author chaieb Mon, 25 Feb 2008 11:59:57 +0100 changeset 26125 345465cc9e79 parent 26124 2514f0ade8bc child 26126 f536ac0f92ca
More primality theorems
--- a/src/HOL/Library/Primes.thy	Mon Feb 25 11:27:27 2008 +0100
+++ b/src/HOL/Library/Primes.thy	Mon Feb 25 11:59:57 2008 +0100
@@ -1,13 +1,13 @@
(*  Title:      HOL/Library/Primes.thy
ID:         \$Id\$
-    Author:     Christophe Tabacznyj and Lawrence C Paulson
+    Author:     Amine Chaieb Christophe Tabacznyj and Lawrence C Paulson
*)

header {* Primality on nat *}

theory Primes
+imports GCD Parity
begin

definition
@@ -45,4 +45,999 @@
lemma prime_dvd_power_two: "prime p ==> p dvd m\<twosuperior> ==> p dvd m"
by (rule prime_dvd_square) (simp_all add: power2_eq_square)

+
+lemma exp_eq_1:"(x::nat)^n = 1 \<longleftrightarrow> x = 1 \<or> n = 0" by (induct n, auto)
+lemma exp_mono_lt: "(x::nat) ^ (Suc n) < y ^ (Suc n) \<longleftrightarrow> x < y"
+  using power_less_imp_less_base[of x "Suc n" y] power_strict_mono[of x y "Suc n"]
+    by auto
+lemma exp_mono_le: "(x::nat) ^ (Suc n) \<le> y ^ (Suc n) \<longleftrightarrow> x \<le> y"
+  by (simp only: linorder_not_less[symmetric] exp_mono_lt)
+
+lemma exp_mono_eq: "(x::nat) ^ Suc n = y ^ Suc n \<longleftrightarrow> x = y"
+using power_inject_base[of x n y] by auto
+
+
+lemma even_square: assumes e: "even (n::nat)" shows "\<exists>x. n ^ 2 = 4*x"
+proof-
+  from e have "2 dvd n" by presburger
+  then obtain k where k: "n = 2*k" using dvd_def by auto
+  hence "n^2 = 4* (k^2)" by (simp add: power2_eq_square)
+  thus ?thesis by blast
+qed
+
+lemma odd_square: assumes e: "odd (n::nat)" shows "\<exists>x. n ^ 2 = 4*x + 1"
+proof-
+  from e have np: "n > 0" by presburger
+  from e have "2 dvd (n - 1)" by presburger
+  then obtain k where "n - 1 = 2*k" using dvd_def by auto
+  hence k: "n = 2*k + 1"  using e by presburger
+  hence "n^2 = 4* (k^2 + k) + 1" by algebra
+  thus ?thesis by blast
+qed
+
+lemma diff_square: "(x::nat)^2 - y^2 = (x+y)*(x - y)"
+proof-
+  have "x \<le> y \<or> y \<le> x" by (rule nat_le_linear)
+  moreover
+  {assume le: "x \<le> y"
+    hence "x ^2 \<le> y^2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def)
+    with le have ?thesis by simp }
+  moreover
+  {assume le: "y \<le> x"
+    hence le2: "y ^2 \<le> x^2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def)
+    from le have "\<exists>z. y + z = x" by presburger
+    then obtain z where z: "x = y + z" by blast
+    from le2 have "\<exists>z. x^2 = y^2 + z" by presburger
+    then obtain z2 where z2: "x^2 = y^2 + z2"  by blast
+    from z z2 have ?thesis apply simp by algebra }
+  ultimately show ?thesis by blast
+qed
+
+(* Elementary theory of divisibility                                         *)
+lemma divides_ge: "(a::nat) dvd b \<Longrightarrow> b = 0 \<or> a \<le> b" unfolding dvd_def by auto
+lemma divides_antisym: "(x::nat) dvd y \<and> y dvd x \<longleftrightarrow> x = y"
+  using dvd_anti_sym[of x y] by auto
+
+lemma divides_add_revr: assumes da: "(d::nat) dvd a" and dab:"d dvd (a + b)"
+  shows "d dvd b"
+proof-
+  from da obtain k where k:"a = d*k" by (auto simp add: dvd_def)
+  from dab obtain k' where k': "a + b = d*k'" by (auto simp add: dvd_def)
+  from k k' have "b = d *(k' - k)" by (simp add : diff_mult_distrib2)
+  thus ?thesis unfolding dvd_def by blast
+qed
+
+declare nat_mult_dvd_cancel_disj[presburger]
+lemma nat_mult_dvd_cancel_disj'[presburger]:
+  "(m\<Colon>nat)*k dvd n*k \<longleftrightarrow> k = 0 \<or> m dvd n" unfolding mult_commute[of m k] mult_commute[of n k] by presburger
+
+lemma divides_mul_l: "(a::nat) dvd b ==> (c * a) dvd (c * b)"
+  by presburger
+
+lemma divides_mul_r: "(a::nat) dvd b ==> (a * c) dvd (b * c)" by presburger
+lemma divides_cases: "(n::nat) dvd m ==> m = 0 \<or> m = n \<or> 2 * n <= m"
+  by (auto simp add: dvd_def)
+lemma divides_le: "m dvd n ==> m <= n \<or> n = (0::nat)" by (auto simp add: dvd_def)
+
+lemma divides_div_not: "(x::nat) = (q * n) + r \<Longrightarrow> 0 < r \<Longrightarrow> r < n ==> ~(n dvd x)"
+  fix k assume H: "0 < r" "r < n" "q * n + r = n * k"
+  from H(3) have r: "r = n* (k -q)" by(simp add: diff_mult_distrib2 mult_commute)
+  {assume "k - q = 0" with r H(1) have False by simp}
+  moreover
+  {assume "k - q \<noteq> 0" with r have "r \<ge> n" by auto
+    with H(2) have False by simp}
+  ultimately show False by blast
+qed
+lemma divides_exp: "(x::nat) dvd y ==> x ^ n dvd y ^ n"
+  by (auto simp add: power_mult_distrib dvd_def)
+
+lemma divides_exp2: "n \<noteq> 0 \<Longrightarrow> (x::nat) ^ n dvd y \<Longrightarrow> x dvd y"
+  by (induct n ,auto simp add: dvd_def)
+
+fun fact :: "nat \<Rightarrow> nat" where
+  "fact 0 = 1"
+| "fact (Suc n) = Suc n * fact n"
+
+lemma fact_lt: "0 < fact n" by(induct n, simp_all)
+lemma fact_le: "fact n \<ge> 1" using fact_lt[of n] by simp
+lemma fact_mono: assumes le: "m \<le> n" shows "fact m \<le> fact n"
+proof-
+  from le have "\<exists>i. n = m+i" by presburger
+  then obtain i where i: "n = m+i" by blast
+  have "fact m \<le> fact (m + i)"
+  proof(induct m)
+    case 0 thus ?case using fact_le[of i] by simp
+  next
+    case (Suc m)
+    have "fact (Suc m) = Suc m * fact m" by simp
+    have th1: "Suc m \<le> Suc (m + i)" by simp
+    from mult_le_mono[of "Suc m" "Suc (m+i)" "fact m" "fact (m+i)", OF th1 Suc.hyps]
+    show ?case by simp
+  qed
+  thus ?thesis using i by simp
+qed
+
+lemma divides_fact: "1 <= p \<Longrightarrow> p <= n ==> p dvd fact n"
+proof(induct n arbitrary: p)
+  case 0 thus ?case by simp
+next
+  case (Suc n p)
+  from Suc.prems have "p = Suc n \<or> p \<le> n" by presburger
+  moreover
