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(* Title: HOL/Library/Pocklington.thy
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ID: $Id:
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Author: Amine Chaieb
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*)
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header {* Pocklington's Theorem for Primes *}
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theory Pocklington
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imports List Primes
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begin
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definition modeq:: "nat => nat => nat => bool" ("(1[_ = _] '(mod _'))")
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where "[a = b] (mod p) == ((a mod p) = (b mod p))"
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definition modneq:: "nat => nat => nat => bool" ("(1[_ \<noteq> _] '(mod _'))")
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where "[a \<noteq> b] (mod p) == ((a mod p) \<noteq> (b mod p))"
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lemma modeq_trans:
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"\<lbrakk> [a = b] (mod p); [b = c] (mod p) \<rbrakk> \<Longrightarrow> [a = c] (mod p)"
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by (simp add:modeq_def)
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lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
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proof
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assume H: "x mod n = y mod n"
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hence "x mod n - y mod n = 0" by simp
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hence "(x mod n - y mod n) mod n = 0" by simp
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hence "(x - y) mod n = 0" by (simp add: zmod_zdiff1_eq[symmetric])
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thus "n dvd x - y" by (simp add: zdvd_iff_zmod_eq_0)
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next
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assume H: "n dvd x - y"
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then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
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hence "x = n*k + y" by simp
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hence "x mod n = (n*k + y) mod n" by simp
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thus "x mod n = y mod n" by (simp add: zmod_zadd_left_eq)
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qed
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lemma nat_mod_lemma: assumes xyn: "[x = y] (mod n)" and xy:"y \<le> x"
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shows "\<exists>q. x = y + n * q"
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proof-
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from xy have th: "int x - int y = int (x - y)" by presburger
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from xyn have "int x mod int n = int y mod int n"
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by (simp add: modeq_def zmod_int[symmetric])
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hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])
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hence "n dvd x - y" by (simp add: th zdvd_int)
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then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
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qed
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lemma nat_mod: "[x = y] (mod n) \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
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(is "?lhs = ?rhs")
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proof
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assume H: "[x = y] (mod n)"
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{assume xy: "x \<le> y"
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from H have th: "[y = x] (mod n)" by (simp add: modeq_def)
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from nat_mod_lemma[OF th xy] have ?rhs
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apply clarify apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
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moreover
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{assume xy: "y \<le> x"
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from nat_mod_lemma[OF H xy] have ?rhs
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apply clarify apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
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ultimately show ?rhs using linear[of x y] by blast
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next
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assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
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hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
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thus ?lhs by (simp add: modeq_def)
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qed
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(* Lemmas about previously defined terms. *)
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lemma prime: "prime p \<longleftrightarrow> p \<noteq> 0 \<and> p\<noteq>1 \<and> (\<forall>m. 0 < m \<and> m < p \<longrightarrow> coprime p m)"
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(is "?lhs \<longleftrightarrow> ?rhs")
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proof-
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{assume "p=0 \<or> p=1" hence ?thesis using prime_0 prime_1 by (cases "p=0", simp_all)}
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moreover
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{assume p0: "p\<noteq>0" "p\<noteq>1"
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{assume H: "?lhs"
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{fix m assume m: "m > 0" "m < p"
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{assume "m=1" hence "coprime p m" by simp}
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moreover
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{assume "p dvd m" hence "p \<le> m" using dvd_imp_le m by blast with m(2)
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have "coprime p m" by simp}
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ultimately have "coprime p m" using prime_coprime[OF H, of m] by blast}
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hence ?rhs using p0 by auto}
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moreover
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{ assume H: "\<forall>m. 0 < m \<and> m < p \<longrightarrow> coprime p m"
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from prime_factor[OF p0(2)] obtain q where q: "prime q" "q dvd p" by blast
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from prime_ge_2[OF q(1)] have q0: "q > 0" by arith
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from dvd_imp_le[OF q(2)] p0 have qp: "q \<le> p" by arith
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{assume "q = p" hence ?lhs using q(1) by blast}
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moreover
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{assume "q\<noteq>p" with qp have qplt: "q < p" by arith
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from H[rule_format, of q] qplt q0 have "coprime p q" by arith
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with coprime_prime[of p q q] q have False by simp hence ?lhs by blast}
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ultimately have ?lhs by blast}
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ultimately have ?thesis by blast}
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ultimately show ?thesis by (cases"p=0 \<or> p=1", auto)
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qed
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lemma finite_number_segment: "card { m. 0 < m \<and> m < n } = n - 1"
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proof-
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have "{ m. 0 < m \<and> m < n } = {1..<n}" by auto
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thus ?thesis by simp
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qed
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lemma coprime_mod: assumes n: "n \<noteq> 0" shows "coprime (a mod n) n \<longleftrightarrow> coprime a n"
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using n dvd_mod_iff[of _ n a] by (auto simp add: coprime)
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(* Congruences. *)
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lemma cong_mod_01[simp,presburger]:
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"[x = y] (mod 0) \<longleftrightarrow> x = y" "[x = y] (mod 1)" "[x = 0] (mod n) \<longleftrightarrow> n dvd x"
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by (simp_all add: modeq_def, presburger)
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lemma cong_sub_cases:
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"[x = y] (mod n) \<longleftrightarrow> (if x <= y then [y - x = 0] (mod n) else [x - y = 0] (mod n))"
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apply (auto simp add: nat_mod)
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apply (rule_tac x="q2" in exI)
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apply (rule_tac x="q1" in exI, simp)
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apply (rule_tac x="q2" in exI)
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apply (rule_tac x="q1" in exI, simp)
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apply (rule_tac x="q1" in exI)
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apply (rule_tac x="q2" in exI, simp)
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apply (rule_tac x="q1" in exI)
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apply (rule_tac x="q2" in exI, simp)
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done
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lemma cong_mult_lcancel: assumes an: "coprime a n" and axy:"[a * x = a * y] (mod n)"
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shows "[x = y] (mod n)"
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proof-
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{assume "a = 0" with an axy coprime_0'[of n] have ?thesis by (simp add: modeq_def) }
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moreover
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{assume az: "a\<noteq>0"
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{assume xy: "x \<le> y" hence axy': "a*x \<le> a*y" by simp
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with axy cong_sub_cases[of "a*x" "a*y" n] have "[a*(y - x) = 0] (mod n)"
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by (simp only: if_True diff_mult_distrib2)
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hence th: "n dvd a*(y -x)" by simp
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from coprime_divprod[OF th] an have "n dvd y - x"
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by (simp add: coprime_commute)
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hence ?thesis using xy cong_sub_cases[of x y n] by simp}
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moreover
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{assume H: "\<not>x \<le> y" hence xy: "y \<le> x" by arith
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from H az have axy': "\<not> a*x \<le> a*y" by auto
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with axy H cong_sub_cases[of "a*x" "a*y" n] have "[a*(x - y) = 0] (mod n)"
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by (simp only: if_False diff_mult_distrib2)
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hence th: "n dvd a*(x - y)" by simp
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from coprime_divprod[OF th] an have "n dvd x - y"
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by (simp add: coprime_commute)
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hence ?thesis using xy cong_sub_cases[of x y n] by simp}
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ultimately have ?thesis by blast}
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ultimately show ?thesis by blast
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qed
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lemma cong_mult_rcancel: assumes an: "coprime a n" and axy:"[x*a = y*a] (mod n)"
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shows "[x = y] (mod n)"
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using cong_mult_lcancel[OF an axy[unfolded mult_commute[of _a]]] .
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lemma cong_refl: "[x = x] (mod n)" by (simp add: modeq_def)
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lemma eq_imp_cong: "a = b \<Longrightarrow> [a = b] (mod n)" by (simp add: cong_refl)
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lemma cong_commute: "[x = y] (mod n) \<longleftrightarrow> [y = x] (mod n)"
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by (auto simp add: modeq_def)
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lemma cong_trans[trans]: "[x = y] (mod n) \<Longrightarrow> [y = z] (mod n) \<Longrightarrow> [x = z] (mod n)"
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by (simp add: modeq_def)
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lemma cong_add: assumes xx': "[x = x'] (mod n)" and yy':"[y = y'] (mod n)"
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shows "[x + y = x' + y'] (mod n)"
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proof-
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have "(x + y) mod n = (x mod n + y mod n) mod n"
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by (simp add: mod_add_left_eq[of x y n] mod_add_right_eq[of "x mod n" y n])
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also have "\<dots> = (x' mod n + y' mod n) mod n" using xx' yy' modeq_def by simp
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also have "\<dots> = (x' + y') mod n"
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by (simp add: mod_add_left_eq[of x' y' n] mod_add_right_eq[of "x' mod n" y' n])
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finally show ?thesis unfolding modeq_def .
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qed
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lemma cong_mult: assumes xx': "[x = x'] (mod n)" and yy':"[y = y'] (mod n)"
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shows "[x * y = x' * y'] (mod n)"
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proof-
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have "(x * y) mod n = (x mod n) * (y mod n) mod n"
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by (simp add: mod_mult1_eq'[of x y n] mod_mult1_eq[of "x mod n" y n])
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also have "\<dots> = (x' mod n) * (y' mod n) mod n" using xx'[unfolded modeq_def] yy'[unfolded modeq_def] by simp
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also have "\<dots> = (x' * y') mod n"
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by (simp add: mod_mult1_eq'[of x' y' n] mod_mult1_eq[of "x' mod n" y' n])
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finally show ?thesis unfolding modeq_def .
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qed
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lemma cong_exp: "[x = y] (mod n) \<Longrightarrow> [x^k = y^k] (mod n)"
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by (induct k, auto simp add: cong_refl cong_mult)
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lemma cong_sub: assumes xx': "[x = x'] (mod n)" and yy': "[y = y'] (mod n)"
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and yx: "y <= x" and yx': "y' <= x'"
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shows "[x - y = x' - y'] (mod n)"
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proof-
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{ fix x a x' a' y b y' b'
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have "(x::nat) + a = x' + a' \<Longrightarrow> y + b = y' + b' \<Longrightarrow> y <= x \<Longrightarrow> y' <= x'
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\<Longrightarrow> (x - y) + (a + b') = (x' - y') + (a' + b)" by arith}
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note th = this
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from xx' yy' obtain q1 q2 q1' q2' where q12: "x + n*q1 = x'+n*q2"
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and q12': "y + n*q1' = y'+n*q2'" unfolding nat_mod by blast+
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from th[OF q12 q12' yx yx']
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have "(x - y) + n*(q1 + q2') = (x' - y') + n*(q2 + q1')"
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by (simp add: right_distrib)
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thus ?thesis unfolding nat_mod by blast
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qed
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lemma cong_mult_lcancel_eq: assumes an: "coprime a n"
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shows "[a * x = a * y] (mod n) \<longleftrightarrow> [x = y] (mod n)" (is "?lhs \<longleftrightarrow> ?rhs")
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proof
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assume H: "?rhs" from cong_mult[OF cong_refl[of a n] H] show ?lhs .
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next
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assume H: "?lhs" hence H': "[x*a = y*a] (mod n)" by (simp add: mult_commute)
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from cong_mult_rcancel[OF an H'] show ?rhs .
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qed
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lemma cong_mult_rcancel_eq: assumes an: "coprime a n"
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shows "[x * a = y * a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
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using cong_mult_lcancel_eq[OF an, of x y] by (simp add: mult_commute)
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lemma cong_add_lcancel_eq: "[a + x = a + y] (mod n) \<longleftrightarrow> [x = y] (mod n)"
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by (simp add: nat_mod)
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lemma cong_add_rcancel_eq: "[x + a = y + a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
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by (simp add: nat_mod)
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lemma cong_add_rcancel: "[x + a = y + a] (mod n) \<Longrightarrow> [x = y] (mod n)"
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by (simp add: nat_mod)
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lemma cong_add_lcancel: "[a + x = a + y] (mod n) \<Longrightarrow> [x = y] (mod n)"
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by (simp add: nat_mod)
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lemma cong_add_lcancel_eq_0: "[a + x = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
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by (simp add: nat_mod)
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lemma cong_add_rcancel_eq_0: "[x + a = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
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by (simp add: nat_mod)
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lemma cong_imp_eq: assumes xn: "x < n" and yn: "y < n" and xy: "[x = y] (mod n)"
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shows "x = y"
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using xy[unfolded modeq_def mod_less[OF xn] mod_less[OF yn]] .
