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 f; clear f;
111 generalize in match f1; clear f1;
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;
254 definition equality_morphism_of_asymmetric_areflexive_transitive_relation:
255 ∀(A: Type)(Aeq: relation A)(trans: transitive ? Aeq).
256 let ASetoidClass1 := AsymmetricAreflexive Contravariant Aeq in
257 let ASetoidClass2 := AsymmetricAreflexive Covariant Aeq in
258 (Morphism_Theory (cons ASetoidClass1 (singl ASetoidClass2)) Impl_Relation_Class).
261 unfold make_compatibility_goal; simpl; unfold impl; auto.
264 definition equality_morphism_of_asymmetric_reflexive_transitive_relation:
265 ∀(A: Type)(Aeq: relation A)(refl: reflexive ? Aeq)(trans: transitive ? Aeq).
266 let ASetoidClass1 := AsymmetricReflexive Contravariant refl in
267 let ASetoidClass2 := AsymmetricReflexive Covariant refl in
268 (Morphism_Theory (cons ASetoidClass1 (singl ASetoidClass2)) Impl_Relation_Class).
271 unfold make_compatibility_goal; simpl; unfold impl; eauto.
274 (* iff AS A RELATION *)
276 Add Relation Prop iff
277 reflexivity proved by iff_refl
278 symmetry proved by iff_sym
279 transitivity proved by iff_trans
282 (* every predicate is morphism from Leibniz+ to Iff_Relation_Class *)
283 definition morphism_theory_of_predicate :
285 let In' := list_of_Leibniz_of_list_of_types In in
286 function_type_of_morphism_signature In' Iff_Relation_Class →
287 Morphism_Theory In' Iff_Relation_Class.
290 induction In; unfold make_compatibility_goal; simpl.
291 intro; apply iff_refl.
292 intro; apply (IHIn (X x)).
295 (* impl AS A RELATION *)
297 Theorem impl_trans: transitive ? impl.
298 hnf; unfold impl; tauto.
301 Add Relation Prop impl
302 reflexivity proved by impl_refl
303 transitivity proved by impl_trans
306 (* THE CIC PART OF THE REFLEXIVE TACTIC (SETOID REWRITE) *)
308 inductive rewrite_direction : Type :=
312 Implicit Type dir: rewrite_direction.
314 definition variance_of_argument_class : Argument_Class → option variance.
323 definition opposite_direction :=
326 Left2Right => Right2Left
327 | Right2Left => Left2Right
330 Lemma opposite_direction_idempotent:
331 ∀dir. (opposite_direction (opposite_direction dir)) = dir.
332 destruct dir; reflexivity.
335 inductive check_if_variance_is_respected :
336 option variance → rewrite_direction → rewrite_direction → Prop
338 MSNone : ∀dir dir'. check_if_variance_is_respected None dir dir'
339 | MSCovariant : ∀dir. check_if_variance_is_respected (Some Covariant) dir dir
342 check_if_variance_is_respected (Some Contravariant) dir (opposite_direction dir).
344 definition relation_class_of_reflexive_relation_class:
345 Reflexive_Relation_Class → Relation_Class.
347 exact (SymmetricReflexive ? s r).
348 exact (AsymmetricReflexive tt r).
352 definition relation_class_of_areflexive_relation_class:
353 Areflexive_Relation_Class → Relation_Class.
355 exact (SymmetricAreflexive ? s).
356 exact (AsymmetricAreflexive tt Aeq).
359 definition carrier_of_reflexive_relation_class :=
360 fun R => carrier_of_relation_class (relation_class_of_reflexive_relation_class R).
362 definition carrier_of_areflexive_relation_class :=
363 fun R => carrier_of_relation_class (relation_class_of_areflexive_relation_class R).
365 definition relation_of_areflexive_relation_class :=
366 fun R => relation_of_relation_class (relation_class_of_areflexive_relation_class R).
