Deplacement des setoides.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@1992 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 75 insertions, 0 deletions

diff --git a/theories/Setoids/Setoid.v b/theories/Setoids/Setoid.v
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+(***********************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
+(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
+(*   \VV/  *************************************************************)
+(*    //   *      This file is distributed under the terms of the      *)
+(*         *       GNU Lesser General Public License Version 2.1       *)
+(***********************************************************************)
+
+(* $Id$: *)
+
+Require Export Setoid_replace.
+
+Section Setoid.
+
+Variable A : Type.
+Variable Aeq : A -> A -> Prop.
+
+Record Setoid_Theory : Prop :=
+{ Seq_refl : (x:A) (Aeq x x);
+  Seq_sym : (x,y:A) (Aeq x y) -> (Aeq y x);
+  Seq_trans : (x,y,z:A) (Aeq x y) -> (Aeq y z) -> (Aeq x z)
+}.
+
+End Setoid.
+
+Definition Prop_S : (Setoid_Theory Prop iff).
+Split; [Exact iff_refl | Exact iff_sym | Exact iff_trans].
+Save.
+
+Add Setoid Prop iff Prop_S.
+
+Hint prop_set : setoid := Resolve (Seq_refl Prop iff Prop_S).
+Hint prop_set : setoid := Resolve (Seq_sym Prop iff Prop_S).
+Hint prop_set : setoid := Resolve (Seq_trans Prop iff Prop_S).
+
+Add Morphism or : or_ext.
+Intros.
+Inversion H1.
+Left.
+Inversion H.
+Apply (H3 H2).
+
+Right.
+Inversion H0.
+Apply (H3 H2).
+Save.
+
+Add Morphism and : and_ext.
+Intros.
+Inversion H1.
+Split.
+Inversion H.
+Apply (H4 H2).
+
+Inversion H0.
+Apply (H4 H3).
+Save.
+
+Add Morphism not : not_ext.
+Red ; Intros.
+Apply H0.
+Inversion H.
+Apply (H3 H1).
+Save.
+
+Definition fleche [A,B:Prop] := A -> B.
+
+Add Morphism fleche : fleche_ext.
+Unfold fleche.
+Intros.
+Inversion H0.
+Inversion H.
+Apply (H3 (H1 (H6 H2))).
+Save.
+