# Theory Pocklington

```(*  Title:      HOL/Number_Theory/Pocklington.thy
Author:     Amine Chaieb, Manuel Eberl
*)

section ‹Pocklington's Theorem for Primes›

theory Pocklington
imports Residues
begin

subsection ‹Lemmas about previously defined terms›

lemma prime_nat_iff'': "prime (p::nat) ⟷ p ≠ 0 ∧ p ≠ 1 ∧ (∀m. 0 < m ∧ m < p ⟶ coprime p m)"
apply (auto simp add: prime_nat_iff)
apply (rule coprimeI)
apply (auto dest: nat_dvd_not_less simp add: ac_simps)
apply (metis One_nat_def dvd_1_iff_1 dvd_pos_nat gcd_nat.order_iff is_unit_gcd linorder_neqE_nat nat_dvd_not_less)
done

lemma finite_number_segment: "card { m. 0 < m ∧ m < n } = n - 1"
proof -
have "{ m. 0 < m ∧ m < n } = {1..<n}" by auto
then show ?thesis by simp
qed

subsection ‹Some basic theorems about solving congruences›

lemma cong_solve:
fixes n :: nat
assumes an: "coprime a n"
shows "∃x. [a * x = b] (mod n)"
proof (cases "a = 0")
case True
with an show ?thesis
by (simp add: cong_def)
next
case False
from bezout_add_strong_nat [OF this]
obtain d x y where dxy: "d dvd a" "d dvd n" "a * x = n * y + d" by blast
from dxy(1,2) have d1: "d = 1"
using assms coprime_common_divisor [of a n d] by simp
with dxy(3) have "a * x * b = (n * y + 1) * b"
by simp
then have "a * (x * b) = n * (y * b) + b"
by (auto simp: algebra_simps)
then have "a * (x * b) mod n = (n * (y * b) + b) mod n"
by simp
then have "a * (x * b) mod n = b mod n"
then have "[a * (x * b) = b] (mod n)"
by (simp only: cong_def)
then show ?thesis by blast
qed

lemma cong_solve_unique:
fixes n :: nat
assumes an: "coprime a n" and nz: "n ≠ 0"
shows "∃!x. x < n ∧ [a * x = b] (mod n)"
proof -
from cong_solve[OF an] obtain x where x: "[a * x = b] (mod n)"
by blast
let ?P = "λx. x < n ∧ [a * x = b] (mod n)"
let ?x = "x mod n"
from x have *: "[a * ?x = b] (mod n)"
by (simp add: cong_def mod_mult_right_eq[of a x n])
from mod_less_divisor[ of n x] nz * have Px: "?P ?x" by simp
have "y = ?x" if Py: "y < n" "[a * y = b] (mod n)" for y
proof -
from Py(2) * have "[a * y = a * ?x] (mod n)"
by (simp add: cong_def)
then have "[y = ?x] (mod n)"
by (metis an cong_mult_lcancel_nat)
with mod_less[OF Py(1)] mod_less_divisor[ of n x] nz
show ?thesis
by (simp add: cong_def)
qed
with Px show ?thesis by blast
qed

lemma cong_solve_unique_nontrivial:
fixes p :: nat
assumes p: "prime p"
and pa: "coprime p a"
and x0: "0 < x"
and xp: "x < p"
shows "∃!y. 0 < y ∧ y < p ∧ [x * y = a] (mod p)"
proof -
from pa have ap: "coprime a p"
by (simp add: ac_simps)
from x0 xp p have px: "coprime x p"
by (auto simp add: prime_nat_iff'' ac_simps)
obtain y where y: "y < p" "[x * y = a] (mod p)" "∀z. z < p ∧ [x * z = a] (mod p) ⟶ z = y"
by (metis cong_solve_unique neq0_conv p prime_gt_0_nat px)
have "y ≠ 0"
proof
assume "y = 0"
with y(2) have "p dvd a"
using cong_dvd_iff by auto
with not_prime_1 p pa show False
by (auto simp add: gcd_nat.order_iff)
qed
with y show ?thesis
by blast
qed

lemma cong_unique_inverse_prime:
fixes p :: nat
assumes "prime p" and "0 < x" and "x < p"
shows "∃!y. 0 < y ∧ y < p ∧ [x * y = 1] (mod p)"
by (rule cong_solve_unique_nontrivial) (use assms in simp_all)

lemma chinese_remainder_coprime_unique:
fixes a :: nat
assumes ab: "coprime a b" and az: "a ≠ 0" and bz: "b ≠ 0"
and ma: "coprime m a" and nb: "coprime n b"
shows "∃!x. coprime x (a * b) ∧ x < a * b ∧ [x = m] (mod a) ∧ [x = n] (mod b)"
proof -
let ?P = "λx. x < a * b ∧ [x = m] (mod a) ∧ [x = n] (mod b)"
from binary_chinese_remainder_unique_nat[OF ab az bz]
obtain x where x: "x < a * b" "[x = m] (mod a)" "[x = n] (mod b)" "∀y. ?P y ⟶ y = x"
by blast
from ma nb x have "coprime x a" "coprime x b"
using cong_imp_coprime cong_sym by blast+
then have "coprime x (a*b)"
by simp
with x show ?thesis
by blast
qed

subsection ‹Lucas's theorem›

lemma lucas_coprime_lemma:
fixes n :: nat
assumes m: "m ≠ 0" and am: "[a^m = 1] (mod n)"
shows "coprime a n"
proof -
consider "n = 1" | "n = 0" | "n > 1" by arith
then show ?thesis
proof cases
case 1
then show ?thesis by simp
next
case 2
with am m show ?thesis
by simp
next
case 3
from m obtain m' where m': "m = Suc m'" by (cases m) blast+
have "d = 1" if d: "d dvd a" "d dvd n" for d
proof -
from am mod_less[OF ‹n > 1›] have am1: "a^m mod n = 1"
by (simp add: cong_def)
from dvd_mult2[OF d(1), of "a^m'"] have dam: "d dvd a^m"
by (simp add: m')
from dvd_mod_iff[OF d(2), of "a^m"] dam am1 show ?thesis
by simp
qed
then show ?thesis
by (auto intro: coprimeI)
qed
qed

lemma lucas_weak:
fixes n :: nat
assumes n: "n ≥ 2"
and an: "[a ^ (n - 1) = 1] (mod n)"
and nm: "∀m. 0 < m ∧ m < n - 1 ⟶ ¬ [a ^ m = 1] (mod n)"
shows "prime n"
proof (rule totient_imp_prime)
show "totient n = n - 1"
proof (rule ccontr)
have "[a ^ totient n = 1] (mod n)"
by (rule euler_theorem, rule lucas_coprime_lemma [of "n - 1"]) (use n an in auto)
moreover assume "totient n ≠ n - 1"
then have "totient n > 0" "totient n < n - 1"
using ‹n ≥ 2› and totient_less[of n] by simp_all
ultimately show False
using nm by auto
qed
qed (use n in auto)

lemma nat_exists_least_iff: "(∃(n::nat). P n) ⟷ (∃n. P n ∧ (∀m < n. ¬ P m))"
