The Coq Standard Library

# Library Coq.Numbers.Cyclic.DoubleCyclic.DoubleDiv

Set Implicit Arguments.
Require Import ZArith.
Require Import BigNumPrelude.
Require Import DoubleType.
Require Import DoubleBase.
Require Import DoubleDivn1.
Require Import DoubleAdd.
Require Import DoubleSub.
Local Open Scope Z_scope.
Ltac zarith := auto with zarith.
Section POS_MOD.
Variable w:Type.
Variable w_0 : w.
Variable w_digits : positive.
Variable w_zdigits : w.
Variable w_WW : w -> w -> zn2z w.
Variable w_pos_mod : w -> w -> w.
Variable w_compare : w -> w -> comparison.
Variable ww_compare : zn2z w -> zn2z w -> comparison.
Variable w_0W : w -> zn2z w.
Variable low: zn2z w -> w.
Variable ww_sub: zn2z w -> zn2z w -> zn2z w.
Variable ww_zdigits : zn2z w.
Definition ww_pos_mod p x :=
let zdigits := w_0W w_zdigits in
match x with
| W0 => W0
| WW xh xl =>
match ww_compare p zdigits with
| Eq => w_WW w_0 xl
| Lt => w_WW w_0 (w_pos_mod (low p) xl)
| Gt =>
match ww_compare p ww_zdigits with
| Lt =>
let n := low (ww_sub p zdigits) in
w_WW (w_pos_mod n xh) xl
| _ => x
end
end
end.
Variable w_to_Z : w -> Z.
Notation wB := (base w_digits).
Notation wwB := (base (ww_digits w_digits)).
Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99).
Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
Variable spec_w_0 : [|w_0|] = 0.
Variable spec_to_Z : forall x, 0 <= [|x|] < wB.
Variable spec_to_w_Z : forall x, 0 <= [[x]] < wwB.
Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
Variable spec_pos_mod : forall w p,
[|w_pos_mod p w|] = [|w|] mod (2 ^ [|p|]).
Variable spec_w_0W : forall l, [[w_0W l]] = [|l|].
Variable spec_ww_compare : forall x y,
ww_compare x y = Z.compare [[x]] [[y]].
Variable spec_ww_sub: forall x y,
[[ww_sub x y]] = ([[x]] - [[y]]) mod wwB.
Variable spec_zdigits : [| w_zdigits |] = Zpos w_digits.
Variable spec_low: forall x, [| low x|] = [[x]] mod wB.
Variable spec_ww_zdigits : [[ww_zdigits]] = 2 * [|w_zdigits|].
Variable spec_ww_digits : ww_digits w_digits = xO w_digits.
Hint Rewrite spec_w_0 spec_w_WW : w_rewrite.
Lemma spec_ww_pos_mod : forall w p,
[[ww_pos_mod p w]] = [[w]] mod (2 ^ [[p]]).
End POS_MOD.
Section DoubleDiv32.
Variable w : Type.
Variable w_0 : w.
Variable w_Bm1 : w.
Variable w_Bm2 : w.
Variable w_WW : w -> w -> zn2z w.
Variable w_compare : w -> w -> comparison.
Variable w_add_c : w -> w -> carry w.
Variable w_add_carry_c : w -> w -> carry w.
Variable w_add : w -> w -> w.
Variable w_add_carry : w -> w -> w.
Variable w_pred : w -> w.
Variable w_sub : w -> w -> w.
Variable w_mul_c : w -> w -> zn2z w.
Variable w_div21 : w -> w -> w -> w*w.
Variable ww_sub_c : zn2z w -> zn2z w -> carry (zn2z w).