+  {assume "p = Suc n" hence ?case  by (simp only: fact.simps dvd_triv_left)}
+  moreover
+  {assume "p \<le> n"
+    with Suc.prems(1) Suc.hyps have th: "p dvd fact n" by simp
+    from dvd_mult[OF th] have ?case by (simp only: fact.simps) }
+  ultimately show ?case by blast
+qed
+
+declare dvd_triv_left[presburger]
+declare dvd_triv_right[presburger]
+lemma divides_rexp:
+  "x dvd y \<Longrightarrow> (x::nat) dvd (y^(Suc n))" by (simp add: dvd_mult2[of x y])
+
+(* The Bezout theorem is a bit ugly for N; it'd be easier for Z              *)
+lemma ind_euclid:
+  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
+  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
+  shows "P a b"
+proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
+  fix n a b
+  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
+  have "a = b \<or> a < b \<or> b < a" by arith
+  moreover {assume eq: "a= b"
+    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq by simp}
+  moreover
+  {assume lt: "a < b"
+    hence "a + b - a < n \<or> a = 0"  using H(2) by arith
+    moreover
+    {assume "a =0" with z c have "P a b" by blast }
+    moreover
+    {assume ab: "a + b - a < n"
+      have th0: "a + b - a = a + (b - a)" using lt by arith
+      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
+      have "P a b" by (simp add: th0[symmetric])}
+    ultimately have "P a b" by blast}
+  moreover
+  {assume lt: "a > b"
+    hence "b + a - b < n \<or> b = 0"  using H(2) by arith
+    moreover
+    {assume "b =0" with z c have "P a b" by blast }
+    moreover
+    {assume ab: "b + a - b < n"
+      have th0: "b + a - b = b + (a - b)" using lt by arith
+      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
+      have "P b a" by (simp add: th0[symmetric])
+      hence "P a b" using c by blast }
+    ultimately have "P a b" by blast}
+ultimately  show "P a b" by blast
+qed
+
+lemma bezout_lemma:
+  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
+  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and> (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
+using ex
+apply clarsimp
+apply (case_tac "a * x = b * y + d" , simp_all)
+apply (rule_tac x="x + y" in exI)
+apply (rule_tac x="y" in exI)
+apply algebra
+apply (rule_tac x="x" in exI)
+apply (rule_tac x="x + y" in exI)
+apply algebra
+done
+
+lemma bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
+apply(induct a b rule: ind_euclid)
+apply blast
+apply clarify
+apply clarsimp
+apply (rule_tac x="d" in exI)
+apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
+apply (rule_tac x="x+y" in exI)
+apply (rule_tac x="y" in exI)
+apply algebra
+apply (rule_tac x="x" in exI)
+apply (rule_tac x="x+y" in exI)
+apply algebra
+done
+
+lemma bezout: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x - b * y = d \<or> b * x - a * y = d)"
+apply clarsimp
+apply (rule_tac x="d" in exI, simp)
+apply (rule_tac x="x" in exI)
+apply (rule_tac x="y" in exI)
+apply auto
+done
+
+(* We can get a stronger version with a nonzeroness assumption.              *)
+
+lemma bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
+  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
+proof-
+  from nz have ap: "a > 0" by simp
+ have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or> (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
+ moreover
+ {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
+   from H have ?thesis by blast }
+ moreover
+ {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
+   {assume b0: "b = 0" with H  have ?thesis by simp}
+   moreover
+   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
+     from divides_le[OF H(2)] b have "d < b \<or> d = b" using le_less by blast
+     moreover
+     {assume db: "d=b"
+       from prems have ?thesis apply simp
+	 apply (rule exI[where x = b], simp)
+	 apply (rule exI[where x = b])
+	by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
+    moreover
+    {assume db: "d < b"
+	{assume "x=0" hence ?thesis  using prems by simp }
+	moreover
+	{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
+
+	  from db have "d \<le> b - 1" by simp
+	  hence "d*b \<le> b*(b - 1)" by simp
+	  with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
+	  have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
+	  from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)" by simp
+	  hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
+	    by (simp only: mult_assoc right_distrib)
+	  hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)" by algebra
+	  hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
+	  hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
+	    by (simp only: diff_add_assoc[OF dble, of d, symmetric])
+	  hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
+	    by (simp only: diff_mult_distrib2 add_commute mult_ac)
+	  hence ?thesis using H(1,2)
+	    apply -
+	    apply (rule exI[where x=d], simp)
+	    apply (rule exI[where x="(b - 1) * y"])
+	    by (rule exI[where x="x*(b - 1) - d"], simp)}
+	ultimately have ?thesis by blast}
+    ultimately have ?thesis by blast}
+  ultimately have ?thesis by blast}
+ ultimately show ?thesis by blast
+qed
+
+(* Greatest common divisor.                                                  *)
+lemma gcd_unique: "d dvd a\<and>d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd(a,b)"
+proof(auto)
+  assume H: "d dvd a" "d dvd b" "\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d"
+  from H(3)[rule_format] gcd_dvd1[of a b] gcd_dvd2[of a b]
+  have th: "gcd (a,b) dvd d" by blast
+  from dvd_anti_sym[OF th gcd_greatest[OF H(1,2)]]  show "d = gcd (a,b)" by blast
