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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(*            Benjamin Gregoire, Laurent Thery, INRIA, 2007             *)
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(** * Signature and specification of bounded integers *)
(** This file specifies [d]-bit integers as [Z/nZ] with [n=2^d] *)
Set Implicit Arguments.
Local Open Scope Z_scope.
(** First, a description via an operator record and a spec record. *)
Module ZnZ.
#[universes(template)]
Class Ops (t:Type) := MkOps {
(* Conversion functions with Z *)
digits : positive;
zdigits: t;
to_Z   : t -> Z;
of_pos : positive -> N * t; (* Euclidean division by [2^digits] *)
head0  : t -> t; (* number of digits 0 in front of the number *)
tail0  : t -> t; (* number of digits 0 at the bottom of the number *)
(* Basic numbers *)
zero : t;
one  : t;
minus_one : t;  (* [2^digits-1], which is equivalent to [-1] *)
(* Comparison *)
compare     : t -> t -> comparison;
eq0         : t -> bool;
(* Basic arithmetic operations *)
opp_c       : t -> carry t;
opp         : t -> t;
opp_carry   : t -> t; (* the carry is known to be -1 *)
succ_c      : t -> carry t;
add_c       : t -> t -> carry t;
add_carry_c : t -> t -> carry t;
succ        : t -> t;
add         : t -> t -> t;
add_carry   : t -> t -> t;
pred_c      : t -> carry t;
sub_c       : t -> t -> carry t;
sub_carry_c : t -> t -> carry t;
pred        : t -> t;
sub         : t -> t -> t;
sub_carry   : t -> t -> t;
mul_c       : t -> t -> zn2z t;
mul         : t -> t -> t;
square_c    : t -> zn2z t;
(* Special divisions operations *)
div21       : t -> t -> t -> t*t;
div_gt      : t -> t -> t * t; (* specialized version of [div] *)
div         : t -> t -> t * t;
modulo_gt   : t -> t -> t; (* specialized version of [mod] *)
modulo      : t -> t -> t;
gcd_gt      : t -> t -> t; (* specialized version of [gcd] *)
gcd         : t -> t -> t;
(* [add_mul_div p i j] is a combination of the [(digits-p)] low bits of [i] above the [p] high bits of [j]: [add_mul_div p i j = i*2^p+j/2^(digits-p)] *)
add_mul_div : t -> t -> t -> t;
(* [pos_mod p i] is [i mod 2^p] *)
pos_mod     : t -> t -> t;
is_even     : t -> bool;
(* square root *)
sqrt2       : t -> t -> t * carry t;
sqrt        : t -> t;
(* bitwise operations *)
lor         : t -> t -> t;
land        : t -> t -> t;
lxor        : t -> t -> t }.
Section Specs.
Context {t : Set}{ops : Ops t}.
Notation "[| x |]" := (to_Z x)  (at level 0, x at level 99).
Let wB := base digits.
Notation "[+| c |]" :=
(interp_carry 1 wB to_Z c)  (at level 0, c at level 99).
Notation "[-| c |]" :=
(interp_carry (-1) wB to_Z c)  (at level 0, c at level 99).
Notation "[|| x ||]" :=
(zn2z_to_Z wB to_Z x)  (at level 0, x at level 99).
Class Specs := MkSpecs {
(* Conversion functions with Z *)
spec_to_Z   : forall x, 0 <= [| x |] < wB;
spec_of_pos : forall p,
Zpos p = (Z.of_N (fst (of_pos p)))*wB + [|(snd (of_pos p))|];
spec_zdigits : [| zdigits |] = Zpos digits;
spec_more_than_1_digit: 1 < Zpos digits;
(* Basic numbers *)
spec_0   : [|zero|] = 0;
spec_1   : [|one|] = 1;
spec_m1 : [|minus_one|] = wB - 1;
(* Comparison *)
spec_compare : forall x y, compare x y = ([|x|] ?= [|y|]);
(* NB: the spec of [eq0] is deliberately partial, see DoubleCyclic where [eq0 x = true <-> x = W0] *)
spec_eq0 : forall x, eq0 x = true -> [|x|] = 0;
(* Basic arithmetic operations *)
spec_opp_c : forall x, [-|opp_c x|] = -[|x|];
spec_opp : forall x, [|opp x|] = (-[|x|]) mod wB;
spec_opp_carry : forall x, [|opp_carry x|] = wB - [|x|] - 1;
spec_succ_c : forall x, [+|succ_c x|] = [|x|] + 1;