Definition w_div32_body a1 a2 a3 b1 b2 :=
match w_compare a1 b1 with
| Lt =>
let (q,r) := w_div21 a1 a2 b1 in
match ww_sub_c (w_WW r a3) (w_mul_c q b2) with
| C0 r1 => (q, r1)
| C1 r1 =>
let q := w_pred q in
ww_add_c_cont w_WW w_add_c w_add_carry_c
(fun r2=>(w_pred q, ww_add w_add_c w_add w_add_carry r2 (WW b1 b2)))
(fun r2 => (q, r2))
r1 (WW b1 b2)
end
| Eq =>
ww_add_c_cont w_WW w_add_c w_add_carry_c
(fun r => (w_Bm2, ww_add w_add_c w_add w_add_carry r (WW b1 b2)))
(fun r => (w_Bm1, r))
(WW (w_sub a2 b2) a3) (WW b1 b2)
| Gt => (w_0, W0)
end.
Definition w_div32 a1 a2 a3 b1 b2 :=
Eval lazy beta iota delta [ww_add_c_cont ww_add w_div32_body] in
w_div32_body a1 a2 a3 b1 b2.
Variable w_digits : positive.
Variable w_to_Z : w -> Z.
Notation wB := (base w_digits).
Notation wwB := (base (ww_digits w_digits)).
Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99).
Notation "[+| c |]" :=
(interp_carry 1 wB w_to_Z c) (at level 0, c at level 99).
Notation "[-| c |]" :=
(interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99).
Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
Notation "[-[ c ]]" :=
(interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c)
(at level 0, c at level 99).
Variable spec_w_0 : [|w_0|] = 0.
Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1.
Variable spec_w_Bm2 : [|w_Bm2|] = wB - 2.
Variable spec_to_Z : forall x, 0 <= [|x|] < wB.
Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
Variable spec_compare :
forall x y, w_compare x y = Z.compare [|x|] [|y|].
Variable spec_w_add_c : forall x y, [+|w_add_c x y|] = [|x|] + [|y|].
Variable spec_w_add_carry_c :
forall x y, [+|w_add_carry_c x y|] = [|x|] + [|y|] + 1.
Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB.
Variable spec_w_add_carry :
forall x y, [|w_add_carry x y|] = ([|x|] + [|y|] + 1) mod wB.
Variable spec_pred : forall x, [|w_pred x|] = ([|x|] - 1) mod wB.
Variable spec_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB.
Variable spec_mul_c : forall x y, [[ w_mul_c x y ]] = [|x|] * [|y|].
Variable spec_div21 : forall a1 a2 b,
wB/2 <= [|b|] ->
[|a1|] < [|b|] ->
let (q,r) := w_div21 a1 a2 b in
[|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\
0 <= [|r|] < [|b|].
Variable spec_ww_sub_c : forall x y, [-[ww_sub_c x y]] = [[x]] - [[y]].
Ltac Spec_w_to_Z x :=
let H:= fresh "HH" in
assert (H:= spec_to_Z x).
Ltac Spec_ww_to_Z x :=
let H:= fresh "HH" in
assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x).
Theorem wB_div2: forall x, wB/2 <= x -> wB <= 2 * x.
Lemma Zmult_lt_0_reg_r_2 : forall n m : Z, 0 <= n -> 0 < m * n -> 0 < m.
Theorem spec_w_div32 : forall a1 a2 a3 b1 b2,
wB/2 <= [|b1|] ->
[[WW a1 a2]] < [[WW b1 b2]] ->
let (q,r) := w_div32 a1 a2 a3 b1 b2 in
[|a1|] * wwB + [|a2|] * wB + [|a3|] =
[|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\
0 <= [[r]] < [|b1|] * wB + [|b2|].
End DoubleDiv32.
Section DoubleDiv21.
Variable w : Type.
Variable w_0 : w.
Variable w_0W : w -> zn2z w.
Variable w_div32 : w -> w -> w -> w -> w -> w * zn2z w.
Variable ww_1 : zn2z w.
Variable ww_compare : zn2z w -> zn2z w -> comparison.
Variable ww_sub : zn2z w -> zn2z w -> zn2z w.
