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".
23 (* DEFINITIONS OF Relation_Class AND n-ARY Morphism_Theory *)
25 (* X will be used to distinguish covariant arguments whose type is an *)
26 (* Asymmetric* relation from contravariant arguments of the same type *)
27 inductive X_Relation_Class (X: Type) : Type ≝
29 ∀A,Aeq. symmetric A Aeq → reflexive ? Aeq → X_Relation_Class X
30 | AsymmetricReflexive : X → ∀A,Aeq. reflexive A Aeq → X_Relation_Class X
31 | SymmetricAreflexive : ∀A,Aeq. symmetric A Aeq → X_Relation_Class X
32 | AsymmetricAreflexive : X → ∀A.∀Aeq : relation A. X_Relation_Class X
33 | Leibniz : Type → X_Relation_Class X.
35 inductive variance : Set ≝
37 | Contravariant : variance.
39 definition Argument_Class ≝ X_Relation_Class variance.
40 definition Relation_Class ≝ X_Relation_Class unit.
42 inductive Reflexive_Relation_Class : Type :=
44 ∀A,Aeq. symmetric A Aeq → reflexive ? Aeq → Reflexive_Relation_Class
46 ∀A,Aeq. reflexive A Aeq → Reflexive_Relation_Class
47 | RLeibniz : Type → Reflexive_Relation_Class.
49 inductive Areflexive_Relation_Class : Type :=
50 | ASymmetric : ∀A,Aeq. symmetric A Aeq → Areflexive_Relation_Class
51 | AAsymmetric : ∀A.∀Aeq : relation A. Areflexive_Relation_Class.
53 definition relation_class_of_argument_class : Argument_Class → Relation_Class.
57 [ apply (SymmetricReflexive ? ? ? H H1)
58 | apply (AsymmetricReflexive ? something ? ? H)
59 | apply (SymmetricAreflexive ? ? ? H)
60 | apply (AsymmetricAreflexive ? something ? r)
61 | apply (Leibniz ? T1)
65 definition carrier_of_relation_class : ∀X. X_Relation_Class X → Type.
71 definition relation_of_relation_class :
72 ∀X,R. carrier_of_relation_class X R → carrier_of_relation_class X R → Prop.
76 [1,2: intros 4; apply r
77 |3,4: intros 3; apply r
82 lemma about_carrier_of_relation_class_and_relation_class_of_argument_class :
84 carrier_of_relation_class ? (relation_class_of_argument_class R) =
85 carrier_of_relation_class ? R.
91 inductive nelistT (A : Type) : Type :=
93 | cons : A → nelistT A → nelistT A.
95 definition Arguments := nelistT Argument_Class.
97 definition function_type_of_morphism_signature :
98 Arguments → Relation_Class → Type.
101 [ exact (carrier_of_relation_class ? t → carrier_of_relation_class ? Out)
102 | exact (carrier_of_relation_class ? t → T)
106 definition make_compatibility_goal_aux:
107 ∀In,Out.∀f,g:function_type_of_morphism_signature In Out.Prop.
109 elim In (a); simplify in f f1;
110 generalize in match f1; clear f1;
111 generalize in match f; clear f;
112 [ elim a; simplify in f f1;
113 [ exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
115 [ exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
116 | exact (∀x1,x2. r x2 x1 → relation_of_relation_class ? Out (f x1) (f1 x2))
118 | exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
120 [ exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
121 | exact (∀x1,x2. r x2 x1 → relation_of_relation_class ? Out (f x1) (f1 x2))
123 | exact (∀x. relation_of_relation_class ? Out (f x) (f1 x))
126 ((carrier_of_relation_class ? t → function_type_of_morphism_signature n Out) →
127 (carrier_of_relation_class ? t → function_type_of_morphism_signature n Out) →
129 elim t; simplify in f f1;
130 [ exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
132 [ exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
133 | exact (∀x1,x2. r x2 x1 → R (f x1) (f1 x2))
135 | exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
137 [ exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
138 | exact (∀x1,x2. r x2 x1 → R (f x1) (f1 x2))
140 | exact (∀x. R (f x) (f1 x))
145 definition make_compatibility_goal :=
146 λIn,Out,f. make_compatibility_goal_aux In Out f f.