by (metis ex_least_nat_le not_less0)

lemma nat_exists_least_iff': "(∃(n::nat). P n) ⟷ P (Least P) ∧ (∀m < (Least P). ¬ P m)"
(is "?lhs ⟷ ?rhs")
proof
show ?lhs if ?rhs
using that by blast
show ?rhs if ?lhs
proof -
from ‹?lhs› obtain n where n: "P n" by blast
let ?x = "Least P"
have "¬ P m" if "m < ?x" for m
by (rule not_less_Least[OF that])
with LeastI_ex[OF ‹?lhs›] show ?thesis
by blast
qed
qed

theorem lucas:
assumes n2: "n ≥ 2" and an1: "[a^(n - 1) = 1] (mod n)"
and pn: "∀p. prime p ∧ p dvd n - 1 ⟶ [a^((n - 1) div p) ≠ 1] (mod n)"
shows "prime n"
proof-
from n2 have n01: "n ≠ 0" "n ≠ 1" "n - 1 ≠ 0"
by arith+
from mod_less_divisor[of n 1] n01 have onen: "1 mod n = 1"
by simp
from lucas_coprime_lemma[OF n01(3) an1] cong_imp_coprime an1
have an: "coprime a n" "coprime (a ^ (n - 1)) n"
using ‹n ≥ 2› by simp_all
have False if H0: "∃m. 0 < m ∧ m < n - 1 ∧ [a ^ m = 1] (mod n)" (is "∃m. ?P m")
proof -
from H0[unfolded nat_exists_least_iff[of ?P]] obtain m where
m: "0 < m" "m < n - 1" "[a ^ m = 1] (mod n)" "∀k <m. ¬?P k"
by blast
have False if nm1: "(n - 1) mod m > 0"
proof -
from mod_less_divisor[OF m(1)] have th0:"(n - 1) mod m < m" by blast
let ?y = "a^ ((n - 1) div m * m)"
note mdeq = div_mult_mod_eq[of "(n - 1)" m]
have yn: "coprime ?y n"
using an(1) by (cases "(n - Suc 0) div m * m = 0") auto
have "?y mod n = (a^m)^((n - 1) div m) mod n"
by (simp add: algebra_simps power_mult)
also have "… = (a^m mod n)^((n - 1) div m) mod n"
using power_mod[of "a^m" n "(n - 1) div m"] by simp
also have "… = 1" using m(3)[unfolded cong_def onen] onen
by (metis power_one)
finally have *: "?y mod n = 1"  .
have **: "[?y * a ^ ((n - 1) mod m) = ?y* 1] (mod n)"
using an1[unfolded cong_def onen] onen
div_mult_mod_eq[of "(n - 1)" m, symmetric]
by (simp add:power_add[symmetric] cong_def * del: One_nat_def)
have "[a ^ ((n - 1) mod m) = 1] (mod n)"
by (metis cong_mult_rcancel_nat mult.commute ** yn)
with m(4)[rule_format, OF th0] nm1
less_trans[OF mod_less_divisor[OF m(1), of "n - 1"] m(2)] show ?thesis
by blast
qed
then have "(n - 1) mod m = 0" by auto
then have mn: "m dvd n - 1" by presburger
then obtain r where r: "n - 1 = m * r"
unfolding dvd_def by blast
from n01 r m(2) have r01: "r ≠ 0" "r ≠ 1" by auto
obtain p where p: "prime p" "p dvd r"
by (metis prime_factor_nat r01(2))
then have th: "prime p ∧ p dvd n - 1"
unfolding r by (auto intro: dvd_mult)
from r have "(a ^ ((n - 1) div p)) mod n = (a^(m*r div p)) mod n"
by (simp add: power_mult)
also have "… = (a^(m*(r div p))) mod n"
using div_mult1_eq[of m r p] p(2)[unfolded dvd_eq_mod_eq_0] by simp
also have "… = ((a^m)^(r div p)) mod n"
by (simp add: power_mult)
also have "… = ((a^m mod n)^(r div p)) mod n"
using power_mod ..
also from m(3) onen have "… = 1"
by (simp add: cong_def)
finally have "[(a ^ ((n - 1) div p))= 1] (mod n)"
using onen by (simp add: cong_def)
with pn th show ?thesis by blast
qed
then have "∀m. 0 < m ∧ m < n - 1 ⟶ ¬ [a ^ m = 1] (mod n)"
by blast
then show ?thesis by (rule lucas_weak[OF n2 an1])
qed

subsection ‹Definition of the order of a number mod ‹n››

definition "ord n a = (if coprime n a then Least (λd. d > 0 ∧ [a ^d = 1] (mod n)) else 0)"

text ‹This has the expected properties.›

lemma coprime_ord:
fixes n::nat
assumes "coprime n a"
shows "ord n a > 0 ∧ [a ^(ord n a) = 1] (mod n) ∧ (∀m. 0 < m ∧ m < ord n a ⟶ [a^ m ≠ 1] (mod n))"
proof-
let ?P = "λd. 0 < d ∧ [a ^ d = 1] (mod n)"
from bigger_prime[of a] obtain p where p: "prime p" "a < p"
by blast
from assms have o: "ord n a = Least ?P"
by (simp add: ord_def)
have ex: "∃m>0. ?P m"
proof (cases "n ≥ 2")
case True
moreover from assms have "coprime a n"
by (simp add: ac_simps)
then have "[a ^ totient n = 1] (mod n)"
by (rule euler_theorem)
ultimately show ?thesis
by (auto intro: exI [where x = "totient n"])
next
case False
then have "n = 0 ∨ n = 1"
by auto
with assms show ?thesis
by auto
qed
from nat_exists_least_iff'[of ?P] ex assms show ?thesis
unfolding o[symmetric] by auto
qed

text ‹With the special value ‹0› for non-coprime case, it's more convenient.›
lemma ord_works: "[a ^ (ord n a) = 1] (mod n) ∧ (∀m. 0 < m ∧ m < ord n a ⟶ ¬ [a^ m = 1] (mod n))"
for n :: nat
by (cases "coprime n a") (use coprime_ord[of n a] in ‹auto simp add: ord_def cong_def›)

lemma ord: "[a^(ord n a) = 1] (mod n)"
for n :: nat
using ord_works by blast

lemma ord_minimal: "0 < m ⟹ m < ord n a ⟹ ¬ [a^m = 1] (mod n)"
for n :: nat
using ord_works by blast

lemma ord_eq_0: "ord n a = 0 ⟷ ¬ coprime n a"
for n :: nat
by (cases "coprime n a") (simp add: coprime_ord, simp add: ord_def)

lemma divides_rexp: "x dvd y ⟹ x dvd (y ^ Suc n)"
for x y :: nat
by (simp add: dvd_mult2[of x y])

lemma ord_divides:"[a ^ d = 1] (mod n) ⟷ ord n a dvd d"
(is "?lhs ⟷ ?rhs")
for n :: nat
proof
assume ?rhs
then obtain k where "d = ord n a * k"
unfolding dvd_def by blast
then have "[a ^ d = (a ^ (ord n a) mod n)^k] (mod n)"
by (simp add : cong_def power_mult power_mod)
also have "[(a ^ (ord n a) mod n)^k = 1] (mod n)"
using ord[of a n, unfolded cong_def]
by (simp add: cong_def power_mod)
finally show ?lhs .
