author chaieb Mon, 25 Feb 2008 12:04:09 +0100 changeset 26126 f536ac0f92ca parent 26125 345465cc9e79 child 26127 70ef56eb650a
Pocklington's Primality criterion
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Pocklington.thy	Mon Feb 25 12:04:09 2008 +0100
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+(*  Title:      HOL/Library/Pocklington.thy
+    ID:         \$Id:
+    Author:     Amine Chaieb
+*)
+
+header {* Pocklington's Theorem for Primes *}
+
+
+theory Pocklington
+imports List Primes
+begin
+
+definition modeq:: "nat => nat => nat => bool"    ("(1[_ = _] '(mod _'))")
+  where "[a = b] (mod p) == ((a mod p) = (b mod p))"
+
+definition modneq:: "nat => nat => nat => bool"    ("(1[_ \<noteq> _] '(mod _'))")
+  where "[a \<noteq> b] (mod p) == ((a mod p) \<noteq> (b mod p))"
+
+lemma modeq_trans:
+  "\<lbrakk> [a = b] (mod p); [b = c] (mod p) \<rbrakk> \<Longrightarrow> [a = c] (mod p)"
+
+lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
+proof
+  assume H: "x mod n = y mod n"
+  hence "x mod n - y mod n = 0" by simp
+  hence "(x mod n - y mod n) mod n = 0" by simp
+  hence "(x - y) mod n = 0" by (simp add: zmod_zdiff1_eq[symmetric])
+  thus "n dvd x - y" by (simp add: zdvd_iff_zmod_eq_0)
+next
+  assume H: "n dvd x - y"
+  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
+  hence "x = n*k + y" by simp
+  hence "x mod n = (n*k + y) mod n" by simp
+  thus "x mod n = y mod n" by (simp add: zmod_zadd_left_eq)
+qed
+
+lemma nat_mod_lemma: assumes xyn: "[x = y] (mod n)" and xy:"y \<le> x"
+  shows "\<exists>q. x = y + n * q"
+proof-
+  from xy have th: "int x - int y = int (x - y)" by presburger
+  from xyn have "int x mod int n = int y mod int n"
+    by (simp add: modeq_def zmod_int[symmetric])
+  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])
+  hence "n dvd x - y" by (simp add: th zdvd_int)
+  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
+qed
+
+lemma nat_mod: "[x = y] (mod n) \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
+  (is "?lhs = ?rhs")
+proof
+  assume H: "[x = y] (mod n)"
+  {assume xy: "x \<le> y"
+    from H have th: "[y = x] (mod n)" by (simp add: modeq_def)
+    from nat_mod_lemma[OF th xy] have ?rhs
+      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
+  moreover
+  {assume xy: "y \<le> x"
+    from nat_mod_lemma[OF H xy] have ?rhs
+      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
+  ultimately  show ?rhs using linear[of x y] by blast
+next
+  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
+  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
+  thus  ?lhs by (simp add: modeq_def)
+qed
+
+(* 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 using prime_0 prime_1 by (cases "p=0", simp_all)}
+  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" using prime_coprime[OF H, of m] by blast}
+      hence ?rhs using p0 by auto}
+    moreover
+    { assume H: "\<forall>m. 0 < m \<and> m < p \<longrightarrow> coprime p m"
+      from prime_factor[OF p0(2)] obtain q where q: "prime q" "q dvd p" by blast
+      from prime_ge_2[OF q(1)] have q0: "q > 0" by arith
+      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
+	with coprime_prime[of p q q] q have False by simp hence ?lhs by blast}
+      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
+
+lemma coprime_mod: assumes n: "n \<noteq> 0" shows "coprime (a mod n) n \<longleftrightarrow> coprime a n"
+  using n dvd_mod_iff[of _ n a] by (auto simp add: coprime)
+
+(* Congruences.                                                              *)
+
+lemma cong_mod_01[simp,presburger]:
+  "[x = y] (mod 0) \<longleftrightarrow> x = y" "[x = y] (mod 1)" "[x = 0] (mod n) \<longleftrightarrow> n dvd x"
+  by (simp_all add: modeq_def, presburger)
+
+lemma cong_sub_cases:
+  "[x = y] (mod n) \<longleftrightarrow> (if x <= y then [y - x = 0] (mod n) else [x - y = 0] (mod n))"
+apply (rule_tac x="q2" in exI)
+apply (rule_tac x="q1" in exI, simp)
+apply (rule_tac x="q2" in exI)
+apply (rule_tac x="q1" in exI, simp)
+apply (rule_tac x="q1" in exI)
+apply (rule_tac x="q2" in exI, simp)
+apply (rule_tac x="q1" in exI)
+apply (rule_tac x="q2" in exI, simp)
+done
+
+lemma cong_mult_lcancel: assumes an: "coprime a n" and axy:"[a * x = a * y] (mod n)"
+  shows "[x = y] (mod n)"
+proof-
+  {assume "a = 0" with an axy coprime_0'[of n] have ?thesis by (simp add: modeq_def) }
+  moreover
+  {assume az: "a\<noteq>0"
+    {assume xy: "x \<le> y" hence axy': "a*x \<le> a*y" by simp
+      with axy cong_sub_cases[of "a*x" "a*y" n]  have "[a*(y - x) = 0] (mod n)"
+	by (simp only: if_True diff_mult_distrib2)
+      hence th: "n dvd a*(y -x)" by simp
+      from coprime_divprod[OF th] an have "n dvd y - x"
+      hence ?thesis using xy cong_sub_cases[of x y n] by simp}
+    moreover
+    {assume H: "\<not>x \<le> y" hence xy: "y \<le> x"  by arith
+      from H az have axy': "\<not> a*x \<le> a*y" by auto
+      with axy H cong_sub_cases[of "a*x" "a*y" n]  have "[a*(x - y) = 0] (mod n)"
+	by (simp only: if_False diff_mult_distrib2)
+      hence th: "n dvd a*(x - y)" by simp
+      from coprime_divprod[OF th] an have "n dvd x - y"
+      hence ?thesis using xy cong_sub_cases[of x y n] by simp}
+    ultimately have ?thesis by blast}
+  ultimately show ?thesis by blast
+qed
+
+lemma cong_mult_rcancel: assumes an: "coprime a n" and axy:"[x*a = y*a] (mod n)"
+  shows "[x = y] (mod n)"
+  using cong_mult_lcancel[OF an axy[unfolded mult_commute[of _a]]] .
