From 3d0f2b7ecfb78308bbb17d135fcceefd121f7624 Mon Sep 17 00:00:00 2001 From: notin Date: Thu, 27 Sep 2007 15:19:14 +0000 Subject: Découpage de Setoid.v MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10147 85f007b7-540e-0410-9357-904b9bb8a0f7 --- theories/Setoids/Setoid.v | 656 +---------------------------------------- theories/Setoids/Setoid_Prop.v | 79 +++++ theories/Setoids/Setoid_tac.v | 596 +++++++++++++++++++++++++++++++++++++ 3 files changed, 678 insertions(+), 653 deletions(-) create mode 100644 theories/Setoids/Setoid_Prop.v create mode 100644 theories/Setoids/Setoid_tac.v (limited to 'theories/Setoids') diff --git a/theories/Setoids/Setoid.v b/theories/Setoids/Setoid.v index 508a0231e..da999dfe7 100644 --- a/theories/Setoids/Setoid.v +++ b/theories/Setoids/Setoid.v @@ -9,660 +9,10 @@ (*i $Id$: i*) -Require Export Relation_Definitions. +Require Export Setoid_tac. +Require Export Setoid_Prop. -Set Implicit Arguments. - -(** * Definitions of [Relation_Class] and n-ary [Morphism_Theory] *) - -(* X will be used to distinguish covariant arguments whose type is an *) -(* Asymmetric* relation from contravariant arguments of the same type *) -Inductive X_Relation_Class (X: Type) : Type := - SymmetricReflexive : - forall A Aeq, symmetric A Aeq -> reflexive _ Aeq -> X_Relation_Class X - | AsymmetricReflexive : X -> forall A Aeq, reflexive A Aeq -> X_Relation_Class X - | SymmetricAreflexive : forall A Aeq, symmetric A Aeq -> X_Relation_Class X - | AsymmetricAreflexive : X -> forall A (Aeq : relation A), X_Relation_Class X - | Leibniz : Type -> X_Relation_Class X. - -Inductive variance : Set := - Covariant - | Contravariant. - -Definition Argument_Class := X_Relation_Class variance. -Definition Relation_Class := X_Relation_Class unit. - -Inductive Reflexive_Relation_Class : Type := - RSymmetric : - forall A Aeq, symmetric A Aeq -> reflexive _ Aeq -> Reflexive_Relation_Class - | RAsymmetric : - forall A Aeq, reflexive A Aeq -> Reflexive_Relation_Class - | RLeibniz : Type -> Reflexive_Relation_Class. - -Inductive Areflexive_Relation_Class : Type := - | ASymmetric : forall A Aeq, symmetric A Aeq -> Areflexive_Relation_Class - | AAsymmetric : forall A (Aeq : relation A), Areflexive_Relation_Class. - -Implicit Type Hole Out: Relation_Class. - -Definition relation_class_of_argument_class : Argument_Class -> Relation_Class. - destruct 1. - exact (SymmetricReflexive _ s r). - exact (AsymmetricReflexive tt r). - exact (SymmetricAreflexive _ s). - exact (AsymmetricAreflexive tt Aeq). - exact (Leibniz _ T). -Defined. - -Definition carrier_of_relation_class : forall X, X_Relation_Class X -> Type. - destruct 1. - exact A. - exact A. - exact A. - exact A. - exact T. -Defined. - -Definition relation_of_relation_class : - forall X R, @carrier_of_relation_class X R -> carrier_of_relation_class R -> Prop. - destruct R. - exact Aeq. - exact Aeq. - exact Aeq. - exact Aeq. - exact (@eq T). -Defined. - -Lemma about_carrier_of_relation_class_and_relation_class_of_argument_class : - forall R, - carrier_of_relation_class (relation_class_of_argument_class R) = - carrier_of_relation_class R. - destruct R; reflexivity. -Defined. - -Inductive nelistT (A : Type) : Type := - singl : A -> nelistT A - | necons : A -> nelistT A -> nelistT A. - -Definition Arguments := nelistT Argument_Class. - -Implicit Type In: Arguments. - -Definition function_type_of_morphism_signature : - Arguments -> Relation_Class -> Type. - intros In Out. - induction In. - exact (carrier_of_relation_class a -> carrier_of_relation_class Out). - exact (carrier_of_relation_class a -> IHIn). -Defined. - -Definition make_compatibility_goal_aux: - forall In Out - (f g: function_type_of_morphism_signature In Out), Prop. - intros; induction In; simpl in f, g. - induction a; simpl in f, g. - exact (forall x1 x2, Aeq x1 x2 -> relation_of_relation_class Out (f x1) (g x2)). - destruct x. - exact (forall x1 x2, Aeq x1 x2 -> relation_of_relation_class Out (f x1) (g x2)). - exact (forall x1 x2, Aeq x2 x1 -> relation_of_relation_class Out (f x1) (g x2)). - exact (forall x1 x2, Aeq x1 x2 -> relation_of_relation_class Out (f x1) (g x2)). - destruct x. - exact (forall x1 x2, Aeq x1 x2 -> relation_of_relation_class Out (f x1) (g x2)). - exact (forall x1 x2, Aeq x2 x1 -> relation_of_relation_class Out (f x1) (g x2)). - exact (forall x, relation_of_relation_class Out (f x) (g x)). - induction a; simpl in f, g. - exact (forall x1 x2, Aeq x1 x2 -> IHIn (f x1) (g x2)). - destruct x. - exact (forall x1 x2, Aeq x1 x2 -> IHIn (f x1) (g x2)). - exact (forall x1 x2, Aeq x2 x1 -> IHIn (f x1) (g x2)). - exact (forall x1 x2, Aeq x1 x2 -> IHIn (f x1) (g x2)). - destruct x. - exact (forall x1 x2, Aeq x1 x2 -> IHIn (f x1) (g x2)). - exact (forall x1 x2, Aeq x2 x1 -> IHIn (f x1) (g x2)). - exact (forall x, IHIn (f x) (g x)). -Defined. - -Definition make_compatibility_goal := - (fun In Out f => make_compatibility_goal_aux In Out f f). - -Record Morphism_Theory In Out : Type := - { Function : function_type_of_morphism_signature In Out; - Compat : make_compatibility_goal In Out Function }. - -(** The [iff] relation class *) - -Definition Iff_Relation_Class : Relation_Class. - eapply (@SymmetricReflexive unit _ iff). - exact iff_sym. - exact iff_refl. -Defined. - -(** The [impl] relation class *) - -Definition impl (A B: Prop) := A -> B. - -Theorem impl_refl: reflexive _ impl. -Proof. - hnf; unfold impl; tauto. -Qed. - -Definition Impl_Relation_Class : Relation_Class. - eapply (@AsymmetricReflexive unit tt _ impl). - exact impl_refl. -Defined. - -(** Every function is a morphism from Leibniz+ to Leibniz *) - -Definition list_of_Leibniz_of_list_of_types: nelistT Type -> Arguments. - induction 1. - exact (singl (Leibniz _ a)). - exact (necons (Leibniz _ a) IHX). -Defined. - -Definition morphism_theory_of_function : - forall (In: nelistT Type) (Out: Type), - let In' := list_of_Leibniz_of_list_of_types In in - let Out' := Leibniz _ Out in - function_type_of_morphism_signature In' Out' -> - Morphism_Theory In' Out'. - intros. - exists X. - induction In; unfold make_compatibility_goal; simpl. - reflexivity. - intro; apply (IHIn (X x)). -Defined. - -(** Every predicate is a morphism from Leibniz+ to Iff_Relation_Class *) - -Definition morphism_theory_of_predicate : - forall (In: nelistT Type), - let In' := list_of_Leibniz_of_list_of_types In in - function_type_of_morphism_signature In' Iff_Relation_Class -> - Morphism_Theory In' Iff_Relation_Class. - intros. - exists X. - induction In; unfold make_compatibility_goal; simpl. - intro; apply iff_refl. - intro; apply (IHIn (X x)). -Defined. - -(** * Utility functions to prove that every transitive relation is a morphism *) - -Definition equality_morphism_of_symmetric_areflexive_transitive_relation: - forall (A: Type)(Aeq: relation A)(sym: symmetric _ Aeq)(trans: transitive _ Aeq), - let ASetoidClass := SymmetricAreflexive _ sym in - (Morphism_Theory (necons ASetoidClass (singl ASetoidClass)) Iff_Relation_Class). - intros. - exists Aeq. - unfold make_compatibility_goal; simpl; split; eauto. -Defined. - -Definition equality_morphism_of_symmetric_reflexive_transitive_relation: - forall (A: Type)(Aeq: relation A)(refl: reflexive _ Aeq)(sym: symmetric _ Aeq) - (trans: transitive _ Aeq), let ASetoidClass := SymmetricReflexive _ sym refl in - (Morphism_Theory (necons ASetoidClass (singl ASetoidClass)) Iff_Relation_Class). - intros. - exists Aeq. - unfold make_compatibility_goal; simpl; split; eauto. -Defined. - -Definition equality_morphism_of_asymmetric_areflexive_transitive_relation: - forall (A: Type)(Aeq: relation A)(trans: transitive _ Aeq), - let ASetoidClass1 := AsymmetricAreflexive Contravariant Aeq in - let ASetoidClass2 := AsymmetricAreflexive Covariant Aeq in - (Morphism_Theory (necons ASetoidClass1 (singl ASetoidClass2)) Impl_Relation_Class). - intros. - exists Aeq. - unfold make_compatibility_goal; simpl; unfold impl; eauto. -Defined. - -Definition equality_morphism_of_asymmetric_reflexive_transitive_relation: - forall (A: Type)(Aeq: relation A)(refl: reflexive _ Aeq)(trans: transitive _ Aeq), - let ASetoidClass1 := AsymmetricReflexive Contravariant refl in - let ASetoidClass2 := AsymmetricReflexive Covariant refl in - (Morphism_Theory (necons ASetoidClass1 (singl ASetoidClass2)) Impl_Relation_Class). - intros. - exists Aeq. - unfold make_compatibility_goal; simpl; unfold impl; eauto. -Defined. - -(** * A few examples on [iff] *) - -(** [iff] as a relation *) - -Add Relation Prop iff - reflexivity proved by iff_refl - symmetry proved by iff_sym - transitivity proved by iff_trans -as iff_relation. - -(** [impl] as a relation *) - -Theorem impl_trans: transitive _ impl. -Proof. - hnf; unfold impl; tauto. -Qed. - -Add Relation Prop impl - reflexivity proved by impl_refl - transitivity proved by impl_trans -as impl_relation. - -(** [impl] is a morphism *) - -Add Morphism impl with signature iff ==> iff ==> iff as Impl_Morphism. -Proof. - unfold impl; tauto. -Qed. - -(** [and] is a morphism *) - -Add Morphism and with signature iff ==> iff ==> iff as And_Morphism. - tauto. -Qed. - -(** [or] is a morphism *) - -Add Morphism or with signature iff ==> iff ==> iff as Or_Morphism. -Proof. - tauto. -Qed. - -(** [not] is a morphism *) - -Add Morphism not with signature iff ==> iff as Not_Morphism. -Proof. - tauto. -Qed. - -(** The same examples on [impl] *) - -Add Morphism and with signature impl ++> impl ++> impl as And_Morphism2. -Proof. - unfold impl; tauto. -Qed. - -Add Morphism or with signature impl ++> impl ++> impl as Or_Morphism2. -Proof. - unfold impl; tauto. -Qed. - -Add Morphism not with signature impl --> impl as Not_Morphism2. -Proof. - unfold impl; tauto. -Qed. - -(** * The CIC part of the reflexive tactic ([setoid_rewrite]) *) - -Inductive rewrite_direction : Type := - | Left2Right - | Right2Left. - -Implicit Type dir: rewrite_direction. - -Definition variance_of_argument_class : Argument_Class -> option variance. - destruct 1. - exact None. - exact (Some v). - exact None. - exact (Some v). - exact None. -Defined. - -Definition opposite_direction := - fun dir => - match dir with - | Left2Right => Right2Left - | Right2Left => Left2Right - end. - -Lemma opposite_direction_idempotent: - forall dir, (opposite_direction (opposite_direction dir)) = dir. -Proof. - destruct dir; reflexivity. -Qed. - -Inductive check_if_variance_is_respected : - option variance -> rewrite_direction -> rewrite_direction -> Prop := - | MSNone : forall dir dir', check_if_variance_is_respected None dir dir' - | MSCovariant : forall dir, check_if_variance_is_respected (Some Covariant) dir dir - | MSContravariant : - forall dir, - check_if_variance_is_respected (Some Contravariant) dir (opposite_direction dir). - -Definition relation_class_of_reflexive_relation_class: - Reflexive_Relation_Class -> Relation_Class. - induction 1. - exact (SymmetricReflexive _ s r). - exact (AsymmetricReflexive tt r). - exact (Leibniz _ T). -Defined. - -Definition relation_class_of_areflexive_relation_class: - Areflexive_Relation_Class -> Relation_Class. - induction 1. - exact (SymmetricAreflexive _ s). - exact (AsymmetricAreflexive tt Aeq). -Defined. - -Definition carrier_of_reflexive_relation_class := - fun R => carrier_of_relation_class (relation_class_of_reflexive_relation_class R). - -Definition carrier_of_areflexive_relation_class := - fun R => carrier_of_relation_class (relation_class_of_areflexive_relation_class R). - -Definition relation_of_areflexive_relation_class := - fun R => relation_of_relation_class (relation_class_of_areflexive_relation_class R). - -Inductive Morphism_Context Hole dir : Relation_Class -> rewrite_direction -> Type := - | App : - forall In Out dir', - Morphism_Theory In Out -> Morphism_Context_List Hole dir dir' In -> - Morphism_Context Hole dir Out dir' - | ToReplace : Morphism_Context Hole dir Hole dir - | ToKeep : - forall S dir', - carrier_of_reflexive_relation_class S -> - Morphism_Context Hole dir (relation_class_of_reflexive_relation_class S) dir' - | ProperElementToKeep : - forall S dir' (x: carrier_of_areflexive_relation_class S), - relation_of_areflexive_relation_class S x x -> - Morphism_Context Hole dir (relation_class_of_areflexive_relation_class S) dir' -with Morphism_Context_List Hole dir : - rewrite_direction -> Arguments -> Type -:= - fcl_singl : - forall S dir' dir'', - check_if_variance_is_respected (variance_of_argument_class S) dir' dir'' -> - Morphism_Context Hole dir (relation_class_of_argument_class S) dir' -> - Morphism_Context_List Hole dir dir'' (singl S) - | fcl_cons : - forall S L dir' dir'', - check_if_variance_is_respected (variance_of_argument_class S) dir' dir'' -> - Morphism_Context Hole dir (relation_class_of_argument_class S) dir' -> - Morphism_Context_List Hole dir dir'' L -> - Morphism_Context_List Hole dir dir'' (necons S L). - -Scheme Morphism_Context_rect2 := Induction for Morphism_Context Sort Type -with Morphism_Context_List_rect2 := Induction for Morphism_Context_List Sort Type. - -Definition product_of_arguments : Arguments -> Type. - induction 1. - exact (carrier_of_relation_class a). - exact (prod (carrier_of_relation_class a) IHX). -Defined. - -Definition get_rewrite_direction: rewrite_direction -> Argument_Class -> rewrite_direction. - intros dir R. - destruct (variance_of_argument_class R). - destruct v. - exact dir. (* covariant *) - exact (opposite_direction dir). (* contravariant *) - exact dir. (* symmetric relation *) -Defined. - -Definition directed_relation_of_relation_class: - forall dir (R: Relation_Class), - carrier_of_relation_class R -> carrier_of_relation_class R -> Prop. - destruct 1. - exact (@relation_of_relation_class unit). - intros; exact (relation_of_relation_class _ X0 X). -Defined. - -Definition directed_relation_of_argument_class: - forall dir (R: Argument_Class), - carrier_of_relation_class R -> carrier_of_relation_class R -> Prop. - intros dir R. - rewrite <- - (about_carrier_of_relation_class_and_relation_class_of_argument_class R). - exact (directed_relation_of_relation_class dir (relation_class_of_argument_class R)). -Defined. - - -Definition relation_of_product_of_arguments: - forall dir In, - product_of_arguments In -> product_of_arguments In -> Prop. - induction In. - simpl. - exact (directed_relation_of_argument_class (get_rewrite_direction dir a) a). - - simpl; intros. - destruct X; destruct X0. - apply and. - exact - (directed_relation_of_argument_class (get_rewrite_direction dir a) a c c0). - exact (IHIn p p0). -Defined. - -Definition apply_morphism: - forall In Out (m: function_type_of_morphism_signature In Out) - (args: product_of_arguments In), carrier_of_relation_class Out. - intros. - induction In. - exact (m args). - simpl in m, args. - destruct args. - exact (IHIn (m c) p). -Defined. - -Theorem apply_morphism_compatibility_Right2Left: - forall In Out (m1 m2: function_type_of_morphism_signature In Out) - (args1 args2: product_of_arguments In), - make_compatibility_goal_aux _ _ m1 m2 -> - relation_of_product_of_arguments Right2Left _ args1 args2 -> - directed_relation_of_relation_class Right2Left _ - (apply_morphism _ _ m2 args1) - (apply_morphism _ _ m1 args2). - induction In; intros. - simpl in m1, m2, args1, args2, H0 |- *. - destruct a; simpl in H; hnf in H0. - apply H; exact H0. - destruct v; simpl in H0; apply H; exact H0. - apply H; exact H0. - destruct v; simpl in H0; apply H; exact H0. - rewrite H0; apply H; exact H0. - - simpl in m1, m2, args1, args2, H0 |- *. - destruct args1; destruct args2; simpl. - destruct H0. - simpl in H. - destruct a; simpl in H. - apply IHIn. - apply H; exact H0. - exact H1. - destruct v. - apply IHIn. - apply H; exact H0. - exact H1. - apply IHIn. - apply H; exact H0. - exact H1. - apply IHIn. - apply H; exact H0. - exact H1. - destruct v. - apply IHIn. - apply H; exact H0. - exact H1. - apply IHIn. - apply H; exact H0. - exact H1. - rewrite H0; apply IHIn. - apply H. - exact H1. -Qed. - -Theorem