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diff --git a/theories/Numbers/Integer/Abstract/ZDivEucl.v b/theories/Numbers/Integer/Abstract/ZDivEucl.v new file mode 100644 index 00000000..bcd16fec --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZDivEucl.v @@ -0,0 +1,605 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(** * Euclidean Division for integers, Euclid convention + + We use here the "usual" formulation of the Euclid Theorem + [forall a b, b<>0 -> exists b q, a = b*q+r /\ 0 < r < |b| ] + + The outcome of the modulo function is hence always positive. + This corresponds to convention "E" in the following paper: + + R. Boute, "The Euclidean definition of the functions div and mod", + ACM Transactions on Programming Languages and Systems, + Vol. 14, No.2, pp. 127-144, April 1992. + + See files [ZDivTrunc] and [ZDivFloor] for others conventions. +*) + +Require Import ZAxioms ZProperties NZDiv. + +Module Type ZDivSpecific (Import Z : ZAxiomsExtSig')(Import DM : DivMod' Z). + Axiom mod_always_pos : forall a b, 0 <= a mod b < abs b. +End ZDivSpecific. + +Module Type ZDiv (Z:ZAxiomsExtSig) + := 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). + +(** 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. rewrite <- (abs_eq b) at 3 by now apply lt_le_incl. + apply mod_always_pos. + Qed. + End ZD. + Module Import NZDivP := NZDivPropFunct Z ZP ZD. + +(** Another formulation of the main equation *) + +Lemma mod_eq : + forall a b, b~=0 -> a mod b == a - b*(a/b). +Proof. +intros. +rewrite <- add_move_l. +symmetry. now apply div_mod. +Qed. + +Ltac pos_or_neg a := + let LT := fresh "LT" in + let LE := fresh "LE" in + destruct (le_gt_cases 0 a) as [LE|LT]; [|rewrite <- opp_pos_neg in LT]. + +(** Uniqueness theorems *) + +Theorem div_mod_unique : forall b q1 q2 r1 r2 : t, + 0<=r1<abs b -> 0<=r2<abs b -> + b*q1+r1 == b*q2+r2 -> q1 == q2 /\ r1 == r2. +Proof. +intros b q1 q2 r1 r2 Hr1 Hr2 EQ. +pos_or_neg b. +rewrite abs_eq in * by trivial. +apply div_mod_unique with b; trivial. +rewrite abs_neq' in * by auto using lt_le_incl. +rewrite eq_sym_iff. apply div_mod_unique with (-b); trivial. +rewrite 2 mul_opp_l. +rewrite add_move_l, sub_opp_r. +rewrite <-add_assoc. +symmetry. rewrite add_move_l, sub_opp_r. +now rewrite (add_comm r2), (add_comm r1). +Qed. + +Theorem div_unique: + forall a b q r, 0<=r<abs b -> a == b*q + r -> q == a/b. +Proof. +intros a b q r Hr EQ. +assert (Hb : b~=0). + pos_or_neg b. + rewrite abs_eq in Hr; intuition; order. + rewrite <- opp_0, eq_opp_r. rewrite abs_neq' in Hr; intuition; order. +destruct (div_mod_unique b q (a/b) r (a mod b)); trivial. +now apply mod_always_pos. +now rewrite <- div_mod. +Qed. + +Theorem mod_unique: + forall a b q r, 0<=r<abs b -> a == b*q + r -> r == a mod b. +Proof. +intros a b q r Hr EQ. +assert (Hb : b~=0). + pos_or_neg b. + rewrite abs_eq in Hr; intuition; order. + rewrite <- opp_0, eq_opp_r. rewrite abs_neq' in Hr; intuition; order. +destruct (div_mod_unique b q (a/b) r (a mod b)); trivial. +now apply mod_always_pos. +now