(** * Decidability and Enumerability *)
Require Export Shared.Prelim.
(* ** Definitions *)
Definition compl X p := fun x : X => ~ p x.
Definition decidable {X} (p : X -> Prop) := exists f, forall x, p x <-> f x = true.
Definition enumerable {X} (p : X -> Prop) := exists f, forall x, p x <-> exists n : nat, f n = Some x.
Definition discrete X := decidable (fun '(x,y) => x = y :> X).
Definition enumerable__T X := exists f : nat -> option X, forall x, exists n, f n = Some x.
(* *** more practical type-theoretic characterisations *)

Lemma decidable_iff X p :
decidable p <-> inhabited (forall x : X, dec (p x)).