(* Title: HOL/Old_Number_Theory/Primes.thy

Author: Amine Chaieb, Christophe Tabacznyj and Lawrence C Paulson

Copyright 1996 University of Cambridge

*)

header {* Primality on nat *}

theory Primes

imports Complex_Main Legacy_GCD

begin

definition coprime :: "nat => nat => bool"

where "coprime m n <-> gcd m n = 1"

definition prime :: "nat => bool"

where "prime p <-> (1 < p ∧ (∀m. m dvd p --> m = 1 ∨ m = p))"

lemma two_is_prime: "prime 2"

apply (auto simp add: prime_def)

apply (case_tac m)

apply (auto dest!: dvd_imp_le)

done

lemma prime_imp_relprime: "prime p ==> ¬ p dvd n ==> gcd p n = 1"

apply (auto simp add: prime_def)

apply (metis gcd_dvd1 gcd_dvd2)

done

text {*

This theorem leads immediately to a proof of the uniqueness of

factorization. If @{term p} divides a product of primes then it is

one of those primes.

*}

lemma prime_dvd_mult: "prime p ==> p dvd m * n ==> p dvd m ∨ p dvd n"

by (blast intro: relprime_dvd_mult prime_imp_relprime)

lemma prime_dvd_square: "prime p ==> p dvd m^Suc (Suc 0) ==> p dvd m"

by (auto dest: prime_dvd_mult)

lemma prime_dvd_power_two: "prime p ==> p dvd m⇧^{2}==> p dvd m"

by (rule prime_dvd_square) (simp_all add: power2_eq_square)

lemma exp_eq_1:"(x::nat)^n = 1 <-> x = 1 ∨ n = 0"

by (induct n, auto)

lemma exp_mono_lt: "(x::nat) ^ (Suc n) < y ^ (Suc n) <-> x < y"

by(metis linorder_not_less not_less0 power_le_imp_le_base power_less_imp_less_base)

lemma exp_mono_le: "(x::nat) ^ (Suc n) ≤ y ^ (Suc n) <-> x ≤ y"

by (simp only: linorder_not_less[symmetric] exp_mono_lt)

lemma exp_mono_eq: "(x::nat) ^ Suc n = y ^ Suc n <-> x = y"

using power_inject_base[of x n y] by auto

lemma even_square: assumes e: "even (n::nat)" shows "∃x. n⇧^{2}= 4*x"

proof-

from e have "2 dvd n" by presburger

then obtain k where k: "n = 2*k" using dvd_def by auto

hence "n⇧^{2}= 4 * k⇧^{2}" by (simp add: power2_eq_square)

thus ?thesis by blast

qed

lemma odd_square: assumes e: "odd (n::nat)" shows "∃x. n⇧^{2}= 4*x + 1"

proof-

from e have np: "n > 0" by presburger

from e have "2 dvd (n - 1)" by presburger

then obtain k where "n - 1 = 2*k" using dvd_def by auto

hence k: "n = 2*k + 1" using e by presburger

hence "n⇧^{2}= 4* (k⇧^{2}+ k) + 1" by algebra

thus ?thesis by blast

qed

lemma diff_square: "(x::nat)⇧^{2}- y⇧^{2}= (x+y)*(x - y)"

proof-

have "x ≤ y ∨ y ≤ x" by (rule nat_le_linear)

moreover

{assume le: "x ≤ y"

hence "x⇧^{2}≤ y⇧^{2}" by (simp only: numeral_2_eq_2 exp_mono_le Let_def)

with le have ?thesis by simp }

moreover

{assume le: "y ≤ x"

hence le2: "y⇧^{2}≤ x⇧^{2}" by (simp only: numeral_2_eq_2 exp_mono_le Let_def)

from le have "∃z. y + z = x" by presburger

then obtain z where z: "x = y + z" by blast

from le2 have "∃z. x⇧^{2}= y⇧^{2}+ z" by presburger

then obtain z2 where z2: "x⇧^{2}= y⇧^{2}+ z2" by blast

from z z2 have ?thesis by simp algebra }

ultimately show ?thesis by blast

qed

text {* Elementary theory of divisibility *}

lemma divides_ge: "(a::nat) dvd b ==> b = 0 ∨ a ≤ b" unfolding dvd_def by auto

lemma divides_antisym: "(x::nat) dvd y ∧ y dvd x <-> x = y"

using dvd_antisym[of x y] by auto

lemma divides_add_revr: assumes da: "(d::nat) dvd a" and dab:"d dvd (a + b)"

shows "d dvd b"

proof-

from da obtain k where k:"a = d*k" by (auto simp add: dvd_def)

from dab obtain k' where k': "a + b = d*k'" by (auto simp add: dvd_def)

from k k' have "b = d *(k' - k)" by (simp add : diff_mult_distrib2)

thus ?thesis unfolding dvd_def by blast

qed

declare nat_mult_dvd_cancel_disj[presburger]

lemma nat_mult_dvd_cancel_disj'[presburger]:

