src/HOL/Old_Number_Theory/Primes.thy
author wenzelm
Sat Oct 10 16:26:23 2015 +0200 (2015-10-10)
changeset 61382 efac889fccbc
parent 61076 bdc1e2f0a86a
child 62348 9a5f43dac883
permissions -rw-r--r--
isabelle update_cartouches;
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(*  Title:      HOL/Old_Number_Theory/Primes.thy
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    Author:     Amine Chaieb, Christophe Tabacznyj and Lawrence C Paulson
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    Copyright   1996  University of Cambridge
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*)
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section \<open>Primality on nat\<close>
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theory Primes
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imports Complex_Main Legacy_GCD
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begin
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definition coprime :: "nat => nat => bool"
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  where "coprime m n \<longleftrightarrow> gcd m n = 1"
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definition prime :: "nat \<Rightarrow> bool"
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  where "prime p \<longleftrightarrow> (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
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lemma two_is_prime: "prime 2"
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  apply (auto simp add: prime_def)
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  apply (case_tac m)
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   apply (auto dest!: dvd_imp_le)
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  done
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lemma prime_imp_relprime: "prime p ==> \<not> p dvd n ==> gcd p n = 1"
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  apply (auto simp add: prime_def)
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  apply (metis gcd_dvd1 gcd_dvd2)
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  done
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text \<open>
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  This theorem leads immediately to a proof of the uniqueness of
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  factorization.  If @{term p} divides a product of primes then it is
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  one of those primes.
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\<close>
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lemma prime_dvd_mult: "prime p ==> p dvd m * n ==> p dvd m \<or> p dvd n"
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  by (blast intro: relprime_dvd_mult prime_imp_relprime)
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lemma prime_dvd_square: "prime p ==> p dvd m^Suc (Suc 0) ==> p dvd m"
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  by (auto dest: prime_dvd_mult)
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lemma prime_dvd_power_two: "prime p ==> p dvd m\<^sup>2 ==> p dvd m"
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  by (rule prime_dvd_square) (simp_all add: power2_eq_square)
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lemma exp_eq_1:"(x::nat)^n = 1 \<longleftrightarrow> x = 1 \<or> n = 0"
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by (induct n, auto)
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lemma exp_mono_lt: "(x::nat) ^ (Suc n) < y ^ (Suc n) \<longleftrightarrow> x < y"
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by(metis linorder_not_less not_less0 power_le_imp_le_base power_less_imp_less_base)
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lemma exp_mono_le: "(x::nat) ^ (Suc n) \<le> y ^ (Suc n) \<longleftrightarrow> x \<le> y"
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by (simp only: linorder_not_less[symmetric] exp_mono_lt)
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lemma exp_mono_eq: "(x::nat) ^ Suc n = y ^ Suc n \<longleftrightarrow> x = y"
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using power_inject_base[of x n y] by auto
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lemma even_square: assumes e: "even (n::nat)" shows "\<exists>x. n\<^sup>2 = 4*x"
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proof-
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  from e have "2 dvd n" by presburger
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  then obtain k where k: "n = 2*k" using dvd_def by auto
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  hence "n\<^sup>2 = 4 * k\<^sup>2" by (simp add: power2_eq_square)
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  thus ?thesis by blast
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qed
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lemma odd_square: assumes e: "odd (n::nat)" shows "\<exists>x. n\<^sup>2 = 4*x + 1"
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proof-
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  from e have np: "n > 0" by presburger
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  from e have "2 dvd (n - 1)" by presburger
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  then obtain k where "n - 1 = 2 * k" ..
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  hence k: "n = 2*k + 1"  using e by presburger 
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  hence "n\<^sup>2 = 4* (k\<^sup>2 + k) + 1" by algebra   
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  thus ?thesis by blast
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qed
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lemma diff_square: "(x::nat)\<^sup>2 - y\<^sup>2 = (x+y)*(x - y)" 
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proof-
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  have "x \<le> y \<or> y \<le> x" by (rule nat_le_linear)
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  moreover
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  {assume le: "x \<le> y"
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    hence "x\<^sup>2 \<le> y\<^sup>2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def)
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    with le have ?thesis by simp }
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  moreover
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  {assume le: "y \<le> x"
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    hence le2: "y\<^sup>2 \<le> x\<^sup>2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def)
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    from le have "\<exists>z. y + z = x" by presburger
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    then obtain z where z: "x = y + z" by blast 
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    from le2 have "\<exists>z. x\<^sup>2 = y\<^sup>2 + z" by presburger
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    then obtain z2 where z2: "x\<^sup>2 = y\<^sup>2 + z2" by blast
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    from z z2 have ?thesis by simp algebra }
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  ultimately show ?thesis by blast  
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qed
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text \<open>Elementary theory of divisibility\<close>
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lemma divides_ge: "(a::nat) dvd b \<Longrightarrow> b = 0 \<or> a \<le> b" unfolding dvd_def by auto
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lemma divides_antisym: "(x::nat) dvd y \<and> y dvd x \<longleftrightarrow> x = y"
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  using dvd_antisym[of x y] by auto
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lemma divides_add_revr: assumes da: "(d::nat) dvd a" and dab:"d dvd (a + b)"
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  shows "d dvd b"
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proof-
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  from da obtain k where k:"a = d*k" by (auto simp add: dvd_def)
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  from dab obtain k' where k': "a + b = d*k'" by (auto simp add: dvd_def)
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  from k k' have "b = d *(k' - k)" by (simp add : diff_mult_distrib2)
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  thus ?thesis unfolding dvd_def by blast
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qed
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declare nat_mult_dvd_cancel_disj[presburger]
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lemma nat_mult_dvd_cancel_disj'[presburger]: 
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  "(m::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 
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lemma divides_mul_l: "(a::nat) dvd b ==> (c * a) dvd (c * b)"
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  by presburger
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lemma divides_mul_r: "(a::nat) dvd b ==> (a * c) dvd (b * c)" by presburger
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lemma divides_cases: "(n::nat) dvd m ==> m = 0 \<or> m = n \<or> 2 * n <= m" 
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  by (auto simp add: dvd_def)
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lemma divides_div_not: "(x::nat) = (q * n) + r \<Longrightarrow> 0 < r \<Longrightarrow> r < n ==> ~(n dvd x)"
