-
Notifications
You must be signed in to change notification settings - Fork 43
/
CSetoids.v
1187 lines (930 loc) · 30.9 KB
/
CSetoids.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
(* Copyright © 1998-2008
* Henk Barendregt
* Luís Cruz-Filipe
* Herman Geuvers
* Mariusz Giero
* Rik van Ginneken
* Dimitri Hendriks
* Sébastien Hinderer
* Cezary Kaliszyk
* Bart Kirkels
* Pierre Letouzey
* Iris Loeb
* Lionel Mamane
* Milad Niqui
* Russell O’Connor
* Randy Pollack
* Nickolay V. Shmyrev
* Bas Spitters
* Dan Synek
* Freek Wiedijk
* Jan Zwanenburg
*
* This work is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This work is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License along
* with this work; if not, write to the Free Software Foundation, Inc.,
* 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
*)
(** printing != %\ensuremath{\mathrel\#}% *)
(** printing == %\ensuremath{\equiv}% #≡# *)
(** printing [=] %\ensuremath{\equiv}% #≡# *)
(** printing [~=] %\ensuremath{\mathrel{\not\equiv}}% #≠# *)
(** printing [#] %\ensuremath{\mathrel\#}% *)
(** printing ex_unq %\ensuremath{\exists^1}% #∃<sup>1</sup># *)
(** printing [o] %\ensuremath\circ% #⋅# *)
(** printing [-C-] %\ensuremath\diamond% *)
(**
* Setoids
Definition of a constructive setoid with apartness,
i.e.%\% a set with an equivalence relation and an apartness relation compatible with it.
*)
Require Import CornTac.
Require Export CLogic.
Require Export Step.
Require Export RSetoid.
Delimit Scope corn_scope with corn.
Open Scope corn_scope.
Definition Relation := Trelation.
Implicit Arguments Treflexive [A].
Implicit Arguments Creflexive [A].
Implicit Arguments Tsymmetric [A].
Implicit Arguments Csymmetric [A].
Implicit Arguments Ttransitive [A].
Implicit Arguments Ctransitive [A].
(* begin hide *)
Set Implicit Arguments.
Unset Strict Implicit.
(* end hide *)
(**
** Relations necessary for Setoids
%\begin{convention}% Let [A:Type].
%\end{convention}%
Notice that their type depends on the main logical connective.
*)
Section Properties_of_relations.
Variable A : Type.
Definition irreflexive (R : Crelation A) : Prop := forall x : A, Not (R x x).
Definition cotransitive (R : Crelation A) : CProp := forall x y : A,
R x y -> forall z : A, R x z or R z y.
Definition tight_apart (eq : Relation A) (ap : Crelation A) : Prop := forall x y : A,
Not (ap x y) <-> eq x y.
Definition antisymmetric (R : Crelation A) : Prop := forall x y : A,
R x y -> Not (R y x).
End Properties_of_relations.
(* begin hide *)
Set Strict Implicit.
Unset Implicit Arguments.
(* end hide *)
(**
** Definition of Setoid
Apartness, being the main relation, needs to be [CProp]-valued. Equality,
as it is characterized by a negative statement, lives in [Prop]. *)
Record is_CSetoid (A : Type) (eq : Relation A) (ap : Crelation A) : CProp :=
{ax_ap_irreflexive : irreflexive ap;
ax_ap_symmetric : Csymmetric ap;
ax_ap_cotransitive : cotransitive ap;
ax_ap_tight : tight_apart eq ap}.
Record CSetoid : Type := makeCSetoid
{cs_crr :> RSetoid;
cs_ap : Crelation cs_crr;
cs_proof : is_CSetoid cs_crr (@st_eq cs_crr) cs_ap}.
Notation cs_eq := st_eq (only parsing).
Implicit Arguments cs_ap [c].
Infix "[=]" := cs_eq (at level 70, no associativity).
Infix "[#]" := cs_ap (at level 70, no associativity).
(* End_SpecReals *)
Definition cs_neq (S : CSetoid) : Relation S := fun x y : S => ~ x [=] y.
Implicit Arguments cs_neq [S].
Infix "[~=]" := cs_neq (at level 70, no associativity).
(**
%\begin{nameconvention}%
In the names of lemmas, we refer to [ [=] ] by [eq], [ [~=] ] by
[neq], and [ [#] ] by [ap].
%\end{nameconvention}%
** Setoid axioms
We want concrete lemmas that state the axiomatic properties of a setoid.
