(************************************************************************) (* * The Coq Proof Assistant / The Coq Development Team *) (* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) (* (a | - b). Proof. apply Z.divide_opp_r. Qed. Lemma Zdivide_opp_r_rev a b : (a | - b) -> (a | b). Proof. apply Z.divide_opp_r. Qed. Lemma Zdivide_opp_l a b : (a | b) -> (- a | b). Proof. apply Z.divide_opp_l. Qed. Lemma Zdivide_opp_l_rev a b : (- a | b) -> (a | b). Proof. apply Z.divide_opp_l. Qed. Theorem Zdivide_Zabs_l a b : (Z.abs a | b) -> (a | b). Proof. apply Z.divide_abs_l. Qed. Theorem Zdivide_Zabs_inv_l a b : (a | b) -> (Z.abs a | b). Proof. apply Z.divide_abs_l. Qed. Hint Resolve Z.divide_refl Z.divide_1_l Z.divide_0_r: zarith. Hint Resolve Z.mul_divide_mono_l Z.mul_divide_mono_r: zarith. Hint Resolve Z.divide_add_r Zdivide_opp_r Zdivide_opp_r_rev Zdivide_opp_l Zdivide_opp_l_rev Z.divide_sub_r Z.divide_mul_l Z.divide_mul_r Z.divide_factor_l Z.divide_factor_r: zarith. (** Auxiliary result. *) Lemma Zmult_one x y : x >= 0 -> x * y = 1 -> x = 1. Proof. Z.swap_greater. apply Z.eq_mul_1_nonneg. Qed. (** Only [1] and [-1] divide [1]. *) Notation Zdivide_1 := Z.divide_1_r (only parsing). (** If [a] divides [b] and [b] divides [a] then [a] is [b] or [-b]. *) Notation Zdivide_antisym := Z.divide_antisym (compat "8.6"). Notation Zdivide_trans := Z.divide_trans (compat "8.6"). (** If [a] divides [b] and [b<>0] then [|a| <= |b|]. *) Lemma Zdivide_bounds a b : (a | b) -> b <> 0 -> Z.abs a <= Z.abs b. Proof. intros H Hb. rewrite <- Z.divide_abs_l, <- Z.divide_abs_r in H. apply Z.abs_pos in Hb. now apply Z.divide_pos_le. Qed. (** [Z.divide] can be expressed using [Z.modulo]. *) Lemma Zmod_divide : forall a b, b<>0 -> a mod b = 0 -> (b | a). Proof. apply Z.mod_divide. Qed. Lemma Zdivide_mod : forall a b, (b | a) -> a mod b = 0. Proof. intros a b (c,->); apply Z_mod_mult. Qed. (** [Z.divide] is hence decidable *) Lemma Zdivide_dec a b : {(a | b)} + {~ (a | b)}. Proof. destruct (Z.eq_dec a 0) as [Ha|Ha]. destruct (Z.eq_dec b 0) as [Hb|Hb]. left; subst; apply Z.divide_0_r. right. subst. contradict Hb. now apply Z.divide_0_l. destruct (Z.eq_dec (b mod a) 0). left. now apply Z.mod_divide. right. now rewrite <- Z.mod_divide. Defined. Theorem Zdivide_Zdiv_eq a b : 0 < a -> (a | b) -> b = a * (b / a). Proof. intros Ha H. rewrite (Z.div_mod b a) at 1; auto with zarith. rewrite Zdivide_mod; auto with zarith. Qed. Theorem Zdivide_Zdiv_eq_2 a b c : 0 < a -> (a | b) -> (c * b) / a = c * (b / a). Proof. intros. apply Z.divide_div_mul_exact; auto with zarith. Qed. Theorem Zdivide_le: forall a b : Z, 0 <= a -> 0 < b -> (a | b) -> a <= b. Proof. intros. now apply Z.divide_pos_le. Qed. Theorem Zdivide_Zdiv_lt_pos a b : 1 < a -> 0 < b -> (a | b) -> 0 < b / a < b . Proof. intros H1 H2 H3; split. apply Z.mul_pos_cancel_l with a; auto with zarith. rewrite <- Zdivide_Zdiv_eq; auto