"(m::nat)*k dvd n*k <-> k = 0 ∨ m dvd n" unfolding mult_commute[of m k] mult_commute[of n k] by presburger

lemma divides_mul_l: "(a::nat) dvd b ==> (c * a) dvd (c * b)"

by presburger

lemma divides_mul_r: "(a::nat) dvd b ==> (a * c) dvd (b * c)" by presburger

lemma divides_cases: "(n::nat) dvd m ==> m = 0 ∨ m = n ∨ 2 * n <= m"

by (auto simp add: dvd_def)

lemma divides_div_not: "(x::nat) = (q * n) + r ==> 0 < r ==> r < n ==> ~(n dvd x)"

proof(auto simp add: dvd_def)

fix k assume H: "0 < r" "r < n" "q * n + r = n * k"

from H(3) have r: "r = n* (k -q)" by(simp add: diff_mult_distrib2 mult_commute)

{assume "k - q = 0" with r H(1) have False by simp}

moreover

{assume "k - q ≠ 0" with r have "r ≥ n" by auto

with H(2) have False by simp}

ultimately show False by blast

qed

lemma divides_exp: "(x::nat) dvd y ==> x ^ n dvd y ^ n"

by (auto simp add: power_mult_distrib dvd_def)

lemma divides_exp2: "n ≠ 0 ==> (x::nat) ^ n dvd y ==> x dvd y"

by (induct n ,auto simp add: dvd_def)

fun fact :: "nat => nat" where

"fact 0 = 1"

| "fact (Suc n) = Suc n * fact n"

lemma fact_lt: "0 < fact n" by(induct n, simp_all)

lemma fact_le: "fact n ≥ 1" using fact_lt[of n] by simp

lemma fact_mono: assumes le: "m ≤ n" shows "fact m ≤ fact n"

proof-

from le have "∃i. n = m+i" by presburger

then obtain i where i: "n = m+i" by blast

have "fact m ≤ fact (m + i)"

proof(induct m)

case 0 thus ?case using fact_le[of i] by simp

next

case (Suc m)

have "fact (Suc m) = Suc m * fact m" by simp

have th1: "Suc m ≤ Suc (m + i)" by simp

from mult_le_mono[of "Suc m" "Suc (m+i)" "fact m" "fact (m+i)", OF th1 Suc.hyps]

show ?case by simp

qed

thus ?thesis using i by simp

qed

lemma divides_fact: "1 <= p ==> p <= n ==> p dvd fact n"

proof(induct n arbitrary: p)

case 0 thus ?case by simp

next

case (Suc n p)

from Suc.prems have "p = Suc n ∨ p ≤ n" by presburger

moreover

{assume "p = Suc n" hence ?case by (simp only: fact.simps dvd_triv_left)}

moreover

{assume "p ≤ n"

with Suc.prems(1) Suc.hyps have th: "p dvd fact n" by simp

from dvd_mult[OF th] have ?case by (simp only: fact.simps) }

ultimately show ?case by blast

qed

declare dvd_triv_left[presburger]

declare dvd_triv_right[presburger]

lemma divides_rexp:

"x dvd y ==> (x::nat) dvd (y^(Suc n))" by (simp add: dvd_mult2[of x y])

text {* Coprimality *}

lemma coprime: "coprime a b <-> (∀d. d dvd a ∧ d dvd b <-> d = 1)"

using gcd_unique[of 1 a b, simplified] by (auto simp add: coprime_def)

lemma coprime_commute: "coprime a b <-> coprime b a" by (simp add: coprime_def gcd_commute)

lemma coprime_bezout: "coprime a b <-> (∃x y. a * x - b * y = 1 ∨ b * x - a * y = 1)"

using coprime_def gcd_bezout by auto

lemma coprime_divprod: "d dvd a * b ==> coprime d a ==> d dvd b"

using relprime_dvd_mult_iff[of d a b] by (auto simp add: coprime_def mult_commute)

lemma coprime_1[simp]: "coprime a 1" by (simp add: coprime_def)

lemma coprime_1'[simp]: "coprime 1 a" by (simp add: coprime_def)

lemma coprime_Suc0[simp]: "coprime a (Suc 0)" by (simp add: coprime_def)

lemma coprime_Suc0'[simp]: "coprime (Suc 0) a" by (simp add: coprime_def)

lemma gcd_coprime:

assumes z: "gcd a b ≠ 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"

shows "coprime a' b'"

proof-

let ?g = "gcd a b"

