(** * Decidable predicates *)
Definition dec (X: Prop) : Type := {X} + {~ X}.
Coercion dec2bool P (d: dec P) := if d then true else false.
Existing Class dec.
Definition Dec (X: Prop) (d: dec X) : dec X := d.
Arguments Dec X {d}.
Notation Decb X := (dec2bool (Dec X)).

Lemma Dec_reflect_eq (X Y: Prop) (d: dec X) (e: dec Y) :
Decb X = Decb Y <->  (X <-> Y).