--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Number_Theory/Pocklington.thy Tue Feb 04 21:28:38 2014 +0000
@@ -0,0 +1,815 @@
+(* Title: HOL/Number_Theory/Pocklington.thy
+ Author: Amine Chaieb
+*)
+
+header {* Pocklington's Theorem for Primes *}
+
+theory Pocklington
+imports Residues
+begin
+
+subsection{*Lemmas about previously defined terms*}
+
+lemma prime:
+ "prime p \<longleftrightarrow> p \<noteq> 0 \<and> p\<noteq>1 \<and> (\<forall>m. 0 < m \<and> m < p \<longrightarrow> coprime p m)"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof-
+ {assume "p=0 \<or> p=1" hence ?thesis
+ by (metis one_not_prime_nat zero_not_prime_nat)}
+ moreover
+ {assume p0: "p\<noteq>0" "p\<noteq>1"
+ {assume H: "?lhs"
+ {fix m assume m: "m > 0" "m < p"
+ {assume "m=1" hence "coprime p m" by simp}
+ moreover
+ {assume "p dvd m" hence "p \<le> m" using dvd_imp_le m by blast with m(2)
+ have "coprime p m" by simp}
+ ultimately have "coprime p m"
+ by (metis H prime_imp_coprime_nat)}
+ hence ?rhs using p0 by auto}
+ moreover
+ { assume H: "\<forall>m. 0 < m \<and> m < p \<longrightarrow> coprime p m"
+ obtain q where q: "prime q" "q dvd p"
+ by (metis p0(2) prime_factor_nat)
+ have q0: "q > 0"
+ by (metis prime_gt_0_nat q(1))
+ from dvd_imp_le[OF q(2)] p0 have qp: "q \<le> p" by arith
+ {assume "q = p" hence ?lhs using q(1) by blast}
+ moreover
+ {assume "q\<noteq>p" with qp have qplt: "q < p" by arith
+ from H[rule_format, of q] qplt q0 have "coprime p q" by arith
+ hence ?lhs
+ by (metis gcd_semilattice_nat.inf_absorb2 one_not_prime_nat q(1) q(2))}
+ ultimately have ?lhs by blast}
+ ultimately have ?thesis by blast}
+ ultimately show ?thesis by (cases"p=0 \<or> p=1", auto)
+qed
+
+lemma finite_number_segment: "card { m. 0 < m \<and> m < n } = n - 1"
+proof-
+ have "{ m. 0 < m \<and> m < n } = {1..<n}" by auto
+ thus ?thesis by simp
+qed
+
+
+subsection{*Some basic theorems about solving congruences*}
+
+lemma cong_solve:
+ fixes n::nat assumes an: "coprime a n" shows "\<exists>x. [a * x = b] (mod n)"
+proof-
+ {assume "a=0" hence ?thesis using an by (simp add: cong_nat_def)}
+ moreover
+ {assume az: "a\<noteq>0"
+ from bezout_add_strong_nat[OF az, of n]
+ 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"
+ by (metis assms coprime_nat)
+ hence "a*x*b = (n*y + 1)*b" using dxy(3) by simp
+ hence "a*(x*b) = n*(y*b) + b"
+ by (auto simp add: algebra_simps)
+ hence "a*(x*b) mod n = (n*(y*b) + b) mod n" by simp
+ hence "a*(x*b) mod n = b mod n" by (simp add: mod_add_left_eq)
+ hence "[a*(x*b) = b] (mod n)" unfolding cong_nat_def .
+ hence ?thesis by blast}
+ultimately show ?thesis by blast
+qed
+
+lemma cong_solve_unique:
+ fixes n::nat assumes an: "coprime a n" and nz: "n \<noteq> 0"
+ shows "\<exists>!x. x < n \<and> [a * x = b] (mod n)"
+proof-
+ let ?P = "\<lambda>x. x < n \<and> [a * x = b] (mod n)"
+ from cong_solve[OF an] obtain x where x: "[a*x = b] (mod n)" by blast
+ let ?x = "x mod n"
+ from x have th: "[a * ?x = b] (mod n)"
+ by (simp add: cong_nat_def mod_mult_right_eq[of a x n])
+ from mod_less_divisor[ of n x] nz th have Px: "?P ?x" by simp
+ {fix y assume Py: "y < n" "[a * y = b] (mod n)"
+ from Py(2) th have "[a * y = a*?x] (mod n)" by (simp add: cong_nat_def)
+ hence "[y = ?x] (mod n)"
+ by (metis an cong_mult_lcancel_nat)
+ with mod_less[OF Py(1)] mod_less_divisor[ of n x] nz
+ have "y = ?x" by (simp add: cong_nat_def)}
+ with Px show ?thesis by blast
+qed
+
+lemma cong_solve_unique_nontrivial:
+ assumes p: "prime p" and pa: "coprime p a" and x0: "0 < x" and xp: "x < p"
+ shows "\<exists>!y. 0 < y \<and> y < p \<and> [x * y = a] (mod p)"
+proof-
+ from pa have ap: "coprime a p"
+ by (metis gcd_nat.commute)
+ have px:"coprime x p"