%\begin{convention}%
Let [S] be a setoid.
%\end{convention}%
*)
(* Begin_SpecReals *)
Section CSetoid_axioms.
Variable S : CSetoid.
Lemma CSetoid_is_CSetoid : is_CSetoid S (cs_eq (r:=S)) (cs_ap (c:=S)).
Proof cs_proof S.
Lemma ap_irreflexive : irreflexive (cs_ap (c:=S)).
Proof.
elim CSetoid_is_CSetoid; auto.
Qed.
Lemma ap_symmetric : Csymmetric (cs_ap (c:=S)).
Proof.
elim CSetoid_is_CSetoid; auto.
Qed.
Lemma ap_cotransitive : cotransitive (cs_ap (c:=S)).
Proof.
elim CSetoid_is_CSetoid; auto.
Qed.
Lemma ap_tight : tight_apart (cs_eq (r:=S)) (cs_ap (c:=S)).
Proof.
elim CSetoid_is_CSetoid; auto.
Qed.
End CSetoid_axioms.
(**
** Setoid basics%\label{section:setoid-basics}%
%\begin{convention}% Let [S] be a setoid.
%\end{convention}%
*)
Lemma is_CSetoid_Setoid : forall S eq ap, is_CSetoid S eq ap -> Setoid_Theory S eq.
Proof.
intros S eq ap p.
destruct p.
split.
firstorder.
intros a b. red in ax_ap_tight0 .
repeat rewrite <- ax_ap_tight0.
firstorder.
intros a b c. red in ax_ap_tight0 .
repeat rewrite <- ax_ap_tight0.
intros H H0 H1.
destruct (ax_ap_cotransitive0 _ _ H1 b); auto.
Qed.
Definition Build_CSetoid (X:Type) (eq:Relation X) (ap:Crelation X) (p:is_CSetoid X eq ap) : CSetoid.
Proof.
exists (Build_RSetoid (is_CSetoid_Setoid _ _ _ p)) ap.
assumption.
Defined.
Section CSetoid_basics.
Variable S : CSetoid.
(* End_SpecReals *)
(**
In `there exists a unique [a:S] such that %\ldots%#...#', we now mean unique with respect to the setoid equality. We use [ex_unq] to denote unique existence.
*)
Definition ex_unq (P : S -> CProp) := {x : S | forall y : S, P y -> x [=] y | P x}.
Lemma eq_reflexive : Treflexive (cs_eq (r:=S)).
Proof.
intro x.
reflexivity.
Qed.
Lemma eq_symmetric : Tsymmetric (cs_eq (r:=S)).
Proof.
intro x; intros y H.
symmetry; assumption.
Qed.
Lemma eq_transitive : Ttransitive (cs_eq (r:=S)).
Proof.
intro x; intros y z H H0.
transitivity y; assumption.
Qed.
(**
%\begin{shortcoming}%
The lemma [eq_reflexive] above is convertible to
[eq_reflexive_unfolded] below. We need the second version too,
because the first cannot be applied when an instance of reflexivity is needed.
(``I have complained bitterly about this.'' RP)
%\end{shortcoming}%
tes
%\begin{nameconvention}%
If lemma [a] is just an unfolding of lemma [b], the name of [a] is the name
[b] with the suffix ``[_unfolded]''.
%\end{nameconvention}%
*)
Lemma eq_reflexive_unfolded : forall x : S, x [=] x.
Proof eq_reflexive.
Lemma eq_symmetric_unfolded : forall x y : S, x [=] y -> y [=] x.
Proof eq_symmetric.
Lemma eq_transitive_unfolded : forall x y z : S, x [=] y -> y [=] z -> x [=] z.
Proof eq_transitive.
Lemma eq_wdl : forall x y z : S, x [=] y -> x [=] z -> z [=] y.
Proof.
intros. now apply (eq_transitive _ x);[apply: eq_symmetric|].
Qed.
Lemma ap_irreflexive_unfolded : forall x : S, Not (x [#] x).
Proof ap_irreflexive S.
Lemma ap_cotransitive_unfolded : forall a b : S, a [#] b -> forall c : S, a [#] c or c [#] b.
Proof.
intros a b H c.
exact (ap_cotransitive _ _ _ H c).
Qed.
Lemma ap_symmetric_unfolded : forall x y : S, x [#] y -> y [#] x.
Proof ap_symmetric S.