+
+lemma cong_refl: "[x = x] (mod n)" by (simp add: modeq_def)
+
+lemma eq_imp_cong: "a = b \<Longrightarrow> [a = b] (mod n)" by (simp add: cong_refl)
+
+lemma cong_commute: "[x = y] (mod n) \<longleftrightarrow> [y = x] (mod n)"
+  by (auto simp add: modeq_def)
+
+lemma cong_trans[trans]: "[x = y] (mod n) \<Longrightarrow> [y = z] (mod n) \<Longrightarrow> [x = z] (mod n)"
+
+lemma cong_add: assumes xx': "[x = x'] (mod n)" and yy':"[y = y'] (mod n)"
+  shows "[x + y = x' + y'] (mod n)"
+proof-
+  have "(x + y) mod n = (x mod n + y mod n) mod n"
+  also have "\<dots> = (x' mod n + y' mod n) mod n" using xx' yy' modeq_def by simp
+  also have "\<dots> = (x' + y') mod n"
+  finally show ?thesis unfolding modeq_def .
+qed
+
+lemma cong_mult: assumes xx': "[x = x'] (mod n)" and yy':"[y = y'] (mod n)"
+  shows "[x * y = x' * y'] (mod n)"
+proof-
+  have "(x * y) mod n = (x mod n) * (y mod n) mod n"
+    by (simp add: mod_mult1_eq'[of x y n] mod_mult1_eq[of "x mod n" y n])
+  also have "\<dots> = (x' mod n) * (y' mod n) mod n" using xx'[unfolded modeq_def] yy'[unfolded modeq_def] by simp
+  also have "\<dots> = (x' * y') mod n"
+    by (simp add: mod_mult1_eq'[of x' y' n] mod_mult1_eq[of "x' mod n" y' n])
+  finally show ?thesis unfolding modeq_def .
+qed
+
+lemma cong_exp: "[x = y] (mod n) \<Longrightarrow> [x^k = y^k] (mod n)"
+  by (induct k, auto simp add: cong_refl cong_mult)
+lemma cong_sub: assumes xx': "[x = x'] (mod n)" and yy': "[y = y'] (mod n)"
+  and yx: "y <= x" and yx': "y' <= x'"
+  shows "[x - y = x' - y'] (mod n)"
+proof-
+  { fix x a x' a' y b y' b'
+    have "(x::nat) + a = x' + a' \<Longrightarrow> y + b = y' + b' \<Longrightarrow> y <= x \<Longrightarrow> y' <= x'
+      \<Longrightarrow> (x - y) + (a + b') = (x' - y') + (a' + b)" by arith}
+  note th = this
+  from xx' yy' obtain q1 q2 q1' q2' where q12: "x + n*q1 = x'+n*q2"
+    and q12': "y + n*q1' = y'+n*q2'" unfolding nat_mod by blast+
+  from th[OF q12 q12' yx yx']
+  have "(x - y) + n*(q1 + q2') = (x' - y') + n*(q2 + q1')"
+  thus ?thesis unfolding nat_mod by blast
+qed
+
+lemma cong_mult_lcancel_eq: assumes an: "coprime a n"
+  shows "[a * x = a * y] (mod n) \<longleftrightarrow> [x = y] (mod n)" (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+  assume H: "?rhs" from cong_mult[OF cong_refl[of a n] H] show ?lhs .
+next
+  assume H: "?lhs" hence H': "[x*a = y*a] (mod n)" by (simp add: mult_commute)
+  from cong_mult_rcancel[OF an H'] show ?rhs  .
+qed
+
+lemma cong_mult_rcancel_eq: assumes an: "coprime a n"
+  shows "[x * a = y * a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
+using cong_mult_lcancel_eq[OF an, of x y] by (simp add: mult_commute)
+
+lemma cong_add_lcancel_eq: "[a + x = a + y] (mod n) \<longleftrightarrow> [x = y] (mod n)"
+
+lemma cong_add_rcancel_eq: "[x + a = y + a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
+
+lemma cong_add_rcancel: "[x + a = y + a] (mod n) \<Longrightarrow> [x = y] (mod n)"
+
+lemma cong_add_lcancel: "[a + x = a + y] (mod n) \<Longrightarrow> [x = y] (mod n)"
+
+lemma cong_add_lcancel_eq_0: "[a + x = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
+
+lemma cong_add_rcancel_eq_0: "[x + a = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
+
+lemma cong_imp_eq: assumes xn: "x < n" and yn: "y < n" and xy: "[x = y] (mod n)"
+  shows "x = y"
+  using xy[unfolded modeq_def mod_less[OF xn] mod_less[OF yn]] .
