(** * 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)).
Hint Resolve Dec_auto Dec_auto_not.
Hint Extern 4 =>  (* Improves type class inference *)
match goal with
| [  |- dec ((fun _ => _) _) ] => cbn
end : typeclass_instances.
Tactic Notation "decide" constr(p) :=
destruct (Dec p).
Tactic Notation "decide" constr(p) "as" simple_intropattern(i) :=
destruct (Dec p) as i.
Tactic Notation "decide" "_" :=
destruct (Dec _).
Tactic Notation "have" constr(E) := let X := fresh "E" in decide E as [X|X]; subst; try congruence; try omega; clear X.

Lemma Dec_false P {H : dec P} : dec2bool (Dec P) = false -> ~P.