rewrite <- div_mod. +Qed. + +(** Sign rules *) + +Lemma div_opp_r : forall a b, b~=0 -> a/(-b) == -(a/b). +Proof. +intros. symmetry. +apply div_unique with (a mod b). +rewrite abs_opp; apply mod_always_pos. +rewrite mul_opp_opp; now apply div_mod. +Qed. + +Lemma mod_opp_r : forall a b, b~=0 -> a mod (-b) == a mod b. +Proof. +intros. symmetry. +apply mod_unique with (-(a/b)). +rewrite abs_opp; apply mod_always_pos. +rewrite mul_opp_opp; now apply div_mod. +Qed. + +Lemma div_opp_l_z : forall a b, b~=0 -> a mod b == 0 -> + (-a)/b == -(a/b). +Proof. +intros a b Hb Hab. symmetry. +apply div_unique with (-(a mod b)). +rewrite Hab, opp_0. split; [order|]. +pos_or_neg b; [rewrite abs_eq | rewrite abs_neq']; order. +now rewrite mul_opp_r, <-opp_add_distr, <-div_mod. +Qed. + +Lemma div_opp_l_nz : forall a b, b~=0 -> a mod b ~= 0 -> + (-a)/b == -(a/b)-sgn b. +Proof. +intros a b Hb Hab. symmetry. +apply div_unique with (abs b -(a mod b)). +rewrite lt_sub_lt_add_l. +rewrite <- le_add_le_sub_l. nzsimpl. +rewrite <- (add_0_l (abs b)) at 2. +rewrite <- add_lt_mono_r. +destruct (mod_always_pos a b); intuition order. +rewrite <- 2 add_opp_r, mul_add_distr_l, 2 mul_opp_r. +rewrite sgn_abs. +rewrite add_shuffle2, add_opp_diag_l; nzsimpl. +rewrite <-opp_add_distr, <-div_mod; order. +Qed. + +Lemma mod_opp_l_z : forall a b, b~=0 -> a mod b == 0 -> + (-a) mod b == 0. +Proof. +intros a b Hb Hab. symmetry. +apply mod_unique with (-(a/b)). +split; [order|now rewrite abs_pos]. +now rewrite <-opp_0, <-Hab, mul_opp_r, <-opp_add_distr, <-div_mod. +Qed. + +Lemma mod_opp_l_nz : forall a b, b~=0 -> a mod b ~= 0 -> + (-a) mod b == abs b - (a mod b). +Proof. +intros a b Hb Hab. symmetry. +apply mod_unique with (-(a/b)-sgn b). +rewrite lt_sub_lt_add_l. +rewrite <- le_add_le_sub_l. nzsimpl. +rewrite <- (add_0_l (abs b)) at 2. +rewrite <- add_lt_mono_r. +destruct (mod_always_pos a b); intuition order. +rewrite <- 2 add_opp_r, mul_add_distr_l, 2 mul_opp_r. +rewrite sgn_abs. +rewrite add_shuffle2, add_opp_diag_l; nzsimpl. +rewrite <-opp_add_distr, <-div_mod; order. +Qed. + +Lemma div_opp_opp_z : forall a b, b~=0 -> a mod b == 0 -> + (-a)/(-b) == a/b. +Proof. +intros. now rewrite div_opp_r, div_opp_l_z, opp_involutive. +Qed. + +Lemma div_opp_opp_nz : forall a b, b~=0 -> a mod b ~= 0 -> + (-a)/(-b) == a/b + sgn(b). +Proof. +intros. rewrite div_opp_r, div_opp_l_nz by trivial. +now rewrite opp_sub_distr, opp_involutive. +Qed. + +Lemma mod_opp_opp_z : forall a b, b~=0 -> a mod b == 0 -> + (-a) mod (-b) == 0. +Proof. +intros. now rewrite mod_opp_r, mod_opp_l_z. +Qed. + +Lemma mod_opp_opp_nz : forall a b, b~=0 -> a mod b ~= 0 -> + (-a) mod (-b) == abs b - a mod b. +Proof. +intros. now rewrite mod_opp_r, mod_opp_l_nz. +Qed. + +(** A division by itself returns 1 *) + +Lemma div_same : forall a, a~=0 -> a/a == 1. +Proof. +intros. symmetry. apply div_unique with 0. +split; [order|now rewrite abs_pos]. +now nzsimpl. +Qed. + +Lemma mod_same : forall a, a~=0 -> a mod a == 0. +Proof. +intros. +rewrite