(**
We would like to write
[[
Lemma eq_equiv_not_ap : forall (x y:S), (x [=] y) iff Not (x [#] y).
]]
In Coq, however, this lemma cannot be easily applied.
Therefore we have to split the lemma into the following two lemmas [eq_imp_not_ap] and [not_ap_imp_eq].
For this we should fix the Prop CProp problem.
*)
Lemma eq_imp_not_ap : forall x y : S, x [=] y -> Not (x [#] y).
Proof.
intros x y.
elim (ap_tight S x y).
intros H1 H2.
assumption.
Qed.
Lemma not_ap_imp_eq : forall x y : S, Not (x [#] y) -> x [=] y.
Proof.
intros x y.
elim (ap_tight S x y).
intros H1 H2.
assumption.
Qed.
Lemma neq_imp_notnot_ap : forall x y : S, x [~=] y -> ~ Not (x [#] y).
Proof.
intros x y H H0.
now apply: H; apply: not_ap_imp_eq.
Qed.
Lemma notnot_ap_imp_neq : forall x y : S, ~ Not (x [#] y) -> x [~=] y.
Proof.
intros x y H H0.
now apply H; apply eq_imp_not_ap.
Qed.
Lemma ap_imp_neq : forall x y : S, x [#] y -> x [~=] y.
Proof.
intros x y H H1.
now apply (eq_imp_not_ap _ _ H1).
Qed.
Lemma not_neq_imp_eq : forall x y : S, ~ x [~=] y -> x [=] y.
Proof.
intros x y H.
apply: not_ap_imp_eq.
intros H0.
apply: H. now apply: ap_imp_neq.
Qed.
Lemma eq_imp_not_neq : forall x y : S, x [=] y -> ~ x [~=] y.
Proof.
intros x y H H0. easy.
Qed.
End CSetoid_basics.
Section product_csetoid.
(**
** The product of setoids *)
Definition prod_ap (A B : CSetoid) (c d : prodT A B) : CProp.
Proof.
destruct c as [a b], d as [a0 b0].
exact (cs_ap (c:=A) a a0 or cs_ap (c:=B) b b0).
Defined.
Definition prod_eq (A B : CSetoid) (c d : prodT A B) : Prop.
Proof.
destruct c as [a b], d as [a0 b0].
exact (a [=] a0 /\ b [=] b0).
Defined.
Lemma prodcsetoid_is_CSetoid : forall A B : CSetoid,
is_CSetoid (prodT A B) (prod_eq A B) (prod_ap A B).
Proof.
(* Can be shortened *)
intros A B.
apply (Build_is_CSetoid _ (prod_eq A B) (prod_ap A B)).
intros x. case x. intros c c0 H.
elim H.
intros H1.
now apply: (ap_irreflexive A _ H1).
apply (ap_irreflexive B _ ).
intros x y. case x. case y.
intros c c0 c1 c2 H.
elim H.
intros.
left.
now apply ap_symmetric.
intros.
right.
now apply ap_symmetric.
intros x y. case x. case y.
intros c c0 c1 c2 H z. case z.
intros c3 c4.
generalize H.
intros.
elim H.
intros.
cut (c1 [#] c3 or c3 [#] c).
intros [H1|H2].
left.
now left.
intros.
right.
now left.
now apply: ap_cotransitive.
intros.
cut (c2 [#] c4 or c4 [#] c0).
intros [H1|H2].
left; now right.
now right;right.
now apply: ap_cotransitive.
intros x y. case x. case y.
intros c c0 c1 c2.
split.
intros.
split.
apply not_ap_imp_eq.
intros H1. now cut (c1 [#] c or c2 [#] c0);[|left].
apply not_ap_imp_eq. intros H1. now cut (c1 [#] c or c2 [#] c0);[|right].
intros.
elim H.
intros H0 H1 H2.
now elim H2;apply eq_imp_not_ap.
Qed.
Definition ProdCSetoid (A B : CSetoid) : CSetoid := Build_CSetoid
(prodT A B) (prod_eq A B) (prod_ap A B) (prodcsetoid_is_CSetoid A B).
End product_csetoid.
Implicit Arguments ex_unq [S].
Hint Resolve eq_reflexive_unfolded ap_irreflexive_unfolded: algebra_r.
Hint Resolve eq_symmetric_unfolded ap_symmetric_unfolded: algebra_s.
Hint Resolve eq_transitive_unfolded ap_cotransitive_unfolded: algebra_c.