368 inductive Morphism_Context Hole dir : Relation_Class → rewrite_direction → Type :=
371 Morphism_Theory In Out → Morphism_Context_List Hole dir dir' In →
372 Morphism_Context Hole dir Out dir'
373 | ToReplace : Morphism_Context Hole dir Hole dir
376 carrier_of_reflexive_relation_class S →
377 Morphism_Context Hole dir (relation_class_of_reflexive_relation_class S) dir'
378 | ProperElementToKeep :
379 ∀S dir' (x: carrier_of_areflexive_relation_class S).
380 relation_of_areflexive_relation_class S x x →
381 Morphism_Context Hole dir (relation_class_of_areflexive_relation_class S) dir'
382 with Morphism_Context_List Hole dir :
383 rewrite_direction → Arguments → Type
387 check_if_variance_is_respected (variance_of_argument_class S) dir' dir'' →
388 Morphism_Context Hole dir (relation_class_of_argument_class S) dir' →
389 Morphism_Context_List Hole dir dir'' (singl S)
392 check_if_variance_is_respected (variance_of_argument_class S) dir' dir'' →
393 Morphism_Context Hole dir (relation_class_of_argument_class S) dir' →
394 Morphism_Context_List Hole dir dir'' L →
395 Morphism_Context_List Hole dir dir'' (cons S L).
397 Scheme Morphism_Context_rect2 := Induction for Morphism_Context Sort Type
398 with Morphism_Context_List_rect2 := Induction for Morphism_Context_List Sort Type.
400 definition product_of_arguments : Arguments → Type.
402 exact (carrier_of_relation_class a).
403 exact (prod (carrier_of_relation_class a) IHX).
406 definition get_rewrite_direction: rewrite_direction → Argument_Class → rewrite_direction.
408 destruct (variance_of_argument_class R).
410 exact dir. (* covariant *)
411 exact (opposite_direction dir). (* contravariant *)
412 exact dir. (* symmetric relation *)
415 definition directed_relation_of_relation_class:
416 ∀dir (R: Relation_Class).
417 carrier_of_relation_class R → carrier_of_relation_class R → Prop.
419 exact (@relation_of_relation_class unit).
420 intros; exact (relation_of_relation_class ? X0 X).
423 definition directed_relation_of_argument_class:
424 ∀dir (R: Argument_Class).
425 carrier_of_relation_class R → carrier_of_relation_class R → Prop.
428 (about_carrier_of_relation_class_and_relation_class_of_argument_class R).
429 exact (directed_relation_of_relation_class dir (relation_class_of_argument_class R)).
433 definition relation_of_product_of_arguments:
435 product_of_arguments In → product_of_arguments In → Prop.
438 exact (directed_relation_of_argument_class (get_rewrite_direction dir a) a).
441 destruct X; destruct X0.
444 (directed_relation_of_argument_class (get_rewrite_direction dir a) a c c0).
448 definition apply_morphism:
449 ∀In Out (m: function_type_of_morphism_signature In Out)
450 (args: product_of_arguments In). carrier_of_relation_class Out.
456 exact (IHIn (m c) p).
459 Theorem apply_morphism_compatibility_Right2Left:
460 ∀In Out (m1 m2: function_type_of_morphism_signature In Out)
461 (args1 args2: product_of_arguments In).
462 make_compatibility_goal_aux ? ? m1 m2 →
463 relation_of_product_of_arguments Right2Left ? args1 args2 →
464 directed_relation_of_relation_class Right2Left ?
465 (apply_morphism ? ? m2 args1)
466 (apply_morphism ? ? m1 args2).
467 induction In; intros.
468 simpl in m1. m2. args1. args2. H0 |- *.
469 destruct a; simpl in H; hnf in H0.
471 destruct v; simpl in H0; apply H; exact H0.
473 destruct v; simpl in H0; apply H; exact H0.
474 rewrite H0; apply H; exact H0.
476 simpl in m1. m2. args1. args2. H0 |- *.
477 destruct args1; destruct args2; simpl.
480 destruct a; simpl in H.
501 rewrite H0; apply IHIn.