+qed
+
+lemma gcd_eq: assumes H: "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd u \<and> d dvd v"
+  shows "gcd (x,y) = gcd(u,v)"
+proof-
+  from H have "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd gcd (u,v)" by simp
+  with gcd_unique[of "gcd(u,v)" x y]  show ?thesis by auto
+qed
+
+lemma bezout_gcd: "\<exists>x y. a * x - b * y = gcd(a,b) \<or> b * x - a * y = gcd(a,b)"
+proof-
+  let ?g = "gcd (a,b)"
+  from bezout[of a b] obtain d x y where d: "d dvd a" "d dvd b" "a * x - b * y = d \<or> b * x - a * y = d" by blast
+  from d(1,2) have "d dvd ?g" by simp
+  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
+  from d(3) have "(a * x - b * y)*k = d*k \<or> (b * x - a * y)*k = d*k" by blast
+  hence "a * x * k - b * y*k = d*k \<or> b * x * k - a * y*k = d*k"
+    by (simp only: diff_mult_distrib)
+  hence "a * (x * k) - b * (y*k) = ?g \<or> b * (x * k) - a * (y*k) = ?g"
+    by (simp add: k mult_assoc)
+  thus ?thesis by blast
+qed
+
+lemma bezout_gcd_strong: assumes a: "a \<noteq> 0"
+  shows "\<exists>x y. a * x = b * y + gcd(a,b)"
+proof-
+  let ?g = "gcd (a,b)"
+  from bezout_add_strong[OF a, of b]
+  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
+  from d(1,2) have "d dvd ?g" by simp
+  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
+  from d(3) have "a * x * k = (b * y + d) *k " by auto
+  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
+  thus ?thesis by blast
+qed
+
+lemma gcd_mult_distrib: "gcd(a * c, b * c) = c * gcd(a,b)"
+
+lemma gcd_bezout: "(\<exists>x y. a * x - b * y = d \<or> b * x - a * y = d) \<longleftrightarrow> gcd(a,b) dvd d"
+  (is "?lhs \<longleftrightarrow> ?rhs")
+proof-
+  let ?g = "gcd (a,b)"
+  {assume H: ?rhs then obtain k where k: "d = ?g*k" unfolding dvd_def by blast
+    from bezout_gcd[of a b] obtain x y where xy: "a * x - b * y = ?g \<or> b * x - a * y = ?g"
+      by blast
+    hence "(a * x - b * y)*k = ?g*k \<or> (b * x - a * y)*k = ?g*k" by auto
+    hence "a * x*k - b * y*k = ?g*k \<or> b * x * k - a * y*k = ?g*k"
+      by (simp only: diff_mult_distrib)
+    hence "a * (x*k) - b * (y*k) = d \<or> b * (x * k) - a * (y*k) = d"
+      by (simp add: k[symmetric] mult_assoc)
+    hence ?lhs by blast}
+  moreover
+  {fix x y assume H: "a * x - b * y = d \<or> b * x - a * y = d"
+    have dv: "?g dvd a*x" "?g dvd b * y" "?g dvd b*x" "?g dvd a * y"
+      using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
+    from dvd_diff[OF dv(1,2)] dvd_diff[OF dv(3,4)] H
+    have ?rhs by auto}
+  ultimately show ?thesis by blast
+qed
+
+lemma gcd_bezout_sum: assumes H:"a * x + b * y = d" shows "gcd(a,b) dvd d"
+proof-
+  let ?g = "gcd (a,b)"
+    have dv: "?g dvd a*x" "?g dvd b * y"
+      using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
+    show ?thesis by auto
+qed
+
+lemma gcd_mult': "gcd(b,a * b) = b"
+by (simp add: gcd_mult mult_commute[of a b])
+
+lemma gcd_add: "gcd(a + b,b) = gcd(a,b)" "gcd(b + a,b) = gcd(a,b)" "gcd(a,a + b) = gcd(a,b)" "gcd(a,b + a) = gcd(a,b)"
+
+lemma gcd_sub: "b <= a ==> gcd(a - b,b) = gcd(a,b)" "a <= b ==> gcd(a,b - a) = gcd(a,b)"
+proof-
+  {fix a b assume H: "b \<le> (a::nat)"
+    hence th: "a - b + b = a" by arith
+    from gcd_add(1)[of "a - b" b] th  have "gcd(a - b,b) = gcd(a,b)" by simp}
+  note th = this
+{
+  assume ab: "b \<le> a"
+  from th[OF ab] show "gcd (a - b, b) = gcd (a, b)" by blast
+next
+  assume ab: "a \<le> b"
+  from th[OF ab] show "gcd (a,b - a) = gcd (a, b)"
+qed
+
+(* Coprimality                                                               *)
+
+lemma coprime: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
+using gcd_unique[of 1 a b, simplified] by (auto simp add: coprime_def)
+lemma coprime_commute: "coprime a b \<longleftrightarrow> coprime b a" by (simp add: coprime_def gcd_commute)
+
+lemma coprime_bezout: "coprime a b \<longleftrightarrow> (\<exists>x y. a * x - b * y = 1 \<or> b * x - a * y = 1)"
+using coprime_def gcd_bezout by auto
+
+lemma coprime_divprod: "d dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
+  using relprime_dvd_mult_iff[of d a b] by (auto simp add: coprime_def mult_commute)
+
+lemma coprime_1[simp]: "coprime a 1" by (simp add: coprime_def)
+lemma coprime_1'[simp]: "coprime 1 a" by (simp add: coprime_def)
+lemma coprime_Suc0[simp]: "coprime a (Suc 0)" by (simp add: coprime_def)
+lemma coprime_Suc0'[simp]: "coprime (Suc 0) a" by (simp add: coprime_def)
+
+lemma gcd_coprime:
+  assumes z: "gcd(a,b) \<noteq> 0" and a: "a = a' * gcd(a,b)" and b: "b = b' * gcd(a,b)"
+  shows    "coprime a' b'"
+proof-
+  let ?g = "gcd(a,b)"
+  {assume bz: "a = 0" from b bz z a have ?thesis by (simp add: gcd_zero coprime_def)}
+  moreover
+  {assume az: "a\<noteq> 0"
+    from z have z': "?g > 0" by simp
+    from bezout_gcd_strong[OF az, of b]
+    obtain x y where xy: "a*x = b*y + ?g" by blast
+    from xy a b have "?g * a'*x = ?g * (b'*y + 1)" by (simp add: ring_simps)
+    hence "?g * (a'*x) = ?g * (b'*y + 1)" by (simp add: mult_assoc)
+    hence "a'*x = (b'*y + 1)"
+      by (simp only: nat_mult_eq_cancel1[OF z'])
+    hence "a'*x - b'*y = 1" by simp
+    with coprime_bezout[of a' b'] have ?thesis by auto}
+  ultimately show ?thesis by blast
+qed
+lemma coprime_0: "coprime d 0 \<longleftrightarrow> d = 1" by (simp add: coprime_def)
+lemma coprime_mul: assumes da: "coprime d a" and db: "coprime d b"
+  shows "coprime d (a * b)"
+proof-
+  from da have th: "gcd(a, d) = 1" by (simp add: coprime_def gcd_commute)
+  from gcd_mult_cancel[of a d b, OF th] db[unfolded coprime_def] have "gcd (d, a*b) = 1"
+  thus ?thesis unfolding coprime_def .