Declare Left Step eq_wdl.
Declare Right Step eq_transitive_unfolded.
(**
** Relations and predicates
Here we define the notions of well-definedness and strong extensionality
on predicates and relations.
%\begin{convention}% Let [S] be a setoid.
%\end{convention}%
%\begin{nameconvention}%
- ``well-defined'' is abbreviated to [well_def] (or [wd]).
- ``strongly extensional'' is abbreviated to [strong_ext] (or [strext]).
%\end{nameconvention}%
*)
Section CSetoid_relations_and_predicates.
Variable S : CSetoid.
(**
*** Predicates
At this stage, we consider [CProp]- and [Prop]-valued predicates on setoids.
%\begin{convention}% Let [P] be a predicate on (the carrier of) [S].
%\end{convention}%
*)
Section CSetoidPredicates.
Variable P : S -> CProp.
Definition pred_strong_ext : CProp := forall x y : S, P x -> P y or x [#] y.
Definition pred_wd : CProp := forall x y : S, P x -> x [=] y -> P y.
End CSetoidPredicates.
Record wd_pred : Type :=
{wdp_pred :> S -> CProp;
wdp_well_def : pred_wd wdp_pred}.
Record CSetoid_predicate : Type :=
{csp_pred :> S -> CProp;
csp_strext : pred_strong_ext csp_pred}.
Lemma csp_wd : forall P : CSetoid_predicate, pred_wd P.
Proof.
intro P.
intro x; intros y H H0.
elim (csp_strext P x y H).
auto.
set (eq_imp_not_ap _ _ _ H0); contradiction.
Qed.
(** Similar, with [Prop] instead of [CProp]. *)
Section CSetoidPPredicates.
Variable P : S -> Prop.
Definition pred_strong_ext' : CProp := forall x y : S, P x -> P y or x [#] y.
Definition pred_wd' : Prop := forall x y : S, P x -> x [=] y -> P y.
End CSetoidPPredicates.
(**
*** Definition of a setoid predicate *)
Record CSetoid_predicate' : Type :=
{csp'_pred :> S -> Prop;
csp'_strext : pred_strong_ext' csp'_pred}.
Lemma csp'_wd : forall P : CSetoid_predicate', pred_wd' P.
Proof.
intro P.
intro x; intros y H H0.
elim (csp'_strext P x y H).
auto.
set (eq_imp_not_ap _ _ _ H0); contradiction.
Qed.
(**
*** Relations
%\begin{convention}%
Let [R] be a relation on (the carrier of) [S].
%\end{convention}% *)
Section CsetoidRelations.
Variable R : S -> S -> Prop.
Definition rel_wdr : Prop := forall x y z : S, R x y -> y [=] z -> R x z.
Definition rel_wdl : Prop := forall x y z : S, R x y -> x [=] z -> R z y.
Definition rel_strext : CProp := forall x1 x2 y1 y2 : S,
R x1 y1 -> (x1 [#] x2 or y1 [#] y2) or R x2 y2.
Definition rel_strext_lft : CProp := forall x1 x2 y : S, R x1 y -> x1 [#] x2 or R x2 y.
Definition rel_strext_rht : CProp := forall x y1 y2 : S, R x y1 -> y1 [#] y2 or R x y2.
Lemma rel_strext_imp_lftarg : rel_strext -> rel_strext_lft.
Proof.
intros H x1 x2 y H0.
generalize (H x1 x2 y y).
intros H1.
elim (H1 H0);[|auto].
intros [H2|H3];[auto|].
elim (ap_irreflexive S _ H3).
Qed.
Lemma rel_strext_imp_rhtarg : rel_strext -> rel_strext_rht.
Proof.
intros H x y1 y2 H0.
generalize (H x x y1 y2 H0). intros [[H1|H2]|H3]; auto.
elim (ap_irreflexive _ _ H1).
Qed.
Lemma rel_strextarg_imp_strext :
rel_strext_rht -> rel_strext_lft -> rel_strext.
Proof.
intros H H0 x1 x2 y1 y2 H1.
elim (H x1 y1 y2 H1); intro H2;[|elim (H0 x1 x2 y2 H2)];auto.
Qed.
End CsetoidRelations.
(**
*** Definition of a setoid relation
The type of relations over a setoid. *)
Record CSetoid_relation : Type :=
{csr_rel :> S -> S -> Prop;
csr_wdr : rel_wdr csr_rel;
csr_wdl : rel_wdl csr_rel;
csr_strext : rel_strext csr_rel}.