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proof(auto simp add: dvd_def)
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  fix k assume H: "0 < r" "r < n" "q * n + r = n * k"
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  from H(3) have r: "r = n* (k -q)" by(simp add: diff_mult_distrib2 mult.commute)
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  {assume "k - q = 0" with r H(1) have False by simp}
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  moreover
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  {assume "k - q \<noteq> 0" with r have "r \<ge> n" by auto
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    with H(2) have False by simp}
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  ultimately show False by blast
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qed
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lemma divides_exp: "(x::nat) dvd y ==> x ^ n dvd y ^ n"
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  by (auto simp add: power_mult_distrib dvd_def)
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lemma divides_exp2: "n \<noteq> 0 \<Longrightarrow> (x::nat) ^ n dvd y \<Longrightarrow> x dvd y" 
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  by (induct n ,auto simp add: dvd_def)
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fun fact :: "nat \<Rightarrow> nat" where
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  "fact 0 = 1"
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| "fact (Suc n) = Suc n * fact n"
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lemma fact_lt: "0 < fact n" by(induct n, simp_all)
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lemma fact_le: "fact n \<ge> 1" using fact_lt[of n] by simp 
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lemma fact_mono: assumes le: "m \<le> n" shows "fact m \<le> fact n"
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proof-
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  from le have "\<exists>i. n = m+i" by presburger
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  then obtain i where i: "n = m+i" by blast 
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  have "fact m \<le> fact (m + i)"
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  proof(induct m)
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    case 0 thus ?case using fact_le[of i] by simp
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  next
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    case (Suc m)
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    have "fact (Suc m) = Suc m * fact m" by simp
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    have th1: "Suc m \<le> Suc (m + i)" by simp
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    from mult_le_mono[of "Suc m" "Suc (m+i)" "fact m" "fact (m+i)", OF th1 Suc.hyps]
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    show ?case by simp
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  qed
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  thus ?thesis using i by simp
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qed
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lemma divides_fact: "1 <= p \<Longrightarrow> p <= n ==> p dvd fact n"
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proof(induct n arbitrary: p)
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  case 0 thus ?case by simp
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next
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  case (Suc n p)
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  from Suc.prems have "p = Suc n \<or> p \<le> n" by presburger 
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  moreover
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  {assume "p = Suc n" hence ?case  by (simp only: fact.simps dvd_triv_left)}
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  moreover
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  {assume "p \<le> n"
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    with Suc.prems(1) Suc.hyps have th: "p dvd fact n" by simp
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    from dvd_mult[OF th] have ?case by (simp only: fact.simps) }
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  ultimately show ?case by blast
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qed
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declare dvd_triv_left[presburger]
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declare dvd_triv_right[presburger]
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lemma divides_rexp: 
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  "x dvd y \<Longrightarrow> (x::nat) dvd (y^(Suc n))" by (simp add: dvd_mult2[of x y])
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text \<open>Coprimality\<close>
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lemma coprime: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
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using gcd_unique[of 1 a b, simplified] by (auto simp add: coprime_def)
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lemma coprime_commute: "coprime a b \<longleftrightarrow> coprime b a" by (simp add: coprime_def gcd_commute)
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lemma coprime_bezout: "coprime a b \<longleftrightarrow> (\<exists>x y. a * x - b * y = 1 \<or> b * x - a * y = 1)"
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using coprime_def gcd_bezout by auto
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lemma coprime_divprod: "d dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
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  using relprime_dvd_mult_iff[of d a b] by (auto simp add: coprime_def mult.commute)
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lemma coprime_1[simp]: "coprime a 1" by (simp add: coprime_def)
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lemma coprime_1'[simp]: "coprime 1 a" by (simp add: coprime_def)
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lemma coprime_Suc0[simp]: "coprime a (Suc 0)" by (simp add: coprime_def)
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lemma coprime_Suc0'[simp]: "coprime (Suc 0) a" by (simp add: coprime_def)
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lemma gcd_coprime: 
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  assumes z: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b" 
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  shows    "coprime a' b'"
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proof-
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  let ?g = "gcd a b"
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  {assume bz: "a = 0" from b bz z a have ?thesis by (simp add: gcd_zero coprime_def)}
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  moreover 
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  {assume az: "a\<noteq> 0" 
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    from z have z': "?g > 0" by simp
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    from bezout_gcd_strong[OF az, of b] 
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    obtain x y where xy: "a*x = b*y + ?g" by blast
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    from xy a b have "?g * a'*x = ?g * (b'*y + 1)" by (simp add: algebra_simps)
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    hence "?g * (a'*x) = ?g * (b'*y + 1)" by (simp add: mult.assoc)
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    hence "a'*x = (b'*y + 1)"
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      by (simp only: nat_mult_eq_cancel1[OF z']) 
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    hence "a'*x - b'*y = 1" by simp
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    with coprime_bezout[of a' b'] have ?thesis by auto}
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  ultimately show ?thesis by blast
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qed
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lemma coprime_0: "coprime d 0 \<longleftrightarrow> d = 1" by (simp add: coprime_def)
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lemma coprime_mul: assumes da: "coprime d a" and db: "coprime d b"
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  shows "coprime d (a * b)"
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proof-
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  from da have th: "gcd a d = 1" by (simp add: coprime_def gcd_commute)
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  from gcd_mult_cancel[of a d b, OF th] db[unfolded coprime_def] have "gcd d (a*b) = 1"
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    by (simp add: gcd_commute)
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  thus ?thesis unfolding coprime_def .
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qed
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lemma coprime_lmul2: assumes dab: "coprime d (a * b)" shows "coprime d b"
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using dab unfolding coprime_bezout
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apply clarsimp
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apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all)
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apply (rule_tac x="x" in exI)
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apply (rule_tac x="a*y" in exI)