Definition ww_div21 a1 a2 b :=
match a1 with
| W0 =>
match ww_compare a2 b with
| Gt => (ww_1, ww_sub a2 b)
| Eq => (ww_1, W0)
| Lt => (W0, a2)
end
| WW a1h a1l =>
match a2 with
| W0 =>
match b with
| W0 => (W0, W0)
| WW b1 b2 =>
let (q1, r) := w_div32 a1h a1l w_0 b1 b2 in
match r with
| W0 => (WW q1 w_0, W0)
| WW r1 r2 =>
let (q2, s) := w_div32 r1 r2 w_0 b1 b2 in
(WW q1 q2, s)
end
end
| WW a2h a2l =>
match b with
| W0 => (W0, W0)
| WW b1 b2 =>
let (q1, r) := w_div32 a1h a1l a2h b1 b2 in
match r with
| W0 => (WW q1 w_0, w_0W a2l)
| WW r1 r2 =>
let (q2, s) := w_div32 r1 r2 a2l b1 b2 in
(WW q1 q2, s)
end
end
end
end.
Variable w_digits : positive.
Variable w_to_Z : w -> Z.
Notation wB := (base w_digits).
Notation wwB := (base (ww_digits w_digits)).
Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99).
Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
Notation "[-[ c ]]" :=
(interp_carry (-1) wwB (ww_to_Z w_digits w_to_Z) c)
(at level 0, c at level 99).
Variable spec_w_0 : [|w_0|] = 0.
Variable spec_to_Z : forall x, 0 <= [|x|] < wB.
Variable spec_w_0W : forall l, [[w_0W l]] = [|l|].
Variable spec_w_div32 : forall a1 a2 a3 b1 b2,
wB/2 <= [|b1|] ->
[[WW a1 a2]] < [[WW b1 b2]] ->
let (q,r) := w_div32 a1 a2 a3 b1 b2 in
[|a1|] * wwB + [|a2|] * wB + [|a3|] =
[|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\
0 <= [[r]] < [|b1|] * wB + [|b2|].
Variable spec_ww_1 : [[ww_1]] = 1.
Variable spec_ww_compare : forall x y,
ww_compare x y = Z.compare [[x]] [[y]].
Variable spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB.
Theorem wwB_div: wwB = 2 * (wwB / 2).
Ltac Spec_w_to_Z x :=
let H:= fresh "HH" in
assert (H:= spec_to_Z x).
Ltac Spec_ww_to_Z x :=
let H:= fresh "HH" in
assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x).
Theorem spec_ww_div21 : forall a1 a2 b,
wwB/2 <= [[b]] ->
[[a1]] < [[b]] ->
let (q,r) := ww_div21 a1 a2 b in
[[a1]] *wwB+[[a2]] = [[q]] * [[b]] + [[r]] /\ 0 <= [[r]] < [[b]].
End DoubleDiv21.
Section DoubleDivGt.
Variable w : Type.
Variable w_digits : positive.
Variable w_0 : w.
Variable w_WW : w -> w -> zn2z w.
Variable w_0W : w -> zn2z w.
Variable w_compare : w -> w -> comparison.
Variable w_eq0 : w -> bool.
Variable w_opp_c : w -> carry w.
Variable w_opp w_opp_carry : w -> w.
Variable w_sub_c : w -> w -> carry w.
Variable w_sub w_sub_carry : w -> w -> w.
Variable w_div_gt : w -> w -> w*w.
Variable w_mod_gt : w -> w -> w.
Variable w_gcd_gt : w -> w -> w.
Variable w_add_mul_div : w -> w -> w -> w.
Variable w_head0 : w -> w.
Variable w_div21 : w -> w -> w -> w * w.
Variable w_div32 : w -> w -> w -> w -> w -> w * zn2z w.
Variable _ww_zdigits : zn2z w.
Variable ww_1 : zn2z w.
Variable ww_add_mul_div : zn2z w -> zn2z w -> zn2z w -> zn2z w.
Variable w_zdigits : w.