(**
*** [CProp] Relations
%\begin{convention}%
Let [R] be a relation on (the carrier of) [S].
%\end{convention}%
*)
Section CCsetoidRelations.
Variable R : S -> S -> CProp.
Definition Crel_wdr : CProp := forall x y z : S, R x y -> y [=] z -> R x z.
Definition Crel_wdl : CProp := forall x y z : S, R x y -> x [=] z -> R z y.
Definition Crel_strext : CProp := forall x1 x2 y1 y2 : S,
R x1 y1 -> R x2 y2 or x1 [#] x2 or y1 [#] y2.
Definition Crel_strext_lft : CProp := forall x1 x2 y : S, R x1 y -> R x2 y or x1 [#] x2.
Definition Crel_strext_rht : CProp := forall x y1 y2 : S, R x y1 -> R x y2 or y1 [#] y2.
Lemma Crel_strext_imp_lftarg : Crel_strext -> Crel_strext_lft.
Proof.
intros H x1 x2 y H0. generalize (H x1 x2 y y).
intros [H1|H2];auto.
case H2. auto. intro H3. elim (ap_irreflexive _ _ H3).
Qed.
Lemma Crel_strext_imp_rhtarg : Crel_strext -> Crel_strext_rht.
Proof.
intros H x y1 y2 H0.
generalize (H x x y1 y2 H0). intros [H1|H2];auto.
case H2; auto. intro H3. elim (ap_irreflexive _ _ H3).
Qed.
Lemma Crel_strextarg_imp_strext :
Crel_strext_rht -> Crel_strext_lft -> Crel_strext.
Proof.
intros H H0 x1 x2 y1 y2 H1.
elim (H x1 y1 y2 H1); auto.
intro H2.
elim (H0 x1 x2 y2 H2); auto.
Qed.
End CCsetoidRelations.
(**
*** Definition of a [CProp] setoid relation
The type of relations over a setoid. *)
Record CCSetoid_relation : Type :=
{Ccsr_rel :> S -> S -> CProp;
Ccsr_strext : Crel_strext Ccsr_rel}.
Lemma Ccsr_wdr : forall R : CCSetoid_relation, Crel_wdr R.
Proof.
intro R.
intros x y z H H0.
elim (Ccsr_strext R x x y z H);auto.
intros [H1|H2]. elim (ap_irreflexive _ _ H1).
set (eq_imp_not_ap _ _ _ H0). contradiction.
Qed.
Lemma Ccsr_wdl : forall R : CCSetoid_relation, Crel_wdl R.
Proof.
intros R x y z H H0.
elim (Ccsr_strext R x z y y H);auto.
intros [H1|H2]; [set (eq_imp_not_ap _ _ _ H0); contradiction| elim (ap_irreflexive _ _ H2)].
Qed.
Lemma ap_wdr : Crel_wdr (cs_ap (c:=S)).
Proof.
intros x y z H H0.
generalize (eq_imp_not_ap _ _ _ H0); intro H1.
elim (ap_cotransitive _ _ _ H z); intro H2.
assumption.
elim H1.
now apply: ap_symmetric.
Qed.
Lemma ap_wdl : Crel_wdl (cs_ap (c:=S)).
Proof.
intros x y z H H0.
generalize (ap_wdr y x z); intro H1.
apply ap_symmetric.
now apply H1;[apply ap_symmetric|].
Qed.
Lemma ap_wdr_unfolded : forall x y z : S, x [#] y -> y [=] z -> x [#] z.
Proof ap_wdr.
Lemma ap_wdl_unfolded : forall x y z : S, x [#] y -> x [=] z -> z [#] y.
Proof ap_wdl.
Lemma ap_strext : Crel_strext (cs_ap (c:=S)).
Proof.
intros x1 x2 y1 y2 H.
case (ap_cotransitive _ _ _ H x2); intro H0;auto.
case (ap_cotransitive _ _ _ H0 y2); intro H1;auto.
right; right.
now apply ap_symmetric.
Qed.
Definition predS_well_def (P : S -> CProp) : CProp := forall x y : S,
P x -> x [=] y -> P y.
End CSetoid_relations_and_predicates.
Declare Left Step ap_wdl_unfolded.
Declare Right Step ap_wdr_unfolded.
