Module Type ZBitsProp
(Import A : ZAxiomsSig')
(Import B : ZMulOrderProp A)
(Import C : ZParityProp A B)
(Import D : ZSgnAbsProp A B)
(Import E : ZPowProp A B C D)
(Import F : ZDivProp A B D)
(Import G : NZLog2Prop A A A B E).
Include BoolEqualityFacts A.
Ltac order_nz := try apply pow_nonzero; order'.
Ltac order_pos' := try apply abs_nonneg; order_pos.
Hint Rewrite div_0_l mod_0_l div_1_r mod_1_r : nz.
(** Some properties of power and division *)
(** An injection from bits [true] and [false] to numbers 1 and 0. We declare it as a (local) coercion for shorter statements. *)
Definition b2z (b:bool) := if b then 1 else 0.
Local Coercion b2z : bool >-> t.
Instance b2z_wd : Proper (Logic.eq ==> eq) b2z := _.

Lemma exists_div2 a : exists a' (b:bool), a == 2*a' + b.