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Require Import Morphisms BinInt ZDivEucl.
Local Open Scope Z_scope.

Definitions of division for binary integers, Euclid convention.

In this convention, the remainder is always positive. For other conventions, see Z.div and Z.quot in file BinIntDef. To avoid collision with the other divisions, we place this one under a module.

Module ZEuclid.

 Definition modulo a b := Z.modulo a (Z.abs b).
 Definition div a b := (Z.sgn b) * (Z.div a (Z.abs b)).

 Instance mod_wd : Proper (eq==>eq==>eq) modulo.Proper (eq ==> eq ==> eq) modulo
 Proof.Proper (eq ==> eq ==> eq) modulo congruence. Qed.
 Instance div_wd : Proper (eq==>eq==>eq) div.Proper (eq ==> eq ==> eq) div
 Proof.Proper (eq ==> eq ==> eq) div congruence. Qed.

 Theorem div_mod a b : b<>0 -> a = b*(div a b) + modulo a b.a, b:Z
b <> 0 -> a = b * div a b + modulo a b
 Proof.a, b:Z
b <> 0 -> a = b * div a b + modulo a b
  intros Hb.a, b:Z
Hb:b <> 0
a = b * div a b + modulo a b unfold div, modulo.a, b:Z
Hb:b <> 0
a = b * (Z.sgn b * (a / Z.abs b)) + a mod Z.abs b
  rewrite Z.mul_assoc.a, b:Z
Hb:b <> 0
a = b * Z.sgn b * (a / Z.abs b) + a mod Z.abs b rewrite Z.sgn_abs.a, b:Z
Hb:b <> 0
a = Z.abs b * (a / Z.abs b) + a mod Z.abs b apply Z.div_mod.a, b:Z
Hb:b <> 0
Z.abs b <> 0
  now destruct b.
 Qed.

 Lemma mod_always_pos a b : b<>0 -> 0 <= modulo a b < Z.abs b.a, b:Z
b <> 0 -> 0 <= modulo a b < Z.abs b
 Proof.a, b:Z
b <> 0 -> 0 <= modulo a b < Z.abs b
  intros Hb.a, b:Z
Hb:b <> 0
0 <= modulo a b < Z.abs b unfold modulo.a, b:Z
Hb:b <> 0
0 <= a mod Z.abs b < Z.abs b
  apply Z.mod_pos_bound.a, b:Z
Hb:b <> 0
0 < Z.abs b
  destruct b; compute; trivial.a:Z
Hb:0 <> 0
Eq = Lt now destruct Hb.
 Qed.

 Lemma mod_bound_pos a b : 0<=a -> 0<b -> 0 <= modulo a b < b.a, b:Z
0 <= a -> 0 < b -> 0 <= modulo a b < b
 Proof.a, b:Z
0 <= a -> 0 < b -> 0 <= modulo a b < b
  intros _ Hb.a, b:Z
Hb:0 < b
0 <= modulo a b < b rewrite <- (Z.abs_eq b) at 3 by Z.order.a, b:Z
Hb:0 < b
0 <= modulo a b < Z.abs b
  apply mod_always_pos.a, b:Z
Hb:0 < b
b <> 0 Z.order.
 Qed.

 Include ZEuclidProp Z Z Z.

End ZEuclid.