Qed.
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.

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