+ by (metis gcd_nat.commute p prime x0 xp)
+ obtain y where y: "y < p" "[x * y = a] (mod p)" "\<forall>z. z < p \<and> [x * z = a] (mod p) \<longrightarrow> z = y"
+ by (metis cong_solve_unique neq0_conv p prime_gt_0_nat px)
+ {assume y0: "y = 0"
+ with y(2) have th: "p dvd a"
+ by (metis cong_dvd_eq_nat gcd_lcm_complete_lattice_nat.top_greatest mult_0_right)
+ have False
+ by (metis gcd_nat.absorb1 one_not_prime_nat p pa th)}
+ with y show ?thesis unfolding Ex1_def using neq0_conv by blast
+qed
+
+lemma cong_unique_inverse_prime:
+ fixes p::nat
+ assumes p: "prime p" and x0: "0 < x" and xp: "x < p"
+ shows "\<exists>!y. 0 < y \<and> y < p \<and> [x * y = 1] (mod p)"
+by (metis cong_solve_unique_nontrivial gcd_lcm_complete_lattice_nat.inf_bot_left gcd_nat.commute assms)
+
+lemma chinese_remainder_coprime_unique:
+ fixes a::nat
+ assumes ab: "coprime a b" and az: "a \<noteq> 0" and bz: "b \<noteq> 0"
+ and ma: "coprime m a" and nb: "coprime n b"
+ shows "\<exists>!x. coprime x (a * b) \<and> x < a * b \<and> [x = m] (mod a) \<and> [x = n] (mod b)"
+proof-
+ let ?P = "\<lambda>x. x < a * b \<and> [x = m] (mod a) \<and> [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)"
+ "\<forall>y. ?P y \<longrightarrow> y = x" by blast
+ from ma nb x
+ have "coprime x a" "coprime x b"
+ by (metis cong_gcd_eq_nat)+
+ then have "coprime x (a*b)"
+ by (metis coprime_mul_eq_nat)
+ with x show ?thesis by blast
+qed
+
+
+subsection{*Lucas's theorem*}
+
+lemma phi_limit_strong: "phi(n) \<le> n - 1"
+proof -
+ have "phi n = card {x. 0 < x \<and> x < int n \<and> coprime x (int n)}"
+ by (simp add: phi_def)
+ also have "... \<le> card {0 <..< int n}"
+ by (rule card_mono) auto
+ also have "... = card {0 <..< n}"
+ by (simp add: transfer_nat_int_set_functions)
+ also have "... \<le> n - 1"
+ by (metis card_greaterThanLessThan le_refl One_nat_def)
+ finally show ?thesis .
+qed
+
+lemma phi_lowerbound_1: assumes n: "n \<ge> 2"
+ shows "phi n \<ge> 1"
+proof -
+ have "1 \<le> card {0::int <.. 1}"
+ by auto
+ also have "... \<le> card {x. 0 < x \<and> x < n \<and> coprime x n}"
+ apply (rule card_mono) using assms
+ apply (auto simp add: )
+ by (metis dual_order.antisym gcd_1_int gcd_int.commute int_one_le_iff_zero_less)
+ also have "... = phi n"
+ by (simp add: phi_def)
+ finally show ?thesis .
+qed
+
+lemma phi_lowerbound_1_nat: assumes n: "n \<ge> 2"
+ shows "phi(int n) \<ge> 1"
+by (metis n nat_le_iff nat_numeral phi_lowerbound_1)
+
+lemma euler_theorem_nat:
+ fixes m::nat
+ assumes "coprime a m"
+ shows "[a ^ phi m = 1] (mod m)"
+by (metis assms le0 euler_theorem [transferred])
+
+lemma lucas_coprime_lemma:
+ fixes n::nat
+ assumes m: "m\<noteq>0" and am: "[a^m = 1] (mod n)"
+ shows "coprime a n"
+proof-
+ {assume "n=1" hence ?thesis by simp}
+ moreover
+ {assume "n = 0" hence ?thesis using am m
+ by (metis am cong_0_nat gcd_nat.right_neutral power_eq_one_eq_nat)}
+ moreover
+ {assume n: "n\<noteq>0" "n\<noteq>1"
+ from m obtain m' where m': "m = Suc m'" by (cases m, blast+)
+ {fix d
+ assume d: "d dvd a" "d dvd n"
+ from n have n1: "1 < n" by arith
+ from am mod_less[OF n1] have am1: "a^m mod n = 1" unfolding cong_nat_def by simp
+ 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
+ have "d = 1" by simp }
+ hence ?thesis by auto
+ }
+ ultimately show ?thesis by blast
+qed
+
+lemma lucas_weak:
+ fixes n::nat
+ assumes n: "n \<ge> 2" and an:"[a^(n - 1) = 1] (mod n)"
+ and nm: "\<forall>m. 0 <m \<and> m < n - 1 \<longrightarrow> \<not> [a^m = 1] (mod n)"
+ shows "prime n"
+proof-
+ from n have n1: "n \<noteq> 1" "n\<noteq>0" "n - 1 \<noteq> 0" "n - 1 > 0" "n - 1 < n" by arith+
+ from lucas_coprime_lemma[OF n1(3) an] have can: "coprime a n" .