+qed
+lemma coprime_lmul2: assumes dab: "coprime d (a * b)" shows "coprime d b"
+using prems unfolding coprime_bezout
+apply clarsimp
+apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all)
+apply (rule_tac x="x" in exI)
+apply (rule_tac x="a*y" in exI)
+apply (rule_tac x="a*x" in exI)
+apply (rule_tac x="y" in exI)
+done
+
+lemma coprime_rmul2: "coprime d (a * b) \<Longrightarrow> coprime d a"
+unfolding coprime_bezout
+apply clarsimp
+apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all)
+apply (rule_tac x="x" in exI)
+apply (rule_tac x="b*y" in exI)
+apply (rule_tac x="b*x" in exI)
+apply (rule_tac x="y" in exI)
+done
+lemma coprime_mul_eq: "coprime d (a * b) \<longleftrightarrow> coprime d a \<and>  coprime d b"
+  using coprime_rmul2[of d a b] coprime_lmul2[of d a b] coprime_mul[of d a b]
+  by blast
+
+lemma gcd_coprime_exists:
+  assumes nz: "gcd(a,b) \<noteq> 0"
+  shows "\<exists>a' b'. a = a' * gcd(a,b) \<and> b = b' * gcd(a,b) \<and> coprime a' b'"
+proof-
+  let ?g = "gcd (a,b)"
+  from gcd_dvd1[of a b] gcd_dvd2[of a b]
+  obtain a' b' where "a = ?g*a'"  "b = ?g*b'" unfolding dvd_def by blast
+  hence ab': "a = a'*?g" "b = b'*?g" by algebra+
+  from ab' gcd_coprime[OF nz ab'] show ?thesis by blast
+qed
+
+lemma coprime_exp: "coprime d a ==> coprime d (a^n)"
+  by(induct n, simp_all add: coprime_mul)
+
+lemma coprime_exp_imp: "coprime a b ==> coprime (a ^n) (b ^n)"
+  by (induct n, simp_all add: coprime_mul_eq coprime_commute coprime_exp)
+lemma coprime_refl[simp]: "coprime n n \<longleftrightarrow> n = 1" by (simp add: coprime_def)
+lemma coprime_plus1[simp]: "coprime (n + 1) n"
+  apply (rule exI[where x=1])
+  apply (rule exI[where x=1])
+  apply simp
+  done
+lemma coprime_minus1: "n \<noteq> 0 ==> coprime (n - 1) n"
+  using coprime_plus1[of "n - 1"] coprime_commute[of "n - 1" n] by auto
+
+lemma bezout_gcd_pow: "\<exists>x y. a ^n * x - b ^ n * y = gcd(a,b) ^ n \<or> b ^ n * x - a ^ n * y = gcd(a,b) ^ n"
+proof-
+  let ?g = "gcd (a,b)"
+  {assume z: "?g = 0" hence ?thesis
+      apply (cases n, simp)
+      apply arith
+      apply (simp only: z power_0_Suc)
+      apply (rule exI[where x=0])
+      apply (rule exI[where x=0])
+      by simp}
+  moreover
+  {assume z: "?g \<noteq> 0"
+    from gcd_dvd1[of a b] gcd_dvd2[of a b] obtain a' b' where
+      ab': "a = a'*?g" "b = b'*?g" unfolding dvd_def by (auto simp add: mult_ac)
+    hence ab'': "?g*a' = a" "?g * b' = b" by algebra+
+    from coprime_exp_imp[OF gcd_coprime[OF z ab'], unfolded coprime_bezout, of n]
+    obtain x y where "a'^n * x - b'^n * y = 1 \<or> b'^n * x - a'^n * y = 1"  by blast
+    hence "?g^n * (a'^n * x - b'^n * y) = ?g^n \<or> ?g^n*(b'^n * x - a'^n * y) = ?g^n"
+      using z by auto
+    then have "a^n * x - b^n * y = ?g^n \<or> b^n * x - a^n * y = ?g^n"
+      using z ab'' by (simp only: power_mult_distrib[symmetric]
+	diff_mult_distrib2 mult_assoc[symmetric])
+    hence  ?thesis by blast }
+  ultimately show ?thesis by blast
+qed
+lemma gcd_exp: "gcd (a^n, b^n) = gcd(a,b)^n"
+proof-
+  let ?g = "gcd(a^n,b^n)"
+  let ?gn = "gcd(a,b)^n"
+  {fix e assume H: "e dvd a^n" "e dvd b^n"
+    from bezout_gcd_pow[of a n b] obtain x y
+      where xy: "a ^ n * x - b ^ n * y = ?gn \<or> b ^ n * x - a ^ n * y = ?gn" by blast
+    from dvd_diff [OF dvd_mult2[OF H(1), of x] dvd_mult2[OF H(2), of y]]
+      dvd_diff [OF dvd_mult2[OF H(2), of x] dvd_mult2[OF H(1), of y]] xy
+    have "e dvd ?gn" by (cases "a ^ n * x - b ^ n * y = gcd (a, b) ^ n", simp_all)}
+  hence th:  "\<forall>e. e dvd a^n \<and> e dvd b^n \<longrightarrow> e dvd ?gn" by blast
+  from divides_exp[OF gcd_dvd1[of a b], of n] divides_exp[OF gcd_dvd2[of a b], of n] th
+    gcd_unique have "?gn = ?g" by blast thus ?thesis by simp
+qed
+
+lemma coprime_exp2:  "coprime (a ^ Suc n) (b^ Suc n) \<longleftrightarrow> coprime a b"
+by (simp only: coprime_def gcd_exp exp_eq_1) simp
+
+lemma division_decomp: assumes dc: "(a::nat) dvd b * c"
+  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
+proof-
+  let ?g = "gcd (a,b)"
+  {assume "?g = 0" with dc have ?thesis apply (simp add: gcd_zero)
+      apply (rule exI[where x="0"])
+      by (rule exI[where x="c"], simp)}
+  moreover
+  {assume z: "?g \<noteq> 0"
+    from gcd_coprime_exists[OF z]
+    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast
+    from gcd_dvd2[of a b] have thb: "?g dvd b" .
