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Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZDivTrunc.v')
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZDivTrunc.v | 596 |
1 files changed, 348 insertions, 248 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZDivTrunc.v b/theories/Numbers/Integer/Abstract/ZDivTrunc.v index 069d8a8d..bd8b6ce2 100644 --- a/theories/Numbers/Integer/Abstract/ZDivTrunc.v +++ b/theories/Numbers/Integer/Abstract/ZDivTrunc.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 (Trunc convention) We use here the convention known as Trunc, or Round-Toward-Zero, @@ -24,25 +26,24 @@ See files [ZDivFloor] and [ZDivEucl] for others conventions. *) -Require Import ZAxioms ZProperties NZDiv. - -Module Type ZDivSpecific (Import Z:ZAxiomsSig')(Import DM : DivMod' Z). - Axiom mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b. - Axiom mod_opp_l : forall a b, b ~= 0 -> (-a) mod b == - (a mod b). - Axiom mod_opp_r : forall a b, b ~= 0 -> a mod (-b) == a mod b. -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 ZQuotProp + (Import A : ZAxiomsSig') + (Import B : ZMulOrderProp A) + (Import C : ZSgnAbsProp A B). (** We benefit from what already exists for NZ *) - Module Import NZDivP := NZDivPropFunct Z ZP Z. + Module Import Private_Div. + Module Quot2Div <: NZDiv A. + Definition div := quot. + Definition modulo := A.rem. + Definition div_wd := quot_wd. + Definition mod_wd := rem_wd. + Definition div_mod := quot_rem. + Definition mod_bound_pos := rem_bound_pos. + End Quot2Div. + Module NZQuot := Nop <+ NZDivProp A Quot2Div B. + End Private_Div. Ltac pos_or_neg a := let LT := fresh "LT" in @@ -51,175 +52,274 @@ Ltac pos_or_neg a := (** Another formulation of the main equation *) -Lemma mod_eq : - forall a b, b~=0 -> a mod b == a - b*(a/b). +Lemma rem_eq : + forall a b, b~=0 -> a rem b == a - b*(a÷b). Proof. intros. rewrite <- add_move_l. -symmetry. now apply div_mod. +symmetry. now apply quot_rem. Qed. (** A few sign rules (simple ones) *) -Lemma mod_opp_opp : forall a b, b ~= 0 -> (-a) mod (-b) == - (a mod b). -Proof. intros. now rewrite mod_opp_r, mod_opp_l. Qed. +Lemma rem_opp_opp : forall a b, b ~= 0 -> (-a) rem (-b) == - (a rem b). +Proof. intros. now rewrite rem_opp_r, rem_opp_l. Qed. -Lemma div_opp_l : forall a b, b ~= 0 -> (-a)/b == -(a/b). +Lemma quot_opp_l : forall a b, b ~= 0 -> (-a)÷b == -(a÷b). Proof. intros. rewrite <- (mul_cancel_l _ _ b) by trivial. -rewrite <- (add_cancel_r _ _ ((-a) mod b)). -now rewrite <- div_mod, mod_opp_l, mul_opp_r, <- opp_add_distr, <- div_mod. +rewrite <- (add_cancel_r _ _ ((-a) rem b)). +now rewrite <- quot_rem, rem_opp_l, mul_opp_r, <- opp_add_distr, <- quot_rem. Qed. -Lemma div_opp_r : forall a b, b ~= 0 -> a/(-b) == -(a/b). +Lemma quot_opp_r : forall a b, b ~= 0 -> a÷(-b) == -(a÷b). Proof. intros. assert (-b ~= 0) by (now rewrite eq_opp_l, opp_0). rewrite <- (mul_cancel_l _ _ (-b)) by trivial. -rewrite <- (add_cancel_r _ _ (a mod (-b))). -now rewrite <- div_mod, mod_opp_r, mul_opp_opp, <- div_mod. -Qed. - -Lemma div_opp_opp : forall a b, b ~= 0 -> (-a)/(-b) == a/b. -Proof. intros. now rewrite div_opp_r, div_opp_l, opp_involutive. Qed. - -(** The sign of [a mod b] is the one of [a] *) - -(* TODO: a proper sgn function and theory *) - -Lemma mod_sign : forall a b, b~=0 -> 0 <= (a mod b) * a. -Proof. -assert (Aux : forall a b, 0<b -> 0 <= (a mod b) * a). - intros. pos_or_neg a. - apply mul_nonneg_nonneg; trivial. now destruct (mod_bound a b). - rewrite <- mul_opp_opp, <- mod_opp_l by order. - apply mul_nonneg_nonneg; try order. destruct (mod_bound (-a) b); order. -intros. pos_or_neg b. apply Aux; order. -rewrite <- mod_opp_r by order. apply Aux; order. +rewrite <- (add_cancel_r _ _ (a rem (-b))). +now rewrite <- quot_rem, rem_opp_r, mul_opp_opp, <- quot_rem. Qed. +Lemma quot_opp_opp : forall a b, b ~= 0 -> (-a)÷(-b) == a÷b. +Proof. intros. now rewrite quot_opp_r, quot_opp_l, opp_involutive. Qed. (** Uniqueness