1.1 --- a/NEWS Tue Sep 01 14:13:34 2009 +0200
1.2 +++ b/NEWS Tue Sep 01 15:39:33 2009 +0200
1.3 @@ -18,6 +18,15 @@
1.4
1.5 *** HOL ***
1.6
1.7 +* Reorganization of number theory:
1.8 + * former session NumberTheory now names Old_Number_Theory; former session NewNumberTheory
1.9 + named NumberTheory;
1.10 + * split off prime number ingredients from theory GCD to theory Number_Theory/Primes;
1.11 + * moved legacy theories Legacy_GCD and Primes from Library/ to Old_Number_Theory/;
1.12 + * moved theory Pocklington from Library/ to Old_Number_Theory/;
1.13 + * removed various references to Old_Number_Theory from HOL distribution.
1.14 +INCOMPATIBILITY.
1.15 +
1.16 * New testing tool "Mirabelle" for automated (proof) tools. Applies
1.17 several tools and tactics like sledgehammer, metis, or quickcheck, to
1.18 every proof step in a theory. To be used in batch mode via the
2.1 --- a/src/HOL/Extraction/Euclid.thy Tue Sep 01 14:13:34 2009 +0200
2.2 +++ b/src/HOL/Extraction/Euclid.thy Tue Sep 01 15:39:33 2009 +0200
2.3 @@ -1,5 +1,4 @@
2.4 (* Title: HOL/Extraction/Euclid.thy
2.5 - ID: $Id$
2.6 Author: Markus Wenzel, TU Muenchen
2.7 Freek Wiedijk, Radboud University Nijmegen
2.8 Stefan Berghofer, TU Muenchen
2.9 @@ -8,7 +7,7 @@
2.10 header {* Euclid's theorem *}
2.11
2.12 theory Euclid
2.13 -imports "~~/src/HOL/NumberTheory/Factorization" Util Efficient_Nat
2.14 +imports "~~/src/HOL/Old_Number_Theory/Factorization" Util Efficient_Nat
2.15 begin
2.16
2.17 text {*
3.1 --- a/src/HOL/Extraction/ROOT.ML Tue Sep 01 14:13:34 2009 +0200
3.2 +++ b/src/HOL/Extraction/ROOT.ML Tue Sep 01 15:39:33 2009 +0200
3.3 @@ -1,5 +1,4 @@
3.4 (* Title: HOL/Extraction/ROOT.ML
3.5 - ID: $Id$
3.6
3.7 Examples for program extraction in Higher-Order Logic.
3.8 *)
3.9 @@ -8,5 +7,5 @@
3.10 warning "HOL proof terms required for running extraction examples"
3.11 else
3.12 (Proofterm.proofs := 2;
3.13 - no_document use_thys ["Efficient_Nat", "~~/src/HOL/NumberTheory/Factorization"];
3.14 + no_document use_thys ["Efficient_Nat", "~~/src/HOL/Old_Number_Theory/Factorization"];
3.15 use_thys ["Greatest_Common_Divisor", "Warshall", "Higman", "Pigeonhole", "Euclid"]);
4.1 --- a/src/HOL/GCD.thy Tue Sep 01 14:13:34 2009 +0200
4.2 +++ b/src/HOL/GCD.thy Tue Sep 01 15:39:33 2009 +0200
4.3 @@ -1,11 +1,9 @@
4.4 -(* Title: GCD.thy
4.5 - Authors: Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
4.6 +(* Authors: Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
4.7 Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow
4.8
4.9
4.10 -This file deals with the functions gcd and lcm, and properties of
4.11 -primes. Definitions and lemmas are proved uniformly for the natural
4.12 -numbers and integers.
4.13 +This file deals with the functions gcd and lcm. Definitions and
4.14 +lemmas are proved uniformly for the natural numbers and integers.
4.15
4.16 This file combines and revises a number of prior developments.
4.17
4.18 @@ -52,11 +50,6 @@
4.19
4.20 end
4.21
4.22 -class prime = one +
4.23 -
4.24 -fixes
4.25 - prime :: "'a \<Rightarrow> bool"
4.26 -
4.27
4.28 (* definitions for the natural numbers *)
4.29
4.30 @@ -80,20 +73,6 @@
4.31 end
4.32
4.33
4.34 -instantiation nat :: prime
4.35 -
4.36 -begin
4.37 -
4.38 -definition
4.39 - prime_nat :: "nat \<Rightarrow> bool"
4.40 -where
4.41 - [code del]: "prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
4.42 -
4.43 -instance proof qed
4.44 -
4.45 -end
4.46 -
4.47 -
4.48 (* definitions for the integers *)
4.49
4.50 instantiation int :: gcd
4.51 @@ -115,28 +94,13 @@
4.52 end
4.53
4.54
4.55 -instantiation int :: prime
4.56 -
4.57 -begin
4.58 -
4.59 -definition
4.60 - prime_int :: "int \<Rightarrow> bool"
4.61 -where
4.62 - [code del]: "prime_int p = prime (nat p)"
4.63 -
4.64 -instance proof qed
4.65 -
4.66 -end
4.67 -
4.68 -
4.69 subsection {* Set up Transfer *}
4.70
4.71
4.72 lemma transfer_nat_int_gcd:
4.73 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
4.74 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
4.75 - "(x::int) >= 0 \<Longrightarrow> prime (nat x) = prime x"
4.76 - unfolding gcd_int_def lcm_int_def prime_int_def
4.77 + unfolding gcd_int_def lcm_int_def
4.78 by auto
4.79
4.80 lemma transfer_nat_int_gcd_closures:
4.81 @@ -150,8 +114,7 @@
4.82 lemma transfer_int_nat_gcd:
4.83 "gcd (int x) (int y) = int (gcd x y)"
4.84 "lcm (int x) (int y) = int (lcm x y)"
4.85 - "prime (int x) = prime x"
4.86 - by (unfold gcd_int_def lcm_int_def prime_int_def, auto)
4.87 + by (unfold gcd_int_def lcm_int_def, auto)
4.88
4.89 lemma transfer_int_nat_gcd_closures:
4.90 "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
4.91 @@ -1003,20 +966,6 @@
4.92 apply (auto simp add: gcd_mult_cancel_int)
4.93 done
4.94
4.95 -lemma prime_odd_nat: "prime (p::nat) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
4.96 - unfolding prime_nat_def
4.97 - apply (subst even_mult_two_ex)
4.98 - apply clarify
4.99 - apply (drule_tac x = 2 in spec)
4.100 - apply auto
4.101 -done
4.102 -
4.103 -lemma prime_odd_int: "prime (p::int) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
4.104 - unfolding prime_int_def
4.105 - apply (frule prime_odd_nat)
4.106 - apply (auto simp add: even_nat_def)
4.107 -done
4.108 -
4.109 lemma coprime_common_divisor_nat: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
4.110 x dvd b \<Longrightarrow> x = 1"
4.111 apply (subgoal_tac "x dvd gcd a b")
4.112 @@ -1753,327 +1702,4 @@
4.113 show ?thesis by(simp add: Gcd_def fold_set gcd_commute_int)
4.114 qed
4.115
4.116 -
4.117 -subsection {* Primes *}
4.118 -
4.119 -(* FIXME Is there a better way to handle these, rather than making them elim rules? *)
4.120 -
4.121 -lemma prime_ge_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 0"
4.122 - by (unfold prime_nat_def, auto)
4.123 -
4.124 -lemma prime_gt_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p > 0"
4.125 - by (unfold prime_nat_def, auto)
4.126 -
4.127 -lemma prime_ge_1_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 1"
4.128 - by (unfold prime_nat_def, auto)
4.129 -
4.130 -lemma prime_gt_1_nat [elim]: "prime (p::nat) \<Longrightarrow> p > 1"
4.131 - by (unfold prime_nat_def, auto)
4.132 -
4.133 -lemma prime_ge_Suc_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= Suc 0"
4.134 - by (unfold prime_nat_def, auto)
4.135 -
4.136 -lemma prime_gt_Suc_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p > Suc 0"
4.137 - by (unfold prime_nat_def, auto)
4.138 -
4.139 -lemma prime_ge_2_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 2"
4.140 - by (unfold prime_nat_def, auto)
4.141 -
4.142 -lemma prime_ge_0_int [elim]: "prime (p::int) \<Longrightarrow> p >= 0"
4.143 - by (unfold prime_int_def prime_nat_def) auto
4.144 -
4.145 -lemma prime_gt_0_int [elim]: "prime (p::int) \<Longrightarrow> p > 0"
4.146 - by (unfold prime_int_def prime_nat_def, auto)
4.147 -
4.148 -lemma prime_ge_1_int [elim]: "prime (p::int) \<Longrightarrow> p >= 1"
4.149 - by (unfold prime_int_def prime_nat_def, auto)
4.150 -
4.151 -lemma prime_gt_1_int [elim]: "prime (p::int) \<Longrightarrow> p > 1"
4.152 - by (unfold prime_int_def prime_nat_def, auto)
4.153 -
4.154 -lemma prime_ge_2_int [elim]: "prime (p::int) \<Longrightarrow> p >= 2"
4.155 - by (unfold prime_int_def prime_nat_def, auto)
4.156 -
4.157 -
4.158 -lemma prime_int_altdef: "prime (p::int) = (1 < p \<and> (\<forall>m \<ge> 0. m dvd p \<longrightarrow>
4.159 - m = 1 \<or> m = p))"
4.160 - using prime_nat_def [transferred]
4.161 - apply (case_tac "p >= 0")
4.162 - by (blast, auto simp add: prime_ge_0_int)
4.163 -
4.164 -lemma prime_imp_coprime_nat: "prime (p::nat) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
4.165 - apply (unfold prime_nat_def)
4.166 - apply (metis gcd_dvd1_nat gcd_dvd2_nat)
4.167 - done
4.168 -
4.169 -lemma prime_imp_coprime_int: "prime (p::int) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
4.170 - apply (unfold prime_int_altdef)
4.171 - apply (metis gcd_dvd1_int gcd_dvd2_int gcd_ge_0_int)
4.172 - done
4.173 -
4.174 -lemma prime_dvd_mult_nat: "prime (p::nat) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
4.175 - by (blast intro: coprime_dvd_mult_nat prime_imp_coprime_nat)
4.176 -
4.177 -lemma prime_dvd_mult_int: "prime (p::int) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
4.178 - by (blast intro: coprime_dvd_mult_int prime_imp_coprime_int)
4.179 -
4.180 -lemma prime_dvd_mult_eq_nat [simp]: "prime (p::nat) \<Longrightarrow>
4.181 - p dvd m * n = (p dvd m \<or> p dvd n)"
4.182 - by (rule iffI, rule prime_dvd_mult_nat, auto)
4.183 -
4.184 -lemma prime_dvd_mult_eq_int [simp]: "prime (p::int) \<Longrightarrow>
4.185 - p dvd m * n = (p dvd m \<or> p dvd n)"
4.186 - by (rule iffI, rule prime_dvd_mult_int, auto)
4.187 -
4.188 -lemma not_prime_eq_prod_nat: "(n::nat) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
4.189 - EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
4.190 - unfolding prime_nat_def dvd_def apply auto
4.191 - by(metis mult_commute linorder_neq_iff linorder_not_le mult_1 n_less_n_mult_m one_le_mult_iff less_imp_le_nat)
4.192 -
4.193 -lemma not_prime_eq_prod_int: "(n::int) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
4.194 - EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
4.195 - unfolding prime_int_altdef dvd_def
4.196 - apply auto
4.197 - by(metis div_mult_self1_is_id div_mult_self2_is_id int_div_less_self int_one_le_iff_zero_less zero_less_mult_pos zless_le)
4.198 -
4.199 -lemma prime_dvd_power_nat [rule_format]: "prime (p::nat) -->
4.200 - n > 0 --> (p dvd x^n --> p dvd x)"
4.201 - by (induct n rule: nat_induct, auto)
4.202 -
4.203 -lemma prime_dvd_power_int [rule_format]: "prime (p::int) -->
4.204 - n > 0 --> (p dvd x^n --> p dvd x)"
4.205 - apply (induct n rule: nat_induct, auto)
4.206 - apply (frule prime_ge_0_int)
4.207 - apply auto
4.208 -done
4.209 -
4.210 -subsubsection{* Make prime naively executable *}
4.211 -
4.212 -lemma zero_not_prime_nat [simp]: "~prime (0::nat)"
4.213 - by (simp add: prime_nat_def)
4.214 -
4.215 -lemma zero_not_prime_int [simp]: "~prime (0::int)"
4.216 - by (simp add: prime_int_def)
4.217 -
4.218 -lemma one_not_prime_nat [simp]: "~prime (1::nat)"
4.219 - by (simp add: prime_nat_def)
4.220 -
4.221 -lemma Suc_0_not_prime_nat [simp]: "~prime (Suc 0)"
4.222 - by (simp add: prime_nat_def One_nat_def)
4.223 -
4.224 -lemma one_not_prime_int [simp]: "~prime (1::int)"
4.225 - by (simp add: prime_int_def)
4.226 -
4.227 -lemma prime_nat_code[code]:
4.228 - "prime(p::nat) = (p > 1 & (ALL n : {1<..<p}. ~(n dvd p)))"
4.229 -apply(simp add: Ball_def)
4.230 -apply (metis less_not_refl prime_nat_def dvd_triv_right not_prime_eq_prod_nat)
4.231 -done
4.232 -
4.233 -lemma prime_nat_simp:
4.234 - "prime(p::nat) = (p > 1 & (list_all (%n. ~ n dvd p) [2..<p]))"
4.235 -apply(simp only:prime_nat_code list_ball_code greaterThanLessThan_upt)
4.236 -apply(simp add:nat_number One_nat_def)
4.237 -done
4.238 -
4.239 -lemmas prime_nat_simp_number_of[simp] = prime_nat_simp[of "number_of m", standard]
4.240 -
4.241 -lemma prime_int_code[code]:
4.242 - "prime(p::int) = (p > 1 & (ALL n : {1<..<p}. ~(n dvd p)))" (is "?L = ?R")
4.243 -proof
4.244 - assume "?L" thus "?R"
4.245 - by (clarsimp simp: prime_gt_1_int) (metis int_one_le_iff_zero_less prime_int_altdef zless_le)
4.246 -next
4.247 - assume "?R" thus "?L" by (clarsimp simp:Ball_def) (metis dvdI not_prime_eq_prod_int)
4.248 -qed
4.249 -
4.250 -lemma prime_int_simp:
4.251 - "prime(p::int) = (p > 1 & (list_all (%n. ~ n dvd p) [2..p - 1]))"
4.252 -apply(simp only:prime_int_code list_ball_code greaterThanLessThan_upto)
4.253 -apply simp
4.254 -done
4.255 -
4.256 -lemmas prime_int_simp_number_of[simp] = prime_int_simp[of "number_of m", standard]
4.257 -
4.258 -lemma two_is_prime_nat [simp]: "prime (2::nat)"
4.259 -by simp
4.260 -
4.261 -lemma two_is_prime_int [simp]: "prime (2::int)"
4.262 -by simp
4.263 -
4.264 -text{* A bit of regression testing: *}
4.265 -
4.266 -lemma "prime(97::nat)"
4.267 -by simp
4.268 -
4.269 -lemma "prime(97::int)"
4.270 -by simp
4.271 -
4.272 -lemma "prime(997::nat)"
4.273 -by eval
4.274 -
4.275 -lemma "prime(997::int)"
4.276 -by eval
4.277 -
4.278 -
4.279 -lemma prime_imp_power_coprime_nat: "prime (p::nat) \<Longrightarrow> ~ p dvd a \<Longrightarrow> coprime a (p^m)"
4.280 - apply (rule coprime_exp_nat)
4.281 - apply (subst gcd_commute_nat)
4.282 - apply (erule (1) prime_imp_coprime_nat)
4.283 -done
4.284 -
4.285 -lemma prime_imp_power_coprime_int: "prime (p::int) \<Longrightarrow> ~ p dvd a \<Longrightarrow> coprime a (p^m)"
4.286 - apply (rule coprime_exp_int)
4.287 - apply (subst gcd_commute_int)
4.288 - apply (erule (1) prime_imp_coprime_int)
4.289 -done
4.290 -
4.291 -lemma primes_coprime_nat: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
4.292 - apply (rule prime_imp_coprime_nat, assumption)
4.293 - apply (unfold prime_nat_def, auto)
4.294 -done
4.295 -
4.296 -lemma primes_coprime_int: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
4.297 - apply (rule prime_imp_coprime_int, assumption)
4.298 - apply (unfold prime_int_altdef, clarify)
4.299 - apply (drule_tac x = q in spec)
4.300 - apply (drule_tac x = p in spec)
4.301 - apply auto
4.302 -done
4.303 -
4.304 -lemma primes_imp_powers_coprime_nat: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow> coprime (p^m) (q^n)"
4.305 - by (rule coprime_exp2_nat, rule primes_coprime_nat)
4.306 -
4.307 -lemma primes_imp_powers_coprime_int: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow> coprime (p^m) (q^n)"
4.308 - by (rule coprime_exp2_int, rule primes_coprime_int)
4.309 -
4.310 -lemma prime_factor_nat: "n \<noteq> (1::nat) \<Longrightarrow> \<exists> p. prime p \<and> p dvd n"
4.311 - apply (induct n rule: nat_less_induct)
4.312 - apply (case_tac "n = 0")
4.313 - using two_is_prime_nat apply blast
4.314 - apply (case_tac "prime n")
4.315 - apply blast
4.316 - apply (subgoal_tac "n > 1")
4.317 - apply (frule (1) not_prime_eq_prod_nat)
4.318 - apply (auto intro: dvd_mult dvd_mult2)
4.319 -done
4.320 -
4.321 -(* An Isar version:
4.322 -
4.323 -lemma prime_factor_b_nat:
4.324 - fixes n :: nat
4.325 - assumes "n \<noteq> 1"
4.326 - shows "\<exists>p. prime p \<and> p dvd n"
4.327 -
4.328 -using `n ~= 1`
4.329 -proof (induct n rule: less_induct_nat)
4.330 - fix n :: nat
4.331 - assume "n ~= 1" and
4.332 - ih: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)"
4.333 - thus "\<exists>p. prime p \<and> p dvd n"
4.334 - proof -
4.335 - {
4.336 - assume "n = 0"
4.337 - moreover note two_is_prime_nat
4.338 - ultimately have ?thesis
4.339 - by (auto simp del: two_is_prime_nat)
4.340 - }
4.341 - moreover
4.342 - {
4.343 - assume "prime n"
4.344 - hence ?thesis by auto
4.345 - }
4.346 - moreover
4.347 - {
4.348 - assume "n ~= 0" and "~ prime n"
4.349 - with `n ~= 1` have "n > 1" by auto
4.350 - with `~ prime n` and not_prime_eq_prod_nat obtain m k where
4.351 - "n = m * k" and "1 < m" and "m < n" by blast
4.352 - with ih obtain p where "prime p" and "p dvd m" by blast
4.353 - with `n = m * k` have ?thesis by auto
4.354 - }
4.355 - ultimately show ?thesis by blast
4.356 - qed
4.357 -qed
4.358 -
4.359 -*)
4.360 -
4.361 -text {* One property of coprimality is easier to prove via prime factors. *}
4.362 -
4.363 -lemma prime_divprod_pow_nat:
4.364 - assumes p: "prime (p::nat)" and ab: "coprime a b" and pab: "p^n dvd a * b"
4.365 - shows "p^n dvd a \<or> p^n dvd b"
4.366 -proof-
4.367 - {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
4.368 - apply (cases "n=0", simp_all)
4.369 - apply (cases "a=1", simp_all) done}
4.370 - moreover
4.371 - {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
4.372 - then obtain m where m: "n = Suc m" by (cases n, auto)
4.373 - from n have "p dvd p^n" by (intro dvd_power, auto)
4.374 - also note pab
4.375 - finally have pab': "p dvd a * b".
4.376 - from prime_dvd_mult_nat[OF p pab']
4.377 - have "p dvd a \<or> p dvd b" .
4.378 - moreover
4.379 - {assume pa: "p dvd a"
4.380 - have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
4.381 - from coprime_common_divisor_nat [OF ab, OF pa] p have "\<not> p dvd b" by auto
4.382 - with p have "coprime b p"
4.383 - by (subst gcd_commute_nat, intro prime_imp_coprime_nat)
4.384 - hence pnb: "coprime (p^n) b"
4.385 - by (subst gcd_commute_nat, rule coprime_exp_nat)
4.386 - from coprime_divprod_nat[OF pnba pnb] have ?thesis by blast }
4.387 - moreover
4.388 - {assume pb: "p dvd b"
4.389 - have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
4.390 - from coprime_common_divisor_nat [OF ab, of p] pb p have "\<not> p dvd a"
4.391 - by auto
4.392 - with p have "coprime a p"
4.393 - by (subst gcd_commute_nat, intro prime_imp_coprime_nat)
4.394 - hence pna: "coprime (p^n) a"
4.395 - by (subst gcd_commute_nat, rule coprime_exp_nat)
4.396 - from coprime_divprod_nat[OF pab pna] have ?thesis by blast }
4.397 - ultimately have ?thesis by blast}
4.398 - ultimately show ?thesis by blast
4.399 -qed
4.400 -
4.401 -subsection {* Infinitely many primes *}
4.402 -
4.403 -lemma next_prime_bound: "\<exists>(p::nat). prime p \<and> n < p \<and> p <= fact n + 1"
4.404 -proof-
4.405 - have f1: "fact n + 1 \<noteq> 1" using fact_ge_one_nat [of n] by arith
4.406 - from prime_factor_nat [OF f1]
4.407 - obtain p where "prime p" and "p dvd fact n + 1" by auto
4.408 - hence "p \<le> fact n + 1"
4.409 - by (intro dvd_imp_le, auto)
4.410 - {assume "p \<le> n"
4.411 - from `prime p` have "p \<ge> 1"
4.412 - by (cases p, simp_all)
4.413 - with `p <= n` have "p dvd fact n"
4.414 - by (intro dvd_fact_nat)
4.415 - with `p dvd fact n + 1` have "p dvd fact n + 1 - fact n"
4.416 - by (rule dvd_diff_nat)
4.417 - hence "p dvd 1" by simp
4.418 - hence "p <= 1" by auto
4.419 - moreover from `prime p` have "p > 1" by auto
4.420 - ultimately have False by auto}
4.421 - hence "n < p" by arith
4.422 - with `prime p` and `p <= fact n + 1` show ?thesis by auto
4.423 -qed
4.424 -
4.425 -lemma bigger_prime: "\<exists>p. prime p \<and> p > (n::nat)"
4.426 -using next_prime_bound by auto
4.427 -
4.428 -lemma primes_infinite: "\<not> (finite {(p::nat). prime p})"
4.429 -proof
4.430 - assume "finite {(p::nat). prime p}"
4.431 - with Max_ge have "(EX b. (ALL x : {(p::nat). prime p}. x <= b))"
4.432 - by auto
4.433 - then obtain b where "ALL (x::nat). prime x \<longrightarrow> x <= b"
4.434 - by auto
4.435 - with bigger_prime [of b] show False by auto
4.436 -qed
4.437 -
4.438 -
4.439 end
5.1 --- a/src/HOL/Import/HOL4Compat.thy Tue Sep 01 14:13:34 2009 +0200
5.2 +++ b/src/HOL/Import/HOL4Compat.thy Tue Sep 01 15:39:33 2009 +0200
5.3 @@ -3,7 +3,7 @@
5.4 *)
5.5
5.6 theory HOL4Compat
5.7 -imports HOL4Setup Complex_Main Primes ContNotDenum
5.8 +imports HOL4Setup Complex_Main "~~/src/HOL/Old_Number_Theory/Primes" ContNotDenum
5.9 begin
5.10
5.11 no_notation differentiable (infixl "differentiable" 60)
6.1 --- a/src/HOL/IsaMakefile Tue Sep 01 14:13:34 2009 +0200
6.2 +++ b/src/HOL/IsaMakefile Tue Sep 01 15:39:33 2009 +0200
6.3 @@ -34,8 +34,8 @@
6.4 HOL-Modelcheck \
6.5 HOL-NanoJava \
6.6 HOL-Nominal-Examples \
6.7 - HOL-NewNumberTheory \
6.8 - HOL-NumberTheory \
6.9 + HOL-Number_Theory \
6.10 + HOL-Old_Number_Theory \
6.11 HOL-Prolog \
6.12 HOL-SET-Protocol \
6.13 HOL-SizeChange \
6.14 @@ -282,11 +282,13 @@
6.15 Complex.thy \
6.16 Deriv.thy \
6.17 Fact.thy \
6.18 + GCD.thy \
6.19 Integration.thy \
6.20 Lim.thy \
6.21 Limits.thy \
6.22 Ln.thy \
6.23 Log.thy \
6.24 + Lubs.thy \
6.25 MacLaurin.thy \
6.26 NatTransfer.thy \
6.27 NthRoot.thy \
6.28 @@ -294,9 +296,7 @@
6.29 Series.thy \
6.30 Taylor.thy \
6.31 Transcendental.thy \
6.32 - GCD.thy \
6.33 Parity.thy \
6.34 - Lubs.thy \
6.35 PReal.thy \
6.36 Rational.thy \
6.37 RComplete.thy \
6.38 @@ -330,10 +330,10 @@
6.39 Library/Finite_Cartesian_Product.thy Library/FrechetDeriv.thy \
6.40 Library/Fraction_Field.thy Library/Fundamental_Theorem_Algebra.thy \
6.41 Library/Inner_Product.thy Library/Kleene_Algebra.thy \
6.42 - Library/Lattice_Syntax.thy Library/Legacy_GCD.thy \
6.43 + Library/Lattice_Syntax.thy \
6.44 Library/Library.thy Library/List_Prefix.thy Library/List_Set.thy \
6.45 Library/State_Monad.thy Library/Nat_Int_Bij.thy Library/Multiset.thy \
6.46 - Library/Permutation.thy Library/Primes.thy Library/Pocklington.thy \
6.47 + Library/Permutation.thy \
6.48 Library/Quotient.thy Library/Quicksort.thy Library/Nat_Infinity.thy \
6.49 Library/Word.thy Library/README.html Library/Continuity.thy \
6.50 Library/Order_Relation.thy Library/Nested_Environment.thy \
6.51 @@ -487,38 +487,39 @@
6.52 @cd Import/HOLLight; $(ISABELLE_TOOL) usedir -b $(OUT)/HOL HOLLight
6.53
6.54
6.55 -## HOL-NewNumberTheory
6.56 +## HOL-Number_Theory
6.57
6.58 -HOL-NewNumberTheory: HOL $(LOG)/HOL-NewNumberTheory.gz
6.59 +HOL-Number_Theory: HOL $(LOG)/HOL-Number_Theory.gz
6.60
6.61 -$(LOG)/HOL-NewNumberTheory.gz: $(OUT)/HOL $(ALGEBRA_DEPENDENCIES) \
6.62 +$(LOG)/HOL-Number_Theory.gz: $(OUT)/HOL $(ALGEBRA_DEPENDENCIES) \
6.63 Library/Multiset.thy \
6.64 - NewNumberTheory/Binomial.thy \
6.65 - NewNumberTheory/Cong.thy \
6.66 - NewNumberTheory/Fib.thy \
6.67 - NewNumberTheory/MiscAlgebra.thy \
6.68 - NewNumberTheory/Residues.thy \
6.69 - NewNumberTheory/UniqueFactorization.thy \
6.70 - NewNumberTheory/ROOT.ML
6.71 - @$(ISABELLE_TOOL) usedir -g true $(OUT)/HOL NewNumberTheory
6.72 + Number_Theory/Binomial.thy \
6.73 + Number_Theory/Cong.thy \
6.74 + Number_Theory/Fib.thy \
6.75 + Number_Theory/MiscAlgebra.thy \
6.76 + Number_Theory/Number_Theory.thy \
6.77 + Number_Theory/Residues.thy \
6.78 + Number_Theory/UniqueFactorization.thy \
6.79 + Number_Theory/ROOT.ML
6.80 + @$(ISABELLE_TOOL) usedir -g true $(OUT)/HOL Number_Theory
6.81
6.82
6.83 -## HOL-NumberTheory
6.84 +## HOL-Old_Number_Theory
6.85
6.86 -HOL-NumberTheory: HOL $(LOG)/HOL-NumberTheory.gz
6.87 +HOL-Old_Number_Theory: HOL $(LOG)/HOL-Old_Number_Theory.gz
6.88
6.89 -$(LOG)/HOL-NumberTheory.gz: $(OUT)/HOL Library/Permutation.thy \
6.90 - Library/Primes.thy NumberTheory/Fib.thy \
6.91 - NumberTheory/Factorization.thy NumberTheory/BijectionRel.thy \
6.92 - NumberTheory/Chinese.thy NumberTheory/EulerFermat.thy \
6.93 - NumberTheory/IntFact.thy NumberTheory/IntPrimes.thy \
6.94 - NumberTheory/WilsonBij.thy NumberTheory/WilsonRuss.thy \
6.95 - NumberTheory/Finite2.thy NumberTheory/Int2.thy \
6.96 - NumberTheory/EvenOdd.thy NumberTheory/Residues.thy \
6.97 - NumberTheory/Euler.thy NumberTheory/Gauss.thy \
6.98 - NumberTheory/Quadratic_Reciprocity.thy Library/Infinite_Set.thy \
6.99 - NumberTheory/ROOT.ML
6.100 - @$(ISABELLE_TOOL) usedir -g true $(OUT)/HOL NumberTheory
6.101 +$(LOG)/HOL-Old_Number_Theory.gz: $(OUT)/HOL Library/Permutation.thy \
6.102 + Old_Number_Theory/Primes.thy Old_Number_Theory/Fib.thy \
6.103 + Old_Number_Theory/Factorization.thy Old_Number_Theory/BijectionRel.thy \
6.104 + Old_Number_Theory/Chinese.thy Old_Number_Theory/EulerFermat.thy \
6.105 + Old_Number_Theory/IntFact.thy Old_Number_Theory/IntPrimes.thy \
6.106 + Old_Number_Theory/WilsonBij.thy Old_Number_Theory/WilsonRuss.thy \
6.107 + Old_Number_Theory/Finite2.thy Old_Number_Theory/Int2.thy \
6.108 + Old_Number_Theory/EvenOdd.thy Old_Number_Theory/Residues.thy \
6.109 + Old_Number_Theory/Euler.thy Old_Number_Theory/Gauss.thy \
6.110 + Old_Number_Theory/Quadratic_Reciprocity.thy Library/Infinite_Set.thy \
6.111 + Old_Number_Theory/Legacy_GCD.thy Old_Number_Theory/Pocklington.thy Old_Number_Theory/ROOT.ML
6.112 + @$(ISABELLE_TOOL) usedir -g true $(OUT)/HOL Old_Number_Theory
6.113
6.114
6.115 ## HOL-Hoare
6.116 @@ -573,7 +574,7 @@
6.117 Library/FuncSet.thy \
6.118 Library/Multiset.thy \
6.119 Library/Permutation.thy \
6.120 - Library/Primes.thy \
6.121 + Number_Theory/Primes.thy \
6.122 Algebra/AbelCoset.thy \
6.123 Algebra/Bij.thy \
6.124 Algebra/Congruence.thy \
6.125 @@ -876,7 +877,7 @@
6.126 HOL-ex: HOL $(LOG)/HOL-ex.gz
6.127
6.128 $(LOG)/HOL-ex.gz: $(OUT)/HOL Library/Commutative_Ring.thy \
6.129 - Library/Primes.thy ex/Abstract_NAT.thy ex/Antiquote.thy \
6.130 + Number_Theory/Primes.thy ex/Abstract_NAT.thy ex/Antiquote.thy \
6.131 ex/Arith_Examples.thy \
6.132 ex/Arithmetic_Series_Complex.thy ex/BT.thy ex/BinEx.thy \
6.133 ex/Binary.thy ex/CTL.thy ex/Chinese.thy ex/Classical.thy \
7.1 --- a/src/HOL/Isar_examples/ROOT.ML Tue Sep 01 14:13:34 2009 +0200
7.2 +++ b/src/HOL/Isar_examples/ROOT.ML Tue Sep 01 15:39:33 2009 +0200
7.3 @@ -4,7 +4,7 @@
7.4 Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
7.5 *)
7.6
7.7 -no_document use_thys ["../NumberTheory/Primes", "../NumberTheory/Fibonacci"];
7.8 +no_document use_thys ["../Old_Number_Theory/Primes", "../Old_Number_Theory/Fibonacci"];
7.9
7.10 use_thys [
7.11 "Basic_Logic",
8.1 --- a/src/HOL/Library/Legacy_GCD.thy Tue Sep 01 14:13:34 2009 +0200
8.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
8.3 @@ -1,787 +0,0 @@
8.4 -(* Title: HOL/GCD.thy
8.5 - Author: Christophe Tabacznyj and Lawrence C Paulson
8.6 - Copyright 1996 University of Cambridge
8.7 -*)
8.8 -
8.9 -header {* The Greatest Common Divisor *}
8.10 -
8.11 -theory Legacy_GCD
8.12 -imports Main
8.13 -begin
8.14 -
8.15 -text {*
8.16 - See \cite{davenport92}. \bigskip
8.17 -*}
8.18 -
8.19 -subsection {* Specification of GCD on nats *}
8.20 -
8.21 -definition
8.22 - is_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where -- {* @{term gcd} as a relation *}
8.23 - [code del]: "is_gcd m n p \<longleftrightarrow> p dvd m \<and> p dvd n \<and>
8.24 - (\<forall>d. d dvd m \<longrightarrow> d dvd n \<longrightarrow> d dvd p)"
8.25 -
8.26 -text {* Uniqueness *}
8.27 -
8.28 -lemma is_gcd_unique: "is_gcd a b m \<Longrightarrow> is_gcd a b n \<Longrightarrow> m = n"
8.29 - by (simp add: is_gcd_def) (blast intro: dvd_anti_sym)
8.30 -
8.31 -text {* Connection to divides relation *}
8.32 -
8.33 -lemma is_gcd_dvd: "is_gcd a b m \<Longrightarrow> k dvd a \<Longrightarrow> k dvd b \<Longrightarrow> k dvd m"
8.34 - by (auto simp add: is_gcd_def)
8.35 -
8.36 -text {* Commutativity *}
8.37 -
8.38 -lemma is_gcd_commute: "is_gcd m n k = is_gcd n m k"
8.39 - by (auto simp add: is_gcd_def)
8.40 -
8.41 -
8.42 -subsection {* GCD on nat by Euclid's algorithm *}
8.43 -
8.44 -fun
8.45 - gcd :: "nat => nat => nat"
8.46 -where
8.47 - "gcd m n = (if n = 0 then m else gcd n (m mod n))"
8.48 -lemma gcd_induct [case_names "0" rec]:
8.49 - fixes m n :: nat
8.50 - assumes "\<And>m. P m 0"
8.51 - and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
8.52 - shows "P m n"
8.53 -proof (induct m n rule: gcd.induct)
8.54 - case (1 m n) with assms show ?case by (cases "n = 0") simp_all
8.55 -qed
8.56 -
8.57 -lemma gcd_0 [simp, algebra]: "gcd m 0 = m"
8.58 - by simp
8.59 -
8.60 -lemma gcd_0_left [simp,algebra]: "gcd 0 m = m"
8.61 - by simp
8.62 -
8.63 -lemma gcd_non_0: "n > 0 \<Longrightarrow> gcd m n = gcd n (m mod n)"
8.64 - by simp
8.65 -
8.66 -lemma gcd_1 [simp, algebra]: "gcd m (Suc 0) = Suc 0"
8.67 - by simp
8.68 -
8.69 -lemma nat_gcd_1_right [simp, algebra]: "gcd m 1 = 1"
8.70 - unfolding One_nat_def by (rule gcd_1)
8.71 -
8.72 -declare gcd.simps [simp del]
8.73 -
8.74 -text {*
8.75 - \medskip @{term "gcd m n"} divides @{text m} and @{text n}. The
8.76 - conjunctions don't seem provable separately.
8.77 -*}
8.78 -
8.79 -lemma gcd_dvd1 [iff, algebra]: "gcd m n dvd m"
8.80 - and gcd_dvd2 [iff, algebra]: "gcd m n dvd n"
8.81 - apply (induct m n rule: gcd_induct)
8.82 - apply (simp_all add: gcd_non_0)
8.83 - apply (blast dest: dvd_mod_imp_dvd)
8.84 - done
8.85 -
8.86 -text {*
8.87 - \medskip Maximality: for all @{term m}, @{term n}, @{term k}
8.88 - naturals, if @{term k} divides @{term m} and @{term k} divides
8.89 - @{term n} then @{term k} divides @{term "gcd m n"}.
8.90 -*}
8.91 -
8.92 -lemma gcd_greatest: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
8.93 - by (induct m n rule: gcd_induct) (simp_all add: gcd_non_0 dvd_mod)
8.94 -
8.95 -text {*
8.96 - \medskip Function gcd yields the Greatest Common Divisor.
8.97 -*}
8.98 -
8.99 -lemma is_gcd: "is_gcd m n (gcd m n) "
8.100 - by (simp add: is_gcd_def gcd_greatest)
8.101 -
8.102 -
8.103 -subsection {* Derived laws for GCD *}
8.104 -
8.105 -lemma gcd_greatest_iff [iff, algebra]: "k dvd gcd m n \<longleftrightarrow> k dvd m \<and> k dvd n"
8.106 - by (blast intro!: gcd_greatest intro: dvd_trans)
8.107 -
8.108 -lemma gcd_zero[algebra]: "gcd m n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
8.109 - by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff)
8.110 -
8.111 -lemma gcd_commute: "gcd m n = gcd n m"
8.112 - apply (rule is_gcd_unique)
8.113 - apply (rule is_gcd)
8.114 - apply (subst is_gcd_commute)
8.115 - apply (simp add: is_gcd)
8.116 - done
8.117 -
8.118 -lemma gcd_assoc: "gcd (gcd k m) n = gcd k (gcd m n)"
8.119 - apply (rule is_gcd_unique)
8.120 - apply (rule is_gcd)
8.121 - apply (simp add: is_gcd_def)
8.122 - apply (blast intro: dvd_trans)
8.123 - done
8.124 -
8.125 -lemma gcd_1_left [simp, algebra]: "gcd (Suc 0) m = Suc 0"
8.126 - by (simp add: gcd_commute)
8.127 -
8.128 -lemma nat_gcd_1_left [simp, algebra]: "gcd 1 m = 1"
8.129 - unfolding One_nat_def by (rule gcd_1_left)
8.130 -
8.131 -text {*
8.132 - \medskip Multiplication laws
8.133 -*}
8.134 -
8.135 -lemma gcd_mult_distrib2: "k * gcd m n = gcd (k * m) (k * n)"
8.136 - -- {* \cite[page 27]{davenport92} *}
8.137 - apply (induct m n rule: gcd_induct)
8.138 - apply simp
8.139 - apply (case_tac "k = 0")
8.140 - apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2)
8.141 - done
8.142 -
8.143 -lemma gcd_mult [simp, algebra]: "gcd k (k * n) = k"
8.144 - apply (rule gcd_mult_distrib2 [of k 1 n, simplified, symmetric])
8.145 - done
8.146 -
8.147 -lemma gcd_self [simp, algebra]: "gcd k k = k"
8.148 - apply (rule gcd_mult [of k 1, simplified])
8.149 - done
8.150 -
8.151 -lemma relprime_dvd_mult: "gcd k n = 1 ==> k dvd m * n ==> k dvd m"
8.152 - apply (insert gcd_mult_distrib2 [of m k n])
8.153 - apply simp
8.154 - apply (erule_tac t = m in ssubst)
8.155 - apply simp
8.156 - done
8.157 -
8.158 -lemma relprime_dvd_mult_iff: "gcd k n = 1 ==> (k dvd m * n) = (k dvd m)"
8.159 - by (auto intro: relprime_dvd_mult dvd_mult2)
8.160 -
8.161 -lemma gcd_mult_cancel: "gcd k n = 1 ==> gcd (k * m) n = gcd m n"
8.162 - apply (rule dvd_anti_sym)
8.163 - apply (rule gcd_greatest)
8.164 - apply (rule_tac n = k in relprime_dvd_mult)
8.165 - apply (simp add: gcd_assoc)
8.166 - apply (simp add: gcd_commute)
8.167 - apply (simp_all add: mult_commute)
8.168 - done
8.169 -
8.170 -
8.171 -text {* \medskip Addition laws *}
8.172 -
8.173 -lemma gcd_add1 [simp, algebra]: "gcd (m + n) n = gcd m n"
8.174 - by (cases "n = 0") (auto simp add: gcd_non_0)
8.175 -
8.176 -lemma gcd_add2 [simp, algebra]: "gcd m (m + n) = gcd m n"
8.177 -proof -
8.178 - have "gcd m (m + n) = gcd (m + n) m" by (rule gcd_commute)
8.179 - also have "... = gcd (n + m) m" by (simp add: add_commute)
8.180 - also have "... = gcd n m" by simp
8.181 - also have "... = gcd m n" by (rule gcd_commute)
8.182 - finally show ?thesis .
8.183 -qed
8.184 -
8.185 -lemma gcd_add2' [simp, algebra]: "gcd m (n + m) = gcd m n"
8.186 - apply (subst add_commute)
8.187 - apply (rule gcd_add2)
8.188 - done
8.189 -
8.190 -lemma gcd_add_mult[algebra]: "gcd m (k * m + n) = gcd m n"
8.191 - by (induct k) (simp_all add: add_assoc)
8.192 -
8.193 -lemma gcd_dvd_prod: "gcd m n dvd m * n"
8.194 - using mult_dvd_mono [of 1] by auto
8.195 -
8.196 -text {*
8.197 - \medskip Division by gcd yields rrelatively primes.
