Library Coq.Numbers.Cyclic.DoubleCyclic.DoubleBase


Set Implicit Arguments.

Require Import ZArith Ndigits.
Require Import BigNumPrelude.
Require Import DoubleType.

Local Open Scope Z_scope.


Section DoubleBase.
 Variable w : Type.
 Variable w_0 : w.
 Variable w_1 : w.
 Variable w_Bm1 : w.
 Variable w_WW : w -> w -> zn2z w.
 Variable w_0W : w -> zn2z w.
 Variable w_digits : positive.
 Variable w_zdigits: w.
 Variable w_add: w -> w -> zn2z w.
 Variable w_to_Z : w -> Z.
 Variable w_compare : w -> w -> comparison.

 Definition ww_digits := xO w_digits.

 Definition ww_zdigits := w_add w_zdigits w_zdigits.

 Definition ww_to_Z := zn2z_to_Z (base w_digits) w_to_Z.

 Definition ww_1 := WW w_0 w_1.

 Definition ww_Bm1 := WW w_Bm1 w_Bm1.

 Definition ww_WW xh xl : zn2z (zn2z w) :=
  match xh, xl with
  | W0, W0 => W0
  | _, _ => WW xh xl
  end.

 Definition ww_W0 h : zn2z (zn2z w) :=
  match h with
  | W0 => W0
  | _ => WW h W0
  end.

 Definition ww_0W l : zn2z (zn2z w) :=
  match l with
  | W0 => W0
  | _ => WW W0 l
  end.

 Definition double_WW (n:nat) :=
  match n return word w n -> word w n -> word w (S n) with
  | O => w_WW
  | S n =>
    fun (h l : zn2z (word w n)) =>
     match h, l with
     | W0, W0 => W0
     | _, _ => WW h l
     end
  end.

 Definition double_wB n := base (w_digits << n).

 Fixpoint double_to_Z (n:nat) : word w n -> Z :=
  match n return word w n -> Z with
  | O => w_to_Z
  | S n => zn2z_to_Z (double_wB n) (double_to_Z n)
  end.

 Fixpoint extend_aux (n:nat) (x:zn2z w) {struct n}: word w (S n) :=
  match n return word w (S n) with
  | O => x
  | S n1 => WW W0 (extend_aux n1 x)
  end.

 Definition extend (n:nat) (x:w) : word w (S n) :=
  let r := w_0W x in
  match r with
  | W0 => W0
  | _ => extend_aux n r
  end.

 Definition double_0 n : word w n :=
   match n return word w n with
   | O => w_0
   | S _ => W0
   end.

 Definition double_split (n:nat) (x:zn2z (word w n)) :=
  match x with
  | W0 =>
    match n return word w n * word w n with
    | O => (w_0,w_0)
    | S _ => (W0, W0)
    end
  | WW h l => (h,l)
  end.

 Definition ww_compare x y :=
  match x, y with
  | W0, W0 => Eq
  | W0, WW yh yl =>
    match w_compare w_0 yh with
    | Eq => w_compare w_0 yl
    | _ => Lt
    end
  | WW xh xl, W0 =>
    match w_compare xh w_0 with
    | Eq => w_compare xl w_0
    | _ => Gt
    end
  | WW xh xl, WW yh yl =>
    match w_compare xh yh with
    | Eq => w_compare xl yl
    | Lt => Lt
    | Gt => Gt
    end
  end.

 Fixpoint get_low (n : nat) {struct n}:
  word w n -> w :=
  match n return (word w n -> w) with
  | 0%nat => fun x => x
  | S n1 =>
      fun x =>
      match x with
      | W0 => w_0
      | WW _ x1 => get_low n1 x1
      end
  end.

 Section DoubleProof.
  Notation wB := (base w_digits).
  Notation wwB := (base ww_digits).
  Notation "[| x |]" := (w_to_Z x) (at level 0, x at level 99).
  Notation "[[ x ]]" := (ww_to_Z x) (at level 0, x at level 99).
  Notation "[+[ c ]]" :=
   (interp_carry 1 wwB ww_to_Z c) (at level 0, c at level 99).
  Notation "[-[ c ]]" :=
   (interp_carry (-1) wwB ww_to_Z c) (at level 0, c at level 99).
  Notation "[! n | x !]" := (double_to_Z n x) (at level 0, x at level 99).

  Variable spec_w_0 : [|w_0|] = 0.
  Variable spec_w_1 : [|w_1|] = 1.
  Variable spec_w_Bm1 : [|w_Bm1|] = wB - 1.
  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_to_Z : forall x, 0 <= [|x|] < wB.
  Variable spec_w_compare : forall x y,
     w_compare x y = Z.compare [|x|] [|y|].

  Lemma wwB_wBwB : wwB = wB^2.

  Lemma spec_ww_1 : [[ww_1]] = 1.

  Lemma spec_ww_Bm1 : [[ww_Bm1]] = wwB - 1.

  Lemma lt_0_wB : 0 < wB.

  Lemma lt_0_wwB : 0 < wwB.

  Lemma wB_pos: 1 < wB.

  Lemma wwB_pos: 1 < wwB.

  Theorem wB_div_2: 2 * (wB / 2) = wB.

  Theorem wwB_div_2 : wwB / 2 = wB / 2 * wB.

  Lemma mod_wwB : forall z x,
   (z*wB + [|x|]) mod wwB = (z mod wB)*wB + [|x|].

  Lemma wB_div : forall x y, ([|x|] * wB + [|y|]) / wB = [|x|].

  Lemma wB_div_plus : forall x y p,
   0 <= p ->
   ([|x|]*wB + [|y|]) / 2^(Zpos w_digits + p) = [|x|] / 2^p.

  Lemma lt_wB_wwB : wB < wwB.

  Lemma w_to_Z_wwB : forall x, x < wB -> x < wwB.

  Lemma spec_ww_to_Z : forall x, 0 <= [[x]] < wwB.

  Lemma double_wB_wwB : forall n, double_wB n * double_wB n = double_wB (S n).

  Lemma double_wB_pos:
   forall n, 0 <= double_wB n.

  Lemma double_wB_more_digits:
  forall n, wB <= double_wB n.

  Lemma spec_double_to_Z :
   forall n (x:word w n), 0 <= [!n | x!] < double_wB n.

  Lemma spec_get_low:
  forall n x,
     [!n | x!] < wB -> [|get_low n x|] = [!n | x!].

  Lemma spec_double_WW : forall n (h l : word w n),
    [!S n|double_WW n h l!] = [!n|h!] * double_wB n + [!n|l!].

  Lemma spec_extend_aux : forall n x, [!S n|extend_aux n x!] = [[x]].

  Lemma spec_extend : forall n x, [!S n|extend n x!] = [|x|].

  Lemma spec_double_0 : forall n, [!n|double_0 n!] = 0.

  Lemma spec_double_split : forall n x,
   let (h,l) := double_split n x in
   [!S n|x!] = [!n|h!] * double_wB n + [!n|l!].

  Lemma wB_lex_inv: forall a b c d,
      a < c ->
      a * wB + [|b|] < c * wB + [|d|].

  Ltac comp2ord := match goal with
   | |- Lt = (?x ?= ?y) => symmetry; change (x < y)
   | |- Gt = (?x ?= ?y) => symmetry; change (x > y); apply Z.lt_gt
  end.

  Lemma spec_ww_compare : forall x y,
    ww_compare x y = Z.compare [[x]] [[y]].

 End DoubleProof.

End DoubleBase.