+
+lemma cong_divides_modulus: "[x = y] (mod m) \<Longrightarrow> n dvd m ==> [x = y] (mod n)"
+  apply (auto simp add: nat_mod dvd_def)
+  apply (rule_tac x="k*q1" in exI)
+  apply (rule_tac x="k*q2" in exI)
+  by simp
+
+lemma cong_0_divides: "[x = 0] (mod n) \<longleftrightarrow> n dvd x" by simp
+
+lemma cong_1_divides:"[x = 1] (mod n) ==> n dvd x - 1"
+  apply (cases "x\<le>1", simp_all)
+  using cong_sub_cases[of x 1 n] by auto
+
+lemma cong_divides: "[x = y] (mod n) \<Longrightarrow> n dvd x \<longleftrightarrow> n dvd y"
+apply (auto simp add: nat_mod dvd_def)
+apply (rule_tac x="k + q1 - q2" in exI, simp add: add_mult_distrib2 diff_mult_distrib2)
+apply (rule_tac x="k + q2 - q1" in exI, simp add: add_mult_distrib2 diff_mult_distrib2)
+done
+
+lemma cong_coprime: assumes xy: "[x = y] (mod n)"
+  shows "coprime n x \<longleftrightarrow> coprime n y"
+proof-
+  {assume "n=0" hence ?thesis using xy by simp}
+  moreover
+  {assume nz: "n \<noteq> 0"
+  have "coprime n x \<longleftrightarrow> coprime (x mod n) n"
+    by (simp add: coprime_mod[OF nz, of x] coprime_commute[of n x])
+  also have "\<dots> \<longleftrightarrow> coprime (y mod n) n" using xy[unfolded modeq_def] by simp
+  also have "\<dots> \<longleftrightarrow> coprime y n" by (simp add: coprime_mod[OF nz, of y])
+  finally have ?thesis by (simp add: coprime_commute) }
+ultimately show ?thesis by blast
+qed
+
+lemma cong_mod: "~(n = 0) \<Longrightarrow> [a mod n = a] (mod n)" by (simp add: modeq_def)
+
+lemma mod_mult_cong: "~(a = 0) \<Longrightarrow> ~(b = 0)
+  \<Longrightarrow> [x mod (a * b) = y] (mod a) \<longleftrightarrow> [x = y] (mod a)"
+
+lemma cong_mod_mult: "[x = y] (mod n) \<Longrightarrow> m dvd n \<Longrightarrow> [x = y] (mod m)"
+  apply (auto simp add: nat_mod dvd_def)
+  apply (rule_tac x="k*q1" in exI)
+  apply (rule_tac x="k*q2" in exI, simp)
+  done
+
+(* Some things when we know more about the order.                            *)
+
+lemma cong_le: "y <= x \<Longrightarrow> [x = y] (mod n) \<longleftrightarrow> (\<exists>q. x = q * n + y)"
+  using nat_mod_lemma[of x y n]
+  apply auto
+  apply (rule_tac x="q" in exI)
+  apply (rule_tac x="q + q" in exI)
+  by (auto simp: ring_simps)
+
+lemma cong_to_1: "[a = 1] (mod n) \<longleftrightarrow> a = 0 \<and> n = 1 \<or> (\<exists>m. a = 1 + m * n)"
+proof-
+  {assume "n = 0 \<or> n = 1\<or> a = 0 \<or> a = 1" hence ?thesis
+      apply (cases "n=0", simp_all add: cong_commute)
+      apply (cases "n=1", simp_all add: cong_commute modeq_def)
+      apply arith
+      by (cases "a=1", simp_all add: modeq_def cong_commute)}
+  moreover
+  {assume n: "n\<noteq>0" "n\<noteq>1" and a:"a\<noteq>0" "a \<noteq> 1" hence a': "a \<ge> 1" by simp
+    hence ?thesis using cong_le[OF a', of n] by auto }
+  ultimately show ?thesis by auto
+qed
+
+(* Some basic theorems about solving congruences.                            *)
+
+
+lemma cong_solve: assumes an: "coprime a n" shows "\<exists>x. [a * x = b] (mod n)"
+proof-
+  {assume "a=0" hence ?thesis using an by (simp add: modeq_def)}
+  moreover
+  {assume az: "a\<noteq>0"
+  from bezout_add_strong[OF az, of n]
+  obtain d x y where dxy: "d dvd a" "d dvd n" "a*x = n*y + d" by blast
+  from an[unfolded coprime, rule_format, of d] dxy(1,2) have d1: "d = 1" by blast
+  hence "a*x*b = (n*y + 1)*b" using dxy(3) by simp
+  hence "a*(x*b) = n*(y*b) + b" by algebra
+  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 modeq_def .
+  hence ?thesis by blast}
+ultimately  show ?thesis by blast
+qed
+
+lemma cong_solve_unique: 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: modeq_def mod_mult1_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: modeq_def)
+    hence "[y = ?x] (mod n)" by (simp add: cong_mult_lcancel_eq[OF an])
+    with mod_less[OF Py(1)] mod_less_divisor[ of n x] nz
+    have "y = ?x" by (simp add: modeq_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 p have p1: "p > 1" using prime_ge_2[OF p] by arith
+  hence p01: "p \<noteq> 0" "p \<noteq> 1" by arith+
+  from pa have ap: "coprime a p" by (simp add: coprime_commute)
+  from prime_coprime[OF p, of x] dvd_imp_le[of p x] x0 xp have px:"coprime x p"
+    by (auto simp add: coprime_commute)
+  from cong_solve_unique[OF px p01(1)]
+  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 blast
+  {assume y0: "y = 0"
+    with y(2) have th: "p dvd a" by (simp add: cong_commute[of 0 a p])
+    with p coprime_prime[OF pa, of p] have False by simp}
+  with y show ?thesis unfolding Ex1_def using neq0_conv by blast
+qed
+lemma cong_unique_inverse_prime:
+  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)"
+  using cong_solve_unique_nontrivial[OF p coprime_1[of p] x0 xp] .
+
+(* Forms of the Chinese remainder theorem.                                   *)
+
+lemma cong_chinese:
+  assumes ab: "coprime a b" and  xya: "[x = y] (mod a)"
+  and xyb: "[x = y] (mod b)"
+  shows "[x = y] (mod a*b)"
+  using ab xya xyb
+  by (simp add: cong_sub_cases[of x y a] cong_sub_cases[of x y b]
+    cong_sub_cases[of x y "a*b"])
+(cases "x \<le> y", simp_all add: divides_mul[of a _ b])
+
+lemma chinese_remainder_unique:
+  assumes ab: "coprime a b" and az: "a \<noteq> 0" and bz: "b\<noteq>0"
+  shows "\<exists>!x. x < a * b \<and> [x = m] (mod a) \<and> [x = n] (mod b)"
+proof-
+  from az bz have abpos: "a*b > 0" by simp
+  from chinese_remainder[OF ab az bz] obtain x q1 q2 where