with zarith. now apply Z.div_lt. Qed. Lemma Zmod_div_mod n m a: 0 < n -> 0 < m -> (n | m) -> a mod n = (a mod m) mod n. Proof. intros H1 H2 (p,Hp). rewrite (Z.div_mod a m) at 1; auto with zarith. rewrite Hp at 1. rewrite Z.mul_shuffle0, Z.add_comm, Z.mod_add; auto with zarith. Qed. Lemma Zmod_divide_minus a b c: 0 < b -> a mod b = c -> (b | a - c). Proof. intros H H1. apply Z.mod_divide; auto with zarith. rewrite Zminus_mod; auto with zarith. rewrite H1. rewrite <- (Z.mod_small c b) at 1. rewrite Z.sub_diag, Z.mod_0_l; auto with zarith. subst. now apply Z.mod_pos_bound. Qed. Lemma Zdivide_mod_minus a b c: 0 <= c < b -> (b | a - c) -> a mod b = c. Proof. intros (H1, H2) H3. assert (0 < b) by Z.order. replace a with ((a - c) + c); auto with zarith. rewrite Z.add_mod; auto with zarith. rewrite (Zdivide_mod (a-c) b); try rewrite Z.add_0_l; auto with zarith. rewrite Z.mod_mod; try apply Zmod_small; auto with zarith. Qed. (** * Greatest common divisor (gcd). *) (** There is no unicity of the gcd; hence we define the predicate [Zis_gcd a b g] expressing that [g] 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 g:Z) : Prop := Zis_gcd_intro : (g | a) -> (g | b) -> (forall x, (x | a) -> (x | b) -> (x | g)) -> Zis_gcd a b g. (** Trivial properties of [gcd] *) Lemma Zis_gcd_sym : forall a b d, Zis_gcd a b d -> Zis_gcd b a d. Proof. induction 1; constructor; intuition. Qed. Lemma Zis_gcd_0 : forall a, Zis_gcd a 0 a. Proof. constructor; auto with zarith. Qed. Lemma Zis_gcd_1 : forall a, Zis_gcd a 1 1. Proof. constructor; auto with zarith. Qed. Lemma Zis_gcd_refl : forall a, Zis_gcd a a a. Proof. constructor; auto with zarith. Qed. Lemma Zis_gcd_minus : forall a b d, Zis_gcd a (- b) d -> Zis_gcd b a d. Proof. induction 1; constructor; intuition. Qed. Lemma Zis_gcd_opp : forall a b d, Zis_gcd a b d -> Zis_gcd b a (- d). Proof. induction 1; constructor; intuition. Qed. Lemma Zis_gcd_0_abs a : Zis_gcd 0 a (Z.abs a). Proof. apply Zabs_ind. intros; apply Zis_gcd_sym; apply Zis_gcd_0; auto. intros; apply Zis_gcd_opp; apply Zis_gcd_0; auto. Qed. Hint Resolve Zis_gcd_sym Zis_gcd_0 Zis_gcd_minus Zis_gcd_opp: zarith. Theorem Zis_gcd_unique: forall a b c d : Z, Zis_gcd a b c -> Zis_gcd a b d -> c = d \/ c = (- d). Proof. intros a b c d [Hc1 Hc2 Hc3] [Hd1 Hd2 Hd3]. assert (c|d) by auto. assert (d|c) by auto. apply Z.divide_antisym; auto. Qed. (** * 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(u3,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. apply Zlt_0_rec. clear v3 Hv3; intros. destruct (Z_zerop x) as [Heq|Hneq]. apply Euclid_intro with (u := u1) (v := u2) (d := u3). assumption. apply H3. rewrite Heq; 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. intro eq; pattern u3 at 2; 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 H1; rewrite H2; trivial. ring. intros; apply H3. 