diff options
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZDivCoq.v | 390 | ||||
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZDivMath.v | 396 | ||||
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZDivOcaml.v | 541 | ||||
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZMulOrder.v | 6 | ||||
| -rw-r--r-- | theories/Numbers/NatInt/NZAdd.v | 12 | ||||
| -rw-r--r-- | theories/Numbers/NatInt/NZAxioms.v | 5 | ||||
| -rw-r--r-- | theories/Numbers/NatInt/NZDiv.v | 528 | ||||
| -rw-r--r-- | theories/Numbers/NatInt/NZMulOrder.v | 5 | ||||
| -rw-r--r-- | theories/Numbers/Natural/Abstract/NDiv.v | 249 | ||||
| -rw-r--r-- | theories/Numbers/Natural/Peano/NPeano.v | 75 | ||||
| -rw-r--r-- | theories/Numbers/vo.itarget | 5 | ||||
| -rw-r--r-- | theories/ZArith/ZOdiv.v | 171 |
12 files changed, 2249 insertions, 134 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZDivCoq.v b/theories/Numbers/Integer/Abstract/ZDivCoq.v new file mode 100644 index 0000000000..402d520d5a --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZDivCoq.v @@ -0,0 +1,390 @@ +(************************************************************************) +(* 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 + + We use here the historical convention of Coq : + [a = bq+r /\ 0 < |r| < |b| /\ Sign(r) = Sgn(b)] + *) + +Require Import ZAxioms ZProperties NZDiv. + +Open Scope NumScope. + +Module Type ZDiv (Import Z : ZAxiomsSig). + + Parameter Inline div : t -> t -> t. + Parameter Inline modulo : t -> t -> t. + + Infix "/" := div : NumScope. + Infix "mod" := modulo (at level 40, no associativity) : NumScope. + + Instance div_wd : Proper (eq==>eq==>eq) div. + Instance mod_wd : Proper (eq==>eq==>eq) modulo. + + Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b). + 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 ZDiv. + +Module Type ZDivSig := ZAxiomsSig <+ ZDiv. + +Module ZDivPropFunct (Import Z : ZDivSig). + (* TODO: en faire un arg du foncteur + comprendre le bug de SearchAbout *) + Module Import ZP := ZPropFunct Z. + +(** We benefit from what already exists for NZ *) + + Module Z' <: NZDivSig. + Include Z. + Lemma mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b. + Proof. intros; apply mod_pos_bound; auto. Qed. + End Z'. + Module Import NZDivP := NZDivPropFunct Z'. + +(** 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. apply div_mod; auto. +Qed. + +(** A few sign rules (simple ones) *) + +Lemma div_mod_opp_opp : forall a b, b~=0 -> + (-a/-b) == a/b /\ (-a) mod (-b) == -(a mod b). +Proof. +intros a b Hb. +assert (-b ~= 0). + contradict Hb. rewrite eq_opp_l, opp_0 in Hb; auto. +assert (EQ := opp_involutive a). +rewrite (div_mod a b) in EQ at 2; auto. +rewrite (div_mod (-a) (-b)) in EQ; auto. + +destruct (lt_ge_cases 0 b). +rewrite opp_add_distr in EQ. +rewrite <- mul_opp_l, opp_involutive in EQ. +destruct (div_mod_unique b (-a/-b) (a/b) (-(-a mod -b)) (a mod b)); auto. +rewrite <- (opp_involutive b) at 3. +rewrite <- opp_lt_mono. +rewrite opp_nonneg_nonpos. +destruct (mod_neg_bound (-a) (-b)); auto. +rewrite opp_neg_pos; auto. +apply mod_pos_bound; auto. +split; auto. +rewrite eq_opp_r; auto. + +rewrite eq_opp_l in EQ. +rewrite opp_add_distr in EQ. +rewrite <- mul_opp_l in EQ. +destruct (div_mod_unique (-b) (-a/-b) (a/b) (-a mod -b) (-(a mod b))); auto. +apply mod_pos_bound; auto. +rewrite opp_pos_neg; order. +rewrite <- opp_lt_mono. +rewrite opp_nonneg_nonpos. +destruct (mod_neg_bound a b); intuition; order. +Qed. + +Lemma div_opp_opp : forall a b, b~=0 -> -a/-b == a/b. +Proof. +intros; destruct (div_mod_opp_opp a b); auto. +Qed. + +Lemma mod_opp_opp : forall a b, b~=0 -> (-a) mod (-b) == - (a mod b). +Proof. +intros; destruct (div_mod_opp_opp a b); auto. +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; auto. +rewrite <- opp_inj_wd in EQ. +rewrite 2 opp_add_distr in EQ. rewrite <- 2 mul_opp_l in EQ. +rewrite <- (opp_inj_wd r1 r2). +apply div_mod_unique with (-b); auto. +rewrite <- opp_lt_mono, opp_nonneg_nonpos; intuition. +rewrite <- opp_lt_mono, opp_nonneg_nonpos; intuition. +Qed. + +Theorem div_unique: + forall a b q r, (0<=r<b \/ b<r<=0) -> a == b*q + r -> q == a/b. +Proof. +intros a b q r Hr EQ. +assert (Hb : b~=0) by (destruct Hr; intuition; order). +rewrite (div_mod a b Hb) in EQ. +destruct (div_mod_unique b (a/b) q (a mod b) r); auto. +destruct Hr; [left; apply mod_pos_bound|right; apply mod_neg_bound]; + intuition order. +Qed. + +Theorem div_unique_pos: + forall a b q r, 0<=r<b -> a == b*q + r -> q == a/b. +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. +Proof. intros; apply div_unique with r; auto. Qed. + +Theorem mod_unique: + forall a b q r, (0<=r<b \/ b<r<=0) -> a == b*q + r -> r == a mod b. +Proof. +intros a b q r Hr EQ. +assert (Hb : b~=0) by (destruct Hr; intuition; order). +rewrite (div_mod a b Hb) in EQ. +destruct (div_mod_unique b (a/b) q (a mod b) r); auto. +destruct Hr; [left; apply mod_pos_bound|right; apply mod_neg_bound]; + intuition order. +Qed. + +Theorem mod_unique_pos: + forall a b q r, 0<=r<b -> a == b*q + r -> r == a mod b. +Proof. intros; apply mod_unique with q; auto. Qed. + +Theorem mod_unique_neg: + forall a b q r, b<r<=0 -> a == b*q + r -> r == a mod b. +Proof. intros; apply mod_unique with q; auto. Qed. + + +(** A division by itself returns 1 *) + +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]. + +Lemma div_same : forall a, a~=0 -> a/a == 1. +Proof. +intros. pos_or_neg a. apply div_same; order. +rewrite <- div_opp_opp; auto. apply div_same; auto. +Qed. + +Lemma mod_same : forall a, a~=0 -> a mod a == 0. +Proof. +intros. rewrite mod_eq, div_same; auto. 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; auto. apply div_0_l; auto. +Qed. + +Lemma mod_0_l: forall a, a~=0 -> 0 mod a == 0. +Proof. +intros; rewrite mod_eq, div_0_l; nzsimpl; auto. +Qed. + +Lemma div_1_r: forall a, a/1 == a. +Proof. +intros. symmetry. apply div_unique with 0. left. split; order || apply lt_0_1. +nzsimpl; auto. +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. symmetry. apply div_unique with 0. +destruct (lt_ge_cases 0 b); [left|right]; split; order. +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; auto. 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. + +(* A REVOIR APRES LA REGLE DES SIGNES +Lemma div_small_iff : forall a b, b~=0 -> (a/b==0 <-> 0<=a<b \/ b<a<=0). +intros. apply div_small_iff; auto'. Qed. + +Lemma mod_small_iff : forall a b, b~=0 -> (a mod b == a <-> a<b). +Proof. intros. apply mod_small_iff; auto'. Qed. + +Lemma div_str_pos_iff : forall a b, b~=0 -> (0<a/b <-> b<=a). +Proof. intros. apply div_str_pos_iff; auto'. 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. + +(* STILL TODO !! + +(** [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. destruct (le_gt_cases 0 a). +apply div_le; auto. +destruct (lt_ge_cases 0 b). +apply le_trans with 0. + admit. (* !!! *) +apply div_pos; order. +Admitted. (* !!! *) + +Lemma mul_div_le : forall a b, b~=0 -> b*(a/b) <= a. +Proof. intros. apply mul_div_le; auto'. Qed. + +Lemma mul_succ_div_gt: forall a b, b~=0 -> a < b*(S (a/b)). +Proof. intros; apply mul_succ_div_gt; auto'. Qed. + +(** The previous inequality 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. apply div_exact; auto'. Qed. + +(** Some additionnal inequalities about div. *) + +Theorem div_lt_upper_bound: + forall a b q, b~=0 -> a < b*q -> a/b < q. +Proof. intros. apply div_lt_upper_bound; auto'. Qed. + +Theorem div_le_upper_bound: + forall a b q, b~=0 -> a <= b*q -> a/b <= q. +Proof. intros; apply div_le_upper_bound; auto'. Qed. + +Theorem div_le_lower_bound: + forall a b q, b~=0 -> b*q <= a -> q <= a/b. +Proof. intros; apply div_le_lower_bound; auto'. Qed. + +(** A division respects opposite monotonicity for the divisor *) + +Lemma div_le_compat_l: forall p q r, 0<q<r -> p/r <= p/q. +Proof. intros. apply div_le_compat_l. auto'. auto. 