Library Coq.Numbers.Cyclic.ZModulo.ZModulo


Type Z viewed modulo a particular constant corresponds to Z/nZ

as defined abstractly in CyclicAxioms.
Even if the construction provided here is not reused for building the efficient arbitrary precision numbers, it provides a simple implementation of CyclicAxioms, hence ensuring its coherence.

Set Implicit Arguments.

Require Import Bool.
Require Import ZArith.
Require Import Znumtheory.
Require Import Zpow_facts.
Require Import DoubleType.
Require Import CyclicAxioms.

Local Open Scope Z_scope.

Section ZModulo.

 Variable digits : positive.
 Hypothesis digits_ne_1 : digits <> 1%positive.

 Definition wB := base digits.

 Definition t := Z.
 Definition zdigits := Zpos digits.
 Definition to_Z x := x mod wB.

 Notation "[| x |]" := (to_Z x) (at level 0, x at level 99).

 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).

 Lemma spec_more_than_1_digit: 1 < Zpos digits.
 Let digits_gt_1 := spec_more_than_1_digit.

 Lemma wB_pos : wB > 0.
 Hint Resolve wB_pos : core.

 Lemma spec_to_Z_1 : forall x, 0 <= [|x|].

 Lemma spec_to_Z_2 : forall x, [|x|] < wB.
 Hint Resolve spec_to_Z_1 spec_to_Z_2 : core.

 Lemma spec_to_Z : forall x, 0 <= [|x|] < wB.

 Definition of_pos x :=
   let (q,r) := Z.pos_div_eucl x wB in (N_of_Z q, r).

 Lemma spec_of_pos : forall p,
   Zpos p = (Z.of_N (fst (of_pos p)))*wB + [|(snd (of_pos p))|].

 Lemma spec_zdigits : [|zdigits|] = Zpos digits.

 Definition zero := 0.
 Definition one := 1.
 Definition minus_one := wB - 1.

 Lemma spec_0 : [|zero|] = 0.

 Lemma spec_1 : [|one|] = 1.

 Lemma spec_Bm1 : [|minus_one|] = wB - 1.

 Definition compare x y := Z.compare [|x|] [|y|].

 Lemma spec_compare : forall x y,
   compare x y = Z.compare [|x|] [|y|].

 Definition eq0 x :=
   match [|x|] with Z0 => true | _ => false end.

 Lemma spec_eq0 : forall x, eq0 x = true -> [|x|] = 0.

 Definition opp_c x :=
   if eq0 x then C0 0 else C1 (- x).
 Definition opp x := - x.
 Definition opp_carry x := - x - 1.

 Lemma spec_opp_c : forall x, [-|opp_c x|] = -[|x|].

 Lemma spec_opp : forall x, [|opp x|] = (-[|x|]) mod wB.

 Lemma spec_opp_carry : forall x, [|opp_carry x|] = wB - [|x|] - 1.

 Definition succ_c x :=
  let y := Z.succ x in
  if eq0 y then C1 0 else C0 y.

 Definition add_c x y :=
  let z := [|x|] + [|y|] in
  if Z_lt_le_dec z wB then C0 z else C1 (z-wB).

 Definition add_carry_c x y :=
  let z := [|x|]+[|y|]+1 in
  if Z_lt_le_dec z wB then C0 z else C1 (z-wB).

 Definition succ := Z.succ.
 Definition add := Z.add.
 Definition add_carry x y := x + y + 1.

 Lemma Zmod_equal :
  forall x y z, z>0 -> (x-y) mod z = 0 -> x mod z = y mod z.

 Lemma spec_succ_c : forall x, [+|succ_c x|] = [|x|] + 1.

 Lemma spec_add_c : forall x y, [+|add_c x y|] = [|x|] + [|y|].

 Lemma spec_add_carry_c : forall x y, [+|add_carry_c x y|] = [|x|] + [|y|] + 1.

 Lemma spec_succ : forall x, [|succ x|] = ([|x|] + 1) mod wB.

 Lemma spec_add : forall x y, [|add x y|] = ([|x|] + [|y|]) mod wB.

 Lemma spec_add_carry :
         forall x y, [|add_carry x y|] = ([|x|] + [|y|] + 1) mod wB.

 Definition pred_c x :=
  if eq0 x then C1 (wB-1) else C0 (x-1).

 Definition sub_c x y :=
  let z := [|x|]-[|y|] in
  if Z_lt_le_dec z 0 then C1 (wB+z) else C0 z.

