From 11104cdcb1e53cd83768d2ce9858829b457e2d65 Mon Sep 17 00:00:00 2001 From: sacerdot Date: Tue, 7 Sep 2004 14:44:57 +0000 Subject: * The Coq part of the reflexive tactic setoid_rewrite is generalized to asymmetric relations thanks to the introduction of morphisms that are covariant or contravariant in their arguments. * The ML part of the tactic is updated only for backward compatibility: it is still not possible to benefit from the asymmetric relations and relative morphisms. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@6070 85f007b7-540e-0410-9357-904b9bb8a0f7 --- theories/Setoids/Setoid.v | 406 +++++++++++++++++++++++++++++++++------------- 1 file changed, 292 insertions(+), 114 deletions(-) (limited to 'theories/Setoids') diff --git a/theories/Setoids/Setoid.v b/theories/Setoids/Setoid.v index bf54f0567..d98f94419 100644 --- a/theories/Setoids/Setoid.v +++ b/theories/Setoids/Setoid.v @@ -19,26 +19,62 @@ Definition is_reflexive (A: Type) (Aeq: A -> A -> Prop) : Prop := Definition is_symmetric (A: Type) (Aeq: A -> A -> Prop) : Prop := forall (x y:A), Aeq x y -> Aeq y x. -Inductive Relation_Class : Type := - Reflexive : forall A Aeq, (@is_reflexive A Aeq) -> Relation_Class - | Leibniz : Type -> Relation_Class. +(* X will be used to distinguish covariant arguments whose type is an *) +(* AsymmetricReflexive relation from covariant arguments of the same type*) +Inductive X_Relation_Class (X: Type) : Type := + SymmetricReflexive : + forall A Aeq, @is_symmetric A Aeq -> is_reflexive Aeq -> X_Relation_Class X + | AsymmetricReflexive : X -> forall A Aeq, @is_reflexive A Aeq -> 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. Implicit Type Hole Out: Relation_Class. -Definition carrier_of_relation_class : Relation_Class -> Type. - intro; case X; intros. +Definition relation_class_of_argument_class : Argument_Class -> Relation_Class. + destruct 1. + exact (SymmetricReflexive _ i i0). + exact (AsymmetricReflexive tt i). + exact (Leibniz _ T). +Defined. + +Definition carrier_of_relation_class : forall X, X_Relation_Class X -> Type. + destruct 1. + 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 (@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) - | cons : A -> (nelistT A) -> (nelistT A). + singl : A -> nelistT A + | cons : A -> nelistT A -> nelistT A. + +Definition Arguments := nelistT Argument_Class. -Implicit Type In: (nelistT Relation_Class). +Implicit Type In: Arguments. Definition function_type_of_morphism_signature : - (nelistT Relation_Class) -> Relation_Class -> Type. + Arguments -> Relation_Class -> Type. intros In Out. induction In. exact (carrier_of_relation_class a -> carrier_of_relation_class Out). @@ -49,14 +85,18 @@ 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; destruct Out; simpl in f, g. - exact (forall (x1 x2: A), (Aeq x1 x2) -> (Aeq0 (f x1) (g x2))). - exact (forall (x1 x2: A), (Aeq x1 x2) -> f x1 = g x2). - exact (forall (x: T), (Aeq (f x) (g x))). - exact (forall (x: T), f x = g x). + 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 x, relation_of_relation_class Out (f x) (g x)). induction a; simpl in f, g. - exact (forall (x1 x2: A), (Aeq x1 x2) -> IHIn (f x1) (g x2)). - exact (forall (x: T), IHIn (f x) (g x)). + 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 := @@ -66,18 +106,17 @@ Record Morphism_Theory In Out : Type := {Function : function_type_of_morphism_signature In Out; Compat : make_compatibility_goal In Out Function}. -Definition list_of_Leibniz_of_list_of_types: - nelistT Type -> nelistT Relation_Class. +Definition list_of_Leibniz_of_list_of_types: nelistT Type -> Arguments. induction 1. - exact (singl (Leibniz a)). - exact (cons (Leibniz a) IHX). + exact (singl (Leibniz _ a)). + exact (cons (Leibniz _ a) IHX). Defined. (* every function is a morphism from Leibniz+ to Leibniz *) 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 + let Out' := Leibniz _ Out in function_type_of_morphism_signature In' Out' -> Morphism_Theory In' Out'. intros. @@ -92,7 +131,8 @@ Defined. Add Relation Prop iff reflexivity proved by iff_refl symmetry proved by iff_sym. Definition Prop_Relation_Class : Relation_Class. - eapply (@Reflexive _ iff). + eapply (@SymmetricReflexive unit _ iff). + exact iff_sym. exact iff_refl. Defined. @@ -111,26 +151,63 @@ Defined. (* THE CIC PART OF THE REFLEXIVE TACTIC (SETOID REWRITE) *) -Inductive Morphism_Context Hole : Relation_Class -> Type := - App : forall In Out, - Morphism_Theory In Out -> Morphism_Context_List Hole In -> - Morphism_Context Hole Out - | Toreplace : Morphism_Context Hole Hole +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. +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. + 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). + +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: Relation_Class), - carrier_of_relation_class S -> Morphism_Context Hole S - | Imp : - Morphism_Context Hole Prop_Relation_Class -> - Morphism_Context Hole Prop_Relation_Class -> - Morphism_Context Hole Prop_Relation_Class -with Morphism_Context_List Hole: nelistT Relation_Class -> Type := + forall S dir', + carrier_of_relation_class S -> Morphism_Context Hole dir S dir' +with Morphism_Context_List Hole dir : + rewrite_direction -> Arguments -> Type +:= fcl_singl : - forall (S: Relation_Class), Morphism_Context Hole S -> - Morphism_Context_List Hole (singl S) + 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: Relation_Class) (L: nelistT Relation_Class), - Morphism_Context Hole S -> Morphism_Context_List Hole L -> - Morphism_Context_List Hole (cons S L). + 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'' (cons S L). Scheme Morphism_Context_rect2 := Induction for Morphism_Context Sort Type with Morphism_Context_List_rect2 := Induction for Morphism_Context_List Sort Type. @@ -138,34 +215,57 @@ with Morphism_Context_List_rect2 := Induction for Morphism_Context_List Sort Typ Inductive prodT (A B: Type) : Type := pairT : A -> B -> prodT A B. -Definition product_of_relation_class_list : nelistT Relation_Class -> Type. +Definition product_of_arguments : Arguments -> Type. induction 1. exact (carrier_of_relation_class a). exact (prodT (carrier_of_relation_class a) IHX). Defined. -Definition relation_of_relation_class: - forall Out, - carrier_of_relation_class Out -> carrier_of_relation_class Out -> Prop. - destruct Out. - exact Aeq. - exact (@eq T). +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 relation_of_product_of_relation_class_list: - forall In, - product_of_relation_class_list In -> product_of_relation_class_list In -> Prop. +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. - exact (relation_of_relation_class a). + simpl. + exact (directed_relation_of_argument_class (get_rewrite_direction dir a) a). simpl; intros. destruct X; destruct X0. - exact (relation_of_relation_class a c c0 /\ IHIn p p0). + 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_relation_class_list In), carrier_of_relation_class Out. + (args: product_of_arguments In), carrier_of_relation_class Out. intros. induction In. exact (m args). @@ -174,23 +274,57 @@ Definition apply_morphism: exact (IHIn (m c) p). Defined. -Theorem apply_morphism_compatibility: +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. + 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. + 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_relation_class_list In), + (args1 args2: product_of_arguments In), make_compatibility_goal_aux _ _ m1 m2 -> - relation_of_product_of_relation_class_list _ args1 args2 -> - relation_of_relation_class _ + relation_of_product_of_arguments Left2Right _ args1 args2 -> + directed_relation_of_relation_class Left2Right _ (apply_morphism _ _ m1 args1) (apply_morphism _ _ m2 args2). - intros. - induction In. - simpl; simpl in m1, m2, args1, args2, H0. - destruct a; destruct Out. - apply H; exact H0. - simpl; apply H; exact H0. - simpl; rewrite H0; apply H. - simpl; rewrite H0; apply H. - simpl in args1, args2, H0. + 