Module Type NBitsProp
(Import A : NAxiomsSig')
(Import B : NSubProp A)
(Import C : NParityProp A B)
(Import D : NPowProp A B C)
(Import E : NDivProp A B)
(Import F : NLog2Prop A B C D).
Include BoolEqualityFacts A.
Ltac order_nz := try apply pow_nonzero; order'.
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 b2n (b:bool) := if b then 1 else 0.
Local Coercion b2n : bool >-> t.
Instance b2n_proper : Proper (Logic.eq ==> eq) b2n.
(** We can compact [testbit_odd_0] [testbit_even_0] [testbit_even_succ] [testbit_odd_succ] in only two lemmas. *)
(** Alternative characterisations of [testbit] *)
(** This concise equation could have been taken as specification for testbit in the interface, but it would have been hard to implement with little initial knowledge about div and mod *)
(** This characterisation that uses only basic operations and power was initially taken as specification for testbit. We describe [a] as having a low part and a high part, with the corresponding bit in the middle. This characterisation is moderatly complex to implement, but also moderately usable... *)
(** Results about the injection [b2n] *)
(** The specification of testbit by low and high parts is complete *)
(** All bits of number 0 are 0 *)
(** Various ways to refer to the lowest bit of a number *)
(** Hence testing a bit is equivalent to shifting and testing parity *)
(** [log2] gives the highest nonzero bit *)
(** Hence the number of bits of [a] is [1+log2 a] (see [Pos.size_nat] and [Pos.size]). *)
(** Testing bits after division or multiplication by a power of two *)
(** Selecting the low part of a number can be done by a modulo *)
(** We now prove that having the same bits implies equality. For that we use a notion of equality over functional streams of bits. *)
Definition eqf (f g:t -> bool) := forall n:t, f n = g n.
Instance eqf_equiv : Equivalence eqf.
Local Infix "===" := eqf (at level 70, no associativity).
Instance testbit_eqf : Proper (eq==>eqf) testbit.
(** Only zero corresponds to the always-false stream. *)
(** If two numbers produce the same stream of bits, they are equal. *)
Hint Rewrite lxor_spec lor_spec land_spec ldiff_spec bits_0 : bitwise.
Ltac bitwise := apply bits_inj; intros ?m; autorewrite with bitwise.
(** The streams of bits that correspond to a natural numbers are exactly the ones that are always 0 after some point *)

Lemma are_bits : forall (f:t->bool), Proper (eq==>Logic.eq) f ->
((exists n, f === testbit n) <->
(exists k, forall m, k<=m -> f m = false)).