{assume bz: "a = 0" from b bz z a have ?thesis by (simp add: gcd_zero coprime_def)}

moreover

{assume az: "a≠ 0"

from z have z': "?g > 0" by simp

from bezout_gcd_strong[OF az, of b]

obtain x y where xy: "a*x = b*y + ?g" by blast

from xy a b have "?g * a'*x = ?g * (b'*y + 1)" by (simp add: algebra_simps)

hence "?g * (a'*x) = ?g * (b'*y + 1)" by (simp add: mult_assoc)

hence "a'*x = (b'*y + 1)"

by (simp only: nat_mult_eq_cancel1[OF z'])

hence "a'*x - b'*y = 1" by simp

with coprime_bezout[of a' b'] have ?thesis by auto}

ultimately show ?thesis by blast

qed

lemma coprime_0: "coprime d 0 <-> d = 1" by (simp add: coprime_def)

lemma coprime_mul: assumes da: "coprime d a" and db: "coprime d b"

shows "coprime d (a * b)"

proof-

from da have th: "gcd a d = 1" by (simp add: coprime_def gcd_commute)

from gcd_mult_cancel[of a d b, OF th] db[unfolded coprime_def] have "gcd d (a*b) = 1"

by (simp add: gcd_commute)

thus ?thesis unfolding coprime_def .

qed

lemma coprime_lmul2: assumes dab: "coprime d (a * b)" shows "coprime d b"

using dab unfolding coprime_bezout

apply clarsimp

apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all)

apply (rule_tac x="x" in exI)

apply (rule_tac x="a*y" in exI)

apply (simp add: mult_ac)

apply (rule_tac x="a*x" in exI)

apply (rule_tac x="y" in exI)

apply (simp add: mult_ac)

done

lemma coprime_rmul2: "coprime d (a * b) ==> coprime d a"

unfolding coprime_bezout

apply clarsimp

apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all)

apply (rule_tac x="x" in exI)

apply (rule_tac x="b*y" in exI)

apply (simp add: mult_ac)

apply (rule_tac x="b*x" in exI)

apply (rule_tac x="y" in exI)

apply (simp add: mult_ac)

done

lemma coprime_mul_eq: "coprime d (a * b) <-> coprime d a ∧ coprime d b"

using coprime_rmul2[of d a b] coprime_lmul2[of d a b] coprime_mul[of d a b]

by blast

lemma gcd_coprime_exists:

assumes nz: "gcd a b ≠ 0"

shows "∃a' b'. a = a' * gcd a b ∧ b = b' * gcd a b ∧ coprime a' b'"

proof-

let ?g = "gcd a b"

from gcd_dvd1[of a b] gcd_dvd2[of a b]

obtain a' b' where "a = ?g*a'" "b = ?g*b'" unfolding dvd_def by blast

hence ab': "a = a'*?g" "b = b'*?g" by algebra+

from ab' gcd_coprime[OF nz ab'] show ?thesis by blast

qed

lemma coprime_exp: "coprime d a ==> coprime d (a^n)"

by(induct n, simp_all add: coprime_mul)

lemma coprime_exp_imp: "coprime a b ==> coprime (a ^n) (b ^n)"

by (induct n, simp_all add: coprime_mul_eq coprime_commute coprime_exp)

lemma coprime_refl[simp]: "coprime n n <-> n = 1" by (simp add: coprime_def)

lemma coprime_plus1[simp]: "coprime (n + 1) n"

apply (simp add: coprime_bezout)

apply (rule exI[where x=1])

apply (rule exI[where x=1])

apply simp

done

lemma coprime_minus1: "n ≠ 0 ==> coprime (n - 1) n"

using coprime_plus1[of "n - 1"] coprime_commute[of "n - 1" n] by auto

lemma bezout_gcd_pow: "∃x y. a ^n * x - b ^ n * y = gcd a b ^ n ∨ b ^ n * x - a ^ n * y = gcd a b ^ n"

proof-

let ?g = "gcd a b"

{assume z: "?g = 0" hence ?thesis

apply (cases n, simp)

apply arith

apply (simp only: z power_0_Suc)

apply (rule exI[where x=0])

apply (rule exI[where x=0])

apply simp

done }

moreover

{assume z: "?g ≠ 0"

from gcd_dvd1[of a b] gcd_dvd2[of a b] obtain a' b' where

ab': "a = a'*?g" "b = b'*?g" unfolding dvd_def by (auto simp add: mult_ac)

hence ab'': "?g*a' = a" "?g * b' = b" by algebra+

from coprime_exp_imp[OF gcd_coprime[OF z ab'], unfolded coprime_bezout, of n]

obtain x y where "a'^n * x - b'^n * y = 1 ∨ b'^n * x - a'^n * y = 1" by blast

hence "?g^n * (a'^n * x - b'^n * y) = ?g^n ∨ ?g^n*(b'^n * x - a'^n * y) = ?g^n"

using z by auto

then have "a^n * x - b^n * y = ?g^n ∨ b^n * x - a^n * y = ?g^n"

using z ab'' by (simp only: power_mult_distrib[symmetric]

diff_mult_distrib2 mult_assoc[symmetric])

hence ?thesis by blast }

ultimately show ?thesis by blast

qed

lemma gcd_exp: "gcd (a^n) (b^n) = gcd a b^n"

proof-

let ?g = "gcd (a^n) (b^n)"

let ?gn = "gcd a b^n"

