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Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZDivMath.v')
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diff --git a/theories/Numbers/Integer/Abstract/ZDivMath.v b/theories/Numbers/Integer/Abstract/ZDivMath.v deleted file mode 100644 index dfc9ee4bc7..0000000000 --- a/theories/Numbers/Integer/Abstract/ZDivMath.v +++ /dev/null @@ -1,396 +0,0 @@ -(************************************************************************) -(* 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. - |