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lemma cong_divides_modulus: "[x = y] (mod m) \<Longrightarrow> n dvd m ==> [x = y] (mod n)"
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apply (auto simp add: nat_mod dvd_def)
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apply (rule_tac x="k*q1" in exI)
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apply (rule_tac x="k*q2" in exI)
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by simp
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lemma cong_0_divides: "[x = 0] (mod n) \<longleftrightarrow> n dvd x" by simp
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lemma cong_1_divides:"[x = 1] (mod n) ==> n dvd x - 1"
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apply (cases "x\<le>1", simp_all)
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using cong_sub_cases[of x 1 n] by auto
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lemma cong_divides: "[x = y] (mod n) \<Longrightarrow> n dvd x \<longleftrightarrow> n dvd y"
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apply (auto simp add: nat_mod dvd_def)
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apply (rule_tac x="k + q1 - q2" in exI, simp add: add_mult_distrib2 diff_mult_distrib2)
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apply (rule_tac x="k + q2 - q1" in exI, simp add: add_mult_distrib2 diff_mult_distrib2)
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done
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lemma cong_coprime: assumes xy: "[x = y] (mod n)"
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shows "coprime n x \<longleftrightarrow> coprime n y"
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proof-
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{assume "n=0" hence ?thesis using xy by simp}
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moreover
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{assume nz: "n \<noteq> 0"
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have "coprime n x \<longleftrightarrow> coprime (x mod n) n"
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by (simp add: coprime_mod[OF nz, of x] coprime_commute[of n x])
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also have "\<dots> \<longleftrightarrow> coprime (y mod n) n" using xy[unfolded modeq_def] by simp
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also have "\<dots> \<longleftrightarrow> coprime y n" by (simp add: coprime_mod[OF nz, of y])
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finally have ?thesis by (simp add: coprime_commute) }
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ultimately show ?thesis by blast
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qed
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lemma cong_mod: "~(n = 0) \<Longrightarrow> [a mod n = a] (mod n)" by (simp add: modeq_def)
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lemma mod_mult_cong: "~(a = 0) \<Longrightarrow> ~(b = 0)
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\<Longrightarrow> [x mod (a * b) = y] (mod a) \<longleftrightarrow> [x = y] (mod a)"
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by (simp add: modeq_def mod_mult2_eq mod_add_left_eq)
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lemma cong_mod_mult: "[x = y] (mod n) \<Longrightarrow> m dvd n \<Longrightarrow> [x = y] (mod m)"
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apply (auto simp add: nat_mod dvd_def)
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apply (rule_tac x="k*q1" in exI)
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apply (rule_tac x="k*q2" in exI, simp)
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done
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(* Some things when we know more about the order. *)
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lemma cong_le: "y <= x \<Longrightarrow> [x = y] (mod n) \<longleftrightarrow> (\<exists>q. x = q * n + y)"
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using nat_mod_lemma[of x y n]
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apply auto
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apply (simp add: nat_mod)
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apply (rule_tac x="q" in exI)
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apply (rule_tac x="q + q" in exI)
|
|
294 |
by (auto simp: ring_simps)
|
|
295 |
|
|
296 |
lemma cong_to_1: "[a = 1] (mod n) \<longleftrightarrow> a = 0 \<and> n = 1 \<or> (\<exists>m. a = 1 + m * n)"
|
|
297 |
proof-
|
|
298 |
{assume "n = 0 \<or> n = 1\<or> a = 0 \<or> a = 1" hence ?thesis
|
|
299 |
apply (cases "n=0", simp_all add: cong_commute)
|
|
300 |
apply (cases "n=1", simp_all add: cong_commute modeq_def)
|
|
301 |
apply arith
|
|
302 |
by (cases "a=1", simp_all add: modeq_def cong_commute)}
|
|
303 |
moreover
|
|
304 |
{assume n: "n\<noteq>0" "n\<noteq>1" and a:"a\<noteq>0" "a \<noteq> 1" hence a': "a \<ge> 1" by simp
|
|
305 |
hence ?thesis using cong_le[OF a', of n] by auto }
|
|
306 |
ultimately show ?thesis by auto
|
|
307 |
qed
|
|
308 |
|
|
309 |
(* Some basic theorems about solving congruences. *)
|
|
310 |
|
|
311 |
|
|
312 |
lemma cong_solve: assumes an: "coprime a n" shows "\<exists>x. [a * x = b] (mod n)"
|
|
313 |
proof-
|
|
314 |
{assume "a=0" hence ?thesis using an by (simp add: modeq_def)}
|
|
315 |
moreover
|
|
316 |
{assume az: "a\<noteq>0"
|
|
317 |
from bezout_add_strong[OF az, of n]
|
|
318 |
obtain d x y where dxy: "d dvd a" "d dvd n" "a*x = n*y + d" by blast
|
|
319 |
from an[unfolded coprime, rule_format, of d] dxy(1,2) have d1: "d = 1" by blast
|
|
320 |
hence "a*x*b = (n*y + 1)*b" using dxy(3) by simp
|
|
321 |
hence "a*(x*b) = n*(y*b) + b" by algebra
|
|
322 |
hence "a*(x*b) mod n = (n*(y*b) + b) mod n" by simp
|
|
323 |
hence "a*(x*b) mod n = b mod n" by (simp add: mod_add_left_eq)
|
|
324 |
hence "[a*(x*b) = b] (mod n)" unfolding modeq_def .
|
|
325 |
hence ?thesis by blast}
|
|
326 |
ultimately show ?thesis by blast
|
|
327 |
qed
|
|
328 |
|
|
329 |
lemma cong_solve_unique: assumes an: "coprime a n" and nz: "n \<noteq> 0"
|
|
330 |
shows "\<exists>!x. x < n \<and> [a * x = b] (mod n)"
|
|
331 |
proof-
|
|
332 |
let ?P = "\<lambda>x. x < n \<and> [a * x = b] (mod n)"
|
|
333 |
from cong_solve[OF an] obtain x where x: "[a*x = b] (mod n)" by blast
|
|
334 |
let ?x = "x mod n"
|
|
335 |
from x have th: "[a * ?x = b] (mod n)"
|
|
336 |
by (simp add: modeq_def mod_mult1_eq[of a x n])
|
|
337 |
from mod_less_divisor[ of n x] nz th have Px: "?P ?x" by simp
|
|
338 |
{fix y assume Py: "y < n" "[a * y = b] (mod n)"
|
|
339 |
from Py(2) th have "[a * y = a*?x] (mod n)" by (simp add: modeq_def)
|
|
340 |
hence "[y = ?x] (mod n)" by (simp add: cong_mult_lcancel_eq[OF an])
|
|
341 |
with mod_less[OF Py(1)] mod_less_divisor[ of n x] nz
|
|
342 |
have "y = ?x" by (simp add: modeq_def)}
|
|
343 |
with Px show ?thesis by blast
|
|
344 |
qed
|
|
345 |
|
|
346 |
lemma cong_solve_unique_nontrivial:
|
|
347 |
assumes p: "prime p" and pa: "coprime p a" and x0: "0 < x" and xp: "x < p"
|
|
348 |
shows "\<exists>!y. 0 < y \<and> y < p \<and> [x * y = a] (mod p)"
|
|
349 |
proof-
|
|
350 |
from p have p1: "p > 1" using prime_ge_2[OF p] by arith
|
|
351 |
hence p01: "p \<noteq> 0" "p \<noteq> 1" by arith+
|
|
352 |
from pa have ap: "coprime a p" by (simp add: coprime_commute)
|
|
353 |
from prime_coprime[OF p, of x] dvd_imp_le[of p x] x0 xp have px:"coprime x p"
|
|
354 |
by (auto simp add: coprime_commute)
|
|
355 |
from cong_solve_unique[OF px p01(1)]
|
|
356 |
obtain y where y: "y < p" "[x * y = a] (mod p)" "\<forall>z. z < p \<and> [x * z = a] (mod p) \<longrightarrow> z = y" by blast
|
|
357 |
{assume y0: "y = 0"
|
|
358 |
with y(2) have th: "p dvd a" by (simp add: cong_commute[of 0 a p])
|
|
359 |
with p coprime_prime[OF pa, of p] have False by simp}
|
|
360 |
with y show ?thesis unfolding Ex1_def using neq0_conv by blast
|
|
361 |
qed
|
|
362 |
lemma cong_unique_inverse_prime:
|
|
363 |
assumes p: "prime p" and x0: "0 < x" and xp: "x < p"
|
|
364 |
shows "\<exists>!y. 0 < y \<and> y < p \<and> [x * y = 1] (mod p)"
|
|
365 |
using cong_solve_unique_nontrivial[OF p coprime_1[of p] x0 xp] .