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. + rewrite H0; apply H; exact H0. + + simpl in m1, m2, args1, args2, H0 |- *. destruct args1; destruct args2; simpl. destruct H0. simpl in H. @@ -198,89 +332,128 @@ Theorem apply_morphism_compatibility: 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. Definition interp : - forall Hole Out, carrier_of_relation_class Hole -> - Morphism_Context Hole Out -> carrier_of_relation_class Out. - intros Hole Out H t. + 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 (fun S _ => carrier_of_relation_class S) - (fun L fcl => product_of_relation_class_list L)); + (@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 -> X0). - exact X. - split; [ exact X | exact X0 ]. + 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 (L: nelistT Relation_Class), carrier_of_relation_class Hole -> - Morphism_Context_List Hole L -> product_of_relation_class_list L. - intros Hole L H t. + 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 (fun S _ => carrier_of_relation_class S) - (fun L fcl => product_of_relation_class_list L)); + (@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 -> X0). - exact X. - split; [ exact X | exact X0 ]. + 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 Out (E1 E2: carrier_of_relation_class Hole) - (E: Morphism_Context Hole Out), - (relation_of_relation_class Hole E1 E2) -> - (relation_of_relation_class Out (interp E1 E) (interp E2 E)). + 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)). intros. elim E using - (@Morphism_Context_rect2 Hole - (fun S E => relation_of_relation_class S (interp E1 E) (interp E2 E)) - (fun L fcl => - relation_of_product_of_relation_class_list _ - (interp_relation_class_list E1 fcl) - (interp_relation_class_list E2 fcl))); - intros. - change (relation_of_relation_class Out0 + (@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))). - apply apply_morphism_compatibility. - exact (Compat m). - exact H0. + 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 S; destruct dir'0; simpl. + apply i0. + apply i0. apply i. - simpl; reflexivity. - - change - (relation_of_relation_class Prop_Relation_Class - (interp E1 m -> interp E1 m0) (interp E2 m -> interp E2 m0)). - simpl; simpl in H0, H1. - tauto. - - exact H0. + apply i. + reflexivity. + reflexivity. + + 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 i; 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 - (relation_of_relation_class _ (interp E1 m) (interp E2 m) /\ - relation_of_product_of_relation_class_list _ + (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. - exact H0. + clear m0 H1; destruct S; simpl in H0 |- *; unfold get_rewrite_direction; simpl. + destruct dir''; destruct dir'0; (exact H0 || hnf; apply i; 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. @@ -291,24 +464,29 @@ Record Setoid_Theory (A: Type) (Aeq: A -> A -> Prop) : Prop := Seq_sym : forall x y:A, Aeq x y -> Aeq y x; Seq_trans : forall x y z:A, Aeq x y -> Aeq y z -> Aeq x z}. -Definition relation_class_of_setoid_theory: +Definition argument_class_of_setoid_theory: forall (A: Type) (Aeq: A -> A -> Prop), - Setoid_Theory Aeq -> Relation_Class. + Setoid_Theory Aeq -> Argument_Class. intros. - apply (@Reflexive _ Aeq). + apply (@SymmetricReflexive variance _ Aeq). + exact (Seq_sym H). exact (Seq_refl H). Defined. +Definition relation_class_of_setoid_theory := + fun A Aeq Setoid => + relation_class_of_argument_class + (@argument_class_of_setoid_theory A Aeq Setoid). + Definition equality_morphism_of_setoid_theory: forall (A: Type) (Aeq: A -> A -> Prop) (ST: Setoid_Theory Aeq), - let ASetoidClass := relation_class_of_setoid_theory ST in + let ASetoidClass := argument_class_of_setoid_theory ST in (Morphism_Theory (cons ASetoidClass (singl ASetoidClass)) Prop_Relation_Class). intros. exists Aeq. pose (sym := Seq_sym ST); clearbody sym. pose (trans := Seq_trans ST); clearbody trans. - (*CSC: symmetry and transitivity used here *) unfold make_compatibility_goal; simpl; split; eauto. Defined. -- cgit v1.2.3