1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti. C.Sacerdoti Coen. *)
8 (* ||A|| E.Tassi. S.Zacchiroli *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 (* Code ported from the Coq theorem prover by Claudio Sacerdoti Coen *)
16 (* Original author: Claudio Sacerdoti Coen. for the Coq system *)
18 set "baseuri" "cic:/matita/technicalities/setoids".
20 include "datatypes/constructors.ma".
21 include "logic/connectives2.ma".
22 include "logic/coimplication.ma".
24 (* DEFINITIONS OF Relation_Class AND n-ARY Morphism_Theory *)
26 (* X will be used to distinguish covariant arguments whose type is an *)
27 (* Asymmetric* relation from contravariant arguments of the same type *)
28 inductive X_Relation_Class (X: Type) : Type ≝
30 ∀A,Aeq. symmetric A Aeq → reflexive ? Aeq → X_Relation_Class X
31 | AsymmetricReflexive : X → ∀A,Aeq. reflexive A Aeq → X_Relation_Class X
32 | SymmetricAreflexive : ∀A,Aeq. symmetric A Aeq → X_Relation_Class X
33 | AsymmetricAreflexive : X → ∀A.∀Aeq : relation A. X_Relation_Class X
34 | Leibniz : Type → X_Relation_Class X.
36 inductive variance : Set ≝
38 | Contravariant : variance.
40 definition Argument_Class ≝ X_Relation_Class variance.
41 definition Relation_Class ≝ X_Relation_Class unit.
43 inductive Reflexive_Relation_Class : Type :=
45 ∀A,Aeq. symmetric A Aeq → reflexive ? Aeq → Reflexive_Relation_Class
47 ∀A,Aeq. reflexive A Aeq → Reflexive_Relation_Class
48 | RLeibniz : Type → Reflexive_Relation_Class.
50 inductive Areflexive_Relation_Class : Type :=
51 | ASymmetric : ∀A,Aeq. symmetric A Aeq → Areflexive_Relation_Class
52 | AAsymmetric : ∀A.∀Aeq : relation A. Areflexive_Relation_Class.
54 definition relation_class_of_argument_class : Argument_Class → Relation_Class.
56 [ apply (SymmetricReflexive ? ? ? H H1)
57 | apply (AsymmetricReflexive ? something ? ? H)
58 | apply (SymmetricAreflexive ? ? ? H)
59 | apply (AsymmetricAreflexive ? something ? r)
64 definition carrier_of_relation_class : ∀X. X_Relation_Class X → Type.
65 intros (X x); cases x (A o o o o A o o A o o o A o A); exact A.
68 definition relation_of_relation_class:
69 ∀X,R. carrier_of_relation_class X R → carrier_of_relation_class X R → Prop.
70 intros 2; cases R; simplify; [1,2,3,4: assumption | apply (eq T) ]
73 lemma about_carrier_of_relation_class_and_relation_class_of_argument_class :
75 carrier_of_relation_class ? (relation_class_of_argument_class R) =
76 carrier_of_relation_class ? R.
77 intro; cases R; reflexivity.
80 inductive nelistT (A : Type) : Type :=
82 | cons : A → nelistT A → nelistT A.
84 definition Arguments := nelistT Argument_Class.
86 definition function_type_of_morphism_signature :
87 Arguments → Relation_Class → Type.
88 intros (In Out); elim In;
89 [ exact (carrier_of_relation_class ? t → carrier_of_relation_class ? Out)
90 | exact (carrier_of_relation_class ? t → T)
94 definition make_compatibility_goal_aux:
95 ∀In,Out.∀f,g:function_type_of_morphism_signature In Out.Prop.
