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diff --git a/theories/Numbers/Integer/Abstract/ZDivTrunc.v b/theories/Numbers/Integer/Abstract/ZDivTrunc.v new file mode 100644 index 00000000..3200ba2a --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZDivTrunc.v @@ -0,0 +1,532 @@ +(************************************************************************) +(* 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 (Trunc convention) + + We use here the convention known as Trunc, or Round-Toward-Zero, + where [a/b] is the integer with the largest absolute value to + be between zero and the exact fraction. It can be summarized by: + + [a = bq+r /\ 0 <= |r| < |b| /\ Sign(r) = Sign(a)] + + This is the convention of Ocaml and many other systems (C, ASM, ...). + This convention is named "T" 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 [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). + +(** We benefit from what already exists for NZ *) + + Module Import NZDivP := NZDivPropFunct Z ZP Z. + +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]. + +(** 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. + +(** 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 div_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. +Qed. + +Lemma div_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. +Qed. + + +(** Uniqueness theorems *) + +Theorem div_mod_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. +rewrite <- (opp_inj_wd r1 r2). +apply 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 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. + +(** A division by itself returns 1 *) + +Lemma div_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. +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. +rewrite <- div_opp_opp, 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. pos_or_neg a. now apply div_1_r. +apply opp_inj. rewrite <- div_opp_l. apply 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. +Proof. +intros. rewrite mod_eq, div_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 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. 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. +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 -> 0<b -> a mod b <= a. +Proof. exact mod_le. 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 <-> 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 (abs_eq a), (abs_neq' b); intuition; order. +rewrite <- opp_inj_wd, opp_0, <- div_opp_l, div_small_iff by order. + rewrite (abs_neq' a), (abs_eq b); intuition; order. +rewrite <- div_opp_opp, 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). +Proof. +intros. rewrite mod_eq, <- div_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. + +(** 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. pos_or_neg a. apply 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. +Qed. + +(** With this choice of division, + rounding of div is always done toward zero: *) + +Lemma mul_div_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. +split. +apply mul_nonneg_nonneg; [|apply div_pos]; order. +apply mul_div_le; order. +Qed. + +Lemma mul_div_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 in *. +destruct (mul_div_le (-a) b); tauto. +Qed. + +(** 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. + +(** Similar results with negative numbers *) + +Lemma mul_pred_div_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_nonneg_nonpos in *. +now apply mul_succ_div_gt. +Qed. + +Lemma mul_pred_div_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 <- opp_pos_neg in *. +now apply mul_succ_div_gt. +Qed. + +Lemma mul_succ_div_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_nonneg_nonpos, <- opp_pos_neg in *. +now apply mul_succ_div_gt. +Qed. + +(** 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 mod_eq by order. rewrite sub_move_r; nzsimpl; tauto. +Qed. + +(** Some additionnal inequalities about div. *) + +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 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 *) + +(** 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] *) + +Lemma mod_add : forall a b c, c~=0 -> 0 <= (a+b*c)*a -> + (a + b * c) mod c == a mod 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. + rewrite <- mul_opp_opp in *. + apply 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. +Qed. + +Lemma div_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. +Qed. + +Lemma div_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. +Qed. + +(** Cancellations. *) + +Lemma div_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. + rewrite <- neq_mul_0; intuition order. +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. + 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. +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. +Proof. +intros. rewrite !(mul_comm c); now apply div_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. +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). +Qed. + +Lemma mul_mod_distr_l: forall a b c, b~=0 -> c~=0 -> + (c*a) mod (c*b) == c * (a mod b). +Proof. +intros; rewrite !(mul_comm c); now apply mul_mod_distr_r. +Qed. + +(** Operations modulo. *) + +Theorem mod_mod: forall a n, n~=0 -> + (a mod n) mod n == a mod 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. +Qed. + +Lemma mul_mod_idemp_l : forall a b n, n~=0 -> + ((a mod n)*b) mod n == (a*b) mod 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. +assert (Aux2 : forall a b n, 0<=a -> n~=0 -> + ((a mod n)*b) mod n == (a*b) mod n). + intros. pos_or_neg b. now apply Aux1. + apply opp_inj. rewrite <-2 mod_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 Aux2; order. +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. + +(** 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] + for any a and b. + For instance, take (8 + (-10)) mod 3 = -2 whereas + (8 mod 3 + (-10 mod 3)) mod 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. +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. +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. +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. +Proof. +intros. rewrite !(add_comm a). apply add_mod_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. +Proof. +intros a b n Hn Hab. rewrite add_mod_idemp_l, add_mod_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)]; + 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. +Qed. + + +(** 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). +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. + rewrite <- neq_mul_0; intuition order. +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 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. +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). +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. + |