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apply (simp add: ac_simps)
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apply (rule_tac x="a*x" in exI)
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apply (rule_tac x="y" in exI)
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apply (simp add: ac_simps)
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done
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lemma coprime_rmul2: "coprime d (a * b) \<Longrightarrow> coprime d a"
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unfolding coprime_bezout
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apply clarsimp
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apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all)
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apply (rule_tac x="x" in exI)
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apply (rule_tac x="b*y" in exI)
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apply (simp add: ac_simps)
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apply (rule_tac x="b*x" in exI)
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apply (rule_tac x="y" in exI)
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apply (simp add: ac_simps)
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done
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lemma coprime_mul_eq: "coprime d (a * b) \<longleftrightarrow> coprime d a \<and>  coprime d b"
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  using coprime_rmul2[of d a b] coprime_lmul2[of d a b] coprime_mul[of d a b] 
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  by blast
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lemma gcd_coprime_exists:
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  assumes nz: "gcd a b \<noteq> 0" 
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  shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
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proof-
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  let ?g = "gcd a b"
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  from gcd_dvd1[of a b] gcd_dvd2[of a b] 
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  obtain a' b' where "a = ?g*a'"  "b = ?g*b'" unfolding dvd_def by blast
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  hence ab': "a = a'*?g" "b = b'*?g" by algebra+
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  from ab' gcd_coprime[OF nz ab'] show ?thesis by blast
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qed
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lemma coprime_exp: "coprime d a ==> coprime d (a^n)" 
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  by(induct n, simp_all add: coprime_mul)
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lemma coprime_exp_imp: "coprime a b ==> coprime (a ^n) (b ^n)"
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  by (induct n, simp_all add: coprime_mul_eq coprime_commute coprime_exp)
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lemma coprime_refl[simp]: "coprime n n \<longleftrightarrow> n = 1" by (simp add: coprime_def)
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lemma coprime_plus1[simp]: "coprime (n + 1) n"
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  apply (simp add: coprime_bezout)
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  apply (rule exI[where x=1])
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  apply (rule exI[where x=1])
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  apply simp
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  done
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lemma coprime_minus1: "n \<noteq> 0 ==> coprime (n - 1) n"
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  using coprime_plus1[of "n - 1"] coprime_commute[of "n - 1" n] by auto
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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"
chaieb@26125
   278
proof-
haftmann@27556
   279
  let ?g = "gcd a b"
chaieb@26125
   280
  {assume z: "?g = 0" hence ?thesis 
chaieb@26125
   281
      apply (cases n, simp)
chaieb@26125
   282
      apply arith
chaieb@26125
   283
      apply (simp only: z power_0_Suc)
chaieb@26125
   284
      apply (rule exI[where x=0])
chaieb@26125
   285
      apply (rule exI[where x=0])
wenzelm@41541
   286
      apply simp
wenzelm@41541
   287
      done }
chaieb@26125
   288
  moreover
chaieb@26125
   289
  {assume z: "?g \<noteq> 0"
chaieb@26125
   290
    from gcd_dvd1[of a b] gcd_dvd2[of a b] obtain a' b' where
haftmann@57514
   291
      ab': "a = a'*?g" "b = b'*?g" unfolding dvd_def by (auto simp add: ac_simps)
chaieb@26125
   292
    hence ab'': "?g*a' = a" "?g * b' = b" by algebra+
chaieb@26125
   293
    from coprime_exp_imp[OF gcd_coprime[OF z ab'], unfolded coprime_bezout, of n]
chaieb@26125
   294
    obtain x y where "a'^n * x - b'^n * y = 1 \<or> b'^n * x - a'^n * y = 1"  by blast
chaieb@26125
   295
    hence "?g^n * (a'^n * x - b'^n * y) = ?g^n \<or> ?g^n*(b'^n * x - a'^n * y) = ?g^n"
chaieb@26125
   296
      using z by auto 
chaieb@26125
   297
    then have "a^n * x - b^n * y = ?g^n \<or> b^n * x - a^n * y = ?g^n"
chaieb@26125
   298
      using z ab'' by (simp only: power_mult_distrib[symmetric] 
haftmann@57512
   299
        diff_mult_distrib2 mult.assoc[symmetric])
chaieb@26125
   300
    hence  ?thesis by blast }
chaieb@26125
   301
  ultimately show ?thesis by blast
chaieb@26125
   302
qed
chaieb@27567
   303
chaieb@27567
   304
lemma gcd_exp: "gcd (a^n) (b^n) = gcd a b^n"
chaieb@26125
   305
proof-
haftmann@27556
   306
  let ?g = "gcd (a^n) (b^n)"
chaieb@27567
   307
  let ?gn = "gcd a b^n"
chaieb@26125
   308
  {fix e assume H: "e dvd a^n" "e dvd b^n"
chaieb@26125
   309
    from bezout_gcd_pow[of a n b] obtain x y 
chaieb@26125
   310
      where xy: "a ^ n * x - b ^ n * y = ?gn \<or> b ^ n * x - a ^ n * y = ?gn" by blast
nipkow@31952
   311
    from dvd_diff_nat [OF dvd_mult2[OF H(1), of x] dvd_mult2[OF H(2), of y]]
nipkow@31952
   312
      dvd_diff_nat [OF dvd_mult2[OF H(2), of x] dvd_mult2[OF H(1), of y]] xy
haftmann@27556
   313
    have "e dvd ?gn" by (cases "a ^ n * x - b ^ n * y = gcd a b ^ n", simp_all)}
chaieb@26125
   314
  hence th:  "\<forall>e. e dvd a^n \<and> e dvd b^n \<longrightarrow> e dvd ?gn" by blast
chaieb@26125
   315
  from divides_exp[OF gcd_dvd1[of a b], of n] divides_exp[OF gcd_dvd2[of a b], of n] th
chaieb@26125
   316
    gcd_unique have "?gn = ?g" by blast thus ?thesis by simp 
chaieb@26125
   317
qed
chaieb@26125
   318
chaieb@26125
   319
lemma coprime_exp2:  "coprime (a ^ Suc n) (b^ Suc n) \<longleftrightarrow> coprime a b"
chaieb@26125
   320
by (simp only: coprime_def gcd_exp exp_eq_1) simp
chaieb@26125
   321
chaieb@26125
   322
lemma division_decomp: assumes dc: "(a::nat) dvd b * c"
chaieb@26125
   323
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
chaieb@26125
   324
proof-
haftmann@27556
   325
  let ?g = "gcd a b"
chaieb@26125
   326
  {assume "?g = 0" with dc have ?thesis apply (simp add: gcd_zero)
chaieb@26125
   327
      apply (rule exI[where x="0"])
chaieb@26125
   328
      by (rule exI[where x="c"], simp)}
chaieb@26125
   329
  moreover
chaieb@26125
   330
  {assume z: "?g \<noteq> 0"
chaieb@26125
   331
    from gcd_coprime_exists[OF z]
chaieb@26125
   332
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast
chaieb@26125
   333
    from gcd_dvd2[of a b] have thb: "?g dvd b" .
chaieb@26125
   334
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast  
chaieb@26125
   335
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
chaieb@26125
   336
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
haftmann@57512
   337
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc)
chaieb@26125
   338
    with z have th_1: "a' dvd b'*c" by simp
chaieb@26125
   339
    from coprime_divprod[OF th_1 ab'(3)] have thc: "a' dvd c" . 
chaieb@26125
   340
    from ab' have "a = ?g*a'" by algebra
chaieb@26125
   341
    with thb thc have ?thesis by blast }
chaieb@26125
   342
  ultimately show ?thesis by blast
chaieb@26125
   343
qed
chaieb@26125
   344
chaieb@26125
   345
lemma nat_power_eq_0_iff: "(m::nat) ^ n = 0 \<longleftrightarrow> n \<noteq> 0 \<and> m = 0" by (induct n, auto)
chaieb@26125
   346
chaieb@26125
   347
lemma divides_rev: assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0" shows "a dvd b"
chaieb@26125
   348
proof-
haftmann@27556
   349
  let ?g = "gcd a b"
chaieb@26125
   350
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
chaieb@26125
   351
  {assume "?g = 0" with ab n have ?thesis by (simp add: gcd_zero)}
chaieb@26125
   352
  moreover
chaieb@26125
   353
  {assume z: "?g \<noteq> 0"
wenzelm@41541
   354
    hence zn: "?g ^ n \<noteq> 0" using n by simp
chaieb@26125
   355
    from gcd_coprime_exists[OF z] 
chaieb@26125
   356
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast
chaieb@26125
   357
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n" by (simp add: ab'(1,2)[symmetric])
haftmann@57512
   358
    hence "?g^n*a'^n dvd ?g^n *b'^n" by (simp only: power_mult_distrib mult.commute)
chaieb@26125
   359
    with zn z n have th0:"a'^n dvd b'^n" by (auto simp add: nat_power_eq_0_iff)
chaieb@26125
   360
    have "a' dvd a'^n" by (simp add: m)
chaieb@26125
   361
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
haftmann@57512
   362
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
chaieb@26125
   363
    from coprime_divprod[OF th1 coprime_exp[OF ab'(3), of m]]
chaieb@26125
   364
    have "a' dvd b'" .