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. apply mod_add; auto'. Qed. + +Lemma div_add : forall a b c, c~=0 -> + (a + b * c) / c == a / c + b. +Proof. intros. apply div_add; auto'. Qed. + +Lemma div_add_l: forall a b c, b~=0 -> + (a * b + c) / b == a + c / b. +Proof. intros. apply div_add_l; auto'. Qed. + +(** Cancellations. *) + +Lemma div_mul_cancel_r : forall a b c, b~=0 -> c~=0 -> + (a*c)/(b*c) == a/b. +Proof. intros. apply div_mul_cancel_r; auto'. Qed. + +Lemma div_mul_cancel_l : forall a b c, b~=0 -> c~=0 -> + (c*a)/(c*b) == a/b. +Proof. intros. apply div_mul_cancel_l; auto'. 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. apply mul_mod_distr_l; auto'. 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. apply mul_mod_distr_r; auto'. Qed. + +(** Operations modulo. *) + +Theorem mod_mod: forall a n, n~=0 -> + (a mod n) mod n == a mod n. +Proof. intros. apply mod_mod; auto'. Qed. + +Lemma mul_mod_idemp_l : forall a b n, n~=0 -> + ((a mod n)*b) mod n == (a*b) mod n. +Proof. intros. apply mul_mod_idemp_l; auto'. Qed. + +Lemma mul_mod_idemp_r : forall a b n, n~=0 -> + (a*(b mod n)) mod n == (a*b) mod n. +Proof. intros. apply mul_mod_idemp_r; auto'. Qed. + +Theorem mul_mod: forall a b n, n~=0 -> + (a * b) mod n == ((a mod n) * (b mod n)) mod n. +Proof. intros. apply mul_mod; auto'. Qed. + +Lemma add_mod_idemp_l : forall a b n, n~=0 -> + ((a mod n)+b) mod n == (a+b) mod n. +Proof. intros. apply add_mod_idemp_l; auto'. Qed. + +Lemma add_mod_idemp_r : forall a b n, n~=0 -> + (a+(b mod n)) mod n == (a+b) mod n. +Proof. intros. apply add_mod_idemp_r; auto'. Qed. + +Theorem add_mod: forall a b n, n~=0 -> + (a+b) mod n == (a mod n + b mod n) mod n. +Proof. intros. apply add_mod; auto'. Qed. + +Lemma div_div : forall a b c, b~=0 -> c~=0 -> + (a/b)/c == a/(b*c). +Proof. intros. apply div_div; auto'. Qed. + +(** A last inequality: *) + +Theorem div_mul_le: + forall a b c, b~=0 -> c*(a/b) <= (c*a)/b. +Proof. intros. apply div_mul_le; auto'. 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. apply mod_divides; auto'. Qed. +*) + +End ZDivPropFunct. + diff --git a/theories/Numbers/Integer/Abstract/ZDivMath.v b/theories/Numbers/Integer/Abstract/ZDivMath.v new file mode 100644 index 0000000000..dfc9ee4bc7 --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZDivMath.v @@ -0,0 +1,396 @@ +(************************************************************************) +(* 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 + + We use here the "mathematical" convention, i.e. Round-Toward-Bottom : + [a = bq+r /\ 0 < r < |b| ] + *) + +Require Import ZAxioms ZProperties NZDiv. + +Open Scope NumScope. + +Module Type ZDiv (Import Z : ZAxiomsSig). + + Parameter Inline div : t -> t -> t. + Parameter Inline modulo : t -> t -> t. + + Infix "/" := div : NumScope. + Infix "mod" := modulo (at level 40, no associativity) : NumScope. + + Instance div_wd : Proper (eq==>eq==>eq) div. + Instance mod_wd : Proper (eq==>eq==>eq) modulo. + + Definition abs z := max z (-z). + + Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b). + Axiom mod_always_pos : forall a b, 0 <= a mod b < abs b. + +End ZDiv. + +Module Type ZDivSig := ZAxiomsSig <+ ZDiv. + +Module ZDivPropFunct (Import Z : ZDivSig). + (* TODO: en faire un arg du foncteur + comprendre le bug de SearchAbout *) + Module Import ZP := ZPropFunct Z. + +(** We benefit from what already exists for NZ *) + + Module Z' <: NZDivSig. + Include Z. + Lemma mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b. + Proof. + intros. rewrite <- (max_l b (-b)) at 3. + apply mod_always_pos. + apply le_trans with 0; [ rewrite opp_nonpos_nonneg |]; order. + Qed. + End Z'. + Module Import NZDivP := NZDivPropFunct Z'. + +(** 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. apply div_mod; auto. +Qed. + +(* STILL TODO ... + +(** A few sign rules (simple ones) *) + +Lemma div_mod_opp_opp : forall a b, b~=0 -> + (-a/-b) == a/b /\ (-a) mod (-b) == -(a mod b). +Proof. +intros a b Hb. +assert (-b ~= 0). + contradict Hb. rewrite eq_opp_l, opp_0 in Hb; auto. +assert (EQ := opp_involutive a). +rewrite (div_mod a b) in EQ at 2; auto. +rewrite (div_mod (-a) (-b)) in EQ; auto. + +destruct (lt_ge_cases 0 b). +rewrite opp_add_distr in EQ. +rewrite <- mul_opp_l, opp_involutive in EQ. +destruct (div_mod_unique b (-a/-b) (a/b) (-(-a mod -b)) (a mod b)); auto. +rewrite <- (opp_involutive b) at 3. +rewrite <- opp_lt_mono. +rewrite opp_nonneg_nonpos. +destruct (mod_neg_bound (-a) (-b)); auto. +rewrite opp_neg_pos; auto. +apply mod_pos_bound; auto. +split; auto. +rewrite eq_opp_r; auto. + +rewrite eq_opp_l in EQ. +rewrite opp_add_distr in EQ. +rewrite <- mul_opp_l in EQ. +destruct (div_mod_unique (-b) (-a/-b) (a/b) (-a mod -b) (-(a mod b))); auto. +apply mod_pos_bound; auto. +rewrite opp_pos_neg; order. +rewrite <- opp_lt_mono. +rewrite opp_nonneg_nonpos. +destruct (mod_neg_bound a b); intuition; order. +Qed. + +Lemma div_opp_opp : forall a b, b~=0 -> -a/-b == a/b. +Proof. +intros; destruct (div_mod_opp_opp a b); auto. +Qed. + +Lemma mod_opp_opp : forall a b, b~=0 -> (-a) mod (-b) == - (a mod b). +Proof. +intros; destruct (div_mod_opp_opp a b); auto. +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; auto. +rewrite <- opp_inj_wd in EQ. +rewrite 2 opp_add_distr in EQ. rewrite <- 2 mul_opp_l in EQ. +rewrite <- (opp_inj_wd r1 r2). +apply div_mod_unique with (-b); auto. +rewrite <- opp_lt_mono, opp_nonneg_nonpos; intuition. +rewrite <- opp_lt_mono, opp_nonneg_nonpos; intuition. +Qed. + +Theorem div_unique: + forall a b q r, (0<=r<b \/ b<r<=0) -> a == b*q + r -> q == a/b. +Proof. +intros a b q r Hr EQ. +assert (Hb : b~=0) by (destruct Hr; intuition; order). +rewrite (div_mod a b Hb) in EQ. +destruct (div_mod_unique b (a/b) q (a mod b) r); auto. +destruct Hr; [left; apply mod_pos_bound|right; apply mod_neg_bound]; + intuition order. +Qed. + +Theorem div_unique_pos: + forall a b q r, 0<=r<b -> a == b*q + r -> q == a/b. +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. +Proof. intros; apply div_unique with r; auto. Qed. + +Theorem mod_unique: + forall a b q r, (0<=r<b \/ b<r<=0) -> a == b*q + r -> r == a mod b. +Proof. +intros a b q r Hr EQ. +assert (Hb : b~=0) by (destruct Hr; intuition; order). +rewrite (div_mod a b Hb) in EQ. +destruct (div_mod_unique b (a/b) q (a mod b) r); auto. +destruct Hr; [left; apply mod_pos_bound|right; apply mod_neg_bound]; + intuition order. +Qed. + +Theorem mod_unique_pos: + forall a b q r, 0<=r<b -> a == b*q + r -> r == a mod b. +Proof. intros; apply mod_unique with q; auto. Qed. + +Theorem mod_unique_neg: + forall a b q r, b<r<=0 -> a == b*q + r -> r == a mod b. +Proof. intros; apply mod_unique with q; auto. Qed. + + +(** A division by itself returns 1 *) + +Ltac pos_or_neg a := + destruct (le_gt_cases 0 a) as [LE|LT]; [|rewrite <- opp_pos_neg in LT]. + +Lemma div_same : forall a, a~=0 -> a/a == 1. +Proof. +intros. pos_or_neg a. apply div_same; order. +rewrite <- div_opp_opp; auto. apply div_same; auto. +Qed. + +Lemma mod_same : forall a, a~=0 -> a mod a == 0. +Proof. +intros. rewrite mod_eq, div_same; auto. 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; auto. apply div_0_l; auto. +Qed. + +Lemma mod_0_l: forall a, a~=0 -> 0 mod a == 0. +Proof. +intros; rewrite mod_eq, div_0_l; nzsimpl; auto. +Qed. + +Lemma div_1_r: forall a, a/1 == a. +Proof. +intros. symmetry. apply div_unique with 0. left. split; order || apply lt_0_1. +nzsimpl; auto. +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. symmetry. apply div_unique with 0. +destruct (lt_ge_cases 0 b); [left|right]; split; order. +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; auto. 