+    xq12: "x = m + q1 * a" "x = n + q2 * b" by blast
+  let ?w = "x mod (a*b)"
+  have wab: "?w < a*b" by (simp add: mod_less_divisor[OF abpos])
+  from xq12(1) have "?w mod a = ((m + q1 * a) mod (a*b)) mod a" by simp
+  also have "\<dots> = m mod a" apply (simp add: mod_mult2_eq)
+    by simp
+  finally have th1: "[?w = m] (mod a)" by (simp add: modeq_def)
+  from xq12(2) have "?w mod b = ((n + q2 * b) mod (a*b)) mod b" by simp
+  also have "\<dots> = ((n + q2 * b) mod (b*a)) mod b" by (simp add: mult_commute)
+  also have "\<dots> = n mod b" apply (simp add: mod_mult2_eq)
+    by simp
+  finally have th2: "[?w = n] (mod b)" by (simp add: modeq_def)
+  {fix y assume H: "y < a*b" "[y = m] (mod a)" "[y = n] (mod b)"
+    with th1 th2 have H': "[y = ?w] (mod a)" "[y = ?w] (mod b)"
+    from cong_chinese[OF ab H'] mod_less[OF H(1)] mod_less[OF wab]
+    have "y = ?w" by (simp add: modeq_def)}
+  with th1 th2 wab show ?thesis by blast
+qed
+
+lemma chinese_remainder_coprime_unique:
+  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 chinese_remainder_unique[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 cong_coprime[OF x(2)] cong_coprime[OF x(3)]
+  have "coprime x a" "coprime x b" by (simp_all add: coprime_commute)
+  with coprime_mul[of x a b] have "coprime x (a*b)" by simp
+  with x show ?thesis by blast
+qed
+
+(* Euler totient function.                                                   *)
+
+definition phi_def: "\<phi> n = card { m. 0 < m \<and> m <= n \<and> coprime m n }"
+lemma phi_0[simp]: "\<phi> 0 = 0"
+  unfolding phi_def by (auto simp add: card_eq_0_iff)
+
+lemma phi_finite[simp]: "finite ({ m. 0 < m \<and> m <= n \<and> coprime m n })"
+proof-
+  have "{ m. 0 < m \<and> m <= n \<and> coprime m n } \<subseteq> {0..n}" by auto
+  thus ?thesis by (auto intro: finite_subset)
+qed
+
+declare coprime_1[presburger]
+lemma phi_1[simp]: "\<phi> 1 = 1"
+proof-
+  {fix m
+    have "0 < m \<and> m <= 1 \<and> coprime m 1 \<longleftrightarrow> m = 1" by presburger }
+  thus ?thesis by (simp add: phi_def)
+qed
+
+lemma [simp]: "\<phi> (Suc 0) = Suc 0" using phi_1 by simp
+
+lemma phi_alt: "\<phi>(n) = card { m. coprime m n \<and> m < n}"
+proof-
+  {assume "n=0 \<or> n=1" hence ?thesis by (cases "n=0", simp_all)}
+  moreover
+  {assume n: "n\<noteq>0" "n\<noteq>1"
+    {fix m
+      from n have "0 < m \<and> m <= n \<and> coprime m n \<longleftrightarrow> coprime m n \<and> m < n"
+	apply (cases "m = 0", simp_all)
+	apply (cases "m = 1", simp_all)
+	apply (cases "m = n", auto)
+	done }
+    hence ?thesis unfolding phi_def by simp}
+  ultimately show ?thesis by auto
+qed
+
+lemma phi_finite_lemma[simp]: "finite {m. coprime m n \<and>  m < n}" (is "finite ?S")
+  by (rule finite_subset[of "?S" "{0..n}"], auto)
+
+lemma phi_another: assumes n: "n\<noteq>1"
+  shows "\<phi> n = card {m. 0 < m \<and> m < n \<and> coprime m n }"
+proof-
+  {fix m
+    from n have "0 < m \<and> m < n \<and> coprime m n \<longleftrightarrow> coprime m n \<and> m < n"
+      by (cases "m=0", auto)}
+  thus ?thesis unfolding phi_alt by auto
+qed
+
+lemma phi_limit: "\<phi> n \<le> n"
+proof-
+  have "{ m. coprime m n \<and> m < n} \<subseteq> {0 ..<n}" by auto
+  with card_mono[of "{0 ..<n}" "{ m. coprime m n \<and> m < n}"]
+  show ?thesis unfolding phi_alt by auto
+qed
+
+lemma stupid[simp]: "{m. (0::nat) < m \<and> m < n} = {1..<n}"
+  by auto
+
+lemma phi_limit_strong: assumes n: "n\<noteq>1"
+  shows "\<phi>(n) \<le> n - 1"
+proof-
+  show ?thesis
+    unfolding phi_another[OF n] finite_number_segment[of n, symmetric]
+    by (rule card_mono[of "{m. 0 < m \<and> m < n}" "{m. 0 < m \<and> m < n \<and> coprime m n}"], auto)
+qed
+
+lemma phi_lowerbound_1_strong: assumes n: "n \<ge> 1"
+  shows "\<phi>(n) \<ge> 1"
+proof-
+  let ?S = "{ m. 0 < m \<and> m <= n \<and> coprime m n }"
+  from card_0_eq[of ?S] n have "\<phi> n \<noteq> 0" unfolding phi_alt
+    apply auto
+    apply (cases "n=1", simp_all)
+    apply (rule exI[where x=1], simp)
+    done
+  thus ?thesis by arith
+qed
+
+lemma phi_lowerbound_1: "2 <= n ==> 1 <= \<phi>(n)"
+  using phi_lowerbound_1_strong[of n] by auto
+
+lemma phi_lowerbound_2: assumes n: "3 <= n" shows "2 <= \<phi> (n)"
+proof-
+  let ?S = "{ m. 0 < m \<and> m <= n \<and> coprime m n }"
+  have inS: "{1, n - 1} \<subseteq> ?S" using n coprime_plus1[of "n - 1"]
+    by (auto simp add: coprime_commute)
+  from n have c2: "card {1, n - 1} = 2" by (auto simp add: card_insert_if)
+  from card_mono[of ?S "{1, n - 1}", simplified inS c2] show ?thesis
+    unfolding phi_def by auto
+qed
+
+lemma phi_prime: "\<phi> n = n - 1 \<and> n\<noteq>0 \<and> n\<noteq>1 \<longleftrightarrow> prime n"
+proof-
+  {assume "n=0 \<or> n=1" hence ?thesis by (cases "n=1", simp_all)}
+  moreover
+  {assume n: "n\<noteq>0" "n\<noteq>1"
+    let ?S = "{m. 0 < m \<and> m < n}"
+    have fS: "finite ?S" by simp
+    let ?S' = "{m. 0 < m \<and> m < n \<and> coprime m n}"
+    have fS':"finite ?S'" apply (rule finite_subset[of ?S' ?S]) by auto
+    {assume H: "\<phi> n = n - 1 \<and> n\<noteq>0 \<and> n\<noteq>1"
+      hence ceq: "card ?S' = card ?S"
+      using n finite_number_segment[of n] phi_another[OF n(2)] by simp
+      {fix m assume m: "0 < m" "m < n" "\<not> coprime m n"
+	hence mS': "m \<notin> ?S'" by auto
+	have "insert m ?S' \<le> ?S" using m by auto
+	from m have "card (insert m ?S') \<le> card ?S"
