Coq.Logic.Decidable

# Module Coq.Logic.Decidable

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Properties of decidable propositions

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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`.

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