Definition ww_div_gt_aux ah al bh bl :=
Eval lazy beta iota delta [ww_sub ww_opp] in
let p := w_head0 bh in
match w_compare p w_0 with
| Gt =>
let b1 := w_add_mul_div p bh bl in
let b2 := w_add_mul_div p bl w_0 in
let a1 := w_add_mul_div p w_0 ah in
let a2 := w_add_mul_div p ah al in
let a3 := w_add_mul_div p al w_0 in
let (q,r) := w_div32 a1 a2 a3 b1 b2 in
(WW w_0 q, ww_add_mul_div
(ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c
w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r)
| _ => (ww_1, ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c
w_opp w_sub w_sub_carry (WW ah al) (WW bh bl))
end.
Definition ww_div_gt a b :=
Eval lazy beta iota delta [ww_div_gt_aux double_divn1
double_divn1_p double_divn1_p_aux double_divn1_0 double_divn1_0_aux
double_split double_0 double_WW] in
match a, b with
| W0, _ => (W0, W0)
| _, W0 => (W0, W0)
| WW ah al, WW bh bl =>
if w_eq0 ah then
let (q,r) := w_div_gt al bl in
(WW w_0 q, w_0W r)
else
match w_compare w_0 bh with
| Eq =>
let(q,r):=
double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21
w_compare w_sub 1 a bl in
(q, w_0W r)
| Lt => ww_div_gt_aux ah al bh bl
| Gt => (W0, W0)
end
end.
Definition ww_mod_gt_aux ah al bh bl :=
Eval lazy beta iota delta [ww_sub ww_opp] in
let p := w_head0 bh in
match w_compare p w_0 with
| Gt =>
let b1 := w_add_mul_div p bh bl in
let b2 := w_add_mul_div p bl w_0 in
let a1 := w_add_mul_div p w_0 ah in
let a2 := w_add_mul_div p ah al in
let a3 := w_add_mul_div p al w_0 in
let (q,r) := w_div32 a1 a2 a3 b1 b2 in
ww_add_mul_div (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c
w_opp w_sub w_sub_carry _ww_zdigits (w_0W p)) W0 r
| _ =>
ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c
w_opp w_sub w_sub_carry (WW ah al) (WW bh bl)
end.
Definition ww_mod_gt a b :=
Eval lazy beta iota delta [ww_mod_gt_aux double_modn1
double_modn1_p double_modn1_p_aux double_modn1_0 double_modn1_0_aux
double_split double_0 double_WW snd] in
match a, b with
| W0, _ => W0
| _, W0 => W0
| WW ah al, WW bh bl =>
if w_eq0 ah then w_0W (w_mod_gt al bl)
else
match w_compare w_0 bh with
| Eq =>
w_0W (double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21
w_compare w_sub 1 a bl)
| Lt => ww_mod_gt_aux ah al bh bl
| Gt => W0
end
end.
Definition ww_gcd_gt_body (cont: w->w->w->w->zn2z w) (ah al bh bl: w) :=
Eval lazy beta iota delta [ww_mod_gt_aux double_modn1
double_modn1_p double_modn1_p_aux double_modn1_0 double_modn1_0_aux
double_split double_0 double_WW snd] in
match w_compare w_0 bh with
| Eq =>
match w_compare w_0 bl with
| Eq => WW ah al
| Lt =>
let m := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21
w_compare w_sub 1 (WW ah al) bl in
WW w_0 (w_gcd_gt bl m)
| Gt => W0
end
| Lt =>
let m := ww_mod_gt_aux ah al bh bl in
match m with
| W0 => WW bh bl
| WW mh ml =>
match w_compare w_0 mh with
| Eq =>
match w_compare w_0 ml with
| Eq => WW bh bl
| _ =>
let r := double_modn1 w_zdigits w_0 w_head0 w_add_mul_div w_div21
w_compare w_sub 1 (WW bh bl) ml in
WW w_0 (w_gcd_gt ml r)
end
| Lt =>
let r := ww_mod_gt_aux bh bl mh ml in
match r with
| W0 => m
| WW rh rl => cont mh ml rh rl
end
| Gt => W0
end
end
| Gt => W0
end.