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. + +(* A REVOIR APRES LA REGLE DES SIGNES +Lemma div_small_iff : forall a b, b~=0 -> (a/b==0 <-> 0<=a<b \/ b<a<=0). +intros. apply div_small_iff; auto'. Qed. + +Lemma mod_small_iff : forall a b, b~=0 -> (a mod b == a <-> a<b). +Proof. intros. apply mod_small_iff; auto'. Qed. + +Lemma div_str_pos_iff : forall a b, b~=0 -> (0<a/b <-> b<=a). +Proof. intros. apply div_str_pos_iff; auto'. 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. + +(* STILL TODO !! + +(** [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. destruct (le_gt_cases 0 a). +apply div_le_mono; auto. +destruct (lt_ge_cases 0 b). +apply le_trans with 0. + admit. (* !!! *) +apply div_pos; order. +Admitted. (* !!! *) + +Lemma mul_div_le : forall a b, b~=0 -> b*(a/b) <= a. +Proof. intros. apply mul_div_le; auto'. Qed. + +Lemma mul_succ_div_gt: forall a b, b~=0 -> a < b*(S (a/b)). +Proof. intros; apply mul_succ_div_gt; auto'. Qed. + +(** The previous inequality 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. apply div_exact; auto'. Qed. + +(** Some additionnal inequalities about div. *) + +Theorem div_lt_upper_bound: + forall a b q, b~=0 -> a < b*q -> a/b < q. +Proof. intros. apply div_lt_upper_bound; auto'. Qed. + +Theorem div_le_upper_bound: + forall a b q, b~=0 -> a <= b*q -> a/b <= q. +Proof. intros; apply div_le_upper_bound; auto'. Qed. + +Theorem div_le_lower_bound: + forall a b q, b~=0 -> b*q <= a -> q <= a/b. +Proof. intros; apply div_le_lower_bound; auto'. Qed. + +(** A division respects opposite monotonicity for the divisor *) + +Lemma div_le_compat_l: forall p q r, 0<q<r -> p/r <= p/q. +Proof. intros. apply div_le_compat_l. auto'. auto. 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. apply mod_add; auto'. Qed. + +Lemma div_add : forall a b c, c~=0 -> + (a + b * c) / c == a / c + b. +Proof. intros. apply div_add; auto'. Qed. + +Lemma div_add_l: forall a b c, b~=0 -> + (a * b + c) / b == a + c / b. +Proof. intros. apply div_add_l; auto'. Qed. + +(** Cancellations. *) + +Lemma div_mul_cancel_r : forall a b c, b~=0 -> c~=0 -> + (a*c)/(b*c) == a/b. +Proof. intros. apply div_mul_cancel_r; auto'. Qed. + +Lemma div_mul_cancel_l : forall a b c, b~=0 -> c~=0 -> + (c*a)/(c*b) == a/b. +Proof. intros. apply div_mul_cancel_l; auto'. 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. apply mul_mod_distr_l; auto'. 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. apply mul_mod_distr_r; auto'. Qed. + +(** Operations modulo. *) + +Theorem mod_mod: forall a n, n~=0 -> + (a mod n) mod n == a mod n. +Proof. intros. apply mod_mod; auto'. Qed. + +Lemma mul_mod_idemp_l : forall a b n, n~=0 -> + ((a mod n)*b) mod n == (a*b) mod n. +Proof. intros. apply mul_mod_idemp_l; auto'. Qed. + +Lemma mul_mod_idemp_r : forall a b n, n~=0 -> + (a*(b mod n)) mod n == (a*b) mod n. +Proof. intros. apply mul_mod_idemp_r; auto'. Qed. + +Theorem mul_mod: forall a b n, n~=0 -> + (a * b) mod n == ((a mod n) * (b mod n)) mod n. +Proof. intros. apply mul_mod; auto'. Qed. + +Lemma add_mod_idemp_l : forall a b n, n~=0 -> + ((a mod n)+b) mod n == (a+b) mod n. +Proof. intros. apply add_mod_idemp_l; auto'. Qed. + +Lemma add_mod_idemp_r : forall a b n, n~=0 -> + (a+(b mod n)) mod n == (a+b) mod n. +Proof. intros. apply add_mod_idemp_r; auto'. Qed. + +Theorem add_mod: forall a b n, n~=0 -> + (a+b) mod n == (a mod n + b mod n) mod n. +Proof. intros. apply add_mod; auto'. Qed. + +Lemma div_div : forall a b c, b~=0 -> c~=0 -> + (a/b)/c == a/(b*c). +Proof. intros. apply div_div; auto'. Qed. + +(** A last inequality: *) + +Theorem div_mul_le: + forall a b c, b~=0 -> c*(a/b) <= (c*a)/b. +Proof. intros. apply div_mul_le; auto'. 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. apply mod_divides; auto'. Qed. +*) +*) + +End ZDivPropFunct. + diff --git a/theories/Numbers/Integer/Abstract/ZDivOcaml.v b/theories/Numbers/Integer/Abstract/ZDivOcaml.v new file mode 100644 index 0000000000..2f68da9330 --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZDivOcaml.v @@ -0,0 +1,541 @@ +(************************************************************************) +(* 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 + + We use here the convention of Ocaml and many other system (C, ASM, ...), + i.e. Round-Toward-Zero : + + [a = bq+r /\ 0 < |r| < |b| /\ Sign(r) = Sgn(a)] +*) + +Require Import ZAxioms ZProperties NZDiv. + +Open Scope NumScope. + +Module Type ZDiv (Import Z : ZAxiomsSig). + + Parameter Inline div : t -> t -> t. + Parameter Inline modulo : t -> t -> t. + + Infix "/" := div : NumScope. + Infix "mod" := modulo (at level 40, no associativity) : NumScope. + + Instance div_wd : Proper (eq==>eq==>eq) div. + Instance mod_wd : Proper (eq==>eq==>eq) modulo. + + Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b). + 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 ZDiv. + +Module Type ZDivSig := ZAxiomsSig <+ ZDiv. + +Module ZDivPropFunct (Import Z : ZDivSig). + (* TODO: en faire un arg du foncteur + comprendre le bug de SearchAbout *) + Module Import ZP := ZPropFunct Z. + +(** We benefit from what already exists for NZ *) + + Module Import NZDivP := NZDivPropFunct 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. apply div_mod; auto. +Qed. + +(** A few sign rules (simple ones) *) + +Lemma mod_opp_opp : forall a b, b ~= 0 -> (-a) mod (-b) == - (a mod b). +Proof. intros. rewrite mod_opp_r, mod_opp_l; auto. Qed. + +Lemma div_opp_l : forall a b, b ~= 0 -> (-a)/b == -(a/b). +Proof. +intros. +rewrite <- (mul_cancel_l _ _ b); auto. +rewrite <- (add_cancel_r _ _ ((-a) mod b)); auto. +rewrite <- div_mod; auto. +rewrite mod_opp_l, mul_opp_r, <- opp_add_distr, <- div_mod; auto. +Qed. + +Lemma div_opp_r : forall a b, b ~= 0 -> a/(-b) == -(a/b). +Proof. +intros. +assert (-b ~= 0) by (rewrite eq_opp_l, opp_0; auto). +rewrite <- (mul_cancel_l _ _ (-b)); auto. +rewrite <- (add_cancel_r _ _ (a mod (-b))); auto. +rewrite <- div_mod; auto. +rewrite mod_opp_r, mul_opp_opp, <- div_mod; auto. +Qed. + +Lemma div_opp_opp : forall a b, b ~= 0 -> (-a)/(-b) == a/b. +Proof. intros. rewrite div_opp_r, div_opp_l, opp_involutive; auto. 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; auto. destruct (mod_bound a b); auto. + rewrite <- mul_opp_opp, <- mod_opp_l by order. + apply mul_nonneg_nonneg; try order. destruct (mod_bound (-a) b); try 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; auto. +rewrite <- opp_inj_wd in EQ. +rewrite 2 opp_add_distr in EQ. rewrite <- 2 mul_opp_l in EQ. +rewrite <- (opp_inj_wd r1 r2). +apply div_mod_unique with (-b); auto. +rewrite <- opp_lt_mono, opp_nonneg_nonpos; intuition. +rewrite <- opp_lt_mono, opp_nonneg_nonpos; intuition. +Qed. + +Theorem div_unique: + forall a b q r, 0<=a -> 0<=r<b -> a == b*q + r -> q == a/b. +Proof. intros; apply div_unique with r; auto. Qed. + +Theorem mod_unique: + forall a b q r, 0<=a -> 0<=r<b -> a == b*q + r -> r == a mod b. +Proof. intros; apply mod_unique with q; auto. 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; auto. apply div_same; auto. +Qed. + +Lemma mod_same : forall a, a~=0 -> a mod a == 0. +Proof. +intros. rewrite mod_eq, div_same; auto. 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; auto. apply div_0_l; auto. +Qed. + +Lemma mod_0_l: forall a, a~=0 -> 0 mod a == 0. +Proof. +intros; rewrite mod_eq, div_0_l; nzsimpl; auto. +Qed. + +Lemma div_1_r: forall a, a/1 == a. +Proof. +intros. pos_or_neg a. apply div_1_r; auto. +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; auto. 