Addition of FunExt.lp, PropExt.lp and some theorems (#35)

Addition of some of the encodings necessary in the verification of Leo-III proofs, specifically:
- Addition of FunExt.lp with an axiom for functional extensionality
- Addition of PropExt.lp with an axiom for propositional extensionality and a number of theorems proving simplifications and equalities used for the conversion between propositional and equational literals
- Addition of nths to list.lp, along with a number of theorems
- Addition of a theorem to bool.lp

Author: melanie-taprogge

Bool.lp

@@ -34,6 +34,16 @@ with istrue false ↪ ⊥;
 
 coerce_rule coerce 𝔹 Prop $x ↪ istrue $x;
 
+symbol istrue=true [x] : π (istrue x) → π (x = true)≔
+begin
+  assume x h;
+  refine ∨ₑ (case_𝔹 x) _ _
+      { assume h1; refine h1 }
+      { assume h1;
+      have H1: π (istrue false) { rewrite eq_sym h1; refine h };
+      refine ⊥ₑ H1 };
+end;
+
 // non confusion of constructors
 
 opaque symbol false≠true : π (false ≠ true) ≔

CHANGES.md

@@ -3,6 +3,15 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/),
 and this project adheres to [Semantic Versioning](https://semver.org/).
 
+
+## Unreleased
+
+### Added
+
+- FunExt: Axiom for functional extensionality
+- PropExt: Axiom for propositional extensionality and related theorems
+- List: add nths
+
 ## 1.2.0 (2025-02-04)
 
 ### Added

FunExt.lp

@@ -0,0 +1,4 @@
+require open Stdlib.Set Stdlib.Prop Stdlib.Eq Stdlib.FOL Stdlib.HOL;
+
+symbol funExt [a b] (f g : τ (a ⤳ b)) : π(`∀ x, f x = g x) → π(f = g);
+

List.lp

@@ -38,6 +38,9 @@ seqn n x_0 ... x_n-1 == the sequence of the x_i; can be partially applied.
  ** Random access:
         nth x0 s i == the item i of s (numbered from 0), or x0 if s does
                       not have at least i+1 items (i.e., size x <= i)
+      nths x0 s ks == the sequence that is formed when taking for each 
+                      index i in n the i-th item of ks (numbered from 0), 
+                      or x0 if s has fewer than i+1 items (i.e., size s <= i). 
               s`_i == standard notation for nth x0 s i for a default x0,
                       e.g., 0 for rings.
   set_nth x0 s i y == s where item i has been changed to y; if s does not
@@ -1083,6 +1086,21 @@ begin
   assume a beq x l hrefl; simplify; rewrite hrefl; apply ⊤ᵢ;
 end;
 
+opaque symbol mem_tail [a] (beq: τ a → τ a → 𝔹) (n m: τ a) (l: 𝕃 a) : 
+  π (∈ beq n l) → π (∈ beq n (m ⸬ l)) ≔
+begin
+  assume a beq n m; induction
+    {assume h1; refine ⊥ₑ h1}
+    {assume n0 l h1 h2;
+    have H0: π (beq n n0) →  π (beq n m or (beq n n0 or ∈ beq n l))
+      {assume h3;
+      refine orᵢ₂ (beq n m) [beq n n0 or ∈ beq n l] (orᵢ₁ [beq n n0] (∈ beq n l) h3)};
+    have H1: π (∈ beq n l) →  π (beq n m or (beq n n0 or ∈ beq n l))
+      {assume h3;
+      refine orᵢ₂ (beq n m) [beq n n0 or ∈ beq n l] (orᵢ₂ (beq n n0) [∈ beq n l] h3)};
+    refine orₑ [beq n n0] [∈ beq n l] (beq n m or (beq n n0 or ∈ beq n l)) h2 H0 H1}
+end;
+
 opaque symbol mem_take [a] beq n l (x:τ a) :
   π (∈ beq x (take n l)) → π (∈ beq x l) ≔
 begin
@@ -1591,6 +1609,64 @@ begin
   }
 end;
 
+// nths
+
+symbol nths [a : Set] : τ a → 𝕃 a → 𝕃 nat → 𝕃 a;
+
+rule nths $b $l $n ↪ map (nth $b $l) $n;
+
+opaque symbol mapS_iota (m n : τ nat) : 
+  π (map (+1) (iota n m) = iota (n +1) m)≔
+begin
+  induction
+    { induction
+      { refine eq_refl □ }
+      { assume n h; refine h } }
+    { induction
+      { assume h n; refine eq_refl (n +1 ⸬ □) }
+      { simplify; assume n0 h0 h1 m; rewrite h1 (m +1);
+      refine eq_refl (m +1 ⸬ m +1 +1 ⸬ iota (m +1 +1 +1) n0) } }
+end;
+
+opaque symbol indexes_cons [a : Set] (x : τ a) (l : 𝕃 a) : 
+  π (indexes (x ⸬ l) = 0 ⸬ (map (+1) (indexes l)))≔
+    begin
+      assume a x;
+      induction
+        { refine eq_refl (0 ⸬ □) }
+        { simplify;
+        assume y l h;
+        rewrite mapS_iota;
+        refine eq_refl (0 ⸬ 1 ⸬ iota 2 (size l)) }
+end;
+
+opaque symbol nths_shift_cons [a : Set] (x b : τ a) (l : 𝕃 a) (n : 𝕃 nat) : 
+  π ((nths b (x ⸬ l) (map (+1) n)) = (nths b l n))≔
+begin
+  assume a x b l; induction
+    { refine eq_refl □ }
+    { assume n ll h1;
+    have H1: π (nths b (x ⸬ l) (map (+1) (n ⸬ ll)) =
+      nth b (x ⸬ l) (n +1) ⸬ nths b (x ⸬ l) (map (+1) ll))
+      { refine eq_refl (nth b l n ⸬ map (nth b (x ⸬ l)) (map (+1) ll)) };
+    rewrite H1; rewrite h1;
+    refine eq_refl (nths b l (n ⸬ ll)) }
+end;
+
+opaque symbol nths_indexes_id [a : Set] (b : τ a) (l : 𝕃 a) : 
+  π (nths b l (indexes l) = l)≔
+begin
+  assume a b; induction
+  { refine eq_refl □ }
+  { assume x l h1;
+    rewrite indexes_cons x l;
+    have H1: π (nths b (x ⸬ l) (0 ⸬ map (+1) (indexes l)) =
+      x ⸬ nths b (x ⸬ l) (map (+1) (indexes l)))
+    { refine eq_refl (x ⸬ map (nth b (x ⸬ l)) (map (+1) (indexes l))) };
+    rewrite H1; rewrite nths_shift_cons x b l (indexes l); rewrite h1;
+    refine eq_refl (x ⸬ l) }
+end;
+
 // sumn
 
 symbol sumn : 𝕃 nat → ℕ;