forked from SSProve/ssprove
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathRulesStateProb.v
1686 lines (1507 loc) · 57.7 KB
/
RulesStateProb.v
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
From Coq Require Import RelationClasses Morphisms Utf8.
From Mon Require Import SPropMonadicStructures SpecificationMonads MonadExamples
SPropBase FiniteProbabilities.
From Relational Require Import OrderEnrichedCategory
OrderEnrichedRelativeMonadExamples Commutativity GenericRulesSimple.
Set Warnings "-notation-overridden,-ambiguous-paths".
From mathcomp Require Import all_ssreflect all_algebra reals distr realsum
finmap.set finmap.finmap xfinmap.
Set Warnings "notation-overridden,ambiguous-paths".
From Crypt Require Import Axioms ChoiceAsOrd SubDistr Couplings Theta_dens
Theta_exCP LaxComp FreeProbProg RelativeMonadMorph_prod
StateTransformingLaxMorph choice_type.
Import SPropNotations.
Import Num.Theory.
#[local] Open Scope ring_scope.
#[local] Definition ops_StP (S : choiceType) :=
@ops_StP S.
#[local] Definition ar_StP (S : choiceType) :=
@ar_StP S.
(* free monad *)
Definition FrStP (S : choiceType) :=
@FrStP S.
#[local] Definition pure {S : choiceType} {A : ord_choiceType} (a : A) :=
ord_relmon_unit (FrStP S) A a.
#[local] Definition bindF {S : choiceType} {A B : ord_choiceType}
(f : TypeCat ⦅ choice_incl A; FrStP S B ⦆ ) (m : FrStP S A) :=
ord_relmon_bind (FrStP S) f m.
Definition retF {S : choiceType} {A : choiceType} (a : A) :=
retrFree (ops_StP S) (ar_StP S) A a.
(* morphism *)
Definition θ {S1 S2 : choiceType} :=
@thetaFstdex S1 S2.
Definition θ0 {S : choiceType} {A : ord_choiceType} (c : FrStP S A) :=
@unaryIntState S A c.
(* spec monad *)
#[local] Definition WrelSt {S1 S2 : choiceType} :=
rlmm_codomain (@θ S1 S2).
(* Rem.: this spec monad is a ordered relative monad, while previously we were using an ordered monad *)
Definition retW {A1 A2 : ord_choiceType} {S1 S2 : choiceType} (a : A1 * A2) :
Base.dfst (@WrelSt S1 S2 ⟨ A1, A2 ⟩).
Proof.
apply (ord_relmon_unit WrelSt).
simpl.
exact: a.
Defined.
Definition bindW {A1 A2 B1 B2 : ord_choiceType} {S1 S2 : choiceType}
(w : Base.dfst (@WrelSt S1 S2 ⟨ A1, A2 ⟩))
(f : A1 * A2 → Base.dfst (@WrelSt S1 S2 ⟨ B1, B2 ⟩)) :
Base.dfst (@WrelSt S1 S2 ⟨ B1, B2 ⟩).
Proof.
unshelve eapply (ord_relmon_bind WrelSt).
- simpl. exact: npair A1 A2.
- simpl.
exists f.
move => [a1 a2] [b1 b2] Hleq.
inversion Hleq.
by move => [s1 s2] π H.
- exact: w.
Defined.
Import OrderEnrichedRelativeMonadExamplesNotation.
Definition semantic_judgement (A1 A2 : ord_choiceType) {S1 S2 : choiceType}
(c1 : FrStP S1 A1) (c2 : FrStP S2 A2)
(w : Base.dfst (WrelSt ⟨ A1, A2 ⟩)) : Prop :=
(θ ⟨A1,A2⟩)∙1 ⟨c1,c2⟩ ≤ w.
Definition fromPrePost {A1 A2 : ord_choiceType} {S1 S2: choiceType}
(pre : (S1 * S2) → Prop)
(post : (A1 * S1) → (A2 * S2) → Prop) :
Base.dfst (@WrelSt S1 S2 ⟨ A1, A2 ⟩).
Proof.
simpl.
unshelve econstructor.
move=> [is1 is2]. unshelve econstructor.
move=> myPost.
exact (
pre (is1,is2) ∧
∀ as1 as2, (post as1 as2) → myPost (as1, as2)
).
move => x y Hxy [H1 H2].
split.
- assumption.
- move => as1 as2 post12. apply: Hxy. by apply: H2.
move => x y Heq π.
by rewrite Heq.
Defined.
Declare Scope rsemantic_scope.
Delimit Scope rsemantic_scope with rsem.
Module RSemanticNotation.
Notation "⊨ c1 ≈ c2 [{ w }]" :=
(semantic_judgement _ _ c1 c2 w) : rsemantic_scope.
Notation "⊨ ⦃ pre ⦄ c1 ≈ c2 ⦃ post ⦄" :=
(semantic_judgement _ _ c1 c2 (fromPrePost pre post))
: rsemantic_scope.
End RSemanticNotation.
Import RSemanticNotation.
#[local] Open Scope rsemantic_scope.
Import finmap.set finmap.finmap xfinmap.
Open Scope fset_scope.
Definition d_inv {A1 A2 : choiceType} (d : SDistr (F_choice_prod ⟨ A1, A2 ⟩)) :
SDistr (F_choice_prod ⟨ A2, A1 ⟩) :=
dswap d.
Lemma d_inv_coupling {A1 A2} {c1 : SDistr A1} {c2 : SDistr A2}
(d : SDistr (F_choice_prod ⟨ A1, A2 ⟩ )) (d_coupling : coupling d c1 c2) :
coupling (d_inv d) c2 c1.
Proof.
unfold coupling. split.
- unfold lmg. unfold d_inv.
apply distr_ext. move=> x. erewrite (dfst_dswap d).
destruct d_coupling as [lH rH]. rewrite -rH. unfold rmg. reflexivity.
- unfold rmg. unfold d_inv.
apply distr_ext. move=> x. erewrite (dsnd_dswap d).
destruct d_coupling as [lH rH]. rewrite -lH. unfold lmg. reflexivity.
Qed.
Theorem inv_rule {A1 A2 : ord_choiceType} {S1 S2 : choiceType} {P Q}
(c1 : FrStP S1 A1) (c2 : FrStP S2 A2)
(H : ⊨ ⦃ P ⦄ c1 ≈ c2 ⦃ Q ⦄ ) :
⊨ ⦃ λ '(st1,st2), P (st2, st1) ⦄ c2 ≈ c1 ⦃ λ as1 as2, Q as2 as1 ⦄.
