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(*         *      The Rocq Prover / The Rocq Development Team           *)
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(** Properties of decidable propositions *)
(** Note: the following definition of [decidable] can be used to express the notion of decidability in computability theory only in an axiom-free Coq (since a proof of [forall n, decidable (P n)], for [P] a countable class of problems, induces by extraction the existence of a (total) recursive algorithm [f] such that [f(n)] evaluates to [true] iff [P n], and since, conversely, a Coq proof of termination of a recursive algorithm deciding [P n] can be extracted in turn into a proof of [forall n, decidable (P n)]). In the presence of axioms such as [Classical_prop.classic], it would take a different meaning. *)
Definition decidable (P:Prop) := P \/ ~ P.

Theorem not_iff :
forall A B:Prop, decidable A -> decidable B ->
~ (A <-> B) -> (A /\ ~ B) \/ (~ A /\ B).