chaieb@26125
   365
    hence "a'*?g dvd b'*?g" by simp
chaieb@26125
   366
    with ab'(1,2)  have ?thesis by simp }
chaieb@26125
   367
  ultimately show ?thesis by blast
chaieb@26125
   368
qed
chaieb@26125
   369
chaieb@26125
   370
lemma divides_mul: assumes mr: "m dvd r" and nr: "n dvd r" and mn:"coprime m n" 
chaieb@26125
   371
  shows "m * n dvd r"
chaieb@26125
   372
proof-
chaieb@26125
   373
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
chaieb@26125
   374
    unfolding dvd_def by blast
haftmann@57512
   375
  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
chaieb@26125
   376
  hence "m dvd n'" using relprime_dvd_mult_iff[OF mn[unfolded coprime_def]] by simp
chaieb@26125
   377
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
chaieb@26125
   378
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@26125
   379
qed
chaieb@26125
   380
wenzelm@26144
   381
wenzelm@61382
   382
text \<open>A binary form of the Chinese Remainder Theorem.\<close>
chaieb@26125
   383
chaieb@26125
   384
lemma chinese_remainder: assumes ab: "coprime a b" and a:"a \<noteq> 0" and b:"b \<noteq> 0"
chaieb@26125
   385
  shows "\<exists>x q1 q2. x = u + q1 * a \<and> x = v + q2 * b"
chaieb@26125
   386
proof-
chaieb@26125
   387
  from bezout_add_strong[OF a, of b] bezout_add_strong[OF b, of a]
chaieb@26125
   388
  obtain d1 x1 y1 d2 x2 y2 where dxy1: "d1 dvd a" "d1 dvd b" "a * x1 = b * y1 + d1" 
chaieb@26125
   389
    and dxy2: "d2 dvd b" "d2 dvd a" "b * x2 = a * y2 + d2" by blast
chaieb@26125
   390
  from gcd_unique[of 1 a b, simplified ab[unfolded coprime_def], simplified] 
chaieb@26125
   391
    dxy1(1,2) dxy2(1,2) have d12: "d1 = 1" "d2 =1" by auto
chaieb@26125
   392
  let ?x = "v * a * x1 + u * b * x2"
chaieb@26125
   393
  let ?q1 = "v * x1 + u * y2"
chaieb@26125
   394
  let ?q2 = "v * y1 + u * x2"
chaieb@26125
   395
  from dxy2(3)[simplified d12] dxy1(3)[simplified d12] 
chaieb@26125
   396
  have "?x = u + ?q1 * a" "?x = v + ?q2 * b" by algebra+ 
chaieb@26125
   397
  thus ?thesis by blast
chaieb@26125
   398
qed
chaieb@26125
   399
wenzelm@61382
   400
text \<open>Primality\<close>
wenzelm@26144
   401
wenzelm@61382
   402
text \<open>A few useful theorems about primes\<close>
chaieb@26125
   403
chaieb@26125
   404
lemma prime_0[simp]: "~prime 0" by (simp add: prime_def)
chaieb@26125
   405
lemma prime_1[simp]: "~ prime 1"  by (simp add: prime_def)
chaieb@26125
   406
lemma prime_Suc0[simp]: "~ prime (Suc 0)"  by (simp add: prime_def)
chaieb@26125
   407
chaieb@26125
   408
lemma prime_ge_2: "prime p ==> p \<ge> 2" by (simp add: prime_def)
chaieb@26125
   409
lemma prime_factor: assumes n: "n \<noteq> 1" shows "\<exists> p. prime p \<and> p dvd n"
chaieb@26125
   410
using n
chaieb@26125
   411
proof(induct n rule: nat_less_induct)
chaieb@26125
   412
  fix n
chaieb@26125
   413
  assume H: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)" "n \<noteq> 1"
chaieb@26125
   414
  let ?ths = "\<exists>p. prime p \<and> p dvd n"
chaieb@26125
   415
  {assume "n=0" hence ?ths using two_is_prime by auto}
chaieb@26125
   416
  moreover
chaieb@26125
   417
  {assume nz: "n\<noteq>0" 
chaieb@26125
   418
    {assume "prime n" hence ?ths by - (rule exI[where x="n"], simp)}
chaieb@26125
   419
    moreover
chaieb@26125
   420
    {assume n: "\<not> prime n"
chaieb@26125
   421
      with nz H(2) 
chaieb@26125
   422
      obtain k where k:"k dvd n" "k \<noteq> 1" "k \<noteq> n" by (auto simp add: prime_def) 
chaieb@26125
   423
      from dvd_imp_le[OF k(1)] nz k(3) have kn: "k < n" by simp
chaieb@26125
   424
      from H(1)[rule_format, OF kn k(2)] obtain p where p: "prime p" "p dvd k" by blast
chaieb@26125
   425
      from dvd_trans[OF p(2) k(1)] p(1) have ?ths by blast}
chaieb@26125
   426
    ultimately have ?ths by blast}
chaieb@26125
   427
  ultimately show ?ths by blast
chaieb@26125
   428
qed
chaieb@26125
   429
chaieb@26125
   430
lemma prime_factor_lt: assumes p: "prime p" and n: "n \<noteq> 0" and npm:"n = p * m"
chaieb@26125
   431
  shows "m < n"
chaieb@26125
   432
proof-
chaieb@26125
   433
  {assume "m=0" with n have ?thesis by simp}
chaieb@26125
   434
  moreover
chaieb@26125
   435
  {assume m: "m \<noteq> 0"
chaieb@26125
   436
    from npm have mn: "m dvd n" unfolding dvd_def by auto
chaieb@26125
   437
    from npm m have "n \<noteq> m" using p by auto
chaieb@26125
   438
    with dvd_imp_le[OF mn] n have ?thesis by simp}
chaieb@26125
   439
  ultimately show ?thesis by blast
chaieb@26125
   440
qed
chaieb@26125
   441
chaieb@26125
   442
lemma euclid_bound: "\<exists>p. prime p \<and> n < p \<and>  p <= Suc (fact n)"
chaieb@26125
   443
proof-
chaieb@26125
   444
  have f1: "fact n + 1 \<noteq> 1" using fact_le[of n] by arith 
chaieb@26125
   445
  from prime_factor[OF f1] obtain p where p: "prime p" "p dvd fact n + 1" by blast
chaieb@26125
   446
  from dvd_imp_le[OF p(2)] have pfn: "p \<le> fact n + 1" by simp
chaieb@26125
   447
  {assume np: "p \<le> n"
chaieb@26125
   448
    from p(1) have p1: "p \<ge> 1" by (cases p, simp_all)
chaieb@26125
   449
    from divides_fact[OF p1 np] have pfn': "p dvd fact n" .