Proof.
move => [st1 st2] /=. move => π [H1 H2] /=.
specialize (H (st2, st1) (fun '(as1, as2) => π (as2, as1))).
simpl in H.
destruct H as [d [d_coupling Hd]].
split; auto.
exists (@d_inv _ _ d).
split.
by apply: d_inv_coupling.
move => as2 as1 H'.
apply: Hd.
rewrite /d_inv /= in H'. destruct d. simpl in *.
rewrite dswapE in H'. cbn in H'. assumption.
Qed.
(* GENERIC MONADIC RULES *)
Theorem ret_rule {A1 A2 : ord_choiceType} {S1 S2 : choiceType}
(a1 : A1) (a2 : A2):
⊨ @pure S1 A1 a1 ≈ @pure S2 A2 a2 [{ retW (a1, a2) }].
Proof.
rewrite /semantic_judgement /θ.
unfold "≤". simpl.
rewrite /MonoCont_order //=. move => [ss1 ss2] πa1a2 /=.
exists (SDistr_unit (F_choice_prod (npair (prod_choiceType A1 S1) (prod_choiceType A2 S2)))
((a1, ss1), (a2, ss2))).
split.
- rewrite /SubDistr.SDistr_obligation_1 /=.
by apply SDistr_unit_F_choice_prod_coupling.
- move => b1 b2 Hb1b2 /=.
by rewrite -(distr_get _ _ Hb1b2).
Qed.
Theorem weaken_rule {A1 A2 : ord_choiceType} {S1 S2 : choiceType}
{d1 : FrStP S1 A1}
{d2 : FrStP S2 A2} :
∀ w w', (⊨ d1 ≈ d2 [{ w }]) → w ≤ w' → (⊨ d1 ≈ d2 [{ w' }] ).
Proof.
rewrite /semantic_judgement.
by etransitivity.
Qed.
Theorem bind_rule {A1 A2 B1 B2 : ord_choiceType} {S1 S2 : choiceType}
{f1 : A1 → FrStP S1 B1}
{f2 : A2 → FrStP S2 B2}
(m1 : FrStP S1 A1)
(m2 : FrStP S2 A2)
(wm : Base.dfst (WrelSt ⟨ A1, A2 ⟩))
(judge_wm : ⊨ m1 ≈ m2 [{ wm }])
(wf : (A1 * A2) → Base.dfst (WrelSt ⟨ B1, B2 ⟩))
(judge_wf : ∀ a1 a2, ⊨ (f1 a1) ≈ (f2 a2) [{ (wf (a1, a2)) }]) :
⊨ (bindF f1 m1 ) ≈ (bindF f2 m2) [{ bindW wm wf }].
Proof.
move => [st1 st2].
etransitivity.
rewrite /bindF /=.
by apply (rlmm_law2 _ _ _ _ θ ⟨ A1, A2 ⟩ ⟨ B1, B2 ⟩ ⟨ f1 , f2 ⟩ ⟨ m1 , m2 ⟩ (st1, st2)).
rewrite /semantic_judgement in judge_wm, judge_wf.
destruct A1 as [A1 chA1]. destruct A2 as [A2 chA2].
destruct B1 as [B1 chB1]. destruct B2 as [B2 chB2].
simpl in *.
apply (@omon_bind WProp (A1 * S1 * (A2 * S2)) (B1 * S1 * (B2 * S2)) _ _ (judge_wm (st1, st2))).
move => [[a1 st1'] [a2 st2']].
by apply: (judge_wf a1 a2).
Qed.
Theorem bind_rule_pp {A1 A2 B1 B2 : ord_choiceType} {S1 S2 : choiceType}
{f1 : A1 → FrStP S1 B1}
{f2 : A2 → FrStP S2 B2}
(m1 : FrStP S1 A1)
(m2 : FrStP S2 A2)
(pre : S1 * S2 → Prop)
(middle : (A1 * S1) → (A2 * S2) → Prop)
(post : (B1 * S1) → (B2 * S2) → Prop)
(judge_wm : ⊨ ⦃ pre ⦄ m1 ≈ m2 ⦃ middle ⦄)
(judge_wf : ∀ a1 a2,
⊨ ⦃ λ '(s1, s2), middle (a1, s1) (a2, s2) ⦄
f1 a1 ≈ f2 a2
⦃ post ⦄ ) :
⊨ ⦃ pre ⦄ (bindrFree _ _ m1 f1 ) ≈ (bindrFree _ _ m2 f2) ⦃ post ⦄.
Proof.
destruct S1, S2, A1, A2, B1, B2.
eapply weaken_rule.
- apply bind_rule with (wf := (fun '(a1, a2) => fromPrePost (fun '(s1, s2) => middle (a1, s1) (a2, s2)) post)).
+ exact judge_wm.
+ exact judge_wf.
- cbv. intuition.
Qed.
(* Pre-condition manipulating rules *)
Theorem pre_weaken_rule {A1 A2 : ord_choiceType} {S1 S2 : choiceType}
{d1 : FrStP S1 A1}
{d2 : FrStP S2 A2} :
∀ (pre pre' : S1 * S2 → Prop) post,
(⊨ ⦃ pre ⦄ d1 ≈ d2 ⦃ post ⦄) →
(∀ st1 st2, pre' (st1, st2) → pre (st1, st2) ) →
(⊨ ⦃ pre' ⦄ d1 ≈ d2 ⦃ post ⦄).
Proof.
move => w w' post Hjudg Hleq. move => [st1 st2].
move => π [H1 H2]. simpl in π.
apply: Hjudg.
rewrite /fromPrePost /=.
split.
- by apply: Hleq.
- assumption.
Qed.
Theorem pre_hypothesis_rule {A1 A2 : ord_choiceType} {S1 S2 : choiceType}
{d1 : FrStP S1 A1}
{d2 : FrStP S2 A2} :
∀ (pre : S1 * S2 → Prop) post,
(∀ st1 st2,
pre (st1, st2) →
⊨ ⦃ (λ st, st.1 = st1 ∧ st.2 = st2 ) ⦄ d1 ≈ d2 ⦃ post ⦄
) →
(⊨ ⦃ pre ⦄ d1 ≈ d2 ⦃ post ⦄).
Proof.
move => pre post Hjudg. move => [st1 st2].
move => π [H1 H2] /=. simpl in π.
apply: (Hjudg st1 st2 H1 (st1, st2)).
by rewrite /fromPrePost /=.
Qed.