theorems *) -Theorem div_mod_unique : forall b q1 q2 r1 r2 : t, +Theorem quot_rem_unique : forall b q1 q2 r1 r2 : t, (0<=r1<b \/ b<r1<=0) -> (0<=r2<b \/ b<r2<=0) -> b*q1+r1 == b*q2+r2 -> q1 == q2 /\ r1 == r2. Proof. intros b q1 q2 r1 r2 Hr1 Hr2 EQ. destruct Hr1; destruct Hr2; try (intuition; order). -apply div_mod_unique with b; trivial. +apply NZQuot.div_mod_unique with b; trivial. rewrite <- (opp_inj_wd r1 r2). -apply div_mod_unique with (-b); trivial. +apply NZQuot.div_mod_unique with (-b); trivial. rewrite <- opp_lt_mono, opp_nonneg_nonpos; tauto. rewrite <- opp_lt_mono, opp_nonneg_nonpos; tauto. now rewrite 2 mul_opp_l, <- 2 opp_add_distr, opp_inj_wd. Qed. -Theorem div_unique: - forall a b q r, 0<=a -> 0<=r<b -> a == b*q + r -> q == a/b. -Proof. intros; now apply div_unique with r. Qed. +Theorem quot_unique: + forall a b q r, 0<=a -> 0<=r<b -> a == b*q + r -> q == a÷b. +Proof. intros; now apply NZQuot.div_unique with r. Qed. -Theorem mod_unique: - forall a b q r, 0<=a -> 0<=r<b -> a == b*q + r -> r == a mod b. -Proof. intros; now apply mod_unique with q. Qed. +Theorem rem_unique: + forall a b q r, 0<=a -> 0<=r<b -> a == b*q + r -> r == a rem b. +Proof. intros; now apply NZQuot.mod_unique with q. Qed. (** A division by itself returns 1 *) -Lemma div_same : forall a, a~=0 -> a/a == 1. +Lemma quot_same : forall a, a~=0 -> a÷a == 1. Proof. -intros. pos_or_neg a. apply div_same; order. -rewrite <- div_opp_opp by trivial. now apply div_same. +intros. pos_or_neg a. apply NZQuot.div_same; order. +rewrite <- quot_opp_opp by trivial. now apply NZQuot.div_same. Qed. -Lemma mod_same : forall a, a~=0 -> a mod a == 0. +Lemma rem_same : forall a, a~=0 -> a rem a == 0. Proof. -intros. rewrite mod_eq, div_same by trivial. nzsimpl. apply sub_diag. +intros. rewrite rem_eq, quot_same by trivial. nzsimpl. apply sub_diag. Qed. (** A division of a small number by a bigger one yields zero. *) -Theorem div_small: forall a b, 0<=a<b -> a/b == 0. -Proof. exact div_small. Qed. +Theorem quot_small: forall a b, 0<=a<b -> a÷b == 0. +Proof. exact NZQuot.div_small. Qed. -(** Same situation, in term of modulo: *) +(** Same situation, in term of remulo: *) -Theorem mod_small: forall a b, 0<=a<b -> a mod b == a. -Proof. exact mod_small. Qed. +Theorem rem_small: forall a b, 0<=a<b -> a rem b == a. +Proof. exact NZQuot.mod_small. Qed. (** * Basic values of divisions and modulo. *) -Lemma div_0_l: forall a, a~=0 -> 0/a == 0. +Lemma quot_0_l: forall a, a~=0 -> 0÷a == 0. Proof. -intros. pos_or_neg a. apply div_0_l; order. -rewrite <- div_opp_opp, opp_0 by trivial. now apply div_0_l. +intros. pos_or_neg a. apply NZQuot.div_0_l; order. +rewrite <- quot_opp_opp, opp_0 by trivial. now apply NZQuot.div_0_l. Qed. -Lemma mod_0_l: forall a, a~=0 -> 0 mod a == 0. +Lemma rem_0_l: forall a, a~=0 -> 0 rem a == 0. Proof. -intros; rewrite mod_eq, div_0_l; now nzsimpl. +intros; rewrite rem_eq, quot_0_l; now nzsimpl. Qed. -Lemma div_1_r: forall a, a/1 == a. +Lemma quot_1_r: forall a, a÷1 == a. Proof. -intros. pos_or_neg a. now apply div_1_r. -apply opp_inj. rewrite <- div_opp_l. apply div_1_r; order. +intros. pos_or_neg a. now apply NZQuot.div_1_r. +apply opp_inj. rewrite <- quot_opp_l. apply NZQuot.div_1_r; order. intro EQ; symmetry in EQ; revert EQ; apply lt_neq, lt_0_1. Qed. -Lemma mod_1_r: forall a, a mod 1 == 0. +Lemma rem_1_r: forall a, a rem 1 == 0. Proof. -intros. rewrite mod_eq, div_1_r; nzsimpl; auto using sub_diag. +intros. rewrite rem_eq, quot_1_r; nzsimpl; auto using sub_diag. intro EQ; symmetry in EQ; revert EQ; apply lt_neq; apply lt_0_1. Qed. -Lemma div_1_l: forall a, 1<a -> 1/a == 0. -Proof. exact div_1_l. Qed. +Lemma quot_1_l: forall a, 1<a -> 1÷a == 0. +Proof. exact NZQuot.div_1_l. Qed. + +Lemma rem_1_l: forall a, 1<a -> 1 rem a == 1. +Proof. exact NZQuot.mod_1_l. Qed. -Lemma mod_1_l: forall a, 1<a -> 1 mod a == 1. -Proof. exact mod_1_l. Qed. +Lemma quot_mul : forall a b, b~=0 -> (a*b)÷b == a. +Proof. +intros. pos_or_neg a; pos_or_neg