The Coq Standard Library

# Library Coq.Numbers.Cyclic.DoubleCyclic.DoubleCyclic

Set Implicit Arguments.
Require Import ZArith.
Require Import BigNumPrelude.
Require Import DoubleType.
Require Import DoubleBase.
Require Import DoubleAdd.
Require Import DoubleSub.
Require Import DoubleMul.
Require Import DoubleSqrt.
Require Import DoubleLift.
Require Import DoubleDivn1.
Require Import DoubleDiv.
Require Import CyclicAxioms.
Local Open Scope Z_scope.
Section Z_2nZ.
Context {t : Type}{ops : ZnZ.Ops t}.
Let w_digits := ZnZ.digits.
Let w_zdigits := ZnZ.zdigits.
Let w_to_Z := ZnZ.to_Z.
Let w_of_pos := ZnZ.of_pos.
Let w_head0 := ZnZ.head0.
Let w_tail0 := ZnZ.tail0.
Let w_0 := ZnZ.zero.
Let w_1 := ZnZ.one.
Let w_Bm1 := ZnZ.minus_one.
Let w_compare := ZnZ.compare.
Let w_eq0 := ZnZ.eq0.
Let w_opp_c := ZnZ.opp_c.
Let w_opp := ZnZ.opp.
Let w_opp_carry := ZnZ.opp_carry.
Let w_succ_c := ZnZ.succ_c.
Let w_add_c := ZnZ.add_c.
Let w_add_carry_c := ZnZ.add_carry_c.
Let w_succ := ZnZ.succ.
Let w_add := ZnZ.add.
Let w_add_carry := ZnZ.add_carry.
Let w_pred_c := ZnZ.pred_c.
Let w_sub_c := ZnZ.sub_c.
Let w_sub_carry_c := ZnZ.sub_carry_c.
Let w_pred := ZnZ.pred.
Let w_sub := ZnZ.sub.
Let w_sub_carry := ZnZ.sub_carry.
Let w_mul_c := ZnZ.mul_c.
Let w_mul := ZnZ.mul.
Let w_square_c := ZnZ.square_c.
Let w_div21 := ZnZ.div21.
Let w_div_gt := ZnZ.div_gt.
Let w_div := ZnZ.div.
Let w_mod_gt := ZnZ.modulo_gt.
Let w_mod := ZnZ.modulo.
Let w_gcd_gt := ZnZ.gcd_gt.
Let w_gcd := ZnZ.gcd.
Let w_add_mul_div := ZnZ.add_mul_div.
Let w_pos_mod := ZnZ.pos_mod.
Let w_is_even := ZnZ.is_even.
Let w_sqrt2 := ZnZ.sqrt2.
Let w_sqrt := ZnZ.sqrt.
Let _zn2z := zn2z t.
Let wB := base w_digits.
Let w_Bm2 := w_pred w_Bm1.
Let ww_1 := ww_1 w_0 w_1.
Let ww_Bm1 := ww_Bm1 w_Bm1.
Let w_add2 a b := match w_add_c a b with C0 p => WW w_0 p | C1 p => WW w_1 p end.
Let _ww_digits := xO w_digits.
Let _ww_zdigits := w_add2 w_zdigits w_zdigits.
Let to_Z := zn2z_to_Z wB w_to_Z.
Let w_W0 := ZnZ.WO.
Let w_0W := ZnZ.OW.
Let w_WW := ZnZ.WW.
Let ww_of_pos p :=
match w_of_pos p with
| (N0, l) => (N0, WW w_0 l)
| (Npos ph, l) =>
let (n,h) := w_of_pos ph in (n, w_WW h l)
end.
Let head0 :=
Eval lazy beta delta [ww_head0] in
ww_head0 w_0 w_0W w_compare w_head0 w_add2 w_zdigits _ww_zdigits.
Let tail0 :=
Eval lazy beta delta [ww_tail0] in
ww_tail0 w_0 w_0W w_compare w_tail0 w_add2 w_zdigits _ww_zdigits.