+ from euler_theorem_nat[OF can] have afn: "[a ^ phi n = 1] (mod n)"
+ by auto
+ {assume "phi n \<noteq> n - 1"
+ with phi_limit_strong phi_lowerbound_1_nat [OF n]
+ have c:"phi n > 0 \<and> phi n < n - 1"
+ by (metis gr0I leD less_linear not_one_le_zero)
+ from nm[rule_format, OF c] afn have False ..}
+ hence "phi n = n - 1" by blast
+ with prime_phi phi_prime n1(1,2) show ?thesis
+ by auto
+qed
+
+lemma nat_exists_least_iff: "(\<exists>(n::nat). P n) \<longleftrightarrow> (\<exists>n. P n \<and> (\<forall>m < n. \<not> P m))"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ assume ?rhs thus ?lhs by blast
+next
+ assume H: ?lhs then obtain n where n: "P n" by blast
+ let ?x = "Least P"
+ {fix m assume m: "m < ?x"
+ from not_less_Least[OF m] have "\<not> P m" .}
+ with LeastI_ex[OF H] show ?rhs by blast
+qed
+
+lemma nat_exists_least_iff': "(\<exists>(n::nat). P n) \<longleftrightarrow> (P (Least P) \<and> (\<forall>m < (Least P). \<not> P m))"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof-
+ {assume ?rhs hence ?lhs by blast}
+ moreover
+ { assume H: ?lhs then obtain n where n: "P n" by blast
+ let ?x = "Least P"
+ {fix m assume m: "m < ?x"
+ from not_less_Least[OF m] have "\<not> P m" .}
+ with LeastI_ex[OF H] have ?rhs by blast}
+ ultimately show ?thesis by blast
+qed
+
+theorem lucas:
+ assumes n2: "n \<ge> 2" and an1: "[a^(n - 1) = 1] (mod n)"
+ and pn: "\<forall>p. prime p \<and> p dvd n - 1 \<longrightarrow> [a^((n - 1) div p) \<noteq> 1] (mod n)"
+ shows "prime n"
+proof-
+ from n2 have n01: "n\<noteq>0" "n\<noteq>1" "n - 1 \<noteq> 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_nat an1
+ have an: "coprime a n" "coprime (a^(n - 1)) n"
+ by (auto simp add: coprime_exp_nat gcd_nat.commute)
+ {assume H0: "\<exists>m. 0 < m \<and> m < n - 1 \<and> [a ^ m = 1] (mod n)" (is "EX m. ?P m")
+ from H0[unfolded nat_exists_least_iff[of ?P]] obtain m where
+ m: "0 < m" "m < n - 1" "[a ^ m = 1] (mod n)" "\<forall>k <m. \<not>?P k" by blast
+ {assume nm1: "(n - 1) mod m > 0"
+ 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 = mod_div_equality[of "(n - 1)" m]
+ have yn: "coprime ?y n"
+ by (metis an(1) coprime_exp_nat gcd_nat.commute)
+ have "?y mod n = (a^m)^((n - 1) div m) mod n"
+ by (simp add: algebra_simps power_mult)
+ also have "\<dots> = (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 "\<dots> = 1" using m(3)[unfolded cong_nat_def onen] onen
+ by (metis power_one)
+ finally have th3: "?y mod n = 1" .
+ have th2: "[?y * a ^ ((n - 1) mod m) = ?y* 1] (mod n)"
+ using an1[unfolded cong_nat_def onen] onen
+ mod_div_equality[of "(n - 1)" m, symmetric]
+ by (simp add:power_add[symmetric] cong_nat_def th3 del: One_nat_def)
+ have th1: "[a ^ ((n - 1) mod m) = 1] (mod n)"
+ by (metis cong_mult_rcancel_nat nat_mult_commute th2 yn)
+ from m(4)[rule_format, OF th0] nm1
+ less_trans[OF mod_less_divisor[OF m(1), of "n - 1"] m(2)] th1
+ have False by blast }
+ hence "(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\<noteq>0" "r\<noteq>1" by - (rule ccontr, simp)+
+ obtain p where p: "prime p" "p dvd r"
+ by (metis prime_factor_nat r01(2))
+ hence th: "prime p \<and> p dvd n - 1" unfolding r by (auto intro: dvd_mult)
+ have "(a ^ ((n - 1) div p)) mod n = (a^(m*r div p)) mod n" using r
+ by (simp add: power_mult)
+ also have "\<dots> = (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 "\<dots> = ((a^m)^(r div p)) mod n" by (simp add: power_mult)
+ also have "\<dots> = ((a^m mod n)^(r div p)) mod n" using power_mod[of "a^m" "n" "r div p" ] ..
+ also have "\<dots> = 1" using m(3) onen by (simp add: cong_nat_def power_Suc_0)
+ finally have "[(a ^ ((n - 1) div p))= 1] (mod n)"
+ using onen by (simp add: cong_nat_def)
+ with pn[rule_format, OF th] have False by blast}
+ hence th: "\<forall>m. 0 < m \<and> m < n - 1 \<longrightarrow> \<not> [a ^ m = 1] (mod n)" by blast
+ from lucas_weak[OF n2 an1 th] show ?thesis .