+	  by - (rule card_mono[of ?S "insert m ?S'"], auto)
+	hence False
+	  unfolding card_insert_disjoint[of "?S'" m, OF fS' mS'] ceq
+	  by simp }
+      hence "\<forall>m. 0 <m \<and> m < n \<longrightarrow> coprime m n" by blast
+      hence "prime n" unfolding prime using n by (simp add: coprime_commute)}
+    moreover
+    {assume H: "prime n"
+      hence "?S = ?S'" unfolding prime using n
+	by (auto simp add: coprime_commute)
+      hence "card ?S = card ?S'" by simp
+      hence "\<phi> n = n - 1" unfolding phi_another[OF n(2)] by simp}
+    ultimately have ?thesis using n by blast}
+  ultimately show ?thesis by (cases "n=0") blast+
+qed
+
+(* Multiplicativity property.                                                *)
+
+lemma phi_multiplicative: assumes ab: "coprime a b"
+  shows "\<phi> (a * b) = \<phi> a * \<phi> b"
+proof-
+  {assume "a = 0 \<or> b = 0 \<or> a = 1 \<or> b = 1"
+    hence ?thesis
+      by (cases "a=0", simp, cases "b=0", simp, cases"a=1", simp_all) }
+  moreover
+  {assume a: "a\<noteq>0" "a\<noteq>1" and b: "b\<noteq>0" "b\<noteq>1"
+    hence ab0: "a*b \<noteq> 0" by simp
+    let ?S = "\<lambda>k. {m. coprime m k \<and> m < k}"
+    let ?f = "\<lambda>x. (x mod a, x mod b)"
+    have eq: "?f ` (?S (a*b)) = (?S a \<times> ?S b)"
+    proof-
+      {fix x assume x:"x \<in> ?S (a*b)"
+	hence x': "coprime x (a*b)" "x < a*b" by simp_all
+	hence xab: "coprime x a" "coprime x b" by (simp_all add: coprime_mul_eq)
+	from mod_less_divisor a b have xab':"x mod a < a" "x mod b < b" by auto
+	from xab xab' have "?f x \<in> (?S a \<times> ?S b)"
+	  by (simp add: coprime_mod[OF a(1)] coprime_mod[OF b(1)])}
+      moreover
+      {fix x y assume x: "x \<in> ?S a" and y: "y \<in> ?S b"
+	hence x': "coprime x a" "x < a" and y': "coprime y b" "y < b" by simp_all
+	from chinese_remainder_coprime_unique[OF ab a(1) b(1) x'(1) y'(1)]
+	obtain z where z: "coprime z (a * b)" "z < a * b" "[z = x] (mod a)"
+	  "[z = y] (mod b)" by blast
+	hence "(x,y) \<in> ?f ` (?S (a*b))"
+	  using y'(2) mod_less_divisor[of b y] x'(2) mod_less_divisor[of a x]
+	  by (auto simp add: image_iff modeq_def)}
+      ultimately show ?thesis by auto
+    qed
+    have finj: "inj_on ?f (?S (a*b))"
+      unfolding inj_on_def
+    proof(clarify)
+      fix x y assume H: "coprime x (a * b)" "x < a * b" "coprime y (a * b)"
+	"y < a * b" "x mod a = y mod a" "x mod b = y mod b"
+      hence cp: "coprime x a" "coprime x b" "coprime y a" "coprime y b"
+      from chinese_remainder_coprime_unique[OF ab a(1) b(1) cp(3,4)] H
+      show "x = y" unfolding modeq_def by blast
+    qed
+    from card_image[OF finj, unfolded eq] have ?thesis
+      unfolding phi_alt by simp }
+  ultimately show ?thesis by auto
+qed
+
+(* Fermat's Little theorem / Fermat-Euler theorem.                           *)
+
+lemma (in comm_monoid_mult) fold_related:
+  assumes Re: "R e e"
+  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
+  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
+  shows "R (fold (op *) h e S) (fold (op *) g e S)"
+  using prems
+  by -(rule finite_subset_induct,auto)
+
+
+lemma nproduct_mod:
+  assumes fS: "finite S" and n0: "n \<noteq> 0"
+  shows "[setprod (\<lambda>m. a(m) mod n) S = setprod a S] (mod n)"
+proof-
+  have th1:"[1 = 1] (mod n)" by (simp add: modeq_def)
+  from cong_mult
+  have th3:"\<forall>x1 y1 x2 y2.
+    [x1 = x2] (mod n) \<and> [y1 = y2] (mod n) \<longrightarrow> [x1 * y1 = x2 * y2] (mod n)"
+    by blast
+  have th4:"\<forall>x\<in>S. [a x mod n = a x] (mod n)" by (simp add: modeq_def)
+  from fold_related[where h="(\<lambda>m. a(m) mod n)" and g=a, OF th1 th3 fS, OF th4] show ?thesis unfolding setprod_def by (simp add: fS)
+qed
+
+lemma nproduct_cmul:
+  assumes fS:"finite S"
+  shows "setprod (\<lambda>m. (c::'a::{comm_monoid_mult,recpower})* a(m)) S = c ^ (card S) * setprod a S"
+unfolding setprod_timesf setprod_constant[OF fS, of c] ..
+
+lemma coprime_nproduct:
+  assumes fS: "finite S" and Sn: "\<forall>x\<in>S. coprime n (a x)"
+  shows "coprime n (setprod a S)"
+  using fS Sn
+unfolding setprod_def
+apply -
+apply (rule finite_subset_induct)
+
+lemma (in comm_monoid_mult)
+  fold_eq_general:
+  assumes fS: "finite S"
+  and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y"
+  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
+  shows "fold (op *) f1 e S = fold (op *) f2 e S'"
+proof-
+  from h f12 have hS: "h ` S = S'" by auto
+  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
+    from f12 h H  have "x = y" by auto }
+  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
+  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
+  from hS have "fold (op *) f2 e S' = fold (op *) f2 e (h ` S)" by simp
+  also have "\<dots> = fold (op *) (f2 o h) e S"
+    using fold_reindex[OF fS hinj, of f2 e] .
+  also have "\<dots> = fold (op *) f1 e S " using th fold_cong[OF fS, of "f2 o h" f1 e]
+    by blast
+  finally show ?thesis ..
+qed
+
+lemma fermat_little: assumes an: "coprime a n"
+  shows "[a ^ (\<phi> n) = 1] (mod n)"
+proof-
+  {assume "n=0" hence ?thesis by simp}
+  moreover
+  {assume "n=1" hence ?thesis by (simp add: modeq_def)}
+  moreover
+  {assume nz: "n \<noteq> 0" and n1: "n \<noteq> 1"
+    let ?S = "{m. coprime m n \<and> m < n}"
+    let ?P = "\<Prod> ?S"
+    have fS: "finite ?S" by simp
+    have cardfS: "\<phi> n = card ?S" unfolding phi_alt ..