next
assume ?lhs
show ?rhs
proof (cases "coprime n a")
case prem: False
then have o: "ord n a = 0" by (simp add: ord_def)
show ?thesis
proof (cases d)
case 0
with o prem show ?thesis by (simp add: cong_def)
next
case (Suc d')
then have d0: "d ≠ 0" by simp
from prem obtain p where p: "p dvd n" "p dvd a" "p ≠ 1"
by (auto elim: not_coprimeE)
from ‹?lhs› obtain q1 q2 where q12: "a ^ d + n * q1 = 1 + n * q2"
using prem d0 lucas_coprime_lemma
by (auto elim: not_coprimeE simp add: ac_simps)
then have "a ^ d + n * q1 - n * q2 = 1" by simp
with dvd_diff_nat [OF dvd_add [OF divides_rexp]]  dvd_mult2 Suc p have "p dvd 1"
by metis
with p(3) have False by simp
then show ?thesis ..
qed
next
case H: True
let ?o = "ord n a"
let ?q = "d div ord n a"
let ?r = "d mod ord n a"
have eqo: "[(a^?o)^?q = 1] (mod n)"
using cong_pow ord_works by fastforce
from H have onz: "?o ≠ 0" by (simp add: ord_eq_0)
then have opos: "?o > 0" by simp
from div_mult_mod_eq[of d "ord n a"] ‹?lhs›
have "[a^(?o*?q + ?r) = 1] (mod n)"
by (simp add: cong_def mult.commute)
then have "[(a^?o)^?q * (a^?r) = 1] (mod n)"
then have th: "[a^?r = 1] (mod n)"
using eqo mod_mult_left_eq[of "(a^?o)^?q" "a^?r" n]
by (simp add: cong_def del: One_nat_def) (metis mod_mult_left_eq nat_mult_1)
show ?thesis
proof (cases "?r = 0")
case True
then show ?thesis by (simp add: dvd_eq_mod_eq_0)
next
case False
with mod_less_divisor[OF opos, of d] have r0o:"?r >0 ∧ ?r < ?o" by simp
from conjunct2[OF ord_works[of a n], rule_format, OF r0o] th
show ?thesis by blast
qed
qed
qed

lemma order_divides_totient:
"ord n a dvd totient n" if "coprime n a"
using that euler_theorem [of a n]
by (simp add: ord_divides [symmetric] ac_simps)

lemma order_divides_expdiff:
fixes n::nat and a::nat assumes na: "coprime n a"
shows "[a^d = a^e] (mod n) ⟷ [d = e] (mod (ord n a))"
proof -
have th: "[a^d = a^e] (mod n) ⟷ [d = e] (mod (ord n a))"
if na: "coprime n a" and ed: "(e::nat) ≤ d"
for n a d e :: nat
proof -
from na ed have "∃c. d = e + c" by presburger
then obtain c where c: "d = e + c" ..
from na have an: "coprime a n"
by (simp add: ac_simps)
then have aen: "coprime (a ^ e) n"
by (cases "e > 0") simp_all
from an have acn: "coprime (a ^ c) n"
by (cases "c > 0") simp_all
from c have "[a^d = a^e] (mod n) ⟷ [a^(e + c) = a^(e + 0)] (mod n)"
by simp
also have "… ⟷ [a^e* a^c = a^e *a^0] (mod n)" by (simp add: power_add)
also have  "… ⟷ [a ^ c = 1] (mod n)"
using cong_mult_lcancel_nat [OF aen, of "a^c" "a^0"] by simp
also have "… ⟷ ord n a dvd c"
by (simp only: ord_divides)
also have "… ⟷ [e + c = e + 0] (mod ord n a)"
by (auto simp add: cong_altdef_nat)
finally show ?thesis
using c by simp
qed
consider "e ≤ d" | "d ≤ e" by arith
then show ?thesis
proof cases
case 1
with na show ?thesis by (rule th)
next
case 2
from th[OF na this] show ?thesis
by (metis cong_sym)
qed
qed

lemma ord_not_coprime [simp]: "¬coprime n a ⟹ ord n a = 0"
by (simp add: ord_def)

lemma ord_1 [simp]: "ord 1 n = 1"
proof -
have "(LEAST k. k > 0) = (1 :: nat)"
by (rule Least_equality) auto
thus ?thesis by (simp add: ord_def)
qed

lemma ord_1_right [simp]: "ord (n::nat) 1 = 1"
using ord_divides[of 1 1 n] by simp

lemma ord_Suc_0_right [simp]: "ord (n::nat) (Suc 0) = 1"
using ord_divides[of 1 1 n] by simp

lemma ord_0_nat [simp]: "ord 0 (n :: nat) = (if n = 1 then 1 else 0)"
proof -
have "(LEAST k. k > 0) = (1 :: nat)"
by (rule Least_equality) auto
thus ?thesis by (auto simp: ord_def)
qed

lemma ord_0_right_nat [simp]: "ord (n :: nat) 0 = (if n = 1 then 1 else 0)"
proof -
have "(LEAST k. k > 0) = (1 :: nat)"
by (rule Least_equality) auto
thus ?thesis by (auto simp: ord_def)
qed

lemma ord_divides': "[a ^ d = Suc 0] (mod n) = (ord n a dvd d)"
using ord_divides[of a d n] by simp

lemma ord_Suc_0 [simp]: "ord (Suc 0) n = 1"
using ord_1[where 'a = nat] by (simp del: ord_1)

lemma ord_mod [simp]: "ord n (k mod n) = ord n k"
by (cases "n = 0") (auto simp add: ord_def cong_def power_mod)

lemma ord_gt_0_iff [simp]: "ord (n::nat) x > 0 ⟷ coprime n x"
using ord_eq_0[of n x] by auto

lemma ord_eq_Suc_0_iff: "ord n (x::nat) = Suc 0 ⟷ [x = 1] (mod n)"
using ord_divides[of x 1 n] by (auto simp: ord_divides')

lemma ord_cong:
assumes "[k1 = k2] (mod n)"
shows   "ord n k1 = ord n k2"
proof -
have "ord n (k1 mod n) = ord n (k2 mod n)"
by (simp only: assms[unfolded cong_def])
thus ?thesis by simp
qed

lemma ord_nat_code [code_unfold]:
"ord n a =
(if n = 0 then if a = 1 then 1 else 0 else
if coprime n a then Min (Set.filter (λk. [a ^ k = 1] (mod n)) {0<..n}) else 0)"
proof (cases "coprime n a ∧ n > 0")
case True
define A where "A = {k∈{0<..n}. [a ^ k = 1] (mod n)}"
define k where "k = (LEAST k. k > 0 ∧ [a ^ k = 1] (mod n))"
have totient: "totient n ∈ A"
using euler_theorem[of a n] True