chaieb@26125
   450
    from divides_add_revr[OF pfn' p(2)] p(1) have False by simp}
chaieb@26125
   451
  hence "n < p" by arith
chaieb@26125
   452
  with p(1) pfn show ?thesis by auto
chaieb@26125
   453
qed
chaieb@26125
   454
chaieb@26125
   455
lemma euclid: "\<exists>p. prime p \<and> p > n" using euclid_bound by auto
nipkow@31044
   456
chaieb@26125
   457
lemma primes_infinite: "\<not> (finite {p. prime p})"
nipkow@31044
   458
apply(simp add: finite_nat_set_iff_bounded_le)
nipkow@31044
   459
apply (metis euclid linorder_not_le)
nipkow@31044
   460
done
chaieb@26125
   461
chaieb@26125
   462
lemma coprime_prime: assumes ab: "coprime a b"
chaieb@26125
   463
  shows "~(prime p \<and> p dvd a \<and> p dvd b)"
chaieb@26125
   464
proof
chaieb@26125
   465
  assume "prime p \<and> p dvd a \<and> p dvd b"
chaieb@26125
   466
  thus False using ab gcd_greatest[of p a b] by (simp add: coprime_def)
chaieb@26125
   467
qed
chaieb@26125
   468
lemma coprime_prime_eq: "coprime a b \<longleftrightarrow> (\<forall>p. ~(prime p \<and> p dvd a \<and> p dvd b))" 
chaieb@26125
   469
  (is "?lhs = ?rhs")
chaieb@26125
   470
proof-
chaieb@26125
   471
  {assume "?lhs" with coprime_prime  have ?rhs by blast}
chaieb@26125
   472
  moreover
chaieb@26125
   473
  {assume r: "?rhs" and c: "\<not> ?lhs"
chaieb@26125
   474
    then obtain g where g: "g\<noteq>1" "g dvd a" "g dvd b" unfolding coprime_def by blast
chaieb@26125
   475
    from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast
chaieb@26125
   476
    from dvd_trans [OF p(2) g(2)] dvd_trans [OF p(2) g(3)] 
chaieb@26125
   477
    have "p dvd a" "p dvd b" . with p(1) r have False by blast}
chaieb@26125
   478
  ultimately show ?thesis by blast
chaieb@26125
   479
qed
chaieb@26125
   480
chaieb@26125
   481
lemma prime_coprime: assumes p: "prime p" 
chaieb@26125
   482
  shows "n = 1 \<or> p dvd n \<or> coprime p n"
chaieb@26125
   483
using p prime_imp_relprime[of p n] by (auto simp add: coprime_def)
chaieb@26125
   484
chaieb@26125
   485
lemma prime_coprime_strong: "prime p \<Longrightarrow> p dvd n \<or> coprime p n"
chaieb@26125
   486
  using prime_coprime[of p n] by auto
chaieb@26125
   487
chaieb@26125
   488
declare  coprime_0[simp]
chaieb@26125
   489
chaieb@26125
   490
lemma coprime_0'[simp]: "coprime 0 d \<longleftrightarrow> d = 1" by (simp add: coprime_commute[of 0 d])
chaieb@26125
   491
lemma coprime_bezout_strong: assumes ab: "coprime a b" and b: "b \<noteq> 1"
chaieb@26125
   492
  shows "\<exists>x y. a * x = b * y + 1"
chaieb@26125
   493
proof-
chaieb@26125
   494
  from ab b have az: "a \<noteq> 0" by - (rule ccontr, auto)
chaieb@26125
   495
  from bezout_gcd_strong[OF az, of b] ab[unfolded coprime_def]
chaieb@26125
   496
  show ?thesis by auto
chaieb@26125
   497
qed
chaieb@26125
   498
chaieb@26125
   499
lemma bezout_prime: assumes p: "prime p"  and pa: "\<not> p dvd a"
chaieb@26125
   500
  shows "\<exists>x y. a*x = p*y + 1"
chaieb@26125
   501
proof-
chaieb@26125
   502
  from p have p1: "p \<noteq> 1" using prime_1 by blast 
chaieb@26125
   503
  from prime_coprime[OF p, of a] p1 pa have ap: "coprime a p" 
chaieb@26125
   504
    by (auto simp add: coprime_commute)
chaieb@26125
   505
  from coprime_bezout_strong[OF ap p1] show ?thesis . 
chaieb@26125
   506
qed
chaieb@26125
   507
lemma prime_divprod: assumes p: "prime p" and pab: "p dvd a*b"
chaieb@26125
   508
  shows "p dvd a \<or> p dvd b"
chaieb@26125
   509
proof-
chaieb@26125
   510
  {assume "a=1" hence ?thesis using pab by simp }
chaieb@26125
   511
  moreover
chaieb@26125
   512
  {assume "p dvd a" hence ?thesis by blast}
chaieb@26125
   513
  moreover
chaieb@26125
   514
  {assume pa: "coprime p a" from coprime_divprod[OF pab pa]  have ?thesis .. }
chaieb@26125
   515
  ultimately show ?thesis using prime_coprime[OF p, of a] by blast
chaieb@26125
   516
qed
chaieb@26125
   517
chaieb@26125
   518
lemma prime_divprod_eq: assumes p: "prime p"
chaieb@26125
   519
  shows "p dvd a*b \<longleftrightarrow> p dvd a \<or> p dvd b"
chaieb@26125
   520
using p prime_divprod dvd_mult dvd_mult2 by auto
chaieb@26125
   521
chaieb@26125
   522
lemma prime_divexp: assumes p:"prime p" and px: "p dvd x^n"
chaieb@26125
   523
  shows "p dvd x"
chaieb@26125
   524
using px
chaieb@26125
   525
proof(induct n)
chaieb@26125
   526
  case 0 thus ?case by simp
chaieb@26125
   527
next
chaieb@26125
   528
  case (Suc n) 
chaieb@26125
   529
  hence th: "p dvd x*x^n" by simp
chaieb@26125
   530
  {assume H: "p dvd x^n"
chaieb@26125
   531
    from Suc.hyps[OF H] have ?case .}
chaieb@26125
   532
  with prime_divprod[OF p th] show ?case by blast
chaieb@26125
   533
qed
chaieb@26125
   534
chaieb@26125
   535
lemma prime_divexp_n: "prime p \<Longrightarrow> p dvd x^n \<Longrightarrow> p^n dvd x^n"
chaieb@26125
   536
  using prime_divexp[of p x n] divides_exp[of p x n] by blast
chaieb@26125
   537
chaieb@26125
   538
lemma coprime_prime_dvd_ex: assumes xy: "\<not>coprime x y"
chaieb@26125
   539
  shows "\<exists>p. prime p \<and> p dvd x \<and> p dvd y"
chaieb@26125
   540
proof-
chaieb@26125
   541
  from xy[unfolded coprime_def] obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd y" 
chaieb@26125
   542
    by blast
chaieb@26125
   543
  from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast
chaieb@26125
   544
  from g(2,3) dvd_trans[OF p(2)] p(1) show ?thesis by auto
chaieb@26125
   545
qed
chaieb@26125
   546
lemma coprime_sos: assumes xy: "coprime x y" 
wenzelm@53077
   547
  shows "coprime (x * y) (x\<^sup>2 + y\<^sup>2)"
chaieb@26125
   548
proof-
wenzelm@53077
   549
  {assume c: "\<not> coprime (x * y) (x\<^sup>2 + y\<^sup>2)"
chaieb@26125
   550
    from coprime_prime_dvd_ex[OF c] obtain p 
wenzelm@53077
   551
      where p: "prime p" "p dvd x*y" "p dvd x\<^sup>2 + y\<^sup>2" by blast
chaieb@26125
   552
    {assume px: "p dvd x"
haftmann@27651
   553
      from dvd_mult[OF px, of x] p(3) 
wenzelm@53077
   554
        obtain r s where "x * x = p * r" and "x\<^sup>2 + y\<^sup>2 = p * s"
haftmann@27651
   555
          by (auto elim!: dvdE)
wenzelm@53077
   556
        then have "y\<^sup>2 = p * (s - r)" 
haftmann@27651
   557
          by (auto simp add: power2_eq_square diff_mult_distrib2)
wenzelm@53077
   558
        then have "p dvd y\<^sup>2" ..