 Definition sub_carry_c x y :=
  let z := [|x|]-[|y|]-1 in
  if Z_lt_le_dec z 0 then C1 (wB+z) else C0 z.

 Definition pred := Z.pred.
 Definition sub := Z.sub.
 Definition sub_carry x y := x - y - 1.

 Lemma spec_pred_c : forall x, [-|pred_c x|] = [|x|] - 1.

 Lemma spec_sub_c : forall x y, [-|sub_c x y|] = [|x|] - [|y|].

 Lemma spec_sub_carry_c : forall x y, [-|sub_carry_c x y|] = [|x|] - [|y|] - 1.

 Lemma spec_pred : forall x, [|pred x|] = ([|x|] - 1) mod wB.

 Lemma spec_sub : forall x y, [|sub x y|] = ([|x|] - [|y|]) mod wB.

 Lemma spec_sub_carry :
  forall x y, [|sub_carry x y|] = ([|x|] - [|y|] - 1) mod wB.

 Definition mul_c x y :=
  let (h,l) := Z.div_eucl ([|x|]*[|y|]) wB in
  if eq0 h then if eq0 l then W0 else WW h l else WW h l.

 Definition mul := Z.mul.

 Definition square_c x := mul_c x x.

 Lemma spec_mul_c : forall x y, [|| mul_c x y ||] = [|x|] * [|y|].

 Lemma spec_mul : forall x y, [|mul x y|] = ([|x|] * [|y|]) mod wB.

 Lemma spec_square_c : forall x, [|| square_c x||] = [|x|] * [|x|].

 Definition div x y := Z.div_eucl [|x|] [|y|].

 Lemma spec_div : forall a b, 0 < [|b|] ->
      let (q,r) := div a b in
      [|a|] = [|q|] * [|b|] + [|r|] /\
      0 <= [|r|] < [|b|].

 Definition div_gt := div.

 Lemma 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|].

 Definition modulo x y := [|x|] mod [|y|].
 Definition modulo_gt x y := [|x|] mod [|y|].

 Lemma spec_modulo : forall a b, 0 < [|b|] ->
      [|modulo a b|] = [|a|] mod [|b|].

 Lemma spec_modulo_gt : forall a b, [|a|] > [|b|] -> 0 < [|b|] ->
      [|modulo_gt a b|] = [|a|] mod [|b|].

 Definition gcd x y := Z.gcd [|x|] [|y|].
 Definition gcd_gt x y := Z.gcd [|x|] [|y|].

 Lemma Zgcd_bound : forall a b, 0<=a -> 0<=b -> Z.gcd a b <= Z.max a b.

 Lemma spec_gcd : forall a b, Zis_gcd [|a|] [|b|] [|gcd a b|].

 Lemma spec_gcd_gt : forall a b, [|a|] > [|b|] ->
      Zis_gcd [|a|] [|b|] [|gcd_gt a b|].

 Definition div21 a1 a2 b :=
  Z.div_eucl ([|a1|]*wB+[|a2|]) [|b|].

 Lemma 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|].

 Definition add_mul_div p x y :=
   ([|x|] * (2 ^ [|p|]) + [|y|] / (2 ^ ((Zpos digits) - [|p|]))).
 Lemma 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.

 Definition pos_mod p w := [|w|] mod (2 ^ [|p|]).
 Lemma spec_pos_mod : forall w p,
       [|pos_mod p w|] = [|w|] mod (2 ^ [|p|]).

 Definition is_even x :=
   if Z.eq_dec ([|x|] mod 2) 0 then true else false.

 Lemma spec_is_even : forall x,
      if is_even x then [|x|] mod 2 = 0 else [|x|] mod 2 = 1.

 Definition sqrt x := Z.sqrt [|x|].
 Lemma spec_sqrt : forall x,
       [|sqrt x|] ^ 2 <= [|x|] < ([|sqrt x|] + 1) ^ 2.

 Definition sqrt2 x y :=
  let z := [|x|]*wB+[|y|] in
  match z with
   | Z0 => (0, C0 0)
   | Zpos p =>
      let (s,r) := Z.sqrtrem (Zpos p) in
      (s, if Z_lt_le_dec r wB then C0 r else C1 (r-wB))
   | Zneg _ => (0, C0 0)
  end.

 Lemma spec_sqrt2 : forall x y,
       wB/ 4 <= [|x|] ->
       let (s,r) := sqrt2 x y in
          [||WW x y||] = [|s|] ^ 2 + [+|r|] /\
          [+|r|] <= 2 * [|s|].