apply_morphism_compatibility_Left2Right: - forall In Out (m1 m2: function_type_of_morphism_signature In Out) - (args1 args2: product_of_arguments In), - make_compatibility_goal_aux _ _ m1 m2 -> - relation_of_product_of_arguments Left2Right _ args1 args2 -> - directed_relation_of_relation_class Left2Right _ - (apply_morphism _ _ m1 args1) - (apply_morphism _ _ m2 args2). -Proof. - induction In; intros. - simpl in m1, m2, args1, args2, H0 |- *. - destruct a; simpl in H; hnf in H0. - apply H; exact H0. - destruct v; simpl in H0; apply H; exact H0. - apply H; exact H0. - destruct v; simpl in H0; apply H; exact H0. - rewrite H0; apply H; exact H0. - - simpl in m1, m2, args1, args2, H0 |- *. - destruct args1; destruct args2; simpl. - destruct H0. - simpl in H. - destruct a; simpl in H. - apply IHIn. - apply H; exact H0. - exact H1. - destruct v. - apply IHIn. - apply H; exact H0. - exact H1. - apply IHIn. - apply H; exact H0. - exact H1. - apply IHIn. - apply H; exact H0. - exact H1. - apply IHIn. - destruct v; simpl in H, H0; apply H; exact H0. - exact H1. - rewrite H0; apply IHIn. - apply H. - exact H1. -Qed. - -Definition interp : - forall Hole dir Out dir', carrier_of_relation_class Hole -> - Morphism_Context Hole dir Out dir' -> carrier_of_relation_class Out. - intros Hole dir Out dir' H t. - elim t using - (@Morphism_Context_rect2 Hole dir (fun S _ _ => carrier_of_relation_class S) - (fun _ L fcl => product_of_arguments L)); - intros. - exact (apply_morphism _ _ (Function m) X). - exact H. - exact c. - exact x. - simpl; - rewrite <- - (about_carrier_of_relation_class_and_relation_class_of_argument_class S); - exact X. - split. - rewrite <- - (about_carrier_of_relation_class_and_relation_class_of_argument_class S); - exact X. - exact X0. -Defined. - -(* CSC: interp and interp_relation_class_list should be mutually defined, since - the proof term of each one contains the proof term of the other one. However - I cannot do that interactively (I should write the Fix by hand) *) -Definition interp_relation_class_list : - forall Hole dir dir' (L: Arguments), carrier_of_relation_class Hole -> - Morphism_Context_List Hole dir dir' L -> product_of_arguments L. - intros Hole dir dir' L H t. - elim t using - (@Morphism_Context_List_rect2 Hole dir (fun S _ _ => carrier_of_relation_class S) - (fun _ L fcl => product_of_arguments L)); - intros. - exact (apply_morphism _ _ (Function m) X). - exact H. - exact c. - exact x. - simpl; - rewrite <- - (about_carrier_of_relation_class_and_relation_class_of_argument_class S); - exact X. - split. - rewrite <- - (about_carrier_of_relation_class_and_relation_class_of_argument_class S); - exact X. - exact X0. -Defined. - -Theorem setoid_rewrite: - forall Hole dir Out dir' (E1 E2: carrier_of_relation_class Hole) - (E: Morphism_Context Hole dir Out dir'), - (directed_relation_of_relation_class dir Hole E1 E2) -> - (directed_relation_of_relation_class dir' Out (interp E1 E) (interp E2 E)). -Proof. - intros. - elim E using - (@Morphism_Context_rect2 Hole dir - (fun S dir'' E => directed_relation_of_relation_class dir'' S (interp E1 E) (interp E2 E)) - (fun dir'' L fcl => - relation_of_product_of_arguments dir'' _ - (interp_relation_class_list E1 fcl) - (interp_relation_class_list E2 fcl))); intros. - change (directed_relation_of_relation_class dir'0 Out0 - (apply_morphism _ _ (Function m) (interp_relation_class_list E1 m0)) - (apply_morphism _ _ (Function m) (interp_relation_class_list E2 m0))). - destruct dir'0. - apply apply_morphism_compatibility_Left2Right. - exact (Compat m). - exact H0. - apply apply_morphism_compatibility_Right2Left. - exact (Compat m). - exact H0. - - exact H. - - unfold interp, Morphism_Context_rect2. - (* CSC: reflexivity used here *) - destruct S; destruct dir'0; simpl; (apply r || reflexivity). - - destruct dir'0; exact r. - - destruct S; unfold directed_relation_of_argument_class; simpl in H0 |- *; - unfold get_rewrite_direction; simpl. - destruct dir'0; destruct dir''; - (exact H0 || - unfold directed_relation_of_argument_class; simpl; apply s; exact H0). - (* the following mess with generalize/clear/intros is to help Coq resolving *) - (* second order unification problems. *) - generalize m c H0; clear H0 m c; inversion c; - generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros; - (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3). - destruct dir'0; destruct dir''; - (exact H0 || - unfold directed_relation_of_argument_class; simpl; apply s; exact H0). - (* the following mess with generalize/clear/intros is to help Coq resolving *) - (* second order unification problems. *) - generalize m c H0; clear H0 m c; inversion c; - generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros; - (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3). - destruct dir'0; destruct dir''; (exact H0 || hnf; symmetry; exact H0). - - change - (directed_relation_of_argument_class (get_rewrite_direction dir'' S) S - (eq_rect _ (fun T : Type => T) (interp E1 m) _ - (about_carrier_of_relation_class_and_relation_class_of_argument_class S)) - (eq_rect _ (fun T : Type => T) (interp E2 m) _ - (about_carrier_of_relation_class_and_relation_class_of_argument_class S)) /\ - relation_of_product_of_arguments dir'' _ - (interp_relation_class_list E1 m0) (interp_relation_class_list