97 elim In (a); simplify in f f1;
98 generalize in match f1; clear f1;
99 generalize in match f; clear f;
100 [ elim a; simplify in f f1;
101 [ exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
103 [ exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
104 | exact (∀x1,x2. r x2 x1 → relation_of_relation_class ? Out (f x1) (f1 x2))
106 | exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
108 [ exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
109 | exact (∀x1,x2. r x2 x1 → relation_of_relation_class ? Out (f x1) (f1 x2))
111 | exact (∀x. relation_of_relation_class ? Out (f x) (f1 x))
114 ((carrier_of_relation_class ? t → function_type_of_morphism_signature n Out) →
115 (carrier_of_relation_class ? t → function_type_of_morphism_signature n Out) →
117 elim t; simplify in f f1;
118 [1,3: exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
120 [1,3: exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
121 |2,4: exact (∀x1,x2. r x2 x1 → R (f x1) (f1 x2))
123 | exact (∀x. R (f x) (f1 x))
128 definition make_compatibility_goal :=
129 λIn,Out,f. make_compatibility_goal_aux In Out f f.
131 record Morphism_Theory (In: Arguments) (Out: Relation_Class) : Type :=
132 { Function : function_type_of_morphism_signature In Out;
133 Compat : make_compatibility_goal In Out Function
136 definition list_of_Leibniz_of_list_of_types: nelistT Type → Arguments.
139 [ apply (singl ? (Leibniz ? t))
140 | apply (cons ? (Leibniz ? t) a)
144 (* every function is a morphism from Leibniz+ to Leibniz *)
145 definition morphism_theory_of_function :
146 ∀In: nelistT Type.∀Out: Type.
147 let In' := list_of_Leibniz_of_list_of_types In in
148 let Out' := Leibniz ? Out in
149 function_type_of_morphism_signature In' Out' →
150 Morphism_Theory In' Out'.
152 apply (mk_Morphism_Theory ? ? f);
153 unfold In' in f; clear In';
154 unfold Out' in f; clear Out';
155 generalize in match f; clear f;
157 [ unfold make_compatibility_goal;
168 (* THE iff RELATION CLASS *)
170 definition Iff_Relation_Class : Relation_Class.
171 apply (SymmetricReflexive unit ? iff);
172 [ exact symmetric_iff
173 | exact reflexive_iff
177 (* THE impl RELATION CLASS *)
179 definition impl \def \lambda A,B:Prop. A → B.
181 theorem impl_refl: reflexive ? impl.
189 definition Impl_Relation_Class : Relation_Class.
190 unfold Relation_Class;
191 apply (AsymmetricReflexive unit something ? impl);
195 (* UTILITY FUNCTIONS TO PROVE THAT EVERY TRANSITIVE RELATION IS A MORPHISM *)
197 definition equality_morphism_of_symmetric_areflexive_transitive_relation:
198 ∀A: Type.∀Aeq: relation A.∀sym: symmetric ? Aeq.∀trans: transitive ? Aeq.
199 let ASetoidClass := SymmetricAreflexive ? ? ? sym in
200 (Morphism_Theory (cons ? ASetoidClass (singl ? ASetoidClass))
203 apply mk_Morphism_Theory;
205 | unfold make_compatibility_goal;
209 unfold transitive in H;
210 unfold symmetric in sym;
216 definition equality_morphism_of_symmetric_reflexive_transitive_relation:
217 ∀A: Type.∀Aeq: relation A.∀refl: reflexive ? Aeq.∀sym: symmetric ? Aeq.
218 ∀trans: transitive ? Aeq.
219 let ASetoidClass := SymmetricReflexive ? ? ? sym refl in
220 (Morphism_Theory (cons ? ASetoidClass (singl ? ASetoidClass)) Iff_Relation_Class).
222 apply mk_Morphism_Theory;
228 unfold transitive in H;
229 unfold symmetric in sym;
234 definition equality_morphism_of_asymmetric_areflexive_transitive_relation:
235 ∀A: Type.∀Aeq: relation A.∀trans: transitive ? Aeq.
236 let ASetoidClass1 := AsymmetricAreflexive ? Contravariant ? Aeq in
237 let ASetoidClass2 := AsymmetricAreflexive ? Covariant ? Aeq in
238 (Morphism_Theory (cons ? ASetoidClass1 (singl ? ASetoidClass2)) Impl_Relation_Class).
240 apply mk_Morphism_Theory;
251 definition equality_morphism_of_asymmetric_reflexive_transitive_relation:
252 ∀A: Type.∀Aeq: relation A.∀refl: reflexive ? Aeq.∀trans: transitive ? Aeq.
