Set Implicit Arguments.
Hint Extern 4 => exact _ : core.
Ltac inv H := inversion H; subst; clear H.
Definition dec (X: Prop) : Type := {X} + {~ X}.
Coercion dec2bool P (d: dec P) := if d then true else false.
Definition is_true (b : bool) := b = true.
Existing Class dec.
Definition Dec (X: Prop) (d: dec X) : dec X := d.
Arguments Dec X {d}.

Lemma Dec_reflect (X: Prop) (d: dec X) :
is_true (Dec X) <-> X.