+    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
+    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
+    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
+    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
+    with z have th_1: "a' dvd b'*c" by simp
+    from coprime_divprod[OF th_1 ab'(3)] have thc: "a' dvd c" .
+    from ab' have "a = ?g*a'" by algebra
+    with thb thc have ?thesis by blast }
+  ultimately show ?thesis by blast
+qed
+
+lemma nat_power_eq_0_iff: "(m::nat) ^ n = 0 \<longleftrightarrow> n \<noteq> 0 \<and> m = 0" by (induct n, auto)
+
+lemma divides_rev: assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0" shows "a dvd b"
+proof-
+  let ?g = "gcd (a,b)"
+  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
+  {assume "?g = 0" with ab n have ?thesis by (simp add: gcd_zero)}
+  moreover
+  {assume z: "?g \<noteq> 0"
+    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
+    from gcd_coprime_exists[OF z]
+    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast
+    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n" by (simp add: ab'(1,2)[symmetric])
+    hence "?g^n*a'^n dvd ?g^n *b'^n" by (simp only: power_mult_distrib mult_commute)
+    with zn z n have th0:"a'^n dvd b'^n" by (auto simp add: nat_power_eq_0_iff)
+    have "a' dvd a'^n" by (simp add: m)
+    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
+    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
+    from coprime_divprod[OF th1 coprime_exp[OF ab'(3), of m]]
+    have "a' dvd b'" .
+    hence "a'*?g dvd b'*?g" by simp
+    with ab'(1,2)  have ?thesis by simp }
+  ultimately show ?thesis by blast
+qed
+
+lemma divides_mul: assumes mr: "m dvd r" and nr: "n dvd r" and mn:"coprime m n"
+  shows "m * n dvd r"
+proof-
+  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
+    unfolding dvd_def by blast
+  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
+  hence "m dvd n'" using relprime_dvd_mult_iff[OF mn[unfolded coprime_def]] by simp
+  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
+  from n' k show ?thesis unfolding dvd_def by auto
+qed
+
+(* A binary form of the Chinese Remainder Theorem.                           *)
+
+lemma chinese_remainder: assumes ab: "coprime a b" and a:"a \<noteq> 0" and b:"b \<noteq> 0"
+  shows "\<exists>x q1 q2. x = u + q1 * a \<and> x = v + q2 * b"
+proof-
+  obtain d1 x1 y1 d2 x2 y2 where dxy1: "d1 dvd a" "d1 dvd b" "a * x1 = b * y1 + d1"
+    and dxy2: "d2 dvd b" "d2 dvd a" "b * x2 = a * y2 + d2" by blast
+  from gcd_unique[of 1 a b, simplified ab[unfolded coprime_def], simplified]
+    dxy1(1,2) dxy2(1,2) have d12: "d1 = 1" "d2 =1" by auto
+  let ?x = "v * a * x1 + u * b * x2"
+  let ?q1 = "v * x1 + u * y2"
+  let ?q2 = "v * y1 + u * x2"
+  from dxy2(3)[simplified d12] dxy1(3)[simplified d12]
+  have "?x = u + ?q1 * a" "?x = v + ?q2 * b" by algebra+
+  thus ?thesis by blast
+qed
+
+(* Primality                                                                 *)
+(* A few useful theorems about primes                                        *)
+
+lemma prime_0[simp]: "~prime 0" by (simp add: prime_def)
+lemma prime_1[simp]: "~ prime 1"  by (simp add: prime_def)
+lemma prime_Suc0[simp]: "~ prime (Suc 0)"  by (simp add: prime_def)
+
+lemma prime_ge_2: "prime p ==> p \<ge> 2" by (simp add: prime_def)
+lemma prime_factor: assumes n: "n \<noteq> 1" shows "\<exists> p. prime p \<and> p dvd n"
+using n
+proof(induct n rule: nat_less_induct)
+  fix n
+  assume H: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)" "n \<noteq> 1"
+  let ?ths = "\<exists>p. prime p \<and> p dvd n"
+  {assume "n=0" hence ?ths using two_is_prime by auto}
+  moreover
+  {assume nz: "n\<noteq>0"
+    {assume "prime n" hence ?ths by - (rule exI[where x="n"], simp)}
+    moreover
+    {assume n: "\<not> prime n"
+      with nz H(2)
+      obtain k where k:"k dvd n" "k \<noteq> 1" "k \<noteq> n" by (auto simp add: prime_def)
+      from dvd_imp_le[OF k(1)] nz k(3) have kn: "k < n" by simp
+      from H(1)[rule_format, OF kn k(2)] obtain p where p: "prime p" "p dvd k" by blast
+      from dvd_trans[OF p(2) k(1)] p(1) have ?ths by blast}
+    ultimately have ?ths by blast}
+  ultimately show ?ths by blast
+qed
+
+lemma prime_factor_lt: assumes p: "prime p" and n: "n \<noteq> 0" and npm:"n = p * m"
+  shows "m < n"
+proof-
+  {assume "m=0" with n have ?thesis by simp}
+  moreover
+  {assume m: "m \<noteq> 0"
+    from npm have mn: "m dvd n" unfolding dvd_def by auto
+    from npm m have "n \<noteq> m" using p by auto
+    with dvd_imp_le[OF mn] n have ?thesis by simp}
+  ultimately show ?thesis by blast
+qed
+
+lemma euclid_bound: "\<exists>p. prime p \<and> n < p \<and>  p <= Suc (fact n)"
+proof-
+  have f1: "fact n + 1 \<noteq> 1" using fact_le[of n] by arith
+  from prime_factor[OF f1] obtain p where p: "prime p" "p dvd fact n + 1" by blast
+  from dvd_imp_le[OF p(2)] have pfn: "p \<le> fact n + 1" by simp
+  {assume np: "p \<le> n"
+    from p(1) have p1: "p \<ge> 1" by (cases p, simp_all)
+    from divides_fact[OF p1 np] have pfn': "p dvd fact n" .
