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Diffstat (limited to 'theories7/Logic/Decidable.v')
| -rw-r--r-- | theories7/Logic/Decidable.v | 58 |
1 files changed, 58 insertions, 0 deletions
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. + |