section ‹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" ..
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: ac_simps)
apply (rule_tac x="a*x" in exI)
apply (rule_tac x="y" in exI)
apply (simp add: ac_simps)
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: ac_simps)
apply (rule_tac x="b*x" in exI)
apply (rule_tac x="y" in exI)
apply (simp add: ac_simps)
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: ac_simps)
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 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 ⟹ ∀a∈A. ∀b∈B. Primes.coprime (f a) (g b) ⟹
inj_on (λ(a, b). f a * g b) (A × B)"
apply (auto simp add: inj_on_def)
apply (metis coprime_def dvd_antisym dvd_triv_left relprime_dvd_mult_iff)
apply (metis coprime_commute coprime_divprod dvd_antisym dvd_triv_right)
done
declare power_Suc0[simp del]
end