+    {fix m assume m: "m \<in> ?S"
+      hence "coprime m n" by simp
+      with coprime_mul[of n a m] an have "coprime (a*m) n"
+    hence Sn: "\<forall>m\<in> ?S. coprime (a*m) n " by blast
+    from coprime_nproduct[OF fS, of n "\<lambda>m. m"] have nP:"coprime ?P n"
+    have Paphi: "[?P*a^ (\<phi> n) = ?P*1] (mod n)"
+    proof-
+      let ?h = "\<lambda>m. m mod n"
+      {fix m assume mS: "m\<in> ?S"
+	hence "?h m \<in> ?S" by simp}
+      hence hS: "?h ` ?S = ?S"by (auto simp add: image_iff)
+      have "a\<noteq>0" using an n1 nz apply- apply (rule ccontr) by simp
+      hence inj: "inj_on (op * a) ?S" unfolding inj_on_def by simp
+
+      have eq0: "fold op * (?h \<circ> op * a) 1 {m. coprime m n \<and> m < n} =
+     fold op * (\<lambda>m. m) 1 {m. coprime m n \<and> m < n}"
+      proof (rule fold_eq_general[where h="?h o (op * a)"])
+	show "finite ?S" using fS .
+      next
+	{fix y assume yS: "y \<in> ?S" hence y: "coprime y n" "y < n" by simp_all
+	  from cong_solve_unique[OF an nz, of y]
+	  obtain x where x:"x < n" "[a * x = y] (mod n)" "\<forall>z. z < n \<and> [a * z = y] (mod n) \<longrightarrow> z=x" by blast
+	  from cong_coprime[OF x(2)] y(1)
+	  have xm: "coprime x n" by (simp add: coprime_mul_eq coprime_commute)
+	  {fix z assume "z \<in> ?S" "(?h \<circ> op * a) z = y"
+	    hence z: "coprime z n" "z < n" "(?h \<circ> op * a) z = y" by simp_all
+	    from x(3)[rule_format, of z] z(2,3) have "z=x"
+	      unfolding modeq_def mod_less[OF y(2)] by simp}
+	  with xm x(1,2) have "\<exists>!x. x \<in> ?S \<and> (?h \<circ> op * a) x = y"
+	    unfolding modeq_def mod_less[OF y(2)] by auto }
+	thus "\<forall>y\<in>{m. coprime m n \<and> m < n}.
+       \<exists>!x. x \<in> {m. coprime m n \<and> m < n} \<and> ((\<lambda>m. m mod n) \<circ> op * a) x = y" by blast
+      next
+	{fix x assume xS: "x\<in> ?S"
+	  hence x: "coprime x n" "x < n" by simp_all
+	  with an have "coprime (a*x) n"
+	    by (simp add: coprime_mul_eq[of n a x] coprime_commute)
+	  hence "?h (a*x) \<in> ?S" using nz
+	    by (simp add: coprime_mod[OF nz] mod_less_divisor)}
+	thus " \<forall>x\<in>{m. coprime m n \<and> m < n}.
+       ((\<lambda>m. m mod n) \<circ> op * a) x \<in> {m. coprime m n \<and> m < n} \<and>
+       ((\<lambda>m. m mod n) \<circ> op * a) x = ((\<lambda>m. m mod n) \<circ> op * a) x" by simp
+      qed
+      from nproduct_mod[OF fS nz, of "op * a"]
+      have "[(setprod (op *a) ?S) = (setprod (?h o (op * a)) ?S)] (mod n)"
+	unfolding o_def
+      also have "[setprod (?h o (op * a)) ?S = ?P ] (mod n)"
+	using eq0 fS an by (simp add: setprod_def modeq_def o_def)
+      finally show "[?P*a^ (\<phi> n) = ?P*1] (mod n)"
+	unfolding cardfS mult_commute[of ?P "a^ (card ?S)"]
+	  nproduct_cmul[OF fS, symmetric] mult_1_right by simp
+    qed
+    from cong_mult_lcancel[OF nP Paphi] have ?thesis . }
+  ultimately show ?thesis by blast
+qed
+
+lemma fermat_little_prime: assumes p: "prime p" and ap: "coprime a p"
+  shows "[a^ (p - 1) = 1] (mod p)"
+  using fermat_little[OF ap] p[unfolded phi_prime[symmetric]]
+by simp
+
+
+(* Lucas's theorem.                                                          *)
+
+lemma lucas_coprime_lemma:
+  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 exp_eq_1[of a m] by simp}
+  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 modeq_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 unfolding coprime by auto
+  }
+  ultimately show ?thesis by blast
+qed
+
+lemma lucas_weak:
+  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 fermat_little[OF can] have afn: "[a ^ \<phi> n = 1] (mod n)" .
+  {assume "\<phi> n \<noteq> n - 1"
+    with phi_limit_strong[OF n1(1)] phi_lowerbound_1[OF n]
+    have c:"\<phi> n > 0 \<and> \<phi> n < n - 1" by arith
+    from nm[rule_format, OF c] afn have False ..}
+  hence "\<phi> n = n - 1" by blast
+  with phi_prime[of n] n1(1,2) show ?thesis by simp
+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
+
+lemma power_mod: "((x::nat) mod m)^n mod m = x^n mod m"
+proof(induct n)
+  case 0 thus ?case by simp
+next
+  case (Suc n)
+  have "(x mod m)^(Suc n) mod m = ((x mod m) * (((x mod m) ^ n) mod m)) mod m"
+  also have "\<dots> = ((x mod m) * (x^n mod m)) mod m" using Suc.hyps by simp
+  also have "\<dots> = x^(Suc n) mod m"
+    by (simp add: mod_mult1_eq'[symmetric] mod_mult1_eq[symmetric])
+  finally show ?case .
+qed
+
+lemma 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> \<not> [a^((n - 1) div p) = 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_coprime[OF an1]
+  have an: "coprime a n" "coprime (a^(n - 1)) n" by (simp_all add: coprime_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]
+      from coprime_exp[OF an(1)[unfolded coprime_commute[of a n]],
+	of "(n - 1) div m * m"]
+      have yn: "coprime ?y n" by (simp add: coprime_commute)
+      have "?y mod n = (a^m)^((n - 1) div m) mod n"
+	by (simp add: ring_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 modeq_def onen] onen by simp
+      finally have th3: "?y mod n = 1"  .
