(* for good measure, we will define exponentiation for nat and prove that lognat is correct *)
Fixpoint expnat(a e : nat) : nat :=
  match e with
    | 0 => 1
    | (S e') => a * (expnat a e')
  end.

Theorem expnat_2_monotonic : forall e1 e2, e1 <= e2 -> (expnat 2 e1 <= expnat 2 e2).