fol]
+q:126 [in hydras.Ackermann.fol]
+q:13 [in hydras.Prelude.First_toggle]
+q:130 [in hydras.Ackermann.folProp]
+q:131 [in hydras.Ackermann.fol]
+q:132 [in hydras.Ackermann.folProp]
+q:14 [in hydras.Prelude.First_toggle]
+q:14 [in additions.More_on_positive]
+q:14 [in additions.Trace_exercise]
+q:15 [in hydras.Prelude.More_Arith]
+q:15 [in hydras.Prelude.First_toggle]
+Q:15 [in hydras.Schutte.MoreEpsilonIota]
+Q:15 [in hydras.Schutte.PartialFun]
+q:16 [in additions.More_on_positive]
+Q:162 [in hydras.Ackermann.fol]
+q:17 [in hydras.Hydra.Omega2_Small]
+q:172 [in hydras.rpo.term]
+q:173 [in hydras.Ackermann.fol]
+q:174 [in hydras.Ackermann.fol]
+q:176 [in hydras.Ackermann.fol]
+q:18 [in additions.More_on_positive]
+q:187 [in hydras.Ackermann.fol]
+q:19 [in hydras.Prelude.MoreLists]
+q:193 [in hydras.Ackermann.fol]
+q:21 [in Goedel.rosser]
+q:22 [in hydras.Prelude.Simple_LexProd]
+q:23 [in additions.AM]
+q:24 [in Goedel.rosser]
+q:24 [in hydras.Prelude.Simple_LexProd]
+Q:240 [in hydras.Ackermann.fol]
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+q:259 [in hydras.Ackermann.fol]
+q:264 [in hydras.Ackermann.subAll]
+q:265 [in hydras.Ackermann.subAll]
+q:265 [in hydras.Ackermann.fol]
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+q:27 [in hydras.Schutte.GRelations]
+q:277 [in hydras.Ackermann.fol]
+q:279 [in hydras.Ackermann.fol]
+Q:280 [in gaia_hydras.nfwfgaia]
+q:286 [in hydras.Ackermann.fol]
+q:287 [in hydras.Ackermann.fol]
+q:29 [in hydras.Prelude.More_Arith]
+Q:290 [in hydras.Ackermann.fol]
+q:291 [in additions.Euclidean_Chains]
+q:296 [in additions.Euclidean_Chains]
+q:3 [in hydras.Prelude.LibHyps_Experiments]
+q:3 [in hydras.MoreAck.PrimRecExamples]
+q:30 [in hydras.MoreAck.Ack]
+q:301 [in additions.Euclidean_Chains]
+q:305 [in hydras.Ackermann.fol]
+q:308 [in hydras.Ackermann.fol]
+q:309 [in hydras.Ackermann.fol]
+q:31 [in hydras.Schutte.GRelations]
+q:315 [in hydras.Ackermann.fol]
+q:316 [in hydras.Ackermann.fol]
+q:317 [in additions.Euclidean_Chains]
+q:32 [in Goedel.rosserPA]
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+q:329 [in hydras.Ackermann.fol]
+q:332 [in hydras.Ackermann.fol]
+Q:336 [in hydras.Ackermann.fol]
+q:34 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+q:35 [in hydras.Prelude.More_Arith]
+q:35 [in Goedel.rosserPA]
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+q:43 [in hydras.Prelude.More_Arith]
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+Q:447 [in hydras.Gamma0.Gamma0]
+q:46 [in hydras.Prelude.More_Arith]
+q:48 [in additions.More_on_positive]
+q:49 [in additions.FirstSteps]
+Q:504 [in gaia_hydras.nfwfgaia]
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+q:95 [in hydras.Ackermann.folProp]
+q:97 [in hydras.Ackermann.codeSubFormula]
+q:98 [in hydras.Prelude.MoreLists]
+

R

+RDB:19 [in hydras.Prelude.DecPreOrder_Instances]
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+recs:115 [in hydras.Ackermann.codeFreeVar]
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+recs:119 [in hydras.Ackermann.codeFreeVar]
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+recs:121 [in hydras.Ackermann.wellFormed]
+recs:122 [in hydras.Ackermann.checkPrf]
+recs:122 [in hydras.Ackermann.codePA]
+recs:1221 [in hydras.Ackermann.codeSubFormula]
+recs:1224 [in hydras.Ackermann.codeSubFormula]
+recs:123 [in hydras.Ackermann.wellFormed]
+recs:124 [in hydras.Ackermann.checkPrf]
+recs:124 [in hydras.Ackermann.codePA]
+recs:1240 [in hydras.Ackermann.codeSubFormula]
+recs:1243 [in hydras.Ackermann.codeSubFormula]
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S

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+

T

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+T:44 [in hydras.Ackermann.folProp]
+t:441 [in hydras.Ackermann.folProp]
+T:45 [in hydras.Ackermann.folLogic]
+t:46 [in hydras.rpo.term]
+t:46 [in hydras.Ackermann.codeFreeVar]
+t:47 [in hydras.rpo.term]
+T:47 [in hydras.Ackermann.LNN]
+T:47 [in hydras.Ackermann.LNT]
+t:47 [in hydras.Ackermann.wellFormed]
+t:474 [in hydras.Epsilon0.Paths]
+t:48 [in hydras.Ackermann.codeFreeVar]
+t:48 [in hydras.Ackermann.subProp]
+t:48 [in hydras.Ackermann.fol]
+T:48 [in hydras.Ackermann.folLogic]
+T:48 [in hydras.Ackermann.folReplace]
+T:48 [in hydras.MoreAck.FolExamples]
+T:49 [in hydras.Ackermann.folProof]
+t:49 [in hydras.Ackermann.folProp]
+t:5 [in hydras.Ackermann.codeFreeVar]
+t:5 [in hydras.Ackermann.codeSubTerm]
+t:50 [in hydras.Ackermann.codeFreeVar]
+T:50 [in hydras.Ackermann.LNN]
+t:50 [in hydras.Ackermann.model]
+T:51 [in hydras.Ackermann.folProof]
+T:51 [in hydras.Ackermann.LNT]
+t:51 [in hydras.Ackermann.fol]
+T:51 [in hydras.Ackermann.folLogic]
+t:52 [in hydras.Ackermann.codeFreeVar]
+t:52 [in hydras.Ackermann.codeSubTerm]
+t:52 [in hydras.Ackermann.wellFormed]
+T:53 [in hydras.Ackermann.folProof]
+t:53 [in hydras.Ackermann.folProp]
+T:53 [in hydras.Ackermann.folReplace]
+t:534 [in hydras.Epsilon0.Paths]
+t:54 [in hydras.Ackermann.codeFreeVar]
+T:54 [in hydras.Ackermann.LNN]
+T:54 [in hydras.Ackermann.folLogic]
+t:543 [in hydras.Epsilon0.Paths]
+T:55 [in hydras.Ackermann.LNT]
+t:55 [in hydras.Ackermann.wellFormed]
+t:55 [in hydras.Ackermann.fol]
+t:56 [in hydras.Ackermann.codeFreeVar]
+t:56 [in hydras.Ackermann.codeSubTerm]
+T:56 [in hydras.Ackermann.folLogic3]
+t:57 [in hydras.Ackermann.wellFormed]
+T:57 [in hydras.Ackermann.folLogic]
+t:57 [in hydras.rpo.rpo]
+t:58 [in hydras.Ackermann.subAll]
+t:58 [in hydras.Ackermann.codeFreeVar]
+T:58 [in hydras.Ackermann.LNN]
+T:58 [in hydras.Ackermann.LNT]
+T:58 [in hydras.Ackermann.folReplace]
+t:59 [in hydras.rpo.rpo]
+T:6 [in hydras.Ackermann.folLogic2]
+T:6 [in hydras.Ackermann.wConsistent]
+T:6 [in hydras.Ackermann.folReplace]
+t:60 [in hydras.Ackermann.codeFreeVar]
+t:60 [in hydras.Ackermann.codeSubTerm]
+t:60 [in hydras.Ackermann.wellFormed]
+T:60 [in hydras.Ackermann.folLogic]
+t:61 [in hydras.Ackermann.codeFreeVar]
+T:61 [in hydras.Ackermann.LNN]
+T:61 [in hydras.Ackermann.LNT]
+t:61 [in hydras.rpo.rpo]
+t:62 [in hydras.Ackermann.codeFreeVar]
+t:63 [in hydras.Ackermann.codeFreeVar]
+T:63 [in hydras.Ackermann.folLogic]
+T:63 [in hydras.Ackermann.folReplace]
+t:64 [in hydras.Ackermann.codeFreeVar]
+t:64 [in hydras.Ackermann.codeSubTerm]
+T:64 [in hydras.Ackermann.LNN]
+T:64 [in hydras.Ackermann.LNT]
+t:64 [in hydras.Ackermann.wellFormed]
+t:65 [in hydras.Ackermann.codeFreeVar]
+T:65 [in hydras.Ackermann.folLogic3]
+T:65 [in Goedel.rosser]
+t:66 [in hydras.Ackermann.codeFreeVar]
+t:66 [in hydras.Ackermann.wellFormed]
+T:66 [in hydras.Ackermann.folLogic]
+t:67 [in hydras.Ackermann.codeFreeVar]
+T:67 [in hydras.Ackermann.LNN]
+T:67 [in hydras.Ackermann.LNT]
+T:67 [in hydras.Ackermann.folReplace]
+t:68 [in hydras.Ackermann.codeFreeVar]
+t:68 [in hydras.Ackermann.codeSubTerm]
+T:68 [in hydras.Ackermann.LNN2LNT]
+t:68 [in hydras.Ackermann.wellFormed]
+t:69 [in hydras.Gamma0.T2]
+t:69 [in hydras.Ackermann.codeFreeVar]
+t:69 [in hydras.Ackermann.folLogic]
+t:692 [in hydras.Ackermann.codeSubFormula]
+t:695 [in hydras.Ackermann.codeSubFormula]
+t:698 [in hydras.Ackermann.codeSubFormula]
+t:7 [in hydras.Ackermann.codeFreeVar]
+T:7 [in hydras.Ackermann.folLogic]
+T:7 [in hydras.MoreAck.FolExamples]
+t:70 [in hydras.Schutte.Correctness_E0]
+t:70 [in hydras.Ackermann.codeFreeVar]
+T:70 [in hydras.Ackermann.LNN]
+T:70 [in hydras.Ackermann.LNT]
+t:70 [in hydras.Ackermann.wellFormed]
+T:70 [in hydras.Ackermann.folLogic]
+t:701 [in hydras.Ackermann.codeSubFormula]
+t:704 [in hydras.Ackermann.codeSubFormula]
+t:707 [in hydras.Ackermann.codeSubFormula]
+t:71 [in hydras.Ackermann.model]
+t:71 [in hydras.Ackermann.subProp]
+T:71 [in hydras.Ackermann.folReplace]
+t:710 [in hydras.Ackermann.codeSubFormula]
+t:713 [in hydras.Ackermann.codeSubFormula]
+t:716 [in hydras.Ackermann.codeSubFormula]
+t:719 [in hydras.Ackermann.codeSubFormula]
+t:72 [in hydras.Ackermann.codeFreeVar]
+t:72 [in hydras.Ackermann.codeSubTerm]
+t:72 [in hydras.Ackermann.wellFormed]
+t:722 [in hydras.Ackermann.codeSubFormula]
+t:725 [in hydras.Ackermann.codeSubFormula]
+t:727 [in hydras.Ackermann.codeSubFormula]
+t:729 [in hydras.Ackermann.codeSubFormula]
+t:73 [in hydras.Schutte.Correctness_E0]
+t:73 [in hydras.Ackermann.subAll]
+T:73 [in hydras.Ackermann.LNN]
+T:73 [in hydras.Ackermann.LNT]
+T:73 [in hydras.Ackermann.folLogic]
+t:731 [in hydras.Ackermann.codeSubFormula]
+t:733 [in hydras.Ackermann.codeSubFormula]
+t:735 [in hydras.Ackermann.codeSubFormula]
+t:737 [in hydras.Ackermann.codeSubFormula]
+t:739 [in hydras.Ackermann.codeSubFormula]
+t:74 [in hydras.Ackermann.codeFreeVar]
+t:74 [in hydras.Ackermann.wellFormed]
+T:74 [in Goedel.rosser]
+t:741 [in hydras.Ackermann.codeSubFormula]
+t:743 [in hydras.Ackermann.codeSubFormula]
+t:745 [in hydras.Ackermann.codeSubFormula]
+t:747 [in hydras.Ackermann.codeSubFormula]
+t:749 [in hydras.Ackermann.codeSubFormula]
+T:75 [in hydras.Ackermann.folLogic3]
+t:750 [in hydras.Ackermann.codeSubFormula]
+t:751 [in hydras.Ackermann.codeSubFormula]
+t:752 [in hydras.Ackermann.codeSubFormula]
+t:753 [in hydras.Ackermann.codeSubFormula]
+t:754 [in hydras.Ackermann.codeSubFormula]
+t:755 [in hydras.Epsilon0.T1]
+t:76 [in hydras.Schutte.Correctness_E0]
+t:76 [in hydras.Ackermann.codeFreeVar]
+t:76 [in hydras.Ackermann.codeSubTerm]
+T:76 [in hydras.Ackermann.LNN]
+T:76 [in hydras.Ackermann.LNT]
+t:76 [in hydras.Ackermann.wellFormed]
+t:76 [in hydras.Ackermann.folLogic]
+T:76 [in hydras.Ackermann.folReplace]
+t:76 [in hydras.rpo.rpo]
+t:761 [in hydras.Ackermann.codeSubFormula]
+t:763 [in hydras.Ackermann.codeSubFormula]
+t:765 [in hydras.Ackermann.codeSubFormula]
+t:767 [in hydras.Ackermann.codeSubFormula]
+t:769 [in hydras.Ackermann.codeSubFormula]
+T:77 [in hydras.Ackermann.folLogic]
+t:771 [in hydras.Ackermann.codeSubFormula]
+t:773 [in hydras.Ackermann.codeSubFormula]
+t:775 [in hydras.Ackermann.codeSubFormula]
+t:777 [in hydras.Ackermann.codeSubFormula]
+t:779 [in hydras.Ackermann.codeSubFormula]
+t:78 [in hydras.Ackermann.codeFreeVar]
+t:78 [in hydras.Ackermann.wellFormed]
+t:782 [in hydras.Ackermann.codeSubFormula]
+t:785 [in hydras.Ackermann.codeSubFormula]
+t:787 [in hydras.Ackermann.codeSubFormula]
+t:789 [in hydras.Ackermann.codeSubFormula]
+T:79 [in hydras.Ackermann.LNN]
+t:79 [in hydras.Ackermann.LNT]
+t:791 [in hydras.Ackermann.codeSubFormula]
+t:793 [in hydras.Ackermann.codeSubFormula]
+t:795 [in hydras.Ackermann.codeSubFormula]
+t:797 [in hydras.Ackermann.codeSubFormula]
+t:799 [in hydras.Ackermann.codeSubFormula]
+t:8 [in hydras.Ackermann.codeFreeVar]
+t:8 [in hydras.MoreAck.AckNotPR]
+t:8 [in hydras.Ackermann.PA]
+T:8 [in hydras.Ackermann.Deduction]
+t:80 [in hydras.Ackermann.codeFreeVar]
+t:80 [in hydras.Ackermann.codeSubTerm]
+T:80 [in hydras.Ackermann.LNT]
+t:80 [in hydras.Ackermann.wellFormed]
+t:80 [in hydras.rpo.rpo]
+t:800 [in hydras.Ackermann.codeSubFormula]
+t:801 [in hydras.Ackermann.codeSubFormula]
+t:802 [in hydras.Ackermann.codeSubFormula]
+t:803 [in hydras.Ackermann.codeSubFormula]
+t:804 [in hydras.Ackermann.codeSubFormula]
+t:805 [in hydras.Ackermann.codeSubFormula]
+t:806 [in hydras.Ackermann.codeSubFormula]
+t:807 [in hydras.Ackermann.codeSubFormula]
+t:809 [in hydras.Ackermann.codeSubFormula]
+T:81 [in hydras.Ackermann.folLogic]
+T:81 [in hydras.Ackermann.folReplace]
+t:811 [in hydras.Ackermann.codeSubFormula]
+t:811 [in gaia_hydras.nfwfgaia]
+t:813 [in hydras.Ackermann.codeSubFormula]
+t:815 [in hydras.Ackermann.codeSubFormula]
+t:817 [in hydras.Ackermann.codeSubFormula]
+t:82 [in hydras.Ackermann.codeFreeVar]
+t:82 [in hydras.Ackermann.LNN]
+t:82 [in hydras.Ackermann.wellFormed]
+t:83 [in hydras.rpo.more_list]
+T:83 [in hydras.Ackermann.LNN]
+T:83 [in hydras.Ackermann.LNT]
+t:83 [in hydras.Ackermann.codeSubFormula]
+t:830 [in hydras.Ackermann.codeSubFormula]
+t:833 [in hydras.Ackermann.codeSubFormula]
+t:84 [in hydras.rpo.term]
+t:84 [in hydras.Ackermann.codeFreeVar]
+t:84 [in hydras.Ackermann.codeSubTerm]
+t:84 [in hydras.Ackermann.wellFormed]
+T:84 [in hydras.Ackermann.folLogic]
+t:84 [in hydras.rpo.rpo]
+t:85 [in additions.Addition_Chains]
+t:858 [in hydras.Ackermann.codeSubFormula]
+t:86 [in hydras.Ackermann.codeFreeVar]
+T:86 [in hydras.Ackermann.LNN]
+t:86 [in hydras.Ackermann.LNT]
+t:86 [in hydras.Ackermann.wellFormed]
+T:86 [in hydras.Ackermann.folReplace]
+T:87 [in hydras.Ackermann.LNT]
+t:87 [in hydras.Ackermann.folLogic3]
+t:88 [in hydras.Ackermann.codeFreeVar]
+t:88 [in hydras.Ackermann.codeSubTerm]
+t:88 [in hydras.Ackermann.wellFormed]
+T:88 [in hydras.Ackermann.folLogic]
+t:88 [in hydras.Gamma0.Gamma0]
+t:89 [in hydras.Ackermann.LNN]
+t:89 [in hydras.Ackermann.codeSubFormula]
+T:89 [in hydras.Ackermann.folLogic3]
+t:89 [in additions.Addition_Chains]
+t:897 [in hydras.Ackermann.codeSubFormula]
+t:9 [in hydras.Ackermann.codeFreeVar]
+t:9 [in hydras.Ackermann.codeSubTerm]
+T:9 [in hydras.Ackermann.wConsistent]
+t:9 [in hydras.rpo.rpo]
+t:90 [in hydras.Ackermann.codeFreeVar]
+T:90 [in hydras.Ackermann.LNN]
+t:90 [in hydras.Ackermann.wellFormed]
+T:90 [in hydras.Ackermann.folLogic]
+t:90 [in hydras.Gamma0.Gamma0]
+t:91 [in hydras.rpo.term]
+t:91 [in hydras.Hydra.O2H]
+T:91 [in hydras.Ackermann.LNT]
+t:91 [in hydras.rpo.rpo]
+t:92 [in hydras.Ackermann.codeFreeVar]
+t:92 [in hydras.Ackermann.codeSubTerm]
+t:92 [in hydras.rpo.more_list]
+t:92 [in hydras.Ackermann.wellFormed]
+t:92 [in hydras.Gamma0.Gamma0]
+t:93 [in hydras.Ackermann.wellFormed]
+T:93 [in hydras.Ackermann.folLogic]
+T:93 [in hydras.Ackermann.folReplace]
+t:94 [in hydras.Ackermann.codeFreeVar]
+t:94 [in hydras.Ackermann.codeSubTerm]
+T:94 [in hydras.Ackermann.LNN]
+T:94 [in hydras.Ackermann.LNT]
+t:94 [in hydras.Ackermann.subProp]
+t:94 [in hydras.Ackermann.wellFormed]
+t:94 [in additions.Addition_Chains]
+T:94 [in hydras.Ackermann.folReplace]
+t:94 [in hydras.MoreAck.PrimRecExamples]
+t:94 [in hydras.rpo.rpo]
+t:95 [in hydras.rpo.term]
+t:95 [in hydras.Ackermann.wellFormed]
+t:96 [in hydras.Ackermann.subAll]
+t:96 [in hydras.Ackermann.codeFreeVar]
+t:96 [in hydras.Ackermann.codeSubTerm]
+t:96 [in hydras.Ackermann.wellFormed]
+T:96 [in hydras.Ackermann.folLogic]
+T:97 [in hydras.Ackermann.LNN]
+t:97 [in hydras.Ackermann.wellFormed]
+t:98 [in hydras.Ackermann.codeFreeVar]
+t:98 [in hydras.Ackermann.codeSubTerm]
+T:98 [in hydras.Ackermann.LNT]
+t:98 [in hydras.Ackermann.wellFormed]
+t:98 [in hydras.rpo.rpo]
+t:99 [in hydras.Ackermann.wellFormed]
+T:99 [in hydras.Ackermann.folLogic]
+T:99 [in hydras.Ackermann.folLogic3]
+T:99 [in hydras.Ackermann.folReplace]
+t:99 [in hydras.MoreAck.PrimRecExamples]
+

U

+ui:187 [in hydras.rpo.term]
+ui:325 [in hydras.rpo.rpo]
+ui:328 [in hydras.rpo.rpo]
+ui:369 [in hydras.rpo.term]
+u':693 [in gaia_hydras.nfwfgaia]
+u1:107 [in hydras.rpo.rpo]
+u1:114 [in hydras.rpo.rpo]
+u1:121 [in hydras.rpo.rpo]
+u1:89 [in hydras.rpo.list_permut]
+u1:97 [in hydras.rpo.list_permut]
+u2:109 [in hydras.rpo.rpo]
+u2:116 [in hydras.rpo.rpo]
+u2:122 [in hydras.rpo.rpo]
+u2:90 [in hydras.rpo.list_permut]
+u2:98 [in hydras.rpo.list_permut]
+u3:91 [in hydras.rpo.list_permut]
+u3:99 [in hydras.rpo.list_permut]
+u4:100 [in hydras.rpo.list_permut]
+u4:92 [in hydras.rpo.list_permut]
+u:1202 [in gaia_hydras.nfwfgaia]
+u:1213 [in gaia_hydras.nfwfgaia]
+u:123 [in hydras.rpo.rpo]
+u:1238 [in gaia_hydras.nfwfgaia]
+u:1242 [in gaia_hydras.nfwfgaia]
+u:1245 [in gaia_hydras.nfwfgaia]
+u:1265 [in gaia_hydras.nfwfgaia]
+u:1281 [in gaia_hydras.nfwfgaia]
+u:130 [in hydras.Epsilon0.Paths]
+U:136 [in hydras.Ackermann.LNN2LNT]
+u:136 [in hydras.Epsilon0.Paths]
+u:137 [in hydras.Epsilon0.Paths]
+u:143 [in hydras.Schutte.Schutte_basics]
+u:143 [in hydras.Epsilon0.Paths]
+u:147 [in hydras.rpo.term]
+U:15 [in hydras.Ackermann.LNT]
+u:150 [in hydras.rpo.term]
+u:150 [in hydras.Epsilon0.Paths]
+u:1512 [in gaia_hydras.nfwfgaia]
+u:1515 [in gaia_hydras.nfwfgaia]
+u:1518 [in gaia_hydras.nfwfgaia]
+u:155 [in hydras.Epsilon0.Paths]
+u:161 [in hydras.Schutte.Ordering_Functions]
+u:162 [in hydras.rpo.term]
+u:170 [in hydras.rpo.term]
+u:175 [in hydras.rpo.term]
+u:1788 [in gaia_hydras.nfwfgaia]
+U:18 [in hydras.Ackermann.LNN]
+u:180 [in hydras.rpo.term]
+u:183 [in hydras.rpo.rpo]
+u:189 [in hydras.rpo.term]
+u:196 [in hydras.rpo.rpo]
+u:2041 [in gaia_hydras.nfwfgaia]
+u:2052 [in gaia_hydras.nfwfgaia]
+u:2080 [in gaia_hydras.nfwfgaia]
+u:2085 [in gaia_hydras.nfwfgaia]
+U:24 [in hydras.Prelude.DecPreOrder_Instances]
+U:25 [in hydras.Prelude.DecPreOrder_Instances]
+u:273 [in hydras.rpo.term]
+u:276 [in hydras.rpo.term]
+u:326 [in hydras.rpo.term]
+u:328 [in hydras.rpo.term]
+U:34 [in hydras.Schutte.Countable]
+u:340 [in hydras.rpo.term]
+u:348 [in hydras.rpo.term]
+u:354 [in hydras.rpo.term]
+u:358 [in hydras.rpo.term]
+u:371 [in hydras.rpo.term]
+u:390 [in gaia_hydras.nfwfgaia]
+u:413 [in gaia_hydras.nfwfgaia]
+U:48 [in hydras.Schutte.Schutte_basics]
+U:5 [in hydras.Ackermann.folLogic]
+U:62 [in hydras.Schutte.AP]
+U:67 [in hydras.Schutte.PartialFun]
+u:679 [in gaia_hydras.nfwfgaia]
+u:686 [in gaia_hydras.nfwfgaia]
+u:688 [in gaia_hydras.nfwfgaia]
+U:69 [in hydras.Schutte.PartialFun]
+u:691 [in gaia_hydras.nfwfgaia]
+u:701 [in gaia_hydras.nfwfgaia]
+u:711 [in gaia_hydras.nfwfgaia]
+u:719 [in gaia_hydras.nfwfgaia]
+u:745 [in gaia_hydras.nfwfgaia]
+u:77 [in hydras.rpo.rpo]
+u:799 [in gaia_hydras.nfwfgaia]
+U:80 [in hydras.Schutte.PartialFun]
+u:81 [in hydras.rpo.rpo]
+u:810 [in gaia_hydras.nfwfgaia]
+u:814 [in gaia_hydras.nfwfgaia]
+U:86 [in hydras.Schutte.PartialFun]
+u:93 [in hydras.rpo.term]
+u:97 [in hydras.rpo.term]
+u:996 [in gaia_hydras.nfwfgaia]
+

V

+value:103 [in hydras.Ackermann.model]
+value:107 [in hydras.Ackermann.model]
+value:111 [in hydras.Ackermann.model]
+value:115 [in hydras.Ackermann.model]
+value:127 [in hydras.Ackermann.model]
+value:131 [in hydras.Ackermann.model]
+value:135 [in hydras.Ackermann.model]
+value:139 [in hydras.Ackermann.model]
+value:142 [in hydras.Ackermann.model]
+value:146 [in hydras.Ackermann.model]
+value:159 [in hydras.Ackermann.model]
+value:165 [in hydras.Ackermann.model]
+value:17 [in hydras.Ackermann.model]
+value:21 [in hydras.Ackermann.model]
+value:222 [in hydras.Ackermann.model]
+value:32 [in hydras.Ackermann.model]
+value:39 [in hydras.Ackermann.model]
+value:43 [in hydras.Ackermann.model]
+value:70 [in hydras.Ackermann.model]
+value:80 [in hydras.Ackermann.model]
+value:86 [in hydras.Ackermann.model]
+Var:11 [in hydras.Prelude.WfVariant]
+Var:159 [in hydras.Hydra.Hydra_Definitions]
+Var:27 [in gaia_hydras.GHydra]
+Var:3 [in hydras.Hydra.Epsilon0_Needed_Std]
+Var:31 [in gaia_hydras.GHydra]
+Var:34 [in gaia_hydras.GHydra]
+Var:4 [in hydras.Hydra.Epsilon0_Needed_Free]
+Var:5 [in hydras.Hydra.Epsilon0_Needed_Generic]
+vs:103 [in Goedel.PRrepresentable]
+vs:108 [in Goedel.PRrepresentable]
+vs:114 [in Goedel.PRrepresentable]
+v':694 [in gaia_hydras.nfwfgaia]
+v0:107 [in hydras.Ackermann.codeSubFormula]
+v0:1286 [in hydras.Ackermann.codeSubFormula]
+v0:1290 [in hydras.Ackermann.codeSubFormula]
+v0:20 [in hydras.Ackermann.codeSubTerm]
+v0:24 [in hydras.Ackermann.codeSubTerm]
+v0:42 [in hydras.Ackermann.codeSubTerm]
+v0:46 [in hydras.Ackermann.codeSubTerm]
+v1:102 [in hydras.Ackermann.subProp]
+v1:11 [in hydras.Epsilon0.Epsilon0rpo]
+v1:11 [in hydras.Ackermann.folLogic2]
+v1:110 [in hydras.Ackermann.subProp]
+v1:121 [in hydras.Ackermann.subAll]
+v1:122 [in hydras.Ackermann.subAll]
+v1:124 [in hydras.Ackermann.subProp]
+v1:129 [in hydras.Ackermann.subAll]
+v1:129 [in hydras.Ackermann.subProp]
+v1:130 [in hydras.Ackermann.subAll]
+v1:131 [in hydras.Ackermann.subAll]
+v1:132 [in hydras.Ackermann.subAll]
+v1:133 [in hydras.Ackermann.subAll]
+v1:134 [in hydras.Ackermann.subAll]
+v1:135 [in hydras.Ackermann.subAll]
+v1:136 [in hydras.Ackermann.subAll]
+v1:137 [in hydras.Ackermann.subAll]
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+v:63 [in hydras.Ackermann.expressible]
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+v:66 [in hydras.Ackermann.folReplace]
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+v:68 [in Goedel.codeSysPrf]
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+v:7 [in hydras.Ackermann.PA]
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+v:70 [in hydras.Prelude.MoreVectors]
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+v:72 [in hydras.Ackermann.folLogic]
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+v:75 [in hydras.Ackermann.subAll]
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+v:78 [in hydras.Ackermann.codeSubTerm]
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+v:78 [in hydras.Ackermann.LNT]
+v:79 [in Goedel.codeSysPrf]
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+v:8 [in hydras.Prelude.MoreVectors]
+v:80 [in hydras.Ackermann.folLogic]
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+v:803 [in gaia_hydras.nfwfgaia]
+v:805 [in gaia_hydras.nfwfgaia]
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+V:81 [in hydras.Schutte.PartialFun]
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+v:82 [in hydras.Ackermann.LNT]
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+v:82 [in Goedel.codeSysPrf]
+v:82 [in hydras.Prelude.MoreVectors]
+v:83 [in hydras.Ackermann.model]
+v:83 [in hydras.Ackermann.folLogic]
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+v:86 [in hydras.Ackermann.codeSubTerm]
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+v:87 [in hydras.Ackermann.subProp]
+V:87 [in hydras.Schutte.PartialFun]
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+v:879 [in hydras.Ackermann.codeSubFormula]
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+v:88 [in hydras.Ackermann.model]
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+

W

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+w:998 [in gaia_hydras.nfwfgaia]
+

X

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+x:1016 [in gaia_hydras.nfwfgaia]
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+x:1089 [in gaia_hydras.nfwfgaia]
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+x:1095 [in gaia_hydras.nfwfgaia]
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+x:11 [in hydras.Ackermann.Languages]
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+x:1119 [in gaia_hydras.nfwfgaia]
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+x:1139 [in gaia_hydras.nfwfgaia]
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+x:115 [in gaia_hydras.nfwfgaia]
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+x:119 [in gaia_hydras.nfwfgaia]
+x:1197 [in gaia_hydras.nfwfgaia]
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+x:12 [in hydras.Schutte.MoreEpsilonIota]
+x:12 [in additions.Wf_transparent]
+x:12 [in hydras.Prelude.MoreDecidable]
+x:12 [in hydras.Ackermann.folLogic3]
+x:12 [in hydras.Ackermann.wConsistent]
+x:12 [in Goedel.rosser]
+x:12 [in hydras.Schutte.Countable]
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+x:1206 [in gaia_hydras.nfwfgaia]
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+x:1217 [in gaia_hydras.nfwfgaia]
+x:1218 [in gaia_hydras.nfwfgaia]
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+x:1220 [in gaia_hydras.nfwfgaia]
+x:1222 [in gaia_hydras.nfwfgaia]
+x:1224 [in gaia_hydras.nfwfgaia]
+x:1229 [in gaia_hydras.nfwfgaia]
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+x:1250 [in gaia_hydras.nfwfgaia]
+x:1259 [in gaia_hydras.nfwfgaia]
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+x:1267 [in gaia_hydras.nfwfgaia]
+x:1268 [in gaia_hydras.nfwfgaia]
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+X:129 [in hydras.Ackermann.cPair]
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+x:13 [in hydras.Prelude.Iterates]
+x:13 [in additions.FirstSteps]
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+x:13 [in hydras.Ackermann.PAtheory]
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+x:13 [in additions.Naive]
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+x:1311 [in gaia_hydras.nfwfgaia]
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+x:134 [in hydras.Prelude.Merge_Sort]
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+x:138 [in hydras.Ackermann.codeFreeVar]
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+x:1394 [in gaia_hydras.nfwfgaia]
+x:1397 [in gaia_hydras.nfwfgaia]
+x:14 [in gaia_hydras.T1Choice]
+x:14 [in additions.AM]
+x:14 [in additions.Compatibility]
+x:14 [in Goedel.rosserPA]
+x:14 [in additions.Monoid_def]
+x:14 [in gaia_hydras.ON_gfinite]
+x:14 [in hydras.Ackermann.Deduction]
+X:14 [in hydras.Schutte.Schutte_basics]
+x:14 [in gaia_hydras.onType]
+x:14 [in hydras.Ackermann.NN]
+x:14 [in hydras.Schutte.Countable]
+x:140 [in hydras.Ackermann.codeFreeVar]
+x:140 [in hydras.Prelude.Merge_Sort]
+x:140 [in hydras.Schutte.Schutte_basics]
+x:140 [in gaia_hydras.nfwfgaia]
+x:1405 [in gaia_hydras.nfwfgaia]
+x:1409 [in hydras.Ackermann.codeSubFormula]
+x:1409 [in gaia_hydras.nfwfgaia]
+x:141 [in hydras.Ackermann.codeFreeVar]
+X:141 [in hydras.Schutte.Schutte_basics]
+x:141 [in gaia_hydras.nfwfgaia]
+x:1411 [in gaia_hydras.nfwfgaia]
+x:1414 [in hydras.Ackermann.codeSubFormula]
+x:1414 [in gaia_hydras.nfwfgaia]
+x:1416 [in hydras.Ackermann.codeSubFormula]
+x:142 [in hydras.Ackermann.codeFreeVar]
+x:142 [in gaia_hydras.nfwfgaia]
+x:1421 [in hydras.Ackermann.codeSubFormula]
+x:1426 [in hydras.Ackermann.codeSubFormula]
+x:143 [in hydras.Ackermann.codeFreeVar]
+x:1438 [in gaia_hydras.nfwfgaia]
+x:1439 [in gaia_hydras.nfwfgaia]
+x:144 [in hydras.Ackermann.codeFreeVar]
+x:144 [in hydras.Prelude.Merge_Sort]
+x:144 [in additions.Euclidean_Chains]
+X:144 [in hydras.Schutte.Schutte_basics]
+x:144 [in hydras.OrdinalNotations.ON_Generic]
+x:1440 [in gaia_hydras.nfwfgaia]
+x:145 [in hydras.Ackermann.codeFreeVar]
+x:1451 [in gaia_hydras.nfwfgaia]
+x:1452 [in gaia_hydras.nfwfgaia]
+x:1453 [in gaia_hydras.nfwfgaia]
+x:1454 [in gaia_hydras.nfwfgaia]
+x:1455 [in gaia_hydras.nfwfgaia]
+x:146 [in hydras.Ackermann.codeFreeVar]
+x:146 [in gaia_hydras.T1Bridge]
+X:146 [in hydras.Schutte.Schutte_basics]
+x:1467 [in gaia_hydras.nfwfgaia]
+x:147 [in hydras.Prelude.MoreLists]
+x:148 [in hydras.Prelude.Merge_Sort]
+x:148 [in gaia_hydras.T1Bridge]
+x:148 [in hydras.Ackermann.cPair]
+X:148 [in hydras.Schutte.Schutte_basics]
+x:148 [in gaia_hydras.nfwfgaia]
+x:1486 [in gaia_hydras.nfwfgaia]
+x:149 [in hydras.Epsilon0.Paths]
+x:15 [in hydras.Prelude.Iterates]
+x:15 [in additions.Compatibility]
+x:15 [in additions.Pow_variant]
+x:15 [in additions.Pow]
+x:15 [in hydras.Ackermann.PAtheory]
+x:15 [in hydras.MoreAck.AckNotPR]
+x:15 [in hydras.Ackermann.subProp]
+x:15 [in Goedel.rosserPA]
+x:15 [in hydras.Ackermann.cPair]
+x:15 [in gaia_hydras.ON_gfinite]
+X:15 [in hydras.Schutte.Well_Orders]
+x:15 [in additions.Addition_Chains]
+x:15 [in Goedel.rosser]
+x:150 [in gaia_hydras.nfwfgaia]
+x:151 [in gaia_hydras.nfwfgaia]
+x:152 [in hydras.Prelude.Merge_Sort]
+x:152 [in hydras.Ackermann.cPair]
+x:152 [in additions.Euclidean_Chains]
+x:152 [in hydras.Schutte.Schutte_basics]
+x:152 [in hydras.Epsilon0.Paths]
+x:153 [in hydras.Schutte.Schutte_basics]
+x:1539 [in gaia_hydras.nfwfgaia]
+x:154 [in hydras.Prelude.Merge_Sort]
+x:154 [in hydras.Schutte.Schutte_basics]
+x:154 [in hydras.Epsilon0.Paths]
+X:155 [in hydras.Prelude.MoreLists]
+X:155 [in hydras.Schutte.Schutte_basics]
+x:156 [in hydras.Ackermann.folProp]
+x:1567 [in gaia_hydras.nfwfgaia]
+x:157 [in additions.Euclidean_Chains]
+x:157 [in hydras.Schutte.Schutte_basics]
+x:157 [in hydras.OrdinalNotations.ON_Generic]
+x:1570 [in hydras.Ackermann.codeSubFormula]
+x:1571 [in gaia_hydras.nfwfgaia]
+x:1572 [in hydras.Ackermann.codeSubFormula]
+x:1574 [in hydras.Ackermann.codeSubFormula]
+x:1576 [in hydras.Ackermann.codeSubFormula]
+x:1578 [in hydras.Ackermann.codeSubFormula]
+X:158 [in hydras.Prelude.MoreLists]
+x:158 [in hydras.Ackermann.folProp]
+x:1580 [in hydras.Ackermann.codeSubFormula]
+x:1582 [in hydras.Ackermann.codeSubFormula]
+x:1584 [in hydras.Ackermann.codeSubFormula]
+x:1586 [in hydras.Ackermann.codeSubFormula]
+x:1588 [in hydras.Ackermann.codeSubFormula]
+x:159 [in hydras.Prelude.Merge_Sort]
+X:159 [in hydras.Schutte.Schutte_basics]
+x:1590 [in hydras.Ackermann.codeSubFormula]
+x:1592 [in hydras.Ackermann.codeSubFormula]
+x:1594 [in gaia_hydras.nfwfgaia]
+x:1597 [in gaia_hydras.nfwfgaia]
+x:16 [in gaia_hydras.T1Choice]
+x:16 [in hydras.Prelude.Iterates]
+x:16 [in hydras.Ackermann.PAtheory]
+x:16 [in hydras.Schutte.Critical]
+x:16 [in gaia_hydras.T1Bridge]
+x:16 [in additions.Monoid_def]
+x:16 [in hydras.Prelude.STDPP_compat]
+x:16 [in Goedel.goedel1]
+x:16 [in hydras.Ackermann.NN]
+x:16 [in hydras.Schutte.Countable]
+x:160 [in hydras.Ackermann.folProp]
+x:1600 [in gaia_hydras.nfwfgaia]
+x:161 [in Goedel.PRrepresentable]
+X:161 [in hydras.Prelude.MoreLists]
+X:161 [in hydras.Schutte.Schutte_basics]
+x:162 [in hydras.Ackermann.folProp]
+x:162 [in hydras.Ackermann.cPair]
+x:162 [in gaia_hydras.nfwfgaia]
+x:163 [in Goedel.PRrepresentable]
+x:163 [in hydras.Prelude.Merge_Sort]
+x:163 [in hydras.Ackermann.cPair]
+x:163 [in hydras.Schutte.Schutte_basics]
+x:163 [in gaia_hydras.nfwfgaia]
+x:163 [in hydras.Ackermann.codePA]
+x:164 [in Goedel.PRrepresentable]
+x:164 [in hydras.Ackermann.folProp]
+x:164 [in hydras.Schutte.Schutte_basics]
+x:164 [in gaia_hydras.nfwfgaia]
+x:164 [in hydras.Ackermann.codePA]
+x:165 [in hydras.Schutte.Schutte_basics]
+x:165 [in gaia_hydras.nfwfgaia]
+x:1655 [in gaia_hydras.nfwfgaia]
+x:1656 [in gaia_hydras.nfwfgaia]
+x:1658 [in gaia_hydras.nfwfgaia]
+x:166 [in Goedel.PRrepresentable]
+x:166 [in hydras.Ackermann.folProp]
+x:166 [in gaia_hydras.nfwfgaia]
+x:1663 [in gaia_hydras.nfwfgaia]
+x:1666 [in gaia_hydras.nfwfgaia]
+x:1668 [in gaia_hydras.nfwfgaia]
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+x:167 [in hydras.Prelude.Merge_Sort]
+x:167 [in gaia_hydras.nfwfgaia]
+x:1679 [in gaia_hydras.nfwfgaia]
+x:168 [in Goedel.PRrepresentable]
+x:168 [in hydras.Prelude.MoreLists]
+x:168 [in hydras.Ackermann.folProp]
+X:168 [in hydras.Schutte.Schutte_basics]
+x:1681 [in gaia_hydras.nfwfgaia]
+x:1682 [in gaia_hydras.nfwfgaia]
+x:1683 [in gaia_hydras.nfwfgaia]
+x:1684 [in gaia_hydras.nfwfgaia]
+x:1697 [in gaia_hydras.nfwfgaia]
+x:1698 [in gaia_hydras.nfwfgaia]
+x:17 [in additions.FirstSteps]
+x:17 [in hydras.Ackermann.PA]
+x:17 [in hydras.Ackermann.cPair]
+x:17 [in gaia_hydras.ON_gfinite]
+x:17 [in hydras.Ackermann.Deduction]
+x:17 [in hydras.Ackermann.NNtheory]
+x:17 [in hydras.OrdinalNotations.ON_Generic]
+x:17 [in gaia_hydras.onType]
+x:17 [in hydras.Ackermann.folLogic3]
+x:17 [in Goedel.goedel1]
+x:170 [in hydras.Schutte.Schutte_basics]
+x:1701 [in gaia_hydras.nfwfgaia]
+x:1704 [in gaia_hydras.nfwfgaia]
+x:1707 [in gaia_hydras.nfwfgaia]
+x:1708 [in gaia_hydras.nfwfgaia]
+x:1709 [in gaia_hydras.nfwfgaia]
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+x:171 [in hydras.Ackermann.codeFreeVar]
+x:171 [in hydras.Schutte.Schutte_basics]
+x:1710 [in gaia_hydras.nfwfgaia]
+x:1711 [in gaia_hydras.nfwfgaia]
+x:1712 [in gaia_hydras.nfwfgaia]
+x:1715 [in gaia_hydras.nfwfgaia]
+x:1716 [in gaia_hydras.nfwfgaia]
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+x:172 [in hydras.Schutte.Schutte_basics]
+x:1723 [in gaia_hydras.nfwfgaia]
+x:1724 [in gaia_hydras.nfwfgaia]
+x:1725 [in gaia_hydras.nfwfgaia]
+x:1726 [in gaia_hydras.nfwfgaia]
+x:1728 [in gaia_hydras.nfwfgaia]
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+x:1730 [in gaia_hydras.nfwfgaia]
+x:1732 [in gaia_hydras.nfwfgaia]
+x:1734 [in gaia_hydras.nfwfgaia]
+x:1744 [in gaia_hydras.nfwfgaia]
+x:1748 [in gaia_hydras.nfwfgaia]
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+x:175 [in additions.Euclidean_Chains]
+x:1755 [in gaia_hydras.nfwfgaia]
+x:1756 [in gaia_hydras.nfwfgaia]
+x:1761 [in gaia_hydras.nfwfgaia]
+x:1766 [in gaia_hydras.nfwfgaia]
+x:1767 [in gaia_hydras.nfwfgaia]
+x:177 [in Goedel.PRrepresentable]
+x:177 [in hydras.Prelude.Iterates]
+x:177 [in hydras.Ackermann.fol]
+x:1770 [in gaia_hydras.nfwfgaia]
+x:178 [in hydras.Ackermann.subAll]
+x:1780 [in gaia_hydras.nfwfgaia]
+x:1781 [in gaia_hydras.nfwfgaia]
+x:1787 [in gaia_hydras.nfwfgaia]
+x:179 [in Goedel.PRrepresentable]
+x:179 [in hydras.Ackermann.fol]
+x:1790 [in gaia_hydras.nfwfgaia]
+x:18 [in gaia_hydras.T1Choice]
+x:18 [in hydras.solutions_exercises.MinPR2]
+x:18 [in additions.Euclidean_Chains]
+x:18 [in hydras.Ackermann.Deduction]
+x:18 [in hydras.Schutte.PartialFun]
+x:18 [in hydras.Prelude.MoreDecidable]
+x:18 [in hydras.Ackermann.NN]
+x:18 [in hydras.Schutte.Countable]
+X:18 [in hydras.Prelude.MoreVectors]
+x:180 [in hydras.Prelude.Iterates]
+x:180 [in hydras.Ackermann.fol]
+x:181 [in Goedel.PRrepresentable]
+x:181 [in hydras.Ackermann.subAll]
+x:181 [in hydras.Prelude.Iterates]
+x:181 [in hydras.Ackermann.fol]
+X:182 [in hydras.Prelude.MoreLists]
+x:183 [in Goedel.PRrepresentable]
+x:183 [in hydras.Prelude.Iterates]
+x:183 [in hydras.OrdinalNotations.ON_Generic]
+x:184 [in hydras.Ackermann.subAll]
+x:184 [in hydras.Ackermann.folProp]
+x:1846 [in gaia_hydras.nfwfgaia]
+x:185 [in Goedel.PRrepresentable]
+x:185 [in hydras.Ackermann.codeFreeVar]
+x:186 [in hydras.Ackermann.codeFreeVar]
+X:186 [in hydras.Prelude.MoreLists]
+x:187 [in Goedel.PRrepresentable]
+x:187 [in hydras.Ackermann.subAll]
+x:1873 [in gaia_hydras.nfwfgaia]
+x:188 [in hydras.Ackermann.model]
+x:1882 [in gaia_hydras.nfwfgaia]
+x:189 [in Goedel.PRrepresentable]
+x:189 [in hydras.Prelude.Iterates]
+x:189 [in hydras.Prelude.MoreLists]
+x:1890 [in gaia_hydras.nfwfgaia]
+x:1891 [in gaia_hydras.nfwfgaia]
+x:1892 [in gaia_hydras.nfwfgaia]
+x:1896 [in gaia_hydras.nfwfgaia]
+x:1899 [in gaia_hydras.nfwfgaia]
+x:19 [in additions.Compatibility]
+x:19 [in hydras.Prelude.Restriction]
+x:19 [in hydras.Schutte.GRelations]
+x:19 [in hydras.solutions_exercises.MorePRExamples]
+x:19 [in hydras.Prelude.MoreOrders]
+x:19 [in hydras.Ackermann.Deduction]
+x:19 [in hydras.Ackermann.folLogic3]
+x:19 [in additions.Addition_Chains]
+x:19 [in hydras.Ackermann.NN]
+x:190 [in hydras.Ackermann.subAll]
+x:190 [in hydras.Prelude.MoreLists]
+x:190 [in hydras.Ackermann.model]
+x:1904 [in gaia_hydras.nfwfgaia]
+x:1908 [in gaia_hydras.nfwfgaia]
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+x:191 [in hydras.Ackermann.subAll]
+x:1913 [in gaia_hydras.nfwfgaia]
+x:1917 [in gaia_hydras.nfwfgaia]
+x:192 [in hydras.Ackermann.subAll]
+x:1922 [in gaia_hydras.nfwfgaia]
+x:1928 [in gaia_hydras.nfwfgaia]
+x:193 [in hydras.rpo.term]
+x:193 [in Goedel.PRrepresentable]
+x:193 [in hydras.Ackermann.subAll]
+x:193 [in hydras.Prelude.MoreLists]
+x:1934 [in gaia_hydras.nfwfgaia]
+x:1938 [in gaia_hydras.nfwfgaia]
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+x:194 [in additions.Addition_Chains]
+x:1943 [in gaia_hydras.nfwfgaia]
+x:1948 [in gaia_hydras.nfwfgaia]
+x:195 [in Goedel.PRrepresentable]
+x:195 [in hydras.Ackermann.subAll]
+x:1954 [in gaia_hydras.nfwfgaia]
+x:196 [in hydras.Ackermann.subAll]
+x:196 [in hydras.Prelude.Iterates]
+x:196 [in hydras.OrdinalNotations.ON_Generic]
+x:1960 [in gaia_hydras.nfwfgaia]
+x:197 [in Goedel.PRrepresentable]
+x:197 [in additions.Addition_Chains]
+x:1972 [in gaia_hydras.nfwfgaia]
+x:1978 [in gaia_hydras.nfwfgaia]
+x:198 [in hydras.Ackermann.subAll]
+x:1983 [in gaia_hydras.nfwfgaia]
+x:1988 [in gaia_hydras.nfwfgaia]
+x:199 [in Goedel.PRrepresentable]
+x:199 [in hydras.Ackermann.codeFreeVar]
+x:1994 [in gaia_hydras.nfwfgaia]
+x:1999 [in gaia_hydras.nfwfgaia]
+x:2 [in hydras.MoreAck.AckNotPR]
+x:2 [in Goedel.rosserPA]
+X:2 [in hydras.Ackermann.Deduction]
+x:2 [in hydras.Prelude.MoreDecidable]
+x:20 [in hydras.Prelude.DecPreOrder_Instances]
+x:20 [in hydras.Prelude.Iterates]
+x:20 [in hydras.Ackermann.subProp]
+x:20 [in gaia_hydras.T1Bridge]
+x:20 [in hydras.Ackermann.Deduction]
+X:20 [in hydras.Schutte.Schutte_basics]
+x:200 [in hydras.Ackermann.subAll]
+x:200 [in hydras.Ackermann.codeFreeVar]
+x:200 [in gaia_hydras.nfwfgaia]
+x:201 [in Goedel.PRrepresentable]
+x:201 [in hydras.Ackermann.subAll]
+x:201 [in additions.Euclidean_Chains]
+x:2010 [in gaia_hydras.nfwfgaia]
+x:2016 [in gaia_hydras.nfwfgaia]
+x:2017 [in gaia_hydras.nfwfgaia]
+x:2018 [in gaia_hydras.nfwfgaia]
+x:202 [in hydras.Ackermann.subAll]
+x:202 [in hydras.Prelude.Iterates]
+x:2020 [in gaia_hydras.nfwfgaia]
+x:203 [in Goedel.PRrepresentable]
+x:203 [in hydras.Ackermann.subAll]
+x:203 [in hydras.Prelude.Iterates]
+x:2033 [in gaia_hydras.nfwfgaia]
+x:2034 [in gaia_hydras.nfwfgaia]
+x:2035 [in gaia_hydras.nfwfgaia]
+x:204 [in hydras.Ackermann.subAll]
+x:204 [in hydras.Prelude.MoreLists]
+x:2040 [in gaia_hydras.nfwfgaia]
+x:2045 [in gaia_hydras.nfwfgaia]
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+x:205 [in hydras.Ackermann.subAll]
+x:2057 [in gaia_hydras.nfwfgaia]
+x:2058 [in gaia_hydras.nfwfgaia]
+x:206 [in hydras.Ackermann.subAll]
+x:206 [in hydras.Prelude.Iterates]
+x:2068 [in gaia_hydras.nfwfgaia]
+x:207 [in Goedel.PRrepresentable]
+x:2076 [in gaia_hydras.nfwfgaia]
+x:208 [in hydras.Prelude.Iterates]
+x:208 [in hydras.Prelude.MoreLists]
+x:2081 [in gaia_hydras.nfwfgaia]
+x:2089 [in gaia_hydras.nfwfgaia]
+x:209 [in Goedel.PRrepresentable]
+x:2095 [in gaia_hydras.nfwfgaia]
+x:2096 [in gaia_hydras.nfwfgaia]
+x:2098 [in gaia_hydras.nfwfgaia]
+x:21 [in additions.AM]
+x:21 [in additions.Pow_variant]
+x:21 [in additions.Pow]
+x:21 [in hydras.Ackermann.Deduction]
+x:21 [in hydras.Schutte.Schutte_basics]
+x:21 [in hydras.OrdinalNotations.ON_Generic]
+x:21 [in hydras.Ackermann.folLogic3]
+x:21 [in hydras.MoreAck.FolExamples]
+x:2100 [in gaia_hydras.nfwfgaia]
+x:2103 [in gaia_hydras.nfwfgaia]
+x:2104 [in gaia_hydras.nfwfgaia]
+x:2106 [in gaia_hydras.nfwfgaia]
+x:211 [in Goedel.PRrepresentable]
+x:213 [in Goedel.PRrepresentable]
+x:214 [in hydras.Prelude.MoreLists]
+x:214 [in hydras.Ackermann.model]
+x:214 [in additions.Euclidean_Chains]
+x:215 [in Goedel.PRrepresentable]
+x:216 [in hydras.Ackermann.model]
+x:216 [in additions.Euclidean_Chains]
+X:216 [in hydras.Schutte.Schutte_basics]
+x:217 [in Goedel.PRrepresentable]
+x:217 [in hydras.Ackermann.cPair]
+x:218 [in hydras.Prelude.Merge_Sort]
+x:218 [in hydras.Schutte.Schutte_basics]
+x:219 [in Goedel.PRrepresentable]
+x:219 [in hydras.Ackermann.cPair]
+x:22 [in hydras.Prelude.DecPreOrder_Instances]
+x:22 [in hydras.Prelude.Iterates]
+x:22 [in hydras.Prelude.MoreLists]
+x:22 [in additions.Compatibility]
+x:22 [in hydras.Prelude.Restriction]
+x:22 [in Goedel.rosserPA]
+x:22 [in hydras.Ackermann.fol]
+x:22 [in hydras.Schutte.Well_Orders]
+x:22 [in hydras.Ackermann.folLogic3]
+x:22 [in hydras.Schutte.Ordering_Functions]
+x:22 [in Goedel.rosser]
+x:22 [in hydras.Schutte.Countable]
+x:220 [in hydras.Prelude.Merge_Sort]
+x:220 [in hydras.Schutte.Schutte_basics]
+x:221 [in Goedel.PRrepresentable]
+x:221 [in hydras.Prelude.Iterates]
+x:222 [in hydras.Prelude.Merge_Sort]
+x:222 [in hydras.Schutte.Schutte_basics]
+x:223 [in hydras.Ackermann.cPair]
+x:223 [in hydras.Schutte.Schutte_basics]
+x:224 [in hydras.Prelude.Iterates]
+x:224 [in hydras.Prelude.Merge_Sort]
+x:225 [in Goedel.PRrepresentable]
+x:225 [in hydras.Prelude.Iterates]
+x:225 [in hydras.Schutte.Schutte_basics]
+x:226 [in hydras.Prelude.Merge_Sort]
+x:227 [in hydras.Ackermann.cPair]
+X:227 [in hydras.Schutte.Schutte_basics]
+x:228 [in hydras.Prelude.Iterates]
+x:228 [in hydras.Prelude.Merge_Sort]
+X:228 [in hydras.Schutte.Schutte_basics]
+x:229 [in hydras.Ackermann.codePA]
+x:23 [in hydras.Ackermann.folProp]
+x:23 [in gaia_hydras.T1Bridge]
+x:23 [in Goedel.rosserPA]
+x:23 [in additions.Euclidean_Chains]
+x:23 [in hydras.OrdinalNotations.ON_Generic]
+x:23 [in hydras.Ackermann.NN]
+x:23 [in hydras.MoreAck.FolExamples]
+x:230 [in hydras.Prelude.Merge_Sort]
+x:230 [in hydras.Ackermann.codePA]
+x:231 [in hydras.Ackermann.cPair]
+x:232 [in Goedel.PRrepresentable]
+x:232 [in hydras.Prelude.Merge_Sort]
+x:234 [in hydras.Prelude.Merge_Sort]
+x:235 [in additions.AM]
+x:235 [in hydras.Ackermann.folProp]
+x:236 [in hydras.Prelude.Merge_Sort]
+x:237 [in hydras.Ackermann.folProp]
+x:238 [in hydras.Prelude.Merge_Sort]
+x:239 [in hydras.Ackermann.folProp]
+x:24 [in additions.Compatibility]
+x:24 [in hydras.Prelude.Sort_spec]
+x:24 [in hydras.Ackermann.subProp]
+x:24 [in hydras.Schutte.Countable]
+x:241 [in hydras.Ackermann.folProp]
+x:242 [in additions.AM]
+x:243 [in hydras.Ackermann.folProp]
+x:245 [in hydras.Ackermann.subAll]
+x:245 [in hydras.Ackermann.folProp]
+x:245 [in additions.Addition_Chains]
+x:246 [in hydras.Ackermann.subAll]
+x:247 [in hydras.Ackermann.folProp]
+x:248 [in hydras.Ackermann.cPair]
+x:248 [in additions.Addition_Chains]
+x:249 [in Goedel.PRrepresentable]
+x:25 [in hydras.Prelude.Restriction]
+x:25 [in Goedel.fixPoint]
+x:25 [in hydras.Ackermann.cPair]
+x:25 [in hydras.Schutte.Well_Orders]
+x:25 [in additions.Addition_Chains]
+x:25 [in hydras.Ackermann.NN]
+x:25 [in hydras.MoreAck.FolExamples]
+x:250 [in hydras.Ackermann.cPair]
+x:250 [in gaia_hydras.nfwfgaia]
+x:251 [in Goedel.PRrepresentable]
+x:251 [in additions.Addition_Chains]
+x:253 [in Goedel.PRrepresentable]
+x:254 [in hydras.Ackermann.cPair]
+x:255 [in Goedel.PRrepresentable]
+x:256 [in hydras.Ackermann.cPair]
+x:257 [in Goedel.PRrepresentable]
+x:258 [in hydras.Ackermann.cPair]
+x:258 [in additions.Addition_Chains]
+x:259 [in Goedel.PRrepresentable]
+x:259 [in hydras.Ackermann.subAll]
+x:259 [in additions.Addition_Chains]
+x:26 [in hydras.Ackermann.folProp]
+x:26 [in hydras.MoreAck.AckNotPR]
+x:26 [in hydras.Schutte.Critical]
+x:26 [in hydras.Epsilon0.F_omega]
+x:26 [in hydras.Prelude.MoreOrders]
+x:26 [in Goedel.rosserPA]
+x:26 [in hydras.OrdinalNotations.ON_Finite]
+x:26 [in Goedel.rosser]
+x:260 [in hydras.Ackermann.subAll]
+x:261 [in hydras.Ackermann.subAll]
+x:261 [in hydras.Ackermann.cPair]
+x:261 [in additions.Addition_Chains]
+x:262 [in hydras.Ackermann.subAll]
+x:263 [in hydras.Ackermann.subAll]
+x:263 [in hydras.Ackermann.folProp]
+x:264 [in Goedel.PRrepresentable]
+x:266 [in hydras.Ackermann.subAll]
+x:267 [in hydras.Ackermann.subAll]
+x:268 [in Goedel.PRrepresentable]
+x:268 [in hydras.Ackermann.subAll]
+x:269 [in hydras.Ackermann.subAll]
+x:27 [in hydras.Prelude.MoreLists]
+x:27 [in hydras.Prelude.Sort_spec]
+x:27 [in additions.More_on_positive]
+x:27 [in gaia_hydras.T1Bridge]
+x:27 [in hydras.Ackermann.cPair]
+x:27 [in hydras.Ackermann.fol]
+x:27 [in hydras.Ackermann.NN]
+x:270 [in hydras.Ackermann.subAll]
+X:270 [in additions.Euclidean_Chains]
+x:270 [in gaia_hydras.nfwfgaia]
+x:271 [in hydras.Ackermann.subAll]
+x:271 [in gaia_hydras.nfwfgaia]
+x:272 [in hydras.Ackermann.subAll]
+x:272 [in additions.Addition_Chains]
+x:273 [in hydras.Ackermann.subAll]
+x:274 [in hydras.Ackermann.subAll]
+x:274 [in additions.Addition_Chains]
+x:275 [in Goedel.PRrepresentable]
+x:275 [in hydras.Ackermann.subAll]
+x:276 [in hydras.Ackermann.subAll]
+X:276 [in additions.Euclidean_Chains]
+x:276 [in gaia_hydras.nfwfgaia]
+x:277 [in hydras.Ackermann.subAll]
+x:278 [in hydras.Ackermann.subAll]
+x:279 [in hydras.Ackermann.subAll]
+x:28 [in additions.Pow_variant]
+x:28 [in additions.Pow]
+x:28 [in hydras.Ackermann.subProp]
+x:28 [in hydras.Ackermann.PA]
+x:28 [in additions.Monoid_def]
+X:28 [in hydras.Schutte.Schutte_basics]
+x:28 [in hydras.OrdinalNotations.ON_Generic]
+x:28 [in hydras.Schutte.Well_Orders]
+x:28 [in hydras.Ackermann.NN]
+x:280 [in hydras.Ackermann.subAll]
+x:280 [in hydras.Ackermann.folProp]
+x:281 [in hydras.Ackermann.subAll]
+x:281 [in gaia_hydras.nfwfgaia]
+x:284 [in hydras.Ackermann.folProp]
+x:285 [in gaia_hydras.nfwfgaia]
+x:286 [in hydras.Ackermann.folProp]
+x:287 [in gaia_hydras.nfwfgaia]
+x:288 [in hydras.Ackermann.folProp]
+x:288 [in hydras.Ackermann.fol]
+x:289 [in hydras.Ackermann.fol]
+x:29 [in Goedel.fixPoint]
+x:29 [in hydras.Ackermann.folProp]
+x:29 [in additions.More_on_positive]
+x:29 [in hydras.MoreAck.AckNotPR]
+x:29 [in hydras.Epsilon0.Large_Sets]
+x:29 [in hydras.Schutte.Critical]
+x:29 [in hydras.Epsilon0.F_omega]
+x:29 [in hydras.Ackermann.cPair]
+x:29 [in hydras.Ackermann.fol]
+X:29 [in hydras.Prelude.MoreVectors]
+x:290 [in hydras.rpo.term]
+x:290 [in hydras.Ackermann.folProp]
+x:292 [in hydras.Ackermann.subAll]
+x:292 [in hydras.Ackermann.folProp]
+x:293 [in hydras.Ackermann.subAll]
+x:293 [in additions.Addition_Chains]
+x:294 [in hydras.Ackermann.folProp]
+x:297 [in gaia_hydras.nfwfgaia]
+x:3 [in Goedel.PRrepresentable]
+x:3 [in hydras.Ackermann.PAconsistent]
+x:3 [in hydras.Ackermann.LNN]
+x:3 [in hydras.Ackermann.LNT]
+x:3 [in hydras.MoreAck.Iterate_compat]
+x:30 [in Goedel.PRrepresentable]
+x:30 [in hydras.Prelude.Restriction]
+x:30 [in hydras.Ackermann.folProp]
+x:30 [in gaia_hydras.T1Bridge]
+x:30 [in hydras.Ackermann.PA]
+x:30 [in additions.Monoid_instances]
+X:30 [in hydras.Schutte.Schutte_basics]
+x:300 [in gaia_hydras.nfwfgaia]
+x:301 [in Goedel.PRrepresentable]
+x:301 [in hydras.Ackermann.subAll]
+x:302 [in hydras.Ackermann.subAll]
+x:302 [in gaia_hydras.nfwfgaia]
+x:303 [in hydras.Ackermann.subAll]
+x:304 [in hydras.Ackermann.subAll]
+x:305 [in hydras.Ackermann.subAll]
+x:306 [in hydras.Ackermann.subAll]
+x:307 [in hydras.Ackermann.subAll]
+x:308 [in hydras.Ackermann.subAll]
+x:309 [in hydras.Ackermann.subAll]
+x:31 [in hydras.Schutte.Critical]
+x:31 [in hydras.solutions_exercises.MultisetWf]
+x:31 [in hydras.Epsilon0.F_omega]
+x:31 [in hydras.Ackermann.cPair]
+x:31 [in hydras.Schutte.Well_Orders]
+x:310 [in hydras.Ackermann.subAll]
+x:311 [in hydras.Ackermann.subAll]
+x:311 [in gaia_hydras.nfwfgaia]
+x:312 [in hydras.Ackermann.subAll]
+x:313 [in hydras.Ackermann.subAll]
+x:314 [in gaia_hydras.nfwfgaia]
+x:315 [in gaia_hydras.nfwfgaia]
+x:316 [in gaia_hydras.nfwfgaia]
+x:317 [in gaia_hydras.nfwfgaia]
+x:318 [in hydras.Ackermann.subAll]
+x:318 [in gaia_hydras.nfwfgaia]
+x:319 [in gaia_hydras.nfwfgaia]
+x:32 [in hydras.Ackermann.folLogic2]
+x:32 [in hydras.Ackermann.PA]
+x:32 [in hydras.Prelude.MoreOrders]
+x:32 [in additions.Monoid_def]
+x:32 [in additions.Monoid_instances]
+x:32 [in additions.fib]
+x:32 [in hydras.Ackermann.NN]
+x:320 [in gaia_hydras.nfwfgaia]
+x:322 [in gaia_hydras.nfwfgaia]
+x:323 [in hydras.Ackermann.subAll]
+x:324 [in hydras.Ackermann.subAll]
+x:324 [in gaia_hydras.nfwfgaia]
+x:325 [in hydras.Ackermann.subAll]
+x:326 [in hydras.Ackermann.subAll]
+x:327 [in hydras.Ackermann.subAll]
+x:328 [in hydras.Ackermann.subAll]
+x:329 [in gaia_hydras.nfwfgaia]
+x:33 [in hydras.Ackermann.subProp]
+x:33 [in hydras.Epsilon0.F_omega]
+x:33 [in hydras.Ackermann.PA]
+x:33 [in Goedel.rosserPA]
+x:33 [in hydras.Ackermann.fol]
+x:33 [in additions.fib]
+X:33 [in hydras.Schutte.Well_Orders]
+x:33 [in Goedel.rosser]
+x:330 [in hydras.Ackermann.subAll]
+x:330 [in gaia_hydras.nfwfgaia]
+x:332 [in gaia_hydras.nfwfgaia]
+x:333 [in hydras.Ackermann.subAll]
+x:333 [in hydras.Ackermann.fol]
+x:334 [in hydras.Ackermann.fol]
+x:335 [in hydras.Ackermann.fol]
+x:339 [in gaia_hydras.nfwfgaia]
+x:34 [in Goedel.PRrepresentable]
+x:34 [in additions.Pow_variant]
+x:34 [in additions.Pow]
+x:34 [in hydras.Epsilon0.Large_Sets]
+x:34 [in hydras.Schutte.Critical]
+x:34 [in additions.Monoid_def]
+x:34 [in additions.Monoid_instances]
+x:34 [in hydras.Ackermann.NN]
+x:34 [in Goedel.rosser]
+x:34 [in gaia_hydras.nfwfgaia]
+x:341 [in gaia_hydras.nfwfgaia]
+x:342 [in gaia_hydras.nfwfgaia]
+x:343 [in gaia_hydras.nfwfgaia]
+x:345 [in gaia_hydras.nfwfgaia]
+x:346 [in hydras.Ackermann.primRec]
+x:347 [in gaia_hydras.nfwfgaia]
+x:349 [in gaia_hydras.nfwfgaia]
+x:35 [in hydras.Prelude.Sort_spec]
+x:35 [in hydras.Epsilon0.F_omega]
+x:35 [in hydras.Prelude.MoreOrders]
+x:350 [in hydras.Ackermann.primRec]
+x:351 [in gaia_hydras.nfwfgaia]
+x:354 [in hydras.Ackermann.subAll]
+x:355 [in hydras.Ackermann.subAll]
+x:355 [in hydras.Ackermann.primRec]
+x:358 [in hydras.Ackermann.primRec]
+x:36 [in hydras.Ackermann.folLogic2]
+x:36 [in hydras.Schutte.Critical]
+x:36 [in hydras.Ackermann.fol]
+X:36 [in hydras.Schutte.Well_Orders]
+x:36 [in hydras.Ackermann.NN]
+x:360 [in hydras.Ackermann.subAll]
+x:361 [in hydras.Ackermann.subAll]
+x:362 [in hydras.Ackermann.subAll]
+x:363 [in hydras.Ackermann.subAll]
+x:363 [in hydras.Ackermann.primRec]
+x:365 [in hydras.Ackermann.subAll]
+x:365 [in gaia_hydras.nfwfgaia]
+x:366 [in hydras.Ackermann.subAll]
+x:368 [in hydras.Ackermann.subAll]
+x:369 [in hydras.Ackermann.primRec]
+x:37 [in hydras.Prelude.Restriction]
+x:37 [in additions.FirstSteps]
+x:37 [in hydras.Ackermann.PA]
+x:37 [in additions.Monoid_instances]
+X:37 [in hydras.Schutte.Schutte_basics]
+x:37 [in additions.fib]
+x:37 [in hydras.Ackermann.NN]
+x:370 [in gaia_hydras.nfwfgaia]
+x:374 [in hydras.Ackermann.subAll]
+x:375 [in hydras.Ackermann.subAll]
+x:376 [in Goedel.PRrepresentable]
+x:376 [in hydras.Ackermann.primRec]
+x:377 [in hydras.Ackermann.subAll]
+x:378 [in Goedel.PRrepresentable]
+x:378 [in hydras.Ackermann.subAll]
+x:378 [in hydras.Ackermann.fol]
+x:378 [in gaia_hydras.nfwfgaia]
+x:379 [in hydras.Ackermann.fol]
+x:379 [in gaia_hydras.nfwfgaia]
+x:38 [in hydras.Prelude.MoreLists]
+x:38 [in hydras.Ackermann.subProp]
+x:38 [in Goedel.rosserPA]
+x:38 [in additions.Monoid_instances]
+x:38 [in gaia_hydras.nfwfgaia]
+x:380 [in Goedel.PRrepresentable]
+x:380 [in hydras.Ackermann.subAll]
+x:380 [in hydras.Ackermann.primRec]
+x:382 [in Goedel.PRrepresentable]
+x:382 [in gaia_hydras.nfwfgaia]
+x:383 [in gaia_hydras.nfwfgaia]
+x:386 [in hydras.Ackermann.primRec]
+x:389 [in gaia_hydras.nfwfgaia]
+x:39 [in hydras.Ackermann.PA]
+x:39 [in hydras.Prelude.MoreOrders]
+x:39 [in hydras.Ackermann.fol]
+x:39 [in gaia_hydras.nfwfgaia]
+x:392 [in gaia_hydras.nfwfgaia]
+x:393 [in hydras.Ackermann.primRec]
+x:399 [in gaia_hydras.nfwfgaia]
+x:4 [in hydras.Prelude.Iterates]
+x:4 [in hydras.Ackermann.NN2PA]
+x:4 [in additions.FirstSteps]
+X:4 [in hydras.Schutte.Lub]
+x:4 [in hydras.Ackermann.Deduction]
+x:4 [in gaia_hydras.onType]
+x:4 [in hydras.Ackermann.wConsistent]
+x:4 [in hydras.MoreAck.PrimRecExamples]
+x:40 [in additions.AM]
+x:40 [in hydras.Prelude.MoreLists]
+x:40 [in additions.Compatibility]
+x:40 [in hydras.Schutte.Critical]
+x:40 [in gaia_hydras.T1Bridge]
+x:40 [in hydras.Ackermann.primRec]
+x:40 [in hydras.OrdinalNotations.ON_Generic]
+X:40 [in hydras.Schutte.Well_Orders]
+x:40 [in hydras.Ackermann.NN]
+x:400 [in hydras.Ackermann.primRec]
+x:401 [in hydras.Ackermann.folProp]
+x:408 [in hydras.Ackermann.primRec]
+x:41 [in additions.Compatibility]
+x:41 [in hydras.Ackermann.PA]
+x:41 [in hydras.Prelude.MoreOrders]
+x:41 [in Goedel.rosserPA]
+x:41 [in hydras.Ackermann.primRec]
+x:41 [in hydras.Schutte.Ordering_Functions]
+x:411 [in hydras.Epsilon0.T1]
+x:412 [in hydras.Ackermann.primRec]
+x:42 [in hydras.Prelude.Iterates]
+x:42 [in hydras.Ackermann.model]
+x:42 [in hydras.Ackermann.PA]
+x:42 [in hydras.Ackermann.NN]
+x:42 [in hydras.Schutte.Ordering_Functions]
+X:42 [in hydras.MoreAck.FolExamples]
+x:423 [in hydras.Ackermann.fol]
+x:425 [in hydras.Ackermann.primRec]
+x:426 [in hydras.Ackermann.fol]
+x:429 [in hydras.Ackermann.folProp]
+x:43 [in additions.FirstSteps]
+x:43 [in hydras.Ackermann.subProp]
+x:43 [in hydras.OrdinalNotations.ON_Generic]
+X:43 [in hydras.Schutte.Well_Orders]
+x:431 [in hydras.Ackermann.primRec]
+x:434 [in hydras.Ackermann.folProp]
+x:434 [in gaia_hydras.nfwfgaia]
+x:435 [in gaia_hydras.nfwfgaia]
+x:438 [in hydras.Gamma0.Gamma0]
+x:439 [in hydras.Ackermann.primRec]
+x:44 [in additions.Compatibility]
+x:44 [in hydras.Prelude.Restriction]
+x:44 [in additions.Pow_variant]
+x:44 [in additions.Pow]
+x:44 [in hydras.Prelude.Sort_spec]
+x:44 [in hydras.Schutte.Well_Orders]
+x:44 [in hydras.Ackermann.NN]
+x:440 [in hydras.Ackermann.folProp]
+x:440 [in hydras.Gamma0.Gamma0]
+x:443 [in hydras.Gamma0.Gamma0]
+x:447 [in hydras.Ackermann.primRec]
+x:448 [in hydras.Gamma0.Gamma0]
+x:45 [in hydras.Ackermann.NN]
+x:452 [in hydras.Ackermann.primRec]
+x:46 [in hydras.Prelude.MoreLists]
+x:46 [in hydras.Ackermann.folProp]
+x:46 [in hydras.Ackermann.PA]
+x:46 [in hydras.OrdinalNotations.ON_Generic]
+x:46 [in hydras.MoreAck.FolExamples]
+x:460 [in hydras.Ackermann.primRec]
+x:465 [in hydras.Ackermann.primRec]
+x:469 [in hydras.Ackermann.primRec]
+x:47 [in hydras.Ackermann.model]
+x:47 [in hydras.Ackermann.subProp]
+x:47 [in hydras.Schutte.Critical]
+x:47 [in gaia_hydras.T1Bridge]
+X:47 [in hydras.Schutte.Well_Orders]
+x:473 [in hydras.Ackermann.primRec]
+x:473 [in additions.Euclidean_Chains]
+x:475 [in additions.Euclidean_Chains]
+x:477 [in additions.Euclidean_Chains]
+x:479 [in additions.Euclidean_Chains]
+x:48 [in additions.AM]
+x:48 [in hydras.Ackermann.folProp]
+x:48 [in hydras.Ackermann.PA]
+x:480 [in additions.Euclidean_Chains]
+x:481 [in hydras.Ackermann.primRec]
+x:486 [in hydras.Ackermann.primRec]
+X:49 [in hydras.Schutte.Schutte_basics]
+x:49 [in hydras.MoreAck.PrimRecExamples]
+x:491 [in hydras.Ackermann.primRec]
+x:496 [in hydras.Ackermann.primRec]
+x:5 [in hydras.solutions_exercises.MinPR2]
+x:5 [in hydras.Ackermann.PAconsistent]
+x:5 [in additions.Compatibility]
+x:5 [in hydras.Ackermann.LNN]
+x:5 [in hydras.Ackermann.LNT]
+x:5 [in hydras.MoreAck.AckNotPR]
+x:5 [in hydras.Ackermann.subProp]
+x:5 [in additions.Euclidean_Chains]
+x:5 [in additions.Naive]
+X:5 [in hydras.Ackermann.Deduction]
+x:5 [in hydras.Prelude.STDPP_compat]
+X:5 [in hydras.Schutte.Well_Orders]
+x:5 [in additions.Addition_Chains]
+x:5 [in hydras.Ackermann.NN]
+x:5 [in hydras.rpo.closure]
+x:5 [in hydras.Ackermann.Languages]
+x:50 [in hydras.Ackermann.PA]
+X:50 [in hydras.Schutte.Schutte_basics]
+x:500 [in hydras.Ackermann.primRec]
+x:500 [in gaia_hydras.nfwfgaia]
+x:503 [in gaia_hydras.nfwfgaia]
+x:506 [in gaia_hydras.nfwfgaia]
+x:507 [in gaia_hydras.nfwfgaia]
+x:509 [in gaia_hydras.nfwfgaia]
+x:51 [in hydras.Ackermann.model]
+x:51 [in hydras.Ackermann.subProp]
+x:51 [in hydras.Ackermann.PA]
+x:515 [in gaia_hydras.nfwfgaia]
+x:52 [in hydras.Ackermann.folProp]
+x:520 [in gaia_hydras.nfwfgaia]
+x:525 [in gaia_hydras.nfwfgaia]
+x:528 [in gaia_hydras.nfwfgaia]
+x:53 [in additions.Pow_variant]
+x:53 [in additions.Pow]
+x:532 [in gaia_hydras.nfwfgaia]
+x:535 [in gaia_hydras.nfwfgaia]
+x:536 [in gaia_hydras.nfwfgaia]
+x:537 [in gaia_hydras.nfwfgaia]
+x:538 [in gaia_hydras.nfwfgaia]
+x:539 [in gaia_hydras.nfwfgaia]
+x:54 [in hydras.Ackermann.fol]
+x:54 [in hydras.MoreAck.PrimRecExamples]
+x:540 [in gaia_hydras.nfwfgaia]
+x:541 [in gaia_hydras.nfwfgaia]
+x:547 [in gaia_hydras.nfwfgaia]
+x:55 [in hydras.Ackermann.model]
+X:55 [in hydras.Schutte.Well_Orders]
+x:551 [in gaia_hydras.nfwfgaia]
+x:554 [in gaia_hydras.nfwfgaia]
+x:56 [in hydras.Ackermann.subProp]
+x:56 [in Goedel.rosserPA]
+x:566 [in hydras.Ackermann.primRec]
+x:567 [in hydras.Ackermann.primRec]
+x:568 [in hydras.Ackermann.primRec]
+x:569 [in hydras.Ackermann.primRec]
+x:57 [in hydras.MoreAck.AckNotPR]
+x:57 [in hydras.Ackermann.fol]
+x:57 [in hydras.MoreAck.PrimRecExamples]
+x:570 [in hydras.Ackermann.primRec]
+x:570 [in gaia_hydras.nfwfgaia]
+x:571 [in hydras.Ackermann.primRec]
+x:572 [in hydras.Ackermann.primRec]
+x:573 [in hydras.Ackermann.primRec]
+x:573 [in gaia_hydras.nfwfgaia]
+x:577 [in gaia_hydras.nfwfgaia]
+x:581 [in hydras.Ackermann.primRec]
+x:582 [in hydras.Ackermann.primRec]
+x:583 [in gaia_hydras.nfwfgaia]
+x:584 [in gaia_hydras.nfwfgaia]
+x:586 [in gaia_hydras.nfwfgaia]
+x:588 [in gaia_hydras.nfwfgaia]
+x:59 [in hydras.Ackermann.model]
+x:59 [in hydras.Ackermann.fol]
+x:59 [in hydras.OrdinalNotations.ON_Generic]
+x:59 [in Goedel.rosser]
+x:591 [in gaia_hydras.nfwfgaia]
+x:592 [in gaia_hydras.nfwfgaia]
+x:594 [in gaia_hydras.nfwfgaia]
+x:6 [in Goedel.PRrepresentable]
+x:6 [in hydras.Prelude.DecPreOrder_Instances]
+x:6 [in hydras.Prelude.More_Arith]
+x:6 [in hydras.Ackermann.LNN]
+x:6 [in additions.Pow_variant]
+x:6 [in hydras.Schutte.Lub]
+x:6 [in additions.Pow]
+x:6 [in hydras.Schutte.Critical]
+x:6 [in hydras.Prelude.MoreOrders]
+x:60 [in hydras.Ackermann.fol]
+X:60 [in hydras.Schutte.Well_Orders]
+x:60 [in Goedel.rosser]
+x:605 [in gaia_hydras.nfwfgaia]
+x:606 [in gaia_hydras.nfwfgaia]
+x:607 [in gaia_hydras.nfwfgaia]
+x:608 [in hydras.Gamma0.Gamma0]
+x:61 [in additions.Pow_variant]
+x:61 [in additions.Pow]
+x:61 [in hydras.Ackermann.subProp]
+x:61 [in hydras.Schutte.PartialFun]
+x:61 [in hydras.MoreAck.PrimRecExamples]
+x:611 [in hydras.Gamma0.Gamma0]
+x:612 [in gaia_hydras.nfwfgaia]
+x:616 [in gaia_hydras.nfwfgaia]
+x:617 [in gaia_hydras.nfwfgaia]
+x:618 [in gaia_hydras.nfwfgaia]
+x:619 [in gaia_hydras.nfwfgaia]
+x:62 [in hydras.Ackermann.folProp]
+x:621 [in gaia_hydras.nfwfgaia]
+x:623 [in gaia_hydras.nfwfgaia]
+x:63 [in hydras.Prelude.Fuel]
+x:63 [in Goedel.rosserPA]
+x:63 [in hydras.Schutte.PartialFun]
+x:636 [in gaia_hydras.nfwfgaia]
+x:64 [in additions.Pow_variant]
+x:64 [in additions.Pow]
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+z:269 [in additions.Euclidean_Chains]
+z:27 [in Goedel.PRrepresentable]
+z:27 [in hydras.Schutte.Well_Orders]
+z:276 [in Goedel.PRrepresentable]
+z:283 [in additions.Euclidean_Chains]
+z:285 [in additions.Euclidean_Chains]
+z:287 [in additions.Euclidean_Chains]
+z:289 [in additions.Euclidean_Chains]
+z:30 [in additions.Monoid_def]
+z:30 [in hydras.Schutte.Well_Orders]
+z:341 [in Goedel.PRrepresentable]
+z:343 [in Goedel.PRrepresentable]
+z:346 [in Goedel.PRrepresentable]
+z:347 [in Goedel.PRrepresentable]
+z:348 [in Goedel.PRrepresentable]
+z:349 [in Goedel.PRrepresentable]
+z:35 [in gaia_hydras.nfwfgaia]
+z:36 [in hydras.Prelude.MoreOrders]
+z:36 [in additions.Monoid_instances]
+z:366 [in Goedel.PRrepresentable]
+z:367 [in Goedel.PRrepresentable]
+z:368 [in Goedel.PRrepresentable]
+z:369 [in Goedel.PRrepresentable]
+z:372 [in Goedel.PRrepresentable]
+z:373 [in Goedel.PRrepresentable]
+z:374 [in Goedel.PRrepresentable]
+z:375 [in Goedel.PRrepresentable]
+z:38 [in gaia_hydras.T1Bridge]
+z:382 [in hydras.Ackermann.primRec]
+z:388 [in hydras.Ackermann.primRec]
+z:395 [in hydras.Ackermann.primRec]
+z:402 [in hydras.Ackermann.primRec]
+z:402 [in gaia_hydras.nfwfgaia]
+z:403 [in gaia_hydras.nfwfgaia]
+z:404 [in gaia_hydras.nfwfgaia]
+z:405 [in gaia_hydras.nfwfgaia]
+z:406 [in gaia_hydras.nfwfgaia]
+z:410 [in hydras.Ackermann.primRec]
+z:417 [in gaia_hydras.nfwfgaia]
+z:418 [in gaia_hydras.nfwfgaia]
+z:419 [in gaia_hydras.nfwfgaia]
+z:420 [in gaia_hydras.nfwfgaia]
+z:421 [in gaia_hydras.nfwfgaia]
+Z:43 [in hydras.Ackermann.code]
+z:43 [in hydras.Ackermann.fol]
+z:454 [in gaia_hydras.nfwfgaia]
+z:455 [in gaia_hydras.nfwfgaia]
+z:456 [in gaia_hydras.nfwfgaia]
+z:46 [in hydras.Ackermann.fol]
+z:49 [in hydras.Ackermann.fol]
+z:5 [in Goedel.PRrepresentable]
+z:511 [in gaia_hydras.nfwfgaia]
+z:512 [in gaia_hydras.nfwfgaia]
+z:513 [in gaia_hydras.nfwfgaia]
+z:52 [in hydras.Ackermann.fol]
+z:57 [in hydras.Schutte.Schutte_basics]
+z:578 [in gaia_hydras.nfwfgaia]
+z:58 [in hydras.Schutte.Schutte_basics]
+z:589 [in gaia_hydras.nfwfgaia]
+z:62 [in gaia_hydras.nfwfgaia]
+z:625 [in gaia_hydras.nfwfgaia]
+z:63 [in hydras.MoreAck.PrimRecExamples]
+z:631 [in gaia_hydras.nfwfgaia]
+z:632 [in gaia_hydras.nfwfgaia]
+z:633 [in gaia_hydras.nfwfgaia]
+z:634 [in gaia_hydras.nfwfgaia]
+z:635 [in gaia_hydras.nfwfgaia]
+z:638 [in gaia_hydras.nfwfgaia]
+z:639 [in gaia_hydras.nfwfgaia]
+z:640 [in gaia_hydras.nfwfgaia]
+z:641 [in gaia_hydras.nfwfgaia]
+z:642 [in gaia_hydras.nfwfgaia]
+z:643 [in gaia_hydras.nfwfgaia]
+z:646 [in gaia_hydras.nfwfgaia]
+z:649 [in gaia_hydras.nfwfgaia]
+z:7 [in hydras.MoreAck.AckNotPR]
+z:7 [in additions.Euclidean_Chains]
+z:7 [in hydras.Prelude.STDPP_compat]
+z:730 [in hydras.Epsilon0.T1]
+z:741 [in gaia_hydras.nfwfgaia]
+z:742 [in gaia_hydras.nfwfgaia]
+z:743 [in gaia_hydras.nfwfgaia]
+z:744 [in gaia_hydras.nfwfgaia]
+z:796 [in gaia_hydras.nfwfgaia]
+z:798 [in gaia_hydras.nfwfgaia]
+z:8 [in Goedel.PRrepresentable]
+z:8 [in additions.Addition_Chains]
+z:801 [in gaia_hydras.nfwfgaia]
+z:804 [in gaia_hydras.nfwfgaia]
+z:808 [in gaia_hydras.nfwfgaia]
+z:812 [in gaia_hydras.nfwfgaia]
+z:818 [in gaia_hydras.nfwfgaia]
+z:821 [in gaia_hydras.nfwfgaia]
+z:825 [in gaia_hydras.nfwfgaia]
+z:828 [in gaia_hydras.nfwfgaia]
+z:833 [in gaia_hydras.nfwfgaia]
+z:86 [in hydras.Ackermann.fol]
+z:88 [in hydras.Ackermann.fol]
+z:9 [in hydras.rpo.closure]
+z:91 [in hydras.Prelude.Iterates]
+z:91 [in gaia_hydras.nfwfgaia]
+z:94 [in hydras.Epsilon0.Large_Sets]
+z:96 [in hydras.Schutte.Schutte_basics]
+z:97 [in hydras.Prelude.Iterates]
+z:97 [in hydras.Epsilon0.Large_Sets]
+z:97 [in hydras.Schutte.Schutte_basics]
+z:98 [in hydras.Epsilon0.Large_Sets]
+z:98 [in hydras.Ackermann.fol]
+z:98 [in hydras.MoreAck.PrimRecExamples]
+z:99 [in hydras.Epsilon0.Large_Sets]
+z:99 [in hydras.Ackermann.fol]
+z:99 [in additions.Addition_Chains]
+