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. + +(** TODO: TO MIGRATE LATER *) +Definition abs z := max z (-z). +Lemma abs_pos : forall z, 0<=z -> abs z == z. +Proof. +intros; apply max_l. apply le_trans with 0; auto. +rewrite opp_nonpos_nonneg; auto. +Qed. +Lemma abs_neg : forall z, 0<=-z -> abs z == -z. +Proof. +intros; apply max_r. apply le_trans with 0; auto. +rewrite <- opp_nonneg_nonpos; auto. +Qed. + +Lemma eq_sym_iff : forall x y, x==y <-> y==x. +Proof. +intros; split; symmetry; auto. +Qed. + +(** END TODO *) + +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_pos; intuition; order. +rewrite <- opp_inj_wd, opp_0, <- div_opp_r, div_small_iff by order. + rewrite (abs_pos a), (abs_neg b); intuition; order. +rewrite <- opp_inj_wd, opp_0, <- div_opp_l, div_small_iff by order. + rewrite (abs_neg a), (abs_pos b); intuition; order. +rewrite <- div_opp_opp, div_small_iff by order. + rewrite (abs_neg a), (abs_neg 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. intuition. +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. +destruct (mul_div_le (-a) b); auto. +rewrite opp_nonneg_nonpos; auto. +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. + +(** Some previous inequalities are 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; intuition. +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; auto. rewrite mul_comm; auto. +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; auto. rewrite mul_comm; auto. +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 auto. +rewrite <- (add_cancel_r _ _ ((a+b*c) mod c)). +rewrite <- div_mod; auto. +rewrite mod_add; auto. +rewrite mul_add_distr_l, add_shuffle0, <-div_mod, mul_comm; auto. +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). intros; apply div_add; auto. +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); apply div_mul_cancel_r; auto. +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; intuition). +rewrite ! mod_eq by auto. +rewrite div_mul_cancel_r by order. +rewrite mul_sub_distr_r, <- !mul_assoc, (mul_comm (a/b) c); auto. +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); apply mul_mod_distr_r; auto. +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 auto. 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. apply Aux1; auto. + 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. apply Aux2; auto. +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). apply mul_mod_idemp_l; auto. +Qed. + +Theorem mul_mod: forall a b n, n~=0 -> + (a * b) mod n == ((a mod n) * (b mod n)) mod n. +Proof. +intros. rewrite mul_mod_idemp_l, mul_mod_idemp_r; auto. +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)]. +apply Aux; auto. +apply opp_inj. rewrite <-2 mod_opp_l, 2 opp_add_distr, <-mod_opp_l by order. +apply Aux; auto. +rewrite opp_nonneg_nonpos; auto. +rewrite opp_nonneg_nonpos; auto. +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; auto. +rewrite mul_comm; auto. +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; auto. +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; auto. 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; auto. + 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; intuition 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. + pos_or_neg a; pos_or_neg b. + apply mod_divides; order. + rewrite <- mod_opp_r, mod_divides by order. + split; intros (c,Hc); exists (-c). + rewrite mul_opp_r, <- mul_opp_l; auto. + rewrite mul_opp_opp; auto. + rewrite <- opp_inj_wd, opp_0, <- mod_opp_l, mod_divides by order. + split; intros (c,Hc); exists (-c). + rewrite mul_opp_r, eq_opp_r; auto. + rewrite mul_opp_r, opp_inj_wd; auto. + rewrite <- opp_inj_wd, opp_0, <- mod_opp_opp, mod_divides by order. + split; intros (c,Hc); exists c. + rewrite <-opp_inj_wd, <- mul_opp_l; auto. + rewrite mul_opp_l, opp_inj_wd; auto. +Qed. + +End ZDivPropFunct. + diff --git a/theories/Numbers/Integer/Abstract/ZMulOrder.v b/theories/Numbers/Integer/Abstract/ZMulOrder.v index 4f11dcc5c0..a9cf0dc4d3 100644 --- a/theories/Numbers/Integer/Abstract/ZMulOrder.v +++ b/theories/Numbers/Integer/Abstract/ZMulOrder.v @@ -34,12 +34,6 @@ now apply mul_le_mono_nonpos_l. apply mul_le_mono_nonpos_r; [now apply le_trans with q | assumption]. Qed. -Theorem mul_nonneg_nonneg : forall n m, 0 <= n -> 0 <= m -> 0 <= n * m. -Proof. -intros n m H1 H2. -rewrite <- (mul_0_l m). now apply mul_le_mono_nonneg_r. -Qed. - Theorem mul_nonpos_nonpos : forall n m, n <= 0 -> m <= 0 -> 0 <= n * m. Proof. intros n m H1 H2. diff --git a/theories/Numbers/NatInt/NZAdd.v b/theories/Numbers/NatInt/NZAdd.v index 579d2197f5..785abb5c7f 100644 --- a/theories/Numbers/NatInt/NZAdd.v +++ b/theories/Numbers/NatInt/NZAdd.v @@ -67,18 +67,20 @@ Proof. intros n m p. rewrite (add_comm n p), (add_comm m p). apply add_cancel_l. Qed. +Theorem add_shuffle0 : forall n m p, n+m+p == n+p+m. +Proof. +intros n m p. rewrite <- 2 add_assoc, add_cancel_l. apply add_comm. +Qed. + Theorem add_shuffle1 : forall n m p q, (n + m) + (p + q) == (n + p) + (m + q). Proof. -intros n m p q. -rewrite <- (add_assoc n m), <- (add_assoc n p), add_cancel_l. -rewrite 2 add_assoc, add_cancel_r. now apply add_comm. +intros n m p q. rewrite 2 add_assoc, add_cancel_r. apply add_shuffle0. Qed. Theorem add_shuffle2 : forall n m p q, (n + m) + (p + q) == (n + q) + (m + p). Proof. intros n m p q. -rewrite <- (add_assoc n m), <- (add_assoc n q), add_cancel_l. -rewrite add_assoc. now apply add_comm. +rewrite 2 add_assoc, add_shuffle0, add_cancel_r. apply add_shuffle0. Qed. Theorem sub_1_r : forall n, n - 1 == P n. diff --git a/theories/Numbers/NatInt/NZAxioms.v b/theories/Numbers/NatInt/NZAxioms.v index 20a85e2ef5..c0f3093291 100644 --- a/theories/Numbers/NatInt/NZAxioms.v +++ b/theories/Numbers/NatInt/NZAxioms.v @@ -102,6 +102,11 @@ Notation "x <= y" := (le x y) : NumScope. Notation "x > y" := (lt y x) (only parsing) : NumScope. Notation "x >= y" := (le y x) (only parsing) : NumScope. +Notation "x < y < z" := (x<y /\ y<z) : NumScope. +Notation "x <= y <= z" := (x<=y /\ y<=z) : NumScope. +Notation "x <= y < z" := (x<=y /\ y<z) : NumScope. +Notation "x < y <= z" := (x<y /\ y<=z) : NumScope. + Instance lt_wd : Proper (eq ==> eq ==> iff) lt. (** Compatibility of [le] can be proved later from [lt_wd] and [lt_eq_cases] *) diff --git a/theories/Numbers/NatInt/NZDiv.v b/theories/Numbers/NatInt/NZDiv.v new file mode 100644 index 0000000000..9ea654cc92 --- /dev/null +++ b/theories/Numbers/NatInt/NZDiv.v @@ -0,0 +1,528 @@ +(************************************************************************) +(* 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 *) + +Require Import NZAxioms NZMulOrder. + +Open Scope NumScope. + +Module Type NZDiv (Import NZ : NZOrdAxiomsSig). + + Parameter Inline div : t -> t -> t. + Parameter Inline modulo : t -> t -> t. + + Infix "/" := div : NumScope. + Infix "mod" := modulo (at level 40, no associativity) : NumScope. + + Instance div_wd : Proper (eq==>eq==>eq) div. + Instance mod_wd : Proper (eq==>eq==>eq) modulo. + + Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b). + Axiom mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b. + +End NZDiv. + +Module Type NZDivSig := NZOrdAxiomsSig <+ NZDiv. + +Module NZDivPropFunct (Import NZ : NZDivSig). + (* TODO: a transformer un jour en arg funct puis include *) + Module Import P := NZMulOrderPropFunct NZ. + +(** Uniqueness theorems *) + +Theorem div_mod_unique : + forall b q1 q2 r1 r2, 0<=r1<b -> 0<=r2<b -> + b*q1+r1 == b*q2+r2 -> q1 == q2 /\ r1 == r2. +Proof. +intros b. +assert (U : forall q1 q2 r1 r2, + b*q1+r1 == b*q2+r2 -> 0<=r1<b -> 0<=r2 -> q1<q2 -> False). + intros q1 q2 r1 r2 EQ LT Hr1 Hr2. + contradict EQ. + apply lt_neq. + apply lt_le_trans with (b*q1+b). + rewrite <- add_lt_mono_l; intuition. + apply le_trans with (b*q2). + rewrite mul_comm, <- mul_succ_l, mul_comm. + apply mul_le_mono_nonneg_l; intuition; try order. + rewrite