b. apply NZQuot.div_mul; order. +rewrite <- quot_opp_opp, <- mul_opp_r by order. apply NZQuot.div_mul; order. +rewrite <- opp_inj_wd, <- quot_opp_l, <- mul_opp_l by order. +apply NZQuot.div_mul; order. +rewrite <- opp_inj_wd, <- quot_opp_r, <- mul_opp_opp by order. +apply NZQuot.div_mul; order. +Qed. -Lemma div_mul : forall a b, b~=0 -> (a*b)/b == a. +Lemma rem_mul : forall a b, b~=0 -> (a*b) rem b == 0. Proof. -intros. pos_or_neg a; pos_or_neg b. apply div_mul; order. -rewrite <- div_opp_opp, <- mul_opp_r by order. apply div_mul; order. -rewrite <- opp_inj_wd, <- div_opp_l, <- mul_opp_l by order. apply div_mul; order. -rewrite <- opp_inj_wd, <- div_opp_r, <- mul_opp_opp by order. apply div_mul; order. +intros. rewrite rem_eq, quot_mul by trivial. rewrite mul_comm; apply sub_diag. Qed. -Lemma mod_mul : forall a b, b~=0 -> (a*b) mod b == 0. +Theorem quot_unique_exact a b q: b~=0 -> a == b*q -> q == a÷b. Proof. -intros. rewrite mod_eq, div_mul by trivial. rewrite mul_comm; apply sub_diag. + intros Hb H. rewrite H, mul_comm. symmetry. now apply quot_mul. Qed. -(** * Order results about mod and div *) +(** The sign of [a rem b] is the one of [a] (when it's not null) *) + +Lemma rem_nonneg : forall a b, b~=0 -> 0 <= a -> 0 <= a rem b. +Proof. + intros. pos_or_neg b. destruct (rem_bound_pos a b); order. + rewrite <- rem_opp_r; trivial. + destruct (rem_bound_pos a (-b)); trivial. +Qed. + +Lemma rem_nonpos : forall a b, b~=0 -> a <= 0 -> a rem b <= 0. +Proof. + intros a b Hb Ha. + apply opp_nonneg_nonpos. apply opp_nonneg_nonpos in Ha. + rewrite <- rem_opp_l by trivial. now apply rem_nonneg. +Qed. + +Lemma rem_sign_mul : forall a b, b~=0 -> 0 <= (a rem b) * a. +Proof. +intros a b Hb. destruct (le_ge_cases 0 a). + apply mul_nonneg_nonneg; trivial. now apply rem_nonneg. + apply mul_nonpos_nonpos; trivial. now apply rem_nonpos. +Qed. + +Lemma rem_sign_nz : forall a b, b~=0 -> a rem b ~= 0 -> + sgn (a rem b) == sgn a. +Proof. +intros a b Hb H. destruct (lt_trichotomy 0 a) as [LT|[EQ|LT]]. +rewrite 2 sgn_pos; try easy. + generalize (rem_nonneg a b Hb (lt_le_incl _ _ LT)). order. +now rewrite <- EQ, rem_0_l, sgn_0. +rewrite 2 sgn_neg; try easy. + generalize (rem_nonpos a b Hb (lt_le_incl _ _ LT)). order. +Qed. + +Lemma rem_sign : forall a b, a~=0 -> b~=0 -> sgn (a rem b) ~= -sgn a. +Proof. +intros a b Ha Hb H. +destruct (eq_decidable (a rem b) 0) as [EQ|NEQ]. +apply Ha, sgn_null_iff, opp_inj. now rewrite <- H, opp_0, EQ, sgn_0. +apply Ha, sgn_null_iff. apply eq_mul_0_l with 2; try order'. nzsimpl'. +apply add_move_0_l. rewrite <- H. symmetry. now apply rem_sign_nz. +Qed. + +(** Operations and absolute value *) + +Lemma rem_abs_l : forall a b, b ~= 0 -> (abs a) rem b == abs (a rem b). +Proof. +intros a b Hb. destruct (le_ge_cases 0 a) as [LE|LE]. +rewrite 2 abs_eq; try easy. now apply rem_nonneg. +rewrite 2 abs_neq, rem_opp_l; try easy. now apply rem_nonpos. +Qed. + +Lemma rem_abs_r : forall a b, b ~= 0 -> a rem (abs b) == a rem b. +Proof. +intros a b Hb. destruct (le_ge_cases 0 b). +now rewrite abs_eq. now rewrite abs_neq, ?rem_opp_r. +Qed. + +Lemma rem_abs : forall a b, b ~= 0 -> (abs a) rem (abs b) == abs (a rem b). +Proof. +intros. now rewrite rem_abs_r, rem_abs_l. +Qed. + +Lemma quot_abs_l : forall a b, b ~= 0 -> (abs a)÷b == (sgn a)*(a÷b). +Proof. +intros a b Hb. destruct (lt_trichotomy 0 a) as [LT|[EQ|LT]]. +rewrite abs_eq, sgn_pos by order. now nzsimpl. +rewrite <- EQ, abs_0, quot_0_l; trivial. now nzsimpl. +rewrite abs_neq, quot_opp_l, sgn_neg by order. + rewrite mul_opp_l. now nzsimpl. +Qed. + +Lemma quot_abs_r : forall a b, b ~= 0 -> a÷(abs b) == (sgn b)*(a÷b). +Proof. +intros a b Hb. destruct (lt_trichotomy 0 b) as [LT|[EQ|LT]]. +rewrite abs_eq, sgn_pos by order. now nzsimpl. +order. +rewrite abs_neq, quot_opp_r, sgn_neg by order. + rewrite mul_opp_l. now nzsimpl. +Qed. + +Lemma quot_abs : forall a b, b ~= 0 -> (abs a)÷(abs b) == abs (a÷b). +Proof. +intros a b Hb. +pos_or_neg a; [rewrite (abs_eq