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 (Z.divide_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; auto with zarith. intros x Ha Hb. elim (Zis_gcd_bezout a b d H). intros u v Huv. elim Ha; intros a' Ha'. elim Hb; intros b' Hb'. apply Zdivide_intro with (u * a' + v * b'). rewrite <- Huv. replace (c * (u * a + v * b)) with (u * (c * a) + v * (c * b)). rewrite Ha'; rewrite Hb'; ring. ring. 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 (Z.divide_antisym b d). split; auto with zarith. rewrite H4 in H3. rewrite Z.mul_comm in H3. apply Z.mul_reg_l with d; auto with zarith. intros; omega. apply Gauss with a. rewrite H3. auto with zarith. red; auto with zarith. apply Gauss with c. rewrite Z.mul_comm. rewrite <- H3. auto with zarith. red; 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). Proof. 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. destruct H1. destruct H1 as (a',H1). destruct H3 as (b',H3). replace (a/g) with a'; [|rewrite H1; rewrite Z_div_mult; auto with zarith]. replace (b/g) with b'; [|rewrite H3; rewrite Z_div_mult; auto with zarith]. constructor. exists a'; auto with zarith. exists b'; auto with zarith. intros x (xa,H5) (xb,H6). destruct (H4 (x*g)) as (x',Hx'). exists xa; rewrite Z.mul_assoc; rewrite <- H5; auto. exists xb; rewrite Z.mul_assoc; rewrite <- H6; auto. replace g with (1*g) in Hx'; auto with zarith. do 2 rewrite Z.mul_assoc in Hx'. apply Z.mul_reg_r in Hx'; trivial. rewrite Z.mul_1_r in Hx'. exists x'; auto with zarith. Qed. Theorem rel_prime_sym: forall a b, rel_prime a b -> rel_prime b a. Proof. intros a b H; auto with zarith. red; apply Zis_gcd_sym; auto with zarith. Qed. Theorem rel_prime_div: forall p q r, rel_prime p q -> (r | p) -> rel_prime r q. Proof. intros p q r H (u, H1); subst. inversion_clear H as [H1 H2 H3]. red; apply Zis_gcd_intro; try apply Z.divide_1_l. intros x H4 H5; apply H3; auto. apply Z.divide_mul_r; auto. Qed. Theorem rel_prime_1: forall n, rel_prime 1 n. Proof. intros n; red; apply Zis_gcd_intro; auto. exists 1; auto with zarith. exists n; auto with zarith. Qed. Theorem not_rel_prime_0: forall n, 1 < n -> ~ rel_prime 0 n. Proof. intros n H H1; absurd (n = 1 \/ n = -1). intros [H2 | H2]; subst; contradict H; auto with zarith. case (Zis_gcd_unique 0 n n 1); auto. apply Zis_gcd_intro; auto. exists 0; auto with zarith. exists 1; auto with zarith. Qed. Theorem rel_prime_mod: forall p q, 0 < q -> rel_prime p q -> rel_prime (p mod q) q. Proof. intros p q H H0. assert (H1: Bezout p q 1). apply rel_prime_bezout; auto. inversion_clear H1 as [q1 r1 H2]. apply bezout_rel_prime. apply Bezout_intro with q1 (r1 + q1 * (p / q)). rewrite <- H2. pattern p at 3; rewrite (Z_div_mod_eq p q); try ring; auto with zarith. Qed. Theorem rel_prime_mod_rev: forall p q, 0 < q -> rel_prime (p mod q) q -> rel_prime p q. Proof. intros p q H H0. rewrite (Z_div_mod_eq p q); auto with zarith; red. apply Zis_gcd_sym; apply Zis_gcd_for_euclid2; auto with zarith. Qed. Theorem Zrel_prime_neq_mod_0: forall a b, 1 < b -> rel_prime a b -> a mod b <> 0. Proof. intros a b H H1 H2. case (not_rel_prime_0 _ H). rewrite <- H2. apply rel_prime_mod; auto with zarith. 