le_succ_l; auto. + rewrite <- (add_0_r (b*q2)) at 1. + rewrite <- add_le_mono_l; intuition. + +intros q1 q2 r1 r2 Hr1 Hr2 EQ; destruct (lt_trichotomy q1 q2) as [LT|[EQ'|GT]]. +elim (U q1 q2 r1 r2); intuition. +split; auto. rewrite EQ' in EQ. rewrite add_cancel_l in EQ; auto. +elim (U q2 q1 r2 r1); intuition. +Qed. + +Theorem div_unique: + forall a b q r, 0<=a -> 0<=r<b -> + a == b*q + r -> q == a/b. +Proof. +intros a b q r Ha (Hb,Hr) EQ. +rewrite (div_mod a b) in EQ by order. +destruct (div_mod_unique b (a/b) q (a mod b) r); auto. +apply mod_bound; order. +Qed. + +Theorem mod_unique: + forall a b q r, 0<=a -> 0<=r<b -> + a == b*q + r -> r == a mod b. +Proof. +intros a b q r Ha (Hb,Hr) EQ. +rewrite (div_mod a b) in EQ by order. +destruct (div_mod_unique b (a/b) q (a mod b) r); auto. +apply mod_bound; order. +Qed. + + +(** A division by itself returns 1 *) + +Lemma div_same : forall a, 0<a -> a/a == 1. +Proof. +intros. symmetry. +apply div_unique with 0; intuition; try order. +nzsimpl; auto. +Qed. + +Lemma mod_same : forall a, 0<a -> a mod a == 0. +Proof. +intros. symmetry. +apply mod_unique with 1; intuition; try order. +nzsimpl; auto. +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. +intros. symmetry. +apply div_unique with a; intuition; try order. +nzsimpl; auto. +Qed. + +(** Same situation, in term of modulo: *) + +Theorem mod_small: forall a b, 0<=a<b -> a mod b == a. +Proof. +intros. symmetry. +apply mod_unique with 0; intuition; try order. +nzsimpl; auto. +Qed. + +(** * Basic values of divisions and modulo. *) + +Lemma div_0_l: forall a, 0<a -> 0/a == 0. +Proof. +intros; apply div_small; split; order. +Qed. + +Lemma mod_0_l: forall a, 0<a -> 0 mod a == 0. +Proof. +intros; apply mod_small; split; order. +Qed. + +Lemma div_1_r: forall a, 0<=a -> a/1 == a. +Proof. +intros. symmetry. +apply div_unique with 0; try split; try order; try apply lt_0_1. +nzsimpl; auto. +Qed. + +Lemma mod_1_r: forall a, 0<=a -> a mod 1 == 0. +Proof. +intros. symmetry. +apply mod_unique with a; try split; try order; try apply lt_0_1. +nzsimpl; auto. +Qed. + +Lemma div_1_l: forall a, 1<a -> 1/a == 0. +Proof. +intros; apply div_small; split; auto. apply le_succ_diag_r. +Qed. + +Lemma mod_1_l: forall a, 1<a -> 1 mod a == 1. +Proof. +intros; apply mod_small; split; auto. apply le_succ_diag_r. +Qed. + +Lemma div_mul : forall a b, 0<=a -> 0<b -> (a*b)/b == a. +Proof. +intros; symmetry. +apply div_unique with 0; try split; try order. +apply mul_nonneg_nonneg; order. +nzsimpl; apply mul_comm. +Qed. + +Lemma mod_mul : forall a b, 0<=a -> 0<b -> (a*b) mod b == 0. +Proof. +intros; symmetry. +apply mod_unique with a; try split; try order. +apply mul_nonneg_nonneg; order. +nzsimpl; apply mul_comm. +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. +intros. destruct (le_gt_cases b a). +apply le_trans with b; auto. +apply lt_le_incl. destruct (mod_bound a b); auto. +rewrite lt_eq_cases; right. +apply mod_small; auto. +Qed. + + +(* Division of positive numbers is positive. *) + +Lemma div_pos: forall a b, 0<=a -> 0<b -> 0 <= a/b. +Proof. +intros. +rewrite (mul_le_mono_pos_l _ _ b); auto; nzsimpl. +rewrite (add_le_mono_r _ _ (a mod b)). +rewrite <- div_mod by order. +nzsimpl. +apply mod_le; auto. +Qed. + +Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b. +Proof. +intros a b (Hb,Hab). +assert (LE : 0 <= a/b) by (apply div_pos; order). +assert (MOD : a mod b < b) by (destruct (mod_bound a b); order). +rewrite lt_eq_cases in LE; destruct LE as [LT|EQ]; auto. +exfalso; revert Hab. +rewrite (div_mod a b), <-EQ; nzsimpl; order. +Qed. + +Lemma div_small_iff : forall a b, 0<=a -> 0<b -> (a/b==0 <-> a<b). +Proof. +intros a b Ha Hb; split; intros Hab. +destruct (lt_ge_cases a b); auto. +symmetry in Hab. contradict Hab. apply lt_neq, div_str_pos; auto. +apply div_small; auto. +Qed. + +Lemma mod_small_iff : forall a b, 0<=a -> 0<b -> (a mod b == a <-> a<b). +Proof. +intros a b Ha Hb. split; intros H; auto using mod_small. +rewrite <- div_small_iff; auto. +rewrite <- (mul_cancel_l _ _ b) by order. +rewrite <- (add_cancel_r _ _ (a mod b)). +rewrite <- div_mod, H by order. nzsimpl; auto. +Qed. + +Lemma div_str_pos_iff : forall a b, 0<=a -> 0<b -> (0<a/b <-> b<=a). +Proof. +intros a b Ha Hb; split; intros Hab. +destruct (lt_ge_cases a b) as [LT|LE]; auto. +rewrite <- div_small_iff in LT; order. +apply div_str_pos; auto. +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. +intros. +assert (0 < b) by (apply lt_trans with 1; auto using lt_0_1). +destruct (lt_ge_cases a b). +rewrite div_small; try split; order. +rewrite (div_mod a b) at 2; [|order]. +apply lt_le_trans with (b*(a/b)). +rewrite <- (mul_1_l (a/b)) at 1. +rewrite <- mul_lt_mono_pos_r; auto. +apply div_str_pos; auto. +rewrite <- (add_0_r (b*(a/b))) at 1. +rewrite <- add_le_mono_l. destruct (mod_bound a b); order. +Qed. + +(** [le] is compatible with a positive division. *) + +Lemma div_le_mono : forall a b c, 0<c -> 0<=a<=b -> a/c <= b/c. +Proof. +intros a b c Hc (Ha,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; auto. +rewrite (div_mod b c) at 1; [| order]. +rewrite <- add_assoc, <- add_le_mono_l. +apply le_trans with (c+0). +nzsimpl; destruct (mod_bound b c); order. +rewrite <- add_le_mono_l. destruct (mod_bound a c); order. +Qed. + +(** The following two properties could be used as specification of div *) + +Lemma mul_div_le : forall a b, 0<=a -> 0<b -> b*(a/b) <= a. +Proof. +intros. +rewrite (add_le_mono_r _ _ (a mod b)), <- div_mod by order. +rewrite <- (add_0_r a) at 1. +rewrite <- add_le_mono_l. destruct (mod_bound a b); order. +Qed. + +Lemma mul_succ_div_gt : forall a b, 0<=a -> 0<b -> a < b*(S (a/b)). +Proof. +intros. +rewrite (div_mod a b) at 1; [|order]. +rewrite (mul_succ_r). +rewrite <- add_lt_mono_l. +destruct (mod_bound a b); auto. +Qed. + + +(** The previous inequality is exact iff the modulo is zero. *) + +Lemma div_exact : forall a b, 0<=a -> 0<b -> (a == b*(a/b) <-> a mod b == 0). +Proof. +intros. rewrite (div_mod a b) at 1; [|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<=a -> 0<b -> a < b*q -> a/b < q. +Proof. +intros. +rewrite (mul_lt_mono_pos_l b) by order. +apply le_lt_trans with a; auto. +apply mul_div_le; auto. +Qed. + +Theorem div_le_upper_bound: + forall a b q, 0<=a -> 0<b -> a <= b*q -> a/b <= q. +Proof. +intros. +rewrite (mul_le_mono_pos_l _ _ b) by order. +apply le_trans with a; auto. +apply mul_div_le; auto. +Qed. + +Theorem div_le_lower_bound: + forall a b q, 0<=a -> 0<b -> b*q <= a -> q <= a/b. +Proof. +intros a b q Ha Hb H. +destruct (lt_ge_cases 0 q). +rewrite <- (div_mul q b); try order. +apply div_le_mono; auto. +rewrite mul_comm; split; auto. +apply lt_le_incl, mul_pos_pos; auto. +apply le_trans with 0; auto; apply div_pos; auto. +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. + intros p q r Hp (Hq,Hqr). + apply div_le_lower_bound; auto. + rewrite (div_mod p r) at 2; [|order]. + apply le_trans with (r*(p/r)). + apply mul_le_mono_nonneg_r; try order. + apply div_pos; order. + rewrite <- (add_0_r (r*(p/r))) at 1. + rewrite <- add_le_mono_l. destruct (mod_bound p r); order. +Qed. + + +(** * Relations between usual operations and mod and div *) + +Lemma mod_add : forall a b c, 0<=a -> 0<=a+b*c -> 0<c -> + (a + b * c) mod c == a mod c. +Proof. + intros. + symmetry. + apply mod_unique with (a/c+b); auto. + apply mod_bound; auto. + rewrite mul_add_distr_l, add_shuffle0, <- div_mod by order. + rewrite mul_comm; auto. +Qed. + +Lemma div_add : forall a b c, 0<=a -> 0<=a+b*c -> 0<c -> + (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. + rewrite mul_comm; auto. +Qed. + +Lemma div_add_l: forall a b c, 0<=c -> 0<=a*b+c -> 0<b -> + (a * b + c) / b == a + c / b. +Proof. + intros a b c. rewrite (add_comm _ c), (add_comm a). intros. apply div_add; auto. +Qed. + +(** Cancellations. *) + +Lemma div_mul_cancel_r : forall a b c, 0<=a -> 0<b -> 0<c -> + (a*c)/(b*c) == a/b. +Proof. + intros. + symmetry. + apply div_unique with ((a mod b)*c). + apply mul_nonneg_nonneg; order. + split. + apply mul_nonneg_nonneg; destruct (mod_bound a b); order. + rewrite <- mul_lt_mono_pos_r; auto. destruct (mod_bound a b); auto. + rewrite (div_mod a b) at 1; [|order]. + rewrite mul_add_distr_r. + rewrite add_cancel_r. + rewrite <- 2 mul_assoc. rewrite (mul_comm c); auto. +Qed. + +Lemma div_mul_cancel_l : forall a b c, 0<=a -> 0<b -> 0<c -> + (c*a)/(c*b) == a/b. +Proof. + intros. + rewrite (mul_comm c a), (mul_comm c b); apply div_mul_cancel_r; auto. +Qed. + +Lemma mul_mod_distr_l: forall a b c, 0<=a -> 0<b -> 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; auto. + rewrite <- mul_assoc, <- mul_add_distr_l, mul_cancel_l by order. + apply div_mod; order. + intro EQ; symmetry in EQ; revert EQ. apply lt_neq, mul_pos_pos; auto. +Qed. + +Lemma mul_mod_distr_r: forall a b c, 0<=a -> 0<b -> 0<c -> + (a*c) mod (b*c) == (a mod b) * c. +Proof. + intros. + rewrite (mul_comm a c), (mul_comm b c); rewrite mul_mod_distr_l; auto. + apply mul_comm. +Qed. + +(** Operations modulo. *) + +Theorem mod_mod: forall a n, 0<=a -> 0<n -> + (a mod n) mod n == a mod n. +Proof. + intros. destruct (mod_bound a n); auto. rewrite mod_small_iff; auto. +Qed. + +Lemma mul_mod_idemp_l : forall a b n, 0<=a -> 0<=b -> 0<n -> + ((a mod n)*b) mod n == (a*b) mod n. +Proof. + intros a b n Ha Hb Hn. symmetry. + generalize (mul_nonneg_nonneg _ _ Ha Hb). + rewrite (div_mod a n) at 1 2; [|order]. + rewrite add_comm, (mul_comm n), (mul_comm _ b). + rewrite mul_add_distr_l, mul_assoc. + intros. rewrite mod_add; auto. + rewrite mul_comm; auto. + apply mul_nonneg_nonneg; destruct (mod_bound a n); auto. +Qed. + +Lemma mul_mod_idemp_r : forall a b n, 0<=a -> 0<=b -> 0<n -> + (a*(b mod n)) mod n == (a*b) mod n. +Proof. + intros. rewrite !(mul_comm a). apply mul_mod_idemp_l; auto. +Qed. + +Theorem mul_mod: forall a b n, 0<=a -> 0<=b -> 0<n -> + (a * b) mod n == ((a mod n) * (b mod n)) mod n. +Proof. + intros. rewrite mul_mod_idemp_l, mul_mod_idemp_r; auto. + destruct (mod_bound b n); auto. +Qed. + +Lemma add_mod_idemp_l : forall a b n, 0<=a -> 0<=b -> 0<n -> + ((a mod n)+b) mod n == (a+b) mod n. +Proof. + intros a b n Ha Hb Hn. symmetry. + generalize (add_nonneg_nonneg _ _ Ha Hb). + rewrite (div_mod a n) at 1 2; [|order]. + rewrite <- add_assoc, add_comm, mul_comm. + intros. rewrite mod_add; auto. + apply add_nonneg_nonneg; auto. destruct (mod_bound a n); auto. +Qed. + +Lemma add_mod_idemp_r : forall a b n, 0<=a -> 0<=b -> 0<n -> + (a+(b mod n)) mod n == (a+b) mod n. +Proof. + intros. rewrite !(add_comm a). apply add_mod_idemp_l; auto. +Qed. + +Theorem add_mod: forall a b n, 0<=a -> 0<=b -> 0<n -> + (a+b) mod n == (a mod n + b mod n) mod n. +Proof. + intros. rewrite add_mod_idemp_l, add_mod_idemp_r; auto. + destruct (mod_bound b n); auto. +Qed. + +Lemma div_div : forall a b c, 0<=a -> 0<b -> 0<c -> + (a/b)/c == a/(b*c). +Proof. + intros a b c Ha Hb Hc. + apply div_unique with (b*((a/b) mod c) + a mod b); auto. + (* begin 0<= ... <b*c *) + destruct (mod_bound (a/b) c), (mod_bound a b); auto using div_pos. + split. + apply add_nonneg_nonneg; auto. + apply mul_nonneg_nonneg; order. + apply lt_le_trans with (b*((a/b) mod c) + b). + rewrite <- add_lt_mono_l; auto. + rewrite <- mul_succ_r, <- mul_le_mono_pos_l, le_succ_l; auto. + (* end 0<= ... < b*c *) + rewrite (div_mod a b) at 1; [|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. + intros. + apply div_le_lower_bound; auto. + apply mul_nonneg_nonneg; auto. + rewrite mul_assoc, (mul_comm b c), <- mul_assoc. + apply mul_le_mono_nonneg_l; auto. + apply mul_div_le; auto. +Qed. + +(** mod is related to divisibility *) + +Lemma mod_divides : forall a b, 0<=a -> 0<b -> + (a mod b == 0 <-> exists c, a == b*c). +Proof. + split. + intros. exists (a/b). rewrite div_exact; auto. + intros (c,Hc). symmetry; apply mod_unique with c; auto. + split; order. + nzsimpl; auto. +Qed. + +End NZDivPropFunct. + diff --git a/theories/Numbers/NatInt/NZMulOrder.v b/theories/Numbers/NatInt/NZMulOrder.v index 306b69022c..b6e4849f5e 100644 --- a/theories/Numbers/NatInt/NZMulOrder.v +++ b/theories/Numbers/NatInt/NZMulOrder.v @@ -202,6 +202,11 @@ Proof. intros; rewrite mul_comm; now apply mul_pos_neg. Qed. +Theorem mul_nonneg_nonneg : forall n m, 0 <= n -> 0 <= m -> 0 <= n*m. +Proof. +intros. rewrite <- (mul_0_l m). apply mul_le_mono_nonneg; order. +Qed. + Theorem lt_1_mul_pos : forall n m, 1 < n -> 0 < m -> 1 < n * m. Proof. intros n m H1 H2. apply -> (mul_lt_mono_pos_r m) in H1. diff --git a/theories/Numbers/Natural/Abstract/NDiv.v b/theories/Numbers/Natural/Abstract/NDiv.v new file mode 100644 index 0000000000..9ff9c08cf2 --- /dev/null +++ b/theories/Numbers/Natural/Abstract/NDiv.v @@ -0,0 +1,249 @@ +(************************************************************************) +(* 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 *) + +Require Import NAxioms NProperties NZDiv. + +Open Scope NumScope. + +Module Type NDiv (Import N : NAxiomsSig). + + Parameter Inline div : t -> t -> t. + Parameter Inline modulo : t -> t -> t. + + Infix "/" := div : NumScope. + Infix "mod" := modulo (at level 40, no associativity) : NumScope. + + Instance div_wd : Proper (eq==>eq==>eq) div. + Instance mod_wd : Proper (eq==>eq==>eq) modulo. + + Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b). + Axiom mod_upper_bound : forall a b, b ~= 0 -> a mod b < b. + +End NDiv. + +Module Type NDivSig := NAxiomsSig <+ NDiv. + +Module NDivPropFunct (Import N : NDivSig). + Module Import NP := NPropFunct N. + +(** We benefit from what already exists for NZ *) + + Module N' <: NZDivSig. + Include N. + Lemma mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b. + Proof. split. apply le_0_l. apply mod_upper_bound. order. Qed. + End N'. + Module Import NZDivP := NZDivPropFunct N'. + + Ltac auto' := try rewrite <- neq_0_lt_0; auto using le_0_l. + +(** Let's now state again theorems, but without useless hypothesis. *) + +(** Uniqueness theorems *) + +Theorem div_mod_unique : + forall b q1 q2 r1 r2, r1<b -> r2<b -> + b*q1+r1 == b*q2+r2 -> q1 == q2 /\ r1 == r2. +Proof. intros. apply div_mod_unique with b; auto'. Qed. + +Theorem div_unique: + forall a b q r, r<b -> a == b*q + r -> q == a/b. +Proof. intros; apply div_unique with r; auto'. Qed. + +Theorem mod_unique: + forall a b q r, r<b -> a == b*q + r -> r == a mod b. +Proof. intros. apply mod_unique with q; auto'. Qed. + +(** A division by itself returns 1 *) + +Lemma div_same : forall a, a~=0 -> a/a == 1. +Proof. intros. apply div_same; auto'. Qed. + +Lemma mod_same : forall a, a~=0 -> a mod a == 0. +Proof. intros. apply mod_same; auto'. Qed. + +(** A division of a small number by a bigger one yields zero. *) + +Theorem div_small: forall a b, a<b -> a/b == 0. +Proof. intros. apply div_small; auto'. Qed. + +(** Same situation, in term of modulo: *) + +Theorem mod_small: forall a b, a<b -> a mod b == a. +Proof. intros. apply mod_small; auto'. Qed. + +(** * Basic values of divisions and modulo. *) + +Lemma div_0_l: forall a, a~=0 -> 0/a == 0. +Proof. intros. apply div_0_l; auto'. Qed. + +Lemma mod_0_l: forall a, a~=0 -> 0 mod a == 0. +Proof. intros. apply mod_0_l; auto'. Qed. + +Lemma div_1_r: forall a, a/1 == a. +Proof. intros. apply div_1_r; auto'. Qed. + +Lemma mod_1_r: forall a, a mod 1 == 0. +Proof. intros. apply mod_1_r; auto'. 