|
|
366 |
|
|
367 |
(* Forms of the Chinese remainder theorem. *)
|
|
368 |
|
|
369 |
lemma cong_chinese:
|
|
370 |
assumes ab: "coprime a b" and xya: "[x = y] (mod a)"
|
|
371 |
and xyb: "[x = y] (mod b)"
|
|
372 |
shows "[x = y] (mod a*b)"
|
|
373 |
using ab xya xyb
|
|
374 |
by (simp add: cong_sub_cases[of x y a] cong_sub_cases[of x y b]
|
|
375 |
cong_sub_cases[of x y "a*b"])
|
|
376 |
(cases "x \<le> y", simp_all add: divides_mul[of a _ b])
|
|
377 |
|
|
378 |
lemma chinese_remainder_unique:
|
|
379 |
assumes ab: "coprime a b" and az: "a \<noteq> 0" and bz: "b\<noteq>0"
|
|
380 |
shows "\<exists>!x. x < a * b \<and> [x = m] (mod a) \<and> [x = n] (mod b)"
|
|
381 |
proof-
|
|
382 |
from az bz have abpos: "a*b > 0" by simp
|
|
383 |
from chinese_remainder[OF ab az bz] obtain x q1 q2 where
|
|
384 |
xq12: "x = m + q1 * a" "x = n + q2 * b" by blast
|
|
385 |
let ?w = "x mod (a*b)"
|
|
386 |
have wab: "?w < a*b" by (simp add: mod_less_divisor[OF abpos])
|
|
387 |
from xq12(1) have "?w mod a = ((m + q1 * a) mod (a*b)) mod a" by simp
|
|
388 |
also have "\<dots> = m mod a" apply (simp add: mod_mult2_eq)
|
|
389 |
apply (subst mod_add_left_eq)
|
|
390 |
by simp
|
|
391 |
finally have th1: "[?w = m] (mod a)" by (simp add: modeq_def)
|
|
392 |
from xq12(2) have "?w mod b = ((n + q2 * b) mod (a*b)) mod b" by simp
|
|
393 |
also have "\<dots> = ((n + q2 * b) mod (b*a)) mod b" by (simp add: mult_commute)
|
|
394 |
also have "\<dots> = n mod b" apply (simp add: mod_mult2_eq)
|
|
395 |
apply (subst mod_add_left_eq)
|
|
396 |
by simp
|
|
397 |
finally have th2: "[?w = n] (mod b)" by (simp add: modeq_def)
|
|
398 |
{fix y assume H: "y < a*b" "[y = m] (mod a)" "[y = n] (mod b)"
|
|
399 |
with th1 th2 have H': "[y = ?w] (mod a)" "[y = ?w] (mod b)"
|
|
400 |
by (simp_all add: modeq_def)
|
|
401 |
from cong_chinese[OF ab H'] mod_less[OF H(1)] mod_less[OF wab]
|
|
402 |
have "y = ?w" by (simp add: modeq_def)}
|
|
403 |
with th1 th2 wab show ?thesis by blast
|
|
404 |
qed
|
|
405 |
|
|
406 |
lemma chinese_remainder_coprime_unique:
|
|
407 |
assumes ab: "coprime a b" and az: "a \<noteq> 0" and bz: "b \<noteq> 0"
|
|
408 |
and ma: "coprime m a" and nb: "coprime n b"
|
|
409 |
shows "\<exists>!x. coprime x (a * b) \<and> x < a * b \<and> [x = m] (mod a) \<and> [x = n] (mod b)"
|
|
410 |
proof-
|
|
411 |
let ?P = "\<lambda>x. x < a * b \<and> [x = m] (mod a) \<and> [x = n] (mod b)"
|
|
412 |
from chinese_remainder_unique[OF ab az bz]
|
|
413 |
obtain x where x: "x < a * b" "[x = m] (mod a)" "[x = n] (mod b)"
|
|
414 |
"\<forall>y. ?P y \<longrightarrow> y = x" by blast
|
|
415 |
from ma nb cong_coprime[OF x(2)] cong_coprime[OF x(3)]
|
|
416 |
have "coprime x a" "coprime x b" by (simp_all add: coprime_commute)
|
|
417 |
with coprime_mul[of x a b] have "coprime x (a*b)" by simp
|
|
418 |
with x show ?thesis by blast
|
|
419 |
qed
|
|
420 |
|
|
421 |
(* Euler totient function. *)
|
|
422 |
|
|
423 |
definition phi_def: "\<phi> n = card { m. 0 < m \<and> m <= n \<and> coprime m n }"
|
|
424 |
lemma phi_0[simp]: "\<phi> 0 = 0"
|
|
425 |
unfolding phi_def by (auto simp add: card_eq_0_iff)
|
|
426 |
|
|
427 |
lemma phi_finite[simp]: "finite ({ m. 0 < m \<and> m <= n \<and> coprime m n })"
|
|
428 |
proof-
|
|
429 |
have "{ m. 0 < m \<and> m <= n \<and> coprime m n } \<subseteq> {0..n}" by auto
|
|
430 |
thus ?thesis by (auto intro: finite_subset)
|
|
431 |
qed
|
|
432 |
|
|
433 |
declare coprime_1[presburger]
|
|
434 |
lemma phi_1[simp]: "\<phi> 1 = 1"
|
|
435 |
proof-
|
|
436 |
{fix m
|
|
437 |
have "0 < m \<and> m <= 1 \<and> coprime m 1 \<longleftrightarrow> m = 1" by presburger }
|
|
438 |
thus ?thesis by (simp add: phi_def)
|
|
439 |
qed
|
|
440 |
|
|
441 |
lemma [simp]: "\<phi> (Suc 0) = Suc 0" using phi_1 by simp
|
|
442 |
|
|
443 |
lemma phi_alt: "\<phi>(n) = card { m. coprime m n \<and> m < n}"
|
|
444 |
proof-
|
|
445 |
{assume "n=0 \<or> n=1" hence ?thesis by (cases "n=0", simp_all)}
|
|
446 |
moreover
|
|
447 |
{assume n: "n\<noteq>0" "n\<noteq>1"
|
|
448 |
{fix m
|
|
449 |
from n have "0 < m \<and> m <= n \<and> coprime m n \<longleftrightarrow> coprime m n \<and> m < n"
|
|
450 |
apply (cases "m = 0", simp_all)
|
|
451 |
apply (cases "m = 1", simp_all)
|
|
452 |
apply (cases "m = n", auto)
|
|
453 |
done }
|
|
454 |
hence ?thesis unfolding phi_def by simp}
|
|
455 |
ultimately show ?thesis by auto
|
|
456 |
qed
|
|
457 |
|
|
458 |
lemma phi_finite_lemma[simp]: "finite {m. coprime m n \<and> m < n}" (is "finite ?S")
|
|
459 |
by (rule finite_subset[of "?S" "{0..n}"], auto)
|
|
460 |
|
|
461 |
lemma phi_another: assumes n: "n\<noteq>1"
|
|
462 |
shows "\<phi> n = card {m. 0 < m \<and> m < n \<and> coprime m n }"
|
|
463 |
proof-
|
|
464 |
{fix m
|
|
465 |
from n have "0 < m \<and> m < n \<and> coprime m n \<longleftrightarrow> coprime m n \<and> m < n"
|
|
466 |
by (cases "m=0", auto)}
|
|
467 |
thus ?thesis unfolding phi_alt by auto
|
|
468 |
qed
|
|
469 |
|
|
470 |
lemma phi_limit: "\<phi> n \<le> n"
|
|
471 |
proof-
|
|
472 |
have "{ m. coprime m n \<and> m < n} \<subseteq> {0 ..<n}" by auto
|
|
473 |
with card_mono[of "{0 ..<n}" "{ m. coprime m n \<and> m < n}"]
|
|
474 |
show ?thesis unfolding phi_alt by auto
|
|
475 |
qed
|
|
476 |
|
|
477 |
lemma stupid[simp]: "{m. (0::nat) < m \<and> m < n} = {1..<n}"
|
|
478 |
by auto
|
|
479 |
|
|
480 |
lemma phi_limit_strong: assumes n: "n\<noteq>1"
|
|
481 |
shows "\<phi>(n) \<le> n - 1"
|
|
482 |
proof-
|
|
483 |
show ?thesis
|
|
484 |
unfolding phi_another[OF n] finite_number_segment[of n, symmetric]
|
|
485 |
by (rule card_mono[of "{m. 0 < m \<and> m < n}" "{m. 0 < m \<and> m < n \<and> coprime m n}"], auto)
|
|
486 |
qed
|
|
487 |
|
|
488 |
lemma phi_lowerbound_1_strong: assumes n: "n \<ge> 1"
|
|
489 |
shows "\<phi>(n) \<ge> 1"
|
|
490 |
proof-
|
|
491 |
let ?S = "{ m. 0 < m \<and> m <= n \<and> coprime m n }"
|
|
492 |
from card_0_eq[of ?S] n have "\<phi> n \<noteq> 0" unfolding phi_alt
|
|
493 |
apply auto
|
|
494 |
apply (cases "n=1", simp_all)
|
|
495 |
apply (rule exI[where x=1], simp)
|
|
496 |
done
|
|
497 |
thus ?thesis by arith
|
|
498 |
qed
|
|
499 |
|
|
500 |
lemma phi_lowerbound_1: "2 <= n ==> 1 <= \<phi>(n)"
|
|
501 |
using phi_lowerbound_1_strong[of n] by auto
|
|
502 |
|
|
503 |
lemma phi_lowerbound_2: assumes n: "3 <= n" shows "2 <= \<phi> (n)"
|
|
504 |
proof-
|
|
505 |
let ?S = "{ m. 0 < m \<and> m <= n \<and> coprime m n }"
|
|
506 |
have inS: "{1, n - 1} \<subseteq> ?S" using n coprime_plus1[of "n - 1"]
|
|
507 |
by (auto simp add: coprime_commute)
|
|
508 |
from n have c2: "card {1, n - 1} = 2" by (auto simp add: card_insert_if)
|
|
509 |
from card_mono[of ?S "{1, n - 1}", simplified inS c2] show ?thesis
|
|
510 |
unfolding phi_def by auto
|
|
511 |
qed
|
|
512 |
|
|
513 |
lemma phi_prime: "\<phi> n = n - 1 \<and> n\<noteq>0 \<and> n\<noteq>1 \<longleftrightarrow> prime n"
|
|
514 |
proof-
|
|
515 |
{assume "n=0 \<or> n=1" hence ?thesis by (cases "n=1", simp_all)}
|
|
516 |
moreover
|
|
517 |
{assume n: "n\<noteq>0" "n\<noteq>1"
|
|
518 |
let ?S = "{m. 0 < m \<and> m < n}"
|
|
519 |
have fS: "finite ?S" by simp
|
|
520 |
let ?S' = "{m. 0 < m \<and> m < n \<and> coprime m n}"
|
|
521 |
have fS':"finite ?S'" apply (rule finite_subset[of ?S' ?S]) by auto
|
|
522 |
{assume H: "\<phi> n = n - 1 \<and> n\<noteq>0 \<and> n\<noteq>1"
|
|
523 |
hence ceq: "card ?S' = card ?S"
|
|
524 |
using n finite_number_segment[of n] phi_another[OF n(2)] by simp
|
|
525 |
{fix m assume m: "0 < m" "m < n" "\<not> coprime m n"
|
|
526 |
hence mS': "m \<notin> ?S'" by auto
|
|
527 |
have "insert m ?S' \<le> ?S" using m by auto
|
|
528 |
from m have "card (insert m ?S') \<le> card ?S"
|
|
529 |
by - (rule card_mono[of ?S "insert m ?S'"], auto)
|
|
530 |
hence False
|
|
531 |
unfolding card_insert_disjoint[of "?S'" m, OF fS' mS'] ceq
|
|
532 |
by simp }
|
|
533 |
hence "\<forall>m. 0 <m \<and> m < n \<longrightarrow> coprime m n" by blast
|
|
534 |
hence "prime n" unfolding prime using n by (simp add: coprime_commute)}
|
|
535 |
moreover
|
|
536 |
{assume H: "prime n"
|
|
537 |
hence "?S = ?S'" unfolding prime using n
|
|
538 |
by (auto simp add: coprime_commute)
|
|
539 |
hence "card ?S = card ?S'" by simp
|
|
540 |
hence "\<phi> n = n - 1" unfolding phi_another[OF n(2)] by simp}
|
|
541 |
ultimately have ?thesis using n by blast}
|
|
542 |
ultimately show ?thesis by (cases "n=0") blast+
|
|
543 |
qed
|
|
544 |
|
|
545 |
(* Multiplicativity property. *)
|
|
546 |
|
|
547 |
lemma phi_multiplicative: assumes ab: "coprime a b"
|
|
548 |
shows "\<phi> (a * b) = \<phi> a * \<phi> b"
|
|
549 |
proof-
|
|
550 |
{assume "a = 0 \<or> b = 0 \<or> a = 1 \<or> b = 1"
|
|
551 |
hence ?thesis
|
|
552 |
by (cases "a=0", simp, cases "b=0", simp, cases"a=1", simp_all) }
|
|
553 |
moreover
|
|
554 |
{assume a: "a\<noteq>0" "a\<noteq>1" and b: "b\<noteq>0" "b\<noteq>1"
|
|
555 |
hence ab0: "a*b \<noteq> 0" by simp
|
|
556 |
let ?S = "\<lambda>k. {m. coprime m k \<and> m < k}"
|
|
557 |
let ?f = "\<lambda>x. (x mod a, x mod b)"
|
|
558 |
have eq: "?f ` (?S (a*b)) = (?S a \<times> ?S b)"
|
|
559 |
proof-
|
|
560 |
{fix x assume x:"x \<in> ?S (a*b)"
|
|
561 |
hence x': "coprime x (a*b)" "x < a*b" by simp_all
|
|
562 |
hence xab: "coprime x a" "coprime x b" by (simp_all add: coprime_mul_eq)
|
|
563 |
from mod_less_divisor a b have xab':"x mod a < a" "x mod b < b" by auto
|
|
564 |
from xab xab' have "?f x \<in> (?S a \<times> ?S b)"
|
|
565 |
by (simp add: coprime_mod[OF a(1)] coprime_mod[OF b(1)])}
|
|
566 |
moreover
|
|
567 |
{fix x y assume x: "x \<in> ?S a" and y: "y \<in> ?S b"
|
|
568 |
hence x': "coprime x a" "x < a" and y': "coprime y b" "y < b" by simp_all
|
|
569 |
from chinese_remainder_coprime_unique[OF ab a(1) b(1) x'(1) y'(1)]
|
|
570 |
obtain z where z: "coprime z (a * b)" "z < a * b" "[z = x] (mod a)"
|
|
571 |
"[z = y] (mod b)" by blast
|
|
572 |
hence "(x,y) \<in> ?f ` (?S (a*b))"
|
|
573 |
using y'(2) mod_less_divisor[of b y] x'(2) mod_less_divisor[of a x]
|
|
574 |
by (auto simp add: image_iff modeq_def)}
|
|
575 |
ultimately show ?thesis by auto
|
|
576 |
qed
|
|
577 |
have finj: "inj_on ?f (?S (a*b))"
|
|
578 |
unfolding inj_on_def
|
|
579 |
proof(clarify)
|
|
580 |
fix x y assume H: "coprime x (a * b)" "x < a * b" "coprime y (a * b)"
|
|
581 |
"y < a * b" "x mod a = y mod a" "x mod b = y mod b"
|
|
582 |
hence cp: "coprime x a" "coprime x b" "coprime y a" "coprime y b"
|
|
583 |
by (simp_all add: coprime_mul_eq)
|
|
584 |
from chinese_remainder_coprime_unique[OF ab a(1) b(1) cp(3,4)] H
|
|
585 |
show "x = y" unfolding modeq_def by blast
|
|
586 |
qed
|
|
587 |
from card_image[OF finj, unfolded eq] have ?thesis
|
|
588 |
unfolding phi_alt by simp }
|
|
589 |
ultimately show ?thesis by auto
|
|
590 |
qed
|
|
591 |
|
|
592 |
(* Fermat's Little theorem / Fermat-Euler theorem. *)
|
|
593 |
|
|
594 |
lemma (in comm_monoid_mult) fold_related:
|
|
595 |
assumes Re: "R e e"
|
|
596 |
and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
|
|
597 |
and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
|
|
598 |
shows "R (fold (op *) h e S) (fold (op *) g e S)"
|
|
599 |
using prems
|
|
600 |
by -(rule finite_subset_induct,auto)
|
|
601 |
|
|
602 |
|
|
603 |
lemma nproduct_mod:
|
|
604 |
assumes fS: "finite S" and n0: "n \<noteq> 0"
|
|
605 |
shows "[setprod (\<lambda>m. a(m) mod n) S = setprod a S] (mod n)"
|
|
606 |
proof-
|
|
607 |
have th1:"[1 = 1] (mod n)" by (simp add: modeq_def)
|
|
608 |
from cong_mult
|
|
609 |
have th3:"\<forall>x1 y1 x2 y2.