+

Module Index

+

A

+Ackermann [in gaia_hydras.nfwfgaia]
+Alt [in hydras.Epsilon0.Epsilon0rpo]
+Alt [in hydras.MoreAck.Ack]
+Alt [in additions.Demo]
+Alt [in hydras.MoreAck.PrimRecExamples]
+Alt [in hydras.Hydra.Hydra_Definitions]
+Alt.Eps0_rpo [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_alg [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Vars [in hydras.Epsilon0.Epsilon0rpo]
+Alt2 [in hydras.MoreAck.Ack]
+Alt3 [in hydras.MoreAck.Ack]
+

B

+Bad [in additions.FirstSteps]
+Bad [in hydras.Hydra.Hydra_Examples]
+Bad [in additions.Euclidean_Chains]
+Bad [in additions.Monoid_instances]
+BadSubst [in hydras.MoreAck.BadSubst]
+BadSubstF2 [in hydras.MoreAck.BadSubst]
+Bad2 [in additions.Euclidean_Chains]
+Bad3 [in additions.Euclidean_Chains]
+Bad4 [in additions.Euclidean_Chains]
+

C

+CantorOrdinal [in gaia_hydras.nfwfgaia]
+Compat815 [in hydras.Prelude.Compat815]
+

D

+Demo [in additions.Monoid_def]
+Direct_proof [in hydras.Epsilon0.T1]
+

E

+Examples [in hydras.Hydra.Hydra_Examples]
+Examples [in hydras.Ackermann.primRec]
+Examples [in additions.Euclidean_Chains]
+

F

+FixpointDef [in gaia_hydras.GHessenberg]
+FolNotations [in hydras.Ackermann.fol]
+

G

+Gamma0 [in gaia_hydras.nfwfgaia]
+Gamma0_rpo [in hydras.Gamma0.Gamma0]
+Gamma0_alg [in hydras.Gamma0.Gamma0]
+Gamma0_prec [in hydras.Gamma0.Gamma0]
+Gamma0_sig [in hydras.Gamma0.Gamma0]
+G0 [in hydras.Gamma0.Gamma0]
+

I

+iota_demo.Bad [in hydras.Schutte.Schutte_basics]
+iota_demo [in hydras.Schutte.Schutte_basics]
+

L

+LispAbbreviations [in hydras.Ackermann.cPair]
+LNT [in hydras.Ackermann.LNT]
+LO [in hydras.OrdinalNotations.OmegaOmega]
+

M

+Make [in hydras.rpo.term]
+Make [in hydras.rpo.list_set]
+Make [in hydras.rpo.dickson]
+Make [in hydras.rpo.list_permut]
+Make [in hydras.rpo.rpo]
+Make.DecVar [in hydras.rpo.term]
+Make.DS [in hydras.rpo.list_set]
+Make.DS [in hydras.rpo.dickson]
+Make.DS [in hydras.rpo.list_permut]
+Make.F [in hydras.rpo.term]
+Make.LP [in hydras.rpo.list_set]
+Make.LP [in hydras.rpo.dickson]
+Make.LP [in hydras.rpo.rpo]
+Make.P [in hydras.rpo.rpo]
+Make.T [in hydras.rpo.rpo]
+Make.Term_eq_dec [in hydras.rpo.term]
+Make.VSet [in hydras.rpo.term]
+Make.X [in hydras.rpo.term]
+MoreExamples [in hydras.MoreAck.PrimRecExamples]
+M2 [in additions.FirstSteps]
+M2 [in additions.Naive]
+

N

+NNnotations [in hydras.Ackermann.LNN]
+N_mod [in additions.Naive]
+

O

+ONDef [in gaia_hydras.onType]
+ONDef.Exports [in gaia_hydras.onType]
+OO [in hydras.OrdinalNotations.OmegaOmega]
+OpaqueWf [in hydras.Prelude.Fuel]
+

P

+Permut [in hydras.rpo.list_permut]
+Permut.DS [in hydras.rpo.list_permut]
+Precedence [in hydras.rpo.rpo]
+PRNotations [in hydras.Ackermann.primRec]
+

R

+RPO [in hydras.rpo.rpo]
+RPO.LP [in hydras.rpo.rpo]
+RPO.P [in hydras.rpo.rpo]
+RPO.T [in hydras.rpo.rpo]
+

S

+S [in hydras.rpo.decidable_set]
+S [in hydras.rpo.list_set]
+Semibundled [in hydras.Prelude.DecPreOrder]
+Signature [in hydras.rpo.term]
+S.DS [in hydras.rpo.list_set]
+

T

+Term [in hydras.rpo.term]
+Term.F [in hydras.rpo.term]
+Term.Term_eq_dec [in hydras.rpo.term]
+Term.X [in hydras.rpo.term]
+Toy [in hydras.MoreAck.FolExamples]
+

U

+Usual [in Goedel.PRrepresentable]
+

V

+Variables [in hydras.rpo.term]
+Vars [in hydras.Gamma0.Gamma0]
+

W

+Wfsum [in gaia_hydras.nfwfgaia]
+Wf_ex3 [in gaia_hydras.nfwfgaia]
+Wf_ex [in gaia_hydras.nfwfgaia]
+

Z

+Z [in additions.Naive]
+

+

Variable Index

+

A

+Abstract_Properties.alpha [in hydras.Epsilon0.Hprime]
+AB_given.rel_to_fun.Rf [in hydras.Schutte.PartialFun]
+AB_given.f_given.Rf [in hydras.Schutte.PartialFun]
+AB_given.f_given.f [in hydras.Schutte.PartialFun]
+AB_given.DB [in hydras.Schutte.PartialFun]
+AB_given.DA [in hydras.Schutte.PartialFun]
+AB_given.Hb [in hydras.Schutte.PartialFun]
+AB_given.Ha [in hydras.Schutte.PartialFun]
+AB_given.B [in hydras.Schutte.PartialFun]
+AB_given.A [in hydras.Schutte.PartialFun]
+Ack_Properties.Induc_step.Hm [in hydras.MoreAck.Ack]
+Ack_Properties.Induc_step.m [in hydras.MoreAck.Ack]
+Ack_Properties.P [in hydras.MoreAck.Ack]
+Alt.well_founded.R [in hydras.Epsilon0.Epsilon0rpo]
+AP_closed.denM [in hydras.Schutte.AP]
+AP_closed.inhM [in hydras.Schutte.AP]
+AP_closed.OM [in hydras.Schutte.AP]
+AP_closed.M [in hydras.Schutte.AP]
+AP_Unbounded.ksi_fixed.n [in hydras.Schutte.AP]
+AP_Unbounded.ksi_fixed.lt_ksi [in hydras.Schutte.AP]
+AP_Unbounded.ksi_fixed.ksi [in hydras.Schutte.AP]
+AP_Unbounded.beta [in hydras.Schutte.AP]
+AP_Unbounded.seq [in hydras.Schutte.AP]
+AP_Unbounded.alpha [in hydras.Schutte.AP]
+A_def.m [in hydras.OrdinalNotations.ON_Omega2]
+A_given.LtA [in hydras.solutions_exercises.MultisetWf]
+A_given.A [in hydras.solutions_exercises.MultisetWf]
+

B

+BadExample.F [in hydras.MoreAck.BadSubst]
+BadExample.F1 [in hydras.MoreAck.BadSubst]
+Bad.CounterExample.mul [in additions.FirstSteps]
+Bad.CounterExample.one [in additions.FirstSteps]
+battle_length_notPR.m [in hydras.Hydra.Hydra_Theorems]
+battle_length_notPR.h [in hydras.Hydra.Hydra_Theorems]
+battle_length_notPR.alpha [in hydras.Hydra.Hydra_Theorems]
+Battle_length.l [in hydras.Hydra.Battle_length]
+Battle_length.Hk [in hydras.Hydra.Battle_length]
+Battle_length.k [in hydras.Hydra.Battle_length]
+Battle_length.Halpha [in hydras.Hydra.Battle_length]
+Battle_length.alpha [in hydras.Hydra.Battle_length]
+binary_power_proof.E_eq [in additions.Addition_Chains]
+binary_power_proof.E_one [in additions.Addition_Chains]
+binary_power_proof.E_op [in additions.Addition_Chains]
+binary_power_proof.A [in additions.Addition_Chains]
+Bounded.m_decrease [in hydras.Hydra.Epsilon0_Needed_Generic]
+building_ordering_function_by_induction.B [in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1.g_1 [in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1.beta_fixed.beta_B [in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1.beta_fixed.beta [in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1.H_B [in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1.B [in hydras.Schutte.Ordering_Functions]
+

C

+Check_Proof.Prf [in hydras.Ackermann.checkPrf]
+Check_Proof.wellFormedFormula [in hydras.Ackermann.checkPrf]
+Check_Proof.wellFormedTerm [in hydras.Ackermann.checkPrf]
+Check_Proof.Formula [in hydras.Ackermann.checkPrf]
+Check_Proof.Terms [in hydras.Ackermann.checkPrf]
+Check_Proof.Term [in hydras.Ackermann.checkPrf]
+Check_Proof.codeRInj [in hydras.Ackermann.checkPrf]
+Check_Proof.codeFInj [in hydras.Ackermann.checkPrf]
+Check_Proof.codeArityRIsCorrect2 [in hydras.Ackermann.checkPrf]
+Check_Proof.codeArityRIsCorrect1 [in hydras.Ackermann.checkPrf]
+Check_Proof.codeArityFIsCorrect2 [in hydras.Ackermann.checkPrf]
+Check_Proof.codeArityFIsCorrect1 [in hydras.Ackermann.checkPrf]
+Check_Proof.codeArityR [in hydras.Ackermann.checkPrf]
+Check_Proof.codeArityF [in hydras.Ackermann.checkPrf]
+Check_Proof.L [in hydras.Ackermann.checkPrf]
+close.codeFormula [in hydras.Ackermann.codePA]
+close.Formula [in hydras.Ackermann.codePA]
+close.L [in hydras.Ackermann.codePA]
+Code_Free_Vars.Terms [in hydras.Ackermann.codeFreeVar]
+Code_Free_Vars.Term [in hydras.Ackermann.codeFreeVar]
+Code_Free_Vars.System [in hydras.Ackermann.codeFreeVar]
+Code_Free_Vars.Formulas [in hydras.Ackermann.codeFreeVar]
+Code_Free_Vars.Formula [in hydras.Ackermann.codeFreeVar]
+Code_Free_Vars.L [in hydras.Ackermann.codeFreeVar]
+Code_Substitute_Term.Terms [in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Term.Term [in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Term.System [in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Term.Formulas [in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Term.Formula [in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Term.L [in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Formula.codeTerm [in hydras.Ackermann.codeSubFormula]
+Code_Substitute_Formula.codeFormula [in hydras.Ackermann.codeSubFormula]
+Code_Substitute_Formula.L [in hydras.Ackermann.codeSubFormula]
+Code_Term_Formula_Proof.SysPrf [in hydras.Ackermann.code]
+Code_Term_Formula_Proof.Prf [in hydras.Ackermann.code]
+Code_Term_Formula_Proof.Terms [in hydras.Ackermann.code]
+Code_Term_Formula_Proof.Term [in hydras.Ackermann.code]
+Code_Term_Formula_Proof.System [in hydras.Ackermann.code]
+Code_Term_Formula_Proof.Formulas [in hydras.Ackermann.code]
+Code_Term_Formula_Proof.Formula [in hydras.Ackermann.code]
+Code_Term_Formula_Proof.LcodeDef.cR [in hydras.Ackermann.code]
+Code_Term_Formula_Proof.LcodeDef.cF [in hydras.Ackermann.code]
+Code_Term_Formula_Proof.L [in hydras.Ackermann.code]
+code_nat_list.drop [in hydras.Ackermann.cPair]
+code_SysPrf.LNN.expressU2 [in Goedel.codeSysPrf]
+code_SysPrf.LNN.expressU1 [in Goedel.codeSysPrf]
+code_SysPrf.LNN.freeVarfU [in Goedel.codeSysPrf]
+code_SysPrf.LNN.v0 [in Goedel.codeSysPrf]
+code_SysPrf.LNN.fU [in Goedel.codeSysPrf]
+code_SysPrf.LNN.U [in Goedel.codeSysPrf]
+code_SysPrf.LNN.TextendsNN [in Goedel.codeSysPrf]
+code_SysPrf.LNN.T [in Goedel.codeSysPrf]
+code_SysPrf.codeArityRIsCorrect2 [in Goedel.codeSysPrf]
+code_SysPrf.codeArityRIsCorrect1 [in Goedel.codeSysPrf]
+code_SysPrf.codeArityFIsCorrect2 [in Goedel.codeSysPrf]
+code_SysPrf.codeArityFIsCorrect1 [in Goedel.codeSysPrf]
+code_SysPrf.codeArityR [in Goedel.codeSysPrf]
+code_SysPrf.codeArityF [in Goedel.codeSysPrf]
+code_SysPrf.L [in Goedel.codeSysPrf]
+Code_PA.codeFormulaInj [in hydras.Ackermann.codePA]
+Code_PA.codeFormula [in hydras.Ackermann.codePA]
+Code_PA.codeTerm [in hydras.Ackermann.codePA]
+compose2Examples.f [in hydras.MoreAck.PrimRecExamples]
+compose2Examples.f' [in hydras.MoreAck.PrimRecExamples]
+compose2Examples.g [in hydras.MoreAck.PrimRecExamples]
+compose2Examples.g' [in hydras.MoreAck.PrimRecExamples]
+compose2Examples.h [in hydras.MoreAck.PrimRecExamples]
+CompositionProofs.App.cn [in additions.AM]
+CompositionProofs.App.cp [in additions.AM]
+CompositionProofs.App.Hn [in additions.AM]
+CompositionProofs.App.Hp [in additions.AM]
+CompositionProofs.App.n [in additions.AM]
+CompositionProofs.App.p [in additions.AM]
+CompositionProofs.FFK.cp [in additions.AM]
+CompositionProofs.FFK.cq [in additions.AM]
+CompositionProofs.FFK.Hp [in additions.AM]
+CompositionProofs.FFK.Hq [in additions.AM]
+CompositionProofs.FFK.p [in additions.AM]
+CompositionProofs.FFK.q [in additions.AM]
+CompositionProofs.FK.cn [in additions.AM]
+CompositionProofs.FK.Hn [in additions.AM]
+CompositionProofs.FK.n [in additions.AM]
+CompositionProofs.KFK.Hpr [in additions.AM]
+CompositionProofs.KFK.Hq [in additions.AM]
+CompositionProofs.KFK.kpr [in additions.AM]
+CompositionProofs.KFK.mq [in additions.AM]
+CompositionProofs.KFK.p [in additions.AM]
+CompositionProofs.KFK.q [in additions.AM]
+CompositionProofs.KFK.r [in additions.AM]
+Composition.f [in hydras.MoreAck.PrimRecExamples]
+Composition.g [in hydras.MoreAck.PrimRecExamples]
+Composition.h [in hydras.MoreAck.PrimRecExamples]
+Composition.k [in hydras.MoreAck.PrimRecExamples]
+Composition.t [in hydras.MoreAck.PrimRecExamples]
+Composition.u [in hydras.MoreAck.PrimRecExamples]
+Composition.v [in hydras.MoreAck.PrimRecExamples]
+Composition.x [in hydras.MoreAck.PrimRecExamples]
+Composition.y [in hydras.MoreAck.PrimRecExamples]
+Composition.z [in hydras.MoreAck.PrimRecExamples]
+Constant_to_standard_Proof.delta [in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.m [in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.gamma [in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.l [in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.l_def [in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.P [in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.t [in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.Hpa [in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.Hpos [in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.Halpha [in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.n [in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.beta [in hydras.Epsilon0.Paths]
+Constant_to_standard_Proof.alpha [in hydras.Epsilon0.Paths]
+Correctness.L [in hydras.Ackermann.fol]
+Correctness.P [in hydras.Ackermann.fol]
+Correctness.Q [in hydras.Ackermann.fol]
+Correctness.R [in hydras.Ackermann.fol]
+Countable.Countable_bijection.g_bij [in hydras.Schutte.Countable]
+Countable.Countable_bijection.g [in hydras.Schutte.Countable]
+Countable.Countable_bijection.B [in hydras.Schutte.Countable]
+Countable.Countable_bijection.A [in hydras.Schutte.Countable]
+Countable.Countable_bijection.V [in hydras.Schutte.Countable]
+Countable.Definitions.A [in hydras.Schutte.Countable]
+Countable.Definitions.U [in hydras.Schutte.Countable]
+Countable.U [in hydras.Schutte.Countable]
+Counter_Example.is_in_seq [in hydras.solutions_exercises.MultisetWf]
+Counter_Example.Hwf [in hydras.solutions_exercises.MultisetWf]
+Counter_Example.R [in hydras.solutions_exercises.MultisetWf]
+CPair_projections.searchXY [in hydras.Ackermann.cPair]
+Cter_example.l [in hydras.solutions_exercises.schutte_cnf_counter_example]
+Cter_example.alpha [in hydras.solutions_exercises.schutte_cnf_counter_example]
+Cter_example.cnf_lt_epsilon0_iff [in hydras.solutions_exercises.schutte_cnf_counter_example]
+

D

+Deduction_Theorem.list_incl [in hydras.Ackermann.Deduction]
+Deduction_Theorem.SysPrf [in hydras.Ackermann.Deduction]
+Deduction_Theorem.Prf [in hydras.Ackermann.Deduction]
+Deduction_Theorem.Terms [in hydras.Ackermann.Deduction]
+Deduction_Theorem.Term [in hydras.Ackermann.Deduction]
+Deduction_Theorem.System [in hydras.Ackermann.Deduction]
+Deduction_Theorem.Formulas [in hydras.Ackermann.Deduction]
+Deduction_Theorem.Formula [in hydras.Ackermann.Deduction]
+Deduction_Theorem.L [in hydras.Ackermann.Deduction]
+Definitions.A [in additions.FirstSteps]
+Definitions.A [in hydras.Prelude.Simple_LexProd]
+Definitions.B [in hydras.Prelude.Simple_LexProd]
+Definitions.ltA [in hydras.Prelude.Simple_LexProd]
+Definitions.ltB [in hydras.Prelude.Simple_LexProd]
+Definitions.mult [in additions.FirstSteps]
+Definitions.one [in additions.FirstSteps]
+Definitions.pair2sig [in hydras.Prelude.Simple_LexProd]
+Definitions.wfA [in hydras.Prelude.Simple_LexProd]
+Definitions.wfB [in hydras.Prelude.Simple_LexProd]
+Defs.A [in hydras.Prelude.MoreOrders]
+Defs.lt [in hydras.Prelude.MoreOrders]
+depth_rec_demo.a [in hydras.MoreAck.FolExamples]
+depth_rec_demo.P [in hydras.MoreAck.FolExamples]
+depth_rec_demo.L [in hydras.MoreAck.FolExamples]
+Direct_proof.well_foundedness_proof.Acc_strong [in hydras.Epsilon0.T1]
+dom_AckNotPR.Hf [in hydras.MoreAck.AckNotPR]
+dom_AckNotPR.f [in hydras.MoreAck.AckNotPR]
+DS_iota.Proof_case_6.Hn [in hydras.Hydra.O2H]
+DS_iota.Proof_case_6.n [in hydras.Hydra.O2H]
+DS_iota.Proof_case_6.gamma_pos [in hydras.Hydra.O2H]
+DS_iota.Proof_case_6.Hgamma [in hydras.Hydra.O2H]
+DS_iota.Proof_case_6.Hbeta [in hydras.Hydra.O2H]
+DS_iota.Proof_case_6.gamma [in hydras.Hydra.O2H]
+DS_iota.Proof_case_6.beta [in hydras.Hydra.O2H]
+DS_iota.Proof_case_5.Hn [in hydras.Hydra.O2H]
+DS_iota.Proof_case_5.n [in hydras.Hydra.O2H]
+DS_iota.Proof_case_5.Hgamma [in hydras.Hydra.O2H]
+DS_iota.Proof_case_5.gamma [in hydras.Hydra.O2H]
+DS_iota.Proof_case_4.Hn [in hydras.Hydra.O2H]
+DS_iota.Proof_case_4.n [in hydras.Hydra.O2H]
+DS_iota.Proof_case_4.Hlim [in hydras.Hydra.O2H]
+DS_iota.Proof_case_4.Hlambda [in hydras.Hydra.O2H]
+DS_iota.Proof_case_4.lambda [in hydras.Hydra.O2H]
+DS_iota.Hrec [in hydras.Hydra.O2H]
+DS_iota.nonzero [in hydras.Hydra.O2H]
+DS_iota.Halpha [in hydras.Hydra.O2H]
+DS_iota.i [in hydras.Hydra.O2H]
+DS_iota.alpha [in hydras.Hydra.O2H]
+

E

+Equality_Logic_Rules.termsMap [in hydras.Ackermann.folLogic3]
+Equality_Logic_Rules.SysPrf [in hydras.Ackermann.folLogic3]
+Equality_Logic_Rules.Prf [in hydras.Ackermann.folLogic3]
+Equality_Logic_Rules.L [in hydras.Ackermann.folLogic3]
+Equations_for_addition.case3.Hac [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.case2.Hac [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.case1.Hac [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.beta [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.alpha [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.Hnfc [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.Hnfa [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.p [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.n [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.d [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.c [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.b [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.a [in hydras.Schutte.Correctness_E0]
+Equivalence.A [in additions.Compatibility]
+Equivalence.is_power_of [in additions.Compatibility]
+Equivalence.one [in additions.Compatibility]
+Equivalence.op [in additions.Compatibility]
+Equivalence.pos_iter_M [in additions.Compatibility]
+essai.n [in hydras.Epsilon0.T1]
+Examples.f [in hydras.Ackermann.LNT]
+Examples.f1 [in hydras.MoreAck.LNN_Examples]
+Examples.f2 [in hydras.MoreAck.LNN_Examples]
+Examples.f2' [in hydras.MoreAck.LNN_Examples]
+Examples.f3 [in hydras.MoreAck.LNN_Examples]
+Examples.f4 [in hydras.MoreAck.LNN_Examples]
+Examples.t1 [in hydras.MoreAck.LNN_Examples]
+Exs.f [in hydras.MoreAck.PrimRecExamples]
+Exs.fdiv2 [in hydras.MoreAck.PrimRecExamples]
+Exs.ffib [in hydras.MoreAck.PrimRecExamples]
+

F

+First_Order_Logic.Formula_Decidability.consTermsHelp [in hydras.Ackermann.fol]
+First_Order_Logic.Formula_Decidability.nilTermsHelp [in hydras.Ackermann.fol]
+First_Order_Logic.Formula_Decidability.language_eqdec [in hydras.Ackermann.fol]
+First_Order_Logic.L [in hydras.Ackermann.fol]
+Fix.A [in hydras.Prelude.Restriction]
+Fix.E [in hydras.Prelude.Restriction]
+Fix.F [in hydras.Prelude.Restriction]
+Fix.Hwf [in hydras.Prelude.Restriction]
+Fix.P [in hydras.Prelude.Restriction]
+Fix.R [in hydras.Prelude.Restriction]
+Fix.Rwf [in hydras.Prelude.Restriction]
+Fol_Properties.lt_depth [in hydras.Ackermann.folProp]
+Fol_Properties.L [in hydras.Ackermann.folProp]
+F_monotony_l.case_lt.Hd [in hydras.Epsilon0.F_alpha]
+F_monotony_l.case_lt.Hlt [in hydras.Epsilon0.F_alpha]
+F_monotony_l.case_lt.n [in hydras.Epsilon0.F_alpha]
+F_monotony_l.case_eq.Heq [in hydras.Epsilon0.F_alpha]
+F_monotony_l.H'_beta_alpha [in hydras.Epsilon0.F_alpha]
+F_monotony_l.beta [in hydras.Epsilon0.F_alpha]
+F_monotony_l.alpha [in hydras.Epsilon0.F_alpha]
+F_alpha_notPR.H [in hydras.Epsilon0.F_omega]
+F_alpha_notPR.case_lt.Halpha [in hydras.Epsilon0.F_omega]
+F_alpha_notPR.alpha [in hydras.Epsilon0.F_omega]
+

G

+Gamma.All_OK.c [in additions.AM]
+Gamma.All_OK.n [in additions.AM]
+Gamma.gamma [in additions.AM]
+Gamma.gamma [in additions.Euclidean_Chains]
+General_Relations.elagage.to_nat_elagage.R [in hydras.Schutte.GRelations]
+General_Relations.surjection2injection.R_surj [in hydras.Schutte.GRelations]
+General_Relations.surjection2injection.R [in hydras.Schutte.GRelations]
+General_Relations.injection2surjection.R_inj [in hydras.Schutte.GRelations]
+General_Relations.injection2surjection.R [in hydras.Schutte.GRelations]
+General_Relations.DB [in hydras.Schutte.GRelations]
+General_Relations.DA [in hydras.Schutte.GRelations]
+General_Relations.B [in hydras.Schutte.GRelations]
+General_Relations.A [in hydras.Schutte.GRelations]
+General_Relations.Definitions.R [in hydras.Schutte.GRelations]
+General_Relations.Definitions.DB [in hydras.Schutte.GRelations]
+General_Relations.Definitions.DA [in hydras.Schutte.GRelations]
+General_Relations.Definitions.B [in hydras.Schutte.GRelations]
+General_Relations.Definitions.A [in hydras.Schutte.GRelations]
+Generic.A [in hydras.Prelude.Merge_Sort]
+Generic.Merging.Correctness.merge_sort.split_permutation [in hydras.Prelude.Merge_Sort]
+Generic.Merging.Correctness.merge_sort.split_decr [in hydras.Prelude.Merge_Sort]
+Generic.Merging.Correctness.merge_sort.split [in hydras.Prelude.Merge_Sort]
+Generic.Merging.le [in hydras.Prelude.Merge_Sort]
+Goedel's_1st_Incompleteness.expressT2 [in Goedel.goedel1]
+Goedel's_1st_Incompleteness.expressT1 [in Goedel.goedel1]
+Goedel's_1st_Incompleteness.freeVarRepT [in Goedel.goedel1]
+Goedel's_1st_Incompleteness.v0 [in Goedel.goedel1]
+Goedel's_1st_Incompleteness.repT [in Goedel.goedel1]
+Goedel's_1st_Incompleteness.extendsNN [in Goedel.goedel1]
+Goedel's_1st_Incompleteness.T [in Goedel.goedel1]
+Goedel's_2nd_Incompleteness.HBL3 [in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.HBL2 [in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.expressT2 [in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.expressT1 [in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.freeVarRepT [in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.v0 [in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.repT [in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.extendsPA [in Goedel.goedel2]
+Goedel's_2nd_Incompleteness.T [in Goedel.goedel2]
+

H

+Hypos.Hn [in hydras.Prelude.First_toggle]
+Hypos.Hnp [in hydras.Prelude.First_toggle]
+Hypos.Hp [in hydras.Prelude.First_toggle]
+Hypos.R [in hydras.Prelude.First_toggle]
+Hypos.search_toggle [in hydras.Prelude.First_toggle]
+Hypos.spec [in hydras.Prelude.First_toggle]
+H'_F.IHalpha [in hydras.Epsilon0.F_alpha]
+H'_F.alpha [in hydras.Epsilon0.F_alpha]
+H'_F.P [in hydras.Epsilon0.F_alpha]
+H'_omega_cube_3.N [in hydras.Epsilon0.Hprime]
+H'_omega_cube_3.f [in hydras.Epsilon0.Hprime]
+H'_cons.i [in hydras.Epsilon0.Hprime]
+H'_cons.alpha [in hydras.Epsilon0.Hprime]
+

I

+ImpossibilityProof.m [in gaia_hydras.GHydra]
+ImpossibilityProof.mh [in gaia_hydras.GHydra]
+ImpossibilityProof.mu [in gaia_hydras.GHydra]
+ImpossibilityProof.nfMu [in gaia_hydras.GHydra]
+Impossibility_Proof.small_h [in hydras.Hydra.Epsilon0_Needed_Free]
+Impossibility_Proof.big_h [in hydras.Hydra.Epsilon0_Needed_Free]
+Impossibility_Proof.small_h [in hydras.Hydra.Epsilon0_Needed_Std]
+Impossibility_Proof.big_h [in hydras.Hydra.Epsilon0_Needed_Std]
+Impossibility_Proof.small_h [in hydras.Hydra.Omega2_Small]
+Impossibility_Proof.iota [in hydras.Hydra.Omega2_Small]
+Impossibility_Proof.big_h [in hydras.Hydra.Omega2_Small]
+Impossibility_Proof.m [in hydras.Hydra.Omega2_Small]
+Impossibility_Proof.small_h [in hydras.Hydra.Omega_Small]
+Impossibility_Proof.big_h [in hydras.Hydra.Omega_Small]
+Impossibility_Proof.iota [in hydras.Hydra.Omega_Small]
+Impossibility_Proof.m [in hydras.Hydra.Omega_Small]
+Impossibility1.m [in hydras.solutions_exercises.MultisetWf]
+Impossibility1.mDecr [in hydras.solutions_exercises.MultisetWf]
+Impossibility1.x [in hydras.solutions_exercises.MultisetWf]
+Impossibility1.y [in hydras.solutions_exercises.MultisetWf]
+Inclusion_ij.b [in hydras.OrdinalNotations.ON_Finite]
+Inclusion_ij.Hij [in hydras.OrdinalNotations.ON_Finite]
+Inclusion_ij.j [in hydras.OrdinalNotations.ON_Finite]
+Inclusion_ij.i [in hydras.OrdinalNotations.ON_Finite]
+inductive_step.Hn [in hydras.Epsilon0.F_omega]
+inductive_step.n [in hydras.Epsilon0.F_omega]
+Inverse_Image_transp.Rof [in additions.Wf_transparent]
+Inverse_Image_transp.f [in additions.Wf_transparent]
+Inverse_Image_transp.R [in additions.Wf_transparent]
+Inverse_Image_transp.B [in additions.Wf_transparent]
+Inverse_Image_transp.A [in additions.Wf_transparent]
+inversion_of_bijection.f_b [in hydras.Schutte.PartialFun]
+inversion_of_bijection.inv_spec [in hydras.Schutte.PartialFun]
+inversion_of_bijection.f [in hydras.Schutte.PartialFun]
+inversion_of_bijection.DB [in hydras.Schutte.PartialFun]
+inversion_of_bijection.DA [in hydras.Schutte.PartialFun]
+inversion_of_bijection.inhA [in hydras.Schutte.PartialFun]
+inversion_of_bijection.B [in hydras.Schutte.PartialFun]
+inversion_of_bijection.A [in hydras.Schutte.PartialFun]
+

K

+KFK_proof.Hq_prop [in additions.Euclidean_Chains]
+KFK_proof.Hbr_prop [in additions.Euclidean_Chains]
+KFK_proof.Hq [in additions.Euclidean_Chains]
+KFK_proof.Hbr [in additions.Euclidean_Chains]
+KFK_proof.fq [in additions.Euclidean_Chains]
+KFK_proof.kbr [in additions.Euclidean_Chains]
+KFK_proof.r [in additions.Euclidean_Chains]
+KFK_proof.q [in additions.Euclidean_Chains]
+KFK_proof.b [in additions.Euclidean_Chains]
+KP.h' [in hydras.Hydra.KP_example]
+

L

+Lemma_4_4_Proof.H0 [in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.H [in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.Hs [in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.s [in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.n [in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.Hbeta [in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.Halpha [in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.beta [in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_Proof.alpha [in hydras.Epsilon0.Large_Sets]
+Lemma_4_3_Proof.H4 [in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.H1 [in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.H0 [in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.n2 [in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.n1 [in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.n0 [in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.nf2 [in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.nf1 [in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.H00 [in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.beta [in hydras.Epsilon0.Paths]
+Lemma_4_3_Proof.alpha [in hydras.Epsilon0.Paths]
+LExamples.ex1 [in hydras.Ackermann.fol]
+LExamples.ex2 [in hydras.Ackermann.fol]
+LExamples.ex3 [in hydras.Ackermann.fol]
+LExamples.L [in hydras.Ackermann.fol]
+LExamples.P [in hydras.Ackermann.fol]
+LExamples.Q [in hydras.Ackermann.fol]
+lexprod.A [in hydras.Prelude.DecPreOrder_Instances]
+lexprod.B [in hydras.Prelude.DecPreOrder_Instances]
+lexprod.leA [in hydras.Prelude.DecPreOrder_Instances]
+lexprod.leB [in hydras.Prelude.DecPreOrder_Instances]
+lim.f [in hydras.Epsilon0.Large_Sets]
+lim.H [in hydras.Epsilon0.Large_Sets]
+lim.Hlim [in hydras.Epsilon0.Large_Sets]
+lim.Hnf [in hydras.Epsilon0.Large_Sets]
+lim.lambda [in hydras.Epsilon0.Large_Sets]
+List_Remove.Aeq_dec [in hydras.Ackermann.ListExt]
+List_Remove.A [in hydras.Ackermann.ListExt]
+LNN_FixPoint.codeFormula [in Goedel.fixPoint]
+LNT_FixPoint.codeFormula [in Goedel.fixPoint]
+Logic_Rules.SysPrf [in hydras.Ackermann.folLogic]
+Logic_Rules.Prf [in hydras.Ackermann.folLogic]
+Logic_Rules.Terms [in hydras.Ackermann.folLogic]
+Logic_Rules.Term [in hydras.Ackermann.folLogic]
+Logic_Rules.System [in hydras.Ackermann.folLogic]
+Logic_Rules.Formulas [in hydras.Ackermann.folLogic]
+Logic_Rules.Formula [in hydras.Ackermann.folLogic]
+Logic_Rules.L [in hydras.Ackermann.folLogic]
+lt_incl_rpo.H [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.nf2 [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.nf1 [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.Hrec [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.Hsize [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.n2 [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.n1 [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.c2 [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.b2 [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.a2 [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.c1 [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.b1 [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.a1 [in hydras.Gamma0.Gamma0]
+lt_incl_rpo.s [in hydras.Gamma0.Gamma0]
+L_correct_proof.P [in hydras.Epsilon0.L_alpha]
+

M

+Make.DoubleRecursion.Pl2 [in hydras.rpo.term]
+Make.DoubleRecursion.P2 [in hydras.rpo.term]
+Make.Recursion.P [in hydras.rpo.term]
+Make.Recursion.Pl [in hydras.rpo.term]
+Model_Theory.Consistent_Theory.T [in hydras.Ackermann.model]
+Model_Theory.M [in hydras.Ackermann.model]
+Model_Theory.L [in hydras.Ackermann.model]
+MoreOrderType.disp [in gaia_hydras.onType]
+MoreOrderType.Succ_no_limit.Hlim [in gaia_hydras.onType]
+MoreOrderType.Succ_no_limit.Hsucc [in gaia_hydras.onType]
+MoreOrderType.Succ_no_limit.s [in gaia_hydras.onType]
+MoreOrderType.Succ_no_limit.y [in gaia_hydras.onType]
+MoreOrderType.Succ_no_limit.x [in gaia_hydras.onType]
+MoreOrderType.T [in gaia_hydras.onType]
+More_Logic_Rules.SysPrf [in hydras.Ackermann.folLogic2]
+More_Logic_Rules.Prf [in hydras.Ackermann.folLogic2]
+More_Logic_Rules.Terms [in hydras.Ackermann.folLogic2]
+More_Logic_Rules.Term [in hydras.Ackermann.folLogic2]
+More_Logic_Rules.System [in hydras.Ackermann.folLogic2]
+More_Logic_Rules.Formulas [in hydras.Ackermann.folLogic2]
+More_Logic_Rules.Formula [in hydras.Ackermann.folLogic2]
+More_Logic_Rules.L [in hydras.Ackermann.folLogic2]
+M_given.E_eq [in additions.Pow_variant]
+M_given.E_one [in additions.Pow_variant]
+M_given.E_op [in additions.Pow_variant]
+M_given.A [in additions.Pow_variant]
+M_given.E_one [in additions.Pow]
+M_given.A [in additions.Pow]
+M2_def.rt [in additions.Monoid_instances]
+M2_def.mult [in additions.Monoid_instances]
+M2_def.plus [in additions.Monoid_instances]
+M2_def.one [in additions.Monoid_instances]
+M2_def.zero [in additions.Monoid_instances]
+M2_def.A [in additions.Monoid_instances]
+M2.Definitions.A [in additions.Naive]
+M2.Definitions.mult [in additions.Naive]
+M2.Definitions.one [in additions.Naive]
+M2.Definitions.plus [in additions.Naive]
+M2.Definitions.rt [in additions.Naive]
+M2.Definitions.zero [in additions.Naive]
+M2.M2_Definitions.rt [in additions.FirstSteps]
+M2.M2_Definitions.mult [in additions.FirstSteps]
+M2.M2_Definitions.plus [in additions.FirstSteps]
+M2.M2_Definitions.one [in additions.FirstSteps]
+M2.M2_Definitions.zero [in additions.FirstSteps]
+M2.M2_Definitions.A [in additions.FirstSteps]
+

N

+Nmodulo.m [in additions.Monoid_instances]
+Nmodulo.m_gt_1 [in additions.Monoid_instances]
+non_optimality_proof.binary_opt [in additions.Addition_Chains]
+N_mod.m_fixed.m [in additions.Naive]
+

O

+ONDef.Exports.Lemmas.disp [in gaia_hydras.onType]
+ONDef.Exports.Lemmas.U [in gaia_hydras.onType]
+ONDef.Packing.cT [in gaia_hydras.onType]
+ONDef.Packing.disp [in gaia_hydras.onType]
+onFiniteDef.i [in gaia_hydras.onType]
+On_alpha.S2.H [in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.S1.Hbeta [in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.S1.beta [in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.S1.H [in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.HnonZero [in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.Halpha [in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.alpha [in hydras.solutions_exercises.Limit_Infinity]
+On_Iota.unique_P [in hydras.Schutte.MoreEpsilonIota]
+On_Iota.inhA [in hydras.Schutte.MoreEpsilonIota]
+On_Iota.P [in hydras.Schutte.MoreEpsilonIota]
+On_Iota.A [in hydras.Schutte.MoreEpsilonIota]
+ordering_function_unicity.O2 [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.O1 [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.f2 [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.f1 [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.A2 [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.A1 [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity.B [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.O2 [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.O1 [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.f2 [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.f1 [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.A2 [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.A1 [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1.B [in hydras.Schutte.Ordering_Functions]
+

P

+phi_to_psi.alpha [in hydras.Gamma0.Gamma0]
+phi0_mult.f_ok [in hydras.Epsilon0.Large_Sets]
+phi0_mult.f_Sle [in hydras.Epsilon0.Large_Sets]
+phi0_mult.f_mono [in hydras.Epsilon0.Large_Sets]
+phi0_mult.Halpha [in hydras.Epsilon0.Large_Sets]
+phi0_mult.f [in hydras.Epsilon0.Large_Sets]
+phi0_mult.alpha [in hydras.Epsilon0.Large_Sets]
+Primitive_recursion.f [in hydras.MoreAck.PrimRecExamples]
+Primitive_recursion.h [in hydras.MoreAck.PrimRecExamples]
+Primitive_recursion.g [in hydras.MoreAck.PrimRecExamples]
+Primitive_recursion.z [in hydras.MoreAck.PrimRecExamples]
+Primitive_recursion.y [in hydras.MoreAck.PrimRecExamples]
+Primitive_recursion.x [in hydras.MoreAck.PrimRecExamples]
+ProofH.Formula [in hydras.Ackermann.folProof]
+ProofH.Formulas [in hydras.Ackermann.folProof]
+ProofH.L [in hydras.Ackermann.folProof]
+ProofH.System [in hydras.Ackermann.folProof]
+ProofH.Term [in hydras.Ackermann.folProof]
+ProofH.Terms [in hydras.Ackermann.folProof]
+ProofOfLexwf.A [in hydras.solutions_exercises.MultisetWf]
+ProofOfLexwf.Accs [in hydras.solutions_exercises.MultisetWf]
+ProofOfLexwf.HwfA [in hydras.solutions_exercises.MultisetWf]
+ProofOfLexwf.ltA [in hydras.solutions_exercises.MultisetWf]
+Proofs_of_unicity.Proofs.S2.Hgammaalpha [in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.S2.gamma [in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.S1.Halphagamma [in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.S1.gamma [in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.Halphabeta [in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.beta [in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.alpha [in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_lt_succ_le.Proofs.S1.HGammaBeta [in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le.Proofs.S1.gamma [in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le.Proofs.Halphabeta [in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le.Proofs.beta [in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le.Proofs.alpha [in hydras.solutions_exercises.lt_succ_le]
+Proof_of_dist.P [in hydras.Epsilon0.T1]
+Proof_of_mult_nf.Induction.IHbeta [in hydras.Epsilon0.T1]
+Proof_of_mult_nf.Induction.Hbeta [in hydras.Epsilon0.T1]
+Proof_of_mult_nf.Induction.beta [in hydras.Epsilon0.T1]
+Proof_of_mult_nf.P [in hydras.Epsilon0.T1]
+Proof_of_mult_nf.Halpha [in hydras.Epsilon0.T1]
+Proof_of_mult_nf.alpha [in hydras.Epsilon0.T1]
+Proof_of_MinIsPR.minPR [in hydras.solutions_exercises.MinPR2]
+Proof_of_oplus_lt1.H0 [in hydras.Epsilon0.Hessenberg]
+Proof_of_oplus_lt1.n [in hydras.Epsilon0.Hessenberg]
+Proof_of_oplus_lt1.a2 [in hydras.Epsilon0.Hessenberg]
+Proof_of_oplus_lt1.a1 [in hydras.Epsilon0.Hessenberg]
+Proof_of_Ackn_PR.S_step.n [in hydras.MoreAck.AckNotPR]
+Proof_of_4_5_2.HlargeA [in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.HAB1 [in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.HB [in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.HA [in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.Halpha [in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.alpha [in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.B2 [in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.B1 [in hydras.Epsilon0.Large_Sets]
+Proof_of_4_5_2.A [in hydras.Epsilon0.Large_Sets]
+Proof_of_Lemma5.Alpha_positive.closedness.IM [in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.closedness.CM [in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.closedness.HM [in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.closedness.M [in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.Proof_unbounded.gamma [in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.Proof_unbounded.beta [in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.IHalpha [in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.alpha_pos [in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.alpha [in hydras.Schutte.Critical]
+Proof_of_Lemma5.P [in hydras.Schutte.Critical]
+Proof_of_FibIsPR.fib_iter_cPair [in hydras.solutions_exercises.FibonacciPR]
+Proof_of_FibIsPR.fib_step_cPair [in hydras.solutions_exercises.FibonacciPR]
+Proof_of_FibIsPR.fib_iter [in hydras.solutions_exercises.FibonacciPR]
+Proof_of_FibIsPR.fib_step [in hydras.solutions_exercises.FibonacciPR]
+Proof_of_MinIsPR.min_alt [in hydras.solutions_exercises.MinPR]
+Proof_of_H'_mono_l.Limit_case.Hbeta [in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.Succ_case.Hgamma [in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.Succ_case.gamma [in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.H_alpha_beta [in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.beta [in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.alpha [in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.alpha_limit.Hlim [in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.alpha_Succ.alpha_def [in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.alpha_Succ.beta [in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.Halpha [in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.alpha [in hydras.Epsilon0.Hprime]
+proof_of_associativity.g_alpha_beta [in hydras.Schutte.Addition]
+proof_of_associativity.f_alpha_beta [in hydras.Schutte.Addition]
+proof_of_associativity.beta [in hydras.Schutte.Addition]
+proof_of_associativity.alpha [in hydras.Schutte.Addition]
+Proof_of_lt_not_wf.seq_intro.is_in_seq [in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf.seq_intro.Rwf [in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf.seq_intro.R [in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf.seq_intro.seq [in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf.seq_intro.A [in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf.lt_wf [in hydras.solutions_exercises.T1_ltNotWf]
+Properties.The_induction.alpha_limit.Hlim [in hydras.Epsilon0.F_alpha]
+Properties.The_induction.alpha_Succ.alpha_def [in hydras.Epsilon0.F_alpha]
+Properties.The_induction.alpha_Succ.beta [in hydras.Epsilon0.F_alpha]
+Properties.The_induction.Halpha [in hydras.Epsilon0.F_alpha]
+Properties.The_induction.alpha [in hydras.Epsilon0.F_alpha]
+

R

+Refinement_proof.E_eq [in additions.Addition_Chains]
+Refinement_proof.E_one [in additions.Addition_Chains]
+Refinement_proof.E_op [in additions.Addition_Chains]
+Refinement_proof.A [in additions.Addition_Chains]
+Replacement.Formula [in hydras.Ackermann.folReplace]
+Replacement.Formulas [in hydras.Ackermann.folReplace]
+Replacement.L [in hydras.Ackermann.folReplace]
+Replacement.SysPrf [in hydras.Ackermann.folReplace]
+Replacement.System [in hydras.Ackermann.folReplace]
+Replacement.Term [in hydras.Ackermann.folReplace]
+Replacement.Terms [in hydras.Ackermann.folReplace]
+RepresentableExpressible.closedT1 [in hydras.Ackermann.expressible]
+RepresentableExpressible.nn1 [in hydras.Ackermann.expressible]
+RepresentableExpressible.T [in hydras.Ackermann.expressible]
+restricted_recursion.R [in hydras.Prelude.Restriction]
+restricted_recursion.E [in hydras.Prelude.Restriction]
+restricted_recursion.A [in hydras.Prelude.Restriction]
+restricted_recursion.R [in gaia_hydras.nfwfgaia]
+restricted_recursion.P [in gaia_hydras.nfwfgaia]
+restricted_recursion.A [in gaia_hydras.nfwfgaia]
+Rosser's_Incompleteness.expressT2 [in Goedel.rosserPA]
+Rosser's_Incompleteness.expressT1 [in Goedel.rosserPA]
+Rosser's_Incompleteness.freeVarRepT [in Goedel.rosserPA]
+Rosser's_Incompleteness.v0 [in Goedel.rosserPA]
+Rosser's_Incompleteness.repT [in Goedel.rosserPA]
+Rosser's_Incompleteness.extendsPA [in Goedel.rosserPA]
+Rosser's_Incompleteness.T [in Goedel.rosserPA]
+Rosser's_Incompleteness.expressT2 [in Goedel.rosser]
+Rosser's_Incompleteness.expressT1 [in Goedel.rosser]
+Rosser's_Incompleteness.freeVarRepT [in Goedel.rosser]
+Rosser's_Incompleteness.v0 [in Goedel.rosser]
+Rosser's_Incompleteness.repT [in Goedel.rosser]
+Rosser's_Incompleteness.extendsNN [in Goedel.rosser]
+Rosser's_Incompleteness.T [in Goedel.rosser]
+R_given.R [in hydras.Prelude.Sort_spec]
+R_given.A [in hydras.Prelude.Sort_spec]
+

S

+Semantics.A [in additions.AM]
+Semantics.mul [in additions.AM]
+Semantics.one [in additions.AM]
+Sequences.A [in gaia_hydras.nfwfgaia]
+Sequences.R [in gaia_hydras.nfwfgaia]
+Sequences.W [in gaia_hydras.nfwfgaia]
+sqrtIsPR.P [in hydras.solutions_exercises.isqrt]
+sqrtIsPR.Proof_isqrt.n [in hydras.solutions_exercises.isqrt]
+step.F [in hydras.Epsilon0.F_omega]
+step.Hn [in hydras.Epsilon0.F_omega]
+step.n [in hydras.Epsilon0.F_omega]
+SubAllVars.L [in hydras.Ackermann.subAll]
+Substitution_Properties.SysPrf [in hydras.Ackermann.subProp]
+Substitution_Properties.Terms [in hydras.Ackermann.subProp]
+Substitution_Properties.Term [in hydras.Ackermann.subProp]
+Substitution_Properties.System [in hydras.Ackermann.subProp]
+Substitution_Properties.Formulas [in hydras.Ackermann.subProp]
+Substitution_Properties.Formula [in hydras.Ackermann.subProp]
+Substitution_Properties.L [in hydras.Ackermann.subProp]
+succ.beta [in hydras.Epsilon0.Large_Sets]
+succ.f [in hydras.Epsilon0.Large_Sets]
+succ.f_ok [in hydras.Epsilon0.Large_Sets]
+succ.f_Sle [in hydras.Epsilon0.Large_Sets]
+succ.f_mono [in hydras.Epsilon0.Large_Sets]
+succ.Hbeta [in hydras.Epsilon0.Large_Sets]
+S1.H [in hydras.solutions_exercises.is_F_monotonous]
+S1.instance_H [in hydras.solutions_exercises.is_F_monotonous]
+S1.Limit.Hlambda [in hydras.solutions_exercises.F_3]
+S1.Limit.IHlambda [in hydras.solutions_exercises.F_3]
+S1.Limit.lambda [in hydras.solutions_exercises.F_3]
+S1.Limit.S3.Hn [in hydras.solutions_exercises.F_3]
+S1.Limit.S3.n [in hydras.solutions_exercises.F_3]
+S1.P [in hydras.solutions_exercises.F_3]
+S1.Successor.alpha [in hydras.solutions_exercises.F_3]
+S1.Successor.H_alpha [in hydras.solutions_exercises.F_3]
+S1.Successor.S2.Hn [in hydras.solutions_exercises.F_3]
+S1.Successor.S2.n [in hydras.solutions_exercises.F_3]
+S2.a [in additions.Addition_Chains]
+S2.A [in additions.Addition_Chains]
+S2.b [in additions.Addition_Chains]
+S2.B [in additions.Addition_Chains]
+S2.R_true [in additions.Addition_Chains]
+S256.mod256 [in additions.Monoid_instances]
+

T

+Term.DoubleRecursion.Pl2 [in hydras.rpo.term]
+Term.DoubleRecursion.P2 [in hydras.rpo.term]
+Term.Recursion.P [in hydras.rpo.term]
+Term.Recursion.Pl [in hydras.rpo.term]
+the_context.Lt [in hydras.Schutte.Well_Orders]
+the_context.M [in hydras.Schutte.Well_Orders]
+Th13_5.verso.U_fixed.alpha_ [in hydras.Schutte.Ordering_Functions]
+Th13_5.verso.U_fixed.U_inc_A [in hydras.Schutte.Ordering_Functions]
+Th13_5.verso.U_fixed.U_den [in hydras.Schutte.Ordering_Functions]
+Th13_5.verso.U_fixed.U_non_empty [in hydras.Schutte.Ordering_Functions]
+Th13_5.verso.U_fixed.U [in hydras.Schutte.Ordering_Functions]
+Th13_5.verso.B_closed [in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.M_fixed.U [in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.M_fixed.den [in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.M_fixed.ne [in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.M_fixed.inc [in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.M_fixed.M [in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.f_cont [in hydras.Schutte.Ordering_Functions]
+Th13_5.f_ord [in hydras.Schutte.Ordering_Functions]
+Th13_5.f [in hydras.Schutte.Ordering_Functions]
+Th13_5.B [in hydras.Schutte.Ordering_Functions]
+Th13_5.A [in hydras.Schutte.Ordering_Functions]
+Toy.Drinkers_theorem.f [in hydras.MoreAck.FolExamples]
+Toy.OnSubstF.F [in hydras.MoreAck.FolExamples]
+Translate_Proof.AxiomsOK [in hydras.Ackermann.LNN2LNT]
+Translate_Proof.V [in hydras.Ackermann.LNN2LNT]
+Translate_Proof.U [in hydras.Ackermann.LNN2LNT]
+trans_proof.induc [in hydras.Gamma0.Gamma0]
+trans_proof.H23 [in hydras.Gamma0.Gamma0]
+trans_proof.H12 [in hydras.Gamma0.Gamma0]
+trans_proof.n3 [in hydras.Gamma0.Gamma0]
+trans_proof.n2 [in hydras.Gamma0.Gamma0]
+trans_proof.n1 [in hydras.Gamma0.Gamma0]
+trans_proof.c3 [in hydras.Gamma0.Gamma0]
+trans_proof.b3 [in hydras.Gamma0.Gamma0]
+trans_proof.a3 [in hydras.Gamma0.Gamma0]
+trans_proof.c2 [in hydras.Gamma0.Gamma0]
+trans_proof.b2 [in hydras.Gamma0.Gamma0]
+trans_proof.a2 [in hydras.Gamma0.Gamma0]
+trans_proof.c1 [in hydras.Gamma0.Gamma0]
+trans_proof.b1 [in hydras.Gamma0.Gamma0]
+trans_proof.a1 [in hydras.Gamma0.Gamma0]
+

V

+Variants.E [in hydras.Prelude.WfVariant]
+Variants.lt [in hydras.Prelude.WfVariant]
+Variants.m [in hydras.Prelude.WfVariant]
+Variants.T [in hydras.Prelude.WfVariant]
+Variants.tr [in hydras.Prelude.WfVariant]
+

W

+Well_Formed_Term.Well_Formed_Formula.Formula [in hydras.Ackermann.wellFormed]
+Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect2 [in hydras.Ackermann.wellFormed]
+Well_Formed_Term.Well_Formed_Formula.codeArityRIsCorrect1 [in hydras.Ackermann.wellFormed]
+Well_Formed_Term.Well_Formed_Formula.codeArityR [in hydras.Ackermann.wellFormed]
+Well_Formed_Term.Terms [in hydras.Ackermann.wellFormed]
+Well_Formed_Term.Term [in hydras.Ackermann.wellFormed]
+Well_Formed_Term.codeArityFIsCorrect2 [in hydras.Ackermann.wellFormed]
+Well_Formed_Term.codeArityFIsCorrect1 [in hydras.Ackermann.wellFormed]
+Well_Formed_Term.codeArityF [in hydras.Ackermann.wellFormed]
+Well_Formed_Term.L [in hydras.Ackermann.wellFormed]
+well_founded.R [in hydras.Gamma0.Gamma0]
+with_matrices.R [in additions.fib]
+

+

Library Index

+

A

+Ack
+AckNotPR
+Addition
+Addition_Chains
+AM
+AP
+

B

+BadSubst
+Battle_length
+BigBattle
+BinaryStrat
+

C

+Canon
+checkPrf
+closure
+CNF
+code
+codeFreeVar
+codeList
+codeNatToTerm
+codePA
+codeSubFormula
+codeSubTerm
+codeSysPrf
+Comparable
+Compatibility
+Compat815
+Correctness_E0
+Countable
+cPair
+Critical
+