by (auto simp: A_def coprime_commute intro!: Nat.gr0I totient_le)
moreover have "finite A" by (auto simp: A_def)
ultimately have *: "Min A ∈ A" and "∀y. y ∈ A ⟶ Min A ≤ y"
by (auto intro: Min_in)

have "k > 0 ∧ [a ^ k = 1] (mod n)"
unfolding k_def by (rule LeastI[of _ "totient n"]) (use totient in ‹auto simp: A_def›)
moreover have "k ≤ totient n"
unfolding k_def by (intro Least_le) (use totient in ‹auto simp: A_def›)
ultimately have "k ∈ A" using totient_le[of n] by (auto simp: A_def)
hence "Min A ≤ k" by (intro Min_le) (auto simp: ‹finite A›)
moreover from * have "k ≤ Min A"
unfolding k_def by (intro Least_le) (auto simp: A_def)
ultimately show ?thesis using True by (simp add: ord_def k_def A_def Set.filter_def)
qed auto

theorem ord_modulus_mult_coprime:
fixes x :: nat
assumes "coprime m n"
shows   "ord (m * n) x = lcm (ord m x) (ord n x)"
proof (intro dvd_antisym)
have "[x ^ lcm (ord m x) (ord n x) = 1] (mod (m * n))"
using assms by (intro coprime_cong_mult_nat assms) (auto simp: ord_divides')
thus "ord (m * n) x dvd lcm (ord m x) (ord n x)"
by (simp add: ord_divides')
next
show "lcm (ord m x) (ord n x) dvd ord (m * n) x"
proof (intro lcm_least)
show "ord m x dvd ord (m * n) x"
using cong_modulus_mult_nat[of "x ^ ord (m * n) x" 1 m n] assms
by (simp add: ord_divides')
show "ord n x dvd ord (m * n) x"
using cong_modulus_mult_nat[of "x ^ ord (m * n) x" 1 n m] assms
by (simp add: ord_divides' mult.commute)
qed
qed

corollary ord_modulus_prod_coprime:
assumes "finite A" "⋀i j. i ∈ A ⟹ j ∈ A ⟹ i ≠ j ⟹ coprime (f i) (f j)"
shows   "ord (∏i∈A. f i :: nat) x = (LCM i∈A. ord (f i) x)"
using assms by (induction A rule: finite_induct)
(simp, simp, subst ord_modulus_mult_coprime, auto intro!: prod_coprime_right)

lemma ord_power_aux:
fixes m x k a :: nat
defines "l ≡ ord m a"
shows   "ord m (a ^ k) * gcd k l = l"
proof (rule dvd_antisym)
have "[a ^ lcm k l = 1] (mod m)"
unfolding ord_divides by (simp add: l_def)
also have "lcm k l = k * (l div gcd k l)"
by (simp add: lcm_nat_def div_mult_swap)
finally have "ord m (a ^ k) dvd l div gcd k l"
unfolding ord_divides [symmetric] by (simp add: power_mult [symmetric])
thus "ord m (a ^ k) * gcd k l dvd l"
by (cases "l = 0") (auto simp: dvd_div_iff_mult)

have "[(a ^ k) ^ ord m (a ^ k) = 1] (mod m)"
by (rule ord)
also have "(a ^ k) ^ ord m (a ^ k) = a ^ (k * ord m (a ^ k))"
by (simp add: power_mult)
finally have "ord m a dvd k * ord m (a ^ k)"
by (simp add: ord_divides')
hence "l dvd gcd (k * ord m (a ^ k)) (l * ord m (a ^ k))"
by (intro gcd_greatest dvd_triv_left) (auto simp: l_def ord_divides')
also have "gcd (k * ord m (a ^ k)) (l * ord m (a ^ k)) = ord m (a ^ k) * gcd k l"
by (subst gcd_mult_distrib_nat) (auto simp: mult_ac)
finally show "l dvd ord m (a ^ k) * gcd k l" .
qed

theorem ord_power: "coprime m a ⟹ ord m (a ^ k :: nat) = ord m a div gcd k (ord m a)"
using ord_power_aux[of m a k] by (metis div_mult_self_is_m gcd_pos_nat ord_eq_0)

lemma inj_power_mod:
assumes "coprime n (a :: nat)"
shows   "inj_on (λk. a ^ k mod n) {..<ord n a}"
proof
fix k l assume *: "k ∈ {..<ord n a}" "l ∈ {..<ord n a}" "a ^ k mod n = a ^ l mod n"
have "k = l" if "k < l" "l < ord n a" "[a ^ k = a ^ l] (mod n)" for k l
proof -
have "l = k + (l - k)" using that by simp
also have "a ^ … = a ^ k * a ^ (l - k)"
also have "[… = a ^ l * a ^ (l - k)] (mod n)"
using that by (intro cong_mult) auto
finally have "[a ^ l * a ^ (l - k) = a ^ l * 1] (mod n)"
by (simp add: cong_sym_eq)
with assms have "[a ^ (l - k) = 1] (mod n)"
by (subst (asm) cong_mult_lcancel_nat) (auto simp: coprime_commute)
hence "ord n a dvd l - k"
by (simp add: ord_divides')
from dvd_imp_le[OF this] and ‹l < ord n a› have "l - k = 0"
by (cases "l - k = 0") auto
with ‹k < l› show "k = l" by simp
qed
from this[of k l] and this[of l k] and * show "k = l"
by (cases k l rule: linorder_cases) (auto simp: cong_def)
qed

lemma ord_eq_2_iff: "ord n (x :: nat) = 2 ⟷ [x ≠ 1] (mod n) ∧ [x⇧2 = 1] (mod n)"
proof
assume x: "[x ≠ 1] (mod n) ∧ [x⇧2 = 1] (mod n)"
hence "coprime n x"
by (metis coprime_commute lucas_coprime_lemma zero_neq_numeral)
with x have "ord n x dvd 2" "ord n x ≠ 1" "ord n x > 0"
by (auto simp: ord_divides' ord_eq_Suc_0_iff)
thus "ord n x = 2" by (auto dest!: dvd_imp_le simp del: ord_gt_0_iff)
qed (use ord_divides[of _ 2] ord_divides[of _ 1] in auto)

lemma square_mod_8_eq_1_iff: "[x⇧2 = 1] (mod 8) ⟷ odd (x :: nat)"
proof -
have "[x⇧2 = 1] (mod 8) ⟷ ((x mod 8)⇧2 mod 8 = 1)"
by (simp add: power_mod cong_def)
also have "… ⟷ x mod 8 ∈ {1, 3, 5, 7}"
proof
assume x: "(x mod 8)⇧2 mod 8 = 1"
have "x mod 8 ∈ {..<8}" by simp
also have "{..<8} = {0, 1, 2, 3, 4, 5, 6, 7::nat}"
by (simp add: lessThan_nat_numeral lessThan_Suc insert_commute)
finally have x_cases: "x mod 8 ∈ {0, 1, 2, 3, 4, 5, 6, 7}" .