+    from divides_add_revr[OF pfn' p(2)] p(1) have False by simp}
+  hence "n < p" by arith
+  with p(1) pfn show ?thesis by auto
+qed
+
+lemma euclid: "\<exists>p. prime p \<and> p > n" using euclid_bound by auto
+lemma primes_infinite: "\<not> (finite {p. prime p})"
+  fix n f
+  assume H: "Collect prime = f ` {i. i < (n::nat)}"
+  let ?P = "Collect prime"
+  let ?m = "Max ?P"
+  have P0: "?P \<noteq> {}" using two_is_prime by auto
+  from H have fP: "finite ?P" using finite_conv_nat_seg_image by blast
+  from Max_in[OF fP P0]  have "?m \<in> ?P" .
+  from Max_ge[OF fP P0] have contr: "\<forall> p. prime p \<longrightarrow> p \<le> ?m" by blast
+  from euclid[of ?m] obtain q where q: "prime q" "q > ?m" by blast
+  with contr show False by auto
+qed
+
+lemma coprime_prime: assumes ab: "coprime a b"
+  shows "~(prime p \<and> p dvd a \<and> p dvd b)"
+proof
+  assume "prime p \<and> p dvd a \<and> p dvd b"
+  thus False using ab gcd_greatest[of p a b] by (simp add: coprime_def)
+qed
+lemma coprime_prime_eq: "coprime a b \<longleftrightarrow> (\<forall>p. ~(prime p \<and> p dvd a \<and> p dvd b))"
+  (is "?lhs = ?rhs")
+proof-
+  {assume "?lhs" with coprime_prime  have ?rhs by blast}
+  moreover
+  {assume r: "?rhs" and c: "\<not> ?lhs"
+    then obtain g where g: "g\<noteq>1" "g dvd a" "g dvd b" unfolding coprime_def by blast
+    from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast
+    from dvd_trans [OF p(2) g(2)] dvd_trans [OF p(2) g(3)]
+    have "p dvd a" "p dvd b" . with p(1) r have False by blast}
+  ultimately show ?thesis by blast
+qed
+
+lemma prime_coprime: assumes p: "prime p"
+  shows "n = 1 \<or> p dvd n \<or> coprime p n"
+using p prime_imp_relprime[of p n] by (auto simp add: coprime_def)
+
+lemma prime_coprime_strong: "prime p \<Longrightarrow> p dvd n \<or> coprime p n"
+  using prime_coprime[of p n] by auto
+
+declare  coprime_0[simp]
+
+lemma coprime_0'[simp]: "coprime 0 d \<longleftrightarrow> d = 1" by (simp add: coprime_commute[of 0 d])
+lemma coprime_bezout_strong: assumes ab: "coprime a b" and b: "b \<noteq> 1"
+  shows "\<exists>x y. a * x = b * y + 1"
+proof-
+  from ab b have az: "a \<noteq> 0" by - (rule ccontr, auto)
+  from bezout_gcd_strong[OF az, of b] ab[unfolded coprime_def]
+  show ?thesis by auto
+qed
+
+lemma bezout_prime: assumes p: "prime p"  and pa: "\<not> p dvd a"
+  shows "\<exists>x y. a*x = p*y + 1"
+proof-
+  from p have p1: "p \<noteq> 1" using prime_1 by blast
+  from prime_coprime[OF p, of a] p1 pa have ap: "coprime a p"
+    by (auto simp add: coprime_commute)
+  from coprime_bezout_strong[OF ap p1] show ?thesis .
+qed
+lemma prime_divprod: assumes p: "prime p" and pab: "p dvd a*b"
+  shows "p dvd a \<or> p dvd b"
+proof-
+  {assume "a=1" hence ?thesis using pab by simp }
+  moreover
+  {assume "p dvd a" hence ?thesis by blast}
+  moreover
+  {assume pa: "coprime p a" from coprime_divprod[OF pab pa]  have ?thesis .. }
+  ultimately show ?thesis using prime_coprime[OF p, of a] by blast
+qed
+
+lemma prime_divprod_eq: assumes p: "prime p"
+  shows "p dvd a*b \<longleftrightarrow> p dvd a \<or> p dvd b"
+using p prime_divprod dvd_mult dvd_mult2 by auto
+
+lemma prime_divexp: assumes p:"prime p" and px: "p dvd x^n"
+  shows "p dvd x"
+using px
+proof(induct n)
+  case 0 thus ?case by simp
+next
+  case (Suc n)
+  hence th: "p dvd x*x^n" by simp
+  {assume H: "p dvd x^n"
+    from Suc.hyps[OF H] have ?case .}
+  with prime_divprod[OF p th] show ?case by blast
+qed
+
+lemma prime_divexp_n: "prime p \<Longrightarrow> p dvd x^n \<Longrightarrow> p^n dvd x^n"
+  using prime_divexp[of p x n] divides_exp[of p x n] by blast
+
+lemma coprime_prime_dvd_ex: assumes xy: "\<not>coprime x y"
+  shows "\<exists>p. prime p \<and> p dvd x \<and> p dvd y"
+proof-
+  from xy[unfolded coprime_def] obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd y"
+    by blast
+  from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast
+  from g(2,3) dvd_trans[OF p(2)] p(1) show ?thesis by auto
+qed
+lemma coprime_sos: assumes xy: "coprime x y"
+  shows "coprime (x * y) (x^2 + y^2)"
+proof-
+  {assume c: "\<not> coprime (x * y) (x^2 + y^2)"
+    from coprime_prime_dvd_ex[OF c] obtain p
+      where p: "prime p" "p dvd x*y" "p dvd x^2 + y^2" by blast
+    {assume px: "p dvd x"
+      from dvd_mult[OF px, of x] p(3) have "p dvd y^2"
+	unfolding dvd_def
+	apply (auto simp add: power2_eq_square)
+	apply (rule_tac x= "ka - k" in exI)
+      with prime_divexp[OF p(1), of y 2] have py: "p dvd y" .
