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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: Decidable.v,v 1.5.2.1 2004/07/16 19:31:06 herbelin Exp $	 i*)
+
+(** Properties of decidable propositions *)
+
+Definition decidable (P:Prop) := P \/ ~ P.
+
+Theorem dec_not_not : forall P:Prop, decidable P -> (~ P -> False) -> P.
+unfold decidable in |- *; tauto. 
+Qed.
+
+Theorem dec_True : decidable True.
+unfold decidable in |- *; auto.
+Qed.
+
+Theorem dec_False : decidable False.
+unfold decidable, not in |- *; auto.
+Qed.
+
+Theorem dec_or :
+ forall A B:Prop, decidable A -> decidable B -> decidable (A \/ B).
+unfold decidable in |- *; tauto. 
+Qed.
+
+Theorem dec_and :
+ forall A B:Prop, decidable A -> decidable B -> decidable (A /\ B).
+unfold decidable in |- *; tauto. 
+Qed.
+
+Theorem dec_not : forall A:Prop, decidable A -> decidable (~ A).
+unfold decidable in |- *; tauto. 
+Qed.
+
+Theorem dec_imp :
+ forall A B:Prop, decidable A -> decidable B -> decidable (A -> B).
+unfold decidable in |- *; tauto. 
+Qed.
+
+Theorem not_not : forall P:Prop, decidable P -> ~ ~ P -> P.
+unfold decidable in |- *; tauto. Qed.
+
+Theorem not_or : forall A B:Prop, ~ (A \/ B) -> ~ A /\ ~ B.
+tauto. Qed.
+
+Theorem not_and : forall A B:Prop, decidable A -> ~ (A /\ B) -> ~ A \/ ~ B.
+unfold decidable in |- *; tauto. Qed.
+
+Theorem not_imp : forall A B:Prop, decidable A -> ~ (A -> B) -> A /\ ~ B.
+unfold decidable in |- *; tauto.
+Qed.
+
+Theorem imp_simp : forall A B:Prop, decidable A -> (A -> B) -> ~ A \/ B.
+unfold decidable in |- *; tauto.
+Qed.