253 let ASetoidClass1 := AsymmetricReflexive ? Contravariant ? ? refl in
254 let ASetoidClass2 := AsymmetricReflexive ? Covariant ? ? refl in
255 (Morphism_Theory (cons ? ASetoidClass1 (singl ? ASetoidClass2)) Impl_Relation_Class).
257 apply mk_Morphism_Theory;
268 (* iff AS A RELATION *)
270 (*DA PORTARE:Add Relation Prop iff
271 reflexivity proved by iff_refl
272 symmetry proved by iff_sym
273 transitivity proved by iff_trans
276 (* every predicate is morphism from Leibniz+ to Iff_Relation_Class *)
277 definition morphism_theory_of_predicate :
279 let In' := list_of_Leibniz_of_list_of_types In in
280 function_type_of_morphism_signature In' Iff_Relation_Class →
281 Morphism_Theory In' Iff_Relation_Class.
283 apply mk_Morphism_Theory;
285 | generalize in match f; clear f;
286 unfold In'; clear In';
298 (* impl AS A RELATION *)
300 theorem impl_trans: transitive ? impl.
307 (*DA PORTARE: Add Relation Prop impl
308 reflexivity proved by impl_refl
309 transitivity proved by impl_trans
312 (* THE CIC PART OF THE REFLEXIVE TACTIC (SETOID REWRITE) *)
314 inductive rewrite_direction : Type :=
315 Left2Right: rewrite_direction
316 | Right2Left: rewrite_direction.
318 definition variance_of_argument_class : Argument_Class → option variance.
329 definition opposite_direction :=
332 [ Left2Right ⇒ Right2Left
333 | Right2Left ⇒ Left2Right
336 lemma opposite_direction_idempotent:
337 ∀dir. opposite_direction (opposite_direction dir) = dir.
343 inductive check_if_variance_is_respected :
344 option variance → rewrite_direction → rewrite_direction → Prop
346 MSNone : ∀dir,dir'. check_if_variance_is_respected (None ?) dir dir'
347 | MSCovariant : ∀dir. check_if_variance_is_respected (Some ? Covariant) dir dir
350 check_if_variance_is_respected (Some ? Contravariant) dir (opposite_direction dir).
352 definition relation_class_of_reflexive_relation_class:
353 Reflexive_Relation_Class → Relation_Class.
356 [ apply (SymmetricReflexive ? ? ? H H1)
357 | apply (AsymmetricReflexive ? something ? ? H)
358 | apply (Leibniz ? T)
362 definition relation_class_of_areflexive_relation_class:
363 Areflexive_Relation_Class → Relation_Class.
366 [ apply (SymmetricAreflexive ? ? ? H)
367 | apply (AsymmetricAreflexive ? something ? r)
371 definition carrier_of_reflexive_relation_class :=
372 λR.carrier_of_relation_class ? (relation_class_of_reflexive_relation_class R).
374 definition carrier_of_areflexive_relation_class :=
375 λR.carrier_of_relation_class ? (relation_class_of_areflexive_relation_class R).
377 definition relation_of_areflexive_relation_class :=
378 λR.relation_of_relation_class ? (relation_class_of_areflexive_relation_class R).
380 inductive Morphism_Context (Hole: Relation_Class) (dir:rewrite_direction) : Relation_Class → rewrite_direction → Type :=
383 Morphism_Theory In Out → Morphism_Context_List Hole dir dir' In →
384 Morphism_Context Hole dir Out dir'
385 | ToReplace : Morphism_Context Hole dir Hole dir
388 carrier_of_reflexive_relation_class S →
389 Morphism_Context Hole dir (relation_class_of_reflexive_relation_class S) dir'
390 | ProperElementToKeep :
391 ∀S,dir'.∀x: carrier_of_areflexive_relation_class S.
