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diff --git a/theories/ZArith/Znumtheory.v b/theories/ZArith/Znumtheory.v new file mode 100644 index 00000000..715cdc7d --- /dev/null +++ b/theories/ZArith/Znumtheory.v @@ -0,0 +1,640 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) + +(*i $Id: Znumtheory.v,v 1.5.2.1 2004/07/16 19:31:22 herbelin Exp $ i*) + +Require Import ZArith_base. +Require Import ZArithRing. +Require Import Zcomplements. +Require Import Zdiv. +Open Local Scope Z_scope. + +(** This file contains some notions of number theory upon Z numbers: + - a divisibility predicate [Zdivide] + - a gcd predicate [gcd] + - Euclid algorithm [euclid] + - an efficient [Zgcd] function + - a relatively prime predicate [rel_prime] + - a prime predicate [prime] +*) + +(** * Divisibility *) + +Inductive Zdivide (a b:Z) : Prop := + Zdivide_intro : forall q:Z, b = q * a -> Zdivide a b. + +(** Syntax for divisibility *) + +Notation "( a | b )" := (Zdivide a b) (at level 0) : Z_scope. + +(** Results concerning divisibility*) + +Lemma Zdivide_refl : forall a:Z, (a | a). +Proof. +intros; apply Zdivide_intro with 1; ring. +Qed. + +Lemma Zone_divide : forall a:Z, (1 | a). +Proof. +intros; apply Zdivide_intro with a; ring. +Qed. + +Lemma Zdivide_0 : forall a:Z, (a | 0). +Proof. +intros; apply Zdivide_intro with 0; ring. +Qed. + +Hint Resolve Zdivide_refl Zone_divide Zdivide_0: zarith. + +Lemma Zmult_divide_compat_l : forall a b c:Z, (a | b) -> (c * a | c * b). +Proof. +simple induction 1; intros; apply Zdivide_intro with q. +rewrite H0; ring. +Qed. + +Lemma Zmult_divide_compat_r : forall a b c:Z, (a | b) -> (a * c | b * c). +Proof. +intros a b c; rewrite (Zmult_comm a c); rewrite (Zmult_comm b c). +apply Zmult_divide_compat_l; trivial. +Qed. + +Hint Resolve Zmult_divide_compat_l Zmult_divide_compat_r: zarith. + +Lemma Zdivide_plus_r : forall a b c:Z, (a | b) -> (a | c) -> (a | b + c). +Proof. +simple induction 1; intros q Hq; simple induction 1; intros q' Hq'. +apply Zdivide_intro with (q + q'). +rewrite Hq; rewrite Hq'; ring. +Qed. + +Lemma Zdivide_opp_r : forall a b:Z, (a | b) -> (a | - b). +Proof. +simple induction 1; intros; apply Zdivide_intro with (- q). +rewrite H0; ring. +Qed. + +Lemma Zdivide_opp_r_rev : forall a b:Z, (a | - b) -> (a | b). +Proof. +intros; replace b with (- - b). apply Zdivide_opp_r; trivial. ring. +Qed. + +Lemma Zdivide_opp_l : forall a b:Z, (a | b) -> (- a | b). +Proof. +simple induction 1; intros; apply Zdivide_intro with (- q). +rewrite H0; ring. +Qed. + +Lemma Zdivide_opp_l_rev : forall a b:Z, (- a | b) -> (a | b). +Proof. +intros; replace a with (- - a). apply Zdivide_opp_l; trivial. ring. +Qed. + +Lemma Zdivide_minus_l : forall a b c:Z, (a | b) -> (a | c) -> (a | b - c). +Proof. +simple induction 1; intros q Hq; simple induction 1; intros q' Hq'. +apply Zdivide_intro with (q - q'). +rewrite Hq; rewrite Hq'; ring. +Qed. + +Lemma Zdivide_mult_l : forall a b c:Z, (a | b) -> (a | b * c). +Proof. +simple induction 1; intros q Hq; apply Zdivide_intro with (q * c). +rewrite Hq; ring. +Qed. + +Lemma Zdivide_mult_r : forall a b c:Z, (a | c) -> (a | b * c). +Proof. +simple induction 1; intros q Hq; apply Zdivide_intro with (q * b). +rewrite Hq; ring. +Qed. + +Lemma Zdivide_factor_r : forall a b:Z, (a | a * b). +Proof. +intros; apply Zdivide_intro with b; ring. +Qed. + +Lemma Zdivide_factor_l : forall a b:Z, (a | b * a). +Proof. +intros; apply Zdivide_intro with b; ring. +Qed. + +Hint Resolve Zdivide_plus_r Zdivide_opp_r Zdivide_opp_r_rev Zdivide_opp_l + Zdivide_opp_l_rev Zdivide_minus_l Zdivide_mult_l Zdivide_mult_r + Zdivide_factor_r Zdivide_factor_l: zarith. + +(** Auxiliary result. *) + +Lemma Zmult_one : forall x y:Z, x >= 0 -> x * y = 1 -> x = 1. +Proof. +intros x y H H0; destruct (Zmult_1_inversion_l _ _ H0) as [Hpos| Hneg]. + assumption. + rewrite Hneg in H; simpl in H. + contradiction (Zle_not_lt 0 (-1)). + apply Zge_le; assumption. + apply Zorder.Zlt_neg_0. +Qed. + +(** Only [1] and [-1] divide [1]. *) + +Lemma Zdivide_1 : forall x:Z, (x | 1) -> x = 1 \/ x = -1. +Proof. +simple induction 1; intros. +elim (Z_lt_ge_dec 0 x); [ left | right ]. +apply Zmult_one with q; auto with zarith; rewrite H0; ring. +assert (- x = 1); auto with zarith. +apply Zmult_one with (- q); auto with zarith; rewrite H0; ring. +Qed. + +(** If [a] divides [b] and [b] divides [a] then [a] is [b] or [-b]. *) + +Lemma Zdivide_antisym : forall a b:Z, (a | b) -> (b | a) -> a = b \/ a = - b. +Proof. +simple induction 1; intros. +inversion H1. +rewrite H0 in H2; clear H H1. +case (Z_zerop a); intro. +left; rewrite H0; rewrite e; ring. +assert (Hqq0 : q0 * q = 1). +apply Zmult_reg_l with a. +assumption. +ring. +pattern a at 2 in |- *; rewrite H2; ring. +assert (q | 1). +rewrite <- Hqq0; auto with zarith. +elim (Zdivide_1 q H); intros. +rewrite H1 in H0; left; omega. +rewrite H1 in H0; right; omega. +Qed. + +(** If [a] divides [b] and [b<>0] then [|a| <= |b|]. *) + +Lemma Zdivide_bounds : forall a b:Z, (a | b) -> b <> 0 -> Zabs a <= Zabs b. +Proof. +simple induction 1; intros. +assert (Zabs b = Zabs q * Zabs a). + subst; apply Zabs_Zmult. +rewrite H2. +assert (H3 := Zabs_pos q). +assert (H4 := Zabs_pos a). +assert (Zabs q * Zabs a >= 1 * Zabs a); auto with zarith. +apply Zmult_ge_compat; auto with zarith. +elim (Z_lt_ge_dec (Zabs q) 1); [ intros | auto with zarith ]. +assert (Zabs q = 0). + omega. +assert (q = 0). + rewrite <- (Zabs_Zsgn q). +rewrite H5; auto with zarith. +subst q; omega. +Qed. + +(** * Greatest common divisor (gcd). *) + +(** There is no unicity of the gcd; hence we define the predicate [gcd a b d] + expressing that [d] is a gcd of [a] and [b]. + (We show later that the [gcd] is actually unique if we discard its sign.) *) + +Inductive Zis_gcd (a b d:Z) : Prop := + Zis_gcd_intro : + (d | a) -> + (d | b) -> (forall x:Z, (x | a) -> (x | b) -> (x | d)) -> Zis_gcd a b d. + +(** Trivial properties of [gcd] *) + +Lemma Zis_gcd_sym : forall a b d:Z, Zis_gcd a b d -> Zis_gcd b a d. +Proof. +simple induction 1; constructor; intuition. +Qed. + +Lemma Zis_gcd_0 : forall a:Z, Zis_gcd a 0 a. +Proof. +constructor; auto with zarith. +Qed. + +Lemma Zis_gcd_minus : forall a b d:Z, Zis_gcd a (- b) d -> Zis_gcd b a d. +Proof. +simple induction 1; constructor; intuition. +Qed. + +Lemma Zis_gcd_opp : forall a b d:Z, Zis_gcd a b d -> Zis_gcd b a (- d). +Proof. +simple induction 1; constructor; intuition. +Qed. + +Hint Resolve Zis_gcd_sym Zis_gcd_0 Zis_gcd_minus Zis_gcd_opp: zarith. + +(** * Extended Euclid algorithm. *) + +(** Euclid's algorithm to compute the [gcd] mainly relies on + the following property. *) + +Lemma Zis_gcd_for_euclid : + forall a b d q:Z, Zis_gcd b (a - q * b) d -> Zis_gcd a b d. +Proof. +simple induction 1; constructor; intuition. +replace a with (a - q * b + q * b). auto with zarith. ring. +Qed. + +Lemma Zis_gcd_for_euclid2 : + forall b d q r:Z, Zis_gcd r b d -> Zis_gcd b (b * q + r) d. +Proof. +simple induction 1; constructor; intuition. +apply H2; auto. +replace r with (b * q + r - b * q). auto with zarith. ring. +Qed. + +(** We implement the extended version of Euclid's algorithm, + i.e. the one computing Bezout's coefficients as it computes + the [gcd]. We follow the algorithm given in Knuth's + "Art of Computer Programming", vol 2, page 325. *) + +Section extended_euclid_algorithm. + +Variables a b : Z. + +(** The specification of Euclid's algorithm is the existence of + [u], [v] and [d] such that [ua+vb=d] and [(gcd a b d)]. *) + +Inductive Euclid : Set := + Euclid_intro : + forall u v d:Z, u * a + v * b = d -> Zis_gcd a b d -> Euclid. + +(** The recursive part of Euclid's algorithm uses well-founded + recursion of non-negative integers. It maintains 6 integers + [u1,u2,u3,v1,v2,v3] such that the following invariant holds: + [u1*a+u2*b=u3] and [v1*a+v2*b=v3] and [gcd(u2,v3)=gcd(a,b)]. + *) + +Lemma euclid_rec : + forall v3:Z, + 0 <= v3 -> + forall u1 u2 u3 v1 v2:Z, + u1 * a + u2 * b = u3 -> + v1 * a + v2 * b = v3 -> + (forall d:Z, Zis_gcd u3 v3 d -> Zis_gcd a b d) -> Euclid. +Proof. +intros v3 Hv3; generalize Hv3; pattern v3 in |- *. +apply Z_lt_rec. +clear v3 Hv3; intros. +elim (Z_zerop x); intro. +apply Euclid_intro with (u := u1) (v := u2) (d := u3). +assumption. +apply H2. +rewrite a0; auto with zarith. +set (q := u3 / x) in *. +assert (Hq : 0 <= u3 - q * x < x). +replace (u3 - q * x) with (u3 mod x). +apply Z_mod_lt; omega. +assert (xpos : x > 0). omega. +generalize (Z_div_mod_eq u3 x xpos). +unfold q in |- *. +intro eq; pattern u3 at 2 in |- *; rewrite eq; ring. +apply (H (u3 - q * x) Hq (proj1 Hq) v1 v2 x (u1 - q * v1) (u2 - q * v2)). +tauto. +replace ((u1 - q * v1) * a + (u2 - q * v2) * b) with + (u1 * a + u2 * b - q * (v1 * a + v2 * b)). +rewrite H0; rewrite H1; trivial. +ring. +intros; apply H2. +apply Zis_gcd_for_euclid with q; assumption. +assumption. +Qed. + +(** We get Euclid's algorithm by applying [euclid_rec] on + [1,0,a,0,1,b] when [b>=0] and [1,0,a,0,-1,-b] when [b<0]. *) + +Lemma euclid : Euclid. +Proof. +case (Z_le_gt_dec 0 b); intro. +intros; + apply euclid_rec with + (u1 := 1) (u2 := 0) (u3 := a) (v1 := 0) (v2 := 1) (v3 := b); + auto with zarith; ring. +intros; + apply euclid_rec with + (u1 := 1) (u2 := 0) (u3 := a) (v1 := 0) (v2 := -1) (v3 := - b); + auto with zarith; try ring. +Qed. + +End extended_euclid_algorithm. + +Theorem Zis_gcd_uniqueness_apart_sign : + forall a b d d':Z, Zis_gcd a b d -> Zis_gcd a b d' -> d = d' \/ d = - d'. +Proof. +simple induction 1. +intros H1 H2 H3; simple induction 1; intros. +generalize (H3 d' H4 H5); intro Hd'd. +generalize (H6 d H1 H2); intro Hdd'. +exact (Zdivide_antisym d d' Hdd' Hd'd). +Qed. + +(** * Bezout's coefficients *) + +Inductive Bezout (a b d:Z) : Prop := + Bezout_intro : forall u v:Z, u * a + v * b = d -> Bezout a b d. + +(** Existence of Bezout's coefficients for the [gcd] of [a] and [b] *) + +Lemma Zis_gcd_bezout : forall a b d:Z, Zis_gcd a b d -> Bezout a b d. +Proof. +intros a b d Hgcd. +elim (euclid a b); intros u v d0 e g. +generalize (Zis_gcd_uniqueness_apart_sign a b d d0 Hgcd g). +intro H; elim H; clear H; intros. +apply Bezout_intro with u v. +rewrite H; assumption. +apply Bezout_intro with (- u) (- v). +rewrite H; rewrite <- e; ring. +Qed. + +(** gcd of [ca] and [cb] is [c gcd(a,b)]. *) + +Lemma Zis_gcd_mult : + forall a b c d:Z, Zis_gcd a b d -> Zis_gcd (c * a) (c * b) (c * d). +Proof. +intros a b c d; simple induction 1; constructor; intuition. +elim (Zis_gcd_bezout a b d H); intros. +elim H3; intros. +elim H4; intros. +apply Zdivide_intro with (u * q + v * q0). +rewrite <- H5. +replace (c * (u * a + v * b)) with (u * (c * a) + v * (c * b)). +rewrite H6; rewrite H7; ring. +ring. +Qed. + +(** We could obtain a [Zgcd] function via [euclid]. But we propose + here a more direct version of a [Zgcd], with better extraction + (no bezout coeffs). *) + +Definition Zgcd_pos : + forall a:Z, + 0 <= a -> forall b:Z, {g : Z | 0 <= a -> Zis_gcd a b g /\ g >= 0}. +Proof. +intros a Ha. +apply + (Z_lt_rec + (fun a:Z => forall b:Z, {g : Z | 0 <= a -> Zis_gcd a b g /\ g >= 0})); + try assumption. +intro x; case x. +intros _ b; exists (Zabs b). + elim (Z_le_lt_eq_dec _ _ (Zabs_pos b)). + intros H0; split. + apply Zabs_ind. + intros; apply Zis_gcd_sym; apply Zis_gcd_0; auto. + intros; apply Zis_gcd_opp; apply Zis_gcd_0; auto. + auto with zarith. + + intros H0; rewrite <- H0. + rewrite <- (Zabs_Zsgn b); rewrite <- H0; simpl in |- *. + split; [ apply Zis_gcd_0 | idtac ]; auto with zarith. + +intros p Hrec b. +generalize (Z_div_mod b (Zpos p)). +case (Zdiv_eucl b (Zpos p)); intros q r Hqr. +elim Hqr; clear Hqr; intros; auto with zarith. +elim (Hrec r H0 (Zpos p)); intros g Hgkl. +inversion_clear H0. +elim (Hgkl H1); clear Hgkl; intros H3 H4. +exists g; intros. +split; auto. +rewrite H. +apply Zis_gcd_for_euclid2; auto. + +intros p Hrec b. +exists 0; intros. +elim H; auto. +Defined. + +Definition Zgcd_spec : forall a b:Z, {g : Z | Zis_gcd a b g /\ g >= 0}. +Proof. +intros a; case (Z_gt_le_dec 0 a). +intros; assert (0 <= - a). +omega. +elim (Zgcd_pos (- a) H b); intros g Hgkl. +exists g. +intuition. +intros Ha b; elim (Zgcd_pos a Ha b); intros g; exists g; intuition. +Defined. + +Definition Zgcd (a b:Z) := let (g, _) := Zgcd_spec a b in g. + +Lemma Zgcd_is_pos : forall a b:Z, Zgcd a b >= 0. +intros a b; unfold Zgcd in |- *; case (Zgcd_spec