Module Type NZLog2 (A : NZOrdAxiomsSig)(B : Pow A) := Log2 A <+ NZLog2Spec A B.(** * [log2_up] : a binary logarithm that rounds up instead of down *)

(** For once, we define instead of axiomatizing, thanks to log2 *)

Definition log2_up a :=
 match compare 1 a with
  | Lt => S (log2 (P a))
  | _ => 0
 end.

(* Specifications: *)
Lemma log2_up_eqn0 : forall a, a<=1 -> log2_up a == 0.
Lemma log2_up_eqn : forall a, 1<a -> log2_up a == S (log2 (P a)).
Lemma log2_up_spec : forall a, 1<a -> 2^(P (log2_up a)) < a <= 2^(log2_up a).
Lemma log2_up_nonpos : forall a, a<=0 -> log2_up a == 0.