|
|
610 |
[x1 = x2] (mod n) \<and> [y1 = y2] (mod n) \<longrightarrow> [x1 * y1 = x2 * y2] (mod n)"
|
|
611 |
by blast
|
|
612 |
have th4:"\<forall>x\<in>S. [a x mod n = a x] (mod n)" by (simp add: modeq_def)
|
|
613 |
from fold_related[where h="(\<lambda>m. a(m) mod n)" and g=a, OF th1 th3 fS, OF th4] show ?thesis unfolding setprod_def by (simp add: fS)
|
|
614 |
qed
|
|
615 |
|
|
616 |
lemma nproduct_cmul:
|
|
617 |
assumes fS:"finite S"
|
|
618 |
shows "setprod (\<lambda>m. (c::'a::{comm_monoid_mult,recpower})* a(m)) S = c ^ (card S) * setprod a S"
|
|
619 |
unfolding setprod_timesf setprod_constant[OF fS, of c] ..
|
|
620 |
|
|
621 |
lemma coprime_nproduct:
|
|
622 |
assumes fS: "finite S" and Sn: "\<forall>x\<in>S. coprime n (a x)"
|
|
623 |
shows "coprime n (setprod a S)"
|
|
624 |
using fS Sn
|
|
625 |
unfolding setprod_def
|
|
626 |
apply -
|
|
627 |
apply (rule finite_subset_induct)
|
|
628 |
by (auto simp add: coprime_mul)
|
|
629 |
|
|
630 |
lemma (in comm_monoid_mult)
|
|
631 |
fold_eq_general:
|
|
632 |
assumes fS: "finite S"
|
|
633 |
and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y"
|
|
634 |
and f12: "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
|
|
635 |
shows "fold (op *) f1 e S = fold (op *) f2 e S'"
|
|
636 |
proof-
|
|
637 |
from h f12 have hS: "h ` S = S'" by auto
|
|
638 |
{fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
|
|
639 |
from f12 h H have "x = y" by auto }
|
|
640 |
hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
|
|
641 |
from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
|
|
642 |
from hS have "fold (op *) f2 e S' = fold (op *) f2 e (h ` S)" by simp
|
|
643 |
also have "\<dots> = fold (op *) (f2 o h) e S"
|
|
644 |
using fold_reindex[OF fS hinj, of f2 e] .
|
|
645 |
also have "\<dots> = fold (op *) f1 e S " using th fold_cong[OF fS, of "f2 o h" f1 e]
|
|
646 |
by blast
|
|
647 |
finally show ?thesis ..
|
|
648 |
qed
|
|
649 |
|
|
650 |
lemma fermat_little: assumes an: "coprime a n"
|
|
651 |
shows "[a ^ (\<phi> n) = 1] (mod n)"
|
|
652 |
proof-
|
|
653 |
{assume "n=0" hence ?thesis by simp}
|
|
654 |
moreover
|
|
655 |
{assume "n=1" hence ?thesis by (simp add: modeq_def)}
|
|
656 |
moreover
|
|
657 |
{assume nz: "n \<noteq> 0" and n1: "n \<noteq> 1"
|
|
658 |
let ?S = "{m. coprime m n \<and> m < n}"
|
|
659 |
let ?P = "\<Prod> ?S"
|
|
660 |
have fS: "finite ?S" by simp
|
|
661 |
have cardfS: "\<phi> n = card ?S" unfolding phi_alt ..
|
|
662 |
{fix m assume m: "m \<in> ?S"
|
|
663 |
hence "coprime m n" by simp
|
|
664 |
with coprime_mul[of n a m] an have "coprime (a*m) n"
|
|
665 |
by (simp add: coprime_commute)}
|
|
666 |
hence Sn: "\<forall>m\<in> ?S. coprime (a*m) n " by blast
|
|
667 |
from coprime_nproduct[OF fS, of n "\<lambda>m. m"] have nP:"coprime ?P n"
|
|
668 |
by (simp add: coprime_commute)
|
|
669 |
have Paphi: "[?P*a^ (\<phi> n) = ?P*1] (mod n)"
|
|
670 |
proof-
|
|
671 |
let ?h = "\<lambda>m. m mod n"
|
|
672 |
{fix m assume mS: "m\<in> ?S"
|
|
673 |
hence "?h m \<in> ?S" by simp}
|
|
674 |
hence hS: "?h ` ?S = ?S"by (auto simp add: image_iff)
|
|
675 |
have "a\<noteq>0" using an n1 nz apply- apply (rule ccontr) by simp
|
|
676 |
hence inj: "inj_on (op * a) ?S" unfolding inj_on_def by simp
|
|
677 |
|
|
678 |
have eq0: "fold op * (?h \<circ> op * a) 1 {m. coprime m n \<and> m < n} =
|
|
679 |
fold op * (\<lambda>m. m) 1 {m. coprime m n \<and> m < n}"
|
|
680 |
proof (rule fold_eq_general[where h="?h o (op * a)"])
|
|
681 |
show "finite ?S" using fS .
|
|
682 |
next
|
|
683 |
{fix y assume yS: "y \<in> ?S" hence y: "coprime y n" "y < n" by simp_all
|
|
684 |
from cong_solve_unique[OF an nz, of y]
|
|
685 |
obtain x where x:"x < n" "[a * x = y] (mod n)" "\<forall>z. z < n \<and> [a * z = y] (mod n) \<longrightarrow> z=x" by blast
|
|
686 |
from cong_coprime[OF x(2)] y(1)
|
|
687 |
have xm: "coprime x n" by (simp add: coprime_mul_eq coprime_commute)
|
|
688 |
{fix z assume "z \<in> ?S" "(?h \<circ> op * a) z = y"
|
|
689 |
hence z: "coprime z n" "z < n" "(?h \<circ> op * a) z = y" by simp_all
|
|
690 |
from x(3)[rule_format, of z] z(2,3) have "z=x"
|
|
691 |
unfolding modeq_def mod_less[OF y(2)] by simp}
|
|
692 |
with xm x(1,2) have "\<exists>!x. x \<in> ?S \<and> (?h \<circ> op * a) x = y"
|
|
693 |
unfolding modeq_def mod_less[OF y(2)] by auto }
|
|
694 |
thus "\<forall>y\<in>{m. coprime m n \<and> m < n}.
|
|
695 |
\<exists>!x. x \<in> {m. coprime m n \<and> m < n} \<and> ((\<lambda>m. m mod n) \<circ> op * a) x = y" by blast
|
|
696 |
next
|
|
697 |
{fix x assume xS: "x\<in> ?S"
|
|
698 |
hence x: "coprime x n" "x < n" by simp_all
|
|
699 |
with an have "coprime (a*x) n"
|
|
700 |
by (simp add: coprime_mul_eq[of n a x] coprime_commute)
|
|
701 |
hence "?h (a*x) \<in> ?S" using nz
|
|
702 |
by (simp add: coprime_mod[OF nz] mod_less_divisor)}
|
|
703 |
thus " \<forall>x\<in>{m. coprime m n \<and> m < n}.
|
|
704 |
((\<lambda>m. m mod n) \<circ> op * a) x \<in> {m. coprime m n \<and> m < n} \<and>
|
|
705 |
((\<lambda>m. m mod n) \<circ> op * a) x = ((\<lambda>m. m mod n) \<circ> op * a) x" by simp
|
|
706 |
qed
|
|
707 |
from nproduct_mod[OF fS nz, of "op * a"]
|
|
708 |
have "[(setprod (op *a) ?S) = (setprod (?h o (op * a)) ?S)] (mod n)"
|
|
709 |
unfolding o_def
|
|
710 |
by (simp add: cong_commute)
|
|
711 |
also have "[setprod (?h o (op * a)) ?S = ?P ] (mod n)"
|
|
712 |
using eq0 fS an by (simp add: setprod_def modeq_def o_def)
|
|
713 |
finally show "[?P*a^ (\<phi> n) = ?P*1] (mod n)"
|
|
714 |
unfolding cardfS mult_commute[of ?P "a^ (card ?S)"]
|
|
715 |
nproduct_cmul[OF fS, symmetric] mult_1_right by simp
|
|
716 |
qed
|
|
717 |
from cong_mult_lcancel[OF nP Paphi] have ?thesis . }
|
|
718 |
ultimately show ?thesis by blast
|
|
719 |
qed
|
|
720 |
|
|
721 |
lemma fermat_little_prime: assumes p: "prime p" and ap: "coprime a p"
|
|
722 |
shows "[a^ (p - 1) = 1] (mod p)"
|
|
723 |
using fermat_little[OF ap] p[unfolded phi_prime[symmetric]]
|
|
724 |
by simp
|
|
725 |
|
|
726 |
|
|
727 |
(* Lucas's theorem. *)
|
|
728 |
|
|
729 |
lemma lucas_coprime_lemma:
|
|
730 |
assumes m: "m\<noteq>0" and am: "[a^m = 1] (mod n)"
|
|
731 |
shows "coprime a n"
|
|
732 |
proof-
|
|
733 |
{assume "n=1" hence ?thesis by simp}
|
|
734 |
moreover
|
|
735 |
{assume "n = 0" hence ?thesis using am m exp_eq_1[of a m] by simp}
|
|
736 |
moreover
|
|
737 |
{assume n: "n\<noteq>0" "n\<noteq>1"
|
|
738 |
from m obtain m' where m': "m = Suc m'" by (cases m, blast+)
|
|
739 |
{fix d
|
|
740 |
assume d: "d dvd a" "d dvd n"
|
|
741 |
from n have n1: "1 < n" by arith
|
|
742 |
from am mod_less[OF n1] have am1: "a^m mod n = 1" unfolding modeq_def by simp
|
|
743 |
from dvd_mult2[OF d(1), of "a^m'"] have dam:"d dvd a^m" by (simp add: m')
|
|
744 |
from dvd_mod_iff[OF d(2), of "a^m"] dam am1
|
|
745 |
have "d = 1" by simp }
|
|
746 |
hence ?thesis unfolding coprime by auto
|
|
747 |
}
|
|
748 |
ultimately show ?thesis by blast
|
|
749 |
qed
|
|
750 |
|
|
751 |
lemma lucas_weak:
|
|
752 |
assumes n: "n \<ge> 2" and an:"[a^(n - 1) = 1] (mod n)"
|
|
753 |
and nm: "\<forall>m. 0 <m \<and> m < n - 1 \<longrightarrow> \<not> [a^m = 1] (mod n)"
|
|
754 |
shows "prime n"
|
|
755 |
proof-
|
|
756 |
from n have n1: "n \<noteq> 1" "n\<noteq>0" "n - 1 \<noteq> 0" "n - 1 > 0" "n - 1 < n" by arith+
|
|
757 |
from lucas_coprime_lemma[OF n1(3) an] have can: "coprime a n" .
