Library Coq.Numbers.Integer.SpecViaZ.ZSig


Require Import BinInt.

Open Scope Z_scope.

ZSig

Interface of a rich structure about integers. Specifications are written via translation to Z.

Module Type ZType.

 Parameter t : Type.

 Parameter to_Z : t -> Z.

 Definition eq x y := [x] = [y].
 Definition lt x y := [x] < [y].
 Definition le x y := [x] <= [y].

 Parameter of_Z : Z -> t.
 Parameter spec_of_Z: forall x, to_Z (of_Z x) = x.

 Parameter compare : t -> t -> comparison.
 Parameter eqb : t -> t -> bool.
 Parameter ltb : t -> t -> bool.
 Parameter leb : t -> t -> bool.
 Parameter min : t -> t -> t.
 Parameter max : t -> t -> t.
 Parameter zero : t.
 Parameter one : t.
 Parameter two : t.
 Parameter minus_one : t.
 Parameter succ : t -> t.
 Parameter add : t -> t -> t.
 Parameter pred : t -> t.
 Parameter sub : t -> t -> t.
 Parameter opp : t -> t.
 Parameter mul : t -> t -> t.
 Parameter square : t -> t.
 Parameter pow_pos : t -> positive -> t.
 Parameter pow_N : t -> N -> t.
 Parameter pow : t -> t -> t.
 Parameter sqrt : t -> t.
 Parameter log2 : t -> t.
 Parameter div_eucl : t -> t -> t * t.
 Parameter div : t -> t -> t.
 Parameter modulo : t -> t -> t.
 Parameter quot : t -> t -> t.
 Parameter rem : t -> t -> t.
 Parameter gcd : t -> t -> t.
 Parameter sgn : t -> t.
 Parameter abs : t -> t.
 Parameter even : t -> bool.
 Parameter odd : t -> bool.
 Parameter testbit : t -> t -> bool.
 Parameter shiftr : t -> t -> t.
 Parameter shiftl : t -> t -> t.
 Parameter land : t -> t -> t.
 Parameter lor : t -> t -> t.
 Parameter ldiff : t -> t -> t.
 Parameter lxor : t -> t -> t.
 Parameter div2 : t -> t.

 Parameter spec_compare: forall x y, compare x y = ([x] ?= [y]).
 Parameter spec_eqb : forall x y, eqb x y = ([x] =? [y]).
 Parameter spec_ltb : forall x y, ltb x y = ([x] <? [y]).
 Parameter spec_leb : forall x y, leb x y = ([x] <=? [y]).
 Parameter spec_min : forall x y, [min x y] = Z.min [x] [y].
 Parameter spec_max : forall x y, [max x y] = Z.max [x] [y].
 Parameter spec_0: [zero] = 0.
 Parameter spec_1: [one] = 1.
 Parameter spec_2: [two] = 2.
 Parameter spec_m1: [minus_one] = -1.
 Parameter spec_succ: forall n, [succ n] = [n] + 1.
 Parameter spec_add: forall x y, [add x y] = [x] + [y].
 Parameter spec_pred: forall x, [pred x] = [x] - 1.
 Parameter spec_sub: forall x y, [sub x y] = [x] - [y].
 Parameter spec_opp: forall x, [opp x] = - [x].
 Parameter spec_mul: forall x y, [mul x y] = [x] * [y].
 Parameter spec_square: forall x, [square x] = [x] * [x].
 Parameter spec_pow_pos: forall x n, [pow_pos x n] = [x] ^ Zpos n.
 Parameter spec_pow_N: forall x n, [pow_N x n] = [x] ^ Z.of_N n.
 Parameter spec_pow: forall x n, [pow x n] = [x] ^ [n].
 Parameter spec_sqrt: forall x, [sqrt x] = Z.sqrt [x].
 Parameter spec_log2: forall x, [log2 x] = Z.log2 [x].
 Parameter spec_div_eucl: forall x y,
   let (q,r) := div_eucl x y in ([q], [r]) = Z.div_eucl [x] [y].
 Parameter spec_div: forall x y, [div x y] = [x] / [y].
 Parameter spec_modulo: forall x y, [modulo x y] = [x] mod [y].
 Parameter spec_quot: forall x y, [quot x y] = [x] รท [y].
 Parameter spec_rem: forall x y, [rem x y] = Z.rem [x] [y].
 Parameter spec_gcd: forall a b, [gcd a b] = Z.gcd [a] [b].
 Parameter spec_sgn : forall x, [sgn x] = Z.sgn [x].
 Parameter spec_abs : forall x, [abs x] = Z.abs [x].
 Parameter spec_even : forall x, even x = Z.even [x].
 Parameter spec_odd : forall x, odd x = Z.odd [x].
 Parameter spec_testbit: forall x p, testbit x p = Z.testbit [x] [p].
 Parameter spec_shiftr: forall x p, [shiftr x p] = Z.shiftr [x] [p].
 Parameter spec_shiftl: forall x p, [shiftl x p] = Z.shiftl [x] [p].
 Parameter spec_land: forall x y, [land x y] = Z.land [x] [y].
 Parameter spec_lor: forall x y, [lor x y] = Z.lor [x] [y].
 Parameter spec_ldiff: forall x y, [ldiff x y] = Z.ldiff [x] [y].
 Parameter spec_lxor: forall x y, [lxor x y] = Z.lxor [x] [y].
 Parameter spec_div2: forall x, [div2 x] = Z.div2 [x].

End ZType.

Module Type ZType_Notation (Import Z:ZType).
 Notation "[ x ]" := (to_Z x).
 Infix "==" := eq (at level 70).
 Notation "0" := zero.
 Notation "1" := one.
 Notation "2" := two.
 Infix "+" := add.
 Infix "-" := sub.
 Infix "*" := mul.
 Infix "^" := pow.
 Notation "- x" := (opp x).
 Infix "<=" := le.
 Infix "<" := lt.
End ZType_Notation.

Module Type ZType' := ZType <+ ZType_Notation.