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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * 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*)
Section Setoid.
Variable A : Type.
Variable Aeq : A -> A -> Prop.
Record Setoid_Theory : 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.
Definition Prop_S : Setoid_Theory Prop iff.
split; [ exact iff_refl | exact iff_sym | exact iff_trans ].
Qed.
Add Setoid Prop iff Prop_S.
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 or : or_ext.
intros.
inversion H1.
left.
inversion H.
apply (H3 H2).
right.
inversion H0.
apply (H3 H2).
Qed.
Add Morphism and : and_ext.
intros.
inversion H1.
split.
inversion H.
apply (H4 H2).
inversion H0.
apply (H4 H3).
Qed.
Add Morphism not : not_ext.
red in |- *; intros.
apply H0.
inversion H.
apply (H3 H1).
Qed.
Definition fleche (A B:Prop) := A -> B.
Add Morphism fleche : fleche_ext.
unfold fleche in |- *.
intros.
inversion H0.
inversion H.
apply (H3 (H1 (H6 H2))).
Qed.
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