Library Coq.Logic.Decidable

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 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. ```
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