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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 5920 2004-07-16 20:01:26Z herbelin $ 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.
|