+      from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1
+      have False by simp }
+    moreover
+    {assume py: "p dvd y"
+      from dvd_mult[OF py, of y] p(3) have "p dvd x^2"
+	unfolding dvd_def
+	apply (auto simp add: power2_eq_square)
+	apply (rule_tac x= "ka - k" in exI)
+      with prime_divexp[OF p(1), of x 2] have px: "p dvd x" .
+      from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1
+      have False by simp }
+    ultimately have False using prime_divprod[OF p(1,2)] by blast}
+  thus ?thesis by blast
+qed
+
+lemma distinct_prime_coprime: "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
+  unfolding prime_def coprime_prime_eq by blast
+
+lemma prime_coprime_lt: assumes p: "prime p" and x: "0 < x" and xp: "x < p"
+  shows "coprime x p"
+proof-
+  {assume c: "\<not> coprime x p"
+    then obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd p" unfolding coprime_def by blast
+  from dvd_imp_le[OF g(2)] x xp have gp: "g < p" by arith
+  from g(2) x have "g \<noteq> 0" by - (rule ccontr, simp)
+  with g gp p[unfolded prime_def] have False by blast}
+thus ?thesis by blast
+qed
+
+lemma even_dvd[simp]: "even (n::nat) \<longleftrightarrow> 2 dvd n" by presburger
+lemma prime_odd: "prime p \<Longrightarrow> p = 2 \<or> odd p" unfolding prime_def by auto
+
+(* One property of coprimality is easier to prove via prime factors.         *)
+
+lemma prime_divprod_pow:
+  assumes p: "prime p" and ab: "coprime a b" and pab: "p^n dvd a * b"
+  shows "p^n dvd a \<or> p^n dvd b"
+proof-
+  {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
+      apply (cases "n=0", simp_all)
+      apply (cases "a=1", simp_all) done}
+  moreover
+  {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
+    then obtain m where m: "n = Suc m" by (cases n, auto)
+    from divides_exp2[OF n pab] have pab': "p dvd a*b" .
+    from prime_divprod[OF p pab']
+    have "p dvd a \<or> p dvd b" .
+    moreover
+    {assume pa: "p dvd a"
+      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
+      from coprime_prime[OF ab, of p] p pa have "\<not> p dvd b" by blast
+      with prime_coprime[OF p, of b] b
+      have cpb: "coprime b p" using coprime_commute by blast
+      from coprime_exp[OF cpb] have pnb: "coprime (p^n) b"
+      from coprime_divprod[OF pnba pnb] have ?thesis by blast }
+    moreover
+    {assume pb: "p dvd b"
+      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
+      from coprime_prime[OF ab, of p] p pb have "\<not> p dvd a" by blast
+      with prime_coprime[OF p, of a] a
+      have cpb: "coprime a p" using coprime_commute by blast
+      from coprime_exp[OF cpb] have pnb: "coprime (p^n) a"
+      from coprime_divprod[OF pab pnb] have ?thesis by blast }
+    ultimately have ?thesis by blast}
+  ultimately show ?thesis by blast
+qed
+
+lemma nat_mult_eq_one: "(n::nat) * m = 1 \<longleftrightarrow> n = 1 \<and> m = 1" (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+  assume H: "?lhs"
+  hence "n dvd 1" "m dvd 1" unfolding dvd_def by (auto simp add: mult_commute)
+  thus ?rhs by auto
+next
+  assume ?rhs then show ?lhs by auto
+qed
+
+lemma power_Suc0[simp]: "Suc 0 ^ n = Suc 0"
+  unfolding One_nat_def[symmetric] power_one ..
+lemma coprime_pow: assumes ab: "coprime a b" and abcn: "a * b = c ^n"
+  shows "\<exists>r s. a = r^n  \<and> b = s ^n"
+  using ab abcn
+proof(induct c arbitrary: a b rule: nat_less_induct)
+  fix c a b
+  assume H: "\<forall>m<c. \<forall>a b. coprime a b \<longrightarrow> a * b = m ^ n \<longrightarrow> (\<exists>r s. a = r ^ n \<and> b = s ^ n)" "coprime a b" "a * b = c ^ n"
+  let ?ths = "\<exists>r s. a = r^n  \<and> b = s ^n"
+  {assume n: "n = 0"
+    with H(3) power_one have "a*b = 1" by simp
+    hence "a = 1 \<and> b = 1" by simp
+    hence ?ths
+      apply -
+      apply (rule exI[where x=1])
+      apply (rule exI[where x=1])
+      using power_one[of  n]
+      by simp}
+  moreover
+  {assume n: "n \<noteq> 0" then obtain m where m: "n = Suc m" by (cases n, auto)
+    {assume c: "c = 0"
+      with H(3) m H(2) have ?ths apply simp
+	apply (cases "a=0", simp_all)
+	apply (rule exI[where x="0"], simp)
+	apply (rule exI[where x="0"], simp)
+	done}
+    moreover
+    {assume "c=1" with H(3) power_one have "a*b = 1" by simp
+	hence "a = 1 \<and> b = 1" by simp
+	hence ?ths
+      apply -
+      apply (rule exI[where x=1])
+      apply (rule exI[where x=1])
+      using power_one[of  n]
+      by simp}
+  moreover
+  {assume c: "c\<noteq>1" "c \<noteq> 0"
+    from prime_factor[OF c(1)] obtain p where p: "prime p" "p dvd c" by blast
+    from prime_divprod_pow[OF p(1) H(2), unfolded H(3), OF divides_exp[OF p(2), of n]]
+    have pnab: "p ^ n dvd a \<or> p^n dvd b" .