506 Theorem apply_morphism_compatibility_Left2Right:
507 ∀In Out (m1 m2: function_type_of_morphism_signature In Out)
508 (args1 args2: product_of_arguments In).
509 make_compatibility_goal_aux ? ? m1 m2 →
510 relation_of_product_of_arguments Left2Right ? args1 args2 →
511 directed_relation_of_relation_class Left2Right ?
512 (apply_morphism ? ? m1 args1)
513 (apply_morphism ? ? m2 args2).
514 induction In; intros.
515 simpl in m1. m2. args1. args2. H0 |- *.
516 destruct a; simpl in H; hnf in H0.
518 destruct v; simpl in H0; apply H; exact H0.
520 destruct v; simpl in H0; apply H; exact H0.
521 rewrite H0; apply H; exact H0.
523 simpl in m1. m2. args1. args2. H0 |- *.
524 destruct args1; destruct args2; simpl.
527 destruct a; simpl in H.
542 destruct v; simpl in H. H0; apply H; exact H0.
544 rewrite H0; apply IHIn.
550 ∀Hole dir Out dir'. carrier_of_relation_class Hole →
551 Morphism_Context Hole dir Out dir' → carrier_of_relation_class Out.
552 intros Hole dir Out dir' H t.
554 (@Morphism_Context_rect2 Hole dir (fun S ? ? => carrier_of_relation_class S)
555 (fun ? L fcl => product_of_arguments L));
557 exact (apply_morphism ? ? (Function m) X).
563 (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
567 (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
572 (*CSC: interp and interp_relation_class_list should be mutually defined. since
573 the proof term of each one contains the proof term of the other one. However
574 I cannot do that interactively (I should write the Fix by hand) *)
575 definition interp_relation_class_list :
576 ∀Hole dir dir' (L: Arguments). carrier_of_relation_class Hole →
577 Morphism_Context_List Hole dir dir' L → product_of_arguments L.
578 intros Hole dir dir' L H t.
580 (@Morphism_Context_List_rect2 Hole dir (fun S ? ? => carrier_of_relation_class S)
581 (fun ? L fcl => product_of_arguments L));
583 exact (apply_morphism ? ? (Function m) X).
589 (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
593 (about_carrier_of_relation_class_and_relation_class_of_argument_class S);
598 Theorem setoid_rewrite:
599 ∀Hole dir Out dir' (E1 E2: carrier_of_relation_class Hole)
600 (E: Morphism_Context Hole dir Out dir').
601 (directed_relation_of_relation_class dir Hole E1 E2) →
602 (directed_relation_of_relation_class dir' Out (interp E1 E) (interp E2 E)).
605 (@Morphism_Context_rect2 Hole dir
606 (fun S dir'' E => directed_relation_of_relation_class dir'' S (interp E1 E) (interp E2 E))
608 relation_of_product_of_arguments dir'' ?
609 (interp_relation_class_list E1 fcl)
610 (interp_relation_class_list E2 fcl))); intros.
611 change (directed_relation_of_relation_class dir'0 Out0
612 (apply_morphism ? ? (Function m) (interp_relation_class_list E1 m0))
613 (apply_morphism ? ? (Function m) (interp_relation_class_list E2 m0))).
615 apply apply_morphism_compatibility_Left2Right.
618 apply apply_morphism_compatibility_Right2Left.
624 unfold interp. Morphism_Context_rect2.
625 (*CSC: reflexivity used here*)
626 destruct S; destruct dir'0; simpl; (apply r || reflexivity).
628 destruct dir'0; exact r.
630 destruct S; unfold directed_relation_of_argument_class; simpl in H0 |- *;
631 unfold get_rewrite_direction; simpl.
632 destruct dir'0; destruct dir'';
634 unfold directed_relation_of_argument_class; simpl; apply s; exact H0).
635 (* the following mess with generalize/clear/intros is to help Coq resolving *)
636 (* second order unification problems. *)
637 generalize m c H0; clear H0 m c; inversion c;
638 generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros;
639 (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3).
