diff options
Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZDivFloor.v')
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZDivFloor.v | 123 |
1 files changed, 75 insertions, 48 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZDivFloor.v b/theories/Numbers/Integer/Abstract/ZDivFloor.v index efefab81..14003d89 100644 --- a/theories/Numbers/Integer/Abstract/ZDivFloor.v +++ b/theories/Numbers/Integer/Abstract/ZDivFloor.v @@ -1,11 +1,13 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) +Require Import ZAxioms ZMulOrder ZSgnAbs NZDiv. + (** * Euclidean Division for integers (Floor convention) We use here the convention known as Floor, or Round-Toward-Bottom, @@ -24,33 +26,13 @@ See files [ZDivTrunc] and [ZDivEucl] for others conventions. *) -Require Import ZAxioms ZProperties NZDiv. - -Module Type ZDivSpecific (Import Z:ZAxiomsSig')(Import DM : DivMod' Z). - Axiom mod_pos_bound : forall a b, 0 < b -> 0 <= a mod b < b. - Axiom mod_neg_bound : forall a b, b < 0 -> b < a mod b <= 0. -End ZDivSpecific. - -Module Type ZDiv (Z:ZAxiomsSig) - := DivMod Z <+ NZDivCommon Z <+ ZDivSpecific Z. - -Module Type ZDivSig := ZAxiomsExtSig <+ ZDiv. -Module Type ZDivSig' := ZAxiomsExtSig' <+ ZDiv <+ DivModNotation. - -Module ZDivPropFunct (Import Z : ZDivSig')(Import ZP : ZPropSig Z). +Module Type ZDivProp + (Import A : ZAxiomsSig') + (Import B : ZMulOrderProp A) + (Import C : ZSgnAbsProp A B). (** We benefit from what already exists for NZ *) - - Module ZD <: NZDiv Z. - Definition div := div. - Definition modulo := modulo. - Definition div_wd := div_wd. - Definition mod_wd := mod_wd. - Definition div_mod := div_mod. - Lemma mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b. - Proof. intros. now apply mod_pos_bound. Qed. - End ZD. - Module Import NZDivP := NZDivPropFunct Z ZP ZD. +Module Import Private_NZDiv := Nop <+ NZDivProp A A B. (** Another formulation of the main equation *) @@ -62,6 +44,18 @@ rewrite <- add_move_l. symmetry. now apply div_mod. Qed. +(** We have a general bound for absolute values *) + +Lemma mod_bound_abs : + forall a b, b~=0 -> abs (a mod b) < abs b. +Proof. +intros. +destruct (abs_spec b) as [(LE,EQ)|(LE,EQ)]; rewrite EQ. +destruct (mod_pos_bound a b). order. now rewrite abs_eq. +destruct (mod_neg_bound a b). order. rewrite abs_neq; trivial. +now rewrite <- opp_lt_mono. +Qed. + (** Uniqueness theorems *) Theorem div_mod_unique : forall b q1 q2 r1 r2 : t, @@ -94,7 +88,7 @@ Theorem div_unique_pos: Proof. intros; apply div_unique with r; auto. Qed. Theorem div_unique_neg: - forall a b q r, 0<=r<b -> a == b*q + r -> q == a/b. + forall a b q r, b<r<=0 -> a == b*q + r -> q == a/b. Proof. intros; apply div_unique with r; auto. Qed. Theorem mod_unique: @@ -230,11 +224,26 @@ rewrite mod_opp_opp, mod_opp_l_nz by trivial. now rewrite opp_sub_distr, add_comm, add_opp_r. Qed. -(** The sign of [a mod b] is the one of [b] *) +(** The sign of [a mod b] is the one of [b] (when it isn't null) *) + +Lemma mod_sign_nz : forall a b, b~=0 -> a mod b ~= 0 -> + sgn (a mod b) == sgn b. +Proof. +intros a b Hb H. destruct (lt_ge_cases 0 b) as [Hb'|Hb']. +destruct (mod_pos_bound a