a)|rewrite (abs_neq a)]; + try apply opp_nonneg_nonpos; try order. +pos_or_neg b; [rewrite (abs_eq b)|rewrite (abs_neq b)]; + try apply opp_nonneg_nonpos; try order. +rewrite abs_eq; try easy. apply NZQuot.div_pos; order. +rewrite <- abs_opp, <- quot_opp_r, abs_eq; try easy. + apply NZQuot.div_pos; order. +pos_or_neg b; [rewrite (abs_eq b)|rewrite (abs_neq b)]; + try apply opp_nonneg_nonpos; try order. +rewrite <- (abs_opp (_÷_)), <- quot_opp_l, abs_eq; try easy. + apply NZQuot.div_pos; order. +rewrite <- (quot_opp_opp a b), abs_eq; try easy. + apply NZQuot.div_pos; order. +Qed. + +(** We have a general bound for absolute values *) + +Lemma rem_bound_abs : + forall a b, b~=0 -> abs (a rem b) < abs b. +Proof. +intros. rewrite <- rem_abs; trivial. +apply rem_bound_pos. apply abs_nonneg. now apply abs_pos. +Qed. + +(** * Order results about rem and quot *) (** A modulo cannot grow beyond its starting point. *) -Theorem mod_le: forall a b, 0<=a -> 0<b -> a mod b <= a. -Proof. exact mod_le. Qed. +Theorem rem_le: forall a b, 0<=a -> 0<b -> a rem b <= a. +Proof. exact NZQuot.mod_le. Qed. -Theorem div_pos : forall a b, 0<=a -> 0<b -> 0<= a/b. -Proof. exact div_pos. Qed. +Theorem quot_pos : forall a b, 0<=a -> 0<b -> 0<= a÷b. +Proof. exact NZQuot.div_pos. Qed. -Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b. -Proof. exact div_str_pos. Qed. +Lemma quot_str_pos : forall a b, 0<b<=a -> 0 < a÷b. +Proof. exact NZQuot.div_str_pos. Qed. -Lemma div_small_iff : forall a b, b~=0 -> (a/b==0 <-> abs a < abs b). +Lemma quot_small_iff : forall a b, b~=0 -> (a÷b==0 <-> abs a < abs b). Proof. intros. pos_or_neg a; pos_or_neg b. -rewrite div_small_iff; try order. rewrite 2 abs_eq; intuition; order. -rewrite <- opp_inj_wd, opp_0, <- div_opp_r, div_small_iff by order. +rewrite NZQuot.div_small_iff; try order. rewrite 2 abs_eq; intuition; order. +rewrite <- opp_inj_wd, opp_0, <- quot_opp_r, NZQuot.div_small_iff by order. rewrite (abs_eq a), (abs_neq' b); intuition; order. -rewrite <- opp_inj_wd, opp_0, <- div_opp_l, div_small_iff by order. +rewrite <- opp_inj_wd, opp_0, <- quot_opp_l, NZQuot.div_small_iff by order. rewrite (abs_neq' a), (abs_eq b); intuition; order. -rewrite <- div_opp_opp, div_small_iff by order. +rewrite <- quot_opp_opp, NZQuot.div_small_iff by order. rewrite (abs_neq' a), (abs_neq' b); intuition; order. Qed. -Lemma mod_small_iff : forall a b, b~=0 -> (a mod b == a <-> abs a < abs b). +Lemma rem_small_iff : forall a b, b~=0 -> (a rem b == a <-> abs a < abs b). Proof. -intros. rewrite mod_eq, <- div_small_iff by order. +intros. rewrite rem_eq, <- quot_small_iff by order. rewrite sub_move_r, <- (add_0_r a) at 1. rewrite add_cancel_l. rewrite eq_sym_iff, eq_mul_0. tauto. Qed. @@ -227,306 +327,306 @@ Qed. (** As soon as the divisor is strictly greater than 1, the division is strictly decreasing. *) -Lemma div_lt : forall a b, 0<a -> 1<b -> a/b < a. -Proof. exact div_lt. Qed. +Lemma quot_lt : forall a b, 0<a -> 1<b -> a÷b < a. +Proof. exact NZQuot.div_lt. Qed. (** [le] is compatible with a positive division. *) -Lemma div_le_mono : forall a b c, 0<c -> a<=b -> a/c <= b/c. +Lemma quot_le_mono : forall a b c, 0<c -> a<=b -> a÷c <= b÷c. Proof. -intros. pos_or_neg a. apply div_le_mono; auto. +intros. pos_or_neg a. apply NZQuot.div_le_mono; auto. pos_or_neg b. apply le_trans with 0. - rewrite <- opp_nonneg_nonpos, <- div_opp_l by order. - apply div_pos; order. - apply div_pos; order. -rewrite opp_le_mono in *. rewrite <- 2 div_opp_l by order. - apply div_le_mono; intuition; order. + rewrite <- opp_nonneg_nonpos, <- quot_opp_l by order. + apply quot_pos; order. + apply quot_pos; order. +rewrite opp_le_mono in *. rewrite <- 2 quot_opp_l by order. + apply NZQuot.div_le_mono; intuition; order. Qed. (** With this choice of division, - rounding of div is always done toward zero: *) + rounding of quot is always done toward zero: *) -Lemma mul_div_le : forall a b, 0<=a -> b~=0 -> 0 <= b*(a/b) <= a. +Lemma