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. destruct 1; intros. assert (a = - p \/ - p < a < -1 \/ a = -1 \/ a = 0 \/ a = 1 \/ 1 < a < p \/ a = p). { assert (Z.abs a <= Z.abs p) as H2. apply Zdivide_bounds; [ assumption | omega ]. revert H2. pattern (Z.abs a); apply Zabs_ind; pattern (Z.abs p); apply Zabs_ind; intros; omega. } intuition idtac. (* -p < a < -1 *) - absurd (rel_prime (- a) p); intuition. inversion H2. assert (- a | - a) by auto with zarith. assert (- a | p) by auto with zarith. apply H7, Z.divide_1_r in H8; intuition. (* a = 0 *) - inversion H1. subst a; omega. (* 1 < a < p *) - absurd (rel_prime a p); intuition. inversion H2. assert (a | a) by auto with zarith. assert (a | p) by auto with zarith. apply H7, Z.divide_1_r in H8; 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. intros; constructor; intros; auto with zarith. apply prime_divisors in H1; intuition; subst; auto with zarith. - absurd (p | a); auto with zarith. - absurd (p | a); intuition. Qed. Hint Resolve prime_rel_prime: zarith. (** As a consequence, a prime number is relatively prime with smaller numbers *) Theorem rel_prime_le_prime: forall a p, prime p -> 1 <= a < p -> rel_prime a p. Proof. intros a p Hp [H1 H2]. apply rel_prime_sym; apply prime_rel_prime; auto. intros [q Hq]; subst a. case (Z.le_gt_cases q 0); intros Hl. absurd (q * p <= 0 * p); auto with zarith. absurd (1 * p <= q * p); 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. Lemma not_prime_0: ~ prime 0. Proof. intros H1; case (prime_divisors _ H1 2); auto with zarith. Qed. Lemma not_prime_1: ~ prime 1. Proof. intros H1; absurd (1 < 1); auto with zarith. inversion H1; auto. Qed. Lemma prime_2: prime 2. Proof. apply prime_intro; auto with zarith. intros n (H,H'); Z.le_elim H; auto with zarith. - contradict H'; auto with zarith. - subst n. constructor; auto with zarith. Qed. Theorem prime_3: prime 3. Proof. apply prime_intro; auto with zarith. intros n (H,H'); Z.le_elim H; auto with zarith. - replace n with 2 by omega. constructor; auto with zarith. intros x (q,Hq) (q',Hq'). exists (q' - q). ring_simplify. now rewrite <- Hq, <- Hq'. - replace n with 1 by trivial. constructor; auto with zarith. Qed. Theorem prime_ge_2 p : prime p -> 2 <= p. Proof. intros (Hp,_); auto with zarith. Qed. Definition prime' p := 1