a b); tauto. +Qed. + +Lemma Zgcd_is_gcd : forall a b:Z, Zis_gcd a b (Zgcd a b). +intros a b; unfold Zgcd in |- *; case (Zgcd_spec a b); tauto. +Qed. + +(** * Relative primality *) + +Definition rel_prime (a b:Z) : Prop := Zis_gcd a b 1. + +(** Bezout's theorem: [a] and [b] are relatively prime if and + only if there exist [u] and [v] such that [ua+vb = 1]. *) + +Lemma rel_prime_bezout : forall a b:Z, rel_prime a b -> Bezout a b 1. +Proof. +intros a b; exact (Zis_gcd_bezout a b 1). +Qed. + +Lemma bezout_rel_prime : forall a b:Z, Bezout a b 1 -> rel_prime a b. +Proof. +simple induction 1; constructor; auto with zarith. +intros. rewrite <- H0; auto with zarith. +Qed. + +(** Gauss's theorem: if [a] divides [bc] and if [a] and [b] are + relatively prime, then [a] divides [c]. *) + +Theorem Gauss : forall a b c:Z, (a | b * c) -> rel_prime a b -> (a | c). +Proof. +intros. elim (rel_prime_bezout a b H0); intros. +replace c with (c * 1); [ idtac | ring ]. +rewrite <- H1. +replace (c * (u * a + v * b)) with (c * u * a + v * (b * c)); + [ eauto with zarith | ring ]. +Qed. + +(** If [a] is relatively prime to [b] and [c], then it is to [bc] *) + +Lemma rel_prime_mult : + forall a b c:Z, rel_prime a b -> rel_prime a c -> rel_prime a (b * c). +Proof. +intros a b c Hb Hc. +elim (rel_prime_bezout a b Hb); intros. +elim (rel_prime_bezout a c Hc); intros. +apply bezout_rel_prime. +apply Bezout_intro with + (u := u * u0 * a + v0 * c * u + u0 * v * b) (v := v * v0). +rewrite <- H. +replace (u * a + v * b) with ((u * a + v * b) * 1); [ idtac | ring ]. +rewrite <- H0. +ring. +Qed. + +Lemma rel_prime_cross_prod : + forall a b c d:Z, + rel_prime a b -> + rel_prime c d -> b > 0 -> d > 0 -> a * d = b * c -> a = c /\ b = d. +Proof. +intros a b c d; intros. +elim (Zdivide_antisym b d). +split; auto with zarith. +rewrite H4 in H3. +rewrite Zmult_comm in H3. +apply Zmult_reg_l with d; auto with zarith. +intros; omega. +apply Gauss with a. +rewrite H3. +auto with zarith. +red in |- *; auto with zarith. +apply Gauss with c. +rewrite Zmult_comm. +rewrite <- H3. +auto with zarith. +red in |- *; auto with zarith. +Qed. + +(** After factorization by a gcd, the original numbers are relatively prime. *) + +Lemma Zis_gcd_rel_prime : + forall a b g:Z, + b > 0 -> g >= 0 -> Zis_gcd a b g -> rel_prime (a / g) (b / g). +intros a b g; intros. +assert (g <> 0). + intro. + elim H1; intros. + elim H4; intros. + rewrite H2 in H6; subst b; omega. +unfold rel_prime in |- *. +elim (Zgcd_spec (a / g) (b / g)); intros g' [H3 H4]. +assert (H5 := Zis_gcd_mult _ _ g _ H3). +rewrite <- Z_div_exact_2 in H5; auto with zarith. +rewrite <- Z_div_exact_2 in H5; auto with zarith. +elim (Zis_gcd_uniqueness_apart_sign _ _ _ _ H1 H5). +intros; rewrite (Zmult_reg_l 1 g' g); auto with zarith. +intros; rewrite (Zmult_reg_l 1 (- g') g); auto with zarith. +pattern g at 1 in |- *; rewrite H6; ring. + +elim H1; intros. +elim H7; intros. +rewrite H9. +replace (q * g) with (0 + q * g). +rewrite Z_mod_plus. +compute in |- *; auto. +omega. +ring. + +elim H1; intros. +elim H6; intros. +rewrite H9. +replace (q * g) with (0 + q * g). +rewrite Z_mod_plus. +compute in |- *; auto. +omega. +ring. +Qed. + +(** * Primality *) + +Inductive prime (p:Z) : Prop := + prime_intro : + 1 < p -> (forall n:Z, 1 <= n < p -> rel_prime n p) -> prime p. + +(** The sole divisors of a prime number [p] are [-1], [1], [p] and [-p]. *) + +Lemma prime_divisors : + forall p:Z, + prime p -> forall a:Z, (a | p) -> a = -1 \/ a = 1 \/ a = p \/ a = - p. +Proof. +simple induction 1; intros. +assert + (a = - p \/ - p < a < -1 \/ a = -1 \/ a = 0 \/ a = 1 \/ 1 < a < p \/ a = p). +assert (Zabs a <= Zabs p). apply Zdivide_bounds; [ assumption | omega ]. +generalize H3. +pattern (Zabs a) in |- *; apply Zabs_ind; pattern (Zabs p) in |- *; + apply Zabs_ind; intros; omega. +intuition idtac. +(* -p < a < -1 *) +absurd (rel_prime (- a) p); intuition. +inversion H3. +assert (- a | - a); auto with zarith. +assert (- a | p); auto with zarith. +generalize (H8 (- a) H9 H10); intuition idtac. +generalize (Zdivide_1 (- a) H11); intuition. +(* a = 0 *) +inversion H2. subst a; omega. +(* 1 < a < p *) +absurd (rel_prime a p); intuition. +inversion H3. +assert (a | a); auto with zarith. +assert (a | p); auto with zarith. +generalize (H8 a H9 H10); intuition idtac. +generalize (Zdivide_1 a H11); intuition. +Qed. + +(** A prime number is relatively prime with any number it does not divide *) + +Lemma prime_rel_prime : + forall p:Z, prime p -> forall a:Z, ~ (p | a) -> rel_prime p a. +Proof. +simple induction 1; intros. +constructor; intuition. +elim (prime_divisors p H x H3); intuition; subst; auto with zarith. +absurd (p | a); auto with zarith. +absurd (p | a); intuition. +Qed. + +Hint Resolve prime_rel_prime: zarith. + +(** [Zdivide] can be expressed using [Zmod]. *) + +Lemma Zmod_divide : forall a b:Z, b > 0 -> a mod b = 0 -> (b | a). +intros a b H H0. +apply Zdivide_intro with (a / b). +pattern a at 1 in |- *; rewrite (Z_div_mod_eq a b H). +rewrite H0; ring. +Qed. + +Lemma Zdivide_mod : forall a b:Z, b > 0 -> (b | a) -> a mod b = 0. +intros a b; simple destruct 2; intros; subst. +change (q * b) with (0 + q * b) in |- *. +rewrite Z_mod_plus; auto. +Qed. + +(** [Zdivide] is hence decidable *) + +Lemma Zdivide_dec : forall a b:Z, {(a | b)} + {~ (a | b)}. +Proof. +intros a b; elim (Ztrichotomy_inf a 0). +(* a<0 *) +intros H; elim H; intros. +case (Z_eq_dec (b mod - a) 0). +left; apply Zdivide_opp_l_rev; apply Zmod_divide; auto with zarith. +intro H1; right; intro; elim H1; apply Zdivide_mod; auto with zarith. +(* a=0 *) +case (Z_eq_dec b 0); intro. +left; subst; auto with zarith. +right; subst; intro H0; inversion H0; omega. +(* a>0 *) +intro H; case (Z_eq_dec (b mod a) 0). +left; apply Zmod_divide; auto with zarith. +intro H1; right; intro; elim H1; apply Zdivide_mod; auto with zarith. +Qed. + +(** If a prime [p] divides [ab] then it divides either [a] or [b] *) + +Lemma prime_mult : + forall p:Z, prime p -> forall a b:Z, (p | a * b) -> (p | a) \/ (p | b). +Proof. +intro p; simple induction 1; intros. +case (Zdivide_dec p a); intuition. +right; apply Gauss with a; auto with zarith. +Qed. + |