(**
** Functions between setoids
Such functions must preserve the setoid equality
and be strongly extensional w.r.t.%\% the apartness, i.e.%\%
if [f(x,y) [#] f(x1,y1)], then [x [#] x1 + y [#] y1].
For every arity this has to be defined separately.
%\begin{convention}%
Let [S1], [S2] and [S3] be setoids.
%\end{convention}%
First we consider unary functions. *)
Section CSetoid_functions.
Variables S1 S2 S3 : CSetoid.
Section unary_functions.
(**
In the following two definitions,
[f] is a function from (the carrier of) [S1] to
(the carrier of) [S2]. *)
Variable f : S1 -> S2.
Definition fun_wd : Prop := forall x y : S1, x [=] y -> f x [=] f y.
Definition fun_strext : CProp := forall x y : S1, f x [#] f y -> x [#] y.
Lemma fun_strext_imp_wd : fun_strext -> fun_wd.
Proof.
intros H x y H0.
apply not_ap_imp_eq.
intro H1.
generalize (H _ _ H1); intro H2.
now generalize (eq_imp_not_ap _ _ _ H0).
Qed.
End unary_functions.
Record CSetoid_fun : Type :=
{csf_fun :> S1 -> S2;
csf_strext : fun_strext csf_fun}.
Lemma csf_wd : forall f : CSetoid_fun, fun_wd f.
Proof.
intro f.
apply fun_strext_imp_wd.
apply csf_strext.
Qed.
Lemma csf_wd_unfolded: forall (f : CSetoid_fun) (x y : S1), x[=]y -> f x[=]f y.
Proof csf_wd.
Definition Const_CSetoid_fun : S2 -> CSetoid_fun.
Proof.
intro c; apply (Build_CSetoid_fun (fun x : S1 => c)); intros x y H.
elim (ap_irreflexive _ _ H).
Defined.
Section binary_functions.
(**
Now we consider binary functions.
In the following two definitions,
[f] is a function from [S1] and [S2] to [S3].
*)
Variable f : S1 -> S2 -> S3.
Definition bin_fun_wd : Prop := forall x1 x2 y1 y2,
x1 [=] x2 -> y1 [=] y2 -> f x1 y1 [=] f x2 y2.
Definition bin_fun_strext : CProp := forall x1 x2 y1 y2,
f x1 y1 [#] f x2 y2 -> x1 [#] x2 or y1 [#] y2.
Lemma bin_fun_strext_imp_wd : bin_fun_strext -> bin_fun_wd.
Proof.
intros H x1 x2 y1 y2 H0 H1.
apply not_ap_imp_eq.
intro H2.
generalize (H _ _ _ _ H2); intro H3.
elim H3; intro H4.
now set (eq_imp_not_ap _ _ _ H0).
now set (eq_imp_not_ap _ _ _ H1).
Qed.
End binary_functions.
Record CSetoid_bin_fun : Type :=
{csbf_fun :> S1 -> S2 -> S3;
csbf_strext : bin_fun_strext csbf_fun}.
Lemma csbf_wd : forall f : CSetoid_bin_fun, bin_fun_wd f.
Proof.
intro f. apply: bin_fun_strext_imp_wd.
apply csbf_strext.
Qed.
Lemma csbf_wd_unfolded : forall (f : CSetoid_bin_fun) (x x' : S1) (y y' : S2),
x [=] x' -> y [=] y' -> f x y [=] f x' y'.
Proof csbf_wd.
Lemma csf_strext_unfolded : forall (f : CSetoid_fun) (x y : S1), f x [#] f y -> x [#] y.
Proof csf_strext.
End CSetoid_functions.
Lemma bin_fun_is_wd_fun_lft : forall S1 S2 S3 (f : CSetoid_bin_fun S1 S2 S3) (c : S2),
fun_wd _ _ (fun x : S1 => f x c).
Proof.
intros S1 S2 S3 f c x y H.
now apply csbf_wd; [|apply eq_reflexive].
Qed.
Lemma bin_fun_is_wd_fun_rht : forall S1 S2 S3 (f : CSetoid_bin_fun S1 S2 S3) (c : S1),
fun_wd _ _ (fun x : S2 => f c x).
Proof.
intros S1 S2 S3 f c x y H. now apply csbf_wd; [apply eq_reflexive|].
Qed.
Lemma bin_fun_is_strext_fun_lft : forall S1 S2 S3 (f : CSetoid_bin_fun S1 S2 S3) (c : S2),
fun_strext _ _ (fun x : S1 => f x c).