+qed
+
+
+subsection{*Definition of the order of a number mod n (0 in non-coprime case)*}
+
+definition "ord n a = (if coprime n a then Least (\<lambda>d. d > 0 \<and> [a ^d = 1] (mod n)) else 0)"
+
+(* This has the expected properties. *)
+
+lemma coprime_ord:
+ fixes n::nat
+ assumes "coprime n a"
+ shows "ord n a > 0 \<and> [a ^(ord n a) = 1] (mod n) \<and> (\<forall>m. 0 < m \<and> m < ord n a \<longrightarrow> [a^ m \<noteq> 1] (mod n))"
+proof-
+ let ?P = "\<lambda>d. 0 < d \<and> [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)
+ {assume "n=0 \<or> n=1" with assms have "\<exists>m>0. ?P m"
+ by auto}
+ moreover
+ {assume "n\<noteq>0 \<and> n\<noteq>1" hence n2:"n \<ge> 2" by arith
+ from assms have na': "coprime a n"
+ by (metis gcd_nat.commute)
+ from phi_lowerbound_1_nat[OF n2] euler_theorem_nat [OF na']
+ have ex: "\<exists>m>0. ?P m" by - (rule exI[where x="phi n"], auto) }
+ ultimately have ex: "\<exists>m>0. ?P m" by blast
+ from nat_exists_least_iff'[of ?P] ex assms show ?thesis
+ unfolding o[symmetric] by auto
+qed
+
+(* With the special value 0 for non-coprime case, it's more convenient. *)
+lemma ord_works:
+ fixes n::nat
+ shows "[a ^ (ord n a) = 1] (mod n) \<and> (\<forall>m. 0 < m \<and> m < ord n a \<longrightarrow> ~[a^ m = 1] (mod n))"
+apply (cases "coprime n a")
+using coprime_ord[of n a]
+by (blast, simp add: ord_def cong_nat_def)
+
+lemma ord:
+ fixes n::nat
+ shows "[a^(ord n a) = 1] (mod n)" using ord_works by blast
+
+lemma ord_minimal:
+ fixes n::nat
+ shows "0 < m \<Longrightarrow> m < ord n a \<Longrightarrow> ~[a^m = 1] (mod n)"
+ using ord_works by blast
+
+lemma ord_eq_0:
+ fixes n::nat
+ shows "ord n a = 0 \<longleftrightarrow> ~coprime n a"
+by (cases "coprime n a", simp add: coprime_ord, simp add: ord_def)
+
+lemma divides_rexp:
+ "x dvd y \<Longrightarrow> (x::nat) dvd (y^(Suc n))"
+ by (simp add: dvd_mult2[of x y])
+
+lemma ord_divides:
+ fixes n::nat
+ shows "[a ^ d = 1] (mod n) \<longleftrightarrow> ord n a dvd d" (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ assume rh: ?rhs
+ then obtain k where "d = ord n a * k" unfolding dvd_def by blast
+ hence "[a ^ d = (a ^ (ord n a) mod n)^k] (mod n)"
+ by (simp add : cong_nat_def power_mult power_mod)
+ also have "[(a ^ (ord n a) mod n)^k = 1] (mod n)"
+ using ord[of a n, unfolded cong_nat_def]
+ by (simp add: cong_nat_def power_mod)
+ finally show ?lhs .
+next
+ assume lh: ?lhs
+ { assume H: "\<not> coprime n a"
+ hence o: "ord n a = 0" by (simp add: ord_def)
+ {assume d: "d=0" with o H have ?rhs by (simp add: cong_nat_def)}
+ moreover
+ {assume d0: "d\<noteq>0" then obtain d' where d': "d = Suc d'" by (cases d, auto)
+ from H
+ obtain p where p: "p dvd n" "p dvd a" "p \<noteq> 1" by auto
+ from lh
+ obtain q1 q2 where q12:"a ^ d + n * q1 = 1 + n * q2"
+ by (metis H d0 gcd_nat.commute lucas_coprime_lemma)
+ hence "a ^ d + n * q1 - n * q2 = 1" by simp
+ with dvd_diff_nat [OF dvd_add [OF divides_rexp[OF p(2), of d'] dvd_mult2[OF p(1), of q1]] dvd_mult2[OF p(1), of q2]] d'
+ have "p dvd 1" by simp
+ with p(3) have False by simp
+ hence ?rhs ..}
+ ultimately have ?rhs by blast}
+ moreover
+ {assume H: "coprime n a"
+ 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)"
+ by (metis cong_exp_nat ord power_one)
+ from H have onz: "?o \<noteq> 0" by (simp add: ord_eq_0)
+ hence op: "?o > 0" by simp
+ from mod_div_equality[of d "ord n a"] lh
+ have "[a^(?o*?q + ?r) = 1] (mod n)" by (simp add: cong_nat_def mult_commute)
+ hence "[(a^?o)^?q * (a^?r) = 1] (mod n)"
+ by (simp add: cong_nat_def power_mult[symmetric] power_add[symmetric])
+ hence th: "[a^?r = 1] (mod n)"
+ using eqo mod_mult_left_eq[of "(a^?o)^?q" "a^?r" n]
+ apply (simp add: cong_nat_def del: One_nat_def)
+ by (simp add: mod_mult_left_eq[symmetric])
+ {assume r: "?r = 0" hence ?rhs by (simp add: dvd_eq_mod_eq_0)}
+ moreover
+ {assume r: "?r \<noteq> 0"
+ with mod_less_divisor[OF op, of d] have r0o:"?r >0 \<and> ?r < ?o" by simp
