(* ** 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.

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