{fix e assume H: "e dvd a^n" "e dvd b^n"

from bezout_gcd_pow[of a n b] obtain x y

where xy: "a ^ n * x - b ^ n * y = ?gn ∨ b ^ n * x - a ^ n * y = ?gn" by blast

from dvd_diff_nat [OF dvd_mult2[OF H(1), of x] dvd_mult2[OF H(2), of y]]

dvd_diff_nat [OF dvd_mult2[OF H(2), of x] dvd_mult2[OF H(1), of y]] xy

have "e dvd ?gn" by (cases "a ^ n * x - b ^ n * y = gcd a b ^ n", simp_all)}

hence th: "∀e. e dvd a^n ∧ e dvd b^n --> e dvd ?gn" by blast

from divides_exp[OF gcd_dvd1[of a b], of n] divides_exp[OF gcd_dvd2[of a b], of n] th

gcd_unique have "?gn = ?g" by blast thus ?thesis by simp

qed

lemma coprime_exp2: "coprime (a ^ Suc n) (b^ Suc n) <-> coprime a b"

by (simp only: coprime_def gcd_exp exp_eq_1) simp

lemma division_decomp: assumes dc: "(a::nat) dvd b * c"

shows "∃b' c'. a = b' * c' ∧ b' dvd b ∧ c' dvd c"

proof-

let ?g = "gcd a b"

{assume "?g = 0" with dc have ?thesis apply (simp add: gcd_zero)

apply (rule exI[where x="0"])

by (rule exI[where x="c"], simp)}

moreover

{assume z: "?g ≠ 0"

from gcd_coprime_exists[OF z]

obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast

from gcd_dvd2[of a b] have thb: "?g dvd b" .

from ab'(1) have "a' dvd a" unfolding dvd_def by blast

with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp

from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto

hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)

with z have th_1: "a' dvd b'*c" by simp

from coprime_divprod[OF th_1 ab'(3)] have thc: "a' dvd c" .

from ab' have "a = ?g*a'" by algebra

with thb thc have ?thesis by blast }

ultimately show ?thesis by blast

qed

lemma nat_power_eq_0_iff: "(m::nat) ^ n = 0 <-> n ≠ 0 ∧ m = 0" by (induct n, auto)

lemma divides_rev: assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n ≠ 0" shows "a dvd b"

proof-

let ?g = "gcd a b"

from n obtain m where m: "n = Suc m" by (cases n, simp_all)

{assume "?g = 0" with ab n have ?thesis by (simp add: gcd_zero)}

moreover

{assume z: "?g ≠ 0"

hence zn: "?g ^ n ≠ 0" using n by simp

from gcd_coprime_exists[OF z]

obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast

from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n" by (simp add: ab'(1,2)[symmetric])

hence "?g^n*a'^n dvd ?g^n *b'^n" by (simp only: power_mult_distrib mult_commute)

with zn z n have th0:"a'^n dvd b'^n" by (auto simp add: nat_power_eq_0_iff)

have "a' dvd a'^n" by (simp add: m)

with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp

hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)

from coprime_divprod[OF th1 coprime_exp[OF ab'(3), of m]]

have "a' dvd b'" .

hence "a'*?g dvd b'*?g" by simp

with ab'(1,2) have ?thesis by simp }

ultimately show ?thesis by blast

qed

lemma divides_mul: assumes mr: "m dvd r" and nr: "n dvd r" and mn:"coprime m n"

shows "m * n dvd r"

proof-

from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"

unfolding dvd_def by blast

from mr n' have "m dvd n'*n" by (simp add: mult_commute)

hence "m dvd n'" using relprime_dvd_mult_iff[OF mn[unfolded coprime_def]] by simp

then obtain k where k: "n' = m*k" unfolding dvd_def by blast

from n' k show ?thesis unfolding dvd_def by auto

qed

text {* A binary form of the Chinese Remainder Theorem. *}

lemma chinese_remainder: assumes ab: "coprime a b" and a:"a ≠ 0" and b:"b ≠ 0"

shows "∃x q1 q2. x = u + q1 * a ∧ x = v + q2 * b"

proof-

from bezout_add_strong[OF a, of b] bezout_add_strong[OF b, of a]

obtain d1 x1 y1 d2 x2 y2 where dxy1: "d1 dvd a" "d1 dvd b" "a * x1 = b * y1 + d1"

and dxy2: "d2 dvd b" "d2 dvd a" "b * x2 = a * y2 + d2" by blast

from gcd_unique[of 1 a b, simplified ab[unfolded coprime_def], simplified]

dxy1(1,2) dxy2(1,2) have d12: "d1 = 1" "d2 =1" by auto

let ?x = "v * a * x1 + u * b * x2"

let ?q1 = "v * x1 + u * y2"

let ?q2 = "v * y1 + u * x2"

from dxy2(3)[simplified d12] dxy1(3)[simplified d12]

have "?x = u + ?q1 * a" "?x = v + ?q2 * b" by algebra+

thus ?thesis by blast

qed

text {* Primality *}

text {* A few useful theorems about primes *}

lemma prime_0[simp]: "~prime 0" by (simp add: prime_def)

lemma prime_1[simp]: "~ prime 1" by (simp add: prime_def)

lemma prime_Suc0[simp]: "~ prime (Suc 0)" by (simp add: prime_def)

lemma prime_ge_2: "prime p ==> p ≥ 2" by (simp add: prime_def)

lemma prime_factor: assumes n: "n ≠ 1" shows "∃ p. prime p ∧ p dvd n"

using n

proof(induct n rule: nat_less_induct)

fix n

assume H: "∀m<n. m ≠ 1 --> (∃p. prime p ∧ p dvd m)" "n ≠ 1"

let ?ths = "∃p. prime p ∧ p dvd n"