148 record Morphism_Theory (In: Arguments) (Out: Relation_Class) : Type :=
149 { Function : function_type_of_morphism_signature In Out;
150 Compat : make_compatibility_goal In Out Function
153 definition list_of_Leibniz_of_list_of_types: nelistT Type → Arguments.
156 [ apply (singl ? (Leibniz ? t))
157 | apply (cons ? (Leibniz ? t) a)
161 (* every function is a morphism from Leibniz+ to Leibniz *)
162 definition morphism_theory_of_function :
163 ∀In: nelistT Type.∀Out: Type.
164 let In' := list_of_Leibniz_of_list_of_types In in
165 let Out' := Leibniz ? Out in
166 function_type_of_morphism_signature In' Out' →
167 Morphism_Theory In' Out'.
169 apply (mk_Morphism_Theory ? ? f);
170 unfold In' in f; clear In';
171 unfold Out' in f; clear Out';
172 generalize in match f; clear f;
174 [ unfold make_compatibility_goal;
187 (* THE iff RELATION CLASS *)
189 definition Iff_Relation_Class : Relation_Class.
190 apply (SymmetricReflexive unit ? iff);
191 [ exact symmetric_iff
192 | exact reflexive_iff
196 (* THE impl RELATION CLASS *)
198 definition impl \def \lambda A,B:Prop. A → B.
200 theorem impl_refl: reflexive ? impl.
208 definition Impl_Relation_Class : Relation_Class.
209 unfold Relation_Class;
210 apply (AsymmetricReflexive unit something ? impl);
214 (* UTILITY FUNCTIONS TO PROVE THAT EVERY TRANSITIVE RELATION IS A MORPHISM *)
216 definition equality_morphism_of_symmetric_areflexive_transitive_relation:
217 ∀A: Type.∀Aeq: relation A.∀sym: symmetric ? Aeq.∀trans: transitive ? Aeq.
218 let ASetoidClass := SymmetricAreflexive ? ? ? sym in
219 (Morphism_Theory (cons ? ASetoidClass (singl ? ASetoidClass))
222 apply mk_Morphism_Theory;
224 | unfold make_compatibility_goal;
228 unfold transitive in H;
229 unfold symmetric in sym;
235 definition equality_morphism_of_symmetric_reflexive_transitive_relation:
236 ∀A: Type.∀Aeq: relation A.∀refl: reflexive ? Aeq.∀sym: symmetric ? Aeq.
237 ∀trans: transitive ? Aeq.
238 let ASetoidClass := SymmetricReflexive ? ? ? sym refl in
239 (Morphism_Theory (cons ? ASetoidClass (singl ? ASetoidClass)) Iff_Relation_Class).
241 apply mk_Morphism_Theory;
247 unfold transitive in H;
248 unfold symmetric in sym;
253 definition equality_morphism_of_asymmetric_areflexive_transitive_relation:
254 ∀A: Type.∀Aeq: relation A.∀trans: transitive ? Aeq.
255 let ASetoidClass1 := AsymmetricAreflexive ? Contravariant ? Aeq in
256 let ASetoidClass2 := AsymmetricAreflexive ? Covariant ? Aeq in
257 (Morphism_Theory (cons ? ASetoidClass1 (singl ? ASetoidClass2)) Impl_Relation_Class).
259 apply mk_Morphism_Theory;
270 definition equality_morphism_of_asymmetric_reflexive_transitive_relation:
271 ∀A: Type.∀Aeq: relation A.∀refl: reflexive ? Aeq.∀trans: transitive ? Aeq.
272 let ASetoidClass1 := AsymmetricReflexive ? Contravariant ? ? refl in
273 let ASetoidClass2 := AsymmetricReflexive ? Covariant ? ? refl in
274 (Morphism_Theory (cons ? ASetoidClass1 (singl ? ASetoidClass2)) Impl_Relation_Class).
276 apply mk_Morphism_Theory;
287 (* iff AS A RELATION *)
289 (*DA PORTARE:Add Relation Prop iff
290 reflexivity proved by iff_refl
291 symmetry proved by iff_sym
292 transitivity proved by iff_trans
295 (* every predicate is morphism from Leibniz+ to Iff_Relation_Class *)
296 definition morphism_theory_of_predicate :
298 let In' := list_of_Leibniz_of_list_of_types In in
299 function_type_of_morphism_signature In' Iff_Relation_Class →
300 Morphism_Theory In' Iff_Relation_Class.
302 apply mk_Morphism_Theory;
304 | generalize in match f; clear f;
305 unfold In'; clear In';
309 alias id "iff_refl" = "cic:/matita/logic/coimplication/iff_refl.con".