Theorem pre_strong_hypothesis_rule {A1 A2 : ord_choiceType} {S1 S2 : choiceType}
{d1 : FrStP S1 A1}
{d2 : FrStP S2 A2} :
forall (pre : S1 * S2 -> Prop) post, (forall st1 st2, pre (st1, st2)) -> (⊨ ⦃ (fun st => True ) ⦄ d1 ≈ d2 ⦃ post ⦄) ->
(⊨ ⦃ pre ⦄ d1 ≈ d2 ⦃ post ⦄).
Proof.
move => pre post Hpre Hjudg.
by apply (pre_weaken_rule (fun st => True) _).
Qed.
(* Rem.: took around 40s to Qed. *)
(* post-condition manipulating rules *)
(* Rem.: simplified the proof resorting to weaken_rule, should be quickier *)
Theorem post_weaken_rule {A1 A2 : ord_choiceType} {S1 S2 : choiceType}
{d1 : FrStP S1 A1}
{d2 : FrStP S2 A2} :
forall (pre : S1 * S2 -> Prop) (post1 post2 : A1 * S1 -> A2 * S2 -> Prop),
(⊨ ⦃ pre ⦄ d1 ≈ d2 ⦃ post1 ⦄) ->
(forall as1 as2, post1 as1 as2 -> post2 as1 as2) -> (⊨ ⦃ pre ⦄ d1 ≈ d2 ⦃ post2 ⦄).
Proof.
move => pre post1 post2 Hjudg Hleq.
eapply weaken_rule.
- exact Hjudg.
- cbv. intuition.
Qed.
Declare Scope RulesStateProb_scope.
Delimit Scope RulesStateProb_scope with RSP.
Module RSPNotation.
Notation "x <- c1 ;; c2" :=
(bindF (fun x => c2) c1)
(right associativity, at level 84, c1 at next level)
: RulesStateProb_scope.
Notation " x ∈ T <<- c1 ;; c2 " :=
(bindF (fun x : T => c2) c1)
(right associativity, at level 90, c1 at next level)
: RulesStateProb_scope.
Notation "c1 ;; c2" :=
(bindF (fun _ => c2) c1)
(at level 100, right associativity)
: RulesStateProb_scope.
End RSPNotation.
Import RSPNotation.
Open Scope RulesStateProb_scope.
Theorem seq_rule { A1 A2 : ord_choiceType }
{ B1 B2 : ord_choiceType }
{S1 S2 : choiceType}
{f1 : A1 -> FrStP S1 B1}
{f2 : A2 -> FrStP S2 B2}
(m1 : FrStP S1 A1) (m2 : FrStP S2 A2)
(P : S1 * S2 -> Prop) (R : A1 * S1 -> A2 * S2 -> Prop)
(Q : B1 * S1 -> B2 * S2 -> Prop)
(judge1 : ⊨ ⦃ P ⦄ m1 ≈ m2 ⦃ R ⦄ )
(judge2 : forall a1 a2, ⊨ ⦃ (fun st => R (a1, st.1) (a2, st.2)) ⦄ (f1 a1) ≈ (f2 a2) ⦃ Q ⦄ ) :
⊨ ⦃ P ⦄ x ∈ A1 <<- m1 ;; f1 x ≈ x ∈ A2 <<- m2 ;; f2 x ⦃ Q ⦄.
Proof.
have H : ⊨ x ∈ A1 <<- m1;; f1 x ≈ x ∈ A2 <<- m2;; f2 x
[{bindW (fromPrePost P R) (fun a : A1 * A2 => fromPrePost (fun st : S1 * S2 => R (a.1, st.1) (a.2, st.2)) Q)}]
:= (bind_rule m1 m2 (fromPrePost P R) judge1
(fun a => fromPrePost (fun st => R (a.1, st.1) (a.2, st.2)) Q) judge2).
rewrite /fromPrePost.
move => [st1 st2] /=.
move => β [Hbeta1 Hbeta2].
specialize (H (st1, st2) β). simpl in H. destruct H as [HH1 [HH2 HH2']].
split; auto. rewrite /fromPrePost in judge2.
move => [a1 sst1] [a2 sst2] HR.
specialize (judge2 a1 a2).
split; assumption.
exists HH1.
split; assumption.
Qed.
(* Rem.: can we do \sum_ ( c \in C ) ... where C : choiceType? *)
Definition prod_comp { L R } { M : finType } { d1 : SDistr L } {d2 : SDistr M } { d3 : SDistr R } d12 d23
( H12 : coupling d12 d1 d2 ) ( H23 : coupling d23 d2 d3 ): SDistr (F_choice_prod ⟨ L, R ⟩).
Proof.
exists (fun '(l, r) => \sum_ ( m <- (index_enum M) | m \in (dinsupp d2) ) (d12 (l,m) * d23 (m,r)) / (d2 m)).
- admit.
- admit.
- admit.
Admitted.
Definition prod_comp_coupling { L R } { M : finType} { d1 : SDistr L } {d2 : SDistr M } { d3 : SDistr R }
{ d12 d23 } (H12 : coupling d12 d1 d2) (H23 : coupling d23 d2 d3):
coupling (@prod_comp L R M d1 d2 d3 d12 d23 H12 H23) d1 d3.
Proof. Admitted.
(* useful to introduce intermediate games [Formal Certification of Code-Based Cryptographic Proofs, page 24] *)
Theorem comp_rule { A1 A3 : ord_choiceType } { A2 S2 : finType } { S1 S3 : choiceType }
{ P P' } { Q Q'}
(c1 : FrStP S1 A1) (c2 : FrStP S2 A2) (c3 : FrStP S3 A3)
(H12 : ⊨ ⦃ P ⦄ c1 ≈ c2 ⦃ Q ⦄)
(H23 : ⊨ ⦃ P' ⦄ c2 ≈ c3 ⦃ Q' ⦄) :
⊨ ⦃ fun '(s1, s3) => exists s2, P(s1,s2) /\ P'(s2,s3) ⦄
c1 ≈ c3
⦃ fun as1 as3 => exists as2, Q as1 as2 /\ Q' as2 as3⦄.
Proof.
move => [s1 s3].
move => π. simpl in π.
rewrite /fromPrePost /=. move => [[s2 [HP HP']] H].
specialize (H12 (s1, s2) (fun '(as1, as2) => forall as3, Q' as2 as3 -> π (as1, as3)) ). simpl in H12.
destruct H12 as [d12 [coupling12 H12]].
{ split.
assumption.
move => as1 as2 HQ as3 HQ'.
apply: (H as1 as3).
exists as2. split; assumption. }
specialize (H23 (s2, s3) (fun '(as2, as3) => forall as1, Q as1 as2 -> π (as1, as3)) ). simpl in H23.
destruct H23 as [d23 [coupling23 H23]].