D

+decidable_set
+DecPreOrder
+DecPreOrder_Instances
+Deduction
+Demo
+Demo_power
+Dichotomy
+dickson
+

E

+Epsilon0
+Epsilon0rpo
+Epsilon0_Needed_Std
+Epsilon0_Needed_Free
+Epsilon0_Needed_Generic
+Euclidean_Chains
+Example_3PlusOmega
+expressible
+expressibleExamples
+Exp2
+extEqualNat
+E0
+

F

+fib
+FibonacciPR
+Fib2
+FirstSteps
+First_toggle
+fixPoint
+fol
+FolExamples
+folLogic
+folLogic2
+folLogic3
+folProof
+folProp
+folReplace
+Fuel
+F_3
+F_alpha
+F_omega
+

G

+GaiaToHydra
+Gamma0
+GCanon
+ge_omega_iff
+GF_alpha
+GHessenberg
+GHprime
+GHydra
+GLarge_Sets
+GL_alpha
+goedel1
+goedel2
+GPaths
+GPrelude
+GRelations
+

H

+Hessenberg
+Hprime
+HydraGaia_Examples
+Hydra_Termination
+Hydra_Extraction
+Hydra_Lemmas
+Hydra_Definitions
+Hydra_Examples
+Hydra_Theorems
+

I

+isqrt
+is_F_monotonous
+Iterates
+Iterate_compat
+

K

+KP_example
+

L

+Languages
+Large_Sets
+Large_Sets_Examples
+LibHyps_Experiments
+Limit_Infinity
+ListExt
+list_permut
+list_set
+LNN
+LNN_Examples
+LNN2LNT
+LNT
+lt_succ_le
+Lub
+L_alpha
+

M

+Merge_Sort
+MinPR
+MinPR2
+misc
+model
+Monoid_instances
+Monoid_def
+MoreDecidable
+MoreEpsilonIota
+MoreLibHyps
+MoreLists
+MoreOrders
+MorePRExamples
+MoreVectors
+More_Arith
+more_list
+More_on_positive
+MultisetWf
+

N

+Naive
+NewNotations
+nfwfgaia
+NN
+NNtheory
+NN2PA
+

O

+OmegaOmega
+Omega_Small
+Omega2_Small
+OnCodeList
+onType
+ON_Generic
+ON_mult
+ON_Finite
+ON_Omega_plus_omega
+ON_Omega
+ON_Omega2
+ON_plus
+ON_gfinite
+ON_O
+Ordering_Functions
+OrdNotations
+O2H
+

P

+PA
+PAconsistent
+PartialFun
+PAtheory
+Paths
+Pow
+Pow_variant
+predSuccUnicity
+primRec
+PrimRecExamples
+prLogic
+PRrepresentable
+

R

+Restriction
+rosser
+rosserPA
+rpo
+

S

+Schutte
+Schutte_basics
+schutte_cnf_counter_example
+Simple_LexProd
+Sort_spec
+ssrnat_extracts
+STDPP_compat
+Strategies
+subAll
+subProp
+

T

+term
+Trace_exercise
+T1
+T1Bridge
+T1Choice
+T1_ltNotWf
+T2
+T2Bridge
+

W

+wConsistent
+wellFormed
+Well_Orders
+WfVariant
+Wf_transparent
+

+

Lemma Index

+

A

+absurd1 [in hydras.Ackermann.LNN]
+absurd1 [in hydras.Ackermann.LNT]
+absurd1 [in hydras.Ackermann.folLogic]
+Accs_all [in hydras.solutions_exercises.MultisetWf]
+acc_from_to_round_plus [in hydras.Hydra.O2H]
+Acc_nil [in hydras.solutions_exercises.MultisetWf]
+Acc_inverse_image [in additions.Wf_transparent]
+Acc_lemma [in additions.Wf_transparent]
+acc_from_LT [in hydras.Epsilon0.Paths]
+acc_from_trans [in hydras.Epsilon0.Paths]
+acc_trans [in hydras.rpo.closure]
+acc_rec [in gaia_hydras.nfwfgaia]
+acc_imp [in hydras.solutions_exercises.T1_ltNotWf]
+Ackermann.addC_CE [in gaia_hydras.nfwfgaia]
+Ackermann.add_normal [in gaia_hydras.nfwfgaia]
+Ackermann.add_le4 [in gaia_hydras.nfwfgaia]
+Ackermann.add_le3 [in gaia_hydras.nfwfgaia]
+Ackermann.add_inj [in gaia_hydras.nfwfgaia]
+Ackermann.add_le2 [in gaia_hydras.nfwfgaia]
+Ackermann.add_le1 [in gaia_hydras.nfwfgaia]
+Ackermann.add_to_cons [in gaia_hydras.nfwfgaia]
+Ackermann.add_fin_omega [in gaia_hydras.nfwfgaia]
+Ackermann.add_int [in gaia_hydras.nfwfgaia]
+Ackermann.add1Nfin [in gaia_hydras.nfwfgaia]
+Ackermann.all_zeroE [in gaia_hydras.nfwfgaia]
+Ackermann.ap_pr2CE [in gaia_hydras.nfwfgaia]
+Ackermann.ap_pr4 [in gaia_hydras.nfwfgaia]
+Ackermann.ap_pr3 [in gaia_hydras.nfwfgaia]
+Ackermann.ap_pr2 [in gaia_hydras.nfwfgaia]
+Ackermann.ap_pr1 [in gaia_hydras.nfwfgaia]
+Ackermann.ap_limit [in gaia_hydras.nfwfgaia]
+Ackermann.ap_pr0 [in gaia_hydras.nfwfgaia]
+Ackermann.conc1 [in gaia_hydras.nfwfgaia]
+Ackermann.fincP [in gaia_hydras.nfwfgaia]
+Ackermann.finite_ltP [in gaia_hydras.nfwfgaia]
+Ackermann.fooCE [in gaia_hydras.nfwfgaia]
+Ackermann.le_succ_succE [in gaia_hydras.nfwfgaia]
+Ackermann.le_succ_succ [in gaia_hydras.nfwfgaia]
+Ackermann.limit_pr [in gaia_hydras.nfwfgaia]
+Ackermann.limit_pr1 [in gaia_hydras.nfwfgaia]
+Ackermann.limit1 [in gaia_hydras.nfwfgaia]
+Ackermann.limit10 [in gaia_hydras.nfwfgaia]
+Ackermann.limit11 [in gaia_hydras.nfwfgaia]
+Ackermann.limit12a [in gaia_hydras.nfwfgaia]
+Ackermann.limit12b1 [in gaia_hydras.nfwfgaia]
+Ackermann.limit12b2 [in gaia_hydras.nfwfgaia]
+Ackermann.limit12b3 [in gaia_hydras.nfwfgaia]
+Ackermann.limit12b4 [in gaia_hydras.nfwfgaia]
+Ackermann.limit12b5 [in gaia_hydras.nfwfgaia]
+Ackermann.limit2 [in gaia_hydras.nfwfgaia]
+Ackermann.limit3 [in gaia_hydras.nfwfgaia]
+Ackermann.limit4 [in gaia_hydras.nfwfgaia]
+Ackermann.limit5 [in gaia_hydras.nfwfgaia]
+Ackermann.limit6 [in gaia_hydras.nfwfgaia]
+Ackermann.limit7 [in gaia_hydras.nfwfgaia]
+Ackermann.limit8 [in gaia_hydras.nfwfgaia]
+Ackermann.limit9 [in gaia_hydras.nfwfgaia]
+Ackermann.lt_succ_le_4 [in gaia_hydras.nfwfgaia]
+Ackermann.lt_succ_le_3 [in gaia_hydras.nfwfgaia]
+Ackermann.lt_succ_le_2 [in gaia_hydras.nfwfgaia]
+Ackermann.lt_succ_le_1 [in gaia_hydras.nfwfgaia]
+Ackermann.lt_succ_succE [in gaia_hydras.nfwfgaia]
+Ackermann.lt_succ_succ [in gaia_hydras.nfwfgaia]
+Ackermann.lt_omega2 [in gaia_hydras.nfwfgaia]
+Ackermann.lt_omega1 [in gaia_hydras.nfwfgaia]
+Ackermann.lt_epsilon0 [in gaia_hydras.nfwfgaia]
+Ackermann.lt_not_wf [in gaia_hydras.nfwfgaia]
+Ackermann.minus_le [in gaia_hydras.nfwfgaia]
+Ackermann.minus_lt [in gaia_hydras.nfwfgaia]
+Ackermann.nf_add [in gaia_hydras.nfwfgaia]
+Ackermann.nf_sub [in gaia_hydras.nfwfgaia]
+Ackermann.nf_pred [in gaia_hydras.nfwfgaia]
+Ackermann.nf_succ [in gaia_hydras.nfwfgaia]
+Ackermann.nf_finite [in gaia_hydras.nfwfgaia]
+Ackermann.nf_omega [in gaia_hydras.nfwfgaia]
+Ackermann.nf_split [in gaia_hydras.nfwfgaia]
+Ackermann.nf_Wf [in gaia_hydras.nfwfgaia]
+Ackermann.nf_consE [in gaia_hydras.nfwfgaia]
+Ackermann.nf_cons_cons [in gaia_hydras.nfwfgaia]
+Ackermann.nf_int [in gaia_hydras.nfwfgaia]
+Ackermann.nf_psi [in gaia_hydras.nfwfgaia]
+Ackermann.nf_0 [in gaia_hydras.nfwfgaia]
+Ackermann.normal_compose [in gaia_hydras.nfwfgaia]
+Ackermann.normal_limit [in gaia_hydras.nfwfgaia]
+Ackermann.normal_id [in gaia_hydras.nfwfgaia]
+Ackermann.omega_minus_one [in gaia_hydras.nfwfgaia]
+Ackermann.omega_least_inf2 [in gaia_hydras.nfwfgaia]
+Ackermann.omega_least_inf1 [in gaia_hydras.nfwfgaia]
+Ackermann.phiE [in gaia_hydras.nfwfgaia]
+Ackermann.phiE_3 [in gaia_hydras.nfwfgaia]
+Ackermann.phiE_2 [in gaia_hydras.nfwfgaia]
+Ackermann.phiE_1 [in gaia_hydras.nfwfgaia]
+Ackermann.phiL [in gaia_hydras.nfwfgaia]
+Ackermann.phi3v [in gaia_hydras.nfwfgaia]
+Ackermann.pred_succ [in gaia_hydras.nfwfgaia]
+Ackermann.pred_lt [in gaia_hydras.nfwfgaia]
+Ackermann.pred_le [in gaia_hydras.nfwfgaia]
+Ackermann.psi_succ_pr1 [in gaia_hydras.nfwfgaia]
+Ackermann.psi_lt2 [in gaia_hydras.nfwfgaia]
+Ackermann.psi_lt1 [in gaia_hydras.nfwfgaia]
+Ackermann.size_prop [in gaia_hydras.nfwfgaia]
+Ackermann.size_prop1 [in gaia_hydras.nfwfgaia]
+Ackermann.size_psi [in gaia_hydras.nfwfgaia]
+Ackermann.size_d [in gaia_hydras.nfwfgaia]
+Ackermann.size_c [in gaia_hydras.nfwfgaia]
+Ackermann.size_b [in gaia_hydras.nfwfgaia]
+Ackermann.size_a [in gaia_hydras.nfwfgaia]
+Ackermann.split_add [in gaia_hydras.nfwfgaia]
+Ackermann.split_le [in gaia_hydras.nfwfgaia]
+Ackermann.split_pred [in gaia_hydras.nfwfgaia]
+Ackermann.split_succ [in gaia_hydras.nfwfgaia]
+Ackermann.split_is_succ [in gaia_hydras.nfwfgaia]
+Ackermann.split_limit [in gaia_hydras.nfwfgaia]
+Ackermann.split_finite [in gaia_hydras.nfwfgaia]
+Ackermann.sub_pr1rCE [in gaia_hydras.nfwfgaia]
+Ackermann.sub_pr1r [in gaia_hydras.nfwfgaia]
+Ackermann.sub_nz [in gaia_hydras.nfwfgaia]
+Ackermann.sub_pr1 [in gaia_hydras.nfwfgaia]
+Ackermann.sub_pr [in gaia_hydras.nfwfgaia]
+Ackermann.sub_le1 [in gaia_hydras.nfwfgaia]
+Ackermann.sub_1bCE [in gaia_hydras.nfwfgaia]
+Ackermann.sub_1aCE [in gaia_hydras.nfwfgaia]
+Ackermann.sub_int [in gaia_hydras.nfwfgaia]
+Ackermann.sub1a [in gaia_hydras.nfwfgaia]
+Ackermann.sub1b [in gaia_hydras.nfwfgaia]
+Ackermann.sub1Nfin [in gaia_hydras.nfwfgaia]
+Ackermann.succ_psiCE [in gaia_hydras.nfwfgaia]
+Ackermann.succ_prCE [in gaia_hydras.nfwfgaia]
+Ackermann.succ_psi2 [in gaia_hydras.nfwfgaia]
+Ackermann.succ_psi1 [in gaia_hydras.nfwfgaia]
+Ackermann.succ_is_add_one [in gaia_hydras.nfwfgaia]
+Ackermann.succ_psi_lt2 [in gaia_hydras.nfwfgaia]
+Ackermann.succ_psi_lt [in gaia_hydras.nfwfgaia]
+Ackermann.succ_psi [in gaia_hydras.nfwfgaia]
+Ackermann.succ_nz [in gaia_hydras.nfwfgaia]
+Ackermann.succ_inj [in gaia_hydras.nfwfgaia]
+Ackermann.succ_p1 [in gaia_hydras.nfwfgaia]
+Ackermann.succ_pred [in gaia_hydras.nfwfgaia]
+Ackermann.succ_lt [in gaia_hydras.nfwfgaia]
+Ackermann.sup_Oalpha_succ [in gaia_hydras.nfwfgaia]
+Ackermann.sup_Oalpha_limit [in gaia_hydras.nfwfgaia]
+Ackermann.sup_Oalpha_zero [in gaia_hydras.nfwfgaia]
+Ackermann.sup_unique [in gaia_hydras.nfwfgaia]
+Ackermann.TT1T3_inj [in gaia_hydras.nfwfgaia]
+Ackermann.T1succ_nat [in gaia_hydras.nfwfgaia]
+Ackermann.T1T3_surj [in gaia_hydras.nfwfgaia]
+Ackermann.T1T3_inc [in gaia_hydras.nfwfgaia]
+Ackermann.T1T3_lt_epsilon0 [in gaia_hydras.nfwfgaia]
+Ackermann.T3addA [in gaia_hydras.nfwfgaia]
+Ackermann.T3addn0 [in gaia_hydras.nfwfgaia]
+Ackermann.T3addS [in gaia_hydras.nfwfgaia]
+Ackermann.T3add_eq0 [in gaia_hydras.nfwfgaia]
+Ackermann.T3add0n [in gaia_hydras.nfwfgaia]
+Ackermann.T3eqE [in gaia_hydras.nfwfgaia]
+Ackermann.T3eqP [in gaia_hydras.nfwfgaia]
+Ackermann.T3eq_add2l [in gaia_hydras.nfwfgaia]
+Ackermann.T3eq_le [in gaia_hydras.nfwfgaia]
+Ackermann.T3finite_succ [in gaia_hydras.nfwfgaia]
+Ackermann.T3finite1 [in gaia_hydras.nfwfgaia]
+Ackermann.T3finite2 [in gaia_hydras.nfwfgaia]
+Ackermann.T3ge1 [in gaia_hydras.nfwfgaia]
+Ackermann.T3gt1 [in gaia_hydras.nfwfgaia]
+Ackermann.T3lcp0_pr [in gaia_hydras.nfwfgaia]
+Ackermann.T3leNgt [in gaia_hydras.nfwfgaia]
+Ackermann.T3lenn [in gaia_hydras.nfwfgaia]
+Ackermann.T3len0 [in gaia_hydras.nfwfgaia]
+Ackermann.T3leP [in gaia_hydras.nfwfgaia]
+Ackermann.T3le_add2r [in gaia_hydras.nfwfgaia]
+Ackermann.T3le_add2l [in gaia_hydras.nfwfgaia]
+Ackermann.T3le_psi [in gaia_hydras.nfwfgaia]
+Ackermann.T3le_total [in gaia_hydras.nfwfgaia]
+Ackermann.T3le_anti [in gaia_hydras.nfwfgaia]
+Ackermann.T3le_trans [in gaia_hydras.nfwfgaia]
+Ackermann.T3le_lt_trans [in gaia_hydras.nfwfgaia]
+Ackermann.T3le_consE [in gaia_hydras.nfwfgaia]
+Ackermann.T3le_eqVlt [in gaia_hydras.nfwfgaia]
+Ackermann.T3le0n [in gaia_hydras.nfwfgaia]
+Ackermann.T3ltE [in gaia_hydras.nfwfgaia]
+Ackermann.T3ltgtP [in gaia_hydras.nfwfgaia]
+Ackermann.T3ltNge [in gaia_hydras.nfwfgaia]
+Ackermann.T3ltnn [in gaia_hydras.nfwfgaia]
+Ackermann.T3ltn0 [in gaia_hydras.nfwfgaia]
+Ackermann.T3ltP [in gaia_hydras.nfwfgaia]
+Ackermann.T3ltW [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_add2r [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_add2l [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_psi_a [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_psi_c [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_psi_b [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_psi_bc [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_le_trans [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_trans [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_psi' [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_trichotomy [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_anti [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_neAle [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_ne' [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_ne [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_consE [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt_psi [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt0n [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt1 [in gaia_hydras.nfwfgaia]
+Ackermann.T3nf_finite [in gaia_hydras.nfwfgaia]
+Ackermann.T3subnn [in gaia_hydras.nfwfgaia]
+Ackermann.T3subn0 [in gaia_hydras.nfwfgaia]
+Ackermann.T3sub0n [in gaia_hydras.nfwfgaia]
+AckSn_as_PRiterate [in hydras.MoreAck.AckNotPR]
+AckSn_as_iterate [in hydras.MoreAck.AckNotPR]
+Ack_not_PR [in hydras.MoreAck.AckNotPR]
+Ack_iterate_rw [in hydras.Epsilon0.F_omega]
+Ack_strict_mono_l [in hydras.MoreAck.Ack]
+Ack_Sn_plus [in hydras.MoreAck.Ack]
+Ack_mono_r [in hydras.MoreAck.Ack]
+Ack_mono_l [in hydras.MoreAck.Ack]
+Ack_strict_mono [in hydras.MoreAck.Ack]
+Ack_properties [in hydras.MoreAck.Ack]
+Ack_m_mono_weak [in hydras.MoreAck.Ack]
+Ack_4_n [in hydras.MoreAck.Ack]
+Ack_3_n [in hydras.MoreAck.Ack]
+Ack_2_n [in hydras.MoreAck.Ack]
+Ack_1_n [in hydras.MoreAck.Ack]
+Ack_S_S [in hydras.MoreAck.Ack]
+Ack_S_0 [in hydras.MoreAck.Ack]
+Ack_0 [in hydras.MoreAck.Ack]
+addPairwiseEquals [in hydras.Ackermann.folLogic3]
+add_r_0 [in hydras.Hydra.Hydra_Lemmas]
+Alive_standard [in hydras.Hydra.Hydra_Theorems]
+Alive_free [in hydras.Hydra.Hydra_Theorems]
+all_cases [in hydras.Hydra.O2H]
+all_ord_acc [in hydras.Schutte.Schutte_basics]
+Almost_done [in hydras.Hydra.BigBattle]
+alpha_pos [in hydras.Hydra.O2H]
+alpha_lt_beta [in hydras.Schutte.AP]
+alpha_plus_zero [in hydras.Epsilon0.E0]
+alpha_sup [in hydras.Schutte.Ordering_Functions]
+alpha_A [in hydras.Schutte.Ordering_Functions]
+alpha_plus_sup [in hydras.Schutte.Addition]
+alpha_plus_zero [in hydras.Schutte.Addition]
+alt_double_ok [in additions.Demo]
+Alt.add'_ok [in hydras.MoreAck.PrimRecExamples]
+Alt.compose_01 [in hydras.MoreAck.PrimRecExamples]
+Alt.Eps0_prec.prec_transitive [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.prec_antisym [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.prec_dec [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.eq_symbol_dec [in hydras.Epsilon0.Epsilon0rpo]
+Alt.lt_inc_rpo_0 [in hydras.Epsilon0.Epsilon0rpo]
+Alt.lt_subterm2 [in hydras.Epsilon0.Epsilon0rpo]
+Alt.lt_subterm1 [in hydras.Epsilon0.Epsilon0rpo]
+Alt.nat_2_term_mono [in hydras.Epsilon0.Epsilon0rpo]
+Alt.nat_lt_cons [in hydras.Epsilon0.Epsilon0rpo]
+Alt.nf_Acc [in hydras.Epsilon0.Epsilon0rpo]
+Alt.nf_Wf [in hydras.Epsilon0.Epsilon0rpo]
+Alt.rpo_trans [in hydras.Epsilon0.Epsilon0rpo]
+Alt.R_inc_rpo [in hydras.Epsilon0.Epsilon0rpo]
+Alt.R1 [in hydras.Epsilon0.Epsilon0rpo]
+Alt.R2 [in hydras.Epsilon0.Epsilon0rpo]
+Alt.R3 [in hydras.Epsilon0.Epsilon0rpo]
+Alt.R4 [in hydras.Epsilon0.Epsilon0rpo]
+Alt.T1_size3 [in hydras.Epsilon0.Epsilon0rpo]
+Alt.T1_size2 [in hydras.Epsilon0.Epsilon0rpo]
+Alt.T1_size1 [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Vars.eq_variable_dec [in hydras.Epsilon0.Epsilon0rpo]
+Alt.well_founded_rpo [in hydras.Epsilon0.Epsilon0rpo]
+AM_power_Ok [in additions.AM]
+andE1 [in hydras.Ackermann.LNN]
+andE1 [in hydras.Ackermann.LNT]
+andE1 [in hydras.Ackermann.folLogic]
+andE2 [in hydras.Ackermann.LNN]
+andE2 [in hydras.Ackermann.LNT]
+andE2 [in hydras.Ackermann.folLogic]
+andI [in hydras.Ackermann.LNN]
+andI [in hydras.Ackermann.LNT]
+andI [in hydras.Ackermann.folLogic]
+andRelPR [in hydras.Ackermann.primRec]
+andSym [in hydras.Ackermann.LNN]
+andSym [in hydras.Ackermann.LNT]
+andSym [in hydras.Ackermann.folLogic]
+approx_ok [in hydras.Epsilon0.Canon]
+ap_plusR [in hydras.Epsilon0.T1]
+ap_plus [in hydras.Epsilon0.T1]
+ap_cases [in hydras.OrdinalNotations.ON_Omega2]
+AP_plus_AP [in hydras.Schutte.AP]
+AP_to_phi0 [in hydras.Schutte.AP]
+AP_phi0 [in hydras.Schutte.AP]
+AP_o_segment [in hydras.Schutte.AP]
+AP_closed [in hydras.Schutte.AP]
+AP_sup [in hydras.Schutte.AP]
+AP_unbounded [in hydras.Schutte.AP]
+AP_unbounded_0 [in hydras.Schutte.AP]
+AP_mult_fin_r_closed [in hydras.Schutte.AP]
+AP_mult_Sn_closed [in hydras.Schutte.AP]
+AP_plus_closed [in hydras.Schutte.AP]
+AP_finite_eq_one [in hydras.Schutte.AP]
+AP_omega [in hydras.Schutte.AP]
+AP_one [in hydras.Schutte.AP]
+ap_ref [in gaia_hydras.T1Bridge]
+Axm [in hydras.Ackermann.LNN]
+Axm [in hydras.Ackermann.LNT]
+Axm [in hydras.Ackermann.folLogic]
+axp_scheme_length1 [in additions.Addition_Chains]
+axp_correct [in additions.Addition_Chains]
+A_full [in hydras.Schutte.Critical]
+A_Cr [in hydras.Schutte.Critical]
+A_closed [in hydras.Schutte.Ordering_Functions]
+A_denum [in hydras.Schutte.Ordering_Functions]
+A1_A2 [in hydras.Schutte.Ordering_Functions]
+A2_A1 [in hydras.Schutte.Ordering_Functions]
+

B

+Bad.correct_exp_too_strong [in additions.FirstSteps]
+Bad.F3_correct [in additions.Euclidean_Chains]
+Bad.height_lt_size [in hydras.Hydra.Hydra_Examples]
+Bad.mul_not_associative [in additions.FirstSteps]
+Bad.one_not_neutral [in additions.FirstSteps]
+Bad2.Fcompose_correct [in additions.Euclidean_Chains]
+Bad4.F23_ok [in additions.Euclidean_Chains]
+battle_length_std_Hardy [in hydras.Hydra.Hydra_Theorems]
+battle_length_std [in hydras.Hydra.Battle_length]
+battle_example [in hydras.Hydra.KP_example]
+Bbeta_denum [in hydras.Schutte.Ordering_Functions]
+betaEquiv [in Goedel.PRrepresentable]
+betaRepresentable [in Goedel.PRrepresentable]
+betaThm2 [in Goedel.PRrepresentable]
+betaThm3 [in Goedel.PRrepresentable]
+betaThm4 [in Goedel.PRrepresentable]
+betaThm5 [in Goedel.PRrepresentable]
+beta_def [in Goedel.PRrepresentable]
+Bigstep [in hydras.Hydra.BigBattle]
+big_to_small [in hydras.Hydra.Epsilon0_Needed_Free]
+big_to_small [in hydras.Hydra.Omega2_Small]
+big_to_small [in hydras.Hydra.Omega_Small]
+binary_power_mult_ok [in additions.Pow_variant]
+binary_power_mult_ok [in additions.Pow]
+binary_chain_length [in additions.Addition_Chains]
+binary_generator_not_optimal [in additions.Addition_Chains]
+binary_generator_correct [in additions.Addition_Chains]
+binary_correct [in additions.Addition_Chains]
+bin_pow_scheme_length1 [in additions.Addition_Chains]
+bitsP [in additions.fib]
+bits_cat [in additions.fib]
+bool_eq_iff [in hydras.Epsilon0.T1]
+bool_decide_eq_false [in hydras.Prelude.STDPP_compat]
+bool_decide_eq_true [in hydras.Prelude.STDPP_compat]
+boundComputationMonotone [in hydras.Ackermann.codeSubFormula]
+boundedCheck [in hydras.Prelude.MoreDecidable]
+boundedLT [in hydras.Ackermann.NNtheory]
+boundedSearch1 [in hydras.Ackermann.primRec]
+boundedSearch2 [in hydras.Ackermann.primRec]
+boundedSearch3 [in hydras.solutions_exercises.isqrt]
+boundedSearch4 [in hydras.solutions_exercises.isqrt]
+boundMakeTrace [in hydras.Ackermann.codeSubFormula]
+boundSubFormulaHelp [in hydras.Ackermann.codeSubFormula]
+boundSubFormulaHelp1 [in hydras.Ackermann.codeSubFormula]
+boundSubFormulaHelp2 [in hydras.Ackermann.codeSubFormula]
+boundSubTerm [in hydras.Ackermann.codeSubFormula]
+boundSubTerms [in hydras.Ackermann.codeSubFormula]
+but_last_iota_from' [in hydras.Prelude.MoreLists]
+but_last_app [in hydras.Prelude.MoreLists]
+but_last_shift' [in hydras.Prelude.MoreLists]
+but_last_interval [in hydras.Prelude.MoreLists]
+but_last_iota_from [in hydras.Prelude.MoreLists]
+

C

+call_lt_wf [in hydras.Epsilon0.F_alpha]
+CanonSSn [in hydras.Epsilon0.Canon]
+canonS_rel_rounds [in hydras.Hydra.O2H]
+canonS_iota [in hydras.Hydra.O2H]
+canonS_iota_i [in hydras.Hydra.O2H]
+canonS_iota_final [in hydras.Hydra.O2H]
+canonS_iota_6 [in hydras.Hydra.O2H]
+canonS_iota_5 [in hydras.Hydra.O2H]
+canonS_iota_4 [in hydras.Hydra.O2H]
+canonS_ocons_succE [in gaia_hydras.GCanon]
+canonS_zero_inv [in gaia_hydras.GCanon]
+canonS_phi0_succE [in gaia_hydras.GCanon]
+canonS_cons_not_zero [in gaia_hydras.GCanon]
+canonS_lt [in gaia_hydras.GCanon]
+canonS_LE [in gaia_hydras.GCanon]
+CanonS_phi0_Succ [in hydras.Epsilon0.Canon]
+CanonS_lt [in hydras.Epsilon0.Canon]
+CanonS_phi0_lim [in hydras.Epsilon0.Canon]
+canonS_LE [in hydras.Epsilon0.Canon]
+canonS_limit_mono [in hydras.Epsilon0.Canon]
+canonS_limit_lub [in hydras.Epsilon0.Canon]
+canonS_limit_strong [in hydras.Epsilon0.Canon]
+canonS_cons_not_zero [in hydras.Epsilon0.Canon]
+canonS_LT [in hydras.Epsilon0.Canon]
+canonS_zero_inv [in hydras.Epsilon0.Canon]
+canonS_cons_succ_eqn2 [in hydras.Epsilon0.Canon]
+canonS_phi0_succ_eqn [in hydras.Epsilon0.Canon]
+canonS_succ [in hydras.Epsilon0.Canon]
+canonS_lim2 [in hydras.Epsilon0.Canon]
+canonS_lim1 [in hydras.Epsilon0.Canon]
+CanonS_Phi0_Succ_eqn [in hydras.Epsilon0.Paths]
+CanonS_plus_1 [in hydras.Epsilon0.Paths]
+canon_limit_of [in gaia_hydras.GCanon]
+canon_limit_mono [in gaia_hydras.GCanon]
+canon_limit_strong [in gaia_hydras.GCanon]
+canon_lim3 [in gaia_hydras.GCanon]
+canon_lim2 [in gaia_hydras.GCanon]
+canon_tail [in gaia_hydras.GCanon]
+canon_lim1 [in gaia_hydras.GCanon]
+canon_SSn_zero [in gaia_hydras.GCanon]
+canon_lt [in gaia_hydras.GCanon]
+canon_succ [in gaia_hydras.GCanon]
+canon_not_null [in hydras.Epsilon0.Large_Sets]
+Canon_of_limit_not_null [in hydras.Epsilon0.Canon]
+Canon_lt [in hydras.Epsilon0.Canon]
+Canon_Omega [in hydras.Epsilon0.Canon]
+Canon_Succ [in hydras.Epsilon0.Canon]
+canon_limit_mono [in hydras.Epsilon0.Canon]
+canon_limit_strong [in hydras.Epsilon0.Canon]
+canon_lt [in hydras.Epsilon0.Canon]
+canon_LT [in hydras.Epsilon0.Canon]
+canon_SSn_zero [in hydras.Epsilon0.Canon]
+canon_succ [in hydras.Epsilon0.Canon]
+canon_lim3 [in hydras.Epsilon0.Canon]
+canon_lim2 [in hydras.Epsilon0.Canon]
+canon_lim1 [in hydras.Epsilon0.Canon]
+canon_tail [in hydras.Epsilon0.Canon]
+canon_zero [in hydras.Epsilon0.Canon]
+Canon_plus_first_step_lim [in hydras.Epsilon0.Paths]
+Canon_plus_first_step [in hydras.Epsilon0.Paths]
+Canon_mono1 [in hydras.Epsilon0.Paths]
+Canon_plus_inv [in hydras.Epsilon0.Paths]
+canon0_lt [in gaia_hydras.GCanon]
+canon0_phi0_succE [in gaia_hydras.GCanon]
+canon0_LT [in hydras.Epsilon0.Canon]
+canon0_cons_succ_eqn2 [in hydras.Epsilon0.Canon]
+canon0_phi0_succ_eqn [in hydras.Epsilon0.Canon]
+canon0_succ [in hydras.Epsilon0.Canon]
+canon0_lim2 [in hydras.Epsilon0.Canon]
+CantorOrdinal.addC_CE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_simpl3 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_normal [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_le4 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_le3 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_inj [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_le2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_le1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_simpl2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_simpl1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_to_cons [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_fin_omega [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add_int [in gaia_hydras.nfwfgaia]
+CantorOrdinal.add1Nfin [in gaia_hydras.nfwfgaia]
+CantorOrdinal.ap_pr4 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.ap_pr3 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.ap_pr2CE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.ap_pr2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.ap_pr1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.ap_pr0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.cantorCE2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.cantorCE3 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.cantor_CE1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.cantor_unique [in gaia_hydras.nfwfgaia]
+CantorOrdinal.cantor_exists [in gaia_hydras.nfwfgaia]
+CantorOrdinal.div_by_omega_pr [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expF_eq1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expF_eq0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expF_1n [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expF_mul [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expF_add [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expO_add [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expO_eq1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expO_eq0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expO_1n [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expO_n0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expO_mul1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expx_nat [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expx_pnat [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expx0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expx1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.expx1CE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_prod [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_sum [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_FO3 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_FO2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_FO1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_FO [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_consCE1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_int [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_eq1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_eq0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp0nz [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp00 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp1x [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp2omega [in gaia_hydras.nfwfgaia]
+CantorOrdinal.ex_middle_pick [in gaia_hydras.nfwfgaia]
+CantorOrdinal.fincP [in gaia_hydras.nfwfgaia]
+CantorOrdinal.fooCE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.head_lt_cons [in gaia_hydras.nfwfgaia]
+CantorOrdinal.le_succ_succE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.le_succ_succ [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_prop [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_lub [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_unique [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_CE3 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_CE2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_CE1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_unique2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_unique1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_pr [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_pr1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit3 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ_le_4 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ_le_3 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ_le_2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ_le_1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ_succE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ2CE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_succ_succ [in gaia_hydras.nfwfgaia]
+CantorOrdinal.lt_not_wf [in gaia_hydras.nfwfgaia]
+CantorOrdinal.minus_le [in gaia_hydras.nfwfgaia]
+CantorOrdinal.minus_lt [in gaia_hydras.nfwfgaia]
+CantorOrdinal.min_exists [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mulA [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mulnf0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_omega_pr3 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_omega2_pr1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_omega_pr1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_sum_omega [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_omega_limit [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_int_limit [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_fin_omega [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_succ [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_distr [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_phi0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_int [in gaia_hydras.nfwfgaia]
+CantorOrdinal.mul_na [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_exp [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_expF [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_expO [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_rev_prod [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_rev_sum [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_rev_unique [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_rev [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_revCE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_div_by_omega [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_mul [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_toNF [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_omega_plus_n [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_sub [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_add [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_pred [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_succ [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_log [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_phi0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_finite [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_omega [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_split [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_Wf' [in gaia_hydras.nfwfgaia]
+CantorOrdinal.nf_Wf [in gaia_hydras.nfwfgaia]
+CantorOrdinal.normal_compose [in gaia_hydras.nfwfgaia]
+CantorOrdinal.normal_limit [in gaia_hydras.nfwfgaia]
+CantorOrdinal.normal_id [in gaia_hydras.nfwfgaia]
+CantorOrdinal.omega_tower_unbounded [in gaia_hydras.nfwfgaia]
+CantorOrdinal.omega_tower_nf [in gaia_hydras.nfwfgaia]
+CantorOrdinal.omega_minus_one [in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi0_log [in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi0_lt1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi0_le [in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi0_lt [in gaia_hydras.nfwfgaia]
+CantorOrdinal.plus_int_Ox [in gaia_hydras.nfwfgaia]
+CantorOrdinal.pow_mon3 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.pow_mon2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.pow_mon1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.pow_omega [in gaia_hydras.nfwfgaia]
+CantorOrdinal.pred_succ_CE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.pred_succ [in gaia_hydras.nfwfgaia]
+CantorOrdinal.pred_lt [in gaia_hydras.nfwfgaia]
+CantorOrdinal.pred_le [in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_add [in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_limit1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_le [in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_pred [in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_succ [in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_finite [in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_is_succ [in gaia_hydras.nfwfgaia]
+CantorOrdinal.split_limit [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_pr1rCE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_nzCE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_nz [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_pr1r [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_pr1CE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_pr1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_pr [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_le1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_int [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_1bCE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub_1aCE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub1a [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub1b [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sub1Nfin [in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_is_add_one [in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_injCE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_inj [in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_p1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_predCE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_pred [in gaia_hydras.nfwfgaia]
+CantorOrdinal.succ_lt [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sup_Oalpha_limit [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sup_Oalpha_succ [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sup_Oalpha_zero [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sup_unique [in gaia_hydras.nfwfgaia]
+CantorOrdinal.tail_lt_cons [in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_pred [in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_succ [in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_ex2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_ex1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_mon [in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_nf [in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF_nz [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1addA [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1addn0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1addS [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1add_eq0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1add0n [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ap_phi0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1eqE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1eqP [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1eq_mul2l [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1eq_add2l [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1eq_le [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1finite_succ [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1finite1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1finite2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1finite2CE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1F_inj [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ge1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1leNgt [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lenn [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1len0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1leP [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_pmull [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_mul [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_mulCE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_pmulrl [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_pmulrCE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_pmulr [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_mul2r [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_mul2l [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_add2r [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_add2l [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_cons_le [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_trans [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_lt_trans [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_total [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_consE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le_eqVlt [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le0n [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1log_exp2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1log_exp1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1log_exp0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1log_prod [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1log_phi0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltgtP [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltNge [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltnn [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltn0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltP [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltW [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_mul2r [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_mul2l [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_add2r [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_add2l [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_cons_le [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_le_trans [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_trans [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_anti [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_trichotomy [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_neAle [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_ne' [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt_ne [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt0n [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1muln0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1muln1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1muln1_CE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul_omega [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul_eq1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul_eq0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul0n [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul1n [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul1nCE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nat_inc [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nfCE [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_rect [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_finite [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_finite1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_consb [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_consa [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_cons_cons [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf_cons0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1phi0_zero' [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1phi0_zero [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1subnn [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1subn0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1sub0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1succ_nat [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1transfinite_induction_Q [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1transfinite_induction [in gaia_hydras.nfwfgaia]
+Cantor_normal_form [in hydras.Epsilon0.T1]
+cases_for_mult [in hydras.Epsilon0.T1]
+case_Eq [in hydras.Schutte.Correctness_E0]
+case_gt [in hydras.Schutte.Correctness_E0]
+case_lt [in hydras.Schutte.Correctness_E0]
+chain_gen_OK [in additions.AM]
+chain_gen_OK [in additions.Euclidean_Chains]
+checkPrfCorrect1 [in hydras.Ackermann.checkPrf]
+checkPrfCorrect2 [in hydras.Ackermann.checkPrf]
+checkPrfEQnIsPR [in hydras.Ackermann.checkPrf]
+checkTraceCorrect [in hydras.Ackermann.codeSubFormula]
+classification [in hydras.Schutte.Schutte_basics]
+closedNatToTerm [in hydras.Ackermann.LNN]
+closedNatToTerm [in hydras.Ackermann.LNT]
+closedNN [in hydras.Ackermann.NN]
+closedNN1 [in hydras.Ackermann.NN]
+closedPA [in hydras.Ackermann.PA]
+closedPA1 [in hydras.Ackermann.PA]
+closedT [in hydras.Ackermann.expressible]
+Closed_Cr [in hydras.Schutte.Critical]
+cnf_of_epsilon0 [in hydras.Schutte.CNF]
+cnf_lt_epsilon0 [in hydras.Schutte.CNF]
+cnf_exists_unique [in hydras.Schutte.CNF]
+cnf_unicity [in hydras.Schutte.CNF]
+cnf_exists [in hydras.Schutte.CNF]
+cnf_plus [in hydras.Schutte.CNF]
+cnf_plus2 [in hydras.Schutte.CNF]
+cnf_plus1 [in hydras.Schutte.CNF]
+cnf_eq [in hydras.Schutte.CNF]
+cnf_head_eq [in hydras.Schutte.CNF]
+cnf_of_ap [in hydras.Schutte.CNF]
+cnf_Cons [in hydras.Epsilon0.E0]
+cnf_Omega_term [in hydras.Epsilon0.E0]
+cnf_Succ [in hydras.Epsilon0.E0]
+cnf_phi0 [in hydras.Epsilon0.E0]
+cnf_rw [in hydras.Epsilon0.E0]
+codeAndCorrect [in hydras.Ackermann.code]
+codeAppCorrect [in hydras.Ackermann.codeList]
+codeArityLNNFIsCorrect1 [in hydras.Ackermann.Languages]
+codeArityLNNFIsCorrect2 [in hydras.Ackermann.Languages]
+codeArityLNNRIsCorrect1 [in hydras.Ackermann.Languages]
+codeArityLNNRIsCorrect2 [in hydras.Ackermann.Languages]
+codeArityLNTFIsCorrect1 [in hydras.Ackermann.Languages]
+codeArityLNTFIsCorrect2 [in hydras.Ackermann.Languages]
+codeArityLNTRIsCorrect1 [in hydras.Ackermann.Languages]
+codeArityLNTRIsCorrect2 [in hydras.Ackermann.Languages]
+codeCloseCorrect [in hydras.Ackermann.codePA]
+codeCloseListCorrect [in hydras.Ackermann.codePA]
+codeForallCorrect [in hydras.Ackermann.code]
+codeForallIsPR [in hydras.Ackermann.prLogic]
+codeFormulaInj [in hydras.Ackermann.code]
+codeFreeVarFormulaCorrect [in hydras.Ackermann.codeFreeVar]
+codeFreeVarListFormulaCorrect [in hydras.Ackermann.codeFreeVar]
+codeFreeVarTermCorrect [in hydras.Ackermann.codeFreeVar]
+codeFreeVarTermIsPR [in hydras.Ackermann.codeFreeVar]
+codeFreeVarTermsCorrect [in hydras.Ackermann.codeFreeVar]
+codeIffCorrect [in hydras.Ackermann.code]
+codeImpCorrect [in hydras.Ackermann.code]
+codeInCorrect [in hydras.Ackermann.codeList]
+codeInductionSchemaCorrect1 [in hydras.Ackermann.codePA]
+codeInductionSchemaCorrect2 [in hydras.Ackermann.codePA]
+codeInductionSchemaCorrect3 [in hydras.Ackermann.codePA]
+codeLengthCorrect [in hydras.Ackermann.codeList]
+codeListInj [in hydras.Ackermann.cPair]
+codeListRemoveCorrect [in hydras.Ackermann.codeList]
+codeLNNRelationInj [in hydras.Ackermann.Languages]
+codeLNTFunctionInj [in hydras.Ackermann.Languages]
+codeLNTRelationInj [in hydras.Ackermann.Languages]
+codeNatToTermCorrectLNN [in hydras.Ackermann.codeNatToTerm]
+codeNatToTermCorrectLNT [in hydras.Ackermann.codeNatToTerm]
+codeNewVarCorrect [in hydras.Ackermann.codeSubFormula]
+codeNoDupCorrect [in hydras.Ackermann.codeList]
+codeNotCorrect [in hydras.Ackermann.code]
+codeNthCorrect [in hydras.Ackermann.cPair]
+codeNVarsCorrect [in hydras.Ackermann.checkPrf]
+codeOpenCorrect [in hydras.Ackermann.codePA]
+codeOrCorrect [in hydras.Ackermann.code]
+codePAcorrect1 [in hydras.Ackermann.codePA]
+codePAcorrect2 [in hydras.Ackermann.codePA]
+codePAcorrect3 [in hydras.Ackermann.codePA]
+codePrfInjAxm [in hydras.Ackermann.code]
+codeSubFormulaCorrect [in hydras.Ackermann.codeSubFormula]
+codeSubTermCorrect [in hydras.Ackermann.codeSubTerm]
+codeSubTermsCorrect [in hydras.Ackermann.codeSubTerm]
+codeSubTermsIsPR [in hydras.Ackermann.codeSubTerm]
+codeSysPfCorrect [in Goedel.codeSysPrf]
+codeSysPrfCorrect1 [in Goedel.codeSysPrf]
+codeSysPrfCorrect2 [in Goedel.codeSysPrf]
+codeSysPrfCorrect3 [in Goedel.codeSysPrf]
+codeSysPrfNCorrect1 [in Goedel.codeSysPrf]
+codeSysPrfNCorrect2 [in Goedel.codeSysPrf]
+codeSysPrfNCorrect3 [in Goedel.codeSysPrf]
+codeTermFreeVar [in hydras.Ackermann.codeSubFormula]
+codeTermInj [in hydras.Ackermann.code]
+codeTermsInj [in hydras.Ackermann.code]
+coeff_lt [in hydras.Epsilon0.T1]
+coeff_lt [in hydras.Schutte.Correctness_E0]
+compare_reflectR [in hydras.Epsilon0.T1]
+compare_of_phi0 [in hydras.Epsilon0.T1]
+compare_fin_rw [in hydras.Epsilon0.T1]
+compare_lt_iff [in hydras.Epsilon0.T1]
+compare_lt_impl [in hydras.Epsilon0.T1]
+compare_eq_iff [in hydras.Epsilon0.T1]
+compare_refl [in hydras.Epsilon0.T1]
+compare_correct [in hydras.Epsilon0.T1]
+compare_reflect [in hydras.Epsilon0.T1]
+compare_rev [in hydras.Epsilon0.T1]
+compare_cons [in hydras.Epsilon0.T1]
+compare_correct [in hydras.OrdinalNotations.ON_Omega2]
+compare_reflect [in hydras.OrdinalNotations.ON_Omega2]
+compare_reflect [in hydras.Prelude.Comparable]
+compare_trans [in hydras.Prelude.Comparable]
+compare_ge_iff [in hydras.Prelude.Comparable]
+compare_le_iff [in hydras.Prelude.Comparable]
+compare_le_iff_refl [in hydras.Prelude.Comparable]
+compare_gt_not_lt [in hydras.Prelude.Comparable]
+compare_lt_not_gt [in hydras.Prelude.Comparable]
+compare_gt_trans [in hydras.Prelude.Comparable]
+compare_gt_irrefl [in hydras.Prelude.Comparable]
+compare_gt_iff [in hydras.Prelude.Comparable]
+compare_eq_trans [in hydras.Prelude.Comparable]
+compare_refl [in hydras.Prelude.Comparable]
+compare_eq_iff [in hydras.Prelude.Comparable]
+compare_lt_irrefl [in hydras.Prelude.Comparable]
+compare_lt_trans [in hydras.Prelude.Comparable]
+compare_lt_iff [in hydras.Prelude.Comparable]
+compare_reflect [in hydras.OrdinalNotations.ON_O]
+compare_correct [in hydras.OrdinalNotations.ON_O]
+compare_h2g [in gaia_hydras.T1Bridge]
+compare_g2h [in gaia_hydras.T1Bridge]
+compare_ref [in gaia_hydras.T1Bridge]
+compare_correct [in hydras.OrdinalNotations.ON_mult]
+compare_reflect [in hydras.OrdinalNotations.ON_mult]
+compare_correct [in hydras.OrdinalNotations.ON_plus]
+compare_reflect [in hydras.OrdinalNotations.ON_plus]
+compare_reflect [in hydras.OrdinalNotations.ON_Finite]
+compare_correct [in hydras.OrdinalNotations.ON_Finite]
+compare_correct [in hydras.Epsilon0.E0]
+compare_rw_gt [in hydras.Gamma0.Gamma0]
+compare_rw_eq [in hydras.Gamma0.Gamma0]
+compare_rw_lt [in hydras.Gamma0.Gamma0]
+compare_Gt [in hydras.Gamma0.Gamma0]
+compare_Eq [in hydras.Gamma0.Gamma0]
+compare_Lt [in hydras.Gamma0.Gamma0]
+compare_correct [in hydras.Gamma0.Gamma0]
+compare_reflect [in hydras.Gamma0.Gamma0]
+Compat815.le_not_lt [in hydras.Prelude.Compat815]
+Compat815.le_plus_r [in hydras.Prelude.Compat815]
+Compat815.le_lt_n_Sm [in hydras.Prelude.Compat815]
+Compat815.le_lt_or_eq [in hydras.Prelude.Compat815]
+Compat815.le_n_0_eq [in hydras.Prelude.Compat815]
+Compat815.lt_not_le [in hydras.Prelude.Compat815]
+Compat815.lt_n_S [in hydras.Prelude.Compat815]
+Compat815.lt_S_n [in hydras.Prelude.Compat815]
+Compat815.lt_n_Sm_le [in hydras.Prelude.Compat815]
+Compat815.mult_O_le [in hydras.Prelude.Compat815]
+Compat815.n_SSn [in hydras.Prelude.Compat815]
+Compat815.n_SSSn [in hydras.Prelude.Compat815]
+Compat815.plus_Snm_nSm [in hydras.Prelude.Compat815]
+composeSigmaRepresentable [in Goedel.PRrepresentable]
+compose2_3 [in hydras.MoreAck.PrimRecExamples]
+compose2_2 [in hydras.MoreAck.PrimRecExamples]
+compose2_1 [in hydras.MoreAck.PrimRecExamples]
+compose2_0 [in hydras.MoreAck.PrimRecExamples]
+computation_eval_rw [in additions.Addition_Chains]
+computeEvalStrongRecHelp [in hydras.Ackermann.cPair]
+constant_to_standard_path [in gaia_hydras.GPaths]
+constant_to_standard_path [in hydras.Epsilon0.Paths]
+constant_to_standard [in hydras.Epsilon0.Paths]
+constant_to_standard_0 [in hydras.Epsilon0.Paths]
+consTerms [in hydras.Ackermann.fol]
+const_pathS_LT' [in hydras.Epsilon0.Paths]
+const_pathS_eps_zero [in hydras.Epsilon0.Paths]
+const_pathS_first_step [in hydras.Epsilon0.Paths]
+const_pathS_eps_LE_2 [in hydras.Epsilon0.Paths]
+const_pathS_eps_trans [in hydras.Epsilon0.Paths]
+const_pathS_trans [in hydras.Epsilon0.Paths]
+const_pathS_inv_strong [in hydras.Epsilon0.Paths]
+const_pathS_inv [in hydras.Epsilon0.Paths]
+const_pathS_LE [in hydras.Epsilon0.Paths]
+const_path_LT [in hydras.Epsilon0.Paths]
+const_pathS_LT [in hydras.Epsilon0.Paths]
+const_pathS_zero [in hydras.Epsilon0.Paths]
+const_pathS_nf [in hydras.Epsilon0.Paths]
+const_pathS_repeatR [in hydras.Epsilon0.Paths]
+const_pathS_repeat [in hydras.Epsilon0.Paths]
+const_pathSE [in hydras.Epsilon0.Paths]
+cons_def [in hydras.Epsilon0.T1]
+cons_nf [in hydras.Epsilon0.T1]
+cons_standard_path [in hydras.Epsilon0.Paths]
+cons_standard_pathS [in hydras.Epsilon0.Paths]
+cons_lt_epsilon0 [in hydras.Gamma0.Gamma0]
+cons_rw [in hydras.Gamma0.Gamma0]
+contrad [in hydras.solutions_exercises.MultisetWf]
+contrad [in hydras.solutions_exercises.T1_ltNotWf]
+contradiction [in hydras.Ackermann.LNN]
+contradiction [in hydras.Ackermann.LNT]
+contradiction [in hydras.Ackermann.folLogic]
+Contradiction [in hydras.Hydra.Omega_Small]
+correct_app [in additions.AM]
+Cor12 [in gaia_hydras.GPaths]
+Cor12 [in hydras.Epsilon0.Paths]
+Cor12_E0 [in hydras.Epsilon0.Paths]
+Cor12_3 [in hydras.Epsilon0.Paths]
+Cor12_2 [in hydras.Epsilon0.Paths]
+Cor12_1 [in hydras.Epsilon0.Paths]
+countable_members [in hydras.Schutte.Schutte_basics]
+countable_not_Unbounded [in hydras.Schutte.Schutte_basics]
+countable_segment_proper [in hydras.Schutte.Ordering_Functions]
+countable_image [in hydras.Schutte.Countable]
+countable_bij_funR [in hydras.Schutte.Countable]
+countable_bij_fun [in hydras.Schutte.Countable]
+countable_surj [in hydras.Schutte.Countable]
+counter_ex [in hydras.solutions_exercises.schutte_cnf_counter_example]
+cPairInjHelp [in hydras.Ackermann.cPair]
+cPairInj1 [in hydras.Ackermann.cPair]
+cPairInj2 [in hydras.Ackermann.cPair]
+cPairLemma1 [in hydras.Ackermann.cPair]
+cPairLe1 [in hydras.Ackermann.cPair]
+cPairLe1A [in hydras.Ackermann.cPair]
+cPairLe2 [in hydras.Ackermann.cPair]
+cPairLe2A [in hydras.Ackermann.cPair]
+cPairLe3 [in hydras.Ackermann.cPair]
+cPairLt1 [in hydras.Ackermann.cPair]
+cPairLt2 [in hydras.Ackermann.cPair]
+cPairProjections [in hydras.Ackermann.cPair]
+cPairProjectionsHelp [in hydras.Ackermann.cPair]
+cPairProjections1 [in hydras.Ackermann.cPair]
+cPairProjections2 [in hydras.Ackermann.cPair]
+cp1 [in hydras.Ackermann.LNN]
+cp1 [in hydras.Ackermann.LNT]
+cp1 [in hydras.Ackermann.folLogic]
+cp2 [in hydras.Ackermann.LNN]
+cp2 [in hydras.Ackermann.LNT]
+cp2 [in hydras.Ackermann.folLogic]
+Cr_1_iff [in hydras.Schutte.Critical]
+Cr_incl [in hydras.Schutte.Critical]
+Cr_lt [in hydras.Schutte.Critical]
+Cr_pos_iff [in hydras.Schutte.Critical]
+Cr_pos_inv [in hydras.Schutte.Critical]
+Cr_zero_AP [in hydras.Schutte.Critical]
+Cr_zero_inv [in hydras.Schutte.Critical]
+Cr_pos [in hydras.Schutte.Critical]
+Cr_zero [in hydras.Schutte.Critical]
+Cr_inv [in hydras.Schutte.Critical]
+Cr_equation [in hydras.Schutte.Critical]
+Cr_extensional [in hydras.Schutte.Critical]
+cst_correct [in hydras.MoreAck.PrimRecExamples]
+cst0_correct [in hydras.MoreAck.PrimRecExamples]
+cTripleProj [in hydras.Ackermann.cPair]
+cTripleProj1 [in hydras.Ackermann.cPair]
+cTripleProj2 [in hydras.Ackermann.cPair]
+cTripleProj3 [in hydras.Ackermann.cPair]
+C87_ok' [in additions.Addition_Chains]
+

D

+decideAxioms [in Goedel.rosserPA]
+decideAxioms [in Goedel.rosser]
+decide_hltE [in gaia_hydras.T1Bridge]
+decomp [in hydras.Prelude.MoreVectors]
+decompose [in hydras.OrdinalNotations.ON_Omega2]
+decompos2 [in hydras.Prelude.MoreVectors]
+decomp1 [in hydras.Prelude.MoreVectors]
+decomp2 [in hydras.Prelude.MoreVectors]
+decr_seq [in hydras.solutions_exercises.MultisetWf]
+DeductionTheorem [in hydras.Ackermann.Deduction]
+den_U [in hydras.Schutte.Ordering_Functions]
+depthForall [in hydras.Ackermann.fol]
+depthImp1 [in hydras.Ackermann.fol]
+depthImp2 [in hydras.Ackermann.fol]
+depthNot [in hydras.Ackermann.fol]
+dicho_gt [in additions.Dichotomy]
+dicho_lt [in additions.Dichotomy]
+dicho_aux_gt [in additions.Dichotomy]
+dicho_aux_lt [in additions.Dichotomy]
+dicho_aux_le_xIXI [in additions.Dichotomy]
+dicho_aux_le_xIXO [in additions.Dichotomy]
+dicho_aux_le_xOXI [in additions.Dichotomy]
+dicho_aux_le_xOXO [in additions.Dichotomy]
+dicho_aux_xIxI [in additions.Dichotomy]
+dicho_aux_xIxO [in additions.Dichotomy]
+dicho_aux_xOxI [in additions.Dichotomy]
+dicho_aux_xOxO [in additions.Dichotomy]
+diffP [in gaia_hydras.GPrelude]
+Direct_proof.nf_Acc [in hydras.Epsilon0.T1]
+Direct_proof.Acc_implies_Acc_strong [in hydras.Epsilon0.T1]
+Direct_proof.Acc_strong_stronger [in hydras.Epsilon0.T1]
+Direct_proof.acc_impl [in hydras.Epsilon0.T1]
+Direct_proof.wf_LT [in hydras.Epsilon0.T1]
+Direct_proof.Acc_zero [in hydras.Epsilon0.T1]
+div_eucl_unique [in hydras.Prelude.More_Arith]
+div_not_qlt [in hydras.Prelude.More_Arith]
+div_gamma_pos [in additions.Strategies]
+div2_even_plus [in hydras.Prelude.More_Arith]
+div2_incr [in hydras.Prelude.More_Arith]
+div2_of_Even [in hydras.Prelude.More_Arith]
+div2_double_is_id [in hydras.Prelude.More_Arith]
+dominates_iterate [in hydras.Prelude.Iterates]
+dominates_from_le [in hydras.Prelude.Iterates]
+dominates_trans_strong [in hydras.Prelude.Iterates]
+dominates_trans [in hydras.Prelude.Iterates]
+dominates_from_trans [in hydras.Prelude.Iterates]
+dom_AckNotPR [in hydras.MoreAck.AckNotPR]
+Done [in hydras.Hydra.BigBattle]
+doubleIsPR [in hydras.solutions_exercises.MorePRExamples]
+doubleS_law [in hydras.Hydra.BigBattle]
+double_is_even [in hydras.Prelude.More_Arith]
+double_inj [in hydras.Prelude.More_Arith]
+double_plus [in hydras.Prelude.More_Arith]
+double_S [in hydras.Prelude.More_Arith]
+DS_iota_3 [in hydras.Hydra.O2H]
+DS_iota_2 [in hydras.Hydra.O2H]
+DS_iota_1 [in hydras.Hydra.O2H]
+

E

+embedding [in hydras.Schutte.Correctness_E0]
+empty_interval [in hydras.Prelude.MoreLists]
+empty_ordering [in hydras.Schutte.Ordering_Functions]
+epsilon_equiv [in hydras.Schutte.MoreEpsilonIota]
+epsilon_ind [in hydras.Schutte.MoreEpsilonIota]
+epsilon_fxp [in hydras.Gamma0.Gamma0]
+epsilon0_Cr1 [in hydras.Schutte.Critical]
+epsilon0_lfp [in hydras.Schutte.AP]
+epsilon0_AP [in hydras.Schutte.AP]
+epsilon0_fxp [in hydras.Schutte.AP]
+epsilon0_as_lub [in hydras.Gamma0.Gamma0]
+epsilon0_fxp [in hydras.Gamma0.Gamma0]
+eqDepth [in hydras.Ackermann.fol]
+eqLT [in hydras.Ackermann.LNN]
+eqn_simpl2 [in gaia_hydras.nfwfgaia]
+eqn_simpl1 [in gaia_hydras.nfwfgaia]
+eqPlus [in hydras.Ackermann.LNN]
+eqPlus [in hydras.Ackermann.LNT]
+eqRefl [in hydras.Ackermann.LNN]
+eqRefl [in hydras.Ackermann.LNT]
+eqRefl [in hydras.Ackermann.folLogic3]
+eqSucc [in hydras.Ackermann.LNN]
+eqSucc [in hydras.Ackermann.LNT]
+eqSym [in hydras.Ackermann.LNN]
+eqSym [in hydras.Ackermann.LNT]
+eqSym [in hydras.Ackermann.folLogic3]
+eqTimes [in hydras.Ackermann.LNN]
+eqTimes [in hydras.Ackermann.LNT]
+eqTrans [in hydras.Ackermann.LNN]
+eqTrans [in hydras.Ackermann.LNT]
+eqTrans [in hydras.Ackermann.folLogic3]
+equalFunction [in hydras.Ackermann.folLogic3]
+equalRelation [in hydras.Ackermann.folLogic3]
+equiv_not_lt [in hydras.Prelude.DecPreOrder]
+eq_succ_lt [in hydras.Epsilon0.T1]
+eq_succ_LT [in hydras.Epsilon0.T1]
+eq_dec [in hydras.OrdinalNotations.ON_Omega2]
+eq_dec [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+eq_le [in hydras.Schutte.Schutte_basics]
+Euc1 [in hydras.Prelude.More_Arith]
+evalListComp [in hydras.MoreAck.AckNotPR]
+evalListCompose2 [in hydras.MoreAck.AckNotPR]
+evalListPrimrec_S [in hydras.MoreAck.AckNotPR]
+evalListPrimrec_0 [in hydras.MoreAck.AckNotPR]
+evalList_Const [in hydras.MoreAck.AckNotPR]
+evalPrimRecParam [in hydras.Ackermann.primRec]
+evalProjFuncInd [in hydras.Ackermann.primRec]
+evalStrongRecHelpParam [in hydras.Ackermann.cPair]
+evalStrongRecHelp1 [in hydras.Ackermann.cPair]
+evalStrongRecHelp2 [in hydras.Ackermann.cPair]
+even_prod [in hydras.Prelude.More_Arith]
+every_battle_terminates [in hydras.Hydra.Hydra_Termination]
+every_battle_terminates [in gaia_hydras.GHydra]
+exact_log2_spec [in additions.More_on_positive]
+exact_log2xOx0 [in additions.More_on_positive]
+Examples.big_correct''' [in additions.Euclidean_Chains]
+Examples.big_correct' [in additions.Euclidean_Chains]
+Examples.big_correct [in additions.Euclidean_Chains]
+Examples.RM [in additions.Euclidean_Chains]
+exec_equiv [in additions.AM]
+exec_app [in additions.AM]
+existE [in hydras.Ackermann.LNN]
+existE [in hydras.Ackermann.LNT]
+existE [in hydras.Ackermann.folLogic]
+existI [in hydras.Ackermann.LNN]
+existI [in hydras.Ackermann.LNT]
+existI [in hydras.Ackermann.folLogic]
+existSimp [in hydras.Ackermann.LNN]
+existSimp [in hydras.Ackermann.LNT]
+existSimp [in hydras.Ackermann.folLogic]
+existSys [in hydras.Ackermann.LNN]
+existSys [in hydras.Ackermann.LNT]
+existSys [in hydras.Ackermann.folLogic]
+exists_map12_without_repetition [in hydras.rpo.more_list]
+exists_map_without_repetition [in hydras.rpo.more_list]
+ExNotIncl [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+exponents_lt_eval [in hydras.Schutte.CNF]
+exponent_pos_of_nat [in additions.Addition_Chains]
+exponent_pos2nat [in additions.Addition_Chains]
+exponent_nat_neq_0 [in additions.Addition_Chains]
+expressibleAlternate [in hydras.Ackermann.expressible]
+expressT'1 [in Goedel.rosserPA]
+expressT'2 [in Goedel.rosserPA]
+exp_F_eq [in hydras.Epsilon0.T1]
+exp_fin_omega [in hydras.Epsilon0.T1]
+exp_alt_ok [in hydras.solutions_exercises.MorePRExamples]
+exp2S [in hydras.Prelude.Exp2]
+exp2_as_iterate [in hydras.Prelude.Iterates]
+exp2_mono_weak [in hydras.Prelude.Iterates]
+exp2_mono [in hydras.Prelude.Iterates]
+exp2_ge_S [in hydras.Prelude.Iterates]
+exp2_k_mult_pos [in hydras.Epsilon0.Large_Sets]
+exp2_mono1 [in hydras.Hydra.BigBattle]
+exp2_Plus [in additions.Addition_Chains]
+exp2_gt_id [in hydras.Prelude.Exp2]
+exp2_not_zero [in hydras.Prelude.Exp2]
+exp2_positive [in hydras.Prelude.Exp2]
+extendsNN [in Goedel.rosserPA]
+extEqualCompose [in hydras.Ackermann.primRec]
+extEqualCompose2 [in hydras.Ackermann.primRec]
+extEqualOneParamList [in hydras.Ackermann.primRec]
+extEqualPrimRec [in hydras.Ackermann.primRec]
+extEqualRefl [in hydras.Ackermann.extEqualNat]
+extEqualSym [in hydras.Ackermann.extEqualNat]
+extEqualTrans [in hydras.Ackermann.extEqualNat]
+extEqualVectorRefl [in hydras.Ackermann.primRec]
+E0compare_correct [in gaia_hydras.T1Bridge]
+E0fin_cnf [in gaia_hydras.T1Bridge]
+E0g2h_omegaE [in gaia_hydras.T1Bridge]
+E0g2h_plusE [in gaia_hydras.T1Bridge]
+E0g2h_mulE [in gaia_hydras.T1Bridge]
+E0g2h_phi0 [in gaia_hydras.T1Bridge]
+E0g2h_Fin [in gaia_hydras.T1Bridge]
+E0is_succE [in gaia_hydras.T1Bridge]
+E0is_succ_succ [in gaia_hydras.T1Bridge]
+E0lt_wf [in hydras.Epsilon0.E0]
+E0_plus_correct [in hydras.Schutte.Correctness_E0]
+E0_canon_lt [in gaia_hydras.GCanon]
+E0_pred_succK [in gaia_hydras.T1Bridge]
+E0_diffE [in gaia_hydras.T1Bridge]
+E0_eqE [in gaia_hydras.T1Bridge]
+E0_g2h_h2gK [in gaia_hydras.T1Bridge]
+E0_h2g_g2hK [in gaia_hydras.T1Bridge]
+E0_h2g_nf [in gaia_hydras.T1Bridge]
+E0_lt_ge [in hydras.Epsilon0.E0]
+E0_lt_eq_lt [in hydras.Epsilon0.E0]
+E0_not_Lt_zero [in hydras.Epsilon0.E0]
+E0_Lt_Succ_inv [in hydras.Epsilon0.E0]
+E0_Lt_irrefl [in hydras.Epsilon0.E0]
+E0_pred_Lt [in hydras.Epsilon0.E0]
+E0_pred_of_Succ [in hydras.Epsilon0.E0]
+E0_eq_dec [in hydras.Epsilon0.E0]
+E0_eq_iff [in hydras.Epsilon0.E0]
+E0_eq_intro [in hydras.Epsilon0.E0]
+

