Library Coq.Numbers.Integer.BigZ.ZMake


Require Import ZArith.
Require Import BigNumPrelude.
Require Import NSig.
Require Import ZSig.

Open Scope Z_scope.

ZMake

A generic transformation from a structure of natural numbers NSig.NType to a structure of integers ZSig.ZType.

Module Make (NN:NType) <: ZType.

 Inductive t_ :=
  | Pos : NN.t -> t_
  | Neg : NN.t -> t_.

 Definition t := t_.

 Definition zero := Pos NN.zero.
 Definition one := Pos NN.one.
 Definition two := Pos NN.two.
 Definition minus_one := Neg NN.one.

 Definition of_Z x :=
  match x with
  | Zpos x => Pos (NN.of_N (Npos x))
  | Z0 => zero
  | Zneg x => Neg (NN.of_N (Npos x))
  end.

 Definition to_Z x :=
  match x with
  | Pos nx => NN.to_Z nx
  | Neg nx => Z.opp (NN.to_Z nx)
  end.

 Theorem spec_of_Z: forall x, to_Z (of_Z x) = x.

 Definition eq x y := (to_Z x = to_Z y).

 Theorem spec_0: to_Z zero = 0.

 Theorem spec_1: to_Z one = 1.

 Theorem spec_2: to_Z two = 2.

 Theorem spec_m1: to_Z minus_one = -1.

 Definition compare x y :=
  match x, y with
  | Pos nx, Pos ny => NN.compare nx ny
  | Pos nx, Neg ny =>
    match NN.compare nx NN.zero with
    | Gt => Gt
    | _ => NN.compare ny NN.zero
    end
  | Neg nx, Pos ny =>
    match NN.compare NN.zero nx with
    | Lt => Lt
    | _ => NN.compare NN.zero ny
    end
  | Neg nx, Neg ny => NN.compare ny nx
  end.

 Theorem spec_compare :
  forall x y, compare x y = Z.compare (to_Z x) (to_Z y).

 Definition eqb x y :=
  match compare x y with
  | Eq => true
  | _ => false
  end.

 Theorem spec_eqb x y : eqb x y = Z.eqb (to_Z x) (to_Z y).

 Definition lt n m := to_Z n < to_Z m.
 Definition le n m := to_Z n <= to_Z m.

 Definition ltb (x y : t) : bool :=
  match compare x y with
  | Lt => true
  | _ => false
  end.

 Theorem spec_ltb x y : ltb x y = Z.ltb (to_Z x) (to_Z y).

 Definition leb (x y : t) : bool :=
  match compare x y with
  | Gt => false
  | _ => true
  end.

 Theorem spec_leb x y : leb x y = Z.leb (to_Z x) (to_Z y).

 Definition min n m := match compare n m with Gt => m | _ => n end.
 Definition max n m := match compare n m with Lt => m | _ => n end.

 Theorem spec_min : forall n m, to_Z (min n m) = Z.min (to_Z n) (to_Z m).

 Theorem spec_max : forall n m, to_Z (max n m) = Z.max (to_Z n) (to_Z m).

 Definition to_N x :=
  match x with
  | Pos nx => nx
  | Neg nx => nx
  end.

 Definition abs x := Pos (to_N x).

 Theorem spec_abs: forall x, to_Z (abs x) = Z.abs (to_Z x).

 Definition opp x :=
  match x with
  | Pos nx => Neg nx
  | Neg nx => Pos nx
  end.

 Theorem spec_opp: forall x, to_Z (opp x) = - to_Z x.

 Definition succ x :=
  match x with
  | Pos n => Pos (NN.succ n)
  | Neg n =>
    match NN.compare NN.zero n with
    | Lt => Neg (NN.pred n)
    | _ => one
    end
  end.

 Theorem spec_succ: forall n, to_Z (succ n) = to_Z n + 1.