{assume "n=0" hence ?ths using two_is_prime by auto}

moreover

{assume nz: "n≠0"

{assume "prime n" hence ?ths by - (rule exI[where x="n"], simp)}

moreover

{assume n: "¬ prime n"

with nz H(2)

obtain k where k:"k dvd n" "k ≠ 1" "k ≠ n" by (auto simp add: prime_def)

from dvd_imp_le[OF k(1)] nz k(3) have kn: "k < n" by simp

from H(1)[rule_format, OF kn k(2)] obtain p where p: "prime p" "p dvd k" by blast

from dvd_trans[OF p(2) k(1)] p(1) have ?ths by blast}

ultimately have ?ths by blast}

ultimately show ?ths by blast

qed

lemma prime_factor_lt: assumes p: "prime p" and n: "n ≠ 0" and npm:"n = p * m"

shows "m < n"

proof-

{assume "m=0" with n have ?thesis by simp}

moreover

{assume m: "m ≠ 0"

from npm have mn: "m dvd n" unfolding dvd_def by auto

from npm m have "n ≠ m" using p by auto

with dvd_imp_le[OF mn] n have ?thesis by simp}

ultimately show ?thesis by blast

qed

lemma euclid_bound: "∃p. prime p ∧ n < p ∧ p <= Suc (fact n)"

proof-

have f1: "fact n + 1 ≠ 1" using fact_le[of n] by arith

from prime_factor[OF f1] obtain p where p: "prime p" "p dvd fact n + 1" by blast

from dvd_imp_le[OF p(2)] have pfn: "p ≤ fact n + 1" by simp

{assume np: "p ≤ n"

from p(1) have p1: "p ≥ 1" by (cases p, simp_all)

from divides_fact[OF p1 np] have pfn': "p dvd fact n" .

from divides_add_revr[OF pfn' p(2)] p(1) have False by simp}

hence "n < p" by arith

with p(1) pfn show ?thesis by auto

qed

lemma euclid: "∃p. prime p ∧ p > n" using euclid_bound by auto

lemma primes_infinite: "¬ (finite {p. prime p})"

apply(simp add: finite_nat_set_iff_bounded_le)

apply (metis euclid linorder_not_le)

done

lemma coprime_prime: assumes ab: "coprime a b"

shows "~(prime p ∧ p dvd a ∧ p dvd b)"

proof

assume "prime p ∧ p dvd a ∧ p dvd b"

thus False using ab gcd_greatest[of p a b] by (simp add: coprime_def)

qed

lemma coprime_prime_eq: "coprime a b <-> (∀p. ~(prime p ∧ p dvd a ∧ p dvd b))"

(is "?lhs = ?rhs")

proof-

{assume "?lhs" with coprime_prime have ?rhs by blast}

moreover

{assume r: "?rhs" and c: "¬ ?lhs"

then obtain g where g: "g≠1" "g dvd a" "g dvd b" unfolding coprime_def by blast

from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast

from dvd_trans [OF p(2) g(2)] dvd_trans [OF p(2) g(3)]

have "p dvd a" "p dvd b" . with p(1) r have False by blast}

ultimately show ?thesis by blast

qed

lemma prime_coprime: assumes p: "prime p"

shows "n = 1 ∨ p dvd n ∨ coprime p n"

using p prime_imp_relprime[of p n] by (auto simp add: coprime_def)

lemma prime_coprime_strong: "prime p ==> p dvd n ∨ coprime p n"

using prime_coprime[of p n] by auto

declare coprime_0[simp]

lemma coprime_0'[simp]: "coprime 0 d <-> d = 1" by (simp add: coprime_commute[of 0 d])

lemma coprime_bezout_strong: assumes ab: "coprime a b" and b: "b ≠ 1"

shows "∃x y. a * x = b * y + 1"

proof-

from ab b have az: "a ≠ 0" by - (rule ccontr, auto)

from bezout_gcd_strong[OF az, of b] ab[unfolded coprime_def]

show ?thesis by auto

qed

lemma bezout_prime: assumes p: "prime p" and pa: "¬ p dvd a"

shows "∃x y. a*x = p*y + 1"

proof-

from p have p1: "p ≠ 1" using prime_1 by blast

from prime_coprime[OF p, of a] p1 pa have ap: "coprime a p"

by (auto simp add: coprime_commute)

from coprime_bezout_strong[OF ap p1] show ?thesis .