{ split.
assumption.
move => as2 as3 HQ as1 HQ'.
apply: (H as1 as3).
exists as2. split; assumption. }
pose d13 := prod_comp d12 d23 coupling12 coupling23.
exists d13.
split.
by apply: prod_comp_coupling.
move => as1 as3 d13_gt0.
apply: (H as1 as3).
(* by definition of d13*) admit.
Admitted.
(* Lemma bij_pres_summable { A B: choiceType } { d : A -> R } { f : A -> B } {finv : B -> A } *)
(* (kinvf : cancel finv f) (kfinv : cancel f finv) ( H: summable (T:= A) (R:=R) d): *)
(* summable (T:=B) (R:=R) (fun b : B => d (finv b)). *)
(* Admitted. *)
(*CA: not used *)
Definition d__f { A B : ord_choiceType} { d : SDistr A } { f : A -> B } : SDistr B. Admitted.
(*CA's proof sketch
d__f : B -> [0,1]
b ↦ ∑_{a ∈ A: f(a) = b} d(a) // d-measure of the pre-image of b -- in particular if b is not in image(f) then d__f(b) = 0 //
- 0 ≤ d__f (b) because sum of non-negative quantities
- for J ⊆ B, d__f (J) = d (f^-1 (J)) that is finite
- ∑_{b ∈ B} d__f(b) = ∑_{a ∈ A} d(a)
*)
(* CA: old proof for a bijective f *)
(* Proof. *)
(* destruct d as [d Hd1 Hd2 Hd3]. *)
(* unshelve eexists. *)
(* { move => b. exact: d (finv b). } *)
(* - move => b /=. by apply: Hd1. *)
(* - by apply: bij_pres_summable. *)
(* - unshelve erewrite <- reindex_psum. *)
(* { apply: predT. } *)
(* assumption. *)
(* by []. *)
(* exists f. *)
(* -- move => x H. apply: kinvf. *)
(* -- move => x H. apply: kfinv. *)
(* Defined. *)
(*CA: not used *)
Theorem post_conclusion_rule {A0 A1 B : ord_choiceType} { S : choiceType } { pre : S * S -> Prop }
{c0 : FrStP S A0 } { c1 : FrStP S A1 }
{ f0 : A0 -> B } { f1 : A1 -> B } (* (Hbij0 : bijective f0) (Hbij1 : bijective f1) *)
(H : ⊨ ⦃ pre ⦄
(x0 <- c0 ;; retF x0) ≈
(x1 <- c1 ;; retF x1)
⦃ fun '(a0, s0) '(a1, s1) => s0 = s1 /\ f0 a0 = f1 a1 ⦄) :
⊨ ⦃ pre ⦄ x0 <- c0 ;; retF (f0 x0) ≈ x1 <- c1 ;; retF (f1 x1) ⦃ eq ⦄.
Proof.
move => [s0 s1].
specialize (H (s0, s1)).
unfold "≤" in *. simpl. simpl in H.
rewrite /MonoCont_order //=. rewrite /MonoCont_order //= in H.
move => β [hs0 h].
specialize (H (fun '(a0, s0, (a1, s1)) => (β (f0 a0 ,s0, (f1 a1, s1))))).
destruct H as [d [H H']].
split.
- assumption.
- move => [a1 st1] [a2 st2] [Heqa Heqst]. subst.
apply: h. by rewrite Heqst.
- unshelve eexists.
{ unshelve eapply d__f.
exact: F_choice_prod ⟨ F_choice_prod ⟨ A0, S ⟩, F_choice_prod ⟨ A1, S ⟩ ⟩.
exact: d.
move => [[a0 st0] [a1 st1]]. exact: (f0 a0, st0, (f1 a1, st1)). }
split.
{ (*CA: let fs0 : A0 * S -> B * S = fun (a0, st) => (f a0, st)
to prove the lmg it suffices to show that
"θ (x <- c0 ;; ret (f0 x)) : SDistr B * S" =
" d__fs0 (θ (x <- c0 ;; ret x) "
// it will map (b,s) ↦ ∑_{a0 ∈ A0: f(a0) = b} θ (x <- c0; ret x) //
indeed for (b1,st1),
Σ_{(b0, st0)} d' (b0,st0) (b1,st1) =
Σ_{(a0,st0) a1 : f0(a0) = b0 /\ f1(a1) = b1} d (a0,st0) (a1,st1) = [coupling d _ _ ]
Σ_{a1 : f1(a1) = } θ (x <- c1 ;; ret x) (a1, st1) = θ (x <- c1 ;; ret (f1 x)
*) admit. }
move => [b0 st0] [b1 st1] Hgt0.
(* by definition of d' fi^-1(bi) are both non empty
-> exits ai s.t. fi(ai) = bi, i = 1,2
-> specialize H with (a0,st0) (a1,st1) and get the thesis
*)
admit.
Admitted.
(*CA: depends on post_conclusion_rule but is not used *)
Lemma f_preserves_eq { A B : ord_choiceType } { S : choiceType }
{ x y: FrStP S A }
(f : A -> B ) (* (Hbij : bijective f) *)
( H: ⊨ ⦃ fun '(s1, s2) => s1 = s2 ⦄
( X <- x ;; retF X ) ≈
( Y <- y ;; retF Y)
⦃ eq ⦄ ) :
⊨ ⦃ fun '(s1, s2) => s1 = s2 ⦄
(X <- x ;; retF (f X) ) ≈
(Y <- y ;; retF (f Y) )
⦃ eq ⦄.
Proof.
apply: post_conclusion_rule; auto.
unshelve eapply post_weaken_rule. { exact: eq. }
- assumption.
- move => /= [a1 s1] [a2 s2] [H1 H2]. split; by subst.
Qed.
Theorem if_rule {A1 A2 : ord_choiceType} {S1 S2 : choiceType}
(c1 c2 : FrStP S1 A1)
(c1' c2' : FrStP S2 A2)
{b1 b2 : bool}
{pre : S1 * S2 -> Prop} {post : A1 * S1 -> A2 * S2 -> Prop}
{pre_b1b2 : forall st, pre st -> b1 = b2}
{ H1 : ⊨ ⦃ fun st => pre st /\ b1 = true ⦄ c1 ≈ c1' ⦃ post ⦄ }
{ H2 : ⊨ ⦃ fun st => pre st /\ b1 = false ⦄ c2 ≈ c2' ⦃ post ⦄ } :
⊨ ⦃ pre ⦄
(if b1 then c1 else c2) ≈
(if b2 then c1' else c2')
⦃ post ⦄.
