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 fold_left_map : forall (C : Type) a l (k : C -> B), fold_left f (map k l) a = fold_left (fun a b => f a (k b)) l a.