392 relation_of_areflexive_relation_class S x x →
393 Morphism_Context Hole dir (relation_class_of_areflexive_relation_class S) dir'
394 with Morphism_Context_List :
395 rewrite_direction → Arguments → Type
399 check_if_variance_is_respected (variance_of_argument_class S) dir' dir'' →
400 Morphism_Context Hole dir (relation_class_of_argument_class S) dir' →
401 Morphism_Context_List Hole dir dir'' (singl ? S)
404 check_if_variance_is_respected (variance_of_argument_class S) dir' dir'' →
405 Morphism_Context Hole dir (relation_class_of_argument_class S) dir' →
406 Morphism_Context_List Hole dir dir'' L →
407 Morphism_Context_List Hole dir dir'' (cons ? S L).
409 lemma Morphism_Context_rect2:
412 ∀r:Relation_Class.∀r0:rewrite_direction.Morphism_Context Hole dir r r0 → Type.
414 ∀r:rewrite_direction.∀a:Arguments.Morphism_Context_List Hole dir r a → Type.
416 ∀m:Morphism_Theory In Out.∀m0:Morphism_Context_List Hole dir dir' In.
417 P0 dir' In m0 → P Out dir' (App Hole ? ? ? ? m m0)) →
418 P Hole dir (ToReplace Hole dir) →
419 (∀S:Reflexive_Relation_Class.∀dir'.∀c:carrier_of_reflexive_relation_class S.
420 P (relation_class_of_reflexive_relation_class S) dir'
421 (ToKeep Hole dir S dir' c)) →
422 (∀S:Areflexive_Relation_Class.∀dir'.
423 ∀x:carrier_of_areflexive_relation_class S.
424 ∀r:relation_of_areflexive_relation_class S x x.
425 P (relation_class_of_areflexive_relation_class S) dir'
426 (ProperElementToKeep Hole dir S dir' x r)) →
427 (∀S:Argument_Class.∀dir',dir''.
428 ∀c:check_if_variance_is_respected (variance_of_argument_class S) dir' dir''.
429 ∀m:Morphism_Context Hole dir (relation_class_of_argument_class S) dir'.
430 P (relation_class_of_argument_class S) dir' m ->
431 P0 dir'' (singl ? S) (fcl_singl ? ? S ? ? c m)) →
432 (∀S:Argument_Class.∀L:Arguments.∀dir',dir''.
433 ∀c:check_if_variance_is_respected (variance_of_argument_class S) dir' dir''.
434 ∀m:Morphism_Context Hole dir (relation_class_of_argument_class S) dir'.
435 P (relation_class_of_argument_class S) dir' m →
436 ∀m0:Morphism_Context_List Hole dir dir'' L.
437 P0 dir'' L m0 → P0 dir'' (cons ? S L) (fcl_cons ? ? S ? ? ? c m m0)) →
438 ∀r:Relation_Class.∀r0:rewrite_direction.∀m:Morphism_Context Hole dir r r0.
441 λHole,dir,P,P0,f,f0,f1,f2,f3,f4.
443 F (rc:Relation_Class) (r0:rewrite_direction)
444 (m:Morphism_Context Hole dir rc r0) on m : P rc r0 m
446 match m return λrc.λr0.λm0.P rc r0 m0 with
447 [ App In Out dir' m0 m1 ⇒ f In Out dir' m0 m1 (F0 dir' In m1)
449 | ToKeep S dir' c ⇒ f1 S dir' c
450 | ProperElementToKeep S dir' x r1 ⇒ f2 S dir' x r1
453 F0 (r:rewrite_direction) (a:Arguments)
454 (m:Morphism_Context_List Hole dir r a) on m : P0 r a m
456 match m return λr.λa.λm0.P0 r a m0 with
457 [ fcl_singl S dir' dir'' c m0 ⇒
458 f3 S dir' dir'' c m0 (F (relation_class_of_argument_class S) dir' m0)
459 | fcl_cons S L dir' dir'' c m0 m1 ⇒
460 f4 S L dir' dir'' c m0 (F (relation_class_of_argument_class S) dir' m0)
465 lemma Morphism_Context_List_rect2:
468 ∀r:Relation_Class.∀r0:rewrite_direction.Morphism_Context Hole dir r r0 → Type.