Proof.
apply pre_hypothesis_rule. move=> st1 st2 pre_holds.
specialize (pre_b1b2 (st1, st2) pre_holds). subst.
destruct b2 eqn:Hb.
- apply (pre_weaken_rule (fun st => pre st /\ true = true)).
assumption.
rewrite /= => st1' st2' [Heq1 Heq2]. subst.
split; auto.
- apply (pre_weaken_rule (fun st => pre st /\ false = false)).
assumption.
rewrite /= => st1' st2' [Heq1 Heq2]. subst.
split; auto.
Qed.
(* TODO: asymmetric variants of if_rule: if_ruleL and if_ruleR *)
Fixpoint bounded_do_while {S : choiceType} (n : nat) (c : FrStP S bool_choiceType) :
FrStP S bool_choiceType :=
(* false means fuel emptied, true means execution finished *)
match n with
| 0 => retF false
| S n => bindF (fun b => match b with
| false => retF true
| true => bounded_do_while n c
end
) c
end.
Theorem bounded_do_while_rule {A1 A2 : ord_choiceType} {S1 S2 : choiceType}
{n : nat}
(c1 : FrStP S1 bool_choiceType)
(c2 : FrStP S2 bool_choiceType)
{inv : bool -> bool -> (S1 * S2) -> Prop}
{H : ⊨ ⦃ inv true true ⦄ c1 ≈ c2 ⦃ fun bs1 bs2 => (inv bs1.1 bs2.1) (bs1.2, bs2.2) /\ bs1.1 = bs2.1 ⦄ } :
⊨ ⦃ inv true true ⦄
bounded_do_while n c1 ≈ bounded_do_while n c2
⦃ fun ls rs => (ls.1 = false /\ rs.1 = false) \/ (inv false false) (ls.2, rs.2) ⦄.
Proof.
induction n.
- simpl. eapply weaken_rule.
apply ret_rule. simpl. intros [? ?] ?. simpl. cbv. intuition eauto.
- simpl. eapply weaken_rule.
apply bind_rule. apply H.
move => b1 b2. eapply weaken_rule. apply if_rule.
move => st.
instantiate (1 := fun s => inv b1 b2 s /\ b1 = b2).
rewrite /=. move => [hfoo heq]. assumption.
instantiate (1 := fun ls rs => ls.1 = false /\ rs.1 = false \/ (inv false false) (ls.2, rs.2)).
eapply weaken_rule. apply IHn. simpl. intros [? ?] ?. cbv. intuition eauto.
rewrite -H3. rewrite {2}H4. assumption.
eapply weaken_rule. apply ret_rule.
simpl. intros [? ?] ?. cbv. intuition eauto.
apply H2. right. rewrite -H3. rewrite {2}H4. assumption.
instantiate (1 := fun '(b1, b2) => fromPrePost (fun st => (inv b1 b2 st) /\ b1 = b2)
(fun ls rs => ls.1 = false /\ rs.1 = false \/ (inv false false (ls.2, rs.2)))).
move => [st1 st2] /=.
cbv; intuition.
move => [st1 st2] /=. move => β /=.
move => [h1 h2].
split; auto.
move => [b1 s1] [b2 s2] /= [hh1 hh2]. subst.
split; auto.
Qed.
(*TODO: asymmetric variants of bounded_do_while -- Rem.: low priority as not useful for our examples *)
Definition θ_dens { S : choiceType } { X : ord_choiceType } :=
@Theta_dens.unary_theta_dens (F_choice_prod_obj ⟨ X, S ⟩).
Lemma Pr_eq {X Y : ord_choiceType} { S1 S2 : choiceType } {A : pred (X * S1)} {B : pred (Y * S2)}
Ψ ϕ
(c1 : FrStP S1 X) (c2 : FrStP S2 Y)
(H : ⊨ ⦃ Ψ ⦄ c1 ≈ c2 ⦃ ϕ ⦄)
{ s1 s2 } (HPsi : Ψ (s1, s2) )
(Hpost : forall x y, ϕ x y -> (A x) <-> (B y)) :
\P_[ θ_dens (θ0 c1 s1) ] A =
\P_[ θ_dens (θ0 c2 s2) ] B.
Proof.
rewrite /pr /=.
specialize (H (s1,s2) (fun '(a, b) => A a <-> B b)). simpl in H.
destruct H as [d [[H11 H12] H2]].
split; assumption.
rewrite /θ0 /θ_dens /unary_theta_dens /=.
rewrite -H11 -H12.
rewrite /lmg /rmg.
assert ((fun x : X * S1 => (A x)%:R * dfst d x) = (fun x : X *S1 => (A x)%:R * psum (fun w => d (x, w)))) as HeqH11.
{ extensionality k. rewrite dfstE. reflexivity. }
rewrite HeqH11. simpl in HeqH11.
assert ((fun x : X * S1 => (A x)%:R * psum (fun w => d (x, w))) = (fun x : X * S1 => psum (fun w => (A x)%:R * d (x, w)))) as H4.
{ extensionality k. rewrite -psumZ. reflexivity.
case (A k); intuition. by rewrite ler01. }
rewrite H4.
assert ((fun x : Y * S2 => (B x)%:R * dsnd d x) = (fun y : Y * S2 => (B y)%:R * psum (fun w => d (w, y)))) as HeqH12.
{ extensionality K. rewrite dsndE. reflexivity. }
rewrite HeqH12.
unfold F_choice_prod_obj in d.
assert ((fun y : Y * S2 => (B y)%:R * psum (fun w => d (w, y))) = (fun y : Y * S2 => psum (fun w => (B y)%:R * d (w, y)))) as H5.
{ extensionality k. rewrite -psumZ. reflexivity.
case (B k); intuition; by rewrite ler01. }
rewrite H5.
clear H5 H4 HeqH12 HeqH11.
rewrite -(@psum_pair _ _ _ (fun '(x, y) => (A x)%:R * d (x, y))).
rewrite -(@psum_pair_swap _ _ _ (fun '(x, y) => (B y)%:R * d (x, y))).
f_equal.
extensionality k.
destruct k as [x y].
case (0 < d (x, y)) eqn:Hd.
move: Hd. move/idP => Hd.
specialize (H2 _ _ Hd).
case (A x) eqn:Ha.