Proof.
intros S1 S2 S3 f c x y H. cut (x [#] y or c [#] c). intros [H1|H2];auto.
now set (ap_irreflexive _ c H2).
eapply csbf_strext. apply H.
Defined.
Lemma bin_fun_is_strext_fun_rht : forall S1 S2 S3 (f : CSetoid_bin_fun S1 S2 S3) (c : S1),
fun_strext _ _ (fun x : S2 => f c x).
Proof.
intros S1 S2 S3 op c x y H. cut (c [#] c or x [#] y). intro Hv. elim Hv. intro Hf.
generalize (ap_irreflexive _ c Hf). tauto. auto.
eapply csbf_strext. apply H.
Defined.
Definition bin_fun2fun_rht (S1 S2 S3:CSetoid) (f : CSetoid_bin_fun S1 S2 S3) (c : S1) : CSetoid_fun S2 S3 :=
Build_CSetoid_fun _ _ (fun x : S2 => f c x) (bin_fun_is_strext_fun_rht _ _ _ f c).
Definition bin_fun2fun_lft (S1 S2 S3:CSetoid) (f : CSetoid_bin_fun S1 S2 S3) (c : S2) : CSetoid_fun S1 S3 :=
Build_CSetoid_fun _ _ (fun x : S1 => f x c) (bin_fun_is_strext_fun_lft _ _ _ f c).
Hint Resolve csf_wd_unfolded csbf_wd_unfolded csf_strext_unfolded: algebra_c.
Implicit Arguments fun_wd [S1 S2].
Implicit Arguments fun_strext [S1 S2].
(**
** The unary and binary (inner) operations on a csetoid
An operation is a function with domain(s) and co-domain equal.
%\begin{nameconvention}%
The word ``unary operation'' is abbreviated to [un_op];
``binary operation'' is abbreviated to [bin_op].
%\end{nameconvention}%
%\begin{convention}%
Let [S] be a setoid.
%\end{convention}%
*)
Section csetoid_inner_ops.
Variable S : CSetoid.
(** Properties of binary operations *)
Definition commutes (f : S -> S -> S) : Prop := forall x y : S, f x y [=] f y x.
Definition associative (f : S -> S -> S) : Prop := forall x y z : S,
f x (f y z) [=] f (f x y) z.
(** Well-defined unary operations on a setoid. *)
Definition un_op_wd := fun_wd (S1:=S) (S2:=S).
Definition un_op_strext := fun_strext (S1:=S) (S2:=S).
Definition CSetoid_un_op := CSetoid_fun S S.
Definition Build_CSetoid_un_op := Build_CSetoid_fun S S.
Lemma id_strext : un_op_strext (fun x : S => x).
Proof. now repeat intro. Qed.
Lemma id_pres_eq : un_op_wd (fun x : S => x).
Proof. now repeat intro. Qed.
Definition id_un_op := Build_CSetoid_un_op (fun x : S => x) id_strext.
(* begin hide *)
Identity Coercion un_op_fun : CSetoid_un_op >-> CSetoid_fun.
(* end hide *)
Definition cs_un_op_strext := csf_strext S S.
(** Well-defined binary operations on a setoid. *)
Definition bin_op_wd := bin_fun_wd S S S.
Definition bin_op_strext := bin_fun_strext S S S.
Definition CSetoid_bin_op : Type := CSetoid_bin_fun S S S.
Definition Build_CSetoid_bin_op := Build_CSetoid_bin_fun S S S.
Definition cs_bin_op_wd := csbf_wd S S S.
Definition cs_bin_op_strext := csbf_strext S S S.
(* begin hide *)
Identity Coercion bin_op_bin_fun : CSetoid_bin_op >-> CSetoid_bin_fun.
(* end hide *)
Lemma bin_op_is_wd_un_op_lft : forall (op : CSetoid_bin_op) (c : S),
un_op_wd (fun x : S => op x c).
Proof.
apply bin_fun_is_wd_fun_lft.
Qed.
Lemma bin_op_is_wd_un_op_rht : forall (op : CSetoid_bin_op) (c : S),
un_op_wd (fun x : S => op c x).
Proof.
apply bin_fun_is_wd_fun_rht.
Qed.
Lemma bin_op_is_strext_un_op_lft : forall (op : CSetoid_bin_op) (c : S),
un_op_strext (fun x : S => op x c).