b Hb'). rewrite 2 sgn_pos; order. +destruct (mod_neg_bound a b). order. rewrite 2 sgn_neg; order. +Qed. -(* TODO: a proper sgn function and theory *) +Lemma mod_sign : forall a b, b~=0 -> sgn (a mod b) ~= -sgn b. +Proof. +intros a b Hb H. +destruct (eq_decidable (a mod b) 0) as [EQ|NEQ]. +apply Hb, sgn_null_iff, opp_inj. now rewrite <- H, opp_0, EQ, sgn_0. +apply Hb, sgn_null_iff. apply eq_mul_0_l with 2; try order'. nzsimpl'. +apply add_move_0_l. rewrite <- H. symmetry. now apply mod_sign_nz. +Qed. -Lemma mod_sign : forall a b, b~=0 -> (0 <= (a mod b) * b). +Lemma mod_sign_mul : forall a b, b~=0 -> 0 <= (a mod b) * b. Proof. intros. destruct (lt_ge_cases 0 b). apply mul_nonneg_nonneg; destruct (mod_pos_bound a b); order. @@ -307,6 +316,11 @@ Proof. intros. rewrite mod_eq, div_mul by trivial. rewrite mul_comm; apply sub_diag. Qed. +Theorem div_unique_exact a b q: b~=0 -> a == b*q -> q == a/b. +Proof. + intros Hb H. rewrite H, mul_comm. symmetry. now apply div_mul. +Qed. + (** * Order results about mod and div *) (** A modulo cannot grow beyond its starting point. *) @@ -585,15 +599,25 @@ Proof. Qed. (** With the current convention, the following result isn't always - true for negative divisors. For instance - [ 3/(-2)/(-2) = 1 <> 0 = 3 / (-2*-2) ]. *) + true with a negative last divisor. For instance + [ 3/(-2)/(-2) = 1 <> 0 = 3 / (-2*-2) ], or + [ 5/2/(-2) = -1 <> -2 = 5 / (2*-2) ]. *) -Lemma div_div : forall a b c, 0<b -> 0<c -> +Lemma div_div : forall a b c, b~=0 -> 0<c -> (a/b)/c == a/(b*c). Proof. intros a b c Hb Hc. apply div_unique with (b*((a/b) mod c) + a mod b). (* begin 0<= ... <b*c \/ ... *) + apply neg_pos_cases in Hb. destruct Hb as [Hb|Hb]. + right. + destruct (mod_pos_bound (a/b) c), (mod_neg_bound a b); trivial. + split. + apply le_lt_trans with (b*((a/b) mod c) + b). + now rewrite <- mul_succ_r, <- mul_le_mono_neg_l, le_succ_l. + now rewrite <- add_lt_mono_l. + apply add_nonpos_nonpos; trivial. + apply mul_nonpos_nonneg; order. left. destruct (mod_pos_bound (a/b) c), (mod_pos_bound a b); trivial. split. @@ -609,24 +633,27 @@ Proof. apply div_mod; order. Qed. +(** Similarly, the following result doesn't always hold when [c<0]. + For instance [3 mod (-2*-2)) = 3] while + [3 mod (-2) + (-2)*((3/-2) mod -2) = -1]. +*) + +Lemma rem_mul_r : forall a b c, b~=0 -> 0<c -> + a mod (b*c) == a mod b + b*((a/b) mod c). +Proof. + intros a b c Hb Hc. + apply add_cancel_l with (b*c*(a/(b*c))). + rewrite <- div_mod by (apply neq_mul_0; split; order). + rewrite <- div_div by trivial. + rewrite add_assoc, add_shuffle0, <- mul_assoc, <- mul_add_distr_l. + rewrite <- div_mod by order. + apply div_mod; order. +Qed. + (** A last inequality: *) Theorem div_mul_le: forall a b c, 0<=a -> 0<b -> 0<=c -> c*(a/b) <= (c*a)/b. Proof. exact div_mul_le. Qed. -(** mod is related to divisibility *) - -Lemma mod_divides : forall a b, b~=0 -> - (a mod b == 0 <-> exists c, a == b*c). -Proof. -intros a b Hb. split. -intros Hab. exists (a/b). rewrite (div_mod a b Hb) at 1. - rewrite Hab. now nzsimpl. -intros (c,Hc). -rewrite Hc, mul_comm. -now apply mod_mul. -Qed. - -End ZDivPropFunct. - +End ZDivProp. |