chaieb@26125
   559
      with prime_divexp[OF p(1), of y 2] have py: "p dvd y" .
chaieb@26125
   560
      from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1  
chaieb@26125
   561
      have False by simp }
chaieb@26125
   562
    moreover
chaieb@26125
   563
    {assume py: "p dvd y"
haftmann@27651
   564
      from dvd_mult[OF py, of y] p(3)
wenzelm@53077
   565
        obtain r s where "y * y = p * r" and "x\<^sup>2 + y\<^sup>2 = p * s"
haftmann@27651
   566
          by (auto elim!: dvdE)
wenzelm@53077
   567
        then have "x\<^sup>2 = p * (s - r)" 
haftmann@27651
   568
          by (auto simp add: power2_eq_square diff_mult_distrib2)
wenzelm@53077
   569
        then have "p dvd x\<^sup>2" ..
chaieb@26125
   570
      with prime_divexp[OF p(1), of x 2] have px: "p dvd x" .
chaieb@26125
   571
      from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1  
chaieb@26125
   572
      have False by simp }
chaieb@26125
   573
    ultimately have False using prime_divprod[OF p(1,2)] by blast}
chaieb@26125
   574
  thus ?thesis by blast
chaieb@26125
   575
qed
chaieb@26125
   576
chaieb@26125
   577
lemma distinct_prime_coprime: "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
chaieb@26125
   578
  unfolding prime_def coprime_prime_eq by blast
chaieb@26125
   579
chaieb@26125
   580
lemma prime_coprime_lt: assumes p: "prime p" and x: "0 < x" and xp: "x < p"
chaieb@26125
   581
  shows "coprime x p"
chaieb@26125
   582
proof-
chaieb@26125
   583
  {assume c: "\<not> coprime x p"
chaieb@26125
   584
    then obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd p" unfolding coprime_def by blast
chaieb@26125
   585
  from dvd_imp_le[OF g(2)] x xp have gp: "g < p" by arith
chaieb@26125
   586
  from g(2) x have "g \<noteq> 0" by - (rule ccontr, simp)
chaieb@26125
   587
  with g gp p[unfolded prime_def] have False by blast}
chaieb@26125
   588
thus ?thesis by blast
chaieb@26125
   589
qed
chaieb@26125
   590
chaieb@26125
   591
lemma prime_odd: "prime p \<Longrightarrow> p = 2 \<or> odd p" unfolding prime_def by auto
chaieb@26125
   592
wenzelm@26144
   593
wenzelm@61382
   594
text \<open>One property of coprimality is easier to prove via prime factors.\<close>
chaieb@26125
   595
chaieb@26125
   596
lemma prime_divprod_pow: 
chaieb@26125
   597
  assumes p: "prime p" and ab: "coprime a b" and pab: "p^n dvd a * b"
chaieb@26125
   598
  shows "p^n dvd a \<or> p^n dvd b"
chaieb@26125
   599
proof-
chaieb@26125
   600
  {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis 
chaieb@26125
   601
      apply (cases "n=0", simp_all)
chaieb@26125
   602
      apply (cases "a=1", simp_all) done}
chaieb@26125
   603
  moreover
chaieb@26125
   604
  {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1" 
chaieb@26125
   605
    then obtain m where m: "n = Suc m" by (cases n, auto)
chaieb@26125
   606
    from divides_exp2[OF n pab] have pab': "p dvd a*b" .
chaieb@26125
   607
    from prime_divprod[OF p pab'] 
chaieb@26125
   608
    have "p dvd a \<or> p dvd b" .
chaieb@26125
   609
    moreover
chaieb@26125
   610
    {assume pa: "p dvd a"
haftmann@57512
   611
      have pnba: "p^n dvd b*a" using pab by (simp add: mult.commute)
chaieb@26125
   612
      from coprime_prime[OF ab, of p] p pa have "\<not> p dvd b" by blast
chaieb@26125
   613
      with prime_coprime[OF p, of b] b 
chaieb@26125
   614
      have cpb: "coprime b p" using coprime_commute by blast 
chaieb@26125
   615
      from coprime_exp[OF cpb] have pnb: "coprime (p^n) b" 
wenzelm@32960
   616
        by (simp add: coprime_commute)
chaieb@26125
   617
      from coprime_divprod[OF pnba pnb] have ?thesis by blast }
chaieb@26125
   618
    moreover
chaieb@26125
   619
    {assume pb: "p dvd b"
haftmann@57512
   620
      have pnba: "p^n dvd b*a" using pab by (simp add: mult.commute)
chaieb@26125
   621
      from coprime_prime[OF ab, of p] p pb have "\<not> p dvd a" by blast
chaieb@26125
   622
      with prime_coprime[OF p, of a] a
chaieb@26125
   623
      have cpb: "coprime a p" using coprime_commute by blast 
chaieb@26125
   624
      from coprime_exp[OF cpb] have pnb: "coprime (p^n) a" 
wenzelm@32960
   625
        by (simp add: coprime_commute)
chaieb@26125
   626
      from coprime_divprod[OF pab pnb] have ?thesis by blast }
chaieb@26125
   627
    ultimately have ?thesis by blast}
chaieb@26125
   628
  ultimately show ?thesis by blast
chaieb@26125
   629
qed
chaieb@26125
   630
chaieb@26125
   631
lemma nat_mult_eq_one: "(n::nat) * m = 1 \<longleftrightarrow> n = 1 \<and> m = 1" (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@26125
   632
proof
chaieb@26125
   633
  assume H: "?lhs"
haftmann@57512
   634
  hence "n dvd 1" "m dvd 1" unfolding dvd_def by (auto simp add: mult.commute)
chaieb@26125
   635
  thus ?rhs by auto
chaieb@26125
   636
next
chaieb@26125
   637
  assume ?rhs then show ?lhs by auto
chaieb@26125
   638
qed
chaieb@26125
   639
  
wenzelm@41541
   640
lemma power_Suc0: "Suc 0 ^ n = Suc 0" 
chaieb@26125
   641
  unfolding One_nat_def[symmetric] power_one ..