 Definition add x y :=
  match x, y with
  | Pos nx, Pos ny => Pos (NN.add nx ny)
  | Pos nx, Neg ny =>
    match NN.compare nx ny with
    | Gt => Pos (NN.sub nx ny)
    | Eq => zero
    | Lt => Neg (NN.sub ny nx)
    end
  | Neg nx, Pos ny =>
    match NN.compare nx ny with
    | Gt => Neg (NN.sub nx ny)
    | Eq => zero
    | Lt => Pos (NN.sub ny nx)
    end
  | Neg nx, Neg ny => Neg (NN.add nx ny)
  end.

 Theorem spec_add: forall x y, to_Z (add x y) = to_Z x + to_Z y.

 Definition pred x :=
  match x with
  | Pos nx =>
    match NN.compare NN.zero nx with
    | Lt => Pos (NN.pred nx)
    | _ => minus_one
    end
  | Neg nx => Neg (NN.succ nx)
  end.

 Theorem spec_pred: forall x, to_Z (pred x) = to_Z x - 1.

 Definition sub x y :=
  match x, y with
  | Pos nx, Pos ny =>
    match NN.compare nx ny with
    | Gt => Pos (NN.sub nx ny)
    | Eq => zero
    | Lt => Neg (NN.sub ny nx)
    end
  | Pos nx, Neg ny => Pos (NN.add nx ny)
  | Neg nx, Pos ny => Neg (NN.add nx ny)
  | Neg nx, Neg ny =>
    match NN.compare nx ny with
    | Gt => Neg (NN.sub nx ny)
    | Eq => zero
    | Lt => Pos (NN.sub ny nx)
    end
  end.

 Theorem spec_sub: forall x y, to_Z (sub x y) = to_Z x - to_Z y.

 Definition mul x y :=
  match x, y with
  | Pos nx, Pos ny => Pos (NN.mul nx ny)
  | Pos nx, Neg ny => Neg (NN.mul nx ny)
  | Neg nx, Pos ny => Neg (NN.mul nx ny)
  | Neg nx, Neg ny => Pos (NN.mul nx ny)
  end.

 Theorem spec_mul: forall x y, to_Z (mul x y) = to_Z x * to_Z y.

 Definition square x :=
  match x with
  | Pos nx => Pos (NN.square nx)
  | Neg nx => Pos (NN.square nx)
  end.

 Theorem spec_square: forall x, to_Z (square x) = to_Z x * to_Z x.

 Definition pow_pos x p :=
  match x with
  | Pos nx => Pos (NN.pow_pos nx p)
  | Neg nx =>
    match p with
    | xH => x
    | xO _ => Pos (NN.pow_pos nx p)
    | xI _ => Neg (NN.pow_pos nx p)
    end
  end.

 Theorem spec_pow_pos: forall x n, to_Z (pow_pos x n) = to_Z x ^ Zpos n.

 Definition pow_N x n :=
  match n with
  | N0 => one
  | Npos p => pow_pos x p
  end.

 Theorem spec_pow_N: forall x n, to_Z (pow_N x n) = to_Z x ^ Z.of_N n.

 Definition pow x y :=
  match to_Z y with
  | Z0 => one
  | Zpos p => pow_pos x p
  | Zneg p => zero
  end.

 Theorem spec_pow: forall x y, to_Z (pow x y) = to_Z x ^ to_Z y.

 Definition log2 x :=
  match x with
  | Pos nx => Pos (NN.log2 nx)
  | Neg nx => zero
  end.

 Theorem spec_log2: forall x, to_Z (log2 x) = Z.log2 (to_Z x).

 Definition sqrt x :=
  match x with
  | Pos nx => Pos (NN.sqrt nx)
  | Neg nx => Neg NN.zero
  end.

 Theorem spec_sqrt: forall x, to_Z (sqrt x) = Z.sqrt (to_Z x).