470 ∀r:rewrite_direction.∀a:Arguments.Morphism_Context_List Hole dir r a → Type.
472 ∀m:Morphism_Theory In Out.∀m0:Morphism_Context_List Hole dir dir' In.
473 P0 dir' In m0 → P Out dir' (App Hole ? ? ? ? m m0)) →
474 P Hole dir (ToReplace Hole dir) →
475 (∀S:Reflexive_Relation_Class.∀dir'.∀c:carrier_of_reflexive_relation_class S.
476 P (relation_class_of_reflexive_relation_class S) dir'
477 (ToKeep Hole dir S dir' c)) →
478 (∀S:Areflexive_Relation_Class.∀dir'.
479 ∀x:carrier_of_areflexive_relation_class S.
480 ∀r:relation_of_areflexive_relation_class S x x.
481 P (relation_class_of_areflexive_relation_class S) dir'
482 (ProperElementToKeep Hole dir S dir' x r)) →
483 (∀S:Argument_Class.∀dir',dir''.
484 ∀c:check_if_variance_is_respected (variance_of_argument_class S) dir' dir''.
485 ∀m:Morphism_Context Hole dir (relation_class_of_argument_class S) dir'.
486 P (relation_class_of_argument_class S) dir' m ->
487 P0 dir'' (singl ? S) (fcl_singl ? ? S ? ? c m)) →
488 (∀S:Argument_Class.∀L:Arguments.∀dir',dir''.
489 ∀c:check_if_variance_is_respected (variance_of_argument_class S) dir' dir''.
490 ∀m:Morphism_Context Hole dir (relation_class_of_argument_class S) dir'.
491 P (relation_class_of_argument_class S) dir' m →
492 ∀m0:Morphism_Context_List Hole dir dir'' L.
493 P0 dir'' L m0 → P0 dir'' (cons ? S L) (fcl_cons ? ? S ? ? ? c m m0)) →
494 ∀r:rewrite_direction.∀a:Arguments.∀m:Morphism_Context_List Hole dir r a.
497 λHole,dir,P,P0,f,f0,f1,f2,f3,f4.
499 F (rc:Relation_Class) (r0:rewrite_direction)
500 (m:Morphism_Context Hole dir rc r0) on m : P rc r0 m
502 match m return λrc.λr0.λm0.P rc r0 m0 with
503 [ App In Out dir' m0 m1 ⇒ f In Out dir' m0 m1 (F0 dir' In m1)
505 | ToKeep S dir' c ⇒ f1 S dir' c
506 | ProperElementToKeep S dir' x r1 ⇒ f2 S dir' x r1
509 F0 (r:rewrite_direction) (a:Arguments)
510 (m:Morphism_Context_List Hole dir r a) on m : P0 r a m
512 match m return λr.λa.λm0.P0 r a m0 with
513 [ fcl_singl S dir' dir'' c m0 ⇒
514 f3 S dir' dir'' c m0 (F (relation_class_of_argument_class S) dir' m0)
515 | fcl_cons S L dir' dir'' c m0 m1 ⇒
516 f4 S L dir' dir'' c m0 (F (relation_class_of_argument_class S) dir' m0)
521 definition product_of_arguments : Arguments → Type.
524 [ apply (carrier_of_relation_class ? t)
525 | apply (Prod (carrier_of_relation_class ? t) T)
529 definition get_rewrite_direction: rewrite_direction → Argument_Class → rewrite_direction.
531 cases (variance_of_argument_class R) (a);
534 [ exact dir (* covariant *)
535 | exact (opposite_direction dir) (* contravariant *)
540 definition directed_relation_of_relation_class:
541 ∀dir:rewrite_direction.∀R: Relation_Class.
542 carrier_of_relation_class ? R → carrier_of_relation_class ? R → Prop.
545 [ exact (relation_of_relation_class ? ? c c1)
546 | apply (relation_of_relation_class ? ? c1 c)
550 definition directed_relation_of_argument_class:
551 ∀dir:rewrite_direction.∀R: Argument_Class.
552 carrier_of_relation_class ? R → carrier_of_relation_class ? R → Prop.