+ case (B y) eqn: Hb.
reflexivity.
move: H2. intuition. rewrite H. reflexivity. auto.
case (B y) eqn:Hb.
intuition. rewrite H0. reflexivity. auto.
reflexivity.
assert (d (x, y) = 0).
rewrite Order.POrderTheory.lt_def in Hd.
apply Bool.andb_false_iff in Hd.
destruct Hd.
++ move: H. move/eqP. auto.
++ assert (0 <= d (x, y)) as Hn.
{ apply ge0_mu. }
move: H. move/idP. intuition.
by rewrite H !GRing.mulr0.
(* summable B*)
assert ((fun x =>
(nat_of_bool (let '(_, y) := x in B y))%:R * d x) =
(fun '(x, y) => (B y)%:R * d (x, y))) as Heq1.
{ extensionality k. destruct k as [k1 k2].
case (B k2). reflexivity. reflexivity. }
rewrite -Heq1.
pose (@summable_pr R (prod_choiceType (prod_choiceType X S1)
(prod_choiceType Y S2))
(fun '(x, y) => B y) d).
simpl in *. unfold nat_of_bool in s. rewrite /nat_of_bool. exact s.
(* summable A *)
assert ((fun x =>
(nat_of_bool (let '(x, _) := x in A x))%:R * d x) =
(fun '(x, y) => (A x)%:R * d (x, y))) as Heq2.
{ extensionality k. destruct k as [k1 k2].
case (B k2). reflexivity. reflexivity. }
rewrite -Heq2.
pose (@summable_pr R (prod_choiceType (prod_choiceType X S1)
(prod_choiceType Y S2))
(fun '(x, y) => A x) d).
simpl in *. unfold nat_of_bool in s. rewrite /nat_of_bool. exact s.
Qed.
Corollary coupling_eq { A : ord_choiceType } { S : choiceType }
(K1 K2 : FrStP S A )
(ψ : S * S -> Prop)
(H : ⊨ ⦃ ψ ⦄ K1 ≈ K2 ⦃ eq ⦄):
forall s1 s2, ψ (s1, s2) -> θ_dens (θ0 K1 s1) = θ_dens (θ0 K2 s2).
Proof.
move => s1 s2 psi_s1_s2.
apply distr_ext => /= w.
assert (\P_[ θ_dens (θ0 K1 s1) ] (pred1 w) = \P_[ θ_dens (θ0 K2 s2) ] (pred1 w)).
{ apply: (Pr_eq ψ eq); rewrite //= => x y Heq. by subst. }
by repeat rewrite -pr_pred1 in H0.
Qed.
Lemma rewrite_eqDistrL { A1 A2 : ord_choiceType } {S1 S2 : choiceType } { P } { Q }
(c1 c1' : FrStP S1 A1) (c2 : FrStP S2 A2)
(H : ⊨ ⦃ P ⦄ c1 ≈ c2 ⦃ Q ⦄)
(θeq : forall s : S1, θ_dens (θ0 c1 s) = θ_dens (θ0 c1' s) ) :
⊨ ⦃ P ⦄ c1' ≈ c2 ⦃ Q ⦄.
Proof.
move => [s1 s2].
specialize (H (s1, s2)).
specialize (θeq s1).
rewrite /θ0 /θ_dens /= in θeq.
rewrite /θ /= /MonoCont_order /=.
rewrite -θeq.
by apply H.
Qed.
Lemma rewrite_eqDistrR { A1 A2 : ord_choiceType } {S1 S2 : choiceType} { P } { Q }
(c1 : FrStP S1 A1) (c2 c2': FrStP S2 A2)
(H : ⊨ ⦃ P ⦄ c1 ≈ c2 ⦃ Q ⦄)
(θeq : forall s : S2, θ_dens (θ0 c2 s) = θ_dens (θ0 c2' s) ) :
⊨ ⦃ P ⦄ c1 ≈ c2' ⦃ Q ⦄.
Proof.
move => [s1 s2].
specialize (H (s1, s2)).
specialize (θeq s2).
rewrite /θ0 /θ_dens /= in θeq.
rewrite /θ /= /MonoCont_order /=.
rewrite -θeq.
by apply H.
Qed.
Definition coupling_self_SDistr { A } ( d: SDistr A) : SDistr (F_choice_prod ⟨ A, A ⟩) :=
dmargin (fun a => (a,a)) d.
Lemma coupling_self { A } (d : SDistr A) :
coupling (coupling_self_SDistr d) d d.
Proof.
unfold coupling. unfold coupling_self_SDistr. split.
- unfold lmg. unfold dmargin.
apply distr_ext. move=> a.
rewrite dlet_dlet.
have coucou: d a = (\dlet_(y <- d) dunit y) a . rewrite dlet_dunit_id. reflexivity.
rewrite coucou. f_equal. f_equal.
apply boolp.funext. move=> y. apply distr_ext. move=> b. rewrite dlet_unit.
reflexivity.
- unfold rmg. unfold dmargin.
apply distr_ext. move=> a.
rewrite dlet_dlet.
have coucou: d a = (\dlet_(y <- d) dunit y) a . rewrite dlet_dunit_id. reflexivity.
rewrite coucou. f_equal. f_equal.
apply boolp.funext. move=> y. apply distr_ext. move=> b. rewrite dlet_unit.
reflexivity.
Qed.
Lemma aux_lemma0 {A} (c : SDistr A) (a1 a2 : A) :
coupling_self_SDistr c (a1,a2) = (if a1 == a2 then c a1 else 0).
Proof.
destruct (eqType_lem A a1 a2).
- rewrite H. unfold coupling_self_SDistr. rewrite refl_true.
unfold dmargin.
have coucou : c a2 = (\dlet_(x <- c) dunit x) a2.
symmetry. rewrite dlet_dunit_id. reflexivity.
rewrite coucou. f_equal.
Abort.
Lemma aux_domain : forall u v : R, u * v <> 0 -> u <> 0.
Proof.
move=> u v. apply contra_not. move=> H0. rewrite H0.
apply GRing.Theory.mul0r.
Qed.
Lemma aux_lemma { A } {d : SDistr A} :
forall a1 a2, 0 < (coupling_self_SDistr d) (a1,a2) -> a1 = a2.
Proof.
move=> a1 a2. unfold coupling_self_SDistr. rewrite dmargin_psumE.
move=> Hpsum.
have Hpsum' : psum (fun x : A => ((x, x) == (a1, a2))%:R * d x) <> 0.
move=> abs. rewrite -abs in Hpsum. rewrite Order.POrderTheory.ltxx in Hpsum.
discriminate.
clear Hpsum.
eapply neq0_psum in Hpsum'. destruct Hpsum'.
apply aux_domain in H.
destruct (eqType_lem bool_eqType ((x,x) == (a1,a2)) true) as [Houi | Hnon].
move: Houi => /eqP Houi. move: Houi => [H1 H2]. rewrite -H1 -H2. reflexivity.
have Hnon' : (x,x) == (a1,a2) = false.
destruct ((x,x) == (a1,a2)). contradiction. reflexivity.
rewrite Hnon' in H. cbn in H. contradiction.