E2 m0)). - split. - clear m0 H1; destruct S; simpl in H0 |- *; unfold get_rewrite_direction; simpl. - destruct dir''; destruct dir'0; (exact H0 || hnf; apply s; exact H0). - inversion c. - rewrite <- H3; exact H0. - rewrite (opposite_direction_idempotent dir'0); exact H0. - destruct dir''; destruct dir'0; (exact H0 || hnf; apply s; exact H0). - inversion c. - rewrite <- H3; exact H0. - rewrite (opposite_direction_idempotent dir'0); exact H0. - destruct dir''; destruct dir'0; (exact H0 || hnf; symmetry; exact H0). - exact H1. - Qed. - -(** * Miscelenous *) - -(** For backwark compatibility *) +(** For backward compatibility *) Record Setoid_Theory (A: Type) (Aeq: relation A) : Prop := { Seq_refl : forall x:A, Aeq x x; diff --git a/theories/Setoids/Setoid_Prop.v b/theories/Setoids/Setoid_Prop.v new file mode 100644 index 000000000..9c0b4b2b6 --- /dev/null +++ b/theories/Setoids/Setoid_Prop.v @@ -0,0 +1,79 @@ + +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* iff ==> iff as Impl_Morphism. +Proof. + unfold impl; tauto. +Qed. + +(** [and] is a morphism *) + +Add Morphism and with signature iff ==> iff ==> iff as And_Morphism. + tauto. +Qed. + +(** [or] is a morphism *) + +Add Morphism or with signature iff ==> iff ==> iff as Or_Morphism. +Proof. + tauto. +Qed. + +(** [not] is a morphism *) + +Add Morphism not with signature iff ==> iff as Not_Morphism. +Proof. + tauto. +Qed. + +(** The same examples on [impl] *) + +Add Morphism and with signature impl ++> impl ++> impl as And_Morphism2. +Proof. + unfold impl; tauto. +Qed. + +Add Morphism or with signature impl ++> impl ++> impl as Or_Morphism2. +Proof. + unfold impl; tauto. +Qed. + +Add Morphism not with signature impl --> impl as Not_Morphism2. +Proof. + unfold impl; tauto. +Qed. + diff --git a/theories/Setoids/Setoid_tac.v b/theories/Setoids/Setoid_tac.v new file mode 100644 index 000000000..f979ec11c --- /dev/null +++ b/theories/Setoids/Setoid_tac.v @@ -0,0 +1,596 @@ + +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* reflexive _ Aeq -> X_Relation_Class X + | AsymmetricReflexive : X -> forall A Aeq, reflexive A Aeq -> X_Relation_Class X + | SymmetricAreflexive : forall A Aeq, symmetric A Aeq -> X_Relation_Class X + | AsymmetricAreflexive : X -> forall A (Aeq : relation A), X_Relation_Class X + | Leibniz : Type -> X_Relation_Class X. + +Inductive variance : Set := + Covariant + | Contravariant. + +Definition Argument_Class := X_Relation_Class variance. +Definition Relation_Class := X_Relation_Class unit. + +Inductive Reflexive_Relation_Class : Type := + RSymmetric : + forall A Aeq, symmetric A Aeq -> reflexive _ Aeq -> Reflexive_Relation_Class + | RAsymmetric : + forall A Aeq, reflexive A Aeq -> Reflexive_Relation_Class + | RLeibniz : Type -> Reflexive_Relation_Class. + +Inductive Areflexive_Relation_Class : Type := + | ASymmetric : forall A Aeq, symmetric A Aeq -> Areflexive_Relation_Class + | AAsymmetric : forall A (Aeq : relation A), Areflexive_Relation_Class. + +Implicit Type Hole Out: Relation_Class. + +Definition relation_class_of_argument_class : Argument_Class -> Relation_Class. + destruct 1. + exact (SymmetricReflexive _ s r). + exact (AsymmetricReflexive tt r). + exact (SymmetricAreflexive _ s). + exact (AsymmetricAreflexive tt Aeq). + exact (Leibniz _ T). +Defined. + +Definition carrier_of_relation_class : forall X, X_Relation_Class X -> Type. + destruct 1. + exact A. + exact A. + exact A. + exact A. + exact T. +Defined. + +Definition relation_of_relation_class : + forall X R, @carrier_of_relation_class X R -> carrier_of_relation_class R -> Prop. + destruct R. + exact Aeq. + exact Aeq. + exact Aeq. + exact Aeq. + exact (@eq T). +Defined. + +Lemma about_carrier_of_relation_class_and_relation_class_of_argument_class : + forall R, + carrier_of_relation_class (relation_class_of_argument_class R) = + carrier_of_relation_class R. + destruct R; reflexivity. +Defined. + +Inductive nelistT (A : Type) : Type := + singl : A -> nelistT A + | necons : A -> nelistT A -> nelistT A. + +Definition Arguments := nelistT Argument_Class. + +Implicit Type In: Arguments. + +Definition function_type_of_morphism_signature : + Arguments -> Relation_Class -> Type. + intros In Out. + induction In. + exact (carrier_of_relation_class a -> carrier_of_relation_class Out). + exact (carrier_of_relation_class a -> IHIn). +Defined. + +Definition make_compatibility_goal_aux: + forall In Out + (f g: function_type_of_morphism_signature In Out), Prop. + intros; induction In; simpl in f, g. + induction a; simpl in f, g. + exact (forall x1 x2, Aeq x1 x2 -> relation_of_relation_class Out (f x1) (g x2)). + destruct x. + exact (forall x1 x2, Aeq x1 x2 -> relation_of_relation_class Out (f x1) (g x2)). + exact (forall x1 x2, Aeq x2 x1 -> relation_of_relation_class Out (f x1) (g x2)). + exact (forall x1 x2, Aeq x1 x2 -> relation_of_relation_class Out (f x1) (g x2)). + destruct x. + exact (forall x1 x2, Aeq x1 x2 -> relation_of_relation_class Out (f x1) (g x2)). + exact (forall x1 x2, Aeq x2 x1 -> relation_of_relation_class Out (f x1) (g x2)). + exact (forall x, relation_of_relation_class Out (f x) (g x)). + induction a; simpl in f, g. + exact (forall x1 x2, Aeq x1 x2 -> IHIn (f x1) (g x2)). + destruct x. + exact (forall x1 x2, Aeq x1 x2 -> IHIn (f x1) (g x2)). + exact (forall x1 x2, Aeq x2 x1 -> IHIn (f x1) (g x2)). + exact (forall x1 x2, Aeq x1 x2 -> IHIn (f x1) (g x2)). + destruct x. + exact (forall x1 x2, Aeq x1 x2 -> IHIn (f x1) (g x2)). + exact (forall x1 x2, Aeq x2 x1 -> IHIn (f x1) (g x2)). + exact (forall x, IHIn (f x) (g x)). +Defined. + +Definition make_compatibility_goal := + (fun In Out f => make_compatibility_goal_aux In Out f f). + +Record Morphism_Theory In Out : Type := + { Function : function_type_of_morphism_signature In Out; + Compat : make_compatibility_goal In Out Function }. + + +(** The [iff] relation class *) + +Definition Iff_Relation_Class : Relation_Class. + eapply (@SymmetricReflexive unit _ iff). + exact iff_sym. + exact iff_refl. +Defined. + +(** The [impl] relation class *) + +Definition impl (A B: Prop) := A -> B. + +Theorem impl_refl: reflexive _ impl. +Proof. + hnf; unfold impl; tauto. +Qed. + +Definition Impl_Relation_Class : Relation_Class. + eapply (@AsymmetricReflexive unit tt _ impl). + exact impl_refl. +Defined. + +(** Every function is a morphism from Leibniz+ to Leibniz *) + +Definition list_of_Leibniz_of_list_of_types: nelistT Type -> Arguments. + induction 1. + exact (singl (Leibniz _ a)). + exact (necons (Leibniz _ a) IHX). +Defined. + +Definition morphism_theory_of_function : + forall (In: nelistT Type) (Out: Type), + let In' := list_of_Leibniz_of_list_of_types In in + let Out' := Leibniz _ Out in + function_type_of_morphism_signature In' Out' -> + Morphism_Theory In' Out'. + intros. + exists X. + induction In; unfold make_compatibility_goal; simpl. + reflexivity. + intro; apply (IHIn (X x)). +Defined. + +(** Every predicate is a morphism from Leibniz+ to Iff_Relation_Class *) + +Definition morphism_theory_of_predicate : + forall (In: nelistT Type), + let In' := list_of_Leibniz_of_list_of_types In in + function_type_of_morphism_signature In' Iff_Relation_Class -> + Morphism_Theory In' Iff_Relation_Class. + intros. + exists X. + induction In; unfold make_compatibility_goal; simpl. + intro; apply iff_refl. + intro; apply (IHIn (X x)). +Defined. + +(** * Utility functions to prove that every transitive relation is a morphism *) + +Definition equality_morphism_of_symmetric_areflexive_transitive_relation: + forall (A: Type)(Aeq: relation A)(sym: symmetric _ Aeq)(trans: transitive _ Aeq), + let ASetoidClass := SymmetricAreflexive _ sym in + (Morphism_Theory (necons ASetoidClass (singl ASetoidClass)) Iff_Relation_Class). + intros. + exists Aeq. + unfold make_compatibility_goal; simpl; split; eauto. +Defined. + +Definition equality_morphism_of_symmetric_reflexive_transitive_relation: + forall (A: Type)(Aeq: relation A)(refl: reflexive _ Aeq)(sym: symmetric _ Aeq) + (trans: transitive _ Aeq), let ASetoidClass := SymmetricReflexive _ sym refl in + (Morphism_Theory (necons ASetoidClass (singl ASetoidClass)) Iff_Relation_Class). + intros. + exists Aeq. + unfold make_compatibility_goal; simpl; split; eauto. +Defined. + +Definition equality_morphism_of_asymmetric_areflexive_transitive_relation: + forall (A: Type)(Aeq: relation A)(trans: transitive _ Aeq), + let ASetoidClass1 := AsymmetricAreflexive Contravariant Aeq in + let ASetoidClass2 := AsymmetricAreflexive Covariant Aeq in + (Morphism_Theory (necons ASetoidClass1 (singl ASetoidClass2)) Impl_Relation_Class). + intros. + exists Aeq. + unfold make_compatibility_goal; simpl; unfold impl; eauto. +Defined. + +Definition equality_morphism_of_asymmetric_reflexive_transitive_relation: + forall (A: Type)(Aeq: relation A)(refl: reflexive _ Aeq)(trans: transitive _ Aeq), + let ASetoidClass1 := AsymmetricReflexive Contravariant refl in + let ASetoidClass2 := AsymmetricReflexive Covariant refl in + (Morphism_Theory (necons ASetoidClass1 (singl ASetoidClass2)) Impl_Relation_Class). + intros. + exists Aeq. + unfold make_compatibility_goal; simpl; unfold impl; eauto. +Defined. + +(** * The CIC part of the reflexive tactic ([setoid_rewrite]) *) + +Inductive rewrite_direction : Type := + | Left2Right + | Right2Left. + +Implicit Type dir: rewrite_direction. + +Definition variance_of_argument_class : Argument_Class -> option variance. + destruct 1. + exact None. + exact (Some v). + exact None. + exact (Some v). + exact None. +Defined. + +Definition opposite_direction := + fun dir => + match dir with + | Left2Right => Right2Left + | Right2Left => Left2Right + end. + +Lemma opposite_direction_idempotent: + forall dir, (opposite_direction (opposite_direction dir)) = dir. +Proof. + destruct dir; reflexivity. +Qed. + +Inductive check_if_variance_is_respected : + option variance -> rewrite_direction -> rewrite_direction -> Prop := + | MSNone : forall dir dir', check_if_variance_is_respected None dir dir' + | MSCovariant : forall dir, check_if_variance_is_respected (Some Covariant) dir dir + | MSContravariant : + forall dir, + check_if_variance_is_respected (Some Contravariant) dir (opposite_direction dir). + +Definition relation_class_of_reflexive_relation_class: + Reflexive_Relation_Class -> Relation_Class. + induction 1. + exact (SymmetricReflexive _ s r). + exact (AsymmetricReflexive tt r). + exact (Leibniz _ T). +Defined. + +Definition relation_class_of_areflexive_relation_class: + Areflexive_Relation_Class -> Relation_Class. + induction 1. + exact (SymmetricAreflexive _ s). + exact (AsymmetricAreflexive tt Aeq). +Defined. + +Definition