+      have th2: "[?y * a ^ ((n - 1) mod m) = ?y* 1] (mod n)"
+	using an1[unfolded modeq_def onen] onen
+	  mod_div_equality[of "(n - 1)" m, symmetric]
+      from cong_mult_lcancel[of ?y n "a^((n - 1) mod m)" 1, OF yn th2]
+      have th1: "[a ^ ((n - 1) mod m) = 1] (mod n)"  .
+      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)+
+    from prime_factor[OF r01(2)] obtain p where p: "prime p" "p dvd r" by blast
+    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
+    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: modeq_def)
+    finally have "[(a ^ ((n - 1) div p))= 1] (mod n)"
+      using onen by (simp add: modeq_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
+
+(* 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:
+  assumes na: "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> \<not> [a^ m = 1] (mod n))"
+proof-
+  let ?P = "\<lambda>d. 0 < d \<and> [a ^ d = 1] (mod n)"
+  from euclid[of a] obtain p where p: "prime p" "a < p" by blast
+  from na have o: "ord n a = Least ?P" by (simp add: ord_def)
+  {assume "n=0 \<or> n=1" with na have "\<exists>m>0. ?P m" apply auto apply (rule exI[where x=1]) by (simp  add: modeq_def)}
+  moreover
+  {assume "n\<noteq>0 \<and> n\<noteq>1" hence n2:"n \<ge> 2" by arith
+    from na have na': "coprime a n" by (simp add: coprime_commute)
+    from phi_lowerbound_1[OF n2] fermat_little[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 na show ?thesis
+    unfolding o[symmetric] by auto
+qed
+(* With the special value 0 for non-coprime case, it's more convenient.      *)
+lemma ord_works:
+ "[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 modeq_def)
+
+lemma ord: "[a^(ord n a) = 1] (mod n)" using ord_works by blast
+lemma ord_minimal: "0 < m \<Longrightarrow> m < ord n a \<Longrightarrow> ~[a^m = 1] (mod n)"
+  using ord_works by blast
+lemma ord_eq_0: "ord n a = 0 \<longleftrightarrow> ~coprime n a"
+by (cases "coprime n a", simp add: neq0_conv coprime_ord, simp add: neq0_conv ord_def)
+
+lemma ord_divides:
+ "[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 : modeq_def power_mult power_mod)
+  also have "[(a ^ (ord n a) mod n)^k = 1] (mod n)"
+    using ord[of a n, unfolded modeq_def] by (simp add: modeq_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: modeq_def)}
+    moreover
+    {assume d0: "d\<noteq>0" then obtain d' where d': "d = Suc d'" by (cases d, auto)
+      from H[unfolded coprime]
+      obtain p where p: "p dvd n" "p dvd a" "p \<noteq> 1" by auto
+      from lh[unfolded nat_mod]
+      obtain q1 q2 where q12:"a ^ d + n * q1 = 1 + n * q2" by blast
+      hence "a ^ d + n * q1 - n * q2 = 1" by simp
+      with dvd_diff [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"
+    from cong_exp[OF ord[of a n], of ?q]
+    have eqo: "[(a^?o)^?q = 1] (mod n)"  by (simp add: modeq_def)
+    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: modeq_def mult_commute)
+    hence "[(a^?o)^?q * (a^?r) = 1] (mod n)"
+    hence th: "[a^?r = 1] (mod n)"
+      using eqo mod_mult1_eq'[of "(a^?o)^?q" "a^?r" n]
+      apply (simp add: modeq_def del: One_nat_def)
+    {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: "coprime n a \<Longrightarrow> ord n a dvd \<phi> n"
+using ord_divides fermat_little coprime_commute by simp
+lemma order_divides_expdiff:
+  assumes na: "coprime n a"
+  shows "[a^d = a^e] (mod n) \<longleftrightarrow> [d = e] (mod (ord n a))"
+proof-
+  {fix n a d e
+    assume na: "coprime n a" and ed: "(e::nat) \<le> d"
+    hence "\<exists>c. d = e + c" by arith
+    then obtain c where c: "d = e + c" by arith
+    from na have an: "coprime a n" by (simp add: coprime_commute)
+    from coprime_exp[OF na, of e]
+    have aen: "coprime (a^e) n" by (simp add: coprime_commute)
+    from coprime_exp[OF na, of c]
+    have acn: "coprime (a^c) n" by (simp add: coprime_commute)
+    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_eq[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_eq[of e c 0 "ord n a", simplified cong_0_divides]
+      by simp
+    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 (simp add: cong_commute) }
+  ultimately show ?thesis by blast
+qed
+
+(* 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 using prime_0 two_is_prime by auto}
+  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"
+	from prime_factor[OF m(2)] obtain p where
+	  p: "prime p" "p dvd m" by blast
+	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_anti_sym[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^2 \<le> n \<longrightarrow> d = 1)"
+proof-
+  {assume "n=0 \<or> n=1" hence ?thesis using prime_0 prime_1
+    by (auto simp add: nat_power_eq_0_iff)}
+  moreover
+  {assume n: "n\<noteq>0" "n\<noteq>1"
+    hence np: "n > 1" by arith
+    {fix d assume d: "d dvd n" "d^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^2 > n*1" using n
+	  by (simp add: power2_eq_square mult_less_cancel1)
+	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'^2 \<le> n \<longrightarrow> d' = 1"
+      from d n have "d \<noteq> 0" apply - apply (rule ccontr) by simp
+      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^2 \<le> n \<or> e^2 \<le> n" using dp ep
+	by (auto simp add: e power2_eq_square mult_le_cancel_left)
+      moreover
+      {assume h: "d^2 \<le> n"
+	from H[rule_format, of d] h d have "d = 1" by blast}
+      moreover
+      {assume h: "e^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^2 \<le> n)"
+  (is "?lhs \<longleftrightarrow>?rhs")
+proof-
+  {assume "n=0 \<or> n=1" hence ?thesis using prime_0 prime_1 by auto}
+  moreover
+  {assume n: "n\<noteq>0" "n\<noteq>1"
+    {assume H: ?lhs
+      from H[unfolded prime_divisor_sqrt] n
+      have ?rhs  apply clarsimp by (erule_tac x="p" in allE, simp add: prime_1)
+    }
+    moreover
+    {assume H: ?rhs
+      {fix d assume d: "d dvd n" "d^2 \<le> n" "d\<noteq>1"
+	from prime_factor[OF d(3)]
+	obtain p where p: "prime p" "p dvd d" by blast
+	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^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
+(* 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-
+  from pp prime_0 prime_1 have p01: "p \<noteq> 0" "p \<noteq> 1" by - (rule ccontr, simp)+
+  from cong_1_divides[OF an, unfolded nqr, unfolded dvd_def]
+  obtain k where k: "a ^ (q * r) - 1 = n*k" by blast
+  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 modeq_def)}
+  hence a0: "a\<noteq>0" ..