318 (* impl AS A RELATION *)
320 theorem impl_trans: transitive ? impl.
327 (*DA PORTARE: Add Relation Prop impl
328 reflexivity proved by impl_refl
329 transitivity proved by impl_trans
332 (* THE CIC PART OF THE REFLEXIVE TACTIC (SETOID REWRITE) *)
334 inductive rewrite_direction : Type :=
335 Left2Right: rewrite_direction
336 | Right2Left: rewrite_direction.
338 (*definition variance_of_argument_class : Argument_Class → option variance.
347 definition opposite_direction :=
350 Left2Right => Right2Left
351 | Right2Left => Left2Right
354 Lemma opposite_direction_idempotent:
355 ∀dir. (opposite_direction (opposite_direction dir)) = dir.
356 destruct dir; reflexivity.
359 inductive check_if_variance_is_respected :
360 option variance → rewrite_direction → rewrite_direction → Prop
362 MSNone : ∀dir dir'. check_if_variance_is_respected None dir dir'
363 | MSCovariant : ∀dir. check_if_variance_is_respected (Some Covariant) dir dir
366 check_if_variance_is_respected (Some Contravariant) dir (opposite_direction dir).
368 definition relation_class_of_reflexive_relation_class:
369 Reflexive_Relation_Class → Relation_Class.
371 exact (SymmetricReflexive ? s r).
372 exact (AsymmetricReflexive tt r).
376 definition relation_class_of_areflexive_relation_class:
377 Areflexive_Relation_Class → Relation_Class.
379 exact (SymmetricAreflexive ? s).
380 exact (AsymmetricAreflexive tt Aeq).
383 definition carrier_of_reflexive_relation_class :=
384 fun R => carrier_of_relation_class (relation_class_of_reflexive_relation_class R).
386 definition carrier_of_areflexive_relation_class :=
387 fun R => carrier_of_relation_class (relation_class_of_areflexive_relation_class R).
389 definition relation_of_areflexive_relation_class :=
390 fun R => relation_of_relation_class (relation_class_of_areflexive_relation_class R).
392 inductive Morphism_Context Hole dir : Relation_Class → rewrite_direction → Type :=
395 Morphism_Theory In Out → Morphism_Context_List Hole dir dir' In →
396 Morphism_Context Hole dir Out dir'
397 | ToReplace : Morphism_Context Hole dir Hole dir
400 carrier_of_reflexive_relation_class S →
401 Morphism_Context Hole dir (relation_class_of_reflexive_relation_class S) dir'
402 | ProperElementToKeep :
403 ∀S dir' (x: carrier_of_areflexive_relation_class S).
404 relation_of_areflexive_relation_class S x x →
405 Morphism_Context Hole dir (relation_class_of_areflexive_relation_class S) dir'
406 with Morphism_Context_List Hole dir :
407 rewrite_direction → Arguments → Type
411 check_if_variance_is_respected (variance_of_argument_class S) dir' dir'' →
412 Morphism_Context Hole dir (relation_class_of_argument_class S) dir' →
413 Morphism_Context_List Hole dir dir'' (singl S)
416 check_if_variance_is_respected (variance_of_argument_class S) dir' dir'' →
417 Morphism_Context Hole dir (relation_class_of_argument_class S) dir' →
418 Morphism_Context_List Hole dir dir'' L →
419 Morphism_Context_List Hole dir dir'' (cons S L).
421 Scheme Morphism_Context_rect2 := Induction for Morphism_Context Sort Type
422 with Morphism_Context_List_rect2 := Induction for Morphism_Context_List Sort Type.
424 definition product_of_arguments : Arguments → Type.
426 exact (carrier_of_relation_class a).
427 exact (prod (carrier_of_relation_class a) IHX).
430 definition get_rewrite_direction: rewrite_direction → Argument_Class → rewrite_direction.
432 destruct (variance_of_argument_class R).
434 exact dir. (* covariant *)
435 exact (opposite_direction dir). (* contravariant *)
436 exact dir. (* symmetric relation *)
439 definition directed_relation_of_relation_class:
440 ∀dir (R: Relation_Class).
