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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
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.
Proof. congruence. Qed.
Instance div_wd : Proper (eq==>eq==>eq) div.
Proof. congruence. Qed.
Theorem div_mod a b : b<>0 -> a = b*(div a b) + modulo a b.
Proof.
intros Hb. unfold div, modulo.
rewrite Z.mul_assoc. rewrite Z.sgn_abs. apply Z.div_mod.
now destruct b.
Qed.
Lemma mod_always_pos a b : b<>0 -> 0 <= modulo a b < Z.abs b.
Proof.
intros Hb. unfold modulo.
apply Z.mod_pos_bound.
destruct b; compute; trivial. now destruct Hb.
Qed.
Lemma mod_bound_pos a b : 0<=a -> 0<b -> 0 <= modulo a b < b.
Proof.
intros _ Hb. rewrite <- (Z.abs_eq b) at 3 by Z.order.
apply mod_always_pos. Z.order.
Qed.
Include ZEuclidProp Z Z Z.
End ZEuclid.
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