8.198 -*}
8.199 -
8.200 -lemma div_gcd_relprime:
8.201 - assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
8.202 - shows "gcd (a div gcd a b) (b div gcd a b) = 1"
8.203 -proof -
8.204 - let ?g = "gcd a b"
8.205 - let ?a' = "a div ?g"
8.206 - let ?b' = "b div ?g"
8.207 - let ?g' = "gcd ?a' ?b'"
8.208 - have dvdg: "?g dvd a" "?g dvd b" by simp_all
8.209 - have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
8.210 - from dvdg dvdg' obtain ka kb ka' kb' where
8.211 - kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
8.212 - unfolding dvd_def by blast
8.213 - then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'" by simp_all
8.214 - then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
8.215 - by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
8.216 - dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
8.217 - have "?g \<noteq> 0" using nz by (simp add: gcd_zero)
8.218 - then have gp: "?g > 0" by simp
8.219 - from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
8.220 - with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
8.221 -qed
8.222 -
8.223 -
8.224 -lemma gcd_unique: "d dvd a\<and>d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
8.225 -proof(auto)
8.226 - assume H: "d dvd a" "d dvd b" "\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d"
8.227 - from H(3)[rule_format] gcd_dvd1[of a b] gcd_dvd2[of a b]
8.228 - have th: "gcd a b dvd d" by blast
8.229 - from dvd_anti_sym[OF th gcd_greatest[OF H(1,2)]] show "d = gcd a b" by blast
8.230 -qed
8.231 -
8.232 -lemma gcd_eq: assumes H: "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd u \<and> d dvd v"
8.233 - shows "gcd x y = gcd u v"
8.234 -proof-
8.235 - from H have "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd gcd u v" by simp
8.236 - with gcd_unique[of "gcd u v" x y] show ?thesis by auto
8.237 -qed
8.238 -
8.239 -lemma ind_euclid:
8.240 - assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
8.241 - and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
8.242 - shows "P a b"
8.243 -proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
8.244 - fix n a b
8.245 - assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
8.246 - have "a = b \<or> a < b \<or> b < a" by arith
8.247 - moreover {assume eq: "a= b"
8.248 - from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq by simp}
8.249 - moreover
8.250 - {assume lt: "a < b"
8.251 - hence "a + b - a < n \<or> a = 0" using H(2) by arith
8.252 - moreover
8.253 - {assume "a =0" with z c have "P a b" by blast }
8.254 - moreover
8.255 - {assume ab: "a + b - a < n"
8.256 - have th0: "a + b - a = a + (b - a)" using lt by arith
8.257 - from add[rule_format, OF H(1)[rule_format, OF ab th0]]
8.258 - have "P a b" by (simp add: th0[symmetric])}
8.259 - ultimately have "P a b" by blast}
8.260 - moreover
8.261 - {assume lt: "a > b"
8.262 - hence "b + a - b < n \<or> b = 0" using H(2) by arith
8.263 - moreover
8.264 - {assume "b =0" with z c have "P a b" by blast }
8.265 - moreover
8.266 - {assume ab: "b + a - b < n"
8.267 - have th0: "b + a - b = b + (a - b)" using lt by arith
8.268 - from add[rule_format, OF H(1)[rule_format, OF ab th0]]
8.269 - have "P b a" by (simp add: th0[symmetric])
8.270 - hence "P a b" using c by blast }
8.271 - ultimately have "P a b" by blast}
8.272 -ultimately show "P a b" by blast
8.273 -qed
8.274 -
8.275 -lemma bezout_lemma:
8.276 - assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
8.277 - shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and> (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
8.278 -using ex
8.279 -apply clarsimp
8.280 -apply (rule_tac x="d" in exI, simp add: dvd_add)
8.281 -apply (case_tac "a * x = b * y + d" , simp_all)
8.282 -apply (rule_tac x="x + y" in exI)
8.283 -apply (rule_tac x="y" in exI)
8.284 -apply algebra
8.285 -apply (rule_tac x="x" in exI)
8.286 -apply (rule_tac x="x + y" in exI)
8.287 -apply algebra
8.288 -done
8.289 -
8.290 -lemma bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
8.291 -apply(induct a b rule: ind_euclid)
8.292 -apply blast
8.293 -apply clarify
8.294 -apply (rule_tac x="a" in exI, simp add: dvd_add)
8.295 -apply clarsimp
8.296 -apply (rule_tac x="d" in exI)
8.297 -apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
8.298 -apply (rule_tac x="x+y" in exI)
8.299 -apply (rule_tac x="y" in exI)
8.300 -apply algebra
8.301 -apply (rule_tac x="x" in exI)
8.302 -apply (rule_tac x="x+y" in exI)
8.303 -apply algebra
8.304 -done
8.305 -
8.306 -lemma bezout: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x - b * y = d \<or> b * x - a * y = d)"
8.307 -using bezout_add[of a b]
8.308 -apply clarsimp
8.309 -apply (rule_tac x="d" in exI, simp)
8.310 -apply (rule_tac x="x" in exI)
8.311 -apply (rule_tac x="y" in exI)
8.312 -apply auto
8.313 -done
8.314 -
8.315 -
8.316 -text {* We can get a stronger version with a nonzeroness assumption. *}
8.317 -lemma divides_le: "m dvd n ==> m <= n \<or> n = (0::nat)" by (auto simp add: dvd_def)
8.318 -
8.319 -lemma bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
8.320 - shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
8.321 -proof-
8.322 - from nz have ap: "a > 0" by simp
8.323 - from bezout_add[of a b]
8.324 - have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or> (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
8.325 - moreover
8.326 - {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
8.327 - from H have ?thesis by blast }
8.328 - moreover
8.329 - {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
8.330 - {assume b0: "b = 0" with H have ?thesis by simp}
8.331 - moreover
8.332 - {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
8.333 - from divides_le[OF H(2)] b have "d < b \<or> d = b" using le_less by blast
8.334 - moreover
8.335 - {assume db: "d=b"
8.336 - from prems have ?thesis apply simp
8.337 - apply (rule exI[where x = b], simp)
8.338 - apply (rule exI[where x = b])
8.339 - by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
8.340 - moreover
8.341 - {assume db: "d < b"
8.342 - {assume "x=0" hence ?thesis using prems by simp }
8.343 - moreover
8.344 - {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
8.345 -
8.346 - from db have "d \<le> b - 1" by simp
8.347 - hence "d*b \<le> b*(b - 1)" by simp
8.348 - with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
8.349 - have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
8.350 - from H (3) have "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)" by algebra
8.351 - hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
8.352 - hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
8.353 - by (simp only: diff_add_assoc[OF dble, of d, symmetric])
8.354 - hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
8.355 - by (simp only: diff_mult_distrib2 add_commute mult_ac)
8.356 - hence ?thesis using H(1,2)
8.357 - apply -
8.358 - apply (rule exI[where x=d], simp)
8.359 - apply (rule exI[where x="(b - 1) * y"])
8.360 - by (rule exI[where x="x*(b - 1) - d"], simp)}
8.361 - ultimately have ?thesis by blast}
8.362 - ultimately have ?thesis by blast}
8.363 - ultimately have ?thesis by blast}
8.364 - ultimately show ?thesis by blast
8.365 -qed
8.366 -
8.367 -
8.368 -lemma bezout_gcd: "\<exists>x y. a * x - b * y = gcd a b \<or> b * x - a * y = gcd a b"
8.369 -proof-
8.370 - let ?g = "gcd a b"
8.371 - from bezout[of a b] obtain d x y where d: "d dvd a" "d dvd b" "a * x - b * y = d \<or> b * x - a * y = d" by blast
8.372 - from d(1,2) have "d dvd ?g" by simp
8.373 - then obtain k where k: "?g = d*k" unfolding dvd_def by blast
8.374 - from d(3) have "(a * x - b * y)*k = d*k \<or> (b * x - a * y)*k = d*k" by blast
8.375 - hence "a * x * k - b * y*k = d*k \<or> b * x * k - a * y*k = d*k"
8.376 - by (algebra add: diff_mult_distrib)
8.377 - hence "a * (x * k) - b * (y*k) = ?g \<or> b * (x * k) - a * (y*k) = ?g"
8.378 - by (simp add: k mult_assoc)
8.379 - thus ?thesis by blast
8.380 -qed
8.381 -
8.382 -lemma bezout_gcd_strong: assumes a: "a \<noteq> 0"
8.383 - shows "\<exists>x y. a * x = b * y + gcd a b"
8.384 -proof-
8.385 - let ?g = "gcd a b"
8.386 - from bezout_add_strong[OF a, of b]
8.387 - obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
8.388 - from d(1,2) have "d dvd ?g" by simp
8.389 - then obtain k where k: "?g = d*k" unfolding dvd_def by blast
8.390 - from d(3) have "a * x * k = (b * y + d) *k " by algebra
8.391 - hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
8.392 - thus ?thesis by blast
8.393 -qed
8.394 -
8.395 -lemma gcd_mult_distrib: "gcd(a * c) (b * c) = c * gcd a b"
8.396 -by(simp add: gcd_mult_distrib2 mult_commute)
8.397 -
8.398 -lemma gcd_bezout: "(\<exists>x y. a * x - b * y = d \<or> b * x - a * y = d) \<longleftrightarrow> gcd a b dvd d"
8.399 - (is "?lhs \<longleftrightarrow> ?rhs")
8.400 -proof-
8.401 - let ?g = "gcd a b"
8.402 - {assume H: ?rhs then obtain k where k: "d = ?g*k" unfolding dvd_def by blast
8.403 - from bezout_gcd[of a b] obtain x y where xy: "a * x - b * y = ?g \<or> b * x - a * y = ?g"
8.404 - by blast
8.405 - hence "(a * x - b * y)*k = ?g*k \<or> (b * x - a * y)*k = ?g*k" by auto
8.406 - hence "a * x*k - b * y*k = ?g*k \<or> b * x * k - a * y*k = ?g*k"
8.407 - by (simp only: diff_mult_distrib)
8.408 - hence "a * (x*k) - b * (y*k) = d \<or> b * (x * k) - a * (y*k) = d"
8.409 - by (simp add: k[symmetric] mult_assoc)
8.410 - hence ?lhs by blast}
8.411 - moreover
8.412 - {fix x y assume H: "a * x - b * y = d \<or> b * x - a * y = d"
8.413 - have dv: "?g dvd a*x" "?g dvd b * y" "?g dvd b*x" "?g dvd a * y"
8.414 - using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
8.415 - from dvd_diff_nat[OF dv(1,2)] dvd_diff_nat[OF dv(3,4)] H
8.416 - have ?rhs by auto}
8.417 - ultimately show ?thesis by blast
8.418 -qed
8.419 -
8.420 -lemma gcd_bezout_sum: assumes H:"a * x + b * y = d" shows "gcd a b dvd d"
8.421 -proof-
8.422 - let ?g = "gcd a b"
8.423 - have dv: "?g dvd a*x" "?g dvd b * y"
8.424 - using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
8.425 - from dvd_add[OF dv] H
8.426 - show ?thesis by auto
8.427 -qed
8.428 -
8.429 -lemma gcd_mult': "gcd b (a * b) = b"
8.430 -by (simp add: gcd_mult mult_commute[of a b])
8.431 -
8.432 -lemma gcd_add: "gcd(a + b) b = gcd a b"
8.433 - "gcd(b + a) b = gcd a b" "gcd a (a + b) = gcd a b" "gcd a (b + a) = gcd a b"
8.434 -apply (simp_all add: gcd_add1)
8.435 -by (simp add: gcd_commute gcd_add1)
8.436 -
8.437 -lemma gcd_sub: "b <= a ==> gcd(a - b) b = gcd a b" "a <= b ==> gcd a (b - a) = gcd a b"
8.438 -proof-
8.439 - {fix a b assume H: "b \<le> (a::nat)"
8.440 - hence th: "a - b + b = a" by arith
8.441 - from gcd_add(1)[of "a - b" b] th have "gcd(a - b) b = gcd a b" by simp}
8.442 - note th = this
8.443 -{
8.444 - assume ab: "b \<le> a"
8.445 - from th[OF ab] show "gcd (a - b) b = gcd a b" by blast
8.446 -next
8.447 - assume ab: "a \<le> b"
8.448 - from th[OF ab] show "gcd a (b - a) = gcd a b"
8.449 - by (simp add: gcd_commute)}
8.450 -qed
8.451 -
8.452 -
8.453 -subsection {* LCM defined by GCD *}
8.454 -
8.455 -
8.456 -definition
8.457 - lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat"
8.458 -where
8.459 - lcm_def: "lcm m n = m * n div gcd m n"
8.460 -
8.461 -lemma prod_gcd_lcm:
8.462 - "m * n = gcd m n * lcm m n"
8.463 - unfolding lcm_def by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod])
8.464 -
8.465 -lemma lcm_0 [simp]: "lcm m 0 = 0"
8.466 - unfolding lcm_def by simp
8.467 -
8.468 -lemma lcm_1 [simp]: "lcm m 1 = m"
8.469 - unfolding lcm_def by simp
8.470 -
8.471 -lemma lcm_0_left [simp]: "lcm 0 n = 0"
8.472 - unfolding lcm_def by simp
8.473 -
8.474 -lemma lcm_1_left [simp]: "lcm 1 m = m"
8.475 - unfolding lcm_def by simp
8.476 -
8.477 -lemma dvd_pos:
8.478 - fixes n m :: nat
8.479 - assumes "n > 0" and "m dvd n"
8.480 - shows "m > 0"
8.481 -using assms by (cases m) auto
8.482 -
8.483 -lemma lcm_least:
8.484 - assumes "m dvd k" and "n dvd k"
8.485 - shows "lcm m n dvd k"
8.486 -proof (cases k)
8.487 - case 0 then show ?thesis by auto
8.488 -next
8.489 - case (Suc _) then have pos_k: "k > 0" by auto
8.490 - from assms dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
8.491 - with gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp
8.492 - from assms obtain p where k_m: "k = m * p" using dvd_def by blast
8.493 - from assms obtain q where k_n: "k = n * q" using dvd_def by blast
8.494 - from pos_k k_m have pos_p: "p > 0" by auto
8.495 - from pos_k k_n have pos_q: "q > 0" by auto
8.496 - have "k * k * gcd q p = k * gcd (k * q) (k * p)"
8.497 - by (simp add: mult_ac gcd_mult_distrib2)
8.498 - also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
8.499 - by (simp add: k_m [symmetric] k_n [symmetric])
8.500 - also have "\<dots> = k * p * q * gcd m n"
8.501 - by (simp add: mult_ac gcd_mult_distrib2)
8.502 - finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
8.503 - by (simp only: k_m [symmetric] k_n [symmetric])
8.504 - then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
8.505 - by (simp add: mult_ac)
8.506 - with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
8.507 - by simp
8.508 - with prod_gcd_lcm [of m n]
8.509 - have "lcm m n * gcd q p * gcd m n = k * gcd m n"
8.510 - by (simp add: mult_ac)
8.511 - with pos_gcd have "lcm m n * gcd q p = k" by simp
8.512 - then show ?thesis using dvd_def by auto
8.513 -qed
8.514 -
8.515 -lemma lcm_dvd1 [iff]:
8.516 - "m dvd lcm m n"
8.517 -proof (cases m)
8.518 - case 0 then show ?thesis by simp
8.519 -next
8.520 - case (Suc _)
8.521 - then have mpos: "m > 0" by simp
8.522 - show ?thesis
8.523 - proof (cases n)
8.524 - case 0 then show ?thesis by simp
8.525 - next
8.526 - case (Suc _)
8.527 - then have npos: "n > 0" by simp
8.528 - have "gcd m n dvd n" by simp
8.529 - then obtain k where "n = gcd m n * k" using dvd_def by auto
8.530 - then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n" by (simp add: mult_ac)
8.531 - also have "\<dots> = m * k" using mpos npos gcd_zero by simp
8.532 - finally show ?thesis by (simp add: lcm_def)
8.533 - qed
8.534 -qed
8.535 -
8.536 -lemma lcm_dvd2 [iff]:
8.537 - "n dvd lcm m n"
8.538 -proof (cases n)
8.539 - case 0 then show ?thesis by simp
8.540 -next
8.541 - case (Suc _)
8.542 - then have npos: "n > 0" by simp
8.543 - show ?thesis
8.544 - proof (cases m)
8.545 - case 0 then show ?thesis by simp
8.546 - next
8.547 - case (Suc _)
8.548 - then have mpos: "m > 0" by simp
8.549 - have "gcd m n dvd m" by simp
8.550 - then obtain k where "m = gcd m n * k" using dvd_def by auto
8.551 - then have "m * n div gcd m n = (gcd m n * k) * n div gcd m n" by (simp add: mult_ac)
8.552 - also have "\<dots> = n * k" using mpos npos gcd_zero by simp
8.553 - finally show ?thesis by (simp add: lcm_def)
8.554 - qed
8.555 -qed
8.556 -
8.557 -lemma gcd_add1_eq: "gcd (m + k) k = gcd (m + k) m"
8.558 - by (simp add: gcd_commute)
8.559 -
8.560 -lemma gcd_diff2: "m \<le> n ==> gcd n (n - m) = gcd n m"
8.561 - apply (subgoal_tac "n = m + (n - m)")
8.562 - apply (erule ssubst, rule gcd_add1_eq, simp)
8.563 - done
8.564 -
8.565 -
8.566 -subsection {* GCD and LCM on integers *}
8.567 -
8.568 -definition
8.569 - zgcd :: "int \<Rightarrow> int \<Rightarrow> int" where
8.570 - "zgcd i j = int (gcd (nat (abs i)) (nat (abs j)))"
8.571 -
8.572 -lemma zgcd_zdvd1 [iff,simp, algebra]: "zgcd i j dvd i"
8.573 -by (simp add: zgcd_def int_dvd_iff)
8.574 -
8.575 -lemma zgcd_zdvd2 [iff,simp, algebra]: "zgcd i j dvd j"
8.576 -by (simp add: zgcd_def int_dvd_iff)
8.577 -
8.578 -lemma zgcd_pos: "zgcd i j \<ge> 0"
8.579 -by (simp add: zgcd_def)
8.580 -
8.581 -lemma zgcd0 [simp,algebra]: "(zgcd i j = 0) = (i = 0 \<and> j = 0)"
8.582 -by (simp add: zgcd_def gcd_zero)
8.583 -
8.584 -lemma zgcd_commute: "zgcd i j = zgcd j i"
8.585 -unfolding zgcd_def by (simp add: gcd_commute)
8.586 -
8.587 -lemma zgcd_zminus [simp, algebra]: "zgcd (- i) j = zgcd i j"
8.588 -unfolding zgcd_def by simp
8.589 -
8.590 -lemma zgcd_zminus2 [simp, algebra]: "zgcd i (- j) = zgcd i j"
8.591 -unfolding zgcd_def by simp
8.592 -
8.593 - (* should be solved by algebra*)
8.594 -lemma zrelprime_dvd_mult: "zgcd i j = 1 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k"
8.595 - unfolding zgcd_def
8.596 -proof -
8.597 - assume "int (gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>)) = 1" "i dvd k * j"
8.598 - then have g: "gcd (nat \<bar>i\<bar>) (nat \<bar>j\<bar>) = 1" by simp
8.599 - from `i dvd k * j` obtain h where h: "k*j = i*h" unfolding dvd_def by blast
8.600 - have th: "nat \<bar>i\<bar> dvd nat \<bar>k\<bar> * nat \<bar>j\<bar>"
8.601 - unfolding dvd_def
8.602 - by (rule_tac x= "nat \<bar>h\<bar>" in exI, simp add: h nat_abs_mult_distrib [symmetric])
8.603 - from relprime_dvd_mult [OF g th] obtain h' where h': "nat \<bar>k\<bar> = nat \<bar>i\<bar> * h'"
8.604 - unfolding dvd_def by blast
8.605 - from h' have "int (nat \<bar>k\<bar>) = int (nat \<bar>i\<bar> * h')" by simp
8.606 - then have "\<bar>k\<bar> = \<bar>i\<bar> * int h'" by (simp add: int_mult)
8.607 - then show ?thesis
8.608 - apply (subst abs_dvd_iff [symmetric])
8.609 - apply (subst dvd_abs_iff [symmetric])
8.610 - apply (unfold dvd_def)
8.611 - apply (rule_tac x = "int h'" in exI, simp)
8.612 - done
8.613 -qed
8.614 -
8.615 -lemma int_nat_abs: "int (nat (abs x)) = abs x" by arith
8.616 -
8.617 -lemma zgcd_greatest:
8.618 - assumes "k dvd m" and "k dvd n"
8.619 - shows "k dvd zgcd m n"
8.620 -proof -
8.621 - let ?k' = "nat \<bar>k\<bar>"
8.622 - let ?m' = "nat \<bar>m\<bar>"
8.623 - let ?n' = "nat \<bar>n\<bar>"
8.624 - from `k dvd m` and `k dvd n` have dvd': "?k' dvd ?m'" "?k' dvd ?n'"
8.625 - unfolding zdvd_int by (simp_all only: int_nat_abs abs_dvd_iff dvd_abs_iff)
8.626 - from gcd_greatest [OF dvd'] have "int (nat \<bar>k\<bar>) dvd zgcd m n"
8.627 - unfolding zgcd_def by (simp only: zdvd_int)
8.628 - then have "\<bar>k\<bar> dvd zgcd m n" by (simp only: int_nat_abs)
8.629 - then show "k dvd zgcd m n" by simp
8.630 -qed
8.631 -
8.632 -lemma div_zgcd_relprime:
8.633 - assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
8.634 - shows "zgcd (a div (zgcd a b)) (b div (zgcd a b)) = 1"
8.635 -proof -
8.636 - from nz have nz': "nat \<bar>a\<bar> \<noteq> 0 \<or> nat \<bar>b\<bar> \<noteq> 0" by arith
8.637 - let ?g = "zgcd a b"
8.638 - let ?a' = "a div ?g"
8.639 - let ?b' = "b div ?g"
8.640 - let ?g' = "zgcd ?a' ?b'"
8.641 - have dvdg: "?g dvd a" "?g dvd b" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2)
8.642 - have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: zgcd_zdvd1 zgcd_zdvd2)
8.643 - from dvdg dvdg' obtain ka kb ka' kb' where
8.644 - kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
8.645 - unfolding dvd_def by blast
8.646 - then have "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all
8.647 - then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
8.648 - by (auto simp add: zdvd_mult_div_cancel [OF dvdg(1)]
8.649 - zdvd_mult_div_cancel [OF dvdg(2)] dvd_def)
8.650 - have "?g \<noteq> 0" using nz by simp
8.651 - then have gp: "?g \<noteq> 0" using zgcd_pos[where i="a" and j="b"] by arith
8.652 - from zgcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
8.653 - with zdvd_mult_cancel1 [OF gp] have "\<bar>?g'\<bar> = 1" by simp
8.654 - with zgcd_pos show "?g' = 1" by simp
8.655 -qed
8.656 -
8.657 -lemma zgcd_0 [simp, algebra]: "zgcd m 0 = abs m"
8.658 - by (simp add: zgcd_def abs_if)
8.659 -
8.660 -lemma zgcd_0_left [simp, algebra]: "zgcd 0 m = abs m"
8.661 - by (simp add: zgcd_def abs_if)
8.662 -
8.663 -lemma zgcd_non_0: "0 < n ==> zgcd m n = zgcd n (m mod n)"
8.664 - apply (frule_tac b = n and a = m in pos_mod_sign)
8.665 - apply (simp del: pos_mod_sign add: zgcd_def abs_if nat_mod_distrib)
8.666 - apply (auto simp add: gcd_non_0 nat_mod_distrib [symmetric] zmod_zminus1_eq_if)
8.667 - apply (frule_tac a = m in pos_mod_bound)
8.668 - apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2 nat_le_eq_zle)
8.669 - done
8.670 -
8.671 -lemma zgcd_eq: "zgcd m n = zgcd n (m mod n)"
8.672 - apply (case_tac "n = 0", simp add: DIVISION_BY_ZERO)
8.673 - apply (auto simp add: linorder_neq_iff zgcd_non_0)
8.674 - apply (cut_tac m = "-m" and n = "-n" in zgcd_non_0, auto)
8.675 - done
8.676 -
8.677 -lemma zgcd_1 [simp, algebra]: "zgcd m 1 = 1"
8.678 - by (simp add: zgcd_def abs_if)
8.679 -
8.680 -lemma zgcd_0_1_iff [simp, algebra]: "zgcd 0 m = 1 \<longleftrightarrow> \<bar>m\<bar> = 1"
8.681 - by (simp add: zgcd_def abs_if)
8.682 -
8.683 -lemma zgcd_greatest_iff[algebra]: "k dvd zgcd m n = (k dvd m \<and> k dvd n)"
8.684 - by (simp add: zgcd_def abs_if int_dvd_iff dvd_int_iff nat_dvd_iff)
8.685 -
8.686 -lemma zgcd_1_left [simp, algebra]: "zgcd 1 m = 1"
8.687 - by (simp add: zgcd_def gcd_1_left)
8.688 -
8.689 -lemma zgcd_assoc: "zgcd (zgcd k m) n = zgcd k (zgcd m n)"
8.690 - by (simp add: zgcd_def gcd_assoc)
8.691 -
8.692 -lemma zgcd_left_commute: "zgcd k (zgcd m n) = zgcd m (zgcd k n)"
8.693 - apply (rule zgcd_commute [THEN trans])
8.694 - apply (rule zgcd_assoc [THEN trans])
8.695 - apply (rule zgcd_commute [THEN arg_cong])
8.696 - done
8.697 -
8.698 -lemmas zgcd_ac = zgcd_assoc zgcd_commute zgcd_left_commute
8.699 - -- {* addition is an AC-operator *}
8.700 -
8.701 -lemma zgcd_zmult_distrib2: "0 \<le> k ==> k * zgcd m n = zgcd (k * m) (k * n)"
8.702 - by (simp del: minus_mult_right [symmetric]
8.703 - add: minus_mult_right nat_mult_distrib zgcd_def abs_if
8.704 - mult_less_0_iff gcd_mult_distrib2 [symmetric] zmult_int [symmetric])
8.705 -
8.706 -lemma zgcd_zmult_distrib2_abs: "zgcd (k * m) (k * n) = abs k * zgcd m n"
8.707 - by (simp add: abs_if zgcd_zmult_distrib2)
8.708 -
8.709 -lemma zgcd_self [simp]: "0 \<le> m ==> zgcd m m = m"
8.710 - by (cut_tac k = m and m = 1 and n = 1 in zgcd_zmult_distrib2, simp_all)
8.711 -
8.712 -lemma zgcd_zmult_eq_self [simp]: "0 \<le> k ==> zgcd k (k * n) = k"
8.713 - by (cut_tac k = k and m = 1 and n = n in zgcd_zmult_distrib2, simp_all)
8.714 -
8.715 -lemma zgcd_zmult_eq_self2 [simp]: "0 \<le> k ==> zgcd (k * n) k = k"
8.716 - by (cut_tac k = k and m = n and n = 1 in zgcd_zmult_distrib2, simp_all)
8.717 -
8.718 -
8.719 -definition "zlcm i j = int (lcm(nat(abs i)) (nat(abs j)))"
8.720 -
8.721 -lemma dvd_zlcm_self1[simp, algebra]: "i dvd zlcm i j"
8.722 -by(simp add:zlcm_def dvd_int_iff)
8.723 -
8.724 -lemma dvd_zlcm_self2[simp, algebra]: "j dvd zlcm i j"
8.725 -by(simp add:zlcm_def dvd_int_iff)
8.726 -
8.727 -
8.728 -lemma dvd_imp_dvd_zlcm1:
8.729 - assumes "k dvd i" shows "k dvd (zlcm i j)"
8.730 -proof -
8.731 - have "nat(abs k) dvd nat(abs i)" using `k dvd i`
8.732 - by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric])
8.733 - thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
8.734 -qed
8.735 -
8.736 -lemma dvd_imp_dvd_zlcm2:
8.737 - assumes "k dvd j" shows "k dvd (zlcm i j)"
8.738 -proof -
8.739 - have "nat(abs k) dvd nat(abs j)" using `k dvd j`
8.740 - by(simp add:int_dvd_iff[symmetric] dvd_int_iff[symmetric])
8.741 - thus ?thesis by(simp add:zlcm_def dvd_int_iff)(blast intro: dvd_trans)
8.742 -qed
8.743 -
8.744 -
8.745 -lemma zdvd_self_abs1: "(d::int) dvd (abs d)"
8.746 -by (case_tac "d <0", simp_all)
8.747 -
8.748 -lemma zdvd_self_abs2: "(abs (d::int)) dvd d"
8.749 -by (case_tac "d<0", simp_all)
8.750 -
8.751 -(* lcm a b is positive for positive a and b *)
8.752 -
8.753 -lemma lcm_pos:
8.754 - assumes mpos: "m > 0"
8.755 - and npos: "n>0"
8.756 - shows "lcm m n > 0"
8.757 -proof(rule ccontr, simp add: lcm_def gcd_zero)
8.758 -assume h:"m*n div gcd m n = 0"
8.759 -from mpos npos have "gcd m n \<noteq> 0" using gcd_zero by simp
8.760 -hence gcdp: "gcd m n > 0" by simp
8.761 -with h
8.762 -have "m*n < gcd m n"
8.763 - by (cases "m * n < gcd m n") (auto simp add: div_if[OF gcdp, where m="m*n"])
8.764 -moreover
8.765 -have "gcd m n dvd m" by simp
8.766 - with mpos dvd_imp_le have t1:"gcd m n \<le> m" by simp
8.767 - with npos have t1:"gcd m n *n \<le> m*n" by simp
8.768 - have "gcd m n \<le> gcd m n*n" using npos by simp
8.769 - with t1 have "gcd m n \<le> m*n" by arith
8.770 -ultimately show "False" by simp
8.771 -qed
8.772 -
8.773 -lemma zlcm_pos:
8.774 - assumes anz: "a \<noteq> 0"
8.775 - and bnz: "b \<noteq> 0"
8.776 - shows "0 < zlcm a b"
8.777 -proof-
8.778 - let ?na = "nat (abs a)"
8.779 - let ?nb = "nat (abs b)"
8.780 - have nap: "?na >0" using anz by simp
8.781 - have nbp: "?nb >0" using bnz by simp
8.782 - have "0 < lcm ?na ?nb" by (rule lcm_pos[OF nap nbp])
8.783 - thus ?thesis by (simp add: zlcm_def)
8.784 -qed
8.785 -
8.786 -lemma zgcd_code [code]:
8.787 - "zgcd k l = \<bar>if l = 0 then k else zgcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
8.788 - by (simp add: zgcd_def gcd.simps [of "nat \<bar>k\<bar>"] nat_mod_distrib)
8.789 -
8.790 -end
9.1 --- a/src/HOL/Library/Library.thy Tue Sep 01 14:13:34 2009 +0200
9.2 +++ b/src/HOL/Library/Library.thy Tue Sep 01 15:39:33 2009 +0200
9.3 @@ -43,11 +43,9 @@
9.4 OptionalSugar
9.5 Option_ord
9.6 Permutation
9.7 - Pocklington
9.8 Poly_Deriv
9.9 Polynomial
9.10 Preorder
9.11 - Primes
9.12 Product_Vector
9.13 Quicksort
9.14 Quotient
10.1 --- a/src/HOL/Library/Pocklington.thy Tue Sep 01 14:13:34 2009 +0200
10.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
10.3 @@ -1,1263 +0,0 @@
10.4 -(* Title: HOL/Library/Pocklington.thy
10.5 - Author: Amine Chaieb
10.6 -*)
10.7 -
10.8 -header {* Pocklington's Theorem for Primes *}
10.9 -
10.10 -theory Pocklington
10.11 -imports Main Primes
10.12 -begin
10.13 -
10.14 -definition modeq:: "nat => nat => nat => bool" ("(1[_ = _] '(mod _'))")
10.15 - where "[a = b] (mod p) == ((a mod p) = (b mod p))"
10.16 -
10.17 -definition modneq:: "nat => nat => nat => bool" ("(1[_ \<noteq> _] '(mod _'))")
10.18 - where "[a \<noteq> b] (mod p) == ((a mod p) \<noteq> (b mod p))"
10.19 -
10.20 -lemma modeq_trans:
10.21 - "\<lbrakk> [a = b] (mod p); [b = c] (mod p) \<rbrakk> \<Longrightarrow> [a = c] (mod p)"
10.22 - by (simp add:modeq_def)
10.23 -
10.24 -
10.25 -lemma nat_mod_lemma: assumes xyn: "[x = y] (mod n)" and xy:"y \<le> x"
10.26 - shows "\<exists>q. x = y + n * q"
10.27 -using xyn xy unfolding modeq_def using nat_mod_eq_lemma by blast
10.28 -
10.29 -lemma nat_mod[algebra]: "[x = y] (mod n) \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
10.30 -unfolding modeq_def nat_mod_eq_iff ..
10.31 -
10.32 -(* Lemmas about previously defined terms. *)
10.33 -
10.34 -lemma prime: "prime p \<longleftrightarrow> p \<noteq> 0 \<and> p\<noteq>1 \<and> (\<forall>m. 0 < m \<and> m < p \<longrightarrow> coprime p m)"
10.35 - (is "?lhs \<longleftrightarrow> ?rhs")
10.36 -proof-
10.37 - {assume "p=0 \<or> p=1" hence ?thesis using prime_0 prime_1 by (cases "p=0", simp_all)}
10.38 - moreover
10.39 - {assume p0: "p\<noteq>0" "p\<noteq>1"
10.40 - {assume H: "?lhs"
10.41 - {fix m assume m: "m > 0" "m < p"
10.42 - {assume "m=1" hence "coprime p m" by simp}
10.43 - moreover
10.44 - {assume "p dvd m" hence "p \<le> m" using dvd_imp_le m by blast with m(2)
10.45 - have "coprime p m" by simp}
10.46 - ultimately have "coprime p m" using prime_coprime[OF H, of m] by blast}
10.47 - hence ?rhs using p0 by auto}
10.48 - moreover
10.49 - { assume H: "\<forall>m. 0 < m \<and> m < p \<longrightarrow> coprime p m"
10.50 - from prime_factor[OF p0(2)] obtain q where q: "prime q" "q dvd p" by blast
10.51 - from prime_ge_2[OF q(1)] have q0: "q > 0" by arith
10.52 - from dvd_imp_le[OF q(2)] p0 have qp: "q \<le> p" by arith
10.53 - {assume "q = p" hence ?lhs using q(1) by blast}
10.54 - moreover
10.55 - {assume "q\<noteq>p" with qp have qplt: "q < p" by arith
10.56 - from H[rule_format, of q] qplt q0 have "coprime p q" by arith
10.57 - with coprime_prime[of p q q] q have False by simp hence ?lhs by blast}
10.58 - ultimately have ?lhs by blast}
10.59 - ultimately have ?thesis by blast}
10.60 - ultimately show ?thesis by (cases"p=0 \<or> p=1", auto)
10.61 -qed
10.62 -
10.63 -lemma finite_number_segment: "card { m. 0 < m \<and> m < n } = n - 1"
10.64 -proof-
10.65 - have "{ m. 0 < m \<and> m < n } = {1..<n}" by auto
10.66 - thus ?thesis by simp
10.67 -qed
10.68 -
10.69 -lemma coprime_mod: assumes n: "n \<noteq> 0" shows "coprime (a mod n) n \<longleftrightarrow> coprime a n"
10.70 - using n dvd_mod_iff[of _ n a] by (auto simp add: coprime)
10.71 -
10.72 -(* Congruences. *)
10.73 -
10.74 -lemma cong_mod_01[simp,presburger]:
10.75 - "[x = y] (mod 0) \<longleftrightarrow> x = y" "[x = y] (mod 1)" "[x = 0] (mod n) \<longleftrightarrow> n dvd x"
10.76 - by (simp_all add: modeq_def, presburger)
10.77 -
10.78 -lemma cong_sub_cases:
10.79 - "[x = y] (mod n) \<longleftrightarrow> (if x <= y then [y - x = 0] (mod n) else [x - y = 0] (mod n))"
10.80 -apply (auto simp add: nat_mod)
10.81 -apply (rule_tac x="q2" in exI)
10.82 -apply (rule_tac x="q1" in exI, simp)
10.83 -apply (rule_tac x="q2" in exI)
10.84 -apply (rule_tac x="q1" in exI, simp)
10.85 -apply (rule_tac x="q1" in exI)
10.86 -apply (rule_tac x="q2" in exI, simp)
10.87 -apply (rule_tac x="q1" in exI)
10.88 -apply (rule_tac x="q2" in exI, simp)
10.89 -done
10.90 -
10.91 -lemma cong_mult_lcancel: assumes an: "coprime a n" and axy:"[a * x = a * y] (mod n)"
10.92 - shows "[x = y] (mod n)"
10.93 -proof-
10.94 - {assume "a = 0" with an axy coprime_0'[of n] have ?thesis by (simp add: modeq_def) }
10.95 - moreover
10.96 - {assume az: "a\<noteq>0"
10.97 - {assume xy: "x \<le> y" hence axy': "a*x \<le> a*y" by simp
10.98 - with axy cong_sub_cases[of "a*x" "a*y" n] have "[a*(y - x) = 0] (mod n)"
10.99 - by (simp only: if_True diff_mult_distrib2)
10.100 - hence th: "n dvd a*(y -x)" by simp
10.101 - from coprime_divprod[OF th] an have "n dvd y - x"
10.102 - by (simp add: coprime_commute)
10.103 - hence ?thesis using xy cong_sub_cases[of x y n] by simp}
10.104 - moreover
10.105 - {assume H: "\<not>x \<le> y" hence xy: "y \<le> x" by arith
10.106 - from H az have axy': "\<not> a*x \<le> a*y" by auto
10.107 - with axy H cong_sub_cases[of "a*x" "a*y" n] have "[a*(x - y) = 0] (mod n)"
10.108 - by (simp only: if_False diff_mult_distrib2)
10.109 - hence th: "n dvd a*(x - y)" by simp
10.110 - from coprime_divprod[OF th] an have "n dvd x - y"
10.111 - by (simp add: coprime_commute)
10.112 - hence ?thesis using xy cong_sub_cases[of x y n] by simp}
10.113 - ultimately have ?thesis by blast}
10.114 - ultimately show ?thesis by blast
10.115 -qed
10.116 -
10.117 -lemma cong_mult_rcancel: assumes an: "coprime a n" and axy:"[x*a = y*a] (mod n)"
10.118 - shows "[x = y] (mod n)"
10.119 - using cong_mult_lcancel[OF an axy[unfolded mult_commute[of _a]]] .
10.120 -
10.121 -lemma cong_refl: "[x = x] (mod n)" by (simp add: modeq_def)
10.122 -
10.123 -lemma eq_imp_cong: "a = b \<Longrightarrow> [a = b] (mod n)" by (simp add: cong_refl)
10.124 -
10.125 -lemma cong_commute: "[x = y] (mod n) \<longleftrightarrow> [y = x] (mod n)"
10.126 - by (auto simp add: modeq_def)
10.127 -
10.128 -lemma cong_trans[trans]: "[x = y] (mod n) \<Longrightarrow> [y = z] (mod n) \<Longrightarrow> [x = z] (mod n)"
10.129 - by (simp add: modeq_def)
10.130 -
10.131 -lemma cong_add: assumes xx': "[x = x'] (mod n)" and yy':"[y = y'] (mod n)"
10.132 - shows "[x + y = x' + y'] (mod n)"
10.133 -proof-
10.134 - have "(x + y) mod n = (x mod n + y mod n) mod n"
10.135 - by (simp add: mod_add_left_eq[of x y n] mod_add_right_eq[of "x mod n" y n])
10.136 - also have "\<dots> = (x' mod n + y' mod n) mod n" using xx' yy' modeq_def by simp
10.137 - also have "\<dots> = (x' + y') mod n"
10.138 - by (simp add: mod_add_left_eq[of x' y' n] mod_add_right_eq[of "x' mod n" y' n])
10.139 - finally show ?thesis unfolding modeq_def .
10.140 -qed
10.141 -
10.142 -lemma cong_mult: assumes xx': "[x = x'] (mod n)" and yy':"[y = y'] (mod n)"
10.143 - shows "[x * y = x' * y'] (mod n)"
10.144 -proof-
10.145 - have "(x * y) mod n = (x mod n) * (y mod n) mod n"
10.146 - by (simp add: mod_mult_left_eq[of x y n] mod_mult_right_eq[of "x mod n" y n])
10.147 - also have "\<dots> = (x' mod n) * (y' mod n) mod n" using xx'[unfolded modeq_def] yy'[unfolded modeq_def] by simp
10.148 - also have "\<dots> = (x' * y') mod n"
10.149 - by (simp add: mod_mult_left_eq[of x' y' n] mod_mult_right_eq[of "x' mod n" y' n])
10.150 - finally show ?thesis unfolding modeq_def .
10.151 -qed
10.152 -
10.153 -lemma cong_exp: "[x = y] (mod n) \<Longrightarrow> [x^k = y^k] (mod n)"
10.154 - by (induct k, auto simp add: cong_refl cong_mult)
10.155 -lemma cong_sub: assumes xx': "[x = x'] (mod n)" and yy': "[y = y'] (mod n)"
10.156 - and yx: "y <= x" and yx': "y' <= x'"
10.157 - shows "[x - y = x' - y'] (mod n)"
10.158 -proof-
10.159 - { fix x a x' a' y b y' b'
10.160 - have "(x::nat) + a = x' + a' \<Longrightarrow> y + b = y' + b' \<Longrightarrow> y <= x \<Longrightarrow> y' <= x'
10.161 - \<Longrightarrow> (x - y) + (a + b') = (x' - y') + (a' + b)" by arith}
10.162 - note th = this
10.163 - from xx' yy' obtain q1 q2 q1' q2' where q12: "x + n*q1 = x'+n*q2"
10.164 - and q12': "y + n*q1' = y'+n*q2'" unfolding nat_mod by blast+
10.165 - from th[OF q12 q12' yx yx']
10.166 - have "(x - y) + n*(q1 + q2') = (x' - y') + n*(q2 + q1')"
10.167 - by (simp add: right_distrib)
10.168 - thus ?thesis unfolding nat_mod by blast
10.169 -qed
10.170 -
10.171 -lemma cong_mult_lcancel_eq: assumes an: "coprime a n"
10.172 - shows "[a * x = a * y] (mod n) \<longleftrightarrow> [x = y] (mod n)" (is "?lhs \<longleftrightarrow> ?rhs")
10.173 -proof
10.174 - assume H: "?rhs" from cong_mult[OF cong_refl[of a n] H] show ?lhs .
10.175 -next
10.176 - assume H: "?lhs" hence H': "[x*a = y*a] (mod n)" by (simp add: mult_commute)
10.177 - from cong_mult_rcancel[OF an H'] show ?rhs .
10.178 -qed
10.179 -
10.180 -lemma cong_mult_rcancel_eq: assumes an: "coprime a n"
10.181 - shows "[x * a = y * a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
10.182 -using cong_mult_lcancel_eq[OF an, of x y] by (simp add: mult_commute)
10.183 -
10.184 -lemma cong_add_lcancel_eq: "[a + x = a + y] (mod n) \<longleftrightarrow> [x = y] (mod n)"
10.185 - by (simp add: nat_mod)
10.186 -
10.187 -lemma cong_add_rcancel_eq: "[x + a = y + a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
10.188 - by (simp add: nat_mod)
10.189 -
10.190 -lemma cong_add_rcancel: "[x + a = y + a] (mod n) \<Longrightarrow> [x = y] (mod n)"
10.191 - by (simp add: nat_mod)
10.192 -
10.193 -lemma cong_add_lcancel: "[a + x = a + y] (mod n) \<Longrightarrow> [x = y] (mod n)"
10.194 - by (simp add: nat_mod)
10.195 -
10.196 -lemma cong_add_lcancel_eq_0: "[a + x = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
10.197 - by (simp add: nat_mod)
10.198 -
10.199 -lemma cong_add_rcancel_eq_0: "[x + a = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
10.200 - by (simp add: nat_mod)
10.201 -
10.202 -lemma cong_imp_eq: assumes xn: "x < n" and yn: "y < n" and xy: "[x = y] (mod n)"
10.203 - shows "x = y"
10.204 - using xy[unfolded modeq_def mod_less[OF xn] mod_less[OF yn]] .
10.205 -
10.206 -lemma cong_divides_modulus: "[x = y] (mod m) \<Longrightarrow> n dvd m ==> [x = y] (mod n)"
10.207 - apply (auto simp add: nat_mod dvd_def)
10.208 - apply (rule_tac x="k*q1" in exI)
10.209 - apply (rule_tac x="k*q2" in exI)
10.210 - by simp
10.211 -
10.212 -lemma cong_0_divides: "[x = 0] (mod n) \<longleftrightarrow> n dvd x" by simp
10.213 -
10.214 -lemma cong_1_divides:"[x = 1] (mod n) ==> n dvd x - 1"
10.215 - apply (cases "x\<le>1", simp_all)
10.216 - using cong_sub_cases[of x 1 n] by auto
10.217 -
10.218 -lemma cong_divides: "[x = y] (mod n) \<Longrightarrow> n dvd x \<longleftrightarrow> n dvd y"
10.219 -apply (auto simp add: nat_mod dvd_def)
10.220 -apply (rule_tac x="k + q1 - q2" in exI, simp add: add_mult_distrib2 diff_mult_distrib2)
10.221 -apply (rule_tac x="k + q2 - q1" in exI, simp add: add_mult_distrib2 diff_mult_distrib2)
10.222 -done
10.223 -
10.224 -lemma cong_coprime: assumes xy: "[x = y] (mod n)"
10.225 - shows "coprime n x \<longleftrightarrow> coprime n y"
10.226 -proof-
10.227 - {assume "n=0" hence ?thesis using xy by simp}
10.228 - moreover
10.229 - {assume nz: "n \<noteq> 0"
10.230 - have "coprime n x \<longleftrightarrow> coprime (x mod n) n"
10.231 - by (simp add: coprime_mod[OF nz, of x] coprime_commute[of n x])
10.232 - also have "\<dots> \<longleftrightarrow> coprime (y mod n) n" using xy[unfolded modeq_def] by simp
10.233 - also have "\<dots> \<longleftrightarrow> coprime y n" by (simp add: coprime_mod[OF nz, of y])
10.234 - finally have ?thesis by (simp add: coprime_commute) }
10.235 -ultimately show ?thesis by blast
10.236 -qed
10.237 -
10.238 -lemma cong_mod: "~(n = 0) \<Longrightarrow> [a mod n = a] (mod n)" by (simp add: modeq_def)
10.239 -
10.240 -lemma mod_mult_cong: "~(a = 0) \<Longrightarrow> ~(b = 0)
10.241 - \<Longrightarrow> [x mod (a * b) = y] (mod a) \<longleftrightarrow> [x = y] (mod a)"
10.242 - by (simp add: modeq_def mod_mult2_eq mod_add_left_eq)
10.243 -
10.244 -lemma cong_mod_mult: "[x = y] (mod n) \<Longrightarrow> m dvd n \<Longrightarrow> [x = y] (mod m)"
10.245 - apply (auto simp add: nat_mod dvd_def)
10.246 - apply (rule_tac x="k*q1" in exI)
10.247 - apply (rule_tac x="k*q2" in exI, simp)
10.248 - done
10.249 -
10.250 -(* Some things when we know more about the order. *)
10.251 -
10.252 -lemma cong_le: "y <= x \<Longrightarrow> [x = y] (mod n) \<longleftrightarrow> (\<exists>q. x = q * n + y)"
10.253 - using nat_mod_lemma[of x y n]
10.254 - apply auto
10.255 - apply (simp add: nat_mod)
10.256 - apply (rule_tac x="q" in exI)
10.257 - apply (rule_tac x="q + q" in exI)
10.258 - by (auto simp: algebra_simps)
10.259 -
10.260 -lemma cong_to_1: "[a = 1] (mod n) \<longleftrightarrow> a = 0 \<and> n = 1 \<or> (\<exists>m. a = 1 + m * n)"
10.261 -proof-
10.262 - {assume "n = 0 \<or> n = 1\<or> a = 0 \<or> a = 1" hence ?thesis
10.263 - apply (cases "n=0", simp_all add: cong_commute)
10.264 - apply (cases "n=1", simp_all add: cong_commute modeq_def)
10.265 - apply arith
10.266 - by (cases "a=1", simp_all add: modeq_def cong_commute)}
10.267 - moreover
10.268 - {assume n: "n\<noteq>0" "n\<noteq>1" and a:"a\<noteq>0" "a \<noteq> 1" hence a': "a \<ge> 1" by simp
10.269 - hence ?thesis using cong_le[OF a', of n] by auto }
10.270 - ultimately show ?thesis by auto
10.271 -qed
10.272 -
10.273 -(* Some basic theorems about solving congruences. *)
10.274 -
10.275 -
10.276 -lemma cong_solve: assumes an: "coprime a n" shows "\<exists>x. [a * x = b] (mod n)"
10.277 -proof-
10.278 - {assume "a=0" hence ?thesis using an by (simp add: modeq_def)}
10.279 - moreover
10.280 - {assume az: "a\<noteq>0"
10.281 - from bezout_add_strong[OF az, of n]
10.282 - obtain d x y where dxy: "d dvd a" "d dvd n" "a*x = n*y + d" by blast
10.283 - from an[unfolded coprime, rule_format, of d] dxy(1,2) have d1: "d = 1" by blast
10.284 - hence "a*x*b = (n*y + 1)*b" using dxy(3) by simp
10.285 - hence "a*(x*b) = n*(y*b) + b" by algebra
10.286 - hence "a*(x*b) mod n = (n*(y*b) + b) mod n" by simp
10.287 - hence "a*(x*b) mod n = b mod n" by (simp add: mod_add_left_eq)
10.288 - hence "[a*(x*b) = b] (mod n)" unfolding modeq_def .
10.289 - hence ?thesis by blast}
10.290 -ultimately show ?thesis by blast
10.291 -qed
10.292 -
10.293 -lemma cong_solve_unique: assumes an: "coprime a n" and nz: "n \<noteq> 0"
10.294 - shows "\<exists>!x. x < n \<and> [a * x = b] (mod n)"
10.295 -proof-
10.296 - let ?P = "\<lambda>x. x < n \<and> [a * x = b] (mod n)"
10.297 - from cong_solve[OF an] obtain x where x: "[a*x = b] (mod n)" by blast
10.298 - let ?x = "x mod n"
10.299 - from x have th: "[a * ?x = b] (mod n)"
10.300 - by (simp add: modeq_def mod_mult_right_eq[of a x n])
10.301 - from mod_less_divisor[ of n x] nz th have Px: "?P ?x" by simp
10.302 - {fix y assume Py: "y < n" "[a * y = b] (mod n)"
10.303 - from Py(2) th have "[a * y = a*?x] (mod n)" by (simp add: modeq_def)
10.304 - hence "[y = ?x] (mod n)" by (simp add: cong_mult_lcancel_eq[OF an])
10.305 - with mod_less[OF Py(1)] mod_less_divisor[ of n x] nz
10.306 - have "y = ?x" by (simp add: modeq_def)}
10.307 - with Px show ?thesis by blast
10.308 -qed
10.309 -
10.310 -lemma cong_solve_unique_nontrivial:
10.311 - assumes p: "prime p" and pa: "coprime p a" and x0: "0 < x" and xp: "x < p"
10.312 - shows "\<exists>!y. 0 < y \<and> y < p \<and> [x * y = a] (mod p)"
10.313 -proof-
10.314 - from p have p1: "p > 1" using prime_ge_2[OF p] by arith
10.315 - hence p01: "p \<noteq> 0" "p \<noteq> 1" by arith+
10.316 - from pa have ap: "coprime a p" by (simp add: coprime_commute)
10.317 - from prime_coprime[OF p, of x] dvd_imp_le[of p x] x0 xp have px:"coprime x p"
10.318 - by (auto simp add: coprime_commute)
10.319 - from cong_solve_unique[OF px p01(1)]
10.320 - 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
10.321 - {assume y0: "y = 0"
10.322 - with y(2) have th: "p dvd a" by (simp add: cong_commute[of 0 a p])
10.323 - with p coprime_prime[OF pa, of p] have False by simp}
10.324 - with y show ?thesis unfolding Ex1_def using neq0_conv by blast
10.325 -qed
10.326 -lemma cong_unique_inverse_prime:
10.327 - assumes p: "prime p" and x0: "0 < x" and xp: "x < p"
10.328 - shows "\<exists>!y. 0 < y \<and> y < p \<and> [x * y = 1] (mod p)"
10.329 - using cong_solve_unique_nontrivial[OF p coprime_1[of p] x0 xp] .