mul_quot_le : forall a b, 0<=a -> b~=0 -> 0 <= b*(a÷b) <= a. Proof. intros. pos_or_neg b. split. -apply mul_nonneg_nonneg; [|apply div_pos]; order. -apply mul_div_le; order. -rewrite <- mul_opp_opp, <- div_opp_r by order. +apply mul_nonneg_nonneg; [|apply quot_pos]; order. +apply NZQuot.mul_div_le; order. +rewrite <- mul_opp_opp, <- quot_opp_r by order. split. -apply mul_nonneg_nonneg; [|apply div_pos]; order. -apply mul_div_le; order. +apply mul_nonneg_nonneg; [|apply quot_pos]; order. +apply NZQuot.mul_div_le; order. Qed. -Lemma mul_div_ge : forall a b, a<=0 -> b~=0 -> a <= b*(a/b) <= 0. +Lemma mul_quot_ge : forall a b, a<=0 -> b~=0 -> a <= b*(a÷b) <= 0. Proof. intros. -rewrite <- opp_nonneg_nonpos, opp_le_mono, <-mul_opp_r, <-div_opp_l by order. +rewrite <- opp_nonneg_nonpos, opp_le_mono, <-mul_opp_r, <-quot_opp_l by order. rewrite <- opp_nonneg_nonpos in *. -destruct (mul_div_le (-a) b); tauto. +destruct (mul_quot_le (-a) b); tauto. Qed. -(** For positive numbers, considering [S (a/b)] leads to an upper bound for [a] *) +(** For positive numbers, considering [S (a÷b)] leads to an upper bound for [a] *) -Lemma mul_succ_div_gt: forall a b, 0<=a -> 0<b -> a < b*(S (a/b)). -Proof. exact mul_succ_div_gt. Qed. +Lemma mul_succ_quot_gt: forall a b, 0<=a -> 0<b -> a < b*(S (a÷b)). +Proof. exact NZQuot.mul_succ_div_gt. Qed. (** Similar results with negative numbers *) -Lemma mul_pred_div_lt: forall a b, a<=0 -> 0<b -> b*(P (a/b)) < a. +Lemma mul_pred_quot_lt: forall a b, a<=0 -> 0<b -> b*(P (a÷b)) < a. Proof. intros. -rewrite opp_lt_mono, <- mul_opp_r, opp_pred, <- div_opp_l by order. +rewrite opp_lt_mono, <- mul_opp_r, opp_pred, <- quot_opp_l by order. rewrite <- opp_nonneg_nonpos in *. -now apply mul_succ_div_gt. +now apply mul_succ_quot_gt. Qed. -Lemma mul_pred_div_gt: forall a b, 0<=a -> b<0 -> a < b*(P (a/b)). +Lemma mul_pred_quot_gt: forall a b, 0<=a -> b<0 -> a < b*(P (a÷b)). Proof. intros. -rewrite <- mul_opp_opp, opp_pred, <- div_opp_r by order. +rewrite <- mul_opp_opp, opp_pred, <- quot_opp_r by order. rewrite <- opp_pos_neg in *. -now apply mul_succ_div_gt. +now apply mul_succ_quot_gt. Qed. -Lemma mul_succ_div_lt: forall a b, a<=0 -> b<0 -> b*(S (a/b)) < a. +Lemma mul_succ_quot_lt: forall a b, a<=0 -> b<0 -> b*(S (a÷b)) < a. Proof. intros. -rewrite opp_lt_mono, <- mul_opp_l, <- div_opp_opp by order. +rewrite opp_lt_mono, <- mul_opp_l, <- quot_opp_opp by order. rewrite <- opp_nonneg_nonpos, <- opp_pos_neg in *. -now apply mul_succ_div_gt. +now apply mul_succ_quot_gt. Qed. -(** Inequality [mul_div_le] is exact iff the modulo is zero. *) +(** Inequality [mul_quot_le] is exact iff the modulo is zero. *) -Lemma div_exact : forall a b, b~=0 -> (a == b*(a/b) <-> a mod b == 0). +Lemma quot_exact : forall a b, b~=0 -> (a == b*(a÷b) <-> a rem b == 0). Proof. -intros. rewrite mod_eq by order. rewrite sub_move_r; nzsimpl; tauto. +intros. rewrite rem_eq by order. rewrite sub_move_r; nzsimpl; tauto. Qed. -(** Some additionnal inequalities about div. *) +(** Some additionnal inequalities about quot. *) -Theorem div_lt_upper_bound: - forall a b q, 0<=a -> 0<b -> a < b*q -> a/b < q. -Proof. exact div_lt_upper_bound. Qed. +Theorem quot_lt_upper_bound: + forall a b q, 0<=a -> 0<b -> a < b*q -> a÷b < q. +Proof. exact NZQuot.div_lt_upper_bound. Qed. -Theorem div_le_upper_bound: - forall a b q, 0<b -> a <= b*q -> a/b <= q. +Theorem quot_le_upper_bound: + forall a b q, 0<b -> a <= b*q -> a÷b <= q. Proof. intros. -rewrite <- (div_mul q b) by order. -apply div_le_mono; trivial. now rewrite mul_comm. +rewrite <- (quot_mul q b) by order. +apply quot_le_mono; trivial. now rewrite mul_comm. Qed. -Theorem div_le_lower_bound: - forall a b q, 0<b -> b*q <= a -> q <= a/b. +Theorem quot_le_lower_bound: + forall a b q, 0<b -> b*q <= a -> q <= a÷b. Proof. intros. -rewrite <- (div_mul q b) by order. -apply div_le_mono; trivial. now rewrite mul_comm. +rewrite <- (quot_mul q b) by order. +apply quot_le_mono; trivial. now rewrite