Proof. apply bin_fun_is_strext_fun_lft. Defined.
Lemma bin_op_is_strext_un_op_rht : forall (op : CSetoid_bin_op) (c : S),
un_op_strext (fun x : S => op c x).
Proof. apply bin_fun_is_strext_fun_rht. Defined.
Definition bin_op2un_op_rht (op : CSetoid_bin_op) (c : S) : CSetoid_un_op :=
bin_fun2fun_rht _ _ _ op c.
Definition bin_op2un_op_lft (op : CSetoid_bin_op) (c : S) : CSetoid_un_op :=
bin_fun2fun_lft _ _ _ op c.
Lemma un_op_wd_unfolded : forall (op : CSetoid_un_op) (x y : S),
x [=] y -> op x [=] op y.
Proof csf_wd S S.
Lemma un_op_strext_unfolded : forall (op : CSetoid_un_op) (x y : S),
op x [#] op y -> x [#] y.
Proof cs_un_op_strext.
Lemma bin_op_wd_unfolded : forall (op : CSetoid_bin_op) (x1 x2 y1 y2 : S),
x1 [=] x2 -> y1 [=] y2 -> op x1 y1 [=] op x2 y2.
Proof cs_bin_op_wd.
Lemma bin_op_strext_unfolded : forall (op : CSetoid_bin_op) (x1 x2 y1 y2 : S),
op x1 y1 [#] op x2 y2 -> x1 [#] x2 or y1 [#] y2.
Proof cs_bin_op_strext.
End csetoid_inner_ops.
Implicit Arguments commutes [S].
Implicit Arguments associative [S].
(* Needs to be unfolded to be used as a Hint *)
Hint Resolve ap_wdr_unfolded ap_wdl_unfolded bin_op_wd_unfolded un_op_wd_unfolded : algebra_c.
(**
** The binary outer operations on a csetoid
%\begin{convention}%
Let [S1] and [S2] be setoids.
%\end{convention}%
*)
Section csetoid_outer_ops.
Variables S1 S2 : CSetoid.
(**
Well-defined outer operations on a setoid.
*)
Definition outer_op_well_def := bin_fun_wd S1 S2 S2.
Definition outer_op_strext := bin_fun_strext S1 S2 S2.
Definition CSetoid_outer_op : Type := CSetoid_bin_fun S1 S2 S2.
Definition Build_CSetoid_outer_op := Build_CSetoid_bin_fun S1 S2 S2.
Definition csoo_wd := csbf_wd S1 S2 S2.
Definition csoo_strext := csbf_strext S1 S2 S2.
Lemma csoo_wd_unfolded : forall (op : CSetoid_outer_op) x1 x2 y1 y2,
x1 [=] x2 -> y1 [=] y2 -> op x1 y1 [=] op x2 y2.
Proof csoo_wd.
(* begin hide *)
Identity Coercion outer_op_bin_fun : CSetoid_outer_op >-> CSetoid_bin_fun.
(* end hide *)
End csetoid_outer_ops.
Hint Resolve csoo_wd_unfolded: algebra_c.
(**
** Subsetoids
%\begin{convention}%
Let [S] be a setoid, and [P] a predicate on the carrier of [S].
%\end{convention}%
*)
Section SubCSetoids.
Variable S : CSetoid.
Variable P : S -> CProp.
Record subcsetoid_crr : Type :=
{scs_elem :> S;
scs_prf : P scs_elem}.
(** Though [scs_elem] is declared as a coercion, it does not satisfy the
uniform inheritance condition and will not be inserted. However it will
also not be printed, which is handy.
*)
Definition restrict_relation (R : Relation S) : Relation subcsetoid_crr :=
fun a b : subcsetoid_crr =>
match a, b with
| Build_subcsetoid_crr x _, Build_subcsetoid_crr y _ => R x y
end.
Definition Crestrict_relation (R : Crelation S) : Crelation subcsetoid_crr :=
fun a b : subcsetoid_crr =>
match a, b with
| Build_subcsetoid_crr x _, Build_subcsetoid_crr y _ => R x y
end.
Definition subcsetoid_eq : Relation subcsetoid_crr :=
restrict_relation (cs_eq (r:=S)).
Definition subcsetoid_ap : Crelation subcsetoid_crr :=
Crestrict_relation (cs_ap (c:=S)).
Remark subcsetoid_equiv : Tequiv _ subcsetoid_eq.
Proof.
split.