from x have "x mod 8 ∉ {0, 2, 4, 6}"
using x by (auto intro: Nat.gr0I)
with x_cases show "x mod 8 ∈ {1, 3, 5, 7}" by simp
qed auto
also have "… ⟷ odd (x mod 8)"
by (auto elim!: oddE)
also have "… ⟷ odd x"
by presburger
finally show ?thesis .
qed

lemma ord_twopow_aux:
assumes "k ≥ 3" and "odd (x :: nat)"
shows   "[x ^ (2 ^ (k - 2)) = 1] (mod (2 ^ k))"
using assms(1)
proof (induction k rule: dec_induct)
case base
from assms have "[x⇧2 = 1] (mod 8)"
by (subst square_mod_8_eq_1_iff) auto
thus ?case by simp
next
case (step k)
define k' where "k' = k - 2"
have k: "k = Suc (Suc k')"
using ‹k ≥ 3› by (simp add: k'_def)
from ‹k ≥ 3› have "2 * k ≥ Suc k" by presburger

from ‹odd x› have "x > 0" by (intro Nat.gr0I) auto
from step.IH have "2 ^ k dvd (x ^ (2 ^ (k - 2)) - 1)"
by (rule cong_to_1_nat)
then obtain t where "x ^ (2 ^ (k - 2)) - 1 = t * 2 ^ k"
by auto
hence "x ^ (2 ^ (k - 2)) = t * 2 ^ k + 1"
by (metis ‹0 < x› add.commute add_diff_inverse_nat less_one neq0_conv power_eq_0_iff)
hence "(x ^ (2 ^ (k - 2))) ^ 2 = (t * 2 ^ k + 1) ^ 2"
by (rule arg_cong)
hence "[(x ^ (2 ^ (k - 2))) ^ 2 = (t * 2 ^ k + 1) ^ 2] (mod (2 ^ Suc k))"
by simp
also have "(x ^ (2 ^ (k - 2))) ^ 2 = x ^ (2 ^ (k - 1))"
by (simp_all add: power_even_eq[symmetric] power_mult k )
also have "(t * 2 ^ k + 1) ^ 2 = t⇧2 * 2 ^ (2 * k) + t * 2 ^ Suc k + 1"
by (subst power2_eq_square)
(auto simp: algebra_simps k power2_eq_square[of t]
also have "[… = 0 + 0 + 1] (mod 2 ^ Suc k)"
using ‹2 * k ≥ Suc k›
(auto simp: cong_0_iff intro: dvd_mult[OF le_imp_power_dvd] simp del: power_Suc)
finally show ?case by simp
qed

lemma ord_twopow_3_5:
assumes "k ≥ 3" "x mod 8 ∈ {3, 5 :: nat}"
shows   "ord (2 ^ k) x = 2 ^ (k - 2)"
using assms(1)
proof (induction k rule: less_induct)
have "x mod 8 = 3 ∨ x mod 8 = 5" using assms by auto
hence "odd x" by presburger
case (less k)
from ‹k ≥ 3› consider "k = 3" | "k = 4" | "k ≥ 5" by force
thus ?case
proof cases
case 1
thus ?thesis using assms
by (auto simp: ord_eq_2_iff cong_def simp flip: power_mod[of x])
next
case 2
from assms have "x mod 8 = 3 ∨ x mod 8 = 5" by auto
then have x': "x mod 16 = 3 ∨ x mod 16 = 5 ∨ x mod 16 = 11 ∨ x mod 16 = 13"
using mod_double_nat [of x 8] by auto
hence "[x ^ 4 = 1] (mod 16)" using assms
by (auto simp: cong_def simp flip: power_mod[of x])
hence "ord 16 x dvd 2⇧2" by (simp add: ord_divides')
then obtain l where l: "ord 16 x = 2 ^ l" "l ≤ 2"
by (subst (asm) divides_primepow_nat) auto

have "[x ^ 2 ≠ 1] (mod 16)"
using x' by (auto simp: cong_def simp flip: power_mod[of x])
hence "¬ord 16 x dvd 2" by (simp add: ord_divides')
with l have "l = 2"
using le_imp_power_dvd[of l 1 2] by (cases "l ≤ 1") auto
with l show ?thesis by (simp add: ‹k = 4›)
next
case 3
define k' where "k' = k - 2"
have k': "k' ≥ 2" and [simp]: "k = Suc (Suc k')"
using 3 by (simp_all add: k'_def)
have IH: "ord (2 ^ k') x = 2 ^ (k' - 2)" "ord (2 ^ Suc k') x = 2 ^ (k' - 1)"
using less.IH[of k'] less.IH[of "Suc k'"] 3 by simp_all
from IH have cong: "[x ^ (2 ^ (k' - 2)) = 1] (mod (2 ^ k'))"
by (simp_all add: ord_divides')
have notcong: "[x ^ (2 ^ (k' - 2)) ≠ 1] (mod (2 ^ Suc k'))"
proof
assume "[x ^ (2 ^ (k' - 2)) = 1] (mod (2 ^ Suc k'))"
hence "ord (2 ^ Suc k') x dvd 2 ^ (k' - 2)"
by (simp add: ord_divides')
also have "ord (2 ^ Suc k') x = 2 ^ (k' - 1)"
using IH by simp
finally have "k' - 1 ≤ k' - 2"
by (rule power_dvd_imp_le) auto
with ‹k' ≥ 2› show False by simp
qed

have "2 ^ k' + 1 < 2 ^ k' + (2 ^ k' :: nat)"
using one_less_power[of "2::nat" k'] k' by (intro add_strict_left_mono) auto
with cong notcong have cong': "x ^ (2 ^ (k' - 2)) mod 2 ^ Suc k' = 1 + 2 ^ k'"
using mod_double_nat [of ‹x ^ 2 ^ (k' - 2)› ‹2 ^ k'›] k' by (auto simp: cong_def)

hence "x ^ (2 ^ (k' - 2)) mod 2 ^ k = 1 + 2 ^ k' ∨
x ^ (2 ^ (k' - 2)) mod 2 ^ k = 1 + 2 ^ k' + 2 ^ Suc k'"
using mod_double_nat [of ‹x ^ 2 ^ (k' - 2)› ‹2 ^ Suc k'›] by auto
hence eq: "[x ^ 2 ^ (k' - 1) = 1 + 2 ^ (k - 1)] (mod 2 ^ k)"
proof
assume *: "x ^ (2 ^ (k' - 2)) mod (2 ^ k) = 1 + 2 ^ k'"
have "[x ^ (2 ^ (k' - 2)) = x ^ (2 ^ (k' - 2)) mod 2 ^ k] (mod 2 ^ k)"
by simp
also have "[x ^ (2 ^ (k' - 2)) mod (2 ^ k) = 1 + 2 ^ k'] (mod 2 ^ k)"
by (subst *) auto
finally have "[(x ^ 2 ^ (k' - 2)) ^ 2 = (1 + 2 ^ k') ^ 2] (mod 2 ^ k)"
by (rule cong_pow)
hence "[x ^ 2 ^ Suc (k' - 2) = (1 + 2 ^ k') ^ 2] (mod 2 ^ k)"
by (simp add: power_mult [symmetric] power_Suc2 [symmetric] del: power_Suc)
also have "Suc (k' - 2) = k' - 1"
using k' by simp
also have "(1 + 2 ^ k' :: nat)⇧2 = 1 + 2 ^ (k - 1) + 2 ^ (2 * k')"