10.330 -
10.331 -(* Forms of the Chinese remainder theorem. *)
10.332 -
10.333 -lemma cong_chinese:
10.334 - assumes ab: "coprime a b" and xya: "[x = y] (mod a)"
10.335 - and xyb: "[x = y] (mod b)"
10.336 - shows "[x = y] (mod a*b)"
10.337 - using ab xya xyb
10.338 - by (simp add: cong_sub_cases[of x y a] cong_sub_cases[of x y b]
10.339 - cong_sub_cases[of x y "a*b"])
10.340 -(cases "x \<le> y", simp_all add: divides_mul[of a _ b])
10.341 -
10.342 -lemma chinese_remainder_unique:
10.343 - assumes ab: "coprime a b" and az: "a \<noteq> 0" and bz: "b\<noteq>0"
10.344 - shows "\<exists>!x. x < a * b \<and> [x = m] (mod a) \<and> [x = n] (mod b)"
10.345 -proof-
10.346 - from az bz have abpos: "a*b > 0" by simp
10.347 - from chinese_remainder[OF ab az bz] obtain x q1 q2 where
10.348 - xq12: "x = m + q1 * a" "x = n + q2 * b" by blast
10.349 - let ?w = "x mod (a*b)"
10.350 - have wab: "?w < a*b" by (simp add: mod_less_divisor[OF abpos])
10.351 - from xq12(1) have "?w mod a = ((m + q1 * a) mod (a*b)) mod a" by simp
10.352 - also have "\<dots> = m mod a" apply (simp add: mod_mult2_eq)
10.353 - apply (subst mod_add_left_eq)
10.354 - by simp
10.355 - finally have th1: "[?w = m] (mod a)" by (simp add: modeq_def)
10.356 - from xq12(2) have "?w mod b = ((n + q2 * b) mod (a*b)) mod b" by simp
10.357 - also have "\<dots> = ((n + q2 * b) mod (b*a)) mod b" by (simp add: mult_commute)
10.358 - also have "\<dots> = n mod b" apply (simp add: mod_mult2_eq)
10.359 - apply (subst mod_add_left_eq)
10.360 - by simp
10.361 - finally have th2: "[?w = n] (mod b)" by (simp add: modeq_def)
10.362 - {fix y assume H: "y < a*b" "[y = m] (mod a)" "[y = n] (mod b)"
10.363 - with th1 th2 have H': "[y = ?w] (mod a)" "[y = ?w] (mod b)"
10.364 - by (simp_all add: modeq_def)
10.365 - from cong_chinese[OF ab H'] mod_less[OF H(1)] mod_less[OF wab]
10.366 - have "y = ?w" by (simp add: modeq_def)}
10.367 - with th1 th2 wab show ?thesis by blast
10.368 -qed
10.369 -
10.370 -lemma chinese_remainder_coprime_unique:
10.371 - assumes ab: "coprime a b" and az: "a \<noteq> 0" and bz: "b \<noteq> 0"
10.372 - and ma: "coprime m a" and nb: "coprime n b"
10.373 - shows "\<exists>!x. coprime x (a * b) \<and> x < a * b \<and> [x = m] (mod a) \<and> [x = n] (mod b)"
10.374 -proof-
10.375 - let ?P = "\<lambda>x. x < a * b \<and> [x = m] (mod a) \<and> [x = n] (mod b)"
10.376 - from chinese_remainder_unique[OF ab az bz]
10.377 - obtain x where x: "x < a * b" "[x = m] (mod a)" "[x = n] (mod b)"
10.378 - "\<forall>y. ?P y \<longrightarrow> y = x" by blast
10.379 - from ma nb cong_coprime[OF x(2)] cong_coprime[OF x(3)]
10.380 - have "coprime x a" "coprime x b" by (simp_all add: coprime_commute)
10.381 - with coprime_mul[of x a b] have "coprime x (a*b)" by simp
10.382 - with x show ?thesis by blast
10.383 -qed
10.384 -
10.385 -(* Euler totient function. *)
10.386 -
10.387 -definition phi_def: "\<phi> n = card { m. 0 < m \<and> m <= n \<and> coprime m n }"
10.388 -
10.389 -lemma phi_0[simp]: "\<phi> 0 = 0"
10.390 - unfolding phi_def by auto
10.391 -
10.392 -lemma phi_finite[simp]: "finite ({ m. 0 < m \<and> m <= n \<and> coprime m n })"
10.393 -proof-
10.394 - have "{ m. 0 < m \<and> m <= n \<and> coprime m n } \<subseteq> {0..n}" by auto
10.395 - thus ?thesis by (auto intro: finite_subset)
10.396 -qed
10.397 -
10.398 -declare coprime_1[presburger]
10.399 -lemma phi_1[simp]: "\<phi> 1 = 1"
10.400 -proof-
10.401 - {fix m
10.402 - have "0 < m \<and> m <= 1 \<and> coprime m 1 \<longleftrightarrow> m = 1" by presburger }
10.403 - thus ?thesis by (simp add: phi_def)
10.404 -qed
10.405 -
10.406 -lemma [simp]: "\<phi> (Suc 0) = Suc 0" using phi_1 by simp
10.407 -
10.408 -lemma phi_alt: "\<phi>(n) = card { m. coprime m n \<and> m < n}"
10.409 -proof-
10.410 - {assume "n=0 \<or> n=1" hence ?thesis by (cases "n=0", simp_all)}
10.411 - moreover
10.412 - {assume n: "n\<noteq>0" "n\<noteq>1"
10.413 - {fix m
10.414 - from n have "0 < m \<and> m <= n \<and> coprime m n \<longleftrightarrow> coprime m n \<and> m < n"
10.415 - apply (cases "m = 0", simp_all)
10.416 - apply (cases "m = 1", simp_all)
10.417 - apply (cases "m = n", auto)
10.418 - done }
10.419 - hence ?thesis unfolding phi_def by simp}
10.420 - ultimately show ?thesis by auto
10.421 -qed
10.422 -
10.423 -lemma phi_finite_lemma[simp]: "finite {m. coprime m n \<and> m < n}" (is "finite ?S")
10.424 - by (rule finite_subset[of "?S" "{0..n}"], auto)
10.425 -
10.426 -lemma phi_another: assumes n: "n\<noteq>1"
10.427 - shows "\<phi> n = card {m. 0 < m \<and> m < n \<and> coprime m n }"
10.428 -proof-
10.429 - {fix m
10.430 - from n have "0 < m \<and> m < n \<and> coprime m n \<longleftrightarrow> coprime m n \<and> m < n"
10.431 - by (cases "m=0", auto)}
10.432 - thus ?thesis unfolding phi_alt by auto
10.433 -qed
10.434 -
10.435 -lemma phi_limit: "\<phi> n \<le> n"
10.436 -proof-
10.437 - have "{ m. coprime m n \<and> m < n} \<subseteq> {0 ..<n}" by auto
10.438 - with card_mono[of "{0 ..<n}" "{ m. coprime m n \<and> m < n}"]
10.439 - show ?thesis unfolding phi_alt by auto
10.440 -qed
10.441 -
10.442 -lemma stupid[simp]: "{m. (0::nat) < m \<and> m < n} = {1..<n}"
10.443 - by auto
10.444 -
10.445 -lemma phi_limit_strong: assumes n: "n\<noteq>1"
10.446 - shows "\<phi>(n) \<le> n - 1"
10.447 -proof-
10.448 - show ?thesis
10.449 - unfolding phi_another[OF n] finite_number_segment[of n, symmetric]
10.450 - by (rule card_mono[of "{m. 0 < m \<and> m < n}" "{m. 0 < m \<and> m < n \<and> coprime m n}"], auto)
10.451 -qed
10.452 -
10.453 -lemma phi_lowerbound_1_strong: assumes n: "n \<ge> 1"
10.454 - shows "\<phi>(n) \<ge> 1"
10.455 -proof-
10.456 - let ?S = "{ m. 0 < m \<and> m <= n \<and> coprime m n }"
10.457 - from card_0_eq[of ?S] n have "\<phi> n \<noteq> 0" unfolding phi_alt
10.458 - apply auto
10.459 - apply (cases "n=1", simp_all)
10.460 - apply (rule exI[where x=1], simp)
10.461 - done
10.462 - thus ?thesis by arith
10.463 -qed
10.464 -
10.465 -lemma phi_lowerbound_1: "2 <= n ==> 1 <= \<phi>(n)"
10.466 - using phi_lowerbound_1_strong[of n] by auto
10.467 -
10.468 -lemma phi_lowerbound_2: assumes n: "3 <= n" shows "2 <= \<phi> (n)"
10.469 -proof-
10.470 - let ?S = "{ m. 0 < m \<and> m <= n \<and> coprime m n }"
10.471 - have inS: "{1, n - 1} \<subseteq> ?S" using n coprime_plus1[of "n - 1"]
10.472 - by (auto simp add: coprime_commute)
10.473 - from n have c2: "card {1, n - 1} = 2" by (auto simp add: card_insert_if)
10.474 - from card_mono[of ?S "{1, n - 1}", simplified inS c2] show ?thesis
10.475 - unfolding phi_def by auto
10.476 -qed
10.477 -
10.478 -lemma phi_prime: "\<phi> n = n - 1 \<and> n\<noteq>0 \<and> n\<noteq>1 \<longleftrightarrow> prime n"
10.479 -proof-
10.480 - {assume "n=0 \<or> n=1" hence ?thesis by (cases "n=1", simp_all)}
10.481 - moreover
10.482 - {assume n: "n\<noteq>0" "n\<noteq>1"
10.483 - let ?S = "{m. 0 < m \<and> m < n}"
10.484 - have fS: "finite ?S" by simp
10.485 - let ?S' = "{m. 0 < m \<and> m < n \<and> coprime m n}"
10.486 - have fS':"finite ?S'" apply (rule finite_subset[of ?S' ?S]) by auto
10.487 - {assume H: "\<phi> n = n - 1 \<and> n\<noteq>0 \<and> n\<noteq>1"
10.488 - hence ceq: "card ?S' = card ?S"
10.489 - using n finite_number_segment[of n] phi_another[OF n(2)] by simp
10.490 - {fix m assume m: "0 < m" "m < n" "\<not> coprime m n"
10.491 - hence mS': "m \<notin> ?S'" by auto
10.492 - have "insert m ?S' \<le> ?S" using m by auto
10.493 - from m have "card (insert m ?S') \<le> card ?S"
10.494 - by - (rule card_mono[of ?S "insert m ?S'"], auto)
10.495 - hence False
10.496 - unfolding card_insert_disjoint[of "?S'" m, OF fS' mS'] ceq
10.497 - by simp }
10.498 - hence "\<forall>m. 0 <m \<and> m < n \<longrightarrow> coprime m n" by blast
10.499 - hence "prime n" unfolding prime using n by (simp add: coprime_commute)}
10.500 - moreover
10.501 - {assume H: "prime n"
10.502 - hence "?S = ?S'" unfolding prime using n
10.503 - by (auto simp add: coprime_commute)
10.504 - hence "card ?S = card ?S'" by simp
10.505 - hence "\<phi> n = n - 1" unfolding phi_another[OF n(2)] by simp}
10.506 - ultimately have ?thesis using n by blast}
10.507 - ultimately show ?thesis by (cases "n=0") blast+
10.508 -qed
10.509 -
10.510 -(* Multiplicativity property. *)
10.511 -
10.512 -lemma phi_multiplicative: assumes ab: "coprime a b"
10.513 - shows "\<phi> (a * b) = \<phi> a * \<phi> b"
10.514 -proof-
10.515 - {assume "a = 0 \<or> b = 0 \<or> a = 1 \<or> b = 1"
10.516 - hence ?thesis
10.517 - by (cases "a=0", simp, cases "b=0", simp, cases"a=1", simp_all) }
10.518 - moreover
10.519 - {assume a: "a\<noteq>0" "a\<noteq>1" and b: "b\<noteq>0" "b\<noteq>1"
10.520 - hence ab0: "a*b \<noteq> 0" by simp
10.521 - let ?S = "\<lambda>k. {m. coprime m k \<and> m < k}"
10.522 - let ?f = "\<lambda>x. (x mod a, x mod b)"
10.523 - have eq: "?f ` (?S (a*b)) = (?S a \<times> ?S b)"
10.524 - proof-
10.525 - {fix x assume x:"x \<in> ?S (a*b)"
10.526 - hence x': "coprime x (a*b)" "x < a*b" by simp_all
10.527 - hence xab: "coprime x a" "coprime x b" by (simp_all add: coprime_mul_eq)
10.528 - from mod_less_divisor a b have xab':"x mod a < a" "x mod b < b" by auto
10.529 - from xab xab' have "?f x \<in> (?S a \<times> ?S b)"
10.530 - by (simp add: coprime_mod[OF a(1)] coprime_mod[OF b(1)])}
10.531 - moreover
10.532 - {fix x y assume x: "x \<in> ?S a" and y: "y \<in> ?S b"
10.533 - hence x': "coprime x a" "x < a" and y': "coprime y b" "y < b" by simp_all
10.534 - from chinese_remainder_coprime_unique[OF ab a(1) b(1) x'(1) y'(1)]
10.535 - obtain z where z: "coprime z (a * b)" "z < a * b" "[z = x] (mod a)"
10.536 - "[z = y] (mod b)" by blast
10.537 - hence "(x,y) \<in> ?f ` (?S (a*b))"
10.538 - using y'(2) mod_less_divisor[of b y] x'(2) mod_less_divisor[of a x]
10.539 - by (auto simp add: image_iff modeq_def)}
10.540 - ultimately show ?thesis by auto
10.541 - qed
10.542 - have finj: "inj_on ?f (?S (a*b))"
10.543 - unfolding inj_on_def
10.544 - proof(clarify)
10.545 - fix x y assume H: "coprime x (a * b)" "x < a * b" "coprime y (a * b)"
10.546 - "y < a * b" "x mod a = y mod a" "x mod b = y mod b"
10.547 - hence cp: "coprime x a" "coprime x b" "coprime y a" "coprime y b"
10.548 - by (simp_all add: coprime_mul_eq)
10.549 - from chinese_remainder_coprime_unique[OF ab a(1) b(1) cp(3,4)] H
10.550 - show "x = y" unfolding modeq_def by blast
10.551 - qed
10.552 - from card_image[OF finj, unfolded eq] have ?thesis
10.553 - unfolding phi_alt by simp }
10.554 - ultimately show ?thesis by auto
10.555 -qed
10.556 -
10.557 -(* Fermat's Little theorem / Fermat-Euler theorem. *)
10.558 -
10.559 -
10.560 -lemma nproduct_mod:
10.561 - assumes fS: "finite S" and n0: "n \<noteq> 0"
10.562 - shows "[setprod (\<lambda>m. a(m) mod n) S = setprod a S] (mod n)"
10.563 -proof-
10.564 - have th1:"[1 = 1] (mod n)" by (simp add: modeq_def)
10.565 - from cong_mult
10.566 - have th3:"\<forall>x1 y1 x2 y2.
10.567 - [x1 = x2] (mod n) \<and> [y1 = y2] (mod n) \<longrightarrow> [x1 * y1 = x2 * y2] (mod n)"
10.568 - by blast
10.569 - have th4:"\<forall>x\<in>S. [a x mod n = a x] (mod n)" by (simp add: modeq_def)
10.570 - from fold_image_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)
10.571 -qed
10.572 -
10.573 -lemma nproduct_cmul:
10.574 - assumes fS:"finite S"
10.575 - shows "setprod (\<lambda>m. (c::'a::{comm_monoid_mult})* a(m)) S = c ^ (card S) * setprod a S"
10.576 -unfolding setprod_timesf setprod_constant[OF fS, of c] ..
10.577 -
10.578 -lemma coprime_nproduct:
10.579 - assumes fS: "finite S" and Sn: "\<forall>x\<in>S. coprime n (a x)"
10.580 - shows "coprime n (setprod a S)"
10.581 - using fS unfolding setprod_def by (rule finite_subset_induct)
10.582 - (insert Sn, auto simp add: coprime_mul)
10.583 -
10.584 -lemma fermat_little: assumes an: "coprime a n"
10.585 - shows "[a ^ (\<phi> n) = 1] (mod n)"
10.586 -proof-
10.587 - {assume "n=0" hence ?thesis by simp}
10.588 - moreover
10.589 - {assume "n=1" hence ?thesis by (simp add: modeq_def)}
10.590 - moreover
10.591 - {assume nz: "n \<noteq> 0" and n1: "n \<noteq> 1"
10.592 - let ?S = "{m. coprime m n \<and> m < n}"
10.593 - let ?P = "\<Prod> ?S"
10.594 - have fS: "finite ?S" by simp
10.595 - have cardfS: "\<phi> n = card ?S" unfolding phi_alt ..
10.596 - {fix m assume m: "m \<in> ?S"
10.597 - hence "coprime m n" by simp
10.598 - with coprime_mul[of n a m] an have "coprime (a*m) n"
10.599 - by (simp add: coprime_commute)}
10.600 - hence Sn: "\<forall>m\<in> ?S. coprime (a*m) n " by blast
10.601 - from coprime_nproduct[OF fS, of n "\<lambda>m. m"] have nP:"coprime ?P n"
10.602 - by (simp add: coprime_commute)
10.603 - have Paphi: "[?P*a^ (\<phi> n) = ?P*1] (mod n)"
10.604 - proof-
10.605 - let ?h = "\<lambda>m. m mod n"
10.606 - {fix m assume mS: "m\<in> ?S"
10.607 - hence "?h m \<in> ?S" by simp}
10.608 - hence hS: "?h ` ?S = ?S"by (auto simp add: image_iff)
10.609 - have "a\<noteq>0" using an n1 nz apply- apply (rule ccontr) by simp
10.610 - hence inj: "inj_on (op * a) ?S" unfolding inj_on_def by simp
10.611 -
10.612 - have eq0: "fold_image op * (?h \<circ> op * a) 1 {m. coprime m n \<and> m < n} =
10.613 - fold_image op * (\<lambda>m. m) 1 {m. coprime m n \<and> m < n}"
10.614 - proof (rule fold_image_eq_general[where h="?h o (op * a)"])
10.615 - show "finite ?S" using fS .
10.616 - next
10.617 - {fix y assume yS: "y \<in> ?S" hence y: "coprime y n" "y < n" by simp_all
10.618 - from cong_solve_unique[OF an nz, of y]
10.619 - 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
10.620 - from cong_coprime[OF x(2)] y(1)
10.621 - have xm: "coprime x n" by (simp add: coprime_mul_eq coprime_commute)
10.622 - {fix z assume "z \<in> ?S" "(?h \<circ> op * a) z = y"
10.623 - hence z: "coprime z n" "z < n" "(?h \<circ> op * a) z = y" by simp_all
10.624 - from x(3)[rule_format, of z] z(2,3) have "z=x"
10.625 - unfolding modeq_def mod_less[OF y(2)] by simp}
10.626 - with xm x(1,2) have "\<exists>!x. x \<in> ?S \<and> (?h \<circ> op * a) x = y"
10.627 - unfolding modeq_def mod_less[OF y(2)] by auto }
10.628 - thus "\<forall>y\<in>{m. coprime m n \<and> m < n}.
10.629 - \<exists>!x. x \<in> {m. coprime m n \<and> m < n} \<and> ((\<lambda>m. m mod n) \<circ> op * a) x = y" by blast
10.630 - next
10.631 - {fix x assume xS: "x\<in> ?S"
10.632 - hence x: "coprime x n" "x < n" by simp_all
10.633 - with an have "coprime (a*x) n"
10.634 - by (simp add: coprime_mul_eq[of n a x] coprime_commute)
10.635 - hence "?h (a*x) \<in> ?S" using nz
10.636 - by (simp add: coprime_mod[OF nz] mod_less_divisor)}
10.637 - thus " \<forall>x\<in>{m. coprime m n \<and> m < n}.
10.638 - ((\<lambda>m. m mod n) \<circ> op * a) x \<in> {m. coprime m n \<and> m < n} \<and>
10.639 - ((\<lambda>m. m mod n) \<circ> op * a) x = ((\<lambda>m. m mod n) \<circ> op * a) x" by simp
10.640 - qed
10.641 - from nproduct_mod[OF fS nz, of "op * a"]
10.642 - have "[(setprod (op *a) ?S) = (setprod (?h o (op * a)) ?S)] (mod n)"
10.643 - unfolding o_def
10.644 - by (simp add: cong_commute)
10.645 - also have "[setprod (?h o (op * a)) ?S = ?P ] (mod n)"
10.646 - using eq0 fS an by (simp add: setprod_def modeq_def o_def)
10.647 - finally show "[?P*a^ (\<phi> n) = ?P*1] (mod n)"
10.648 - unfolding cardfS mult_commute[of ?P "a^ (card ?S)"]
10.649 - nproduct_cmul[OF fS, symmetric] mult_1_right by simp
10.650 - qed
10.651 - from cong_mult_lcancel[OF nP Paphi] have ?thesis . }
10.652 - ultimately show ?thesis by blast
10.653 -qed
10.654 -
10.655 -lemma fermat_little_prime: assumes p: "prime p" and ap: "coprime a p"
10.656 - shows "[a^ (p - 1) = 1] (mod p)"
10.657 - using fermat_little[OF ap] p[unfolded phi_prime[symmetric]]
10.658 -by simp
10.659 -
10.660 -
10.661 -(* Lucas's theorem. *)
10.662 -
10.663 -lemma lucas_coprime_lemma:
10.664 - assumes m: "m\<noteq>0" and am: "[a^m = 1] (mod n)"
10.665 - shows "coprime a n"
10.666 -proof-
10.667 - {assume "n=1" hence ?thesis by simp}
10.668 - moreover
10.669 - {assume "n = 0" hence ?thesis using am m exp_eq_1[of a m] by simp}
10.670 - moreover
10.671 - {assume n: "n\<noteq>0" "n\<noteq>1"
10.672 - from m obtain m' where m': "m = Suc m'" by (cases m, blast+)
10.673 - {fix d
10.674 - assume d: "d dvd a" "d dvd n"
10.675 - from n have n1: "1 < n" by arith
10.676 - from am mod_less[OF n1] have am1: "a^m mod n = 1" unfolding modeq_def by simp
10.677 - from dvd_mult2[OF d(1), of "a^m'"] have dam:"d dvd a^m" by (simp add: m')
10.678 - from dvd_mod_iff[OF d(2), of "a^m"] dam am1
10.679 - have "d = 1" by simp }
10.680 - hence ?thesis unfolding coprime by auto
10.681 - }
10.682 - ultimately show ?thesis by blast
10.683 -qed
10.684 -
10.685 -lemma lucas_weak:
10.686 - assumes n: "n \<ge> 2" and an:"[a^(n - 1) = 1] (mod n)"
10.687 - and nm: "\<forall>m. 0 <m \<and> m < n - 1 \<longrightarrow> \<not> [a^m = 1] (mod n)"
10.688 - shows "prime n"
10.689 -proof-
10.690 - from n have n1: "n \<noteq> 1" "n\<noteq>0" "n - 1 \<noteq> 0" "n - 1 > 0" "n - 1 < n" by arith+
10.691 - from lucas_coprime_lemma[OF n1(3) an] have can: "coprime a n" .
10.692 - from fermat_little[OF can] have afn: "[a ^ \<phi> n = 1] (mod n)" .
10.693 - {assume "\<phi> n \<noteq> n - 1"
10.694 - with phi_limit_strong[OF n1(1)] phi_lowerbound_1[OF n]
10.695 - have c:"\<phi> n > 0 \<and> \<phi> n < n - 1" by arith
10.696 - from nm[rule_format, OF c] afn have False ..}
10.697 - hence "\<phi> n = n - 1" by blast
10.698 - with phi_prime[of n] n1(1,2) show ?thesis by simp
10.699 -qed
10.700 -
10.701 -lemma nat_exists_least_iff: "(\<exists>(n::nat). P n) \<longleftrightarrow> (\<exists>n. P n \<and> (\<forall>m < n. \<not> P m))"
10.702 - (is "?lhs \<longleftrightarrow> ?rhs")
10.703 -proof
10.704 - assume ?rhs thus ?lhs by blast
10.705 -next
10.706 - assume H: ?lhs then obtain n where n: "P n" by blast
10.707 - let ?x = "Least P"
10.708 - {fix m assume m: "m < ?x"
10.709 - from not_less_Least[OF m] have "\<not> P m" .}
10.710 - with LeastI_ex[OF H] show ?rhs by blast
10.711 -qed
10.712 -
10.713 -lemma nat_exists_least_iff': "(\<exists>(n::nat). P n) \<longleftrightarrow> (P (Least P) \<and> (\<forall>m < (Least P). \<not> P m))"
10.714 - (is "?lhs \<longleftrightarrow> ?rhs")
10.715 -proof-
10.716 - {assume ?rhs hence ?lhs by blast}
10.717 - moreover
10.718 - { assume H: ?lhs then obtain n where n: "P n" by blast
10.719 - let ?x = "Least P"
10.720 - {fix m assume m: "m < ?x"
10.721 - from not_less_Least[OF m] have "\<not> P m" .}
10.722 - with LeastI_ex[OF H] have ?rhs by blast}
10.723 - ultimately show ?thesis by blast
10.724 -qed
10.725 -
10.726 -lemma power_mod: "((x::nat) mod m)^n mod m = x^n mod m"
10.727 -proof(induct n)
10.728 - case 0 thus ?case by simp
10.729 -next
10.730 - case (Suc n)
10.731 - have "(x mod m)^(Suc n) mod m = ((x mod m) * (((x mod m) ^ n) mod m)) mod m"
10.732 - by (simp add: mod_mult_right_eq[symmetric])
10.733 - also have "\<dots> = ((x mod m) * (x^n mod m)) mod m" using Suc.hyps by simp
10.734 - also have "\<dots> = x^(Suc n) mod m"
10.735 - by (simp add: mod_mult_left_eq[symmetric] mod_mult_right_eq[symmetric])
10.736 - finally show ?case .
10.737 -qed
10.738 -
10.739 -lemma lucas:
10.740 - assumes n2: "n \<ge> 2" and an1: "[a^(n - 1) = 1] (mod n)"
10.741 - and pn: "\<forall>p. prime p \<and> p dvd n - 1 \<longrightarrow> \<not> [a^((n - 1) div p) = 1] (mod n)"
10.742 - shows "prime n"
10.743 -proof-
10.744 - from n2 have n01: "n\<noteq>0" "n\<noteq>1" "n - 1 \<noteq> 0" by arith+
10.745 - from mod_less_divisor[of n 1] n01 have onen: "1 mod n = 1" by simp
10.746 - from lucas_coprime_lemma[OF n01(3) an1] cong_coprime[OF an1]
10.747 - have an: "coprime a n" "coprime (a^(n - 1)) n" by (simp_all add: coprime_commute)
10.748 - {assume H0: "\<exists>m. 0 < m \<and> m < n - 1 \<and> [a ^ m = 1] (mod n)" (is "EX m. ?P m")
10.749 - from H0[unfolded nat_exists_least_iff[of ?P]] obtain m where
10.750 - m: "0 < m" "m < n - 1" "[a ^ m = 1] (mod n)" "\<forall>k <m. \<not>?P k" by blast
10.751 - {assume nm1: "(n - 1) mod m > 0"
10.752 - from mod_less_divisor[OF m(1)] have th0:"(n - 1) mod m < m" by blast
10.753 - let ?y = "a^ ((n - 1) div m * m)"
10.754 - note mdeq = mod_div_equality[of "(n - 1)" m]
10.755 - from coprime_exp[OF an(1)[unfolded coprime_commute[of a n]],
10.756 - of "(n - 1) div m * m"]
10.757 - have yn: "coprime ?y n" by (simp add: coprime_commute)
10.758 - have "?y mod n = (a^m)^((n - 1) div m) mod n"
10.759 - by (simp add: algebra_simps power_mult)
10.760 - also have "\<dots> = (a^m mod n)^((n - 1) div m) mod n"
10.761 - using power_mod[of "a^m" n "(n - 1) div m"] by simp
10.762 - also have "\<dots> = 1" using m(3)[unfolded modeq_def onen] onen
10.763 - by (simp add: power_Suc0)
10.764 - finally have th3: "?y mod n = 1" .
10.765 - have th2: "[?y * a ^ ((n - 1) mod m) = ?y* 1] (mod n)"
10.766 - using an1[unfolded modeq_def onen] onen
10.767 - mod_div_equality[of "(n - 1)" m, symmetric]
10.768 - by (simp add:power_add[symmetric] modeq_def th3 del: One_nat_def)
10.769 - from cong_mult_lcancel[of ?y n "a^((n - 1) mod m)" 1, OF yn th2]
10.770 - have th1: "[a ^ ((n - 1) mod m) = 1] (mod n)" .
10.771 - from m(4)[rule_format, OF th0] nm1
10.772 - less_trans[OF mod_less_divisor[OF m(1), of "n - 1"] m(2)] th1
10.773 - have False by blast }
10.774 - hence "(n - 1) mod m = 0" by auto
10.775 - then have mn: "m dvd n - 1" by presburger
10.776 - then obtain r where r: "n - 1 = m*r" unfolding dvd_def by blast
10.777 - from n01 r m(2) have r01: "r\<noteq>0" "r\<noteq>1" by - (rule ccontr, simp)+
10.778 - from prime_factor[OF r01(2)] obtain p where p: "prime p" "p dvd r" by blast
10.779 - hence th: "prime p \<and> p dvd n - 1" unfolding r by (auto intro: dvd_mult)
10.780 - have "(a ^ ((n - 1) div p)) mod n = (a^(m*r div p)) mod n" using r
10.781 - by (simp add: power_mult)
10.782 - 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
10.783 - also have "\<dots> = ((a^m)^(r div p)) mod n" by (simp add: power_mult)
10.784 - also have "\<dots> = ((a^m mod n)^(r div p)) mod n" using power_mod[of "a^m" "n" "r div p" ] ..
10.785 - also have "\<dots> = 1" using m(3) onen by (simp add: modeq_def power_Suc0)
10.786 - finally have "[(a ^ ((n - 1) div p))= 1] (mod n)"
10.787 - using onen by (simp add: modeq_def)
10.788 - with pn[rule_format, OF th] have False by blast}
10.789 - hence th: "\<forall>m. 0 < m \<and> m < n - 1 \<longrightarrow> \<not> [a ^ m = 1] (mod n)" by blast
10.790 - from lucas_weak[OF n2 an1 th] show ?thesis .
10.791 -qed
10.792 -
10.793 -(* Definition of the order of a number mod n (0 in non-coprime case). *)
10.794 -
10.795 -definition "ord n a = (if coprime n a then Least (\<lambda>d. d > 0 \<and> [a ^d = 1] (mod n)) else 0)"
10.796 -
10.797 -(* This has the expected properties. *)
10.798 -
10.799 -lemma coprime_ord:
10.800 - assumes na: "coprime n a"
10.801 - 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))"
10.802 -proof-
10.803 - let ?P = "\<lambda>d. 0 < d \<and> [a ^ d = 1] (mod n)"
10.804 - from euclid[of a] obtain p where p: "prime p" "a < p" by blast
10.805 - from na have o: "ord n a = Least ?P" by (simp add: ord_def)
10.806 - {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)}
10.807 - moreover
10.808 - {assume "n\<noteq>0 \<and> n\<noteq>1" hence n2:"n \<ge> 2" by arith
10.809 - from na have na': "coprime a n" by (simp add: coprime_commute)
10.810 - from phi_lowerbound_1[OF n2] fermat_little[OF na']
10.811 - have ex: "\<exists>m>0. ?P m" by - (rule exI[where x="\<phi> n"], auto) }
10.812 - ultimately have ex: "\<exists>m>0. ?P m" by blast
10.813 - from nat_exists_least_iff'[of ?P] ex na show ?thesis
10.814 - unfolding o[symmetric] by auto
10.815 -qed
10.816 -(* With the special value 0 for non-coprime case, it's more convenient. *)
10.817 -lemma ord_works:
10.818 - "[a ^ (ord n a) = 1] (mod n) \<and> (\<forall>m. 0 < m \<and> m < ord n a \<longrightarrow> ~[a^ m = 1] (mod n))"
10.819 -apply (cases "coprime n a")
10.820 -using coprime_ord[of n a]
10.821 -by (blast, simp add: ord_def modeq_def)
10.822 -
10.823 -lemma ord: "[a^(ord n a) = 1] (mod n)" using ord_works by blast
10.824 -lemma ord_minimal: "0 < m \<Longrightarrow> m < ord n a \<Longrightarrow> ~[a^m = 1] (mod n)"
10.825 - using ord_works by blast
10.826 -lemma ord_eq_0: "ord n a = 0 \<longleftrightarrow> ~coprime n a"
10.827 -by (cases "coprime n a", simp add: neq0_conv coprime_ord, simp add: neq0_conv ord_def)
10.828 -
10.829 -lemma ord_divides:
10.830 - "[a ^ d = 1] (mod n) \<longleftrightarrow> ord n a dvd d" (is "?lhs \<longleftrightarrow> ?rhs")
10.831 -proof
10.832 - assume rh: ?rhs
10.833 - then obtain k where "d = ord n a * k" unfolding dvd_def by blast
10.834 - hence "[a ^ d = (a ^ (ord n a) mod n)^k] (mod n)"
10.835 - by (simp add : modeq_def power_mult power_mod)
10.836 - also have "[(a ^ (ord n a) mod n)^k = 1] (mod n)"
10.837 - using ord[of a n, unfolded modeq_def]
10.838 - by (simp add: modeq_def power_mod power_Suc0)
10.839 - finally show ?lhs .
10.840 -next
10.841 - assume lh: ?lhs
10.842 - { assume H: "\<not> coprime n a"
10.843 - hence o: "ord n a = 0" by (simp add: ord_def)
10.844 - {assume d: "d=0" with o H have ?rhs by (simp add: modeq_def)}
10.845 - moreover
10.846 - {assume d0: "d\<noteq>0" then obtain d' where d': "d = Suc d'" by (cases d, auto)
10.847 - from H[unfolded coprime]
10.848 - obtain p where p: "p dvd n" "p dvd a" "p \<noteq> 1" by auto
10.849 - from lh[unfolded nat_mod]
10.850 - obtain q1 q2 where q12:"a ^ d + n * q1 = 1 + n * q2" by blast
10.851 - hence "a ^ d + n * q1 - n * q2 = 1" by simp
10.852 - with dvd_diff_nat [OF dvd_add [OF divides_rexp[OF p(2), of d'] dvd_mult2[OF p(1), of q1]] dvd_mult2[OF p(1), of q2]] d' have "p dvd 1" by simp
10.853 - with p(3) have False by simp
10.854 - hence ?rhs ..}
10.855 - ultimately have ?rhs by blast}
10.856 - moreover
10.857 - {assume H: "coprime n a"
10.858 - let ?o = "ord n a"
10.859 - let ?q = "d div ord n a"
10.860 - let ?r = "d mod ord n a"
10.861 - from cong_exp[OF ord[of a n], of ?q]
10.862 - have eqo: "[(a^?o)^?q = 1] (mod n)" by (simp add: modeq_def power_Suc0)
10.863 - from H have onz: "?o \<noteq> 0" by (simp add: ord_eq_0)
10.864 - hence op: "?o > 0" by simp
10.865 - from mod_div_equality[of d "ord n a"] lh
10.866 - have "[a^(?o*?q + ?r) = 1] (mod n)" by (simp add: modeq_def mult_commute)
10.867 - hence "[(a^?o)^?q * (a^?r) = 1] (mod n)"
10.868 - by (simp add: modeq_def power_mult[symmetric] power_add[symmetric])
10.869 - hence th: "[a^?r = 1] (mod n)"
10.870 - using eqo mod_mult_left_eq[of "(a^?o)^?q" "a^?r" n]
10.871 - apply (simp add: modeq_def del: One_nat_def)
10.872 - by (simp add: mod_mult_left_eq[symmetric])
10.873 - {assume r: "?r = 0" hence ?rhs by (simp add: dvd_eq_mod_eq_0)}
10.874 - moreover
10.875 - {assume r: "?r \<noteq> 0"
10.876 - with mod_less_divisor[OF op, of d] have r0o:"?r >0 \<and> ?r < ?o" by simp
10.877 - from conjunct2[OF ord_works[of a n], rule_format, OF r0o] th
10.878 - have ?rhs by blast}
10.879 - ultimately have ?rhs by blast}
10.880 - ultimately show ?rhs by blast
10.881 -qed
10.882 -
10.883 -lemma order_divides_phi: "coprime n a \<Longrightarrow> ord n a dvd \<phi> n"
10.884 -using ord_divides fermat_little coprime_commute by simp
10.885 -lemma order_divides_expdiff:
10.886 - assumes na: "coprime n a"
10.887 - shows "[a^d = a^e] (mod n) \<longleftrightarrow> [d = e] (mod (ord n a))"
10.888 -proof-
10.889 - {fix n a d e
10.890 - assume na: "coprime n a" and ed: "(e::nat) \<le> d"
10.891 - hence "\<exists>c. d = e + c" by arith
10.892 - then obtain c where c: "d = e + c" by arith
10.893 - from na have an: "coprime a n" by (simp add: coprime_commute)
10.894 - from coprime_exp[OF na, of e]
10.895 - have aen: "coprime (a^e) n" by (simp add: coprime_commute)
10.896 - from coprime_exp[OF na, of c]
10.897 - have acn: "coprime (a^c) n" by (simp add: coprime_commute)
10.898 - have "[a^d = a^e] (mod n) \<longleftrightarrow> [a^(e + c) = a^(e + 0)] (mod n)"
10.899 - using c by simp
10.900 - also have "\<dots> \<longleftrightarrow> [a^e* a^c = a^e *a^0] (mod n)" by (simp add: power_add)
10.901 - also have "\<dots> \<longleftrightarrow> [a ^ c = 1] (mod n)"
10.902 - using cong_mult_lcancel_eq[OF aen, of "a^c" "a^0"] by simp
10.903 - also have "\<dots> \<longleftrightarrow> ord n a dvd c" by (simp only: ord_divides)
10.904 - also have "\<dots> \<longleftrightarrow> [e + c = e + 0] (mod ord n a)"
10.905 - using cong_add_lcancel_eq[of e c 0 "ord n a", simplified cong_0_divides]
10.906 - by simp
10.907 - finally have "[a^d = a^e] (mod n) \<longleftrightarrow> [d = e] (mod (ord n a))"
10.908 - using c by simp }
10.909 - note th = this
10.910 - have "e \<le> d \<or> d \<le> e" by arith
10.911 - moreover
10.912 - {assume ed: "e \<le> d" from th[OF na ed] have ?thesis .}
10.913 - moreover
10.914 - {assume de: "d \<le> e"
10.915 - from th[OF na de] have ?thesis by (simp add: cong_commute) }
10.916 - ultimately show ?thesis by blast
10.917 -qed
10.918 -
10.919 -(* Another trivial primality characterization. *)
10.920 -
10.921 -lemma prime_prime_factor:
10.922 - "prime n \<longleftrightarrow> n \<noteq> 1\<and> (\<forall>p. prime p \<and> p dvd n \<longrightarrow> p = n)"
10.923 -proof-
10.924 - {assume n: "n=0 \<or> n=1" hence ?thesis using prime_0 two_is_prime by auto}
10.925 - moreover
10.926 - {assume n: "n\<noteq>0" "n\<noteq>1"
10.927 - {assume pn: "prime n"
10.928 -
10.929 - from pn[unfolded prime_def] have "\<forall>p. prime p \<and> p dvd n \<longrightarrow> p = n"
10.930 - using n
10.931 - apply (cases "n = 0 \<or> n=1",simp)
10.932 - by (clarsimp, erule_tac x="p" in allE, auto)}
10.933 - moreover
10.934 - {assume H: "\<forall>p. prime p \<and> p dvd n \<longrightarrow> p = n"
10.935 - from n have n1: "n > 1" by arith
10.936 - {fix m assume m: "m dvd n" "m\<noteq>1"
10.937 - from prime_factor[OF m(2)] obtain p where
10.938 - p: "prime p" "p dvd m" by blast
10.939 - from dvd_trans[OF p(2) m(1)] p(1) H have "p = n" by blast
10.940 - with p(2) have "n dvd m" by simp
10.941 - hence "m=n" using dvd_anti_sym[OF m(1)] by simp }
10.942 - with n1 have "prime n" unfolding prime_def by auto }
10.943 - ultimately have ?thesis using n by blast}
10.944 - ultimately show ?thesis by auto
10.945 -qed
10.946 -
10.947 -lemma prime_divisor_sqrt:
10.948 - "prime n \<longleftrightarrow> n \<noteq> 1 \<and> (\<forall>d. d dvd n \<and> d^2 \<le> n \<longrightarrow> d = 1)"
10.949 -proof-
10.950 - {assume "n=0 \<or> n=1" hence ?thesis using prime_0 prime_1
10.951 - by (auto simp add: nat_power_eq_0_iff)}
10.952 - moreover
10.953 - {assume n: "n\<noteq>0" "n\<noteq>1"
10.954 - hence np: "n > 1" by arith
10.955 - {fix d assume d: "d dvd n" "d^2 \<le> n" and H: "\<forall>m. m dvd n \<longrightarrow> m=1 \<or> m=n"
10.956 - from H d have d1n: "d = 1 \<or> d=n" by blast
10.957 - {assume dn: "d=n"
10.958 - have "n^2 > n*1" using n
10.959 - by (simp add: power2_eq_square mult_less_cancel1)
10.960 - with dn d(2) have "d=1" by simp}
10.961 - with d1n have "d = 1" by blast }
10.962 - moreover
10.963 - {fix d assume d: "d dvd n" and H: "\<forall>d'. d' dvd n \<and> d'^2 \<le> n \<longrightarrow> d' = 1"
10.964 - from d n have "d \<noteq> 0" apply - apply (rule ccontr) by simp
10.965 - hence dp: "d > 0" by simp
10.966 - from d[unfolded dvd_def] obtain e where e: "n= d*e" by blast
10.967 - from n dp e have ep:"e > 0" by simp
10.968 - have "d^2 \<le> n \<or> e^2 \<le> n" using dp ep
10.969 - by (auto simp add: e power2_eq_square mult_le_cancel_left)
10.970 - moreover
10.971 - {assume h: "d^2 \<le> n"
10.972 - from H[rule_format, of d] h d have "d = 1" by blast}
10.973 - moreover
10.974 - {assume h: "e^2 \<le> n"
10.975 - from e have "e dvd n" unfolding dvd_def by (simp add: mult_commute)
10.976 - with H[rule_format, of e] h have "e=1" by simp
10.977 - with e have "d = n" by simp}
10.978 - ultimately have "d=1 \<or> d=n" by blast}
10.979 - ultimately have ?thesis unfolding prime_def using np n(2) by blast}
10.980 - ultimately show ?thesis by auto
10.981 -qed
10.982 -lemma prime_prime_factor_sqrt:
10.983 - "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)"
10.984 - (is "?lhs \<longleftrightarrow>?rhs")
10.985 -proof-
10.986 - {assume "n=0 \<or> n=1" hence ?thesis using prime_0 prime_1 by auto}
10.987 - moreover
10.988 - {assume n: "n\<noteq>0" "n\<noteq>1"
10.989 - {assume H: ?lhs
10.990 - from H[unfolded prime_divisor_sqrt] n
10.991 - have ?rhs apply clarsimp by (erule_tac x="p" in allE, simp add: prime_1)
10.992 - }
10.993 - moreover
10.994 - {assume H: ?rhs
10.995 - {fix d assume d: "d dvd n" "d^2 \<le> n" "d\<noteq>1"
10.996 - from prime_factor[OF d(3)]
10.997 - obtain p where p: "prime p" "p dvd d" by blast
10.998 - from n have np: "n > 0" by arith
10.999 - from d(1) n have "d \<noteq> 0" by - (rule ccontr, auto)
10.1000 - hence dp: "d > 0" by arith
10.1001 - from mult_mono[OF dvd_imp_le[OF p(2) dp] dvd_imp_le[OF p(2) dp]] d(2)
10.1002 - have "p^2 \<le> n" unfolding power2_eq_square by arith
10.1003 - with H n p(1) dvd_trans[OF p(2) d(1)] have False by blast}
10.1004 - with n prime_divisor_sqrt have ?lhs by auto}
10.1005 - ultimately have ?thesis by blast }
10.1006 - ultimately show ?thesis by (cases "n=0 \<or> n=1", auto)
10.1007 -qed
10.1008 -(* Pocklington theorem. *)
10.1009 -
10.1010 -lemma pocklington_lemma:
10.1011 - assumes n: "n \<ge> 2" and nqr: "n - 1 = q*r" and an: "[a^ (n - 1) = 1] (mod n)"
10.1012 - and aq:"\<forall>p. prime p \<and> p dvd q \<longrightarrow> coprime (a^ ((n - 1) div p) - 1) n"
10.1013 - and pp: "prime p" and pn: "p dvd n"
10.1014 - shows "[p = 1] (mod q)"
10.1015 -proof-
10.1016 - from pp prime_0 prime_1 have p01: "p \<noteq> 0" "p \<noteq> 1" by - (rule ccontr, simp)+
10.1017 - from cong_1_divides[OF an, unfolded nqr, unfolded dvd_def]
10.1018 - obtain k where k: "a ^ (q * r) - 1 = n*k" by blast
10.1019 - from pn[unfolded dvd_def] obtain l where l: "n = p*l" by blast
10.1020 - {assume a0: "a = 0"
10.1021 - hence "a^ (n - 1) = 0" using n by (simp add: power_0_left)
10.1022 - with n an mod_less[of 1 n] have False by (simp add: power_0_left modeq_def)}
10.1023 - hence a0: "a\<noteq>0" ..
10.1024 - from n nqr have aqr0: "a ^ (q * r) \<noteq> 0" using a0 by (simp add: neq0_conv)
10.1025 - hence "(a ^ (q * r) - 1) + 1 = a ^ (q * r)" by simp
10.1026 - with k l have "a ^ (q * r) = p*l*k + 1" by simp
10.1027 - hence "a ^ (r * q) + p * 0 = 1 + p * (l*k)" by (simp add: mult_ac)
10.1028 - hence odq: "ord p (a^r) dvd q"
10.1029 - unfolding ord_divides[symmetric] power_mult[symmetric] nat_mod by blast
10.1030 - from odq[unfolded dvd_def] obtain d where d: "q = ord p (a^r) * d" by blast
10.1031 - {assume d1: "d \<noteq> 1"
10.1032 - from prime_factor[OF d1] obtain P where P: "prime P" "P dvd d" by blast
10.1033 - from d dvd_mult[OF P(2), of "ord p (a^r)"] have Pq: "P dvd q" by simp
10.1034 - from aq P(1) Pq have caP:"coprime (a^ ((n - 1) div P) - 1) n" by blast
10.1035 - from Pq obtain s where s: "q = P*s" unfolding dvd_def by blast
10.1036 - have P0: "P \<noteq> 0" using P(1) prime_0 by - (rule ccontr, simp)
10.1037 - from P(2) obtain t where t: "d = P*t" unfolding dvd_def by blast
10.1038 - from d s t P0 have s': "ord p (a^r) * t = s" by algebra
10.1039 - have "ord p (a^r) * t*r = r * ord p (a^r) * t" by algebra
10.1040 - hence exps: "a^(ord p (a^r) * t*r) = ((a ^ r) ^ ord p (a^r)) ^ t"
10.1041 - by (simp only: power_mult)
10.1042 - have "[((a ^ r) ^ ord p (a^r)) ^ t= 1^t] (mod p)"
10.1043 - by (rule cong_exp, rule ord)
10.1044 - then have th: "[((a ^ r) ^ ord p (a^r)) ^ t= 1] (mod p)"
10.1045 - by (simp add: power_Suc0)
10.1046 - from cong_1_divides[OF th] exps have pd0: "p dvd a^(ord p (a^r) * t*r) - 1" by simp
10.1047 - from nqr s s' have "(n - 1) div P = ord p (a^r) * t*r" using P0 by simp
10.1048 - with caP have "coprime (a^(ord p (a^r) * t*r) - 1) n" by simp
10.1049 - with p01 pn pd0 have False unfolding coprime by auto}
10.1050 - hence d1: "d = 1" by blast
10.1051 - hence o: "ord p (a^r) = q" using d by simp
10.1052 - from pp phi_prime[of p] have phip: " \<phi> p = p - 1" by simp
10.1053 - {fix d assume d: "d dvd p" "d dvd a" "d \<noteq> 1"
10.1054 - from pp[unfolded prime_def] d have dp: "d = p" by blast
10.1055 - from n have n12:"Suc (n - 2) = n - 1" by arith
10.1056 - with divides_rexp[OF d(2)[unfolded dp], of "n - 2"]
10.1057 - have th0: "p dvd a ^ (n - 1)" by simp
10.1058 - from n have n0: "n \<noteq> 0" by simp
10.1059 - from d(2) an n12[symmetric] have a0: "a \<noteq> 0"
10.1060 - by - (rule ccontr, simp add: modeq_def)
10.1061 - have th1: "a^ (n - 1) \<noteq> 0" using n d(2) dp a0 by (auto simp add: neq0_conv)
10.1062 - from coprime_minus1[OF th1, unfolded coprime]
10.1063 - dvd_trans[OF pn cong_1_divides[OF an]] th0 d(3) dp
10.1064 - have False by auto}
10.1065 - hence cpa: "coprime p a" using coprime by auto
10.1066 - from coprime_exp[OF cpa, of r] coprime_commute
10.1067 - have arp: "coprime (a^r) p" by blast
10.1068 - from fermat_little[OF arp, simplified ord_divides] o phip
10.1069 - have "q dvd (p - 1)" by simp
10.1070 - then obtain d where d:"p - 1 = q * d" unfolding dvd_def by blast
10.1071 - from prime_0 pp have p0:"p \<noteq> 0" by - (rule ccontr, auto)
10.1072 - from p0 d have "p + q * 0 = 1 + q * d" by simp
10.1073 - with nat_mod[of p 1 q, symmetric]
10.1074 - show ?thesis by blast
10.1075 -qed
10.1076 -
10.1077 -lemma pocklington:
10.1078 - assumes n: "n \<ge> 2" and nqr: "n - 1 = q*r" and sqr: "n \<le> q^2"
10.1079 - and an: "[a^ (n - 1) = 1] (mod n)"
10.1080 - and aq:"\<forall>p. prime p \<and> p dvd q \<longrightarrow> coprime (a^ ((n - 1) div p) - 1) n"
10.1081 - shows "prime n"
10.1082 -unfolding prime_prime_factor_sqrt[of n]
10.1083 -proof-
10.1084 - let ?ths = "n \<noteq> 0 \<and> n \<noteq> 1 \<and> \<not> (\<exists>p. prime p \<and> p dvd n \<and> p\<twosuperior> \<le> n)"
10.1085 - from n have n01: "n\<noteq>0" "n\<noteq>1" by arith+
10.1086 - {fix p assume p: "prime p" "p dvd n" "p^2 \<le> n"
10.1087 - from p(3) sqr have "p^(Suc 1) \<le> q^(Suc 1)" by (simp add: power2_eq_square)
10.1088 - hence pq: "p \<le> q" unfolding exp_mono_le .
10.1089 - from pocklington_lemma[OF n nqr an aq p(1,2)] cong_1_divides
10.1090 - have th: "q dvd p - 1" by blast
10.1091 - have "p - 1 \<noteq> 0"using prime_ge_2[OF p(1)] by arith
10.1092 - with divides_ge[OF th] pq have False by arith }
10.1093 - with n01 show ?ths by blast
10.1094 -qed
10.1095 -
10.1096 -(* Variant for application, to separate the exponentiation. *)
10.1097 -lemma pocklington_alt:
10.1098 - assumes n: "n \<ge> 2" and nqr: "n - 1 = q*r" and sqr: "n \<le> q^2"
10.1099 - and an: "[a^ (n - 1) = 1] (mod n)"
10.1100 - 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)"
10.1101 - shows "prime n"
10.1102 -proof-
10.1103 - {fix p assume p: "prime p" "p dvd q"
10.1104 - from aq[rule_format] p obtain b where
10.1105 - b: "[a^((n - 1) div p) = b] (mod n)" "coprime (b - 1) n" by blast
10.1106 - {assume a0: "a=0"
10.1107 - from n an have "[0 = 1] (mod n)" unfolding a0 power_0_left by auto
10.1108 - hence False using n by (simp add: modeq_def dvd_eq_mod_eq_0[symmetric])}
10.1109 - hence a0: "a\<noteq> 0" ..
10.1110 - hence a1: "a \<ge> 1" by arith
10.1111 - from one_le_power[OF a1] have ath: "1 \<le> a ^ ((n - 1) div p)" .
10.1112 - {assume b0: "b = 0"
10.1113 - from p(2) nqr have "(n - 1) mod p = 0"
10.1114 - apply (simp only: dvd_eq_mod_eq_0[symmetric]) by (rule dvd_mult2, simp)
10.1115 - with mod_div_equality[of "n - 1" p]
10.1116 - have "(n - 1) div p * p= n - 1" by auto
10.1117 - hence eq: "(a^((n - 1) div p))^p = a^(n - 1)"
10.1118 - by (simp only: power_mult[symmetric])
10.1119 - from prime_ge_2[OF p(1)] have pS: "Suc (p - 1) = p" by arith
10.1120 - from b(1) have d: "n dvd a^((n - 1) div p)" unfolding b0 cong_0_divides .
10.1121 - from divides_rexp[OF d, of "p - 1"] pS eq cong_divides[OF an] n
10.1122 - have False by simp}
10.1123 - then have b0: "b \<noteq> 0" ..
10.1124 - hence b1: "b \<ge> 1" by arith
10.1125 - from cong_coprime[OF cong_sub[OF b(1) cong_refl[of 1] ath b1]] b(2) nqr
10.1126 - have "coprime (a ^ ((n - 1) div p) - 1) n" by (simp add: coprime_commute)}
10.1127 - hence th: "\<forall>p. prime p \<and> p dvd q \<longrightarrow> coprime (a ^ ((n - 1) div p) - 1) n "
10.1128 - by blast
10.1129 - from pocklington[OF n nqr sqr an th] show ?thesis .