F

+F [in gaia_hydras.onType]
+fact_alt_ok [in hydras.solutions_exercises.MorePRExamples]
+fact_correct [in hydras.MoreAck.PrimRecExamples]
+Fake_thm [in hydras.solutions_exercises.is_F_monotonous]
+fastexp3P [in additions.fib]
+fbij [in hydras.Schutte.Ordering_Functions]
+Fcompose_correct [in additions.Euclidean_Chains]
+Fcompose_correct_nat [in additions.Euclidean_Chains]
+Fexp2_correct [in additions.Euclidean_Chains]
+Fexp2_nat_correct [in additions.Euclidean_Chains]
+FF [in hydras.Epsilon0.F_omega]
+FFK_correct [in additions.AM]
+FFK_correct [in additions.Euclidean_Chains]
+FFK_correct_nat [in additions.Euclidean_Chains]
+fibmP [in additions.fib]
+fibtP [in additions.fib]
+fibt_aux [in additions.fib]
+fibZ2P [in additions.fib]
+fibZ3P [in additions.fib]
+fib_mul2_OK [in additions.Fib2]
+fib_mul2_OK_0 [in additions.Fib2]
+fib_SSn [in additions.Fib2]
+fib_ind [in additions.Fib2]
+fib_alt_Ok [in hydras.solutions_exercises.FibonacciPR]
+fib_OK0 [in hydras.solutions_exercises.FibonacciPR]
+find_not_mem [in hydras.rpo.more_list]
+finite_ltR [in hydras.Epsilon0.T1]
+finite_lt [in hydras.Epsilon0.T1]
+Finite_ref [in gaia_hydras.T1Bridge]
+finite_not_limit [in hydras.Schutte.Schutte_basics]
+finite_lt_omega [in hydras.Schutte.Schutte_basics]
+finite_inj [in hydras.Schutte.Schutte_basics]
+finite_mono [in hydras.Schutte.Schutte_basics]
+finite_lt_inv [in hydras.Schutte.Schutte_basics]
+finite_is_finite [in hydras.Gamma0.Gamma0]
+finite_plus_infinite [in hydras.Schutte.Addition]
+finord_lt_wf [in gaia_hydras.ON_gfinite]
+finord_compare_correct [in gaia_hydras.ON_gfinite]
+FinS_Succ_eq [in hydras.Epsilon0.E0]
+FinS_eq [in hydras.Epsilon0.E0]
+fin_lt_omega [in hydras.Gamma0.Gamma0]
+FirstIncompletenessA [in Goedel.goedel1]
+FirstIncompletenessA [in Goedel.goedel2]
+FirstIncompletenessB [in Goedel.goedel1]
+FixPointLNN [in Goedel.fixPoint]
+FixPointLNT [in Goedel.fixPoint]
+FixR_F_eq [in hydras.Prelude.Restriction]
+FK_correct [in additions.AM]
+FK_correct [in additions.Euclidean_Chains]
+flatten [in hydras.Epsilon0.Paths]
+flatten_valid_2 [in additions.Addition_Chains]
+flatten_valid [in additions.Addition_Chains]
+flatten_aux_valid [in additions.Addition_Chains]
+forallE [in hydras.Ackermann.LNN]
+forallE [in hydras.Ackermann.LNT]
+forallE [in hydras.Ackermann.folLogic]
+forallI [in hydras.Ackermann.LNN]
+forallI [in hydras.Ackermann.LNT]
+forallI [in hydras.Ackermann.folLogic]
+forallSimp [in hydras.Ackermann.LNN]
+forallSimp [in hydras.Ackermann.LNT]
+forallSimp [in hydras.Ackermann.folLogic]
+forall_weak [in hydras.Prelude.Sort_spec]
+Forall_forall [in hydras.Prelude.MoreVectors]
+Forall_and [in hydras.Prelude.MoreVectors]
+Forall_inv [in hydras.Prelude.MoreVectors]
+Forall2R_iff [in hydras.Prelude.MoreLists]
+Forall2R_cons [in hydras.Prelude.MoreLists]
+Forall2_indR [in hydras.Prelude.MoreLists]
+Forall2_RR [in hydras.Prelude.MoreLists]
+Forall2_R [in hydras.Prelude.MoreLists]
+Forall2_inv [in hydras.Prelude.MoreVectors]
+Formula_depth_ind2 [in hydras.Ackermann.fol]
+Formula_depth_rec2_forall [in hydras.Ackermann.fol]
+Formula_depth_rec2_not [in hydras.Ackermann.fol]
+Formula_depth_rec2_imp [in hydras.Ackermann.fol]
+Formula_depth_rec2rec_nice [in hydras.Ackermann.fol]
+Formula_depth_rec_indep [in hydras.Ackermann.fol]
+formula_eqdec [in hydras.Ackermann.fol]
+freeVarAddExists1 [in Goedel.PRrepresentable]
+freeVarAddExists2 [in Goedel.PRrepresentable]
+freeVarAddForalls1 [in Goedel.PRrepresentable]
+freeVarAddForalls2 [in Goedel.PRrepresentable]
+freeVarClosed [in hydras.Ackermann.folProp]
+freeVarClosedList1 [in hydras.Ackermann.folProp]
+freeVarClosedList2 [in hydras.Ackermann.folProp]
+freeVarCodeSysPf [in Goedel.codeSysPrf]
+freeVarCodeSysPrf [in Goedel.codeSysPrf]
+freeVarCodeSysPrfN [in Goedel.codeSysPrf]
+freeVarG [in Goedel.goedel1]
+freeVarInterpFormula [in hydras.Ackermann.model]
+freeVarInterpRel [in hydras.Ackermann.model]
+freeVarInterpTerm [in hydras.Ackermann.model]
+freeVarListFormulaApp [in hydras.Ackermann.folProp]
+freeVarLT [in hydras.Ackermann.LNN]
+freeVarMap_ext [in hydras.Ackermann.subAll]
+freeVarMap1 [in hydras.Ackermann.subAll]
+freeVarNNHelp [in hydras.Ackermann.model]
+freeVarPlus [in hydras.Ackermann.LNN]
+freeVarPlus [in hydras.Ackermann.LNT]
+freeVarPrimRecPiFormulaHelp1 [in Goedel.PRrepresentable]
+freeVarPrimRecSigmaFormulaHelp1 [in Goedel.PRrepresentable]
+freeVarRepT' [in Goedel.rosserPA]
+freeVarSubAllFormula1 [in hydras.Ackermann.subAll]
+freeVarSubAllFormula2 [in hydras.Ackermann.subAll]
+freeVarSubAllTerms1 [in hydras.Ackermann.subAll]
+freeVarSubAllTerms2 [in hydras.Ackermann.subAll]
+freeVarSubAllTerm1 [in hydras.Ackermann.subAll]
+freeVarSubAllTerm2 [in hydras.Ackermann.subAll]
+freeVarSubFormula1 [in hydras.Ackermann.subProp]
+freeVarSubFormula2 [in hydras.Ackermann.subProp]
+freeVarSubFormula3 [in hydras.Ackermann.subProp]
+freeVarSubFormula4 [in hydras.Ackermann.subProp]
+freeVarSubTerms1 [in hydras.Ackermann.subProp]
+freeVarSubTerms2 [in hydras.Ackermann.subProp]
+freeVarSubTerms3 [in hydras.Ackermann.subProp]
+freeVarSubTerms4 [in hydras.Ackermann.subProp]
+freeVarSubTerm1 [in hydras.Ackermann.subProp]
+freeVarSubTerm2 [in hydras.Ackermann.subProp]
+freeVarSubTerm3 [in hydras.Ackermann.subProp]
+freeVarSubTerm4 [in hydras.Ackermann.subProp]
+freeVarSucc [in hydras.Ackermann.LNN]
+freeVarSucc [in hydras.Ackermann.LNT]
+freeVarTApply [in hydras.Ackermann.folProp]
+freeVarTimes [in hydras.Ackermann.LNN]
+freeVarTimes [in hydras.Ackermann.LNT]
+freeVarZero [in hydras.Ackermann.LNN]
+freeVarZero [in hydras.Ackermann.LNT]
+fstar_S [in hydras.Epsilon0.F_alpha]
+Fstar_S [in hydras.Epsilon0.F_alpha]
+FS_rw [in hydras.Epsilon0.T1]
+fun_le_trans [in hydras.Prelude.Iterates]
+fun_bijection_is_ZL_bijection [in hydras.Schutte.PartialFun]
+fun_bijection_codomain [in hydras.Schutte.PartialFun]
+F_alpha_not_PR [in gaia_hydras.GF_alpha]
+F_alpha_not_PR_E0 [in gaia_hydras.GF_alpha]
+F_restricted_mono_l [in gaia_hydras.GF_alpha]
+F_limE [in gaia_hydras.GF_alpha]
+F_succE [in gaia_hydras.GF_alpha]
+F_alpha_0_eq [in gaia_hydras.GF_alpha]
+F_mono_l [in gaia_hydras.GF_alpha]
+F_zeroE [in gaia_hydras.GF_alpha]
+F_alpha_positive [in gaia_hydras.GF_alpha]
+F_alpha_Succ_le [in gaia_hydras.GF_alpha]
+F_alpha_dom [in gaia_hydras.GF_alpha]
+F_alpha_mono [in gaia_hydras.GF_alpha]
+F_alpha_gt [in gaia_hydras.GF_alpha]
+f_succ_eqn [in hydras.Epsilon0.F_alpha]
+f_lim_eqn [in hydras.Epsilon0.F_alpha]
+f_zero_eqn [in hydras.Epsilon0.F_alpha]
+f_star_iterate [in hydras.Epsilon0.F_alpha]
+f_star_Succ [in hydras.Epsilon0.F_alpha]
+f_eq2 [in hydras.Epsilon0.F_alpha]
+f_star_zero_eqn [in hydras.Epsilon0.F_alpha]
+F_mono_l [in hydras.Epsilon0.F_alpha]
+F_mono_l_0 [in hydras.Epsilon0.F_alpha]
+F_restricted_mono_l [in hydras.Epsilon0.F_alpha]
+F_alpha_dom [in hydras.Epsilon0.F_alpha]
+F_alpha_Succ_le [in hydras.Epsilon0.F_alpha]
+F_alpha_positive [in hydras.Epsilon0.F_alpha]
+F_alpha_gt [in hydras.Epsilon0.F_alpha]
+F_alpha_mono [in hydras.Epsilon0.F_alpha]
+F_One_Zero_ge [in hydras.Epsilon0.F_alpha]
+F_One_Zero_dom [in hydras.Epsilon0.F_alpha]
+F_alpha_0_eq [in hydras.Epsilon0.F_alpha]
+F_succ_eqn [in hydras.Epsilon0.F_alpha]
+F_lim_eqn [in hydras.Epsilon0.F_alpha]
+F_zero_eqn [in hydras.Epsilon0.F_alpha]
+F_star_iterate [in hydras.Epsilon0.F_alpha]
+F_star_Succ [in hydras.Epsilon0.F_alpha]
+F_eq2 [in hydras.Epsilon0.F_alpha]
+F_star_zero_eqn [in hydras.Epsilon0.F_alpha]
+f_ok_inv [in hydras.Epsilon0.Large_Sets]
+F_alpha_notPR_inv [in hydras.Epsilon0.F_omega]
+F_alpha_PR_inv [in hydras.Epsilon0.F_omega]
+F_alpha_not_PR [in hydras.Epsilon0.F_omega]
+F_omega_not_PR [in hydras.Epsilon0.F_omega]
+F_vs_Ack [in hydras.Epsilon0.F_omega]
+F_iterate_rw [in hydras.Epsilon0.F_omega]
+F_alpha_Sn [in hydras.solutions_exercises.F_3]
+F_3_eqn [in hydras.solutions_exercises.F_3]
+f_minoration [in hydras.Epsilon0.Hprime]
+f_sup_commutes [in hydras.Schutte.Ordering_Functions]
+F_not_lim [in hydras.Gamma0.Gamma0]
+F0 [in hydras.Epsilon0.Hprime]
+F1 [in hydras.solutions_exercises.MultisetWf]
+F1 [in hydras.Epsilon0.Hprime]
+F1 [in hydras.Hydra.KP_example]
+F1_correct [in additions.AM]
+F1_correct [in additions.Euclidean_Chains]
+F1_neutral_r [in additions.Euclidean_Chains]
+F1_neutral_l [in additions.Euclidean_Chains]
+F1_simpl [in hydras.Epsilon0.Hprime]
+F2 [in hydras.Epsilon0.F_alpha]
+F2 [in hydras.solutions_exercises.MultisetWf]
+F2 [in hydras.Epsilon0.Hprime]
+F2 [in hydras.Hydra.KP_example]
+F2C_correct [in additions.Euclidean_Chains]
+F2q_correct [in additions.AM]
+F2q_correct_1 [in additions.AM]
+F2q_correct_0 [in additions.AM]
+F2_correct [in additions.Euclidean_Chains]
+F3 [in hydras.Epsilon0.Hprime]
+F3_correct [in additions.AM]
+F3_correct [in additions.Euclidean_Chains]
+F5 [in hydras.Epsilon0.F_alpha]
+F6 [in hydras.Epsilon0.F_alpha]
+F7 [in hydras.Epsilon0.F_alpha]
+F8 [in hydras.Epsilon0.F_alpha]
+F9 [in hydras.Epsilon0.F_alpha]
+F9_correct [in additions.Euclidean_Chains]
+

G

+gamma_positive [in hydras.Epsilon0.Paths]
+Gamma0_prec.prec_transitive [in hydras.Gamma0.Gamma0]
+Gamma0_prec.prec_antisym [in hydras.Gamma0.Gamma0]
+Gamma0_prec.prec_dec [in hydras.Gamma0.Gamma0]
+Gamma0_sig.eq_symbol_dec [in hydras.Gamma0.Gamma0]
+Gamma0.add_le4 [in gaia_hydras.nfwfgaia]
+Gamma0.add_le3 [in gaia_hydras.nfwfgaia]
+Gamma0.add_inj [in gaia_hydras.nfwfgaia]
+Gamma0.add_le2 [in gaia_hydras.nfwfgaia]
+Gamma0.add_le1 [in gaia_hydras.nfwfgaia]
+Gamma0.add_to_cons [in gaia_hydras.nfwfgaia]
+Gamma0.add_fin_omega [in gaia_hydras.nfwfgaia]
+Gamma0.add_int [in gaia_hydras.nfwfgaia]
+Gamma0.add1Nfin [in gaia_hydras.nfwfgaia]
+Gamma0.ap_pr4 [in gaia_hydras.nfwfgaia]
+Gamma0.ap_pr3 [in gaia_hydras.nfwfgaia]
+Gamma0.ap_pr2 [in gaia_hydras.nfwfgaia]
+Gamma0.ap_pr1 [in gaia_hydras.nfwfgaia]
+Gamma0.ap_limit [in gaia_hydras.nfwfgaia]
+Gamma0.ap_pr0 [in gaia_hydras.nfwfgaia]
+Gamma0.le_succ_succE [in gaia_hydras.nfwfgaia]
+Gamma0.le_succ_succ [in gaia_hydras.nfwfgaia]
+Gamma0.limit_pr [in gaia_hydras.nfwfgaia]
+Gamma0.limit_pr1 [in gaia_hydras.nfwfgaia]
+Gamma0.lt_succ_le_4 [in gaia_hydras.nfwfgaia]
+Gamma0.lt_succ_le_3 [in gaia_hydras.nfwfgaia]
+Gamma0.lt_succ_le_2 [in gaia_hydras.nfwfgaia]
+Gamma0.lt_succ_le_1 [in gaia_hydras.nfwfgaia]
+Gamma0.lt_succ_succE [in gaia_hydras.nfwfgaia]
+Gamma0.lt_succ_succ [in gaia_hydras.nfwfgaia]
+Gamma0.lt_tail [in gaia_hydras.nfwfgaia]
+Gamma0.minus_le [in gaia_hydras.nfwfgaia]
+Gamma0.minus_lt [in gaia_hydras.nfwfgaia]
+Gamma0.nf_phi [in gaia_hydras.nfwfgaia]
+Gamma0.nf_add [in gaia_hydras.nfwfgaia]
+Gamma0.nf_sub [in gaia_hydras.nfwfgaia]
+Gamma0.nf_pred [in gaia_hydras.nfwfgaia]
+Gamma0.nf_succ [in gaia_hydras.nfwfgaia]
+Gamma0.nf_split [in gaia_hydras.nfwfgaia]
+Gamma0.nf_finite [in gaia_hydras.nfwfgaia]
+Gamma0.nf_one [in gaia_hydras.nfwfgaia]
+Gamma0.nf_omega [in gaia_hydras.nfwfgaia]
+Gamma0.nf_psi [in gaia_hydras.nfwfgaia]
+Gamma0.no_critical [in gaia_hydras.nfwfgaia]
+Gamma0.omega_minus_one [in gaia_hydras.nfwfgaia]
+Gamma0.omega_least_inf2 [in gaia_hydras.nfwfgaia]
+Gamma0.omega_least_inf1 [in gaia_hydras.nfwfgaia]
+Gamma0.omega_lt_epsilon0 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_inv3 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_inv2 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_inv0 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_ltE [in gaia_hydras.nfwfgaia]
+Gamma0.phi_eqE [in gaia_hydras.nfwfgaia]
+Gamma0.phi_inj1 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_inj [in gaia_hydras.nfwfgaia]
+Gamma0.phi_mono_c [in gaia_hydras.nfwfgaia]
+Gamma0.phi_mono_b [in gaia_hydras.nfwfgaia]
+Gamma0.phi_mono_a [in gaia_hydras.nfwfgaia]
+Gamma0.phi_inv1 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_ab_le2 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_ab_le1 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_spec4c [in gaia_hydras.nfwfgaia]
+Gamma0.phi_spec4b [in gaia_hydras.nfwfgaia]
+Gamma0.phi_spec4a [in gaia_hydras.nfwfgaia]
+Gamma0.phi_spec3 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_spec2 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_spec1 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_principalR [in gaia_hydras.nfwfgaia]
+Gamma0.phi_cases [in gaia_hydras.nfwfgaia]
+Gamma0.phi_succ [in gaia_hydras.nfwfgaia]
+Gamma0.phi_fix2 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_fix1 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_le3 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_le2 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_le1 [in gaia_hydras.nfwfgaia]
+Gamma0.phi_ap [in gaia_hydras.nfwfgaia]
+Gamma0.pred_succ [in gaia_hydras.nfwfgaia]
+Gamma0.pred_lt [in gaia_hydras.nfwfgaia]
+Gamma0.pred_le [in gaia_hydras.nfwfgaia]
+Gamma0.psi_phi1 [in gaia_hydras.nfwfgaia]
+Gamma0.psi_lt2 [in gaia_hydras.nfwfgaia]
+Gamma0.psi_lt1 [in gaia_hydras.nfwfgaia]
+Gamma0.size_prop [in gaia_hydras.nfwfgaia]
+Gamma0.size_prop1 [in gaia_hydras.nfwfgaia]
+Gamma0.split_add [in gaia_hydras.nfwfgaia]
+Gamma0.split_le [in gaia_hydras.nfwfgaia]
+Gamma0.split_pred [in gaia_hydras.nfwfgaia]
+Gamma0.split_succ [in gaia_hydras.nfwfgaia]
+Gamma0.split_is_succ [in gaia_hydras.nfwfgaia]
+Gamma0.split_limit [in gaia_hydras.nfwfgaia]
+Gamma0.split_finite [in gaia_hydras.nfwfgaia]
+Gamma0.sub_pr1r [in gaia_hydras.nfwfgaia]
+Gamma0.sub_nz [in gaia_hydras.nfwfgaia]
+Gamma0.sub_pr1 [in gaia_hydras.nfwfgaia]
+Gamma0.sub_pr [in gaia_hydras.nfwfgaia]
+Gamma0.sub_le1 [in gaia_hydras.nfwfgaia]
+Gamma0.sub_int [in gaia_hydras.nfwfgaia]
+Gamma0.sub1a [in gaia_hydras.nfwfgaia]
+Gamma0.sub1b [in gaia_hydras.nfwfgaia]
+Gamma0.sub1Nfin [in gaia_hydras.nfwfgaia]
+Gamma0.succ_is_add_one [in gaia_hydras.nfwfgaia]
+Gamma0.succ_psi_lt2 [in gaia_hydras.nfwfgaia]
+Gamma0.succ_psi_lt [in gaia_hydras.nfwfgaia]
+Gamma0.succ_psi [in gaia_hydras.nfwfgaia]
+Gamma0.succ_nz [in gaia_hydras.nfwfgaia]
+Gamma0.succ_inj [in gaia_hydras.nfwfgaia]
+Gamma0.succ_p1 [in gaia_hydras.nfwfgaia]
+Gamma0.succ_pred [in gaia_hydras.nfwfgaia]
+Gamma0.succ_lt [in gaia_hydras.nfwfgaia]
+Gamma0.T1succ_nat [in gaia_hydras.nfwfgaia]
+Gamma0.T1T2range1 [in gaia_hydras.nfwfgaia]
+Gamma0.T1T2range2 [in gaia_hydras.nfwfgaia]
+Gamma0.T1T2_inc [in gaia_hydras.nfwfgaia]
+Gamma0.T1T2_inj [in gaia_hydras.nfwfgaia]
+Gamma0.T2addA [in gaia_hydras.nfwfgaia]
+Gamma0.T2addn0 [in gaia_hydras.nfwfgaia]
+Gamma0.T2add_eq0 [in gaia_hydras.nfwfgaia]
+Gamma0.T2add0n [in gaia_hydras.nfwfgaia]
+Gamma0.T2eqE [in gaia_hydras.nfwfgaia]
+Gamma0.T2eqP [in gaia_hydras.nfwfgaia]
+Gamma0.T2eq_add2l [in gaia_hydras.nfwfgaia]
+Gamma0.T2eq_le [in gaia_hydras.nfwfgaia]
+Gamma0.T2finite_succ [in gaia_hydras.nfwfgaia]
+Gamma0.T2finite2 [in gaia_hydras.nfwfgaia]
+Gamma0.T2ge1 [in gaia_hydras.nfwfgaia]
+Gamma0.T2gt1 [in gaia_hydras.nfwfgaia]
+Gamma0.T2leNgt [in gaia_hydras.nfwfgaia]
+Gamma0.T2lenn [in gaia_hydras.nfwfgaia]
+Gamma0.T2len0 [in gaia_hydras.nfwfgaia]
+Gamma0.T2leP [in gaia_hydras.nfwfgaia]
+Gamma0.T2le_add2r [in gaia_hydras.nfwfgaia]
+Gamma0.T2le_add2l [in gaia_hydras.nfwfgaia]
+Gamma0.T2le_psi_b [in gaia_hydras.nfwfgaia]
+Gamma0.T2le_psi1 [in gaia_hydras.nfwfgaia]
+Gamma0.T2le_trans [in gaia_hydras.nfwfgaia]
+Gamma0.T2le_lt_trans [in gaia_hydras.nfwfgaia]
+Gamma0.T2le_total [in gaia_hydras.nfwfgaia]
+Gamma0.T2le_eqVlt [in gaia_hydras.nfwfgaia]
+Gamma0.T2le_consE [in gaia_hydras.nfwfgaia]
+Gamma0.T2le0n [in gaia_hydras.nfwfgaia]
+Gamma0.T2ltE [in gaia_hydras.nfwfgaia]
+Gamma0.T2ltgtP [in gaia_hydras.nfwfgaia]
+Gamma0.T2ltNge [in gaia_hydras.nfwfgaia]
+Gamma0.T2ltnn [in gaia_hydras.nfwfgaia]
+Gamma0.T2ltn0 [in gaia_hydras.nfwfgaia]
+Gamma0.T2ltP [in gaia_hydras.nfwfgaia]
+Gamma0.T2ltW [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_add2r [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_add2l [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_psi_a [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_psi_b [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_le_trans [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_trans [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_psi_aux [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_trichotomy [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_anti [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_neAle [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_ne' [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_ne [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_consE [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt_psi [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt0n [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt1 [in gaia_hydras.nfwfgaia]
+Gamma0.T2nat_inc [in gaia_hydras.nfwfgaia]
+Gamma0.T2nf_finite [in gaia_hydras.nfwfgaia]
+Gamma0.T2nf_consE [in gaia_hydras.nfwfgaia]
+Gamma0.T2nf_cons_cons [in gaia_hydras.nfwfgaia]
+Gamma0.T2subnn [in gaia_hydras.nfwfgaia]
+Gamma0.T2subn0 [in gaia_hydras.nfwfgaia]
+Gamma0.T2sub0n [in gaia_hydras.nfwfgaia]
+gcanon_limit_v2 [in gaia_hydras.GCanon]
+gcanon_zero [in gaia_hydras.GCanon]
+get_predecessor [in hydras.Prelude.More_Arith]
+ge_omega_iff [in hydras.solutions_exercises.ge_omega_iff]
+ge_zero [in hydras.Schutte.Addition]
+ge_o_segment [in hydras.Schutte.Addition]
+gE0le_iff [in gaia_hydras.T1Bridge]
+gE0lt_wf [in gaia_hydras.T1Bridge]
+gE0lt_iff [in gaia_hydras.T1Bridge]
+gE0_eq_intro [in gaia_hydras.T1Bridge]
+GnawS_omega [in hydras.Epsilon0.Large_Sets]
+GnawS_Gnaw [in hydras.Epsilon0.Large_Sets]
+gnawS_path_toS [in hydras.Epsilon0.Paths]
+gnawS_cut2 [in hydras.Epsilon0.Paths]
+gnawS_cut1 [in hydras.Epsilon0.Paths]
+gnawS_SSn [in hydras.Epsilon0.Paths]
+gnawS_tail [in hydras.Epsilon0.Paths]
+gnawS_succ_eqn2 [in hydras.Epsilon0.Paths]
+gnawS_succ_eqn1 [in hydras.Epsilon0.Paths]
+gnawS_lim2 [in hydras.Epsilon0.Paths]
+gnawS_lim1 [in hydras.Epsilon0.Paths]
+gnaws_rw [in hydras.Epsilon0.Paths]
+gnawS_app [in hydras.Epsilon0.Paths]
+gnawS_to_path_toS [in hydras.Epsilon0.Paths]
+gnawS_nf [in hydras.Epsilon0.Paths]
+gnawS_zero [in hydras.Epsilon0.Paths]
+gnawS_gnaw [in hydras.Epsilon0.Paths]
+gnaw_omega_1 [in hydras.Epsilon0.Large_Sets]
+gnaw_omega_n_SSn [in hydras.Epsilon0.Large_Sets]
+Gnaw_omega [in hydras.Epsilon0.Large_Sets]
+Gnaw_GnawS [in hydras.Epsilon0.Large_Sets]
+gnaw_last_step [in hydras.Epsilon0.Large_Sets]
+gnaw_n_R [in hydras.Epsilon0.Large_Sets]
+gnaw_finite [in hydras.Epsilon0.Large_Sets]
+gnaw_finite_1 [in hydras.Epsilon0.Large_Sets]
+gnaw_finite_1_iota [in hydras.Epsilon0.Large_Sets]
+gnaw_path_to [in hydras.Epsilon0.Paths]
+gnaw_app [in hydras.Epsilon0.Paths]
+gnaw_to_path_to [in hydras.Epsilon0.Paths]
+gnaw_rw [in hydras.Epsilon0.Paths]
+gnaw_succ [in hydras.Epsilon0.Paths]
+gnaw_zero [in hydras.Epsilon0.Paths]
+gnaw_nf [in hydras.Epsilon0.Paths]
+gnaw_gnawS [in hydras.Epsilon0.Paths]
+Goedel'sIncompleteness1st [in Goedel.goedel1]
+GS [in Goedel.goedel2]
+g_1_of [in hydras.Schutte.Ordering_Functions]
+g_1_bij [in hydras.Schutte.Ordering_Functions]
+g_bij [in hydras.Schutte.Ordering_Functions]
+g_mono [in hydras.Schutte.Ordering_Functions]
+g_lemma [in hydras.Schutte.Ordering_Functions]
+g_def [in hydras.Schutte.Ordering_Functions]
+g_unic [in hydras.Schutte.Ordering_Functions]
+g_def1 [in hydras.Schutte.Ordering_Functions]
+G0.compare_correct [in hydras.Gamma0.Gamma0]
+G0.lt_wf [in hydras.Gamma0.Gamma0]
+G0.lt_LT_incl [in hydras.Gamma0.Gamma0]
+G0.Lt_wf [in hydras.Gamma0.Gamma0]
+G0.nfb_proof_unicity [in hydras.Gamma0.Gamma0]
+G0.nfb_equiv [in hydras.Gamma0.Gamma0]
+G0.nfb_a [in hydras.Gamma0.Gamma0]
+G0.zero_nfb [in hydras.Gamma0.Gamma0]
+g2h_canon [in gaia_hydras.GCanon]
+g2h_E0zero [in gaia_hydras.T1Bridge]
+g2h_E0_succ [in gaia_hydras.T1Bridge]
+g2h_succ [in gaia_hydras.T1Bridge]
+g2h_plusE [in gaia_hydras.T1Bridge]
+g2h_multE [in gaia_hydras.T1Bridge]
+g2h_zero [in gaia_hydras.T1Bridge]
+g2h_cons [in gaia_hydras.T1Bridge]
+g2h_phi0 [in gaia_hydras.T1Bridge]
+g2h_diffE [in gaia_hydras.T1Bridge]
+g2h_eqE [in gaia_hydras.T1Bridge]
+g2h_h2gK [in gaia_hydras.T1Bridge]
+g2h_eqE [in gaia_hydras.T2Bridge]
+g2h_h2gK [in gaia_hydras.T2Bridge]
+

H

+HBL1 [in Goedel.goedel2]
+hcons_mult_S2 [in hydras.Hydra.Hydra_Lemmas]
+hcons_mult_S1 [in hydras.Hydra.Hydra_Lemmas]
+hcons_mult_S0 [in hydras.Hydra.Hydra_Lemmas]
+hcons_mult_comm [in hydras.Hydra.Hydra_Lemmas]
+hcons_mult_app [in hydras.Hydra.Hydra_Lemmas]
+headbits [in additions.fib]
+head_LT_cons [in hydras.Epsilon0.T1]
+head_lt_cons [in hydras.Epsilon0.T1]
+head_lt [in hydras.Epsilon0.T1]
+head_lt [in hydras.Schutte.Correctness_E0]
+head_no_round [in hydras.Hydra.Hydra_Lemmas]
+head_no_round_n [in hydras.Hydra.Hydra_Lemmas]
+height_lt_size [in hydras.Hydra.Hydra_Examples]
+height_bad [in hydras.Hydra.Omega_Small]
+HFLim [in hydras.Epsilon0.F_alpha]
+HFsucc [in hydras.Epsilon0.F_alpha]
+HF0 [in hydras.Epsilon0.F_alpha]
+hlt_iff [in gaia_hydras.T1Bridge]
+Hmu [in hydras.Hydra.Epsilon0_Needed_Std]
+hmultA [in gaia_hydras.GaiaToHydra]
+hmult_dist [in gaia_hydras.GaiaToHydra]
+hnf_g2h [in gaia_hydras.T1Bridge]
+hs_eq_dec [in hydras.Hydra.Hydra_Definitions]
+hs2lK [in hydras.Hydra.Hydra_Definitions]
+Hvariant_Termination [in hydras.Hydra.Hydra_Lemmas]
+hyper_exp2_S [in hydras.Prelude.Iterates]
+hy_app_l_nil [in hydras.Hydra.O2H]
+hy_app_assoc [in hydras.Hydra.Hydra_Lemmas]
+h_eq_dec [in hydras.Hydra.Hydra_Definitions]
+H'_F [in gaia_hydras.GF_alpha]
+H'_omega_cube_min [in gaia_hydras.GHprime]
+H'_alpha_gt [in gaia_hydras.GHprime]
+H'_alpha_mono [in gaia_hydras.GHprime]
+H'_dom [in gaia_hydras.GHprime]
+H'_omega_double [in gaia_hydras.GHprime]
+H'_omega [in gaia_hydras.GHprime]
+H'_eq3 [in gaia_hydras.GHprime]
+H'_eq2 [in gaia_hydras.GHprime]
+H'_eq1 [in gaia_hydras.GHprime]
+H'_F [in hydras.Epsilon0.F_alpha]
+H'_L_ [in hydras.Epsilon0.L_alpha]
+H'_dom [in hydras.Epsilon0.Hprime]
+H'_mono_l [in hydras.Epsilon0.Hprime]
+H'_mono_l_1 [in hydras.Epsilon0.Hprime]
+H'_mono_l_0 [in hydras.Epsilon0.Hprime]
+H'_alpha_mono_weak [in hydras.Epsilon0.Hprime]
+H'_alpha_ge_id [in hydras.Epsilon0.Hprime]
+H'_restricted_mono_l [in hydras.Epsilon0.Hprime]
+H'_alpha_dom [in hydras.Epsilon0.Hprime]
+H'_alpha_Succ_le [in hydras.Epsilon0.Hprime]
+H'_alpha_gt [in hydras.Epsilon0.Hprime]
+H'_alpha_mono [in hydras.Epsilon0.Hprime]
+H'_non_mono1 [in hydras.Epsilon0.Hprime]
+H'_omega_cube_min [in hydras.Epsilon0.Hprime]
+H'_omega_sqr_min [in hydras.Epsilon0.Hprime]
+H'_Phi0_omega_exact_formula [in hydras.Epsilon0.Hprime]
+H'_Phi0_omega [in hydras.Epsilon0.Hprime]
+H'_omega_cube [in hydras.Epsilon0.Hprime]
+H'_Phi0_Si [in hydras.Epsilon0.Hprime]
+H'_Phi0_succ [in hydras.Epsilon0.Hprime]
+H'_Phi0_succ_0 [in hydras.Epsilon0.Hprime]
+H'_Phi0_succ_1 [in hydras.Epsilon0.Hprime]
+H'_Omega_term [in hydras.Epsilon0.Hprime]
+H'_Fin_iterate [in hydras.Epsilon0.Hprime]
+H'_Omega_term_0 [in hydras.Epsilon0.Hprime]
+H'_Omega_term_1 [in hydras.Epsilon0.Hprime]
+H'_cons [in hydras.Epsilon0.Hprime]
+H'_omega_sqr [in hydras.Epsilon0.Hprime]
+H'_omega_i [in hydras.Epsilon0.Hprime]
+H'_omega_4 [in hydras.Epsilon0.Hprime]
+H'_omega_3 [in hydras.Epsilon0.Hprime]
+H'_omega_double [in hydras.Epsilon0.Hprime]
+H'_Plus_Fin [in hydras.Epsilon0.Hprime]
+H'_omega [in hydras.Epsilon0.Hprime]
+H'_Fin [in hydras.Epsilon0.Hprime]
+H'_eq2 [in hydras.Epsilon0.Hprime]
+H'_eq3 [in hydras.Epsilon0.Hprime]
+H'_eq2_0 [in hydras.Epsilon0.Hprime]
+H'_eq1 [in hydras.Epsilon0.Hprime]
+H'_L_ [in gaia_hydras.GL_alpha]
+h0_h1 [in hydras.Hydra.KP_example]
+h1_h2 [in hydras.Hydra.KP_example]
+h2g_zero [in gaia_hydras.T1Bridge]
+h2g_cons [in gaia_hydras.T1Bridge]
+h2g_diffE [in gaia_hydras.T1Bridge]
+h2g_eqE [in gaia_hydras.T1Bridge]
+h2g_g2hK [in gaia_hydras.T1Bridge]
+h2g_eqE [in gaia_hydras.T2Bridge]
+h2g_g2hK [in gaia_hydras.T2Bridge]
+h2_h3 [in hydras.Hydra.KP_example]
+

I

+id_le_f_alpha [in hydras.Epsilon0.F_alpha]
+iffExist [in hydras.Ackermann.folReplace]
+iffE1 [in hydras.Ackermann.LNN]
+iffE1 [in hydras.Ackermann.LNT]
+iffE1 [in hydras.Ackermann.folLogic]
+iffE2 [in hydras.Ackermann.LNN]
+iffE2 [in hydras.Ackermann.LNT]
+iffE2 [in hydras.Ackermann.folLogic]
+iffI [in hydras.Ackermann.LNN]
+iffI [in hydras.Ackermann.LNT]
+iffI [in hydras.Ackermann.folLogic]
+iffRefl [in hydras.Ackermann.LNN]
+iffRefl [in hydras.Ackermann.LNT]
+iffRefl [in hydras.Ackermann.folLogic]
+iffSym [in hydras.Ackermann.LNN]
+iffSym [in hydras.Ackermann.LNT]
+iffSym [in hydras.Ackermann.folLogic]
+iffTrans [in hydras.Ackermann.LNN]
+iffTrans [in hydras.Ackermann.LNT]
+iffTrans [in hydras.Ackermann.folLogic]
+if_simpl [in gaia_hydras.nfwfgaia]
+image_as_Im [in hydras.Schutte.PartialFun]
+image_B_g_seg [in hydras.Schutte.Ordering_Functions]
+impE [in hydras.Ackermann.LNN]
+impE [in hydras.Ackermann.LNT]
+impE [in hydras.Ackermann.folLogic]
+impForall [in hydras.Ackermann.folReplace]
+impI [in hydras.Ackermann.LNN]
+impI [in hydras.Ackermann.LNT]
+impI [in hydras.Ackermann.folLogic]
+Impossibility_free [in hydras.Hydra.Epsilon0_Needed_Free]
+Impossibility_std [in hydras.Hydra.Epsilon0_Needed_Std]
+Impossibility_std [in gaia_hydras.GHydra]
+Impossibility_free [in gaia_hydras.GHydra]
+Impossible [in hydras.Hydra.Omega2_Small]
+impossible_nat [in hydras.solutions_exercises.MultisetWf]
+impRefl [in hydras.Ackermann.LNN]
+impRefl [in hydras.Ackermann.LNT]
+impRefl [in hydras.Ackermann.folLogic]
+impTrans [in hydras.Ackermann.LNN]
+impTrans [in hydras.Ackermann.LNT]
+impTrans [in hydras.Ackermann.folLogic]
+im_U_f [in hydras.Schutte.Ordering_Functions]
+incl_decomp [in hydras.Prelude.MoreLists]
+incl_inv [in hydras.Prelude.MoreLists]
+Incompleteness [in Goedel.rosser]
+Inc_U_A [in hydras.Schutte.Ordering_Functions]
+induct [in hydras.Ackermann.PA]
+induct_step [in hydras.Schutte.Critical]
+Inh_ord [in hydras.Schutte.Schutte_basics]
+inh_U [in hydras.Schutte.Ordering_Functions]
+inject_lt_epsilon0_ex_unique [in hydras.Schutte.Correctness_E0]
+inject_lt_epsilon0_ex [in hydras.Schutte.Correctness_E0]
+inject_lt_epsilon0_ex_cnf [in hydras.Schutte.Correctness_E0]
+inject_mult_fin_r [in hydras.Schutte.Correctness_E0]
+inject_plus [in hydras.Schutte.Correctness_E0]
+inject_rw [in hydras.Schutte.Correctness_E0]
+inject_lt_epsilon0 [in hydras.Schutte.Correctness_E0]
+inject_monoR [in hydras.Schutte.Correctness_E0]
+inject_injective [in hydras.Schutte.Correctness_E0]
+inject_mono [in hydras.Schutte.Correctness_E0]
+inject_mono_0 [in hydras.Schutte.Correctness_E0]
+inject_of_phi0 [in hydras.Schutte.Correctness_E0]
+inject_of_omega [in hydras.Schutte.Correctness_E0]
+inject_of_finite [in hydras.Schutte.Correctness_E0]
+inject_of_zero [in hydras.Schutte.Correctness_E0]
+inject_of_finite_pos [in hydras.Schutte.Correctness_E0]
+interval_lt_not_In [in hydras.Prelude.MoreLists]
+interval_sorted_ge [in hydras.Prelude.MoreLists]
+interval_unroll [in hydras.Prelude.MoreLists]
+interval_app [in hydras.Prelude.MoreLists]
+interval_singleton [in hydras.Prelude.MoreLists]
+interval_not_empty_iff [in hydras.Prelude.MoreLists]
+interval_not_empty [in hydras.Prelude.MoreLists]
+interval_length [in hydras.Prelude.MoreLists]
+interval_def [in gaia_hydras.GPaths]
+inv_Pi [in hydras.solutions_exercises.FibonacciPR]
+inv_fun_bij [in hydras.Schutte.PartialFun]
+inv_composeR [in hydras.Schutte.PartialFun]
+inv_compose [in hydras.Schutte.PartialFun]
+inv_trans [in hydras.rpo.closure]
+In_betaFormula_subst_2_1 [in Goedel.PRrepresentable]
+In_betaFormula_subst_2 [in Goedel.PRrepresentable]
+In_betaFormula [in Goedel.PRrepresentable]
+In_betaFormula_subst_1 [in Goedel.PRrepresentable]
+In_betaFormula_subst_1_2 [in Goedel.PRrepresentable]
+In_betaFormula_subst_1_2_0 [in Goedel.PRrepresentable]
+In_sorted_ge_inv [in hydras.Prelude.MoreLists]
+in_remove [in hydras.rpo.more_list]
+in_map_in [in hydras.rpo.more_list]
+in_in_map [in hydras.rpo.more_list]
+In_freeVarListFormulaE [in hydras.Ackermann.folProp]
+In_freeVarListFormula [in hydras.Ackermann.folProp]
+in_remove_neq [in hydras.Ackermann.ListExt]
+In_add2 [in hydras.MoreAck.FolExamples]
+In_add1 [in hydras.MoreAck.FolExamples]
+In_cases [in hydras.Prelude.MoreVectors]
+iotas_succ [in hydras.Hydra.O2H]
+iota_phi0 [in hydras.Hydra.O2H]
+iota_of_succ [in hydras.Hydra.O2H]
+iota_tail [in hydras.Hydra.O2H]
+iota_rw4 [in hydras.Hydra.O2H]
+iota_rw3 [in hydras.Hydra.O2H]
+iota_rw2 [in hydras.Hydra.O2H]
+iota_rw1 [in hydras.Hydra.O2H]
+iota_succ_round [in hydras.Hydra.O2H]
+iota_succ_round_n [in hydras.Hydra.O2H]
+iota_succ_R1 [in hydras.Hydra.O2H]
+iota_iotas [in hydras.Hydra.O2H]
+iota_from_lt_not_In [in hydras.Prelude.MoreLists]
+iota_from_sorted_ge [in hydras.Prelude.MoreLists]
+iota_from_unroll [in hydras.Prelude.MoreLists]
+iota_from_plus [in hydras.Prelude.MoreLists]
+iota_from_app [in hydras.Prelude.MoreLists]
+iota_adapt [in gaia_hydras.GPaths]
+iota_ind [in hydras.Schutte.MoreEpsilonIota]
+iota_eq [in hydras.Schutte.MoreEpsilonIota]
+iota_spec_1 [in hydras.Schutte.MoreEpsilonIota]
+iota_fun_ind [in hydras.Schutte.PartialFun]
+iota_fun_e [in hydras.Schutte.PartialFun]
+iota_demo.Bad.le_bottom_zero [in hydras.Schutte.Schutte_basics]
+iota_demo.Bad.bottom_eq [in hydras.Schutte.Schutte_basics]
+iota_demo.Bad.le_zero_bottom [in hydras.Schutte.Schutte_basics]
+iota_demo.zero_le [in hydras.Schutte.Schutte_basics]
+iota_demo.R [in hydras.Schutte.Schutte_basics]
+iota_mono [in hydras.OrdinalNotations.ON_Finite]
+iota_compare_commute [in hydras.OrdinalNotations.ON_Finite]
+is_limit_phi0 [in hydras.Schutte.AP]
+is_in_seq_not_Acc [in hydras.solutions_exercises.MultisetWf]
+is_limit_sup_members [in hydras.Schutte.Schutte_basics]
+is_limit_omega [in hydras.Schutte.Schutte_basics]
+is_limit_ab [in hydras.Gamma0.Gamma0]
+is_limit_intro [in hydras.Gamma0.Gamma0]
+is_limit_cons_inv [in hydras.Gamma0.Gamma0]
+is_limit_not_succ [in hydras.Gamma0.Gamma0]
+is_finite_finite [in hydras.Gamma0.Gamma0]
+iterate_ge_diag' [in hydras.Prelude.Iterates]
+iterate_gt_diag' [in hydras.Prelude.Iterates]
+iterate_ge_from [in hydras.Prelude.Iterates]
+iterate_ext2 [in hydras.Prelude.Iterates]
+iterate_dom_prop [in hydras.Prelude.Iterates]
+iterate_mono_1 [in hydras.Prelude.Iterates]
+iterate_ge'' [in hydras.Prelude.Iterates]
+iterate_ge' [in hydras.Prelude.Iterates]
+iterate_Sge [in hydras.Prelude.Iterates]
+iterate_ge [in hydras.Prelude.Iterates]
+iterate_mono [in hydras.Prelude.Iterates]
+iterate_mono2 [in hydras.Prelude.Iterates]
+iterate_le_np_le [in hydras.Prelude.Iterates]
+iterate_le_n_Sn [in hydras.Prelude.Iterates]
+iterate_lt_from [in hydras.Prelude.Iterates]
+iterate_lt [in hydras.Prelude.Iterates]
+iterate_le [in hydras.Prelude.Iterates]
+iterate_ext [in hydras.Prelude.Iterates]
+iterate_rw [in hydras.Prelude.Iterates]
+iterate_S_eqn2 [in hydras.Prelude.Iterates]
+iterate_S_eqn [in hydras.Prelude.Iterates]
+iterate_compat3 [in hydras.Prelude.Iterates]
+iterate_compat2 [in hydras.Prelude.Iterates]
+iterate_compat [in hydras.Prelude.Iterates]
+iterate_comm [in hydras.Prelude.Iterates]
+iterate_nat_rec [in hydras.MoreAck.AckNotPR]
+iterate_extEqual [in hydras.MoreAck.Iterate_compat]
+iterate_compat [in hydras.MoreAck.Iterate_compat]
+iterate2_mono2 [in hydras.Prelude.Iterates]
+iterate2_mono3 [in hydras.Prelude.Iterates]
+iterate2_mono_weak [in hydras.Prelude.Iterates]
+iterate2_mono [in hydras.Prelude.Iterates]
+iter_combine [in additions.fib]
+iter_comm [in additions.fib]
+iter_mul [in additions.fib]
+I_i_wf [in gaia_hydras.onType]
+

K

+Kchain_correct_conv [in additions.Euclidean_Chains]
+KFF_correct [in additions.AM]
+KFF_correct [in additions.Euclidean_Chains]
+KFF_correct_nat [in additions.Euclidean_Chains]
+KFK_correct [in additions.AM]
+KFK_correct [in additions.Euclidean_Chains]
+KFK_proper [in additions.Euclidean_Chains]
+KFK_correct_nat [in additions.Euclidean_Chains]
+KP_5_iii [in hydras.Epsilon0.Paths]
+KP_arrowS_zero [in hydras.Epsilon0.Paths]
+ksi_plus_beta_eq [in hydras.Schutte.AP]
+ksi_plus_beta [in hydras.Schutte.AP]
+ksi_plus_seq_n' [in hydras.Schutte.AP]
+ksi_plus_seq_n [in hydras.Schutte.AP]
+KS_thm_2_4 [in gaia_hydras.GPaths]
+KS_thm_2_4_E0 [in hydras.Epsilon0.Paths]
+KS_thm_2_4 [in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma5 [in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma4 [in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma3 [in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma3_0 [in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma2 [in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma1' [in hydras.Epsilon0.Paths]
+KS_thm_2_4_lemma1 [in hydras.Epsilon0.Paths]
+K2F_correct [in additions.Euclidean_Chains]
+K2F_correct_nat [in additions.Euclidean_Chains]
+k3_1_proper [in additions.Euclidean_Chains]
+k3_1_correct [in additions.Euclidean_Chains]
+k7_3_correct [in additions.Euclidean_Chains]
+

L

+L [in hydras.Epsilon0.F_omega]
+L [in hydras.solutions_exercises.F_3]
+L [in additions.Addition_Chains]
+largeb_n_R [in hydras.Epsilon0.Large_Sets]
+largeb_n [in hydras.Epsilon0.Large_Sets]
+largeb_finite [in hydras.Epsilon0.Large_Sets]
+Largeb_Sb [in hydras.Epsilon0.Large_Sets]
+largeb_Sb [in hydras.Epsilon0.Large_Sets]
+LargeSb_b [in hydras.Epsilon0.Large_Sets]
+largeSb_b [in hydras.Epsilon0.Large_Sets]
+largeS_iff [in hydras.Epsilon0.Large_Sets]
+large_omega_iff [in hydras.Epsilon0.Large_Sets]
+large_omega_2 [in hydras.Epsilon0.Large_Sets]
+large_omega_1 [in hydras.Epsilon0.Large_Sets]
+large_n_iff [in hydras.Epsilon0.Large_Sets]
+large_iff [in hydras.Epsilon0.Large_Sets]
+Least_zero [in hydras.OrdinalNotations.ON_Omega2]
+least_AP [in hydras.Schutte.AP]
+Least_is_0 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Least_not_Limit [in hydras.Prelude.MoreOrders]
+Least_not_Succ [in hydras.Prelude.MoreOrders]
+least_member_of_eq [in hydras.Schutte.Well_Orders]
+least_member_ex_unique [in hydras.Schutte.Well_Orders]
+least_member_unicity [in hydras.Schutte.Well_Orders]
+least_member_glb [in hydras.Schutte.Well_Orders]
+least_member_lower_bound [in hydras.Schutte.Well_Orders]
+Lemma_4_5_2 [in hydras.Epsilon0.Large_Sets]
+Lemma_4_5_1 [in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_2 [in hydras.Epsilon0.Large_Sets]
+Lemma_4_4_0 [in hydras.Epsilon0.Paths]
+Lemma_4_4_1 [in hydras.Epsilon0.Paths]
+Lemma_4_3 [in hydras.Epsilon0.Paths]
+Lemma_4_3_0 [in hydras.Epsilon0.Paths]
+Lemma2_6_1 [in gaia_hydras.GPaths]
+Lemma2_6_1_E0 [in hydras.Epsilon0.Paths]
+Lemma2_6_1 [in hydras.Epsilon0.Paths]
+Lemma4_4 [in hydras.Epsilon0.Large_Sets]
+Lemma5 [in hydras.Schutte.Critical]
+Lemma5_7 [in hydras.Schutte.Critical]
+Lemma5_2 [in hydras.Schutte.Critical]
+Lemma5_1 [in hydras.Schutte.Critical]
+Lemma5_04 [in hydras.Schutte.Critical]
+Lemma5_03 [in hydras.Schutte.Critical]
+Lemma5_02 [in hydras.Schutte.Critical]
+Lemma5_01 [in hydras.Schutte.Critical]
+Lemma5_0 [in hydras.Schutte.Critical]
+lengthTerms [in hydras.Ackermann.wellFormed]
+length_abnc [in hydras.Gamma0.T2]
+length_ab [in hydras.Gamma0.T2]
+length_psi [in hydras.Gamma0.T2]
+length_n [in hydras.Gamma0.T2]
+length_c [in hydras.Gamma0.T2]
+length_b [in hydras.Gamma0.T2]
+length_a [in hydras.Gamma0.T2]
+length_map [in hydras.rpo.more_list]
+lexwf [in hydras.solutions_exercises.MultisetWf]
+lexwf [in hydras.solutions_exercises.MultisetWf]
+lex_strict_intro_left [in hydras.Prelude.DecPreOrder_Instances]
+lex_strict_intro_right [in hydras.Prelude.DecPreOrder_Instances]
+lex_inv_right [in hydras.Prelude.DecPreOrder_Instances]
+lex_inv_left [in hydras.Prelude.DecPreOrder_Instances]
+lex_of_equiv [in hydras.Prelude.DecPreOrder_Instances]
+lex_nat_wf [in hydras.MoreAck.Ack]
+lex_trans [in hydras.rpo.rpo]
+LE_LT_eq_dec [in hydras.Epsilon0.T1]
+LE_r [in hydras.Epsilon0.T1]
+le_lt_LT [in hydras.Epsilon0.T1]
+LE_phi0 [in hydras.Epsilon0.T1]
+LE_LT_trans [in hydras.Epsilon0.T1]
+LE_antisym [in hydras.Epsilon0.T1]
+LE_trans [in hydras.Epsilon0.T1]
+LE_refl [in hydras.Epsilon0.T1]
+LE_zero [in hydras.Epsilon0.T1]
+LE_le [in hydras.Epsilon0.T1]
+LE_nf_r [in hydras.Epsilon0.T1]
+LE_nf_l [in hydras.Epsilon0.T1]
+le_phi0 [in hydras.Epsilon0.T1]
+le_tail [in hydras.Epsilon0.T1]
+le_zero_inv [in hydras.Epsilon0.T1]
+le_inv [in hydras.Epsilon0.T1]
+le_eq_lt_dec [in hydras.Epsilon0.T1]
+le_0 [in hydras.OrdinalNotations.ON_Omega2]
+le_intror [in hydras.OrdinalNotations.ON_Omega2]
+le_min_bd [in hydras.Prelude.Comparable]
+le_min_a [in hydras.Prelude.Comparable]
+le_max_b [in hydras.Prelude.Comparable]
+le_max_a [in hydras.Prelude.Comparable]
+le_not_gt [in hydras.Prelude.Comparable]
+le_lt_trans [in hydras.Prelude.Comparable]
+le_trans [in hydras.Prelude.Comparable]
+le_refl [in hydras.Prelude.Comparable]
+le_lt_equiv_dec [in hydras.Prelude.DecPreOrder]
+le_lt_weak [in hydras.Prelude.DecPreOrder]
+le_lt_trans [in hydras.Prelude.DecPreOrder]
+le_not_gt [in hydras.Prelude.DecPreOrder]
+le_phi0 [in hydras.Schutte.AP]
+le_lt_trans [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+le_0 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+le_intror [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+le_introl [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+le_S_Ack [in hydras.MoreAck.Ack]
+LE_ref [in gaia_hydras.T1Bridge]
+le_ref [in gaia_hydras.T1Bridge]
+le_lt_eq [in hydras.Prelude.MoreOrders]
+le_alpha_zero [in hydras.Schutte.Schutte_basics]
+le_sup_members [in hydras.Schutte.Schutte_basics]
+le_disj [in hydras.Schutte.Schutte_basics]
+le_not_gt [in hydras.Schutte.Schutte_basics]
+le_eq_or_lt [in hydras.Schutte.Schutte_basics]
+le_antisym [in hydras.Schutte.Schutte_basics]
+le_trans [in hydras.Schutte.Schutte_basics]
+le_lt_trans [in hydras.Schutte.Schutte_basics]
+le_refl [in hydras.Schutte.Schutte_basics]
+Le_refl [in hydras.Epsilon0.E0]
+Le_trans [in hydras.Epsilon0.E0]
+le_lt_eq_dec [in hydras.Epsilon0.E0]
+Le_iff [in hydras.Epsilon0.E0]
+Le_Lt_trans [in hydras.Schutte.Well_Orders]
+Le_antisym [in hydras.Schutte.Well_Orders]
+Le_refl [in hydras.Schutte.Well_Orders]
+le_plus_l [in hydras.Gamma0.Gamma0]
+le_plus_r [in hydras.Gamma0.Gamma0]
+le_b_phi_ab [in hydras.Gamma0.Gamma0]
+le_one_cons [in hydras.Gamma0.Gamma0]
+le_cons_tail [in hydras.Gamma0.Gamma0]
+le_trans [in hydras.Gamma0.Gamma0]
+le_lt_trans [in hydras.Gamma0.Gamma0]
+le_inv_nc [in hydras.Gamma0.Gamma0]
+le_psi_term_le [in hydras.Gamma0.Gamma0]
+le_zero_alpha [in hydras.Gamma0.Gamma0]
+le_a_mult_Sn_a [in hydras.Schutte.Addition]
+le_plus_r [in hydras.Schutte.Addition]
+le_plus_l [in hydras.Schutte.Addition]
+LF1 [in gaia_hydras.GF_alpha]
+LF1 [in hydras.Epsilon0.F_alpha]
+LF2 [in gaia_hydras.GF_alpha]
+LF2 [in hydras.Epsilon0.F_alpha]
+LF2_0 [in hydras.Epsilon0.F_alpha]
+LF2' [in gaia_hydras.GF_alpha]
+LF2' [in hydras.Epsilon0.F_alpha]
+LF3 [in hydras.solutions_exercises.F_3]
+LF3_2 [in gaia_hydras.GF_alpha]
+LF3_2 [in hydras.Epsilon0.F_alpha]
+liftCloseFrom [in hydras.Ackermann.subAll]
+limitb_limit [in hydras.OrdinalNotations.ON_Omega2]
+LimitNotSucc [in hydras.Prelude.Comparable]
+limit_is_lub [in hydras.OrdinalNotations.ON_Omega2]
+limit_is_lub_0 [in hydras.OrdinalNotations.ON_Omega2]
+limit_no_R1 [in hydras.Hydra.O2H]
+limit_no_head [in hydras.Hydra.O2H]
+Limit_Infinity [in hydras.solutions_exercises.Limit_Infinity]
+limit_canonS_not_zero [in gaia_hydras.GCanon]
+limit_iff [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+limit_is_omega [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Limit_gt_Zero [in hydras.Epsilon0.E0]
+Limit_not_Zero [in hydras.Epsilon0.E0]
+limit_rounds [in hydras.Hydra.Omega2_Small]
+limit_rounds_0 [in hydras.Hydra.Omega2_Small]
+limit_plus_fin_ok [in hydras.Gamma0.Gamma0]
+limit_plus_fin_inv0 [in hydras.Gamma0.Gamma0]
+limit_plus_fin_lim [in hydras.Gamma0.Gamma0]
+limit_plus_fin_plus [in hydras.Gamma0.Gamma0]
+list_size_size_eq [in hydras.rpo.more_list]
+list_size_fold [in hydras.rpo.more_list]
+list_size_app [in hydras.rpo.more_list]
+list_size_tl_compat [in hydras.rpo.more_list]
+list_app_length [in hydras.rpo.more_list]
+LL [in hydras.Epsilon0.F_alpha]
+LNN_eqdec [in hydras.Ackermann.LNN]
+LNN2LNT_subFormula [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_freeVarF2 [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_freeVarF1 [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_freeVarF [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_exist [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_iff [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_and [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_or [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_subTerms [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_subTerm [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_freeVarTs [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_freeVarT [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_natToTerm [in hydras.Ackermann.LNN2LNT]
+LNotPR [in hydras.Hydra.Hydra_Theorems]
+LNT_eqdec [in hydras.Ackermann.LNT]
+LNT2LNN_subFormula [in hydras.Ackermann.LNN2LNT]
+LNT2LNN_freeVarF [in hydras.Ackermann.LNN2LNT]
+LNT2LNN_freeVarT [in hydras.Ackermann.LNN2LNT]
+LNT2LNN_subTerm [in hydras.Ackermann.LNN2LNT]
+LNT2LNN_natToTerm [in hydras.Ackermann.LNN2LNT]
+LNT2LNT_formula [in hydras.Ackermann.LNN2LNT]
+LNT2LNT_term [in hydras.Ackermann.LNN2LNT]
+LocallySorted_inv_In [in hydras.Prelude.Sort_spec]
+LocallySorted_trans [in hydras.Prelude.Sort_spec]
+LocallySorted_cons' [in hydras.Prelude.Sort_spec]
+LocallySorted_cons [in hydras.Prelude.Sort_spec]
+LO.compare_correct [in hydras.OrdinalNotations.OmegaOmega]
+LO.compare_reflect [in hydras.OrdinalNotations.OmegaOmega]
+LO.compare_rev [in hydras.OrdinalNotations.OmegaOmega]
+LO.compare_ref [in hydras.OrdinalNotations.OmegaOmega]
+LO.cons_nf [in hydras.OrdinalNotations.OmegaOmega]
+LO.eq_ref [in hydras.OrdinalNotations.OmegaOmega]
+LO.fin_nf [in hydras.OrdinalNotations.OmegaOmega]
+LO.limitb_ref [in hydras.OrdinalNotations.OmegaOmega]
+LO.lt_trans [in hydras.OrdinalNotations.OmegaOmega]
+LO.lt_irrefl [in hydras.OrdinalNotations.OmegaOmega]
+LO.lt_ref [in hydras.OrdinalNotations.OmegaOmega]
+LO.mult_plus_distr_l [in hydras.OrdinalNotations.OmegaOmega]
+LO.mult_nf [in hydras.OrdinalNotations.OmegaOmega]
+LO.mult_ref [in hydras.OrdinalNotations.OmegaOmega]
+LO.nf_ref [in hydras.OrdinalNotations.OmegaOmega]
+LO.nf_inv3 [in hydras.OrdinalNotations.OmegaOmega]
+LO.nf_inv2 [in hydras.OrdinalNotations.OmegaOmega]
+LO.phi0_nf [in hydras.OrdinalNotations.OmegaOmega]
+LO.phi0_ref [in hydras.OrdinalNotations.OmegaOmega]
+LO.plus_nf [in hydras.OrdinalNotations.OmegaOmega]
+LO.plus_ref [in hydras.OrdinalNotations.OmegaOmega]
+LO.single_nf [in hydras.OrdinalNotations.OmegaOmega]
+LO.succb_ref [in hydras.OrdinalNotations.OmegaOmega]
+LO.succ_nf [in hydras.OrdinalNotations.OmegaOmega]
+LO.succ_ref [in hydras.OrdinalNotations.OmegaOmega]
+LO.zero_nf [in hydras.OrdinalNotations.OmegaOmega]
+LS [in hydras.solutions_exercises.FibonacciPR]
+LS [in hydras.Hydra.BigBattle]
+ltBoolFalse [in hydras.Ackermann.primRec]
+ltBoolTrue [in hydras.Ackermann.primRec]
+Ltb_ij [in hydras.OrdinalNotations.ON_Finite]
+ltn_add_ll [in gaia_hydras.nfwfgaia]
+ltn_add_el [in gaia_hydras.nfwfgaia]
+ltn_add_le [in gaia_hydras.nfwfgaia]
+ltn_simpl2 [in gaia_hydras.nfwfgaia]
+ltn_simpl1 [in gaia_hydras.nfwfgaia]
+lt_succ_le_2' [in hydras.Epsilon0.T1]
+LT_succ_LT_eq_dec [in hydras.Epsilon0.T1]
+lt_one [in hydras.Epsilon0.T1]
+lt_cons_phi0_inv [in hydras.Epsilon0.T1]
+LT_eq_LT_dec [in hydras.Epsilon0.T1]
+LT_succ [in hydras.Epsilon0.T1]
+LT_add [in hydras.Epsilon0.T1]
+lt_plus_r [in hydras.Epsilon0.T1]
+lt_plus_l [in hydras.Epsilon0.T1]
+LT_of_finite [in hydras.Epsilon0.T1]
+lt_omega_inv [in hydras.Epsilon0.T1]
+LT_one [in hydras.Epsilon0.T1]
+LT_succ_LE_2 [in hydras.Epsilon0.T1]
+LT_succ_LE [in hydras.Epsilon0.T1]
+lt_succ_le [in hydras.Epsilon0.T1]
+lt_succ_le_2 [in hydras.Epsilon0.T1]
+LT_succ_LE_R [in hydras.Epsilon0.T1]
+lt_succ_le_R [in hydras.Epsilon0.T1]
+LT_cons_0 [in hydras.Epsilon0.T1]
+LT_inv_strong [in hydras.Epsilon0.T1]
+LT_inv [in hydras.Epsilon0.T1]
+LT_LE_trans [in hydras.Epsilon0.T1]
+LT_irrefl [in hydras.Epsilon0.T1]
+LT_trans [in hydras.Epsilon0.T1]
+LT_lt [in hydras.Epsilon0.T1]
+LT_nf_r [in hydras.Epsilon0.T1]
+LT_nf_l [in hydras.Epsilon0.T1]
+lt_a_phi0_b_iff [in hydras.Epsilon0.T1]
+lt_a_phi0_b_def [in hydras.Epsilon0.T1]
+lt_a_phi0_b_phi0R [in hydras.Epsilon0.T1]
+lt_a_phi0_b_phi0 [in hydras.Epsilon0.T1]
+lt_a_phi0_b_intro [in hydras.Epsilon0.T1]
+lt_a_phi0_b_inv1 [in hydras.Epsilon0.T1]
+lt_a_phi0_b_inv [in hydras.Epsilon0.T1]
+lt_a_phi0_a [in hydras.Epsilon0.T1]
+lt_succ [in hydras.Epsilon0.T1]
+lt_inv_head [in hydras.Epsilon0.T1]
+lt_trans [in hydras.Epsilon0.T1]
+lt_fin_iff [in hydras.Epsilon0.T1]
+lt_inv_tail [in hydras.Epsilon0.T1]
+lt_inv_coeff_dec [in hydras.Epsilon0.T1]
+lt_inv_coeff [in hydras.Epsilon0.T1]
+lt_inv [in hydras.Epsilon0.T1]
+lt_irrefl [in hydras.Epsilon0.T1]
+lt_inv_strong [in hydras.Epsilon0.T1]
+lt_omega [in hydras.OrdinalNotations.ON_Omega2]
+lt_eq_lt_dec [in hydras.OrdinalNotations.ON_Omega2]
+lt_succ [in hydras.OrdinalNotations.ON_Omega2]
+lt_succ_le [in hydras.OrdinalNotations.ON_Omega2]
+lt_lt_Sn [in hydras.Prelude.More_Arith]
+LT_to_round_plus [in hydras.Hydra.O2H]
+Lt_n_F_Zero_n [in hydras.Epsilon0.F_alpha]
+lt_eq_lt [in hydras.Prelude.Comparable]
+lt_incl_le [in hydras.Prelude.Comparable]
+lt_le_trans [in hydras.Prelude.Comparable]
+lt_not_ge [in hydras.Prelude.Comparable]
+lt_not_gt [in hydras.Prelude.Comparable]
+lt_wf [in hydras.OrdinalNotations.ON_O]
+LT_to_standard_path [in gaia_hydras.GPaths]
+LT_path_to [in gaia_hydras.GPaths]
+lt_a_phi0_b_oplus [in hydras.Epsilon0.Hessenberg]
+lt_le_trans [in hydras.Prelude.DecPreOrder]
+lt_not_equiv [in hydras.Prelude.DecPreOrder]
+lt_not_ge [in hydras.Prelude.DecPreOrder]
+lt_irreflexive [in hydras.Prelude.DecPreOrder]
+lt_S_2i [in additions.More_on_positive]
+lt_phi0 [in hydras.Schutte.AP]
+lt_beta_exists [in hydras.Schutte.AP]
+lt_omega [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+lt_succ [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+lt_succ_le [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+lt_le_trans [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+LT_inv [in hydras.solutions_exercises.MultisetWf]
+lt_wf_double_ind [in hydras.MoreAck.Ack]
+lt_wf_ind [in hydras.MoreAck.Ack]
+LT_ref [in gaia_hydras.T1Bridge]
+lt_ref [in gaia_hydras.T1Bridge]
+lt_eq_lt_dec [in hydras.OrdinalNotations.ON_mult]
+lt_wf [in hydras.OrdinalNotations.ON_mult]
+LT_to_standard_battle [in hydras.Hydra.Epsilon0_Needed_Std]
+lt_eq_lt_dec [in hydras.OrdinalNotations.ON_plus]
+lt_wf [in hydras.OrdinalNotations.ON_plus]
+lt_omega_finite [in hydras.Schutte.Schutte_basics]
+lt_omega_limit_lt_exists_lt [in hydras.Schutte.Schutte_basics]
+lt_omega_limit [in hydras.Schutte.Schutte_basics]
+lt_sup_exists_lt [in hydras.Schutte.Schutte_basics]
+lt_sup_exists_leq [in hydras.Schutte.Schutte_basics]
+lt_succ_lt [in hydras.Schutte.Schutte_basics]
+lt_succ_le_2 [in hydras.Schutte.Schutte_basics]
+lt_succ_le [in hydras.Schutte.Schutte_basics]
+lt_succ [in hydras.Schutte.Schutte_basics]
+lt_or_ge [in hydras.Schutte.Schutte_basics]
+lt_irrefl [in hydras.Schutte.Schutte_basics]
+lt_le_trans [in hydras.Schutte.Schutte_basics]
+lt_trans [in hydras.Schutte.Schutte_basics]
+lt_le [in hydras.Schutte.Schutte_basics]
+lt_wf [in hydras.OrdinalNotations.ON_Finite]
+lt_omega_inv [in hydras.Epsilon0.E0]
+Lt_Le_incl [in hydras.Epsilon0.E0]
+Lt_trans [in hydras.Epsilon0.E0]
+lt_Succ_le_2 [in hydras.Epsilon0.E0]
+lt_Succ_inv [in hydras.Epsilon0.E0]
+Lt_Succ [in hydras.Epsilon0.E0]
+Lt_Succ_Le [in hydras.Epsilon0.E0]
+Lt_not_Gt [in hydras.Schutte.Well_Orders]
+Lt_Le_trans [in hydras.Schutte.Well_Orders]
+Lt_connect [in hydras.Schutte.Well_Orders]
+LT_to_standard_path [in hydras.Epsilon0.Paths]
+LT_acc_from [in hydras.Epsilon0.Paths]
+LT_path_to [in hydras.Epsilon0.Paths]
+LT_path_toS [in hydras.Epsilon0.Paths]
+lt_not_gt [in hydras.Gamma0.Gamma0]
+lt_epsilon0_succ [in hydras.Gamma0.Gamma0]
+lt_epsilon0_trans [in hydras.Gamma0.Gamma0]
+lt_epsilon0_okR [in hydras.Gamma0.Gamma0]
+lt_epsilon0_ok [in hydras.Gamma0.Gamma0]
+lt_a_phi_ab [in hydras.Gamma0.Gamma0]
+lt_inc_rpo_0 [in hydras.Gamma0.Gamma0]
+lt_rpo_cons_cons [in hydras.Gamma0.Gamma0]
+lt_subterm1 [in hydras.Gamma0.Gamma0]
+lt_succ_le [in hydras.Gamma0.Gamma0]
+lt_succ [in hydras.Gamma0.Gamma0]
+lt_compatR [in hydras.Gamma0.Gamma0]
+lt_compat [in hydras.Gamma0.Gamma0]
+lt_omega_is_fin [in hydras.Gamma0.Gamma0]
+lt_omega_inv [in hydras.Gamma0.Gamma0]
+lt_cons_omega_inv [in hydras.Gamma0.Gamma0]
+lt_one_inv [in hydras.Gamma0.Gamma0]
+lt_tail [in hydras.Gamma0.Gamma0]
+lt_tail0 [in hydras.Gamma0.Gamma0]
+lt_alpha_cons [in hydras.Gamma0.Gamma0]
+lt_alpha_psi [in hydras.Gamma0.Gamma0]
+lt_beta_cons [in hydras.Gamma0.Gamma0]
+lt_beta_psi [in hydras.Gamma0.Gamma0]
+lt_le_trans [in hydras.Gamma0.Gamma0]
+lt_trans [in hydras.Gamma0.Gamma0]
+lt_trans0 [in hydras.Gamma0.Gamma0]
+lt_than_psi [in hydras.Gamma0.Gamma0]
+lt_irr [in hydras.Gamma0.Gamma0]
+lt_wf [in gaia_hydras.nfwfgaia]
+lt_not_wf [in hydras.solutions_exercises.T1_ltNotWf]
+LT1 [in hydras.Epsilon0.T1]
+LT2 [in hydras.Epsilon0.T1]
+LT3 [in hydras.Epsilon0.T1]
+LT4 [in hydras.Epsilon0.T1]
+lub_mono [in hydras.Gamma0.Gamma0]
+lub_unicity [in hydras.Gamma0.Gamma0]
+Lvar [in hydras.Hydra.Epsilon0_Needed_Std]
+L_omega_cube_3_eq [in hydras.Epsilon0.Large_Sets]
+L_omega_cube_eqn [in hydras.Epsilon0.Large_Sets]
+L_omega_cube_ok [in hydras.Epsilon0.Large_Sets]
+L_omega_square_times_ok [in hydras.Epsilon0.Large_Sets]
+L_phi0_mult_Sle [in hydras.Epsilon0.Large_Sets]
+L_phi0_mult_smono [in hydras.Epsilon0.Large_Sets]
+L_phi0_mult_ok [in hydras.Epsilon0.Large_Sets]
+L_omega_square_ok [in hydras.Epsilon0.Large_Sets]
+L_omega_square_smono [in hydras.Epsilon0.Large_Sets]
+L_omega_square_Sle [in hydras.Epsilon0.Large_Sets]
+L_omega_square_eqn [in hydras.Epsilon0.Large_Sets]
+L_omega_square_eqn1 [in hydras.Epsilon0.Large_Sets]
+L_omega_mult_eqn [in hydras.Epsilon0.Large_Sets]
+L_omega_mult_ok [in hydras.Epsilon0.Large_Sets]
+L_omega_mult_Sle [in hydras.Epsilon0.Large_Sets]
+L_omega_ok [in hydras.Epsilon0.Large_Sets]
+L_omega_smono [in hydras.Epsilon0.Large_Sets]
+L_omega_Sle [in hydras.Epsilon0.Large_Sets]
+L_fin_ok [in hydras.Epsilon0.Large_Sets]
+L_S_succ_rw [in hydras.Epsilon0.Large_Sets]
+L_fin_smono [in hydras.Epsilon0.Large_Sets]
+L_finS_S_le [in hydras.Epsilon0.Large_Sets]
+L_lim_ok [in hydras.Epsilon0.Large_Sets]
+L_succ_ok [in hydras.Epsilon0.Large_Sets]
+L_succ_Sle [in hydras.Epsilon0.Large_Sets]
+L_succ_mono [in hydras.Epsilon0.Large_Sets]
+L_spec_unicity [in hydras.Epsilon0.Large_Sets]
+L_pos_inv [in hydras.Epsilon0.Large_Sets]
+L_zero_inv [in hydras.Epsilon0.Large_Sets]
+L_spec_compat [in hydras.Epsilon0.Large_Sets]
+L_spec_inv2 [in hydras.Epsilon0.Large_Sets]
+L_correct [in hydras.Epsilon0.L_alpha]
+L_ok [in hydras.Epsilon0.L_alpha]
+L_ok_lim [in hydras.Epsilon0.L_alpha]
+L_ok_succ [in hydras.Epsilon0.L_alpha]
+L_ok0 [in hydras.Epsilon0.L_alpha]
+L_succ_ok [in hydras.Epsilon0.L_alpha]
+L_ge_S [in hydras.Epsilon0.L_alpha]
+L_ge_id [in hydras.Epsilon0.L_alpha]
+L_omega [in hydras.Epsilon0.L_alpha]
+L_finite [in hydras.Epsilon0.L_alpha]
+L_lim_eqn [in hydras.Epsilon0.L_alpha]
+L_succ_eqn [in hydras.Epsilon0.L_alpha]
+L_eq2 [in hydras.Epsilon0.L_alpha]
+L_zero_eqn [in hydras.Epsilon0.L_alpha]
+l_std_ok [in hydras.Hydra.Battle_length]
+L_0_3 [in hydras.Hydra.BigBattle]
+L_2_3 [in hydras.Hydra.BigBattle]
+L_0_2 [in hydras.Hydra.BigBattle]
+l_std_ok [in gaia_hydras.GHydra]
+l_stdE [in gaia_hydras.GHydra]
+L_fin_ok [in gaia_hydras.GLarge_Sets]
+L_correct [in gaia_hydras.GL_alpha]
+L_pos_inv [in gaia_hydras.GL_alpha]
+L_spec1 [in gaia_hydras.GL_alpha]
+L_spec0 [in gaia_hydras.GL_alpha]
+L_ge_S [in gaia_hydras.GL_alpha]
+L_ge_id [in gaia_hydras.GL_alpha]
+L_omega [in gaia_hydras.GL_alpha]
+L_finite [in gaia_hydras.GL_alpha]
+L_limE [in gaia_hydras.GL_alpha]
+L_succE [in gaia_hydras.GL_alpha]
+L_eq2 [in gaia_hydras.GL_alpha]
+L_zeroE [in gaia_hydras.GL_alpha]
+L0 [in hydras.Epsilon0.T1]
+L0 [in additions.AM]
+L0 [in hydras.solutions_exercises.FibonacciPR]
+L0_95 [in hydras.Hydra.BigBattle]
+L00 [in hydras.Epsilon0.F_omega]
+L02 [in hydras.Epsilon0.F_omega]
+L03 [in hydras.Epsilon0.T1]
+L04 [in hydras.solutions_exercises.F_3]
+L04' [in hydras.solutions_exercises.F_3]
+L05 [in hydras.Epsilon0.T1]
+L06 [in hydras.Hydra.Battle_length]
+L1 [in hydras.Epsilon0.T1]
+L1 [in hydras.solutions_exercises.predSuccUnicity]
+L1 [in hydras.MoreAck.expressibleExamples]
+L1 [in hydras.solutions_exercises.Limit_Infinity]
+L1 [in hydras.solutions_exercises.FibonacciPR]
+L1 [in hydras.solutions_exercises.lt_succ_le]
+L1' [in gaia_hydras.T1Bridge]
+L10 [in hydras.Epsilon0.T1]
+L10 [in hydras.Hydra.BigBattle]
+L11 [in hydras.Epsilon0.T1]
+L12 [in hydras.Epsilon0.T1]
+L13 [in hydras.Epsilon0.T1]
+L14 [in hydras.Epsilon0.T1]
+L2 [in hydras.Epsilon0.T1]
+L2 [in hydras.solutions_exercises.predSuccUnicity]
+L2 [in hydras.MoreAck.expressibleExamples]
+L2 [in hydras.solutions_exercises.Limit_Infinity]
+L2 [in hydras.Epsilon0.F_omega]
+L2 [in hydras.solutions_exercises.FibonacciPR]
+L2 [in hydras.solutions_exercises.F_3]
+L2 [in additions.Addition_Chains]
+l2hsK [in hydras.Hydra.Hydra_Definitions]
+L2_95_S [in hydras.Hydra.BigBattle]
+L2_95 [in hydras.Hydra.BigBattle]
+L2_6_2 [in hydras.Epsilon0.Paths]
+L22 [in hydras.Hydra.BigBattle]
+L3 [in hydras.Epsilon0.T1]
+L3 [in hydras.MoreAck.expressibleExamples]
+L3 [in hydras.solutions_exercises.Limit_Infinity]
+L3 [in hydras.solutions_exercises.F_3]
+L3a [in hydras.Schutte.Ordering_Functions]
+L3_u [in hydras.Schutte.Ordering_Functions]
+L4 [in hydras.Epsilon0.T1]
+L4 [in hydras.MoreAck.expressibleExamples]
+L4 [in hydras.solutions_exercises.Limit_Infinity]
+L4 [in hydras.Hydra.BigBattle]
+L4 [in hydras.solutions_exercises.F_3]
+L46 [in hydras.Hydra.BigBattle]
+L5 [in hydras.Epsilon0.T1]
+L5 [in hydras.MoreAck.expressibleExamples]
+L5 [in hydras.MoreAck.Ack]
+L6 [in hydras.Epsilon0.T1]
+L6 [in hydras.MoreAck.expressibleExamples]
+L7 [in hydras.Epsilon0.T1]
+L8 [in hydras.Epsilon0.T1]
+L87'' [in additions.Addition_Chains]
+L9 [in hydras.Epsilon0.T1]
+L94 [in hydras.Hydra.BigBattle]
+L95 [in hydras.Hydra.BigBattle]
+

