Ajout lemmes arithmetiques

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@1557 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 47 insertions, 0 deletions

diff --git a/theories/Logic/Decidable.v b/theories/Logic/Decidable.v
new file mode 100644
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+++ b/theories/Logic/Decidable.v
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+Definition decidable := [P:Prop] P \/ ~P.
+
+Theorem dec_not_not : (P:Prop)(decidable P) -> (~P -> False) -> P.
+Unfold decidable; Tauto. 
+Save.
+
+Theorem dec_True: (decidable True).
+Unfold decidable; Auto.
+Save.
+
+Theorem dec_False: (decidable False).
+Unfold decidable not; Auto.
+Save.
+
+Theorem dec_or: (A,B:Prop)(decidable A) -> (decidable B) -> (decidable (A\/B)).
+Unfold decidable; Tauto. 
+Save.
+
+Theorem dec_and: (A,B:Prop)(decidable A) -> (decidable B) ->(decidable (A/\B)).
+Unfold decidable; Tauto. 
+Save.
+
+Theorem dec_not: (A:Prop)(decidable A) -> (decidable ~A).
+Unfold decidable; Tauto. 
+Save.
+
+Theorem dec_imp: (A,B:Prop)(decidable A) -> (decidable B) ->(decidable (A->B)).
+Unfold decidable; Tauto. 
+Save.
+
+Theorem not_not : (P:Prop)(decidable P) -> (~(~P)) -> P.
+Unfold decidable; Tauto. Save.
+
+Theorem not_or : (A,B:Prop) ~(A\/B) -> ~A /\ ~B.
+Tauto. Save.
+
+Theorem not_and : (A,B:Prop) (decidable A) -> ~(A/\B) -> ~A \/ ~B.
+Unfold decidable; Tauto. Save.
+
+Theorem not_imp : (A,B:Prop) (decidable A) -> ~(A -> B) -> A /\ ~B.
+Unfold decidable;Tauto.
+Save.
+
+Theorem imp_simp : (A,B:Prop) (decidable A) -> (A -> B) -> ~A \/ B.
+Unfold decidable; Tauto.
+Save.
+