qed

lemma prime_divprod: assumes p: "prime p" and pab: "p dvd a*b"

shows "p dvd a ∨ p dvd b"

proof-

{assume "a=1" hence ?thesis using pab by simp }

moreover

{assume "p dvd a" hence ?thesis by blast}

moreover

{assume pa: "coprime p a" from coprime_divprod[OF pab pa] have ?thesis .. }

ultimately show ?thesis using prime_coprime[OF p, of a] by blast

qed

lemma prime_divprod_eq: assumes p: "prime p"

shows "p dvd a*b <-> p dvd a ∨ p dvd b"

using p prime_divprod dvd_mult dvd_mult2 by auto

lemma prime_divexp: assumes p:"prime p" and px: "p dvd x^n"

shows "p dvd x"

using px

proof(induct n)

case 0 thus ?case by simp

next

case (Suc n)

hence th: "p dvd x*x^n" by simp

{assume H: "p dvd x^n"

from Suc.hyps[OF H] have ?case .}

with prime_divprod[OF p th] show ?case by blast

qed

lemma prime_divexp_n: "prime p ==> p dvd x^n ==> p^n dvd x^n"

using prime_divexp[of p x n] divides_exp[of p x n] by blast

lemma coprime_prime_dvd_ex: assumes xy: "¬coprime x y"

shows "∃p. prime p ∧ p dvd x ∧ p dvd y"

proof-

from xy[unfolded coprime_def] obtain g where g: "g ≠ 1" "g dvd x" "g dvd y"

by blast

from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast

from g(2,3) dvd_trans[OF p(2)] p(1) show ?thesis by auto

qed

lemma coprime_sos: assumes xy: "coprime x y"

shows "coprime (x * y) (x⇧^{2}+ y⇧^{2})"

proof-

{assume c: "¬ coprime (x * y) (x⇧^{2}+ y⇧^{2})"

from coprime_prime_dvd_ex[OF c] obtain p

where p: "prime p" "p dvd x*y" "p dvd x⇧^{2}+ y⇧^{2}" by blast

{assume px: "p dvd x"

from dvd_mult[OF px, of x] p(3)

obtain r s where "x * x = p * r" and "x⇧^{2}+ y⇧^{2}= p * s"

by (auto elim!: dvdE)

then have "y⇧^{2}= p * (s - r)"

by (auto simp add: power2_eq_square diff_mult_distrib2)

then have "p dvd y⇧^{2}" ..

with prime_divexp[OF p(1), of y 2] have py: "p dvd y" .

from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1

have False by simp }

moreover

{assume py: "p dvd y"

from dvd_mult[OF py, of y] p(3)

obtain r s where "y * y = p * r" and "x⇧^{2}+ y⇧^{2}= p * s"

by (auto elim!: dvdE)

then have "x⇧^{2}= p * (s - r)"

by (auto simp add: power2_eq_square diff_mult_distrib2)

then have "p dvd x⇧^{2}" ..

with prime_divexp[OF p(1), of x 2] have px: "p dvd x" .

from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1

have False by simp }

ultimately have False using prime_divprod[OF p(1,2)] by blast}

thus ?thesis by blast

qed

lemma distinct_prime_coprime: "prime p ==> prime q ==> p ≠ q ==> coprime p q"

unfolding prime_def coprime_prime_eq by blast

lemma prime_coprime_lt: assumes p: "prime p" and x: "0 < x" and xp: "x < p"

shows "coprime x p"

proof-

{assume c: "¬ coprime x p"

then obtain g where g: "g ≠ 1" "g dvd x" "g dvd p" unfolding coprime_def by blast

from dvd_imp_le[OF g(2)] x xp have gp: "g < p" by arith

from g(2) x have "g ≠ 0" by - (rule ccontr, simp)

with g gp p[unfolded prime_def] have False by blast}

thus ?thesis by blast

qed

lemma even_dvd[simp]: "even (n::nat) <-> 2 dvd n" by presburger

lemma prime_odd: "prime p ==> p = 2 ∨ odd p" unfolding prime_def by auto

text {* One property of coprimality is easier to prove via prime factors. *}

lemma prime_divprod_pow:

assumes p: "prime p" and ab: "coprime a b" and pab: "p^n dvd a * b"

shows "p^n dvd a ∨ p^n dvd b"

proof-

{assume "n = 0 ∨ a = 1 ∨ b = 1" with pab have ?thesis

apply (cases "n=0", simp_all)

apply (cases "a=1", simp_all) done}

moreover

{assume n: "n ≠ 0" and a: "a≠1" and b: "b≠1"

then obtain m where m: "n = Suc m" by (cases n, auto)

from divides_exp2[OF n pab] have pab': "p dvd a*b" .

from prime_divprod[OF p pab']

have "p dvd a ∨ p dvd b" .

