Ltac do_bool :=
  repeat match goal with
    | [ H : beq_nat _ _ = true |- _ ] => apply beq_nat_true in H
    | [ H : beq_nat _ _ = false |- _ ] => apply beq_nat_false in H
    | [ H : andb _ _ = true |- _ ] => apply Bool.andb_true_iff in H
    | [ H : andb _ _ = false |- _ ] => apply Bool.andb_false_iff in H
    | [ H : orb _ _ = true |- _ ] => apply Bool.orb_prop in H
    | [ H : negb _ = true |- _ ] => apply Bool.negb_true_iff in H
    | [ H : negb _ = false |- _ ] => apply Bool.negb_false_iff in H
    | [ H : PeanoNat.Nat.ltb _ _ = true |- _ ] => apply ltb_true_lt in H
    | [ H : PeanoNat.Nat.ltb _ _ = false |- _ ] => apply ltb_false_le in H
    | [ H : leb _ _ = true |- _ ] => apply leb_true_le in H
    | [ H : leb _ _ = false |- _ ] => apply leb_false_lt in H
    | [ |- andb _ _ = true ]=> apply Bool.andb_true_iff
    | [ |- andb _ _ = false ] => apply Bool.andb_false_iff
    | [ |- leb _ _ = true ] => apply leb_correct
    | [ |-  _ <> false ] => apply Bool.not_false_iff_true
    | [ |- beq_nat _ _ = false ] => apply beq_nat_false_iff
    | [ |- beq_nat _ _ = true ] => apply beq_nat_true_iff
  end.Definition null {A : Type} (xs : list A) : bool :=
  match xs with
    | [] => true
    | _ => false
  end.

(* Specifications: *)
Lemma null_sound : forall A (l : list A), null l = true -> l = [].
Lemma null_false_neq_nil : forall A (l : list A), null l = false -> l <> [].