+ from conjunct2[OF ord_works[of a n], rule_format, OF r0o] th
+ have ?rhs by blast}
+ ultimately have ?rhs by blast}
+ ultimately show ?rhs by blast
+qed
+
+lemma order_divides_phi:
+ fixes n::nat shows "coprime n a \<Longrightarrow> ord n a dvd phi n"
+ by (metis ord_divides euler_theorem_nat gcd_nat.commute)
+
+lemma order_divides_expdiff:
+ fixes n::nat and a::nat assumes na: "coprime n a"
+ shows "[a^d = a^e] (mod n) \<longleftrightarrow> [d = e] (mod (ord n a))"
+proof-
+ {fix n::nat and a::nat and d::nat and e::nat
+ assume na: "coprime n a" and ed: "(e::nat) \<le> d"
+ hence "\<exists>c. d = e + c" by presburger
+ then obtain c where c: "d = e + c" by presburger
+ from na have an: "coprime a n"
+ by (metis gcd_nat.commute)
+ have aen: "coprime (a^e) n"
+ by (metis coprime_exp_nat gcd_nat.commute na)
+ have acn: "coprime (a^c) n"
+ by (metis coprime_exp_nat gcd_nat.commute na)
+ have "[a^d = a^e] (mod n) \<longleftrightarrow> [a^(e + c) = a^(e + 0)] (mod n)"
+ using c by simp
+ also have "\<dots> \<longleftrightarrow> [a^e* a^c = a^e *a^0] (mod n)" by (simp add: power_add)
+ also have "\<dots> \<longleftrightarrow> [a ^ c = 1] (mod n)"
+ using cong_mult_lcancel_nat [OF aen, of "a^c" "a^0"] by simp
+ also have "\<dots> \<longleftrightarrow> ord n a dvd c" by (simp only: ord_divides)
+ also have "\<dots> \<longleftrightarrow> [e + c = e + 0] (mod ord n a)"
+ using cong_add_lcancel_nat
+ by (metis cong_dvd_eq_nat dvd_0_right cong_dvd_modulus_nat cong_mult_self_nat nat_mult_1)
+ finally have "[a^d = a^e] (mod n) \<longleftrightarrow> [d = e] (mod (ord n a))"
+ using c by simp }
+ note th = this
+ have "e \<le> d \<or> d \<le> e" by arith
+ moreover
+ {assume ed: "e \<le> d" from th[OF na ed] have ?thesis .}
+ moreover
+ {assume de: "d \<le> e"
+ from th[OF na de] have ?thesis
+ by (metis cong_sym_nat)}
+ ultimately show ?thesis by blast
+qed
+
+subsection{*Another trivial primality characterization*}
+
+lemma prime_prime_factor:
+ "prime n \<longleftrightarrow> n \<noteq> 1\<and> (\<forall>p. prime p \<and> p dvd n \<longrightarrow> p = n)"
+proof-
+ {assume n: "n=0 \<or> n=1"
+ hence ?thesis
+ by (metis bigger_prime dvd_0_right one_not_prime_nat zero_not_prime_nat) }
+ moreover
+ {assume n: "n\<noteq>0" "n\<noteq>1"
+ {assume pn: "prime n"
+
+ from pn[unfolded prime_def] have "\<forall>p. prime p \<and> p dvd n \<longrightarrow> p = n"
+ using n
+ apply (cases "n = 0 \<or> n=1",simp)
+ by (clarsimp, erule_tac x="p" in allE, auto)}
+ moreover
+ {assume H: "\<forall>p. prime p \<and> p dvd n \<longrightarrow> p = n"
+ from n have n1: "n > 1" by arith
+ {fix m assume m: "m dvd n" "m\<noteq>1"
+ then obtain p where p: "prime p" "p dvd m"
+ by (metis prime_factor_nat)
+ from dvd_trans[OF p(2) m(1)] p(1) H have "p = n" by blast
+ with p(2) have "n dvd m" by simp
+ hence "m=n" using dvd_antisym[OF m(1)] by simp }
+ with n1 have "prime n" unfolding prime_def by auto }
+ ultimately have ?thesis using n by blast}
+ ultimately show ?thesis by auto
+qed
+
+lemma prime_divisor_sqrt:
+ "prime n \<longleftrightarrow> n \<noteq> 1 \<and> (\<forall>d. d dvd n \<and> d\<^sup>2 \<le> n \<longrightarrow> d = 1)"
+proof -
+ {assume "n=0 \<or> n=1" hence ?thesis
+ by (metis dvd.order_refl le_refl one_not_prime_nat power_zero_numeral zero_not_prime_nat)}
+ moreover
+ {assume n: "n\<noteq>0" "n\<noteq>1"
+ hence np: "n > 1" by arith
+ {fix d assume d: "d dvd n" "d\<^sup>2 \<le> n" and H: "\<forall>m. m dvd n \<longrightarrow> m=1 \<or> m=n"
+ from H d have d1n: "d = 1 \<or> d=n" by blast
+ {assume dn: "d=n"
+ have "n\<^sup>2 > n*1" using n by (simp add: power2_eq_square)
+ with dn d(2) have "d=1" by simp}
+ with d1n have "d = 1" by blast }
+ moreover
+ {fix d assume d: "d dvd n" and H: "\<forall>d'. d' dvd n \<and> d'\<^sup>2 \<le> n \<longrightarrow> d' = 1"
+ from d n have "d \<noteq> 0"
+ by (metis dvd_0_left_iff)
+ hence 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