10.1130 -qed
10.1131 -
10.1132 -(* Prime factorizations. *)
10.1133 -
10.1134 -definition "primefact ps n = (foldr op * ps 1 = n \<and> (\<forall>p\<in> set ps. prime p))"
10.1135 -
10.1136 -lemma primefact: assumes n: "n \<noteq> 0"
10.1137 - shows "\<exists>ps. primefact ps n"
10.1138 -using n
10.1139 -proof(induct n rule: nat_less_induct)
10.1140 - fix n assume H: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>ps. primefact ps m)" and n: "n\<noteq>0"
10.1141 - let ?ths = "\<exists>ps. primefact ps n"
10.1142 - {assume "n = 1"
10.1143 - hence "primefact [] n" by (simp add: primefact_def)
10.1144 - hence ?ths by blast }
10.1145 - moreover
10.1146 - {assume n1: "n \<noteq> 1"
10.1147 - with n have n2: "n \<ge> 2" by arith
10.1148 - from prime_factor[OF n1] obtain p where p: "prime p" "p dvd n" by blast
10.1149 - from p(2) obtain m where m: "n = p*m" unfolding dvd_def by blast
10.1150 - from n m have m0: "m > 0" "m\<noteq>0" by auto
10.1151 - from prime_ge_2[OF p(1)] have "1 < p" by arith
10.1152 - with m0 m have mn: "m < n" by auto
10.1153 - from H[rule_format, OF mn m0(2)] obtain ps where ps: "primefact ps m" ..
10.1154 - from ps m p(1) have "primefact (p#ps) n" by (simp add: primefact_def)
10.1155 - hence ?ths by blast}
10.1156 - ultimately show ?ths by blast
10.1157 -qed
10.1158 -
10.1159 -lemma primefact_contains:
10.1160 - assumes pf: "primefact ps n" and p: "prime p" and pn: "p dvd n"
10.1161 - shows "p \<in> set ps"
10.1162 - using pf p pn
10.1163 -proof(induct ps arbitrary: p n)
10.1164 - case Nil thus ?case by (auto simp add: primefact_def)
10.1165 -next
10.1166 - case (Cons q qs p n)
10.1167 - from Cons.prems[unfolded primefact_def]
10.1168 - 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
10.1169 - {assume "p dvd q"
10.1170 - with p(1) q(1) have "p = q" unfolding prime_def by auto
10.1171 - hence ?case by simp}
10.1172 - moreover
10.1173 - { assume h: "p dvd foldr op * qs 1"
10.1174 - from q(3) have pqs: "primefact qs (foldr op * qs 1)"
10.1175 - by (simp add: primefact_def)
10.1176 - from Cons.hyps[OF pqs p(1) h] have ?case by simp}
10.1177 - ultimately show ?case using prime_divprod[OF p] by blast
10.1178 -qed
10.1179 -
10.1180 -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)
10.1181 -
10.1182 -(* Variant of Lucas theorem. *)
10.1183 -
10.1184 -lemma lucas_primefact:
10.1185 - assumes n: "n \<ge> 2" and an: "[a^(n - 1) = 1] (mod n)"
10.1186 - and psn: "foldr op * ps 1 = n - 1"
10.1187 - and psp: "list_all (\<lambda>p. prime p \<and> \<not> [a^((n - 1) div p) = 1] (mod n)) ps"
10.1188 - shows "prime n"
10.1189 -proof-
10.1190 - {fix p assume p: "prime p" "p dvd n - 1" "[a ^ ((n - 1) div p) = 1] (mod n)"
10.1191 - from psn psp have psn1: "primefact ps (n - 1)"
10.1192 - by (auto simp add: list_all_iff primefact_variant)
10.1193 - from p(3) primefact_contains[OF psn1 p(1,2)] psp
10.1194 - have False by (induct ps, auto)}
10.1195 - with lucas[OF n an] show ?thesis by blast
10.1196 -qed
10.1197 -
10.1198 -(* Variant of Pocklington theorem. *)
10.1199 -
10.1200 -lemma mod_le: assumes n: "n \<noteq> (0::nat)" shows "m mod n \<le> m"
10.1201 -proof-
10.1202 - from mod_div_equality[of m n]
10.1203 - have "\<exists>x. x + m mod n = m" by blast
10.1204 - then show ?thesis by auto
10.1205 -qed
10.1206 -
10.1207 -
10.1208 -lemma pocklington_primefact:
10.1209 - assumes n: "n \<ge> 2" and qrn: "q*r = n - 1" and nq2: "n \<le> q^2"
10.1210 - and arnb: "(a^r) mod n = b" and psq: "foldr op * ps 1 = q"
10.1211 - and bqn: "(b^q) mod n = 1"
10.1212 - and psp: "list_all (\<lambda>p. prime p \<and> coprime ((b^(q div p)) mod n - 1) n) ps"
10.1213 - shows "prime n"
10.1214 -proof-
10.1215 - from bqn psp qrn
10.1216 - have bqn: "a ^ (n - 1) mod n = 1"
10.1217 - 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
10.1218 - by (simp_all add: power_mult[symmetric] algebra_simps)
10.1219 - from n have n0: "n > 0" by arith
10.1220 - from mod_div_equality[of "a^(n - 1)" n]
10.1221 - mod_less_divisor[OF n0, of "a^(n - 1)"]
10.1222 - have an1: "[a ^ (n - 1) = 1] (mod n)"
10.1223 - unfolding nat_mod bqn
10.1224 - apply -
10.1225 - apply (rule exI[where x="0"])
10.1226 - apply (rule exI[where x="a^(n - 1) div n"])
10.1227 - by (simp add: algebra_simps)
10.1228 - {fix p assume p: "prime p" "p dvd q"
10.1229 - from psp psq have pfpsq: "primefact ps q"
10.1230 - by (auto simp add: primefact_variant list_all_iff)
10.1231 - from psp primefact_contains[OF pfpsq p]
10.1232 - have p': "coprime (a ^ (r * (q div p)) mod n - 1) n"
10.1233 - by (simp add: list_all_iff)
10.1234 - from prime_ge_2[OF p(1)] have p01: "p \<noteq> 0" "p \<noteq> 1" "p =Suc(p - 1)" by arith+
10.1235 - from div_mult1_eq[of r q p] p(2)
10.1236 - have eq1: "r* (q div p) = (n - 1) div p"
10.1237 - unfolding qrn[symmetric] dvd_eq_mod_eq_0 by (simp add: mult_commute)
10.1238 - have ath: "\<And>a (b::nat). a <= b \<Longrightarrow> a \<noteq> 0 ==> 1 <= a \<and> 1 <= b" by arith
10.1239 - from n0 have n00: "n \<noteq> 0" by arith
10.1240 - from mod_le[OF n00]
10.1241 - have th10: "a ^ ((n - 1) div p) mod n \<le> a ^ ((n - 1) div p)" .
10.1242 - {assume "a ^ ((n - 1) div p) mod n = 0"
10.1243 - then obtain s where s: "a ^ ((n - 1) div p) = n*s"
10.1244 - unfolding mod_eq_0_iff by blast
10.1245 - hence eq0: "(a^((n - 1) div p))^p = (n*s)^p" by simp
10.1246 - from qrn[symmetric] have qn1: "q dvd n - 1" unfolding dvd_def by auto
10.1247 - from dvd_trans[OF p(2) qn1] div_mod_equality'[of "n - 1" p]
10.1248 - have npp: "(n - 1) div p * p = n - 1" by (simp add: dvd_eq_mod_eq_0)
10.1249 - with eq0 have "a^ (n - 1) = (n*s)^p"
10.1250 - by (simp add: power_mult[symmetric])
10.1251 - hence "1 = (n*s)^(Suc (p - 1)) mod n" using bqn p01 by simp
10.1252 - also have "\<dots> = 0" by (simp add: mult_assoc)
10.1253 - finally have False by simp }
10.1254 - then have th11: "a ^ ((n - 1) div p) mod n \<noteq> 0" by auto
10.1255 - have th1: "[a ^ ((n - 1) div p) mod n = a ^ ((n - 1) div p)] (mod n)"
10.1256 - unfolding modeq_def by simp
10.1257 - from cong_sub[OF th1 cong_refl[of 1]] ath[OF th10 th11]
10.1258 - have th: "[a ^ ((n - 1) div p) mod n - 1 = a ^ ((n - 1) div p) - 1] (mod n)"
10.1259 - by blast
10.1260 - from cong_coprime[OF th] p'[unfolded eq1]
10.1261 - have "coprime (a ^ ((n - 1) div p) - 1) n" by (simp add: coprime_commute) }
10.1262 - with pocklington[OF n qrn[symmetric] nq2 an1]
10.1263 - show ?thesis by blast
10.1264 -qed
10.1265 -
10.1266 -end
11.1 --- a/src/HOL/Library/Primes.thy Tue Sep 01 14:13:34 2009 +0200
11.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
11.3 @@ -1,828 +0,0 @@
11.4 -(* Title: HOL/Library/Primes.thy
11.5 - Author: Amine Chaieb, Christophe Tabacznyj and Lawrence C Paulson
11.6 - Copyright 1996 University of Cambridge
11.7 -*)
11.8 -
11.9 -header {* Primality on nat *}
11.10 -
11.11 -theory Primes
11.12 -imports Complex_Main Legacy_GCD
11.13 -begin
11.14 -
11.15 -hide (open) const GCD.gcd GCD.coprime GCD.prime
11.16 -
11.17 -definition
11.18 - coprime :: "nat => nat => bool" where
11.19 - "coprime m n \<longleftrightarrow> gcd m n = 1"
11.20 -
11.21 -definition
11.22 - prime :: "nat \<Rightarrow> bool" where
11.23 - [code del]: "prime p \<longleftrightarrow> (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
11.24 -
11.25 -
11.26 -lemma two_is_prime: "prime 2"
11.27 - apply (auto simp add: prime_def)
11.28 - apply (case_tac m)
11.29 - apply (auto dest!: dvd_imp_le)
11.30 - done
11.31 -
11.32 -lemma prime_imp_relprime: "prime p ==> \<not> p dvd n ==> gcd p n = 1"
11.33 - apply (auto simp add: prime_def)
11.34 - apply (metis One_nat_def gcd_dvd1 gcd_dvd2)
11.35 - done
11.36 -
11.37 -text {*
11.38 - This theorem leads immediately to a proof of the uniqueness of
11.39 - factorization. If @{term p} divides a product of primes then it is
11.40 - one of those primes.
11.41 -*}
11.42 -
11.43 -lemma prime_dvd_mult: "prime p ==> p dvd m * n ==> p dvd m \<or> p dvd n"
11.44 - by (blast intro: relprime_dvd_mult prime_imp_relprime)
11.45 -
11.46 -lemma prime_dvd_square: "prime p ==> p dvd m^Suc (Suc 0) ==> p dvd m"
11.47 - by (auto dest: prime_dvd_mult)
11.48 -
11.49 -lemma prime_dvd_power_two: "prime p ==> p dvd m\<twosuperior> ==> p dvd m"
11.50 - by (rule prime_dvd_square) (simp_all add: power2_eq_square)
11.51 -
11.52 -
11.53 -lemma exp_eq_1:"(x::nat)^n = 1 \<longleftrightarrow> x = 1 \<or> n = 0"
11.54 -by (induct n, auto)
11.55 -
11.56 -lemma exp_mono_lt: "(x::nat) ^ (Suc n) < y ^ (Suc n) \<longleftrightarrow> x < y"
11.57 -by(metis linorder_not_less not_less0 power_le_imp_le_base power_less_imp_less_base)
11.58 -
11.59 -lemma exp_mono_le: "(x::nat) ^ (Suc n) \<le> y ^ (Suc n) \<longleftrightarrow> x \<le> y"
11.60 -by (simp only: linorder_not_less[symmetric] exp_mono_lt)
11.61 -
11.62 -lemma exp_mono_eq: "(x::nat) ^ Suc n = y ^ Suc n \<longleftrightarrow> x = y"
11.63 -using power_inject_base[of x n y] by auto
11.64 -
11.65 -
11.66 -lemma even_square: assumes e: "even (n::nat)" shows "\<exists>x. n ^ 2 = 4*x"
11.67 -proof-
11.68 - from e have "2 dvd n" by presburger
11.69 - then obtain k where k: "n = 2*k" using dvd_def by auto
11.70 - hence "n^2 = 4* (k^2)" by (simp add: power2_eq_square)
11.71 - thus ?thesis by blast
11.72 -qed
11.73 -
11.74 -lemma odd_square: assumes e: "odd (n::nat)" shows "\<exists>x. n ^ 2 = 4*x + 1"
11.75 -proof-
11.76 - from e have np: "n > 0" by presburger
11.77 - from e have "2 dvd (n - 1)" by presburger
11.78 - then obtain k where "n - 1 = 2*k" using dvd_def by auto
11.79 - hence k: "n = 2*k + 1" using e by presburger
11.80 - hence "n^2 = 4* (k^2 + k) + 1" by algebra
11.81 - thus ?thesis by blast
11.82 -qed
11.83 -
11.84 -lemma diff_square: "(x::nat)^2 - y^2 = (x+y)*(x - y)"
11.85 -proof-
11.86 - have "x \<le> y \<or> y \<le> x" by (rule nat_le_linear)
11.87 - moreover
11.88 - {assume le: "x \<le> y"
11.89 - hence "x ^2 \<le> y^2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def)
11.90 - with le have ?thesis by simp }
11.91 - moreover
11.92 - {assume le: "y \<le> x"
11.93 - hence le2: "y ^2 \<le> x^2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def)
11.94 - from le have "\<exists>z. y + z = x" by presburger
11.95 - then obtain z where z: "x = y + z" by blast
11.96 - from le2 have "\<exists>z. x^2 = y^2 + z" by presburger
11.97 - then obtain z2 where z2: "x^2 = y^2 + z2" by blast
11.98 - from z z2 have ?thesis apply simp by algebra }
11.99 - ultimately show ?thesis by blast
11.100 -qed
11.101 -
11.102 -text {* Elementary theory of divisibility *}
11.103 -lemma divides_ge: "(a::nat) dvd b \<Longrightarrow> b = 0 \<or> a \<le> b" unfolding dvd_def by auto
11.104 -lemma divides_antisym: "(x::nat) dvd y \<and> y dvd x \<longleftrightarrow> x = y"
11.105 - using dvd_anti_sym[of x y] by auto
11.106 -
11.107 -lemma divides_add_revr: assumes da: "(d::nat) dvd a" and dab:"d dvd (a + b)"
11.108 - shows "d dvd b"
11.109 -proof-
11.110 - from da obtain k where k:"a = d*k" by (auto simp add: dvd_def)
11.111 - from dab obtain k' where k': "a + b = d*k'" by (auto simp add: dvd_def)
11.112 - from k k' have "b = d *(k' - k)" by (simp add : diff_mult_distrib2)
11.113 - thus ?thesis unfolding dvd_def by blast
11.114 -qed
11.115 -
11.116 -declare nat_mult_dvd_cancel_disj[presburger]
11.117 -lemma nat_mult_dvd_cancel_disj'[presburger]:
11.118 - "(m\<Colon>nat)*k dvd n*k \<longleftrightarrow> k = 0 \<or> m dvd n" unfolding mult_commute[of m k] mult_commute[of n k] by presburger
11.119 -
11.120 -lemma divides_mul_l: "(a::nat) dvd b ==> (c * a) dvd (c * b)"
11.121 - by presburger
11.122 -
11.123 -lemma divides_mul_r: "(a::nat) dvd b ==> (a * c) dvd (b * c)" by presburger
11.124 -lemma divides_cases: "(n::nat) dvd m ==> m = 0 \<or> m = n \<or> 2 * n <= m"
11.125 - by (auto simp add: dvd_def)
11.126 -
11.127 -lemma divides_div_not: "(x::nat) = (q * n) + r \<Longrightarrow> 0 < r \<Longrightarrow> r < n ==> ~(n dvd x)"
11.128 -proof(auto simp add: dvd_def)
11.129 - fix k assume H: "0 < r" "r < n" "q * n + r = n * k"
11.130 - from H(3) have r: "r = n* (k -q)" by(simp add: diff_mult_distrib2 mult_commute)
11.131 - {assume "k - q = 0" with r H(1) have False by simp}
11.132 - moreover
11.133 - {assume "k - q \<noteq> 0" with r have "r \<ge> n" by auto
11.134 - with H(2) have False by simp}
11.135 - ultimately show False by blast
11.136 -qed
11.137 -lemma divides_exp: "(x::nat) dvd y ==> x ^ n dvd y ^ n"
11.138 - by (auto simp add: power_mult_distrib dvd_def)
11.139 -
11.140 -lemma divides_exp2: "n \<noteq> 0 \<Longrightarrow> (x::nat) ^ n dvd y \<Longrightarrow> x dvd y"
11.141 - by (induct n ,auto simp add: dvd_def)
11.142 -
11.143 -fun fact :: "nat \<Rightarrow> nat" where
11.144 - "fact 0 = 1"
11.145 -| "fact (Suc n) = Suc n * fact n"
11.146 -
11.147 -lemma fact_lt: "0 < fact n" by(induct n, simp_all)
11.148 -lemma fact_le: "fact n \<ge> 1" using fact_lt[of n] by simp
11.149 -lemma fact_mono: assumes le: "m \<le> n" shows "fact m \<le> fact n"
11.150 -proof-
11.151 - from le have "\<exists>i. n = m+i" by presburger
11.152 - then obtain i where i: "n = m+i" by blast
11.153 - have "fact m \<le> fact (m + i)"
11.154 - proof(induct m)
11.155 - case 0 thus ?case using fact_le[of i] by simp
11.156 - next
11.157 - case (Suc m)
11.158 - have "fact (Suc m) = Suc m * fact m" by simp
11.159 - have th1: "Suc m \<le> Suc (m + i)" by simp
11.160 - from mult_le_mono[of "Suc m" "Suc (m+i)" "fact m" "fact (m+i)", OF th1 Suc.hyps]
11.161 - show ?case by simp
11.162 - qed
11.163 - thus ?thesis using i by simp
11.164 -qed
11.165 -
11.166 -lemma divides_fact: "1 <= p \<Longrightarrow> p <= n ==> p dvd fact n"
11.167 -proof(induct n arbitrary: p)
11.168 - case 0 thus ?case by simp
11.169 -next
11.170 - case (Suc n p)
11.171 - from Suc.prems have "p = Suc n \<or> p \<le> n" by presburger
11.172 - moreover
11.173 - {assume "p = Suc n" hence ?case by (simp only: fact.simps dvd_triv_left)}
11.174 - moreover
11.175 - {assume "p \<le> n"
11.176 - with Suc.prems(1) Suc.hyps have th: "p dvd fact n" by simp
11.177 - from dvd_mult[OF th] have ?case by (simp only: fact.simps) }
11.178 - ultimately show ?case by blast
11.179 -qed
11.180 -
11.181 -declare dvd_triv_left[presburger]
11.182 -declare dvd_triv_right[presburger]
11.183 -lemma divides_rexp:
11.184 - "x dvd y \<Longrightarrow> (x::nat) dvd (y^(Suc n))" by (simp add: dvd_mult2[of x y])
11.185 -
11.186 -text {* Coprimality *}
11.187 -
11.188 -lemma coprime: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
11.189 -using gcd_unique[of 1 a b, simplified] by (auto simp add: coprime_def)
11.190 -lemma coprime_commute: "coprime a b \<longleftrightarrow> coprime b a" by (simp add: coprime_def gcd_commute)
11.191 -
11.192 -lemma coprime_bezout: "coprime a b \<longleftrightarrow> (\<exists>x y. a * x - b * y = 1 \<or> b * x - a * y = 1)"
11.193 -using coprime_def gcd_bezout by auto
11.194 -
11.195 -lemma coprime_divprod: "d dvd a * b \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
11.196 - using relprime_dvd_mult_iff[of d a b] by (auto simp add: coprime_def mult_commute)
11.197 -
11.198 -lemma coprime_1[simp]: "coprime a 1" by (simp add: coprime_def)
11.199 -lemma coprime_1'[simp]: "coprime 1 a" by (simp add: coprime_def)
11.200 -lemma coprime_Suc0[simp]: "coprime a (Suc 0)" by (simp add: coprime_def)
11.201 -lemma coprime_Suc0'[simp]: "coprime (Suc 0) a" by (simp add: coprime_def)
11.202 -
11.203 -lemma gcd_coprime:
11.204 - assumes z: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
11.205 - shows "coprime a' b'"
11.206 -proof-
11.207 - let ?g = "gcd a b"
11.208 - {assume bz: "a = 0" from b bz z a have ?thesis by (simp add: gcd_zero coprime_def)}
11.209 - moreover
11.210 - {assume az: "a\<noteq> 0"
11.211 - from z have z': "?g > 0" by simp
11.212 - from bezout_gcd_strong[OF az, of b]
11.213 - obtain x y where xy: "a*x = b*y + ?g" by blast
11.214 - from xy a b have "?g * a'*x = ?g * (b'*y + 1)" by (simp add: algebra_simps)
11.215 - hence "?g * (a'*x) = ?g * (b'*y + 1)" by (simp add: mult_assoc)
11.216 - hence "a'*x = (b'*y + 1)"
11.217 - by (simp only: nat_mult_eq_cancel1[OF z'])
11.218 - hence "a'*x - b'*y = 1" by simp
11.219 - with coprime_bezout[of a' b'] have ?thesis by auto}
11.220 - ultimately show ?thesis by blast
11.221 -qed
11.222 -lemma coprime_0: "coprime d 0 \<longleftrightarrow> d = 1" by (simp add: coprime_def)
11.223 -lemma coprime_mul: assumes da: "coprime d a" and db: "coprime d b"
11.224 - shows "coprime d (a * b)"
11.225 -proof-
11.226 - from da have th: "gcd a d = 1" by (simp add: coprime_def gcd_commute)
11.227 - from gcd_mult_cancel[of a d b, OF th] db[unfolded coprime_def] have "gcd d (a*b) = 1"
11.228 - by (simp add: gcd_commute)
11.229 - thus ?thesis unfolding coprime_def .
11.230 -qed
11.231 -lemma coprime_lmul2: assumes dab: "coprime d (a * b)" shows "coprime d b"
11.232 -using prems unfolding coprime_bezout
11.233 -apply clarsimp
11.234 -apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all)
11.235 -apply (rule_tac x="x" in exI)
11.236 -apply (rule_tac x="a*y" in exI)
11.237 -apply (simp add: mult_ac)
11.238 -apply (rule_tac x="a*x" in exI)
11.239 -apply (rule_tac x="y" in exI)
11.240 -apply (simp add: mult_ac)
11.241 -done
11.242 -
11.243 -lemma coprime_rmul2: "coprime d (a * b) \<Longrightarrow> coprime d a"
11.244 -unfolding coprime_bezout
11.245 -apply clarsimp
11.246 -apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all)
11.247 -apply (rule_tac x="x" in exI)
11.248 -apply (rule_tac x="b*y" in exI)
11.249 -apply (simp add: mult_ac)
11.250 -apply (rule_tac x="b*x" in exI)
11.251 -apply (rule_tac x="y" in exI)
11.252 -apply (simp add: mult_ac)
11.253 -done
11.254 -lemma coprime_mul_eq: "coprime d (a * b) \<longleftrightarrow> coprime d a \<and> coprime d b"
11.255 - using coprime_rmul2[of d a b] coprime_lmul2[of d a b] coprime_mul[of d a b]
11.256 - by blast
11.257 -
11.258 -lemma gcd_coprime_exists:
11.259 - assumes nz: "gcd a b \<noteq> 0"
11.260 - shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
11.261 -proof-
11.262 - let ?g = "gcd a b"
11.263 - from gcd_dvd1[of a b] gcd_dvd2[of a b]
11.264 - obtain a' b' where "a = ?g*a'" "b = ?g*b'" unfolding dvd_def by blast
11.265 - hence ab': "a = a'*?g" "b = b'*?g" by algebra+
11.266 - from ab' gcd_coprime[OF nz ab'] show ?thesis by blast
11.267 -qed
11.268 -
11.269 -lemma coprime_exp: "coprime d a ==> coprime d (a^n)"
11.270 - by(induct n, simp_all add: coprime_mul)
11.271 -
11.272 -lemma coprime_exp_imp: "coprime a b ==> coprime (a ^n) (b ^n)"
11.273 - by (induct n, simp_all add: coprime_mul_eq coprime_commute coprime_exp)
11.274 -lemma coprime_refl[simp]: "coprime n n \<longleftrightarrow> n = 1" by (simp add: coprime_def)
11.275 -lemma coprime_plus1[simp]: "coprime (n + 1) n"
11.276 - apply (simp add: coprime_bezout)
11.277 - apply (rule exI[where x=1])
11.278 - apply (rule exI[where x=1])
11.279 - apply simp
11.280 - done
11.281 -lemma coprime_minus1: "n \<noteq> 0 ==> coprime (n - 1) n"
11.282 - using coprime_plus1[of "n - 1"] coprime_commute[of "n - 1" n] by auto
11.283 -
11.284 -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"
11.285 -proof-
11.286 - let ?g = "gcd a b"
11.287 - {assume z: "?g = 0" hence ?thesis
11.288 - apply (cases n, simp)
11.289 - apply arith
11.290 - apply (simp only: z power_0_Suc)
11.291 - apply (rule exI[where x=0])
11.292 - apply (rule exI[where x=0])
11.293 - by simp}
11.294 - moreover
11.295 - {assume z: "?g \<noteq> 0"
11.296 - from gcd_dvd1[of a b] gcd_dvd2[of a b] obtain a' b' where
11.297 - ab': "a = a'*?g" "b = b'*?g" unfolding dvd_def by (auto simp add: mult_ac)
11.298 - hence ab'': "?g*a' = a" "?g * b' = b" by algebra+
11.299 - from coprime_exp_imp[OF gcd_coprime[OF z ab'], unfolded coprime_bezout, of n]
11.300 - obtain x y where "a'^n * x - b'^n * y = 1 \<or> b'^n * x - a'^n * y = 1" by blast
11.301 - hence "?g^n * (a'^n * x - b'^n * y) = ?g^n \<or> ?g^n*(b'^n * x - a'^n * y) = ?g^n"
11.302 - using z by auto
11.303 - then have "a^n * x - b^n * y = ?g^n \<or> b^n * x - a^n * y = ?g^n"
11.304 - using z ab'' by (simp only: power_mult_distrib[symmetric]
11.305 - diff_mult_distrib2 mult_assoc[symmetric])
11.306 - hence ?thesis by blast }
11.307 - ultimately show ?thesis by blast
11.308 -qed
11.309 -
11.310 -lemma gcd_exp: "gcd (a^n) (b^n) = gcd a b^n"
11.311 -proof-
11.312 - let ?g = "gcd (a^n) (b^n)"
11.313 - let ?gn = "gcd a b^n"
11.314 - {fix e assume H: "e dvd a^n" "e dvd b^n"
11.315 - from bezout_gcd_pow[of a n b] obtain x y
11.316 - where xy: "a ^ n * x - b ^ n * y = ?gn \<or> b ^ n * x - a ^ n * y = ?gn" by blast
11.317 - from dvd_diff_nat [OF dvd_mult2[OF H(1), of x] dvd_mult2[OF H(2), of y]]
11.318 - dvd_diff_nat [OF dvd_mult2[OF H(2), of x] dvd_mult2[OF H(1), of y]] xy
11.319 - have "e dvd ?gn" by (cases "a ^ n * x - b ^ n * y = gcd a b ^ n", simp_all)}
11.320 - hence th: "\<forall>e. e dvd a^n \<and> e dvd b^n \<longrightarrow> e dvd ?gn" by blast
11.321 - from divides_exp[OF gcd_dvd1[of a b], of n] divides_exp[OF gcd_dvd2[of a b], of n] th
11.322 - gcd_unique have "?gn = ?g" by blast thus ?thesis by simp
11.323 -qed
11.324 -
11.325 -lemma coprime_exp2: "coprime (a ^ Suc n) (b^ Suc n) \<longleftrightarrow> coprime a b"
11.326 -by (simp only: coprime_def gcd_exp exp_eq_1) simp
11.327 -
11.328 -lemma division_decomp: assumes dc: "(a::nat) dvd b * c"
11.329 - shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
11.330 -proof-
11.331 - let ?g = "gcd a b"
11.332 - {assume "?g = 0" with dc have ?thesis apply (simp add: gcd_zero)
11.333 - apply (rule exI[where x="0"])
11.334 - by (rule exI[where x="c"], simp)}
11.335 - moreover
11.336 - {assume z: "?g \<noteq> 0"
11.337 - from gcd_coprime_exists[OF z]
11.338 - obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast
11.339 - from gcd_dvd2[of a b] have thb: "?g dvd b" .
11.340 - from ab'(1) have "a' dvd a" unfolding dvd_def by blast
11.341 - with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
11.342 - from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
11.343 - hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
11.344 - with z have th_1: "a' dvd b'*c" by simp
11.345 - from coprime_divprod[OF th_1 ab'(3)] have thc: "a' dvd c" .
11.346 - from ab' have "a = ?g*a'" by algebra
11.347 - with thb thc have ?thesis by blast }
11.348 - ultimately show ?thesis by blast
11.349 -qed
11.350 -
11.351 -lemma nat_power_eq_0_iff: "(m::nat) ^ n = 0 \<longleftrightarrow> n \<noteq> 0 \<and> m = 0" by (induct n, auto)
11.352 -
11.353 -lemma divides_rev: assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0" shows "a dvd b"
11.354 -proof-
11.355 - let ?g = "gcd a b"
11.356 - from n obtain m where m: "n = Suc m" by (cases n, simp_all)
11.357 - {assume "?g = 0" with ab n have ?thesis by (simp add: gcd_zero)}
11.358 - moreover
11.359 - {assume z: "?g \<noteq> 0"
11.360 - hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
11.361 - from gcd_coprime_exists[OF z]
11.362 - obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast
11.363 - from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n" by (simp add: ab'(1,2)[symmetric])
11.364 - hence "?g^n*a'^n dvd ?g^n *b'^n" by (simp only: power_mult_distrib mult_commute)
11.365 - with zn z n have th0:"a'^n dvd b'^n" by (auto simp add: nat_power_eq_0_iff)
11.366 - have "a' dvd a'^n" by (simp add: m)
11.367 - with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
11.368 - hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
11.369 - from coprime_divprod[OF th1 coprime_exp[OF ab'(3), of m]]
11.370 - have "a' dvd b'" .
11.371 - hence "a'*?g dvd b'*?g" by simp
11.372 - with ab'(1,2) have ?thesis by simp }
11.373 - ultimately show ?thesis by blast
11.374 -qed
11.375 -
11.376 -lemma divides_mul: assumes mr: "m dvd r" and nr: "n dvd r" and mn:"coprime m n"
11.377 - shows "m * n dvd r"
11.378 -proof-
11.379 - from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
11.380 - unfolding dvd_def by blast
11.381 - from mr n' have "m dvd n'*n" by (simp add: mult_commute)
11.382 - hence "m dvd n'" using relprime_dvd_mult_iff[OF mn[unfolded coprime_def]] by simp
11.383 - then obtain k where k: "n' = m*k" unfolding dvd_def by blast
11.384 - from n' k show ?thesis unfolding dvd_def by auto
11.385 -qed
11.386 -
11.387 -
11.388 -text {* A binary form of the Chinese Remainder Theorem. *}
11.389 -
11.390 -lemma chinese_remainder: assumes ab: "coprime a b" and a:"a \<noteq> 0" and b:"b \<noteq> 0"
11.391 - shows "\<exists>x q1 q2. x = u + q1 * a \<and> x = v + q2 * b"
11.392 -proof-
11.393 - from bezout_add_strong[OF a, of b] bezout_add_strong[OF b, of a]
11.394 - obtain d1 x1 y1 d2 x2 y2 where dxy1: "d1 dvd a" "d1 dvd b" "a * x1 = b * y1 + d1"
11.395 - and dxy2: "d2 dvd b" "d2 dvd a" "b * x2 = a * y2 + d2" by blast
11.396 - from gcd_unique[of 1 a b, simplified ab[unfolded coprime_def], simplified]
11.397 - dxy1(1,2) dxy2(1,2) have d12: "d1 = 1" "d2 =1" by auto
11.398 - let ?x = "v * a * x1 + u * b * x2"
11.399 - let ?q1 = "v * x1 + u * y2"
11.400 - let ?q2 = "v * y1 + u * x2"
11.401 - from dxy2(3)[simplified d12] dxy1(3)[simplified d12]
11.402 - have "?x = u + ?q1 * a" "?x = v + ?q2 * b" by algebra+
11.403 - thus ?thesis by blast
11.404 -qed
11.405 -
11.406 -text {* Primality *}
11.407 -
11.408 -text {* A few useful theorems about primes *}
11.409 -
11.410 -lemma prime_0[simp]: "~prime 0" by (simp add: prime_def)
11.411 -lemma prime_1[simp]: "~ prime 1" by (simp add: prime_def)
11.412 -lemma prime_Suc0[simp]: "~ prime (Suc 0)" by (simp add: prime_def)
11.413 -
11.414 -lemma prime_ge_2: "prime p ==> p \<ge> 2" by (simp add: prime_def)
11.415 -lemma prime_factor: assumes n: "n \<noteq> 1" shows "\<exists> p. prime p \<and> p dvd n"
11.416 -using n
11.417 -proof(induct n rule: nat_less_induct)
11.418 - fix n
11.419 - assume H: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)" "n \<noteq> 1"
11.420 - let ?ths = "\<exists>p. prime p \<and> p dvd n"
11.421 - {assume "n=0" hence ?ths using two_is_prime by auto}
11.422 - moreover
11.423 - {assume nz: "n\<noteq>0"
11.424 - {assume "prime n" hence ?ths by - (rule exI[where x="n"], simp)}
11.425 - moreover
11.426 - {assume n: "\<not> prime n"
11.427 - with nz H(2)
11.428 - obtain k where k:"k dvd n" "k \<noteq> 1" "k \<noteq> n" by (auto simp add: prime_def)
11.429 - from dvd_imp_le[OF k(1)] nz k(3) have kn: "k < n" by simp
11.430 - from H(1)[rule_format, OF kn k(2)] obtain p where p: "prime p" "p dvd k" by blast
11.431 - from dvd_trans[OF p(2) k(1)] p(1) have ?ths by blast}
11.432 - ultimately have ?ths by blast}
11.433 - ultimately show ?ths by blast
11.434 -qed
11.435 -
11.436 -lemma prime_factor_lt: assumes p: "prime p" and n: "n \<noteq> 0" and npm:"n = p * m"
11.437 - shows "m < n"
11.438 -proof-
11.439 - {assume "m=0" with n have ?thesis by simp}
11.440 - moreover
11.441 - {assume m: "m \<noteq> 0"
11.442 - from npm have mn: "m dvd n" unfolding dvd_def by auto
11.443 - from npm m have "n \<noteq> m" using p by auto
11.444 - with dvd_imp_le[OF mn] n have ?thesis by simp}
11.445 - ultimately show ?thesis by blast
11.446 -qed
11.447 -
11.448 -lemma euclid_bound: "\<exists>p. prime p \<and> n < p \<and> p <= Suc (fact n)"
11.449 -proof-
11.450 - have f1: "fact n + 1 \<noteq> 1" using fact_le[of n] by arith
11.451 - from prime_factor[OF f1] obtain p where p: "prime p" "p dvd fact n + 1" by blast
11.452 - from dvd_imp_le[OF p(2)] have pfn: "p \<le> fact n + 1" by simp
11.453 - {assume np: "p \<le> n"
11.454 - from p(1) have p1: "p \<ge> 1" by (cases p, simp_all)
11.455 - from divides_fact[OF p1 np] have pfn': "p dvd fact n" .
11.456 - from divides_add_revr[OF pfn' p(2)] p(1) have False by simp}
11.457 - hence "n < p" by arith
11.458 - with p(1) pfn show ?thesis by auto
11.459 -qed
11.460 -
11.461 -lemma euclid: "\<exists>p. prime p \<and> p > n" using euclid_bound by auto
11.462 -
11.463 -lemma primes_infinite: "\<not> (finite {p. prime p})"
11.464 -apply(simp add: finite_nat_set_iff_bounded_le)
11.465 -apply (metis euclid linorder_not_le)
11.466 -done
11.467 -
11.468 -lemma coprime_prime: assumes ab: "coprime a b"
11.469 - shows "~(prime p \<and> p dvd a \<and> p dvd b)"
11.470 -proof
11.471 - assume "prime p \<and> p dvd a \<and> p dvd b"
11.472 - thus False using ab gcd_greatest[of p a b] by (simp add: coprime_def)
11.473 -qed
11.474 -lemma coprime_prime_eq: "coprime a b \<longleftrightarrow> (\<forall>p. ~(prime p \<and> p dvd a \<and> p dvd b))"
11.475 - (is "?lhs = ?rhs")
11.476 -proof-
11.477 - {assume "?lhs" with coprime_prime have ?rhs by blast}
11.478 - moreover
11.479 - {assume r: "?rhs" and c: "\<not> ?lhs"
11.480 - then obtain g where g: "g\<noteq>1" "g dvd a" "g dvd b" unfolding coprime_def by blast
11.481 - from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast
11.482 - from dvd_trans [OF p(2) g(2)] dvd_trans [OF p(2) g(3)]
11.483 - have "p dvd a" "p dvd b" . with p(1) r have False by blast}
11.484 - ultimately show ?thesis by blast
11.485 -qed
11.486 -
11.487 -lemma prime_coprime: assumes p: "prime p"
11.488 - shows "n = 1 \<or> p dvd n \<or> coprime p n"
11.489 -using p prime_imp_relprime[of p n] by (auto simp add: coprime_def)
11.490 -
11.491 -lemma prime_coprime_strong: "prime p \<Longrightarrow> p dvd n \<or> coprime p n"
11.492 - using prime_coprime[of p n] by auto
11.493 -
11.494 -declare coprime_0[simp]
11.495 -
11.496 -lemma coprime_0'[simp]: "coprime 0 d \<longleftrightarrow> d = 1" by (simp add: coprime_commute[of 0 d])
11.497 -lemma coprime_bezout_strong: assumes ab: "coprime a b" and b: "b \<noteq> 1"
11.498 - shows "\<exists>x y. a * x = b * y + 1"
11.499 -proof-
11.500 - from ab b have az: "a \<noteq> 0" by - (rule ccontr, auto)
11.501 - from bezout_gcd_strong[OF az, of b] ab[unfolded coprime_def]
11.502 - show ?thesis by auto
11.503 -qed
11.504 -
11.505 -lemma bezout_prime: assumes p: "prime p" and pa: "\<not> p dvd a"
11.506 - shows "\<exists>x y. a*x = p*y + 1"
11.507 -proof-
11.508 - from p have p1: "p \<noteq> 1" using prime_1 by blast
11.509 - from prime_coprime[OF p, of a] p1 pa have ap: "coprime a p"
11.510 - by (auto simp add: coprime_commute)
11.511 - from coprime_bezout_strong[OF ap p1] show ?thesis .
11.512 -qed
11.513 -lemma prime_divprod: assumes p: "prime p" and pab: "p dvd a*b"
11.514 - shows "p dvd a \<or> p dvd b"
11.515 -proof-
11.516 - {assume "a=1" hence ?thesis using pab by simp }
11.517 - moreover
11.518 - {assume "p dvd a" hence ?thesis by blast}
11.519 - moreover
11.520 - {assume pa: "coprime p a" from coprime_divprod[OF pab pa] have ?thesis .. }
11.521 - ultimately show ?thesis using prime_coprime[OF p, of a] by blast
11.522 -qed
11.523 -
11.524 -lemma prime_divprod_eq: assumes p: "prime p"
11.525 - shows "p dvd a*b \<longleftrightarrow> p dvd a \<or> p dvd b"
11.526 -using p prime_divprod dvd_mult dvd_mult2 by auto
11.527 -
11.528 -lemma prime_divexp: assumes p:"prime p" and px: "p dvd x^n"
11.529 - shows "p dvd x"
11.530 -using px
11.531 -proof(induct n)
11.532 - case 0 thus ?case by simp
11.533 -next
11.534 - case (Suc n)
11.535 - hence th: "p dvd x*x^n" by simp
11.536 - {assume H: "p dvd x^n"
11.537 - from Suc.hyps[OF H] have ?case .}
11.538 - with prime_divprod[OF p th] show ?case by blast
11.539 -qed
11.540 -
11.541 -lemma prime_divexp_n: "prime p \<Longrightarrow> p dvd x^n \<Longrightarrow> p^n dvd x^n"
11.542 - using prime_divexp[of p x n] divides_exp[of p x n] by blast
11.543 -
11.544 -lemma coprime_prime_dvd_ex: assumes xy: "\<not>coprime x y"
11.545 - shows "\<exists>p. prime p \<and> p dvd x \<and> p dvd y"
11.546 -proof-
11.547 - from xy[unfolded coprime_def] obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd y"
11.548 - by blast
11.549 - from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast
11.550 - from g(2,3) dvd_trans[OF p(2)] p(1) show ?thesis by auto
11.551 -qed
11.552 -lemma coprime_sos: assumes xy: "coprime x y"
11.553 - shows "coprime (x * y) (x^2 + y^2)"
11.554 -proof-
11.555 - {assume c: "\<not> coprime (x * y) (x^2 + y^2)"
11.556 - from coprime_prime_dvd_ex[OF c] obtain p
11.557 - where p: "prime p" "p dvd x*y" "p dvd x^2 + y^2" by blast
11.558 - {assume px: "p dvd x"
11.559 - from dvd_mult[OF px, of x] p(3)
11.560 - obtain r s where "x * x = p * r" and "x^2 + y^2 = p * s"
11.561 - by (auto elim!: dvdE)
11.562 - then have "y^2 = p * (s - r)"
11.563 - by (auto simp add: power2_eq_square diff_mult_distrib2)
11.564 - then have "p dvd y^2" ..
11.565 - with prime_divexp[OF p(1), of y 2] have py: "p dvd y" .
11.566 - from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1
11.567 - have False by simp }
11.568 - moreover
11.569 - {assume py: "p dvd y"
11.570 - from dvd_mult[OF py, of y] p(3)
11.571 - obtain r s where "y * y = p * r" and "x^2 + y^2 = p * s"
11.572 - by (auto elim!: dvdE)
11.573 - then have "x^2 = p * (s - r)"
11.574 - by (auto simp add: power2_eq_square diff_mult_distrib2)
11.575 - then have "p dvd x^2" ..
11.576 - with prime_divexp[OF p(1), of x 2] have px: "p dvd x" .
11.577 - from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1
11.578 - have False by simp }
11.579 - ultimately have False using prime_divprod[OF p(1,2)] by blast}
11.580 - thus ?thesis by blast
11.581 -qed
11.582 -
11.583 -lemma distinct_prime_coprime: "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
11.584 - unfolding prime_def coprime_prime_eq by blast
11.585 -
11.586 -lemma prime_coprime_lt: assumes p: "prime p" and x: "0 < x" and xp: "x < p"
11.587 - shows "coprime x p"
11.588 -proof-
11.589 - {assume c: "\<not> coprime x p"
11.590 - then obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd p" unfolding coprime_def by blast
11.591 - from dvd_imp_le[OF g(2)] x xp have gp: "g < p" by arith
11.592 - from g(2) x have "g \<noteq> 0" by - (rule ccontr, simp)
11.593 - with g gp p[unfolded prime_def] have False by blast}
11.594 -thus ?thesis by blast
11.595 -qed
11.596 -
11.597 -lemma even_dvd[simp]: "even (n::nat) \<longleftrightarrow> 2 dvd n" by presburger
11.598 -lemma prime_odd: "prime p \<Longrightarrow> p = 2 \<or> odd p" unfolding prime_def by auto
11.599 -
11.600 -
11.601 -text {* One property of coprimality is easier to prove via prime factors. *}
11.602 -
11.603 -lemma prime_divprod_pow:
11.604 - assumes p: "prime p" and ab: "coprime a b" and pab: "p^n dvd a * b"
11.605 - shows "p^n dvd a \<or> p^n dvd b"
11.606 -proof-
11.607 - {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
11.608 - apply (cases "n=0", simp_all)
11.609 - apply (cases "a=1", simp_all) done}
11.610 - moreover
11.611 - {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
11.612 - then obtain m where m: "n = Suc m" by (cases n, auto)
11.613 - from divides_exp2[OF n pab] have pab': "p dvd a*b" .
11.614 - from prime_divprod[OF p pab']
11.615 - have "p dvd a \<or> p dvd b" .
11.616 - moreover
11.617 - {assume pa: "p dvd a"
11.618 - have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
11.619 - from coprime_prime[OF ab, of p] p pa have "\<not> p dvd b" by blast
11.620 - with prime_coprime[OF p, of b] b
11.621 - have cpb: "coprime b p" using coprime_commute by blast
11.622 - from coprime_exp[OF cpb] have pnb: "coprime (p^n) b"
11.623 - by (simp add: coprime_commute)
11.624 - from coprime_divprod[OF pnba pnb] have ?thesis by blast }
11.625 - moreover
11.626 - {assume pb: "p dvd b"
11.627 - have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
11.628 - from coprime_prime[OF ab, of p] p pb have "\<not> p dvd a" by blast
11.629 - with prime_coprime[OF p, of a] a
11.630 - have cpb: "coprime a p" using coprime_commute by blast
11.631 - from coprime_exp[OF cpb] have pnb: "coprime (p^n) a"
11.632 - by (simp add: coprime_commute)
11.633 - from coprime_divprod[OF pab pnb] have ?thesis by blast }
11.634 - ultimately have ?thesis by blast}
11.635 - ultimately show ?thesis by blast
11.636 -qed
11.637 -
11.638 -lemma nat_mult_eq_one: "(n::nat) * m = 1 \<longleftrightarrow> n = 1 \<and> m = 1" (is "?lhs \<longleftrightarrow> ?rhs")
11.639 -proof
11.640 - assume H: "?lhs"
11.641 - hence "n dvd 1" "m dvd 1" unfolding dvd_def by (auto simp add: mult_commute)
11.642 - thus ?rhs by auto
11.643 -next
11.644 - assume ?rhs then show ?lhs by auto
11.645 -qed
11.646 -
11.647 -lemma power_Suc0[simp]: "Suc 0 ^ n = Suc 0"
11.648 - unfolding One_nat_def[symmetric] power_one ..
11.649 -lemma coprime_pow: assumes ab: "coprime a b" and abcn: "a * b = c ^n"
11.650 - shows "\<exists>r s. a = r^n \<and> b = s ^n"
11.651 - using ab abcn
11.652 -proof(induct c arbitrary: a b rule: nat_less_induct)
11.653 - fix c a b
11.654 - 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"
11.655 - let ?ths = "\<exists>r s. a = r^n \<and> b = s ^n"
11.656 - {assume n: "n = 0"
11.657 - with H(3) power_one have "a*b = 1" by simp
11.658 - hence "a = 1 \<and> b = 1" by simp
11.659 - hence ?ths
11.660 - apply -
11.661 - apply (rule exI[where x=1])
11.662 - apply (rule exI[where x=1])
11.663 - using power_one[of n]
11.664 - by simp}
11.665 - moreover
11.666 - {assume n: "n \<noteq> 0" then obtain m where m: "n = Suc m" by (cases n, auto)
11.667 - {assume c: "c = 0"
11.668 - with H(3) m H(2) have ?ths apply simp
11.669 - apply (cases "a=0", simp_all)
11.670 - apply (rule exI[where x="0"], simp)
11.671 - apply (rule exI[where x="0"], simp)
11.672 - done}
11.673 - moreover
11.674 - {assume "c=1" with H(3) power_one have "a*b = 1" by simp
11.675 - hence "a = 1 \<and> b = 1" by simp
11.676 - hence ?ths
11.677 - apply -
11.678 - apply (rule exI[where x=1])
11.679 - apply (rule exI[where x=1])
11.680 - using power_one[of n]
11.681 - by simp}
11.682 - moreover
11.683 - {assume c: "c\<noteq>1" "c \<noteq> 0"
11.684 - from prime_factor[OF c(1)] obtain p where p: "prime p" "p dvd c" by blast
11.685 - from prime_divprod_pow[OF p(1) H(2), unfolded H(3), OF divides_exp[OF p(2), of n]]
11.686 - have pnab: "p ^ n dvd a \<or> p^n dvd b" .