mul_comm. Qed. (** A division respects opposite monotonicity for the divisor *) -Lemma div_le_compat_l: forall p q r, 0<=p -> 0<q<=r -> p/r <= p/q. -Proof. exact div_le_compat_l. Qed. +Lemma quot_le_compat_l: forall p q r, 0<=p -> 0<q<=r -> p÷r <= p÷q. +Proof. exact NZQuot.div_le_compat_l. Qed. -(** * Relations between usual operations and mod and div *) +(** * Relations between usual operations and rem and quot *) (** Unlike with other division conventions, some results here aren't always valid, and need to be restricted. For instance - [(a+b*c) mod c <> a mod c] for [a=9,b=-5,c=2] *) + [(a+b*c) rem c <> a rem c] for [a=9,b=-5,c=2] *) -Lemma mod_add : forall a b c, c~=0 -> 0 <= (a+b*c)*a -> - (a + b * c) mod c == a mod c. +Lemma rem_add : forall a b c, c~=0 -> 0 <= (a+b*c)*a -> + (a + b * c) rem c == a rem c. Proof. -assert (forall a b c, c~=0 -> 0<=a -> 0<=a+b*c -> (a+b*c) mod c == a mod c). - intros. pos_or_neg c. apply mod_add; order. - rewrite <- (mod_opp_r a), <- (mod_opp_r (a+b*c)) by order. +assert (forall a b c, c~=0 -> 0<=a -> 0<=a+b*c -> (a+b*c) rem c == a rem c). + intros. pos_or_neg c. apply NZQuot.mod_add; order. + rewrite <- (rem_opp_r a), <- (rem_opp_r (a+b*c)) by order. rewrite <- mul_opp_opp in *. - apply mod_add; order. + apply NZQuot.mod_add; order. intros a b c Hc Habc. destruct (le_0_mul _ _ Habc) as [(Habc',Ha)|(Habc',Ha)]. auto. apply opp_inj. revert Ha Habc'. rewrite <- 2 opp_nonneg_nonpos. -rewrite <- 2 mod_opp_l, opp_add_distr, <- mul_opp_l by order. auto. +rewrite <- 2 rem_opp_l, opp_add_distr, <- mul_opp_l by order. auto. Qed. -Lemma div_add : forall a b c, c~=0 -> 0 <= (a+b*c)*a -> - (a + b * c) / c == a / c + b. +Lemma quot_add : forall a b c, c~=0 -> 0 <= (a+b*c)*a -> + (a + b * c) ÷ c == a ÷ c + b. Proof. intros. rewrite <- (mul_cancel_l _ _ c) by trivial. -rewrite <- (add_cancel_r _ _ ((a+b*c) mod c)). -rewrite <- div_mod, mod_add by trivial. -now rewrite mul_add_distr_l, add_shuffle0, <-div_mod, mul_comm. +rewrite <- (add_cancel_r _ _ ((a+b*c) rem c)). +rewrite <- quot_rem, rem_add by trivial. +now rewrite mul_add_distr_l, add_shuffle0, <-quot_rem, mul_comm. Qed. -Lemma div_add_l: forall a b c, b~=0 -> 0 <= (a*b+c)*c -> - (a * b + c) / b == a + c / b. +Lemma quot_add_l: forall a b c, b~=0 -> 0 <= (a*b+c)*c -> + (a * b + c) ÷ b == a + c ÷ b. Proof. - intros a b c. rewrite add_comm, (add_comm a). now apply div_add. + intros a b c. rewrite add_comm, (add_comm a). now apply quot_add. Qed. (** Cancellations. *) -Lemma div_mul_cancel_r : forall a b c, b~=0 -> c~=0 -> - (a*c)/(b*c) == a/b. +Lemma quot_mul_cancel_r : forall a b c, b~=0 -> c~=0 -> + (a*c)÷(b*c) == a÷b. Proof. -assert (Aux1 : forall a b c, 0<=a -> 0<b -> c~=0 -> (a*c)/(b*c) == a/b). - intros. pos_or_neg c. apply div_mul_cancel_r; order. - rewrite <- div_opp_opp, <- 2 mul_opp_r. apply div_mul_cancel_r; order. +assert (Aux1 : forall a b c, 0<=a -> 0<b -> c~=0 -> (a*c)÷(b*c) == a÷b). + intros. pos_or_neg c. apply NZQuot.div_mul_cancel_r; order. + rewrite <- quot_opp_opp, <- 2 mul_opp_r. apply NZQuot.div_mul_cancel_r; order. rewrite <- neq_mul_0; intuition order. -assert (Aux2 : forall a b c, 0<=a -> b~=0 -> c~=0 -> (a*c)/(b*c) == a/b). +assert (Aux2 : forall a b c, 0<=a -> b~=0 -> c~=0 -> (a*c)÷(b*c) == a÷b). intros. pos_or_neg b. apply Aux1; order. - apply opp_inj. rewrite <- 2 div_opp_r, <- mul_opp_l; try order. apply Aux1; order. + apply opp_inj. rewrite <- 2 quot_opp_r, <- mul_opp_l; try order. apply Aux1; order. rewrite <- neq_mul_0; intuition order. intros. pos_or_neg a. apply Aux2; order. -apply opp_inj. rewrite <- 2 div_opp_l, <- mul_opp_l; try order. apply Aux2; order. +apply opp_inj. rewrite <- 2 quot_opp_l, <- mul_opp_l; try order. apply Aux2; order. rewrite <- neq_mul_0; intuition order. Qed. -Lemma div_mul_cancel_l : forall a b c, b~=0 -> c~=0 -> - (c*a)/(c*b) == a/b. +Lemma quot_mul_cancel_l : forall a b c, b~=0 -> c~=0 -> + (c*a)÷(c*b) == a÷b. Proof. -intros. rewrite !