|
|
758 |
from fermat_little[OF can] have afn: "[a ^ \<phi> n = 1] (mod n)" .
|
|
759 |
{assume "\<phi> n \<noteq> n - 1"
|
|
760 |
with phi_limit_strong[OF n1(1)] phi_lowerbound_1[OF n]
|
|
761 |
have c:"\<phi> n > 0 \<and> \<phi> n < n - 1" by arith
|
|
762 |
from nm[rule_format, OF c] afn have False ..}
|
|
763 |
hence "\<phi> n = n - 1" by blast
|
|
764 |
with phi_prime[of n] n1(1,2) show ?thesis by simp
|
|
765 |
qed
|
|
766 |
|
|
767 |
lemma nat_exists_least_iff: "(\<exists>(n::nat). P n) \<longleftrightarrow> (\<exists>n. P n \<and> (\<forall>m < n. \<not> P m))"
|
|
768 |
(is "?lhs \<longleftrightarrow> ?rhs")
|
|
769 |
proof
|
|
770 |
assume ?rhs thus ?lhs by blast
|
|
771 |
next
|
|
772 |
assume H: ?lhs then obtain n where n: "P n" by blast
|
|
773 |
let ?x = "Least P"
|
|
774 |
{fix m assume m: "m < ?x"
|
|
775 |
from not_less_Least[OF m] have "\<not> P m" .}
|
|
776 |
with LeastI_ex[OF H] show ?rhs by blast
|
|
777 |
qed
|
|
778 |
|
|
779 |
lemma nat_exists_least_iff': "(\<exists>(n::nat). P n) \<longleftrightarrow> (P (Least P) \<and> (\<forall>m < (Least P). \<not> P m))"
|
|
780 |
(is "?lhs \<longleftrightarrow> ?rhs")
|
|
781 |
proof-
|
|
782 |
{assume ?rhs hence ?lhs by blast}
|
|
783 |
moreover
|
|
784 |
{ assume H: ?lhs then obtain n where n: "P n" by blast
|
|
785 |
let ?x = "Least P"
|
|
786 |
{fix m assume m: "m < ?x"
|
|
787 |
from not_less_Least[OF m] have "\<not> P m" .}
|
|
788 |
with LeastI_ex[OF H] have ?rhs by blast}
|
|
789 |
ultimately show ?thesis by blast
|
|
790 |
qed
|
|
791 |
|
|
792 |
lemma power_mod: "((x::nat) mod m)^n mod m = x^n mod m"
|
|
793 |
proof(induct n)
|
|
794 |
case 0 thus ?case by simp
|
|
795 |
next
|
|
796 |
case (Suc n)
|
|
797 |
have "(x mod m)^(Suc n) mod m = ((x mod m) * (((x mod m) ^ n) mod m)) mod m"
|
|
798 |
by (simp add: mod_mult1_eq[symmetric])
|
|
799 |
also have "\<dots> = ((x mod m) * (x^n mod m)) mod m" using Suc.hyps by simp
|
|
800 |
also have "\<dots> = x^(Suc n) mod m"
|
|
801 |
by (simp add: mod_mult1_eq'[symmetric] mod_mult1_eq[symmetric])
|
|
802 |
finally show ?case .
|
|
803 |
qed
|
|
804 |
|
|
805 |
lemma lucas:
|
|
806 |
assumes n2: "n \<ge> 2" and an1: "[a^(n - 1) = 1] (mod n)"
|
|
807 |
and pn: "\<forall>p. prime p \<and> p dvd n - 1 \<longrightarrow> \<not> [a^((n - 1) div p) = 1] (mod n)"
|
|
808 |
shows "prime n"
|
|
809 |
proof-
|
|
810 |
from n2 have n01: "n\<noteq>0" "n\<noteq>1" "n - 1 \<noteq> 0" by arith+
|
|
811 |
from mod_less_divisor[of n 1] n01 have onen: "1 mod n = 1" by simp
|
|
812 |
from lucas_coprime_lemma[OF n01(3) an1] cong_coprime[OF an1]
|
|
813 |
have an: "coprime a n" "coprime (a^(n - 1)) n" by (simp_all add: coprime_commute)
|
|
814 |
{assume H0: "\<exists>m. 0 < m \<and> m < n - 1 \<and> [a ^ m = 1] (mod n)" (is "EX m. ?P m")
|
|
815 |
from H0[unfolded nat_exists_least_iff[of ?P]] obtain m where
|
|
816 |
m: "0 < m" "m < n - 1" "[a ^ m = 1] (mod n)" "\<forall>k <m. \<not>?P k" by blast
|
|
817 |
{assume nm1: "(n - 1) mod m > 0"
|
|
818 |
from mod_less_divisor[OF m(1)] have th0:"(n - 1) mod m < m" by blast
|
|
819 |
let ?y = "a^ ((n - 1) div m * m)"
|
|
820 |
note mdeq = mod_div_equality[of "(n - 1)" m]
|
|
821 |
from coprime_exp[OF an(1)[unfolded coprime_commute[of a n]],
|
|
822 |
of "(n - 1) div m * m"]
|
|
823 |
have yn: "coprime ?y n" by (simp add: coprime_commute)
|
|
824 |
have "?y mod n = (a^m)^((n - 1) div m) mod n"
|
|
825 |
by (simp add: ring_simps power_mult)
|
|
826 |
also have "\<dots> = (a^m mod n)^((n - 1) div m) mod n"
|
|
827 |
using power_mod[of "a^m" n "(n - 1) div m"] by simp
|
26158
|
828 |
also have "\<dots> = 1" using m(3)[unfolded modeq_def onen] onen
|
|
829 |
by (simp add: power_Suc0)
|
26126
|
830 |
finally have th3: "?y mod n = 1" .
|
|
831 |
have th2: "[?y * a ^ ((n - 1) mod m) = ?y* 1] (mod n)"
|
|
832 |
using an1[unfolded modeq_def onen] onen
|
|
833 |
mod_div_equality[of "(n - 1)" m, symmetric]
|
|
834 |
by (simp add:power_add[symmetric] modeq_def th3 del: One_nat_def)
|
|
835 |
from cong_mult_lcancel[of ?y n "a^((n - 1) mod m)" 1, OF yn th2]
|
|
836 |
have th1: "[a ^ ((n - 1) mod m) = 1] (mod n)" .
|
|
837 |
from m(4)[rule_format, OF th0] nm1
|
|
838 |
less_trans[OF mod_less_divisor[OF m(1), of "n - 1"] m(2)] th1
|
|
839 |
have False by blast }
|
|
840 |
hence "(n - 1) mod m = 0" by auto
|
|
841 |
then have mn: "m dvd n - 1" by presburger
|
|
842 |
then obtain r where r: "n - 1 = m*r" unfolding dvd_def by blast
|
|
843 |
from n01 r m(2) have r01: "r\<noteq>0" "r\<noteq>1" by - (rule ccontr, simp)+
|
|
844 |
from prime_factor[OF r01(2)] obtain p where p: "prime p" "p dvd r" by blast
|
|
845 |
hence th: "prime p \<and> p dvd n - 1" unfolding r by (auto intro: dvd_mult)
|
|
846 |
have "(a ^ ((n - 1) div p)) mod n = (a^(m*r div p)) mod n" using r
|
|
847 |
by (simp add: power_mult)
|
|
848 |
also have "\<dots> = (a^(m*(r div p))) mod n" using div_mult1_eq[of m r p] p(2)[unfolded dvd_eq_mod_eq_0] by simp
|
|
849 |
also have "\<dots> = ((a^m)^(r div p)) mod n" by (simp add: power_mult)
|
|
850 |
also have "\<dots> = ((a^m mod n)^(r div p)) mod n" using power_mod[of "a^m" "n" "r div p" ] ..
|
26158
|
851 |
also have "\<dots> = 1" using m(3) onen by (simp add: modeq_def power_Suc0)
|
26126
|
852 |
finally have "[(a ^ ((n - 1) div p))= 1] (mod n)"
|
|
853 |
using onen by (simp add: modeq_def)
|
|
854 |
with pn[rule_format, OF th] have False by blast}
|
|
855 |
hence th: "\<forall>m. 0 < m \<and> m < n - 1 \<longrightarrow> \<not> [a ^ m = 1] (mod n)" by blast
|
|
856 |
from lucas_weak[OF n2 an1 th] show ?thesis .
|
|
857 |
qed
|
|
858 |
|
|
859 |
(* Definition of the order of a number mod n (0 in non-coprime case). *)
|
|
860 |
|
|
861 |
definition "ord n a = (if coprime n a then Least (\<lambda>d. d > 0 \<and> [a ^d = 1] (mod n)) else 0)"
|
|
862 |
|
|
863 |
(* This has the expected properties. *)
|
|
864 |
|
|
865 |
lemma coprime_ord:
|
|
866 |
assumes na: "coprime n a"
|
|
867 |
shows "ord n a > 0 \<and> [a ^(ord n a) = 1] (mod n) \<and> (\<forall>m. 0 < m \<and> m < ord n a \<longrightarrow> \<not> [a^ m = 1] (mod n))"
|
|
868 |
proof-
|
|
869 |
let ?P = "\<lambda>d. 0 < d \<and> [a ^ d = 1] (mod n)"
|
|
870 |
from euclid[of a] obtain p where p: "prime p" "a < p" by blast
|
|
871 |
from na have o: "ord n a = Least ?P" by (simp add: ord_def)
|
|
872 |
{assume "n=0 \<or> n=1" with na have "\<exists>m>0. ?P m" apply auto apply (rule exI[where x=1]) by (simp add: modeq_def)}
|
|
873 |
moreover
|
|
874 |
{assume "n\<noteq>0 \<and> n\<noteq>1" hence n2:"n \<ge> 2" by arith
|
|
875 |
from na have na': "coprime a n" by (simp add: coprime_commute)
|
|
876 |
from phi_lowerbound_1[OF n2] fermat_little[OF na']
|
|
877 |
have ex: "\<exists>m>0. ?P m" by - (rule exI[where x="\<phi> n"], auto) }
|
|
878 |
ultimately have ex: "\<exists>m>0. ?P m" by blast
|
|
879 |
from nat_exists_least_iff'[of ?P] ex na show ?thesis
|
|
880 |
unfolding o[symmetric] by auto
|
|
881 |
qed
|
|
882 |
(* With the special value 0 for non-coprime case, it's more convenient. *)
|
|
883 |
lemma ord_works:
|
|
884 |
"[a ^ (ord n a) = 1] (mod n) \<and> (\<forall>m. 0 < m \<and> m < ord n a \<longrightarrow> ~[a^ m = 1] (mod n))"
|
|
885 |
apply (cases "coprime n a")
|
|
886 |
using coprime_ord[of n a]
|
|
887 |
by (blast, simp add: ord_def modeq_def)
|
|
888 |
|
|
889 |
lemma ord: "[a^(ord n a) = 1] (mod n)" using ord_works by blast
|
|
890 |
lemma ord_minimal: "0 < m \<Longrightarrow> m < ord n a \<Longrightarrow> ~[a^m = 1] (mod n)"
|
|
891 |
using ord_works by blast
|
|
892 |
lemma ord_eq_0: "ord n a = 0 \<longleftrightarrow> ~coprime n a"
|
|
893 |
by (cases "coprime n a", simp add: neq0_conv coprime_ord, simp add: neq0_conv ord_def)
|
|
894 |
|
|
895 |
lemma ord_divides:
|
|
896 |
"[a ^ d = 1] (mod n) \<longleftrightarrow> ord n a dvd d" (is "?lhs \<longleftrightarrow> ?rhs")
|
|
897 |
proof
|
|
898 |
assume rh: ?rhs
|
|
899 |
then obtain k where "d = ord n a * k" unfolding dvd_def by blast
|
|
900 |
hence "[a ^ d = (a ^ (ord n a) mod n)^k] (mod n)"
|
|
901 |
by (simp add : modeq_def power_mult power_mod)
|
|
902 |
also have "[(a ^ (ord n a) mod n)^k = 1] (mod n)"
|
26158
|
903 |
using ord[of a n, unfolded modeq_def]
|
|
904 |
by (simp add: modeq_def power_mod power_Suc0)
|
26126
|
905 |
finally show ?lhs .