mod_eq, div_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. + +(** Same situation, in term of modulo: *) + +Theorem mod_small: forall a b, 0<=a<b -> a mod b == a. +Proof. exact mod_small. Qed. + +(** * Basic values of divisions and modulo. *) + +Lemma div_0_l: forall a, a~=0 -> 0/a == 0. +Proof. +intros. pos_or_neg a. apply div_0_l; order. +apply opp_inj. rewrite <- div_opp_r, opp_0 by trivial. now apply div_0_l. +Qed. + +Lemma mod_0_l: forall a, a~=0 -> 0 mod a == 0. +Proof. +intros; rewrite mod_eq, div_0_l; now nzsimpl. +Qed. + +Lemma div_1_r: forall a, a/1 == a. +Proof. +intros. symmetry. apply div_unique with 0. +assert (H:=lt_0_1); rewrite abs_pos; intuition; order. +now nzsimpl. +Qed. + +Lemma mod_1_r: forall a, a mod 1 == 0. +Proof. +intros. rewrite mod_eq, div_1_r; nzsimpl; auto using sub_diag. +apply neq_sym, 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 mod_1_l: forall a, 1<a -> 1 mod a == 1. +Proof. exact mod_1_l. Qed. + +Lemma div_mul : forall a b, b~=0 -> (a*b)/b == a. +Proof. +intros. symmetry. apply div_unique with 0. +split; [order|now rewrite abs_pos]. +nzsimpl; apply mul_comm. +Qed. + +Lemma mod_mul : forall a b, b~=0 -> (a*b) mod b == 0. +Proof. +intros. rewrite mod_eq, div_mul by trivial. rewrite mul_comm; apply sub_diag. +Qed. + +(** * Order results about mod and div *) + +(** A modulo cannot grow beyond its starting point. *) + +Theorem mod_le: forall a b, 0<=a -> b~=0 -> a mod b <= a. +Proof. +intros. pos_or_neg b. apply mod_le; order. +rewrite <- mod_opp_r by trivial. apply mod_le; order. +Qed. + +Theorem div_pos : forall a b, 0<=a -> 0<b -> 0<= a/b. +Proof. exact div_pos. Qed. + +Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b. +Proof. exact div_str_pos. Qed. + +Lemma div_small_iff : forall a b, b~=0 -> (a/b==0 <-> 0<=a<abs b). +Proof. +intros a b Hb. +split. +intros EQ. +rewrite (div_mod a b Hb), EQ; nzsimpl. +apply mod_always_pos. +intros. pos_or_neg b. +apply div_small. +now rewrite <- (abs_eq b). +apply opp_inj; rewrite opp_0, <- div_opp_r by trivial. +apply div_small. +rewrite <- (abs_neq' b) by order. trivial. +Qed. + +Lemma mod_small_iff : forall a b, b~=0 -> (a mod b == a <-> 0<=a<abs b). +Proof. +intros. +rewrite <- div_small_iff, mod_eq by trivial. +rewrite sub_move_r, <- (add_0_r a) at 1. rewrite add_cancel_l. +rewrite eq_sym_iff, eq_mul_0. tauto. +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. + +(** [le] is compatible with a positive division. *) + +Lemma div_le_mono : forall a b c, 0<c -> a<=b -> a/c <= b/c. +Proof. +intros a b c Hc Hab. +rewrite lt_eq_cases in Hab. destruct Hab as [LT|EQ]; + [|rewrite EQ; order]. +rewrite <- lt_succ_r. +rewrite (mul_lt_mono_pos_l c) by order. +nzsimpl. +rewrite (add_lt_mono_r _ _ (a mod c)). +rewrite <- div_mod by order. +apply lt_le_trans with b; trivial. +rewrite (div_mod b c) at 1 by order. +rewrite <- add_assoc, <- add_le_mono_l. +apply le_trans with (c+0). +nzsimpl; destruct (mod_always_pos b c); try order. +rewrite abs_eq in *; order. +rewrite <- add_le_mono_l. destruct (mod_always_pos a