11.687 - from p(2) obtain l where l: "c = p*l" unfolding dvd_def by blast
11.688 - have pn0: "p^n \<noteq> 0" using n prime_ge_2 [OF p(1)] by (simp add: neq0_conv)
11.689 - {assume pa: "p^n dvd a"
11.690 - then obtain k where k: "a = p^n * k" unfolding dvd_def by blast
11.691 - from l have "l dvd c" by auto
11.692 - with dvd_imp_le[of l c] c have "l \<le> c" by auto
11.693 - moreover {assume "l = c" with l c have "p = 1" by simp with p have False by simp}
11.694 - ultimately have lc: "l < c" by arith
11.695 - from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" b]]]
11.696 - have kb: "coprime k b" by (simp add: coprime_commute)
11.697 - from H(3) l k pn0 have kbln: "k * b = l ^ n"
11.698 - by (auto simp add: power_mult_distrib)
11.699 - from H(1)[rule_format, OF lc kb kbln]
11.700 - obtain r s where rs: "k = r ^n" "b = s^n" by blast
11.701 - from k rs(1) have "a = (p*r)^n" by (simp add: power_mult_distrib)
11.702 - with rs(2) have ?ths by blast }
11.703 - moreover
11.704 - {assume pb: "p^n dvd b"
11.705 - then obtain k where k: "b = p^n * k" unfolding dvd_def by blast
11.706 - from l have "l dvd c" by auto
11.707 - with dvd_imp_le[of l c] c have "l \<le> c" by auto
11.708 - moreover {assume "l = c" with l c have "p = 1" by simp with p have False by simp}
11.709 - ultimately have lc: "l < c" by arith
11.710 - from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" a]]]
11.711 - have kb: "coprime k a" by (simp add: coprime_commute)
11.712 - from H(3) l k pn0 n have kbln: "k * a = l ^ n"
11.713 - by (simp add: power_mult_distrib mult_commute)
11.714 - from H(1)[rule_format, OF lc kb kbln]
11.715 - obtain r s where rs: "k = r ^n" "a = s^n" by blast
11.716 - from k rs(1) have "b = (p*r)^n" by (simp add: power_mult_distrib)
11.717 - with rs(2) have ?ths by blast }
11.718 - ultimately have ?ths using pnab by blast}
11.719 - ultimately have ?ths by blast}
11.720 -ultimately show ?ths by blast
11.721 -qed
11.722 -
11.723 -text {* More useful lemmas. *}
11.724 -lemma prime_product:
11.725 - assumes "prime (p * q)"
11.726 - shows "p = 1 \<or> q = 1"
11.727 -proof -
11.728 - from assms have
11.729 - "1 < p * q" and P: "\<And>m. m dvd p * q \<Longrightarrow> m = 1 \<or> m = p * q"
11.730 - unfolding prime_def by auto
11.731 - from `1 < p * q` have "p \<noteq> 0" by (cases p) auto
11.732 - then have Q: "p = p * q \<longleftrightarrow> q = 1" by auto
11.733 - have "p dvd p * q" by simp
11.734 - then have "p = 1 \<or> p = p * q" by (rule P)
11.735 - then show ?thesis by (simp add: Q)
11.736 -qed
11.737 -
11.738 -lemma prime_exp: "prime (p^n) \<longleftrightarrow> prime p \<and> n = 1"
11.739 -proof(induct n)
11.740 - case 0 thus ?case by simp
11.741 -next
11.742 - case (Suc n)
11.743 - {assume "p = 0" hence ?case by simp}
11.744 - moreover
11.745 - {assume "p=1" hence ?case by simp}
11.746 - moreover
11.747 - {assume p: "p \<noteq> 0" "p\<noteq>1"
11.748 - {assume pp: "prime (p^Suc n)"
11.749 - hence "p = 1 \<or> p^n = 1" using prime_product[of p "p^n"] by simp
11.750 - with p have n: "n = 0"
11.751 - by (simp only: exp_eq_1 ) simp
11.752 - with pp have "prime p \<and> Suc n = 1" by simp}
11.753 - moreover
11.754 - {assume n: "prime p \<and> Suc n = 1" hence "prime (p^Suc n)" by simp}
11.755 - ultimately have ?case by blast}
11.756 - ultimately show ?case by blast
11.757 -qed
11.758 -
11.759 -lemma prime_power_mult:
11.760 - assumes p: "prime p" and xy: "x * y = p ^ k"
11.761 - shows "\<exists>i j. x = p ^i \<and> y = p^ j"
11.762 - using xy
11.763 -proof(induct k arbitrary: x y)
11.764 - case 0 thus ?case apply simp by (rule exI[where x="0"], simp)
11.765 -next
11.766 - case (Suc k x y)
11.767 - from Suc.prems have pxy: "p dvd x*y" by auto
11.768 - from prime_divprod[OF p pxy] have pxyc: "p dvd x \<or> p dvd y" .
11.769 - from p have p0: "p \<noteq> 0" by - (rule ccontr, simp)
11.770 - {assume px: "p dvd x"
11.771 - then obtain d where d: "x = p*d" unfolding dvd_def by blast
11.772 - from Suc.prems d have "p*d*y = p^Suc k" by simp
11.773 - hence th: "d*y = p^k" using p0 by simp
11.774 - from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast
11.775 - with d have "x = p^Suc i" by simp
11.776 - with ij(2) have ?case by blast}
11.777 - moreover
11.778 - {assume px: "p dvd y"
11.779 - then obtain d where d: "y = p*d" unfolding dvd_def by blast
11.780 - from Suc.prems d have "p*d*x = p^Suc k" by (simp add: mult_commute)
11.781 - hence th: "d*x = p^k" using p0 by simp
11.782 - from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast
11.783 - with d have "y = p^Suc i" by simp
11.784 - with ij(2) have ?case by blast}
11.785 - ultimately show ?case using pxyc by blast
11.786 -qed
11.787 -
11.788 -lemma prime_power_exp: assumes p: "prime p" and n:"n \<noteq> 0"
11.789 - and xn: "x^n = p^k" shows "\<exists>i. x = p^i"
11.790 - using n xn
11.791 -proof(induct n arbitrary: k)
11.792 - case 0 thus ?case by simp
11.793 -next
11.794 - case (Suc n k) hence th: "x*x^n = p^k" by simp
11.795 - {assume "n = 0" with prems have ?case apply simp
11.796 - by (rule exI[where x="k"],simp)}
11.797 - moreover
11.798 - {assume n: "n \<noteq> 0"
11.799 - from prime_power_mult[OF p th]
11.800 - obtain i j where ij: "x = p^i" "x^n = p^j"by blast
11.801 - from Suc.hyps[OF n ij(2)] have ?case .}
11.802 - ultimately show ?case by blast
11.803 -qed
11.804 -
11.805 -lemma divides_primepow: assumes p: "prime p"
11.806 - shows "d dvd p^k \<longleftrightarrow> (\<exists> i. i \<le> k \<and> d = p ^i)"
11.807 -proof
11.808 - assume H: "d dvd p^k" then obtain e where e: "d*e = p^k"
11.809 - unfolding dvd_def apply (auto simp add: mult_commute) by blast
11.810 - from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast
11.811 - from prime_ge_2[OF p] have p1: "p > 1" by arith
11.812 - from e ij have "p^(i + j) = p^k" by (simp add: power_add)
11.813 - hence "i + j = k" using power_inject_exp[of p "i+j" k, OF p1] by simp
11.814 - hence "i \<le> k" by arith
11.815 - with ij(1) show "\<exists>i\<le>k. d = p ^ i" by blast
11.816 -next
11.817 - {fix i assume H: "i \<le> k" "d = p^i"
11.818 - hence "\<exists>j. k = i + j" by arith
11.819 - then obtain j where j: "k = i + j" by blast
11.820 - hence "p^k = p^j*d" using H(2) by (simp add: power_add)
11.821 - hence "d dvd p^k" unfolding dvd_def by auto}
11.822 - thus "\<exists>i\<le>k. d = p ^ i \<Longrightarrow> d dvd p ^ k" by blast
11.823 -qed
11.824 -
11.825 -lemma coprime_divisors: "d dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow> coprime d e"
11.826 - by (auto simp add: dvd_def coprime)
11.827 -
11.828 -declare power_Suc0[simp del]
11.829 -declare even_dvd[simp del]
11.830 -
11.831 -end
12.1 --- a/src/HOL/NSA/Examples/NSPrimes.thy Tue Sep 01 14:13:34 2009 +0200
12.2 +++ b/src/HOL/NSA/Examples/NSPrimes.thy Tue Sep 01 15:39:33 2009 +0200
12.3 @@ -7,7 +7,7 @@
12.4 header{*The Nonstandard Primes as an Extension of the Prime Numbers*}
12.5
12.6 theory NSPrimes
12.7 -imports "~~/src/HOL/NumberTheory/Factorization" Hyperreal
12.8 +imports "~~/src/HOL/Old_Number_Theory/Factorization" Hyperreal
12.9 begin
12.10
12.11 text{*These can be used to derive an alternative proof of the infinitude of
13.1 --- a/src/HOL/NewNumberTheory/Binomial.thy Tue Sep 01 14:13:34 2009 +0200
13.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
13.3 @@ -1,373 +0,0 @@
13.4 -(* Title: Binomial.thy
13.5 - Authors: Lawrence C. Paulson, Jeremy Avigad, Tobias Nipkow
13.6 -
13.7 -
13.8 -Defines the "choose" function, and establishes basic properties.
13.9 -
13.10 -The original theory "Binomial" was by Lawrence C. Paulson, based on
13.11 -the work of Andy Gordon and Florian Kammueller. The approach here,
13.12 -which derives the definition of binomial coefficients in terms of the
13.13 -factorial function, is due to Jeremy Avigad. The binomial theorem was
13.14 -formalized by Tobias Nipkow.
13.15 -
13.16 -*)
13.17 -
13.18 -
13.19 -header {* Binomial *}
13.20 -
13.21 -theory Binomial
13.22 -imports Cong Fact
13.23 -begin
13.24 -
13.25 -
13.26 -subsection {* Main definitions *}
13.27 -
13.28 -class binomial =
13.29 -
13.30 -fixes
13.31 - binomial :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "choose" 65)
13.32 -
13.33 -(* definitions for the natural numbers *)
13.34 -
13.35 -instantiation nat :: binomial
13.36 -
13.37 -begin
13.38 -
13.39 -fun
13.40 - binomial_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
13.41 -where
13.42 - "binomial_nat n k =
13.43 - (if k = 0 then 1 else
13.44 - if n = 0 then 0 else
13.45 - (binomial (n - 1) k) + (binomial (n - 1) (k - 1)))"
13.46 -
13.47 -instance proof qed
13.48 -
13.49 -end
13.50 -
13.51 -(* definitions for the integers *)
13.52 -
13.53 -instantiation int :: binomial
13.54 -
13.55 -begin
13.56 -
13.57 -definition
13.58 - binomial_int :: "int => int \<Rightarrow> int"
13.59 -where
13.60 - "binomial_int n k = (if n \<ge> 0 \<and> k \<ge> 0 then int (binomial (nat n) (nat k))
13.61 - else 0)"
13.62 -instance proof qed
13.63 -
13.64 -end
13.65 -
13.66 -
13.67 -subsection {* Set up Transfer *}
13.68 -
13.69 -lemma transfer_nat_int_binomial:
13.70 - "(n::int) >= 0 \<Longrightarrow> k >= 0 \<Longrightarrow> binomial (nat n) (nat k) =
13.71 - nat (binomial n k)"
13.72 - unfolding binomial_int_def
13.73 - by auto
13.74 -
13.75 -lemma transfer_nat_int_binomial_closure:
13.76 - "n >= (0::int) \<Longrightarrow> k >= 0 \<Longrightarrow> binomial n k >= 0"
13.77 - by (auto simp add: binomial_int_def)
13.78 -
13.79 -declare TransferMorphism_nat_int[transfer add return:
13.80 - transfer_nat_int_binomial transfer_nat_int_binomial_closure]
13.81 -
13.82 -lemma transfer_int_nat_binomial:
13.83 - "binomial (int n) (int k) = int (binomial n k)"
13.84 - unfolding fact_int_def binomial_int_def by auto
13.85 -
13.86 -lemma transfer_int_nat_binomial_closure:
13.87 - "is_nat n \<Longrightarrow> is_nat k \<Longrightarrow> binomial n k >= 0"
13.88 - by (auto simp add: binomial_int_def)
13.89 -
13.90 -declare TransferMorphism_int_nat[transfer add return:
13.91 - transfer_int_nat_binomial transfer_int_nat_binomial_closure]
13.92 -
13.93 -
13.94 -subsection {* Binomial coefficients *}
13.95 -
13.96 -lemma choose_zero_nat [simp]: "(n::nat) choose 0 = 1"
13.97 - by simp
13.98 -
13.99 -lemma choose_zero_int [simp]: "n \<ge> 0 \<Longrightarrow> (n::int) choose 0 = 1"
13.100 - by (simp add: binomial_int_def)
13.101 -
13.102 -lemma zero_choose_nat [rule_format,simp]: "ALL (k::nat) > n. n choose k = 0"
13.103 - by (induct n rule: induct'_nat, auto)
13.104 -
13.105 -lemma zero_choose_int [rule_format,simp]: "(k::int) > n \<Longrightarrow> n choose k = 0"
13.106 - unfolding binomial_int_def apply (case_tac "n < 0")
13.107 - apply force
13.108 - apply (simp del: binomial_nat.simps)
13.109 -done
13.110 -
13.111 -lemma choose_reduce_nat: "(n::nat) > 0 \<Longrightarrow> 0 < k \<Longrightarrow>
13.112 - (n choose k) = ((n - 1) choose k) + ((n - 1) choose (k - 1))"
13.113 - by simp
13.114 -
13.115 -lemma choose_reduce_int: "(n::int) > 0 \<Longrightarrow> 0 < k \<Longrightarrow>
13.116 - (n choose k) = ((n - 1) choose k) + ((n - 1) choose (k - 1))"
13.117 - unfolding binomial_int_def apply (subst choose_reduce_nat)
13.118 - apply (auto simp del: binomial_nat.simps
13.119 - simp add: nat_diff_distrib)
13.120 -done
13.121 -
13.122 -lemma choose_plus_one_nat: "((n::nat) + 1) choose (k + 1) =
13.123 - (n choose (k + 1)) + (n choose k)"
13.124 - by (simp add: choose_reduce_nat)
13.125 -
13.126 -lemma choose_Suc_nat: "(Suc n) choose (Suc k) =
13.127 - (n choose (Suc k)) + (n choose k)"
13.128 - by (simp add: choose_reduce_nat One_nat_def)
13.129 -
13.130 -lemma choose_plus_one_int: "n \<ge> 0 \<Longrightarrow> k \<ge> 0 \<Longrightarrow> ((n::int) + 1) choose (k + 1) =
13.131 - (n choose (k + 1)) + (n choose k)"
13.132 - by (simp add: binomial_int_def choose_plus_one_nat nat_add_distrib del: binomial_nat.simps)
13.133 -
13.134 -declare binomial_nat.simps [simp del]
13.135 -
13.136 -lemma choose_self_nat [simp]: "((n::nat) choose n) = 1"
13.137 - by (induct n rule: induct'_nat, auto simp add: choose_plus_one_nat)
13.138 -
13.139 -lemma choose_self_int [simp]: "n \<ge> 0 \<Longrightarrow> ((n::int) choose n) = 1"
13.140 - by (auto simp add: binomial_int_def)
13.141 -
13.142 -lemma choose_one_nat [simp]: "(n::nat) choose 1 = n"
13.143 - by (induct n rule: induct'_nat, auto simp add: choose_reduce_nat)
13.144 -
13.145 -lemma choose_one_int [simp]: "n \<ge> 0 \<Longrightarrow> (n::int) choose 1 = n"
13.146 - by (auto simp add: binomial_int_def)
13.147 -
13.148 -lemma plus_one_choose_self_nat [simp]: "(n::nat) + 1 choose n = n + 1"
13.149 - apply (induct n rule: induct'_nat, force)
13.150 - apply (case_tac "n = 0")
13.151 - apply auto
13.152 - apply (subst choose_reduce_nat)
13.153 - apply (auto simp add: One_nat_def)
13.154 - (* natdiff_cancel_numerals introduces Suc *)
13.155 -done
13.156 -
13.157 -lemma Suc_choose_self_nat [simp]: "(Suc n) choose n = Suc n"
13.158 - using plus_one_choose_self_nat by (simp add: One_nat_def)
13.159 -
13.160 -lemma plus_one_choose_self_int [rule_format, simp]:
13.161 - "(n::int) \<ge> 0 \<longrightarrow> n + 1 choose n = n + 1"
13.162 - by (auto simp add: binomial_int_def nat_add_distrib)
13.163 -
13.164 -(* bounded quantification doesn't work with the unicode characters? *)
13.165 -lemma choose_pos_nat [rule_format]: "ALL k <= (n::nat).
13.166 - ((n::nat) choose k) > 0"
13.167 - apply (induct n rule: induct'_nat)
13.168 - apply force
13.169 - apply clarify
13.170 - apply (case_tac "k = 0")
13.171 - apply force
13.172 - apply (subst choose_reduce_nat)
13.173 - apply auto
13.174 -done
13.175 -
13.176 -lemma choose_pos_int: "n \<ge> 0 \<Longrightarrow> k >= 0 \<Longrightarrow> k \<le> n \<Longrightarrow>
13.177 - ((n::int) choose k) > 0"
13.178 - by (auto simp add: binomial_int_def choose_pos_nat)
13.179 -
13.180 -lemma binomial_induct [rule_format]: "(ALL (n::nat). P n n) \<longrightarrow>
13.181 - (ALL n. P (n + 1) 0) \<longrightarrow> (ALL n. (ALL k < n. P n k \<longrightarrow> P n (k + 1) \<longrightarrow>
13.182 - P (n + 1) (k + 1))) \<longrightarrow> (ALL k <= n. P n k)"
13.183 - apply (induct n rule: induct'_nat)
13.184 - apply auto
13.185 - apply (case_tac "k = 0")
13.186 - apply auto
13.187 - apply (case_tac "k = n + 1")
13.188 - apply auto
13.189 - apply (drule_tac x = n in spec) back back
13.190 - apply (drule_tac x = "k - 1" in spec) back back back
13.191 - apply auto
13.192 -done
13.193 -
13.194 -lemma choose_altdef_aux_nat: "(k::nat) \<le> n \<Longrightarrow>
13.195 - fact k * fact (n - k) * (n choose k) = fact n"
13.196 - apply (rule binomial_induct [of _ k n])
13.197 - apply auto
13.198 -proof -
13.199 - fix k :: nat and n
13.200 - assume less: "k < n"
13.201 - assume ih1: "fact k * fact (n - k) * (n choose k) = fact n"
13.202 - hence one: "fact (k + 1) * fact (n - k) * (n choose k) = (k + 1) * fact n"
13.203 - by (subst fact_plus_one_nat, auto)
13.204 - assume ih2: "fact (k + 1) * fact (n - (k + 1)) * (n choose (k + 1)) =
13.205 - fact n"
13.206 - with less have "fact (k + 1) * fact ((n - (k + 1)) + 1) *
13.207 - (n choose (k + 1)) = (n - k) * fact n"
13.208 - by (subst (2) fact_plus_one_nat, auto)
13.209 - with less have two: "fact (k + 1) * fact (n - k) * (n choose (k + 1)) =
13.210 - (n - k) * fact n" by simp
13.211 - have "fact (k + 1) * fact (n - k) * (n + 1 choose (k + 1)) =
13.212 - fact (k + 1) * fact (n - k) * (n choose (k + 1)) +
13.213 - fact (k + 1) * fact (n - k) * (n choose k)"
13.214 - by (subst choose_reduce_nat, auto simp add: ring_simps)
13.215 - also note one
13.216 - also note two
13.217 - also with less have "(n - k) * fact n + (k + 1) * fact n= fact (n + 1)"
13.218 - apply (subst fact_plus_one_nat)
13.219 - apply (subst left_distrib [symmetric])
13.220 - apply simp
13.221 - done
13.222 - finally show "fact (k + 1) * fact (n - k) * (n + 1 choose (k + 1)) =
13.223 - fact (n + 1)" .
13.224 -qed
13.225 -
13.226 -lemma choose_altdef_nat: "(k::nat) \<le> n \<Longrightarrow>
13.227 - n choose k = fact n div (fact k * fact (n - k))"
13.228 - apply (frule choose_altdef_aux_nat)
13.229 - apply (erule subst)
13.230 - apply (simp add: mult_ac)
13.231 -done
13.232 -
13.233 -
13.234 -lemma choose_altdef_int:
13.235 - assumes "(0::int) <= k" and "k <= n"
13.236 - shows "n choose k = fact n div (fact k * fact (n - k))"
13.237 -
13.238 - apply (subst tsub_eq [symmetric], rule prems)
13.239 - apply (rule choose_altdef_nat [transferred])
13.240 - using prems apply auto
13.241 -done
13.242 -
13.243 -lemma choose_dvd_nat: "(k::nat) \<le> n \<Longrightarrow> fact k * fact (n - k) dvd fact n"
13.244 - unfolding dvd_def apply (frule choose_altdef_aux_nat)
13.245 - (* why don't blast and auto get this??? *)
13.246 - apply (rule exI)
13.247 - apply (erule sym)
13.248 -done
13.249 -
13.250 -lemma choose_dvd_int:
13.251 - assumes "(0::int) <= k" and "k <= n"
13.252 - shows "fact k * fact (n - k) dvd fact n"
13.253 -
13.254 - apply (subst tsub_eq [symmetric], rule prems)
13.255 - apply (rule choose_dvd_nat [transferred])
13.256 - using prems apply auto
13.257 -done
13.258 -
13.259 -(* generalizes Tobias Nipkow's proof to any commutative semiring *)
13.260 -theorem binomial: "(a+b::'a::{comm_ring_1,power})^n =
13.261 - (SUM k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P n")
13.262 -proof (induct n rule: induct'_nat)
13.263 - show "?P 0" by simp
13.264 -next
13.265 - fix n
13.266 - assume ih: "?P n"
13.267 - have decomp: "{0..n+1} = {0} Un {n+1} Un {1..n}"
13.268 - by auto
13.269 - have decomp2: "{0..n} = {0} Un {1..n}"
13.270 - by auto
13.271 - have decomp3: "{1..n+1} = {n+1} Un {1..n}"
13.272 - by auto
13.273 - have "(a+b)^(n+1) =
13.274 - (a+b) * (SUM k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
13.275 - using ih by (simp add: power_plus_one)
13.276 - also have "... = a*(SUM k=0..n. of_nat (n choose k) * a^k * b^(n-k)) +
13.277 - b*(SUM k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
13.278 - by (rule distrib)
13.279 - also have "... = (SUM k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) +
13.280 - (SUM k=0..n. of_nat (n choose k) * a^k * b^(n-k+1))"
13.281 - by (subst (1 2) power_plus_one, simp add: setsum_right_distrib mult_ac)
13.282 - also have "... = (SUM k=0..n. of_nat (n choose k) * a^k * b^(n+1-k)) +
13.283 - (SUM k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n+1-k))"
13.284 - by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le
13.285 - power_Suc ring_simps One_nat_def del:setsum_cl_ivl_Suc)
13.286 - also have "... = a^(n+1) + b^(n+1) +
13.287 - (SUM k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) +
13.288 - (SUM k=1..n. of_nat (n choose k) * a^k * b^(n+1-k))"
13.289 - by (simp add: decomp2 decomp3)
13.290 - also have
13.291 - "... = a^(n+1) + b^(n+1) +
13.292 - (SUM k=1..n. of_nat(n+1 choose k) * a^k * b^(n+1-k))"
13.293 - by (auto simp add: ring_simps setsum_addf [symmetric]
13.294 - choose_reduce_nat)
13.295 - also have "... = (SUM k=0..n+1. of_nat (n+1 choose k) * a^k * b^(n+1-k))"
13.296 - using decomp by (simp add: ring_simps)
13.297 - finally show "?P (n + 1)" by simp
13.298 -qed
13.299 -
13.300 -lemma set_explicit: "{S. S = T \<and> P S} = (if P T then {T} else {})"
13.301 - by auto
13.302 -
13.303 -lemma card_subsets_nat [rule_format]:
13.304 - fixes S :: "'a set"
13.305 - assumes "finite S"
13.306 - shows "ALL k. card {T. T \<le> S \<and> card T = k} = card S choose k"
13.307 - (is "?P S")
13.308 -using `finite S`
13.309 -proof (induct set: finite)
13.310 - show "?P {}" by (auto simp add: set_explicit)
13.311 - next fix x :: "'a" and F
13.312 - assume iassms: "finite F" "x ~: F"
13.313 - assume ih: "?P F"
13.314 - show "?P (insert x F)" (is "ALL k. ?Q k")
13.315 - proof
13.316 - fix k
13.317 - show "card {T. T \<subseteq> (insert x F) \<and> card T = k} =
13.318 - card (insert x F) choose k" (is "?Q k")
13.319 - proof (induct k rule: induct'_nat)
13.320 - from iassms have "{T. T \<le> (insert x F) \<and> card T = 0} = {{}}"
13.321 - apply auto
13.322 - apply (subst (asm) card_0_eq)
13.323 - apply (auto elim: finite_subset)
13.324 - done
13.325 - thus "?Q 0"
13.326 - by auto
13.327 - next fix k
13.328 - show "?Q (k + 1)"
13.329 - proof -
13.330 - from iassms have fin: "finite (insert x F)" by auto
13.331 - hence "{ T. T \<subseteq> insert x F \<and> card T = k + 1} =
13.332 - {T. T \<le> F & card T = k + 1} Un
13.333 - {T. T \<le> insert x F & x : T & card T = k + 1}"
13.334 - by (auto intro!: subsetI)
13.335 - with iassms fin have "card ({T. T \<le> insert x F \<and> card T = k + 1}) =
13.336 - card ({T. T \<subseteq> F \<and> card T = k + 1}) +
13.337 - card ({T. T \<subseteq> insert x F \<and> x : T \<and> card T = k + 1})"
13.338 - apply (subst card_Un_disjoint [symmetric])
13.339 - apply auto
13.340 - (* note: nice! Didn't have to say anything here *)
13.341 - done
13.342 - also from ih have "card ({T. T \<subseteq> F \<and> card T = k + 1}) =
13.343 - card F choose (k+1)" by auto
13.344 - also have "card ({T. T \<subseteq> insert x F \<and> x : T \<and> card T = k + 1}) =
13.345 - card ({T. T <= F & card T = k})"
13.346 - proof -
13.347 - let ?f = "%T. T Un {x}"
13.348 - from iassms have "inj_on ?f {T. T <= F & card T = k}"
13.349 - unfolding inj_on_def by (auto intro!: subsetI)
13.350 - hence "card ({T. T <= F & card T = k}) =
13.351 - card(?f ` {T. T <= F & card T = k})"
13.352 - by (rule card_image [symmetric])
13.353 - also from iassms fin have "?f ` {T. T <= F & card T = k} =
13.354 - {T. T \<subseteq> insert x F \<and> x : T \<and> card T = k + 1}"
13.355 - unfolding image_def
13.356 - (* I can't figure out why this next line takes so long *)
13.357 - apply auto
13.358 - apply (frule (1) finite_subset, force)
13.359 - apply (rule_tac x = "xa - {x}" in exI)
13.360 - apply (subst card_Diff_singleton)
13.361 - apply (auto elim: finite_subset)
13.362 - done
13.363 - finally show ?thesis by (rule sym)
13.364 - qed
13.365 - also from ih have "card ({T. T <= F & card T = k}) = card F choose k"
13.366 - by auto
13.367 - finally have "card ({T. T \<le> insert x F \<and> card T = k + 1}) =
13.368 - card F choose (k + 1) + (card F choose k)".
13.369 - with iassms choose_plus_one_nat show ?thesis
13.370 - by auto
13.371 - qed
13.372 - qed
13.373 - qed
13.374 -qed
13.375 -
13.376 -end
14.1 --- a/src/HOL/NewNumberTheory/Cong.thy Tue Sep 01 14:13:34 2009 +0200
14.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
14.3 @@ -1,1091 +0,0 @@
14.4 -(* Title: HOL/Library/Cong.thy
14.5 - ID:
14.6 - Authors: Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
14.7 - Thomas M. Rasmussen, Jeremy Avigad
14.8 -
14.9 -
14.10 -Defines congruence (notation: [x = y] (mod z)) for natural numbers and
14.11 -integers.
14.12 -
14.13 -This file combines and revises a number of prior developments.
14.14 -
14.15 -The original theories "GCD" and "Primes" were by Christophe Tabacznyj
14.16 -and Lawrence C. Paulson, based on \cite{davenport92}. They introduced
14.17 -gcd, lcm, and prime for the natural numbers.
14.18 -
14.19 -The original theory "IntPrimes" was by Thomas M. Rasmussen, and
14.20 -extended gcd, lcm, primes to the integers. Amine Chaieb provided
14.21 -another extension of the notions to the integers, and added a number
14.22 -of results to "Primes" and "GCD".
14.23 -
14.24 -The original theory, "IntPrimes", by Thomas M. Rasmussen, defined and
14.25 -developed the congruence relations on the integers. The notion was
14.26 -extended to the natural numbers by Chiaeb. Jeremy Avigad combined
14.27 -these, revised and tidied them, made the development uniform for the
14.28 -natural numbers and the integers, and added a number of new theorems.
14.29 -
14.30 -*)
14.31 -
14.32 -
14.33 -header {* Congruence *}
14.34 -
14.35 -theory Cong
14.36 -imports GCD
14.37 -begin
14.38 -
14.39 -subsection {* Turn off One_nat_def *}
14.40 -
14.41 -lemma induct'_nat [case_names zero plus1, induct type: nat]:
14.42 - "\<lbrakk> P (0::nat); !!n. P n \<Longrightarrow> P (n + 1)\<rbrakk> \<Longrightarrow> P n"
14.43 -by (erule nat_induct) (simp add:One_nat_def)
14.44 -
14.45 -lemma cases_nat [case_names zero plus1, cases type: nat]:
14.46 - "P (0::nat) \<Longrightarrow> (!!n. P (n + 1)) \<Longrightarrow> P n"
14.47 -by(metis induct'_nat)
14.48 -
14.49 -lemma power_plus_one [simp]: "(x::'a::power)^(n + 1) = x * x^n"
14.50 -by (simp add: One_nat_def)
14.51 -
14.52 -lemma power_eq_one_eq_nat [simp]:
14.53 - "((x::nat)^m = 1) = (m = 0 | x = 1)"
14.54 -by (induct m, auto)
14.55 -
14.56 -lemma card_insert_if' [simp]: "finite A \<Longrightarrow>
14.57 - card (insert x A) = (if x \<in> A then (card A) else (card A) + 1)"
14.58 -by (auto simp add: insert_absorb)
14.59 -
14.60 -(* why wasn't card_insert_if a simp rule? *)
14.61 -declare card_insert_disjoint [simp del]
14.62 -
14.63 -lemma nat_1' [simp]: "nat 1 = 1"
14.64 -by simp
14.65 -
14.66 -(* For those annoying moments where Suc reappears, use Suc_eq_plus1 *)
14.67 -
14.68 -declare nat_1 [simp del]
14.69 -declare add_2_eq_Suc [simp del]
14.70 -declare add_2_eq_Suc' [simp del]
14.71 -
14.72 -
14.73 -declare mod_pos_pos_trivial [simp]
14.74 -
14.75 -
14.76 -subsection {* Main definitions *}
14.77 -
14.78 -class cong =
14.79 -
14.80 -fixes
14.81 - cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" ("(1[_ = _] '(mod _'))")
14.82 -
14.83 -begin
14.84 -
14.85 -abbreviation
14.86 - notcong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" ("(1[_ \<noteq> _] '(mod _'))")
14.87 -where
14.88 - "notcong x y m == (~cong x y m)"
14.89 -
14.90 -end
14.91 -
14.92 -(* definitions for the natural numbers *)
14.93 -
14.94 -instantiation nat :: cong
14.95 -
14.96 -begin
14.97 -
14.98 -definition
14.99 - cong_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"
14.100 -where
14.101 - "cong_nat x y m = ((x mod m) = (y mod m))"
14.102 -
14.103 -instance proof qed
14.104 -
14.105 -end
14.106 -
14.107 -
14.108 -(* definitions for the integers *)
14.109 -
14.110 -instantiation int :: cong
14.111 -
14.112 -begin
14.113 -
14.114 -definition
14.115 - cong_int :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool"
14.116 -where
14.117 - "cong_int x y m = ((x mod m) = (y mod m))"
14.118 -
14.119 -instance proof qed
14.120 -
14.121 -end
14.122 -
14.123 -
14.124 -subsection {* Set up Transfer *}
14.125 -
14.126 -
14.127 -lemma transfer_nat_int_cong:
14.128 - "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> m >= 0 \<Longrightarrow>
14.129 - ([(nat x) = (nat y)] (mod (nat m))) = ([x = y] (mod m))"
14.130 - unfolding cong_int_def cong_nat_def
14.131 - apply (auto simp add: nat_mod_distrib [symmetric])
14.132 - apply (subst (asm) eq_nat_nat_iff)
14.133 - apply (case_tac "m = 0", force, rule pos_mod_sign, force)+
14.134 - apply assumption
14.135 -done
14.136 -
14.137 -declare TransferMorphism_nat_int[transfer add return:
14.138 - transfer_nat_int_cong]
14.139 -
14.140 -lemma transfer_int_nat_cong:
14.141 - "[(int x) = (int y)] (mod (int m)) = [x = y] (mod m)"
14.142 - apply (auto simp add: cong_int_def cong_nat_def)
14.143 - apply (auto simp add: zmod_int [symmetric])
14.144 -done
14.145 -
14.146 -declare TransferMorphism_int_nat[transfer add return:
14.147 - transfer_int_nat_cong]
14.148 -
14.149 -
14.150 -subsection {* Congruence *}
14.151 -
14.152 -(* was zcong_0, etc. *)
14.153 -lemma cong_0_nat [simp, presburger]: "([(a::nat) = b] (mod 0)) = (a = b)"
14.154 - by (unfold cong_nat_def, auto)
14.155 -
14.156 -lemma cong_0_int [simp, presburger]: "([(a::int) = b] (mod 0)) = (a = b)"
14.157 - by (unfold cong_int_def, auto)
14.158 -
14.159 -lemma cong_1_nat [simp, presburger]: "[(a::nat) = b] (mod 1)"
14.160 - by (unfold cong_nat_def, auto)
14.161 -
14.162 -lemma cong_Suc_0_nat [simp, presburger]: "[(a::nat) = b] (mod Suc 0)"
14.163 - by (unfold cong_nat_def, auto simp add: One_nat_def)
14.164 -
14.165 -lemma cong_1_int [simp, presburger]: "[(a::int) = b] (mod 1)"
14.166 - by (unfold cong_int_def, auto)
14.167 -
14.168 -lemma cong_refl_nat [simp]: "[(k::nat) = k] (mod m)"
14.169 - by (unfold cong_nat_def, auto)
14.170 -
14.171 -lemma cong_refl_int [simp]: "[(k::int) = k] (mod m)"
14.172 - by (unfold cong_int_def, auto)
14.173 -
14.174 -lemma cong_sym_nat: "[(a::nat) = b] (mod m) \<Longrightarrow> [b = a] (mod m)"
14.175 - by (unfold cong_nat_def, auto)
14.176 -
14.177 -lemma cong_sym_int: "[(a::int) = b] (mod m) \<Longrightarrow> [b = a] (mod m)"
14.178 - by (unfold cong_int_def, auto)
14.179 -
14.180 -lemma cong_sym_eq_nat: "[(a::nat) = b] (mod m) = [b = a] (mod m)"
14.181 - by (unfold cong_nat_def, auto)
14.182 -
14.183 -lemma cong_sym_eq_int: "[(a::int) = b] (mod m) = [b = a] (mod m)"
14.184 - by (unfold cong_int_def, auto)
14.185 -
14.186 -lemma cong_trans_nat [trans]:
14.187 - "[(a::nat) = b] (mod m) \<Longrightarrow> [b = c] (mod m) \<Longrightarrow> [a = c] (mod m)"
14.188 - by (unfold cong_nat_def, auto)
14.189 -
14.190 -lemma cong_trans_int [trans]:
14.191 - "[(a::int) = b] (mod m) \<Longrightarrow> [b = c] (mod m) \<Longrightarrow> [a = c] (mod m)"
14.192 - by (unfold cong_int_def, auto)
14.193 -
14.194 -lemma cong_add_nat:
14.195 - "[(a::nat) = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a + c = b + d] (mod m)"
14.196 - apply (unfold cong_nat_def)
14.197 - apply (subst (1 2) mod_add_eq)
14.198 - apply simp
14.199 -done
14.200 -
14.201 -lemma cong_add_int:
14.202 - "[(a::int) = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a + c = b + d] (mod m)"
14.203 - apply (unfold cong_int_def)
14.204 - apply (subst (1 2) mod_add_left_eq)
14.205 - apply (subst (1 2) mod_add_right_eq)
14.206 - apply simp
14.207 -done
14.208 -
14.209 -lemma cong_diff_int:
14.210 - "[(a::int) = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a - c = b - d] (mod m)"
14.211 - apply (unfold cong_int_def)
14.212 - apply (subst (1 2) mod_diff_eq)
14.213 - apply simp
14.214 -done
14.215 -
14.216 -lemma cong_diff_aux_int:
14.217 - "(a::int) >= c \<Longrightarrow> b >= d \<Longrightarrow> [(a::int) = b] (mod m) \<Longrightarrow>
14.218 - [c = d] (mod m) \<Longrightarrow> [tsub a c = tsub b d] (mod m)"
14.219 - apply (subst (1 2) tsub_eq)
14.220 - apply (auto intro: cong_diff_int)
14.221 -done;
14.222 -
14.223 -lemma cong_diff_nat:
14.224 - assumes "(a::nat) >= c" and "b >= d" and "[a = b] (mod m)" and
14.225 - "[c = d] (mod m)"
14.226 - shows "[a - c = b - d] (mod m)"
14.227 -
14.228 - using prems by (rule cong_diff_aux_int [transferred]);
14.229 -
14.230 -lemma cong_mult_nat:
14.231 - "[(a::nat) = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a * c = b * d] (mod m)"
14.232 - apply (unfold cong_nat_def)
14.233 - apply (subst (1 2) mod_mult_eq)
14.234 - apply simp
14.235 -done
14.236 -
14.237 -lemma cong_mult_int:
14.238 - "[(a::int) = b] (mod m) \<Longrightarrow> [c = d] (mod m) \<Longrightarrow> [a * c = b * d] (mod m)"
14.239 - apply (unfold cong_int_def)
14.240 - apply (subst (1 2) zmod_zmult1_eq)
14.241 - apply (subst (1 2) mult_commute)
14.242 - apply (subst (1 2) zmod_zmult1_eq)
14.243 - apply simp
14.244 -done
14.245 -
14.246 -lemma cong_exp_nat: "[(x::nat) = y] (mod n) \<Longrightarrow> [x^k = y^k] (mod n)"
14.247 - apply (induct k)
14.248 - apply (auto simp add: cong_refl_nat cong_mult_nat)
14.249 -done
14.250 -
14.251 -lemma cong_exp_int: "[(x::int) = y] (mod n) \<Longrightarrow> [x^k = y^k] (mod n)"
14.252 - apply (induct k)
14.253 - apply (auto simp add: cong_refl_int cong_mult_int)
14.254 -done
14.255 -
14.256 -lemma cong_setsum_nat [rule_format]:
14.257 - "(ALL x: A. [((f x)::nat) = g x] (mod m)) \<longrightarrow>
14.258 - [(SUM x:A. f x) = (SUM x:A. g x)] (mod m)"
14.259 - apply (case_tac "finite A")
14.260 - apply (induct set: finite)
14.261 - apply (auto intro: cong_add_nat)
14.262 -done
14.263 -
14.264 -lemma cong_setsum_int [rule_format]:
14.265 - "(ALL x: A. [((f x)::int) = g x] (mod m)) \<longrightarrow>
14.266 - [(SUM x:A. f x) = (SUM x:A. g x)] (mod m)"
14.267 - apply (case_tac "finite A")
14.268 - apply (induct set: finite)
14.269 - apply (auto intro: cong_add_int)
14.270 -done
14.271 -
14.272 -lemma cong_setprod_nat [rule_format]:
14.273 - "(ALL x: A. [((f x)::nat) = g x] (mod m)) \<longrightarrow>
14.274 - [(PROD x:A. f x) = (PROD x:A. g x)] (mod m)"
14.275 - apply (case_tac "finite A")
14.276 - apply (induct set: finite)
14.277 - apply (auto intro: cong_mult_nat)
14.278 -done
14.279 -
14.280 -lemma cong_setprod_int [rule_format]:
14.281 - "(ALL x: A. [((f x)::int) = g x] (mod m)) \<longrightarrow>
14.282 - [(PROD x:A. f x) = (PROD x:A. g x)] (mod m)"
14.283 - apply (case_tac "finite A")
14.284 - apply (induct set: finite)
14.285 - apply (auto intro: cong_mult_int)
14.286 -done
14.287 -
14.288 -lemma cong_scalar_nat: "[(a::nat)= b] (mod m) \<Longrightarrow> [a * k = b * k] (mod m)"
14.289 - by (rule cong_mult_nat, simp_all)
14.290 -
14.291 -lemma cong_scalar_int: "[(a::int)= b] (mod m) \<Longrightarrow> [a * k = b * k] (mod m)"
14.292 - by (rule cong_mult_int, simp_all)
14.293 -
14.294 -lemma cong_scalar2_nat: "[(a::nat)= b] (mod m) \<Longrightarrow> [k * a = k * b] (mod m)"
14.295 - by (rule cong_mult_nat, simp_all)
14.296 -
14.297 -lemma cong_scalar2_int: "[(a::int)= b] (mod m) \<Longrightarrow> [k * a = k * b] (mod m)"
14.298 - by (rule cong_mult_int, simp_all)
14.299 -
14.300 -lemma cong_mult_self_nat: "[(a::nat) * m = 0] (mod m)"
14.301 - by (unfold cong_nat_def, auto)
14.302 -
14.303 -lemma cong_mult_self_int: "[(a::int) * m = 0] (mod m)"
14.304 - by (unfold cong_int_def, auto)
14.305 -
14.306 -lemma cong_eq_diff_cong_0_int: "[(a::int) = b] (mod m) = [a - b = 0] (mod m)"
14.307 - apply (rule iffI)
14.308 - apply (erule cong_diff_int [of a b m b b, simplified])
14.309 - apply (erule cong_add_int [of "a - b" 0 m b b, simplified])
14.310 -done
14.311 -
14.312 -lemma cong_eq_diff_cong_0_aux_int: "a >= b \<Longrightarrow>
14.313 - [(a::int) = b] (mod m) = [tsub a b = 0] (mod m)"
14.314 - by (subst tsub_eq, assumption, rule cong_eq_diff_cong_0_int)
14.315 -
14.316 -lemma cong_eq_diff_cong_0_nat:
14.317 - assumes "(a::nat) >= b"
14.318 - shows "[a = b] (mod m) = [a - b = 0] (mod m)"
14.319 -
14.320 - using prems by (rule cong_eq_diff_cong_0_aux_int [transferred])
14.321 -
14.322 -lemma cong_diff_cong_0'_nat:
14.323 - "[(x::nat) = y] (mod n) \<longleftrightarrow>
14.324 - (if x <= y then [y - x = 0] (mod n) else [x - y = 0] (mod n))"
14.325 - apply (case_tac "y <= x")
14.326 - apply (frule cong_eq_diff_cong_0_nat [where m = n])
14.327 - apply auto [1]
14.328 - apply (subgoal_tac "x <= y")
14.329 - apply (frule cong_eq_diff_cong_0_nat [where m = n])
14.330 - apply (subst cong_sym_eq_nat)
14.331 - apply auto
14.332 -done
14.333 -
14.334 -lemma cong_altdef_nat: "(a::nat) >= b \<Longrightarrow> [a = b] (mod m) = (m dvd (a - b))"
14.335 - apply (subst cong_eq_diff_cong_0_nat, assumption)
14.336 - apply (unfold cong_nat_def)
14.337 - apply (simp add: dvd_eq_mod_eq_0 [symmetric])
14.338 -done
14.339 -
14.340 -lemma cong_altdef_int: "[(a::int) = b] (mod m) = (m dvd (a - b))"
14.341 - apply (subst cong_eq_diff_cong_0_int)
14.342 - apply (unfold cong_int_def)
14.343 - apply (simp add: dvd_eq_mod_eq_0 [symmetric])
14.344 -done
14.345 -
14.346 -lemma cong_abs_int: "[(x::int) = y] (mod abs m) = [x = y] (mod m)"
14.347 - by (simp add: cong_altdef_int)
14.348 -
14.349 -lemma cong_square_int:
14.350 - "\<lbrakk> prime (p::int); 0 < a; [a * a = 1] (mod p) \<rbrakk>
14.351 - \<Longrightarrow> [a = 1] (mod p) \<or> [a = - 1] (mod p)"
14.352 - apply (simp only: cong_altdef_int)
14.353 - apply (subst prime_dvd_mult_eq_int [symmetric], assumption)
14.354 - (* any way around this? *)
14.355 - apply (subgoal_tac "a * a - 1 = (a - 1) * (a - -1)")
14.356 - apply (auto simp add: ring_simps)
14.357 -done
14.358 -
14.359 -lemma cong_mult_rcancel_int:
14.360 - "coprime k (m::int) \<Longrightarrow> [a * k = b * k] (mod m) = [a = b] (mod m)"
14.361 - apply (subst (1 2) cong_altdef_int)
14.362 - apply (subst left_diff_distrib [symmetric])
14.363 - apply (rule coprime_dvd_mult_iff_int)
14.364 - apply (subst gcd_commute_int, assumption)
14.365 -done
14.366 -
14.367 -lemma cong_mult_rcancel_nat:
14.368 - assumes "coprime k (m::nat)"
14.369 - shows "[a * k = b * k] (mod m) = [a = b] (mod m)"
14.370 -
14.371 - apply (rule cong_mult_rcancel_int [transferred])
14.372 - using prems apply auto
14.373 -done
14.374 -
14.375 -lemma cong_mult_lcancel_nat:
14.376 - "coprime k (m::nat) \<Longrightarrow> [k * a = k * b ] (mod m) = [a = b] (mod m)"
14.377 - by (simp add: mult_commute cong_mult_rcancel_nat)
14.378 -
14.379 -lemma cong_mult_lcancel_int:
14.380 - "coprime k (m::int) \<Longrightarrow> [k * a = k * b] (mod m) = [a = b] (mod m)"
14.381 - by (simp add: mult_commute cong_mult_rcancel_int)
14.382 -
14.383 -(* was zcong_zgcd_zmult_zmod *)
14.384 -lemma coprime_cong_mult_int:
14.385 - "[(a::int) = b] (mod m) \<Longrightarrow> [a = b] (mod n) \<Longrightarrow> coprime m n
14.386 - \<Longrightarrow> [a = b] (mod m * n)"
14.387 - apply (simp only: cong_altdef_int)
14.388 - apply (erule (2) divides_mult_int)
14.389 -done
14.390 -
14.391 -lemma coprime_cong_mult_nat:
14.392 - assumes "[(a::nat) = b] (mod m)" and "[a = b] (mod n)" and "coprime m n"
14.393 - shows "[a = b] (mod m * n)"
14.394 -
14.395 - apply (rule coprime_cong_mult_int [transferred])
14.396 - using prems apply auto
14.397 -done
14.398 -
14.399 -lemma cong_less_imp_eq_nat: "0 \<le> (a::nat) \<Longrightarrow>
14.400 - a < m \<Longrightarrow> 0 \<le> b \<Longrightarrow> b < m \<Longrightarrow> [a = b] (mod m) \<Longrightarrow> a = b"
14.401 - by (auto simp add: cong_nat_def mod_pos_pos_trivial)
14.402 -
14.403 -lemma cong_less_imp_eq_int: "0 \<le> (a::int) \<Longrightarrow>
14.404 - a < m \<Longrightarrow> 0 \<le> b \<Longrightarrow> b < m \<Longrightarrow> [a = b] (mod m) \<Longrightarrow> a = b"
14.405 - by (auto simp add: cong_int_def mod_pos_pos_trivial)
14.406 -
14.407 -lemma cong_less_unique_nat:
14.408 - "0 < (m::nat) \<Longrightarrow> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m))"
14.409 - apply auto
14.410 - apply (rule_tac x = "a mod m" in exI)
14.411 - apply (unfold cong_nat_def, auto)
14.412 -done
14.413 -
14.414 -lemma cong_less_unique_int:
14.415 - "0 < (m::int) \<Longrightarrow> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m))"
14.416 - apply auto
14.417 - apply (rule_tac x = "a mod m" in exI)
14.418 - apply (unfold cong_int_def, auto simp add: mod_pos_pos_trivial)
14.419 -done
14.420 -
14.421 -lemma cong_iff_lin_int: "([(a::int) = b] (mod m)) = (\<exists>k. b = a + m * k)"
14.422 - apply (auto simp add: cong_altdef_int dvd_def ring_simps)
14.423 - apply (rule_tac [!] x = "-k" in exI, auto)