 Lemma two_p_power2 : forall x, x>=0 -> two_p x = 2 ^ x.

 Definition head0 x :=
   match [| x |] with
   | Z0 => zdigits
   | Zneg _ => 0
   | (Zpos _) as p => zdigits - Z.log2 p - 1
   end.

 Lemma spec_head00: forall x, [|x|] = 0 -> [|head0 x|] = Zpos digits.

 Lemma spec_head0 : forall x, 0 < [|x|] ->
         wB/ 2 <= 2 ^ ([|head0 x|]) * [|x|] < wB.

 Fixpoint Ptail p := match p with
  | xO p => (Ptail p)+1
  | _ => 0
 end.

 Lemma Ptail_pos : forall p, 0 <= Ptail p.
 Hint Resolve Ptail_pos : core.

 Lemma Ptail_bounded : forall p d, Zpos p < 2^(Zpos d) -> Ptail p < Zpos d.

 Definition tail0 x :=
  match [|x|] with
   | Z0 => zdigits
   | Zpos p => Ptail p
   | Zneg _ => 0
  end.

 Lemma spec_tail00: forall x, [|x|] = 0 -> [|tail0 x|] = Zpos digits.

 Lemma spec_tail0 : forall x, 0 < [|x|] ->
         exists y, 0 <= y /\ [|x|] = (2 * y + 1) * (2 ^ [|tail0 x|]).

 Definition lor := Z.lor.
 Definition land := Z.land.
 Definition lxor := Z.lxor.

 Lemma spec_lor x y : [|lor x y|] = Z.lor [|x|] [|y|].

 Lemma spec_land x y : [|land x y|] = Z.land [|x|] [|y|].

 Lemma spec_lxor x y : [|lxor x y|] = Z.lxor [|x|] [|y|].

Let's now group everything in two records

 Instance zmod_ops : ZnZ.Ops Z := ZnZ.MkOps
    (digits : positive)
    (zdigits: t)
    (to_Z : t -> Z)
    (of_pos : positive -> N * t)
    (head0 : t -> t)
    (tail0 : t -> t)

    (zero : t)
    (one : t)
    (minus_one : t)

    (compare : t -> t -> comparison)
    (eq0 : t -> bool)

    (opp_c : t -> carry t)
    (opp : t -> t)
    (opp_carry : t -> t)

    (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)

    (div21 : t -> t -> t -> t*t)
    (div_gt : t -> t -> t * t)
    (div : t -> t -> t * t)

    (modulo_gt : t -> t -> t)
    (modulo : t -> t -> t)

    (gcd_gt : t -> t -> t)
    (gcd : t -> t -> t)
    (add_mul_div : t -> t -> t -> t)
    (pos_mod : t -> t -> t)

    (is_even : t -> bool)
    (sqrt2 : t -> t -> t * carry t)
    (sqrt : t -> t)
    (lor : t -> t -> t)
    (land : t -> t -> t)
    (lxor : t -> t -> t).

 Instance zmod_specs : ZnZ.Specs zmod_ops := ZnZ.MkSpecs
    spec_to_Z
    spec_of_pos
    spec_zdigits
    spec_more_than_1_digit

    spec_0
    spec_1
    spec_Bm1

    spec_compare
    spec_eq0

    spec_opp_c
    spec_opp
    spec_opp_carry

    spec_succ_c
    spec_add_c
    spec_add_carry_c
    spec_succ
    spec_add
    spec_add_carry

    spec_pred_c
    spec_sub_c
    spec_sub_carry_c
    spec_pred
    spec_sub
    spec_sub_carry

    spec_mul_c
    spec_mul
    spec_square_c

    spec_div21
    spec_div_gt
    spec_div

    spec_modulo_gt
    spec_modulo

    spec_gcd_gt
    spec_gcd

    spec_head00
    spec_head0
    spec_tail00
    spec_tail0

    spec_add_mul_div
    spec_pos_mod

    spec_is_even
    spec_sqrt2
    spec_sqrt
    spec_lor
    spec_land
    spec_lxor.

End ZModulo.

A modular version of the previous construction.

Module Type PositiveNotOne.
 Parameter p : positive.
 Axiom not_one : p <> 1%positive.
End PositiveNotOne.

Module ZModuloCyclicType (P:PositiveNotOne) <: CyclicType.
 Definition t := Z.
 Instance ops : ZnZ.Ops t := zmod_ops P.p.
 Instance specs : ZnZ.Specs ops := zmod_specs P.not_one.
End ZModuloCyclicType.