M

+majorAnyPR [in hydras.MoreAck.AckNotPR]
+majorProjection [in hydras.MoreAck.AckNotPR]
+majorPR1 [in hydras.MoreAck.AckNotPR]
+majorPR2 [in hydras.MoreAck.AckNotPR]
+majorPR2_strict [in hydras.MoreAck.AckNotPR]
+majorSucc [in hydras.MoreAck.AckNotPR]
+majorZero [in hydras.MoreAck.AckNotPR]
+makeTraceCorrect [in hydras.Ackermann.codeSubFormula]
+makeTraceForallNice [in hydras.Ackermann.codeSubFormula]
+makeTraceImpNice [in hydras.Ackermann.codeSubFormula]
+makeTraceNotNice [in hydras.Ackermann.codeSubFormula]
+makeTrace1 [in hydras.Ackermann.codeSubFormula]
+makeTrace2 [in hydras.Ackermann.codeSubFormula]
+make_chain_correct [in additions.Euclidean_Chains]
+Make.acc_build [in hydras.rpo.rpo]
+Make.acc_lex_drop_proof [in hydras.rpo.rpo]
+Make.ac_syntactic [in hydras.rpo.list_permut]
+Make.ac_syntactic_aux [in hydras.rpo.list_permut]
+Make.add_comm [in hydras.rpo.list_set]
+Make.add_12 [in hydras.rpo.list_set]
+Make.add_2 [in hydras.rpo.list_set]
+Make.add_1 [in hydras.rpo.list_set]
+Make.add_prf [in hydras.rpo.list_set]
+Make.cardinal_eq_set [in hydras.rpo.list_set]
+Make.cardinal_union [in hydras.rpo.list_set]
+Make.cardinal_union_inter_12 [in hydras.rpo.list_set]
+Make.cardinal_union_2 [in hydras.rpo.list_set]
+Make.cardinal_union_1 [in hydras.rpo.list_set]
+Make.cardinal_subset [in hydras.rpo.list_set]
+Make.cons_permut_in [in hydras.rpo.list_permut]
+Make.context_multiset_extension_step_app2 [in hydras.rpo.dickson]
+Make.context_trans_clos_multiset_extension_step_app1 [in hydras.rpo.dickson]
+Make.context_multiset_extension_step_app1 [in hydras.rpo.dickson]
+Make.context_list_permut_app2 [in hydras.rpo.list_permut]
+Make.context_list_permut_app1 [in hydras.rpo.list_permut]
+Make.context_list_permut_cons [in hydras.rpo.list_permut]
+Make.DecVar.eq_A_dec [in hydras.rpo.term]
+Make.dickson [in hydras.rpo.dickson]
+Make.dickson_aux3 [in hydras.rpo.dickson]
+Make.dickson_aux2 [in hydras.rpo.dickson]
+Make.dickson_aux1 [in hydras.rpo.dickson]
+Make.empty_subst_is_id_list [in hydras.rpo.term]
+Make.empty_subst_is_id [in hydras.rpo.term]
+Make.eq_term_dec [in hydras.rpo.term]
+Make.eq_set_list_permut_support [in hydras.rpo.list_set]
+Make.eq_set_trans [in hydras.rpo.list_set]
+Make.eq_set_sym [in hydras.rpo.list_set]
+Make.eq_set_refl [in hydras.rpo.list_set]
+Make.eq_set_dec [in hydras.rpo.list_set]
+Make.filter_union [in hydras.rpo.list_set]
+Make.filter_2 [in hydras.rpo.list_set]
+Make.filter_2_list [in hydras.rpo.list_set]
+Make.filter_1 [in hydras.rpo.list_set]
+Make.filter_1_list [in hydras.rpo.list_set]
+Make.included_remove_red [in hydras.rpo.list_set]
+Make.included_filter_aux [in hydras.rpo.list_set]
+Make.inter_12 [in hydras.rpo.list_set]
+Make.inter_12_aux [in hydras.rpo.list_set]
+Make.inter_2 [in hydras.rpo.list_set]
+Make.inter_2_aux [in hydras.rpo.list_set]
+Make.inter_1 [in hydras.rpo.list_set]
+Make.inter_1_aux [in hydras.rpo.list_set]
+Make.in_permut_in [in hydras.rpo.list_permut]
+Make.in_mult_S [in hydras.rpo.list_permut]
+Make.in_sn_sn [in hydras.rpo.rpo]
+Make.is_a_pos_exists_subtem [in hydras.rpo.term]
+Make.lex1 [in hydras.rpo.rpo]
+Make.lex1_bis [in hydras.rpo.rpo]
+Make.lex2 [in hydras.rpo.rpo]
+Make.lex3 [in hydras.rpo.rpo]
+Make.list_permut_acc [in hydras.rpo.dickson]
+Make.list_permut_multiset_extension_step_2 [in hydras.rpo.dickson]
+Make.list_permut_multiset_extension_step_1 [in hydras.rpo.dickson]
+Make.list_permut_dec [in hydras.rpo.list_permut]
+Make.list_permut_length_2 [in hydras.rpo.list_permut]
+Make.list_permut_length_1 [in hydras.rpo.list_permut]
+Make.list_permut_map [in hydras.rpo.list_permut]
+Make.list_permut_size [in hydras.rpo.list_permut]
+Make.list_permut_length [in hydras.rpo.list_permut]
+Make.list_permut_remove_hd [in hydras.rpo.list_permut]
+Make.list_permut_app_app [in hydras.rpo.list_permut]
+Make.list_permut_add_cons_inside [in hydras.rpo.list_permut]
+Make.list_permut_add_inside [in hydras.rpo.list_permut]
+Make.list_permut_nil [in hydras.rpo.list_permut]
+Make.list_permut_trans [in hydras.rpo.list_permut]
+Make.list_permut_sym [in hydras.rpo.list_permut]
+Make.list_permut_refl [in hydras.rpo.list_permut]
+Make.list_permut_map_acc [in hydras.rpo.rpo]
+Make.mem_dec [in hydras.rpo.list_set]
+Make.multiplicity_app [in hydras.rpo.list_permut]
+Make.multiset_closure [in hydras.rpo.dickson]
+Make.out_mult_O [in hydras.rpo.list_permut]
+Make.o_size3_trans [in hydras.rpo.rpo]
+Make.projection_list_of_SN_terms [in hydras.rpo.rpo]
+Make.remove_red_included [in hydras.rpo.list_set]
+Make.remove_context_list_permut_app2 [in hydras.rpo.list_permut]
+Make.remove_context_list_permut_cons [in hydras.rpo.list_permut]
+Make.replace_at_pos_list_replace_at_pos_in_subterm [in hydras.rpo.term]
+Make.replace_at_pos_is_replace_at_pos2 [in hydras.rpo.term]
+Make.replace_at_pos_is_replace_at_pos1 [in hydras.rpo.term]
+Make.replace_at_pos_unfold [in hydras.rpo.term]
+Make.rpo_add_context [in hydras.rpo.rpo]
+Make.rpo_subst [in hydras.rpo.rpo]
+Make.rpo_mul_trans_clos [in hydras.rpo.rpo]
+Make.rpo_lex_rest_same_length [in hydras.rpo.rpo]
+Make.rpo_trans [in hydras.rpo.rpo]
+Make.rpo_closure [in hydras.rpo.rpo]
+Make.rpo_subterm [in hydras.rpo.rpo]
+Make.rpo_lex_same_length [in hydras.rpo.rpo]
+Make.size_subterm_at_pos [in hydras.rpo.term]
+Make.size_direct_subterm [in hydras.rpo.term]
+Make.size_ge_one [in hydras.rpo.term]
+Make.size_unfold [in hydras.rpo.term]
+Make.subset_cardinal_not_eq_not_eq_set [in hydras.rpo.list_set]
+Make.subset_subset_union [in hydras.rpo.list_set]
+Make.subset_compat [in hydras.rpo.list_set]
+Make.subset_compat_2 [in hydras.rpo.list_set]
+Make.subset_compat_1 [in hydras.rpo.list_set]
+Make.subset_filter [in hydras.rpo.list_set]
+Make.subset_inter_2 [in hydras.rpo.list_set]
+Make.subset_inter_1 [in hydras.rpo.list_set]
+Make.subset_union_2 [in hydras.rpo.list_set]
+Make.subset_union_1 [in hydras.rpo.list_set]
+Make.subset_dec [in hydras.rpo.list_set]
+Make.subst_comp_is_subst_comp [in hydras.rpo.term]
+Make.subst_comp_is_subst_comp_aux2 [in hydras.rpo.term]
+Make.subst_comp_is_subst_comp_aux1 [in hydras.rpo.term]
+Make.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos [in hydras.rpo.term]
+Make.two_cases [in hydras.rpo.dickson]
+Make.two_cases_rpo [in hydras.rpo.rpo]
+Make.union_compat_eq_set [in hydras.rpo.list_set]
+Make.union_compat_subset_2 [in hydras.rpo.list_set]
+Make.union_compat_subset_1 [in hydras.rpo.list_set]
+Make.union_assoc [in hydras.rpo.list_set]
+Make.union_comm [in hydras.rpo.list_set]
+Make.union_empty_2 [in hydras.rpo.list_set]
+Make.union_empty_1 [in hydras.rpo.list_set]
+Make.union_12 [in hydras.rpo.list_set]
+Make.union_12_aux [in hydras.rpo.list_set]
+Make.union_2 [in hydras.rpo.list_set]
+Make.union_2_aux [in hydras.rpo.list_set]
+Make.union_1 [in hydras.rpo.list_set]
+Make.union_1_aux [in hydras.rpo.list_set]
+Make.well_formed_apply_subst [in hydras.rpo.term]
+Make.well_formed_fold [in hydras.rpo.term]
+Make.well_formed_unfold [in hydras.rpo.term]
+Make.wf_rpo [in hydras.rpo.rpo]
+Make.wf_rpo_term [in hydras.rpo.rpo]
+Make.wf_on_mul_rest [in hydras.rpo.rpo]
+Make.wf_on_lex_rest [in hydras.rpo.rpo]
+Make.wf_on_rest [in hydras.rpo.rpo]
+Make.wf_size3 [in hydras.rpo.rpo]
+Make.wf_size2 [in hydras.rpo.rpo]
+Make.wf_size [in hydras.rpo.rpo]
+Make.without_red_permut [in hydras.rpo.list_set]
+Make.without_red_remove_not_common [in hydras.rpo.list_set]
+Make.without_red_remove_not_common_aux [in hydras.rpo.list_set]
+Make.without_red_add_without_red [in hydras.rpo.list_set]
+Make.without_red_singleton [in hydras.rpo.list_set]
+Make.without_red_nil [in hydras.rpo.list_set]
+Make.without_red_add [in hydras.rpo.list_set]
+Make.without_red_remove [in hydras.rpo.list_set]
+Make.without_red_remove_red [in hydras.rpo.list_set]
+Make.without_red_filter_aux [in hydras.rpo.list_set]
+mapListLemma [in hydras.Ackermann.codeSubFormula]
+map_app [in hydras.rpo.more_list]
+map_map [in hydras.rpo.more_list]
+maxApp [in hydras.Ackermann.codeSubFormula]
+maxLemma [in hydras.Ackermann.codeSubFormula]
+maxLemma2 [in hydras.Ackermann.codeSubFormula]
+maxLemma3 [in hydras.Ackermann.codeSubFormula]
+maxSubTerm [in hydras.Ackermann.codeSubFormula]
+maxSubTerms [in hydras.Ackermann.codeSubFormula]
+maxVarFreeVar [in hydras.Ackermann.codeSubFormula]
+max_nf [in hydras.Epsilon0.T1]
+max_le_regL [in hydras.Prelude.More_Arith]
+max_le_regR [in hydras.Prelude.More_Arith]
+max_le_plus [in hydras.Prelude.More_Arith]
+max_refl [in hydras.Prelude.Comparable]
+max_ge_b [in hydras.Prelude.Comparable]
+max_ge_a [in hydras.Prelude.Comparable]
+max_comm [in hydras.Prelude.Comparable]
+max_dec [in hydras.Prelude.Comparable]
+max_v_tl [in hydras.Prelude.MoreVectors]
+max_v_ge [in hydras.Prelude.MoreVectors]
+max_v_lub [in hydras.Prelude.MoreVectors]
+max_v_2 [in hydras.Prelude.MoreVectors]
+membersOk [in hydras.solutions_exercises.OnCodeList]
+membersOk' [in hydras.solutions_exercises.OnCodeList]
+members_omega [in hydras.Schutte.Schutte_basics]
+members_eq [in hydras.Schutte.Schutte_basics]
+members_proper [in hydras.Schutte.Ordering_Functions]
+mem_head_mult_inv [in hydras.Hydra.O2H]
+merge_sort_correct [in hydras.Prelude.Merge_Sort]
+merge_permutation [in hydras.Prelude.Merge_Sort]
+merge_LocallySorted [in hydras.Prelude.Merge_Sort]
+merge_Forall [in hydras.Prelude.Merge_Sort]
+merge_equation [in hydras.Prelude.Merge_Sort]
+merge_rect [in hydras.Prelude.Merge_Sort]
+minimize1 [in Goedel.PRrepresentable]
+minoration [in hydras.Hydra.BigBattle]
+minoration_3 [in hydras.Hydra.BigBattle]
+minoration_2 [in hydras.Hydra.BigBattle]
+minoration_1 [in hydras.Hydra.BigBattle]
+minoration_0 [in hydras.Hydra.BigBattle]
+minPR_correct [in hydras.solutions_exercises.MinPR2]
+minus_le [in hydras.Epsilon0.T1]
+minus_a_a [in hydras.Epsilon0.T1]
+minus_lt [in hydras.Epsilon0.T1]
+minus_semi_assoc [in hydras.Prelude.More_Arith]
+minus_exists [in hydras.Schutte.Addition]
+min_refl [in hydras.Prelude.Comparable]
+min_le_b [in hydras.Prelude.Comparable]
+min_le_ad [in hydras.Prelude.Comparable]
+min_dec [in hydras.Prelude.Comparable]
+min_comm [in hydras.Prelude.Comparable]
+min_max_iff [in hydras.Prelude.Comparable]
+min_alt_correct [in hydras.solutions_exercises.MinPR]
+mlargeS_iff [in hydras.Epsilon0.Large_Sets]
+mlarge_omega [in hydras.Epsilon0.Large_Sets]
+mlarge_FS [in hydras.Epsilon0.Large_Sets]
+mlarge_iff [in hydras.Epsilon0.Large_Sets]
+mlarge_unshift [in hydras.Epsilon0.Large_Sets]
+mlarge_unicity [in hydras.Epsilon0.Large_Sets]
+mlarge_unicity [in gaia_hydras.GLarge_Sets]
+ml_1 [in hydras.Gamma0.Gamma0]
+ml_psi [in hydras.Gamma0.Gamma0]
+ModelConsistent [in hydras.Ackermann.model]
+mono_weak [in hydras.Prelude.Iterates]
+mono_injective [in hydras.Prelude.Iterates]
+mono_le [in hydras.Prelude.Iterates]
+mono_F_Zero [in hydras.Epsilon0.F_alpha]
+mono_seq_weak2 [in hydras.Schutte.AP]
+mono_seq2 [in hydras.Schutte.AP]
+mono_seq [in hydras.Schutte.AP]
+MPSys [in hydras.MoreAck.FolExamples]
+ms_eqn3 [in hydras.Hydra.Hydra_Termination]
+ms_nf [in hydras.Hydra.Hydra_Termination]
+ms_eqn2 [in hydras.Hydra.Hydra_Termination]
+multE4 [in gaia_hydras.T1Bridge]
+multE5 [in gaia_hydras.T1Bridge]
+multLemma1 [in hydras.Ackermann.wellFormed]
+multLemma2 [in hydras.Ackermann.wellFormed]
+mult_plus_distr_l [in hydras.Epsilon0.T1]
+mult_mono [in hydras.Epsilon0.T1]
+mult_nf [in hydras.Epsilon0.T1]
+mult_Sn_add [in hydras.Epsilon0.T1]
+mult_0_a [in hydras.Epsilon0.T1]
+mult_nf_fin [in hydras.Epsilon0.T1]
+mult_a_1 [in hydras.Epsilon0.T1]
+mult_1_a [in hydras.Epsilon0.T1]
+mult_a_0 [in hydras.Epsilon0.T1]
+mult_compat [in hydras.Epsilon0.T1]
+mult_fin_omega [in hydras.Epsilon0.T1]
+mult_Sn_dist [in hydras.Schutte.Correctness_E0]
+mult_lt_lt [in hydras.Prelude.More_Arith]
+mult_ref [in gaia_hydras.T1Bridge]
+mult_ref0 [in gaia_hydras.T1Bridge]
+mult_mod_associative [in additions.Monoid_instances]
+mult_plus_distr_l [in hydras.Epsilon0.E0]
+mult_fin_rw [in hydras.Epsilon0.E0]
+mult_Sn_mono3 [in hydras.Schutte.Addition]
+mult_Sn_mono2 [in hydras.Schutte.Addition]
+mult_fin_r_mono [in hydras.Schutte.Addition]
+mult_fin_r_one [in hydras.Schutte.Addition]
+mult_correct [in hydras.MoreAck.PrimRecExamples]
+mult_eqn1 [in hydras.MoreAck.PrimRecExamples]
+mu_beta_h [in hydras.Hydra.Epsilon0_Needed_Generic]
+mu_positive [in hydras.Hydra.Epsilon0_Needed_Generic]
+mVariant [in gaia_hydras.GHydra]
+my_pow_m2lmul [in additions.fib]
+my_pow_m3lmul [in additions.fib]
+my_powP [in additions.fib]
+m_lt [in hydras.Hydra.Epsilon0_Needed_Free]
+m_variant_LT [in hydras.Hydra.Epsilon0_Needed_Free]
+m_ge [in hydras.Hydra.Epsilon0_Needed_Free]
+m_dominates_Ack [in hydras.Hydra.Hydra_Theorems]
+m_dominates_Ack_from_3 [in hydras.Hydra.Hydra_Theorems]
+m_ge_Ack [in hydras.Hydra.Hydra_Theorems]
+m_ge_F_omega [in hydras.Hydra.Hydra_Theorems]
+m_eqn [in hydras.Hydra.Hydra_Theorems]
+m_ms [in hydras.Hydra.Hydra_Termination]
+m_nf [in hydras.Hydra.Hydra_Termination]
+m_lt [in hydras.Hydra.Epsilon0_Needed_Std]
+m_decrease [in hydras.Hydra.Epsilon0_Needed_Std]
+m_ge [in hydras.Hydra.Epsilon0_Needed_Std]
+m_neq_0 [in additions.Monoid_instances]
+m_nf [in gaia_hydras.GHydra]
+m_ref [in gaia_hydras.GHydra]
+m_strict_mono [in hydras.Hydra.Hydra_Lemmas]
+m_ge_generic [in hydras.Hydra.Epsilon0_Needed_Generic]
+m_ge_0 [in hydras.Hydra.Epsilon0_Needed_Generic]
+m_ge [in hydras.Hydra.Omega2_Small]
+m_lt [in hydras.Hydra.Omega2_Small]
+m_big_h_not_null [in hydras.Hydra.Omega2_Small]
+m_def [in hydras.Epsilon0.Paths]
+m_ge [in hydras.Hydra.Omega_Small]
+m_lt [in hydras.Hydra.Omega_Small]
+m2lfibP [in additions.fib]
+m2lmulP [in additions.fib]
+M2_eq_intros [in additions.Monoid_instances]
+M2.Id2_neutral [in additions.Naive]
+M2.M2_mult_assoc [in additions.Naive]
+M2.power_of_plus [in additions.Naive]
+m3lfibP [in additions.fib]
+m3lmulP [in additions.fib]
+

N

+nAnd [in hydras.Ackermann.LNN]
+nAnd [in hydras.Ackermann.LNT]
+nAnd [in hydras.Ackermann.folLogic]
+natLE [in hydras.Ackermann.NNtheory]
+natLT [in hydras.Ackermann.NNtheory]
+natNE [in hydras.Ackermann.NNtheory]
+natPlus [in hydras.Ackermann.NNtheory]
+natTimes [in hydras.Ackermann.NNtheory]
+nat_double_or_s_double [in hydras.Prelude.More_Arith]
+nat_power_ok [in additions.Compatibility]
+nat_eqb_false [in hydras.Ackermann.primRec]
+nat_2_term_mono [in hydras.Gamma0.Gamma0]
+nat_lt_psi [in hydras.Gamma0.Gamma0]
+nat_lt_cons [in hydras.Gamma0.Gamma0]
+nb_occ_app [in hydras.rpo.more_list]
+nested_Ack_bound [in hydras.MoreAck.Ack]
+neutral_r [in additions.Fib2]
+neutral_l [in additions.Fib2]
+newVar1 [in hydras.Ackermann.folProp]
+newVar2 [in hydras.Ackermann.folProp]
+nExist [in hydras.Ackermann.LNN]
+nExist [in hydras.Ackermann.LNT]
+nExist [in hydras.Ackermann.folLogic]
+next_fib [in additions.Fib2]
+nForall [in hydras.Ackermann.LNN]
+nForall [in hydras.Ackermann.LNT]
+nForall [in hydras.Ackermann.folLogic]
+nf_LT_right [in hydras.Epsilon0.T1]
+nf_exp_F [in hydras.Epsilon0.T1]
+nf_LT_iff [in hydras.Epsilon0.T1]
+nf_omega_omega [in hydras.Epsilon0.T1]
+nf_phi0 [in hydras.Epsilon0.T1]
+nf_omega [in hydras.Epsilon0.T1]
+nf_Wf [in hydras.Epsilon0.T1]
+nf_tail_lt [in hydras.Epsilon0.T1]
+nf_cons_LT [in hydras.Epsilon0.T1]
+nf_omega_tower [in hydras.Epsilon0.T1]
+nf_coeff_irrelevance [in hydras.Epsilon0.T1]
+nf_intro' [in hydras.Epsilon0.T1]
+nf_intro [in hydras.Epsilon0.T1]
+nf_tail_lt_nf [in hydras.Epsilon0.T1]
+nf_of_finite [in hydras.Epsilon0.T1]
+nf_fin [in hydras.Epsilon0.T1]
+nf_FS [in hydras.Epsilon0.T1]
+nf_b_cons_eq [in hydras.Epsilon0.T1]
+nf_cons_iff [in hydras.Epsilon0.T1]
+nf_cons_inv [in hydras.Epsilon0.T1]
+nf_inv3 [in hydras.Epsilon0.T1]
+nf_inv2 [in hydras.Epsilon0.T1]
+nf_inv1 [in hydras.Epsilon0.T1]
+nf_epsilon [in hydras.Gamma0.T2]
+nf_epsilon0 [in hydras.Gamma0.T2]
+nf_omega [in hydras.Gamma0.T2]
+nf_fin [in hydras.Gamma0.T2]
+nf_bounded [in hydras.Schutte.CNF]
+NF_Acc [in hydras.solutions_exercises.MultisetWf]
+NF_inv2 [in hydras.solutions_exercises.MultisetWf]
+NF_inv1 [in hydras.solutions_exercises.MultisetWf]
+nf_ref [in gaia_hydras.T1Bridge]
+nf_canon [in hydras.Epsilon0.Canon]
+nf_proof_unicity [in hydras.Epsilon0.E0]
+nf_m [in hydras.Hydra.Epsilon0_Needed_Generic]
+nf_intro [in hydras.Gamma0.Gamma0]
+nf_coeff_irrelevance [in hydras.Gamma0.Gamma0]
+nf_Wf [in hydras.Gamma0.Gamma0]
+nf_c2 [in hydras.Gamma0.Gamma0]
+nf_c1 [in hydras.Gamma0.Gamma0]
+nf_b2 [in hydras.Gamma0.Gamma0]
+nf_b1 [in hydras.Gamma0.Gamma0]
+nf_a2 [in hydras.Gamma0.Gamma0]
+nf_a1 [in hydras.Gamma0.Gamma0]
+nf_subterm [in hydras.Gamma0.Gamma0]
+nf_fin_inv [in hydras.Gamma0.Gamma0]
+nf_inv_tail [in hydras.Gamma0.Gamma0]
+nf_c [in hydras.Gamma0.Gamma0]
+nf_b [in hydras.Gamma0.Gamma0]
+nf_a [in hydras.Gamma0.Gamma0]
+nilTerms [in hydras.Ackermann.fol]
+nImp [in hydras.Ackermann.LNN]
+nImp [in hydras.Ackermann.LNT]
+nImp [in hydras.Ackermann.folLogic]
+nnE [in hydras.Ackermann.LNN]
+nnE [in hydras.Ackermann.LNT]
+nnE [in hydras.Ackermann.folLogic]
+nnI [in hydras.Ackermann.LNN]
+nnI [in hydras.Ackermann.LNT]
+nnI [in hydras.Ackermann.folLogic]
+nnPlusNotNeeded [in hydras.Ackermann.NNtheory]
+nn1 [in hydras.Ackermann.NN]
+nn2 [in hydras.Ackermann.NN]
+NN2PA [in hydras.Ackermann.NN2PA]
+nn3 [in hydras.Ackermann.NN]
+nn4 [in hydras.Ackermann.NN]
+nn5 [in hydras.Ackermann.NN]
+nn6 [in hydras.Ackermann.NN]
+nn7 [in hydras.Ackermann.NN]
+NN72PA [in hydras.Ackermann.PAtheory]
+nn8 [in hydras.Ackermann.NN]
+NN82PA [in hydras.Ackermann.PAtheory]
+nn9 [in hydras.Ackermann.NN]
+NN92PA [in hydras.Ackermann.PAtheory]
+noMiddle [in hydras.Ackermann.LNN]
+noMiddle [in hydras.Ackermann.LNT]
+noMiddle [in hydras.Ackermann.folLogic]
+none_nb_occ_O [in hydras.rpo.more_list]
+Non_denum [in hydras.Schutte.Schutte_basics]
+nOr [in hydras.Ackermann.LNN]
+nOr [in hydras.Ackermann.LNT]
+nOr [in hydras.Ackermann.folLogic]
+normal_phi0 [in hydras.Schutte.AP]
+normal_plus_alpha [in hydras.Schutte.Addition]
+notBoundedForall [in Goedel.PRrepresentable]
+notCon2wNotCon [in hydras.Ackermann.wConsistent]
+notInFreeVarSys [in hydras.Ackermann.folProp]
+notRelPR [in hydras.Ackermann.primRec]
+not_LT_zero [in hydras.Epsilon0.T1]
+not_zero_lt [in hydras.Epsilon0.T1]
+not_zero_gt_0 [in hydras.Epsilon0.T1]
+not_lt_zero [in hydras.Epsilon0.T1]
+not_double_is_s_double [in hydras.Prelude.More_Arith]
+not_le_gt [in hydras.Prelude.DecPreOrder]
+not_le_ge [in hydras.Prelude.DecPreOrder]
+not_AP_inv2 [in hydras.Schutte.CNF]
+not_AP_inv_0 [in hydras.Schutte.CNF]
+not_acc [in hydras.solutions_exercises.MultisetWf]
+not_countable_unbounded [in hydras.Schutte.Schutte_basics]
+Not_Unbounded_countable [in hydras.Schutte.Schutte_basics]
+Not_Unbounded_bounded [in hydras.Schutte.Schutte_basics]
+not_is_succ_limit [in hydras.Schutte.Schutte_basics]
+not_is_limit_succ [in hydras.Schutte.Schutte_basics]
+not_is_limit_zero [in hydras.Schutte.Schutte_basics]
+not_is_succ_zero [in hydras.Schutte.Schutte_basics]
+not_lt_zero [in hydras.Schutte.Schutte_basics]
+not_gt_le [in hydras.Schutte.Schutte_basics]
+not_lt_zero [in hydras.Gamma0.Gamma0]
+not_decreasing [in gaia_hydras.nfwfgaia]
+not_decreasing [in hydras.solutions_exercises.T1_ltNotWf]
+not_decreasing_aux [in hydras.solutions_exercises.T1_ltNotWf]
+not_acc [in hydras.solutions_exercises.T1_ltNotWf]
+Npos_power_compat [in additions.Compatibility]
+Npos_gt_0 [in additions.More_on_positive]
+Npos_diff_zero [in additions.More_on_positive]
+nth_error_ok_in [in hydras.rpo.more_list]
+nth_error_map [in hydras.rpo.more_list]
+N_size_gt [in additions.Compatibility]
+N_pow_compat [in additions.Compatibility]
+N_pow_power [in additions.Compatibility]
+N_bpow_commute [in additions.Pow_variant]
+N_bpow_ok_R [in additions.Pow_variant]
+N_bpow_ok [in additions.Pow_variant]
+N_bpow_commute [in additions.Pow]
+N_bpow_ok_R [in additions.Pow]
+N_bpow_ok [in additions.Pow]
+N_le_mul_pos [in additions.More_on_positive]
+N_pos_div_eucl_q0 [in additions.More_on_positive]
+N_pos_div_eucl_rest [in additions.More_on_positive]
+N_pos_div_eucl_divides [in additions.More_on_positive]
+N_pos_N2pos [in additions.More_on_positive]
+N_le_1_pos [in additions.More_on_positive]
+N_0_le_n [in additions.More_on_positive]
+N_simpl [in hydras.Epsilon0.Hprime]
+N2pos_lt_switch2 [in additions.More_on_positive]
+N2pos_lt_switch [in additions.More_on_positive]
+N2pos_pos [in additions.More_on_positive]
+

O

+of_image [in hydras.Schutte.Ordering_Functions]
+of_beta' [in hydras.Schutte.Ordering_Functions]
+of_u [in hydras.Schutte.Addition]
+of_g [in hydras.Schutte.Addition]
+OK87 [in additions.Euclidean_Chains]
+omeganf [in gaia_hydras.T1Bridge]
+omega_limit [in hydras.Epsilon0.T1]
+omega_term_def [in hydras.Epsilon0.T1]
+omega_term_plus_rw [in hydras.Epsilon0.T1]
+omega_ap [in hydras.OrdinalNotations.ON_Omega2]
+Omega_limit_limitb [in hydras.OrdinalNotations.ON_Omega2]
+omega_is_limit [in hydras.OrdinalNotations.ON_Omega2]
+omega_mult_mlarge_0 [in hydras.Epsilon0.Large_Sets]
+omega_tower_mono [in hydras.Schutte.AP]
+omega_eqn [in hydras.Schutte.AP]
+omega_second_AP [in hydras.Schutte.AP]
+Omega_as_lub [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+omega_not_succ [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+omega_is_limit [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+omega_ref [in gaia_hydras.T1Bridge]
+Omega_limit_Limit [in hydras.Prelude.MoreOrders]
+Omega_limit_not_Succ [in hydras.Prelude.MoreOrders]
+omega_limit_least_gt [in hydras.Schutte.Schutte_basics]
+omega_limit_least [in hydras.Schutte.Schutte_basics]
+Omega_term_plus [in hydras.Epsilon0.E0]
+omega_lt_wf [in gaia_hydras.onType]
+omega_lt_epsilon [in hydras.Gamma0.Gamma0]
+omega_lt_epsilon0 [in hydras.Gamma0.Gamma0]
+ONDef.Exports.trichoP [in gaia_hydras.onType]
+ONDef.Exports.wf [in gaia_hydras.onType]
+one_ref [in gaia_hydras.T1Bridge]
+one_mod_neutral_r [in additions.Monoid_instances]
+one_mod_neutral_l [in additions.Monoid_instances]
+one_plus_infinite [in hydras.Schutte.Addition]
+one_plus_omega [in hydras.Schutte.Addition]
+OO.lt_wf [in hydras.OrdinalNotations.OmegaOmega]
+OO.lt_embed [in hydras.OrdinalNotations.OmegaOmega]
+OO.nf_proof_unicity [in hydras.OrdinalNotations.OmegaOmega]
+OO.OO_eq_intro [in hydras.OrdinalNotations.OmegaOmega]
+OpaqueWf.lt_wf [in hydras.Prelude.Fuel]
+openClosed [in hydras.Ackermann.folLogic]
+oplusA [in gaia_hydras.GHessenberg]
+oplusa0 [in gaia_hydras.GHessenberg]
+oplusC [in gaia_hydras.GHessenberg]
+oplusE [in gaia_hydras.GHessenberg]
+oplus_lt_phi0 [in hydras.Epsilon0.Hessenberg]
+oplus_of_phi0 [in hydras.Epsilon0.Hessenberg]
+oplus_of_phi0_0 [in hydras.Epsilon0.Hessenberg]
+oplus_strict_mono_bi [in hydras.Epsilon0.Hessenberg]
+oplus_strict_mono_LT_r [in hydras.Epsilon0.Hessenberg]
+oplus_strict_mono_LT_l [in hydras.Epsilon0.Hessenberg]
+oplus_strict_mono_l [in hydras.Epsilon0.Hessenberg]
+oplus_strict_mono_r [in hydras.Epsilon0.Hessenberg]
+oplus_strict_mono_0 [in hydras.Epsilon0.Hessenberg]
+oplus_le2 [in hydras.Epsilon0.Hessenberg]
+oplus_le1 [in hydras.Epsilon0.Hessenberg]
+oplus_lt2 [in hydras.Epsilon0.Hessenberg]
+oplus_lt1 [in hydras.Epsilon0.Hessenberg]
+oplus_lt_0 [in hydras.Epsilon0.Hessenberg]
+oplus_assoc [in hydras.Epsilon0.Hessenberg]
+oplus_assoc_0 [in hydras.Epsilon0.Hessenberg]
+oplus_lt_rw2 [in hydras.Epsilon0.Hessenberg]
+oplus_comm [in hydras.Epsilon0.Hessenberg]
+oplus_comm_0 [in hydras.Epsilon0.Hessenberg]
+oplus_nf [in hydras.Epsilon0.Hessenberg]
+oplus_nf_0 [in hydras.Epsilon0.Hessenberg]
+oplus_bounded_phi0 [in hydras.Epsilon0.Hessenberg]
+oplus_cons_cons [in hydras.Epsilon0.Hessenberg]
+oplus_eqn [in hydras.Epsilon0.Hessenberg]
+oplus_compare_Gt [in hydras.Epsilon0.Hessenberg]
+oplus_gt_rw [in hydras.Epsilon0.Hessenberg]
+oplus_eq_rw [in hydras.Epsilon0.Hessenberg]
+oplus_lt_rw [in hydras.Epsilon0.Hessenberg]
+oplus_compare_Lt [in hydras.Epsilon0.Hessenberg]
+oplus_0_l [in hydras.Epsilon0.Hessenberg]
+oplus_0_r [in hydras.Epsilon0.Hessenberg]
+oPlus_rw [in hydras.Epsilon0.E0]
+oplus_lt_phi0 [in gaia_hydras.GHessenberg]
+oplus_strict_mono_r [in gaia_hydras.GHessenberg]
+oplus_strict_mono_l [in gaia_hydras.GHessenberg]
+oplus_lt2 [in gaia_hydras.GHessenberg]
+oplus_lt1 [in gaia_hydras.GHessenberg]
+oplus_nf [in gaia_hydras.GHessenberg]
+oplus0b [in gaia_hydras.GHessenberg]
+optimal_upper_bound [in additions.Addition_Chains]
+ordering_function_least_least [in hydras.Schutte.Ordering_Functions]
+ordering_unbounded_unbounded [in hydras.Schutte.Ordering_Functions]
+ordering_function_ex [in hydras.Schutte.Ordering_Functions]
+ordering_segment_ex_unique [in hydras.Schutte.Ordering_Functions]
+ordering_segments_of_B [in hydras.Schutte.Ordering_Functions]
+ordering_function_seg_unicity [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity [in hydras.Schutte.Ordering_Functions]
+ordering_le [in hydras.Schutte.Ordering_Functions]
+ordering_function_seg [in hydras.Schutte.Ordering_Functions]
+ordering_function_mono_weakR [in hydras.Schutte.Ordering_Functions]
+Ordering_bijection [in hydras.Schutte.Ordering_Functions]
+ordering_function_monoR [in hydras.Schutte.Ordering_Functions]
+ordering_function_mono_weak [in hydras.Schutte.Ordering_Functions]
+ordering_function_mono [in hydras.Schutte.Ordering_Functions]
+ordering_function_In [in hydras.Schutte.Ordering_Functions]
+ordinal_ok [in hydras.Schutte.Schutte_basics]
+ordinal_segment [in hydras.Schutte.Ordering_Functions]
+ord_eq [in hydras.Schutte.Ordering_Functions]
+ord_ok [in hydras.Schutte.Ordering_Functions]
+orE [in hydras.Ackermann.LNN]
+orE [in hydras.Ackermann.LNT]
+orE [in hydras.Ackermann.folLogic]
+orI1 [in hydras.Ackermann.LNN]
+orI1 [in hydras.Ackermann.LNT]
+orI1 [in hydras.Ackermann.folLogic]
+orI2 [in hydras.Ackermann.LNN]
+orI2 [in hydras.Ackermann.LNT]
+orI2 [in hydras.Ackermann.folLogic]
+orRelPR [in hydras.Ackermann.primRec]
+orSym [in hydras.Ackermann.LNN]
+orSym [in hydras.Ackermann.LNT]
+orSym [in hydras.Ackermann.folLogic]
+orSys [in hydras.Ackermann.LNN]
+orSys [in hydras.Ackermann.LNT]
+orSys [in hydras.Ackermann.folLogic]
+o_finite_mult_mono [in hydras.Epsilon0.Hessenberg]
+o_finite_mult_lt_phi0_1 [in hydras.Epsilon0.Hessenberg]
+o_finite_mult_rw [in hydras.Epsilon0.Hessenberg]
+o_finite_mult_nf [in hydras.Epsilon0.Hessenberg]
+o_finite_mult_S_rw [in hydras.Hydra.Hydra_Termination]
+o2iota [in hydras.Hydra.Epsilon0_Needed_Std]
+o2iota_1 [in hydras.Hydra.Epsilon0_Needed_Std]
+o2iota_0 [in hydras.Hydra.Epsilon0_Needed_Std]
+

P

+PAboundedLT [in hydras.Ackermann.NN2PA]
+PAconsistent [in hydras.Ackermann.PAconsistent]
+PAdec [in hydras.Ackermann.PA]
+PAIncomplete [in Goedel.rosserPA]
+paPlusSym [in hydras.Ackermann.PAtheory]
+param_correctness_for_free [in additions.Addition_Chains]
+param_correctness [in additions.Addition_Chains]
+param_correctness_nat [in additions.Addition_Chains]
+param_correctness_aux [in additions.Addition_Chains]
+path_to_round_plus [in hydras.Hydra.O2H]
+path_toS_round_plus [in hydras.Hydra.O2H]
+path_toS_trace [in hydras.Hydra.O2H]
+path_to_LT [in gaia_hydras.GPaths]
+path_to_iff1 [in gaia_hydras.GPaths]
+path_to_inv1 [in gaia_hydras.GPaths]
+path_to_omega_mult [in hydras.Epsilon0.Large_Sets]
+path_to_standard_equiv [in hydras.Epsilon0.Paths]
+path_toS_standardS_equiv [in hydras.Epsilon0.Paths]
+path_to_S_standard_pathS [in hydras.Epsilon0.Paths]
+path_to_S_iota_from [in hydras.Epsilon0.Paths]
+path_to_LT [in hydras.Epsilon0.Paths]
+path_toS_LT [in hydras.Epsilon0.Paths]
+path_to_gnaw [in hydras.Epsilon0.Paths]
+path_to_mult [in hydras.Epsilon0.Paths]
+path_toS_mult [in hydras.Epsilon0.Paths]
+path_to_tail [in hydras.Epsilon0.Paths]
+path_toS_tail [in hydras.Epsilon0.Paths]
+path_to_not_In_zero [in hydras.Epsilon0.Paths]
+path_to_zero [in hydras.Epsilon0.Paths]
+path_toS_zero [in hydras.Epsilon0.Paths]
+path_toS_zero_inv [in hydras.Epsilon0.Paths]
+path_toS_zero_one [in hydras.Epsilon0.Paths]
+path_toS_zero_but_last [in hydras.Epsilon0.Paths]
+path_to_appR [in hydras.Epsilon0.Paths]
+path_toS_appR [in hydras.Epsilon0.Paths]
+path_to_decompose [in hydras.Epsilon0.Paths]
+path_toS_decompose [in hydras.Epsilon0.Paths]
+path_to_app [in hydras.Epsilon0.Paths]
+path_toS_app [in hydras.Epsilon0.Paths]
+path_toS_gnawS [in hydras.Epsilon0.Paths]
+path_acc_from [in hydras.Epsilon0.Paths]
+path_toS_nf [in hydras.Epsilon0.Paths]
+path_to_path_toS_iff [in hydras.Epsilon0.Paths]
+path_to_path_toS [in hydras.Epsilon0.Paths]
+path_toS_path_to [in hydras.Epsilon0.Paths]
+path_to_interval_inv_le [in hydras.Epsilon0.Paths]
+path_to_not_nil [in hydras.Epsilon0.Paths]
+paZeroOrSucc [in hydras.Ackermann.PAtheory]
+PA_Succ [in hydras.Epsilon0.Hprime]
+PA_Zero [in hydras.Epsilon0.Hprime]
+pa1 [in hydras.Ackermann.PA]
+pa2 [in hydras.Ackermann.PA]
+pa3 [in hydras.Ackermann.PA]
+pa4 [in hydras.Ackermann.PA]
+pa5 [in hydras.Ackermann.PA]
+pa6 [in hydras.Ackermann.PA]
+PC_Zero [in hydras.Epsilon0.Hprime]
+PD_Zero [in hydras.Epsilon0.Hprime]
+PF87 [in additions.Euclidean_Chains]
+phi_mono_RR [in hydras.Gamma0.Gamma0]
+phi_psi [in hydras.Gamma0.Gamma0]
+phi_to_psi_6 [in hydras.Gamma0.Gamma0]
+phi_to_psi_5 [in hydras.Gamma0.Gamma0]
+phi_to_psi_4 [in hydras.Gamma0.Gamma0]
+phi_to_psi_3 [in hydras.Gamma0.Gamma0]
+phi_to_psi_2 [in hydras.Gamma0.Gamma0]
+phi_to_psi_1 [in hydras.Gamma0.Gamma0]
+phi_inj_r [in hydras.Gamma0.Gamma0]
+phi_mono_weak_r [in hydras.Gamma0.Gamma0]
+phi_mono_r [in hydras.Gamma0.Gamma0]
+phi_of_psi_plus_fin [in hydras.Gamma0.Gamma0]
+phi_alpha_zero_gt_alpha [in hydras.Gamma0.Gamma0]
+phi_principalR [in hydras.Gamma0.Gamma0]
+phi_spec1 [in hydras.Gamma0.Gamma0]
+phi_le_ge [in hydras.Gamma0.Gamma0]
+phi_le [in hydras.Gamma0.Gamma0]
+phi_fix [in hydras.Gamma0.Gamma0]
+phi_of_any_cons [in hydras.Gamma0.Gamma0]
+phi_nf [in hydras.Gamma0.Gamma0]
+phi_cases [in hydras.Gamma0.Gamma0]
+phi_cases' [in hydras.Gamma0.Gamma0]
+phi_cases_aux [in hydras.Gamma0.Gamma0]
+phi_of_psi_succ [in hydras.Gamma0.Gamma0]
+phi_alpha_zero [in hydras.Gamma0.Gamma0]
+phi_principal [in hydras.Gamma0.Gamma0]
+phi_to_psi [in hydras.Gamma0.Gamma0]
+phi_of_psi [in hydras.Gamma0.Gamma0]
+phi0_eq [in hydras.Epsilon0.T1]
+phi0_eq_bad [in hydras.Epsilon0.T1]
+phi0_plus_mult [in hydras.Epsilon0.T1]
+phi0_mono [in hydras.Epsilon0.T1]
+phi0_mono_strict_LT [in hydras.Epsilon0.T1]
+phi0_mono_strict [in hydras.Epsilon0.T1]
+phi0_ltR [in hydras.Epsilon0.T1]
+phi0_lt [in hydras.Epsilon0.T1]
+phi0_mult_plus_lt_phi0R [in hydras.Schutte.Correctness_E0]
+phi0_mult_plus_lt_phi0 [in hydras.Schutte.Correctness_E0]
+phi0_mult_lt_phi0 [in hydras.Schutte.Correctness_E0]
+phi0_well_named [in hydras.Schutte.Critical]
+phi0_lt_epsilon0_R [in hydras.Schutte.AP]
+phi0_lt_epsilon0 [in hydras.Schutte.AP]
+phi0_of_limit [in hydras.Schutte.AP]
+phi0_sup [in hydras.Schutte.AP]
+phi0_alpha_phi0_beta [in hydras.Schutte.AP]
+phi0_positive [in hydras.Schutte.AP]
+phi0_inj [in hydras.Schutte.AP]
+phi0_mono_R_weak [in hydras.Schutte.AP]
+phi0_mono_R [in hydras.Schutte.AP]
+phi0_mono_weak [in hydras.Schutte.AP]
+phi0_mono [in hydras.Schutte.AP]
+phi0_zero [in hydras.Schutte.AP]
+phi0_elim [in hydras.Schutte.AP]
+phi0_ordering [in hydras.Schutte.AP]
+phi0_ref [in gaia_hydras.T1Bridge]
+phi0_mono [in hydras.Epsilon0.E0]
+Phi0_def [in hydras.Epsilon0.Hprime]
+plus_cons_cons_eqn [in hydras.Epsilon0.T1]
+plus_smono_LT_r [in hydras.Epsilon0.T1]
+plus_compat [in hydras.Epsilon0.T1]
+plus_left_absorb [in hydras.Epsilon0.T1]
+plus_is_zero [in hydras.Epsilon0.T1]
+plus_nf [in hydras.Epsilon0.T1]
+plus_nf0 [in hydras.Epsilon0.T1]
+plus_cons_cons_rw3 [in hydras.Epsilon0.T1]
+plus_cons_cons_rw2 [in hydras.Epsilon0.T1]
+plus_cons_cons_rw1 [in hydras.Epsilon0.T1]
+plus_zero_r [in hydras.Epsilon0.T1]
+plus_zero [in hydras.Epsilon0.T1]
+plus_alpha_mult_phi0 [in hydras.Schutte.Correctness_E0]
+plus_compat [in hydras.OrdinalNotations.ON_Omega2]
+plus_2 [in hydras.Prelude.More_Arith]
+plus_lt_phi0 [in hydras.Schutte.AP]
+plus_ref [in gaia_hydras.T1Bridge]
+Plus_rw [in hydras.Epsilon0.E0]
+plus_mono_l_weak [in hydras.Gamma0.Gamma0]
+plus_mono_r [in hydras.Gamma0.Gamma0]
+plus_nf [in hydras.Gamma0.Gamma0]
+plus_alpha_0 [in hydras.Gamma0.Gamma0]
+plus_mono_bi [in hydras.Schutte.Addition]
+plus_mono_weak_l [in hydras.Schutte.Addition]
+plus_assoc' [in hydras.Schutte.Addition]
+plus_assoc3 [in hydras.Schutte.Addition]
+plus_assoc2 [in hydras.Schutte.Addition]
+plus_assoc1 [in hydras.Schutte.Addition]
+plus_FF [in hydras.Schutte.Addition]
+plus_limit [in hydras.Schutte.Addition]
+plus_reg_r [in hydras.Schutte.Addition]
+plus_mono_r_weak [in hydras.Schutte.Addition]
+plus_of_succ [in hydras.Schutte.Addition]
+plus_mono_r [in hydras.Schutte.Addition]
+plus_elim [in hydras.Schutte.Addition]
+plus_ordering [in hydras.Schutte.Addition]
+plus_correct [in hydras.MoreAck.PrimRecExamples]
+positive_4step_ind [in additions.More_on_positive]
+Pos_to_nat_diff_0 [in additions.AM]
+Pos_pow_compat [in additions.Compatibility]
+Pos_pow_power [in additions.Compatibility]
+Pos_iter_ok [in additions.Compatibility]
+Pos_iter_op_ok [in additions.Compatibility]
+Pos_iter_op_ok_0 [in additions.Compatibility]
+Pos_bpow_ok_R [in additions.Pow_variant]
+Pos_bpow_ok [in additions.Pow_variant]
+Pos_bpow_of_bpow [in additions.Pow]
+Pos_bpow_of_plus [in additions.Pow]
+Pos_bpow_ok_R [in additions.Pow]
+Pos_bpow_ok [in additions.Pow]
+pos_div_eucl_quotient_lt [in additions.More_on_positive]
+pos_div_eucl_quotient_pos [in additions.More_on_positive]
+pos_gt_3 [in additions.More_on_positive]
+pos_lt_wf [in additions.More_on_positive]
+pos_lt_mul [in additions.More_on_positive]
+pos_le_mul [in additions.More_on_positive]
+Pos_to_nat_neq_0 [in additions.More_on_positive]
+Pos2Nat_morph [in additions.Compatibility]
+Pos2Nat_le_n_pn [in additions.More_on_positive]
+Pos2Nat_le_1_p [in additions.More_on_positive]
+pos2N_inj_add [in additions.More_on_positive]
+pos2N_inj_lt [in additions.More_on_positive]
+power_of_1 [in hydras.Prelude.More_Arith]
+power_of_mult [in additions.Pow_variant]
+power_of_square [in additions.Pow_variant]
+power_of_power [in additions.Pow_variant]
+power_commute_with_x [in additions.Pow_variant]
+power_commute [in additions.Pow_variant]
+power_of_plus [in additions.Pow_variant]
+power_eq3 [in additions.Pow_variant]
+power_eq2 [in additions.Pow_variant]
+power_eq1 [in additions.Pow_variant]
+power_of_mult [in additions.Pow]
+power_of_square [in additions.Pow]
+power_of_power_comm [in additions.Pow]
+power_of_power [in additions.Pow]
+power_commute_with_x [in additions.Pow]
+power_commute [in additions.Pow]
+power_of_plus [in additions.Pow]
+power_eq3 [in additions.Pow]
+power_eq2 [in additions.Pow]
+power_eq1 [in additions.Pow]
+power_R_is_a_refinement [in additions.Addition_Chains]
+power_R_1 [in additions.Addition_Chains]
+power_R_Mult [in additions.Addition_Chains]
+pow3Min [in hydras.Ackermann.codeSubFormula]
+pow3Monotone [in hydras.Ackermann.codeSubFormula]
+PO1 [in additions.Euclidean_Chains]
+PO2 [in additions.Euclidean_Chains]
+PO3 [in additions.Euclidean_Chains]
+PO4 [in additions.Euclidean_Chains]
+PO6 [in additions.Euclidean_Chains]
+PO8 [in additions.Euclidean_Chains]
+PO9 [in additions.Euclidean_Chains]
+Predecessor_unicity [in hydras.solutions_exercises.predSuccUnicity]
+pred_nf [in hydras.Epsilon0.T1]
+pred_LT [in hydras.Epsilon0.T1]
+pred_of_limitR [in hydras.Epsilon0.T1]
+pred_of_limit [in hydras.Epsilon0.T1]
+pred_of_succ [in hydras.Epsilon0.T1]
+pred_of_power [in hydras.Prelude.More_Arith]
+pred_of_succ [in hydras.Gamma0.Gamma0]
+pred_of_limit [in hydras.Gamma0.Gamma0]
+pred_of_cons' [in hydras.Gamma0.Gamma0]
+pred_of_cons [in hydras.Gamma0.Gamma0]
+preserveValue [in hydras.Ackermann.model]
+primRecRepresentable [in Goedel.PRrepresentable]
+primRecRepresentable1 [in Goedel.PRrepresentable]
+primRecSigmaRepresentable [in Goedel.PRrepresentable]
+PrimRec_2_S [in hydras.MoreAck.PrimRecExamples]
+PrimRec_2_0 [in hydras.MoreAck.PrimRecExamples]
+PrimRec_1_S [in hydras.MoreAck.PrimRecExamples]
+PrimRec_1_0 [in hydras.MoreAck.PrimRecExamples]
+PrimRec_0_S [in hydras.MoreAck.PrimRecExamples]
+PrimRec_0_0 [in hydras.MoreAck.PrimRecExamples]
+Progressive_Acc [in hydras.Schutte.Schutte_basics]
+projectionListApplyParam [in hydras.Ackermann.primRec]
+projectionListId [in hydras.Ackermann.primRec]
+projectionListInd [in hydras.Ackermann.primRec]
+projRepresentable [in Goedel.PRrepresentable]
+proj_le_max [in hydras.MoreAck.AckNotPR]
+Proper_A [in hydras.Schutte.Ordering_Functions]
+proper_of_proper [in hydras.Schutte.Ordering_Functions]
+proper_members [in hydras.Schutte.Ordering_Functions]
+Proposition_2_3a [in hydras.Epsilon0.Paths]
+prop_map12_without_repetition [in hydras.rpo.more_list]
+prop_map_without_repetition [in hydras.rpo.more_list]
+psi_eq [in hydras.Gamma0.T2]
+psi_principal [in hydras.Gamma0.Gamma0]
+psi_lt_epsilon0 [in hydras.Gamma0.Gamma0]
+psi_lt_head [in hydras.Gamma0.Gamma0]
+psi_le_cons [in hydras.Gamma0.Gamma0]
+PZero [in hydras.Epsilon0.F_alpha]
+PZero [in hydras.Epsilon0.Hprime]
+P_3 [in hydras.solutions_exercises.F_3]
+P_alpha [in hydras.Epsilon0.Hprime]
+P_alpha_0 [in hydras.Epsilon0.Hprime]
+P_well_founded_induction_type [in gaia_hydras.nfwfgaia]
+P0 [in hydras.MoreAck.Ack]
+

Q

+QA0 [in hydras.Epsilon0.F_alpha]
+QD0 [in hydras.Epsilon0.F_alpha]
+

R

+RAlim [in hydras.Epsilon0.F_alpha]
+RAlim [in hydras.Epsilon0.Hprime]
+RB [in hydras.Epsilon0.F_alpha]
+RB [in hydras.Epsilon0.Hprime]
+RBlim [in hydras.Epsilon0.F_alpha]
+RBlim [in hydras.Epsilon0.Hprime]
+RC [in hydras.Epsilon0.F_alpha]
+RC [in hydras.Epsilon0.Hprime]
+RClim [in hydras.Epsilon0.F_alpha]
+RClim [in hydras.Epsilon0.Hprime]
+RD [in hydras.Epsilon0.F_alpha]
+RD [in hydras.Epsilon0.Hprime]
+RDlim [in hydras.Epsilon0.F_alpha]
+RDlim [in hydras.Epsilon0.Hprime]
+RE [in hydras.Epsilon0.F_alpha]
+RE [in hydras.Epsilon0.Hprime]
+reachable_S [in hydras.Hydra.BigBattle]
+rebindExist [in hydras.Ackermann.folLogic2]
+rebindForall [in hydras.Ackermann.folLogic2]
+reduceAddExists [in Goedel.PRrepresentable]
+reduceAddExistsOneWay [in Goedel.PRrepresentable]
+reduceAddForalls [in Goedel.PRrepresentable]
+reduceAnd [in hydras.Ackermann.folReplace]
+reduceCloseList [in hydras.Ackermann.folReplace]
+reduceExist [in hydras.Ackermann.folReplace]
+reduceForall [in hydras.Ackermann.folReplace]
+reduceIff [in hydras.Ackermann.folReplace]
+reduceIfThenElse [in hydras.Ackermann.folReplace]
+reduceImp [in hydras.Ackermann.folReplace]
+reduceNot [in hydras.Ackermann.folReplace]
+reduceOr [in hydras.Ackermann.folReplace]
+reduceSub [in hydras.Ackermann.folReplace]
+reduceSubAll [in hydras.Ackermann.subAll]
+reduce_lt_plus [in hydras.Epsilon0.T1]
+reduce_assoc_list [in hydras.rpo.more_list]
+reduce1stCompose [in hydras.Ackermann.primRec]
+reduce2ndCompose [in hydras.Ackermann.primRec]
+refines1_R [in gaia_hydras.T1Bridge]
+refines2_R [in gaia_hydras.T1Bridge]
+RElim [in hydras.Epsilon0.F_alpha]
+RElim [in hydras.Epsilon0.Hprime]
+remove_first_sumand [in hydras.Epsilon0.T1]
+remove_heads_r_free [in hydras.Hydra.Hydra_Lemmas]
+remove_heads_r [in hydras.Hydra.Hydra_Lemmas]
+Rem0 [in hydras.Epsilon0.Paths]
+Rem01 [in hydras.Epsilon0.Paths]
+Rem02 [in hydras.Epsilon0.Paths]
+Rem03 [in hydras.Epsilon0.Paths]
+Rem04 [in hydras.Epsilon0.Paths]
+Rem05 [in hydras.Epsilon0.Paths]
+Rem06 [in hydras.Epsilon0.Paths]
+Rem08 [in hydras.Epsilon0.Paths]
+Rem09 [in hydras.Epsilon0.Paths]
+rem1 [in hydras.Hydra.O2H]
+Rem1 [in hydras.MoreAck.Ack]
+Rem1 [in hydras.Epsilon0.Paths]
+Rem10 [in hydras.Epsilon0.Paths]
+Rem11 [in hydras.Epsilon0.Paths]
+rem2 [in hydras.Hydra.O2H]
+Rem2 [in hydras.MoreAck.Ack]
+Rem2 [in hydras.Epsilon0.Paths]
+rem3 [in hydras.Hydra.O2H]
+Rem3 [in hydras.MoreAck.Ack]
+Rem3 [in hydras.Hydra.Epsilon0_Needed_Std]
+Rem4 [in hydras.Hydra.Epsilon0_Needed_Std]
+rem6 [in hydras.Hydra.O2H]
+rem61 [in hydras.Hydra.O2H]
+rem62 [in hydras.Hydra.O2H]
+replaceCompose2 [in hydras.Ackermann.primRec]
+ReplaceFormulaTermMonotone [in hydras.Ackermann.codeSubFormula]
+ReplaceFormulaTermSub [in hydras.Ackermann.codeSubFormula]
+ReplaceTermsTermMonotone [in hydras.Ackermann.codeSubFormula]
+ReplaceTermsTermSub [in hydras.Ackermann.codeSubFormula]
+ReplaceTermTermMonotone [in hydras.Ackermann.codeSubFormula]
+ReplaceTermTermsTermMonotone [in hydras.Ackermann.codeSubFormula]
+ReplaceTermTermSub [in hydras.Ackermann.codeSubFormula]
+RepresentableAlternate [in hydras.Ackermann.expressible]
+RepresentableHalfHelp [in hydras.Ackermann.expressible]
+RepresentableHalf1Alternate [in hydras.Ackermann.expressible]
+RepresentableHalf2Alternate [in hydras.Ackermann.expressible]
+Representable_ext [in hydras.Ackermann.expressible]
+Representable2Expressible [in hydras.Ackermann.expressible]
+restrict [in hydras.Schutte.Ordering_Functions]
+restriction_fwd [in hydras.Prelude.Restriction]
+Rosser'sIncompleteness [in Goedel.rosserPA]
+Rosser'sIncompleteness [in Goedel.rosser]
+rounds_free_equiv2 [in hydras.Hydra.Hydra_Lemmas]
+rounds_free_equiv1 [in hydras.Hydra.Hydra_Lemmas]
+rounds_trans [in hydras.Hydra.Hydra_Lemmas]
+round_decr [in hydras.Hydra.Hydra_Termination]
+round_n_inv [in hydras.Hydra.Hydra_Lemmas]
+round_n_remove_h [in hydras.Hydra.Hydra_Lemmas]
+round_plus_trans [in hydras.Hydra.Hydra_Lemmas]
+round_S [in hydras.Hydra.Omega_Small]
+round_star_intro [in hydras.Hydra.Hydra_Definitions]
+RP [in hydras.Epsilon0.F_alpha]
+RP [in hydras.Epsilon0.Hprime]
+rpo_7_1 [in hydras.Gamma0.Gamma0]
+rpo_6_4 [in hydras.Gamma0.Gamma0]
+rpo_6_1 [in hydras.Gamma0.Gamma0]
+rpo_5_4 [in hydras.Gamma0.Gamma0]
+rpo_5_1 [in hydras.Gamma0.Gamma0]
+rpo_5_3 [in hydras.Gamma0.Gamma0]
+rpo_5_2 [in hydras.Gamma0.Gamma0]
+rpo_4_4 [in hydras.Gamma0.Gamma0]
+rpo_4_1 [in hydras.Gamma0.Gamma0]
+rpo_4_3 [in hydras.Gamma0.Gamma0]
+rpo_4_2 [in hydras.Gamma0.Gamma0]
+rpo_3_4 [in hydras.Gamma0.Gamma0]
+rpo_3_1 [in hydras.Gamma0.Gamma0]
+rpo_3_3 [in hydras.Gamma0.Gamma0]
+rpo_3_2 [in hydras.Gamma0.Gamma0]
+rpo_2_4 [in hydras.Gamma0.Gamma0]
+rpo_2_1 [in hydras.Gamma0.Gamma0]
+rpo_2_3 [in hydras.Gamma0.Gamma0]
+rpo_2_2 [in hydras.Gamma0.Gamma0]
+rpo_trans [in hydras.Gamma0.Gamma0]
+Rwf [in hydras.Prelude.First_toggle]
+R_nat_elaguee_domain [in hydras.Schutte.GRelations]
+R_nat_elaguee_fun [in hydras.Schutte.GRelations]
+R_inv_inj [in hydras.Schutte.GRelations]
+R_inv_surj [in hydras.Schutte.GRelations]
+R_pred_Sn [in hydras.Gamma0.Gamma0]
+R_inc_rpo [in hydras.Gamma0.Gamma0]
+R0 [in hydras.MoreAck.Ack]
+R0 [in hydras.Hydra.Battle_length]
+R0 [in gaia_hydras.onType]
+R0 [in hydras.Epsilon0.Hprime]
+R00 [in hydras.Epsilon0.F_omega]
+R00 [in hydras.solutions_exercises.isqrt]
+R01 [in hydras.solutions_exercises.isqrt]
+R02 [in hydras.solutions_exercises.isqrt]
+R1 [in hydras.solutions_exercises.is_F_monotonous]
+R1 [in hydras.Epsilon0.F_alpha]
+R1 [in hydras.MoreAck.AckNotPR]
+R1 [in hydras.Epsilon0.Large_Sets]
+R1 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+R1 [in hydras.MoreAck.Ack]
+R1 [in hydras.solutions_exercises.schutte_cnf_counter_example]
+R1 [in gaia_hydras.onType]
+R1 [in hydras.solutions_exercises.F_3]
+R1 [in hydras.Epsilon0.Hprime]
+R1 [in hydras.Gamma0.Gamma0]
+R1 [in hydras.MoreAck.FolExamples]
+R1_mem_head [in hydras.Hydra.O2H]
+R1_R2 [in hydras.Hydra.O2H]
+R1_hcons_mult [in hydras.Hydra.O2H]
+R1_hcons [in hydras.Hydra.O2H]
+R1_decr [in hydras.Hydra.Hydra_Termination]
+R1_decr_0 [in hydras.Hydra.Hydra_Termination]
+R1_remove_r_i [in hydras.Hydra.Hydra_Lemmas]
+R1_app [in hydras.Hydra.Hydra_Lemmas]
+R1_iff [in hydras.Hydra.Hydra_Lemmas]
+R1_aux [in hydras.Schutte.Ordering_Functions]
+R12 [in hydras.Epsilon0.Paths]
+R13 [in hydras.Epsilon0.Paths]
+R14 [in hydras.Epsilon0.Paths]
+R15 [in hydras.Epsilon0.Paths]
+R16 [in hydras.Epsilon0.Paths]
+R17 [in hydras.Epsilon0.Paths]
+R18 [in hydras.Epsilon0.Paths]
+R19 [in hydras.Epsilon0.Paths]
+R2 [in hydras.solutions_exercises.is_F_monotonous]
+R2 [in hydras.Epsilon0.Large_Sets]
+R2 [in hydras.MoreAck.Ack]
+R2 [in hydras.solutions_exercises.schutte_cnf_counter_example]
+R2 [in hydras.Hydra.Battle_length]
+R2 [in hydras.Epsilon0.Hprime]
+R2 [in hydras.Schutte.Ordering_Functions]
+R2 [in hydras.Gamma0.Gamma0]
+R2_decr [in hydras.Hydra.Hydra_Termination]
+R2_decr_0 [in hydras.Hydra.Hydra_Termination]
+R2_app [in hydras.Hydra.Hydra_Lemmas]
+R2_iff [in hydras.Hydra.Hydra_Lemmas]
+R20 [in hydras.Epsilon0.Paths]
+R21 [in hydras.Epsilon0.Paths]
+R22 [in hydras.Epsilon0.Paths]
+R23 [in hydras.Epsilon0.Paths]
+R24 [in hydras.Epsilon0.Paths]
+R25 [in hydras.Epsilon0.Paths]
+R26 [in hydras.Epsilon0.Paths]
+R27 [in hydras.Epsilon0.Paths]
+R28 [in hydras.Epsilon0.Paths]
+R29 [in hydras.Epsilon0.Paths]
+R3 [in hydras.solutions_exercises.is_F_monotonous]
+R3 [in hydras.MoreAck.Ack]
+R3 [in hydras.solutions_exercises.schutte_cnf_counter_example]
+R3 [in hydras.Hydra.Battle_length]
+R3 [in hydras.solutions_exercises.F_3]
+R3 [in hydras.Epsilon0.Hprime]
+R3 [in hydras.Schutte.Ordering_Functions]
+R3 [in hydras.Gamma0.Gamma0]
+R3' [in hydras.solutions_exercises.F_3]
+R30 [in hydras.Epsilon0.Paths]
+R31 [in hydras.Epsilon0.Paths]
+R31_0 [in hydras.Epsilon0.Paths]
+R4 [in hydras.solutions_exercises.is_F_monotonous]
+R4 [in hydras.Hydra.Battle_length]
+R4 [in hydras.solutions_exercises.F_3]
+R4 [in hydras.Epsilon0.Hprime]
+R4 [in hydras.Schutte.Ordering_Functions]
+R4 [in hydras.Gamma0.Gamma0]
+R4_3_4 [in hydras.Epsilon0.Paths]
+R4_3_3 [in hydras.Epsilon0.Paths]
+R4_3_2 [in hydras.Epsilon0.Paths]
+R4_3_1 [in hydras.Epsilon0.Paths]
+R4' [in hydras.Schutte.Ordering_Functions]
+R4'' [in hydras.Schutte.Ordering_Functions]
+R42 [in hydras.Schutte.Ordering_Functions]
+R5 [in hydras.Epsilon0.F_omega]
+R5 [in hydras.MoreAck.Ack]
+R5 [in hydras.Hydra.Battle_length]
+R5 [in hydras.solutions_exercises.F_3]
+R5 [in hydras.Epsilon0.Hprime]
+R5 [in hydras.Schutte.Ordering_Functions]
+R5 [in hydras.Gamma0.Gamma0]
+R6 [in hydras.solutions_exercises.is_F_monotonous]
+R6 [in hydras.Hydra.Battle_length]
+R6 [in hydras.Schutte.Ordering_Functions]
+R7 [in hydras.Schutte.Ordering_Functions]
+