554 rewrite < (about_carrier_of_relation_class_and_relation_class_of_argument_class R) in c c1;
555 exact (directed_relation_of_relation_class dir (relation_class_of_argument_class R) c c1).
559 definition relation_of_product_of_arguments:
560 ∀dir:rewrite_direction.∀In.
561 product_of_arguments In → product_of_arguments In → Prop.
566 exact (directed_relation_of_argument_class (get_rewrite_direction r t) t)
568 change in p with (Prod (carrier_of_relation_class variance t) (product_of_arguments n));
569 change in p1 with (Prod (carrier_of_relation_class variance t) (product_of_arguments n));
574 (directed_relation_of_argument_class (get_rewrite_direction r t) t c c1)
580 definition apply_morphism:
581 ∀In,Out.∀m: function_type_of_morphism_signature In Out.
582 ∀args: product_of_arguments In. carrier_of_relation_class ? Out.
586 | change in p with (Prod (carrier_of_relation_class variance t) (product_of_arguments n));
588 change in f1 with (carrier_of_relation_class variance t → function_type_of_morphism_signature n Out);
589 exact (f ? (f1 t1) t2)
593 theorem apply_morphism_compatibility_Right2Left:
594 ∀In,Out.∀m1,m2: function_type_of_morphism_signature In Out.
595 ∀args1,args2: product_of_arguments In.
596 make_compatibility_goal_aux ? ? m1 m2 →
597 relation_of_product_of_arguments Right2Left ? args1 args2 →
598 directed_relation_of_relation_class Right2Left ?
599 (apply_morphism ? ? m2 args1)
600 (apply_morphism ? ? m1 args2).
603 [ simplify in m1 m2 args1 args2 ⊢ %;
605 (directed_relation_of_argument_class
606 (get_rewrite_direction Right2Left t) t args1 args2);
607 generalize in match H1; clear H1;
608 generalize in match H; clear H;
609 generalize in match args2; clear args2;
610 generalize in match args1; clear args1;
611 generalize in match m2; clear m2;
612 generalize in match m1; clear m1;
614 [ intros (T1 r Hs Hr m1 m2 args1 args2 H H1);
617 | intros 8 (v T1 r Hr m1 m2 args1 args2);
637 (carrier_of_relation_class variance t →
638 function_type_of_morphism_signature n Out);
640 (carrier_of_relation_class variance t →
641 function_type_of_morphism_signature n Out);
643 ((carrier_of_relation_class ? t) × (product_of_arguments n));
645 ((carrier_of_relation_class ? t) × (product_of_arguments n));
646 generalize in match H2; clear H2;
647 elim args2 0; clear args2;
648 elim args1; clear args1;
651 (relation_of_product_of_arguments Right2Left n t2 t4);
653 (relation_of_relation_class unit Out (apply_morphism n Out (m1 t3) t4)
654 (apply_morphism n Out (m2 t1) t2));
655 generalize in match H3; clear H3;
656 generalize in match t3; clear t3;
657 generalize in match t1; clear t1;
658 generalize in match H1; clear H1;
659 generalize in match m2; clear m2;
660 generalize in match m1; clear m1;
662 [ intros (T1 r Hs Hr m1 m2 H1 t1 t3 H3);
665 (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
668 [ intros (T1 r Hr m1 m2 H1 t1 t3 H3);
671 (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
672 | intros (T1 r Hr m1 m2 H1 t1 t3 H3);
675 (∀x1,x2:T1.r x2 x1 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
677 | intros (T1 r Hs m1 m2 H1 t1 t3 H3);
680 (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
683 [ intros (T1 r m1 m2 H1 t1 t3 H3);
686 (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
687 | intros (T1 r m1 m2 H1 t1 t3 H3);
690 (∀x1,x2:T1.r x2 x1 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
692 | intros (T m1 m2 H1 t1 t3 H3);
695 (∀x:T. make_compatibility_goal_aux n Out (m1 x) (m2 x));
714 theorem apply_morphism_compatibility_Left2Right:
715 ∀In,Out.∀m1,m2: function_type_of_morphism_signature In Out.
716 ∀args1,args2: product_of_arguments In.