Qed.
Lemma reflexivity_rule { A : ord_choiceType } { S : choiceType }
(c : FrStP S A):
⊨ ⦃ fun '(s1, s2) => s1 = s2 ⦄ c ≈ c ⦃ eq ⦄.
Proof.
move => [st1 st] /=. move => α [H1 H2] /=. subst.
exists (coupling_self_SDistr (θ_dens (θ0 c st))).
split.
- exact: coupling_self.
- move => [a1 s1] [a2 s2] H.
apply: H2. apply: aux_lemma H.
Qed.
Definition dsym { A B : ord_choiceType } { S1 S2 : choiceType } (d : SDistr_carrier
(F_choice_prod_obj
⟨ Choice.Pack {| Choice.base := prod_eqMixin B S2; Choice.mixin := prod_choiceMixin B S2 |},
Choice.Pack {| Choice.base := prod_eqMixin A S1; Choice.mixin := prod_choiceMixin A S1 |} ⟩)) :
SDistr_carrier
(F_choice_prod_obj
⟨ Choice.Pack {| Choice.base := prod_eqMixin A S1; Choice.mixin := prod_choiceMixin A S1 |},
Choice.Pack {| Choice.base := prod_eqMixin B S2; Choice.mixin := prod_choiceMixin B S2 |} ⟩) :=
dswap d.
Lemma dsym_coupling { A B : ord_choiceType } { S1 S2 : choiceType } { d : SDistr_carrier
(F_choice_prod_obj
⟨ Choice.Pack {| Choice.base := prod_eqMixin B S2; Choice.mixin := prod_choiceMixin B S2 |},
Choice.Pack {| Choice.base := prod_eqMixin A S1; Choice.mixin := prod_choiceMixin A S1 |} ⟩) }
{d1 d2 }
(Hcoupling : coupling d d1 d2) : coupling (dsym d) d2 d1.
Proof.
rewrite /dsym. destruct Hcoupling as [dfst_d dsnd_d]. unfold coupling, lmg, rmg in *.
subst. split.
- apply: distr_ext. exact: dfst_dswap d.
- apply: distr_ext. exact: dsnd_dswap d.
Qed.
Lemma symmetry_rule { A B : ord_choiceType } { S1 S2 : choiceType } { pre post }
(c1 : FrStP S1 A) (c2 : FrStP S2 B)
(H: ⊨ ⦃ fun '(s2, s1) => pre (s1, s2) ⦄ c2 ≈ c1 ⦃ fun '(b,s2) '(a,s1) => post (a,s1) (b,s2) ⦄ ):
⊨ ⦃ pre ⦄ c1 ≈ c2 ⦃ post ⦄.
Proof.
move => [s1 s2]. move => /= π.
move => [Hpre H'].
specialize (H (s2,s1) (fun '(a,s1,(b,s2)) => π ((b,s2,(a,s1))))).
cbn in H.
destruct H as [d' [H1 H2]].
- rewrite /=. split.
-- assumption.
-- move => [a h1] [b h2] Hpost /=. apply: (H' (b, h2) (a, h1) Hpost).
simpl in d', H1, H2. exists (dswap d'). split.
- exact: dsym_coupling.
-- move => [b h2] [a h1] Hdsym. apply: (H2 (a, h1) (b, h2)).
apply msupp.
have Heq: dswap (dswap d') = d'. { apply: distr_ext. exact: (dswapK d'). }
rewrite -Heq.
apply dinsupp_swap.
apply /dinsuppP.
rewrite lt0r in Hdsym.
move /andP: Hdsym. move => [Hd1 Hd2].
apply /eqP. assumption.
Qed.
Theorem swap_rule { A1 A2 : ord_choiceType } { S : choiceType } { I : S * S -> Prop } {post : A1 * S -> A2 * S -> Prop }
(c1 : FrStP S A1) (c2 : FrStP S A2)
(Hinv1 : ⊨ ⦃ I ⦄ c1 ≈ c2 ⦃ fun '(a1, s1) '(a2, s2) => I (s1, s2) /\ post (a1,s1) (a2,s2) ⦄ )
(Hinv2 : ⊨ ⦃ I ⦄ c2 ≈ c1 ⦃ fun '(a2, s2) '(a1, s1) => I (s1, s2) /\ post (a1,s1) (a2,s2) ⦄ ):
⊨ ⦃ I ⦄ (c1 ;; c2) ≈ (c2 ;; c1) ⦃ fun '(a2, s2) '(a1, s1) => I (s1, s2) /\ post (a1,s1) (a2,s2) ⦄ .
Proof.
apply: seq_rule.
- exact: Hinv1.
- move => a1 a2.
apply: pre_weaken_rule.
{ apply: post_weaken_rule.
exact: Hinv2.
move => [a1' s1] [a2' s2] [HI HQ].
split; assumption. }
move => st1 st2 /= [HI HQ].
assumption.
Qed.
(*Rem.: don't worry too much about indexes and order, in most cases predicates will be symmetric *)
Theorem swap_ruleL { A1 A2 B : ord_choiceType } { S : choiceType }
{ pre I : S * S -> Prop }
{ post : A2 * S -> A1 * S -> Prop }
(l : FrStP S B) (c1 : FrStP S A1) (c2 : FrStP S A2)
(HL : ⊨ ⦃ pre ⦄ l ≈ l ⦃ fun '(b1, s1) '(b2, s2) => I (s1, s2) ⦄)
(Hinv1 : ⊨ ⦃ I ⦄ c1 ≈ c2 ⦃ fun '(a1, s1) '(a2, s2) => I (s1, s2) /\ post (a2, s2) (a1, s1) ⦄ )
(Hinv2 : ⊨ ⦃ I ⦄ c2 ≈ c1 ⦃ fun '(a2, s2) '(a1, s1) => I (s1, s2) /\ post (a2, s2) (a1, s1) ⦄ ):
⊨ ⦃ pre ⦄ (l ;; c1 ;; c2) ≈ (l ;; c2 ;; c1) ⦃ post ⦄ .
Proof.
apply: seq_rule.
exact: HL.
move => a1 a2 /=.
unshelve apply: post_weaken_rule.