Let ww_WW := Eval lazy beta delta [ww_WW] in (@ww_WW t).
Let ww_0W := Eval lazy beta delta [ww_0W] in (@ww_0W t).
Let ww_W0 := Eval lazy beta delta [ww_W0] in (@ww_W0 t).
Let compare :=
Eval lazy beta delta[ww_compare] in ww_compare w_0 w_compare.
Let eq0 (x:zn2z t) :=
match x with
| W0 => true
| _ => false
end.
Let opp_c :=
Eval lazy beta delta [ww_opp_c] in ww_opp_c w_0 w_opp_c w_opp_carry.
Let opp :=
Eval lazy beta delta [ww_opp] in ww_opp w_0 w_opp_c w_opp_carry w_opp.
Let opp_carry :=
Eval lazy beta delta [ww_opp_carry] in ww_opp_carry w_WW ww_Bm1 w_opp_carry.
Let succ_c :=
Eval lazy beta delta [ww_succ_c] in ww_succ_c w_0 ww_1 w_succ_c.
Let add_c :=
Eval lazy beta delta [ww_add_c] in ww_add_c w_WW w_add_c w_add_carry_c.
Let add_carry_c :=
Eval lazy beta iota delta [ww_add_carry_c ww_succ_c] in
ww_add_carry_c w_0 w_WW ww_1 w_succ_c w_add_c w_add_carry_c.
Let succ :=
Eval lazy beta delta [ww_succ] in ww_succ w_W0 ww_1 w_succ_c w_succ.
Let add :=
Eval lazy beta delta [ww_add] in ww_add w_add_c w_add w_add_carry.
Let add_carry :=
Eval lazy beta iota delta [ww_add_carry ww_succ] in
ww_add_carry w_W0 ww_1 w_succ_c w_add_carry_c w_succ w_add w_add_carry.
Let pred_c :=
Eval lazy beta delta [ww_pred_c] in ww_pred_c w_Bm1 w_WW ww_Bm1 w_pred_c.
Let sub_c :=
Eval lazy beta iota delta [ww_sub_c ww_opp_c] in
ww_sub_c w_0 w_WW w_opp_c w_opp_carry w_sub_c w_sub_carry_c.
Let sub_carry_c :=
Eval lazy beta iota delta [ww_sub_carry_c ww_pred_c ww_opp_carry] in
ww_sub_carry_c w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c w_sub_c w_sub_carry_c.
Let pred :=
Eval lazy beta delta [ww_pred] in ww_pred w_Bm1 w_WW ww_Bm1 w_pred_c w_pred.
Let sub :=
Eval lazy beta iota delta [ww_sub ww_opp] in
ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry.
Let sub_carry :=
Eval lazy beta iota delta [ww_sub_carry ww_pred ww_opp_carry] in
ww_sub_carry w_Bm1 w_WW ww_Bm1 w_opp_carry w_pred_c w_sub_carry_c w_pred
w_sub w_sub_carry.
Let mul_c :=
Eval lazy beta iota delta [ww_mul_c double_mul_c] in
ww_mul_c w_0 w_1 w_WW w_W0 w_mul_c add_c add add_carry.
Let karatsuba_c :=
Eval lazy beta iota delta [ww_karatsuba_c double_mul_c kara_prod] in
ww_karatsuba_c w_0 w_1 w_WW w_W0 w_compare w_add w_sub w_mul_c
add_c add add_carry sub_c sub.
Let mul :=
Eval lazy beta delta [ww_mul] in
ww_mul w_W0 w_add w_mul_c w_mul add.
Let square_c :=
Eval lazy beta delta [ww_square_c] in
ww_square_c w_0 w_1 w_WW w_W0 w_mul_c w_square_c add_c add add_carry.
Let div32 :=
Eval lazy beta iota delta [w_div32] in
w_div32 w_0 w_Bm1 w_Bm2 w_WW w_compare w_add_c w_add_carry_c
w_add w_add_carry w_pred w_sub w_mul_c w_div21 sub_c.