14.424 -done
14.425 -
14.426 -lemma cong_iff_lin_nat: "([(a::nat) = b] (mod m)) =
14.427 - (\<exists>k1 k2. b + k1 * m = a + k2 * m)"
14.428 - apply (rule iffI)
14.429 - apply (case_tac "b <= a")
14.430 - apply (subst (asm) cong_altdef_nat, assumption)
14.431 - apply (unfold dvd_def, auto)
14.432 - apply (rule_tac x = k in exI)
14.433 - apply (rule_tac x = 0 in exI)
14.434 - apply (auto simp add: ring_simps)
14.435 - apply (subst (asm) cong_sym_eq_nat)
14.436 - apply (subst (asm) cong_altdef_nat)
14.437 - apply force
14.438 - apply (unfold dvd_def, auto)
14.439 - apply (rule_tac x = 0 in exI)
14.440 - apply (rule_tac x = k in exI)
14.441 - apply (auto simp add: ring_simps)
14.442 - apply (unfold cong_nat_def)
14.443 - apply (subgoal_tac "a mod m = (a + k2 * m) mod m")
14.444 - apply (erule ssubst)back
14.445 - apply (erule subst)
14.446 - apply auto
14.447 -done
14.448 -
14.449 -lemma cong_gcd_eq_int: "[(a::int) = b] (mod m) \<Longrightarrow> gcd a m = gcd b m"
14.450 - apply (subst (asm) cong_iff_lin_int, auto)
14.451 - apply (subst add_commute)
14.452 - apply (subst (2) gcd_commute_int)
14.453 - apply (subst mult_commute)
14.454 - apply (subst gcd_add_mult_int)
14.455 - apply (rule gcd_commute_int)
14.456 -done
14.457 -
14.458 -lemma cong_gcd_eq_nat:
14.459 - assumes "[(a::nat) = b] (mod m)"
14.460 - shows "gcd a m = gcd b m"
14.461 -
14.462 - apply (rule cong_gcd_eq_int [transferred])
14.463 - using prems apply auto
14.464 -done
14.465 -
14.466 -lemma cong_imp_coprime_nat: "[(a::nat) = b] (mod m) \<Longrightarrow> coprime a m \<Longrightarrow>
14.467 - coprime b m"
14.468 - by (auto simp add: cong_gcd_eq_nat)
14.469 -
14.470 -lemma cong_imp_coprime_int: "[(a::int) = b] (mod m) \<Longrightarrow> coprime a m \<Longrightarrow>
14.471 - coprime b m"
14.472 - by (auto simp add: cong_gcd_eq_int)
14.473 -
14.474 -lemma cong_cong_mod_nat: "[(a::nat) = b] (mod m) =
14.475 - [a mod m = b mod m] (mod m)"
14.476 - by (auto simp add: cong_nat_def)
14.477 -
14.478 -lemma cong_cong_mod_int: "[(a::int) = b] (mod m) =
14.479 - [a mod m = b mod m] (mod m)"
14.480 - by (auto simp add: cong_int_def)
14.481 -
14.482 -lemma cong_minus_int [iff]: "[(a::int) = b] (mod -m) = [a = b] (mod m)"
14.483 - by (subst (1 2) cong_altdef_int, auto)
14.484 -
14.485 -lemma cong_zero_nat [iff]: "[(a::nat) = b] (mod 0) = (a = b)"
14.486 - by (auto simp add: cong_nat_def)
14.487 -
14.488 -lemma cong_zero_int [iff]: "[(a::int) = b] (mod 0) = (a = b)"
14.489 - by (auto simp add: cong_int_def)
14.490 -
14.491 -(*
14.492 -lemma mod_dvd_mod_int:
14.493 - "0 < (m::int) \<Longrightarrow> m dvd b \<Longrightarrow> (a mod b mod m) = (a mod m)"
14.494 - apply (unfold dvd_def, auto)
14.495 - apply (rule mod_mod_cancel)
14.496 - apply auto
14.497 -done
14.498 -
14.499 -lemma mod_dvd_mod:
14.500 - assumes "0 < (m::nat)" and "m dvd b"
14.501 - shows "(a mod b mod m) = (a mod m)"
14.502 -
14.503 - apply (rule mod_dvd_mod_int [transferred])
14.504 - using prems apply auto
14.505 -done
14.506 -*)
14.507 -
14.508 -lemma cong_add_lcancel_nat:
14.509 - "[(a::nat) + x = a + y] (mod n) \<longleftrightarrow> [x = y] (mod n)"
14.510 - by (simp add: cong_iff_lin_nat)
14.511 -
14.512 -lemma cong_add_lcancel_int:
14.513 - "[(a::int) + x = a + y] (mod n) \<longleftrightarrow> [x = y] (mod n)"
14.514 - by (simp add: cong_iff_lin_int)
14.515 -
14.516 -lemma cong_add_rcancel_nat: "[(x::nat) + a = y + a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
14.517 - by (simp add: cong_iff_lin_nat)
14.518 -
14.519 -lemma cong_add_rcancel_int: "[(x::int) + a = y + a] (mod n) \<longleftrightarrow> [x = y] (mod n)"
14.520 - by (simp add: cong_iff_lin_int)
14.521 -
14.522 -lemma cong_add_lcancel_0_nat: "[(a::nat) + x = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
14.523 - by (simp add: cong_iff_lin_nat)
14.524 -
14.525 -lemma cong_add_lcancel_0_int: "[(a::int) + x = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
14.526 - by (simp add: cong_iff_lin_int)
14.527 -
14.528 -lemma cong_add_rcancel_0_nat: "[x + (a::nat) = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
14.529 - by (simp add: cong_iff_lin_nat)
14.530 -
14.531 -lemma cong_add_rcancel_0_int: "[x + (a::int) = a] (mod n) \<longleftrightarrow> [x = 0] (mod n)"
14.532 - by (simp add: cong_iff_lin_int)
14.533 -
14.534 -lemma cong_dvd_modulus_nat: "[(x::nat) = y] (mod m) \<Longrightarrow> n dvd m \<Longrightarrow>
14.535 - [x = y] (mod n)"
14.536 - apply (auto simp add: cong_iff_lin_nat dvd_def)
14.537 - apply (rule_tac x="k1 * k" in exI)
14.538 - apply (rule_tac x="k2 * k" in exI)
14.539 - apply (simp add: ring_simps)
14.540 -done
14.541 -
14.542 -lemma cong_dvd_modulus_int: "[(x::int) = y] (mod m) \<Longrightarrow> n dvd m \<Longrightarrow>
14.543 - [x = y] (mod n)"
14.544 - by (auto simp add: cong_altdef_int dvd_def)
14.545 -
14.546 -lemma cong_dvd_eq_nat: "[(x::nat) = y] (mod n) \<Longrightarrow> n dvd x \<longleftrightarrow> n dvd y"
14.547 - by (unfold cong_nat_def, auto simp add: dvd_eq_mod_eq_0)
14.548 -
14.549 -lemma cong_dvd_eq_int: "[(x::int) = y] (mod n) \<Longrightarrow> n dvd x \<longleftrightarrow> n dvd y"
14.550 - by (unfold cong_int_def, auto simp add: dvd_eq_mod_eq_0)
14.551 -
14.552 -lemma cong_mod_nat: "(n::nat) ~= 0 \<Longrightarrow> [a mod n = a] (mod n)"
14.553 - by (simp add: cong_nat_def)
14.554 -
14.555 -lemma cong_mod_int: "(n::int) ~= 0 \<Longrightarrow> [a mod n = a] (mod n)"
14.556 - by (simp add: cong_int_def)
14.557 -
14.558 -lemma mod_mult_cong_nat: "(a::nat) ~= 0 \<Longrightarrow> b ~= 0
14.559 - \<Longrightarrow> [x mod (a * b) = y] (mod a) \<longleftrightarrow> [x = y] (mod a)"
14.560 - by (simp add: cong_nat_def mod_mult2_eq mod_add_left_eq)
14.561 -
14.562 -lemma neg_cong_int: "([(a::int) = b] (mod m)) = ([-a = -b] (mod m))"
14.563 - apply (simp add: cong_altdef_int)
14.564 - apply (subst dvd_minus_iff [symmetric])
14.565 - apply (simp add: ring_simps)
14.566 -done
14.567 -
14.568 -lemma cong_modulus_neg_int: "([(a::int) = b] (mod m)) = ([a = b] (mod -m))"
14.569 - by (auto simp add: cong_altdef_int)
14.570 -
14.571 -lemma mod_mult_cong_int: "(a::int) ~= 0 \<Longrightarrow> b ~= 0
14.572 - \<Longrightarrow> [x mod (a * b) = y] (mod a) \<longleftrightarrow> [x = y] (mod a)"
14.573 - apply (case_tac "b > 0")
14.574 - apply (simp add: cong_int_def mod_mod_cancel mod_add_left_eq)
14.575 - apply (subst (1 2) cong_modulus_neg_int)
14.576 - apply (unfold cong_int_def)
14.577 - apply (subgoal_tac "a * b = (-a * -b)")
14.578 - apply (erule ssubst)
14.579 - apply (subst zmod_zmult2_eq)
14.580 - apply (auto simp add: mod_add_left_eq)
14.581 -done
14.582 -
14.583 -lemma cong_to_1_nat: "([(a::nat) = 1] (mod n)) \<Longrightarrow> (n dvd (a - 1))"
14.584 - apply (case_tac "a = 0")
14.585 - apply force
14.586 - apply (subst (asm) cong_altdef_nat)
14.587 - apply auto
14.588 -done
14.589 -
14.590 -lemma cong_0_1_nat: "[(0::nat) = 1] (mod n) = (n = 1)"
14.591 - by (unfold cong_nat_def, auto)
14.592 -
14.593 -lemma cong_0_1_int: "[(0::int) = 1] (mod n) = ((n = 1) | (n = -1))"
14.594 - by (unfold cong_int_def, auto simp add: zmult_eq_1_iff)
14.595 -
14.596 -lemma cong_to_1'_nat: "[(a::nat) = 1] (mod n) \<longleftrightarrow>
14.597 - a = 0 \<and> n = 1 \<or> (\<exists>m. a = 1 + m * n)"
14.598 - apply (case_tac "n = 1")
14.599 - apply auto [1]
14.600 - apply (drule_tac x = "a - 1" in spec)
14.601 - apply force
14.602 - apply (case_tac "a = 0")
14.603 - apply (auto simp add: cong_0_1_nat) [1]
14.604 - apply (rule iffI)
14.605 - apply (drule cong_to_1_nat)
14.606 - apply (unfold dvd_def)
14.607 - apply auto [1]
14.608 - apply (rule_tac x = k in exI)
14.609 - apply (auto simp add: ring_simps) [1]
14.610 - apply (subst cong_altdef_nat)
14.611 - apply (auto simp add: dvd_def)
14.612 -done
14.613 -
14.614 -lemma cong_le_nat: "(y::nat) <= x \<Longrightarrow> [x = y] (mod n) \<longleftrightarrow> (\<exists>q. x = q * n + y)"
14.615 - apply (subst cong_altdef_nat)
14.616 - apply assumption
14.617 - apply (unfold dvd_def, auto simp add: ring_simps)
14.618 - apply (rule_tac x = k in exI)
14.619 - apply auto
14.620 -done
14.621 -
14.622 -lemma cong_solve_nat: "(a::nat) \<noteq> 0 \<Longrightarrow> EX x. [a * x = gcd a n] (mod n)"
14.623 - apply (case_tac "n = 0")
14.624 - apply force
14.625 - apply (frule bezout_nat [of a n], auto)
14.626 - apply (rule exI, erule ssubst)
14.627 - apply (rule cong_trans_nat)
14.628 - apply (rule cong_add_nat)
14.629 - apply (subst mult_commute)
14.630 - apply (rule cong_mult_self_nat)
14.631 - prefer 2
14.632 - apply simp
14.633 - apply (rule cong_refl_nat)
14.634 - apply (rule cong_refl_nat)
14.635 -done
14.636 -
14.637 -lemma cong_solve_int: "(a::int) \<noteq> 0 \<Longrightarrow> EX x. [a * x = gcd a n] (mod n)"
14.638 - apply (case_tac "n = 0")
14.639 - apply (case_tac "a \<ge> 0")
14.640 - apply auto
14.641 - apply (rule_tac x = "-1" in exI)
14.642 - apply auto
14.643 - apply (insert bezout_int [of a n], auto)
14.644 - apply (rule exI)
14.645 - apply (erule subst)
14.646 - apply (rule cong_trans_int)
14.647 - prefer 2
14.648 - apply (rule cong_add_int)
14.649 - apply (rule cong_refl_int)
14.650 - apply (rule cong_sym_int)
14.651 - apply (rule cong_mult_self_int)
14.652 - apply simp
14.653 - apply (subst mult_commute)
14.654 - apply (rule cong_refl_int)
14.655 -done
14.656 -
14.657 -lemma cong_solve_dvd_nat:
14.658 - assumes a: "(a::nat) \<noteq> 0" and b: "gcd a n dvd d"
14.659 - shows "EX x. [a * x = d] (mod n)"
14.660 -proof -
14.661 - from cong_solve_nat [OF a] obtain x where
14.662 - "[a * x = gcd a n](mod n)"
14.663 - by auto
14.664 - hence "[(d div gcd a n) * (a * x) = (d div gcd a n) * gcd a n] (mod n)"
14.665 - by (elim cong_scalar2_nat)
14.666 - also from b have "(d div gcd a n) * gcd a n = d"
14.667 - by (rule dvd_div_mult_self)
14.668 - also have "(d div gcd a n) * (a * x) = a * (d div gcd a n * x)"
14.669 - by auto
14.670 - finally show ?thesis
14.671 - by auto
14.672 -qed
14.673 -
14.674 -lemma cong_solve_dvd_int:
14.675 - assumes a: "(a::int) \<noteq> 0" and b: "gcd a n dvd d"
14.676 - shows "EX x. [a * x = d] (mod n)"
14.677 -proof -
14.678 - from cong_solve_int [OF a] obtain x where
14.679 - "[a * x = gcd a n](mod n)"
14.680 - by auto
14.681 - hence "[(d div gcd a n) * (a * x) = (d div gcd a n) * gcd a n] (mod n)"
14.682 - by (elim cong_scalar2_int)
14.683 - also from b have "(d div gcd a n) * gcd a n = d"
14.684 - by (rule dvd_div_mult_self)
14.685 - also have "(d div gcd a n) * (a * x) = a * (d div gcd a n * x)"
14.686 - by auto
14.687 - finally show ?thesis
14.688 - by auto
14.689 -qed
14.690 -
14.691 -lemma cong_solve_coprime_nat: "coprime (a::nat) n \<Longrightarrow>
14.692 - EX x. [a * x = 1] (mod n)"
14.693 - apply (case_tac "a = 0")
14.694 - apply force
14.695 - apply (frule cong_solve_nat [of a n])
14.696 - apply auto
14.697 -done
14.698 -
14.699 -lemma cong_solve_coprime_int: "coprime (a::int) n \<Longrightarrow>
14.700 - EX x. [a * x = 1] (mod n)"
14.701 - apply (case_tac "a = 0")
14.702 - apply auto
14.703 - apply (case_tac "n \<ge> 0")
14.704 - apply auto
14.705 - apply (subst cong_int_def, auto)
14.706 - apply (frule cong_solve_int [of a n])
14.707 - apply auto
14.708 -done
14.709 -
14.710 -lemma coprime_iff_invertible_nat: "m > (1::nat) \<Longrightarrow> coprime a m =
14.711 - (EX x. [a * x = 1] (mod m))"
14.712 - apply (auto intro: cong_solve_coprime_nat)
14.713 - apply (unfold cong_nat_def, auto intro: invertible_coprime_nat)
14.714 -done
14.715 -
14.716 -lemma coprime_iff_invertible_int: "m > (1::int) \<Longrightarrow> coprime a m =
14.717 - (EX x. [a * x = 1] (mod m))"
14.718 - apply (auto intro: cong_solve_coprime_int)
14.719 - apply (unfold cong_int_def)
14.720 - apply (auto intro: invertible_coprime_int)
14.721 -done
14.722 -
14.723 -lemma coprime_iff_invertible'_int: "m > (1::int) \<Longrightarrow> coprime a m =
14.724 - (EX x. 0 <= x & x < m & [a * x = 1] (mod m))"
14.725 - apply (subst coprime_iff_invertible_int)
14.726 - apply auto
14.727 - apply (auto simp add: cong_int_def)
14.728 - apply (rule_tac x = "x mod m" in exI)
14.729 - apply (auto simp add: mod_mult_right_eq [symmetric])
14.730 -done
14.731 -
14.732 -
14.733 -lemma cong_cong_lcm_nat: "[(x::nat) = y] (mod a) \<Longrightarrow>
14.734 - [x = y] (mod b) \<Longrightarrow> [x = y] (mod lcm a b)"
14.735 - apply (case_tac "y \<le> x")
14.736 - apply (auto simp add: cong_altdef_nat lcm_least_nat) [1]
14.737 - apply (rule cong_sym_nat)
14.738 - apply (subst (asm) (1 2) cong_sym_eq_nat)
14.739 - apply (auto simp add: cong_altdef_nat lcm_least_nat)
14.740 -done
14.741 -
14.742 -lemma cong_cong_lcm_int: "[(x::int) = y] (mod a) \<Longrightarrow>
14.743 - [x = y] (mod b) \<Longrightarrow> [x = y] (mod lcm a b)"
14.744 - by (auto simp add: cong_altdef_int lcm_least_int) [1]
14.745 -
14.746 -lemma cong_cong_coprime_nat: "coprime a b \<Longrightarrow> [(x::nat) = y] (mod a) \<Longrightarrow>
14.747 - [x = y] (mod b) \<Longrightarrow> [x = y] (mod a * b)"
14.748 - apply (frule (1) cong_cong_lcm_nat)back
14.749 - apply (simp add: lcm_nat_def)
14.750 -done
14.751 -
14.752 -lemma cong_cong_coprime_int: "coprime a b \<Longrightarrow> [(x::int) = y] (mod a) \<Longrightarrow>
14.753 - [x = y] (mod b) \<Longrightarrow> [x = y] (mod a * b)"
14.754 - apply (frule (1) cong_cong_lcm_int)back
14.755 - apply (simp add: lcm_altdef_int cong_abs_int abs_mult [symmetric])
14.756 -done
14.757 -
14.758 -lemma cong_cong_setprod_coprime_nat [rule_format]: "finite A \<Longrightarrow>
14.759 - (ALL i:A. (ALL j:A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))) \<longrightarrow>
14.760 - (ALL i:A. [(x::nat) = y] (mod m i)) \<longrightarrow>
14.761 - [x = y] (mod (PROD i:A. m i))"
14.762 - apply (induct set: finite)
14.763 - apply auto
14.764 - apply (rule cong_cong_coprime_nat)
14.765 - apply (subst gcd_commute_nat)
14.766 - apply (rule setprod_coprime_nat)
14.767 - apply auto
14.768 -done
14.769 -
14.770 -lemma cong_cong_setprod_coprime_int [rule_format]: "finite A \<Longrightarrow>
14.771 - (ALL i:A. (ALL j:A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))) \<longrightarrow>
14.772 - (ALL i:A. [(x::int) = y] (mod m i)) \<longrightarrow>
14.773 - [x = y] (mod (PROD i:A. m i))"
14.774 - apply (induct set: finite)
14.775 - apply auto
14.776 - apply (rule cong_cong_coprime_int)
14.777 - apply (subst gcd_commute_int)
14.778 - apply (rule setprod_coprime_int)
14.779 - apply auto
14.780 -done
14.781 -
14.782 -lemma binary_chinese_remainder_aux_nat:
14.783 - assumes a: "coprime (m1::nat) m2"
14.784 - shows "EX b1 b2. [b1 = 1] (mod m1) \<and> [b1 = 0] (mod m2) \<and>
14.785 - [b2 = 0] (mod m1) \<and> [b2 = 1] (mod m2)"
14.786 -proof -
14.787 - from cong_solve_coprime_nat [OF a]
14.788 - obtain x1 where one: "[m1 * x1 = 1] (mod m2)"
14.789 - by auto
14.790 - from a have b: "coprime m2 m1"
14.791 - by (subst gcd_commute_nat)
14.792 - from cong_solve_coprime_nat [OF b]
14.793 - obtain x2 where two: "[m2 * x2 = 1] (mod m1)"
14.794 - by auto
14.795 - have "[m1 * x1 = 0] (mod m1)"
14.796 - by (subst mult_commute, rule cong_mult_self_nat)
14.797 - moreover have "[m2 * x2 = 0] (mod m2)"
14.798 - by (subst mult_commute, rule cong_mult_self_nat)
14.799 - moreover note one two
14.800 - ultimately show ?thesis by blast
14.801 -qed
14.802 -
14.803 -lemma binary_chinese_remainder_aux_int:
14.804 - assumes a: "coprime (m1::int) m2"
14.805 - shows "EX b1 b2. [b1 = 1] (mod m1) \<and> [b1 = 0] (mod m2) \<and>
14.806 - [b2 = 0] (mod m1) \<and> [b2 = 1] (mod m2)"
14.807 -proof -
14.808 - from cong_solve_coprime_int [OF a]
14.809 - obtain x1 where one: "[m1 * x1 = 1] (mod m2)"
14.810 - by auto
14.811 - from a have b: "coprime m2 m1"
14.812 - by (subst gcd_commute_int)
14.813 - from cong_solve_coprime_int [OF b]
14.814 - obtain x2 where two: "[m2 * x2 = 1] (mod m1)"
14.815 - by auto
14.816 - have "[m1 * x1 = 0] (mod m1)"
14.817 - by (subst mult_commute, rule cong_mult_self_int)
14.818 - moreover have "[m2 * x2 = 0] (mod m2)"
14.819 - by (subst mult_commute, rule cong_mult_self_int)
14.820 - moreover note one two
14.821 - ultimately show ?thesis by blast
14.822 -qed
14.823 -
14.824 -lemma binary_chinese_remainder_nat:
14.825 - assumes a: "coprime (m1::nat) m2"
14.826 - shows "EX x. [x = u1] (mod m1) \<and> [x = u2] (mod m2)"
14.827 -proof -
14.828 - from binary_chinese_remainder_aux_nat [OF a] obtain b1 b2
14.829 - where "[b1 = 1] (mod m1)" and "[b1 = 0] (mod m2)" and
14.830 - "[b2 = 0] (mod m1)" and "[b2 = 1] (mod m2)"
14.831 - by blast
14.832 - let ?x = "u1 * b1 + u2 * b2"
14.833 - have "[?x = u1 * 1 + u2 * 0] (mod m1)"
14.834 - apply (rule cong_add_nat)
14.835 - apply (rule cong_scalar2_nat)
14.836 - apply (rule `[b1 = 1] (mod m1)`)
14.837 - apply (rule cong_scalar2_nat)
14.838 - apply (rule `[b2 = 0] (mod m1)`)
14.839 - done
14.840 - hence "[?x = u1] (mod m1)" by simp
14.841 - have "[?x = u1 * 0 + u2 * 1] (mod m2)"
14.842 - apply (rule cong_add_nat)
14.843 - apply (rule cong_scalar2_nat)
14.844 - apply (rule `[b1 = 0] (mod m2)`)
14.845 - apply (rule cong_scalar2_nat)
14.846 - apply (rule `[b2 = 1] (mod m2)`)
14.847 - done
14.848 - hence "[?x = u2] (mod m2)" by simp
14.849 - with `[?x = u1] (mod m1)` show ?thesis by blast
14.850 -qed
14.851 -
14.852 -lemma binary_chinese_remainder_int:
14.853 - assumes a: "coprime (m1::int) m2"
14.854 - shows "EX x. [x = u1] (mod m1) \<and> [x = u2] (mod m2)"
14.855 -proof -
14.856 - from binary_chinese_remainder_aux_int [OF a] obtain b1 b2
14.857 - where "[b1 = 1] (mod m1)" and "[b1 = 0] (mod m2)" and
14.858 - "[b2 = 0] (mod m1)" and "[b2 = 1] (mod m2)"
14.859 - by blast
14.860 - let ?x = "u1 * b1 + u2 * b2"
14.861 - have "[?x = u1 * 1 + u2 * 0] (mod m1)"
14.862 - apply (rule cong_add_int)
14.863 - apply (rule cong_scalar2_int)
14.864 - apply (rule `[b1 = 1] (mod m1)`)
14.865 - apply (rule cong_scalar2_int)
14.866 - apply (rule `[b2 = 0] (mod m1)`)
14.867 - done
14.868 - hence "[?x = u1] (mod m1)" by simp
14.869 - have "[?x = u1 * 0 + u2 * 1] (mod m2)"
14.870 - apply (rule cong_add_int)
14.871 - apply (rule cong_scalar2_int)
14.872 - apply (rule `[b1 = 0] (mod m2)`)
14.873 - apply (rule cong_scalar2_int)
14.874 - apply (rule `[b2 = 1] (mod m2)`)
14.875 - done
14.876 - hence "[?x = u2] (mod m2)" by simp
14.877 - with `[?x = u1] (mod m1)` show ?thesis by blast
14.878 -qed
14.879 -
14.880 -lemma cong_modulus_mult_nat: "[(x::nat) = y] (mod m * n) \<Longrightarrow>
14.881 - [x = y] (mod m)"
14.882 - apply (case_tac "y \<le> x")
14.883 - apply (simp add: cong_altdef_nat)
14.884 - apply (erule dvd_mult_left)
14.885 - apply (rule cong_sym_nat)
14.886 - apply (subst (asm) cong_sym_eq_nat)
14.887 - apply (simp add: cong_altdef_nat)
14.888 - apply (erule dvd_mult_left)
14.889 -done
14.890 -
14.891 -lemma cong_modulus_mult_int: "[(x::int) = y] (mod m * n) \<Longrightarrow>
14.892 - [x = y] (mod m)"
14.893 - apply (simp add: cong_altdef_int)
14.894 - apply (erule dvd_mult_left)
14.895 -done
14.896 -
14.897 -lemma cong_less_modulus_unique_nat:
14.898 - "[(x::nat) = y] (mod m) \<Longrightarrow> x < m \<Longrightarrow> y < m \<Longrightarrow> x = y"
14.899 - by (simp add: cong_nat_def)
14.900 -
14.901 -lemma binary_chinese_remainder_unique_nat:
14.902 - assumes a: "coprime (m1::nat) m2" and
14.903 - nz: "m1 \<noteq> 0" "m2 \<noteq> 0"
14.904 - shows "EX! x. x < m1 * m2 \<and> [x = u1] (mod m1) \<and> [x = u2] (mod m2)"
14.905 -proof -
14.906 - from binary_chinese_remainder_nat [OF a] obtain y where
14.907 - "[y = u1] (mod m1)" and "[y = u2] (mod m2)"
14.908 - by blast
14.909 - let ?x = "y mod (m1 * m2)"
14.910 - from nz have less: "?x < m1 * m2"
14.911 - by auto
14.912 - have one: "[?x = u1] (mod m1)"
14.913 - apply (rule cong_trans_nat)
14.914 - prefer 2
14.915 - apply (rule `[y = u1] (mod m1)`)
14.916 - apply (rule cong_modulus_mult_nat)
14.917 - apply (rule cong_mod_nat)
14.918 - using nz apply auto
14.919 - done
14.920 - have two: "[?x = u2] (mod m2)"
14.921 - apply (rule cong_trans_nat)
14.922 - prefer 2
14.923 - apply (rule `[y = u2] (mod m2)`)
14.924 - apply (subst mult_commute)
14.925 - apply (rule cong_modulus_mult_nat)
14.926 - apply (rule cong_mod_nat)
14.927 - using nz apply auto
14.928 - done
14.929 - have "ALL z. z < m1 * m2 \<and> [z = u1] (mod m1) \<and> [z = u2] (mod m2) \<longrightarrow>
14.930 - z = ?x"
14.931 - proof (clarify)
14.932 - fix z
14.933 - assume "z < m1 * m2"
14.934 - assume "[z = u1] (mod m1)" and "[z = u2] (mod m2)"
14.935 - have "[?x = z] (mod m1)"
14.936 - apply (rule cong_trans_nat)
14.937 - apply (rule `[?x = u1] (mod m1)`)
14.938 - apply (rule cong_sym_nat)
14.939 - apply (rule `[z = u1] (mod m1)`)
14.940 - done
14.941 - moreover have "[?x = z] (mod m2)"
14.942 - apply (rule cong_trans_nat)
14.943 - apply (rule `[?x = u2] (mod m2)`)
14.944 - apply (rule cong_sym_nat)
14.945 - apply (rule `[z = u2] (mod m2)`)
14.946 - done
14.947 - ultimately have "[?x = z] (mod m1 * m2)"
14.948 - by (auto intro: coprime_cong_mult_nat a)
14.949 - with `z < m1 * m2` `?x < m1 * m2` show "z = ?x"
14.950 - apply (intro cong_less_modulus_unique_nat)
14.951 - apply (auto, erule cong_sym_nat)
14.952 - done
14.953 - qed
14.954 - with less one two show ?thesis
14.955 - by auto
14.956 - qed
14.957 -
14.958 -lemma chinese_remainder_aux_nat:
14.959 - fixes A :: "'a set" and
14.960 - m :: "'a \<Rightarrow> nat"
14.961 - assumes fin: "finite A" and
14.962 - cop: "ALL i : A. (ALL j : A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))"
14.963 - shows "EX b. (ALL i : A.
14.964 - [b i = 1] (mod m i) \<and> [b i = 0] (mod (PROD j : A - {i}. m j)))"
14.965 -proof (rule finite_set_choice, rule fin, rule ballI)
14.966 - fix i
14.967 - assume "i : A"
14.968 - with cop have "coprime (PROD j : A - {i}. m j) (m i)"
14.969 - by (intro setprod_coprime_nat, auto)
14.970 - hence "EX x. [(PROD j : A - {i}. m j) * x = 1] (mod m i)"
14.971 - by (elim cong_solve_coprime_nat)
14.972 - then obtain x where "[(PROD j : A - {i}. m j) * x = 1] (mod m i)"
14.973 - by auto
14.974 - moreover have "[(PROD j : A - {i}. m j) * x = 0]
14.975 - (mod (PROD j : A - {i}. m j))"
14.976 - by (subst mult_commute, rule cong_mult_self_nat)
14.977 - ultimately show "\<exists>a. [a = 1] (mod m i) \<and> [a = 0]
14.978 - (mod setprod m (A - {i}))"
14.979 - by blast
14.980 -qed
14.981 -
14.982 -lemma chinese_remainder_nat:
14.983 - fixes A :: "'a set" and
14.984 - m :: "'a \<Rightarrow> nat" and
14.985 - u :: "'a \<Rightarrow> nat"
14.986 - assumes
14.987 - fin: "finite A" and
14.988 - cop: "ALL i:A. (ALL j : A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))"
14.989 - shows "EX x. (ALL i:A. [x = u i] (mod m i))"
14.990 -proof -
14.991 - from chinese_remainder_aux_nat [OF fin cop] obtain b where
14.992 - bprop: "ALL i:A. [b i = 1] (mod m i) \<and>
14.993 - [b i = 0] (mod (PROD j : A - {i}. m j))"
14.994 - by blast
14.995 - let ?x = "SUM i:A. (u i) * (b i)"
14.996 - show "?thesis"
14.997 - proof (rule exI, clarify)
14.998 - fix i
14.999 - assume a: "i : A"
14.1000 - show "[?x = u i] (mod m i)"
14.1001 - proof -
14.1002 - from fin a have "?x = (SUM j:{i}. u j * b j) +
14.1003 - (SUM j:A-{i}. u j * b j)"
14.1004 - by (subst setsum_Un_disjoint [symmetric], auto intro: setsum_cong)
14.1005 - hence "[?x = u i * b i + (SUM j:A-{i}. u j * b j)] (mod m i)"
14.1006 - by auto
14.1007 - also have "[u i * b i + (SUM j:A-{i}. u j * b j) =
14.1008 - u i * 1 + (SUM j:A-{i}. u j * 0)] (mod m i)"
14.1009 - apply (rule cong_add_nat)
14.1010 - apply (rule cong_scalar2_nat)
14.1011 - using bprop a apply blast
14.1012 - apply (rule cong_setsum_nat)
14.1013 - apply (rule cong_scalar2_nat)
14.1014 - using bprop apply auto
14.1015 - apply (rule cong_dvd_modulus_nat)
14.1016 - apply (drule (1) bspec)
14.1017 - apply (erule conjE)
14.1018 - apply assumption
14.1019 - apply (rule dvd_setprod)
14.1020 - using fin a apply auto
14.1021 - done
14.1022 - finally show ?thesis
14.1023 - by simp
14.1024 - qed
14.1025 - qed
14.1026 -qed
14.1027 -
14.1028 -lemma coprime_cong_prod_nat [rule_format]: "finite A \<Longrightarrow>
14.1029 - (ALL i: A. (ALL j: A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))) \<longrightarrow>
14.1030 - (ALL i: A. [(x::nat) = y] (mod m i)) \<longrightarrow>
14.1031 - [x = y] (mod (PROD i:A. m i))"
14.1032 - apply (induct set: finite)
14.1033 - apply auto
14.1034 - apply (erule (1) coprime_cong_mult_nat)
14.1035 - apply (subst gcd_commute_nat)
14.1036 - apply (rule setprod_coprime_nat)
14.1037 - apply auto
14.1038 -done
14.1039 -
14.1040 -lemma chinese_remainder_unique_nat:
14.1041 - fixes A :: "'a set" and
14.1042 - m :: "'a \<Rightarrow> nat" and
14.1043 - u :: "'a \<Rightarrow> nat"
14.1044 - assumes
14.1045 - fin: "finite A" and
14.1046 - nz: "ALL i:A. m i \<noteq> 0" and
14.1047 - cop: "ALL i:A. (ALL j : A. i \<noteq> j \<longrightarrow> coprime (m i) (m j))"
14.1048 - shows "EX! x. x < (PROD i:A. m i) \<and> (ALL i:A. [x = u i] (mod m i))"
14.1049 -proof -
14.1050 - from chinese_remainder_nat [OF fin cop] obtain y where
14.1051 - one: "(ALL i:A. [y = u i] (mod m i))"
14.1052 - by blast
14.1053 - let ?x = "y mod (PROD i:A. m i)"
14.1054 - from fin nz have prodnz: "(PROD i:A. m i) \<noteq> 0"
14.1055 - by auto
14.1056 - hence less: "?x < (PROD i:A. m i)"
14.1057 - by auto
14.1058 - have cong: "ALL i:A. [?x = u i] (mod m i)"
14.1059 - apply auto
14.1060 - apply (rule cong_trans_nat)
14.1061 - prefer 2
14.1062 - using one apply auto
14.1063 - apply (rule cong_dvd_modulus_nat)
14.1064 - apply (rule cong_mod_nat)
14.1065 - using prodnz apply auto
14.1066 - apply (rule dvd_setprod)
14.1067 - apply (rule fin)
14.1068 - apply assumption
14.1069 - done
14.1070 - have unique: "ALL z. z < (PROD i:A. m i) \<and>
14.1071 - (ALL i:A. [z = u i] (mod m i)) \<longrightarrow> z = ?x"
14.1072 - proof (clarify)
14.1073 - fix z
14.1074 - assume zless: "z < (PROD i:A. m i)"
14.1075 - assume zcong: "(ALL i:A. [z = u i] (mod m i))"
14.1076 - have "ALL i:A. [?x = z] (mod m i)"
14.1077 - apply clarify
14.1078 - apply (rule cong_trans_nat)
14.1079 - using cong apply (erule bspec)
14.1080 - apply (rule cong_sym_nat)
14.1081 - using zcong apply auto
14.1082 - done
14.1083 - with fin cop have "[?x = z] (mod (PROD i:A. m i))"
14.1084 - by (intro coprime_cong_prod_nat, auto)
14.1085 - with zless less show "z = ?x"
14.1086 - apply (intro cong_less_modulus_unique_nat)
14.1087 - apply (auto, erule cong_sym_nat)
14.1088 - done
14.1089 - qed
14.1090 - from less cong unique show ?thesis
14.1091 - by blast
14.1092 -qed
14.1093 -
14.1094 -end
15.1 --- a/src/HOL/NewNumberTheory/Fib.thy Tue Sep 01 14:13:34 2009 +0200
15.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
15.3 @@ -1,319 +0,0 @@
15.4 -(* Title: Fib.thy
15.5 - Authors: Lawrence C. Paulson, Jeremy Avigad
15.6 -
15.7 -
15.8 -Defines the fibonacci function.
15.9 -
15.10 -The original "Fib" is due to Lawrence C. Paulson, and was adapted by
15.11 -Jeremy Avigad.
15.12 -*)
15.13 -
15.14 -
15.15 -header {* Fib *}
15.16 -
15.17 -theory Fib
15.18 -imports Binomial
15.19 -begin
15.20 -
15.21 -
15.22 -subsection {* Main definitions *}
15.23 -
15.24 -class fib =
15.25 -
15.26 -fixes
15.27 - fib :: "'a \<Rightarrow> 'a"
15.28 -
15.29 -
15.30 -(* definition for the natural numbers *)
15.31 -
15.32 -instantiation nat :: fib
15.33 -
15.34 -begin
15.35 -
15.36 -fun
15.37 - fib_nat :: "nat \<Rightarrow> nat"
15.38 -where
15.39 - "fib_nat n =
15.40 - (if n = 0 then 0 else
15.41 - (if n = 1 then 1 else
15.42 - fib (n - 1) + fib (n - 2)))"
15.43 -
15.44 -instance proof qed
15.45 -
15.46 -end
15.47 -
15.48 -(* definition for the integers *)
15.49 -
15.50 -instantiation int :: fib
15.51 -
15.52 -begin
15.53 -
15.54 -definition
15.55 - fib_int :: "int \<Rightarrow> int"
15.56 -where
15.57 - "fib_int n = (if n >= 0 then int (fib (nat n)) else 0)"
15.58 -
15.59 -instance proof qed
15.60 -
15.61 -end
15.62 -
15.63 -
15.64 -subsection {* Set up Transfer *}
15.65 -
15.66 -
15.67 -lemma transfer_nat_int_fib:
15.68 - "(x::int) >= 0 \<Longrightarrow> fib (nat x) = nat (fib x)"
15.69 - unfolding fib_int_def by auto
15.70 -
15.71 -lemma transfer_nat_int_fib_closure:
15.72 - "n >= (0::int) \<Longrightarrow> fib n >= 0"
15.73 - by (auto simp add: fib_int_def)
15.74 -
15.75 -declare TransferMorphism_nat_int[transfer add return:
15.76 - transfer_nat_int_fib transfer_nat_int_fib_closure]
15.77 -
15.78 -lemma transfer_int_nat_fib:
15.79 - "fib (int n) = int (fib n)"
15.80 - unfolding fib_int_def by auto
15.81 -
15.82 -lemma transfer_int_nat_fib_closure:
15.83 - "is_nat n \<Longrightarrow> fib n >= 0"
15.84 - unfolding fib_int_def by auto
15.85 -
15.86 -declare TransferMorphism_int_nat[transfer add return:
15.87 - transfer_int_nat_fib transfer_int_nat_fib_closure]
15.88 -
15.89 -
15.90 -subsection {* Fibonacci numbers *}
15.91 -
15.92 -lemma fib_0_nat [simp]: "fib (0::nat) = 0"
15.93 - by simp
15.94 -
15.95 -lemma fib_0_int [simp]: "fib (0::int) = 0"
15.96 - unfolding fib_int_def by simp
15.97 -
15.98 -lemma fib_1_nat [simp]: "fib (1::nat) = 1"
15.99 - by simp
15.100 -
15.101 -lemma fib_Suc_0_nat [simp]: "fib (Suc 0) = Suc 0"
15.102 - by simp
15.103 -
15.104 -lemma fib_1_int [simp]: "fib (1::int) = 1"
15.105 - unfolding fib_int_def by simp
15.106 -
15.107 -lemma fib_reduce_nat: "(n::nat) >= 2 \<Longrightarrow> fib n = fib (n - 1) + fib (n - 2)"
15.108 - by simp
15.109 -
15.110 -declare fib_nat.simps [simp del]
15.111 -
15.112 -lemma fib_reduce_int: "(n::int) >= 2 \<Longrightarrow> fib n = fib (n - 1) + fib (n - 2)"
15.113 - unfolding fib_int_def
15.114 - by (auto simp add: fib_reduce_nat nat_diff_distrib)
15.115 -
15.116 -lemma fib_neg_int [simp]: "(n::int) < 0 \<Longrightarrow> fib n = 0"
15.117 - unfolding fib_int_def by auto
15.118 -
15.119 -lemma fib_2_nat [simp]: "fib (2::nat) = 1"
15.120 - by (subst fib_reduce_nat, auto)
15.121 -
15.122 -lemma fib_2_int [simp]: "fib (2::int) = 1"
15.123 - by (subst fib_reduce_int, auto)
15.124 -
15.125 -lemma fib_plus_2_nat: "fib ((n::nat) + 2) = fib (n + 1) + fib n"
15.126 - by (subst fib_reduce_nat, auto simp add: One_nat_def)
15.127 -(* the need for One_nat_def is due to the natdiff_cancel_numerals
15.128 - procedure *)
15.129 -
15.130 -lemma fib_induct_nat: "P (0::nat) \<Longrightarrow> P (1::nat) \<Longrightarrow>
15.131 - (!!n. P n \<Longrightarrow> P (n + 1) \<Longrightarrow> P (n + 2)) \<Longrightarrow> P n"
15.132 - apply (atomize, induct n rule: nat_less_induct)
15.133 - apply auto
15.134 - apply (case_tac "n = 0", force)
15.135 - apply (case_tac "n = 1", force)
15.136 - apply (subgoal_tac "n >= 2")
15.137 - apply (frule_tac x = "n - 1" in spec)
15.138 - apply (drule_tac x = "n - 2" in spec)
15.139 - apply (drule_tac x = "n - 2" in spec)
15.140 - apply auto
15.141 - apply (auto simp add: One_nat_def) (* again, natdiff_cancel *)
15.142 -done
15.143 -
15.144 -lemma fib_add_nat: "fib ((n::nat) + k + 1) = fib (k + 1) * fib (n + 1) +
15.145 - fib k * fib n"
15.146 - apply (induct n rule: fib_induct_nat)
15.147 - apply auto
15.148 - apply (subst fib_reduce_nat)
15.149 - apply (auto simp add: ring_simps)
15.150 - apply (subst (1 3 5) fib_reduce_nat)
15.151 - apply (auto simp add: ring_simps Suc_eq_plus1)
15.152 -(* hmmm. Why doesn't "n + (1 + (1 + k))" simplify to "n + k + 2"? *)
15.153 - apply (subgoal_tac "n + (k + 2) = n + (1 + (1 + k))")
15.154 - apply (erule ssubst) back back
15.155 - apply (erule ssubst) back
15.156 - apply auto
15.157 -done
15.158 -
15.159 -lemma fib_add'_nat: "fib (n + Suc k) = fib (Suc k) * fib (Suc n) +
15.160 - fib k * fib n"
15.161 - using fib_add_nat by (auto simp add: One_nat_def)
15.162 -
15.163 -
15.164 -(* transfer from nats to ints *)
15.165 -lemma fib_add_int [rule_format]: "(n::int) >= 0 \<Longrightarrow> k >= 0 \<Longrightarrow>
15.166 - fib (n + k + 1) = fib (k + 1) * fib (n + 1) +
15.167 - fib k * fib n "
15.168 -
15.169 - by (rule fib_add_nat [transferred])
15.170 -
15.171 -lemma fib_neq_0_nat: "(n::nat) > 0 \<Longrightarrow> fib n ~= 0"
15.172 - apply (induct n rule: fib_induct_nat)
15.173 - apply (auto simp add: fib_plus_2_nat)
15.174 -done
15.175 -
15.176 -lemma fib_gr_0_nat: "(n::nat) > 0 \<Longrightarrow> fib n > 0"
15.177 - by (frule fib_neq_0_nat, simp)
15.178 -
15.179 -lemma fib_gr_0_int: "(n::int) > 0 \<Longrightarrow> fib n > 0"
15.180 - unfolding fib_int_def by (simp add: fib_gr_0_nat)
15.181 -
15.182 -text {*
15.183 - \medskip Concrete Mathematics, page 278: Cassini's identity. The proof is
15.184 - much easier using integers, not natural numbers!
15.185 -*}
15.186 -
15.187 -lemma fib_Cassini_aux_int: "fib (int n + 2) * fib (int n) -
15.188 - (fib (int n + 1))^2 = (-1)^(n + 1)"
15.189 - apply (induct n)
15.190 - apply (auto simp add: ring_simps power2_eq_square fib_reduce_int
15.191 - power_add)
15.192 -done
15.193 -
15.194 -lemma fib_Cassini_int: "n >= 0 \<Longrightarrow> fib (n + 2) * fib n -
15.195 - (fib (n + 1))^2 = (-1)^(nat n + 1)"
15.196 - by (insert fib_Cassini_aux_int [of "nat n"], auto)
15.197 -
15.198 -(*
15.199 -lemma fib_Cassini'_int: "n >= 0 \<Longrightarrow> fib (n + 2) * fib n =
15.200 - (fib (n + 1))^2 + (-1)^(nat n + 1)"
15.201 - by (frule fib_Cassini_int, simp)
15.202 -*)
15.203 -
15.204 -lemma fib_Cassini'_int: "n >= 0 \<Longrightarrow> fib ((n::int) + 2) * fib n =
15.205 - (if even n then tsub ((fib (n + 1))^2) 1
15.206 - else (fib (n + 1))^2 + 1)"
15.207 - apply (frule fib_Cassini_int, auto simp add: pos_int_even_equiv_nat_even)
15.208 - apply (subst tsub_eq)
15.209 - apply (insert fib_gr_0_int [of "n + 1"], force)
15.210 - apply auto
15.211 -done
15.212 -
15.213 -lemma fib_Cassini_nat: "fib ((n::nat) + 2) * fib n =
15.214 - (if even n then (fib (n + 1))^2 - 1
15.215 - else (fib (n + 1))^2 + 1)"
15.216 -
15.217 - by (rule fib_Cassini'_int [transferred, of n], auto)
15.218 -
15.219 -
15.220 -text {* \medskip Toward Law 6.111 of Concrete Mathematics *}
15.221 -
15.222 -lemma coprime_fib_plus_1_nat: "coprime (fib (n::nat)) (fib (n + 1))"
15.223 - apply (induct n rule: fib_induct_nat)
15.224 - apply auto
15.225 - apply (subst (2) fib_reduce_nat)
15.226 - apply (auto simp add: Suc_eq_plus1) (* again, natdiff_cancel *)
15.227 - apply (subst add_commute, auto)
15.228 - apply (subst gcd_commute_nat, auto simp add: ring_simps)
15.229 -done
15.230 -
15.231 -lemma coprime_fib_Suc_nat: "coprime (fib n) (fib (Suc n))"
15.232 - using coprime_fib_plus_1_nat by (simp add: One_nat_def)
15.233 -
15.234 -lemma coprime_fib_plus_1_int:
15.235 - "n >= 0 \<Longrightarrow> coprime (fib (n::int)) (fib (n + 1))"
15.236 - by (erule coprime_fib_plus_1_nat [transferred])
15.237 -
15.238 -lemma gcd_fib_add_nat: "gcd (fib (m::nat)) (fib (n + m)) = gcd (fib m) (fib n)"
15.239 - apply (simp add: gcd_commute_nat [of "fib m"])
15.240 - apply (rule cases_nat [of _ m])
15.241 - apply simp
15.242 - apply (subst add_assoc [symmetric])
15.243 - apply (simp add: fib_add_nat)
15.244 - apply (subst gcd_commute_nat)
15.245 - apply (subst mult_commute)
15.246 - apply (subst gcd_add_mult_nat)
15.247 - apply (subst gcd_commute_nat)
15.248 - apply (rule gcd_mult_cancel_nat)
15.249 - apply (rule coprime_fib_plus_1_nat)
15.250 -done
15.251 -
15.252 -lemma gcd_fib_add_int [rule_format]: "m >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow>
15.253 - gcd (fib (m::int)) (fib (n + m)) = gcd (fib m) (fib n)"
15.254 - by (erule gcd_fib_add_nat [transferred])
15.255 -
15.256 -lemma gcd_fib_diff_nat: "(m::nat) \<le> n \<Longrightarrow>
15.257 - gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
15.258 - by (simp add: gcd_fib_add_nat [symmetric, of _ "n-m"])
15.259 -
15.260 -lemma gcd_fib_diff_int: "0 <= (m::int) \<Longrightarrow> m \<le> n \<Longrightarrow>
15.261 - gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
15.262 - by (simp add: gcd_fib_add_int [symmetric, of _ "n-m"])
15.263 -
15.264 -lemma gcd_fib_mod_nat: "0 < (m::nat) \<Longrightarrow>
15.265 - gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
15.266 -proof (induct n rule: less_induct)
15.267 - case (less n)
15.268 - from less.prems have pos_m: "0 < m" .
15.269 - show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
15.270 - proof (cases "m < n")
15.271 - case True note m_n = True
15.272 - then have m_n': "m \<le> n" by auto
15.273 - with pos_m have pos_n: "0 < n" by auto
15.274 - with pos_m m_n have diff: "n - m < n" by auto
15.275 - have "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib ((n - m) mod m))"
15.276 - by (simp add: mod_if [of n]) (insert m_n, auto)
15.277 - also have "\<dots> = gcd (fib m) (fib (n - m))"
15.278 - by (simp add: less.hyps diff pos_m)
15.279 - also have "\<dots> = gcd (fib m) (fib n)" by (simp add: gcd_fib_diff_nat m_n')
15.280 - finally show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)" .
15.281 - next
15.282 - case False then show "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
15.283 - by (cases "m = n") auto
15.284 - qed
15.285 -qed
15.286 -
15.287 -lemma gcd_fib_mod_int:
15.288 - assumes "0 < (m::int)" and "0 <= n"
15.289 - shows "gcd (fib m) (fib (n mod m)) = gcd (fib m) (fib n)"
15.290 -
15.291 - apply (rule gcd_fib_mod_nat [transferred])
15.292 - using prems apply auto
15.293 -done
15.294 -
15.295 -lemma fib_gcd_nat: "fib (gcd (m::nat) n) = gcd (fib m) (fib n)"
15.296 - -- {* Law 6.111 *}
15.297 - apply (induct m n rule: gcd_nat_induct)
15.298 - apply (simp_all add: gcd_non_0_nat gcd_commute_nat gcd_fib_mod_nat)
15.299 -done
15.300 -
15.301 -lemma fib_gcd_int: "m >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow>
15.302 - fib (gcd (m::int) n) = gcd (fib m) (fib n)"
15.303 - by (erule fib_gcd_nat [transferred])
15.304 -
15.305 -lemma atMost_plus_one_nat: "{..(k::nat) + 1} = insert (k + 1) {..k}"
15.306 - by auto
15.307 -
15.308 -theorem fib_mult_eq_setsum_nat:
15.309 - "fib ((n::nat) + 1) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
15.310 - apply (induct n)
15.311 - apply (auto simp add: atMost_plus_one_nat fib_plus_2_nat ring_simps)
15.312 -done
15.313 -
15.314 -theorem fib_mult_eq_setsum'_nat:
15.315 - "fib (Suc n) * fib n = (\<Sum>k \<in> {..n}. fib k * fib k)"
15.316 - using fib_mult_eq_setsum_nat by (simp add: One_nat_def)
15.317 -
15.318 -theorem fib_mult_eq_setsum_int [rule_format]:
15.319 - "n >= 0 \<Longrightarrow> fib ((n::int) + 1) * fib n = (\<Sum>k \<in> {0..n}. fib k * fib k)"
15.320 - by (erule fib_mult_eq_setsum_nat [transferred])
15.321 -
15.322 -end
16.1 --- a/src/HOL/NewNumberTheory/MiscAlgebra.thy Tue Sep 01 14:13:34 2009 +0200
16.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
16.3 @@ -1,355 +0,0 @@
16.4 -(* Title: MiscAlgebra.thy
16.5 - Author: Jeremy Avigad
16.6 -
16.7 -These are things that can be added to the Algebra library.