Fixpoint ww_gcd_gt_aux
(p:positive) (cont: w -> w -> w -> w -> zn2z w) (ah al bh bl : w)
{struct p} : zn2z w :=
ww_gcd_gt_body
(fun mh ml rh rl => match p with
| xH => cont mh ml rh rl
| xO p => ww_gcd_gt_aux p (ww_gcd_gt_aux p cont) mh ml rh rl
| xI p => ww_gcd_gt_aux p (ww_gcd_gt_aux p cont) mh ml rh rl
end) ah al bh bl.
Variable w_to_Z : w -> Z.
Notation wB := (base w_digits).
Notation wwB := (base (ww_digits w_digits)).
Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99).
Notation "[-| c |]" :=
(interp_carry (-1) wB w_to_Z c) (at level 0, c at level 99).
Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
Variable spec_w_0 : [|w_0|] = 0.
Variable spec_to_Z : forall x, 0 <= [|x|] < wB.
Variable spec_to_w_Z : forall x, 0 <= [[x]] < wwB.
Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|].
Variable spec_w_0W : forall l, [[w_0W l]] = [|l|].
Variable spec_compare :
forall x y, w_compare x y = Z.compare [|x|] [|y|].
Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0.
Variable spec_opp_c : forall x, [-|w_opp_c x|] = -[|x|].
Variable spec_opp : forall x, [|w_opp x|] = (-[|x|]) mod wB.
Variable spec_opp_carry : forall x, [|w_opp_carry x|] = wB - [|x|] - 1.
Variable spec_sub_c : forall x y, [-|w_sub_c x y|] = [|x|] - [|y|].
Variable spec_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB.
Variable spec_sub_carry :
forall x y, [|w_sub_carry x y|] = ([|x|] - [|y|] - 1) mod wB.
Variable spec_div_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] ->
let (q,r) := w_div_gt a b in
[|a|] = [|q|] * [|b|] + [|r|] /\
0 <= [|r|] < [|b|].
Variable spec_mod_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] ->
[|w_mod_gt a b|] = [|a|] mod [|b|].
Variable spec_gcd_gt : forall a b, [|a|] > [|b|] ->
Zis_gcd [|a|] [|b|] [|w_gcd_gt a b|].
Variable spec_add_mul_div : forall x y p,
[|p|] <= Zpos w_digits ->
[| w_add_mul_div p x y |] =
([|x|] * (2 ^ ([|p|])) +
[|y|] / (2 ^ ((Zpos w_digits) - [|p|]))) mod wB.
Variable spec_head0 : forall x, 0 < [|x|] ->
wB/ 2 <= 2 ^ [|w_head0 x|] * [|x|] < wB.
Variable spec_div21 : forall a1 a2 b,
wB/2 <= [|b|] ->
[|a1|] < [|b|] ->
let (q,r) := w_div21 a1 a2 b in
[|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\
0 <= [|r|] < [|b|].
Variable spec_w_div32 : forall a1 a2 a3 b1 b2,
wB/2 <= [|b1|] ->
[[WW a1 a2]] < [[WW b1 b2]] ->
let (q,r) := w_div32 a1 a2 a3 b1 b2 in
[|a1|] * wwB + [|a2|] * wB + [|a3|] =
[|q|] * ([|b1|] * wB + [|b2|]) + [[r]] /\
0 <= [[r]] < [|b1|] * wB + [|b2|].
Variable spec_w_zdigits: [|w_zdigits|] = Zpos w_digits.
Variable spec_ww_digits_ : [[_ww_zdigits]] = Zpos (xO w_digits).
Variable spec_ww_1 : [[ww_1]] = 1.
Variable spec_ww_add_mul_div : forall x y p,
[[p]] <= Zpos (xO w_digits) ->
[[ ww_add_mul_div p x y ]] =
([[x]] * (2^[[p]]) +
[[y]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB.
Ltac Spec_w_to_Z x :=
let H:= fresh "HH" in
assert (H:= spec_to_Z x).
Ltac Spec_ww_to_Z x :=
let H:= fresh "HH" in
assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x).