+  from n nqr have aqr0: "a ^ (q * r) \<noteq> 0" using a0 by (simp add: neq0_conv)
+  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] nat_mod  by blast
+  from odq[unfolded dvd_def] obtain d where d: "q = ord p (a^r) * d" by blast
+  {assume d1: "d \<noteq> 1"
+    from prime_factor[OF d1] obtain P where P: "prime P" "P dvd d" by blast
+    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) prime_0 by - (rule ccontr, simp)
+    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 algebra
+    have "ord p (a^r) * t*r = r * ord p (a^r) * t" by algebra
+    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 (rule cong_exp, rule ord)
+    then have th: "[((a ^ r) ^ ord p (a^r)) ^ t= 1] (mod p)" by simp
+    from cong_1_divides[OF th] exps have pd0: "p dvd a^(ord p (a^r) * t*r) - 1" by simp
+    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 have False unfolding coprime 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 n12:"Suc (n - 2) = n - 1" by arith
+    with divides_rexp[OF d(2)[unfolded dp], of "n - 2"]
+    have th0: "p dvd a ^ (n - 1)" by simp
+    from n have n0: "n \<noteq> 0" by simp
+    from d(2) an n12[symmetric] have a0: "a \<noteq> 0"
+      by - (rule ccontr, simp add: modeq_def)
+    have th1: "a^ (n - 1) \<noteq> 0" using n d(2) dp a0 by (auto simp add: neq0_conv)
+    from coprime_minus1[OF th1, unfolded coprime]
+      dvd_trans[OF pn cong_1_divides[OF an]] th0 d(3) dp
+    have False by auto}
+  hence cpa: "coprime p a" using coprime by auto
+  from coprime_exp[OF cpa, of r] coprime_commute
+  have arp: "coprime (a^r) p" by blast
+  from fermat_little[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
+  from prime_0 pp have p0:"p \<noteq> 0" by -  (rule ccontr, auto)
+  from p0 d have "p + q * 0 = 1 + q * d" by simp
+  with nat_mod[of p 1 q, symmetric]
+  show ?thesis by blast
+qed
+
+lemma pocklington:
+  assumes n: "n \<ge> 2" and nqr: "n - 1 = q*r" and sqr: "n \<le> q^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\<twosuperior> \<le> n)"
+  from n have n01: "n\<noteq>0" "n\<noteq>1" by arith+
+  {fix p assume p: "prime p" "p dvd n" "p^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" unfolding exp_mono_le .
+    from pocklington_lemma[OF n nqr an aq p(1,2)]  cong_1_divides
+    have th: "q dvd p - 1" by blast
+    have "p - 1 \<noteq> 0"using prime_ge_2[OF p(1)] by arith
+    with divides_ge[OF th] pq have False by arith }
+  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^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: modeq_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"
+	apply (simp only: dvd_eq_mod_eq_0[symmetric]) by (rule dvd_mult2, simp)
+      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])
+      from prime_ge_2[OF p(1)] have pS: "Suc (p - 1) = p" by arith
+      from b(1) have d: "n dvd a^((n - 1) div p)" unfolding b0 cong_0_divides .
+      from divides_rexp[OF d, of "p - 1"] pS eq cong_divides[OF an] n
+      have False by simp}
+    then have b0: "b \<noteq> 0" ..
+    hence b1: "b \<ge> 1" by arith
+    from cong_coprime[OF cong_sub[OF b(1) cong_refl[of 1] ath b1]] b(2) nqr
+    have "coprime (a ^ ((n - 1) div p) - 1) n" by (simp add: coprime_commute)}
+  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
+
+(* Prime factorizations.                                                     *)
+
+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
+    from prime_factor[OF n1] obtain p where p: "prime p" "p dvd n" by blast
+    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
+    from prime_ge_2[OF p(1)] have "1 < p" by arith
+    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)"
+    from Cons.hyps[OF pqs p(1) h] have ?case by simp}
+  ultimately show ?case using prime_divprod[OF p] by blast
+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 mod_le: assumes n: "n \<noteq> (0::nat)" shows "m mod n \<le> m"
+proof-
+    from mod_div_equality[of m n]
+    have "\<exists>x. x + m mod n = m" by blast
+    then show ?thesis by auto
+qed
+
+
+lemma pocklington_primefact:
+  assumes n: "n \<ge> 2" and qrn: "q*r = n - 1" and nq2: "n \<le> q^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] ring_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)"
+    unfolding nat_mod bqn
+    apply -
+    apply (rule exI[where x="0"])
+    apply (rule exI[where x="a^(n - 1) div n"])
+  {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"
+    from prime_ge_2[OF p(1)] have p01: "p \<noteq> 0" "p \<noteq> 1" "p =Suc(p - 1)" by arith+
+    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
+    from n0 have n00: "n \<noteq> 0" by arith
+    from mod_le[OF n00]
+    have th10: "a ^ ((n - 1) div p) mod n \<le> a ^ ((n - 1) div p)" .
+    {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"
+      hence "1 = (n*s)^(Suc (p - 1)) mod n" using bqn p01 by simp
+      also have "\<dots> = 0" by (simp add: mult_assoc mod_mult_self_is_0)
+      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 modeq_def by simp
+    from cong_sub[OF th1 cong_refl[of 1]]  ath[OF th10 th11]
+    have th: "[a ^ ((n - 1) div p) mod n - 1 = a ^ ((n - 1) div p) - 1] (mod n)"
+      by blast
+    from cong_coprime[OF th] p'[unfolded eq1]
+    have "coprime (a ^ ((n - 1) div p) - 1) n" by (simp add: coprime_commute) }
+  with pocklington[OF n qrn[symmetric] nq2 an1]
+  show ?thesis by blast
+qed
+
+
+end```