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(*              The Compcert verified compiler                         *)
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(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
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(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
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(** Correctness of instruction selection for integer division *)
Local Open Scope cminorsel_scope.
(** * Main approximation theorems *)
Section Z_DIV_MUL.
Variable N: Z.      (**r number of relevant bits *)
Hypothesis N_pos: N >= 0.
Variable d: Z.      (**r divisor *)
Hypothesis d_pos: d > 0.
(** This is theorem 4.2 from Granlund and Montgomery, PLDI 1994. *)

(* Specifications: *)
Lemma Zdiv_mul_pos:
forall m l,
l >= 0 ->
two_p (N+l) <= m * d <= two_p (N+l) + two_p l ->
forall n,
0 <= n < two_p N ->
Z.div n d = Z.div (m * n) (two_p (N + l)).
Lemma Zdiv_unique_2:
forall x y q, y > 0 -> 0 < y * q - x <= y -> Z.div x y = q - 1.
Lemma Zdiv_mul_opp:
forall m l,
l >= 0 ->
two_p (N+l) < m * d <= two_p (N+l) + two_p l ->
forall n,
0 < n <= two_p N ->
Z.div n d = - Z.div (m * (-n)) (two_p (N + l)) - 1.