+ have "d\<^sup>2 \<le> n \<or> e\<^sup>2 \<le> n" using dp ep
+ by (auto simp add: e power2_eq_square mult_le_cancel_left)
+ moreover
+ {assume h: "d\<^sup>2 \<le> n"
+ from H[rule_format, of d] h d have "d = 1" by blast}
+ moreover
+ {assume h: "e\<^sup>2 \<le> n"
+ from e have "e dvd n" unfolding dvd_def by (simp add: mult_commute)
+ with H[rule_format, of e] h have "e=1" by simp
+ with e have "d = n" by simp}
+ ultimately have "d=1 \<or> d=n" by blast}
+ ultimately have ?thesis unfolding prime_def using np n(2) by blast}
+ ultimately show ?thesis by auto
+qed
+
+lemma prime_prime_factor_sqrt:
+ "prime n \<longleftrightarrow> n \<noteq> 0 \<and> n \<noteq> 1 \<and> \<not> (\<exists>p. prime p \<and> p dvd n \<and> p\<^sup>2 \<le> n)"
+ (is "?lhs \<longleftrightarrow>?rhs")
+proof-
+ {assume "n=0 \<or> n=1"
+ hence ?thesis
+ by (metis one_not_prime_nat zero_not_prime_nat)}
+ moreover
+ {assume n: "n\<noteq>0" "n\<noteq>1"
+ {assume H: ?lhs
+ from H[unfolded prime_divisor_sqrt] n
+ have ?rhs
+ apply clarsimp
+ apply (erule_tac x="p" in allE)
+ apply simp
+ done
+ }
+ moreover
+ {assume H: ?rhs
+ {fix d assume d: "d dvd n" "d\<^sup>2 \<le> n" "d\<noteq>1"
+ then obtain p where p: "prime p" "p dvd d"
+ by (metis prime_factor_nat)
+ from n have np: "n > 0" by arith
+ from d(1) n have "d \<noteq> 0" by - (rule ccontr, auto)
+ hence dp: "d > 0" by arith
+ from mult_mono[OF dvd_imp_le[OF p(2) dp] dvd_imp_le[OF p(2) dp]] d(2)
+ have "p\<^sup>2 \<le> n" unfolding power2_eq_square by arith
+ with H n p(1) dvd_trans[OF p(2) d(1)] have False by blast}
+ with n prime_divisor_sqrt have ?lhs by auto}
+ ultimately have ?thesis by blast }
+ ultimately show ?thesis by (cases "n=0 \<or> n=1", auto)
+qed
+
+
+subsection{*Pocklington theorem*}
+
+lemma pocklington_lemma:
+ assumes n: "n \<ge> 2" and nqr: "n - 1 = q*r" and an: "[a^ (n - 1) = 1] (mod n)"
+ and aq:"\<forall>p. prime p \<and> p dvd q \<longrightarrow> 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 \<noteq> 0" "p \<noteq> 1" using pp one_not_prime_nat zero_not_prime_nat by auto
+ 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
+ {assume a0: "a = 0"
+ hence "a^ (n - 1) = 0" using n by (simp add: power_0_left)
+ with n an mod_less[of 1 n] have False by (simp add: power_0_left cong_nat_def)}
+ hence a0: "a\<noteq>0" ..
+ from n nqr have aqr0: "a ^ (q * r) \<noteq> 0" using a0 by simp
+ hence "(a ^ (q * r) - 1) + 1 = a ^ (q * r)" by simp
+ with k l have "a ^ (q * r) = p*l*k + 1" by simp
+ hence "a ^ (r * q) + p * 0 = 1 + p * (l*k)" by (simp add: mult_ac)
+ hence 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
+ {assume d1: "d \<noteq> 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
+ have P0: "P \<noteq> 0" using P(1)
+ by (metis zero_not_prime_nat)
+ 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 nat_mult_assoc)
+ have "ord p (a^r) * t*r = r * ord p (a^r) * t"
+ by (metis mult_assoc mult_commute)
+ hence exps: "a^(ord p (a^r) * t*r) = ((a ^ r) ^ ord p (a^r)) ^ t"
+ by (simp only: power_mult)
+ have "[((a ^ r) ^ ord p (a^r)) ^ t= 1^t] (mod p)"
+ by (metis cong_exp_nat ord)
+ then have th: "[((a ^ r) ^ ord p (a^r)) ^ t= 1] (mod p)"
+ by simp
+ have pd0: "p dvd a^(ord p (a^r) * t*r) - 1"
+ by (metis cong_to_1_nat exps th)
+ 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_nat have False
+ by auto}
+ hence d1: "d = 1" by blast
+ hence o: "ord p (a^r) = q" using d by simp
+ from pp phi_prime[of p] have phip: "phi p = p - 1" by simp
+ {fix d assume d: "d dvd p" "d dvd a" "d \<noteq> 1"
+ from pp[unfolded prime_def] d have dp: "d = p" by blast
+ from n have "n \<noteq> 0" by simp
+ then have False using d
+ by (metis coprime_minus_one_nat dp lucas_coprime_lemma an coprime_nat
+ gcd_lcm_complete_lattice_nat.top_greatest pn)}
+ hence cpa: "coprime p a" by auto
+ have arp: "coprime (a^r) p"
+ by (metis coprime_exp_nat cpa gcd_nat.commute)
