Module Type NZLog2 (A : NZOrdAxiomsSig)(B : Pow A) := Log2 A <+ NZLog2Spec A B.
(** Derived properties of logarithm *)
Module Type NZLog2Prop
(Import A : NZOrdAxiomsSig')
(Import B : NZPow' A)
(Import C : NZLog2 A B)
(Import D : NZMulOrderProp A)
(Import E : NZPowProp A B D).
(** log2 is always non-negative *)

(* Specifications: *)
Lemma log2_nonneg : forall a, 0 <= log2 a.
Ltac order_pos :=
((apply add_pos_pos || apply add_nonneg_nonneg ||
apply mul_pos_pos || apply mul_nonneg_nonneg ||
apply pow_nonneg || apply pow_pos_nonneg ||
apply log2_nonneg || apply (le_le_succ_r 0));
order_pos) (* in case of success of an apply, we recurse *)
|| order'.  (* otherwise *)
(** A tactic for proving positivity and non-negativity *)
(** The spec of log2 indeed determines it *)
Lemma log2_unique : forall a b, 0<=b -> 2^b<=a<2^(S b) -> log2 a == b.
#[global]
Instance log2_wd : Proper (eq==>eq) log2.