by (subst power2_eq_square) (simp add: algebra_simps flip: power_add)
also have "(2 ^ k :: nat) dvd 2 ^ (2 * k')"
using k' by (intro le_imp_power_dvd) auto
hence "[1 + 2 ^ (k - 1) + 2 ^ (2 * k') = 1 + 2 ^ (k - 1) + (0 :: nat)] (mod 2 ^ k)"
by (intro cong_add) (auto simp: cong_0_iff)
finally show "[x ^ 2 ^ (k' - 1) = 1 + 2 ^ (k - 1)] (mod 2 ^ k)"
by simp
next
assume *: "x ^ (2 ^ (k' - 2)) mod 2 ^ k = 1 + 2 ^ k' + 2 ^ Suc k'"
have "[x ^ (2 ^ (k' - 2)) = x ^ (2 ^ (k' - 2)) mod 2 ^ k] (mod 2 ^ k)"
by simp
also have "[x ^ (2 ^ (k' - 2)) mod (2 ^ k) = 1 + 3 * 2 ^ k'] (mod 2 ^ k)"
by (subst *) auto
finally have "[(x ^ 2 ^ (k' - 2)) ^ 2 = (1 + 3 * 2 ^ k') ^ 2] (mod 2 ^ k)"
by (rule cong_pow)
hence "[x ^ 2 ^ Suc (k' - 2) = (1 + 3 * 2 ^ k') ^ 2] (mod 2 ^ k)"
by (simp add: power_mult [symmetric] power_Suc2 [symmetric] del: power_Suc)
also have "Suc (k' - 2) = k' - 1"
using k' by simp
also have "(1 + 3 * 2 ^ k' :: nat)⇧2 = 1 + 2 ^ (k - 1) + 2 ^ k + 9 * 2 ^ (2 * k')"
by (subst power2_eq_square) (simp add: algebra_simps flip: power_add)
also have "(2 ^ k :: nat) dvd 9 * 2 ^ (2 * k')"
using k' by (intro dvd_mult le_imp_power_dvd) auto
hence "[1 + 2 ^ (k - 1) + 2 ^ k + 9 * 2 ^ (2 * k') = 1 + 2 ^ (k - 1) + 0 + (0 :: nat)]
(mod 2 ^ k)"
by (intro cong_add) (auto simp: cong_0_iff)
finally show "[x ^ 2 ^ (k' - 1) = 1 + 2 ^ (k - 1)] (mod 2 ^ k)"
by simp
qed

have notcong': "[x ^ 2 ^ (k - 3) ≠ 1] (mod 2 ^ k)"
proof
assume "[x ^ 2 ^ (k - 3) = 1] (mod 2 ^ k)"
hence "[x ^ 2 ^ (k' - 1) - x ^ 2 ^ (k' - 1) = 1 + 2 ^ (k - 1) - 1] (mod 2 ^ k)"
by (intro cong_diff_nat eq) auto
hence "[2 ^ (k - 1) = (0 :: nat)] (mod 2 ^ k)"
by (simp add: cong_sym_eq)
hence "2 ^ k dvd 2 ^ (k - 1)"
by (simp add: cong_0_iff)
hence "k ≤ k - 1"
by (rule power_dvd_imp_le) auto
thus False by simp
qed

have "[x ^ 2 ^ (k - 2) = 1] (mod 2 ^ k)"
using ord_twopow_aux[of k x] ‹odd x› ‹k ≥ 3› by simp
hence "ord (2 ^ k) x dvd 2 ^ (k - 2)"
by (simp add: ord_divides')
then obtain l where l: "l ≤ k - 2" "ord (2 ^ k) x = 2 ^ l"
using divides_primepow_nat[of 2 "ord (2 ^ k) x" "k - 2"] by auto

from notcong' have "¬ord (2 ^ k) x dvd 2 ^ (k - 3)"
by (simp add: ord_divides')
with l have "l = k - 2"
using le_imp_power_dvd[of l "k - 3" 2] by (cases "l ≤ k - 3") auto
with l show ?thesis by simp
qed
qed

lemma ord_4_3 [simp]: "ord 4 (3::nat) = 2"
proof -
have "[3 ^ 2 = (1 :: nat)] (mod 4)"
by (simp add: cong_def)
hence "ord 4 (3::nat) dvd 2"
by (subst (asm) ord_divides) auto
hence "ord 4 (3::nat) ≤ 2"
by (intro dvd_imp_le) auto
moreover have "ord 4 (3::nat) ≠ 1"
by (auto simp: ord_eq_Suc_0_iff cong_def)
moreover have "ord 4 (3::nat) ≠ 0"
by (auto simp: gcd_non_0_nat coprime_iff_gcd_eq_1)
ultimately show "ord 4 (3 :: nat) = 2"
by linarith
qed

lemma elements_with_ord_1: "n > 0 ⟹ {x∈totatives n. ord n x = Suc 0} = {1}"
by (auto simp: ord_eq_Suc_0_iff cong_def totatives_less)

lemma residue_prime_has_primroot:
fixes p :: nat
assumes "prime p"
shows "∃a∈totatives p. ord p a = p - 1"
proof -
from residue_prime_mult_group_has_gen[OF assms]
obtain a where a: "a ∈ {1..p-1}" "{1..p-1} = {a ^ i mod p |i. i ∈ UNIV}" by blast
from a have "coprime p a"
using a assms by (intro prime_imp_coprime) (auto dest: dvd_imp_le)
with a(1) have "a ∈ totatives p" by (auto simp: totatives_def coprime_commute)

have "p - 1 = card {1..p-1}" by simp
also have "{1..p-1} = {a ^ i mod p |i. i ∈ UNIV}" by fact
also have "{a ^ i mod p |i. i ∈ UNIV} = (λi. a ^ i mod p) ` {..<ord p a}"
proof (intro equalityI subsetI)
fix x assume "x ∈ {a ^ i mod p |i. i ∈ UNIV}"
then obtain i where [simp]: "x = a ^ i mod p" by auto

have "[a ^ i = a ^ (i mod ord p a)] (mod p)"
using ‹coprime p a› by (subst order_divides_expdiff) auto
hence "∃j. a ^ i mod p = a ^ j mod p ∧ j < ord p a"
using ‹coprime p a› by (intro exI[of _ "i mod ord p a"]) (auto simp: cong_def)
thus "x ∈ (λi. a ^ i mod p) ` {..<ord p a}"
by auto
qed auto
also have "card … = ord p a"
using inj_power_mod[OF ‹coprime p a›] by (subst card_image) auto
finally show ?thesis using ‹a ∈ totatives p›
by auto
qed

subsection ‹Another trivial primality characterization›

lemma prime_prime_factor: "prime n ⟷ n ≠ 1 ∧ (∀p. prime p ∧ p dvd n ⟶ p = n)"
(is "?lhs ⟷ ?rhs")
for n :: nat
proof (cases "n = 0 ∨ n = 1")
case True
then show ?thesis
by (metis bigger_prime dvd_0_right not_prime_1 not_prime_0)
next
case False
show ?thesis
proof
assume "prime n"
then show ?rhs
by (metis not_prime_1 prime_nat_iff)
next
assume ?rhs
with False show "prime n"
by (auto simp: prime_nat_iff) (metis One_nat_def prime_factor_nat prime_nat_iff)
qed
qed