~ (n|p)). Lemma Z_0_1_more x : 0<=x -> x=0 \/ x=1 \/ 1 prime p. Proof. split; intros (Hp,H). - (* prime -> prime' *) constructor; trivial; intros n Hn. constructor; auto with zarith; intros x Hxn Hxp. rewrite <- Z.divide_abs_l in Hxn, Hxp |- *. assert (Hx := Z.abs_nonneg x). set (y:=Z.abs x) in *; clearbody y; clear x; rename y into x. destruct (Z_0_1_more x Hx) as [->|[->|Hx']]. + exfalso. apply Z.divide_0_l in Hxn. omega. + now exists 1. + elim (H x); auto. split; trivial. apply Z.le_lt_trans with n; auto with zarith. apply Z.divide_pos_le; auto with zarith. - (* prime' -> prime *) constructor; trivial. intros n Hn Hnp. case (Zis_gcd_unique n p n 1); auto with zarith. constructor; auto with zarith. apply H; auto with zarith. Qed. Theorem square_not_prime: forall a, ~ prime (a * a). Proof. intros a Ha. rewrite <- (Z.abs_square a) in Ha. assert (H:=Z.abs_nonneg a). set (b:=Z.abs a) in *; clearbody b; clear a; rename b into a. rewrite <- prime_alt in Ha; destruct Ha as (Ha,Ha'). assert (H' : 1 < a) by now apply (Z.square_lt_simpl_nonneg 1). apply (Ha' a). + split; trivial. rewrite <- (Z.mul_1_l a) at 1. apply Z.mul_lt_mono_pos_r; omega. + exists a; auto. Qed. Theorem prime_div_prime: forall p q, prime p -> prime q -> (p | q) -> p = q. Proof. intros p q H H1 H2; assert (Hp: 0 < p); try apply Z.lt_le_trans with 2; try apply prime_ge_2; auto with zarith. assert (Hq: 0 < q); try apply Z.lt_le_trans with 2; try apply prime_ge_2; auto with zarith. case prime_divisors with (2 := H2); auto. intros H4; contradict Hp; subst; auto with zarith. intros [H4| [H4 | H4]]; subst; auto. contradict H; auto; apply not_prime_1. contradict Hp; auto with zarith. Qed. (** we now prove that [Z.gcd] is indeed a gcd in the sense of [Zis_gcd]. *) Notation Zgcd_is_pos := Z.gcd_nonneg (only parsing). Lemma Zgcd_is_gcd : forall a b, Zis_gcd a b (Z.gcd a b). Proof. constructor. apply Z.gcd_divide_l. apply Z.gcd_divide_r. apply Z.gcd_greatest. Qed. Theorem Zgcd_spec : forall x y : Z, {z : Z | Zis_gcd x y z /\ 0 <= z}. Proof. intros x y; exists (Z.gcd x y). split; [apply Zgcd_is_gcd | apply Z.gcd_nonneg]. Qed. Theorem Zdivide_Zgcd: forall p q r : Z, (p | q) -> (p | r) -> (p | Z.gcd q r). Proof. intros. now apply Z.gcd_greatest. Qed. Theorem Zis_gcd_gcd: forall a b c : Z, 0 <= c -> Zis_gcd a b c -> Z.gcd a b = c. Proof. intros a b c H1 H2. case (Zis_gcd_uniqueness_apart_sign a b c (Z.gcd a b)); auto. apply Zgcd_is_gcd; auto. Z.le_elim H1. - generalize (Z.gcd_nonneg a b); auto with zarith. - subst. now case (Z.gcd a b). Qed. Notation Zgcd_inv_0_l := Z.gcd_eq_0_l (only parsing). Notation Zgcd_inv_0_r := Z.gcd_eq_0_r (only parsing). Theorem Zgcd_div_swap0 : forall a b : Z, 0 < Z.gcd a b -> 0 < b -> (a / Z.gcd a b) * b = a * (b/Z.gcd a b). Proof. intros a b Hg Hb. assert (F := Zgcd_is_gcd a b); inversion F as [F1 F2 F3]. pattern b at 2; rewrite (Zdivide_Zdiv_eq (Z.gcd a b) b); auto. repeat rewrite Z.mul_assoc; f_equal. rewrite Z.mul_comm. rewrite <- Zdivide_Zdiv_eq; auto. Qed. Theorem Zgcd_div_swap : forall a b c : Z, 0 < Z.gcd a b -> 0 < b -> (c * a) / Z.gcd a b * b = c * a * (b/Z.gcd a b). Proof. intros a b c Hg Hb. assert (F := Zgcd_is_gcd a b); inversion F as [F1 F2 F3]. pattern b at 2; rewrite (Zdivide_Zdiv_eq (Z.gcd a b) b); auto. repeat rewrite Z.mul_assoc; f_equal. rewrite Zdivide_Zdiv_eq_2; auto. repeat rewrite <- Z.mul_assoc; f_equal. rewrite Z.mul_comm. rewrite <- Zdivide_Zdiv_eq; auto. Qed. Notation Zgcd_comm := Z.gcd_comm (compat "8.6"). Lemma Zgcd_ass a b c : Z.gcd (Z.gcd a b) c = Z.gcd a (Z.gcd b c). Proof. symmetry. apply Z.gcd_assoc. Qed. Notation Zgcd_Zabs := Z.gcd_abs_l (only parsing). Notation Zgcd_0 := Z.gcd_0_r (only parsing). Notation Zgcd_1 := Z.gcd_1_r (only parsing). Hint Resolve Z.gcd_0_r Z.gcd_1_r : zarith. Theorem Zgcd_1_rel_prime : forall a b, Z.gcd a b = 1 <-> rel_prime a b. Proof. unfold rel_prime; split; intro H. rewrite <- H; apply Zgcd_is_gcd. case (Zis_gcd_unique a b (Z.gcd a b) 1); auto. apply Zgcd_is_gcd. intros H2; absurd (0 <= Z.gcd a b); auto with zarith. generalize (Z.gcd_nonneg a b); auto with zarith. Qed. Definition rel_prime_dec: forall a b, { rel_prime a b }+{ ~ rel_prime a b }. Proof. intros a b; case (Z.eq_dec (Z.gcd a b) 1); intros H1. left; apply -> Zgcd_1_rel_prime; auto. right; contradict H1; apply <- Zgcd_1_rel_prime; auto. Defined. Definition prime_dec_aux: forall p m, { forall n, 1 < n < m -> rel_prime n p } + { exists n, 1 < n < m /\ ~ rel_prime n p }. Proof. intros p m. case (Z_lt_dec 1 m); intros H1; [ | left; intros; exfalso; omega ]. pattern m; apply natlike_rec; auto with zarith. left; intros; exfalso; omega. intros x Hx IH; destruct IH as [F|E]. destruct (rel_prime_dec x p) as [Y|N]. left; intros n [HH1 HH2]. rewrite Z.lt_succ_r in HH2. Z.le_elim HH2; subst; auto with zarith. - case (Z_lt_dec 1 x); intros HH1. * right; exists x; split; auto with zarith. * left; intros n [HHH1 HHH2]; contradict HHH1; auto with zarith. - right; destruct E as (n,((H0,H2),H3)); exists n; auto with zarith. Defined. Definition prime_dec: forall p, { prime p }+{ ~ prime p }. Proof. intros p; case (Z_lt_dec 1 p); intros H1. + case (prime_dec_aux p p); intros H2. * left; apply prime_intro; auto. intros n (Hn1,Hn2). Z.le_elim Hn1; auto; subst n. constructor; auto with zarith. * right; intros H3; inversion_clear H3 as [Hp1 Hp2]. case H2; intros n [Hn1 Hn2]; case Hn2; auto with zarith. + right; intros H3; inversion_clear H3 as [Hp1 Hp2]; case H1; auto. Defined. Theorem not_prime_divide: forall p, 1 < p -> ~ prime p -> exists n, 1 < n < p /\ (n | p). Proof. intros p Hp Hp1. case (prime_dec_aux p p); intros H1. - elim Hp1; constructor; auto. intros n (Hn1,Hn2). Z.le_elim Hn1; auto with zarith. subst n; constructor; auto with zarith. - case H1; intros n (Hn1,Hn2). destruct (Z_0_1_more _ (Z.gcd_nonneg n p)) as [H|[H|H]]. + exfalso. apply Z.gcd_eq_0_l in H. omega. + elim Hn2. red. rewrite <- H. apply Zgcd_is_gcd. + exists (Z.gcd n p); split; [ split; auto | apply Z.gcd_divide_r ]. apply Z.le_lt_trans with n; auto with zarith. apply Z.divide_pos_le; auto with zarith. apply Z.gcd_divide_l. Qed.