spec_add_c  : forall x y, [+|add_c x y|] = [|x|] + [|y|];
spec_add_carry_c : forall x y, [+|add_carry_c x y|] = [|x|] + [|y|] + 1;
spec_succ : forall x, [|succ x|] = ([|x|] + 1) mod wB;
spec_add : forall x y, [|add x y|] = ([|x|] + [|y|]) mod wB;
spec_add_carry :
forall x y, [|add_carry x y|] = ([|x|] + [|y|] + 1) mod wB;
spec_pred_c : forall x, [-|pred_c x|] = [|x|] - 1;
spec_sub_c : forall x y, [-|sub_c x y|] = [|x|] - [|y|];
spec_sub_carry_c : forall x y, [-|sub_carry_c x y|] = [|x|] - [|y|] - 1;
spec_pred : forall x, [|pred x|] = ([|x|] - 1) mod wB;
spec_sub : forall x y, [|sub x y|] = ([|x|] - [|y|]) mod wB;
spec_sub_carry :
forall x y, [|sub_carry x y|] = ([|x|] - [|y|] - 1) mod wB;
spec_mul_c : forall x y, [|| mul_c x y ||] = [|x|] * [|y|];
spec_mul : forall x y, [|mul x y|] = ([|x|] * [|y|]) mod wB;
spec_square_c : forall x, [|| square_c x||] = [|x|] * [|x|];
(* Special divisions operations *)
spec_div21 : forall a1 a2 b,
wB/2 <= [|b|] ->
[|a1|] < [|b|] ->
let (q,r) := div21 a1 a2 b in
[|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\
0 <= [|r|] < [|b|];
spec_div_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] ->
let (q,r) := div_gt a b in
[|a|] = [|q|] * [|b|] + [|r|] /\
0 <= [|r|] < [|b|];
spec_div : forall a b, 0 < [|b|] ->
let (q,r) := div a b in
[|a|] = [|q|] * [|b|] + [|r|] /\
0 <= [|r|] < [|b|];
spec_modulo_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] ->
[|modulo_gt a b|] = [|a|] mod [|b|];
spec_modulo :  forall a b, 0 < [|b|] ->
[|modulo a b|] = [|a|] mod [|b|];
spec_gcd_gt : forall a b, [|a|] > [|b|] ->
Zis_gcd [|a|] [|b|] [|gcd_gt a b|];
spec_gcd : forall a b, Zis_gcd [|a|] [|b|] [|gcd a b|];
(* shift operations *)
spec_head00:  forall x, [|x|] = 0 -> [|head0 x|] = Zpos digits;
spec_head0  : forall x,  0 < [|x|] ->
wB/ 2 <= 2 ^ ([|head0 x|]) * [|x|] < wB;
spec_tail00:  forall x, [|x|] = 0 -> [|tail0 x|] = Zpos digits;
spec_tail0  : forall x, 0 < [|x|] ->
exists y, 0 <= y /\ [|x|] = (2 * y + 1) * (2 ^ [|tail0 x|]) ;
spec_add_mul_div : forall x y p,
[|p|] <= Zpos digits ->
[| add_mul_div p x y |] =
([|x|] * (2 ^ [|p|]) +
[|y|] / (2 ^ ((Zpos digits) - [|p|]))) mod wB;
spec_pos_mod : forall w p,
[|pos_mod p w|] = [|w|] mod (2 ^ [|p|]);
(* sqrt *)
spec_is_even : forall x,
if is_even x then [|x|] mod 2 = 0 else [|x|] mod 2 = 1;
spec_sqrt2 : forall x y,
wB/ 4 <= [|x|] ->
let (s,r) := sqrt2 x y in
[||WW x y||] = [|s|] ^ 2 + [+|r|] /\
[+|r|] <= 2 * [|s|];
spec_sqrt : forall x,
[|sqrt x|] ^ 2 <= [|x|] < ([|sqrt x|] + 1) ^ 2;
spec_lor : forall x y, [|lor x y|] = Z.lor [|x|] [|y|];
spec_land : forall x y, [|land x y|] = Z.land [|x|] [|y|];
spec_lxor : forall x y, [|lxor x y|] = Z.lxor [|x|] [|y|]
}.
End Specs.
Arguments Specs {t} ops.
(** Generic construction of double words *)
Section WW.
Context {t : Set}{ops : Ops t}{specs : Specs ops}.
Let wB := base digits.
Definition WO' (eq0:t->bool) zero h :=
if eq0 h then W0 else WW h zero.
Definition WO := Eval lazy beta delta [WO'] in
let eq0 := ZnZ.eq0 in
let zero := ZnZ.zero in
WO' eq0 zero.
Definition OW' (eq0:t->bool) zero l :=
if eq0 l then W0 else WW zero l.
Definition OW := Eval lazy beta delta [OW'] in
let eq0 := ZnZ.eq0 in
let zero := ZnZ.zero in
OW' eq0 zero.
Definition WW'  (eq0:t->bool) zero h l :=
if eq0 h then OW' eq0 zero l else WW h l.
Definition WW := Eval lazy beta delta [WW' OW'] in
let eq0 := ZnZ.eq0 in
let zero := ZnZ.zero in
WW' eq0 zero.

(* Specifications: *)
Lemma spec_WO : forall h,
zn2z_to_Z wB to_Z (WO h) = (to_Z h)*wB.
Lemma spec_OW : forall l,
zn2z_to_Z wB to_Z (OW l) = to_Z l.
Lemma spec_WW : forall h l,
zn2z_to_Z wB to_Z (WW h l) = (to_Z h)*wB + to_Z l.