S

+SA1 [in hydras.Schutte.Ordering_Functions]
+SA2 [in hydras.Schutte.Ordering_Functions]
+searchProof [in Goedel.rosserPA]
+searchProof [in Goedel.rosser]
+SecondIncompletness [in Goedel.goedel2]
+segment_lt_closed [in hydras.Schutte.Ordering_Functions]
+segment_the_ordering_segment [in hydras.Schutte.Ordering_Functions]
+segment_unbounded [in hydras.Schutte.Ordering_Functions]
+segment_lt [in hydras.Schutte.Ordering_Functions]
+self_lt_free [in hydras.Hydra.Epsilon0_Needed_Free]
+self_lt_standard [in hydras.Hydra.Epsilon0_Needed_Std]
+seq_mono_inj [in hydras.Schutte.Schutte_basics]
+seq_mono_intro [in hydras.Schutte.Schutte_basics]
+seq_range_countable [in hydras.Schutte.Countable]
+shift_unshift [in hydras.Prelude.MoreLists]
+shift_no_zero [in hydras.Prelude.MoreLists]
+shift_interval [in hydras.Prelude.MoreLists]
+shift_iota_from [in hydras.Prelude.MoreLists]
+sig_eq_intro [in hydras.OrdinalNotations.ON_Finite]
+simple_last_app1 [in hydras.Prelude.MoreLists]
+simple_last_app [in hydras.Prelude.MoreLists]
+simple_last_correct [in hydras.Prelude.MoreLists]
+single_nf [in hydras.Epsilon0.T1]
+smallestExists [in hydras.Prelude.MoreDecidable]
+small_lemma [in hydras.Epsilon0.Paths]
+SmNotPR [in hydras.Hydra.Hydra_Theorems]
+smono_Sle [in hydras.Prelude.Iterates]
+some_nb_occ_Sn [in hydras.rpo.more_list]
+sorted_ge_suffix [in hydras.Prelude.MoreLists]
+sorted_max_2 [in hydras.Prelude.MoreLists]
+sorted_cut [in hydras.Prelude.MoreLists]
+sorted_max_1 [in hydras.Prelude.MoreLists]
+sorted_not_in_tail [in hydras.Prelude.MoreLists]
+sorted_In [in hydras.Prelude.MoreLists]
+sorted_ge_prefix [in hydras.Prelude.MoreLists]
+sorted_ge_iff [in hydras.Prelude.MoreLists]
+sorted_ge_iff0 [in hydras.Prelude.MoreLists]
+Sorted_mono [in hydras.Prelude.MoreLists]
+sorted_head [in hydras.Prelude.MoreLists]
+sorted_tail' [in hydras.Prelude.MoreLists]
+sorted_tail [in hydras.Prelude.MoreLists]
+sorted_le [in hydras.Prelude.MoreLists]
+sorted_inv_gt [in hydras.Prelude.MoreLists]
+sorted_ge_not_In [in hydras.Prelude.MoreLists]
+sorted_ge_trans [in hydras.Prelude.MoreLists]
+sorted_ge_Forall [in hydras.Prelude.MoreLists]
+sorted_lt [in hydras.Schutte.CNF]
+sorted_lt_lt_2 [in hydras.Schutte.CNF]
+sorted_lt_lt [in hydras.Schutte.CNF]
+sorted_tail [in hydras.Schutte.CNF]
+split_list_app_cons [in hydras.rpo.more_list]
+split_permutation [in hydras.Prelude.Merge_Sort]
+split_decr [in hydras.Prelude.Merge_Sort]
+split'_permutation [in hydras.Prelude.Merge_Sort]
+split'_aux_eq [in hydras.Prelude.Merge_Sort]
+split'_decr [in hydras.Prelude.Merge_Sort]
+split'_aux_length_fst [in hydras.Prelude.Merge_Sort]
+split'_aux_length_preserve [in hydras.Prelude.Merge_Sort]
+sp_mergesort_OK [in hydras.Prelude.Merge_Sort]
+sqrt_correct [in hydras.solutions_exercises.isqrt]
+sqr_eqn [in additions.Pow_variant]
+sqr_eqn [in additions.Pow]
+stable_mergesort_OK [in hydras.Prelude.Merge_Sort]
+standard_gnaw_iota_from [in gaia_hydras.GPaths]
+standard_battle_head [in hydras.Hydra.Hydra_Lemmas]
+standard_incl_free [in hydras.Hydra.Hydra_Lemmas]
+standard_gnaw_2_zero [in hydras.Epsilon0.Paths]
+standard_path_to_zero [in hydras.Epsilon0.Paths]
+standard_gnaw_to_zero [in hydras.Epsilon0.Paths]
+standard_pathS_app [in hydras.Epsilon0.Paths]
+standard_path_equiv_2 [in hydras.Epsilon0.Paths]
+standard_path_equiv_1 [in hydras.Epsilon0.Paths]
+standard_gnaw_nf [in hydras.Epsilon0.Paths]
+standard_gnaw_to_path [in hydras.Epsilon0.Paths]
+standard_gnaw_S_zero [in hydras.Epsilon0.Paths]
+standard_path_compose [in hydras.Epsilon0.Paths]
+standard_path_lt2 [in hydras.Epsilon0.Paths]
+standard_path_zero [in hydras.Epsilon0.Paths]
+standard_path_to_nf [in hydras.Epsilon0.Paths]
+standard_path_LE [in hydras.Epsilon0.Paths]
+standard_gnaw_plus [in hydras.Epsilon0.Paths]
+standard_gnaw_iota_from [in hydras.Epsilon0.Paths]
+standard_gnaw_zero [in hydras.Epsilon0.Paths]
+standard_path_path_to [in hydras.Epsilon0.Paths]
+standard_pathS_path_toS [in hydras.Epsilon0.Paths]
+standard_path_origin [in hydras.Epsilon0.Paths]
+standard_path_toS_zero [in hydras.Epsilon0.Paths]
+standard_path_to_le_inv [in hydras.Epsilon0.Paths]
+standard_path_toS_le_inv [in hydras.Epsilon0.Paths]
+standard_path_unshift [in hydras.Epsilon0.Paths]
+standard_path_unshift_0 [in hydras.Epsilon0.Paths]
+standard_path_shift [in hydras.Epsilon0.Paths]
+steps_rounds [in hydras.Hydra.BigBattle]
+step_rounds [in hydras.Hydra.BigBattle]
+step_to_battle [in hydras.Hydra.Omega2_Small]
+strict_lub_T1limit [in hydras.Epsilon0.T1]
+strict_lub_unique [in hydras.Epsilon0.T1]
+strict_mono_iterate_S [in hydras.Prelude.Iterates]
+strict_lub_ref [in gaia_hydras.T1Bridge]
+subAddExistsNice [in Goedel.PRrepresentable]
+subAddForallsNice [in Goedel.PRrepresentable]
+subAllCloseFrom [in hydras.Ackermann.subAll]
+subAllCloseFrom1 [in hydras.Ackermann.subAll]
+subAllFormulaId [in hydras.Ackermann.subAll]
+subAllFormula_ext [in hydras.Ackermann.subAll]
+subAllnVars1 [in hydras.Ackermann.folLogic3]
+subAllnVars2 [in hydras.Ackermann.folLogic3]
+subAllSubAllFormula [in hydras.Ackermann.subAll]
+subAllSubAllTerm [in hydras.Ackermann.subAll]
+subAllSubAllTerms [in hydras.Ackermann.subAll]
+subAllSubFormula [in hydras.Ackermann.subAll]
+subAllTermId [in hydras.Ackermann.subAll]
+subAllTermsId [in hydras.Ackermann.subAll]
+subAllTerms_ext [in hydras.Ackermann.subAll]
+subAllTerm_ext [in hydras.Ackermann.subAll]
+subFormulaAnd [in hydras.Ackermann.folProp]
+subFormulaDepth [in hydras.Ackermann.folProp]
+subFormulaEqual [in hydras.Ackermann.folProp]
+subFormulaExch [in hydras.Ackermann.subProp]
+subFormulaExist [in hydras.Ackermann.folProp]
+subFormulaExist2 [in hydras.Ackermann.folProp]
+subFormulaForall [in hydras.Ackermann.folProp]
+subFormulaForall2 [in hydras.Ackermann.folProp]
+subFormulaId [in hydras.Ackermann.folProp]
+subFormulaIff [in hydras.Ackermann.folProp]
+subFormulaIfThenElse [in hydras.Ackermann.folProp]
+subFormulaImp [in hydras.Ackermann.folProp]
+subFormulaMinimize [in Goedel.PRrepresentable]
+subFormulaNil [in hydras.Ackermann.subProp]
+subFormulaNot [in hydras.Ackermann.folProp]
+subFormulaNTE [in hydras.Ackermann.subProp]
+subFormulaNTEHelp [in hydras.Ackermann.subProp]
+subFormulaOr [in hydras.Ackermann.folProp]
+subFormulaRelation [in hydras.Ackermann.folProp]
+subFormulaTrans [in hydras.Ackermann.subProp]
+subInterpFormula [in hydras.Ackermann.model]
+subInterpFormula1 [in hydras.Ackermann.model]
+subInterpFormula2 [in hydras.Ackermann.model]
+subInterpRel [in hydras.Ackermann.model]
+subInterpTerm [in hydras.Ackermann.model]
+subNNHelp [in hydras.Ackermann.model]
+SubON_least [in hydras.OrdinalNotations.ON_Generic]
+SubON_limit [in hydras.OrdinalNotations.ON_Generic]
+SubON_successor [in hydras.OrdinalNotations.ON_Generic]
+SubON_inj [in hydras.OrdinalNotations.ON_Generic]
+SubON_mono [in hydras.OrdinalNotations.ON_Generic]
+substExHC [in hydras.Ackermann.folProp]
+substituteFormulaForallNice [in hydras.Ackermann.folProp]
+substituteFormulaImpNice [in hydras.Ackermann.folProp]
+substituteFormulaNotNice [in hydras.Ackermann.folProp]
+subSubAllFormula [in hydras.Ackermann.subAll]
+subSubAllTerm [in hydras.Ackermann.subAll]
+subSubAllTerms [in hydras.Ackermann.subAll]
+subSubFormula [in hydras.Ackermann.folLogic2]
+subSubTerm [in hydras.Ackermann.folLogic2]
+subSubTerms [in hydras.Ackermann.folLogic2]
+subTermApply [in hydras.Ackermann.folProp]
+subTermExch [in hydras.Ackermann.subProp]
+subTermId [in hydras.Ackermann.folProp]
+subTermNil [in hydras.Ackermann.subProp]
+subTermsExch [in hydras.Ackermann.subProp]
+subTermsId [in hydras.Ackermann.folProp]
+subTermsNil [in hydras.Ackermann.subProp]
+subTermsTrans [in hydras.Ackermann.subProp]
+subTermTrans [in hydras.Ackermann.subProp]
+subTermVar1 [in hydras.Ackermann.folProp]
+subTermVar2 [in hydras.Ackermann.folProp]
+subterm_lt [in hydras.Gamma0.Gamma0]
+subToSubAll [in hydras.Ackermann.subAll]
+subWithEquals [in hydras.Ackermann.folLogic3]
+subWithEqualsTerm [in hydras.Ackermann.folLogic3]
+subWithEqualsTerms [in hydras.Ackermann.folLogic3]
+succb_correct [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Succb_Succ [in hydras.Epsilon0.E0]
+succb_not_lim [in hydras.Gamma0.Gamma0]
+Successor_succ [in hydras.OrdinalNotations.ON_Omega2]
+Successor_not [in hydras.OrdinalNotations.ON_Omega2]
+Successor_inv [in hydras.OrdinalNotations.ON_Omega2]
+Successor_unicity [in hydras.solutions_exercises.predSuccUnicity]
+Successor_succ [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Successor_inv4 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Successor_inv3 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Successor_inv2 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Successor_inv1 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+succRepresentable [in Goedel.PRrepresentable]
+succ_rw1 [in hydras.Epsilon0.T1]
+succ_cons' [in hydras.Epsilon0.T1]
+succ_cons [in hydras.Epsilon0.T1]
+succ_lt_limit [in hydras.Epsilon0.T1]
+succ_not_limit [in hydras.Epsilon0.T1]
+succ_compat [in hydras.Epsilon0.T1]
+succ_compatS [in hydras.Epsilon0.T1]
+succ_injective [in hydras.Epsilon0.T1]
+succ_monomorphism [in hydras.Epsilon0.T1]
+succ_strict_monoR [in hydras.Epsilon0.T1]
+succ_mono [in hydras.Epsilon0.T1]
+succ_strict_mono_LT [in hydras.Epsilon0.T1]
+succ_strict_mono [in hydras.Epsilon0.T1]
+succ_of_plus_finite [in hydras.Epsilon0.T1]
+succ_is_plus_one [in hydras.Epsilon0.T1]
+succ_nf [in hydras.Epsilon0.T1]
+succ_lt_a_phi0_b [in hydras.Epsilon0.T1]
+succ_is_succ [in hydras.Epsilon0.T1]
+succ_not_zero [in hydras.Epsilon0.T1]
+succ_is_plus_1 [in hydras.OrdinalNotations.ON_Omega2]
+succ_ok [in hydras.OrdinalNotations.ON_Omega2]
+succ_correct [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+succ_ref [in gaia_hydras.T1Bridge]
+succ_zero_diff [in hydras.Schutte.Schutte_basics]
+succ_injection [in hydras.Schutte.Schutte_basics]
+succ_monoR [in hydras.Schutte.Schutte_basics]
+succ_mono [in hydras.Schutte.Schutte_basics]
+succ_ok [in hydras.Schutte.Schutte_basics]
+succ_correct [in hydras.Epsilon0.E0]
+Succ_of_cons [in hydras.Epsilon0.E0]
+Succ_rw [in hydras.Epsilon0.E0]
+Succ_lt_T1limit [in hydras.Epsilon0.E0]
+Succ_not_T1limit [in hydras.Epsilon0.E0]
+Succ_not_Zero [in hydras.Epsilon0.E0]
+Succ_Succb [in hydras.Epsilon0.E0]
+Succ_inj [in hydras.Epsilon0.E0]
+succ_rounds [in hydras.Hydra.Omega2_Small]
+succ_limit_dec [in hydras.Gamma0.Gamma0]
+succ_nf [in hydras.Gamma0.Gamma0]
+succ_as_plus [in hydras.Gamma0.Gamma0]
+succ_of_cons [in hydras.Gamma0.Gamma0]
+succ_lt_le [in hydras.Gamma0.Gamma0]
+succ_is_plus_1 [in hydras.Schutte.Addition]
+sumToN1 [in hydras.Ackermann.cPair]
+sumToN2 [in hydras.Ackermann.cPair]
+SuperbigStep [in hydras.Hydra.BigBattle]
+supM_gt0 [in hydras.Schutte.AP]
+sup_members_disj [in hydras.Schutte.Schutte_basics]
+sup_members_not_succ [in hydras.Schutte.Schutte_basics]
+sup_members_succ [in hydras.Schutte.Schutte_basics]
+sup_eq_intro [in hydras.Schutte.Schutte_basics]
+sup_mono [in hydras.Schutte.Schutte_basics]
+sup_of_leq [in hydras.Schutte.Schutte_basics]
+sup_least_upper_bound [in hydras.Schutte.Schutte_basics]
+sup_upper_bound [in hydras.Schutte.Schutte_basics]
+sup_ok1 [in hydras.Schutte.Schutte_basics]
+sup_spec_unicity [in hydras.Schutte.Schutte_basics]
+sup_unicity [in hydras.Schutte.Schutte_basics]
+sup_exists [in hydras.Schutte.Schutte_basics]
+sup_M_in_B [in hydras.Schutte.Ordering_Functions]
+switch5IsPR [in hydras.Ackermann.codeSubFormula]
+sysExtend [in hydras.Ackermann.LNN]
+sysExtend [in hydras.Ackermann.LNT]
+sysExtend [in hydras.Ackermann.folLogic]
+SysPrf_rephrase [in hydras.Ackermann.Deduction]
+sysWeaken [in hydras.Ackermann.LNN]
+sysWeaken [in hydras.Ackermann.LNT]
+sysWeaken [in hydras.Ackermann.folLogic]
+S_pred_rw [in hydras.Prelude.Iterates]
+s_decr [in hydras.solutions_exercises.T1_ltNotWf]
+S0_mem_head [in hydras.Hydra.O2H]
+S0_decr [in hydras.Hydra.Hydra_Termination]
+S0_decr_0 [in hydras.Hydra.Hydra_Termination]
+S0_remove_r_i [in hydras.Hydra.Hydra_Lemmas]
+S0_remove_r [in hydras.Hydra.Hydra_Lemmas]
+S0_app [in hydras.Hydra.Hydra_Lemmas]
+S1_decr [in hydras.Hydra.Hydra_Termination]
+S1_decr_0 [in hydras.Hydra.Hydra_Termination]
+S1_app [in hydras.Hydra.Hydra_Lemmas]
+S2_app [in hydras.Hydra.Hydra_Lemmas]
+S2_iff [in hydras.Hydra.Hydra_Lemmas]
+

T

+tail_LT_cons [in hydras.Epsilon0.T1]
+tail_lt_cons [in hydras.Epsilon0.T1]
+tail_lt [in hydras.Epsilon0.T1]
+Termination_strong [in hydras.Hydra.Hydra_Lemmas]
+terms_eqdec [in hydras.Ackermann.fol]
+term_eqdec [in hydras.Ackermann.fol]
+Theorem_4_5 [in hydras.Epsilon0.Large_Sets]
+Theorem_4_5 [in gaia_hydras.GLarge_Sets]
+the_least_ok [in hydras.Schutte.Schutte_basics]
+the_least_unicity [in hydras.Schutte.Well_Orders]
+TH_packed [in hydras.Epsilon0.F_alpha]
+th_In [in hydras.Schutte.Ordering_Functions]
+TH_13_6R [in hydras.Schutte.Ordering_Functions]
+TH_13_6 [in hydras.Schutte.Ordering_Functions]
+Th_13_5_2 [in hydras.Schutte.Ordering_Functions]
+Th_13_5_1 [in hydras.Schutte.Ordering_Functions]
+th_14_6 [in hydras.Gamma0.Gamma0]
+th_14_5 [in hydras.Gamma0.Gamma0]
+Th13_8_1 [in hydras.Schutte.Critical]
+Th13_8 [in hydras.Schutte.Critical]
+tower2_alt_ok [in hydras.solutions_exercises.MorePRExamples]
+Toy.ded1 [in hydras.MoreAck.FolExamples]
+Toy.ded2 [in hydras.MoreAck.FolExamples]
+Toy.ded3 [in hydras.MoreAck.FolExamples]
+Toy.drinkers_thm [in hydras.MoreAck.FolExamples]
+Toy.D0 [in hydras.MoreAck.FolExamples]
+Toy.D01 [in hydras.MoreAck.FolExamples]
+Toy.eq_refl [in hydras.MoreAck.FolExamples]
+Toy.MP' [in hydras.MoreAck.FolExamples]
+Toy.peirce [in hydras.MoreAck.FolExamples]
+Toy.PrfEx2 [in hydras.MoreAck.FolExamples]
+Tprf2T'prf [in Goedel.rosserPA]
+trace_to_std [in hydras.Hydra.O2H]
+trace_to_std_0 [in hydras.Hydra.O2H]
+trace_to_round_plus [in hydras.Hydra.O2H]
+transfinite_induction [in hydras.Schutte.Schutte_basics]
+transitionP [in gaia_hydras.GPaths]
+translateLT1 [in hydras.Ackermann.LNN2LNT]
+translatePrf [in hydras.Ackermann.LNN2LNT]
+translateProof [in hydras.Ackermann.LNN2LNT]
+trans_aux [in hydras.Gamma0.Gamma0]
+trans_clos_is_trans [in hydras.rpo.closure]
+TreeT1K [in gaia_hydras.T1Choice]
+trichotomy [in hydras.Schutte.Schutte_basics]
+tricho_aux [in hydras.Gamma0.Gamma0]
+tricho_lt_7 [in hydras.Gamma0.Gamma0]
+tricho_lt_5 [in hydras.Gamma0.Gamma0]
+tricho_lt_4' [in hydras.Gamma0.Gamma0]
+tricho_lt_4 [in hydras.Gamma0.Gamma0]
+tricho_lt_3 [in hydras.Gamma0.Gamma0]
+tricho_lt_2' [in hydras.Gamma0.Gamma0]
+tricho_lt_2 [in hydras.Gamma0.Gamma0]
+t_0_nil [in hydras.Prelude.MoreVectors]
+T'prf2Tprf [in Goedel.rosserPA]
+t0_empty [in hydras.OrdinalNotations.ON_Finite]
+T1addA [in hydras.Epsilon0.T1]
+T1add_not_monotonous_l [in hydras.Epsilon0.T1]
+T1compare_correct [in gaia_hydras.T1Bridge]
+T1eqE [in gaia_hydras.T1Bridge]
+T1eq_h2g [in gaia_hydras.T1Bridge]
+T1eq_refl [in gaia_hydras.T1Bridge]
+T1is_succ_iff [in hydras.Epsilon0.T1]
+T1is_succ_def [in hydras.Epsilon0.T1]
+T1is_succ_ref [in gaia_hydras.T1Bridge]
+T1le_asym [in gaia_hydras.T1Choice]
+T1le_iff [in gaia_hydras.T1Bridge]
+T1limit_succ_tail [in hydras.Epsilon0.T1]
+T1limit_not_zero [in hydras.Epsilon0.T1]
+T1limit_succ [in hydras.Epsilon0.T1]
+T1limit_cases [in hydras.Epsilon0.T1]
+T1limit_ref [in gaia_hydras.T1Bridge]
+T1limit_canonS_not_zero [in hydras.Epsilon0.Canon]
+T1limit_plus [in hydras.Epsilon0.E0]
+T1limit_phi0 [in hydras.Epsilon0.E0]
+T1limit_Omega_term [in hydras.Epsilon0.E0]
+T1ltE [in gaia_hydras.T1Choice]
+T1lt_iff [in gaia_hydras.T1Bridge]
+T1maxE [in gaia_hydras.T1Choice]
+T1minE [in gaia_hydras.T1Choice]
+T1mulE4 [in gaia_hydras.T1Bridge]
+T1mulE5 [in gaia_hydras.T1Bridge]
+T1mul_not_monotonous [in hydras.Epsilon0.T1]
+T1mul_cons_cons_E [in gaia_hydras.T1Bridge]
+T1mul_a0E [in gaia_hydras.T1Bridge]
+T1nf_canon [in gaia_hydras.GCanon]
+T1_wf [in hydras.Epsilon0.T1]
+T1_to_T2_monoR [in hydras.Gamma0.Gamma0]
+T1_to_T2_mono [in hydras.Gamma0.Gamma0]
+T1_to_T2_lt [in hydras.Gamma0.Gamma0]
+T1_to_T2_inj [in hydras.Gamma0.Gamma0]
+T12Tree_inj [in gaia_hydras.T1Choice]
+T2_size_psi [in hydras.Gamma0.Gamma0]
+T2_size4 [in hydras.Gamma0.Gamma0]
+T2_size3 [in hydras.Gamma0.Gamma0]
+T2_size2 [in hydras.Gamma0.Gamma0]
+T2_size1 [in hydras.Gamma0.Gamma0]
+

U

+unbounded [in hydras.Schutte.Schutte_basics]
+Unbounded_Cr [in hydras.Schutte.Critical]
+Unbounded_not_countable [in hydras.Schutte.Schutte_basics]
+Unbounded_ge [in hydras.Schutte.Addition]
+unique_decomposition [in hydras.Epsilon0.T1]
+unique_decomposition [in hydras.OrdinalNotations.ON_Omega2]
+unshift_pred [in hydras.Prelude.MoreLists]
+unshift_shift [in hydras.Prelude.MoreLists]
+unshift_interval_pred [in hydras.Prelude.MoreLists]
+unshift_interval [in hydras.Prelude.MoreLists]
+unshift_iota_from [in hydras.Prelude.MoreLists]
+unshift_not_nil [in hydras.Prelude.MoreLists]
+unshift_app [in hydras.Prelude.MoreLists]
+unshift_but_last [in hydras.Prelude.MoreLists]
+

V

+Variant_termination [in hydras.Prelude.WfVariant]
+Variant_lt_standard [in hydras.Hydra.Hydra_Theorems]
+Variant_LT_standard [in hydras.Hydra.Hydra_Theorems]
+Variant_lt_free [in hydras.Hydra.Hydra_Theorems]
+Variant_LT_free_0 [in hydras.Hydra.Hydra_Theorems]
+variant_mono_free [in hydras.Hydra.Hydra_Lemmas]
+Vars.eq_variable_dec [in hydras.Gamma0.Gamma0]
+Vid_eq [in hydras.Prelude.MoreVectors]
+

W

+wCon2Con [in hydras.Ackermann.wConsistent]
+wellFormedFormulaCorrect1 [in hydras.Ackermann.wellFormed]
+wellFormedFormulaCorrect2 [in hydras.Ackermann.wellFormed]
+wellFormedTermCorrect1 [in hydras.Ackermann.wellFormed]
+wellFormedTermCorrect2 [in hydras.Ackermann.wellFormed]
+wellFormedTermsCorrect1 [in hydras.Ackermann.wellFormed]
+wellFormedTermsCorrect2 [in hydras.Ackermann.wellFormed]
+wellFormedTermTermsCorrect2 [in hydras.Ackermann.wellFormed]
+well_founded_length [in hydras.rpo.more_list]
+well_founded_rpo [in hydras.Gamma0.Gamma0]
+Wfsum.f_correct [in gaia_hydras.nfwfgaia]
+Wfsum.f_eqn [in gaia_hydras.nfwfgaia]
+Wfsum.f_spec_simp [in gaia_hydras.nfwfgaia]
+Wfsum.lt_dec [in gaia_hydras.nfwfgaia]
+Wfsum.psum_exten [in gaia_hydras.nfwfgaia]
+wf_inverse_image_transparent [in additions.Wf_transparent]
+wf_incl_transparent [in additions.Wf_transparent]
+wf_measure [in hydras.OrdinalNotations.ON_Generic]
+wf_ltn [in gaia_hydras.onType]
+wf_trans [in hydras.rpo.closure]
+Wf_ex3.f_correct [in gaia_hydras.nfwfgaia]
+Wf_ex3.f_eqn [in gaia_hydras.nfwfgaia]
+Wf_ex3.odd_dec [in gaia_hydras.nfwfgaia]
+Wf_ex3.f0c [in gaia_hydras.nfwfgaia]
+Wf_ex3.f0b [in gaia_hydras.nfwfgaia]
+Wf_ex3.f0a [in gaia_hydras.nfwfgaia]
+Wf_ex3.f_spec_simp2 [in gaia_hydras.nfwfgaia]
+Wf_ex3.f_spec_simp1 [in gaia_hydras.nfwfgaia]
+Wf_ex3.f_spec_simp [in gaia_hydras.nfwfgaia]
+Wf_ex3.lte_wf [in gaia_hydras.nfwfgaia]
+Wf_ex.f_correct [in gaia_hydras.nfwfgaia]
+Wf_ex.f_eqn [in gaia_hydras.nfwfgaia]
+Wf_ex.f0 [in gaia_hydras.nfwfgaia]
+Wf_ex.f_spec_simp [in gaia_hydras.nfwfgaia]
+wf_lexico [in hydras.Prelude.Simple_LexProd]
+wf_lex [in hydras.rpo.rpo]
+

X

+xPairDef [in hydras.Ackermann.cPair]
+

Z

+zeroRepresentable [in Goedel.PRrepresentable]
+zero_le [in hydras.Epsilon0.T1]
+zero_nf [in hydras.Epsilon0.T1]
+zero_lt [in hydras.Epsilon0.T1]
+zero_lt [in hydras.Schutte.Correctness_E0]
+zero_le [in hydras.OrdinalNotations.ON_Omega2]
+zero_lt_beta [in hydras.Schutte.AP]
+zero_ref [in gaia_hydras.T1Bridge]
+zero_lt_omega [in hydras.Schutte.Schutte_basics]
+zero_lt_succ [in hydras.Schutte.Schutte_basics]
+zero_or_positive [in hydras.Schutte.Schutte_basics]
+zero_or_greater [in hydras.Schutte.Schutte_basics]
+zero_le [in hydras.Schutte.Schutte_basics]
+zero_not_lim [in hydras.Gamma0.Gamma0]
+zero_lt_succ [in hydras.Gamma0.Gamma0]
+zero_plus_alpha [in hydras.Schutte.Addition]
+ZLS_dec [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+ZtoRD [in additions.fib]
+ZtoRM [in additions.fib]
+Z_pow_compat [in additions.Compatibility]
+Z_pow_compat_pos [in additions.Compatibility]
+Z.power_of_plus [in additions.Naive]
+Z.power_S [in additions.Naive]
+

+

Constructor Index

+

A

+Ackermann.CompareT3Eq [in gaia_hydras.nfwfgaia]
+Ackermann.CompareT3Gt [in gaia_hydras.nfwfgaia]
+Ackermann.CompareT3Lt [in gaia_hydras.nfwfgaia]
+Ackermann.cons [in gaia_hydras.nfwfgaia]
+Ackermann.T3GeqNotGtn [in gaia_hydras.nfwfgaia]
+Ackermann.T3GeqNotLtn [in gaia_hydras.nfwfgaia]
+Ackermann.T3GtnNotLeq [in gaia_hydras.nfwfgaia]
+Ackermann.T3LtnNotGeq [in gaia_hydras.nfwfgaia]
+Ackermann.zero [in gaia_hydras.nfwfgaia]
+Add [in additions.Trace_exercise]
+Alt.Eps0_prec.Mul [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.Lex [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.Free [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.C [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.AC [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.ord_cons [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.ord_zero [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.nat_S [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.nat_0 [in hydras.Epsilon0.Epsilon0rpo]
+Alt.hnode [in hydras.Hydra.Hydra_Definitions]
+apply [in hydras.Ackermann.fol]
+ap_intro [in hydras.Epsilon0.T1]
+ap_intro [in hydras.Gamma0.T2]
+assoc [in hydras.Prelude.STDPP_compat]
+atomic [in hydras.Ackermann.fol]
+AXM [in hydras.Ackermann.folProof]
+A_node [in additions.Addition_Chains]
+

B

+Bad3.Fchain_proper_bad_prf [in additions.Euclidean_Chains]
+Build_OF [in hydras.Schutte.Ordering_Functions]
+

C

+CantorOrdinal.CompareT1Eq [in gaia_hydras.nfwfgaia]
+CantorOrdinal.CompareT1Gt [in gaia_hydras.nfwfgaia]
+CantorOrdinal.CompareT1Lt [in gaia_hydras.nfwfgaia]
+CantorOrdinal.cons [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1GeqNotGtn [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1GeqNotLtn [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1GtnNotLeq [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1LtnNotGeq [in gaia_hydras.nfwfgaia]
+CantorOrdinal.zero [in gaia_hydras.nfwfgaia]
+compare [in hydras.Prelude.Comparable]
+composeFunc [in hydras.Ackermann.primRec]
+cons [in hydras.Epsilon0.T1]
+cons_succ [in hydras.Gamma0.T2]
+CP [in hydras.Ackermann.folProof]
+

D

+decide [in hydras.Prelude.STDPP_compat]
+decide_rel [in hydras.Prelude.STDPP_compat]
+

E

+equal [in hydras.Ackermann.fol]
+equiv [in additions.Monoid_def]
+EQ1 [in hydras.Ackermann.folProof]
+EQ2 [in hydras.Ackermann.folProof]
+EQ3 [in hydras.Ackermann.folProof]
+EQ4 [in hydras.Ackermann.folProof]
+EQ5 [in hydras.Ackermann.folProof]
+

F

+FA1 [in hydras.Ackermann.folProof]
+FA2 [in hydras.Ackermann.folProof]
+FA3 [in hydras.Ackermann.folProof]
+Fchain_proper_prf [in additions.Euclidean_Chains]
+finite_succ [in hydras.Gamma0.T2]
+Fkont_proper_prf [in additions.Euclidean_Chains]
+FO [in hydras.Prelude.Fuel]
+forallH [in hydras.Ackermann.fol]
+Forall2R_last [in hydras.Prelude.MoreLists]
+Forall2R_nil [in hydras.Prelude.MoreLists]
+FS [in hydras.Prelude.Fuel]
+fun_bij_i [in hydras.Schutte.PartialFun]
+fun_inj_i [in hydras.Schutte.PartialFun]
+

G

+Gamma0_prec.Mul [in hydras.Gamma0.Gamma0]
+Gamma0_prec.Lex [in hydras.Gamma0.Gamma0]
+Gamma0_sig.Free [in hydras.Gamma0.Gamma0]
+Gamma0_sig.C [in hydras.Gamma0.Gamma0]
+Gamma0_sig.AC [in hydras.Gamma0.Gamma0]
+Gamma0_sig.ord_cons [in hydras.Gamma0.Gamma0]
+Gamma0_sig.ord_psi [in hydras.Gamma0.Gamma0]
+Gamma0_sig.ord_zero [in hydras.Gamma0.Gamma0]
+Gamma0_sig.nat_S [in hydras.Gamma0.Gamma0]
+Gamma0_sig.nat_0 [in hydras.Gamma0.Gamma0]
+Gamma0.CompareT2Eq [in gaia_hydras.nfwfgaia]
+Gamma0.CompareT2Gt [in gaia_hydras.nfwfgaia]
+Gamma0.CompareT2Lt [in gaia_hydras.nfwfgaia]
+Gamma0.cons [in gaia_hydras.nfwfgaia]
+Gamma0.T2GeqNotGtn [in gaia_hydras.nfwfgaia]
+Gamma0.T2GeqNotLtn [in gaia_hydras.nfwfgaia]
+Gamma0.T2GtnNotLeq [in gaia_hydras.nfwfgaia]
+Gamma0.T2LtnNotGeq [in gaia_hydras.nfwfgaia]
+Gamma0.zero [in gaia_hydras.nfwfgaia]
+gcons [in hydras.Gamma0.T2]
+gcons_lt_e0 [in hydras.Gamma0.T2]
+gcons_nf [in hydras.Gamma0.T2]
+GEN [in hydras.Ackermann.folProof]
+gen_K [in additions.Euclidean_Chains]
+gen_F [in additions.Euclidean_Chains]
+G0.mkg0 [in hydras.Gamma0.Gamma0]
+

H

+hcons [in hydras.Hydra.Hydra_Definitions]
+hh_2 [in hydras.Hydra.O2H]
+hh_1 [in hydras.Hydra.O2H]
+hnil [in hydras.Hydra.Hydra_Definitions]
+

I

+impH [in hydras.Ackermann.fol]
+IMP1 [in hydras.Ackermann.folProof]
+IMP2 [in hydras.Ackermann.folProof]
+InHWit [in hydras.Schutte.MoreEpsilonIota]
+Init [in additions.Trace_exercise]
+is_limit_cons [in hydras.Gamma0.T2]
+is_limit_0 [in hydras.Gamma0.T2]
+is_pr_rel [in hydras.Ackermann.primRec]
+is_pr [in hydras.Ackermann.primRec]
+

K

+Kchain_proper_prf [in additions.Euclidean_Chains]
+

L

+language [in hydras.Ackermann.fol]
+lex_2 [in hydras.Prelude.Simple_LexProd]
+lex_1 [in hydras.Prelude.Simple_LexProd]
+lex1 [in hydras.Prelude.DecPreOrder_Instances]
+lex1 [in hydras.solutions_exercises.MultisetWf]
+lex2 [in hydras.Prelude.DecPreOrder_Instances]
+lex2 [in hydras.solutions_exercises.MultisetWf]
+lex3 [in hydras.solutions_exercises.MultisetWf]
+lex4 [in hydras.solutions_exercises.MultisetWf]
+limit_step [in hydras.Hydra.Omega2_Small]
+limit_plus_F_cons [in hydras.Gamma0.Gamma0]
+limit_plus_F_0 [in hydras.Gamma0.Gamma0]
+LO.ap_intro [in hydras.OrdinalNotations.OmegaOmega]
+LT_right [in hydras.Epsilon0.T1]
+LT_middle [in hydras.Epsilon0.T1]
+LT_left [in hydras.Epsilon0.T1]
+lt_a_phi0_b_c [in hydras.Epsilon0.T1]
+lt_a_phi0_b_z [in hydras.Epsilon0.T1]
+lt_right [in hydras.Epsilon0.T1]
+lt_middle [in hydras.Epsilon0.T1]
+lt_left [in hydras.Epsilon0.T1]
+lt_7 [in hydras.Gamma0.T2]
+lt_6 [in hydras.Gamma0.T2]
+lt_5 [in hydras.Gamma0.T2]
+lt_4 [in hydras.Gamma0.T2]
+lt_3 [in hydras.Gamma0.T2]
+lt_2 [in hydras.Gamma0.T2]
+lt_1 [in hydras.Gamma0.T2]
+LT_ [in hydras.Ackermann.Languages]
+L_spec1 [in hydras.Epsilon0.Large_Sets]
+L_spec0 [in hydras.Epsilon0.Large_Sets]
+

M

+Make.Eq [in hydras.rpo.rpo]
+Make.List_mul_rest_step [in hydras.rpo.rpo]
+Make.List_mul_rest [in hydras.rpo.rpo]
+Make.List_eq_rest [in hydras.rpo.rpo]
+Make.List_gt_rest [in hydras.rpo.rpo]
+Make.List_mul [in hydras.rpo.rpo]
+Make.List_eq [in hydras.rpo.rpo]
+Make.List_gt [in hydras.rpo.rpo]
+Make.Lt [in hydras.rpo.rpo]
+Make.mk_set [in hydras.rpo.list_set]
+Make.mk_sn [in hydras.rpo.rpo]
+Make.rmv_case [in hydras.rpo.dickson]
+Make.Subterm [in hydras.rpo.rpo]
+Make.Term [in hydras.rpo.term]
+Make.Top_eq_mul [in hydras.rpo.rpo]
+Make.Top_eq_lex [in hydras.rpo.rpo]
+Make.Top_gt [in hydras.rpo.rpo]
+Make.Var [in hydras.rpo.term]
+marked0 [in hydras.Prelude.Sort_spec]
+marked1 [in hydras.Prelude.Sort_spec]
+mkE0 [in gaia_hydras.T1Bridge]
+mkord [in hydras.Epsilon0.E0]
+mkP [in hydras.Epsilon0.F_alpha]
+mkP [in hydras.Epsilon0.Hprime]
+mkQ [in hydras.Epsilon0.F_alpha]
+mks [in hydras.Hydra.BigBattle]
+model [in hydras.Ackermann.model]
+MP [in hydras.Ackermann.folProof]
+MUL [in additions.AM]
+Mult [in additions.Addition_Chains]
+mult_op [in additions.Monoid_def]
+Mul_node [in additions.Addition_Chains]
+M2.mat [in additions.FirstSteps]
+M2.mat [in additions.Naive]
+

N

+node [in hydras.Hydra.Hydra_Definitions]
+notH [in hydras.Ackermann.fol]
+

O

+Ok [in hydras.Epsilon0.Large_Sets_Examples]
+ONDef.Mixin [in gaia_hydras.onType]
+ONDef.Pack [in gaia_hydras.onType]
+One_node [in additions.Addition_Chains]
+OO.mkord [in hydras.OrdinalNotations.OmegaOmega]
+

P

+path_toS_cons [in hydras.Epsilon0.Paths]
+path_toS_1 [in hydras.Epsilon0.Paths]
+path_to_cons [in hydras.Epsilon0.Paths]
+path_to_1 [in hydras.Epsilon0.Paths]
+Plus_ [in hydras.Ackermann.Languages]
+PP_omega [in hydras.Epsilon0.T1]
+PP_exp [in hydras.Epsilon0.T1]
+PP_mult [in hydras.Epsilon0.T1]
+PP_add [in hydras.Epsilon0.T1]
+PP_fin [in hydras.Epsilon0.T1]
+PRcons [in hydras.Ackermann.primRec]
+Precedence.Lex [in hydras.rpo.rpo]
+Precedence.Mul [in hydras.rpo.rpo]
+primRecFunc [in hydras.Ackermann.primRec]
+PRnil [in hydras.Ackermann.primRec]
+projFunc [in hydras.Ackermann.primRec]
+PUSH [in additions.AM]
+

R

+rel_surj_i [in hydras.Schutte.GRelations]
+rel_inj_i [in hydras.Schutte.GRelations]
+rel_bij_i [in hydras.Schutte.PartialFun]
+rel_inj_i [in hydras.Schutte.PartialFun]
+Remaining [in hydras.Epsilon0.Large_Sets_Examples]
+result_equiv_success [in additions.AM]
+result_equiv_fail [in additions.AM]
+Return [in additions.Addition_Chains]
+rounds_n [in hydras.Hydra.Hydra_Definitions]
+rounds_1 [in hydras.Hydra.Hydra_Definitions]
+RPO.Eq [in hydras.rpo.rpo]
+RPO.List_mul [in hydras.rpo.rpo]
+RPO.List_eq [in hydras.rpo.rpo]
+RPO.List_gt [in hydras.rpo.rpo]
+RPO.Lt [in hydras.rpo.rpo]
+RPO.Subterm [in hydras.rpo.rpo]
+RPO.Top_eq_mul [in hydras.rpo.rpo]
+RPO.Top_eq_lex [in hydras.rpo.rpo]
+RPO.Top_gt [in hydras.rpo.rpo]
+R1_intro [in hydras.Hydra.Hydra_Definitions]
+R2_intro_2 [in hydras.Hydra.Hydra_Definitions]
+R2_intro [in hydras.Hydra.Hydra_Definitions]
+

S

+Signature.AC [in hydras.rpo.term]
+Signature.C [in hydras.rpo.term]
+Signature.Free [in hydras.rpo.term]
+single_nf [in hydras.Gamma0.T2]
+sorted_ge_cons [in hydras.Prelude.MoreLists]
+sorted_ge_one [in hydras.Prelude.MoreLists]
+sorted_ge_nil [in hydras.Prelude.MoreLists]
+SQR [in additions.AM]
+stack_equivn [in additions.AM]
+stack_equiv0 [in additions.AM]
+stdS_S [in hydras.Epsilon0.Paths]
+stdS_1 [in hydras.Epsilon0.Paths]
+std_S [in hydras.Epsilon0.Paths]
+std_1 [in hydras.Epsilon0.Paths]
+steps_S [in hydras.Hydra.BigBattle]
+steps1 [in hydras.Hydra.BigBattle]
+step1 [in hydras.Hydra.BigBattle]
+step2 [in hydras.Hydra.BigBattle]
+step3 [in hydras.Hydra.BigBattle]
+subterm_trans [in hydras.Gamma0.Gamma0]
+subterm_c [in hydras.Gamma0.Gamma0]
+subterm_b [in hydras.Gamma0.Gamma0]
+subterm_a [in hydras.Gamma0.Gamma0]
+succFunc [in hydras.Ackermann.primRec]
+succ_finite [in hydras.Gamma0.T2]
+succ_step [in hydras.Hydra.Omega2_Small]
+Succ_ [in hydras.Ackermann.Languages]
+SWAP [in additions.AM]
+S.mk_set [in hydras.rpo.list_set]
+S0_rest [in hydras.Hydra.Hydra_Definitions]
+S0_first [in hydras.Hydra.Hydra_Definitions]
+S1_next [in hydras.Hydra.Hydra_Definitions]
+S1_first [in hydras.Hydra.Hydra_Definitions]
+S2_next [in hydras.Hydra.Hydra_Definitions]
+S2_first [in hydras.Hydra.Hydra_Definitions]
+

T

+Tcons [in hydras.Ackermann.fol]
+Term.Term [in hydras.rpo.term]
+Term.Var [in hydras.rpo.term]
+Times_ [in hydras.Ackermann.Languages]
+Tnil [in hydras.Ackermann.fol]
+Too_far [in hydras.Epsilon0.Large_Sets_Examples]
+totalness [in hydras.Prelude.DecPreOrder]
+Toy.a_ [in hydras.MoreAck.FolExamples]
+Toy.A_ [in hydras.MoreAck.FolExamples]
+Toy.b_ [in hydras.MoreAck.FolExamples]
+Toy.B_ [in hydras.MoreAck.FolExamples]
+Toy.C_ [in hydras.MoreAck.FolExamples]
+Toy.f_ [in hydras.MoreAck.FolExamples]
+Toy.g_ [in hydras.MoreAck.FolExamples]
+Toy.h_ [in hydras.MoreAck.FolExamples]
+Toy.P_ [in hydras.MoreAck.FolExamples]
+Toy.Q_ [in hydras.MoreAck.FolExamples]
+Toy.R_ [in hydras.MoreAck.FolExamples]
+trace_toS [in hydras.Hydra.Hydra_Definitions]
+trace_to1 [in hydras.Hydra.Hydra_Definitions]
+t_trans [in hydras.rpo.closure]
+t_step [in hydras.rpo.closure]
+

V

+var [in hydras.Ackermann.fol]
+

Z

+zero [in hydras.Epsilon0.T1]
+zero [in hydras.Gamma0.T2]
+zeroFunc [in hydras.Ackermann.primRec]
+zero_lt_e0 [in hydras.Gamma0.T2]
+zero_nf [in hydras.Gamma0.T2]
+zero_finite [in hydras.Gamma0.T2]
+Zero_ [in hydras.Ackermann.Languages]
+

+

Axiom Index

+

A

+AX1 [in hydras.Schutte.Schutte_basics]
+AX2 [in hydras.Schutte.Schutte_basics]
+AX3 [in hydras.Schutte.Schutte_basics]
+

F

+f [in hydras.Prelude.LibHyps_Experiments]
+

G

+g [in hydras.Prelude.LibHyps_Experiments]
+

H

+h [in hydras.Prelude.LibHyps_Experiments]
+

I

+inh_Ord [in hydras.Schutte.Schutte_basics]
+

L

+lt [in hydras.Schutte.Schutte_basics]
+

O

+Ord [in hydras.Schutte.Schutte_basics]
+

P

+P [in hydras.Prelude.LibHyps_Experiments]
+Precedence.A [in hydras.rpo.rpo]
+Precedence.prec [in hydras.rpo.rpo]
+Precedence.prec_transitive [in hydras.rpo.rpo]
+Precedence.prec_antisym [in hydras.rpo.rpo]
+Precedence.prec_dec [in hydras.rpo.rpo]
+Precedence.status [in hydras.rpo.rpo]
+

R

+RPO.rpo_add_context [in hydras.rpo.rpo]
+RPO.rpo_subst [in hydras.rpo.rpo]
+RPO.rpo_trans [in hydras.rpo.rpo]
+RPO.rpo_closure [in hydras.rpo.rpo]
+RPO.wf_rpo [in hydras.rpo.rpo]
+

S

+Signature.arity [in hydras.rpo.term]
+Signature.eq_symbol_dec [in hydras.rpo.term]
+Signature.symb [in hydras.rpo.term]
+S.A [in hydras.rpo.decidable_set]
+S.cardinal_subset [in hydras.rpo.list_set]
+S.eq_A_dec [in hydras.rpo.decidable_set]
+

T

+Term.empty_subst_is_id_list [in hydras.rpo.term]
+Term.empty_subst_is_id [in hydras.rpo.term]
+Term.eq_term_dec [in hydras.rpo.term]
+Term.is_a_pos_exists_subtem [in hydras.rpo.term]
+Term.replace_at_pos_list_replace_at_pos_in_subterm [in hydras.rpo.term]
+Term.replace_at_pos_is_replace_at_pos2 [in hydras.rpo.term]
+Term.replace_at_pos_is_replace_at_pos1 [in hydras.rpo.term]
+Term.replace_at_pos_unfold [in hydras.rpo.term]
+Term.size_subterm_at_pos [in hydras.rpo.term]
+Term.size_direct_subterm [in hydras.rpo.term]
+Term.size_ge_one [in hydras.rpo.term]
+Term.size_unfold [in hydras.rpo.term]
+Term.subst_comp_is_subst_comp [in hydras.rpo.term]
+Term.subst_comp_is_subst_comp_aux1 [in hydras.rpo.term]
+Term.subterm_at_pos_apply_subst_apply_subst_subterm_at_pos [in hydras.rpo.term]
+Term.term_rec8 [in hydras.rpo.term]
+Term.term_rec7 [in hydras.rpo.term]
+Term.term_rec4 [in hydras.rpo.term]
+Term.term_rec3 [in hydras.rpo.term]
+Term.term_rec2 [in hydras.rpo.term]
+Term.well_formed_apply_subst [in hydras.rpo.term]
+Term.well_formed_fold [in hydras.rpo.term]
+Term.well_formed_unfold [in hydras.rpo.term]
+

V

+Variables.eq_variable_dec [in hydras.rpo.term]
+Variables.var [in hydras.rpo.term]
+

+

Inductive Index

+

A

+Ackermann.compare_T3 [in gaia_hydras.nfwfgaia]
+Ackermann.T3 [in gaia_hydras.nfwfgaia]
+Ackermann.T3leq_xor_gtn [in gaia_hydras.nfwfgaia]
+Ackermann.T3ltn_xor_geq [in gaia_hydras.nfwfgaia]
+Alt.Eps0_prec.status_type [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.arity_type [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.symb0 [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Hydra [in hydras.Hydra.Hydra_Definitions]
+Alt.Vars.empty_set [in hydras.Epsilon0.Epsilon0rpo]
+answer [in hydras.Epsilon0.Large_Sets_Examples]
+ap [in hydras.Epsilon0.T1]
+ap [in hydras.Gamma0.T2]
+Assoc [in hydras.Prelude.STDPP_compat]
+

B

+Bad3.Fchain_proper [in additions.Euclidean_Chains]
+

C

+CantorOrdinal.compare_T1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1leq_xor_gtn [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ltn_xor_geq [in gaia_hydras.nfwfgaia]
+Compare [in hydras.Prelude.Comparable]
+computation [in additions.Addition_Chains]
+

D

+Decision [in hydras.Prelude.STDPP_compat]
+

E

+Equiv [in additions.Monoid_def]
+

F

+Fchain_proper [in additions.Euclidean_Chains]
+Fkont_proper [in additions.Euclidean_Chains]
+Forall2R [in hydras.Prelude.MoreLists]
+Formula [in hydras.Ackermann.fol]
+fuel [in hydras.Prelude.Fuel]
+fun_bijection [in hydras.Schutte.PartialFun]
+fun_injection [in hydras.Schutte.PartialFun]
+

G

+Gamma0_prec.status_type [in hydras.Gamma0.Gamma0]
+Gamma0_sig.arity_type [in hydras.Gamma0.Gamma0]
+Gamma0_sig.symb0 [in hydras.Gamma0.Gamma0]
+Gamma0.compare_T2 [in gaia_hydras.nfwfgaia]
+Gamma0.T2 [in gaia_hydras.nfwfgaia]
+Gamma0.T2leq_xor_gtn [in gaia_hydras.nfwfgaia]
+Gamma0.T2ltn_xor_geq [in gaia_hydras.nfwfgaia]
+

H

+Hydra [in hydras.Hydra.Hydra_Definitions]
+Hydrae [in hydras.Hydra.Hydra_Definitions]
+

I

+info [in additions.Trace_exercise]
+InH [in hydras.Schutte.MoreEpsilonIota]
+instr [in additions.AM]
+isPR [in hydras.Ackermann.primRec]
+isPRrel [in hydras.Ackermann.primRec]
+is_limit [in hydras.Gamma0.T2]
+is_successor [in hydras.Gamma0.T2]
+is_finite [in hydras.Gamma0.T2]
+

K

+Kchain_proper [in additions.Euclidean_Chains]
+

L

+lexico [in hydras.Prelude.Simple_LexProd]
+lexpower [in hydras.solutions_exercises.MultisetWf]
+lex_prod [in hydras.Prelude.DecPreOrder_Instances]
+limit_plus_fin [in hydras.Gamma0.Gamma0]
+LNNRelation [in hydras.Ackermann.Languages]
+LNTFunction [in hydras.Ackermann.Languages]
+LNTRelation [in hydras.Ackermann.Languages]
+LO.ap [in hydras.OrdinalNotations.OmegaOmega]
+lt [in hydras.Gamma0.T2]
+LT_cases [in hydras.Epsilon0.T1]
+lt_a_phi0_b [in hydras.Epsilon0.T1]
+lt_cases [in hydras.Epsilon0.T1]
+lt_epsilon0 [in hydras.Gamma0.T2]
+L_spec [in hydras.Epsilon0.Large_Sets]
+

M

+Make.multiset_extension_step [in hydras.rpo.dickson]
+Make.rpo [in hydras.rpo.rpo]
+Make.rpo_mul_rest_step [in hydras.rpo.rpo]
+Make.rpo_mul_rest [in hydras.rpo.rpo]
+Make.rpo_lex_rest [in hydras.rpo.rpo]
+Make.rpo_mul [in hydras.rpo.rpo]
+Make.rpo_lex [in hydras.rpo.rpo]
+Make.rpo_eq [in hydras.rpo.rpo]
+Make.term [in hydras.rpo.term]
+marked [in hydras.Prelude.Sort_spec]
+mem_head [in hydras.Hydra.O2H]
+Monoid_Exp [in additions.Addition_Chains]
+Mult_op [in additions.Monoid_def]
+

N

+nf [in hydras.Gamma0.T2]
+

O

+one_step [in hydras.Hydra.BigBattle]
+

P

+path_toS [in hydras.Epsilon0.Paths]
+path_to [in hydras.Epsilon0.Paths]
+ppT1 [in hydras.Epsilon0.T1]
+Precedence.status_type [in hydras.rpo.rpo]
+Prf [in hydras.Ackermann.folProof]
+PrimRec [in hydras.Ackermann.primRec]
+PrimRecs [in hydras.Ackermann.primRec]
+

R

+RelDecision [in hydras.Prelude.STDPP_compat]
+rel_surjection [in hydras.Schutte.GRelations]
+rel_injection [in hydras.Schutte.GRelations]
+rel_bijection [in hydras.Schutte.PartialFun]
+rel_injection [in hydras.Schutte.PartialFun]
+result_equiv [in additions.AM]
+rounds [in hydras.Hydra.Hydra_Definitions]
+RPO.rpo [in hydras.rpo.rpo]
+RPO.rpo_mul [in hydras.rpo.rpo]
+RPO.rpo_lex [in hydras.rpo.rpo]
+RPO.rpo_eq [in hydras.rpo.rpo]
+R1 [in hydras.Hydra.Hydra_Definitions]
+R2 [in hydras.Hydra.Hydra_Definitions]
+

S

+signature [in additions.Euclidean_Chains]
+Signature.arity_type [in hydras.rpo.term]
+sorted_ge [in hydras.Prelude.MoreLists]
+stack_equiv [in additions.AM]
+standard_path_to [in hydras.Epsilon0.Paths]
+standard_path_toS [in hydras.Epsilon0.Paths]
+step [in hydras.Hydra.Omega2_Small]
+steps [in hydras.Hydra.BigBattle]
+subterm [in hydras.Gamma0.Gamma0]
+S0 [in hydras.Hydra.Hydra_Definitions]
+S1 [in hydras.Hydra.Hydra_Definitions]
+S2 [in hydras.Hydra.Hydra_Definitions]
+

T

+Term [in hydras.Ackermann.fol]
+Terms [in hydras.Ackermann.fol]
+Term.term [in hydras.rpo.term]
+Total [in hydras.Prelude.DecPreOrder]
+Toy.Fun [in hydras.MoreAck.FolExamples]
+Toy.Rel [in hydras.MoreAck.FolExamples]
+trace_to [in hydras.Hydra.Hydra_Definitions]
+trans_clos [in hydras.rpo.closure]
+T1 [in hydras.Epsilon0.T1]
+T2 [in hydras.Gamma0.T2]
+

V

+Vars.empty_set [in hydras.Gamma0.Gamma0]
+

+

Projection Index

+

A

+arityF [in hydras.Ackermann.fol]
+arityR [in hydras.Ackermann.fol]
+assoc [in hydras.Prelude.STDPP_compat]
+

B

+Bad3.Fchain_proper_bad_prf [in additions.Euclidean_Chains]
+battle_ok [in hydras.Hydra.Hydra_Definitions]
+battle_rel [in hydras.Hydra.Hydra_Definitions]
+

C

+cnf [in gaia_hydras.T1Bridge]
+cnf [in hydras.Epsilon0.E0]
+cnf_ok [in hydras.Epsilon0.E0]
+codeFInj [in hydras.Ackermann.code]
+codeRInj [in hydras.Ackermann.code]
+comparable_comp_spec [in hydras.Prelude.Comparable]
+comparable_sto [in hydras.Prelude.Comparable]
+compare [in hydras.Prelude.Comparable]
+c00 [in additions.Monoid_instances]
+c01 [in additions.Monoid_instances]
+c10 [in additions.Monoid_instances]
+c11 [in additions.Monoid_instances]
+

D

+decide [in hydras.Prelude.STDPP_compat]
+decide_rel [in hydras.Prelude.STDPP_compat]
+decr [in hydras.Prelude.WfVariant]
+

E

+Eone_right [in additions.Monoid_def]
+Eone_left [in additions.Monoid_def]
+Eop_comm [in additions.Monoid_def]
+Eop_assoc [in additions.Monoid_def]
+Eop_proper [in additions.Monoid_def]
+equiv [in additions.Monoid_def]
+Eq_equiv [in additions.Monoid_def]
+

F

+Fchain_proper_prf [in additions.Euclidean_Chains]
+Fkont_proper_prf [in additions.Euclidean_Chains]
+func [in hydras.Ackermann.model]
+Functions [in hydras.Ackermann.fol]
+

G

+gamma_gt [in additions.Strategies]
+gamma_lt [in additions.Strategies]
+G0.vnf [in hydras.Gamma0.Gamma0]
+G0.vnf_ok [in hydras.Gamma0.Gamma0]
+

I

+InHWit [in hydras.Schutte.MoreEpsilonIota]
+iso_inv2 [in hydras.OrdinalNotations.ON_Generic]
+iso_inv1 [in hydras.OrdinalNotations.ON_Generic]
+iso_compare [in hydras.OrdinalNotations.ON_Generic]
+is_pr_rel [in hydras.Ackermann.primRec]
+is_pr [in hydras.Ackermann.primRec]
+

K

+Kchain_proper_prf [in additions.Euclidean_Chains]
+

L

+Lt_irreflexive [in hydras.Schutte.Well_Orders]
+Lt_trans [in hydras.Schutte.Well_Orders]
+

M

+Make.is_a_set [in hydras.rpo.list_set]
+Make.sn [in hydras.rpo.rpo]
+Make.support [in hydras.rpo.list_set]
+Make.tt [in hydras.rpo.rpo]
+mult_op [in additions.Monoid_def]
+m_bounded [in hydras.Hydra.Hydra_Definitions]
+M2.c00 [in additions.FirstSteps]
+M2.c00 [in additions.Naive]
+M2.c01 [in additions.FirstSteps]
+M2.c01 [in additions.Naive]
+M2.c10 [in additions.FirstSteps]
+M2.c10 [in additions.Naive]
+M2.c11 [in additions.FirstSteps]
+M2.c11 [in additions.Naive]
+

N

+nh [in hydras.Hydra.BigBattle]
+n1 [in hydras.Hydra.BigBattle]
+n2 [in hydras.Hydra.BigBattle]
+

O

+OF_mono [in hydras.Schutte.Ordering_Functions]
+OF_onto [in hydras.Schutte.Ordering_Functions]
+OF_total [in hydras.Schutte.Ordering_Functions]
+OF_segment [in hydras.Schutte.Ordering_Functions]
+ONDef.type [in gaia_hydras.onType]
+one_right [in additions.Monoid_def]
+one_left [in additions.Monoid_def]
+On_compare_spec [in hydras.OrdinalNotations.ON_Generic]
+ON_correct_onto [in hydras.OrdinalNotations.ON_Generic]
+ON_correct_inj [in hydras.OrdinalNotations.ON_Generic]
+ON_wf [in hydras.OrdinalNotations.ON_Generic]
+ON_comp [in hydras.OrdinalNotations.ON_Generic]
+OO.data [in hydras.OrdinalNotations.OmegaOmega]
+OO.data_ok [in hydras.OrdinalNotations.OmegaOmega]
+op_comm [in additions.Monoid_def]
+op_assoc [in additions.Monoid_def]
+

P

+PA [in hydras.Epsilon0.F_alpha]
+PA [in hydras.Epsilon0.Hprime]
+PB [in hydras.Epsilon0.F_alpha]
+PB [in hydras.Epsilon0.Hprime]
+PC [in hydras.Epsilon0.F_alpha]
+PC [in hydras.Epsilon0.Hprime]
+PD [in hydras.Epsilon0.F_alpha]
+PD [in hydras.Epsilon0.Hprime]
+PE [in hydras.Epsilon0.F_alpha]
+PE [in hydras.Epsilon0.Hprime]
+

Q

+QA [in hydras.Epsilon0.F_alpha]
+QD [in hydras.Epsilon0.F_alpha]
+QE [in hydras.Epsilon0.F_alpha]
+

R

+rel [in hydras.Ackermann.model]
+Relations [in hydras.Ackermann.fol]
+round [in hydras.Hydra.BigBattle]
+

S

+Semibundled.dec_dec [in hydras.Prelude.DecPreOrder]
+Semibundled.total_dec_total [in hydras.Prelude.DecPreOrder]
+Semibundled.total_dec_pre_order [in hydras.Prelude.DecPreOrder]
+Semibundled.total_dec [in hydras.Prelude.DecPreOrder]
+SubON_onto [in hydras.OrdinalNotations.ON_Generic]
+SubON_incl [in hydras.OrdinalNotations.ON_Generic]
+SubON_compare [in hydras.OrdinalNotations.ON_Generic]
+S.is_a_set [in hydras.rpo.list_set]
+S.support [in hydras.rpo.list_set]
+

T

+totalness [in hydras.Prelude.DecPreOrder]
+total_pre_order_total [in hydras.Prelude.DecPreOrder]
+total_pre_order_pre [in hydras.Prelude.DecPreOrder]
+

U

+U [in hydras.Ackermann.model]
+

V

+variant_decr [in hydras.Hydra.Hydra_Definitions]
+

W

+well_order [in hydras.Schutte.Well_Orders]
+wf [in hydras.Prelude.WfVariant]
+

+

Section Index

+

A

+Abstract_Properties [in hydras.Epsilon0.Hprime]
+AB_given.rel_to_fun [in hydras.Schutte.PartialFun]
+AB_given.f_given [in hydras.Schutte.PartialFun]
+AB_given [in hydras.Schutte.PartialFun]
+Ack_Properties.Induc_step [in hydras.MoreAck.Ack]
+Ack_Properties [in hydras.MoreAck.Ack]
+Adaptation [in additions.Compatibility]
+Alt.well_founded [in hydras.Epsilon0.Epsilon0rpo]
+AP_closed [in hydras.Schutte.AP]
+AP_Unbounded.ksi_fixed [in hydras.Schutte.AP]
+AP_Unbounded [in hydras.Schutte.AP]
+Arith_lemmas [in hydras.Prelude.More_Arith]
+A_def [in hydras.OrdinalNotations.ON_Omega2]
+A_given [in hydras.solutions_exercises.MultisetWf]
+

B

+BadExample [in hydras.MoreAck.BadSubst]
+Bad.CounterExample [in additions.FirstSteps]
+bare_syntax [in hydras.MoreAck.LNN_Examples]
+battle_length_notPR [in hydras.Hydra.Hydra_Theorems]
+Battle_length [in hydras.Hydra.Battle_length]
+binary_power_proof [in additions.Addition_Chains]
+Bounded [in hydras.Hydra.Epsilon0_Needed_Generic]
+building_ordering_function_by_induction [in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1.beta_fixed [in hydras.Schutte.Ordering_Functions]
+building_ordering_function_1 [in hydras.Schutte.Ordering_Functions]
+

C

+Canon_examples [in hydras.Epsilon0.Canon]
+CantorOrdinal.AddLocalNotation [in gaia_hydras.nfwfgaia]
+Check_Proof [in hydras.Ackermann.checkPrf]
+close [in hydras.Ackermann.codePA]
+Code_Free_Vars [in hydras.Ackermann.codeFreeVar]
+Code_Substitute_Term [in hydras.Ackermann.codeSubTerm]
+Code_Substitute_Formula [in hydras.Ackermann.codeSubFormula]
+Code_Term_Formula_Proof.codeTermFormDef [in hydras.Ackermann.code]
+Code_Term_Formula_Proof.LcodeDef [in hydras.Ackermann.code]
+Code_Term_Formula_Proof [in hydras.Ackermann.code]
+code_nat_list [in hydras.Ackermann.cPair]
+code_SysPrf.LNN [in Goedel.codeSysPrf]
+code_SysPrf [in Goedel.codeSysPrf]
+Code_PA [in hydras.Ackermann.codePA]
+Comparable [in hydras.Prelude.Comparable]
+compose2Examples [in hydras.MoreAck.PrimRecExamples]
+Composition [in hydras.MoreAck.PrimRecExamples]
+CompositionProofs [in additions.AM]
+CompositionProofs.App [in additions.AM]
+CompositionProofs.FFK [in additions.AM]
+CompositionProofs.FK [in additions.AM]
+CompositionProofs.KFK [in additions.AM]
+Constant_to_standard_Proof [in hydras.Epsilon0.Paths]
+Correctness [in hydras.Ackermann.fol]
+Countable [in hydras.Schutte.Countable]
+Countable.Countable_bijection [in hydras.Schutte.Countable]
+Countable.Countable_seq_range [in hydras.Schutte.Countable]
+Countable.Definitions [in hydras.Schutte.Countable]
+Counter_Example [in hydras.solutions_exercises.MultisetWf]
+CPair_Order [in hydras.Ackermann.cPair]
+CPair_projections [in hydras.Ackermann.cPair]
+CPair_Injectivity [in hydras.Ackermann.cPair]
+Cter_example [in hydras.solutions_exercises.schutte_cnf_counter_example]
+

D

+Deduction_Theorem [in hydras.Ackermann.Deduction]
+Definitions [in additions.FirstSteps]
+Definitions [in hydras.OrdinalNotations.ON_Generic]
+Definitions [in hydras.Prelude.Simple_LexProd]
+Defs [in hydras.Prelude.MoreOrders]
+Defs [in hydras.OrdinalNotations.ON_mult]
+Defs [in hydras.OrdinalNotations.ON_plus]
+depth_rec_demo [in hydras.MoreAck.FolExamples]
+Direct_proof.well_foundedness_proof.First_attempt [in hydras.Epsilon0.T1]
+Direct_proof.well_foundedness_proof [in hydras.Epsilon0.T1]
+dom_AckNotPR [in hydras.MoreAck.AckNotPR]
+DS_iota.Proof_case_6 [in hydras.Hydra.O2H]
+DS_iota.Proof_case_5 [in hydras.Hydra.O2H]
+DS_iota.Proof_case_4 [in hydras.Hydra.O2H]
+DS_iota [in hydras.Hydra.O2H]
+

E

+Equality_Logic_Rules [in hydras.Ackermann.folLogic3]
+Equations_for_addition.case3 [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.case2 [in hydras.Schutte.Correctness_E0]
+Equations_for_addition.case1 [in hydras.Schutte.Correctness_E0]
+Equations_for_addition [in hydras.Schutte.Correctness_E0]
+Equivalence [in additions.Compatibility]
+essai [in hydras.Epsilon0.T1]
+evalList [in hydras.MoreAck.AckNotPR]
+Examples [in hydras.Ackermann.LNT]
+Examples [in hydras.MoreAck.LNN_Examples]
+Exs [in hydras.MoreAck.PrimRecExamples]
+

F

+First_Order_Logic.Formula_Depth_Induction [in hydras.Ackermann.fol]
+First_Order_Logic.Formula_Decidability [in hydras.Ackermann.fol]
+First_Order_Logic [in hydras.Ackermann.fol]
+Fix [in hydras.Prelude.Restriction]
+Fol_Properties.Substitution.Substitution_Properties [in hydras.Ackermann.folProp]
+Fol_Properties.Substitution.Extensions [in hydras.Ackermann.folProp]
+Fol_Properties.Substitution [in hydras.Ackermann.folProp]
+Fol_Properties.Free_Variables [in hydras.Ackermann.folProp]
+Fol_Properties [in hydras.Ackermann.folProp]
+Forall2_right_induction [in hydras.Prelude.MoreLists]
+Free_Variables [in hydras.Ackermann.LNN]
+Free_Variables [in hydras.Ackermann.LNT]
+F_monotony_l.case_lt [in hydras.Epsilon0.F_alpha]
+F_monotony_l.case_eq [in hydras.Epsilon0.F_alpha]
+F_monotony_l [in hydras.Epsilon0.F_alpha]
+F_alpha_notPR.case_lt [in hydras.Epsilon0.F_omega]
+F_alpha_notPR [in hydras.Epsilon0.F_omega]
+F_omega_notPR [in hydras.Epsilon0.F_omega]
+

G

+Gamma [in additions.AM]
+Gamma [in additions.Euclidean_Chains]
+Gamma.All_OK [in additions.AM]
+General_Relations.elagage.to_nat_elagage [in hydras.Schutte.GRelations]
+General_Relations.elagage [in hydras.Schutte.GRelations]
+General_Relations.surjection2injection [in hydras.Schutte.GRelations]
+General_Relations.injection2surjection [in hydras.Schutte.GRelations]
+General_Relations.Definitions [in hydras.Schutte.GRelations]
+General_Relations [in hydras.Schutte.GRelations]
+Generic [in hydras.Prelude.Merge_Sort]
+Generic.Merging [in hydras.Prelude.Merge_Sort]
+Generic.Merging.Correctness [in hydras.Prelude.Merge_Sort]
+Generic.Merging.Correctness.merge_sort [in hydras.Prelude.Merge_Sort]
+Generic.Splitting [in hydras.Prelude.Merge_Sort]
+Goedel's_1st_Incompleteness [in Goedel.goedel1]
+Goedel's_2nd_Incompleteness.Goedel1PA [in Goedel.goedel2]
+Goedel's_2nd_Incompleteness [in Goedel.goedel2]
+

H

+Hypos [in hydras.Prelude.First_toggle]
+H'_F [in hydras.Epsilon0.F_alpha]
+H'_omega_cube_3 [in hydras.Epsilon0.Hprime]
+H'_cons [in hydras.Epsilon0.Hprime]
+

I

+Ignore_Params [in hydras.Ackermann.primRec]
+ImpossibilityProof [in gaia_hydras.GHydra]
+Impossibility_Proof [in hydras.Hydra.Epsilon0_Needed_Free]
+Impossibility_Proof [in hydras.MoreAck.AckNotPR]
+Impossibility_Proof [in hydras.Hydra.Epsilon0_Needed_Std]
+Impossibility_Proof [in hydras.Hydra.Omega2_Small]
+Impossibility_Proof [in hydras.Hydra.Omega_Small]
+Impossibility1 [in hydras.solutions_exercises.MultisetWf]
+Inclusion_ij [in hydras.OrdinalNotations.ON_Finite]
+inductive_step [in hydras.Epsilon0.F_omega]
+Inverse_Image_transp [in additions.Wf_transparent]
+inversion_of_bijection [in hydras.Schutte.PartialFun]
+

K

+KFK_proof [in additions.Euclidean_Chains]
+KP [in hydras.Hydra.KP_example]
+

L

+lemmas_on_length [in hydras.Gamma0.Gamma0]
+Lemma_4_4_Proof [in hydras.Epsilon0.Large_Sets]
+Lemma_4_3_Proof [in hydras.Epsilon0.Paths]
+LExamples [in hydras.Ackermann.fol]
+lexprod [in hydras.Prelude.DecPreOrder_Instances]
+lim [in hydras.Epsilon0.Large_Sets]
+List_Remove [in hydras.Ackermann.ListExt]
+LNN_FixPoint [in Goedel.fixPoint]
+LNT_FixPoint [in Goedel.fixPoint]
+Logic [in hydras.Ackermann.LNN]
+Logic [in hydras.Ackermann.LNT]
+Logic_Rules.Other_Rules [in hydras.Ackermann.folLogic]
+Logic_Rules.Not_Rules [in hydras.Ackermann.folLogic]
+Logic_Rules [in hydras.Ackermann.folLogic]
+lt_incl_rpo [in hydras.Gamma0.Gamma0]
+L_correct_proof [in hydras.Epsilon0.L_alpha]
+

