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authorGravatar Samuel Mimram <smimram@debian.org>2006-04-28 14:59:16 +0000
committerGravatar Samuel Mimram <smimram@debian.org>2006-04-28 14:59:16 +0000
commit3ef7797ef6fc605dfafb32523261fe1b023aeecb (patch)
treead89c6bb57ceee608fcba2bb3435b74e0f57919e /theories/Setoids
parent018ee3b0c2be79eb81b1f65c3f3fa142d24129c8 (diff)
Imported Upstream version 8.0pl3+8.1alphaupstream/8.0pl3+8.1alpha
Diffstat (limited to 'theories/Setoids')
-rw-r--r--theories/Setoids/Setoid.v675
-rw-r--r--theories/Setoids/intro.tex1
2 files changed, 635 insertions, 41 deletions
diff --git a/theories/Setoids/Setoid.v b/theories/Setoids/Setoid.v
index 63f21fed..6ff73438 100644
--- a/theories/Setoids/Setoid.v
+++ b/theories/Setoids/Setoid.v
@@ -1,3 +1,4 @@
+
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
@@ -6,66 +7,658 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Setoid.v,v 1.5.2.1 2004/07/16 19:31:17 herbelin Exp $: i*)
+(*i $Id: Setoid.v 6306 2004-11-16 16:11:10Z sacerdot $: i*)
+
+Require Export Relation_Definitions.
+
+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
+ | cons : 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}.
+
+Definition list_of_Leibniz_of_list_of_types: nelistT Type -> Arguments.
+ induction 1.
+ 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
+ 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.
+
+(* 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.
+ hnf; unfold impl; tauto.
+Qed.
+
+Definition Impl_Relation_Class : Relation_Class.
+ eapply (@AsymmetricReflexive unit tt _ impl).
+ exact impl_refl.
+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 (cons 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 (cons 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 (cons 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 (cons ASetoidClass1 (singl ASetoidClass2)) Impl_Relation_Class).
+ intros.
+ exists Aeq.
+ unfold make_compatibility_goal; simpl; unfold impl; eauto.
+Defined.
+
+(* 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.
+
+(* every predicate is 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.
+
+(* impl AS A RELATION *)
+
+Theorem impl_trans: transitive _ impl.
+ hnf; unfold impl; tauto.
+Qed.
+
+Add Relation Prop impl
+ reflexivity proved by impl_refl
+ transitivity proved by impl_trans
+ as impl_relation.
+
+(* THE CIC PART OF THE REFLEXIVE TACTIC (SETOID REWRITE) *)
-Section Setoid.
+Inductive rewrite_direction : Type :=
+ Left2Right
+ | Right2Left.
-Variable A : Type.
-Variable Aeq : A -> A -> Prop.
+Implicit Type dir: rewrite_direction.
-Record Setoid_Theory : Prop :=
+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.
+ 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'' (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.
+
+Definition product_of_arguments : Arguments -> Type.
+ induction 1.
+ exact (carrier_of_relation_class a).
+ exact (prodT (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).
+ 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)).
+ 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.
+
+(* BEGIN OF UTILITY/BACKWARD COMPATIBILITY PART *)
+
+Record Setoid_Theory (A: Type) (Aeq: relation A) : Prop :=
{Seq_refl : forall x:A, Aeq x x;
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}.
-End Setoid.
+(* END OF UTILITY/BACKWARD COMPATIBILITY PART *)
+
+(* A FEW EXAMPLES ON iff *)
-Definition Prop_S : Setoid_Theory Prop iff.
-split; [ exact iff_refl | exact iff_sym | exact iff_trans ].
+(* impl IS A MORPHISM *)
+
+Add Morphism impl with signature iff ==> iff ==> iff as Impl_Morphism.
+unfold impl; tauto.
Qed.
-Add Setoid Prop iff Prop_S.
+(* and IS A MORPHISM *)
-Hint Resolve (Seq_refl Prop iff Prop_S): setoid.
-Hint Resolve (Seq_sym Prop iff Prop_S): setoid.
-Hint Resolve (Seq_trans Prop iff Prop_S): setoid.
+Add Morphism and with signature iff ==> iff ==> iff as And_Morphism.
+ tauto.
+Qed.
-Add Morphism or : or_ext.
-intros.
-inversion H1.
-left.
-inversion H.
-apply (H3 H2).
+(* or IS A MORPHISM *)
-right.
-inversion H0.
-apply (H3 H2).
+Add Morphism or with signature iff ==> iff ==> iff as Or_Morphism.
+ tauto.
Qed.
-Add Morphism and : and_ext.
-intros.
-inversion H1.
-split.
-inversion H.
-apply (H4 H2).
+(* not IS A MORPHISM *)
-inversion H0.
-apply (H4 H3).
+Add Morphism not with signature iff ==> iff as Not_Morphism.
+ tauto.
Qed.
-Add Morphism not : not_ext.
-red in |- *; intros.
-apply H0.
-inversion H.
-apply (H3 H1).
+(* THE SAME EXAMPLES ON impl *)
+
+Add Morphism and with signature impl ++> impl ++> impl as And_Morphism2.
+ unfold impl; tauto.
Qed.
-Definition fleche (A B:Prop) := A -> B.
+Add Morphism or with signature impl ++> impl ++> impl as Or_Morphism2.
+ unfold impl; tauto.
+Qed.
-Add Morphism fleche : fleche_ext.
-unfold fleche in |- *.
-intros.
-inversion H0.
-inversion H.
-apply (H3 (H1 (H6 H2))).
+Add Morphism not with signature impl --> impl as Not_Morphism2.
+ unfold impl; tauto.
Qed.
+
+(* FOR BACKWARD COMPATIBILITY *)
+Implicit Arguments Setoid_Theory [].
+Implicit Arguments Seq_refl [].
+Implicit Arguments Seq_sym [].
+Implicit Arguments Seq_trans [].
diff --git a/theories/Setoids/intro.tex b/theories/Setoids/intro.tex
new file mode 100644
index 00000000..50cd025d
--- /dev/null
+++ b/theories/Setoids/intro.tex
@@ -0,0 +1 @@
+\section{Setoids}\label{Setoids}