640 destruct dir'0; destruct dir'';
642 unfold directed_relation_of_argument_class; simpl; apply s; exact H0).
643 (* the following mess with generalize/clear/intros is to help Coq resolving *)
644 (* second order unification problems. *)
645 generalize m c H0; clear H0 m c; inversion c;
646 generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros;
647 (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3).
648 destruct dir'0; destruct dir''; (exact H0 || hnf; symmetry; exact H0).
651 (directed_relation_of_argument_class (get_rewrite_direction dir'' S) S
652 (eq_rect ? (fun T : Type => T) (interp E1 m) ?
653 (about_carrier_of_relation_class_and_relation_class_of_argument_class S))
654 (eq_rect ? (fun T : Type => T) (interp E2 m) ?
655 (about_carrier_of_relation_class_and_relation_class_of_argument_class S)) /\
656 relation_of_product_of_arguments dir'' ?
657 (interp_relation_class_list E1 m0) (interp_relation_class_list E2 m0)).
659 clear m0 H1; destruct S; simpl in H0 |- *; unfold get_rewrite_direction; simpl.
660 destruct dir''; destruct dir'0; (exact H0 || hnf; apply s; exact H0).
662 rewrite <- H3; exact H0.
663 rewrite (opposite_direction_idempotent dir'0); exact H0.
664 destruct dir''; destruct dir'0; (exact H0 || hnf; apply s; exact H0).
666 rewrite <- H3; exact H0.
667 rewrite (opposite_direction_idempotent dir'0); exact H0.
668 destruct dir''; destruct dir'0; (exact H0 || hnf; symmetry; exact H0).
672 (* BEGIN OF UTILITY/BACKWARD COMPATIBILITY PART *)
674 record Setoid_Theory (A: Type) (Aeq: relation A) : Prop :=
675 {Seq_refl : ∀x:A. Aeq x x;
676 Seq_sym : ∀x y:A. Aeq x y → Aeq y x;
677 Seq_trans : ∀x y z:A. Aeq x y → Aeq y z → Aeq x z}.
679 (* END OF UTILITY/BACKWARD COMPATIBILITY PART *)
681 (* A FEW EXAMPLES ON iff *)
683 (* impl IS A MORPHISM *)
685 Add Morphism impl with signature iff ==> iff ==> iff as Impl_Morphism.
689 (* and IS A MORPHISM *)
691 Add Morphism and with signature iff ==> iff ==> iff as And_Morphism.
695 (* or IS A MORPHISM *)
697 Add Morphism or with signature iff ==> iff ==> iff as Or_Morphism.
701 (* not IS A MORPHISM *)
703 Add Morphism not with signature iff ==> iff as Not_Morphism.
707 (* THE SAME EXAMPLES ON impl *)
709 Add Morphism and with signature impl ++> impl ++> impl as And_Morphism2.
713 Add Morphism or with signature impl ++> impl ++> impl as Or_Morphism2.
717 Add Morphism not with signature impl -→ impl as Not_Morphism2.
721 (* FOR BACKWARD COMPATIBILITY *)
722 Implicit Arguments Setoid_Theory [].
723 Implicit Arguments Seq_refl [].
724 Implicit Arguments Seq_sym [].
725 Implicit Arguments Seq_trans [].
728 (* Some tactics for manipulating Setoid Theory not officially
729 declared as Setoid. *)
731 Ltac trans_st x := match goal with
732 | H : Setoid_Theory ? ?eqA |- ?eqA ? ? =>
733 apply (Seq_trans ? ? H) with x; auto
736 Ltac sym_st := match goal with
737 | H : Setoid_Theory ? ?eqA |- ?eqA ? ? =>
738 apply (Seq_sym ? ? H); auto
741 Ltac refl_st := match goal with
742 | H : Setoid_Theory ? ?eqA |- ?eqA ? ? =>
743 apply (Seq_refl ? ? H); auto
746 definition gen_st : ∀A : Set. Setoid_Theory ? (@eq A).
747 Proof. constructor; congruence. Qed.