+ from euler_theorem_nat[OF arp, simplified ord_divides] o phip
+ have "q dvd (p - 1)" by simp
+ then obtain d where d:"p - 1 = q * d" unfolding dvd_def by blast
+ have p0:"p \<noteq> 0"
+ by (metis p01(1))
+ from p0 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 \<ge> 2" and nqr: "n - 1 = q*r" and sqr: "n \<le> q\<^sup>2"
+ and an: "[a^ (n - 1) = 1] (mod n)"
+ and aq: "\<forall>p. prime p \<and> p dvd q \<longrightarrow> coprime (a^ ((n - 1) div p) - 1) n"
+ shows "prime n"
+unfolding prime_prime_factor_sqrt[of n]
+proof-
+ let ?ths = "n \<noteq> 0 \<and> n \<noteq> 1 \<and> \<not> (\<exists>p. prime p \<and> p dvd n \<and> p\<^sup>2 \<le> n)"
+ from n have n01: "n\<noteq>0" "n\<noteq>1" by arith+
+ {fix p assume p: "prime p" "p dvd n" "p\<^sup>2 \<le> n"
+ from p(3) sqr have "p^(Suc 1) \<le> q^(Suc 1)" by (simp add: power2_eq_square)
+ hence pq: "p \<le> q"
+ by (metis le0 power_le_imp_le_base)
+ from pocklington_lemma[OF n nqr an aq p(1,2)]
+ have th: "q dvd p - 1"
+ by (metis cong_to_1_nat)
+ have "p - 1 \<noteq> 0" using prime_ge_2_nat [OF p(1)] by arith
+ with pq p have False
+ by (metis Suc_diff_1 gcd_le2_nat gcd_semilattice_nat.inf_absorb1 not_less_eq_eq
+ prime_gt_0_nat th) }
+ with n01 show ?ths by blast
+qed
+
+(* Variant for application, to separate the exponentiation. *)
+lemma pocklington_alt:
+ assumes n: "n \<ge> 2" and nqr: "n - 1 = q*r" and sqr: "n \<le> q\<^sup>2"
+ and an: "[a^ (n - 1) = 1] (mod n)"
+ and aq:"\<forall>p. prime p \<and> p dvd q \<longrightarrow> (\<exists>b. [a^((n - 1) div p) = b] (mod n) \<and> 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
+ {assume a0: "a=0"
+ from n an have "[0 = 1] (mod n)" unfolding a0 power_0_left by auto
+ hence False using n by (simp add: cong_nat_def dvd_eq_mod_eq_0[symmetric])}
+ hence a0: "a\<noteq> 0" ..
+ hence a1: "a \<ge> 1" by arith
+ from one_le_power[OF a1] have ath: "1 \<le> a ^ ((n - 1) div p)" .
+ {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 mod_div_equality[of "n - 1" p]
+ have "(n - 1) div p * p= n - 1" by auto
+ hence eq: "(a^((n - 1) div p))^p = a^(n - 1)"
+ by (simp only: power_mult[symmetric])
+ have "p - 1 \<noteq> 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 (metis b0 diff_0_eq_0 gcd_dvd2_nat gcd_lcm_complete_lattice_nat.inf_bot_left
+ gcd_lcm_complete_lattice_nat.inf_top_left)
+ from divides_rexp[OF d, of "p - 1"] pS eq cong_dvd_eq_nat [OF an] n
+ have False
+ by simp}
+ then have b0: "b \<noteq> 0" ..
+ hence b1: "b \<ge> 1" by arith thm Cong.cong_diff_nat[OF cong_sym_nat [OF b(1)] cong_refl_nat[of 1]]
+ from cong_imp_coprime_nat[OF Cong.cong_diff_nat[OF cong_sym_nat [OF b(1)] cong_refl_nat[of 1] b1]] ath b1 b(2) nqr
+ have "coprime (a ^ ((n - 1) div p) - 1) n"
+ by simp}
+ hence th: "\<forall>p. prime p \<and> p dvd q \<longrightarrow> coprime (a ^ ((n - 1) div p) - 1) n "
+ by blast
+ from pocklington[OF n nqr sqr an th] show ?thesis .
+qed
+
+
+subsection{*Prime factorizations*}
+
+text{*FIXME Some overlap with material in UniqueFactorization, class unique_factorization.*}
+
+definition "primefact ps n = (foldr op * ps 1 = n \<and> (\<forall>p\<in> set ps. prime p))"
+
+lemma primefact: assumes n: "n \<noteq> 0"
+ shows "\<exists>ps. primefact ps n"
+using n
+proof(induct n rule: nat_less_induct)
+ fix n assume H: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>ps. primefact ps m)" and n: "n\<noteq>0"
+ let ?ths = "\<exists>ps. primefact ps n"
+ {assume "n = 1"
+ hence "primefact [] n" by (simp add: primefact_def)
+ hence ?ths by blast }
+ moreover
+ {assume n1: "n \<noteq> 1"
+ with n have n2: "n \<ge> 2" by arith
+ obtain p where p: "prime p" "p dvd n"
+ by (metis n1 prime_factor_nat)
+ from p(2) obtain m where m: "n = p*m" unfolding dvd_def by blast
+ from n m have m0: "m > 0" "m\<noteq>0" by auto
+ have "1 < p"
+ by (metis p(1) prime_nat_def)
+ with m0 m have mn: "m < n" by auto
+ from H[rule_format, OF mn m0(2)] obtain ps where ps: "primefact ps m" ..