717 make_compatibility_goal_aux ? ? m1 m2 →
718 relation_of_product_of_arguments Left2Right ? args1 args2 →
719 directed_relation_of_relation_class Left2Right ?
720 (apply_morphism ? ? m1 args1)
721 (apply_morphism ? ? m2 args2).
723 elim In 0; simplify; intros;
725 (directed_relation_of_argument_class
726 (get_rewrite_direction Left2Right t) t args1 args2);
727 generalize in match H1; clear H1;
728 generalize in match H; clear H;
729 generalize in match args2; clear args2;
730 generalize in match args1; clear args1;
731 generalize in match m2; clear m2;
732 generalize in match m1; clear m1;
734 [ intros (T1 r Hs Hr m1 m2 args1 args2 H H1);
737 | intros 8 (v T1 r Hr m1 m2 args1 args2);
759 (carrier_of_relation_class variance t →
760 function_type_of_morphism_signature n Out);
762 (carrier_of_relation_class variance t →
763 function_type_of_morphism_signature n Out);
765 ((carrier_of_relation_class ? t) × (product_of_arguments n));
767 ((carrier_of_relation_class ? t) × (product_of_arguments n));
768 generalize in match H2; clear H2;
769 elim args2 0; clear args2;
770 elim args1; clear args1;
773 (relation_of_product_of_arguments Left2Right n t2 t4);
775 (relation_of_relation_class unit Out (apply_morphism n Out (m1 t1) t2)
776 (apply_morphism n Out (m2 t3) t4));
777 generalize in match H3; clear H3;
778 generalize in match t3; clear t3;
779 generalize in match t1; clear t1;
780 generalize in match H1; clear H1;
781 generalize in match m2; clear m2;
782 generalize in match m1; clear m1;
784 [ intros (T1 r Hs Hr m1 m2 H1 t1 t3 H3);
786 (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
789 [ intros (T1 r Hr m1 m2 H1 t1 t3 H3);
792 (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
793 | intros (T1 r Hr m1 m2 H1 t1 t3 H3);
796 (∀x1,x2:T1.r x2 x1 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
798 | intros (T1 r Hs m1 m2 H1 t1 t3 H3);
801 (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
804 [ intros (T1 r m1 m2 H1 t1 t3 H3);
807 (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
808 | intros (T1 r m1 m2 H1 t1 t3 H3);
811 (∀x1,x2:T1.r x2 x1 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
813 | intros (T m1 m2 H1 t1 t3 H3);
816 (∀x:T. make_compatibility_goal_aux n Out (m1 x) (m2 x));
836 ∀Hole,dir,Out,dir'. carrier_of_relation_class ? Hole →
837 Morphism_Context Hole dir Out dir' → carrier_of_relation_class ? Out.
838 intros (Hole dir Out dir' H t).
840 (Morphism_Context_rect2 Hole dir (λS,xx,yy. carrier_of_relation_class ? S)
841 (λxx,L,fcl.product_of_arguments L));
845 | exact (apply_morphism ? ? (Function ? ? m) p)
851 (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
855 (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
863 (*CSC: interp and interp_relation_class_list should be mutually defined. since
864 the proof term of each one contains the proof term of the other one. However
865 I cannot do that interactively (I should write the Fix by hand) *)
866 definition interp_relation_class_list :
867 ∀Hole,dir,dir'.∀L: Arguments. carrier_of_relation_class ? Hole →
868 Morphism_Context_List Hole dir dir' L → product_of_arguments L.
869 intros (Hole dir dir' L H t);
871 (Morphism_Context_List_rect2 Hole dir (λS,xx,yy.carrier_of_relation_class ? S)
872 (λxx,L,fcl.product_of_arguments L));
876 | exact (apply_morphism ? ? (Function ? ? m) p)
882 (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
886 (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
894 Theorem setoid_rewrite:
895 ∀Hole dir Out dir' (E1 E2: carrier_of_relation_class Hole)
896 (E: Morphism_Context Hole dir Out dir').
897 (directed_relation_of_relation_class dir Hole E1 E2) →
898 (directed_relation_of_relation_class dir' Out (interp E1 E) (interp E2 E)).