+    from p(2) obtain l where l: "c = p*l" unfolding dvd_def by blast
+    have pn0: "p^n \<noteq> 0" using n prime_ge_2 [OF p(1)] by (simp add: neq0_conv)
+    {assume pa: "p^n dvd a"
+      then obtain k where k: "a = p^n * k" unfolding dvd_def by blast
+      from l have "l dvd c" by auto
+      with dvd_imp_le[of l c] c have "l \<le> c" by auto
+      moreover {assume "l = c" with l c  have "p = 1" by simp with p have False by simp}
+      ultimately have lc: "l < c" by arith
+      from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" b]]]
+      have kb: "coprime k b" by (simp add: coprime_commute)
+      from H(3) l k pn0 have kbln: "k * b = l ^ n"
+	by (auto simp add: power_mult_distrib)
+      from H(1)[rule_format, OF lc kb kbln]
+      obtain r s where rs: "k = r ^n" "b = s^n" by blast
+      from k rs(1) have "a = (p*r)^n" by (simp add: power_mult_distrib)
+      with rs(2) have ?ths by blast }
+    moreover
+    {assume pb: "p^n dvd b"
+      then obtain k where k: "b = p^n * k" unfolding dvd_def by blast
+      from l have "l dvd c" by auto
+      with dvd_imp_le[of l c] c have "l \<le> c" by auto
+      moreover {assume "l = c" with l c  have "p = 1" by simp with p have False by simp}
+      ultimately have lc: "l < c" by arith
+      from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" a]]]
+      have kb: "coprime k a" by (simp add: coprime_commute)
+      from H(3) l k pn0 n have kbln: "k * a = l ^ n"
+	by (simp add: power_mult_distrib mult_commute)
+      from H(1)[rule_format, OF lc kb kbln]
+      obtain r s where rs: "k = r ^n" "a = s^n" by blast
+      from k rs(1) have "b = (p*r)^n" by (simp add: power_mult_distrib)
+      with rs(2) have ?ths by blast }
+    ultimately have ?ths using pnab by blast}
+  ultimately have ?ths by blast}
+ultimately show ?ths by blast
+qed
+
+(* More useful lemmas.                                                       *)
+lemma prime_product:
+  "prime (p*q) \<Longrightarrow> p = 1 \<or> q  = 1" unfolding prime_def by auto
+
+lemma prime_exp: "prime (p^n) \<longleftrightarrow> prime p \<and> n = 1"
+proof(induct n)
+  case 0 thus ?case by simp
+next
+  case (Suc n)
+  {assume "p = 0" hence ?case by simp}
+  moreover
+  {assume "p=1" hence ?case by simp}
+  moreover
+  {assume p: "p \<noteq> 0" "p\<noteq>1"
+    {assume pp: "prime (p^Suc n)"
+      hence "p = 1 \<or> p^n = 1" using prime_product[of p "p^n"] by simp
+      with p have n: "n = 0"
+	by (simp only: exp_eq_1 ) simp
+      with pp have "prime p \<and> Suc n = 1" by simp}
+    moreover
+    {assume n: "prime p \<and> Suc n = 1" hence "prime (p^Suc n)" by simp}
+    ultimately have ?case by blast}
+  ultimately show ?case by blast
+qed
+
+lemma prime_power_mult:
+  assumes p: "prime p" and xy: "x * y = p ^ k"
+  shows "\<exists>i j. x = p ^i \<and> y = p^ j"
+  using xy
+proof(induct k arbitrary: x y)
+  case 0 thus ?case apply simp by (rule exI[where x="0"], simp)
+next
+  case (Suc k x y)
+  from Suc.prems have pxy: "p dvd x*y" by auto
+  from prime_divprod[OF p pxy] have pxyc: "p dvd x \<or> p dvd y" .
+  from p have p0: "p \<noteq> 0" by - (rule ccontr, simp)
+  {assume px: "p dvd x"
+    then obtain d where d: "x = p*d" unfolding dvd_def by blast
+    from Suc.prems d  have "p*d*y = p^Suc k" by simp
+    hence th: "d*y = p^k" using p0 by simp
+    from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast
+    with d have "x = p^Suc i" by simp
+    with ij(2) have ?case by blast}
+  moreover
+  {assume px: "p dvd y"
+    then obtain d where d: "y = p*d" unfolding dvd_def by blast
+    from Suc.prems d  have "p*d*x = p^Suc k" by (simp add: mult_commute)
+    hence th: "d*x = p^k" using p0 by simp
+    from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast
+    with d have "y = p^Suc i" by simp
+    with ij(2) have ?case by blast}
+  ultimately show ?case  using pxyc by blast
+qed
+
+lemma prime_power_exp: assumes p: "prime p" and n:"n \<noteq> 0"
+  and xn: "x^n = p^k" shows "\<exists>i. x = p^i"
+  using n xn
+proof(induct n arbitrary: k)
+  case 0 thus ?case by simp
+next
+  case (Suc n k) hence th: "x*x^n = p^k" by simp
+  {assume "n = 0" with prems have ?case apply simp
+      by (rule exI[where x="k"],simp)}
+  moreover
+  {assume n: "n \<noteq> 0"
+    from prime_power_mult[OF p th]
+    obtain i j where ij: "x = p^i" "x^n = p^j"by blast
+    from Suc.hyps[OF n ij(2)] have ?case .}
+  ultimately show ?case by blast
+qed
+
+lemma divides_primepow: assumes p: "prime p"
+  shows "d dvd p^k \<longleftrightarrow> (\<exists> i. i \<le> k \<and> d = p ^i)"
+proof
+  assume H: "d dvd p^k" then obtain e where e: "d*e = p^k"
+    unfolding dvd_def  apply (auto simp add: mult_commute) by blast
+  from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast
+  from prime_ge_2[OF p] have p1: "p > 1" by arith
+  from e ij have "p^(i + j) = p^k" by (simp add: power_add)
+  hence "i + j = k" using power_inject_exp[of p "i+j" k, OF p1] by simp
+  hence "i \<le> k" by arith
+  with ij(1) show "\<exists>i\<le>k. d = p ^ i" by blast
+next
+  {fix i assume H: "i \<le> k" "d = p^i"
+    hence "\<exists>j. k = i + j" by arith
+    then obtain j where j: "k = i + j" by blast