+ from ps m p(1) have "primefact (p#ps) n" by (simp add: primefact_def)
+ hence ?ths by blast}
+ ultimately show ?ths by blast
+qed
+
+lemma primefact_contains:
+ assumes pf: "primefact ps n" and p: "prime p" and pn: "p dvd n"
+ shows "p \<in> set ps"
+ using pf p pn
+proof(induct ps arbitrary: p n)
+ case Nil thus ?case by (auto simp add: primefact_def)
+next
+ case (Cons q qs p n)
+ from Cons.prems[unfolded primefact_def]
+ have q: "prime q" "q * foldr op * qs 1 = n" "\<forall>p \<in>set qs. prime p" and p: "prime p" "p dvd q * foldr op * qs 1" by simp_all
+ {assume "p dvd q"
+ with p(1) q(1) have "p = q" unfolding prime_def by auto
+ hence ?case by simp}
+ moreover
+ { assume h: "p dvd foldr op * qs 1"
+ from q(3) have pqs: "primefact qs (foldr op * qs 1)"
+ by (simp add: primefact_def)
+ from Cons.hyps[OF pqs p(1) h] have ?case by simp}
+ ultimately show ?case
+ by (metis p prime_dvd_mult_eq_nat)
+qed
+
+lemma primefact_variant: "primefact ps n \<longleftrightarrow> foldr op * ps 1 = n \<and> list_all prime ps"
+ by (auto simp add: primefact_def list_all_iff)
+
+(* Variant of Lucas theorem. *)
+
+lemma lucas_primefact:
+ assumes n: "n \<ge> 2" and an: "[a^(n - 1) = 1] (mod n)"
+ and psn: "foldr op * ps 1 = n - 1"
+ and psp: "list_all (\<lambda>p. prime p \<and> \<not> [a^((n - 1) div p) = 1] (mod n)) ps"
+ shows "prime n"
+proof-
+ {fix p assume p: "prime p" "p dvd n - 1" "[a ^ ((n - 1) div p) = 1] (mod n)"
+ from psn psp have psn1: "primefact ps (n - 1)"
+ by (auto simp add: list_all_iff primefact_variant)
+ from p(3) primefact_contains[OF psn1 p(1,2)] psp
+ have False by (induct ps, auto)}
+ with lucas[OF n an] show ?thesis by blast
+qed
+
+(* Variant of Pocklington theorem. *)
+
+lemma pocklington_primefact:
+ assumes n: "n \<ge> 2" and qrn: "q*r = n - 1" and nq2: "n \<le> q\<^sup>2"
+ and arnb: "(a^r) mod n = b" and psq: "foldr op * ps 1 = q"
+ and bqn: "(b^q) mod n = 1"
+ and psp: "list_all (\<lambda>p. prime p \<and> coprime ((b^(q div p)) mod n - 1) n) ps"
+ shows "prime n"
+proof-
+ from bqn psp qrn
+ have bqn: "a ^ (n - 1) mod n = 1"
+ and psp: "list_all (\<lambda>p. prime p \<and> coprime (a^(r *(q div p)) mod n - 1) n) ps" unfolding arnb[symmetric] power_mod
+ by (simp_all add: power_mult[symmetric] algebra_simps)
+ from n have n0: "n > 0" by arith
+ from mod_div_equality[of "a^(n - 1)" n]
+ mod_less_divisor[OF n0, of "a^(n - 1)"]
+ have an1: "[a ^ (n - 1) = 1] (mod n)"
+ by (metis bqn cong_nat_def mod_mod_trivial)
+ {fix p assume p: "prime p" "p dvd q"
+ from psp psq have pfpsq: "primefact ps q"
+ by (auto simp add: primefact_variant list_all_iff)
+ from psp primefact_contains[OF pfpsq p]
+ have p': "coprime (a ^ (r * (q div p)) mod n - 1) n"
+ by (simp add: list_all_iff)
+ from p prime_def have p01: "p \<noteq> 0" "p \<noteq> 1" "p =Suc(p - 1)"
+ by auto
+ from div_mult1_eq[of r q p] p(2)
+ have eq1: "r* (q div p) = (n - 1) div p"
+ unfolding qrn[symmetric] dvd_eq_mod_eq_0 by (simp add: mult_commute)
+ have ath: "\<And>a (b::nat). a <= b \<Longrightarrow> a \<noteq> 0 ==> 1 <= a \<and> 1 <= b" by arith
+ {assume "a ^ ((n - 1) div p) mod n = 0"
+ then obtain s where s: "a ^ ((n - 1) div p) = n*s"
+ unfolding mod_eq_0_iff by blast
+ hence eq0: "(a^((n - 1) div p))^p = (n*s)^p" by simp
+ from qrn[symmetric] have qn1: "q dvd n - 1" unfolding dvd_def by auto
+ from dvd_trans[OF p(2) qn1] div_mod_equality'[of "n - 1" p]
+ have npp: "(n - 1) div p * p = n - 1" by (simp add: dvd_eq_mod_eq_0)
+ with eq0 have "a^ (n - 1) = (n*s)^p"
+ by (simp add: power_mult[symmetric])
+ hence "1 = (n*s)^(Suc (p - 1)) mod n" using bqn p01 by simp
+ also have "\<dots> = 0" by (simp add: mult_assoc)
+ finally have False by simp }
+ then have th11: "a ^ ((n - 1) div p) mod n \<noteq> 0" by auto
+ have th1: "[a ^ ((n - 1) div p) mod n = a ^ ((n - 1) div p)] (mod n)"
+ unfolding cong_nat_def by simp
+ from th1 ath[OF mod_less_eq_dividend th11]
+ have th: "[a ^ ((n - 1) div p) mod n - 1 = a ^ ((n - 1) div p) - 1] (mod n)"
+ by (metis cong_diff_nat cong_refl_nat)
+ have "coprime (a ^ ((n - 1) div p) - 1) n"
+ by (metis cong_imp_coprime_nat eq1 p' th) }
+ with pocklington[OF n qrn[symmetric] nq2 an1]
+ show ?thesis by blast
+qed
+
+end