moreover

{assume pa: "p dvd a"

have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)

from coprime_prime[OF ab, of p] p pa have "¬ p dvd b" by blast

with prime_coprime[OF p, of b] b

have cpb: "coprime b p" using coprime_commute by blast

from coprime_exp[OF cpb] have pnb: "coprime (p^n) b"

by (simp add: coprime_commute)

from coprime_divprod[OF pnba pnb] have ?thesis by blast }

moreover

{assume pb: "p dvd b"

have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)

from coprime_prime[OF ab, of p] p pb have "¬ p dvd a" by blast

with prime_coprime[OF p, of a] a

have cpb: "coprime a p" using coprime_commute by blast

from coprime_exp[OF cpb] have pnb: "coprime (p^n) a"

by (simp add: coprime_commute)

from coprime_divprod[OF pab pnb] have ?thesis by blast }

ultimately have ?thesis by blast}

ultimately show ?thesis by blast

qed

lemma nat_mult_eq_one: "(n::nat) * m = 1 <-> n = 1 ∧ m = 1" (is "?lhs <-> ?rhs")

proof

assume H: "?lhs"

hence "n dvd 1" "m dvd 1" unfolding dvd_def by (auto simp add: mult_commute)

thus ?rhs by auto

next

assume ?rhs then show ?lhs by auto

qed

lemma power_Suc0: "Suc 0 ^ n = Suc 0"

unfolding One_nat_def[symmetric] power_one ..

lemma coprime_pow: assumes ab: "coprime a b" and abcn: "a * b = c ^n"

shows "∃r s. a = r^n ∧ b = s ^n"

using ab abcn

proof(induct c arbitrary: a b rule: nat_less_induct)

fix c a b

assume H: "∀m<c. ∀a b. coprime a b --> a * b = m ^ n --> (∃r s. a = r ^ n ∧ b = s ^ n)" "coprime a b" "a * b = c ^ n"

let ?ths = "∃r s. a = r^n ∧ b = s ^n"

{assume n: "n = 0"

with H(3) power_one have "a*b = 1" by simp

hence "a = 1 ∧ b = 1" by simp

hence ?ths

apply -

apply (rule exI[where x=1])

apply (rule exI[where x=1])

using power_one[of n]

by simp}

moreover

{assume n: "n ≠ 0" then obtain m where m: "n = Suc m" by (cases n, auto)

{assume c: "c = 0"

with H(3) m H(2) have ?ths apply simp

apply (cases "a=0", simp_all)

apply (rule exI[where x="0"], simp)

apply (rule exI[where x="0"], simp)

done}

moreover

{assume "c=1" with H(3) power_one have "a*b = 1" by simp

hence "a = 1 ∧ b = 1" by simp

hence ?ths

apply -

apply (rule exI[where x=1])

apply (rule exI[where x=1])

using power_one[of n]

by simp}

moreover

{assume c: "c≠1" "c ≠ 0"

from prime_factor[OF c(1)] obtain p where p: "prime p" "p dvd c" by blast

from prime_divprod_pow[OF p(1) H(2), unfolded H(3), OF divides_exp[OF p(2), of n]]

have pnab: "p ^ n dvd a ∨ p^n dvd b" .

from p(2) obtain l where l: "c = p*l" unfolding dvd_def by blast

have pn0: "p^n ≠ 0" using n prime_ge_2 [OF p(1)] by simp

{assume pa: "p^n dvd a"

then obtain k where k: "a = p^n * k" unfolding dvd_def by blast

from l have "l dvd c" by auto

with dvd_imp_le[of l c] c have "l ≤ c" by auto

moreover {assume "l = c" with l c have "p = 1" by simp with p have False by simp}

ultimately have lc: "l < c" by arith

from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" b]]]

have kb: "coprime k b" by (simp add: coprime_commute)

from H(3) l k pn0 have kbln: "k * b = l ^ n"

by (auto simp add: power_mult_distrib)

from H(1)[rule_format, OF lc kb kbln]

obtain r s where rs: "k = r ^n" "b = s^n" by blast

from k rs(1) have "a = (p*r)^n" by (simp add: power_mult_distrib)

with rs(2) have ?ths by blast }

moreover

{assume pb: "p^n dvd b"

then obtain k where k: "b = p^n * k" unfolding dvd_def by blast

from l have "l dvd c" by auto

with dvd_imp_le[of l c] c have "l ≤ c" by auto

moreover {assume "l = c" with l c have "p = 1" by simp with p have False by simp}

ultimately have lc: "l < c" by arith

from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" a]]]

have kb: "coprime k a" by (simp add: coprime_commute)

from H(3) l k pn0 n have kbln: "k * a = l ^ n"

by (simp add: power_mult_distrib mult_commute)

from H(1)[rule_format, OF lc kb kbln]

obtain r s where rs: "k = r ^n" "a = s^n" by blast

from k rs(1) have "b = (p*r)^n" by (simp add: power_mult_distrib)

with rs(2) have ?ths by blast }

ultimately have ?ths using pnab by blast}

ultimately have ?ths by blast}

ultimately show ?ths by blast

qed

text {* More useful lemmas. *}

lemma prime_product:

assumes "prime (p * q)"

shows "p = 1 ∨ q = 1"

proof -

from assms have

"1 < p * q" and P: "!!m. m dvd p * q ==> m = 1 ∨ m = p * q"

unfolding prime_def by auto

from `1 < p * q` have "p ≠ 0" by (cases p) auto

then have Q: "p = p * q <-> q = 1" by auto

have "p dvd p * q" by simp

then have "p = 1 ∨ p = p * q" by (rule P)

then show ?thesis by (simp add: Q)

qed

lemma prime_exp: "prime (p^n) <-> prime p ∧ n = 1"

proof(induct n)

case 0 thus ?case by simp

next

case (Suc n)