Lemma to_Z_div_minus_p : forall x p,
0 < [|p|] < Zpos w_digits ->
0 <= [|x|] / 2 ^ (Zpos w_digits - [|p|]) < 2 ^ [|p|].
Hint Resolve to_Z_div_minus_p : zarith.
Lemma spec_ww_div_gt_aux : forall ah al bh bl,
[[WW ah al]] > [[WW bh bl]] ->
0 < [|bh|] ->
let (q,r) := ww_div_gt_aux ah al bh bl in
[[WW ah al]] = [[q]] * [[WW bh bl]] + [[r]] /\
0 <= [[r]] < [[WW bh bl]].
Lemma spec_ww_div_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] ->
let (q,r) := ww_div_gt a b in
[[a]] = [[q]] * [[b]] + [[r]] /\
0 <= [[r]] < [[b]].
Lemma spec_ww_mod_gt_aux_eq : forall ah al bh bl,
ww_mod_gt_aux ah al bh bl = snd (ww_div_gt_aux ah al bh bl).
Lemma spec_ww_mod_gt_aux : forall ah al bh bl,
[[WW ah al]] > [[WW bh bl]] ->
0 < [|bh|] ->
[[ww_mod_gt_aux ah al bh bl]] = [[WW ah al]] mod [[WW bh bl]].
Lemma spec_w_mod_gt_eq : forall a b, [|a|] > [|b|] -> 0 <[|b|] ->
[|w_mod_gt a b|] = [|snd (w_div_gt a b)|].
Lemma spec_ww_mod_gt_eq : forall a b, [[a]] > [[b]] -> 0 < [[b]] ->
[[ww_mod_gt a b]] = [[snd (ww_div_gt a b)]].
Lemma spec_ww_mod_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] ->
[[ww_mod_gt a b]] = [[a]] mod [[b]].
Lemma Zis_gcd_mod : forall a b d,
0 < b -> Zis_gcd b (a mod b) d -> Zis_gcd a b d.
Lemma spec_ww_gcd_gt_aux_body :
forall ah al bh bl n cont,
[[WW bh bl]] <= 2^n ->
[[WW ah al]] > [[WW bh bl]] ->
(forall xh xl yh yl,
[[WW xh xl]] > [[WW yh yl]] -> [[WW yh yl]] <= 2^(n-1) ->
Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) ->
Zis_gcd [[WW ah al]] [[WW bh bl]] [[ww_gcd_gt_body cont ah al bh bl]].
Lemma spec_ww_gcd_gt_aux :
forall p cont n,
(forall xh xl yh yl,
[[WW xh xl]] > [[WW yh yl]] ->
[[WW yh yl]] <= 2^n ->
Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) ->
forall ah al bh bl , [[WW ah al]] > [[WW bh bl]] ->
[[WW bh bl]] <= 2^(Zpos p + n) ->
Zis_gcd [[WW ah al]] [[WW bh bl]]
[[ww_gcd_gt_aux p cont ah al bh bl]].
End DoubleDivGt.
Section DoubleDiv.
Variable w : Type.
Variable w_digits : positive.
Variable ww_1 : zn2z w.
Variable ww_compare : zn2z w -> zn2z w -> comparison.
Variable ww_div_gt : zn2z w -> zn2z w -> zn2z w * zn2z w.
Variable ww_mod_gt : zn2z w -> zn2z w -> zn2z w.
Definition ww_div a b :=
match ww_compare a b with
| Gt => ww_div_gt a b
| Eq => (ww_1, W0)
| Lt => (W0, a)
end.
Definition ww_mod a b :=
match ww_compare a b with
| Gt => ww_mod_gt a b
| Eq => W0
| Lt => a
end.
Variable w_to_Z : w -> Z.
Notation wB := (base w_digits).
Notation wwB := (base (ww_digits w_digits)).
Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99).
Notation "[[ x ]]" := (ww_to_Z w_digits w_to_Z x)(at level 0, x at level 99).
Variable spec_to_Z : forall x, 0 <= [|x|] < wB.