Let div21 :=
Eval lazy beta iota delta [ww_div21] in
ww_div21 w_0 w_0W div32 ww_1 compare sub.
Let low (p: zn2z t) := match p with WW _ p1 => p1 | _ => w_0 end.
Let add_mul_div :=
Eval lazy beta delta [ww_add_mul_div] in
ww_add_mul_div w_0 w_WW w_W0 w_0W compare w_add_mul_div sub w_zdigits low.
Let div_gt :=
Eval lazy beta delta [ww_div_gt] in
ww_div_gt w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp
w_opp_carry w_sub_c w_sub w_sub_carry
w_div_gt w_add_mul_div w_head0 w_div21 div32 _ww_zdigits ww_1 add_mul_div w_zdigits.
Let div :=
Eval lazy beta delta [ww_div] in ww_div ww_1 compare div_gt.
Let mod_gt :=
Eval lazy beta delta [ww_mod_gt] in
ww_mod_gt w_0 w_WW w_0W w_compare w_eq0 w_opp_c w_opp w_opp_carry w_sub_c w_sub w_sub_carry
w_mod_gt w_add_mul_div w_head0 w_div21 div32 _ww_zdigits add_mul_div w_zdigits.
Let mod_ :=
Eval lazy beta delta [ww_mod] in ww_mod compare mod_gt.
Let pos_mod :=
Eval lazy beta delta [ww_pos_mod] in
ww_pos_mod w_0 w_zdigits w_WW w_pos_mod compare w_0W low sub _ww_zdigits.
Let is_even :=
Eval lazy beta delta [ww_is_even] in ww_is_even w_is_even.
Let sqrt2 :=
Eval lazy beta delta [ww_sqrt2] in
ww_sqrt2 w_is_even w_compare w_0 w_1 w_Bm1 w_0W w_sub w_square_c
w_div21 w_add_mul_div w_zdigits w_add_c w_sqrt2 w_pred pred_c
pred add_c add sub_c add_mul_div.
Let sqrt :=
Eval lazy beta delta [ww_sqrt] in
ww_sqrt w_is_even w_0 w_sub w_add_mul_div w_zdigits
_ww_zdigits w_sqrt2 pred add_mul_div head0 compare low.
Let gcd_gt_fix :=
Eval cbv beta delta [ww_gcd_gt_aux ww_gcd_gt_body] in
ww_gcd_gt_aux w_0 w_WW w_0W w_compare w_opp_c w_opp w_opp_carry
w_sub_c w_sub w_sub_carry w_gcd_gt
w_add_mul_div w_head0 w_div21 div32 _ww_zdigits add_mul_div
w_zdigits.
Let gcd_cont :=
Eval lazy beta delta [gcd_cont] in gcd_cont ww_1 w_1 w_compare.
Let gcd_gt :=
Eval lazy beta delta [ww_gcd_gt] in
ww_gcd_gt w_0 w_eq0 w_gcd_gt _ww_digits gcd_gt_fix gcd_cont.
Let gcd :=
Eval lazy beta delta [ww_gcd] in
ww_gcd compare w_0 w_eq0 w_gcd_gt _ww_digits gcd_gt_fix gcd_cont.
Definition lor (x y : zn2z t) :=
match x, y with
| W0, _ => y
| _, W0 => x
| WW hx lx, WW hy ly => WW (ZnZ.lor hx hy) (ZnZ.lor lx ly)
end.
Definition land (x y : zn2z t) :=
match x, y with
| W0, _ => W0
| _, W0 => W0
| WW hx lx, WW hy ly => WW (ZnZ.land hx hy) (ZnZ.land lx ly)
end.
Definition lxor (x y : zn2z t) :=
match x, y with
| W0, _ => y
| _, W0 => x
| WW hx lx, WW hy ly => WW (ZnZ.lxor hx hy) (ZnZ.lxor lx ly)
end.