{assume "p = 0" hence ?case by simp}

moreover

{assume "p=1" hence ?case by simp}

moreover

{assume p: "p ≠ 0" "p≠1"

{assume pp: "prime (p^Suc n)"

hence "p = 1 ∨ p^n = 1" using prime_product[of p "p^n"] by simp

with p have n: "n = 0"

by (simp only: exp_eq_1 ) simp

with pp have "prime p ∧ Suc n = 1" by simp}

moreover

{assume n: "prime p ∧ Suc n = 1" hence "prime (p^Suc n)" by simp}

ultimately have ?case by blast}

ultimately show ?case by blast

qed

lemma prime_power_mult:

assumes p: "prime p" and xy: "x * y = p ^ k"

shows "∃i j. x = p ^i ∧ y = p^ j"

using xy

proof(induct k arbitrary: x y)

case 0 thus ?case apply simp by (rule exI[where x="0"], simp)

next

case (Suc k x y)

from Suc.prems have pxy: "p dvd x*y" by auto

from prime_divprod[OF p pxy] have pxyc: "p dvd x ∨ p dvd y" .

from p have p0: "p ≠ 0" by - (rule ccontr, simp)

{assume px: "p dvd x"

then obtain d where d: "x = p*d" unfolding dvd_def by blast

from Suc.prems d have "p*d*y = p^Suc k" by simp

hence th: "d*y = p^k" using p0 by simp

from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast

with d have "x = p^Suc i" by simp

with ij(2) have ?case by blast}

moreover

{assume px: "p dvd y"

then obtain d where d: "y = p*d" unfolding dvd_def by blast

from Suc.prems d have "p*d*x = p^Suc k" by (simp add: mult_commute)

hence th: "d*x = p^k" using p0 by simp

from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast

with d have "y = p^Suc i" by simp

with ij(2) have ?case by blast}

ultimately show ?case using pxyc by blast

qed

lemma prime_power_exp: assumes p: "prime p" and n:"n ≠ 0"

and xn: "x^n = p^k" shows "∃i. x = p^i"

using n xn

proof(induct n arbitrary: k)

case 0 thus ?case by simp

next

case (Suc n k) hence th: "x*x^n = p^k" by simp

{assume "n = 0" with Suc have ?case by simp (rule exI[where x="k"], simp)}

moreover

{assume n: "n ≠ 0"

from prime_power_mult[OF p th]

obtain i j where ij: "x = p^i" "x^n = p^j"by blast

from Suc.hyps[OF n ij(2)] have ?case .}

ultimately show ?case by blast

qed

lemma divides_primepow: assumes p: "prime p"

shows "d dvd p^k <-> (∃ i. i ≤ k ∧ d = p ^i)"

proof

assume H: "d dvd p^k" then obtain e where e: "d*e = p^k"

unfolding dvd_def apply (auto simp add: mult_commute) by blast

from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast

from prime_ge_2[OF p] have p1: "p > 1" by arith

from e ij have "p^(i + j) = p^k" by (simp add: power_add)

hence "i + j = k" using power_inject_exp[of p "i+j" k, OF p1] by simp

hence "i ≤ k" by arith

with ij(1) show "∃i≤k. d = p ^ i" by blast

next

{fix i assume H: "i ≤ k" "d = p^i"

hence "∃j. k = i + j" by arith

then obtain j where j: "k = i + j" by blast

hence "p^k = p^j*d" using H(2) by (simp add: power_add)

hence "d dvd p^k" unfolding dvd_def by auto}

thus "∃i≤k. d = p ^ i ==> d dvd p ^ k" by blast

qed

lemma coprime_divisors: "d dvd a ==> e dvd b ==> coprime a b ==> coprime d e"

by (auto simp add: dvd_def coprime)

lemma mult_inj_if_coprime_nat:

"inj_on f A ==> inj_on g B ==> ALL a:A. ALL b:B. coprime (f a) (g b)

==> inj_on (%(a,b). f a * g b::nat) (A × B)"

apply(auto simp add:inj_on_def)

apply(metis coprime_def dvd_triv_left gcd_proj2_if_dvd_nat gcd_semilattice_nat.inf_commute relprime_dvd_mult)

apply(metis coprime_commute coprime_divprod dvd.neq_le_trans dvd_triv_right)

done

declare power_Suc0[simp del]

declare even_dvd[simp del]

end