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. apply div_mul; auto'. Qed. + +Lemma mod_mul : forall a b, b~=0 -> (a*b) mod b == 0. +Proof. intros. apply mod_mul; auto'. Qed. + + +(** * Order results about mod and div *) + +(** A modulo cannot grow beyond its starting point. *) + +Theorem mod_le: forall a b, b~=0 -> a mod b <= a. +Proof. intros. apply mod_le; auto'. 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 <-> a<b). +Proof. intros. apply div_small_iff; auto'. Qed. + +Lemma mod_small_iff : forall a b, b~=0 -> (a mod b == a <-> a<b). +Proof. intros. apply mod_small_iff; auto'. Qed. + +Lemma div_str_pos_iff : forall a b, b~=0 -> (0<a/b <-> b<=a). +Proof. intros. apply div_str_pos_iff; auto'. 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, c~=0 -> a<=b -> a/c <= b/c. +Proof. intros. apply div_le_mono; auto'. Qed. + +Lemma mul_div_le : forall a b, b~=0 -> b*(a/b) <= a. +Proof. intros. apply mul_div_le; auto'. Qed. + +Lemma mul_succ_div_gt: forall a b, b~=0 -> a < b*(S (a/b)). +Proof. intros; apply mul_succ_div_gt; auto'. Qed. + +(** The previous inequality 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. apply div_exact; auto'. Qed. + +(** Some additionnal inequalities about div. *) + +Theorem div_lt_upper_bound: + forall a b q, b~=0 -> a < b*q -> a/b < q. +Proof. intros. apply div_lt_upper_bound; auto'. Qed. + +Theorem div_le_upper_bound: + forall a b q, b~=0 -> a <= b*q -> a/b <= q. +Proof. intros; apply div_le_upper_bound; auto'. Qed. + +Theorem div_le_lower_bound: + forall a b q, b~=0 -> b*q <= a -> q <= a/b. +Proof. intros; apply div_le_lower_bound; auto'. Qed. + +(** A division respects opposite monotonicity for the divisor *) + +Lemma div_le_compat_l: forall p q r, 0<q<=r -> p/r <= p/q. +Proof. intros. apply div_le_compat_l. auto'. auto. 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. apply mod_add; auto'. Qed. + +Lemma div_add : forall a b c, c~=0 -> + (a + b * c) / c == a / c + b. +Proof. intros. apply div_add; auto'. Qed. + +Lemma div_add_l: forall a b c, b~=0 -> + (a * b + c) / b == a + c / b. +Proof. intros. apply div_add_l; auto'. Qed. + +(** Cancellations. *) + +Lemma div_mul_cancel_r : forall a b c, b~=0 -> c~=0 -> + (a*c)/(b*c) == a/b. +Proof. intros. apply div_mul_cancel_r; auto'. Qed. + +Lemma div_mul_cancel_l : forall a b c, b~=0 -> c~=0 -> + (c*a)/(c*b) == a/b. +Proof. intros. apply div_mul_cancel_l; auto'. 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. apply mul_mod_distr_r; auto'. 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. apply mul_mod_distr_l; auto'. Qed. + +(** Operations modulo. *) + +Theorem mod_mod: forall a n, n~=0 -> + (a mod n) mod n == a mod n. +Proof. intros. apply mod_mod; auto'. Qed. + +Lemma mul_mod_idemp_l : forall a b n, n~=0 -> + ((a mod n)*b) mod n == (a*b) mod n. +Proof. intros. apply mul_mod_idemp_l; auto'. Qed. + +Lemma mul_mod_idemp_r : forall a b n, n~=0 -> + (a*(b mod n)) mod n == (a*b) mod n. +Proof. intros. apply mul_mod_idemp_r; auto'. Qed. + +Theorem mul_mod: forall a b n, n~=0 -> + (a * b) mod n == ((a mod n) * (b mod n)) mod n. +Proof. intros. apply mul_mod; auto'. Qed. + +Lemma add_mod_idemp_l : forall a b n, n~=0 -> + ((a mod n)+b) mod n == (a+b) mod n. +Proof. intros. apply add_mod_idemp_l; auto'. Qed. + +Lemma add_mod_idemp_r : forall a b n, n~=0 -> + (a+(b mod n)) mod n == (a+b) mod n. +Proof. intros. apply add_mod_idemp_r; auto'. Qed. + +Theorem add_mod: forall a b n, n~=0 -> + (a+b) mod n == (a mod n + b mod n) mod n. +Proof. intros. apply add_mod; auto'. Qed. + +Lemma div_div : forall a b c, b~=0 -> c~=0 -> + (a/b)/c == a/(b*c). +Proof. intros. apply div_div; auto'. Qed. + +(** A last inequality: *) + +Theorem div_mul_le: + forall a b c, b~=0 -> c*(a/b) <= (c*a)/b. +Proof. intros. apply div_mul_le; auto'. 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. apply mod_divides; auto'. Qed. + +End NDivPropFunct. + diff --git a/theories/Numbers/Natural/Peano/NPeano.v b/theories/Numbers/Natural/Peano/NPeano.v index 8974ef1143..97ac9f8729 100644 --- a/theories/Numbers/Natural/Peano/NPeano.v +++ b/theories/Numbers/Natural/Peano/NPeano.v @@ -10,10 +10,7 @@ (*i $Id$ i*) -Require Import Arith. -Require Import Min. -Require Import Max. -Require Import NAxioms NProperties. +Require Import Arith MinMax NAxioms NProperties. (** * Implementation of [NAxiomsSig] by [nat] *) @@ -164,7 +161,75 @@ Definition max := max. End NPeanoAxiomsMod. -(* Now we apply the largest property functor *) +(** Now we apply the largest property functor *) Module Export NPeanoPropMod := NPropFunct NPeanoAxiomsMod. + + +(** Euclidean Division *) + +Definition divF div x y := if leb y x then S (div (x-y) y) else 0. +Definition modF mod x y := if leb y x then mod (x-y) y else x. +Definition initF (_ _ : nat) := 0. + +Fixpoint loop {A} (F:A->A)(i:A) (n:nat) : A := + match n with + | 0 => i + | S n => F (loop F i n) + end. + +Definition div x y := loop divF initF x x y. +Definition modulo x y := loop modF initF x x y. +Infix "/" := div : nat_scope. +Infix "mod" := modulo (at level 40, no associativity) : nat_scope. + +Lemma div_mod : forall x y, y<>0 -> x = y*(x/y) + x mod y. +Proof. + cut (forall n x y, y<>0 -> x<=n -> + x = y*(loop divF initF n x y) + (loop modF initF n x y)). + intros H x y Hy. apply H; auto. + induction n. + simpl; unfold initF; simpl. intros. nzsimpl. auto with arith. + simpl; unfold divF at 1, modF at 1. + intros. + destruct (leb y x) as [ ]_eqn:L; + [apply leb_complete in L | apply leb_complete_conv in L]. + rewrite mul_succ_r, <- add_assoc, (add_comm y), add_assoc. + rewrite <- IHn; auto. + symmetry; apply sub_add; auto. + rewrite <- NPeanoAxiomsMod.lt_succ_r. + apply lt_le_trans with x; auto. + apply lt_minus; auto. rewrite <- neq_0_lt_0; auto. + nzsimpl; auto. +Qed. + +Lemma mod_upper_bound : forall x y, y<>0 -> x mod y < y. +Proof. + cut (forall n x y, y<>0 -> x<=n -> loop modF initF n x y < y). + intros H x y Hy. apply H; auto. + induction n. + simpl; unfold initF. intros. rewrite <- neq_0_lt_0; auto. + simpl; unfold modF at 1. + intros. + destruct (leb y x) as [ ]_eqn:L; + [apply leb_complete in L | apply leb_complete_conv in L]; auto. + apply IHn; auto. + rewrite <- NPeanoAxiomsMod.lt_succ_r. + apply lt_le_trans with x; auto. + apply lt_minus; auto. rewrite <- neq_0_lt_0; auto. +Qed. + +Require Import NDiv. + +Module NDivMod <: NDivSig. + Include NPeanoAxiomsMod. + Definition div := div. + Definition modulo := modulo. + Definition div_mod := div_mod. + Definition mod_upper_bound := mod_upper_bound. + Program Instance div_wd : Proper (eq==>eq==>eq) div. + Program Instance mod_wd : Proper (eq==>eq==>eq) modulo. +End NDivMod. + +Module Export NDivPropMod := NDivPropFunct NDivMod. diff --git a/theories/Numbers/vo.itarget b/theories/Numbers/vo.itarget index 003f20f2d6..7c29450eab 100644 --- a/theories/Numbers/vo.itarget +++ b/theories/Numbers/vo.itarget @@ -22,6 +22,9 @@ Integer/Abstract/ZLt.vo Integer/Abstract/ZMulOrder.vo Integer/Abstract/ZMul.vo Integer/Abstract/ZProperties.vo +Integer/Abstract/ZDivCoq.vo +Integer/Abstract/ZDivOcaml.vo +Integer/Abstract/ZDivMath.vo Integer/BigZ/BigZ.vo Integer/BigZ/ZMake.vo Integer/Binary/ZBinary.vo @@ -38,6 +41,7 @@ NatInt/NZMul.vo NatInt/NZOrder.vo NatInt/NZProperties.vo NatInt/NZDomain.vo +NatInt/NZDiv.vo Natural/Abstract/NAddOrder.vo Natural/Abstract/NAdd.vo Natural/Abstract/NAxioms.vo @@ -49,6 +53,7 @@ Natural/Abstract/NOrder.vo Natural/Abstract/NStrongRec.vo Natural/Abstract/NSub.vo Natural/Abstract/NProperties.vo +Natural/Abstract/NDiv.vo Natural/BigN/BigN.vo Natural/BigN/Nbasic.vo Natural/BigN/NMake.vo diff --git a/theories/ZArith/ZOdiv.v b/theories/ZArith/ZOdiv.v index 28b664aa48..c73673a858 100644 --- a/theories/ZArith/ZOdiv.v +++ b/theories/ZArith/ZOdiv.v @@ -7,7 +7,8 @@ (************************************************************************) -Require Import BinPos BinNat Nnat ZArith_base ROmega ZArithRing. +Require