 Definition div_eucl x y :=
  match x, y with
  | Pos nx, Pos ny =>
    let (q, r) := NN.div_eucl nx ny in
    (Pos q, Pos r)
  | Pos nx, Neg ny =>
    let (q, r) := NN.div_eucl nx ny in
    if NN.eqb NN.zero r
    then (Neg q, zero)
    else (Neg (NN.succ q), Neg (NN.sub ny r))
  | Neg nx, Pos ny =>
    let (q, r) := NN.div_eucl nx ny in
    if NN.eqb NN.zero r
    then (Neg q, zero)
    else (Neg (NN.succ q), Pos (NN.sub ny r))
  | Neg nx, Neg ny =>
    let (q, r) := NN.div_eucl nx ny in
    (Pos q, Neg r)
  end.

 Ltac break_nonneg x px EQx :=
  let H := fresh "H" in
  assert (H:=NN.spec_pos x);
  destruct (NN.to_Z x) as [|px|px] eqn:EQx;
   [clear H|clear H|elim H; reflexivity].

 Theorem spec_div_eucl: forall x y,
   let (q,r) := div_eucl x y in
   (to_Z q, to_Z r) = Z.div_eucl (to_Z x) (to_Z y).

 Definition div x y := fst (div_eucl x y).

 Definition spec_div: forall x y,
     to_Z (div x y) = to_Z x / to_Z y.

 Definition modulo x y := snd (div_eucl x y).

 Theorem spec_modulo:
   forall x y, to_Z (modulo x y) = to_Z x mod to_Z y.

 Definition quot x y :=
  match x, y with
  | Pos nx, Pos ny => Pos (NN.div nx ny)
  | Pos nx, Neg ny => Neg (NN.div nx ny)
  | Neg nx, Pos ny => Neg (NN.div nx ny)
  | Neg nx, Neg ny => Pos (NN.div nx ny)
  end.

 Definition rem x y :=
  if eqb y zero then x
  else
    match x, y with
      | Pos nx, Pos ny => Pos (NN.modulo nx ny)
      | Pos nx, Neg ny => Pos (NN.modulo nx ny)
      | Neg nx, Pos ny => Neg (NN.modulo nx ny)
      | Neg nx, Neg ny => Neg (NN.modulo nx ny)
    end.

 Lemma spec_quot : forall x y, to_Z (quot x y) = (to_Z x) รท (to_Z y).

 Lemma spec_rem : forall x y,
   to_Z (rem x y) = Z.rem (to_Z x) (to_Z y).

 Definition gcd x y :=
  match x, y with
  | Pos nx, Pos ny => Pos (NN.gcd nx ny)
  | Pos nx, Neg ny => Pos (NN.gcd nx ny)
  | Neg nx, Pos ny => Pos (NN.gcd nx ny)
  | Neg nx, Neg ny => Pos (NN.gcd nx ny)
  end.

 Theorem spec_gcd: forall a b, to_Z (gcd a b) = Z.gcd (to_Z a) (to_Z b).

 Definition sgn x :=
  match compare zero x with
   | Lt => one
   | Eq => zero
   | Gt => minus_one
  end.

 Lemma spec_sgn : forall x, to_Z (sgn x) = Z.sgn (to_Z x).

 Definition even z :=
  match z with
   | Pos n => NN.even n
   | Neg n => NN.even n
  end.

 Definition odd z :=
  match z with
   | Pos n => NN.odd n
   | Neg n => NN.odd n
  end.

 Lemma spec_even : forall z, even z = Z.even (to_Z z).

 Lemma spec_odd : forall z, odd z = Z.odd (to_Z z).

 Definition norm_pos z :=
   match z with
     | Pos _ => z
     | Neg n => if NN.eqb n NN.zero then Pos n else z
   end.

 Definition testbit a n :=
   match norm_pos n, norm_pos a with
     | Pos p, Pos a => NN.testbit a p
     | Pos p, Neg a => negb (NN.testbit (NN.pred a) p)
     | Neg p, _ => false
   end.

