(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

(** 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_imp : forall A B:Prop, decidable A -> ~ (A -> B) -> A /\ ~ B.