{ exact: (fun '(x2, s2) '(x1, s1) => I(s1,s2) /\ (post (x2,s2) (x1,s1))). }
by apply: (@swap_rule A1 A2 S I (fun '(x1,h1) '(x2, h2) => post (x2,h2) (x1,h1)) c1 c2 Hinv1 Hinv2).
move => [a1' s1] [a2' s2] [HI HHL] /=. assumption.
Qed.
Section AuxLemmasSwapRuleR.
Lemma smMonEqu1
{A1 A2 B : ord_choiceType} {S : choiceType}
(r : A1 -> A2 -> FrStP S B) (c1 : FrStP S A1) (c2 : FrStP S A2) :
(a2 ∈ choice_incl A2 <<- c2;; a1 ∈ choice_incl A1 <<- c1;; (r a1 a2))
=
(a ∈ choice_incl (prod_choiceType A1 A2) <<-
(a2 ∈ choice_incl A2 <<- c2;; a1 ∈ choice_incl A1 <<- c1;; retF (a1, a2));;
r a.1 a.2).
Proof.
symmetry.
cbn. unfold FreeProbProg.rFree_obligation_2.
unshelve epose (assoc := (@ord_relmon_law3 _ _ _ (FrStP S) _ _ _ _ _)).
shelve. shelve. shelve.
exact (fun a : A1 * A2 => r a.1 a.2).
exact (fun a2 : A2 =>
bindrFree (@StateTransformingLaxMorph.ops_StP S) (@StateTransformingLaxMorph.ar_StP S) c1
(fun a1 : A1 => retF (a1, a2))).
cbn in assoc. unfold FreeProbProg.rFree_obligation_2 in assoc.
symmetry in assoc. unshelve eapply equal_f in assoc. exact c2. rewrite assoc.
clear assoc.
f_equal. apply boolp.funext. move=> a2.
unshelve epose (assoc := (@ord_relmon_law3 _ _ _ (FrStP S) _ _ _ _ _)).
shelve. shelve. shelve.
exact (fun a : A1 * A2 => r a.1 a.2).
exact (fun a1 : A1 => retF (a1, a2)).
cbn in assoc. unfold FreeProbProg.rFree_obligation_2 in assoc.
symmetry in assoc. unshelve eapply equal_f in assoc. exact c1. rewrite assoc.
reflexivity.
Qed.
Lemma smMonEqu2
{A1 A2 B : ord_choiceType} {S : choiceType}
(r : A1 -> A2 -> FrStP S B) (c1 : FrStP S A1) (c2 : FrStP S A2) :
(a1 ∈ choice_incl A1 <<- c1;; a2 ∈ choice_incl A2 <<- c2;; (r a1 a2))
=
(a ∈ choice_incl (prod_choiceType A1 A2) <<-
(a1 ∈ choice_incl A1 <<- c1;; a2 ∈ choice_incl A2 <<- c2;; retF (a1, a2));;
r a.1 a.2).
Proof.
symmetry.
cbn. unfold FreeProbProg.rFree_obligation_2.
unshelve epose (assoc := (@ord_relmon_law3 _ _ _ (FrStP S) _ _ _ _ _)).
shelve. shelve. shelve.
exact (fun a : A1 * A2 => r a.1 a.2).
exact (fun a1 : A1 =>
bindrFree (@StateTransformingLaxMorph.ops_StP S) (@StateTransformingLaxMorph.ar_StP S) c2
(fun a2 : A2 => retF (a1, a2))).
cbn in assoc. unfold FreeProbProg.rFree_obligation_2 in assoc.
symmetry in assoc. unshelve eapply equal_f in assoc. exact c1. rewrite assoc.
clear assoc.
f_equal. apply boolp.funext. move=> a1.
unshelve epose (assoc := (@ord_relmon_law3 _ _ _ (FrStP S) _ _ _ _ _)).
shelve. shelve. shelve.
exact (fun a : A1 * A2 => r a.1 a.2).
exact (fun a2 : A2 => retF (a1, a2)).
cbn in assoc. unfold FreeProbProg.rFree_obligation_2 in assoc.
symmetry in assoc. unshelve eapply equal_f in assoc. exact c2. rewrite assoc.
reflexivity.
Qed.
Context (S : choiceType).
Let Frp_fld := @Frp.
Lemma theta0_vsbind {P Q : ord_choiceType} (p : FrStP S P) (q : FrStP S Q)
(s : S) :
θ0 (x ∈ P <<- p ;; q) s
=
(ord_relmon_bind Frp_fld)
(fun ps : P * S => let (p,s) := ps in θ0 q s)
(θ0 p s).
Proof.
unfold θ0.
epose (assoc := rlmm_law2 _ _ _ _ (unaryIntState) P Q (fun _ => q) ).
cbn in assoc. specialize (assoc p).
cbn. unshelve eapply equal_f in assoc. exact s.
rewrite [LHS]assoc.
unfold OrderEnrichedRelativeAdjunctionsExamples.ToTheS_obligation_1.
unfold FreeProbProg.rFree_obligation_2.
reflexivity.
Qed.
Lemma some_commutativity
{A1 A2 B : ord_choiceType}
(post : B * S -> B * S -> Prop)
(r : A1 -> A2 -> FrStP S B)
(c1 : FrStP S A1)
(c2 : FrStP S A2)
(HR : forall (a1 : A1) (a2 : A2), ⊨ ⦃ fun '(s2, s1) => s1 = s2 ⦄ r a1 a2 ≈ r a1 a2 ⦃ post ⦄ )
(post_eq : forall bs bs' : B * S, bs = bs' -> post bs bs')
(Hcomm : forall s : S,
θ_dens (θ0 (a1 ∈ choice_incl A1 <<- c1;; a2 ∈ choice_incl A2 <<- c2;; retF (a1, a2)) s) =
θ_dens (θ0 (a2 ∈ choice_incl A2 <<- c2;; a1 ∈ choice_incl A1 <<- c1;; retF (a1, a2)) s) )
(s : S) :
θ_dens
(θ0
(a ∈ choice_incl (prod_choiceType A1 A2) <<-
(a1 ∈ choice_incl A1 <<- c1;; a2 ∈ choice_incl A2 <<- c2;; retF (a1, a2));;
r a.1 a.2) s) =
θ_dens
(θ0
(a ∈ choice_incl (prod_choiceType A1 A2) <<-
(a2 ∈ choice_incl A2 <<- c2;; a1 ∈ choice_incl A1 <<- c1;; retF (a1, a2));;
r a.1 a.2) s).