16.8 -*)
16.9 -
16.10 -theory MiscAlgebra
16.11 -imports
16.12 - "~~/src/HOL/Algebra/Ring"
16.13 - "~~/src/HOL/Algebra/FiniteProduct"
16.14 -begin;
16.15 -
16.16 -(* finiteness stuff *)
16.17 -
16.18 -lemma bounded_set1_int [intro]: "finite {(x::int). a < x & x < b & P x}"
16.19 - apply (subgoal_tac "{x. a < x & x < b & P x} <= {a<..<b}")
16.20 - apply (erule finite_subset)
16.21 - apply auto
16.22 -done
16.23 -
16.24 -
16.25 -(* The rest is for the algebra libraries *)
16.26 -
16.27 -(* These go in Group.thy. *)
16.28 -
16.29 -(*
16.30 - Show that the units in any monoid give rise to a group.
16.31 -
16.32 - The file Residues.thy provides some infrastructure to use
16.33 - facts about the unit group within the ring locale.
16.34 -*)
16.35 -
16.36 -
16.37 -constdefs
16.38 - units_of :: "('a, 'b) monoid_scheme => 'a monoid"
16.39 - "units_of G == (| carrier = Units G,
16.40 - Group.monoid.mult = Group.monoid.mult G,
16.41 - one = one G |)";
16.42 -
16.43 -(*
16.44 -
16.45 -lemma (in monoid) Units_mult_closed [intro]:
16.46 - "x : Units G ==> y : Units G ==> x \<otimes> y : Units G"
16.47 - apply (unfold Units_def)
16.48 - apply (clarsimp)
16.49 - apply (rule_tac x = "xaa \<otimes> xa" in bexI)
16.50 - apply auto
16.51 - apply (subst m_assoc)
16.52 - apply auto
16.53 - apply (subst (2) m_assoc [symmetric])
16.54 - apply auto
16.55 - apply (subst m_assoc)
16.56 - apply auto
16.57 - apply (subst (2) m_assoc [symmetric])
16.58 - apply auto
16.59 -done
16.60 -
16.61 -*)
16.62 -
16.63 -lemma (in monoid) units_group: "group(units_of G)"
16.64 - apply (unfold units_of_def)
16.65 - apply (rule groupI)
16.66 - apply auto
16.67 - apply (subst m_assoc)
16.68 - apply auto
16.69 - apply (rule_tac x = "inv x" in bexI)
16.70 - apply auto
16.71 -done
16.72 -
16.73 -lemma (in comm_monoid) units_comm_group: "comm_group(units_of G)"
16.74 - apply (rule group.group_comm_groupI)
16.75 - apply (rule units_group)
16.76 - apply (insert prems)
16.77 - apply (unfold units_of_def Units_def comm_monoid_def comm_monoid_axioms_def)
16.78 - apply auto;
16.79 -done;
16.80 -
16.81 -lemma units_of_carrier: "carrier (units_of G) = Units G"
16.82 - by (unfold units_of_def, auto)
16.83 -
16.84 -lemma units_of_mult: "mult(units_of G) = mult G"
16.85 - by (unfold units_of_def, auto)
16.86 -
16.87 -lemma units_of_one: "one(units_of G) = one G"
16.88 - by (unfold units_of_def, auto)
16.89 -
16.90 -lemma (in monoid) units_of_inv: "x : Units G ==>
16.91 - m_inv (units_of G) x = m_inv G x"
16.92 - apply (rule sym)
16.93 - apply (subst m_inv_def)
16.94 - apply (rule the1_equality)
16.95 - apply (rule ex_ex1I)
16.96 - apply (subst (asm) Units_def)
16.97 - apply auto
16.98 - apply (erule inv_unique)
16.99 - apply auto
16.100 - apply (rule Units_closed)
16.101 - apply (simp_all only: units_of_carrier [symmetric])
16.102 - apply (insert units_group)
16.103 - apply auto
16.104 - apply (subst units_of_mult [symmetric])
16.105 - apply (subst units_of_one [symmetric])
16.106 - apply (erule group.r_inv, assumption)
16.107 - apply (subst units_of_mult [symmetric])
16.108 - apply (subst units_of_one [symmetric])
16.109 - apply (erule group.l_inv, assumption)
16.110 -done
16.111 -
16.112 -lemma (in group) inj_on_const_mult: "a: (carrier G) ==>
16.113 - inj_on (%x. a \<otimes> x) (carrier G)"
16.114 - by (unfold inj_on_def, auto)
16.115 -
16.116 -lemma (in group) surj_const_mult: "a : (carrier G) ==>
16.117 - (%x. a \<otimes> x) ` (carrier G) = (carrier G)"
16.118 - apply (auto simp add: image_def)
16.119 - apply (rule_tac x = "(m_inv G a) \<otimes> x" in bexI)
16.120 - apply auto
16.121 -(* auto should get this. I suppose we need "comm_monoid_simprules"
16.122 - for mult_ac rewriting. *)
16.123 - apply (subst m_assoc [symmetric])
16.124 - apply auto
16.125 -done
16.126 -
16.127 -lemma (in group) l_cancel_one [simp]: "x : carrier G \<Longrightarrow> a : carrier G \<Longrightarrow>
16.128 - (x \<otimes> a = x) = (a = one G)"
16.129 - apply auto
16.130 - apply (subst l_cancel [symmetric])
16.131 - prefer 4
16.132 - apply (erule ssubst)
16.133 - apply auto
16.134 -done
16.135 -
16.136 -lemma (in group) r_cancel_one [simp]: "x : carrier G \<Longrightarrow> a : carrier G \<Longrightarrow>
16.137 - (a \<otimes> x = x) = (a = one G)"
16.138 - apply auto
16.139 - apply (subst r_cancel [symmetric])
16.140 - prefer 4
16.141 - apply (erule ssubst)
16.142 - apply auto
16.143 -done
16.144 -
16.145 -(* Is there a better way to do this? *)
16.146 -
16.147 -lemma (in group) l_cancel_one' [simp]: "x : carrier G \<Longrightarrow> a : carrier G \<Longrightarrow>
16.148 - (x = x \<otimes> a) = (a = one G)"
16.149 - by (subst eq_commute, simp)
16.150 -
16.151 -lemma (in group) r_cancel_one' [simp]: "x : carrier G \<Longrightarrow> a : carrier G \<Longrightarrow>
16.152 - (x = a \<otimes> x) = (a = one G)"
16.153 - by (subst eq_commute, simp)
16.154 -
16.155 -(* This should be generalized to arbitrary groups, not just commutative
16.156 - ones, using Lagrange's theorem. *)
16.157 -
16.158 -lemma (in comm_group) power_order_eq_one:
16.159 - assumes fin [simp]: "finite (carrier G)"
16.160 - and a [simp]: "a : carrier G"
16.161 - shows "a (^) card(carrier G) = one G"
16.162 -proof -
16.163 - have "(\<Otimes>x:carrier G. x) = (\<Otimes>x:carrier G. a \<otimes> x)"
16.164 - by (subst (2) finprod_reindex [symmetric],
16.165 - auto simp add: Pi_def inj_on_const_mult surj_const_mult)
16.166 - also have "\<dots> = (\<Otimes>x:carrier G. a) \<otimes> (\<Otimes>x:carrier G. x)"
16.167 - by (auto simp add: finprod_multf Pi_def)
16.168 - also have "(\<Otimes>x:carrier G. a) = a (^) card(carrier G)"
16.169 - by (auto simp add: finprod_const)
16.170 - finally show ?thesis
16.171 -(* uses the preceeding lemma *)
16.172 - by auto
16.173 -qed
16.174 -
16.175 -
16.176 -(* Miscellaneous *)
16.177 -
16.178 -lemma (in cring) field_intro2: "\<zero>\<^bsub>R\<^esub> ~= \<one>\<^bsub>R\<^esub> \<Longrightarrow> ALL x : carrier R - {\<zero>\<^bsub>R\<^esub>}.
16.179 - x : Units R \<Longrightarrow> field R"
16.180 - apply (unfold_locales)
16.181 - apply (insert prems, auto)
16.182 - apply (rule trans)
16.183 - apply (subgoal_tac "a = (a \<otimes> b) \<otimes> inv b")
16.184 - apply assumption
16.185 - apply (subst m_assoc)
16.186 - apply (auto simp add: Units_r_inv)
16.187 - apply (unfold Units_def)
16.188 - apply auto
16.189 -done
16.190 -
16.191 -lemma (in monoid) inv_char: "x : carrier G \<Longrightarrow> y : carrier G \<Longrightarrow>
16.192 - x \<otimes> y = \<one> \<Longrightarrow> y \<otimes> x = \<one> \<Longrightarrow> inv x = y"
16.193 - apply (subgoal_tac "x : Units G")
16.194 - apply (subgoal_tac "y = inv x \<otimes> \<one>")
16.195 - apply simp
16.196 - apply (erule subst)
16.197 - apply (subst m_assoc [symmetric])
16.198 - apply auto
16.199 - apply (unfold Units_def)
16.200 - apply auto
16.201 -done
16.202 -
16.203 -lemma (in comm_monoid) comm_inv_char: "x : carrier G \<Longrightarrow> y : carrier G \<Longrightarrow>
16.204 - x \<otimes> y = \<one> \<Longrightarrow> inv x = y"
16.205 - apply (rule inv_char)
16.206 - apply auto
16.207 - apply (subst m_comm, auto)
16.208 -done
16.209 -
16.210 -lemma (in ring) inv_neg_one [simp]: "inv (\<ominus> \<one>) = \<ominus> \<one>"
16.211 - apply (rule inv_char)
16.212 - apply (auto simp add: l_minus r_minus)
16.213 -done
16.214 -
16.215 -lemma (in monoid) inv_eq_imp_eq: "x : Units G \<Longrightarrow> y : Units G \<Longrightarrow>
16.216 - inv x = inv y \<Longrightarrow> x = y"
16.217 - apply (subgoal_tac "inv(inv x) = inv(inv y)")
16.218 - apply (subst (asm) Units_inv_inv)+
16.219 - apply auto
16.220 -done
16.221 -
16.222 -lemma (in ring) Units_minus_one_closed [intro]: "\<ominus> \<one> : Units R"
16.223 - apply (unfold Units_def)
16.224 - apply auto
16.225 - apply (rule_tac x = "\<ominus> \<one>" in bexI)
16.226 - apply auto
16.227 - apply (simp add: l_minus r_minus)
16.228 -done
16.229 -
16.230 -lemma (in monoid) inv_one [simp]: "inv \<one> = \<one>"
16.231 - apply (rule inv_char)
16.232 - apply auto
16.233 -done
16.234 -
16.235 -lemma (in ring) inv_eq_neg_one_eq: "x : Units R \<Longrightarrow> (inv x = \<ominus> \<one>) = (x = \<ominus> \<one>)"
16.236 - apply auto
16.237 - apply (subst Units_inv_inv [symmetric])
16.238 - apply auto
16.239 -done
16.240 -
16.241 -lemma (in monoid) inv_eq_one_eq: "x : Units G \<Longrightarrow> (inv x = \<one>) = (x = \<one>)"
16.242 - apply auto
16.243 - apply (subst Units_inv_inv [symmetric])
16.244 - apply auto
16.245 -done
16.246 -
16.247 -
16.248 -(* This goes in FiniteProduct *)
16.249 -
16.250 -lemma (in comm_monoid) finprod_UN_disjoint:
16.251 - "finite I \<Longrightarrow> (ALL i:I. finite (A i)) \<longrightarrow> (ALL i:I. ALL j:I. i ~= j \<longrightarrow>
16.252 - (A i) Int (A j) = {}) \<longrightarrow>
16.253 - (ALL i:I. ALL x: (A i). g x : carrier G) \<longrightarrow>
16.254 - finprod G g (UNION I A) = finprod G (%i. finprod G g (A i)) I"
16.255 - apply (induct set: finite)
16.256 - apply force
16.257 - apply clarsimp
16.258 - apply (subst finprod_Un_disjoint)
16.259 - apply blast
16.260 - apply (erule finite_UN_I)
16.261 - apply blast
16.262 - apply (fastsimp)
16.263 - apply (auto intro!: funcsetI finprod_closed)
16.264 -done
16.265 -
16.266 -lemma (in comm_monoid) finprod_Union_disjoint:
16.267 - "[| finite C; (ALL A:C. finite A & (ALL x:A. f x : carrier G));
16.268 - (ALL A:C. ALL B:C. A ~= B --> A Int B = {}) |]
16.269 - ==> finprod G f (Union C) = finprod G (finprod G f) C"
16.270 - apply (frule finprod_UN_disjoint [of C id f])
16.271 - apply (unfold Union_def id_def, auto)
16.272 -done
16.273 -
16.274 -lemma (in comm_monoid) finprod_one [rule_format]:
16.275 - "finite A \<Longrightarrow> (ALL x:A. f x = \<one>) \<longrightarrow>
16.276 - finprod G f A = \<one>"
16.277 -by (induct set: finite) auto
16.278 -
16.279 -
16.280 -(* need better simplification rules for rings *)
16.281 -(* the next one holds more generally for abelian groups *)
16.282 -
16.283 -lemma (in cring) sum_zero_eq_neg:
16.284 - "x : carrier R \<Longrightarrow> y : carrier R \<Longrightarrow> x \<oplus> y = \<zero> \<Longrightarrow> x = \<ominus> y"
16.285 - apply (subgoal_tac "\<ominus> y = \<zero> \<oplus> \<ominus> y")
16.286 - apply (erule ssubst)back
16.287 - apply (erule subst)
16.288 - apply (simp add: ring_simprules)+
16.289 -done
16.290 -
16.291 -(* there's a name conflict -- maybe "domain" should be
16.292 - "integral_domain" *)
16.293 -
16.294 -lemma (in Ring.domain) square_eq_one:
16.295 - fixes x
16.296 - assumes [simp]: "x : carrier R" and
16.297 - "x \<otimes> x = \<one>"
16.298 - shows "x = \<one> | x = \<ominus>\<one>"
16.299 -proof -
16.300 - have "(x \<oplus> \<one>) \<otimes> (x \<oplus> \<ominus> \<one>) = x \<otimes> x \<oplus> \<ominus> \<one>"
16.301 - by (simp add: ring_simprules)
16.302 - also with `x \<otimes> x = \<one>` have "\<dots> = \<zero>"
16.303 - by (simp add: ring_simprules)
16.304 - finally have "(x \<oplus> \<one>) \<otimes> (x \<oplus> \<ominus> \<one>) = \<zero>" .
16.305 - hence "(x \<oplus> \<one>) = \<zero> | (x \<oplus> \<ominus> \<one>) = \<zero>"
16.306 - by (intro integral, auto)
16.307 - thus ?thesis
16.308 - apply auto
16.309 - apply (erule notE)
16.310 - apply (rule sum_zero_eq_neg)
16.311 - apply auto
16.312 - apply (subgoal_tac "x = \<ominus> (\<ominus> \<one>)")
16.313 - apply (simp add: ring_simprules)
16.314 - apply (rule sum_zero_eq_neg)
16.315 - apply auto
16.316 - done
16.317 -qed
16.318 -
16.319 -lemma (in Ring.domain) inv_eq_self: "x : Units R \<Longrightarrow>
16.320 - x = inv x \<Longrightarrow> x = \<one> | x = \<ominus> \<one>"
16.321 - apply (rule square_eq_one)
16.322 - apply auto
16.323 - apply (erule ssubst)back
16.324 - apply (erule Units_r_inv)
16.325 -done
16.326 -
16.327 -
16.328 -(*
16.329 - The following translates theorems about groups to the facts about
16.330 - the units of a ring. (The list should be expanded as more things are
16.331 - needed.)
16.332 -*)
16.333 -
16.334 -lemma (in ring) finite_ring_finite_units [intro]: "finite (carrier R) \<Longrightarrow>
16.335 - finite (Units R)"
16.336 - by (rule finite_subset, auto)
16.337 -
16.338 -(* this belongs with MiscAlgebra.thy *)
16.339 -lemma (in monoid) units_of_pow:
16.340 - "x : Units G \<Longrightarrow> x (^)\<^bsub>units_of G\<^esub> (n::nat) = x (^)\<^bsub>G\<^esub> n"
16.341 - apply (induct n)
16.342 - apply (auto simp add: units_group group.is_monoid
16.343 - monoid.nat_pow_0 monoid.nat_pow_Suc units_of_one units_of_mult
16.344 - One_nat_def)
16.345 -done
16.346 -
16.347 -lemma (in cring) units_power_order_eq_one: "finite (Units R) \<Longrightarrow> a : Units R
16.348 - \<Longrightarrow> a (^) card(Units R) = \<one>"
16.349 - apply (subst units_of_carrier [symmetric])
16.350 - apply (subst units_of_one [symmetric])
16.351 - apply (subst units_of_pow [symmetric])
16.352 - apply assumption
16.353 - apply (rule comm_group.power_order_eq_one)
16.354 - apply (rule units_comm_group)
16.355 - apply (unfold units_of_def, auto)
16.356 -done
16.357 -
16.358 -end
17.1 --- a/src/HOL/NewNumberTheory/ROOT.ML Tue Sep 01 14:13:34 2009 +0200
17.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
17.3 @@ -1,1 +0,0 @@
17.4 -use_thys ["Fib","Residues"];
18.1 --- a/src/HOL/NewNumberTheory/Residues.thy Tue Sep 01 14:13:34 2009 +0200
18.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
18.3 @@ -1,466 +0,0 @@
18.4 -(* Title: HOL/Library/Residues.thy
18.5 - ID:
18.6 - Author: Jeremy Avigad
18.7 -
18.8 - An algebraic treatment of residue rings, and resulting proofs of
18.9 - Euler's theorem and Wilson's theorem.
18.10 -*)
18.11 -
18.12 -header {* Residue rings *}
18.13 -
18.14 -theory Residues
18.15 -imports
18.16 - UniqueFactorization
18.17 - Binomial
18.18 - MiscAlgebra
18.19 -begin
18.20 -
18.21 -
18.22 -(*
18.23 -
18.24 - A locale for residue rings
18.25 -
18.26 -*)
18.27 -
18.28 -constdefs
18.29 - residue_ring :: "int => int ring"
18.30 - "residue_ring m == (|
18.31 - carrier = {0..m - 1},
18.32 - mult = (%x y. (x * y) mod m),
18.33 - one = 1,
18.34 - zero = 0,
18.35 - add = (%x y. (x + y) mod m) |)"
18.36 -
18.37 -locale residues =
18.38 - fixes m :: int and R (structure)
18.39 - assumes m_gt_one: "m > 1"
18.40 - defines "R == residue_ring m"
18.41 -
18.42 -context residues begin
18.43 -
18.44 -lemma abelian_group: "abelian_group R"
18.45 - apply (insert m_gt_one)
18.46 - apply (rule abelian_groupI)
18.47 - apply (unfold R_def residue_ring_def)
18.48 - apply (auto simp add: mod_pos_pos_trivial mod_add_right_eq [symmetric]
18.49 - add_ac)
18.50 - apply (case_tac "x = 0")
18.51 - apply force
18.52 - apply (subgoal_tac "(x + (m - x)) mod m = 0")
18.53 - apply (erule bexI)
18.54 - apply auto
18.55 -done
18.56 -
18.57 -lemma comm_monoid: "comm_monoid R"
18.58 - apply (insert m_gt_one)
18.59 - apply (unfold R_def residue_ring_def)
18.60 - apply (rule comm_monoidI)
18.61 - apply auto
18.62 - apply (subgoal_tac "x * y mod m * z mod m = z * (x * y mod m) mod m")
18.63 - apply (erule ssubst)
18.64 - apply (subst zmod_zmult1_eq [symmetric])+
18.65 - apply (simp_all only: mult_ac)
18.66 -done
18.67 -
18.68 -lemma cring: "cring R"
18.69 - apply (rule cringI)
18.70 - apply (rule abelian_group)
18.71 - apply (rule comm_monoid)
18.72 - apply (unfold R_def residue_ring_def, auto)
18.73 - apply (subst mod_add_eq [symmetric])
18.74 - apply (subst mult_commute)
18.75 - apply (subst zmod_zmult1_eq [symmetric])
18.76 - apply (simp add: ring_simps)
18.77 -done
18.78 -
18.79 -end
18.80 -
18.81 -sublocale residues < cring
18.82 - by (rule cring)
18.83 -
18.84 -
18.85 -context residues begin
18.86 -
18.87 -(* These lemmas translate back and forth between internal and
18.88 - external concepts *)
18.89 -
18.90 -lemma res_carrier_eq: "carrier R = {0..m - 1}"
18.91 - by (unfold R_def residue_ring_def, auto)
18.92 -
18.93 -lemma res_add_eq: "x \<oplus> y = (x + y) mod m"
18.94 - by (unfold R_def residue_ring_def, auto)
18.95 -
18.96 -lemma res_mult_eq: "x \<otimes> y = (x * y) mod m"
18.97 - by (unfold R_def residue_ring_def, auto)
18.98 -
18.99 -lemma res_zero_eq: "\<zero> = 0"
18.100 - by (unfold R_def residue_ring_def, auto)
18.101 -
18.102 -lemma res_one_eq: "\<one> = 1"
18.103 - by (unfold R_def residue_ring_def units_of_def residue_ring_def, auto)
18.104 -
18.105 -lemma res_units_eq: "Units R = { x. 0 < x & x < m & coprime x m}"
18.106 - apply (insert m_gt_one)
18.107 - apply (unfold Units_def R_def residue_ring_def)
18.108 - apply auto
18.109 - apply (subgoal_tac "x ~= 0")
18.110 - apply auto
18.111 - apply (rule invertible_coprime_int)
18.112 - apply (subgoal_tac "x ~= 0")
18.113 - apply auto
18.114 - apply (subst (asm) coprime_iff_invertible'_int)
18.115 - apply (rule m_gt_one)
18.116 - apply (auto simp add: cong_int_def mult_commute)
18.117 -done
18.118 -
18.119 -lemma res_neg_eq: "\<ominus> x = (- x) mod m"
18.120 - apply (insert m_gt_one)
18.121 - apply (unfold R_def a_inv_def m_inv_def residue_ring_def)
18.122 - apply auto
18.123 - apply (rule the_equality)
18.124 - apply auto
18.125 - apply (subst mod_add_right_eq [symmetric])
18.126 - apply auto
18.127 - apply (subst mod_add_left_eq [symmetric])
18.128 - apply auto
18.129 - apply (subgoal_tac "y mod m = - x mod m")
18.130 - apply simp
18.131 - apply (subst zmod_eq_dvd_iff)
18.132 - apply auto
18.133 -done
18.134 -
18.135 -lemma finite [iff]: "finite(carrier R)"
18.136 - by (subst res_carrier_eq, auto)
18.137 -
18.138 -lemma finite_Units [iff]: "finite(Units R)"
18.139 - by (subst res_units_eq, auto)
18.140 -
18.141 -(* The function a -> a mod m maps the integers to the
18.142 - residue classes. The following lemmas show that this mapping
18.143 - respects addition and multiplication on the integers. *)
18.144 -
18.145 -lemma mod_in_carrier [iff]: "a mod m : carrier R"
18.146 - apply (unfold res_carrier_eq)
18.147 - apply (insert m_gt_one, auto)
18.148 -done
18.149 -
18.150 -lemma add_cong: "(x mod m) \<oplus> (y mod m) = (x + y) mod m"
18.151 - by (unfold R_def residue_ring_def, auto, arith)
18.152 -
18.153 -lemma mult_cong: "(x mod m) \<otimes> (y mod m) = (x * y) mod m"
18.154 - apply (unfold R_def residue_ring_def, auto)
18.155 - apply (subst zmod_zmult1_eq [symmetric])
18.156 - apply (subst mult_commute)
18.157 - apply (subst zmod_zmult1_eq [symmetric])
18.158 - apply (subst mult_commute)
18.159 - apply auto
18.160 -done
18.161 -
18.162 -lemma zero_cong: "\<zero> = 0"
18.163 - apply (unfold R_def residue_ring_def, auto)
18.164 -done
18.165 -
18.166 -lemma one_cong: "\<one> = 1 mod m"
18.167 - apply (insert m_gt_one)
18.168 - apply (unfold R_def residue_ring_def, auto)
18.169 -done
18.170 -
18.171 -(* revise algebra library to use 1? *)
18.172 -lemma pow_cong: "(x mod m) (^) n = x^n mod m"
18.173 - apply (insert m_gt_one)
18.174 - apply (induct n)
18.175 - apply (auto simp add: nat_pow_def one_cong One_nat_def)
18.176 - apply (subst mult_commute)
18.177 - apply (rule mult_cong)
18.178 -done
18.179 -
18.180 -lemma neg_cong: "\<ominus> (x mod m) = (- x) mod m"
18.181 - apply (rule sym)
18.182 - apply (rule sum_zero_eq_neg)
18.183 - apply auto
18.184 - apply (subst add_cong)
18.185 - apply (subst zero_cong)
18.186 - apply auto
18.187 -done
18.188 -
18.189 -lemma (in residues) prod_cong:
18.190 - "finite A \<Longrightarrow> (\<Otimes> i:A. (f i) mod m) = (PROD i:A. f i) mod m"
18.191 - apply (induct set: finite)
18.192 - apply (auto simp: one_cong mult_cong)
18.193 -done
18.194 -
18.195 -lemma (in residues) sum_cong:
18.196 - "finite A \<Longrightarrow> (\<Oplus> i:A. (f i) mod m) = (SUM i: A. f i) mod m"
18.197 - apply (induct set: finite)
18.198 - apply (auto simp: zero_cong add_cong)
18.199 -done
18.200 -
18.201 -lemma mod_in_res_units [simp]: "1 < m \<Longrightarrow> coprime a m \<Longrightarrow>
18.202 - a mod m : Units R"
18.203 - apply (subst res_units_eq, auto)
18.204 - apply (insert pos_mod_sign [of m a])
18.205 - apply (subgoal_tac "a mod m ~= 0")
18.206 - apply arith
18.207 - apply auto
18.208 - apply (subst (asm) gcd_red_int)
18.209 - apply (subst gcd_commute_int, assumption)
18.210 -done
18.211 -
18.212 -lemma res_eq_to_cong: "((a mod m) = (b mod m)) = [a = b] (mod (m::int))"
18.213 - unfolding cong_int_def by auto
18.214 -
18.215 -(* Simplifying with these will translate a ring equation in R to a
18.216 - congruence. *)
18.217 -
18.218 -lemmas res_to_cong_simps = add_cong mult_cong pow_cong one_cong
18.219 - prod_cong sum_cong neg_cong res_eq_to_cong
18.220 -
18.221 -(* Other useful facts about the residue ring *)
18.222 -
18.223 -lemma one_eq_neg_one: "\<one> = \<ominus> \<one> \<Longrightarrow> m = 2"
18.224 - apply (simp add: res_one_eq res_neg_eq)
18.225 - apply (insert m_gt_one)
18.226 - apply (subgoal_tac "~(m > 2)")
18.227 - apply arith
18.228 - apply (rule notI)
18.229 - apply (subgoal_tac "-1 mod m = m - 1")
18.230 - apply force
18.231 - apply (subst mod_add_self2 [symmetric])
18.232 - apply (subst mod_pos_pos_trivial)
18.233 - apply auto
18.234 -done
18.235 -
18.236 -end
18.237 -
18.238 -
18.239 -(* prime residues *)
18.240 -
18.241 -locale residues_prime =
18.242 - fixes p :: int and R (structure)
18.243 - assumes p_prime [intro]: "prime p"
18.244 - defines "R == residue_ring p"
18.245 -
18.246 -sublocale residues_prime < residues p
18.247 - apply (unfold R_def residues_def)
18.248 - using p_prime apply auto
18.249 -done
18.250 -
18.251 -context residues_prime begin
18.252 -
18.253 -lemma is_field: "field R"
18.254 - apply (rule cring.field_intro2)
18.255 - apply (rule cring)
18.256 - apply (auto simp add: res_carrier_eq res_one_eq res_zero_eq
18.257 - res_units_eq)
18.258 - apply (rule classical)
18.259 - apply (erule notE)
18.260 - apply (subst gcd_commute_int)
18.261 - apply (rule prime_imp_coprime_int)
18.262 - apply (rule p_prime)
18.263 - apply (rule notI)
18.264 - apply (frule zdvd_imp_le)
18.265 - apply auto
18.266 -done
18.267 -
18.268 -lemma res_prime_units_eq: "Units R = {1..p - 1}"
18.269 - apply (subst res_units_eq)
18.270 - apply auto
18.271 - apply (subst gcd_commute_int)
18.272 - apply (rule prime_imp_coprime_int)
18.273 - apply (rule p_prime)
18.274 - apply (rule zdvd_not_zless)
18.275 - apply auto
18.276 -done
18.277 -
18.278 -end
18.279 -
18.280 -sublocale residues_prime < field
18.281 - by (rule is_field)
18.282 -
18.283 -
18.284 -(*
18.285 - Test cases: Euler's theorem and Wilson's theorem.
18.286 -*)
18.287 -
18.288 -
18.289 -subsection{* Euler's theorem *}
18.290 -
18.291 -(* the definition of the phi function *)
18.292 -
18.293 -constdefs
18.294 - phi :: "int => nat"
18.295 - "phi m == card({ x. 0 < x & x < m & gcd x m = 1})"
18.296 -
18.297 -lemma phi_zero [simp]: "phi 0 = 0"
18.298 - apply (subst phi_def)
18.299 -(* Auto hangs here. Once again, where is the simplification rule
18.300 - 1 == Suc 0 coming from? *)
18.301 - apply (auto simp add: card_eq_0_iff)
18.302 -(* Add card_eq_0_iff as a simp rule? delete card_empty_imp? *)
18.303 -done
18.304 -
18.305 -lemma phi_one [simp]: "phi 1 = 0"
18.306 - apply (auto simp add: phi_def card_eq_0_iff)
18.307 -done
18.308 -
18.309 -lemma (in residues) phi_eq: "phi m = card(Units R)"
18.310 - by (simp add: phi_def res_units_eq)
18.311 -
18.312 -lemma (in residues) euler_theorem1:
18.313 - assumes a: "gcd a m = 1"
18.314 - shows "[a^phi m = 1] (mod m)"
18.315 -proof -
18.316 - from a m_gt_one have [simp]: "a mod m : Units R"
18.317 - by (intro mod_in_res_units)
18.318 - from phi_eq have "(a mod m) (^) (phi m) = (a mod m) (^) (card (Units R))"
18.319 - by simp
18.320 - also have "\<dots> = \<one>"
18.321 - by (intro units_power_order_eq_one, auto)
18.322 - finally show ?thesis
18.323 - by (simp add: res_to_cong_simps)
18.324 -qed
18.325 -
18.326 -(* In fact, there is a two line proof!
18.327 -
18.328 -lemma (in residues) euler_theorem1:
18.329 - assumes a: "gcd a m = 1"
18.330 - shows "[a^phi m = 1] (mod m)"
18.331 -proof -
18.332 - have "(a mod m) (^) (phi m) = \<one>"
18.333 - by (simp add: phi_eq units_power_order_eq_one a m_gt_one)
18.334 - thus ?thesis
18.335 - by (simp add: res_to_cong_simps)
18.336 -qed
18.337 -
18.338 -*)
18.339 -
18.340 -(* outside the locale, we can relax the restriction m > 1 *)
18.341 -
18.342 -lemma euler_theorem:
18.343 - assumes "m >= 0" and "gcd a m = 1"
18.344 - shows "[a^phi m = 1] (mod m)"
18.345 -proof (cases)
18.346 - assume "m = 0 | m = 1"
18.347 - thus ?thesis by auto
18.348 -next
18.349 - assume "~(m = 0 | m = 1)"
18.350 - with prems show ?thesis
18.351 - by (intro residues.euler_theorem1, unfold residues_def, auto)
18.352 -qed
18.353 -
18.354 -lemma (in residues_prime) phi_prime: "phi p = (nat p - 1)"
18.355 - apply (subst phi_eq)
18.356 - apply (subst res_prime_units_eq)
18.357 - apply auto
18.358 -done
18.359 -
18.360 -lemma phi_prime: "prime p \<Longrightarrow> phi p = (nat p - 1)"
18.361 - apply (rule residues_prime.phi_prime)
18.362 - apply (erule residues_prime.intro)
18.363 -done
18.364 -
18.365 -lemma fermat_theorem:
18.366 - assumes "prime p" and "~ (p dvd a)"
18.367 - shows "[a^(nat p - 1) = 1] (mod p)"
18.368 -proof -
18.369 - from prems have "[a^phi p = 1] (mod p)"
18.370 - apply (intro euler_theorem)
18.371 - (* auto should get this next part. matching across
18.372 - substitutions is needed. *)
18.373 - apply (frule prime_gt_1_int, arith)
18.374 - apply (subst gcd_commute_int, erule prime_imp_coprime_int, assumption)
18.375 - done
18.376 - also have "phi p = nat p - 1"
18.377 - by (rule phi_prime, rule prems)
18.378 - finally show ?thesis .
18.379 -qed
18.380 -
18.381 -
18.382 -subsection {* Wilson's theorem *}
18.383 -
18.384 -lemma (in field) inv_pair_lemma: "x : Units R \<Longrightarrow> y : Units R \<Longrightarrow>
18.385 - {x, inv x} ~= {y, inv y} \<Longrightarrow> {x, inv x} Int {y, inv y} = {}"
18.386 - apply auto
18.387 - apply (erule notE)
18.388 - apply (erule inv_eq_imp_eq)
18.389 - apply auto
18.390 - apply (erule notE)
18.391 - apply (erule inv_eq_imp_eq)
18.392 - apply auto
18.393 -done
18.394 -
18.395 -lemma (in residues_prime) wilson_theorem1:
18.396 - assumes a: "p > 2"
18.397 - shows "[fact (p - 1) = - 1] (mod p)"
18.398 -proof -
18.399 - let ?InversePairs = "{ {x, inv x} | x. x : Units R - {\<one>, \<ominus> \<one>}}"
18.400 - have UR: "Units R = {\<one>, \<ominus> \<one>} Un (Union ?InversePairs)"
18.401 - by auto
18.402 - have "(\<Otimes>i: Units R. i) =
18.403 - (\<Otimes>i: {\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i: Union ?InversePairs. i)"
18.404 - apply (subst UR)
18.405 - apply (subst finprod_Un_disjoint)
18.406 - apply (auto intro:funcsetI)
18.407 - apply (drule sym, subst (asm) inv_eq_one_eq)
18.408 - apply auto
18.409 - apply (drule sym, subst (asm) inv_eq_neg_one_eq)
18.410 - apply auto
18.411 - done
18.412 - also have "(\<Otimes>i: {\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
18.413 - apply (subst finprod_insert)
18.414 - apply auto
18.415 - apply (frule one_eq_neg_one)
18.416 - apply (insert a, force)
18.417 - done
18.418 - also have "(\<Otimes>i:(Union ?InversePairs). i) =
18.419 - (\<Otimes> A: ?InversePairs. (\<Otimes> y:A. y))"
18.420 - apply (subst finprod_Union_disjoint)
18.421 - apply force
18.422 - apply force
18.423 - apply clarify
18.424 - apply (rule inv_pair_lemma)
18.425 - apply auto
18.426 - done
18.427 - also have "\<dots> = \<one>"
18.428 - apply (rule finprod_one)
18.429 - apply auto
18.430 - apply (subst finprod_insert)
18.431 - apply auto
18.432 - apply (frule inv_eq_self)
18.433 - apply (auto)
18.434 - done
18.435 - finally have "(\<Otimes>i: Units R. i) = \<ominus> \<one>"
18.436 - by simp
18.437 - also have "(\<Otimes>i: Units R. i) = (\<Otimes>i: Units R. i mod p)"
18.438 - apply (rule finprod_cong')
18.439 - apply (auto)
18.440 - apply (subst (asm) res_prime_units_eq)
18.441 - apply auto
18.442 - done
18.443 - also have "\<dots> = (PROD i: Units R. i) mod p"
18.444 - apply (rule prod_cong)
18.445 - apply auto
18.446 - done
18.447 - also have "\<dots> = fact (p - 1) mod p"
18.448 - apply (subst fact_altdef_int)
18.449 - apply (insert prems, force)
18.450 - apply (subst res_prime_units_eq, rule refl)
18.451 - done
18.452 - finally have "fact (p - 1) mod p = \<ominus> \<one>".
18.453 - thus ?thesis
18.454 - by (simp add: res_to_cong_simps)
18.455 -qed
18.456 -
18.457 -lemma wilson_theorem: "prime (p::int) \<Longrightarrow> [fact (p - 1) = - 1] (mod p)"
18.458 - apply (frule prime_gt_1_int)
18.459 - apply (case_tac "p = 2")
18.460 - apply (subst fact_altdef_int, simp)
18.461 - apply (subst cong_int_def)
18.462 - apply simp
18.463 - apply (rule residues_prime.wilson_theorem1)
18.464 - apply (rule residues_prime.intro)
18.465 - apply auto
18.466 -done
18.467 -
18.468 -
18.469 -end
19.1 --- a/src/HOL/NewNumberTheory/UniqueFactorization.thy Tue Sep 01 14:13:34 2009 +0200
19.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
19.3 @@ -1,967 +0,0 @@
19.4 -(* Title: UniqueFactorization.thy
19.5 - ID:
19.6 - Author: Jeremy Avigad
19.7 -
19.8 -
19.9 - Unique factorization for the natural numbers and the integers.
19.10 -
19.11 - Note: there were previous Isabelle formalizations of unique
19.12 - factorization due to Thomas Marthedal Rasmussen, and, building on
19.13 - that, by Jeremy Avigad and David Gray.
19.14 -*)
19.15 -
19.16 -header {* UniqueFactorization *}
19.17 -
19.18 -theory UniqueFactorization
19.19 -imports Cong Multiset
19.20 -begin
19.21 -
19.22 -(* inherited from Multiset *)
19.23 -declare One_nat_def [simp del]
19.24 -
19.25 -(* As a simp or intro rule,
19.26 -
19.27 - prime p \<Longrightarrow> p > 0
19.28 -
19.29 - wreaks havoc here. When the premise includes ALL x :# M. prime x, it
19.30 - leads to the backchaining
19.31 -
19.32 - x > 0
19.33 - prime x
19.34 - x :# M which is, unfortunately,
19.35 - count M x > 0
19.36 -*)
19.37 -
19.38 -
19.39 -(* useful facts *)
19.40 -
19.41 -lemma setsum_Un2: "finite (A Un B) \<Longrightarrow>
19.42 - setsum f (A Un B) = setsum f (A - B) + setsum f (B - A) +
19.43 - setsum f (A Int B)"
19.44 - apply (subgoal_tac "A Un B = (A - B) Un (B - A) Un (A Int B)")
19.45 - apply (erule ssubst)
19.46 - apply (subst setsum_Un_disjoint)
19.47 - apply auto
19.48 - apply (subst setsum_Un_disjoint)
19.49 - apply auto
19.50 -done
19.51 -
19.52 -lemma setprod_Un2: "finite (A Un B) \<Longrightarrow>
19.53 - setprod f (A Un B) = setprod f (A - B) * setprod f (B - A) *
19.54 - setprod f (A Int B)"
19.55 - apply (subgoal_tac "A Un B = (A - B) Un (B - A) Un (A Int B)")
19.56 - apply (erule ssubst)
19.57 - apply (subst setprod_Un_disjoint)
19.58 - apply auto
19.59 - apply (subst setprod_Un_disjoint)
19.60 - apply auto
19.61 -done
19.62 -
19.63 -(* Should this go in Multiset.thy? *)
19.64 -(* TN: No longer an intro-rule; needed only once and might get in the way *)
19.65 -lemma multiset_eqI: "[| !!x. count M x = count N x |] ==> M = N"
19.66 - by (subst multiset_eq_conv_count_eq, blast)
19.67 -
19.68 -(* Here is a version of set product for multisets. Is it worth moving
19.69 - to multiset.thy? If so, one should similarly define msetsum for abelian
19.70 - semirings, using of_nat. Also, is it worth developing bounded quantifiers
19.71 - "ALL i :# M. P i"?
19.72 -*)
19.73 -
19.74 -constdefs
19.75 - msetprod :: "('a => ('b::{power,comm_monoid_mult})) => 'a multiset => 'b"
19.76 - "msetprod f M == setprod (%x. (f x)^(count M x)) (set_of M)"
19.77 -
19.78 -syntax
19.79 - "_msetprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"
19.80 - ("(3PROD _:#_. _)" [0, 51, 10] 10)
19.81 -
19.82 -translations
19.83 - "PROD i :# A. b" == "msetprod (%i. b) A"
19.84 -
19.85 -lemma msetprod_Un: "msetprod f (A+B) = msetprod f A * msetprod f B"
19.86 - apply (simp add: msetprod_def power_add)
19.87 - apply (subst setprod_Un2)
19.88 - apply auto
19.89 - apply (subgoal_tac
19.90 - "(PROD x:set_of A - set_of B. f x ^ count A x * f x ^ count B x) =
19.91 - (PROD x:set_of A - set_of B. f x ^ count A x)")
19.92 - apply (erule ssubst)
19.93 - apply (subgoal_tac
19.94 - "(PROD x:set_of B - set_of A. f x ^ count A x * f x ^ count B x) =
19.95 - (PROD x:set_of B - set_of A. f x ^ count B x)")
19.96 - apply (erule ssubst)
19.97 - apply (subgoal_tac "(PROD x:set_of A. f x ^ count A x) =
19.98 - (PROD x:set_of A - set_of B. f x ^ count A x) *
19.99 - (PROD x:set_of A Int set_of B. f x ^ count A x)")
19.100 - apply (erule ssubst)
19.101 - apply (subgoal_tac "(PROD x:set_of B. f x ^ count B x) =
19.102 - (PROD x:set_of B - set_of A. f x ^ count B x) *
19.103 - (PROD x:set_of A Int set_of B. f x ^ count B x)")
19.104 - apply (erule ssubst)
19.105 - apply (subst setprod_timesf)
19.106 - apply (force simp add: mult_ac)
19.107 - apply (subst setprod_Un_disjoint [symmetric])
19.108 - apply (auto intro: setprod_cong)
19.109 - apply (subst setprod_Un_disjoint [symmetric])
19.110 - apply (auto intro: setprod_cong)
19.111 -done
19.112 -
19.113 -
19.114 -subsection {* unique factorization: multiset version *}
19.115 -
19.116 -lemma multiset_prime_factorization_exists [rule_format]: "n > 0 -->
19.117 - (EX M. (ALL (p::nat) : set_of M. prime p) & n = (PROD i :# M. i))"
19.118 -proof (rule nat_less_induct, clarify)
19.119 - fix n :: nat
19.120 - assume ih: "ALL m < n. 0 < m --> (EX M. (ALL p : set_of M. prime p) & m =
19.121 - (PROD i :# M. i))"
19.122 - assume "(n::nat) > 0"
19.123 - then have "n = 1 | (n > 1 & prime n) | (n > 1 & ~ prime n)"
19.124 - by arith
19.125 - moreover
19.126 - {
19.127 - assume "n = 1"
19.128 - then have "(ALL p : set_of {#}. prime p) & n = (PROD i :# {#}. i)"
19.129 - by (auto simp add: msetprod_def)
19.130 - }
19.131 - moreover
19.132 - {
19.133 - assume "n > 1" and "prime n"
19.134 - then have "(ALL p : set_of {# n #}. prime p) & n = (PROD i :# {# n #}. i)"
19.135 - by (auto simp add: msetprod_def)
19.136 - }
19.137 - moreover
19.138 - {
19.139 - assume "n > 1" and "~ prime n"
19.140 - from prems not_prime_eq_prod_nat
19.141 - obtain m k where "n = m * k & 1 < m & m < n & 1 < k & k < n"
19.142 - by blast
19.143 - with ih obtain Q R where "(ALL p : set_of Q. prime p) & m = (PROD i:#Q. i)"
19.144 - and "(ALL p: set_of R. prime p) & k = (PROD i:#R. i)"
19.145 - by blast
19.146 - hence "(ALL p: set_of (Q + R). prime p) & n = (PROD i :# Q + R. i)"
19.147 - by (auto simp add: prems msetprod_Un set_of_union)
19.148 - then have "EX M. (ALL p : set_of M. prime p) & n = (PROD i :# M. i)"..
19.149 - }
19.150 - ultimately show "EX M. (ALL p : set_of M. prime p) & n = (PROD i::nat:#M. i)"
19.151 - by blast
19.152 -qed
19.153 -
19.154 -lemma multiset_prime_factorization_unique_aux:
19.155 - fixes a :: nat
19.156 - assumes "(ALL p : set_of M. prime p)" and
19.157 - "(ALL p : set_of N. prime p)" and
19.158 - "(PROD i :# M. i) dvd (PROD i:# N. i)"
19.159 - shows
19.160 - "count M a <= count N a"
19.161 -proof cases
19.162 - assume "a : set_of M"
19.163 - with prems have a: "prime a"
19.164 - by auto
19.165 - with prems have "a ^ count M a dvd (PROD i :# M. i)"
19.166 - by (auto intro: dvd_setprod simp add: msetprod_def)
19.167 - also have "... dvd (PROD i :# N. i)"
19.168 - by (rule prems)
19.169 - also have "... = (PROD i : (set_of N). i ^ (count N i))"
19.170 - by (simp add: msetprod_def)
19.171 - also have "... =
19.172 - a^(count N a) * (PROD i : (set_of N - {a}). i ^ (count N i))"
19.173 - proof (cases)
19.174 - assume "a : set_of N"
19.175 - hence b: "set_of N = {a} Un (set_of N - {a})"
19.176 - by auto
19.177 - thus ?thesis
19.178 - by (subst (1) b, subst setprod_Un_disjoint, auto)
19.179 - next
19.180 - assume "a ~: set_of N"
19.181 - thus ?thesis
19.182 - by auto
19.183 - qed
19.184 - finally have "a ^ count M a dvd
19.185 - a^(count N a) * (PROD i : (set_of N - {a}). i ^ (count N i))".
