Ltac InvBooleans :=
  match goal with
  | [ H: _ && _ = true |- _ ] =>
      destruct (andb_prop _ _ H); clear H; InvBooleans
  | [ H: _ || _ = false |- _ ] =>
      destruct (orb_false_elim _ _ H); clear H; InvBooleans
  | [ H: proj_sumbool ?x = true |- _ ] =>
      generalize (proj_sumbool_true _ H); clear H; intro; InvBooleans
  | _ => idtac
  end.

Lemma pred_dec_false: forall (a b: A), ~P -> (if dec then a else b) = b.