Imported Upstream version 8.0pl3+8.1alphaupstream/8.0pl3+8.1alpha

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}