M

+Make.DoubleRecursion [in hydras.rpo.term]
+Make.Recursion [in hydras.rpo.term]
+Model_Theory.Consistent_Theory [in hydras.Ackermann.model]
+Model_Theory [in hydras.Ackermann.model]
+MoreOrderType [in gaia_hydras.onType]
+MoreOrderType.Succ_no_limit [in gaia_hydras.onType]
+More_Logic_Rules [in hydras.Ackermann.folLogic2]
+M_given.Power_of_op [in additions.Pow_variant]
+M_given.About_power [in additions.Pow_variant]
+M_given [in additions.Pow_variant]
+M_given.Power_of_op [in additions.Pow]
+M_given [in additions.Pow]
+M2_def [in additions.Monoid_instances]
+M2.Definitions [in additions.Naive]
+M2.M2_Definitions [in additions.FirstSteps]
+

N

+Nmodulo [in additions.Monoid_instances]
+NN [in hydras.Ackermann.NN]
+non_optimality_proof [in additions.Addition_Chains]
+NotIncl [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+N_mod.m_fixed [in additions.Naive]
+

O

+OA_given [in hydras.OrdinalNotations.ON_O]
+ONDef.Exports.Lemmas [in gaia_hydras.onType]
+ONDef.Packing [in gaia_hydras.onType]
+ONEpsilon0Def [in gaia_hydras.onType]
+onFiniteDef [in gaia_hydras.onType]
+onOmegaDef [in gaia_hydras.onType]
+on_length [in hydras.Gamma0.T2]
+On_alpha.S2 [in hydras.solutions_exercises.Limit_Infinity]
+On_alpha.S1 [in hydras.solutions_exercises.Limit_Infinity]
+On_alpha [in hydras.solutions_exercises.Limit_Infinity]
+On_Iota [in hydras.Schutte.MoreEpsilonIota]
+ordering_function_unicity [in hydras.Schutte.Ordering_Functions]
+ordering_function_unicity_1 [in hydras.Schutte.Ordering_Functions]
+

P

+PA [in hydras.Ackermann.PA]
+phi_to_psi [in hydras.Gamma0.Gamma0]
+phi0_mult [in hydras.Epsilon0.Large_Sets]
+Primitive_Recursive_Representable [in Goedel.PRrepresentable]
+Primitive_recursion [in hydras.MoreAck.PrimRecExamples]
+ProofH [in hydras.Ackermann.folProof]
+ProofH.Example [in hydras.Ackermann.folProof]
+ProofOfLexwf [in hydras.solutions_exercises.MultisetWf]
+Proofs_of_unicity.Proofs.S2 [in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs.S1 [in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity.Proofs [in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_unicity [in hydras.solutions_exercises.predSuccUnicity]
+Proofs_of_lt_succ_le.Proofs.S1 [in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le.Proofs [in hydras.solutions_exercises.lt_succ_le]
+Proofs_of_lt_succ_le [in hydras.solutions_exercises.lt_succ_le]
+Proof_of_dist [in hydras.Epsilon0.T1]
+Proof_of_mult_nf.Induction [in hydras.Epsilon0.T1]
+Proof_of_mult_nf [in hydras.Epsilon0.T1]
+Proof_of_MinIsPR [in hydras.solutions_exercises.MinPR2]
+Proof_of_oplus_lt1 [in hydras.Epsilon0.Hessenberg]
+Proof_of_oplus_assoc [in hydras.Epsilon0.Hessenberg]
+Proof_of_oplus_comm [in hydras.Epsilon0.Hessenberg]
+Proof_of_plus_nf [in hydras.Epsilon0.Hessenberg]
+Proof_of_Ackn_PR.S_step [in hydras.MoreAck.AckNotPR]
+Proof_of_Ackn_PR [in hydras.MoreAck.AckNotPR]
+Proof_of_4_5_2 [in hydras.Epsilon0.Large_Sets]
+Proof_of_Lemma5.Alpha_positive.closedness [in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive.Proof_unbounded [in hydras.Schutte.Critical]
+Proof_of_Lemma5.Alpha_positive [in hydras.Schutte.Critical]
+Proof_of_Lemma5 [in hydras.Schutte.Critical]
+Proof_of_nested_Ack_bound [in hydras.MoreAck.Ack]
+Proof_of_mult_ref [in gaia_hydras.T1Bridge]
+Proof_of_FibIsPR [in hydras.solutions_exercises.FibonacciPR]
+Proof_of_MinIsPR [in hydras.solutions_exercises.MinPR]
+Proof_of_H'_mono_l.Limit_case [in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l.Succ_case [in hydras.Epsilon0.Hprime]
+Proof_of_H'_mono_l [in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.alpha_limit [in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction.alpha_Succ [in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties.The_induction [in hydras.Epsilon0.Hprime]
+Proof_of_Abstract_Properties [in hydras.Epsilon0.Hprime]
+proof_of_associativity [in hydras.Schutte.Addition]
+Proof_of_lt_not_wf.seq_intro [in hydras.solutions_exercises.T1_ltNotWf]
+Proof_of_lt_not_wf [in hydras.solutions_exercises.T1_ltNotWf]
+Properties [in hydras.Epsilon0.F_alpha]
+Properties_of_f_alpha.The_induction [in hydras.Epsilon0.F_alpha]
+Properties_of_f_alpha [in hydras.Epsilon0.F_alpha]
+Properties.The_induction.alpha_limit [in hydras.Epsilon0.F_alpha]
+Properties.The_induction.alpha_Succ [in hydras.Epsilon0.F_alpha]
+Properties.The_induction [in hydras.Epsilon0.F_alpha]
+

R

+Refinement_proof [in additions.Addition_Chains]
+Replacement [in hydras.Ackermann.folReplace]
+RepresentableExpressible [in hydras.Ackermann.expressible]
+restricted_recursion [in hydras.Prelude.Restriction]
+restricted_recursion [in gaia_hydras.nfwfgaia]
+Rosser's_Incompleteness [in Goedel.rosserPA]
+Rosser's_Incompleteness [in Goedel.rosser]
+R_given [in hydras.Prelude.Sort_spec]
+

S

+Semantics [in additions.AM]
+Sequences [in gaia_hydras.nfwfgaia]
+sqrtIsPR [in hydras.solutions_exercises.isqrt]
+sqrtIsPR.Proof_isqrt [in hydras.solutions_exercises.isqrt]
+step [in hydras.Epsilon0.F_omega]
+Strong_Recursion [in hydras.Ackermann.cPair]
+SubAllVars [in hydras.Ackermann.subAll]
+SubAllVars.subAllCloseFrom [in hydras.Ackermann.subAll]
+SubON_properties [in hydras.OrdinalNotations.ON_Generic]
+Substitution_Properties [in hydras.Ackermann.subProp]
+succ [in hydras.Epsilon0.Large_Sets]
+S1 [in hydras.solutions_exercises.is_F_monotonous]
+S1 [in hydras.solutions_exercises.F_3]
+S1 [in additions.Addition_Chains]
+S1.Limit [in hydras.solutions_exercises.F_3]
+S1.Limit.S3 [in hydras.solutions_exercises.F_3]
+S1.Successor [in hydras.solutions_exercises.F_3]
+S1.Successor.S2 [in hydras.solutions_exercises.F_3]
+S2 [in additions.Addition_Chains]
+S256 [in additions.Monoid_instances]
+

T

+Term.DoubleRecursion [in hydras.rpo.term]
+Term.Recursion [in hydras.rpo.term]
+the_context.About_WO [in hydras.Schutte.Well_Orders]
+the_context [in hydras.Schutte.Well_Orders]
+Th13_5.verso.U_fixed [in hydras.Schutte.Ordering_Functions]
+Th13_5.verso [in hydras.Schutte.Ordering_Functions]
+Th13_5.recto.M_fixed [in hydras.Schutte.Ordering_Functions]
+Th13_5.recto [in hydras.Schutte.Ordering_Functions]
+Th13_5 [in hydras.Schutte.Ordering_Functions]
+Toy.Drinkers_theorem [in hydras.MoreAck.FolExamples]
+Toy.OnSubstF [in hydras.MoreAck.FolExamples]
+Toy.PeirceProof [in hydras.MoreAck.FolExamples]
+Toy.PrimedSymbols [in hydras.MoreAck.FolExamples]
+Toy.ProofOfEx3 [in hydras.MoreAck.FolExamples]
+Translate_Proof [in hydras.Ackermann.LNN2LNT]
+trans_proof [in hydras.Gamma0.Gamma0]
+

V

+Variants [in hydras.Prelude.WfVariant]
+

W

+Well_Formed_Term.Well_Formed_Formula [in hydras.Ackermann.wellFormed]
+Well_Formed_Term [in hydras.Ackermann.wellFormed]
+well_founded [in hydras.Gamma0.Gamma0]
+with_matrices [in additions.fib]
+

+

Instance Index

+

A

+Ackn_IsPR [in hydras.MoreAck.AckNotPR]
+Alt.addIsPR [in hydras.MoreAck.PrimRecExamples]
+Alt.addIsPR' [in hydras.MoreAck.PrimRecExamples]
+Alt.const0_NIsPR [in hydras.MoreAck.PrimRecExamples]
+Alt.pi2_5IsPR [in hydras.MoreAck.PrimRecExamples]
+Alt.predIsPR [in hydras.MoreAck.PrimRecExamples]
+Alt.succIsPR [in hydras.MoreAck.PrimRecExamples]
+Alt.zeroIsPR [in hydras.MoreAck.PrimRecExamples]
+

B

+Bad3.Fcompose_proper [in additions.Euclidean_Chains]
+Binary_strat [in additions.BinaryStrat]
+boddIsPR [in hydras.Ackermann.primRec]
+bool_and_binop [in additions.Monoid_def]
+boundComputationIsPR [in hydras.Ackermann.codeSubFormula]
+boundSearchIsPR [in hydras.Ackermann.primRec]
+bVar [in gaia_hydras.GHydra]
+

C

+callIsPR [in hydras.Ackermann.cPair]
+checkPrfAXMIsPR [in hydras.Ackermann.checkPrf]
+checkPrfCPIsPR [in hydras.Ackermann.checkPrf]
+checkPrfEQ1IsPR [in hydras.Ackermann.checkPrf]
+checkPrfEQ2IsPR [in hydras.Ackermann.checkPrf]
+checkPrfEQ3IsPR [in hydras.Ackermann.checkPrf]
+checkPrfEQ4IsPR [in hydras.Ackermann.checkPrf]
+checkPrfEQ5IsPR [in hydras.Ackermann.checkPrf]
+checkPrfFA1IsPR [in hydras.Ackermann.checkPrf]
+checkPrfFA2IsPR [in hydras.Ackermann.checkPrf]
+checkPrfFA3IsPR [in hydras.Ackermann.checkPrf]
+checkPrfGENIsPR [in hydras.Ackermann.checkPrf]
+checkPrfHelpIsPR [in hydras.Ackermann.checkPrf]
+checkPrfIMP1IsPR [in hydras.Ackermann.checkPrf]
+checkPrfIMP2IsPR [in hydras.Ackermann.checkPrf]
+checkPrfIsPR [in hydras.Ackermann.checkPrf]
+checkPrfMPIsPR [in hydras.Ackermann.checkPrf]
+checkTraceIsPR [in hydras.Ackermann.codeSubFormula]
+codeAndIsPR [in hydras.Ackermann.checkPrf]
+codeAppIsPR [in hydras.Ackermann.codeList]
+codeArityLNNRIsPR [in hydras.Ackermann.Languages]
+codeArityLNTFIsPR [in hydras.Ackermann.Languages]
+codeArityLNTRIsPR [in hydras.Ackermann.Languages]
+codeAxmEqHelpIsPR [in hydras.Ackermann.checkPrf]
+codeCloseIsPR [in hydras.Ackermann.codePA]
+codeCloseListIsPR [in hydras.Ackermann.codePA]
+codeFreeVarFormulaIsPR [in hydras.Ackermann.codeFreeVar]
+codeFreeVarListFormulaIsPR [in hydras.Ackermann.codeFreeVar]
+codeFreeVarTermsIsPR [in hydras.Ackermann.codeFreeVar]
+codeFreeVarTermTermsIsPR [in hydras.Ackermann.codeFreeVar]
+codeIffIsPR [in hydras.Ackermann.checkPrf]
+codeImpIsPR [in hydras.Ackermann.prLogic]
+codeInductionSchemaIsPR [in hydras.Ackermann.codePA]
+codeInIsPR [in hydras.Ackermann.codeList]
+codeLengthIsPR [in hydras.Ackermann.codeList]
+codeListRemoveIsPR [in hydras.Ackermann.codeList]
+codeNatToTermIsPR [in hydras.Ackermann.codeNatToTerm]
+codeNewVarIsPR [in hydras.Ackermann.codeSubFormula]
+codeNoDupIsPR [in hydras.Ackermann.codeList]
+codeNotIsPR [in hydras.Ackermann.prLogic]
+codeNthIsPR [in hydras.Ackermann.cPair]
+codeNVars1IsPR [in hydras.Ackermann.checkPrf]
+codeNVars2IsPR [in hydras.Ackermann.checkPrf]
+codeOpenIsPR [in hydras.Ackermann.codePA]
+codeOrIsPR [in hydras.Ackermann.checkPrf]
+codePAIsPR [in hydras.Ackermann.codePA]
+codeSubFormulaIsPR [in hydras.Ackermann.codeSubFormula]
+codeSubTermIsPR [in hydras.Ackermann.codeSubTerm]
+codeSubTermTermsIsPR [in hydras.Ackermann.codeSubTerm]
+comp [in hydras.OrdinalNotations.ON_Finite]
+compare_T1 [in hydras.Epsilon0.T1]
+compare_omega2 [in hydras.OrdinalNotations.ON_Omega2]
+compare_O [in hydras.OrdinalNotations.ON_O]
+compare_nat_nat [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+compare_mult [in hydras.OrdinalNotations.ON_mult]
+compare_plus [in hydras.OrdinalNotations.ON_plus]
+compare_fin [in hydras.OrdinalNotations.ON_Finite]
+compare_E0 [in hydras.Epsilon0.E0]
+compare_nat [in hydras.OrdinalNotations.ON_Omega]
+compare_T2 [in hydras.Gamma0.Gamma0]
+comparison_eq_dec [in hydras.Prelude.DecPreOrder]
+compose1_NIsPR [in hydras.Ackermann.primRec]
+compose1_3IsPR [in hydras.Ackermann.primRec]
+compose1_2IsPR [in hydras.Ackermann.primRec]
+compose1_1IsPR [in hydras.Ackermann.primRec]
+compose2IsPR [in hydras.Ackermann.primRec]
+compose2_4IsPR [in hydras.Ackermann.primRec]
+compose2_3IsPR [in hydras.Ackermann.primRec]
+compose2_2IsPR [in hydras.Ackermann.primRec]
+compose2_1IsPR [in hydras.Ackermann.primRec]
+compose3_3IsPR [in hydras.Ackermann.primRec]
+compose3_2IsPR [in hydras.Ackermann.primRec]
+compose3_1IsPR [in hydras.Ackermann.primRec]
+compose4_3IsPR [in hydras.Ackermann.primRec]
+compose4_2IsPR [in hydras.Ackermann.primRec]
+comp_equiv_equivalence [in additions.Euclidean_Chains]
+Comp_equiv [in additions.Euclidean_Chains]
+Cons [in hydras.Epsilon0.E0]
+const0_NIsPR [in hydras.Ackermann.primRec]
+const1_NIsPR [in hydras.Ackermann.primRec]
+cPairIsPR [in hydras.Ackermann.cPair]
+cPairPi1IsPR [in hydras.Ackermann.cPair]
+cPairPi2IsPR [in hydras.Ackermann.cPair]
+cTripleIsPR [in hydras.Ackermann.cPair]
+cTriplePi1IsPR [in hydras.Ackermann.cPair]
+cTriplePi2IsPR [in hydras.Ackermann.cPair]
+cTriplePi3IsPR [in hydras.Ackermann.cPair]
+

D

+Demo.nat_mult_op [in additions.Monoid_def]
+Dicho_strat [in additions.Dichotomy]
+div2IsPR [in hydras.Ackermann.cPair]
+doubleIsPR [in hydras.MoreAck.PrimRecExamples]
+

E

+Eop_proper [in additions.Pow_variant]
+Eop_proper [in additions.Pow]
+Epsilon0 [in gaia_hydras.T1Bridge]
+Epsilon0 [in hydras.Epsilon0.E0]
+epsilon0_dec [in hydras.Epsilon0.T1]
+epsilon0_pre_order [in hydras.Epsilon0.T1]
+Epsilon0_correct [in hydras.Schutte.Correctness_E0]
+eqIsPR [in hydras.Ackermann.primRec]
+equiv_equiv [in hydras.Prelude.Sort_spec]
+equiv_equiv [in hydras.Prelude.DecPreOrder]
+Equiv_Trans [in additions.Monoid_def]
+Equiv_Sym [in additions.Monoid_def]
+Equiv_Refl [in additions.Monoid_def]
+Equiv_Equiv [in additions.Monoid_def]
+eq_equiv [in additions.Monoid_def]
+evalStrongRecIsPR [in hydras.Ackermann.cPair]
+exec_Proper [in additions.AM]
+expIsPR [in hydras.solutions_exercises.MorePRExamples]
+exp_alt_PR [in hydras.solutions_exercises.MorePRExamples]
+E0add [in hydras.Epsilon0.E0]
+E0compare [in gaia_hydras.T1Bridge]
+E0fin [in hydras.Epsilon0.E0]
+E0finS [in hydras.Epsilon0.E0]
+E0mul [in hydras.Epsilon0.E0]
+E0one [in hydras.Epsilon0.E0]
+E0zero [in hydras.Epsilon0.E0]
+E0_comp [in gaia_hydras.T1Bridge]
+E0_sto [in gaia_hydras.T1Bridge]
+E0_comp [in hydras.Epsilon0.E0]
+E0_sto [in hydras.Epsilon0.E0]
+E0_phi0 [in hydras.Epsilon0.E0]
+E0_succ [in hydras.Epsilon0.E0]
+E0_omega [in hydras.Epsilon0.E0]
+

F

+factIsPR [in hydras.solutions_exercises.MorePRExamples]
+fact_altIsPR [in hydras.solutions_exercises.MorePRExamples]
+Fcompose_proper [in additions.Euclidean_Chains]
+Fexp2_proper [in additions.Euclidean_Chains]
+Fexp2_nat_proper [in additions.Euclidean_Chains]
+FFK_proper [in additions.Euclidean_Chains]
+fibIsPR [in hydras.solutions_exercises.FibonacciPR]
+fibPRIsPR [in hydras.solutions_exercises.FibonacciPR]
+fib_altIsPR [in hydras.solutions_exercises.FibonacciPR]
+filter001IsPR [in hydras.Ackermann.primRec]
+filter0011IsPR [in hydras.Ackermann.primRec]
+filter01IsPR [in hydras.Ackermann.primRec]
+filter010IsPR [in hydras.Ackermann.primRec]
+filter011IsPR [in hydras.Ackermann.primRec]
+filter10IsPR [in hydras.Ackermann.primRec]
+filter100IsPR [in hydras.Ackermann.primRec]
+filter1000IsPR [in hydras.Ackermann.primRec]
+filter101IsPR [in hydras.Ackermann.primRec]
+filter1011IsPR [in hydras.Ackermann.primRec]
+filter110IsPR [in hydras.Ackermann.primRec]
+filter1100IsPR [in hydras.Ackermann.primRec]
+FinOrd [in hydras.OrdinalNotations.ON_Finite]
+finord_ON [in gaia_hydras.ON_gfinite]
+finord_comp [in gaia_hydras.ON_gfinite]
+finord_sto [in gaia_hydras.ON_gfinite]
+finord_compare [in gaia_hydras.ON_gfinite]
+FinOrd_Omega [in hydras.OrdinalNotations.ON_Omega]
+FK_proper [in additions.Euclidean_Chains]
+free [in hydras.Hydra.Hydra_Definitions]
+F_n_PR [in hydras.Epsilon0.F_omega]
+F_0_isPR [in hydras.Epsilon0.F_omega]
+F_incl_ij [in hydras.OrdinalNotations.ON_Finite]
+F1_proper [in additions.Euclidean_Chains]
+F2_proper [in additions.Euclidean_Chains]
+F3_proper [in additions.Euclidean_Chains]
+

G

+gtIsPR [in hydras.Ackermann.primRec]
+G0.compare_G0 [in hydras.Gamma0.Gamma0]
+G0.Finite [in hydras.Gamma0.Gamma0]
+G0.Gamma0 [in hydras.Gamma0.Gamma0]
+G0.Gamma0_comp [in hydras.Gamma0.Gamma0]
+G0.lt_sto [in hydras.Gamma0.Gamma0]
+G0.Omega [in hydras.Gamma0.Gamma0]
+G0.Phi [in hydras.Gamma0.Gamma0]
+G0.Plus [in hydras.Gamma0.Gamma0]
+G0.zero [in hydras.Gamma0.Gamma0]
+

H

+height_var [in hydras.Hydra.Omega_Small]
+hVar [in gaia_hydras.GHydra]
+HVariant [in hydras.Hydra.Hydra_Termination]
+HVariant_0 [in hydras.Hydra.Hydra_Termination]
+

I

+idIsPR [in hydras.Ackermann.primRec]
+If2IsPR [in hydras.solutions_exercises.MinPR2]
+ignoreParamsIsPR [in hydras.Ackermann.primRec]
+Incl [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+indIsPR [in hydras.Ackermann.primRec]
+ind1ParamIsPR [in hydras.Ackermann.primRec]
+ind2ParamIsPR [in hydras.Ackermann.primRec]
+inhNat [in hydras.Schutte.MoreEpsilonIota]
+Inh_Ord_Ord [in hydras.Schutte.Schutte_basics]
+Inh_OSets [in hydras.Schutte.Schutte_basics]
+InH_Ord [in hydras.Schutte.Schutte_basics]
+Int63mult [in additions.Monoid_instances]
+int63_mult_op [in additions.Monoid_instances]
+Inverse_fun [in hydras.Prelude.DecPreOrder_Instances]
+isPRextEqual [in hydras.Ackermann.primRec]
+isPRTrans [in hydras.Ackermann.primRec]
+iSPR_Ack_Sn [in hydras.MoreAck.AckNotPR]
+issqrtIsPR [in hydras.solutions_exercises.isqrt]
+iterateIsPR [in hydras.Ackermann.primRec]
+

K

+KFF_proper [in additions.Euclidean_Chains]
+K2F_proper [in additions.Euclidean_Chains]
+k7_3_proper [in additions.Euclidean_Chains]
+

L

+LcodeLNN [in hydras.Ackermann.codeNatToTerm]
+LcodeLNT [in hydras.Ackermann.codeNatToTerm]
+leIsPR [in hydras.Ackermann.primRec]
+leq_trans [in hydras.Prelude.MoreOrders]
+lex_prod_dec [in hydras.Prelude.DecPreOrder_Instances]
+Lex_nat_wf [in hydras.MoreAck.Ack]
+Le_trans [in hydras.Schutte.Well_Orders]
+List_length [in hydras.Prelude.DecPreOrder_Instances]
+LO.compare_oo [in hydras.OrdinalNotations.OmegaOmega]
+LO.lo_comparable [in hydras.OrdinalNotations.OmegaOmega]
+LO.lo_strorder [in hydras.OrdinalNotations.OmegaOmega]
+LO.plus_assoc [in hydras.OrdinalNotations.OmegaOmega]
+ltIsPR [in hydras.Ackermann.primRec]
+LT_St [in hydras.Epsilon0.T1]
+lt_dec [in hydras.Epsilon0.T1]
+lt_transitive [in hydras.Prelude.DecPreOrder]
+lt_strorder [in hydras.OrdinalNotations.ON_mult]
+lt_strorder [in hydras.OrdinalNotations.ON_plus]
+L_3_plus_omega [in hydras.OrdinalNotations.Example_3PlusOmega]
+

M

+maxIsPR [in hydras.Ackermann.primRec]
+max_assoc [in hydras.Prelude.Comparable]
+minIsPR [in hydras.solutions_exercises.MinPR2]
+minIsPR [in hydras.solutions_exercises.MinPR]
+minPR_PR [in hydras.solutions_exercises.MinPR2]
+minPR_PR [in hydras.solutions_exercises.MinPR]
+minusIndIsPR [in hydras.Ackermann.primRec]
+minusIsPR [in hydras.Ackermann.primRec]
+min_assoc [in hydras.Prelude.Comparable]
+mIsPR [in hydras.Hydra.Hydra_Theorems]
+mod_Equiv [in additions.Monoid_instances]
+mod_op [in additions.Monoid_instances]
+mod_equiv [in additions.Monoid_instances]
+Monoid_EMonoid [in additions.Monoid_def]
+multIndIsPR [in hydras.Ackermann.primRec]
+multIsPR [in hydras.Ackermann.primRec]
+mult_comp [in hydras.OrdinalNotations.ON_mult]
+mult_mod_proper [in additions.Monoid_instances]
+Mult_i [in hydras.Epsilon0.E0]
+Mul2 [in additions.Fib2]
+M2_Monoid [in additions.Monoid_instances]
+M2_op [in additions.Monoid_instances]
+

N

+Natmult [in additions.Monoid_instances]
+Natplus [in additions.Monoid_instances]
+Nat_le_TO [in hydras.Prelude.DecPreOrder_Instances]
+Nat_le_dec [in hydras.Prelude.DecPreOrder_Instances]
+nat_plus_op [in additions.Monoid_instances]
+nat_mult_op [in additions.Monoid_instances]
+neqIsPR [in hydras.Ackermann.primRec]
+Nmod_Monoid [in additions.Monoid_instances]
+NMult [in additions.Monoid_instances]
+notZeroIsPR [in hydras.Ackermann.primRec]
+NPlus [in additions.Monoid_instances]
+N_plus_op [in additions.Monoid_instances]
+N_mult_op [in additions.Monoid_instances]
+

O

+Olt [in hydras.Epsilon0.F_alpha]
+Olt [in hydras.Epsilon0.L_alpha]
+Olt [in hydras.Epsilon0.Hprime]
+Omega [in hydras.OrdinalNotations.ON_Omega]
+Omega_plus_Omega [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Omega_2 [in hydras.Epsilon0.E0]
+Omega_term [in hydras.Epsilon0.E0]
+omega_ok [in hydras.OrdinalNotations.ON_Omega]
+Omega_comp [in hydras.OrdinalNotations.ON_Omega]
+Omega2 [in hydras.OrdinalNotations.ON_Omega2]
+ON_O [in hydras.OrdinalNotations.ON_O]
+ON_O_comp [in hydras.OrdinalNotations.ON_O]
+ON_mult [in hydras.OrdinalNotations.ON_mult]
+ON_plus [in hydras.OrdinalNotations.ON_plus]
+OO.compare_OO [in hydras.OrdinalNotations.OmegaOmega]
+OO.embed [in hydras.OrdinalNotations.OmegaOmega]
+OO.Fin [in hydras.OrdinalNotations.OmegaOmega]
+OO.mult [in hydras.OrdinalNotations.OmegaOmega]
+OO.ON_OO [in hydras.OrdinalNotations.OmegaOmega]
+OO.OO_comp [in hydras.OrdinalNotations.OmegaOmega]
+OO.oo_str [in hydras.OrdinalNotations.OmegaOmega]
+OO.phi0 [in hydras.OrdinalNotations.OmegaOmega]
+OO.plus [in hydras.OrdinalNotations.OmegaOmega]
+OO.Succ [in hydras.OrdinalNotations.OmegaOmega]
+OO.Zero [in hydras.OrdinalNotations.OmegaOmega]
+OO._Omega [in hydras.OrdinalNotations.OmegaOmega]
+Oplus [in hydras.Epsilon0.E0]
+Oplus_assoc [in hydras.Epsilon0.E0]
+

P

+pi1_4IsPR [in hydras.Ackermann.primRec]
+pi1_3IsPR [in hydras.Ackermann.primRec]
+pi1_2IsPR [in hydras.Ackermann.primRec]
+pi2_4IsPR [in hydras.Ackermann.primRec]
+pi2_3IsPR [in hydras.Ackermann.primRec]
+pi2_2IsPR [in hydras.Ackermann.primRec]
+pi3_4IsPR [in hydras.Ackermann.primRec]
+pi3_3IsPR [in hydras.Ackermann.primRec]
+pi4_4IsPR [in hydras.Ackermann.primRec]
+plusIndIsPR [in hydras.Ackermann.primRec]
+plusIsPR [in hydras.Ackermann.primRec]
+plus_assoc [in hydras.OrdinalNotations.ON_Omega2]
+plus_comp [in hydras.OrdinalNotations.ON_plus]
+plus_assoc [in hydras.Epsilon0.E0]
+plus_assoc [in hydras.Schutte.Addition]
+PMult [in additions.Monoid_instances]
+Pos_bpow_proper [in additions.Pow]
+power_proper [in additions.Pow_variant]
+power_proper [in additions.Pow]
+pow3IsPR [in hydras.Ackermann.codeSubFormula]
+PO_lex_prod [in hydras.Prelude.DecPreOrder_Instances]
+predIsPR [in hydras.Ackermann.primRec]
+PrO_Included [in hydras.Prelude.DecPreOrder_Instances]
+P_mult_op [in additions.Monoid_instances]
+P_dec [in hydras.Epsilon0.Paths]
+

R

+RelDecision_Inverse_fun [in hydras.Prelude.DecPreOrder_Instances]
+ReplaceFormulaTermIsPR [in hydras.Ackermann.codeSubFormula]
+ReplaceTermsTermIsPR [in hydras.Ackermann.codeSubFormula]
+ReplaceTermTermIsPR [in hydras.Ackermann.codeSubFormula]
+ReplaceTermTermsTermIsPR [in hydras.Ackermann.codeSubFormula]
+result_equiv_equiv [in additions.AM]
+Return_proper [in additions.Euclidean_Chains]
+R01 [in hydras.Epsilon0.F_omega]
+R02 [in hydras.Epsilon0.F_omega]
+R03 [in hydras.Epsilon0.F_omega]
+R2 [in hydras.MoreAck.AckNotPR]
+

S

+searchXYIsPR [in hydras.Ackermann.cPair]
+Stack_equiv_equiv [in additions.AM]
+Stack_equiv_refl [in additions.AM]
+standard [in hydras.Hydra.Hydra_Definitions]
+sto [in hydras.OrdinalNotations.ON_O]
+sto [in hydras.OrdinalNotations.ON_Finite]
+Strict_lex [in hydras.Prelude.Simple_LexProd]
+string_op [in additions.Monoid_def]
+subStarIsPR [in Goedel.fixPoint]
+succIsPR [in hydras.Ackermann.primRec]
+sumToNIsPR [in hydras.Ackermann.cPair]
+swapIsPR [in hydras.Ackermann.primRec]
+switchIsPR [in hydras.Ackermann.primRec]
+

T

+Total_Inverse_fun [in hydras.Prelude.DecPreOrder_Instances]
+Total_Reflexive [in hydras.Prelude.DecPreOrder]
+tower2IsPR [in hydras.solutions_exercises.MorePRExamples]
+tower2_alt_PR [in hydras.solutions_exercises.MorePRExamples]
+To_lex_prod [in hydras.Prelude.DecPreOrder_Instances]
+Trans_lex [in hydras.Prelude.Simple_LexProd]
+Two [in hydras.Epsilon0.E0]
+Two_strat [in additions.BinaryStrat]
+T1addAssoc [in hydras.Epsilon0.T1]
+T1compare [in gaia_hydras.T1Bridge]
+t1_strorder [in hydras.Epsilon0.T1]
+

V

+var [in hydras.Hydra.Hydra_Termination]
+

W

+wellFormedFormulaIsPR [in hydras.Ackermann.wellFormed]
+wellFormedTermIsPR [in hydras.Ackermann.wellFormed]
+wellFormedTermsIsPR [in hydras.Ackermann.wellFormed]
+wellFormedTermTermsIsPR [in hydras.Ackermann.wellFormed]
+WF [in hydras.OrdinalNotations.ON_Omega2]
+WF [in hydras.Epsilon0.F_alpha]
+WO_ord [in hydras.Schutte.Schutte_basics]
+WO_nat [in hydras.Schutte.Well_Orders]
+

Z

+ZMult [in additions.Monoid_instances]
+ZMult_Abelian [in additions.Monoid_instances]
+Z_le_TO [in hydras.Prelude.DecPreOrder_Instances]
+Z_le_dec [in hydras.Prelude.DecPreOrder_Instances]
+Z_mult_op [in additions.Monoid_instances]
+

+

Abbreviation Index

+

A

+Ackermann.Tf [in gaia_hydras.nfwfgaia]
+

B

+battle [in hydras.Hydra.Hydra_Definitions]
+

C

+canonS [in gaia_hydras.GCanon]
+canonSSn [in hydras.Epsilon0.Canon]
+CantorOrdinal.LT [in gaia_hydras.nfwfgaia]
+CantorOrdinal.Tf [in gaia_hydras.nfwfgaia]
+

E

+epsilon0 [in hydras.Gamma0.T2]
+EqDecision [in hydras.Prelude.STDPP_compat]
+exH' [in hydras.Ackermann.LNT]
+E0succb [in hydras.Epsilon0.E0]
+

F

+F [in hydras.Schutte.Schutte_basics]
+fin [in hydras.Epsilon0.T1]
+Fin [in hydras.Epsilon0.E0]
+FolNotations.exH' [in hydras.Ackermann.fol]
+Formula [in hydras.Ackermann.subAll]
+Formula [in hydras.Ackermann.folProp]
+Formula [in hydras.Ackermann.codeSubFormula]
+Formula [in hydras.Ackermann.folLogic3]
+Formulas [in hydras.Ackermann.subAll]
+Formulas [in hydras.Ackermann.folProp]
+Formulas [in hydras.Ackermann.codeSubFormula]
+Formulas [in hydras.Ackermann.folLogic3]
+freeVarFormula [in hydras.Ackermann.folProp]
+freeVarTerm [in hydras.Ackermann.folProp]
+freeVarTerms [in hydras.Ackermann.folProp]
+FS [in hydras.Epsilon0.T1]
+FS [in hydras.Gamma0.T2]
+F_alpha_ge_S [in hydras.Epsilon0.F_alpha]
+

G

+G0.omega [in hydras.Gamma0.Gamma0]
+G0.phi [in hydras.Gamma0.Gamma0]
+G0.phi0 [in hydras.Gamma0.Gamma0]
+

H

+hbounded_transitionS [in gaia_hydras.GPaths]
+hcanon [in gaia_hydras.GCanon]
+hCanon_plus [in gaia_hydras.GPaths]
+hcnf [in gaia_hydras.T1Bridge]
+hcons [in gaia_hydras.T2Bridge]
+hconst_path [in gaia_hydras.GPaths]
+hconst_pathS [in gaia_hydras.GPaths]
+head [in hydras.Hydra.Hydra_Definitions]
+hE0 [in gaia_hydras.T1Bridge]
+hE0fin [in gaia_hydras.T1Bridge]
+hE0is_succ [in gaia_hydras.T1Bridge]
+hE0le [in gaia_hydras.T1Bridge]
+hE0limit [in gaia_hydras.T1Bridge]
+hE0lt [in gaia_hydras.T1Bridge]
+hE0omega [in gaia_hydras.T1Bridge]
+hE0phi0 [in gaia_hydras.T1Bridge]
+hE0zero [in gaia_hydras.T1Bridge]
+hF_ [in gaia_hydras.GF_alpha]
+hgnaw [in gaia_hydras.GPaths]
+hgnawS [in gaia_hydras.GPaths]
+hlarge [in gaia_hydras.GLarge_Sets]
+hlargeS [in gaia_hydras.GLarge_Sets]
+hL_spec [in gaia_hydras.GLarge_Sets]
+hmlarge [in gaia_hydras.GLarge_Sets]
+hmlargeS [in gaia_hydras.GLarge_Sets]
+hoplus [in gaia_hydras.GHessenberg]
+hpath [in gaia_hydras.GPaths]
+hpathS [in gaia_hydras.GPaths]
+hpath_to [in gaia_hydras.GPaths]
+hstandard_gnaw [in gaia_hydras.GPaths]
+htransition [in gaia_hydras.GPaths]
+hT1 [in gaia_hydras.T1Bridge]
+hT2 [in gaia_hydras.T2Bridge]
+hyd [in hydras.Hydra.BigBattle]
+hyd1 [in hydras.Hydra.Hydra_Definitions]
+hyd2 [in hydras.Hydra.Hydra_Definitions]
+hyd3 [in hydras.Hydra.Hydra_Definitions]
+hyd4 [in hydras.Hydra.KP_example]
+hzero [in gaia_hydras.T2Bridge]
+h1 [in hydras.Hydra.BigBattle]
+h2 [in hydras.Hydra.BigBattle]
+h3 [in hydras.Hydra.BigBattle]
+

I

+In_list_remove2 [in hydras.Ackermann.ListExt]
+

L

+le [in hydras.Epsilon0.T1]
+le [in hydras.Prelude.Comparable]
+LE [in gaia_hydras.T1Bridge]
+limitb [in hydras.Epsilon0.T1]
+Limitb [in hydras.Epsilon0.E0]
+LispAbbreviations.caar [in hydras.Ackermann.cPair]
+LispAbbreviations.caddr [in hydras.Ackermann.cPair]
+LispAbbreviations.cadr [in hydras.Ackermann.cPair]
+LispAbbreviations.car [in hydras.Ackermann.cPair]
+LispAbbreviations.cddddr [in hydras.Ackermann.cPair]
+LispAbbreviations.cdddr [in hydras.Ackermann.cPair]
+LispAbbreviations.cddr [in hydras.Ackermann.cPair]
+LispAbbreviations.cdr [in hydras.Ackermann.cPair]
+LO.cons [in hydras.OrdinalNotations.OmegaOmega]
+LO.FS [in hydras.OrdinalNotations.OmegaOmega]
+LO.omega [in hydras.OrdinalNotations.OmegaOmega]
+LO.phi0 [in hydras.OrdinalNotations.OmegaOmega]
+LT [in hydras.solutions_exercises.MultisetWf]
+LT [in gaia_hydras.T1Bridge]
+lt_irrefl [in hydras.Prelude.Comparable]
+lt_trans [in hydras.Prelude.Comparable]
+

M

+mult [in hydras.Epsilon0.T1]
+Mult [in hydras.Epsilon0.E0]
+

N

+NF [in hydras.solutions_exercises.MultisetWf]
+NNnotations.S_ [in hydras.Ackermann.LNN]
+n2t [in hydras.Ackermann.LNN]
+

O

+omega [in hydras.Epsilon0.T1]
+omega [in hydras.OrdinalNotations.ON_Omega2]
+omega [in hydras.Gamma0.T2]
+omega [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+omega [in hydras.Schutte.Schutte_basics]
+ONDef.Exports.ON [in gaia_hydras.onType]
+ONDef.Exports.on [in gaia_hydras.onType]
+ONDef.Exports.ONMixin [in gaia_hydras.onType]
+one [in hydras.Epsilon0.T1]
+one [in hydras.Gamma0.T2]
+OO.omega [in hydras.OrdinalNotations.OmegaOmega]
+ordinal [in hydras.Schutte.Schutte_basics]
+

P

+path [in gaia_hydras.GPaths]
+phi0 [in hydras.Epsilon0.T1]
+phi0 [in hydras.Schutte.AP]
+plus [in hydras.Epsilon0.T1]
+Plus [in hydras.Epsilon0.E0]
+power [in hydras.Prelude.More_Arith]
+PReval [in hydras.Ackermann.primRec]
+PRevalN [in hydras.Ackermann.primRec]
+PRNotations.pi1_3 [in hydras.Ackermann.primRec]
+PRNotations.pi1_2 [in hydras.Ackermann.primRec]
+PRNotations.pi1_1 [in hydras.Ackermann.primRec]
+PRNotations.pi2_3 [in hydras.Ackermann.primRec]
+PRNotations.pi2_2 [in hydras.Ackermann.primRec]
+PRNotations.pi3_3 [in hydras.Ackermann.primRec]
+PRNotations.PRcomp [in hydras.Ackermann.primRec]
+PRNotations.PRrec [in hydras.Ackermann.primRec]
+

R

+reflection [in Goedel.goedel1]
+rem [in Goedel.PRrepresentable]
+

S

+substF2 [in hydras.Ackermann.folProp]
+substF3 [in hydras.Ackermann.folProp]
+substF4 [in hydras.Ackermann.folProp]
+substF5 [in hydras.Ackermann.folProp]
+substF6 [in hydras.Ackermann.folProp]
+substF7 [in hydras.Ackermann.folProp]
+substF8 [in hydras.Ackermann.folProp]
+substF9 [in hydras.Ackermann.folProp]
+substituteFormula [in hydras.Ackermann.folProp]
+substituteTerm [in hydras.Ackermann.folProp]
+substituteTerms [in hydras.Ackermann.folProp]
+Succ [in hydras.Epsilon0.E0]
+succb [in hydras.Epsilon0.T1]
+SysPrf [in hydras.Ackermann.subAll]
+System [in hydras.Ackermann.subAll]
+System [in hydras.Ackermann.folProp]
+System [in hydras.Ackermann.codeSubFormula]
+System [in hydras.Ackermann.folLogic3]
+S_ [in hydras.Ackermann.LNT]
+

T

+Term [in hydras.Ackermann.subAll]
+Term [in hydras.Ackermann.folProp]
+Term [in hydras.Ackermann.codeSubFormula]
+Term [in hydras.Ackermann.folLogic3]
+Terms [in hydras.Ackermann.subAll]
+Terms [in hydras.Ackermann.folProp]
+Terms [in hydras.Ackermann.codeSubFormula]
+Terms [in hydras.Ackermann.folLogic3]
+Toy.A [in hydras.MoreAck.FolExamples]
+Toy.a [in hydras.MoreAck.FolExamples]
+Toy.B [in hydras.MoreAck.FolExamples]
+Toy.b [in hydras.MoreAck.FolExamples]
+Toy.C [in hydras.MoreAck.FolExamples]
+Toy.f [in hydras.MoreAck.FolExamples]
+Toy.g [in hydras.MoreAck.FolExamples]
+Toy.h [in hydras.MoreAck.FolExamples]
+Toy.P [in hydras.MoreAck.FolExamples]
+Toy.Q [in hydras.MoreAck.FolExamples]
+Toy.R [in hydras.MoreAck.FolExamples]
+T1 [in gaia_hydras.T1Bridge]
+T1omega [in hydras.Epsilon0.T1]
+T2 [in gaia_hydras.T2Bridge]
+

U

+undecidable [in hydras.Ackermann.folProof]
+Usual.β [in Goedel.PRrepresentable]
+

V

+vfourth [in hydras.Prelude.MoreVectors]
+vfst [in hydras.Prelude.MoreVectors]
+vsnd [in hydras.Prelude.MoreVectors]
+vthird [in hydras.Prelude.MoreVectors]
+

Z

+Zero [in hydras.Epsilon0.E0]
+

_

+_Omega [in hydras.Epsilon0.E0]
+

other

+β [in Goedel.PRrepresentable]
+βR [in Goedel.PRrepresentable]
+ω [in hydras.Epsilon0.T1]
+

+

Definition Index

+

A

+A [in hydras.OrdinalNotations.ON_Omega2]
+acc_from [in gaia_hydras.GPaths]
+acc_from [in hydras.Epsilon0.Paths]
+Ack [in hydras.MoreAck.Ack]
+Ack [in hydras.MoreAck.Ack]
+Ackermann.all_zero [in gaia_hydras.nfwfgaia]
+Ackermann.epsilon0 [in gaia_hydras.nfwfgaia]
+Ackermann.limit_of [in gaia_hydras.nfwfgaia]
+Ackermann.limit12_hyp [in gaia_hydras.nfwfgaia]
+Ackermann.lt_psi [in gaia_hydras.nfwfgaia]
+Ackermann.lt_rec [in gaia_hydras.nfwfgaia]
+Ackermann.lt_psi_rec [in gaia_hydras.nfwfgaia]
+Ackermann.normal [in gaia_hydras.nfwfgaia]
+Ackermann.omega [in gaia_hydras.nfwfgaia]
+Ackermann.one [in gaia_hydras.nfwfgaia]
+Ackermann.phi [in gaia_hydras.nfwfgaia]
+Ackermann.phia [in gaia_hydras.nfwfgaia]
+Ackermann.phi_rec [in gaia_hydras.nfwfgaia]
+Ackermann.phi_rec_psi [in gaia_hydras.nfwfgaia]
+Ackermann.phi0 [in gaia_hydras.nfwfgaia]
+Ackermann.phi1 [in gaia_hydras.nfwfgaia]
+Ackermann.phi10 [in gaia_hydras.nfwfgaia]
+Ackermann.phi11 [in gaia_hydras.nfwfgaia]
+Ackermann.phi12a [in gaia_hydras.nfwfgaia]
+Ackermann.phi12b2 [in gaia_hydras.nfwfgaia]
+Ackermann.phi12b3 [in gaia_hydras.nfwfgaia]
+Ackermann.phi12b4 [in gaia_hydras.nfwfgaia]
+Ackermann.phi12b5 [in gaia_hydras.nfwfgaia]
+Ackermann.phi3 [in gaia_hydras.nfwfgaia]
+Ackermann.phi4 [in gaia_hydras.nfwfgaia]
+Ackermann.phi5 [in gaia_hydras.nfwfgaia]
+Ackermann.phi6 [in gaia_hydras.nfwfgaia]
+Ackermann.phi7 [in gaia_hydras.nfwfgaia]
+Ackermann.phi8 [in gaia_hydras.nfwfgaia]
+Ackermann.phi9 [in gaia_hydras.nfwfgaia]
+Ackermann.psi_succ [in gaia_hydras.nfwfgaia]
+Ackermann.size [in gaia_hydras.nfwfgaia]
+Ackermann.sup_of [in gaia_hydras.nfwfgaia]
+Ackermann.T1_T3 [in gaia_hydras.nfwfgaia]
+Ackermann.T3add [in gaia_hydras.nfwfgaia]
+Ackermann.T3ap [in gaia_hydras.nfwfgaia]
+Ackermann.T3bad [in gaia_hydras.nfwfgaia]
+Ackermann.T3eq [in gaia_hydras.nfwfgaia]
+Ackermann.T3finite [in gaia_hydras.nfwfgaia]
+Ackermann.T3is_succ [in gaia_hydras.nfwfgaia]
+Ackermann.T3le [in gaia_hydras.nfwfgaia]
+Ackermann.T3limit [in gaia_hydras.nfwfgaia]
+Ackermann.T3lt [in gaia_hydras.nfwfgaia]
+Ackermann.T3lta [in gaia_hydras.nfwfgaia]
+Ackermann.T3nat [in gaia_hydras.nfwfgaia]
+Ackermann.T3nf [in gaia_hydras.nfwfgaia]
+Ackermann.T3pred [in gaia_hydras.nfwfgaia]
+Ackermann.T3split [in gaia_hydras.nfwfgaia]
+Ackermann.T3sub [in gaia_hydras.nfwfgaia]
+Ackermann.T3succ [in gaia_hydras.nfwfgaia]
+Ackermann.T3_eqType [in gaia_hydras.nfwfgaia]
+Ackermann.T3_eqMixin [in gaia_hydras.nfwfgaia]
+addExists [in Goedel.PRrepresentable]
+addForalls [in Goedel.PRrepresentable]
+add_r [in hydras.Hydra.Hydra_Definitions]
+add_head_r_plus [in hydras.Hydra.Hydra_Definitions]
+add_head_r [in hydras.Hydra.Hydra_Definitions]
+Alive [in hydras.Hydra.Hydra_Definitions]
+alpha_0_eq [in hydras.Epsilon0.T1]
+alpha_0 [in hydras.Epsilon0.T1]
+alpha1 [in hydras.OrdinalNotations.ON_Finite]
+Alt.Ack [in hydras.MoreAck.Ack]
+Alt.add' [in hydras.MoreAck.PrimRecExamples]
+Alt.double [in additions.Demo]
+Alt.Eps0_prec.status [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.prec [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_prec.A [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.arity [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Eps0_sig.symb [in hydras.Epsilon0.Epsilon0rpo]
+Alt.nat_2_term [in hydras.Epsilon0.Epsilon0rpo]
+Alt.transfinite_recursor [in hydras.Epsilon0.Epsilon0rpo]
+Alt.transfinite_recursor_lt [in hydras.Epsilon0.Epsilon0rpo]
+Alt.T1_size [in hydras.Epsilon0.Epsilon0rpo]
+Alt.T1_2_term [in hydras.Epsilon0.Epsilon0rpo]
+Alt.Vars.var [in hydras.Epsilon0.Epsilon0rpo]
+Alt.xpred [in hydras.MoreAck.PrimRecExamples]
+Alt2.Ack [in hydras.MoreAck.Ack]
+Alt2.test1 [in hydras.MoreAck.Ack]
+Alt2.test2 [in hydras.MoreAck.Ack]
+Alt3.ack [in hydras.MoreAck.Ack]
+AM_power [in additions.AM]
+andH [in hydras.Ackermann.fol]
+andRel [in hydras.Ackermann.primRec]
+any_chain_parametric [in additions.Addition_Chains]
+ap [in hydras.OrdinalNotations.ON_Omega2]
+AP [in hydras.Schutte.AP]
+approx [in hydras.Epsilon0.Canon]
+AxmEq4 [in hydras.Ackermann.folProof]
+AxmEq5 [in hydras.Ackermann.folProof]
+axp_scheme [in additions.Addition_Chains]
+A_ [in hydras.Schutte.Critical]
+

B

+bad [in hydras.OrdinalNotations.ON_Finite]
+BadSubstF2.substF [in hydras.MoreAck.BadSubst]
+BadSubst.substF [in hydras.MoreAck.BadSubst]
+bad_term [in hydras.Epsilon0.T1]
+Bad.correct_expt_function [in additions.FirstSteps]
+Bad.fact [in additions.Monoid_instances]
+Bad.Fchain_correct [in additions.Euclidean_Chains]
+Bad.int63_from_nat [in additions.Monoid_instances]
+Bad4.Fplus [in additions.Euclidean_Chains]
+Bad4.F23 [in additions.Euclidean_Chains]
+battle_length [in hydras.Hydra.Hydra_Definitions]
+beta [in Goedel.PRrepresentable]
+betaFormula [in Goedel.PRrepresentable]
+beta_h [in hydras.Hydra.Epsilon0_Needed_Generic]
+beta1 [in hydras.OrdinalNotations.ON_Finite]
+bigarg [in additions.fib]
+bigO [in hydras.OrdinalNotations.ON_Generic]
+big_h [in hydras.Hydra.Epsilon0_Needed_Generic]
+big0 [in hydras.Schutte.Schutte_basics]
+binary_power_mult [in additions.FirstSteps]
+binary_power_mult [in additions.Pow_variant]
+binary_power_mult [in additions.Pow]
+binary_power_mult [in additions.fib]
+binary_chain [in additions.Addition_Chains]
+bin_pow_scheme [in additions.Addition_Chains]
+bits [in additions.fib]
+bodd [in hydras.Ackermann.primRec]
+bool_decide [in hydras.Prelude.STDPP_compat]
+boundComputation [in hydras.Ackermann.codeSubFormula]
+boundedSearch [in hydras.Ackermann.primRec]
+boundedSearchHelp [in hydras.Ackermann.primRec]
+bounded_by [in hydras.Prelude.MoreLists]
+bounded_transitionS [in gaia_hydras.GPaths]
+bounded_transitionS [in hydras.Epsilon0.Paths]
+box [in Goedel.goedel2]
+but_last [in hydras.Prelude.MoreLists]
+B_termination [in hydras.Hydra.Hydra_Definitions]
+

C

+call_lt [in hydras.Epsilon0.F_alpha]
+canon [in hydras.OrdinalNotations.ON_Omega2]
+canon [in gaia_hydras.GCanon]
+Canon [in hydras.Epsilon0.Canon]
+canon [in hydras.Epsilon0.Canon]
+Canon_plus [in gaia_hydras.GF_alpha]
+Canon_plus [in gaia_hydras.GPaths]
+Canon_plus [in hydras.Epsilon0.Paths]
+CantorOrdinal.exp_O [in gaia_hydras.nfwfgaia]
+CantorOrdinal.exp_F [in gaia_hydras.nfwfgaia]
+CantorOrdinal.ex_middle [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_fct [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_of [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_v2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.limit_v1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.normal [in gaia_hydras.nfwfgaia]
+CantorOrdinal.omega_plus_n [in gaia_hydras.nfwfgaia]
+CantorOrdinal.omega_tower [in gaia_hydras.nfwfgaia]
+CantorOrdinal.one [in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi0 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi1 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi2 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.phi3 [in gaia_hydras.nfwfgaia]
+CantorOrdinal.sup [in gaia_hydras.nfwfgaia]
+CantorOrdinal.toNF [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1add [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1ap [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1bad [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1div_by_omega [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1eq [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1exp [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1finite [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1is_succ [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1le [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1limit [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1log [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1lt [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1mul [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nat [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1nf [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1omega [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1pred [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1split [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1sub [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1succ [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1_eqType [in gaia_hydras.nfwfgaia]
+CantorOrdinal.T1_eqMixin [in gaia_hydras.nfwfgaia]
+chain [in additions.Addition_Chains]
+chain_apply [in additions.AM]
+chain_trace [in additions.Trace_exercise]
+chain_type [in additions.Euclidean_Chains]
+chain_generator [in additions.Addition_Chains]
+chain_correct [in additions.Addition_Chains]
+chain_correct_nat [in additions.Addition_Chains]
+chain_apply [in additions.Addition_Chains]
+chain_execute [in additions.Addition_Chains]
+chain_length [in additions.Addition_Chains]
+charFunction [in hydras.Ackermann.extEqualNat]
+checkPrf [in hydras.Ackermann.checkPrf]
+checkPrfAXM [in hydras.Ackermann.checkPrf]
+checkPrfCP [in hydras.Ackermann.checkPrf]
+checkPrfEQ1 [in hydras.Ackermann.checkPrf]
+checkPrfEQ2 [in hydras.Ackermann.checkPrf]
+checkPrfEQ3 [in hydras.Ackermann.checkPrf]
+checkPrfEQ4 [in hydras.Ackermann.checkPrf]
+checkPrfEQ5 [in hydras.Ackermann.checkPrf]
+checkPrfFA1 [in hydras.Ackermann.checkPrf]
+checkPrfFA2 [in hydras.Ackermann.checkPrf]
+checkPrfFA3 [in hydras.Ackermann.checkPrf]
+checkPrfGEN [in hydras.Ackermann.checkPrf]
+checkPrfHelp [in hydras.Ackermann.checkPrf]
+checkPrfIMP1 [in hydras.Ackermann.checkPrf]
+checkPrfIMP2 [in hydras.Ackermann.checkPrf]
+checkPrfMP [in hydras.Ackermann.checkPrf]
+checkSubFormulaTrace [in hydras.Ackermann.codeSubFormula]
+classic_battle [in hydras.Hydra.Hydra_Lemmas]
+clog2 [in additions.Addition_Chains]
+close [in hydras.Ackermann.folProp]
+closed [in hydras.Ackermann.folProp]
+Closed [in hydras.Schutte.Schutte_basics]
+ClosedSystem [in hydras.Ackermann.folProp]
+closeFrom [in hydras.Ackermann.subAll]
+closeList [in hydras.Ackermann.folProp]
+cnf_t [in hydras.Schutte.CNF]
+code [in additions.AM]
+codeAnd [in hydras.Ackermann.code]
+codeApp [in hydras.Ackermann.codeList]
+codeArityLNNR [in hydras.Ackermann.Languages]
+codeArityLNTF [in hydras.Ackermann.Languages]
+codeArityLNTR [in hydras.Ackermann.Languages]
+codeAxmEqHelp [in hydras.Ackermann.checkPrf]
+codeClose [in hydras.Ackermann.codePA]
+codeCloseList [in hydras.Ackermann.codePA]
+codeF [in hydras.Ackermann.code]
+codeFNN [in Goedel.goedel1]
+codeForall [in hydras.Ackermann.code]
+codeFormula [in hydras.Ackermann.code]
+codeFormula [in Goedel.rosserPA]
+codeFormula [in Goedel.rosser]
+codeFreeVarFormula [in hydras.Ackermann.codeFreeVar]
+codeFreeVarListFormula [in hydras.Ackermann.codeFreeVar]
+codeFreeVarTerm [in hydras.Ackermann.codeFreeVar]
+codeFreeVarTerms [in hydras.Ackermann.codeFreeVar]
+codeFreeVarTermTerms [in hydras.Ackermann.codeFreeVar]
+codeIff [in hydras.Ackermann.code]
+codeImp [in hydras.Ackermann.code]
+codeIn [in hydras.Ackermann.codeList]
+codeInductionSchema [in hydras.Ackermann.codePA]
+codeLength [in hydras.Ackermann.codeList]
+codeList [in hydras.Ackermann.cPair]
+codeListRemove [in hydras.Ackermann.codeList]
+codeLNNRelation [in hydras.Ackermann.Languages]
+codeLNTFunction [in hydras.Ackermann.Languages]
+codeLNTRelation [in hydras.Ackermann.Languages]
+codeNatToTerm [in hydras.Ackermann.codeNatToTerm]
+codeNewVar [in hydras.Ackermann.codeSubFormula]
+codeNoDup [in hydras.Ackermann.codeList]
+codeNot [in hydras.Ackermann.code]
+codeNth [in hydras.Ackermann.cPair]
+codeNVars1 [in hydras.Ackermann.checkPrf]
+codeNVars2 [in hydras.Ackermann.checkPrf]
+codeOpen [in hydras.Ackermann.codePA]
+codeOr [in hydras.Ackermann.code]
+codePA [in hydras.Ackermann.codePA]
+codePrf [in hydras.Ackermann.code]
+codePrf [in Goedel.rosserPA]
+codePrf [in Goedel.rosser]
+codeR [in hydras.Ackermann.code]
+codeSubFormula [in hydras.Ackermann.codeSubFormula]
+codeSubTerm [in hydras.Ackermann.codeSubTerm]
+codeSubTerms [in hydras.Ackermann.codeSubTerm]
+codeSubTermTerms [in hydras.Ackermann.codeSubTerm]
+codeSysPf [in Goedel.codeSysPrf]
+codeSysPf [in Goedel.goedel1]
+codeSysPf [in Goedel.goedel2]
+codeSysPfCorrect [in Goedel.goedel1]
+codeSysPfCorrect [in Goedel.goedel2]
+codeSysPrf [in Goedel.rosserPA]
+codeSysPrf [in Goedel.codeSysPrf]
+codeSysPrf [in Goedel.goedel1]
+codeSysPrf [in Goedel.rosser]
+codeSysPrfCorrect1 [in Goedel.rosserPA]
+codeSysPrfCorrect1 [in Goedel.rosser]
+codeSysPrfCorrect2 [in Goedel.rosserPA]
+codeSysPrfCorrect2 [in Goedel.goedel1]
+codeSysPrfCorrect2 [in Goedel.rosser]
+codeSysPrfCorrect3 [in Goedel.rosserPA]
+codeSysPrfCorrect3 [in Goedel.goedel1]
+codeSysPrfCorrect3 [in Goedel.rosser]
+codeSysPrfNCorrect1 [in Goedel.rosserPA]
+codeSysPrfNCorrect1 [in Goedel.rosser]
+codeSysPrfNCorrect2 [in Goedel.rosserPA]
+codeSysPrfNCorrect2 [in Goedel.rosser]
+codeSysPrfNCorrect3 [in Goedel.rosserPA]
+codeSysPrfNCorrect3 [in Goedel.rosser]
+codeSysPrfNot [in Goedel.rosserPA]
+codeSysPrfNot [in Goedel.codeSysPrf]
+codeSysPrfNot [in Goedel.rosser]
+codeTerm [in hydras.Ackermann.code]
+codeTerms [in hydras.Ackermann.code]
+Compat815.ind_0_1_SS [in hydras.Prelude.Compat815]
+composeSigmaFormula [in Goedel.PRrepresentable]
+compose2 [in hydras.Ackermann.primRec]
+computation_equiv [in additions.Euclidean_Chains]
+computation_eval [in additions.Addition_Chains]
+computation_execute [in additions.Addition_Chains]
+computation_length [in additions.Addition_Chains]
+comp128 [in additions.Addition_Chains]
+Con [in Goedel.goedel2]
+config [in additions.AM]
+config_equiv [in additions.AM]
+Consistent [in hydras.Ackermann.folProof]
+const_path [in gaia_hydras.GPaths]
+const_pathS_eps [in hydras.Epsilon0.Paths]
+const_path [in hydras.Epsilon0.Paths]
+const_pathS [in hydras.Epsilon0.Paths]
+continuous [in hydras.Schutte.Schutte_basics]
+correctness_statement [in additions.AM]
+correctness_statement [in additions.Euclidean_Chains]
+correct_generator [in additions.Addition_Chains]
+cPair [in hydras.Ackermann.cPair]
+cPairPi1 [in hydras.Ackermann.cPair]
+cPairPi2 [in hydras.Ackermann.cPair]
+cpower [in additions.Addition_Chains]
+cpower_pos [in additions.Addition_Chains]
+Cr [in hydras.Schutte.Critical]
+Cr_fun [in hydras.Schutte.Critical]
+cTriple [in hydras.Ackermann.cPair]
+cTriplePi1 [in hydras.Ackermann.cPair]
+cTriplePi2 [in hydras.Ackermann.cPair]
+cTriplePi3 [in hydras.Ackermann.cPair]
+C1 [in additions.Addition_Chains]
+c153 [in additions.Fib2]
+C3 [in additions.Addition_Chains]
+C31_7 [in additions.AM]
+C7 [in additions.Addition_Chains]
+C7_ok' [in additions.Addition_Chains]
+C7_ok [in additions.Addition_Chains]
+C87 [in additions.Addition_Chains]
+C87_ok [in additions.Addition_Chains]
+C87' [in additions.Euclidean_Chains]
+

D

+DecidableSet [in Goedel.rosser]
+depth [in hydras.Ackermann.fol]
+dicho [in additions.Dichotomy]
+dominates [in hydras.Prelude.Iterates]
+dominates [in gaia_hydras.T1Bridge]
+dominates_strong [in hydras.Prelude.Iterates]
+dominates_from [in hydras.Prelude.Iterates]
+dominates_strong [in gaia_hydras.T1Bridge]
+dominates_from [in gaia_hydras.T1Bridge]
+double [in hydras.solutions_exercises.MorePRExamples]
+double [in hydras.MoreAck.PrimRecExamples]
+doubleS [in hydras.Hydra.BigBattle]
+

E

+Elements [in hydras.Epsilon0.T1]
+epsilon [in hydras.Gamma0.T2]
+epsilon_0 [in hydras.Epsilon0.T1]
+epsilon0 [in hydras.Schutte.AP]
+equiv [in hydras.Prelude.Sort_spec]
+eval [in hydras.Schutte.CNF]
+evalComposeFunc [in hydras.Ackermann.primRec]
+evalConstFunc [in hydras.Ackermann.primRec]
+evalList [in hydras.Ackermann.primRec]
+evalOneParamList [in hydras.Ackermann.primRec]
+evalPrimRec [in hydras.Ackermann.primRec]
+evalPrimRecFunc [in hydras.Ackermann.primRec]
+evalPrimRecs [in hydras.Ackermann.primRec]
+evalProjFunc [in hydras.Ackermann.primRec]
+evalStrongRec [in hydras.Ackermann.cPair]
+evalStrongRecHelp [in hydras.Ackermann.cPair]
+eval_pp [in hydras.Epsilon0.T1]
+Ex [in hydras.Epsilon0.Large_Sets]
+exact_log2 [in additions.More_on_positive]
+Examples.big_chain [in additions.Euclidean_Chains]
+Examples.binary_int31_power [in additions.Euclidean_Chains]
+Examples.C87' [in additions.Euclidean_Chains]
+Examples.exact_sqrt [in hydras.Ackermann.primRec]
+Examples.exp56789 [in additions.Euclidean_Chains]
+Examples.Exx [in hydras.Hydra.Hydra_Examples]
+Examples.ex_2 [in hydras.Hydra.Hydra_Examples]
+Examples.ex1 [in hydras.Hydra.Hydra_Examples]
+Examples.ex4 [in hydras.Hydra.Hydra_Examples]
+Examples.Ex5 [in hydras.Hydra.Hydra_Examples]
+Examples.fast_int63_power [in additions.Euclidean_Chains]
+Examples.height_not_strictly_decreasing [in hydras.Hydra.Hydra_Examples]
+Examples.hsize_bigger [in hydras.Hydra.Hydra_Examples]
+Examples.Hy [in hydras.Hydra.Hydra_Examples]
+Examples.Hy_Hy''' [in hydras.Hydra.Hydra_Examples]
+Examples.Hy_3 [in hydras.Hydra.Hydra_Examples]
+Examples.Hy_2 [in hydras.Hydra.Hydra_Examples]
+Examples.Hy_1 [in hydras.Hydra.Hydra_Examples]
+Examples.Hy' [in hydras.Hydra.Hydra_Examples]
+Examples.Hy'H'' [in hydras.Hydra.Hydra_Examples]
+Examples.Hy'' [in hydras.Hydra.Hydra_Examples]
+Examples.Hy''_Hy''' [in hydras.Hydra.Hydra_Examples]
+Examples.Hy''' [in hydras.Hydra.Hydra_Examples]
+Examples.M [in additions.Euclidean_Chains]
+Examples.R2_example [in hydras.Hydra.Hydra_Examples]
+Examples.slow_int31_power [in additions.Euclidean_Chains]
+Examples.sqrtHelp [in hydras.Ackermann.primRec]
+Examples.Unnamed_thm [in additions.Euclidean_Chains]
+exec [in additions.AM]
+existH [in hydras.Ackermann.fol]
+exp [in hydras.Epsilon0.T1]
+exp [in hydras.solutions_exercises.MorePRExamples]
+exponents [in additions.Trace_exercise]
+exponents_le [in hydras.Schutte.CNF]
+exponents_lt [in hydras.Schutte.CNF]
+Expressible [in hydras.Ackermann.expressible]
+ExpressibleHelp [in hydras.Ackermann.expressible]
+expressT'1 [in Goedel.goedel2]
+expressT'2 [in Goedel.goedel2]
+exp_F [in hydras.Epsilon0.T1]
+exp_alt [in hydras.solutions_exercises.MorePRExamples]
+exp2 [in additions.Addition_Chains]
+exp2 [in hydras.Prelude.Exp2]
+extendsNN [in Goedel.goedel2]
+extEqual [in hydras.Ackermann.extEqualNat]
+extEqualVector [in hydras.Ackermann.primRec]
+extEqualVectorGeneral [in hydras.Ackermann.primRec]
+extEqual_ex1 [in hydras.MoreAck.PrimRecExamples]
+ex_pos [in gaia_hydras.T1Choice]
+ex_path4 [in gaia_hydras.GPaths]
+ex_path3 [in gaia_hydras.GPaths]
+ex_path2 [in gaia_hydras.GPaths]
+ex_path1 [in gaia_hydras.GPaths]
+ex_pathS2 [in hydras.Epsilon0.Large_Sets_Examples]
+ex_pathS1 [in hydras.Epsilon0.Large_Sets_Examples]
+ex_bool [in additions.Monoid_def]
+ex_string [in additions.Monoid_def]
+ex_path4 [in hydras.Epsilon0.Paths]
+ex_path3 [in hydras.Epsilon0.Paths]
+ex_path2 [in hydras.Epsilon0.Paths]
+ex_path1 [in hydras.Epsilon0.Paths]
+ex0 [in gaia_hydras.GCanon]
+Ex1 [in hydras.Epsilon0.T1]
+Ex1 [in hydras.OrdinalNotations.ON_Omega2]
+ex1 [in hydras.OrdinalNotations.ON_Omega2]
+Ex1 [in hydras.Gamma0.T2]
+ex1 [in gaia_hydras.GCanon]
+Ex1 [in hydras.Schutte.MoreEpsilonIota]
+ex1 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Ex1 [in hydras.solutions_exercises.MultisetWf]
+Ex1 [in hydras.Epsilon0.Large_Sets_Examples]
+Ex1 [in hydras.OrdinalNotations.ON_Finite]
+Ex1 [in gaia_hydras.GaiaToHydra]
+Ex1 [in hydras.MoreAck.PrimRecExamples]
+Ex1 [in hydras.Prelude.Simple_LexProd]
+Ex10 [in hydras.Gamma0.Gamma0]
+Ex2 [in hydras.Epsilon0.T1]
+Ex2 [in hydras.Gamma0.T2]
+Ex2 [in hydras.MoreAck.AckNotPR]
+ex2 [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Ex2 [in hydras.solutions_exercises.MultisetWf]
+Ex2 [in hydras.Epsilon0.Large_Sets_Examples]
+Ex2 [in hydras.MoreAck.PrimRecExamples]
+Ex2 [in hydras.Prelude.Simple_LexProd]
+Ex3 [in hydras.Epsilon0.T1]
+Ex3 [in hydras.Gamma0.T2]
+ex3 [in hydras.Epsilon0.Large_Sets]
+Ex3 [in hydras.solutions_exercises.MultisetWf]
+Ex3 [in hydras.MoreAck.PrimRecExamples]
+Ex4 [in hydras.Epsilon0.T1]
+Ex4 [in hydras.Gamma0.T2]
+Ex4 [in hydras.MoreAck.AckNotPR]
+Ex4 [in hydras.MoreAck.PrimRecExamples]
+Ex42 [in hydras.Schutte.Schutte]
+Ex42 [in hydras.OrdinalNotations.OmegaOmega]
+Ex42 [in hydras.Epsilon0.E0]
+Ex5 [in hydras.Epsilon0.T1]
+Ex5 [in hydras.Gamma0.T2]
+Ex6 [in hydras.Gamma0.T2]
+Ex6 [in hydras.Gamma0.Gamma0]
+Ex7 [in hydras.Gamma0.T2]
+Ex7 [in hydras.Gamma0.Gamma0]
+Ex8 [in hydras.Gamma0.T2]
+Ex8 [in hydras.Gamma0.Gamma0]
+Ex9 [in hydras.Gamma0.Gamma0]
+E0Canon [in gaia_hydras.GCanon]
+E0eqb [in gaia_hydras.T1Bridge]
+E0fin [in gaia_hydras.T1Bridge]
+E0is_succ [in gaia_hydras.T1Bridge]
+E0is_succ [in hydras.Epsilon0.E0]
+E0le [in gaia_hydras.T1Bridge]
+E0le [in hydras.Epsilon0.E0]
+E0limit [in gaia_hydras.T1Bridge]
+E0limit [in hydras.Epsilon0.E0]
+E0lt [in gaia_hydras.T1Bridge]
+E0lt [in hydras.Epsilon0.E0]
+E0mul [in gaia_hydras.T1Bridge]
+E0plus [in gaia_hydras.T1Bridge]
+e0Sub [in gaia_hydras.T1Bridge]
+E0zero [in gaia_hydras.T1Bridge]
+E0_g2h [in gaia_hydras.T1Bridge]
+E0_h2g [in gaia_hydras.T1Bridge]
+E0_phi0 [in gaia_hydras.T1Bridge]
+E0_omega [in gaia_hydras.T1Bridge]
+E0_pred [in gaia_hydras.T1Bridge]
+E0_succ [in gaia_hydras.T1Bridge]
+E0_eqtype [in gaia_hydras.T1Bridge]
+E0_eq_mixin [in gaia_hydras.T1Bridge]
+E0_pred [in hydras.Epsilon0.E0]
+E1 [in hydras.Epsilon0.T1]
+e1 [in hydras.OrdinalNotations.ON_Omega2]
+E2 [in hydras.Epsilon0.T1]
+e2 [in hydras.OrdinalNotations.ON_Omega2]
+