Variable spec_ww_1 : [[ww_1]] = 1.
Variable spec_ww_compare : forall x y,
ww_compare x y = Z.compare [[x]] [[y]].
Variable spec_ww_div_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] ->
let (q,r) := ww_div_gt a b in
[[a]] = [[q]] * [[b]] + [[r]] /\
0 <= [[r]] < [[b]].
Variable spec_ww_mod_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] ->
[[ww_mod_gt a b]] = [[a]] mod [[b]].
Ltac Spec_w_to_Z x :=
let H:= fresh "HH" in
assert (H:= spec_to_Z x).
Ltac Spec_ww_to_Z x :=
let H:= fresh "HH" in
assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x).
Lemma spec_ww_div : forall a b, 0 < [[b]] ->
let (q,r) := ww_div a b in
[[a]] = [[q]] * [[b]] + [[r]] /\
0 <= [[r]] < [[b]].
Lemma spec_ww_mod : forall a b, 0 < [[b]] ->
[[ww_mod a b]] = [[a]] mod [[b]].
Variable w_0 : w.
Variable w_1 : w.
Variable w_compare : w -> w -> comparison.
Variable w_eq0 : w -> bool.
Variable w_gcd_gt : w -> w -> w.
Variable _ww_digits : positive.
Variable spec_ww_digits_ : _ww_digits = xO w_digits.
Variable ww_gcd_gt_fix :
positive -> (w -> w -> w -> w -> zn2z w) ->
w -> w -> w -> w -> zn2z w.
Variable spec_w_0 : [|w_0|] = 0.
Variable spec_w_1 : [|w_1|] = 1.
Variable spec_compare :
forall x y, w_compare x y = Z.compare [|x|] [|y|].
Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0.
Variable spec_gcd_gt : forall a b, [|a|] > [|b|] ->
Zis_gcd [|a|] [|b|] [|w_gcd_gt a b|].
Variable spec_gcd_gt_fix :
forall p cont n,
(forall xh xl yh yl,
[[WW xh xl]] > [[WW yh yl]] ->
[[WW yh yl]] <= 2^n ->
Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]]) ->
forall ah al bh bl , [[WW ah al]] > [[WW bh bl]] ->
[[WW bh bl]] <= 2^(Zpos p + n) ->
Zis_gcd [[WW ah al]] [[WW bh bl]]
[[ww_gcd_gt_fix p cont ah al bh bl]].
Definition gcd_cont (xh xl yh yl:w) :=
match w_compare w_1 yl with
| Eq => ww_1
| _ => WW xh xl
end.
Lemma spec_gcd_cont : forall xh xl yh yl,
[[WW xh xl]] > [[WW yh yl]] ->
[[WW yh yl]] <= 1 ->
Zis_gcd [[WW xh xl]] [[WW yh yl]] [[gcd_cont xh xl yh yl]].
Variable cont : w -> w -> w -> w -> zn2z w.
Variable spec_cont : forall xh xl yh yl,
[[WW xh xl]] > [[WW yh yl]] ->
[[WW yh yl]] <= 1 ->
Zis_gcd [[WW xh xl]] [[WW yh yl]] [[cont xh xl yh yl]].
Definition ww_gcd_gt a b :=
match a, b with
| W0, _ => b
| _, W0 => a
| WW ah al, WW bh bl =>
if w_eq0 ah then (WW w_0 (w_gcd_gt al bl))
else ww_gcd_gt_fix _ww_digits cont ah al bh bl
end.
Definition ww_gcd a b :=
Eval lazy beta delta [ww_gcd_gt] in
match ww_compare a b with
| Gt => ww_gcd_gt a b
| Eq => a
| Lt => ww_gcd_gt b a
end.
Lemma spec_ww_gcd_gt : forall a b, [[a]] > [[b]] ->
Zis_gcd [[a]] [[b]] [[ww_gcd_gt a b]].
Lemma spec_ww_gcd : forall a b, Zis_gcd [[a]] [[b]] [[ww_gcd a b]].
End DoubleDiv.