|
|
906 |
next
|
|
907 |
assume lh: ?lhs
|
|
908 |
{ assume H: "\<not> coprime n a"
|
|
909 |
hence o: "ord n a = 0" by (simp add: ord_def)
|
|
910 |
{assume d: "d=0" with o H have ?rhs by (simp add: modeq_def)}
|
|
911 |
moreover
|
|
912 |
{assume d0: "d\<noteq>0" then obtain d' where d': "d = Suc d'" by (cases d, auto)
|
|
913 |
from H[unfolded coprime]
|
|
914 |
obtain p where p: "p dvd n" "p dvd a" "p \<noteq> 1" by auto
|
|
915 |
from lh[unfolded nat_mod]
|
|
916 |
obtain q1 q2 where q12:"a ^ d + n * q1 = 1 + n * q2" by blast
|
|
917 |
hence "a ^ d + n * q1 - n * q2 = 1" by simp
|
|
918 |
with dvd_diff [OF dvd_add [OF divides_rexp[OF p(2), of d'] dvd_mult2[OF p(1), of q1]] dvd_mult2[OF p(1), of q2]] d' have "p dvd 1" by simp
|
|
919 |
with p(3) have False by simp
|
|
920 |
hence ?rhs ..}
|
|
921 |
ultimately have ?rhs by blast}
|
|
922 |
moreover
|
|
923 |
{assume H: "coprime n a"
|
|
924 |
let ?o = "ord n a"
|
|
925 |
let ?q = "d div ord n a"
|
|
926 |
let ?r = "d mod ord n a"
|
|
927 |
from cong_exp[OF ord[of a n], of ?q]
|
26158
|
928 |
have eqo: "[(a^?o)^?q = 1] (mod n)" by (simp add: modeq_def power_Suc0)
|
26126
|
929 |
from H have onz: "?o \<noteq> 0" by (simp add: ord_eq_0)
|
|
930 |
hence op: "?o > 0" by simp
|
|
931 |
from mod_div_equality[of d "ord n a"] lh
|
|
932 |
have "[a^(?o*?q + ?r) = 1] (mod n)" by (simp add: modeq_def mult_commute)
|
|
933 |
hence "[(a^?o)^?q * (a^?r) = 1] (mod n)"
|
|
934 |
by (simp add: modeq_def power_mult[symmetric] power_add[symmetric])
|
|
935 |
hence th: "[a^?r = 1] (mod n)"
|
|
936 |
using eqo mod_mult1_eq'[of "(a^?o)^?q" "a^?r" n]
|
|
937 |
apply (simp add: modeq_def del: One_nat_def)
|
|
938 |
by (simp add: mod_mult1_eq'[symmetric])
|
|
939 |
{assume r: "?r = 0" hence ?rhs by (simp add: dvd_eq_mod_eq_0)}
|
|
940 |
moreover
|
|
941 |
{assume r: "?r \<noteq> 0"
|
|
942 |
with mod_less_divisor[OF op, of d] have r0o:"?r >0 \<and> ?r < ?o" by simp
|
|
943 |
from conjunct2[OF ord_works[of a n], rule_format, OF r0o] th
|
|
944 |
have ?rhs by blast}
|
|
945 |
ultimately have ?rhs by blast}
|
|
946 |
ultimately show ?rhs by blast
|
|
947 |
qed
|
|
948 |
|
|
949 |
lemma order_divides_phi: "coprime n a \<Longrightarrow> ord n a dvd \<phi> n"
|
|
950 |
using ord_divides fermat_little coprime_commute by simp
|
|
951 |
lemma order_divides_expdiff:
|
|
952 |
assumes na: "coprime n a"
|
|
953 |
shows "[a^d = a^e] (mod n) \<longleftrightarrow> [d = e] (mod (ord n a))"
|
|
954 |
proof-
|
|
955 |
{fix n a d e
|
|
956 |
assume na: "coprime n a" and ed: "(e::nat) \<le> d"
|
|
957 |
hence "\<exists>c. d = e + c" by arith
|
|
958 |
then obtain c where c: "d = e + c" by arith
|
|
959 |
from na have an: "coprime a n" by (simp add: coprime_commute)
|
|
960 |
from coprime_exp[OF na, of e]
|
|
961 |
have aen: "coprime (a^e) n" by (simp add: coprime_commute)
|
|
962 |
from coprime_exp[OF na, of c]
|
|
963 |
have acn: "coprime (a^c) n" by (simp add: coprime_commute)
|
|
964 |
have "[a^d = a^e] (mod n) \<longleftrightarrow> [a^(e + c) = a^(e + 0)] (mod n)"
|
|
965 |
using c by simp
|
|
966 |
also have "\<dots> \<longleftrightarrow> [a^e* a^c = a^e *a^0] (mod n)" by (simp add: power_add)
|
|
967 |
also have "\<dots> \<longleftrightarrow> [a ^ c = 1] (mod n)"
|
|
968 |
using cong_mult_lcancel_eq[OF aen, of "a^c" "a^0"] by simp
|
|
969 |
also have "\<dots> \<longleftrightarrow> ord n a dvd c" by (simp only: ord_divides)
|
|
970 |
also have "\<dots> \<longleftrightarrow> [e + c = e + 0] (mod ord n a)"
|
|
971 |
using cong_add_lcancel_eq[of e c 0 "ord n a", simplified cong_0_divides]
|
|
972 |
by simp
|
|
973 |
finally have "[a^d = a^e] (mod n) \<longleftrightarrow> [d = e] (mod (ord n a))"
|
|
974 |
using c by simp }
|
|
975 |
note th = this
|
|
976 |
have "e \<le> d \<or> d \<le> e" by arith
|
|
977 |
moreover
|
|
978 |
{assume ed: "e \<le> d" from th[OF na ed] have ?thesis .}
|
|
979 |
moreover
|
|
980 |
{assume de: "d \<le> e"
|
|
981 |
from th[OF na de] have ?thesis by (simp add: cong_commute) }
|
|
982 |
ultimately show ?thesis by blast
|
|
983 |
qed
|
|
984 |
|
|
985 |
(* Another trivial primality characterization. *)
|
|
986 |
|
|
987 |
lemma prime_prime_factor:
|
|
988 |
"prime n \<longleftrightarrow> n \<noteq> 1\<and> (\<forall>p. prime p \<and> p dvd n \<longrightarrow> p = n)"
|
|
989 |
proof-
|
|
990 |
{assume n: "n=0 \<or> n=1" hence ?thesis using prime_0 two_is_prime by auto}
|
|
991 |
moreover
|
|
992 |
{assume n: "n\<noteq>0" "n\<noteq>1"
|
|
993 |
{assume pn: "prime n"
|
|
994 |
|
|
995 |
from pn[unfolded prime_def] have "\<forall>p. prime p \<and> p dvd n \<longrightarrow> p = n"
|
|
996 |
using n
|
|
997 |
apply (cases "n = 0 \<or> n=1",simp)
|
|
998 |
by (clarsimp, erule_tac x="p" in allE, auto)}
|
|
999 |
moreover
|
|
1000 |
{assume H: "\<forall>p. prime p \<and> p dvd n \<longrightarrow> p = n"
|
|
1001 |
from n have n1: "n > 1" by arith
|
|
1002 |
{fix m assume m: "m dvd n" "m\<noteq>1"
|
|
1003 |
from prime_factor[OF m(2)] obtain p where
|
|
1004 |
p: "prime p" "p dvd m" by blast
|
|
1005 |
from dvd_trans[OF p(2) m(1)] p(1) H have "p = n" by blast
|
|
1006 |
with p(2) have "n dvd m" by simp
|
|
1007 |
hence "m=n" using dvd_anti_sym[OF m(1)] by simp }
|
|
1008 |
with n1 have "prime n" unfolding prime_def by auto }
|
|
1009 |
ultimately have ?thesis using n by blast}
|
|
1010 |
ultimately show ?thesis by auto
|
|
1011 |
qed
|
|
1012 |
|
|
1013 |
lemma prime_divisor_sqrt:
|
|
1014 |
"prime n \<longleftrightarrow> n \<noteq> 1 \<and> (\<forall>d. d dvd n \<and> d^2 \<le> n \<longrightarrow> d = 1)"
|
|
1015 |
proof-
|
|
1016 |
{assume "n=0 \<or> n=1" hence ?thesis using prime_0 prime_1
|
|
1017 |
by (auto simp add: nat_power_eq_0_iff)}
|
|
1018 |
moreover
|
|
1019 |
{assume n: "n\<noteq>0" "n\<noteq>1"
|
|
1020 |
hence np: "n > 1" by arith
|
|
1021 |
{fix d assume d: "d dvd n" "d^2 \<le> n" and H: "\<forall>m. m dvd n \<longrightarrow> m=1 \<or> m=n"
|
|
1022 |
from H d have d1n: "d = 1 \<or> d=n" by blast
|
|
1023 |
{assume dn: "d=n"
|
|
1024 |
have "n^2 > n*1" using n
|
|
1025 |
by (simp add: power2_eq_square mult_less_cancel1)
|
|
1026 |
with dn d(2) have "d=1" by simp}
|
|
1027 |
with d1n have "d = 1" by blast }
|
|
1028 |
moreover
|
|
1029 |
{fix d assume d: "d dvd n" and H: "\<forall>d'. d' dvd n \<and> d'^2 \<le> n \<longrightarrow> d' = 1"
|
|
1030 |
from d n have "d \<noteq> 0" apply - apply (rule ccontr) by simp
|
|
1031 |
hence dp: "d > 0" by simp
|
|
1032 |
from d[unfolded dvd_def] obtain e where e: "n= d*e" by blast
|
|
1033 |
from n dp e have ep:"e > 0" by simp
|
|
1034 |
have "d^2 \<le> n \<or> e^2 \<le> n" using dp ep
|
|
1035 |
by (auto simp add: e power2_eq_square mult_le_cancel_left)
|
|
1036 |
moreover
|
|
1037 |
{assume h: "d^2 \<le> n"
|
|
1038 |
from H[rule_format, of d] h d have "d = 1" by blast}
|
|
1039 |
moreover
|
|
1040 |
{assume h: "e^2 \<le> n"
|
|
1041 |
from e have "e dvd n" unfolding dvd_def by (simp add: mult_commute)
|
|
1042 |
with H[rule_format, of e] h have "e=1" by simp
|
|
1043 |
with e have "d = n" by simp}
|
|
1044 |
ultimately have "d=1 \<or> d=n" by blast}
|
|
1045 |
ultimately have ?thesis unfolding prime_def using np n(2) by blast}
|
|
1046 |
ultimately show ?thesis by auto
|
|
1047 |
qed
|
|
1048 |
lemma prime_prime_factor_sqrt:
|
|
1049 |
"prime n \<longleftrightarrow> n \<noteq> 0 \<and> n \<noteq> 1 \<and> \<not> (\<exists>p. prime p \<and> p dvd n \<and> p^2 \<le> n)"