lemma prime_divisor_sqrt: "prime n ⟷ n ≠ 1 ∧ (∀d. d dvd n ∧ d⇧2 ≤ n ⟶ d = 1)"
for n :: nat
proof -
consider "n = 0" | "n = 1" | "n ≠ 0" "n ≠ 1" by blast
then show ?thesis
proof cases
case 1
then show ?thesis by simp
next
case 2
then show ?thesis by simp
next
case n: 3
then have np: "n > 1" by arith
{
fix d
assume d: "d dvd n" "d⇧2 ≤ n"
and H: "∀m. m dvd n ⟶ m = 1 ∨ m = n"
from H d have d1n: "d = 1 ∨ d = n" by blast
then have "d = 1"
proof
assume dn: "d = n"
from n have "n⇧2 > n * 1"
by (simp add: power2_eq_square)
with dn d(2) show ?thesis by simp
qed
}
moreover
{
fix d assume d: "d dvd n" and H: "∀d'. d' dvd n ∧ d'⇧2 ≤ n ⟶ d' = 1"
from d n have "d ≠ 0"
by (metis dvd_0_left_iff)
then have dp: "d > 0" by simp
from d[unfolded dvd_def] obtain e where e: "n= d*e" by blast
from n dp e have ep:"e > 0" by simp
from dp ep have "d⇧2 ≤ n ∨ e⇧2 ≤ n"
by (auto simp add: e power2_eq_square mult_le_cancel_left)
then have "d = 1 ∨ d = n"
proof
assume "d⇧2 ≤ n"
with H[rule_format, of d] d have "d = 1" by blast
then show ?thesis ..
next
assume h: "e⇧2 ≤ n"
from e have "e dvd n" by (simp add: dvd_def mult.commute)
with H[rule_format, of e] h have "e = 1" by simp
with e have "d = n" by simp
then show ?thesis ..
qed
}
ultimately show ?thesis
unfolding prime_nat_iff using np n(2) by blast
qed
qed

lemma prime_prime_factor_sqrt:
"prime (n::nat) ⟷ n ≠ 0 ∧ n ≠ 1 ∧ (∄p. prime p ∧ p dvd n ∧ p⇧2 ≤ n)"
(is "?lhs ⟷?rhs")
proof -
consider "n = 0" | "n = 1" | "n ≠ 0" "n ≠ 1"
by blast
then show ?thesis
proof cases
case 1
then show ?thesis by (metis not_prime_0)
next
case 2
then show ?thesis by (metis not_prime_1)
next
case n: 3
show ?thesis
proof
assume ?lhs
from this[unfolded prime_divisor_sqrt] n show ?rhs
by (metis prime_prime_factor)
next
assume ?rhs
{
fix d
assume d: "d dvd n" "d⇧2 ≤ n" "d ≠ 1"
then obtain p where p: "prime p" "p dvd d"
by (metis prime_factor_nat)
from d(1) n have dp: "d > 0"
by (metis dvd_0_left neq0_conv)
from mult_mono[OF dvd_imp_le[OF p(2) dp] dvd_imp_le[OF p(2) dp]] d(2)
have "p⇧2 ≤ n" unfolding power2_eq_square by arith
with ‹?rhs› n p(1) dvd_trans[OF p(2) d(1)] have False
by blast
}
with n prime_divisor_sqrt show ?lhs by auto
qed
qed
qed

subsection ‹Pocklington theorem›

lemma pocklington_lemma:
fixes p :: nat
assumes n: "n ≥ 2" and nqr: "n - 1 = q * r"
and an: "[a^ (n - 1) = 1] (mod n)"
and aq: "∀p. prime p ∧ p dvd q ⟶ coprime (a ^ ((n - 1) div p) - 1) n"
and pp: "prime p" and pn: "p dvd n"
shows "[p = 1] (mod q)"
proof -
have p01: "p ≠ 0" "p ≠ 1"
using pp by (auto intro: prime_gt_0_nat)
obtain k where k: "a ^ (q * r) - 1 = n * k"
by (metis an cong_to_1_nat dvd_def nqr)
from pn[unfolded dvd_def] obtain l where l: "n = p * l"
by blast
have a0: "a ≠ 0"
proof
assume "a = 0"
with n have "a^ (n - 1) = 0"
by (simp add: power_0_left)
with n an mod_less[of 1 n] show False
by (simp add: power_0_left cong_def)
qed
with n nqr have aqr0: "a ^ (q * r) ≠ 0"
by simp
then have "(a ^ (q * r) - 1) + 1  = a ^ (q * r)"
by simp
with k l have "a ^ (q * r) = p * l * k + 1"
by simp
then have "a ^ (r * q) + p * 0 = 1 + p * (l * k)"
by (simp add: ac_simps)
then have odq: "ord p (a^r) dvd q"
unfolding ord_divides[symmetric] power_mult[symmetric]
by (metis an cong_dvd_modulus_nat mult.commute nqr pn)
from odq[unfolded dvd_def] obtain d where d: "q = ord p (a^r) * d"
by blast
have d1: "d = 1"
proof (rule ccontr)
assume d1: "d ≠ 1"
obtain P where P: "prime P" "P dvd d"
by (metis d1 prime_factor_nat)
from d dvd_mult[OF P(2), of "ord p (a^r)"] have Pq: "P dvd q" by simp
from aq P(1) Pq have caP:"coprime (a^ ((n - 1) div P) - 1) n" by blast
from Pq obtain s where s: "q = P*s" unfolding dvd_def by blast
from P(1) have P0: "P ≠ 0"
by (metis not_prime_0)
from P(2) obtain t where t: "d = P*t" unfolding dvd_def by blast
from d s t P0  have s': "ord p (a^r) * t = s"
by (metis mult.commute mult_cancel1 mult.assoc)
have "ord p (a^r) * t*r = r * ord p (a^r) * t"
by (metis mult.assoc mult.commute)
then have exps: "a^(ord p (a^r) * t*r) = ((a ^ r) ^ ord p (a^r)) ^ t"
by (simp only: power_mult)
then have "[((a ^ r) ^ ord p (a^r)) ^ t= 1] (mod p)"
by (metis cong_pow ord power_one)
then have pd0: "p dvd a^(ord p (a^r) * t*r) - 1"
by (metis cong_to_1_nat exps)
from nqr s s' have "(n - 1) div P = ord p (a^r) * t*r"
using P0 by simp
with caP have "coprime (a ^ (ord p (a ^ r) * t * r) - 1) n"
by simp
with p01 pn pd0 coprime_common_divisor [of _ n p] show False
by auto
qed
with d have o: "ord p (a^r) = q" by simp
from pp totient_prime [of p] have totient_eq: "totient p = p - 1"
by simp
{
fix d