wenzelm@41541
   642
chaieb@26125
   643
lemma coprime_pow: assumes ab: "coprime a b" and abcn: "a * b = c ^n"
chaieb@26125
   644
  shows "\<exists>r s. a = r^n  \<and> b = s ^n"
chaieb@26125
   645
  using ab abcn
chaieb@26125
   646
proof(induct c arbitrary: a b rule: nat_less_induct)
chaieb@26125
   647
  fix c a b
chaieb@26125
   648
  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" 
chaieb@26125
   649
  let ?ths = "\<exists>r s. a = r^n  \<and> b = s ^n"
chaieb@26125
   650
  {assume n: "n = 0"
chaieb@26125
   651
    with H(3) power_one have "a*b = 1" by simp
chaieb@26125
   652
    hence "a = 1 \<and> b = 1" by simp
chaieb@26125
   653
    hence ?ths 
chaieb@26125
   654
      apply -
chaieb@26125
   655
      apply (rule exI[where x=1])
chaieb@26125
   656
      apply (rule exI[where x=1])
chaieb@26125
   657
      using power_one[of  n]
chaieb@26125
   658
      by simp}
chaieb@26125
   659
  moreover
chaieb@26125
   660
  {assume n: "n \<noteq> 0" then obtain m where m: "n = Suc m" by (cases n, auto)
chaieb@26125
   661
    {assume c: "c = 0"
chaieb@26125
   662
      with H(3) m H(2) have ?ths apply simp 
wenzelm@32960
   663
        apply (cases "a=0", simp_all) 
wenzelm@32960
   664
        apply (rule exI[where x="0"], simp)
wenzelm@32960
   665
        apply (rule exI[where x="0"], simp)
wenzelm@32960
   666
        done}
chaieb@26125
   667
    moreover
chaieb@26125
   668
    {assume "c=1" with H(3) power_one have "a*b = 1" by simp 
wenzelm@32960
   669
        hence "a = 1 \<and> b = 1" by simp
wenzelm@32960
   670
        hence ?ths 
chaieb@26125
   671
      apply -
chaieb@26125
   672
      apply (rule exI[where x=1])
chaieb@26125
   673
      apply (rule exI[where x=1])
chaieb@26125
   674
      using power_one[of  n]
chaieb@26125
   675
      by simp}
chaieb@26125
   676
  moreover
chaieb@26125
   677
  {assume c: "c\<noteq>1" "c \<noteq> 0"
chaieb@26125
   678
    from prime_factor[OF c(1)] obtain p where p: "prime p" "p dvd c" by blast
chaieb@26125
   679
    from prime_divprod_pow[OF p(1) H(2), unfolded H(3), OF divides_exp[OF p(2), of n]] 
chaieb@26125
   680
    have pnab: "p ^ n dvd a \<or> p^n dvd b" . 
chaieb@26125
   681
    from p(2) obtain l where l: "c = p*l" unfolding dvd_def by blast
wenzelm@41541
   682
    have pn0: "p^n \<noteq> 0" using n prime_ge_2 [OF p(1)] by simp
chaieb@26125
   683
    {assume pa: "p^n dvd a"
chaieb@26125
   684
      then obtain k where k: "a = p^n * k" unfolding dvd_def by blast
chaieb@26125
   685
      from l have "l dvd c" by auto
chaieb@26125
   686
      with dvd_imp_le[of l c] c have "l \<le> c" by auto
chaieb@26125
   687
      moreover {assume "l = c" with l c  have "p = 1" by simp with p have False by simp}
chaieb@26125
   688
      ultimately have lc: "l < c" by arith
chaieb@26125
   689
      from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" b]]]
chaieb@26125
   690
      have kb: "coprime k b" by (simp add: coprime_commute) 
chaieb@26125
   691
      from H(3) l k pn0 have kbln: "k * b = l ^ n" 
wenzelm@32960
   692
        by (auto simp add: power_mult_distrib)
chaieb@26125
   693
      from H(1)[rule_format, OF lc kb kbln]
chaieb@26125
   694
      obtain r s where rs: "k = r ^n" "b = s^n" by blast
chaieb@26125
   695
      from k rs(1) have "a = (p*r)^n" by (simp add: power_mult_distrib)
chaieb@26125
   696
      with rs(2) have ?ths by blast }
chaieb@26125
   697
    moreover
chaieb@26125
   698
    {assume pb: "p^n dvd b"
chaieb@26125
   699
      then obtain k where k: "b = p^n * k" unfolding dvd_def by blast
chaieb@26125
   700
      from l have "l dvd c" by auto
chaieb@26125
   701
      with dvd_imp_le[of l c] c have "l \<le> c" by auto
chaieb@26125
   702
      moreover {assume "l = c" with l c  have "p = 1" by simp with p have False by simp}
chaieb@26125
   703
      ultimately have lc: "l < c" by arith
chaieb@26125
   704
      from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" a]]]
chaieb@26125
   705
      have kb: "coprime k a" by (simp add: coprime_commute) 
chaieb@26125
   706
      from H(3) l k pn0 n have kbln: "k * a = l ^ n" 
haftmann@57512
   707
        by (simp add: power_mult_distrib mult.commute)
chaieb@26125
   708
      from H(1)[rule_format, OF lc kb kbln]
chaieb@26125
   709
      obtain r s where rs: "k = r ^n" "a = s^n" by blast
chaieb@26125
   710
      from k rs(1) have "b = (p*r)^n" by (simp add: power_mult_distrib)
chaieb@26125
   711
      with rs(2) have ?ths by blast }
chaieb@26125
   712
    ultimately have ?ths using pnab by blast}
chaieb@26125
   713
  ultimately have ?ths by blast}
chaieb@26125
   714
ultimately show ?ths by blast
chaieb@26125
   715
qed
chaieb@26125
   716
wenzelm@61382
   717
text \<open>More useful lemmas.\<close>
chaieb@26125
   718
lemma prime_product: 
haftmann@27651
   719
  assumes "prime (p * q)"
haftmann@27651
   720
  shows "p = 1 \<or> q = 1"
haftmann@27651
   721
proof -
haftmann@27651
   722
  from assms have 
haftmann@27651
   723
    "1 < p * q" and P: "\<And>m. m dvd p * q \<Longrightarrow> m = 1 \<or> m = p * q"
haftmann@27651
   724
    unfolding prime_def by auto
wenzelm@61382
   725
  from \<open>1 < p * q\<close> have "p \<noteq> 0" by (cases p) auto
haftmann@27651
   726
  then have Q: "p = p * q \<longleftrightarrow> q = 1" by auto
haftmann@27651
   727
  have "p dvd p * q" by simp
haftmann@27651
   728
  then have "p = 1 \<or> p = p * q" by (rule P)
haftmann@27651
   729
  then show ?thesis by (simp add: Q)
haftmann@27651
   730
qed
chaieb@26125
   731
chaieb@26125
   732
lemma prime_exp: "prime (p^n) \<longleftrightarrow> prime p \<and> n = 1"
chaieb@26125
   733
proof(induct n)
chaieb@26125
   734
  case 0 thus ?case by simp
chaieb@26125
   735
next
chaieb@26125
   736
  case (Suc n)
chaieb@26125
   737
  {assume "p = 0" hence ?case by simp}
chaieb@26125
   738
  moreover
chaieb@26125
   739
  {assume "p=1" hence ?case by simp}
chaieb@26125
   740
  moreover
chaieb@26125
   741
  {assume p: "p \<noteq> 0" "p\<noteq>1"
chaieb@26125
   742
    {assume pp: "prime (p^Suc n)"
chaieb@26125
   743
      hence "p = 1 \<or> p^n = 1" using prime_product[of p "p^n"] by simp
chaieb@26125
   744
      with p have n: "n = 0" 
wenzelm@32960
   745
        by (simp only: exp_eq_1 ) simp
chaieb@26125
   746
      with pp have "prime p \<and> Suc n = 1" by simp}
chaieb@26125
   747
    moreover
chaieb@26125
   748
    {assume n: "prime p \<and> Suc n = 1" hence "prime (p^Suc n)" by simp}
chaieb@26125
   749
    ultimately have ?case by blast}
chaieb@26125
   750
  ultimately show ?case by blast
chaieb@26125
   751
qed
chaieb@26125
   752
chaieb@26125
   753
lemma prime_power_mult: 
chaieb@26125
   754
  assumes p: "prime p" and xy: "x * y = p ^ k"
chaieb@26125
   755
  shows "\<exists>i j. x = p ^i \<and> y = p^ j"
chaieb@26125
   756
  using xy
chaieb@26125
   757
proof(induct k arbitrary: x y)
chaieb@26125
   758
  case 0 thus ?case apply simp by (rule exI[where x="0"], simp)
chaieb@26125
   759
next
chaieb@26125
   760
  case (Suc k x y)
chaieb@26125
   761
  from Suc.prems have pxy: "p dvd x*y" by auto
chaieb@26125
   762
  from prime_divprod[OF p pxy] have pxyc: "p dvd x \<or> p dvd y" .