|
|
1050 |
(is "?lhs \<longleftrightarrow>?rhs")
|
|
1051 |
proof-
|
|
1052 |
{assume "n=0 \<or> n=1" hence ?thesis using prime_0 prime_1 by auto}
|
|
1053 |
moreover
|
|
1054 |
{assume n: "n\<noteq>0" "n\<noteq>1"
|
|
1055 |
{assume H: ?lhs
|
|
1056 |
from H[unfolded prime_divisor_sqrt] n
|
|
1057 |
have ?rhs apply clarsimp by (erule_tac x="p" in allE, simp add: prime_1)
|
|
1058 |
}
|
|
1059 |
moreover
|
|
1060 |
{assume H: ?rhs
|
|
1061 |
{fix d assume d: "d dvd n" "d^2 \<le> n" "d\<noteq>1"
|
|
1062 |
from prime_factor[OF d(3)]
|
|
1063 |
obtain p where p: "prime p" "p dvd d" by blast
|
|
1064 |
from n have np: "n > 0" by arith
|
|
1065 |
from d(1) n have "d \<noteq> 0" by - (rule ccontr, auto)
|
|
1066 |
hence dp: "d > 0" by arith
|
|
1067 |
from mult_mono[OF dvd_imp_le[OF p(2) dp] dvd_imp_le[OF p(2) dp]] d(2)
|
|
1068 |
have "p^2 \<le> n" unfolding power2_eq_square by arith
|
|
1069 |
with H n p(1) dvd_trans[OF p(2) d(1)] have False by blast}
|
|
1070 |
with n prime_divisor_sqrt have ?lhs by auto}
|
|
1071 |
ultimately have ?thesis by blast }
|
|
1072 |
ultimately show ?thesis by (cases "n=0 \<or> n=1", auto)
|
|
1073 |
qed
|
|
1074 |
(* Pocklington theorem. *)
|
|
1075 |
|
|
1076 |
lemma pocklington_lemma:
|
|
1077 |
assumes n: "n \<ge> 2" and nqr: "n - 1 = q*r" and an: "[a^ (n - 1) = 1] (mod n)"
|
|
1078 |
and aq:"\<forall>p. prime p \<and> p dvd q \<longrightarrow> coprime (a^ ((n - 1) div p) - 1) n"
|
|
1079 |
and pp: "prime p" and pn: "p dvd n"
|
|
1080 |
shows "[p = 1] (mod q)"
|
|
1081 |
proof-
|
|
1082 |
from pp prime_0 prime_1 have p01: "p \<noteq> 0" "p \<noteq> 1" by - (rule ccontr, simp)+
|
|
1083 |
from cong_1_divides[OF an, unfolded nqr, unfolded dvd_def]
|
|
1084 |
obtain k where k: "a ^ (q * r) - 1 = n*k" by blast
|
|
1085 |
from pn[unfolded dvd_def] obtain l where l: "n = p*l" by blast
|
|
1086 |
{assume a0: "a = 0"
|
|
1087 |
hence "a^ (n - 1) = 0" using n by (simp add: power_0_left)
|
|
1088 |
with n an mod_less[of 1 n] have False by (simp add: power_0_left modeq_def)}
|
|
1089 |
hence a0: "a\<noteq>0" ..
|
|
1090 |
from n nqr have aqr0: "a ^ (q * r) \<noteq> 0" using a0 by (simp add: neq0_conv)
|
|
1091 |
hence "(a ^ (q * r) - 1) + 1 = a ^ (q * r)" by simp
|
|
1092 |
with k l have "a ^ (q * r) = p*l*k + 1" by simp
|
|
1093 |
hence "a ^ (r * q) + p * 0 = 1 + p * (l*k)" by (simp add: mult_ac)
|
|
1094 |
hence odq: "ord p (a^r) dvd q"
|
|
1095 |
unfolding ord_divides[symmetric] power_mult[symmetric] nat_mod by blast
|
|
1096 |
from odq[unfolded dvd_def] obtain d where d: "q = ord p (a^r) * d" by blast
|
|
1097 |
{assume d1: "d \<noteq> 1"
|
|
1098 |
from prime_factor[OF d1] obtain P where P: "prime P" "P dvd d" by blast
|
|
1099 |
from d dvd_mult[OF P(2), of "ord p (a^r)"] have Pq: "P dvd q" by simp
|
|
1100 |
from aq P(1) Pq have caP:"coprime (a^ ((n - 1) div P) - 1) n" by blast
|
|
1101 |
from Pq obtain s where s: "q = P*s" unfolding dvd_def by blast
|
|
1102 |
have P0: "P \<noteq> 0" using P(1) prime_0 by - (rule ccontr, simp)
|
|
1103 |
from P(2) obtain t where t: "d = P*t" unfolding dvd_def by blast
|
|
1104 |
from d s t P0 have s': "ord p (a^r) * t = s" by algebra
|
|
1105 |
have "ord p (a^r) * t*r = r * ord p (a^r) * t" by algebra
|
|
1106 |
hence exps: "a^(ord p (a^r) * t*r) = ((a ^ r) ^ ord p (a^r)) ^ t"
|
|
1107 |
by (simp only: power_mult)
|
|
1108 |
have "[((a ^ r) ^ ord p (a^r)) ^ t= 1^t] (mod p)"
|
|
1109 |
by (rule cong_exp, rule ord)
|
26158
|
1110 |
then have th: "[((a ^ r) ^ ord p (a^r)) ^ t= 1] (mod p)"
|
|
1111 |
by (simp add: power_Suc0)
|
26126
|
1112 |
from cong_1_divides[OF th] exps have pd0: "p dvd a^(ord p (a^r) * t*r) - 1" by simp
|
|
1113 |
from nqr s s' have "(n - 1) div P = ord p (a^r) * t*r" using P0 by simp
|
|
1114 |
with caP have "coprime (a^(ord p (a^r) * t*r) - 1) n" by simp
|
|
1115 |
with p01 pn pd0 have False unfolding coprime by auto}
|
|
1116 |
hence d1: "d = 1" by blast
|
|
1117 |
hence o: "ord p (a^r) = q" using d by simp
|
|
1118 |
from pp phi_prime[of p] have phip: " \<phi> p = p - 1" by simp
|
|
1119 |
{fix d assume d: "d dvd p" "d dvd a" "d \<noteq> 1"
|
|
1120 |
from pp[unfolded prime_def] d have dp: "d = p" by blast
|
|
1121 |
from n have n12:"Suc (n - 2) = n - 1" by arith
|
|
1122 |
with divides_rexp[OF d(2)[unfolded dp], of "n - 2"]
|
|
1123 |
have th0: "p dvd a ^ (n - 1)" by simp
|
|
1124 |
from n have n0: "n \<noteq> 0" by simp
|
|
1125 |
from d(2) an n12[symmetric] have a0: "a \<noteq> 0"
|
|
1126 |
by - (rule ccontr, simp add: modeq_def)
|
|
1127 |
have th1: "a^ (n - 1) \<noteq> 0" using n d(2) dp a0 by (auto simp add: neq0_conv)
|
|
1128 |
from coprime_minus1[OF th1, unfolded coprime]
|
|
1129 |
dvd_trans[OF pn cong_1_divides[OF an]] th0 d(3) dp
|
|
1130 |
have False by auto}
|
|
1131 |
hence cpa: "coprime p a" using coprime by auto
|
|
1132 |
from coprime_exp[OF cpa, of r] coprime_commute
|
|
1133 |
have arp: "coprime (a^r) p" by blast
|
|
1134 |
from fermat_little[OF arp, simplified ord_divides] o phip
|
|
1135 |
have "q dvd (p - 1)" by simp
|
|
1136 |
then obtain d where d:"p - 1 = q * d" unfolding dvd_def by blast
|
|
1137 |
from prime_0 pp have p0:"p \<noteq> 0" by - (rule ccontr, auto)
|
|
1138 |
from p0 d have "p + q * 0 = 1 + q * d" by simp
|
|
1139 |
with nat_mod[of p 1 q, symmetric]
|
|
1140 |
show ?thesis by blast
|
|
1141 |
qed
|
|
1142 |
|
|
1143 |
lemma pocklington:
|
|
1144 |
assumes n: "n \<ge> 2" and nqr: "n - 1 = q*r" and sqr: "n \<le> q^2"
|
|
1145 |
and an: "[a^ (n - 1) = 1] (mod n)"
|
|
1146 |
and aq:"\<forall>p. prime p \<and> p dvd q \<longrightarrow> coprime (a^ ((n - 1) div p) - 1) n"
|
|
1147 |
shows "prime n"
|
|
1148 |
unfolding prime_prime_factor_sqrt[of n]
|
|
1149 |
proof-
|
|
1150 |
let ?ths = "n \<noteq> 0 \<and> n \<noteq> 1 \<and> \<not> (\<exists>p. prime p \<and> p dvd n \<and> p\<twosuperior> \<le> n)"
|
|
1151 |
from n have n01: "n\<noteq>0" "n\<noteq>1" by arith+
|
|
1152 |
{fix p assume p: "prime p" "p dvd n" "p^2 \<le> n"
|
|
1153 |
from p(3) sqr have "p^(Suc 1) \<le> q^(Suc 1)" by (simp add: power2_eq_square)
|
|
1154 |
hence pq: "p \<le> q" unfolding exp_mono_le .
|
|
1155 |
from pocklington_lemma[OF n nqr an aq p(1,2)] cong_1_divides
|
|
1156 |
have th: "q dvd p - 1" by blast
|
|
1157 |
have "p - 1 \<noteq> 0"using prime_ge_2[OF p(1)] by arith
|
|
1158 |
with divides_ge[OF th] pq have False by arith }
|
|
1159 |
with n01 show ?ths by blast
|
|
1160 |
qed
|
|
1161 |
|
|
1162 |
(* Variant for application, to separate the exponentiation. *)
|
|
1163 |
lemma pocklington_alt:
|
|
1164 |
assumes n: "n \<ge> 2" and nqr: "n - 1 = q*r" and sqr: "n \<le> q^2"
|
|
1165 |
and an: "[a^ (n - 1) = 1] (mod n)"
|
|
1166 |
and aq:"\<forall>p. prime p \<and> p dvd q \<longrightarrow> (\<exists>b. [a^((n - 1) div p) = b] (mod n) \<and> coprime (b - 1) n)"
|
|
1167 |
shows "prime n"
|
|
1168 |
proof-
|
|
1169 |
{fix p assume p: "prime p" "p dvd q"
|
|
1170 |
from aq[rule_format] p obtain b where
|
|
1171 |
b: "[a^((n - 1) div p) = b] (mod n)" "coprime (b - 1) n" by blast
|
|
1172 |
{assume a0: "a=0"
|
|
1173 |
from n an have "[0 = 1] (mod n)" unfolding a0 power_0_left by auto
|
|
1174 |
hence False using n by (simp add: modeq_def dvd_eq_mod_eq_0[symmetric])}
|
|
1175 |
hence a0: "a\<noteq> 0" ..
|
|
1176 |
hence a1: "a \<ge> 1" by arith
|
|
1177 |
from one_le_power[OF a1] have ath: "1 \<le> a ^ ((n - 1) div p)" .