assume d: "d dvd p" "d dvd a" "d ≠ 1"
from pp[unfolded prime_nat_iff] d have dp: "d = p" by blast
from n have "n ≠ 0" by simp
then have False using d dp pn an
by auto (metis One_nat_def Suc_lessI
‹1 < p ∧ (∀m. m dvd p ⟶ m = 1 ∨ m = p)› ‹a ^ (q * r) = p * l * k + 1› add_diff_cancel_left' dvd_diff_nat dvd_power dvd_triv_left gcd_nat.trans nat_dvd_not_less nqr zero_less_diff zero_less_one)
}
then have cpa: "coprime p a"
by (auto intro: coprimeI)
then have arp: "coprime (a ^ r) p"
by (cases "r > 0") (simp_all add: ac_simps)
from euler_theorem [OF arp, simplified ord_divides] o totient_eq have "q dvd (p - 1)"
by simp
then obtain d where d:"p - 1 = q * d"
unfolding dvd_def by blast
have "p ≠ 0"
by (metis p01(1))
with d have "p + q * 0 = 1 + q * d" by simp
then show ?thesis
by (metis cong_iff_lin_nat mult.commute)
qed

theorem pocklington:
assumes n: "n ≥ 2" and nqr: "n - 1 = q * r" and sqr: "n ≤ q⇧2"
and an: "[a^ (n - 1) = 1] (mod n)"
and aq: "∀p. prime p ∧ p dvd q ⟶ coprime (a^ ((n - 1) div p) - 1) n"
shows "prime n"
unfolding prime_prime_factor_sqrt[of n]
proof -
let ?ths = "n ≠ 0 ∧ n ≠ 1 ∧ (∄p. prime p ∧ p dvd n ∧ p⇧2 ≤ n)"
from n have n01: "n ≠ 0" "n ≠ 1" by arith+
{
fix p
assume p: "prime p" "p dvd n" "p⇧2 ≤ n"
from p(3) sqr have "p^(Suc 1) ≤ q^(Suc 1)"
by (simp add: power2_eq_square)
then have pq: "p ≤ q"
by (metis le0 power_le_imp_le_base)
from pocklington_lemma[OF n nqr an aq p(1,2)] have *: "q dvd p - 1"
by (metis cong_to_1_nat)
have "p - 1 ≠ 0"
using prime_ge_2_nat [OF p(1)] by arith
with pq * have False
by (simp add: nat_dvd_not_less)
}
with n01 show ?ths by blast
qed

text ‹Variant for application, to separate the exponentiation.›
lemma pocklington_alt:
assumes n: "n ≥ 2" and nqr: "n - 1 = q * r" and sqr: "n ≤ q⇧2"
and an: "[a^ (n - 1) = 1] (mod n)"
and aq: "∀p. prime p ∧ p dvd q ⟶ (∃b. [a^((n - 1) div p) = b] (mod n) ∧ coprime (b - 1) n)"
shows "prime n"
proof -
{
fix p
assume p: "prime p" "p dvd q"
from aq[rule_format] p obtain b where b: "[a^((n - 1) div p) = b] (mod n)" "coprime (b - 1) n"
by blast
have a0: "a ≠ 0"
proof
assume a0: "a = 0"
from n an have "[0 = 1] (mod n)"
unfolding a0 power_0_left by auto
then show False
using n by (simp add: cong_def dvd_eq_mod_eq_0[symmetric])
qed
then have a1: "a ≥ 1" by arith
from one_le_power[OF a1] have ath: "1 ≤ a ^ ((n - 1) div p)" .
have b0: "b ≠ 0"
proof
assume b0: "b = 0"
from p(2) nqr have "(n - 1) mod p = 0"
by (metis mod_0 mod_mod_cancel mod_mult_self1_is_0)
with div_mult_mod_eq[of "n - 1" p]
have "(n - 1) div p * p= n - 1" by auto
then have eq: "(a^((n - 1) div p))^p = a^(n - 1)"
by (simp only: power_mult[symmetric])
have "p - 1 ≠ 0"
using prime_ge_2_nat [OF p(1)] by arith
then have pS: "Suc (p - 1) = p" by arith
from b have d: "n dvd a^((n - 1) div p)"
unfolding b0 by auto
from divides_rexp[OF d, of "p - 1"] pS eq cong_dvd_iff [OF an] n show False
by simp
qed
then have b1: "b ≥ 1" by arith
from cong_imp_coprime[OF Cong.cong_diff_nat[OF cong_sym [OF b(1)] cong_refl [of 1] b1]]
ath b1 b nqr
have "coprime (a ^ ((n - 1) div p) - 1) n"
by simp
}
then have "∀p. prime p ∧ p dvd q ⟶ coprime (a ^ ((n - 1) div p) - 1) n "
by blast
then show ?thesis by (rule pocklington[OF n nqr sqr an])
qed

subsection ‹Prime factorizations›

(* FIXME some overlap with material in UniqueFactorization, class unique_factorization *)

definition "primefact ps n ⟷ foldr (*) ps 1 = n ∧ (∀p∈ set ps. prime p)"

lemma primefact:
fixes n :: nat
assumes n: "n ≠ 0"
shows "∃ps. primefact ps n"
proof -
obtain xs where xs: "mset xs = prime_factorization n"
using ex_mset [of "prime_factorization n"] by blast
from assms have "n = prod_mset (prime_factorization n)"
by (simp add: prod_mset_prime_factorization)
also have "… = prod_mset (mset xs)" by (simp add: xs)
also have "… = foldr (*) xs 1" by (induct xs) simp_all
finally have "foldr (*) xs 1 = n" ..
moreover from xs have "∀p∈#mset xs. prime p" by auto
ultimately have "primefact xs n" by (auto simp: primefact_def)
then show ?thesis ..
qed

lemma primefact_contains:
fixes p :: nat
assumes pf: "primefact ps n"
and p: "prime p"
and pn: "p dvd n"
shows "p ∈ set ps"
using pf p pn
proof (induct ps arbitrary: p n)
case Nil
then show ?case by (auto simp: primefact_def)
next
case (Cons q qs)
from Cons.prems[unfolded primefact_def]
have q: "prime q" "q * foldr (*) qs 1 = n" "∀p ∈set qs. prime p"
and p: "prime p" "p dvd q * foldr (*) qs 1"
by simp_all
consider "p dvd q" | "p dvd foldr (*) qs 1"
by (metis p prime_dvd_mult_eq_nat)
then show ?case
proof cases
case 1
with p(1) q(1) ```