Global Instance mk_zn2z_ops : ZnZ.Ops (zn2z t) | 1 :=
ZnZ.MkOps _ww_digits _ww_zdigits
to_Z ww_of_pos head0 tail0
W0 ww_1 ww_Bm1
compare eq0
opp_c opp opp_carry
succ_c add_c add_carry_c
succ add add_carry
pred_c sub_c sub_carry_c
pred sub sub_carry
mul_c mul square_c
div21 div_gt div
mod_gt mod_
gcd_gt gcd
add_mul_div
pos_mod
is_even
sqrt2
sqrt
lor
land
lxor.
Global Instance mk_zn2z_ops_karatsuba : ZnZ.Ops (zn2z t) | 2 :=
ZnZ.MkOps _ww_digits _ww_zdigits
to_Z ww_of_pos head0 tail0
W0 ww_1 ww_Bm1
compare eq0
opp_c opp opp_carry
succ_c add_c add_carry_c
succ add add_carry
pred_c sub_c sub_carry_c
pred sub sub_carry
karatsuba_c mul square_c
div21 div_gt div
mod_gt mod_
gcd_gt gcd
add_mul_div
pos_mod
is_even
sqrt2
sqrt
lor
land
lxor.
Context {specs : ZnZ.Specs ops}.
Hint Resolve
ZnZ.spec_to_Z
ZnZ.spec_of_pos
ZnZ.spec_0
ZnZ.spec_1
ZnZ.spec_m1
ZnZ.spec_compare
ZnZ.spec_eq0
ZnZ.spec_opp_c
ZnZ.spec_opp
ZnZ.spec_opp_carry
ZnZ.spec_succ_c
ZnZ.spec_add_c
ZnZ.spec_add_carry_c
ZnZ.spec_succ
ZnZ.spec_add
ZnZ.spec_add_carry
ZnZ.spec_pred_c
ZnZ.spec_sub_c
ZnZ.spec_sub_carry_c
ZnZ.spec_pred
ZnZ.spec_sub
ZnZ.spec_sub_carry
ZnZ.spec_mul_c
ZnZ.spec_mul
ZnZ.spec_square_c
ZnZ.spec_div21
ZnZ.spec_div_gt
ZnZ.spec_div
ZnZ.spec_modulo_gt
ZnZ.spec_modulo
ZnZ.spec_gcd_gt
ZnZ.spec_gcd
ZnZ.spec_head0
ZnZ.spec_tail0
ZnZ.spec_add_mul_div
ZnZ.spec_pos_mod
ZnZ.spec_is_even
ZnZ.spec_sqrt2
ZnZ.spec_sqrt
ZnZ.spec_WO
ZnZ.spec_OW
ZnZ.spec_WW : ZnZ.
Ltac wwauto := unfold ww_to_Z; eauto with ZnZ.
Let wwB := base _ww_digits.
Notation "[| x |]" := (to_Z x) (at level 0, x at level 99).
Notation "[+| c |]" :=
(interp_carry 1 wwB to_Z c) (at level 0, c at level 99).
Notation "[-| c |]" :=
(interp_carry (-1) wwB to_Z c) (at level 0, c at level 99).
Notation "[[ x ]]" := (zn2z_to_Z wwB to_Z x) (at level 0, x at level 99).
Let spec_ww_to_Z : forall x, 0 <= [| x |] < wwB.
Let spec_ww_of_pos : forall p,
Zpos p = (Z.of_N (fst (ww_of_pos p)))*wwB + [|(snd (ww_of_pos p))|].
Let spec_ww_0 : [|W0|] = 0.
Let spec_ww_1 : [|ww_1|] = 1.
Let spec_ww_Bm1 : [|ww_Bm1|] = wwB - 1.
Let spec_ww_compare :
forall x y, compare x y = Z.compare [|x|] [|y|].
Let spec_ww_eq0 : forall x, eq0 x = true -> [|x|] = 0.
Let spec_ww_opp_c : forall x, [-|opp_c x|] = -[|x|].
