From 6b649aba925b6f7462da07599fe67ebb12a3460e Mon Sep 17 00:00:00 2001
From: Samuel Mimram <samuel.mimram@ens-lyon.org>
Date: Wed, 28 Jul 2004 21:54:47 +0000
Subject: Imported Upstream version 8.0pl1

---
 theories7/Logic/Decidable.v | 58 +++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 58 insertions(+)
 create mode 100644 theories7/Logic/Decidable.v

(limited to 'theories7/Logic/Decidable.v')

diff --git a/theories7/Logic/Decidable.v b/theories7/Logic/Decidable.v
new file mode 100644
index 00000000..537b5e88
--- /dev/null
+++ b/theories7/Logic/Decidable.v
@@ -0,0 +1,58 @@
+(************************************************************************)
+(*  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 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $	 i*)
+
+(** Properties of decidable propositions *)
+
+Definition decidable := [P:Prop] P \/ ~P.
+
+Theorem dec_not_not : (P:Prop)(decidable P) -> (~P -> False) -> P.
+Unfold decidable; Tauto. 
+Qed.
+
+Theorem dec_True: (decidable True).
+Unfold decidable; Auto.
+Qed.
+
+Theorem dec_False: (decidable False).
+Unfold decidable not; Auto.
+Qed.
+
+Theorem dec_or: (A,B:Prop)(decidable A) -> (decidable B) -> (decidable (A\/B)).
+Unfold decidable; Tauto. 
+Qed.
+
+Theorem dec_and: (A,B:Prop)(decidable A) -> (decidable B) ->(decidable (A/\B)).
+Unfold decidable; Tauto. 
+Qed.
+
+Theorem dec_not: (A:Prop)(decidable A) -> (decidable ~A).
+Unfold decidable; Tauto. 
+Qed.
+
+Theorem dec_imp: (A,B:Prop)(decidable A) -> (decidable B) ->(decidable (A->B)).
+Unfold decidable; Tauto. 
+Qed.
+
+Theorem not_not : (P:Prop)(decidable P) -> (~(~P)) -> P.
+Unfold decidable; Tauto. Qed.
+
+Theorem not_or : (A,B:Prop) ~(A\/B) -> ~A /\ ~B.
+Tauto. Qed.
+
+Theorem not_and : (A,B:Prop) (decidable A) -> ~(A/\B) -> ~A \/ ~B.
+Unfold decidable; Tauto. Qed.
+
+Theorem not_imp : (A,B:Prop) (decidable A) -> ~(A -> B) -> A /\ ~B.
+Unfold decidable;Tauto.
+Qed.
+
+Theorem imp_simp : (A,B:Prop) (decidable A) -> (A -> B) -> ~A \/ B.
+Unfold decidable; Tauto.
+Qed.
+
-- 
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