441 carrier_of_relation_class R → carrier_of_relation_class R → Prop.
443 exact (@relation_of_relation_class unit).
444 intros; exact (relation_of_relation_class ? X0 X).
447 definition directed_relation_of_argument_class:
448 ∀dir (R: Argument_Class).
449 carrier_of_relation_class R → carrier_of_relation_class R → Prop.
452 (about_carrier_of_relation_class_and_relation_class_of_argument_class R).
453 exact (directed_relation_of_relation_class dir (relation_class_of_argument_class R)).
457 definition relation_of_product_of_arguments:
459 product_of_arguments In → product_of_arguments In → Prop.
462 exact (directed_relation_of_argument_class (get_rewrite_direction dir a) a).
465 destruct X; destruct X0.
468 (directed_relation_of_argument_class (get_rewrite_direction dir a) a c c0).
472 definition apply_morphism:
473 ∀In Out (m: function_type_of_morphism_signature In Out)
474 (args: product_of_arguments In). carrier_of_relation_class Out.
480 exact (IHIn (m c) p).
483 Theorem apply_morphism_compatibility_Right2Left:
484 ∀In Out (m1 m2: function_type_of_morphism_signature In Out)
485 (args1 args2: product_of_arguments In).
486 make_compatibility_goal_aux ? ? m1 m2 →
487 relation_of_product_of_arguments Right2Left ? args1 args2 →
488 directed_relation_of_relation_class Right2Left ?
489 (apply_morphism ? ? m2 args1)
490 (apply_morphism ? ? m1 args2).
491 induction In; intros.
492 simpl in m1. m2. args1. args2. H0 |- *.
493 destruct a; simpl in H; hnf in H0.
495 destruct v; simpl in H0; apply H; exact H0.
497 destruct v; simpl in H0; apply H; exact H0.
498 rewrite H0; apply H; exact H0.
500 simpl in m1. m2. args1. args2. H0 |- *.
501 destruct args1; destruct args2; simpl.
504 destruct a; simpl in H.
525 rewrite H0; apply IHIn.
530 Theorem apply_morphism_compatibility_Left2Right:
531 ∀In Out (m1 m2: function_type_of_morphism_signature In Out)
532 (args1 args2: product_of_arguments In).
533 make_compatibility_goal_aux ? ? m1 m2 →
534 relation_of_product_of_arguments Left2Right ? args1 args2 →
535 directed_relation_of_relation_class Left2Right ?
536 (apply_morphism ? ? m1 args1)
537 (apply_morphism ? ? m2 args2).
538 induction In; intros.
539 simpl in m1. m2. args1. args2. H0 |- *.
540 destruct a; simpl in H; hnf in H0.
542 destruct v; simpl in H0; apply H; exact H0.
544 destruct v; simpl in H0; apply H; exact H0.
545 rewrite H0; apply H; exact H0.
547 simpl in m1. m2. args1. args2. H0 |- *.
548 destruct args1; destruct args2; simpl.
551 destruct a; simpl in H.
566 destruct v; simpl in H. H0; apply H; exact H0.
568 rewrite H0; apply IHIn.
574 ∀Hole dir Out dir'. carrier_of_relation_class Hole →
575 Morphism_Context Hole dir Out dir' → carrier_of_relation_class Out.
576 intros Hole dir Out dir' H t.
578 (@Morphism_Context_rect2 Hole dir (fun S ? ? => carrier_of_relation_class S)
579 (fun ? L fcl => product_of_arguments L));
581 exact (apply_morphism ? ? (Function m) X).
587 (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
591 (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
596 (*CSC: interp and interp_relation_class_list should be mutually defined. since
597 the proof term of each one contains the proof term of the other one. However
598 I cannot do that interactively (I should write the Fix by hand) *)
599 definition interp_relation_class_list :
600 ∀Hole dir dir' (L: Arguments). carrier_of_relation_class Hole →
601 Morphism_Context_List Hole dir dir' L → product_of_arguments L.
602 intros Hole dir dir' L H t.
604 (@Morphism_Context_List_rect2 Hole dir (fun S ? ? => carrier_of_relation_class S)
605 (fun ? L fcl => product_of_arguments L));
607 exact (apply_morphism ? ? (Function m) X).