Let spec_ww_opp : forall x, [|opp x|] = (-[|x|]) mod wwB.
Let spec_ww_opp_carry : forall x, [|opp_carry x|] = wwB - [|x|] - 1.
Let spec_ww_succ_c : forall x, [+|succ_c x|] = [|x|] + 1.
Let spec_ww_add_c : forall x y, [+|add_c x y|] = [|x|] + [|y|].
Let spec_ww_add_carry_c : forall x y, [+|add_carry_c x y|] = [|x|]+[|y|]+1.
Let spec_ww_succ : forall x, [|succ x|] = ([|x|] + 1) mod wwB.
Let spec_ww_add : forall x y, [|add x y|] = ([|x|] + [|y|]) mod wwB.
Let spec_ww_add_carry : forall x y, [|add_carry x y|]=([|x|]+[|y|]+1)mod wwB.
Let spec_ww_pred_c : forall x, [-|pred_c x|] = [|x|] - 1.
Let spec_ww_sub_c : forall x y, [-|sub_c x y|] = [|x|] - [|y|].
Let spec_ww_sub_carry_c : forall x y, [-|sub_carry_c x y|] = [|x|]-[|y|]-1.
Let spec_ww_pred : forall x, [|pred x|] = ([|x|] - 1) mod wwB.
Let spec_ww_sub : forall x y, [|sub x y|] = ([|x|] - [|y|]) mod wwB.
Let spec_ww_sub_carry : forall x y, [|sub_carry x y|]=([|x|]-[|y|]-1) mod wwB.
Let spec_ww_mul_c : forall x y, [[mul_c x y ]] = [|x|] * [|y|].
Let spec_ww_karatsuba_c : forall x y, [[karatsuba_c x y ]] = [|x|] * [|y|].
Let spec_ww_mul : forall x y, [|mul x y|] = ([|x|] * [|y|]) mod wwB.
Let spec_ww_square_c : forall x, [[square_c x]] = [|x|] * [|x|].
Let spec_w_div32 : forall a1 a2 a3 b1 b2,
wB / 2 <= (w_to_Z b1) ->
[|WW a1 a2|] < [|WW b1 b2|] ->
let (q, r) := div32 a1 a2 a3 b1 b2 in
(w_to_Z a1) * wwB + (w_to_Z a2) * wB + (w_to_Z a3) =
(w_to_Z q) * ((w_to_Z b1)*wB + (w_to_Z b2)) + [|r|] /\
0 <= [|r|] < (w_to_Z b1)*wB + w_to_Z b2.
Let spec_ww_div21 : forall a1 a2 b,
wwB/2 <= [|b|] ->
[|a1|] < [|b|] ->
let (q,r) := div21 a1 a2 b in
[|a1|] *wwB+ [|a2|] = [|q|] * [|b|] + [|r|] /\
0 <= [|r|] < [|b|].
Let spec_add2: forall x y,
[|w_add2 x y|] = w_to_Z x + w_to_Z y.
Qed.
Let spec_low: forall x,
w_to_Z (low x) = [|x|] mod wB.
Qed.
Let spec_ww_digits:
[|_ww_zdigits|] = Zpos (xO w_digits).
Let spec_ww_head00 : forall x, [|x|] = 0 -> [|head0 x|] = Zpos _ww_digits.
Let spec_ww_head0 : forall x, 0 < [|x|] ->
wwB/ 2 <= 2 ^ [|head0 x|] * [|x|] < wwB.
Let spec_ww_tail00 : forall x, [|x|] = 0 -> [|tail0 x|] = Zpos _ww_digits.
Let spec_ww_tail0 : forall x, 0 < [|x|] ->
exists y, 0 <= y /\ [|x|] = (2 * y + 1) * 2 ^ [|tail0 x|].
Lemma spec_ww_add_mul_div : forall x y p,
[|p|] <= Zpos _ww_digits ->
[| add_mul_div p x y |] =
([|x|] * (2 ^ [|p|]) +
[|y|] / (2 ^ ((Zpos _ww_digits) - [|p|]))) mod wwB.