(mul_comm c); now apply div_mul_cancel_r. +intros. rewrite !(mul_comm c); now apply quot_mul_cancel_r. Qed. -Lemma mul_mod_distr_r: forall a b c, b~=0 -> c~=0 -> - (a*c) mod (b*c) == (a mod b) * c. +Lemma mul_rem_distr_r: forall a b c, b~=0 -> c~=0 -> + (a*c) rem (b*c) == (a rem b) * c. Proof. intros. assert (b*c ~= 0) by (rewrite <- neq_mul_0; tauto). -rewrite ! mod_eq by trivial. -rewrite div_mul_cancel_r by order. -now rewrite mul_sub_distr_r, <- !mul_assoc, (mul_comm (a/b) c). +rewrite ! rem_eq by trivial. +rewrite quot_mul_cancel_r by order. +now rewrite mul_sub_distr_r, <- !mul_assoc, (mul_comm (a÷b) c). Qed. -Lemma mul_mod_distr_l: forall a b c, b~=0 -> c~=0 -> - (c*a) mod (c*b) == c * (a mod b). +Lemma mul_rem_distr_l: forall a b c, b~=0 -> c~=0 -> + (c*a) rem (c*b) == c * (a rem b). Proof. -intros; rewrite !(mul_comm c); now apply mul_mod_distr_r. +intros; rewrite !(mul_comm c); now apply mul_rem_distr_r. Qed. (** Operations modulo. *) -Theorem mod_mod: forall a n, n~=0 -> - (a mod n) mod n == a mod n. +Theorem rem_rem: forall a n, n~=0 -> + (a rem n) rem n == a rem n. Proof. -intros. pos_or_neg a; pos_or_neg n. apply mod_mod; order. -rewrite <- ! (mod_opp_r _ n) by trivial. apply mod_mod; order. -apply opp_inj. rewrite <- !mod_opp_l by order. apply mod_mod; order. -apply opp_inj. rewrite <- !mod_opp_opp by order. apply mod_mod; order. +intros. pos_or_neg a; pos_or_neg n. apply NZQuot.mod_mod; order. +rewrite <- ! (rem_opp_r _ n) by trivial. apply NZQuot.mod_mod; order. +apply opp_inj. rewrite <- !rem_opp_l by order. apply NZQuot.mod_mod; order. +apply opp_inj. rewrite <- !rem_opp_opp by order. apply NZQuot.mod_mod; order. Qed. -Lemma mul_mod_idemp_l : forall a b n, n~=0 -> - ((a mod n)*b) mod n == (a*b) mod n. +Lemma mul_rem_idemp_l : forall a b n, n~=0 -> + ((a rem n)*b) rem n == (a*b) rem n. Proof. assert (Aux1 : forall a b n, 0<=a -> 0<=b -> n~=0 -> - ((a mod n)*b) mod n == (a*b) mod n). - intros. pos_or_neg n. apply mul_mod_idemp_l; order. - rewrite <- ! (mod_opp_r _ n) by order. apply mul_mod_idemp_l; order. + ((a rem n)*b) rem n == (a*b) rem n). + intros. pos_or_neg n. apply NZQuot.mul_mod_idemp_l; order. + rewrite <- ! (rem_opp_r _ n) by order. apply NZQuot.mul_mod_idemp_l; order. assert (Aux2 : forall a b n, 0<=a -> n~=0 -> - ((a mod n)*b) mod n == (a*b) mod n). + ((a rem n)*b) rem n == (a*b) rem n). intros. pos_or_neg b. now apply Aux1. - apply opp_inj. rewrite <-2 mod_opp_l, <-2 mul_opp_r by order. + apply opp_inj. rewrite <-2 rem_opp_l, <-2 mul_opp_r by order. apply Aux1; order. intros a b n Hn. pos_or_neg a. now apply Aux2. -apply opp_inj. rewrite <-2 mod_opp_l, <-2 mul_opp_l, <-mod_opp_l by order. +apply opp_inj. rewrite <-2 rem_opp_l, <-2 mul_opp_l, <-rem_opp_l by order. apply Aux2; order. Qed. -Lemma mul_mod_idemp_r : forall a b n, n~=0 -> - (a*(b mod n)) mod n == (a*b) mod n. +Lemma mul_rem_idemp_r : forall a b n, n~=0 -> + (a*(b rem n)) rem n == (a*b) rem n. Proof. -intros. rewrite !(mul_comm a). now apply mul_mod_idemp_l. +intros. rewrite !(mul_comm a). now apply mul_rem_idemp_l. Qed. -Theorem mul_mod: forall a b n, n~=0 -> - (a * b) mod n == ((a mod n) * (b mod n)) mod n. +Theorem mul_rem: forall a b n, n~=0 -> + (a * b) rem n == ((a rem n) * (b rem n)) rem n. Proof. -intros. now rewrite mul_mod_idemp_l, mul_mod_idemp_r. +intros. now rewrite mul_rem_idemp_l, mul_rem_idemp_r. Qed. (** addition and modulo Generally speaking, unlike with other conventions, we don't have - [(a+b) mod n = (a mod n + b mod n) mod n] + [(a+b) rem n = (a rem n + b rem n) rem n] for any a and b. - For instance, take (8 + (-10)) mod 3 = -2 whereas - (8 mod 3 + (-10 mod 3)) mod 3 = 1. + For instance, take (8 + (-10)) rem 3 = -2 whereas + (8 rem 3 + (-10 rem 3)) rem 3 = 1. *) -Lemma add_mod_idemp_l : forall a b n, n~=0 -> 0 <= a*b -> - ((a mod n)+b) mod n == (a+b) mod n. +Lemma add_rem_idemp_l : forall a b n, n~=0 -> 0 <= a*b -> + ((a rem n)+b) rem n == (a+b) rem n. Proof. assert (Aux : forall a b n, 0<=a -> 0<=b -> n~=0 -> - ((a mod n)+b) mod n == (a+b) mod n). - intros. pos_or_neg n. apply add_mod_idemp_l; order. - rewrite <- ! (mod_opp_r _ n) by order. apply add_mod_idemp_l; order. + ((a rem n)+b) rem n == (a+b) rem n). + intros. pos_or_neg n. apply NZQuot.add_mod_idemp_l; order. + rewrite <- ! (rem_opp_r _ n) by order. apply NZQuot.add_mod_idemp_l; order. intros a b n Hn Hab. destruct (le_0_mul _ _ Hab) as [(Ha,Hb)|(Ha,Hb)]. now apply Aux. -apply opp_inj. rewrite <-2 mod_opp_l, 2 opp_add_distr, <-mod_opp_l by order. +apply opp_inj. rewrite <-2 rem_opp_l, 2 opp_add_distr, <-rem_opp_l by order. rewrite <- opp_nonneg_nonpos in *. now apply Aux. Qed. -Lemma add_mod_idemp_r : forall a b n, n~=0 -> 0 <= a*b -> - (a+(b mod n)) mod n == (a+b) mod n. +Lemma add_rem_idemp_r : forall a b n, n~=0 -> 0 <= a*b -> + (a+(b rem n)) rem n == (a+b) rem n. Proof. -intros. rewrite !(add_comm a). apply add_mod_idemp_l; trivial. +intros. rewrite !(add_comm a). apply add_rem_idemp_l; trivial. now rewrite mul_comm. Qed. -Theorem add_mod: forall a b n, n~=0 -> 0 <= a*b -> - (a+b) mod n == (a mod n + b mod n) mod n. +Theorem add_rem: forall a b n, n~=0 -> 0 <= a*b -> + (a+b) rem n == (a rem n + b rem n) rem n. Proof. -intros a b n Hn Hab. rewrite add_mod_idemp_l, add_mod_idemp_r; trivial. +intros a b n Hn Hab. rewrite add_rem_idemp_l, add_rem_idemp_r; trivial. reflexivity. destruct (le_0_mul _ _ Hab) as [(Ha,Hb)|(Ha,Hb)]; - destruct (le_0_mul _ _ (mod_sign b n Hn)) as [(Hb',Hm)|(Hb',Hm)]; + destruct (le_0_mul _ _ (rem_sign_mul b n Hn)) as [(Hb',Hm)|(Hb',Hm)]; auto using mul_nonneg_nonneg, mul_nonpos_nonpos. - setoid_replace b with 0 by order. rewrite mod_0_l by order. nzsimpl; order. - setoid_replace b with 0 by order. rewrite mod_0_l by order. nzsimpl; order. + setoid_replace b with 0 by order. rewrite rem_0_l by order. nzsimpl; order. + setoid_replace b with 0 by order. rewrite rem_0_l by order. nzsimpl; order. Qed. +(** Conversely, the following results need less restrictions here. *) -(** Conversely, the following result needs less restrictions here. *) - -Lemma div_div : forall a b c, b~=0 -> c~=0 -> - (a/b)/c == a/(b*c). +Lemma quot_quot : forall a b c, b~=0 -> c~=0 -> + (a÷b)÷c == a÷(b*c). Proof. -assert (Aux1 : forall a b c, 0<=a -> 0<b -> c~=0 -> (a/b)/c == a/(b*c)). - intros. pos_or_neg c. apply div_div; order. - apply opp_inj. rewrite <- 2 div_opp_r, <- mul_opp_r; trivial. - apply div_div; order. +assert (Aux1 : forall a b c, 0<=a -> 0<b -> c~=0 -> (a÷b)÷c == a÷(b*c)). + intros. pos_or_neg c. apply NZQuot.div_div; order. + apply opp_inj. rewrite <- 2 quot_opp_r, <- mul_opp_r; trivial. + apply NZQuot.div_div; order. rewrite <- neq_mul_0; intuition order. -assert (Aux2 : forall a b c, 0<=a -> b~=0 -> c~=0 -> (a/b)/c == a/(b*c)). +assert (Aux2 : forall a b c, 0<=a -> b~=0 -> c~=0 -> (a÷b)÷c == a÷(b*c)). intros. pos_or_neg b. apply Aux1; order. - apply opp_inj. rewrite <- div_opp_l, <- 2 div_opp_r, <- mul_opp_l; trivial. + apply opp_inj. rewrite <- quot_opp_l, <- 2 quot_opp_r, <- mul_opp_l; trivial. apply Aux1; trivial. rewrite <- neq_mul_0; intuition order. intros. pos_or_neg a. apply Aux2; order. -apply opp_inj. rewrite <- 3 div_opp_l; try order. apply Aux2; order. +apply opp_inj. rewrite <- 3 quot_opp_l; try order. apply Aux2; order. rewrite <- neq_mul_0. tauto. 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). +Lemma mod_mul_r : forall a b c, b~=0 -> c~=0 -> + a rem (b*c) == a rem b + b*((a÷b) rem 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. + intros a b c Hb Hc. + apply add_cancel_l with (b*c*(a÷(b*c))). + rewrite <- quot_rem by (apply neq_mul_0; split; order). + rewrite <- quot_quot by trivial. + rewrite add_assoc, add_shuffle0, <- mul_assoc, <- mul_add_distr_l. + rewrite <- quot_rem by order. + apply quot_rem; order. Qed. -End ZDivPropFunct. +(** A last inequality: *) + +Theorem quot_mul_le: + forall a b c, 0<=a -> 0<b -> 0<=c -> c*(a÷b) <= (c*a)÷b. +Proof. exact NZQuot.div_mul_le. Qed. + +End ZQuotProp. |