|
|
1178 |
{assume b0: "b = 0"
|
|
1179 |
from p(2) nqr have "(n - 1) mod p = 0"
|
|
1180 |
apply (simp only: dvd_eq_mod_eq_0[symmetric]) by (rule dvd_mult2, simp)
|
|
1181 |
with mod_div_equality[of "n - 1" p]
|
|
1182 |
have "(n - 1) div p * p= n - 1" by auto
|
|
1183 |
hence eq: "(a^((n - 1) div p))^p = a^(n - 1)"
|
|
1184 |
by (simp only: power_mult[symmetric])
|
|
1185 |
from prime_ge_2[OF p(1)] have pS: "Suc (p - 1) = p" by arith
|
|
1186 |
from b(1) have d: "n dvd a^((n - 1) div p)" unfolding b0 cong_0_divides .
|
|
1187 |
from divides_rexp[OF d, of "p - 1"] pS eq cong_divides[OF an] n
|
|
1188 |
have False by simp}
|
|
1189 |
then have b0: "b \<noteq> 0" ..
|
|
1190 |
hence b1: "b \<ge> 1" by arith
|
|
1191 |
from cong_coprime[OF cong_sub[OF b(1) cong_refl[of 1] ath b1]] b(2) nqr
|
|
1192 |
have "coprime (a ^ ((n - 1) div p) - 1) n" by (simp add: coprime_commute)}
|
|
1193 |
hence th: "\<forall>p. prime p \<and> p dvd q \<longrightarrow> coprime (a ^ ((n - 1) div p) - 1) n "
|
|
1194 |
by blast
|
|
1195 |
from pocklington[OF n nqr sqr an th] show ?thesis .
|
|
1196 |
qed
|
|
1197 |
|
|
1198 |
(* Prime factorizations. *)
|
|
1199 |
|
|
1200 |
definition "primefact ps n = (foldr op * ps 1 = n \<and> (\<forall>p\<in> set ps. prime p))"
|
|
1201 |
|
|
1202 |
lemma primefact: assumes n: "n \<noteq> 0"
|
|
1203 |
shows "\<exists>ps. primefact ps n"
|
|
1204 |
using n
|
|
1205 |
proof(induct n rule: nat_less_induct)
|
|
1206 |
fix n assume H: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>ps. primefact ps m)" and n: "n\<noteq>0"
|
|
1207 |
let ?ths = "\<exists>ps. primefact ps n"
|
|
1208 |
{assume "n = 1"
|
|
1209 |
hence "primefact [] n" by (simp add: primefact_def)
|
|
1210 |
hence ?ths by blast }
|
|
1211 |
moreover
|
|
1212 |
{assume n1: "n \<noteq> 1"
|
|
1213 |
with n have n2: "n \<ge> 2" by arith
|
|
1214 |
from prime_factor[OF n1] obtain p where p: "prime p" "p dvd n" by blast
|
|
1215 |
from p(2) obtain m where m: "n = p*m" unfolding dvd_def by blast
|
|
1216 |
from n m have m0: "m > 0" "m\<noteq>0" by auto
|
|
1217 |
from prime_ge_2[OF p(1)] have "1 < p" by arith
|
|
1218 |
with m0 m have mn: "m < n" by auto
|
|
1219 |
from H[rule_format, OF mn m0(2)] obtain ps where ps: "primefact ps m" ..
|
|
1220 |
from ps m p(1) have "primefact (p#ps) n" by (simp add: primefact_def)
|
|
1221 |
hence ?ths by blast}
|
|
1222 |
ultimately show ?ths by blast
|
|
1223 |
qed
|
|
1224 |
|
|
1225 |
lemma primefact_contains:
|
|
1226 |
assumes pf: "primefact ps n" and p: "prime p" and pn: "p dvd n"
|
|
1227 |
shows "p \<in> set ps"
|
|
1228 |
using pf p pn
|
|
1229 |
proof(induct ps arbitrary: p n)
|
|
1230 |
case Nil thus ?case by (auto simp add: primefact_def)
|
|
1231 |
next
|
|
1232 |
case (Cons q qs p n)
|
|
1233 |
from Cons.prems[unfolded primefact_def]
|
|
1234 |
have q: "prime q" "q * foldr op * qs 1 = n" "\<forall>p \<in>set qs. prime p" and p: "prime p" "p dvd q * foldr op * qs 1" by simp_all
|
|
1235 |
{assume "p dvd q"
|
|
1236 |
with p(1) q(1) have "p = q" unfolding prime_def by auto
|
|
1237 |
hence ?case by simp}
|
|
1238 |
moreover
|
|
1239 |
{ assume h: "p dvd foldr op * qs 1"
|
|
1240 |
from q(3) have pqs: "primefact qs (foldr op * qs 1)"
|
|
1241 |
by (simp add: primefact_def)
|
|
1242 |
from Cons.hyps[OF pqs p(1) h] have ?case by simp}
|
|
1243 |
ultimately show ?case using prime_divprod[OF p] by blast
|
|
1244 |
qed
|
|
1245 |
|
|
1246 |
lemma primefact_variant: "primefact ps n \<longleftrightarrow> foldr op * ps 1 = n \<and> list_all prime ps" by (auto simp add: primefact_def list_all_iff)
|
|
1247 |
|
|
1248 |
(* Variant of Lucas theorem. *)
|
|
1249 |
|
|
1250 |
lemma lucas_primefact:
|
|
1251 |
assumes n: "n \<ge> 2" and an: "[a^(n - 1) = 1] (mod n)"
|
|
1252 |
and psn: "foldr op * ps 1 = n - 1"
|
|
1253 |
and psp: "list_all (\<lambda>p. prime p \<and> \<not> [a^((n - 1) div p) = 1] (mod n)) ps"
|
|
1254 |
shows "prime n"
|
|
1255 |
proof-
|
|
1256 |
{fix p assume p: "prime p" "p dvd n - 1" "[a ^ ((n - 1) div p) = 1] (mod n)"
|
|
1257 |
from psn psp have psn1: "primefact ps (n - 1)"
|
|
1258 |
by (auto simp add: list_all_iff primefact_variant)
|
|
1259 |
from p(3) primefact_contains[OF psn1 p(1,2)] psp
|
|
1260 |
have False by (induct ps, auto)}
|
|
1261 |
with lucas[OF n an] show ?thesis by blast
|
|
1262 |
qed
|
|
1263 |
|
|
1264 |
(* Variant of Pocklington theorem. *)
|
|
1265 |
|
|
1266 |
lemma mod_le: assumes n: "n \<noteq> (0::nat)" shows "m mod n \<le> m"
|
|
1267 |
proof-
|
|
1268 |
from mod_div_equality[of m n]
|
|
1269 |
have "\<exists>x. x + m mod n = m" by blast
|
|
1270 |
then show ?thesis by auto
|
|
1271 |
qed
|
|
1272 |
|
|
1273 |
|
|
1274 |
lemma pocklington_primefact:
|
|
1275 |
assumes n: "n \<ge> 2" and qrn: "q*r = n - 1" and nq2: "n \<le> q^2"
|
|
1276 |
and arnb: "(a^r) mod n = b" and psq: "foldr op * ps 1 = q"
|
|
1277 |
and bqn: "(b^q) mod n = 1"
|
|
1278 |
and psp: "list_all (\<lambda>p. prime p \<and> coprime ((b^(q div p)) mod n - 1) n) ps"
|
|
1279 |
shows "prime n"
|
|
1280 |
proof-
|
|
1281 |
from bqn psp qrn
|
|
1282 |
have bqn: "a ^ (n - 1) mod n = 1"
|
|
1283 |
and psp: "list_all (\<lambda>p. prime p \<and> coprime (a^(r *(q div p)) mod n - 1) n) ps" unfolding arnb[symmetric] power_mod
|
|
1284 |
by (simp_all add: power_mult[symmetric] ring_simps)
|
|
1285 |
from n have n0: "n > 0" by arith
|
|
1286 |
from mod_div_equality[of "a^(n - 1)" n]
|
|
1287 |
mod_less_divisor[OF n0, of "a^(n - 1)"]
|
|
1288 |
have an1: "[a ^ (n - 1) = 1] (mod n)"
|
|
1289 |
unfolding nat_mod bqn
|
|
1290 |
apply -
|
|
1291 |
apply (rule exI[where x="0"])
|
|
1292 |
apply (rule exI[where x="a^(n - 1) div n"])
|
|
1293 |
by (simp add: ring_simps)
|
|
1294 |
{fix p assume p: "prime p" "p dvd q"
|
|
1295 |
from psp psq have pfpsq: "primefact ps q"
|
|
1296 |
by (auto simp add: primefact_variant list_all_iff)
|
|
1297 |
from psp primefact_contains[OF pfpsq p]
|
|
1298 |
have p': "coprime (a ^ (r * (q div p)) mod n - 1) n"
|
|
1299 |
by (simp add: list_all_iff)
|
|
1300 |
from prime_ge_2[OF p(1)] have p01: "p \<noteq> 0" "p \<noteq> 1" "p =Suc(p - 1)" by arith+
|
|
1301 |
from div_mult1_eq[of r q p] p(2)
|
|
1302 |
have eq1: "r* (q div p) = (n - 1) div p"
|
|
1303 |
unfolding qrn[symmetric] dvd_eq_mod_eq_0 by (simp add: mult_commute)
|
|
1304 |
have ath: "\<And>a (b::nat). a <= b \<Longrightarrow> a \<noteq> 0 ==> 1 <= a \<and> 1 <= b" by arith
|
|
1305 |
from n0 have n00: "n \<noteq> 0" by arith
|
|
1306 |
from mod_le[OF n00]
|
|
1307 |
have th10: "a ^ ((n - 1) div p) mod n \<le> a ^ ((n - 1) div p)" .
|
|
1308 |
{assume "a ^ ((n - 1) div p) mod n = 0"
|
|
1309 |
then obtain s where s: "a ^ ((n - 1) div p) = n*s"
|
|
1310 |
unfolding mod_eq_0_iff by blast
|
|
1311 |
hence eq0: "(a^((n - 1) div p))^p = (n*s)^p" by simp
|
|
1312 |
from qrn[symmetric] have qn1: "q dvd n - 1" unfolding dvd_def by auto
|
|
1313 |
from dvd_trans[OF p(2) qn1] div_mod_equality'[of "n - 1" p]
|
|
1314 |
have npp: "(n - 1) div p * p = n - 1" by (simp add: dvd_eq_mod_eq_0)
|
|
1315 |
with eq0 have "a^ (n - 1) = (n*s)^p"
|
|
1316 |
by (simp add: power_mult[symmetric])
|
|
1317 |
hence "1 = (n*s)^(Suc (p - 1)) mod n" using bqn p01 by simp
|
|
1318 |
also have "\<dots> = 0" by (simp add: mult_assoc mod_mult_self_is_0)
|
|
1319 |
finally have False by simp }
|
|
1320 |
then have th11: "a ^ ((n - 1) div p) mod n \<noteq> 0" by auto
|
|
1321 |
have th1: "[a ^ ((n - 1) div p) mod n = a ^ ((n - 1) div p)] (mod n)"
|
|
1322 |
unfolding modeq_def by simp
|
|
1323 |
from cong_sub[OF th1 cong_refl[of 1]] ath[OF th10 th11]
|
|
1324 |
have th: "[a ^ ((n - 1) div p) mod n - 1 = a ^ ((n - 1) div p) - 1] (mod n)"
|
|
1325 |
by blast
|
|
1326 |
from cong_coprime[OF th] p'[unfolded eq1]
|
|
1327 |
have "coprime (a ^ ((n - 1) div p) - 1) n" by (simp add: coprime_commute) }
|
|
1328 |
with pocklington[OF n qrn[symmetric] nq2 an1]
|
|
1329 |
show ?thesis by blast
|
|
1330 |
qed
|
|
1331 |
|
|
1332 |
|
|
1333 |
end
|