F

+f [in hydras.OrdinalNotations.Example_3PlusOmega]
+F [in Goedel.goedel2]
+fact [in hydras.solutions_exercises.MorePRExamples]
+fact_alt [in hydras.solutions_exercises.MorePRExamples]
+Fapply [in additions.Euclidean_Chains]
+fastexp [in additions.fib]
+fastexp2 [in additions.fib]
+fastexp3 [in additions.fib]
+fastexp4 [in additions.fib]
+Fchain [in additions.Euclidean_Chains]
+Fchain_correct [in additions.AM]
+Fchain_correct [in additions.Euclidean_Chains]
+Fchain_correct_nat [in additions.Euclidean_Chains]
+Fcompose [in additions.Euclidean_Chains]
+Fexp2 [in additions.Euclidean_Chains]
+Fexp2_of_nat [in additions.Euclidean_Chains]
+FFK [in additions.AM]
+FFK [in additions.Euclidean_Chains]
+fib [in additions.Fib2]
+fib [in hydras.solutions_exercises.FibonacciPR]
+fib [in additions.fib]
+fibm [in additions.fib]
+fibonacci [in additions.FirstSteps]
+fibonacci [in additions.Naive]
+fibPR [in hydras.solutions_exercises.FibonacciPR]
+fibt [in additions.fib]
+fib_with_chain [in additions.Fib2]
+fib_eucl [in additions.Fib2]
+fib_pos [in additions.Fib2]
+fib_mul2 [in additions.Fib2]
+fib_alt [in hydras.solutions_exercises.FibonacciPR]
+fin [in hydras.OrdinalNotations.ON_Omega2]
+fin [in hydras.Gamma0.T2]
+fin [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+find [in hydras.rpo.more_list]
+finite [in hydras.Schutte.Schutte_basics]
+finord_lt [in gaia_hydras.ON_gfinite]
+first_toggle [in hydras.Prelude.First_toggle]
+FixpointDef.oplus [in gaia_hydras.GHessenberg]
+FixR [in hydras.Prelude.Restriction]
+FixR_F [in hydras.Prelude.Restriction]
+FK [in additions.AM]
+FK [in additions.Euclidean_Chains]
+Fkont [in additions.Euclidean_Chains]
+Fkont_equiv [in additions.Euclidean_Chains]
+flatten [in additions.Addition_Chains]
+flatten_aux [in additions.Addition_Chains]
+fold_left2 [in hydras.rpo.more_list]
+foo [in hydras.Prelude.Fuel]
+Formula [in hydras.Ackermann.LNN]
+Formula [in hydras.Ackermann.LNT]
+Formulas [in hydras.Ackermann.LNN]
+Formulas [in hydras.Ackermann.LNT]
+Formulas [in hydras.Ackermann.fol]
+FormulasToFormula [in Goedel.PRrepresentable]
+FormulasToFuncs [in Goedel.PRrepresentable]
+Formula_depth_ind [in hydras.Ackermann.fol]
+Formula_depth_rec2 [in hydras.Ackermann.fol]
+Formula_depth_rec2rec [in hydras.Ackermann.fol]
+Formula_depth_rec [in hydras.Ackermann.fol]
+Formula_depth_rec_rec [in hydras.Ackermann.fol]
+freeVarCodeSysPrfN [in Goedel.rosserPA]
+freeVarCodeSysPrfN [in Goedel.rosser]
+freeVarF [in hydras.Ackermann.folProp]
+freeVarListFormula [in hydras.Ackermann.folProp]
+freeVarMap [in hydras.Ackermann.subAll]
+freeVarRepT' [in Goedel.goedel2]
+freeVarT [in hydras.Ackermann.folProp]
+freeVarTs [in hydras.Ackermann.folProp]
+fun_le [in hydras.Prelude.Iterates]
+fun_le [in gaia_hydras.T1Bridge]
+fun_restr [in hydras.Schutte.PartialFun]
+fun_inj [in hydras.Schutte.PartialFun]
+fun_onto [in hydras.Schutte.PartialFun]
+fun_codomain [in hydras.Schutte.PartialFun]
+fun_equiv [in hydras.Schutte.Ordering_Functions]
+fun2PR [in hydras.Ackermann.primRec]
+fusion [in additions.Trace_exercise]
+F_ [in gaia_hydras.GF_alpha]
+f_ [in hydras.Epsilon0.F_alpha]
+f_star [in hydras.Epsilon0.F_alpha]
+F_ [in hydras.Epsilon0.F_alpha]
+F_star [in hydras.Epsilon0.F_alpha]
+f0 [in hydras.MoreAck.LNN_Examples]
+F1 [in additions.AM]
+F1 [in additions.Euclidean_Chains]
+f1 [in hydras.MoreAck.FolExamples]
+F144 [in additions.Euclidean_Chains]
+F197887 [in additions.AM]
+F2 [in additions.Euclidean_Chains]
+f2 [in hydras.MoreAck.FolExamples]
+F2C [in additions.Euclidean_Chains]
+F2q [in additions.AM]
+F2q_of_nat [in additions.AM]
+F3 [in additions.AM]
+F3 [in additions.Euclidean_Chains]
+f3 [in hydras.MoreAck.FolExamples]
+F87 [in additions.Euclidean_Chains]
+F9 [in additions.Euclidean_Chains]
+

G

+g [in hydras.OrdinalNotations.Example_3PlusOmega]
+G [in Goedel.goedel1]
+g [in hydras.Schutte.Ordering_Functions]
+G [in Goedel.goedel2]
+gamma_ [in hydras.Schutte.Critical]
+Gamma0 [in hydras.Schutte.Critical]
+Gamma0_prec.status [in hydras.Gamma0.Gamma0]
+Gamma0_prec.prec [in hydras.Gamma0.Gamma0]
+Gamma0_prec.A [in hydras.Gamma0.Gamma0]
+Gamma0_sig.arity [in hydras.Gamma0.Gamma0]
+Gamma0_sig.symb [in hydras.Gamma0.Gamma0]
+Gamma0.epsilon0 [in gaia_hydras.nfwfgaia]
+Gamma0.lt_psi [in gaia_hydras.nfwfgaia]
+Gamma0.lt_rec [in gaia_hydras.nfwfgaia]
+Gamma0.omega [in gaia_hydras.nfwfgaia]
+Gamma0.one [in gaia_hydras.nfwfgaia]
+Gamma0.phi [in gaia_hydras.nfwfgaia]
+Gamma0.psi [in gaia_hydras.nfwfgaia]
+Gamma0.psi_phi [in gaia_hydras.nfwfgaia]
+Gamma0.psi_phi_aux [in gaia_hydras.nfwfgaia]
+Gamma0.size [in gaia_hydras.nfwfgaia]
+Gamma0.T1T2 [in gaia_hydras.nfwfgaia]
+Gamma0.T2add [in gaia_hydras.nfwfgaia]
+Gamma0.T2ap [in gaia_hydras.nfwfgaia]
+Gamma0.T2eq [in gaia_hydras.nfwfgaia]
+Gamma0.T2finite [in gaia_hydras.nfwfgaia]
+Gamma0.T2finite1 [in gaia_hydras.nfwfgaia]
+Gamma0.T2is_succ [in gaia_hydras.nfwfgaia]
+Gamma0.T2le [in gaia_hydras.nfwfgaia]
+Gamma0.T2limit [in gaia_hydras.nfwfgaia]
+Gamma0.T2lt [in gaia_hydras.nfwfgaia]
+Gamma0.T2lta [in gaia_hydras.nfwfgaia]
+Gamma0.T2nat [in gaia_hydras.nfwfgaia]
+Gamma0.T2nf [in gaia_hydras.nfwfgaia]
+Gamma0.T2pred [in gaia_hydras.nfwfgaia]
+Gamma0.T2split [in gaia_hydras.nfwfgaia]
+Gamma0.T2sub [in gaia_hydras.nfwfgaia]
+Gamma0.T2succ [in gaia_hydras.nfwfgaia]
+Gamma0.T2_pr2 [in gaia_hydras.nfwfgaia]
+Gamma0.T2_pr1 [in gaia_hydras.nfwfgaia]
+Gamma0.T2_eqType [in gaia_hydras.nfwfgaia]
+Gamma0.T2_eqMixin [in gaia_hydras.nfwfgaia]
+gamma1 [in hydras.OrdinalNotations.ON_Finite]
+ge [in hydras.Schutte.Schutte_basics]
+get_decomposition [in hydras.Epsilon0.T1]
+gnaw [in gaia_hydras.GPaths]
+Gnaw [in hydras.Epsilon0.Large_Sets]
+gnaw [in hydras.Epsilon0.Paths]
+gnawS [in gaia_hydras.GPaths]
+GnawS [in hydras.Epsilon0.Large_Sets]
+gnawS [in hydras.Epsilon0.Paths]
+GRelation [in hydras.Schutte.GRelations]
+gstrict_lub [in gaia_hydras.T1Bridge]
+gtail [in hydras.Gamma0.T2]
+G0.Ex42 [in hydras.Gamma0.Gamma0]
+G0.le [in hydras.Gamma0.Gamma0]
+G0.lt [in hydras.Gamma0.Gamma0]
+G0.LT [in hydras.Gamma0.Gamma0]
+g2h [in gaia_hydras.T1Bridge]
+g2h [in gaia_hydras.T2Bridge]
+g2h_seq [in gaia_hydras.T1Bridge]
+

H

+half [in additions.BinaryStrat]
+hcons_mult [in hydras.Hydra.Hydra_Definitions]
+height [in hydras.Hydra.Hydra_Definitions]
+hinit [in hydras.Hydra.BigBattle]
+hsize [in hydras.Hydra.Hydra_Definitions]
+hs2l [in hydras.Hydra.Hydra_Definitions]
+Hydrae_rect2 [in hydras.Hydra.Hydra_Definitions]
+Hydra_rect2 [in hydras.Hydra.Hydra_Definitions]
+hyd_mult [in hydras.Hydra.Hydra_Definitions]
+hyper_exp2 [in hydras.Prelude.Iterates]
+hy_app [in hydras.Hydra.Hydra_Definitions]
+h_forall [in hydras.Hydra.Hydra_Definitions]
+H'_ [in gaia_hydras.GHprime]
+H'_succ_fun [in hydras.Epsilon0.Hprime]
+H'_ [in hydras.Epsilon0.Hprime]
+h0 [in hydras.Hydra.KP_example]
+h1 [in hydras.Hydra.KP_example]
+h2 [in hydras.Hydra.KP_example]
+h2g [in gaia_hydras.T1Bridge]
+h2g [in gaia_hydras.T2Bridge]
+h2g_seq [in gaia_hydras.T1Bridge]
+h3 [in hydras.Hydra.KP_example]
+

I

+Id2 [in additions.Monoid_instances]
+iffH [in hydras.Ackermann.fol]
+ifThenElseH [in hydras.Ackermann.fol]
+ignoreParams [in hydras.Ackermann.primRec]
+image [in hydras.Schutte.PartialFun]
+Included_s [in hydras.Prelude.DecPreOrder_Instances]
+Inconsistent [in hydras.Ackermann.folProof]
+independent [in hydras.Ackermann.folProof]
+InductionSchema [in hydras.Ackermann.PA]
+Infinite [in hydras.solutions_exercises.Limit_Infinity]
+inject [in hydras.Schutte.Correctness_E0]
+interpFormula [in hydras.Ackermann.model]
+interpRels [in hydras.Ackermann.model]
+interpTerm [in hydras.Ackermann.model]
+interpTerms [in hydras.Ackermann.model]
+interpTermsVector [in hydras.Ackermann.model]
+interval [in hydras.Prelude.MoreLists]
+inv [in hydras.solutions_exercises.FibonacciPR]
+inv_fun [in hydras.Schutte.PartialFun]
+In_freeVarSys [in hydras.Ackermann.folProp]
+iota [in hydras.Hydra.O2H]
+iota [in hydras.solutions_exercises.MultisetWf]
+iotas [in hydras.Hydra.O2H]
+iota_from [in hydras.Prelude.MoreLists]
+iota_fun [in hydras.Schutte.PartialFun]
+iota_demo.Bad.bottom [in hydras.Schutte.Schutte_basics]
+iota_demo.zero [in hydras.Schutte.Schutte_basics]
+iota_demo.zero [in hydras.Schutte.Schutte_basics]
+iota_ij [in hydras.OrdinalNotations.ON_Finite]
+Iso_same_op [in hydras.OrdinalNotations.ON_Generic]
+Iso_same_fun [in hydras.OrdinalNotations.ON_Generic]
+Iso_same_cst [in hydras.OrdinalNotations.ON_Generic]
+isqrt [in hydras.solutions_exercises.isqrt]
+isqrt_spec [in hydras.solutions_exercises.isqrt]
+is_lub [in hydras.Schutte.Lub]
+is_cnf_of [in hydras.Schutte.CNF]
+is_limit [in hydras.Schutte.Schutte_basics]
+is_succ [in hydras.Schutte.Schutte_basics]
+is_finite [in hydras.Schutte.Schutte_basics]
+is_successor_of [in gaia_hydras.onType]
+iterate [in hydras.Prelude.Iterates]
+iterate [in hydras.Ackermann.primRec]
+i1 [in hydras.OrdinalNotations.ON_Finite]
+i2 [in hydras.OrdinalNotations.ON_Finite]
+

K

+Kchain [in additions.Euclidean_Chains]
+Kchain_correct [in additions.AM]
+Kchain_correct [in additions.Euclidean_Chains]
+Kchain_correct_nat [in additions.Euclidean_Chains]
+KFF [in additions.AM]
+KFF [in additions.Euclidean_Chains]
+KFK [in additions.AM]
+KFK [in additions.Euclidean_Chains]
+Kkont [in additions.Euclidean_Chains]
+Kkont_equiv [in additions.Euclidean_Chains]
+Kkont_proper [in additions.Euclidean_Chains]
+kont_type [in additions.Euclidean_Chains]
+KP_arrowS [in hydras.Epsilon0.Paths]
+k17_7 [in additions.Euclidean_Chains]
+K2F [in additions.Euclidean_Chains]
+k3_1 [in additions.Euclidean_Chains]
+k7_3 [in additions.Euclidean_Chains]
+K99_24 [in additions.AM]
+

L

+language_decidable [in hydras.Ackermann.fol]
+Large [in hydras.Epsilon0.Large_Sets]
+large [in hydras.Epsilon0.Large_Sets]
+large [in gaia_hydras.GLarge_Sets]
+Largeb [in hydras.Epsilon0.Large_Sets]
+largeb [in hydras.Epsilon0.Large_Sets]
+LargeS [in hydras.Epsilon0.Large_Sets]
+largeS [in hydras.Epsilon0.Large_Sets]
+largeS [in gaia_hydras.GLarge_Sets]
+LargeSb [in hydras.Epsilon0.Large_Sets]
+largeSb [in hydras.Epsilon0.Large_Sets]
+large_set_check [in hydras.Epsilon0.Large_Sets_Examples]
+LE [in hydras.Epsilon0.T1]
+le [in hydras.Gamma0.T2]
+le [in hydras.OrdinalNotations.ON_O]
+le [in hydras.OrdinalNotations.ON_mult]
+le [in hydras.OrdinalNotations.ON_plus]
+le [in hydras.Schutte.Schutte_basics]
+Le [in hydras.Schutte.Well_Orders]
+Least [in hydras.Prelude.MoreOrders]
+least_fixpoint [in hydras.Schutte.Well_Orders]
+least_member [in hydras.Schutte.Well_Orders]
+leBool [in hydras.Ackermann.primRec]
+leq [in hydras.Prelude.MoreOrders]
+lex [in hydras.rpo.rpo]
+lexlt [in hydras.solutions_exercises.MultisetWf]
+lexnf [in hydras.solutions_exercises.MultisetWf]
+lex_nat [in hydras.MoreAck.Ack]
+lheight [in hydras.Hydra.Hydra_Definitions]
+lhsize [in hydras.Hydra.Hydra_Definitions]
+Limit [in hydras.Prelude.MoreOrders]
+limitb [in hydras.OrdinalNotations.ON_Omega2]
+limitb [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+limit_of [in gaia_hydras.onType]
+limit_v2 [in gaia_hydras.onType]
+listValues [in hydras.Ackermann.cPair]
+list_rec3 [in hydras.rpo.more_list]
+list_rec2 [in hydras.rpo.more_list]
+list_size [in hydras.rpo.more_list]
+LNN [in hydras.Ackermann.Languages]
+LNNArityF [in hydras.Ackermann.Languages]
+LNNArityR [in hydras.Ackermann.Languages]
+LNN2LNT_formula [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_terms [in hydras.Ackermann.LNN2LNT]
+LNN2LNT_term [in hydras.Ackermann.LNN2LNT]
+LNT [in hydras.Ackermann.Languages]
+LNTFunctionArity [in hydras.Ackermann.Languages]
+LNTRelationR [in hydras.Ackermann.Languages]
+LNT.Plus [in hydras.Ackermann.LNT]
+LNT.Succ [in hydras.Ackermann.LNT]
+LNT.Times [in hydras.Ackermann.LNT]
+LNT.Zero [in hydras.Ackermann.LNT]
+LNT2LNN_formula [in hydras.Ackermann.LNN2LNT]
+LNT2LNN_terms [in hydras.Ackermann.LNN2LNT]
+LNT2LNN_term [in hydras.Ackermann.LNN2LNT]
+LO.fin [in hydras.OrdinalNotations.OmegaOmega]
+LO.limitb [in hydras.OrdinalNotations.OmegaOmega]
+LO.lt [in hydras.OrdinalNotations.OmegaOmega]
+LO.mult [in hydras.OrdinalNotations.OmegaOmega]
+LO.nf [in hydras.OrdinalNotations.OmegaOmega]
+LO.nf_b [in hydras.OrdinalNotations.OmegaOmega]
+LO.plus [in hydras.OrdinalNotations.OmegaOmega]
+LO.refine [in hydras.OrdinalNotations.OmegaOmega]
+LO.succ [in hydras.OrdinalNotations.OmegaOmega]
+LO.succb [in hydras.OrdinalNotations.OmegaOmega]
+LO.t [in hydras.OrdinalNotations.OmegaOmega]
+LO.zero [in hydras.OrdinalNotations.OmegaOmega]
+LT [in hydras.Epsilon0.T1]
+lt [in hydras.Epsilon0.T1]
+lt [in hydras.OrdinalNotations.ON_O]
+lt [in hydras.Prelude.DecPreOrder]
+lt [in hydras.OrdinalNotations.ON_mult]
+lt [in hydras.OrdinalNotations.ON_plus]
+lt [in hydras.OrdinalNotations.ON_Finite]
+ltBool [in hydras.Ackermann.primRec]
+LTFormula [in hydras.Ackermann.LNN2LNT]
+lt_le_dec [in hydras.Epsilon0.T1]
+lt_eq_lt_dec [in hydras.Epsilon0.T1]
+lt_eq_lt_dec [in hydras.Prelude.Comparable]
+lt_depth [in hydras.Ackermann.fol]
+lt_ge_dec [in hydras.Gamma0.Gamma0]
+lt_eq_lt_dec [in hydras.Gamma0.Gamma0]
+lub [in hydras.Gamma0.Gamma0]
+L_3_plus_omega [in hydras.OrdinalNotations.ON_Omega2]
+L_omega_cube [in hydras.Epsilon0.Large_Sets]
+L_omega_square_times [in hydras.Epsilon0.Large_Sets]
+L_phi0_mult [in hydras.Epsilon0.Large_Sets]
+L_omega_square [in hydras.Epsilon0.Large_Sets]
+L_omega_mult [in hydras.Epsilon0.Large_Sets]
+L_omega [in hydras.Epsilon0.Large_Sets]
+L_fin [in hydras.Epsilon0.Large_Sets]
+L_lim [in hydras.Epsilon0.Large_Sets]
+L_succ [in hydras.Epsilon0.Large_Sets]
+L_test [in hydras.Epsilon0.Large_Sets]
+L_ [in hydras.Epsilon0.L_alpha]
+l_std [in hydras.Hydra.Battle_length]
+l_std [in gaia_hydras.GHydra]
+L_spec [in gaia_hydras.GLarge_Sets]
+L_3_plus_omega [in hydras.Epsilon0.E0]
+L_spec [in gaia_hydras.GL_alpha]
+L_ [in gaia_hydras.GL_alpha]
+L_3_plus_omega [in hydras.Schutte.Addition]
+l2hs [in hydras.Hydra.Hydra_Definitions]
+

M

+m [in hydras.Hydra.Hydra_Termination]
+M [in hydras.Hydra.BigBattle]
+m [in gaia_hydras.GHydra]
+majorized [in hydras.MoreAck.AckNotPR]
+majorizedPR [in hydras.MoreAck.AckNotPR]
+majorizedS [in hydras.MoreAck.AckNotPR]
+majorizedSPR [in hydras.MoreAck.AckNotPR]
+makeTrace [in hydras.Ackermann.codeSubFormula]
+makeTraceForall [in hydras.Ackermann.codeSubFormula]
+makeTraceImp [in hydras.Ackermann.codeSubFormula]
+makeTraceNot [in hydras.Ackermann.codeSubFormula]
+make_chain [in additions.AM]
+make_chain [in additions.Euclidean_Chains]
+Make.add [in hydras.rpo.list_set]
+Make.add_without_red [in hydras.rpo.list_set]
+Make.apply_subst [in hydras.rpo.term]
+Make.build_list_of_SN_terms [in hydras.rpo.rpo]
+Make.cardinal [in hydras.rpo.list_set]
+Make.DecVar.A [in hydras.rpo.term]
+Make.direct_subterm [in hydras.rpo.term]
+Make.elt [in hydras.rpo.list_set]
+Make.elt [in hydras.rpo.list_permut]
+Make.empty [in hydras.rpo.list_set]
+Make.eq_set [in hydras.rpo.list_set]
+Make.eq_elt_dec [in hydras.rpo.list_set]
+Make.eq_elt_dec [in hydras.rpo.list_permut]
+Make.filter [in hydras.rpo.list_set]
+Make.filter_aux [in hydras.rpo.list_set]
+Make.inter [in hydras.rpo.list_set]
+Make.is_a_pos [in hydras.rpo.term]
+Make.list_permut [in hydras.rpo.list_permut]
+Make.list_to_multiset [in hydras.rpo.list_permut]
+Make.make_set [in hydras.rpo.list_set]
+Make.map_subst [in hydras.rpo.term]
+Make.mem [in hydras.rpo.list_set]
+Make.o_size3 [in hydras.rpo.rpo]
+Make.o_size2 [in hydras.rpo.rpo]
+Make.o_size [in hydras.rpo.rpo]
+Make.remove_not_common [in hydras.rpo.list_set]
+Make.remove_red [in hydras.rpo.list_set]
+Make.replace_at_pos_list [in hydras.rpo.term]
+Make.replace_at_pos [in hydras.rpo.term]
+Make.rpo_term [in hydras.rpo.rpo]
+Make.rpo_rest [in hydras.rpo.rpo]
+Make.singleton [in hydras.rpo.list_set]
+Make.size [in hydras.rpo.term]
+Make.size2 [in hydras.rpo.rpo]
+Make.size3 [in hydras.rpo.rpo]
+Make.subset [in hydras.rpo.list_set]
+Make.substitution [in hydras.rpo.term]
+Make.subst_comp [in hydras.rpo.term]
+Make.subterm_at_pos [in hydras.rpo.term]
+Make.symbol [in hydras.rpo.term]
+Make.Term_eq_dec.eq_A_dec [in hydras.rpo.term]
+Make.Term_eq_dec.A [in hydras.rpo.term]
+Make.term_rec8 [in hydras.rpo.term]
+Make.term_rec7 [in hydras.rpo.term]
+Make.term_rec4 [in hydras.rpo.term]
+Make.term_rec3 [in hydras.rpo.term]
+Make.term_rec2 [in hydras.rpo.term]
+Make.union [in hydras.rpo.list_set]
+Make.variable [in hydras.rpo.term]
+Make.well_formed_subst [in hydras.rpo.term]
+Make.well_formed_list [in hydras.rpo.term]
+Make.well_formed [in hydras.rpo.term]
+Make.without_red [in hydras.rpo.list_set]
+map_without_repetition [in hydras.rpo.more_list]
+map12_without_repetition [in hydras.rpo.more_list]
+mark [in hydras.Prelude.Sort_spec]
+match2 [in hydras.Prelude.MoreVectors]
+max [in hydras.Prelude.Comparable]
+maximal_critical [in hydras.Schutte.Critical]
+max_v [in hydras.Prelude.MoreVectors]
+measure_lt [in hydras.OrdinalNotations.ON_Generic]
+mem [in hydras.Ackermann.fol]
+members [in hydras.solutions_exercises.OnCodeList]
+members [in hydras.Schutte.Schutte_basics]
+merge [in hydras.Prelude.Merge_Sort]
+merge_rec [in hydras.Prelude.Merge_Sort]
+merge_ind [in hydras.Prelude.Merge_Sort]
+min [in hydras.Prelude.Comparable]
+minimize [in Goedel.PRrepresentable]
+minus [in hydras.Epsilon0.T1]
+mlarge [in hydras.Epsilon0.Large_Sets]
+mlarge [in gaia_hydras.GLarge_Sets]
+mlargeS [in hydras.Epsilon0.Large_Sets]
+mlargeS [in gaia_hydras.GLarge_Sets]
+mod_eq [in additions.Monoid_instances]
+MoreExamples.cst [in hydras.MoreAck.PrimRecExamples]
+MoreExamples.cst0 [in hydras.MoreAck.PrimRecExamples]
+MoreExamples.fact [in hydras.MoreAck.PrimRecExamples]
+MoreExamples.mult [in hydras.MoreAck.PrimRecExamples]
+MoreExamples.plus [in hydras.MoreAck.PrimRecExamples]
+moser_lepper [in hydras.Gamma0.Gamma0]
+ms [in hydras.Hydra.Hydra_Termination]
+ms [in gaia_hydras.GHydra]
+mults_squares [in additions.AM]
+mult_fin_l [in hydras.OrdinalNotations.ON_Omega2]
+mult_fin_r [in hydras.OrdinalNotations.ON_Omega2]
+mult_mod [in additions.Monoid_instances]
+mult_fin_r [in hydras.Schutte.Addition]
+mult_Sn [in hydras.Schutte.Addition]
+mul2 [in additions.Fib2]
+my_pow [in additions.fib]
+m2lfib [in additions.fib]
+m2lmul [in additions.fib]
+m2lmx [in additions.fib]
+m2lpow [in additions.fib]
+M2N [in additions.Monoid_instances]
+M2_mult [in additions.Monoid_instances]
+M2.Id2 [in additions.FirstSteps]
+M2.Id2 [in additions.Naive]
+M2.M2_mult [in additions.FirstSteps]
+M2.M2_mult [in additions.Naive]
+M2.power [in additions.Naive]
+m3lfib [in additions.fib]
+m3lid [in additions.fib]
+m3lmul [in additions.fib]
+m3lmx [in additions.fib]
+m3lpow [in additions.fib]
+m4lfib [in additions.fib]
+m4lmul [in additions.fib]
+m4lmx [in additions.fib]
+m4lval [in additions.fib]
+M7_3 [in additions.AM]
+

N

+N [in hydras.Hydra.BigBattle]
+naryFunc [in hydras.Ackermann.extEqualNat]
+naryFunc [in hydras.Ackermann.model]
+naryIf [in hydras.solutions_exercises.MinPR2]
+naryRel [in hydras.Ackermann.extEqualNat]
+naryRel [in hydras.Ackermann.model]
+natModel [in hydras.Ackermann.PAconsistent]
+natToTerm [in hydras.Ackermann.LNN]
+natToTerm [in hydras.Ackermann.LNT]
+natToTermLNN [in hydras.Ackermann.codeNatToTerm]
+natToTermLNT [in hydras.Ackermann.codeNatToTerm]
+nat_2_term [in hydras.Gamma0.Gamma0]
+nbterms [in hydras.Gamma0.T2]
+nb_occ [in hydras.rpo.more_list]
+newVar [in hydras.Ackermann.folProp]
+next [in hydras.Hydra.BigBattle]
+next_step [in hydras.Hydra.Hydra_Lemmas]
+next_round [in hydras.Hydra.Hydra_Lemmas]
+next_round_dec [in hydras.Hydra.Hydra_Lemmas]
+nf [in hydras.Epsilon0.T1]
+nfb [in hydras.Gamma0.Gamma0]
+nf_Acc [in hydras.Epsilon0.T1]
+nf_rect [in hydras.Epsilon0.T1]
+nf_b [in hydras.Epsilon0.T1]
+NN [in hydras.Ackermann.NN]
+nnHelp [in hydras.Ackermann.model]
+NNnotations.LT [in hydras.Ackermann.LNN]
+NNnotations.Plus [in hydras.Ackermann.LNN]
+NNnotations.Succ [in hydras.Ackermann.LNN]
+NNnotations.Times [in hydras.Ackermann.LNN]
+NNnotations.Zero [in hydras.Ackermann.LNN]
+nnTranslate [in hydras.Ackermann.model]
+NN1 [in hydras.Ackermann.NN]
+NN2 [in hydras.Ackermann.NN]
+NN3 [in hydras.Ackermann.NN]
+NN4 [in hydras.Ackermann.NN]
+NN5 [in hydras.Ackermann.NN]
+NN6 [in hydras.Ackermann.NN]
+NN7 [in hydras.Ackermann.NN]
+NN8 [in hydras.Ackermann.NN]
+NN9 [in hydras.Ackermann.NN]
+non_commutativity_of_plus [in hydras.OrdinalNotations.ON_Omega2]
+normal [in hydras.Schutte.Ordering_Functions]
+notRel [in hydras.Ackermann.primRec]
+notZero [in hydras.Ackermann.primRec]
+nVars [in hydras.Ackermann.folProof]
+N_pow [in additions.Compatibility]
+N_bpow [in additions.FirstSteps]
+N_bpow [in additions.Pow_variant]
+N_bpow [in additions.Pow]
+N_mod.power [in additions.Naive]
+N_mod.mult_mod [in additions.Naive]
+N2pos [in additions.More_on_positive]
+

O

+OK [in additions.AM]
+OK [in additions.Euclidean_Chains]
+omega_omega [in hydras.Epsilon0.T1]
+omega_tower [in hydras.Epsilon0.T1]
+omega_term [in hydras.Epsilon0.T1]
+omega_ex2 [in hydras.Epsilon0.Large_Sets]
+omega_ex1 [in hydras.Epsilon0.Large_Sets]
+omega_square_thrice_eqn [in hydras.Epsilon0.Large_Sets]
+omega_tower [in hydras.Schutte.AP]
+omega_omega_1_3 [in hydras.Epsilon0.Large_Sets_Examples]
+omega_omega_1_4 [in hydras.Epsilon0.Large_Sets_Examples]
+Omega_limit_type [in hydras.Prelude.MoreOrders]
+Omega_limit [in hydras.Prelude.MoreOrders]
+omega_limit [in hydras.Schutte.Schutte_basics]
+om12 [in gaia_hydras.onType]
+om67 [in gaia_hydras.onType]
+ONDef.Exports.tricho [in gaia_hydras.onType]
+ONDef.on_struct [in gaia_hydras.onType]
+onFiniteMixin [in gaia_hydras.onType]
+onFiniteType [in gaia_hydras.onType]
+onOmegaMixin [in gaia_hydras.onType]
+onOmegaType [in gaia_hydras.onType]
+ON_op_ok [in hydras.OrdinalNotations.ON_Generic]
+ON_fun_ok [in hydras.OrdinalNotations.ON_Generic]
+ON_cst_ok [in hydras.OrdinalNotations.ON_Generic]
+ON_le [in hydras.OrdinalNotations.ON_Generic]
+ON_lt [in hydras.OrdinalNotations.ON_Generic]
+ON_compare [in hydras.OrdinalNotations.ON_Generic]
+ON_t [in hydras.OrdinalNotations.ON_Generic]
+OO.le [in hydras.OrdinalNotations.OmegaOmega]
+OO.lt [in hydras.OrdinalNotations.OmegaOmega]
+open [in hydras.Ackermann.PA]
+oplus [in hydras.Epsilon0.Hessenberg]
+oplus [in gaia_hydras.GHessenberg]
+optimal [in additions.Addition_Chains]
+optimal_generator [in additions.Addition_Chains]
+ord [in hydras.Schutte.Ordering_Functions]
+ordering_segment [in hydras.Schutte.Ordering_Functions]
+orH [in hydras.Ackermann.fol]
+orRel [in hydras.Ackermann.primRec]
+o_length [in hydras.rpo.more_list]
+o_finite_mult [in hydras.Epsilon0.Hessenberg]
+o_36_of_42 [in gaia_hydras.ON_gfinite]
+o_33_of_42 [in gaia_hydras.ON_gfinite]
+o_finite_mult [in gaia_hydras.GHessenberg]
+O3O [in hydras.OrdinalNotations.Example_3PlusOmega]
+

P

+PA [in hydras.Prelude.DecPreOrder_Instances]
+PA [in hydras.Ackermann.PA]
+PairwiseEqual [in hydras.Ackermann.folLogic3]
+parametric [in additions.Addition_Chains]
+path [in hydras.Epsilon0.Paths]
+pathS [in gaia_hydras.GPaths]
+pathS [in hydras.Epsilon0.Paths]
+path_tob [in gaia_hydras.GPaths]
+path_to [in gaia_hydras.GPaths]
+PA1 [in hydras.Ackermann.PA]
+PA2 [in hydras.Ackermann.PA]
+PA3 [in hydras.Ackermann.PA]
+PA4 [in hydras.Ackermann.PA]
+PA5 [in hydras.Ackermann.PA]
+PA6 [in hydras.Ackermann.PA]
+PA7 [in hydras.Ackermann.PA]
+PB [in hydras.Prelude.DecPreOrder_Instances]
+Permut.elt [in hydras.rpo.list_permut]
+Permut.eq_elt_dec [in hydras.rpo.list_permut]
+Permut.list_permut [in hydras.rpo.list_permut]
+Permut.list_to_multiset [in hydras.rpo.list_permut]
+phi [in hydras.Schutte.Critical]
+phi [in hydras.Gamma0.Gamma0]
+plus [in hydras.OrdinalNotations.ON_Omega2]
+plus [in hydras.Gamma0.Gamma0]
+plus [in hydras.Schutte.Addition]
+Pos_pow [in additions.Compatibility]
+Pos_bpow [in additions.FirstSteps]
+Pos_bpow [in additions.Pow_variant]
+Pos_bpow [in additions.Pow]
+pos_eq_dec [in additions.More_on_positive]
+power [in additions.FirstSteps]
+power [in additions.Pow_variant]
+power [in additions.Pow]
+power_t [in additions.FirstSteps]
+power_R [in additions.Addition_Chains]
+pow_17 [in additions.Pow_variant]
+pow_17 [in additions.Pow]
+pow3 [in hydras.Ackermann.codeSubFormula]
+pp [in hydras.Epsilon0.T1]
+ppE0 [in gaia_hydras.T1Bridge]
+pp_size [in hydras.Epsilon0.T1]
+pp0 [in hydras.Epsilon0.T1]
+PRcompose1 [in hydras.MoreAck.PrimRecExamples]
+pred [in hydras.Epsilon0.T1]
+pred [in hydras.Gamma0.T2]
+preorder_equiv [in hydras.Prelude.DecPreOrder]
+primRecFormula [in Goedel.PRrepresentable]
+primRecPiFormulaHelp [in Goedel.PRrepresentable]
+primRecsFormula [in Goedel.PRrepresentable]
+primRecSigmaFormula [in Goedel.PRrepresentable]
+primRecSigmaFormulaHelp [in Goedel.PRrepresentable]
+PrimRecs_PrimRec_ind [in hydras.Ackermann.primRec]
+PrimRecs_PrimRec_rec [in hydras.Ackermann.primRec]
+PrimRec_PrimRecs_ind [in hydras.Ackermann.primRec]
+PrimRec_PrimRecs_rec [in hydras.Ackermann.primRec]
+progressive [in hydras.Schutte.Schutte_basics]
+projectionList [in hydras.Ackermann.primRec]
+projectionListPR [in hydras.Ackermann.primRec]
+projFormula [in Goedel.PRrepresentable]
+proper_statement [in additions.Euclidean_Chains]
+proper_segment_of [in hydras.Schutte.Ordering_Functions]
+proper_segment [in hydras.Schutte.Ordering_Functions]
+psi [in hydras.Gamma0.T2]
+psi_term [in hydras.Gamma0.T2]
+ptwise_le [in hydras.Prelude.MoreLists]
+P87 [in additions.Addition_Chains]
+

R

+reachable [in hydras.Hydra.BigBattle]
+reassoc [in hydras.Epsilon0.T1]
+refinesPred [in gaia_hydras.T1Bridge]
+refinesRel [in gaia_hydras.T1Bridge]
+refines0 [in gaia_hydras.T1Bridge]
+refines1 [in gaia_hydras.T1Bridge]
+refines2 [in gaia_hydras.T1Bridge]
+rel_inv [in hydras.Schutte.PartialFun]
+rel_inj [in hydras.Schutte.PartialFun]
+rel_onto [in hydras.Schutte.PartialFun]
+rel_functional [in hydras.Schutte.PartialFun]
+rel_codomain [in hydras.Schutte.PartialFun]
+rel_domain [in hydras.Schutte.PartialFun]
+rel_enumerates [in hydras.Schutte.Countable]
+rel_numbers [in hydras.Schutte.Countable]
+remove [in hydras.rpo.more_list]
+remove_list [in hydras.rpo.more_list]
+rep [in hydras.OrdinalNotations.ON_Generic]
+ReplaceFormulaTerm [in hydras.Ackermann.codeSubFormula]
+ReplaceTermsTerm [in hydras.Ackermann.codeSubFormula]
+ReplaceTermTerm [in hydras.Ackermann.codeSubFormula]
+ReplaceTermTermsTerm [in hydras.Ackermann.codeSubFormula]
+Representable [in Goedel.PRrepresentable]
+Representable [in hydras.Ackermann.expressible]
+RepresentableAlternate [in Goedel.PRrepresentable]
+RepresentableHalf1 [in hydras.Ackermann.expressible]
+RepresentableHalf2 [in hydras.Ackermann.expressible]
+RepresentableHelp [in Goedel.PRrepresentable]
+RepresentableHelp [in hydras.Ackermann.expressible]
+RepresentablesHelp [in Goedel.PRrepresentable]
+Representable_ext [in Goedel.PRrepresentable]
+RepresentsInSelf [in Goedel.rosser]
+restrict [in hydras.Prelude.Restriction]
+restrict [in gaia_hydras.onType]
+restrict [in gaia_hydras.nfwfgaia]
+restrict_build [in hydras.Prelude.Restriction]
+round [in hydras.Hydra.Hydra_Definitions]
+round_t [in hydras.Hydra.Hydra_Definitions]
+round_spec [in hydras.Hydra.Hydra_Definitions]
+round_star [in hydras.Hydra.Hydra_Definitions]
+round_plus [in hydras.Hydra.Hydra_Definitions]
+round_n [in hydras.Hydra.Hydra_Definitions]
+R_nat_elaguee [in hydras.Schutte.GRelations]
+R_inv [in hydras.Schutte.GRelations]
+R1 [in hydras.Schutte.Ordering_Functions]
+r2i [in hydras.Schutte.PartialFun]
+R2_ind2 [in hydras.Hydra.Hydra_Lemmas]
+

S

+s [in hydras.solutions_exercises.Limit_Infinity]
+s [in hydras.solutions_exercises.T1_ltNotWf]
+segment [in hydras.Schutte.Ordering_Functions]
+Sentence [in hydras.Ackermann.LNN]
+Sentence [in hydras.Ackermann.LNT]
+Sentence [in hydras.Ackermann.folProp]
+seq [in hydras.solutions_exercises.MultisetWf]
+seq_mono [in hydras.Schutte.Schutte_basics]
+seq_range [in hydras.Schutte.Countable]
+set_eq [in hydras.Schutte.Schutte_basics]
+shift [in hydras.Prelude.MoreLists]
+signature_measure [in additions.Euclidean_Chains]
+signature_exponent [in additions.Euclidean_Chains]
+simple_last [in hydras.Prelude.MoreLists]
+slowexp [in additions.fib]
+small_h [in hydras.Hydra.Epsilon0_Needed_Generic]
+some [in hydras.Schutte.MoreEpsilonIota]
+some_pos' [in gaia_hydras.T1Choice]
+some_pos [in gaia_hydras.T1Choice]
+some_pos [in hydras.Schutte.MoreEpsilonIota]
+sort [in hydras.Prelude.Sort_spec]
+sorted [in hydras.Schutte.CNF]
+sort_spec [in hydras.Prelude.Sort_spec]
+sort_correct [in hydras.Prelude.Sort_spec]
+sort_rel [in hydras.Prelude.Sort_spec]
+sort_fun_t [in hydras.Prelude.Sort_spec]
+split_list [in hydras.rpo.more_list]
+sp_merge_sort [in hydras.Prelude.Merge_Sort]
+stable [in hydras.Prelude.Sort_spec]
+stable_test [in hydras.Prelude.Sort_spec]
+stable_merge_sort [in hydras.Prelude.Merge_Sort]
+stack [in additions.AM]
+standard_gnaw [in gaia_hydras.GPaths]
+standard_path [in gaia_hydras.GPaths]
+standard_path [in hydras.Epsilon0.Paths]
+standard_pathS [in hydras.Epsilon0.Paths]
+standard_gnaw [in hydras.Epsilon0.Paths]
+strict_lub_maj [in hydras.Epsilon0.T1]
+strict_lub_lub [in hydras.Epsilon0.T1]
+strict_lub [in hydras.Epsilon0.T1]
+strict_mono [in hydras.Prelude.Iterates]
+strict_mono [in gaia_hydras.T1Bridge]
+strongly_critical [in hydras.Schutte.Critical]
+subAllFormula [in hydras.Ackermann.subAll]
+subAllTerm [in hydras.Ackermann.subAll]
+subAllTerms [in hydras.Ackermann.subAll]
+SubON_same_op [in hydras.OrdinalNotations.ON_Generic]
+SubON_same_fun [in hydras.OrdinalNotations.ON_Generic]
+SubON_same_cst [in hydras.OrdinalNotations.ON_Generic]
+subStar [in Goedel.fixPoint]
+substF [in hydras.Ackermann.folProp]
+substituteFormulaForall [in hydras.Ackermann.folProp]
+substituteFormulaHelp [in hydras.Ackermann.folProp]
+substituteFormulaImp [in hydras.Ackermann.folProp]
+substituteFormulaNot [in hydras.Ackermann.folProp]
+substT [in hydras.Ackermann.folProp]
+substTs [in hydras.Ackermann.folProp]
+succ [in hydras.Epsilon0.T1]
+succ [in hydras.OrdinalNotations.ON_Omega2]
+succ [in hydras.Gamma0.T2]
+succ [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+succ [in hydras.Schutte.Schutte_basics]
+succb [in hydras.OrdinalNotations.ON_Omega2]
+succb [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Successor [in hydras.Prelude.MoreOrders]
+succFormula [in Goedel.PRrepresentable]
+succ_spec [in hydras.Schutte.Schutte_basics]
+sumToN [in hydras.Ackermann.cPair]
+sup [in hydras.Schutte.Schutte_basics]
+sup_spec [in hydras.Schutte.Schutte_basics]
+switchPR [in hydras.Ackermann.primRec]
+SysPrf [in hydras.Ackermann.folProof]
+SysPrf [in hydras.Ackermann.LNN]
+SysPrf [in hydras.Ackermann.LNT]
+System [in hydras.Ackermann.LNN]
+System [in hydras.Ackermann.LNT]
+System [in hydras.Ackermann.fol]
+S.cardinal [in hydras.rpo.list_set]
+S.elt [in hydras.rpo.list_set]
+S.eq_elt_dec [in hydras.rpo.list_set]
+S.subset [in hydras.rpo.list_set]
+S.without_red [in hydras.rpo.list_set]
+S2_ind2 [in hydras.Hydra.Hydra_Lemmas]
+

T

+t [in hydras.OrdinalNotations.ON_Omega2]
+t [in hydras.OrdinalNotations.ON_O]
+t [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+t [in hydras.solutions_exercises.MultisetWf]
+t [in hydras.OrdinalNotations.ON_mult]
+t [in hydras.OrdinalNotations.ON_plus]
+t [in hydras.OrdinalNotations.ON_Finite]
+ten [in hydras.Epsilon0.T1]
+Term [in hydras.Ackermann.LNN]
+Term [in hydras.Ackermann.LNT]
+terminates [in hydras.Prelude.WfVariant]
+Termination [in hydras.Hydra.Hydra_Definitions]
+Terms [in hydras.Ackermann.LNN]
+Terms [in hydras.Ackermann.LNT]
+Terms_Term_rec_full [in hydras.Ackermann.fol]
+Terms_Term_rec [in hydras.Ackermann.fol]
+Terms_Term_ind [in hydras.Ackermann.fol]
+Term_Terms_rec_full [in hydras.Ackermann.fol]
+Term_Terms_rec [in hydras.Ackermann.fol]
+Term_Terms_ind [in hydras.Ackermann.fol]
+Term.apply_subst [in hydras.rpo.term]
+Term.direct_subterm [in hydras.rpo.term]
+Term.is_a_pos [in hydras.rpo.term]
+Term.map_subst [in hydras.rpo.term]
+Term.replace_at_pos_list [in hydras.rpo.term]
+Term.replace_at_pos [in hydras.rpo.term]
+Term.size [in hydras.rpo.term]
+Term.substitution [in hydras.rpo.term]
+Term.subst_comp [in hydras.rpo.term]
+Term.subterm_at_pos [in hydras.rpo.term]
+Term.symbol [in hydras.rpo.term]
+Term.variable [in hydras.rpo.term]
+Term.well_formed_subst [in hydras.rpo.term]
+Term.well_formed_list [in hydras.rpo.term]
+Term.well_formed [in hydras.rpo.term]
+test [in hydras.Hydra.BigBattle]
+the [in hydras.Schutte.MoreEpsilonIota]
+the_least [in hydras.Schutte.Well_Orders]
+the_exponent [in additions.Addition_Chains]
+the_exponent_nat [in additions.Addition_Chains]
+the_ordering_segment [in hydras.Schutte.Ordering_Functions]
+tower2 [in hydras.solutions_exercises.MorePRExamples]
+tower2_alt [in hydras.solutions_exercises.MorePRExamples]
+Toy.Adds [in hydras.MoreAck.FolExamples]
+Toy.arityF [in hydras.MoreAck.FolExamples]
+Toy.arityR [in hydras.MoreAck.FolExamples]
+Toy.F1 [in hydras.MoreAck.FolExamples]
+Toy.F2 [in hydras.MoreAck.FolExamples]
+Toy.F3 [in hydras.MoreAck.FolExamples]
+Toy.F4 [in hydras.MoreAck.FolExamples]
+Toy.F5 [in hydras.MoreAck.FolExamples]
+Toy.F6 [in hydras.MoreAck.FolExamples]
+Toy.L [in hydras.MoreAck.FolExamples]
+Toy.Ldec [in hydras.MoreAck.FolExamples]
+Toy.Peirce [in hydras.MoreAck.FolExamples]
+Toy.PrfContrex7 [in hydras.MoreAck.FolExamples]
+Toy.PrfContrex9 [in hydras.MoreAck.FolExamples]
+Toy.PrfEx1 [in hydras.MoreAck.FolExamples]
+Toy.PrfEx10 [in hydras.MoreAck.FolExamples]
+Toy.PrfEx11 [in hydras.MoreAck.FolExamples]
+Toy.PrfEx2' [in hydras.MoreAck.FolExamples]
+Toy.PrfEx3 [in hydras.MoreAck.FolExamples]
+Toy.PrfEx4 [in hydras.MoreAck.FolExamples]
+Toy.PrfEx5 [in hydras.MoreAck.FolExamples]
+Toy.PrfEx6 [in hydras.MoreAck.FolExamples]
+Toy.PrfEx8 [in hydras.MoreAck.FolExamples]
+Toy.PrfEx9 [in hydras.MoreAck.FolExamples]
+Toy.SysPrfEx2 [in hydras.MoreAck.FolExamples]
+Toy.t0 [in hydras.MoreAck.FolExamples]
+Toy.t1 [in hydras.MoreAck.FolExamples]
+Toy.t2 [in hydras.MoreAck.FolExamples]
+Toy.t3 [in hydras.MoreAck.FolExamples]
+Toy.t4 [in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm8 [in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm7 [in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm6 [in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm5 [in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm4 [in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm3 [in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm2 [in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm1 [in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm0 [in hydras.MoreAck.FolExamples]
+Toy.Unnamed_thm [in hydras.MoreAck.FolExamples]
+Tprf2T'prf [in Goedel.goedel2]
+trace [in hydras.Hydra.Hydra_Definitions]
+trace_mult [in additions.Trace_exercise]
+trace_compare [in additions.Trace_exercise]
+transfinite_recursor [in hydras.Epsilon0.T1]
+transfinite_recursor_lt [in hydras.Epsilon0.T1]
+transfinite_induction_Q [in hydras.Gamma0.Gamma0]
+transfinite_induction [in hydras.Gamma0.Gamma0]
+transition [in gaia_hydras.GPaths]
+transition [in hydras.Epsilon0.Paths]
+transitionb [in gaia_hydras.GPaths]
+transition_S [in hydras.Epsilon0.Paths]
+translateLT [in hydras.Ackermann.LNN2LNT]
+Tree2T1 [in gaia_hydras.T1Choice]
+two [in additions.BinaryStrat]
+T' [in Goedel.rosserPA]
+T' [in Goedel.goedel2]
+T1add [in hydras.Epsilon0.T1]
+T1Choice [in gaia_hydras.T1Choice]
+T1F_ [in gaia_hydras.GF_alpha]
+T1is_succ [in hydras.Epsilon0.T1]
+T1leOrderMixin [in gaia_hydras.T1Choice]
+T1limit [in hydras.Epsilon0.T1]
+T1max [in gaia_hydras.T1Choice]
+T1min [in gaia_hydras.T1Choice]
+T1mixin [in gaia_hydras.T1Choice]
+T1mul [in hydras.Epsilon0.T1]
+T1nat [in hydras.Epsilon0.T1]
+T1orderType [in gaia_hydras.T1Choice]
+T1pp [in gaia_hydras.T1Bridge]
+T1toH [in gaia_hydras.GHydra]
+T1_eq_dec [in hydras.Epsilon0.T1]
+T1_le_Mixin [in gaia_hydras.T1Choice]
+T1_to_T2 [in hydras.Gamma0.T2]
+t1_0 [in hydras.MoreAck.LNN_Examples]
+T1_to_T2_R [in hydras.Gamma0.Gamma0]
+t1_0 [in hydras.MoreAck.FolExamples]
+T12Tree [in gaia_hydras.T1Choice]
+t2_length_aux [in hydras.Gamma0.T2]
+t2_length [in hydras.Gamma0.T2]
+T2_size [in hydras.Gamma0.Gamma0]
+T2_2_term [in hydras.Gamma0.Gamma0]
+

U

+Unbounded [in hydras.Schutte.Schutte_basics]
+Unnamed_thm9 [in gaia_hydras.T1Choice]
+Unnamed_thm8 [in gaia_hydras.T1Choice]
+Unnamed_thm7 [in gaia_hydras.T1Choice]
+Unnamed_thm6 [in gaia_hydras.T1Choice]
+Unnamed_thm5 [in gaia_hydras.T1Choice]
+Unnamed_thm5 [in gaia_hydras.T1Choice]
+Unnamed_thm4 [in gaia_hydras.T1Choice]
+Unnamed_thm3 [in gaia_hydras.T1Choice]
+Unnamed_thm2 [in gaia_hydras.T1Choice]
+Unnamed_thm1 [in gaia_hydras.T1Choice]
+Unnamed_thm0 [in gaia_hydras.T1Choice]
+Unnamed_thm [in gaia_hydras.T1Choice]
+Unnamed_thm [in hydras.Prelude.More_Arith]
+Unnamed_thm [in hydras.MoreAck.expressibleExamples]
+Unnamed_thm0 [in hydras.MoreAck.LNN_Examples]
+Unnamed_thm [in hydras.MoreAck.LNN_Examples]
+Unnamed_thm [in additions.Monoid_def]
+Unnamed_thm0 [in hydras.Ackermann.fol]
+Unnamed_thm [in hydras.Ackermann.fol]
+Unnamed_thm [in hydras.Epsilon0.Canon]
+Unnamed_thm [in hydras.OrdinalNotations.ON_Finite]
+Unnamed_thm1 [in hydras.Prelude.LibHyps_Experiments]
+Unnamed_thm0 [in hydras.Prelude.LibHyps_Experiments]
+Unnamed_thm [in hydras.Prelude.LibHyps_Experiments]
+Unnamed_thm1 [in hydras.solutions_exercises.F_3]
+Unnamed_thm0 [in hydras.solutions_exercises.F_3]
+Unnamed_thm [in hydras.solutions_exercises.F_3]
+Unnamed_thm [in additions.Addition_Chains]
+Unnamed_thm [in hydras.Epsilon0.Hprime]
+Unnamed_thm [in hydras.MoreAck.FolExamples]
+Unnamed_thm [in hydras.MoreAck.FolExamples]
+Unnamed_thm2 [in hydras.MoreAck.PrimRecExamples]
+Unnamed_thm1 [in hydras.MoreAck.PrimRecExamples]
+Unnamed_thm0 [in hydras.MoreAck.PrimRecExamples]
+Unnamed_thm [in hydras.MoreAck.PrimRecExamples]
+unshift [in hydras.Prelude.MoreLists]
+updateValue [in hydras.Ackermann.model]
+upper_bound [in hydras.Schutte.Lub]
+Usual.beta [in Goedel.PRrepresentable]
+

V

+varFormula [in hydras.Ackermann.codeSubFormula]
+Vars.var [in hydras.Gamma0.Gamma0]
+vector_nth [in hydras.Prelude.MoreVectors]
+vector_triple_rect [in hydras.Prelude.MoreVectors]
+vector_double_rect [in hydras.Prelude.MoreVectors]
+vect_from_fun [in hydras.Prelude.MoreVectors]
+Vec2_proj [in hydras.Prelude.MoreVectors]
+Vid [in hydras.Prelude.MoreVectors]
+

W

+wConsistent [in hydras.Ackermann.wConsistent]
+wellFormedFormula [in hydras.Ackermann.wellFormed]
+wellFormedTerm [in hydras.Ackermann.wellFormed]
+wellFormedTerms [in hydras.Ackermann.wellFormed]
+wellFormedTermTerms [in hydras.Ackermann.wellFormed]
+well_founded_restriction_rect [in hydras.Prelude.Restriction]
+well_founded_restriction [in hydras.Prelude.Restriction]
+well_founded_P [in gaia_hydras.nfwfgaia]
+Wfsum.extension [in gaia_hydras.nfwfgaia]
+Wfsum.f [in gaia_hydras.nfwfgaia]
+Wfsum.f_spec [in gaia_hydras.nfwfgaia]
+Wfsum.f1 [in gaia_hydras.nfwfgaia]
+Wfsum.f2 [in gaia_hydras.nfwfgaia]
+Wfsum.psum [in gaia_hydras.nfwfgaia]
+Wf_ex3.f [in gaia_hydras.nfwfgaia]
+Wf_ex3.f2 [in gaia_hydras.nfwfgaia]
+Wf_ex3.f1 [in gaia_hydras.nfwfgaia]
+Wf_ex3.f_spec [in gaia_hydras.nfwfgaia]
+Wf_ex3.lte [in gaia_hydras.nfwfgaia]
+Wf_ex.f [in gaia_hydras.nfwfgaia]
+Wf_ex.f2 [in gaia_hydras.nfwfgaia]
+Wf_ex.f1 [in gaia_hydras.nfwfgaia]
+Wf_ex.f_spec [in gaia_hydras.nfwfgaia]
+wInconsistent [in hydras.Ackermann.wConsistent]
+

X

+xPair [in hydras.Ackermann.cPair]
+

Z

+zero [in hydras.OrdinalNotations.ON_Omega2]
+zero [in hydras.Schutte.Schutte_basics]
+zeroFormula [in Goedel.PRrepresentable]
+ZeroLimitSucc_dec [in hydras.OrdinalNotations.ON_Generic]
+zero_limit_succ_dec [in hydras.Epsilon0.T1]
+zero_limit_succ_dec [in hydras.OrdinalNotations.ON_Omega2]
+Zero_limit_succ_dec [in hydras.OrdinalNotations.ON_Omega2]
+Zero_limit_succ_dec [in hydras.OrdinalNotations.ON_Omega_plus_omega]
+Zero_limit_succ_dec [in hydras.OrdinalNotations.ON_Finite]
+Zero_limit_succ_dec [in hydras.Epsilon0.E0]
+Zero_Limit_Succ_dec [in hydras.Epsilon0.E0]
+Zero_limit_succ_dec [in hydras.OrdinalNotations.ON_Omega]
+Zfibt [in additions.fib]
+ZtoR [in additions.fib]
+Z_pow [in additions.Compatibility]
+Z.power [in additions.Naive]
+

_

+_phi0 [in hydras.Schutte.AP]
+_omega [in hydras.Schutte.Schutte_basics]
+_f [in hydras.Schutte.Ordering_Functions]
+_A [in hydras.Schutte.Ordering_Functions]
+

other

+α [in gaia_hydras.HydraGaia_Examples]
+β [in gaia_hydras.HydraGaia_Examples]
+

+

Record Index

+

A

+Abelian_Monoid [in additions.Monoid_def]
+Abelian_EMonoid [in additions.Monoid_def]
+Assoc [in hydras.Prelude.STDPP_compat]
+

B

+Bad3.Fchain_proper [in additions.Euclidean_Chains]
+Battle [in hydras.Hydra.Hydra_Definitions]
+BoundedVariant [in hydras.Hydra.Hydra_Definitions]
+

C

+Comparable [in hydras.Prelude.Comparable]
+Compare [in hydras.Prelude.Comparable]
+

D

+Decision [in hydras.Prelude.STDPP_compat]
+

E

+EMonoid [in additions.Monoid_def]
+Equiv [in additions.Monoid_def]
+E0 [in gaia_hydras.T1Bridge]
+E0 [in hydras.Epsilon0.E0]
+

F

+Fchain_proper [in additions.Euclidean_Chains]
+Fkont_proper [in additions.Euclidean_Chains]
+

G

+G0.G0 [in hydras.Gamma0.Gamma0]
+

H

+Hvariant [in hydras.Hydra.Hydra_Definitions]
+

I

+InH [in hydras.Schutte.MoreEpsilonIota]
+isPR [in hydras.Ackermann.primRec]
+isPRrel [in hydras.Ackermann.primRec]
+

K

+Kchain_proper [in additions.Euclidean_Chains]
+

L

+Language [in hydras.Ackermann.fol]
+Lcode [in hydras.Ackermann.code]
+

M

+Make.SN_term [in hydras.rpo.rpo]
+Make.t [in hydras.rpo.list_set]
+Model [in hydras.Ackermann.model]
+Monoid [in additions.Monoid_def]
+Mult_op [in additions.Monoid_def]
+M2 [in additions.Monoid_instances]
+M2.t [in additions.FirstSteps]
+M2.t [in additions.Naive]
+

O

+ON [in hydras.OrdinalNotations.ON_Generic]
+ONDef.mixin_of [in gaia_hydras.onType]
+ONDef.pack_type [in gaia_hydras.onType]
+ON_correct [in hydras.OrdinalNotations.ON_Generic]
+ON_Iso [in hydras.OrdinalNotations.ON_Generic]
+OO.OO [in hydras.OrdinalNotations.OmegaOmega]
+ordering_function [in hydras.Schutte.Ordering_Functions]
+

P

+P [in hydras.Epsilon0.F_alpha]
+P [in hydras.Epsilon0.Hprime]
+

Q

+Q [in hydras.Epsilon0.F_alpha]
+

R

+RelDecision [in hydras.Prelude.STDPP_compat]
+

S

+Semibundled.TotalDec [in hydras.Prelude.DecPreOrder]
+Semibundled.TotalDecPreOrder [in hydras.Prelude.DecPreOrder]
+state [in hydras.Hydra.BigBattle]
+Strategy [in additions.Strategies]
+SubON [in hydras.OrdinalNotations.ON_Generic]
+S.t [in hydras.rpo.list_set]
+

T

+Total [in hydras.Prelude.DecPreOrder]
+TotalPreOrder [in hydras.Prelude.DecPreOrder]
+

W

+WfVariant [in hydras.Prelude.WfVariant]
+WO [in hydras.Schutte.Well_Orders]
+

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Global IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(28510 entries)
Notation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(182 entries)
Binder IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(20431 entries)
Module IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(94 entries)
Variable IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(764 entries)
Library IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(191 entries)
Lemma IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(3916 entries)
Constructor IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(281 entries)
Axiom IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(52 entries)
Inductive IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(119 entries)
Projection IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(110 entries)
Section IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(284 entries)
Instance IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(366 entries)
Abbreviation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(182 entries)
Definition IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(1486 entries)
Record IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(52 entries)
+
This page has been generated by coqdoc +
+ +
+ + + \ No newline at end of file diff --git a/theories/html/toc.html b/theories/html/toc.html new file mode 100644 index 00000000..292db1fa --- /dev/null +++ b/theories/html/toc.html @@ -0,0 +1,2100 @@ + + + + + +Table of contents + + + + +
+ + + +
+ +
+

Library hydras.OrdinalNotations.ON_Omega

+

Library hydras.OrdinalNotations.ON_Generic

+ +

Library hydras.OrdinalNotations.ON_Finite

+

Library hydras.OrdinalNotations.ON_mult

+ +

Library hydras.OrdinalNotations.ON_Omega_plus_omega

+

Library hydras.OrdinalNotations.ON_plus

+

Library hydras.OrdinalNotations.ON_O

+

Library hydras.OrdinalNotations.ON_Omega2

+ +

Library hydras.OrdinalNotations.Example_3PlusOmega

+

Library hydras.OrdinalNotations.OmegaOmega

+ +

Library hydras.Prelude.ssrnat_extracts

+ +

Library hydras.Prelude.MoreDecidable

+

Library hydras.Prelude.STDPP_compat

+

Library hydras.Prelude.Comparable

+

Library hydras.Prelude.First_toggle

+

Library hydras.Prelude.Simple_LexProd

+

Library hydras.Prelude.DecPreOrder

+

Library hydras.Prelude.Exp2

+

Library hydras.Prelude.Sort_spec

+

Library hydras.Prelude.Merge_Sort

+

Library hydras.Prelude.WfVariant

+

Library hydras.Prelude.More_Arith

+

Library hydras.Prelude.DecPreOrder_Instances

+

Library hydras.Prelude.Fuel

+

Library hydras.Prelude.Iterates

+ +

Library hydras.Prelude.OrdNotations

+

Library hydras.Prelude.MoreLists

+ +

Library hydras.Prelude.Restriction

+

Library hydras.Prelude.MoreOrders

+

Library hydras.Prelude.MoreVectors

+ +

Library hydras.Prelude.Compat815

+

Library hydras.Prelude.MoreLibHyps

+

Library hydras.Prelude.LibHyps_Experiments

+

Library hydras.Epsilon0.Hessenberg

+ +

Library hydras.Epsilon0.Large_Sets

+ +

Library hydras.Epsilon0.Epsilon0rpo

+ +

Library hydras.Epsilon0.Hprime

+ +

Library hydras.Epsilon0.E0

+ +

Library hydras.Epsilon0.T1

+ +

Library hydras.Epsilon0.Paths

+ +

Library hydras.Epsilon0.Canon

+ +

Library hydras.Epsilon0.Large_Sets_Examples

+

Library hydras.Epsilon0.L_alpha

+ +

Library hydras.Epsilon0.Epsilon0

+

Library hydras.Epsilon0.F_alpha

+ +

Library hydras.Epsilon0.F_omega

+ +

Library hydras.Gamma0.T2

+ +

Library hydras.Gamma0.Gamma0

+ +

Library hydras.rpo.decidable_set

+

Library hydras.rpo.dickson

+ +

Library hydras.rpo.list_set

+ +

Library hydras.rpo.list_permut

+ +

Library hydras.rpo.more_list

+ +

Library hydras.rpo.closure

+

Library hydras.rpo.rpo

+ +

Library hydras.rpo.term

+ +

Library hydras.Schutte.MoreEpsilonIota

+

Library hydras.Schutte.CNF

+ +

Library hydras.Schutte.AP

+ +

Library hydras.Schutte.Schutte_basics

+ +

Library hydras.Schutte.Ordering_Functions

+ +

Library hydras.Schutte.Countable

+

Library hydras.Schutte.Critical

+

Library hydras.Schutte.Schutte

+ +

Library hydras.Schutte.Well_Orders

+

Library hydras.Schutte.Addition

+ +

Library hydras.Schutte.Correctness_E0

+

Library hydras.Schutte.PartialFun

+ +

Library hydras.Schutte.Lub

+

Library hydras.Schutte.GRelations

+

Library hydras.Hydra.Battle_length

+

Library hydras.Hydra.Omega_Small

+

Library hydras.Hydra.BigBattle

+ +

Library hydras.Hydra.Epsilon0_Needed_Generic

+

Library hydras.Hydra.Hydra_Extraction

+

Library hydras.Hydra.KP_example

+

Library hydras.Hydra.Hydra_Termination

+ +

Library hydras.Hydra.Hydra_Theorems

+ +

Library hydras.Hydra.Omega2_Small

+ +

Library hydras.Hydra.Hydra_Definitions

+ +

Library hydras.Hydra.Epsilon0_Needed_Free

+ +

Library hydras.Hydra.Hydra_Examples

+

Library hydras.Hydra.Hydra_Lemmas

+ +

Library hydras.Hydra.O2H

+

Library hydras.Hydra.Epsilon0_Needed_Std

+ +

Library hydras.Ackermann.codeNatToTerm

+

Library hydras.Ackermann.code

+ +

Library hydras.Ackermann.extEqualNat

+

Library hydras.Ackermann.folLogic

+

Library hydras.Ackermann.fol

+ +

Library hydras.Ackermann.LNN

+ +

Library hydras.Ackermann.NN2PA

+

Library hydras.Ackermann.PAtheory

+

Library hydras.Ackermann.prLogic

+

Library hydras.Ackermann.wellFormed

+

Library hydras.Ackermann.checkPrf

+

Library hydras.Ackermann.codePA

+

Library hydras.Ackermann.cPair

+ +

Library hydras.Ackermann.folProof

+

Library hydras.Ackermann.Languages

+ +

Library hydras.Ackermann.LNT

+ +

Library hydras.Ackermann.NNtheory

+

Library hydras.Ackermann.PA

+

Library hydras.Ackermann.subAll

+

Library hydras.Ackermann.codeFreeVar

+

Library hydras.Ackermann.codeSubFormula

+

Library hydras.Ackermann.Deduction

+

Library hydras.Ackermann.folLogic2

+

Library hydras.Ackermann.folProp

+

Library hydras.Ackermann.ListExt

+

Library hydras.Ackermann.misc

+

Library hydras.Ackermann.NN

+ +

Library hydras.Ackermann.subProp

+

Library hydras.Ackermann.codeList

+

Library hydras.Ackermann.codeSubTerm

+

Library hydras.Ackermann.expressible

+

Library hydras.Ackermann.folLogic3

+

Library hydras.Ackermann.folReplace

+

Library hydras.Ackermann.LNN2LNT

+ +

Library hydras.Ackermann.model

+

Library hydras.Ackermann.PAconsistent

+

Library hydras.Ackermann.primRec

+ +

Library hydras.Ackermann.wConsistent

+

Library hydras.Ackermann.NewNotations

+

Library hydras.MoreAck.Ack

+ +

Library hydras.MoreAck.AckNotPR

+ +

Library hydras.MoreAck.FolExamples

+ +

Library hydras.MoreAck.Iterate_compat

+

Library hydras.MoreAck.PrimRecExamples

+ +

Library hydras.MoreAck.LNN_Examples

+

Library hydras.MoreAck.BadSubst

+ +

Library hydras.MoreAck.expressibleExamples

+

Library hydras.solutions_exercises.MinPR

+

Library hydras.solutions_exercises.MinPR2

+

Library hydras.solutions_exercises.FibonacciPR

+

Library hydras.solutions_exercises.MorePRExamples

+ +

Library hydras.solutions_exercises.isqrt

+

Library hydras.solutions_exercises.T1_ltNotWf

+

Library hydras.solutions_exercises.predSuccUnicity

+

Library hydras.solutions_exercises.lt_succ_le

+

Library hydras.solutions_exercises.Limit_Infinity

+

Library hydras.solutions_exercises.ge_omega_iff

+

Library hydras.solutions_exercises.schutte_cnf_counter_example

+

Library hydras.solutions_exercises.F_3

+ +

Library hydras.solutions_exercises.is_F_monotonous

+

Library hydras.solutions_exercises.MultisetWf

+

Library hydras.solutions_exercises.OnCodeList

+

Library additions.AM

+ +

Library additions.Addition_Chains

+ +

Library additions.BinaryStrat

+

Library additions.Compatibility

+ +

Library additions.Demo

+

Library additions.Demo_power

+

Library additions.Dichotomy

+

Library additions.Euclidean_Chains

+ +

Library additions.fib

+

Library additions.Fib2

+

Library additions.FirstSteps

+ +

Library additions.Monoid_def

+ +

Library additions.Monoid_instances

+ +

Library additions.More_on_positive

+ +

Library additions.Naive

+ +

Library additions.Pow

+ +

Library additions.Pow_variant

+ +

Library additions.Strategies

+

Library additions.Trace_exercise

+

Library additions.Wf_transparent

+

Library gaia_hydras.T1Bridge

+ +

Library gaia_hydras.GaiaToHydra

+

Library gaia_hydras.nfwfgaia

+ +

Library gaia_hydras.GCanon

+ +

Library gaia_hydras.GF_alpha

+ +

Library gaia_hydras.GL_alpha

+

Library gaia_hydras.GHprime

+ +

Library gaia_hydras.ON_gfinite

+

Library gaia_hydras.GPaths

+ +

Library gaia_hydras.GLarge_Sets

+

Library gaia_hydras.GHessenberg

+

Library gaia_hydras.HydraGaia_Examples

+

Library gaia_hydras.GHydra

+

Library gaia_hydras.T2Bridge

+ +

Library gaia_hydras.GPrelude

+

Library gaia_hydras.T1Choice

+

Library gaia_hydras.onType

+ +

Library Goedel.rosser

+

Library Goedel.goedel1

+

Library Goedel.PRrepresentable

+

Library Goedel.fixPoint

+

Library Goedel.codeSysPrf

+

Library Goedel.rosserPA

+

Library Goedel.goedel2

+
+
This page has been generated by coqdoc +
+ +
+ + + \ No newline at end of file

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