Module NZLog2UpProp
(Import A : NZDecOrdAxiomsSig')
(Import B : NZPow' A)
(Import C : NZLog2 A B)
(Import D : NZMulOrderProp A)
(Import E : NZPowProp A B D)
(Import F : NZLog2Prop A B C D E).
(** * [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.
Instance log2_up_wd : Proper (eq==>eq) log2_up.
(** [log2_up] is always non-negative *)
(** The spec of [log2_up] indeed determines it *)
(** [log2_up] is exact on powers of 2 *)
(** [log2_up] and successors of powers of 2 *)

Lemma log2_up_succ_pow2 : forall a, 0<=a -> log2_up (S (2^a)) == S a.