 Definition shiftl a n :=
   match norm_pos a, n with
     | Pos a, Pos n => Pos (NN.shiftl a n)
     | Pos a, Neg n => Pos (NN.shiftr a n)
     | Neg a, Pos n => Neg (NN.shiftl a n)
     | Neg a, Neg n => Neg (NN.succ (NN.shiftr (NN.pred a) n))
   end.

 Definition shiftr a n := shiftl a (opp n).

 Definition lor a b :=
   match norm_pos a, norm_pos b with
     | Pos a, Pos b => Pos (NN.lor a b)
     | Neg a, Pos b => Neg (NN.succ (NN.ldiff (NN.pred a) b))
     | Pos a, Neg b => Neg (NN.succ (NN.ldiff (NN.pred b) a))
     | Neg a, Neg b => Neg (NN.succ (NN.land (NN.pred a) (NN.pred b)))
   end.

 Definition land a b :=
   match norm_pos a, norm_pos b with
     | Pos a, Pos b => Pos (NN.land a b)
     | Neg a, Pos b => Pos (NN.ldiff b (NN.pred a))
     | Pos a, Neg b => Pos (NN.ldiff a (NN.pred b))
     | Neg a, Neg b => Neg (NN.succ (NN.lor (NN.pred a) (NN.pred b)))
   end.

 Definition ldiff a b :=
   match norm_pos a, norm_pos b with
     | Pos a, Pos b => Pos (NN.ldiff a b)
     | Neg a, Pos b => Neg (NN.succ (NN.lor (NN.pred a) b))
     | Pos a, Neg b => Pos (NN.land a (NN.pred b))
     | Neg a, Neg b => Pos (NN.ldiff (NN.pred b) (NN.pred a))
   end.

 Definition lxor a b :=
   match norm_pos a, norm_pos b with
     | Pos a, Pos b => Pos (NN.lxor a b)
     | Neg a, Pos b => Neg (NN.succ (NN.lxor (NN.pred a) b))
     | Pos a, Neg b => Neg (NN.succ (NN.lxor a (NN.pred b)))
     | Neg a, Neg b => Pos (NN.lxor (NN.pred a) (NN.pred b))
   end.

 Definition div2 x := shiftr x one.

 Lemma Zlnot_alt1 : forall x, -(x+1) = Z.lnot x.

 Lemma Zlnot_alt2 : forall x, Z.lnot (x-1) = -x.

 Lemma Zlnot_alt3 : forall x, Z.lnot (-x) = x-1.

 Lemma spec_norm_pos : forall x, to_Z (norm_pos x) = to_Z x.

 Lemma spec_norm_pos_pos : forall x y, norm_pos x = Neg y ->
  0 < NN.to_Z y.

 Ltac destr_norm_pos x :=
  rewrite <- (spec_norm_pos x);
  let H := fresh in
  let x' := fresh x in
  assert (H := spec_norm_pos_pos x);
  destruct (norm_pos x) as [x'|x'];
   specialize (H x' (eq_refl _)) || clear H.

 Lemma spec_testbit: forall x p, testbit x p = Z.testbit (to_Z x) (to_Z p).

 Lemma spec_shiftl: forall x p, to_Z (shiftl x p) = Z.shiftl (to_Z x) (to_Z p).

 Lemma spec_shiftr: forall x p, to_Z (shiftr x p) = Z.shiftr (to_Z x) (to_Z p).

 Lemma spec_land: forall x y, to_Z (land x y) = Z.land (to_Z x) (to_Z y).

 Lemma spec_lor: forall x y, to_Z (lor x y) = Z.lor (to_Z x) (to_Z y).

 Lemma spec_ldiff: forall x y, to_Z (ldiff x y) = Z.ldiff (to_Z x) (to_Z y).

 Lemma spec_lxor: forall x y, to_Z (lxor x y) = Z.lxor (to_Z x) (to_Z y).

 Lemma spec_div2: forall x, to_Z (div2 x) = Z.div2 (to_Z x).

End Make.