c); order. +Qed. + +(** In this convention, [div] performs Rounding-Toward-Bottom + when divisor is positive, and Rounding-Toward-Top otherwise. + + Since we cannot speak of rational values here, we express this + fact by multiplying back by [b], and this leads to a nice + unique statement. +*) + +Lemma mul_div_le : forall a b, b~=0 -> b*(a/b) <= a. +Proof. +intros. +rewrite (div_mod a b) at 2; trivial. +rewrite <- (add_0_r (b*(a/b))) at 1. +rewrite <- add_le_mono_l. +now destruct (mod_always_pos a b). +Qed. + +(** Giving a reversed bound is slightly more complex *) + +Lemma mul_succ_div_gt: forall a b, 0<b -> a < b*(S (a/b)). +Proof. +intros. +nzsimpl. +rewrite (div_mod a b) at 1; try order. +rewrite <- add_lt_mono_l. +destruct (mod_always_pos a b). +rewrite abs_eq in *; order. +Qed. + +Lemma mul_pred_div_gt: forall a b, b<0 -> a < b*(P (a/b)). +Proof. +intros a b Hb. +rewrite mul_pred_r, <- add_opp_r. +rewrite (div_mod a b) at 1; try order. +rewrite <- add_lt_mono_l. +destruct (mod_always_pos a b). +rewrite <- opp_pos_neg in Hb. rewrite abs_neq' in *; order. +Qed. + +(** NB: The three previous properties could be used as + specifications for [div]. *) + +(** Inequality [mul_div_le] is exact iff the modulo is zero. *) + +Lemma div_exact : forall a b, b~=0 -> (a == b*(a/b) <-> a mod b == 0). +Proof. +intros. +rewrite (div_mod a b) at 1; try order. +rewrite <- (add_0_r (b*(a/b))) at 2. +apply add_cancel_l. +Qed. + +(** Some additionnal inequalities about div. *) + +Theorem div_lt_upper_bound: + forall a b q, 0<b -> a < b*q -> a/b < q. +Proof. +intros. +rewrite (mul_lt_mono_pos_l b) by trivial. +apply le_lt_trans with a; trivial. +apply mul_div_le; order. +Qed. + +Theorem div_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. +Qed. + +Theorem div_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. +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. + +(** * Relations between usual operations and mod and div *) + +Lemma mod_add : forall a b c, c~=0 -> + (a + b * c) mod c == a mod c. +Proof. +intros. +symmetry. +apply mod_unique with (a/c+b); trivial. +now apply mod_always_pos. +rewrite mul_add_distr_l, add_shuffle0, <- div_mod by order. +now rewrite mul_comm. +Qed. + +Lemma div_add : forall a b c, c~=0 -> + (a + b * c) / c == a / c + b. +Proof. +intros. +apply (mul_cancel_l _ _ c); try order. +apply (add_cancel_r _ _ ((a+b*c) mod c)). +rewrite <- div_mod, mod_add by order. +rewrite mul_add_distr_l, add_shuffle0, <- div_mod by order. +now rewrite mul_comm. +Qed. + +Lemma div_add_l: forall a b c, b~=0 -> + (a * b + c) / b == a + c / b. +Proof. + intros a b c. rewrite (add_comm _ c), (add_comm a). + now apply div_add. +Qed. + +(** Cancellations. *) + +(** With the current convention, the following isn't always true + when [c<0]: [-3*-1 / -2*-1 = 3/2 = 1] while [-3/-2 = 2] *) + +Lemma div_mul_cancel_r : forall a b c, b~=0 -> 0<c -> + (a*c)/(b*c) == a/b. +Proof. +intros. +symmetry. +apply div_unique with ((a mod b)*c). +(* ineqs *) +rewrite abs_mul, (abs_eq c) by order. +rewrite <-(mul_0_l c), <-mul_lt_mono_pos_r, <-mul_le_mono_pos_r by trivial. +apply mod_always_pos. +(* equation *) +rewrite (div_mod a b) at 1 by order. +rewrite mul_add_distr_r. +rewrite add_cancel_r. +rewrite <- 2 mul_assoc. now rewrite (mul_comm c). +Qed. + +Lemma div_mul_cancel_l : forall a b c, b~=0 -> 0<c -> + (c*a)/(c*b) == a/b. +Proof. +intros. rewrite !