19.186 - moreover have "coprime (a ^ count M a)
19.187 - (PROD i : (set_of N - {a}). i ^ (count N i))"
19.188 - apply (subst gcd_commute_nat)
19.189 - apply (rule setprod_coprime_nat)
19.190 - apply (rule primes_imp_powers_coprime_nat)
19.191 - apply (insert prems, auto)
19.192 - done
19.193 - ultimately have "a ^ count M a dvd a^(count N a)"
19.194 - by (elim coprime_dvd_mult_nat)
19.195 - with a show ?thesis
19.196 - by (intro power_dvd_imp_le, auto)
19.197 -next
19.198 - assume "a ~: set_of M"
19.199 - thus ?thesis by auto
19.200 -qed
19.201 -
19.202 -lemma multiset_prime_factorization_unique:
19.203 - assumes "(ALL (p::nat) : set_of M. prime p)" and
19.204 - "(ALL p : set_of N. prime p)" and
19.205 - "(PROD i :# M. i) = (PROD i:# N. i)"
19.206 - shows
19.207 - "M = N"
19.208 -proof -
19.209 - {
19.210 - fix a
19.211 - from prems have "count M a <= count N a"
19.212 - by (intro multiset_prime_factorization_unique_aux, auto)
19.213 - moreover from prems have "count N a <= count M a"
19.214 - by (intro multiset_prime_factorization_unique_aux, auto)
19.215 - ultimately have "count M a = count N a"
19.216 - by auto
19.217 - }
19.218 - thus ?thesis by (simp add:multiset_eq_conv_count_eq)
19.219 -qed
19.220 -
19.221 -constdefs
19.222 - multiset_prime_factorization :: "nat => nat multiset"
19.223 - "multiset_prime_factorization n ==
19.224 - if n > 0 then (THE M. ((ALL p : set_of M. prime p) &
19.225 - n = (PROD i :# M. i)))
19.226 - else {#}"
19.227 -
19.228 -lemma multiset_prime_factorization: "n > 0 ==>
19.229 - (ALL p : set_of (multiset_prime_factorization n). prime p) &
19.230 - n = (PROD i :# (multiset_prime_factorization n). i)"
19.231 - apply (unfold multiset_prime_factorization_def)
19.232 - apply clarsimp
19.233 - apply (frule multiset_prime_factorization_exists)
19.234 - apply clarify
19.235 - apply (rule theI)
19.236 - apply (insert multiset_prime_factorization_unique, blast)+
19.237 -done
19.238 -
19.239 -
19.240 -subsection {* Prime factors and multiplicity for nats and ints *}
19.241 -
19.242 -class unique_factorization =
19.243 -
19.244 -fixes
19.245 - multiplicity :: "'a \<Rightarrow> 'a \<Rightarrow> nat" and
19.246 - prime_factors :: "'a \<Rightarrow> 'a set"
19.247 -
19.248 -(* definitions for the natural numbers *)
19.249 -
19.250 -instantiation nat :: unique_factorization
19.251 -
19.252 -begin
19.253 -
19.254 -definition
19.255 - multiplicity_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
19.256 -where
19.257 - "multiplicity_nat p n = count (multiset_prime_factorization n) p"
19.258 -
19.259 -definition
19.260 - prime_factors_nat :: "nat \<Rightarrow> nat set"
19.261 -where
19.262 - "prime_factors_nat n = set_of (multiset_prime_factorization n)"
19.263 -
19.264 -instance proof qed
19.265 -
19.266 -end
19.267 -
19.268 -(* definitions for the integers *)
19.269 -
19.270 -instantiation int :: unique_factorization
19.271 -
19.272 -begin
19.273 -
19.274 -definition
19.275 - multiplicity_int :: "int \<Rightarrow> int \<Rightarrow> nat"
19.276 -where
19.277 - "multiplicity_int p n = multiplicity (nat p) (nat n)"
19.278 -
19.279 -definition
19.280 - prime_factors_int :: "int \<Rightarrow> int set"
19.281 -where
19.282 - "prime_factors_int n = int ` (prime_factors (nat n))"
19.283 -
19.284 -instance proof qed
19.285 -
19.286 -end
19.287 -
19.288 -
19.289 -subsection {* Set up transfer *}
19.290 -
19.291 -lemma transfer_nat_int_prime_factors:
19.292 - "prime_factors (nat n) = nat ` prime_factors n"
19.293 - unfolding prime_factors_int_def apply auto
19.294 - by (subst transfer_int_nat_set_return_embed, assumption)
19.295 -
19.296 -lemma transfer_nat_int_prime_factors_closure: "n >= 0 \<Longrightarrow>
19.297 - nat_set (prime_factors n)"
19.298 - by (auto simp add: nat_set_def prime_factors_int_def)
19.299 -
19.300 -lemma transfer_nat_int_multiplicity: "p >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow>
19.301 - multiplicity (nat p) (nat n) = multiplicity p n"
19.302 - by (auto simp add: multiplicity_int_def)
19.303 -
19.304 -declare TransferMorphism_nat_int[transfer add return:
19.305 - transfer_nat_int_prime_factors transfer_nat_int_prime_factors_closure
19.306 - transfer_nat_int_multiplicity]
19.307 -
19.308 -
19.309 -lemma transfer_int_nat_prime_factors:
19.310 - "prime_factors (int n) = int ` prime_factors n"
19.311 - unfolding prime_factors_int_def by auto
19.312 -
19.313 -lemma transfer_int_nat_prime_factors_closure: "is_nat n \<Longrightarrow>
19.314 - nat_set (prime_factors n)"
19.315 - by (simp only: transfer_nat_int_prime_factors_closure is_nat_def)
19.316 -
19.317 -lemma transfer_int_nat_multiplicity:
19.318 - "multiplicity (int p) (int n) = multiplicity p n"
19.319 - by (auto simp add: multiplicity_int_def)
19.320 -
19.321 -declare TransferMorphism_int_nat[transfer add return:
19.322 - transfer_int_nat_prime_factors transfer_int_nat_prime_factors_closure
19.323 - transfer_int_nat_multiplicity]
19.324 -
19.325 -
19.326 -subsection {* Properties of prime factors and multiplicity for nats and ints *}
19.327 -
19.328 -lemma prime_factors_ge_0_int [elim]: "p : prime_factors (n::int) \<Longrightarrow> p >= 0"
19.329 - by (unfold prime_factors_int_def, auto)
19.330 -
19.331 -lemma prime_factors_prime_nat [intro]: "p : prime_factors (n::nat) \<Longrightarrow> prime p"
19.332 - apply (case_tac "n = 0")
19.333 - apply (simp add: prime_factors_nat_def multiset_prime_factorization_def)
19.334 - apply (auto simp add: prime_factors_nat_def multiset_prime_factorization)
19.335 -done
19.336 -
19.337 -lemma prime_factors_prime_int [intro]:
19.338 - assumes "n >= 0" and "p : prime_factors (n::int)"
19.339 - shows "prime p"
19.340 -
19.341 - apply (rule prime_factors_prime_nat [transferred, of n p])
19.342 - using prems apply auto
19.343 -done
19.344 -
19.345 -lemma prime_factors_gt_0_nat [elim]: "p : prime_factors x \<Longrightarrow> p > (0::nat)"
19.346 - by (frule prime_factors_prime_nat, auto)
19.347 -
19.348 -lemma prime_factors_gt_0_int [elim]: "x >= 0 \<Longrightarrow> p : prime_factors x \<Longrightarrow>
19.349 - p > (0::int)"
19.350 - by (frule (1) prime_factors_prime_int, auto)
19.351 -
19.352 -lemma prime_factors_finite_nat [iff]: "finite (prime_factors (n::nat))"
19.353 - by (unfold prime_factors_nat_def, auto)
19.354 -
19.355 -lemma prime_factors_finite_int [iff]: "finite (prime_factors (n::int))"
19.356 - by (unfold prime_factors_int_def, auto)
19.357 -
19.358 -lemma prime_factors_altdef_nat: "prime_factors (n::nat) =
19.359 - {p. multiplicity p n > 0}"
19.360 - by (force simp add: prime_factors_nat_def multiplicity_nat_def)
19.361 -
19.362 -lemma prime_factors_altdef_int: "prime_factors (n::int) =
19.363 - {p. p >= 0 & multiplicity p n > 0}"
19.364 - apply (unfold prime_factors_int_def multiplicity_int_def)
19.365 - apply (subst prime_factors_altdef_nat)
19.366 - apply (auto simp add: image_def)
19.367 -done
19.368 -
19.369 -lemma prime_factorization_nat: "(n::nat) > 0 \<Longrightarrow>
19.370 - n = (PROD p : prime_factors n. p^(multiplicity p n))"
19.371 - by (frule multiset_prime_factorization,
19.372 - simp add: prime_factors_nat_def multiplicity_nat_def msetprod_def)
19.373 -
19.374 -thm prime_factorization_nat [transferred]
19.375 -
19.376 -lemma prime_factorization_int:
19.377 - assumes "(n::int) > 0"
19.378 - shows "n = (PROD p : prime_factors n. p^(multiplicity p n))"
19.379 -
19.380 - apply (rule prime_factorization_nat [transferred, of n])
19.381 - using prems apply auto
19.382 -done
19.383 -
19.384 -lemma neq_zero_eq_gt_zero_nat: "((x::nat) ~= 0) = (x > 0)"
19.385 - by auto
19.386 -
19.387 -lemma prime_factorization_unique_nat:
19.388 - "S = { (p::nat) . f p > 0} \<Longrightarrow> finite S \<Longrightarrow> (ALL p : S. prime p) \<Longrightarrow>
19.389 - n = (PROD p : S. p^(f p)) \<Longrightarrow>
19.390 - S = prime_factors n & (ALL p. f p = multiplicity p n)"
19.391 - apply (subgoal_tac "multiset_prime_factorization n = Abs_multiset
19.392 - f")
19.393 - apply (unfold prime_factors_nat_def multiplicity_nat_def)
19.394 - apply (simp add: set_of_def count_def Abs_multiset_inverse multiset_def)
19.395 - apply (unfold multiset_prime_factorization_def)
19.396 - apply (subgoal_tac "n > 0")
19.397 - prefer 2
19.398 - apply force
19.399 - apply (subst if_P, assumption)
19.400 - apply (rule the1_equality)
19.401 - apply (rule ex_ex1I)
19.402 - apply (rule multiset_prime_factorization_exists, assumption)
19.403 - apply (rule multiset_prime_factorization_unique)
19.404 - apply force
19.405 - apply force
19.406 - apply force
19.407 - unfolding set_of_def count_def msetprod_def
19.408 - apply (subgoal_tac "f : multiset")
19.409 - apply (auto simp only: Abs_multiset_inverse)
19.410 - unfolding multiset_def apply force
19.411 -done
19.412 -
19.413 -lemma prime_factors_characterization_nat: "S = {p. 0 < f (p::nat)} \<Longrightarrow>
19.414 - finite S \<Longrightarrow> (ALL p:S. prime p) \<Longrightarrow> n = (PROD p:S. p ^ f p) \<Longrightarrow>
19.415 - prime_factors n = S"
19.416 - by (rule prime_factorization_unique_nat [THEN conjunct1, symmetric],
19.417 - assumption+)
19.418 -
19.419 -lemma prime_factors_characterization'_nat:
19.420 - "finite {p. 0 < f (p::nat)} \<Longrightarrow>
19.421 - (ALL p. 0 < f p \<longrightarrow> prime p) \<Longrightarrow>
19.422 - prime_factors (PROD p | 0 < f p . p ^ f p) = {p. 0 < f p}"
19.423 - apply (rule prime_factors_characterization_nat)
19.424 - apply auto
19.425 -done
19.426 -
19.427 -(* A minor glitch:*)
19.428 -
19.429 -thm prime_factors_characterization'_nat
19.430 - [where f = "%x. f (int (x::nat))",
19.431 - transferred direction: nat "op <= (0::int)", rule_format]
19.432 -
19.433 -(*
19.434 - Transfer isn't smart enough to know that the "0 < f p" should
19.435 - remain a comparison between nats. But the transfer still works.
19.436 -*)
19.437 -
19.438 -lemma primes_characterization'_int [rule_format]:
19.439 - "finite {p. p >= 0 & 0 < f (p::int)} \<Longrightarrow>
19.440 - (ALL p. 0 < f p \<longrightarrow> prime p) \<Longrightarrow>
19.441 - prime_factors (PROD p | p >=0 & 0 < f p . p ^ f p) =
19.442 - {p. p >= 0 & 0 < f p}"
19.443 -
19.444 - apply (insert prime_factors_characterization'_nat
19.445 - [where f = "%x. f (int (x::nat))",
19.446 - transferred direction: nat "op <= (0::int)"])
19.447 - apply auto
19.448 -done
19.449 -
19.450 -lemma prime_factors_characterization_int: "S = {p. 0 < f (p::int)} \<Longrightarrow>
19.451 - finite S \<Longrightarrow> (ALL p:S. prime p) \<Longrightarrow> n = (PROD p:S. p ^ f p) \<Longrightarrow>
19.452 - prime_factors n = S"
19.453 - apply simp
19.454 - apply (subgoal_tac "{p. 0 < f p} = {p. 0 <= p & 0 < f p}")
19.455 - apply (simp only:)
19.456 - apply (subst primes_characterization'_int)
19.457 - apply auto
19.458 - apply (auto simp add: prime_ge_0_int)
19.459 -done
19.460 -
19.461 -lemma multiplicity_characterization_nat: "S = {p. 0 < f (p::nat)} \<Longrightarrow>
19.462 - finite S \<Longrightarrow> (ALL p:S. prime p) \<Longrightarrow> n = (PROD p:S. p ^ f p) \<Longrightarrow>
19.463 - multiplicity p n = f p"
19.464 - by (frule prime_factorization_unique_nat [THEN conjunct2, rule_format,
19.465 - symmetric], auto)
19.466 -
19.467 -lemma multiplicity_characterization'_nat: "finite {p. 0 < f (p::nat)} \<longrightarrow>
19.468 - (ALL p. 0 < f p \<longrightarrow> prime p) \<longrightarrow>
19.469 - multiplicity p (PROD p | 0 < f p . p ^ f p) = f p"
19.470 - apply (rule impI)+
19.471 - apply (rule multiplicity_characterization_nat)
19.472 - apply auto
19.473 -done
19.474 -
19.475 -lemma multiplicity_characterization'_int [rule_format]:
19.476 - "finite {p. p >= 0 & 0 < f (p::int)} \<Longrightarrow>
19.477 - (ALL p. 0 < f p \<longrightarrow> prime p) \<Longrightarrow> p >= 0 \<Longrightarrow>
19.478 - multiplicity p (PROD p | p >= 0 & 0 < f p . p ^ f p) = f p"
19.479 -
19.480 - apply (insert multiplicity_characterization'_nat
19.481 - [where f = "%x. f (int (x::nat))",
19.482 - transferred direction: nat "op <= (0::int)", rule_format])
19.483 - apply auto
19.484 -done
19.485 -
19.486 -lemma multiplicity_characterization_int: "S = {p. 0 < f (p::int)} \<Longrightarrow>
19.487 - finite S \<Longrightarrow> (ALL p:S. prime p) \<Longrightarrow> n = (PROD p:S. p ^ f p) \<Longrightarrow>
19.488 - p >= 0 \<Longrightarrow> multiplicity p n = f p"
19.489 - apply simp
19.490 - apply (subgoal_tac "{p. 0 < f p} = {p. 0 <= p & 0 < f p}")
19.491 - apply (simp only:)
19.492 - apply (subst multiplicity_characterization'_int)
19.493 - apply auto
19.494 - apply (auto simp add: prime_ge_0_int)
19.495 -done
19.496 -
19.497 -lemma multiplicity_zero_nat [simp]: "multiplicity (p::nat) 0 = 0"
19.498 - by (simp add: multiplicity_nat_def multiset_prime_factorization_def)
19.499 -
19.500 -lemma multiplicity_zero_int [simp]: "multiplicity (p::int) 0 = 0"
19.501 - by (simp add: multiplicity_int_def)
19.502 -
19.503 -lemma multiplicity_one_nat [simp]: "multiplicity p (1::nat) = 0"
19.504 - by (subst multiplicity_characterization_nat [where f = "%x. 0"], auto)
19.505 -
19.506 -lemma multiplicity_one_int [simp]: "multiplicity p (1::int) = 0"
19.507 - by (simp add: multiplicity_int_def)
19.508 -
19.509 -lemma multiplicity_prime_nat [simp]: "prime (p::nat) \<Longrightarrow> multiplicity p p = 1"
19.510 - apply (subst multiplicity_characterization_nat
19.511 - [where f = "(%q. if q = p then 1 else 0)"])
19.512 - apply auto
19.513 - apply (case_tac "x = p")
19.514 - apply auto
19.515 -done
19.516 -
19.517 -lemma multiplicity_prime_int [simp]: "prime (p::int) \<Longrightarrow> multiplicity p p = 1"
19.518 - unfolding prime_int_def multiplicity_int_def by auto
19.519 -
19.520 -lemma multiplicity_prime_power_nat [simp]: "prime (p::nat) \<Longrightarrow>
19.521 - multiplicity p (p^n) = n"
19.522 - apply (case_tac "n = 0")
19.523 - apply auto
19.524 - apply (subst multiplicity_characterization_nat
19.525 - [where f = "(%q. if q = p then n else 0)"])
19.526 - apply auto
19.527 - apply (case_tac "x = p")
19.528 - apply auto
19.529 -done
19.530 -
19.531 -lemma multiplicity_prime_power_int [simp]: "prime (p::int) \<Longrightarrow>
19.532 - multiplicity p (p^n) = n"
19.533 - apply (frule prime_ge_0_int)
19.534 - apply (auto simp add: prime_int_def multiplicity_int_def nat_power_eq)
19.535 -done
19.536 -
19.537 -lemma multiplicity_nonprime_nat [simp]: "~ prime (p::nat) \<Longrightarrow>
19.538 - multiplicity p n = 0"
19.539 - apply (case_tac "n = 0")
19.540 - apply auto
19.541 - apply (frule multiset_prime_factorization)
19.542 - apply (auto simp add: set_of_def multiplicity_nat_def)
19.543 -done
19.544 -
19.545 -lemma multiplicity_nonprime_int [simp]: "~ prime (p::int) \<Longrightarrow> multiplicity p n = 0"
19.546 - by (unfold multiplicity_int_def prime_int_def, auto)
19.547 -
19.548 -lemma multiplicity_not_factor_nat [simp]:
19.549 - "p ~: prime_factors (n::nat) \<Longrightarrow> multiplicity p n = 0"
19.550 - by (subst (asm) prime_factors_altdef_nat, auto)
19.551 -
19.552 -lemma multiplicity_not_factor_int [simp]:
19.553 - "p >= 0 \<Longrightarrow> p ~: prime_factors (n::int) \<Longrightarrow> multiplicity p n = 0"
19.554 - by (subst (asm) prime_factors_altdef_int, auto)
19.555 -
19.556 -lemma multiplicity_product_aux_nat: "(k::nat) > 0 \<Longrightarrow> l > 0 \<Longrightarrow>
19.557 - (prime_factors k) Un (prime_factors l) = prime_factors (k * l) &
19.558 - (ALL p. multiplicity p k + multiplicity p l = multiplicity p (k * l))"
19.559 - apply (rule prime_factorization_unique_nat)
19.560 - apply (simp only: prime_factors_altdef_nat)
19.561 - apply auto
19.562 - apply (subst power_add)
19.563 - apply (subst setprod_timesf)
19.564 - apply (rule arg_cong2)back back
19.565 - apply (subgoal_tac "prime_factors k Un prime_factors l = prime_factors k Un
19.566 - (prime_factors l - prime_factors k)")
19.567 - apply (erule ssubst)
19.568 - apply (subst setprod_Un_disjoint)
19.569 - apply auto
19.570 - apply (subgoal_tac "(\<Prod>p\<in>prime_factors l - prime_factors k. p ^ multiplicity p k) =
19.571 - (\<Prod>p\<in>prime_factors l - prime_factors k. 1)")
19.572 - apply (erule ssubst)
19.573 - apply (simp add: setprod_1)
19.574 - apply (erule prime_factorization_nat)
19.575 - apply (rule setprod_cong, auto)
19.576 - apply (subgoal_tac "prime_factors k Un prime_factors l = prime_factors l Un
19.577 - (prime_factors k - prime_factors l)")
19.578 - apply (erule ssubst)
19.579 - apply (subst setprod_Un_disjoint)
19.580 - apply auto
19.581 - apply (subgoal_tac "(\<Prod>p\<in>prime_factors k - prime_factors l. p ^ multiplicity p l) =
19.582 - (\<Prod>p\<in>prime_factors k - prime_factors l. 1)")
19.583 - apply (erule ssubst)
19.584 - apply (simp add: setprod_1)
19.585 - apply (erule prime_factorization_nat)
19.586 - apply (rule setprod_cong, auto)
19.587 -done
19.588 -
19.589 -(* transfer doesn't have the same problem here with the right
19.590 - choice of rules. *)
19.591 -
19.592 -lemma multiplicity_product_aux_int:
19.593 - assumes "(k::int) > 0" and "l > 0"
19.594 - shows
19.595 - "(prime_factors k) Un (prime_factors l) = prime_factors (k * l) &
19.596 - (ALL p >= 0. multiplicity p k + multiplicity p l = multiplicity p (k * l))"
19.597 -
19.598 - apply (rule multiplicity_product_aux_nat [transferred, of l k])
19.599 - using prems apply auto
19.600 -done
19.601 -
19.602 -lemma prime_factors_product_nat: "(k::nat) > 0 \<Longrightarrow> l > 0 \<Longrightarrow> prime_factors (k * l) =
19.603 - prime_factors k Un prime_factors l"
19.604 - by (rule multiplicity_product_aux_nat [THEN conjunct1, symmetric])
19.605 -
19.606 -lemma prime_factors_product_int: "(k::int) > 0 \<Longrightarrow> l > 0 \<Longrightarrow> prime_factors (k * l) =
19.607 - prime_factors k Un prime_factors l"
19.608 - by (rule multiplicity_product_aux_int [THEN conjunct1, symmetric])
19.609 -
19.610 -lemma multiplicity_product_nat: "(k::nat) > 0 \<Longrightarrow> l > 0 \<Longrightarrow> multiplicity p (k * l) =
19.611 - multiplicity p k + multiplicity p l"
19.612 - by (rule multiplicity_product_aux_nat [THEN conjunct2, rule_format,
19.613 - symmetric])
19.614 -
19.615 -lemma multiplicity_product_int: "(k::int) > 0 \<Longrightarrow> l > 0 \<Longrightarrow> p >= 0 \<Longrightarrow>
19.616 - multiplicity p (k * l) = multiplicity p k + multiplicity p l"
19.617 - by (rule multiplicity_product_aux_int [THEN conjunct2, rule_format,
19.618 - symmetric])
19.619 -
19.620 -lemma multiplicity_setprod_nat: "finite S \<Longrightarrow> (ALL x : S. f x > 0) \<Longrightarrow>
19.621 - multiplicity (p::nat) (PROD x : S. f x) =
19.622 - (SUM x : S. multiplicity p (f x))"
19.623 - apply (induct set: finite)
19.624 - apply auto
19.625 - apply (subst multiplicity_product_nat)
19.626 - apply auto
19.627 -done
19.628 -
19.629 -(* Transfer is delicate here for two reasons: first, because there is
19.630 - an implicit quantifier over functions (f), and, second, because the
19.631 - product over the multiplicity should not be translated to an integer
19.632 - product.
19.633 -
19.634 - The way to handle the first is to use quantifier rules for functions.
19.635 - The way to handle the second is to turn off the offending rule.
19.636 -*)
19.637 -
19.638 -lemma transfer_nat_int_sum_prod_closure3:
19.639 - "(SUM x : A. int (f x)) >= 0"
19.640 - "(PROD x : A. int (f x)) >= 0"
19.641 - apply (rule setsum_nonneg, auto)
19.642 - apply (rule setprod_nonneg, auto)
19.643 -done
19.644 -
19.645 -declare TransferMorphism_nat_int[transfer
19.646 - add return: transfer_nat_int_sum_prod_closure3
19.647 - del: transfer_nat_int_sum_prod2 (1)]
19.648 -
19.649 -lemma multiplicity_setprod_int: "p >= 0 \<Longrightarrow> finite S \<Longrightarrow>
19.650 - (ALL x : S. f x > 0) \<Longrightarrow>
19.651 - multiplicity (p::int) (PROD x : S. f x) =
19.652 - (SUM x : S. multiplicity p (f x))"
19.653 -
19.654 - apply (frule multiplicity_setprod_nat
19.655 - [where f = "%x. nat(int(nat(f x)))",
19.656 - transferred direction: nat "op <= (0::int)"])
19.657 - apply auto
19.658 - apply (subst (asm) setprod_cong)
19.659 - apply (rule refl)
19.660 - apply (rule if_P)
19.661 - apply auto
19.662 - apply (rule setsum_cong)
19.663 - apply auto
19.664 -done
19.665 -
19.666 -declare TransferMorphism_nat_int[transfer
19.667 - add return: transfer_nat_int_sum_prod2 (1)]
19.668 -
19.669 -lemma multiplicity_prod_prime_powers_nat:
19.670 - "finite S \<Longrightarrow> (ALL p : S. prime (p::nat)) \<Longrightarrow>
19.671 - multiplicity p (PROD p : S. p ^ f p) = (if p : S then f p else 0)"
19.672 - apply (subgoal_tac "(PROD p : S. p ^ f p) =
19.673 - (PROD p : S. p ^ (%x. if x : S then f x else 0) p)")
19.674 - apply (erule ssubst)
19.675 - apply (subst multiplicity_characterization_nat)
19.676 - prefer 5 apply (rule refl)
19.677 - apply (rule refl)
19.678 - apply auto
19.679 - apply (subst setprod_mono_one_right)
19.680 - apply assumption
19.681 - prefer 3
19.682 - apply (rule setprod_cong)
19.683 - apply (rule refl)
19.684 - apply auto
19.685 -done
19.686 -
19.687 -(* Here the issue with transfer is the implicit quantifier over S *)
19.688 -
19.689 -lemma multiplicity_prod_prime_powers_int:
19.690 - "(p::int) >= 0 \<Longrightarrow> finite S \<Longrightarrow> (ALL p : S. prime p) \<Longrightarrow>
19.691 - multiplicity p (PROD p : S. p ^ f p) = (if p : S then f p else 0)"
19.692 -
19.693 - apply (subgoal_tac "int ` nat ` S = S")
19.694 - apply (frule multiplicity_prod_prime_powers_nat [where f = "%x. f(int x)"
19.695 - and S = "nat ` S", transferred])
19.696 - apply auto
19.697 - apply (subst prime_int_def [symmetric])
19.698 - apply auto
19.699 - apply (subgoal_tac "xb >= 0")
19.700 - apply force
19.701 - apply (rule prime_ge_0_int)
19.702 - apply force
19.703 - apply (subst transfer_nat_int_set_return_embed)
19.704 - apply (unfold nat_set_def, auto)
19.705 -done
19.706 -
19.707 -lemma multiplicity_distinct_prime_power_nat: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow>
19.708 - p ~= q \<Longrightarrow> multiplicity p (q^n) = 0"
19.709 - apply (subgoal_tac "q^n = setprod (%x. x^n) {q}")
19.710 - apply (erule ssubst)
19.711 - apply (subst multiplicity_prod_prime_powers_nat)
19.712 - apply auto
19.713 -done
19.714 -
19.715 -lemma multiplicity_distinct_prime_power_int: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow>
19.716 - p ~= q \<Longrightarrow> multiplicity p (q^n) = 0"
19.717 - apply (frule prime_ge_0_int [of q])
19.718 - apply (frule multiplicity_distinct_prime_power_nat [transferred leaving: n])
19.719 - prefer 4
19.720 - apply assumption
19.721 - apply auto
19.722 -done
19.723 -
19.724 -lemma dvd_multiplicity_nat:
19.725 - "(0::nat) < y \<Longrightarrow> x dvd y \<Longrightarrow> multiplicity p x <= multiplicity p y"
19.726 - apply (case_tac "x = 0")
19.727 - apply (auto simp add: dvd_def multiplicity_product_nat)
19.728 -done
19.729 -
19.730 -lemma dvd_multiplicity_int:
19.731 - "(0::int) < y \<Longrightarrow> 0 <= x \<Longrightarrow> x dvd y \<Longrightarrow> p >= 0 \<Longrightarrow>
19.732 - multiplicity p x <= multiplicity p y"
19.733 - apply (case_tac "x = 0")
19.734 - apply (auto simp add: dvd_def)
19.735 - apply (subgoal_tac "0 < k")
19.736 - apply (auto simp add: multiplicity_product_int)
19.737 - apply (erule zero_less_mult_pos)
19.738 - apply arith
19.739 -done
19.740 -
19.741 -lemma dvd_prime_factors_nat [intro]:
19.742 - "0 < (y::nat) \<Longrightarrow> x dvd y \<Longrightarrow> prime_factors x <= prime_factors y"
19.743 - apply (simp only: prime_factors_altdef_nat)
19.744 - apply auto
19.745 - apply (frule dvd_multiplicity_nat)
19.746 - apply auto
19.747 -(* It is a shame that auto and arith don't get this. *)
19.748 - apply (erule order_less_le_trans)back
19.749 - apply assumption
19.750 -done
19.751 -
19.752 -lemma dvd_prime_factors_int [intro]:
19.753 - "0 < (y::int) \<Longrightarrow> 0 <= x \<Longrightarrow> x dvd y \<Longrightarrow> prime_factors x <= prime_factors y"
19.754 - apply (auto simp add: prime_factors_altdef_int)
19.755 - apply (erule order_less_le_trans)
19.756 - apply (rule dvd_multiplicity_int)
19.757 - apply auto
19.758 -done
19.759 -
19.760 -lemma multiplicity_dvd_nat: "0 < (x::nat) \<Longrightarrow> 0 < y \<Longrightarrow>
19.761 - ALL p. multiplicity p x <= multiplicity p y \<Longrightarrow>
19.762 - x dvd y"
19.763 - apply (subst prime_factorization_nat [of x], assumption)
19.764 - apply (subst prime_factorization_nat [of y], assumption)
19.765 - apply (rule setprod_dvd_setprod_subset2)
19.766 - apply force
19.767 - apply (subst prime_factors_altdef_nat)+
19.768 - apply auto
19.769 -(* Again, a shame that auto and arith don't get this. *)
19.770 - apply (drule_tac x = xa in spec, auto)
19.771 - apply (rule le_imp_power_dvd)
19.772 - apply blast
19.773 -done
19.774 -
19.775 -lemma multiplicity_dvd_int: "0 < (x::int) \<Longrightarrow> 0 < y \<Longrightarrow>
19.776 - ALL p >= 0. multiplicity p x <= multiplicity p y \<Longrightarrow>
19.777 - x dvd y"
19.778 - apply (subst prime_factorization_int [of x], assumption)
19.779 - apply (subst prime_factorization_int [of y], assumption)
19.780 - apply (rule setprod_dvd_setprod_subset2)
19.781 - apply force
19.782 - apply (subst prime_factors_altdef_int)+
19.783 - apply auto
19.784 - apply (rule dvd_power_le)
19.785 - apply auto
19.786 - apply (drule_tac x = xa in spec)
19.787 - apply (erule impE)
19.788 - apply auto
19.789 -done
19.790 -
19.791 -lemma multiplicity_dvd'_nat: "(0::nat) < x \<Longrightarrow>
19.792 - \<forall>p. prime p \<longrightarrow> multiplicity p x \<le> multiplicity p y \<Longrightarrow> x dvd y"
19.793 - apply (cases "y = 0")
19.794 - apply auto
19.795 - apply (rule multiplicity_dvd_nat, auto)
19.796 - apply (case_tac "prime p")
19.797 - apply auto
19.798 -done
19.799 -
19.800 -lemma multiplicity_dvd'_int: "(0::int) < x \<Longrightarrow> 0 <= y \<Longrightarrow>
19.801 - \<forall>p. prime p \<longrightarrow> multiplicity p x \<le> multiplicity p y \<Longrightarrow> x dvd y"
19.802 - apply (cases "y = 0")
19.803 - apply auto
19.804 - apply (rule multiplicity_dvd_int, auto)
19.805 - apply (case_tac "prime p")
19.806 - apply auto
19.807 -done
19.808 -
19.809 -lemma dvd_multiplicity_eq_nat: "0 < (x::nat) \<Longrightarrow> 0 < y \<Longrightarrow>
19.810 - (x dvd y) = (ALL p. multiplicity p x <= multiplicity p y)"
19.811 - by (auto intro: dvd_multiplicity_nat multiplicity_dvd_nat)
19.812 -
19.813 -lemma dvd_multiplicity_eq_int: "0 < (x::int) \<Longrightarrow> 0 < y \<Longrightarrow>
19.814 - (x dvd y) = (ALL p >= 0. multiplicity p x <= multiplicity p y)"
19.815 - by (auto intro: dvd_multiplicity_int multiplicity_dvd_int)
19.816 -
19.817 -lemma prime_factors_altdef2_nat: "(n::nat) > 0 \<Longrightarrow>
19.818 - (p : prime_factors n) = (prime p & p dvd n)"
19.819 - apply (case_tac "prime p")
19.820 - apply auto
19.821 - apply (subst prime_factorization_nat [where n = n], assumption)
19.822 - apply (rule dvd_trans)
19.823 - apply (rule dvd_power [where x = p and n = "multiplicity p n"])
19.824 - apply (subst (asm) prime_factors_altdef_nat, force)
19.825 - apply (rule dvd_setprod)
19.826 - apply auto
19.827 - apply (subst prime_factors_altdef_nat)
19.828 - apply (subst (asm) dvd_multiplicity_eq_nat)
19.829 - apply auto
19.830 - apply (drule spec [where x = p])
19.831 - apply auto
19.832 -done
19.833 -
19.834 -lemma prime_factors_altdef2_int:
19.835 - assumes "(n::int) > 0"
19.836 - shows "(p : prime_factors n) = (prime p & p dvd n)"
19.837 -
19.838 - apply (case_tac "p >= 0")
19.839 - apply (rule prime_factors_altdef2_nat [transferred])
19.840 - using prems apply auto
19.841 - apply (auto simp add: prime_ge_0_int prime_factors_ge_0_int)
19.842 -done
19.843 -
19.844 -lemma multiplicity_eq_nat:
19.845 - fixes x and y::nat
19.846 - assumes [arith]: "x > 0" "y > 0" and
19.847 - mult_eq [simp]: "!!p. prime p \<Longrightarrow> multiplicity p x = multiplicity p y"
19.848 - shows "x = y"
19.849 -
19.850 - apply (rule dvd_anti_sym)
19.851 - apply (auto intro: multiplicity_dvd'_nat)
19.852 -done
19.853 -
19.854 -lemma multiplicity_eq_int:
19.855 - fixes x and y::int
19.856 - assumes [arith]: "x > 0" "y > 0" and
19.857 - mult_eq [simp]: "!!p. prime p \<Longrightarrow> multiplicity p x = multiplicity p y"
19.858 - shows "x = y"
19.859 -
19.860 - apply (rule dvd_anti_sym [transferred])
19.861 - apply (auto intro: multiplicity_dvd'_int)
19.862 -done
19.863 -
19.864 -
19.865 -subsection {* An application *}
19.866 -
19.867 -lemma gcd_eq_nat:
19.868 - assumes pos [arith]: "x > 0" "y > 0"
19.869 - shows "gcd (x::nat) y =
19.870 - (PROD p: prime_factors x Un prime_factors y.
19.871 - p ^ (min (multiplicity p x) (multiplicity p y)))"
19.872 -proof -
19.873 - def z == "(PROD p: prime_factors (x::nat) Un prime_factors y.
19.874 - p ^ (min (multiplicity p x) (multiplicity p y)))"
19.875 - have [arith]: "z > 0"
19.876 - unfolding z_def by (rule setprod_pos_nat, auto)
19.877 - have aux: "!!p. prime p \<Longrightarrow> multiplicity p z =
19.878 - min (multiplicity p x) (multiplicity p y)"
19.879 - unfolding z_def
19.880 - apply (subst multiplicity_prod_prime_powers_nat)
19.881 - apply (auto simp add: multiplicity_not_factor_nat)
19.882 - done
19.883 - have "z dvd x"
19.884 - by (intro multiplicity_dvd'_nat, auto simp add: aux)
19.885 - moreover have "z dvd y"
19.886 - by (intro multiplicity_dvd'_nat, auto simp add: aux)
19.887 - moreover have "ALL w. w dvd x & w dvd y \<longrightarrow> w dvd z"
19.888 - apply auto
19.889 - apply (case_tac "w = 0", auto)
19.890 - apply (erule multiplicity_dvd'_nat)
19.891 - apply (auto intro: dvd_multiplicity_nat simp add: aux)
19.892 - done
19.893 - ultimately have "z = gcd x y"
19.894 - by (subst gcd_unique_nat [symmetric], blast)
19.895 - thus ?thesis
19.896 - unfolding z_def by auto
19.897 -qed
19.898 -
19.899 -lemma lcm_eq_nat:
19.900 - assumes pos [arith]: "x > 0" "y > 0"
19.901 - shows "lcm (x::nat) y =
19.902 - (PROD p: prime_factors x Un prime_factors y.
19.903 - p ^ (max (multiplicity p x) (multiplicity p y)))"
19.904 -proof -
19.905 - def z == "(PROD p: prime_factors (x::nat) Un prime_factors y.
19.906 - p ^ (max (multiplicity p x) (multiplicity p y)))"
19.907 - have [arith]: "z > 0"
19.908 - unfolding z_def by (rule setprod_pos_nat, auto)
19.909 - have aux: "!!p. prime p \<Longrightarrow> multiplicity p z =
19.910 - max (multiplicity p x) (multiplicity p y)"
19.911 - unfolding z_def
19.912 - apply (subst multiplicity_prod_prime_powers_nat)
19.913 - apply (auto simp add: multiplicity_not_factor_nat)
19.914 - done
19.915 - have "x dvd z"
19.916 - by (intro multiplicity_dvd'_nat, auto simp add: aux)
19.917 - moreover have "y dvd z"
19.918 - by (intro multiplicity_dvd'_nat, auto simp add: aux)
19.919 - moreover have "ALL w. x dvd w & y dvd w \<longrightarrow> z dvd w"
19.920 - apply auto
19.921 - apply (case_tac "w = 0", auto)
19.922 - apply (rule multiplicity_dvd'_nat)
19.923 - apply (auto intro: dvd_multiplicity_nat simp add: aux)
19.924 - done
19.925 - ultimately have "z = lcm x y"
19.926 - by (subst lcm_unique_nat [symmetric], blast)
19.927 - thus ?thesis
19.928 - unfolding z_def by auto
19.929 -qed
19.930 -
19.931 -lemma multiplicity_gcd_nat:
19.932 - assumes [arith]: "x > 0" "y > 0"
19.933 - shows "multiplicity (p::nat) (gcd x y) =
19.934 - min (multiplicity p x) (multiplicity p y)"
19.935 -
19.936 - apply (subst gcd_eq_nat)
19.937 - apply auto
19.938 - apply (subst multiplicity_prod_prime_powers_nat)
19.939 - apply auto
19.940 -done
19.941 -
19.942 -lemma multiplicity_lcm_nat:
19.943 - assumes [arith]: "x > 0" "y > 0"
19.944 - shows "multiplicity (p::nat) (lcm x y) =
19.945 - max (multiplicity p x) (multiplicity p y)"
19.946 -
19.947 - apply (subst lcm_eq_nat)
19.948 - apply auto
19.949 - apply (subst multiplicity_prod_prime_powers_nat)
19.950 - apply auto
19.951 -done
19.952 -
19.953 -lemma gcd_lcm_distrib_nat: "gcd (x::nat) (lcm y z) = lcm (gcd x y) (gcd x z)"
19.954 - apply (case_tac "x = 0 | y = 0 | z = 0")
19.955 - apply auto
19.956 - apply (rule multiplicity_eq_nat)
19.957 - apply (auto simp add: multiplicity_gcd_nat multiplicity_lcm_nat
19.958 - lcm_pos_nat)
19.959 -done
19.960 -
19.961 -lemma gcd_lcm_distrib_int: "gcd (x::int) (lcm y z) = lcm (gcd x y) (gcd x z)"
19.962 - apply (subst (1 2 3) gcd_abs_int)
19.963 - apply (subst lcm_abs_int)
19.964 - apply (subst (2) abs_of_nonneg)
19.965 - apply force
19.966 - apply (rule gcd_lcm_distrib_nat [transferred])
19.967 - apply auto
19.968 -done
19.969 -
19.970 -end
20.1 --- a/src/HOL/NumberTheory/BijectionRel.thy Tue Sep 01 14:13:34 2009 +0200
20.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
20.3 @@ -1,231 +0,0 @@
20.4 -(* Title: HOL/NumberTheory/BijectionRel.thy
20.5 - ID: $Id$
20.6 - Author: Thomas M. Rasmussen
20.7 - Copyright 2000 University of Cambridge
20.8 -*)
20.9 -
20.10 -header {* Bijections between sets *}
20.11 -
20.12 -theory BijectionRel imports Main begin
20.13 -
20.14 -text {*
20.15 - Inductive definitions of bijections between two different sets and
20.16 - between the same set. Theorem for relating the two definitions.
20.17 -
20.18 - \bigskip
20.19 -*}
20.20 -
20.21 -inductive_set
20.22 - bijR :: "('a => 'b => bool) => ('a set * 'b set) set"
20.23 - for P :: "'a => 'b => bool"
20.24 -where
20.25 - empty [simp]: "({}, {}) \<in> bijR P"
20.26 -| insert: "P a b ==> a \<notin> A ==> b \<notin> B ==> (A, B) \<in> bijR P
20.27 - ==> (insert a A, insert b B) \<in> bijR P"
20.28 -
20.29 -text {*
20.30 - Add extra condition to @{term insert}: @{term "\<forall>b \<in> B. \<not> P a b"}
20.31 - (and similar for @{term A}).
20.32 -*}
20.33 -
20.34 -definition
20.35 - bijP :: "('a => 'a => bool) => 'a set => bool" where
20.36 - "bijP P F = (\<forall>a b. a \<in> F \<and> P a b --> b \<in> F)"
20.37 -
20.38 -definition
20.39 - uniqP :: "('a => 'a => bool) => bool" where
20.40 - "uniqP P = (\<forall>a b c d. P a b \<and> P c d --> (a = c) = (b = d))"
20.41 -
20.42 -definition
20.43 - symP :: "('a => 'a => bool) => bool" where
20.44 - "symP P = (\<forall>a b. P a b = P b a)"
20.45 -
20.46 -inductive_set
20.47 - bijER :: "('a => 'a => bool) => 'a set set"
20.48 - for P :: "'a => 'a => bool"
20.49 -where
20.50 - empty [simp]: "{} \<in> bijER P"
20.51 -| insert1: "P a a ==> a \<notin> A ==> A \<in> bijER P ==> insert a A \<in> bijER P"
20.52 -| insert2: "P a b ==> a \<noteq> b ==> a \<notin> A ==> b \<notin> A ==> A \<in> bijER P
20.53 - ==> insert a (insert b A) \<in> bijER P"
20.54 -
20.55 -
20.56 -text {* \medskip @{term bijR} *}
20.57 -
20.58 -lemma fin_bijRl: "(A, B) \<in> bijR P ==> finite A"
20.59 - apply (erule bijR.induct)
20.60 - apply auto
20.61 - done
20.62 -
20.63 -lemma fin_bijRr: "(A, B) \<in> bijR P ==> finite B"
20.64 - apply (erule bijR.induct)
20.65 - apply auto
20.66 - done
20.67 -
20.68 -lemma aux_induct:
20.69 - assumes major: "finite F"
20.70 - and subs: "F \<subseteq> A"
20.71 - and cases: "P {}"
20.72 - "!!F a. F \<subseteq> A ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
20.73 - shows "P F"
20.74 - using major subs
20.75 - apply (induct set: finite)
20.76 - apply (blast intro: cases)+
20.77 - done
20.78 -
20.79 -
20.80 -lemma inj_func_bijR_aux1:
20.81 - "A \<subseteq> B ==> a \<notin> A ==> a \<in> B ==> inj_on f B ==> f a \<notin> f ` A"
20.82 - apply (unfold inj_on_def)
20.83 - apply auto
20.84 - done
20.85 -
20.86 -lemma inj_func_bijR_aux2:
20.87 - "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A ==> F <= A
20.88 - ==> (F, f ` F) \<in> bijR P"
20.89 - apply (rule_tac F = F and A = A in aux_induct)
20.90 - apply (rule finite_subset)
20.91 - apply auto
20.92 - apply (rule bijR.insert)
20.93 - apply (rule_tac [3] inj_func_bijR_aux1)
20.94 - apply auto
20.95 - done
20.96 -
20.97 -lemma inj_func_bijR:
20.98 - "\<forall>a. a \<in> A --> P a (f a) ==> inj_on f A ==> finite A
20.99 - ==> (A, f ` A) \<in> bijR P"
20.100 - apply (rule inj_func_bijR_aux2)
20.101 - apply auto
20.102 - done
20.103 -
20.104 -
20.105 -text {* \medskip @{term bijER} *}
20.106 -
20.107 -lemma fin_bijER: "A \<in> bijER P ==> finite A"
20.108 - apply (erule bijER.induct)
20.109 - apply auto
20.110 - done
20.111 -
20.112 -lemma aux1:
20.113 - "a \<notin> A ==> a \<notin> B ==> F \<subseteq> insert a A ==> F \<subseteq> insert a B ==> a \<in> F
20.114 - ==> \<exists>C. F = insert a C \<and> a \<notin> C \<and> C <= A \<and> C <= B"
20.115 - apply (rule_tac x = "F - {a}" in exI)
20.116 - apply auto
20.117 - done
20.118 -
20.119 -lemma aux2: "a \<noteq> b ==> a \<notin> A ==> b \<notin> B ==> a \<in> F ==> b \<in> F
20.120 - ==> F \<subseteq> insert a A ==> F \<subseteq> insert b B
20.121 - ==> \<exists>C. F = insert a (insert b C) \<and> a \<notin> C \<and> b \<notin> C \<and> C \<subseteq> A \<and> C \<subseteq> B"
20.122 - apply (rule_tac x = "F - {a, b}" in exI)
20.123 - apply auto
20.124 - done
20.125 -
20.126 -lemma aux_uniq: "uniqP P ==> P a b ==> P c d ==> (a = c) = (b = d)"
20.127 - apply (unfold uniqP_def)
20.128 - apply auto
20.129 - done
20.130 -
20.131 -lemma aux_sym: "symP P ==> P a b = P b a"
20.132 - apply (unfold symP_def)
20.133 - apply auto
20.134 - done
20.135 -
20.136 -lemma aux_in1:
20.137 - "uniqP P ==> b \<notin> C ==> P b b ==> bijP P (insert b C) ==> bijP P C"
20.138 - apply (unfold bijP_def)
20.139 - apply auto
20.140 - apply (subgoal_tac "b \<noteq> a")
20.141 - prefer 2
20.142 - apply clarify
20.143 - apply (simp add: aux_uniq)
20.144 - apply auto
20.145 - done
20.146 -
20.147 -lemma aux_in2:
20.148 - "symP P ==> uniqP P ==> a \<notin> C ==> b \<notin> C ==> a \<noteq> b ==> P a b
20.149 - ==> bijP P (insert a (insert b C)) ==> bijP P C"
20.150 - apply (unfold bijP_def)
20.151 - apply auto
20.152 - apply (subgoal_tac "aa \<noteq> a")
20.153 - prefer 2
20.154 - apply clarify
20.155 - apply (subgoal_tac "aa \<noteq> b")
20.156 - prefer 2
20.157 - apply clarify
20.158 - apply (simp add: aux_uniq)
20.159 - apply (subgoal_tac "ba \<noteq> a")
20.160 - apply auto
20.161 - apply (subgoal_tac "P a aa")
20.162 - prefer 2
20.163 - apply (simp add: aux_sym)
20.164 - apply (subgoal_tac "b = aa")
20.165 - apply (rule_tac [2] iffD1)
20.166 - apply (rule_tac [2] a = a and c = a and P = P in aux_uniq)
20.167 - apply auto
20.168 - done
20.169 -
20.170 -lemma aux_foo: "\<forall>a b. Q a \<and> P a b --> R b ==> P a b ==> Q a ==> R b"
20.171 - apply auto
20.172 - done
20.173 -
20.174 -lemma aux_bij: "bijP P F ==> symP P ==> P a b ==> (a \<in> F) = (b \<in> F)"
20.175 - apply (unfold bijP_def)
20.176 - apply (rule iffI)
20.177 - apply (erule_tac [!] aux_foo)
20.178 - apply simp_all
20.179 - apply (rule iffD2)
20.180 - apply (rule_tac P = P in aux_sym)
20.181 - apply simp_all
20.182 - done
20.183 -
20.184 -
20.185 -lemma aux_bijRER:
20.186 - "(A, B) \<in> bijR P ==> uniqP P ==> symP P
20.187 - ==> \<forall>F. bijP P F \<and> F \<subseteq> A \<and> F \<subseteq> B --> F \<in> bijER P"
20.188 - apply (erule bijR.induct)
20.189 - apply simp
20.190 - apply (case_tac "a = b")
20.191 - apply clarify
20.192 - apply (case_tac "b \<in> F")
20.193 - prefer 2
20.194 - apply (simp add: subset_insert)
20.195 - apply (cut_tac F = F and a = b and A = A and B = B in aux1)
20.196 - prefer 6
20.197 - apply clarify
20.198 - apply (rule bijER.insert1)
20.199 - apply simp_all
20.200 - apply (subgoal_tac "bijP P C")
20.201 - apply simp
20.202 - apply (rule aux_in1)
20.203 - apply simp_all
20.204 - apply clarify
20.205 - apply (case_tac "a \<in> F")
20.206 - apply (case_tac [!] "b \<in> F")
20.207 - apply (cut_tac F = F and a = a and b = b and A = A and B = B
20.208 - in aux2)
20.209 - apply (simp_all add: subset_insert)
20.210 - apply clarify
20.211 - apply (rule bijER.insert2)
20.212 - apply simp_all
20.213 - apply (subgoal_tac "bijP P C")
20.214 - apply simp
20.215 - apply (rule aux_in2)
20.216 - apply simp_all
20.217 - apply (subgoal_tac "b \<in> F")
20.218 - apply (rule_tac [2] iffD1)
20.219 - apply (rule_tac [2] a = a and F = F and P = P in aux_bij)
20.220 - apply (simp_all (no_asm_simp))
20.221 - apply (subgoal_tac [2] "a \<in> F")
20.222 - apply (rule_tac [3] iffD2)
20.223 - apply (rule_tac [3] b = b and F = F and P = P in aux_bij)
20.224 - apply auto
20.225 - done
20.226 -
20.227 -lemma bijR_bijER:
20.228 - "(A, A) \<in> bijR P ==>
20.229 - bijP P A ==> uniqP P ==> symP P ==> A \<in> bijER P"
20.230 - apply (cut_tac A = A and B = A and P = P in aux_bijRER)
20.231 - apply auto
20.232 - done
20.233 -
20.234 -end
21.1 --- a/src/HOL/NumberTheory/Chinese.thy Tue Sep 01 14:13:34 2009 +0200
21.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
21.3 @@ -1,259 +0,0 @@
21.4 -(* Title: HOL/NumberTheory/Chinese.thy
21.5 - ID: $Id$
21.6 - Author: Thomas M. Rasmussen
21.7 - Copyright 2000 University of Cambridge
21.8 -*)
21.9 -
21.10 -header {* The Chinese Remainder Theorem *}
21.11 -
21.12 -theory Chinese
21.13 -imports IntPrimes
21.14 -begin
21.15 -
21.16 -text {*
21.17 - The Chinese Remainder Theorem for an arbitrary finite number of
21.18 - equations. (The one-equation case is included in theory @{text
21.19 - IntPrimes}. Uses functions for indexing.\footnote{Maybe @{term
21.20 -