901 (@Morphism_Context_rect2 Hole dir
902 (fun S dir'' E => directed_relation_of_relation_class dir'' S (interp E1 E) (interp E2 E))
904 relation_of_product_of_arguments dir'' ?
905 (interp_relation_class_list E1 fcl)
906 (interp_relation_class_list E2 fcl))); intros.
907 change (directed_relation_of_relation_class dir'0 Out0
908 (apply_morphism ? ? (Function m) (interp_relation_class_list E1 m0))
909 (apply_morphism ? ? (Function m) (interp_relation_class_list E2 m0))).
911 apply apply_morphism_compatibility_Left2Right.
914 apply apply_morphism_compatibility_Right2Left.
920 unfold interp. Morphism_Context_rect2.
921 (*CSC: reflexivity used here*)
922 destruct S; destruct dir'0; simpl; (apply r || reflexivity).
924 destruct dir'0; exact r.
926 destruct S; unfold directed_relation_of_argument_class; simpl in H0 |- *;
927 unfold get_rewrite_direction; simpl.
928 destruct dir'0; destruct dir'';
930 unfold directed_relation_of_argument_class; simpl; apply s; exact H0).
931 (* the following mess with generalize/clear/intros is to help Coq resolving *)
932 (* second order unification problems. *)
933 generalize m c H0; clear H0 m c; inversion c;
934 generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros;
935 (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3).
936 destruct dir'0; destruct dir'';
938 unfold directed_relation_of_argument_class; simpl; apply s; exact H0).
939 (* the following mess with generalize/clear/intros is to help Coq resolving *)
940 (* second order unification problems. *)
941 generalize m c H0; clear H0 m c; inversion c;
942 generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros;
943 (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3).
944 destruct dir'0; destruct dir''; (exact H0 || hnf; symmetry; exact H0).
947 (directed_relation_of_argument_class (get_rewrite_direction dir'' S) S
948 (eq_rect ? (fun T : Type => T) (interp E1 m) ?
949 (about_carrier_of_relation_class_and_relation_class_of_argument_class S))
950 (eq_rect ? (fun T : Type => T) (interp E2 m) ?
951 (about_carrier_of_relation_class_and_relation_class_of_argument_class S)) /\
952 relation_of_product_of_arguments dir'' ?
953 (interp_relation_class_list E1 m0) (interp_relation_class_list E2 m0)).
955 clear m0 H1; destruct S; simpl in H0 |- *; unfold get_rewrite_direction; simpl.
956 destruct dir''; destruct dir'0; (exact H0 || hnf; apply s; exact H0).
958 rewrite <- H3; exact H0.
959 rewrite (opposite_direction_idempotent dir'0); exact H0.
960 destruct dir''; destruct dir'0; (exact H0 || hnf; apply s; exact H0).
962 rewrite <- H3; exact H0.
963 rewrite (opposite_direction_idempotent dir'0); exact H0.
964 destruct dir''; destruct dir'0; (exact H0 || hnf; symmetry; exact H0).
968 (* A FEW EXAMPLES ON iff *)
970 (* impl IS A MORPHISM *)
972 Add Morphism impl with signature iff ==> iff ==> iff as Impl_Morphism.
973 unfold impl; tautobatch.
976 (* and IS A MORPHISM *)
978 Add Morphism and with signature iff ==> iff ==> iff as And_Morphism.
982 (* or IS A MORPHISM *)
984 Add Morphism or with signature iff ==> iff ==> iff as Or_Morphism.
988 (* not IS A MORPHISM *)
990 Add Morphism not with signature iff ==> iff as Not_Morphism.
994 (* THE SAME EXAMPLES ON impl *)
996 Add Morphism and with signature impl ++> impl ++> impl as And_Morphism2.
997 unfold impl; tautobatch.
1000 Add Morphism or with signature impl ++> impl ++> impl as Or_Morphism2.
1001 unfold impl; tautobatch.
1004 Add Morphism not with signature impl -→ impl as Not_Morphism2.
1005 unfold impl; tautobatch.