chaieb@26125
   763
  from p have p0: "p \<noteq> 0" by - (rule ccontr, simp) 
chaieb@26125
   764
  {assume px: "p dvd x"
chaieb@26125
   765
    then obtain d where d: "x = p*d" unfolding dvd_def by blast
chaieb@26125
   766
    from Suc.prems d  have "p*d*y = p^Suc k" by simp
chaieb@26125
   767
    hence th: "d*y = p^k" using p0 by simp
chaieb@26125
   768
    from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast
chaieb@26125
   769
    with d have "x = p^Suc i" by simp 
chaieb@26125
   770
    with ij(2) have ?case by blast}
chaieb@26125
   771
  moreover 
chaieb@26125
   772
  {assume px: "p dvd y"
chaieb@26125
   773
    then obtain d where d: "y = p*d" unfolding dvd_def by blast
haftmann@57512
   774
    from Suc.prems d  have "p*d*x = p^Suc k" by (simp add: mult.commute)
chaieb@26125
   775
    hence th: "d*x = p^k" using p0 by simp
chaieb@26125
   776
    from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast
chaieb@26125
   777
    with d have "y = p^Suc i" by simp 
chaieb@26125
   778
    with ij(2) have ?case by blast}
chaieb@26125
   779
  ultimately show ?case  using pxyc by blast
chaieb@26125
   780
qed
chaieb@26125
   781
chaieb@26125
   782
lemma prime_power_exp: assumes p: "prime p" and n:"n \<noteq> 0" 
chaieb@26125
   783
  and xn: "x^n = p^k" shows "\<exists>i. x = p^i"
chaieb@26125
   784
  using n xn
chaieb@26125
   785
proof(induct n arbitrary: k)
chaieb@26125
   786
  case 0 thus ?case by simp
chaieb@26125
   787
next
chaieb@26125
   788
  case (Suc n k) hence th: "x*x^n = p^k" by simp
wenzelm@41541
   789
  {assume "n = 0" with Suc have ?case by simp (rule exI[where x="k"], simp)}
chaieb@26125
   790
  moreover
chaieb@26125
   791
  {assume n: "n \<noteq> 0"
chaieb@26125
   792
    from prime_power_mult[OF p th] 
chaieb@26125
   793
    obtain i j where ij: "x = p^i" "x^n = p^j"by blast
chaieb@26125
   794
    from Suc.hyps[OF n ij(2)] have ?case .}
chaieb@26125
   795
  ultimately show ?case by blast
chaieb@26125
   796
qed
chaieb@26125
   797
chaieb@26125
   798
lemma divides_primepow: assumes p: "prime p" 
chaieb@26125
   799
  shows "d dvd p^k \<longleftrightarrow> (\<exists> i. i \<le> k \<and> d = p ^i)"
chaieb@26125
   800
proof
chaieb@26125
   801
  assume H: "d dvd p^k" then obtain e where e: "d*e = p^k" 
haftmann@57512
   802
    unfolding dvd_def  apply (auto simp add: mult.commute) by blast
chaieb@26125
   803
  from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast
chaieb@26125
   804
  from prime_ge_2[OF p] have p1: "p > 1" by arith
chaieb@26125
   805
  from e ij have "p^(i + j) = p^k" by (simp add: power_add)
chaieb@26125
   806
  hence "i + j = k" using power_inject_exp[of p "i+j" k, OF p1] by simp 
chaieb@26125
   807
  hence "i \<le> k" by arith
chaieb@26125
   808
  with ij(1) show "\<exists>i\<le>k. d = p ^ i" by blast
chaieb@26125
   809
next
chaieb@26125
   810
  {fix i assume H: "i \<le> k" "d = p^i"
chaieb@26125
   811
    hence "\<exists>j. k = i + j" by arith
chaieb@26125
   812
    then obtain j where j: "k = i + j" by blast
chaieb@26125
   813
    hence "p^k = p^j*d" using H(2) by (simp add: power_add)
chaieb@26125
   814
    hence "d dvd p^k" unfolding dvd_def by auto}
chaieb@26125
   815
  thus "\<exists>i\<le>k. d = p ^ i \<Longrightarrow> d dvd p ^ k" by blast
chaieb@26125
   816
qed
chaieb@26125
   817
chaieb@26125
   818
lemma coprime_divisors: "d dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow> coprime d e"
chaieb@26125
   819
  by (auto simp add: dvd_def coprime)
chaieb@26125
   820
nipkow@34223
   821
lemma mult_inj_if_coprime_nat:
nipkow@34223
   822
  "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> ALL a:A. ALL b:B. coprime (f a) (g b)
nipkow@34223
   823
   \<Longrightarrow> inj_on (%(a,b). f a * g b::nat) (A \<times> B)"
nipkow@34223
   824
apply(auto simp add:inj_on_def)
nipkow@34223
   825
apply(metis coprime_def dvd_triv_left gcd_proj2_if_dvd_nat gcd_semilattice_nat.inf_commute relprime_dvd_mult)
nipkow@34223
   826
apply(metis coprime_commute coprime_divprod dvd.neq_le_trans dvd_triv_right)
nipkow@34223
   827
done
nipkow@34223
   828
chaieb@26159
   829
declare power_Suc0[simp del]
haftmann@26757
   830
paulson@11363
   831
end