carrier_of_reflexive_relation_class := + fun R => carrier_of_relation_class (relation_class_of_reflexive_relation_class R). + +Definition carrier_of_areflexive_relation_class := + fun R => carrier_of_relation_class (relation_class_of_areflexive_relation_class R). + +Definition relation_of_areflexive_relation_class := + fun R => relation_of_relation_class (relation_class_of_areflexive_relation_class R). + +Inductive Morphism_Context Hole dir : Relation_Class -> rewrite_direction -> Type := + | App : + forall In Out dir', + Morphism_Theory In Out -> Morphism_Context_List Hole dir dir' In -> + Morphism_Context Hole dir Out dir' + | ToReplace : Morphism_Context Hole dir Hole dir + | ToKeep : + forall S dir', + carrier_of_reflexive_relation_class S -> + Morphism_Context Hole dir (relation_class_of_reflexive_relation_class S) dir' + | ProperElementToKeep : + forall S dir' (x: carrier_of_areflexive_relation_class S), + relation_of_areflexive_relation_class S x x -> + Morphism_Context Hole dir (relation_class_of_areflexive_relation_class S) dir' +with Morphism_Context_List Hole dir : + rewrite_direction -> Arguments -> Type +:= + fcl_singl : + forall S dir' dir'', + check_if_variance_is_respected (variance_of_argument_class S) dir' dir'' -> + Morphism_Context Hole dir (relation_class_of_argument_class S) dir' -> + Morphism_Context_List Hole dir dir'' (singl S) + | fcl_cons : + forall S L dir' dir'', + check_if_variance_is_respected (variance_of_argument_class S) dir' dir'' -> + Morphism_Context Hole dir (relation_class_of_argument_class S) dir' -> + Morphism_Context_List Hole dir dir'' L -> + Morphism_Context_List Hole dir dir'' (necons S L). + +Scheme Morphism_Context_rect2 := Induction for Morphism_Context Sort Type +with Morphism_Context_List_rect2 := Induction for Morphism_Context_List Sort Type. + +Definition product_of_arguments : Arguments -> Type. + induction 1. + exact (carrier_of_relation_class a). + exact (prod (carrier_of_relation_class a) IHX). +Defined. + +Definition get_rewrite_direction: rewrite_direction -> Argument_Class -> rewrite_direction. + intros dir R. + destruct (variance_of_argument_class R). + destruct v. + exact dir. (* covariant *) + exact (opposite_direction dir). (* contravariant *) + exact dir. (* symmetric relation *) +Defined. + +Definition directed_relation_of_relation_class: + forall dir (R: Relation_Class), + carrier_of_relation_class R -> carrier_of_relation_class R -> Prop. + destruct 1. + exact (@relation_of_relation_class unit). + intros; exact (relation_of_relation_class _ X0 X). +Defined. + +Definition directed_relation_of_argument_class: + forall dir (R: Argument_Class), + carrier_of_relation_class R -> carrier_of_relation_class R -> Prop. + intros dir R. + rewrite <- + (about_carrier_of_relation_class_and_relation_class_of_argument_class R). + exact (directed_relation_of_relation_class dir (relation_class_of_argument_class R)). +Defined. + + +Definition relation_of_product_of_arguments: + forall dir In, + product_of_arguments In -> product_of_arguments In -> Prop. + induction In. + simpl. + exact (directed_relation_of_argument_class (get_rewrite_direction dir a) a). + + simpl; intros. + destruct X; destruct X0. + apply and. + exact + (directed_relation_of_argument_class (get_rewrite_direction dir a) a c c0). + exact (IHIn p p0). +Defined. + +Definition apply_morphism: + forall In Out (m: function_type_of_morphism_signature In Out) + (args: product_of_arguments In), carrier_of_relation_class Out. + intros. + induction In. + exact (m args). + simpl in m, args. + destruct args. + exact (IHIn (m c) p). +Defined. + +Theorem apply_morphism_compatibility_Right2Left: + forall In Out (m1 m2: function_type_of_morphism_signature In Out) + (args1 args2: product_of_arguments In), + make_compatibility_goal_aux _ _ m1 m2 -> + relation_of_product_of_arguments Right2Left _ args1 args2 -> + directed_relation_of_relation_class Right2Left _ + (apply_morphism _ _ m2 args1) + (apply_morphism _ _ m1 args2). + induction In; intros. + simpl in m1, m2, args1, args2, H0 |- *. + destruct a; simpl in H; hnf in H0. + apply H; exact H0. + destruct v; simpl in H0; apply H; exact H0. + apply H; exact H0. + destruct v; simpl in H0; apply H; exact H0. + rewrite H0; apply H; exact H0. + + simpl in m1, m2, args1, args2, H0 |- *. + destruct args1; destruct args2; simpl. + destruct H0. + simpl in H. + destruct a; simpl in H. + apply IHIn. + apply H; exact H0. + exact H1. + destruct v. + apply IHIn. + apply H; exact H0. + exact H1. + apply IHIn. + apply H; exact H0. + exact H1. + apply IHIn. + apply H; exact H0. + exact H1. + destruct v. + apply IHIn. + apply H; exact H0. + exact H1. + apply IHIn. + apply H; exact H0. + exact H1. + rewrite H0; apply IHIn. + apply H. + exact H1. +Qed. + +Theorem apply_morphism_compatibility_Left2Right: + forall In Out (m1 m2: function_type_of_morphism_signature In Out) + (args1 args2: product_of_arguments In), + make_compatibility_goal_aux _ _ m1 m2 -> + relation_of_product_of_arguments Left2Right _ args1 args2 -> + directed_relation_of_relation_class Left2Right _ + (apply_morphism _ _ m1 args1) + (apply_morphism _ _ m2 args2). +Proof. + induction In; intros. + simpl in m1, m2, args1, args2, H0 |- *. + destruct a; simpl in H; hnf in H0. + apply H; exact H0. + destruct v; simpl in H0; apply H; exact