Let spec_ww_div_gt : forall a b,
[|a|] > [|b|] -> 0 < [|b|] ->
let (q,r) := div_gt a b in
[|a|] = [|q|] * [|b|] + [|r|] /\ 0 <= [|r|] < [|b|].
Let spec_ww_div : forall a b, 0 < [|b|] ->
let (q,r) := div a b in
[|a|] = [|q|] * [|b|] + [|r|] /\
0 <= [|r|] < [|b|].
Let spec_ww_mod_gt : forall a b,
[|a|] > [|b|] -> 0 < [|b|] ->
[|mod_gt a b|] = [|a|] mod [|b|].
Let spec_ww_mod : forall a b, 0 < [|b|] -> [|mod_ a b|] = [|a|] mod [|b|].
Let spec_ww_gcd_gt : forall a b, [|a|] > [|b|] ->
Zis_gcd [|a|] [|b|] [|gcd_gt a b|].
Let spec_ww_gcd : forall a b, Zis_gcd [|a|] [|b|] [|gcd a b|].
Let spec_ww_is_even : forall x,
match is_even x with
true => [|x|] mod 2 = 0
| false => [|x|] mod 2 = 1
end.
Let spec_ww_sqrt2 : forall x y,
wwB/ 4 <= [|x|] ->
let (s,r) := sqrt2 x y in
[[WW x y]] = [|s|] ^ 2 + [+|r|] /\
[+|r|] <= 2 * [|s|].
Let spec_ww_sqrt : forall x,
[|sqrt x|] ^ 2 <= [|x|] < ([|sqrt x|] + 1) ^ 2.
Let wB_pos : 0 < wB.
Hint Transparent ww_to_Z.
Let ww_testbit_high n x y : Z.pos w_digits <= n ->
Z.testbit [|WW x y|] n =
Z.testbit (ZnZ.to_Z x) (n - Z.pos w_digits).
Let ww_testbit_low n x y : 0 <= n < Z.pos w_digits ->
Z.testbit [|WW x y|] n = Z.testbit (ZnZ.to_Z y) n.
Let spec_lor x y : [|lor x y|] = Z.lor [|x|] [|y|].
Let spec_land x y : [|land x y|] = Z.land [|x|] [|y|].
Let spec_lxor x y : [|lxor x y|] = Z.lxor [|x|] [|y|].
Global Instance mk_zn2z_specs : ZnZ.Specs mk_zn2z_ops.
Global Instance mk_zn2z_specs_karatsuba : ZnZ.Specs mk_zn2z_ops_karatsuba.
End Z_2nZ.
Section MulAdd.
Context {t : Type}{ops : ZnZ.Ops t}{specs : ZnZ.Specs ops}.
Definition mul_add:= w_mul_add ZnZ.zero ZnZ.succ ZnZ.add_c ZnZ.mul_c.
Notation "[| x |]" := (ZnZ.to_Z x) (at level 0, x at level 99).
Notation "[|| x ||]" :=
(zn2z_to_Z (base ZnZ.digits) ZnZ.to_Z x) (at level 0, x at level 99).
Lemma spec_mul_add: forall x y z,
let (zh, zl) := mul_add x y z in
[||WW zh zl||] = [|x|] * [|y|] + [|z|].
End MulAdd.
Modular versions of DoubleCyclic
Module DoubleCyclic (C:CyclicType) <: CyclicType.
Definition t := zn2z C.t.
Instance ops : ZnZ.Ops t := mk_zn2z_ops.
Instance specs : ZnZ.Specs ops := mk_zn2z_specs.
End DoubleCyclic.
Module DoubleCyclicKaratsuba (C:CyclicType) <: CyclicType.
Definition t := zn2z C.t.
Definition ops : ZnZ.Ops t := mk_zn2z_ops_karatsuba.
Definition specs : ZnZ.Specs ops := mk_zn2z_specs_karatsuba.
End DoubleCyclicKaratsuba.