Import BinPos BinNat Nnat ZArith_base ROmega ZArithRing + ZBinary ZDivOcaml Morphisms. Require Export ZOdiv_def. Require Zdiv. @@ -243,6 +244,29 @@ Proof. unfold ZOmod, ZOdiv_eucl; destruct Pdiv_eucl; simpl; auto with zarith. Qed. +(** We know enough to prove that [ZOdiv] and [ZOmod] are instances of + one of the abstract Euclidean divisions of Numbers. *) + +Module ZODiv <: ZDiv ZBinAxiomsMod. + Definition div := ZOdiv. + Definition modulo := ZOmod. + Program Instance div_wd : Proper (eq==>eq==>eq) div. + Program Instance mod_wd : Proper (eq==>eq==>eq) modulo. + + Definition div_mod := fun a b (_:b<>0) => ZO_div_mod_eq a b. + Definition mod_bound := ZOmod_lt_pos_pos. + Definition mod_opp_l := fun a b (_:b<>0) => ZOmod_opp_l a b. + Definition mod_opp_r := fun a b (_:b<>0) => ZOmod_opp_r a b. +End ZODiv. + +Module ZODivMod := ZBinAxiomsMod <+ ZODiv. + +(** We hence benefit from generic results about this abstract division. *) + +Module Z. + Include ZDivPropFunct ZODivMod. +End Z. + (** * Unicity results *) Definition Remainder a b r := @@ -291,10 +315,7 @@ Qed. Theorem ZOdiv_unique: forall a b q r, 0 <= a -> 0 <= r < b -> a = b*q + r -> q = a/b. -Proof. - intros; eapply ZOdiv_unique_full; eauto. - red; romega with *. -Qed. +Proof. exact Z.div_unique. Qed. Theorem ZOmod_unique_full: forall a b q r, Remainder a b r -> @@ -306,10 +327,7 @@ Qed. Theorem ZOmod_unique: forall a b q r, 0 <= a -> 0 <= r < b -> a = b*q + r -> r = a mod b. -Proof. - intros; eapply ZOmod_unique_full; eauto. - red; romega with *. -Qed. +Proof. exact Z.mod_unique. Qed. (** * Basic values of divisions and modulo. *) @@ -334,56 +352,37 @@ Proof. Qed. Lemma ZOmod_1_r: forall a, a mod 1 = 0. -Proof. - intros; symmetry; apply ZOmod_unique_full with a; auto with zarith. - rewrite Remainder_equiv; red; simpl; auto with zarith. -Qed. +Proof. exact Z.mod_1_r. Qed. Lemma ZOdiv_1_r: forall a, a/1 = a. -Proof. - intros; symmetry; apply ZOdiv_unique_full with 0; auto with zarith. - rewrite Remainder_equiv; red; simpl; auto with zarith. -Qed. +Proof. exact Z.div_1_r. Qed. Hint Resolve ZOmod_0_l ZOmod_0_r ZOdiv_0_l ZOdiv_0_r ZOdiv_1_r ZOmod_1_r : zarith. Lemma ZOdiv_1_l: forall a, 1 < a -> 1/a = 0. -Proof. - intros; symmetry; apply ZOdiv_unique with 1; auto with zarith. -Qed. +Proof. exact Z.div_1_l. Qed. Lemma ZOmod_1_l: forall a, 1 < a -> 1 mod a = 1. -Proof. - intros; symmetry; apply ZOmod_unique with 0; auto with zarith. -Qed. +Proof. exact Z.mod_1_l. Qed. Lemma ZO_div_same : forall a:Z, a<>0 -> a/a = 1. -Proof. - intros; symmetry; apply ZOdiv_unique_full with 0; auto with *. - rewrite Remainder_equiv; red; simpl; romega with *. -Qed. +Proof. exact Z.div_same. Qed. Lemma ZO_mod_same : forall a, a mod a = 0. Proof. - destruct a; intros; symmetry. - compute; auto. - apply ZOmod_unique with 1; auto with *; romega with *. - apply ZOmod_unique_full with 1; auto with *; red; romega with *. + intros. destruct (Z_eq_dec a 0); subst. apply ZOmod_0_l. + apply Z.mod_same; auto. Qed. Lemma ZO_mod_mult : forall a b, (a*b) mod b = 0. Proof. - intros a b; destruct (Z_eq_dec b 0) as [Hb|Hb]. - subst; simpl; rewrite ZOmod_0_r; auto with zarith. - symmetry; apply ZOmod_unique_full with a; [ red; romega with * | ring ]. + intros. destruct (Z_eq_dec b 0); subst. rewrite Zmult_0_r. apply ZOmod_0_l. + apply Z.mod_mul; auto. Qed. Lemma ZO_div_mult : forall a b:Z, b <> 0 -> (a*b)/b = a. -Proof. - intros; symmetry; apply ZOdiv_unique_full with 0; auto with zarith; - [ red; romega with * | ring]. -Qed. +Proof. exact Z.div_mul. Qed. (** * Order results about ZOmod and ZOdiv *) @@ -391,16 +390,8 @@ Qed. Lemma ZO_div_pos: forall a b, 0 <= a -> 0 <= b -> 0 <= a/b. Proof. - intros. - destruct (Zle_lt_or_eq 0 b H0). - assert (H2:=ZOmod_lt_pos_pos a b H H1). - rewrite (ZO_div_mod_eq a b) in H. - destruct (Z_lt_le_dec (a/b) 0); auto. - assert (b*(a/b) <= -b). - replace (-b) with (b*-1); [ | ring]. - apply Zmult_le_compat_l; auto with zarith. - romega. - subst b; rewrite ZOdiv_0_r; auto. + intros. destruct (Z_eq_dec b 0); subst. rewrite ZOdiv_0_r; auto. + apply Z.div_pos; auto with zarith. Qed. (** As soon as the divisor is greater or equal than 2, @@ -408,97 +399,41 @@ Qed. Lemma ZO_div_lt : forall a b:Z, 0 < a -> 2 <= b -> a/b < a. Proof. - intros. - assert (Hb : 0 < b) by romega. - assert (H1 : 0 <= a/b) by (apply ZO_div_pos; auto with zarith). - assert (H2 : 0 <= a mod b < b) by (apply ZOmod_lt_pos_pos; auto with zarith). - destruct (Zle_lt_or_eq 0 (a/b) H1) as [H3|H3]; [ | rewrite <- H3; auto]. - pattern a at 2; rewrite (ZO_div_mod_eq a b). - apply Zlt_le_trans with (2*(a/b)). - romega. - apply Zle_trans with (b*(a/b)). - apply Zmult_le_compat_r; auto. - romega. + intros. apply Z.div_lt; auto with zarith. Qed. (** A division of a small number by a bigger one yields zero. *) Theorem ZOdiv_small: forall a b, 0 <= a < b -> a/b = 0. -Proof. - intros a b H; apply sym_equal; apply ZOdiv_unique with a; auto with zarith. -Qed. +Proof. exact Z.div_small. Qed. (** Same situation, in term of modulo: *) Theorem ZOmod_small: forall a n, 0 <= a < n -> a mod n = a. -Proof. - intros a b H; apply sym_equal; apply ZOmod_unique with 0; auto with zarith. -Qed. +Proof. exact Z.mod_small. Qed. (** [Zge] is compatible with a positive division. *) -Lemma ZO_div_monotone_pos : forall a b c:Z, 0<=c -> 0<=a<=b -> a/c <= b/c. -Proof. - intros. - destruct H0. - destruct (Zle_lt_or_eq 0 c H); - [ clear H | subst c; do 2 rewrite ZOdiv_0_r; auto]. - generalize (ZO_div_mod_eq a c). - generalize (ZOmod_lt_pos_pos a c H0 H2). - generalize (ZO_div_mod_eq b c). - generalize (ZOmod_lt_pos_pos b c (Zle_trans _ _ _ H0 H1) H2). - intros. - elim (Z_le_gt_dec (a / c) (b / c)); auto with zarith. - intro. - absurd (a - b >= 1). - omega. - replace (a-b) with (c * (a/c-b/c) + a mod c - b mod c) by - (symmetry; pattern a at 1; rewrite H5; pattern b at 1; rewrite H3; ring). - assert (c * (a / c - b / c) >= c * 1). - apply Zmult_ge_compat_l. - omega. - omega. - assert (c * 1 = c). - ring. - omega. -Qed. - Lemma ZO_div_monotone : forall a b c, 0<=c -> a<=b -> a/c <= b/c. -Proof. - intros. - destruct (Z_le_gt_dec 0 a). - apply ZO_div_monotone_pos; auto with zarith. - destruct (Z_le_gt_dec 0 b). - apply Zle_trans with 0. - apply Zle_left_rev. - simpl. - rewrite <- ZOdiv_opp_l. - apply ZO_div_pos; auto with zarith. - apply ZO_div_pos; auto with zarith. - rewrite <-(Zopp_involutive a), (ZOdiv_opp_l (-a)). - rewrite <-(Zopp_involutive b), (ZOdiv_opp_l (-b)). - generalize (ZO_div_monotone_pos (-b) (-a) c H). - romega. +Proof. + intros. destruct (Z_eq_dec c 0); subst. rewrite !ZOdiv_0_r; auto. + apply Z.div_le_mono; auto with zarith. Qed. +(** Compatitility: *) +Lemma ZO_div_monotone_pos : forall a b c:Z, 0<=c -> 0<=a<=b -> a/c <= b/c. +Proof. intros; apply ZO_div_monotone; intuition. Qed. + (** With our choice of division, rounding of (a/b) is always done toward zero: *) Lemma ZO_mult_div_le : forall a b:Z, 0 <= a -> 0 <= b*(a/b) <= a. Proof. - intros a b Ha. - destruct b as [ |b|b]. - simpl; auto with zarith. - split. - apply Zmult_le_0_compat; auto with zarith. - apply ZO_div_pos; auto with zarith. - generalize (ZO_div_mod_eq a (Zpos b)) (ZOmod_lt_pos_pos a (Zpos b) Ha); romega with *. - change (Zneg b) with (-Zpos b); rewrite ZOdiv_opp_r, Zmult_opp_opp. - split. - apply Zmult_le_0_compat; auto with zarith. - apply ZO_div_pos; auto with zarith. - generalize (ZO_div_mod_eq a (Zpos b)) (ZOmod_lt_pos_pos a (Zpos b) Ha); romega with *. + intros. destruct (Z_eq_dec b 0); subst. rewrite !ZOdiv_0_r; auto with zarith. + apply Z.mul_div_le; auto with zarith. Qed. +(** TODO: finish adapting to generic results *) + Lemma ZO_mult_div_ge : forall a b:Z, a <= 0 -> a <= b*(a/b) <= 0. Proof. intros a b Ha. |