H0. + apply H; exact H0. + destruct v; simpl in H0; apply H; exact H0. + rewrite H0; apply H; exact H0. + + simpl in m1, m2, args1, args2, H0 |- *. + destruct args1; destruct args2; simpl. + destruct H0. + simpl in H. + destruct a; simpl in H. + apply IHIn. + apply H; exact H0. + exact H1. + destruct v. + apply IHIn. + apply H; exact H0. + exact H1. + apply IHIn. + apply H; exact H0. + exact H1. + apply IHIn. + apply H; exact H0. + exact H1. + apply IHIn. + destruct v; simpl in H, H0; apply H; exact H0. + exact H1. + rewrite H0; apply IHIn. + apply H. + exact H1. +Qed. + +Definition interp : + forall Hole dir Out dir', carrier_of_relation_class Hole -> + Morphism_Context Hole dir Out dir' -> carrier_of_relation_class Out. + intros Hole dir Out dir' H t. + elim t using + (@Morphism_Context_rect2 Hole dir (fun S _ _ => carrier_of_relation_class S) + (fun _ L fcl => product_of_arguments L)); + intros. + exact (apply_morphism _ _ (Function m) X). + exact H. + exact c. + exact x. + simpl; + rewrite <- + (about_carrier_of_relation_class_and_relation_class_of_argument_class S); + exact X. + split. + rewrite <- + (about_carrier_of_relation_class_and_relation_class_of_argument_class S); + exact X. + exact X0. +Defined. + +(* CSC: interp and interp_relation_class_list should be mutually defined, since + the proof term of each one contains the proof term of the other one. However + I cannot do that interactively (I should write the Fix by hand) *) +Definition interp_relation_class_list : + forall Hole dir dir' (L: Arguments), carrier_of_relation_class Hole -> + Morphism_Context_List Hole dir dir' L -> product_of_arguments L. + intros Hole dir dir' L H t. + elim t using + (@Morphism_Context_List_rect2 Hole dir (fun S _ _ => carrier_of_relation_class S) + (fun _ L fcl => product_of_arguments L)); + intros. + exact (apply_morphism _ _ (Function m) X). + exact H. + exact c. + exact x. + simpl; + rewrite <- + (about_carrier_of_relation_class_and_relation_class_of_argument_class S); + exact X. + split. + rewrite <- + (about_carrier_of_relation_class_and_relation_class_of_argument_class S); + exact X. + exact X0. +Defined. + +Theorem setoid_rewrite: + forall Hole dir Out dir' (E1 E2: carrier_of_relation_class Hole) + (E: Morphism_Context Hole dir Out dir'), + (directed_relation_of_relation_class dir Hole E1 E2) -> + (directed_relation_of_relation_class dir' Out (interp E1 E) (interp E2 E)). +Proof. + intros. + elim E using + (@Morphism_Context_rect2 Hole dir + (fun S dir'' E => directed_relation_of_relation_class dir'' S (interp E1 E) (interp E2 E)) + (fun dir'' L fcl => + relation_of_product_of_arguments dir'' _ + (interp_relation_class_list E1 fcl) + (interp_relation_class_list E2 fcl))); intros. + change (directed_relation_of_relation_class dir'0 Out0 + (apply_morphism _ _ (Function m) (interp_relation_class_list E1 m0)) + (apply_morphism _ _ (Function m) (interp_relation_class_list E2 m0))). + destruct dir'0. + apply apply_morphism_compatibility_Left2Right. + exact (Compat m). + exact H0. + apply apply_morphism_compatibility_Right2Left. + exact (Compat m). + exact H0. + + exact H. + + unfold interp, Morphism_Context_rect2. + (* CSC: reflexivity used here *) + destruct S; destruct dir'0; simpl; (apply r || reflexivity). + + destruct dir'0; exact r. + + destruct S; unfold directed_relation_of_argument_class; simpl in H0 |- *; + unfold get_rewrite_direction; simpl. + destruct dir'0; destruct dir''; + (exact H0 || + unfold directed_relation_of_argument_class; simpl; apply s; exact H0). + (* the following mess with generalize/clear/intros is to help Coq resolving *) + (* second order unification problems. *) + generalize m c H0; clear H0 m c; inversion c; + generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros; + (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3). + destruct dir'0; destruct dir''; + (exact H0 || + unfold directed_relation_of_argument_class; simpl; apply s; exact H0). + (* the following mess with generalize/clear/intros is to help Coq resolving *) + (* second order unification problems. *) + generalize m c H0; clear H0 m c; inversion c; + generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros; + (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3). + destruct dir'0; destruct dir''; (exact H0 || hnf; symmetry; exact H0). + + change + (directed_relation_of_argument_class (get_rewrite_direction dir'' S) S + (eq_rect _ (fun T : Type => T) (interp E1 m) _ + (about_carrier_of_relation_class_and_relation_class_of_argument_class S)) + (eq_rect _ (fun T : Type => T) (interp E2 m) _ + (about_carrier_of_relation_class_and_relation_class_of_argument_class S)) /\ + relation_of_product_of_arguments dir'' _ + (interp_relation_class_list E1 m0) (interp_relation_class_list E2 m0)). + split. + clear m0 H1; destruct S; simpl in H0 |- *; unfold get_rewrite_direction; simpl. + destruct dir''; destruct dir'0; (exact H0 || hnf; apply s; exact H0). + inversion c. + rewrite <- H3; exact H0. + rewrite (opposite_direction_idempotent dir'0); exact H0. + destruct dir''; destruct dir'0; (exact H0 || hnf; apply s; exact H0). + inversion c. + rewrite <- H3; exact H0. + rewrite (opposite_direction_idempotent dir'0); exact H0. + destruct dir''; destruct dir'0; (exact H0 || hnf; symmetry; exact H0). + exact H1. + Qed. -- cgit v1.2.3