(mul_comm c); now apply div_mul_cancel_r. +Qed. + +Lemma mul_mod_distr_l: forall a b c, b~=0 -> 0<c -> + (c*a) mod (c*b) == c * (a mod b). +Proof. +intros. +rewrite <- (add_cancel_l _ _ ((c*b)* ((c*a)/(c*b)))). +rewrite <- div_mod. +rewrite div_mul_cancel_l by trivial. +rewrite <- mul_assoc, <- mul_add_distr_l, mul_cancel_l by order. +apply div_mod; order. +rewrite <- neq_mul_0; intuition; order. +Qed. + +Lemma mul_mod_distr_r: forall a b c, b~=0 -> 0<c -> + (a*c) mod (b*c) == (a mod b) * c. +Proof. + intros. rewrite !(mul_comm _ c); now rewrite mul_mod_distr_l. +Qed. + + +(** Operations modulo. *) + +Theorem mod_mod: forall a n, n~=0 -> + (a mod n) mod n == a mod n. +Proof. +intros. rewrite mod_small_iff by trivial. +now apply mod_always_pos. +Qed. + +Lemma mul_mod_idemp_l : forall a b n, n~=0 -> + ((a mod n)*b) mod n == (a*b) mod n. +Proof. + intros a b n Hn. symmetry. + rewrite (div_mod a n) at 1 by order. + rewrite add_comm, (mul_comm n), (mul_comm _ b). + rewrite mul_add_distr_l, mul_assoc. + rewrite mod_add by trivial. + now rewrite mul_comm. +Qed. + +Lemma mul_mod_idemp_r : forall a b n, n~=0 -> + (a*(b mod n)) mod n == (a*b) mod n. +Proof. + intros. rewrite !(mul_comm a). now apply mul_mod_idemp_l. +Qed. + +Theorem mul_mod: forall a b n, n~=0 -> + (a * b) mod n == ((a mod n) * (b mod n)) mod n. +Proof. + intros. now rewrite mul_mod_idemp_l, mul_mod_idemp_r. +Qed. + +Lemma add_mod_idemp_l : forall a b n, n~=0 -> + ((a mod n)+b) mod n == (a+b) mod n. +Proof. + intros a b n Hn. symmetry. + rewrite (div_mod a n) at 1 by order. + rewrite <- add_assoc, add_comm, mul_comm. + now rewrite mod_add. +Qed. + +Lemma add_mod_idemp_r : forall a b n, n~=0 -> + (a+(b mod n)) mod n == (a+b) mod n. +Proof. + intros. rewrite !(add_comm a). now apply add_mod_idemp_l. +Qed. + +Theorem add_mod: forall a b n, n~=0 -> + (a+b) mod n == (a mod n + b mod n) mod n. +Proof. + intros. now rewrite add_mod_idemp_l, add_mod_idemp_r. +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) ]. *) + +Lemma div_div : forall a b c, 0<b -> 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<= ... <abs(b*c) *) + rewrite abs_mul. + destruct (mod_always_pos (a/b) c), (mod_always_pos a b). + split. + apply add_nonneg_nonneg; trivial. + apply mul_nonneg_nonneg; order. + apply lt_le_trans with (b*((a/b) mod c) + abs b). + now rewrite <- add_lt_mono_l. + rewrite (abs_eq b) by order. + now rewrite <- mul_succ_r, <- mul_le_mono_pos_l, le_succ_l. + (* end 0<= ... < abs(b*c) *) + rewrite (div_mod a b) at 1 by order. + rewrite add_assoc, add_cancel_r. + rewrite <- mul_assoc, <- mul_add_distr_l, mul_cancel_l 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. + |