613 (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
617 (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
622 Theorem setoid_rewrite:
623 ∀Hole dir Out dir' (E1 E2: carrier_of_relation_class Hole)
624 (E: Morphism_Context Hole dir Out dir').
625 (directed_relation_of_relation_class dir Hole E1 E2) →
626 (directed_relation_of_relation_class dir' Out (interp E1 E) (interp E2 E)).
629 (@Morphism_Context_rect2 Hole dir
630 (fun S dir'' E => directed_relation_of_relation_class dir'' S (interp E1 E) (interp E2 E))
632 relation_of_product_of_arguments dir'' ?
633 (interp_relation_class_list E1 fcl)
634 (interp_relation_class_list E2 fcl))); intros.
635 change (directed_relation_of_relation_class dir'0 Out0
636 (apply_morphism ? ? (Function m) (interp_relation_class_list E1 m0))
637 (apply_morphism ? ? (Function m) (interp_relation_class_list E2 m0))).
639 apply apply_morphism_compatibility_Left2Right.
642 apply apply_morphism_compatibility_Right2Left.
648 unfold interp. Morphism_Context_rect2.
649 (*CSC: reflexivity used here*)
650 destruct S; destruct dir'0; simpl; (apply r || reflexivity).
652 destruct dir'0; exact r.
654 destruct S; unfold directed_relation_of_argument_class; simpl in H0 |- *;
655 unfold get_rewrite_direction; simpl.
656 destruct dir'0; destruct dir'';
658 unfold directed_relation_of_argument_class; simpl; apply s; exact H0).
659 (* the following mess with generalize/clear/intros is to help Coq resolving *)
660 (* second order unification problems. *)
661 generalize m c H0; clear H0 m c; inversion c;
662 generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros;
663 (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3).
664 destruct dir'0; destruct dir'';
666 unfold directed_relation_of_argument_class; simpl; apply s; exact H0).
667 (* the following mess with generalize/clear/intros is to help Coq resolving *)
668 (* second order unification problems. *)
669 generalize m c H0; clear H0 m c; inversion c;
670 generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros;
671 (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3).
672 destruct dir'0; destruct dir''; (exact H0 || hnf; symmetry; exact H0).
675 (directed_relation_of_argument_class (get_rewrite_direction dir'' S) S
676 (eq_rect ? (fun T : Type => T) (interp E1 m) ?
677 (about_carrier_of_relation_class_and_relation_class_of_argument_class S))
678 (eq_rect ? (fun T : Type => T) (interp E2 m) ?
679 (about_carrier_of_relation_class_and_relation_class_of_argument_class S)) /\
680 relation_of_product_of_arguments dir'' ?
681 (interp_relation_class_list E1 m0) (interp_relation_class_list E2 m0)).
683 clear m0 H1; destruct S; simpl in H0 |- *; unfold get_rewrite_direction; simpl.
684 destruct dir''; destruct dir'0; (exact H0 || hnf; apply s; exact H0).
686 rewrite <- H3; exact H0.
687 rewrite (opposite_direction_idempotent dir'0); exact H0.
688 destruct dir''; destruct dir'0; (exact H0 || hnf; apply s; exact H0).
690 rewrite <- H3; exact H0.
691 rewrite (opposite_direction_idempotent dir'0); exact H0.
692 destruct dir''; destruct dir'0; (exact H0 || hnf; symmetry; exact H0).
696 (* A FEW EXAMPLES ON iff *)
698 (* impl IS A MORPHISM *)
700 Add Morphism impl with signature iff ==> iff ==> iff as Impl_Morphism.
704 (* and IS A MORPHISM *)
706 Add Morphism and with signature iff ==> iff ==> iff as And_Morphism.
710 (* or IS A MORPHISM *)
712 Add Morphism or with signature iff ==> iff ==> iff as Or_Morphism.
716 (* not IS A MORPHISM *)
718 Add Morphism not with signature iff ==> iff as Not_Morphism.
722 (* THE SAME EXAMPLES ON impl *)
724 Add Morphism and with signature impl ++> impl ++> impl as And_Morphism2.
728 Add Morphism or with signature impl ++> impl ++> impl as Or_Morphism2.
732 Add Morphism not with signature impl -→ impl as Not_Morphism2.