(* An alternative to xorb that behaves somewhat better wrt simplification.    *)
Definition addb b := if b then negb else id.Notation "x \i 'n' S" := (x \in @unkey_pred _ S _ _)
  (at level 70, format "'[hv' x '/ '  \i 'n'  S ']'") : bool_scope.

(* Specifications: *)
Lemma prop_congr : forall b b' : bool, b = b' -> b = b' :> Prop.
Lemma is_true_true : true.
Lemma not_false_is_true : ~ false.
Lemma is_true_locked_true : locked true.