Definition eq_bool_dec : forall a b : bool, {a = b} + {a <> b}.Fixpoint le_bool (a b : nat) {struct b} : bool :=
  match a, b with
  | O, _ => true
  | S a1, S b1 => le_bool a1 b1
  | _, _ => false
  end.

Theorem list_length_ind : forall P : list A -> Prop, (forall l1 : list A, (forall l2 : list A, length l2 < length l1 -> P l2) -> P l1) -> forall l : list A, P l.