--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Proofs/Extraction/Euclid.thy Mon Sep 06 14:18:16 2010 +0200
@@ -0,0 +1,279 @@
+(* Title: HOL/Proofs/Extraction/Euclid.thy
+ Author: Markus Wenzel, TU Muenchen
+ Author: Freek Wiedijk, Radboud University Nijmegen
+ Author: Stefan Berghofer, TU Muenchen
+*)
+
+header {* Euclid's theorem *}
+
+theory Euclid
+imports "~~/src/HOL/Number_Theory/UniqueFactorization" Util Efficient_Nat
+begin
+
+text {*
+A constructive version of the proof of Euclid's theorem by
+Markus Wenzel and Freek Wiedijk \cite{Wenzel-Wiedijk-JAR2002}.
+*}
+
+lemma factor_greater_one1: "n = m * k \<Longrightarrow> m < n \<Longrightarrow> k < n \<Longrightarrow> Suc 0 < m"
+ by (induct m) auto
+
+lemma factor_greater_one2: "n = m * k \<Longrightarrow> m < n \<Longrightarrow> k < n \<Longrightarrow> Suc 0 < k"
+ by (induct k) auto
+
+lemma prod_mn_less_k:
+ "(0::nat) < n ==> 0 < k ==> Suc 0 < m ==> m * n = k ==> n < k"
+ by (induct m) auto
+
+lemma prime_eq: "prime (p::nat) = (1 < p \<and> (\<forall>m. m dvd p \<longrightarrow> 1 < m \<longrightarrow> m = p))"
+ apply (simp add: prime_nat_def)
+ apply (rule iffI)
+ apply blast
+ apply (erule conjE)
+ apply (rule conjI)
+ apply assumption
+ apply (rule allI impI)+
+ apply (erule allE)
+ apply (erule impE)
+ apply assumption
+ apply (case_tac "m=0")
+ apply simp
+ apply (case_tac "m=Suc 0")
+ apply simp
+ apply simp
+ done
+
+lemma prime_eq': "prime (p::nat) = (1 < p \<and> (\<forall>m k. p = m * k \<longrightarrow> 1 < m \<longrightarrow> m = p))"
+ by (simp add: prime_eq dvd_def HOL.all_simps [symmetric] del: HOL.all_simps)
+
+lemma not_prime_ex_mk:
+ assumes n: "Suc 0 < n"
+ shows "(\<exists>m k. Suc 0 < m \<and> Suc 0 < k \<and> m < n \<and> k < n \<and> n = m * k) \<or> prime n"
+proof -
+ {
+ fix k
+ from nat_eq_dec
+ have "(\<exists>m<n. n = m * k) \<or> \<not> (\<exists>m<n. n = m * k)"
+ by (rule search)
+ }
+ hence "(\<exists>k<n. \<exists>m<n. n = m * k) \<or> \<not> (\<exists>k<n. \<exists>m<n. n = m * k)"
+ by (rule search)
+ thus ?thesis
+ proof
+ assume "\<exists>k<n. \<exists>m<n. n = m * k"
+ then obtain k m where k: "k<n" and m: "m<n" and nmk: "n = m * k"
+ by iprover
+ from nmk m k have "Suc 0 < m" by (rule factor_greater_one1)
+ moreover from nmk m k have "Suc 0 < k" by (rule factor_greater_one2)
+ ultimately show ?thesis using k m nmk by iprover
+ next
+ assume "\<not> (\<exists>k<n. \<exists>m<n. n = m * k)"
+ hence A: "\<forall>k<n. \<forall>m<n. n \<noteq> m * k" by iprover
+ have "\<forall>m k. n = m * k \<longrightarrow> Suc 0 < m \<longrightarrow> m = n"
+ proof (intro allI impI)
+ fix m k
+ assume nmk: "n = m * k"
+ assume m: "Suc 0 < m"
+ from n m nmk have k: "0 < k"
+ by (cases k) auto
+ moreover from n have n: "0 < n" by simp
+ moreover note m
+ moreover from nmk have "m * k = n" by simp
+ ultimately have kn: "k < n" by (rule prod_mn_less_k)
+ show "m = n"
+ proof (cases "k = Suc 0")
+ case True
+ with nmk show ?thesis by (simp only: mult_Suc_right)
+ next
+ case False
+ from m have "0 < m" by simp
+ moreover note n
+ moreover from False n nmk k have "Suc 0 < k" by auto
+ moreover from nmk have "k * m = n" by (simp only: mult_ac)
+ ultimately have mn: "m < n" by (rule prod_mn_less_k)
+ with kn A nmk show ?thesis by iprover
+ qed
+ qed
+ with n have "prime n"
+ by (simp only: prime_eq' One_nat_def simp_thms)
+ thus ?thesis ..
+ qed
+qed
+
+lemma dvd_factorial: "0 < m \<Longrightarrow> m \<le> n \<Longrightarrow> m dvd fact (n::nat)"
+proof (induct n rule: nat_induct)
+ case 0
+ then show ?case by simp
+next
+ case (Suc n)
+ from `m \<le> Suc n` show ?case
+ proof (rule le_SucE)
+ assume "m \<le> n"
+ with `0 < m` have "m dvd fact n" by (rule Suc)
+ then have "m dvd (fact n * Suc n)" by (rule dvd_mult2)
+ then show ?thesis by (simp add: mult_commute)
+ next
+ assume "m = Suc n"
+ then have "m dvd (fact n * Suc n)"
+ by (auto intro: dvdI simp: mult_ac)
+ then show ?thesis by (simp add: mult_commute)
+ qed
+qed
+
+lemma dvd_prod [iff]: "n dvd (PROD m\<Colon>nat:#multiset_of (n # ns). m)"
+ by (simp add: msetprod_Un msetprod_singleton)
+
+definition all_prime :: "nat list \<Rightarrow> bool" where
+ "all_prime ps \<longleftrightarrow> (\<forall>p\<in>set ps. prime p)"
+
+lemma all_prime_simps:
+ "all_prime []"
+ "all_prime (p # ps) \<longleftrightarrow> prime p \<and> all_prime ps"
+ by (simp_all add: all_prime_def)
+
+lemma all_prime_append:
+ "all_prime (ps @ qs) \<longleftrightarrow> all_prime ps \<and> all_prime qs"
+ by (simp add: all_prime_def ball_Un)
+
+lemma split_all_prime:
+ assumes "all_prime ms" and "all_prime ns"
+ shows "\<exists>qs. all_prime qs \<and> (PROD m\<Colon>nat:#multiset_of qs. m) =
+ (PROD m\<Colon>nat:#multiset_of ms. m) * (PROD m\<Colon>nat:#multiset_of ns. m)" (is "\<exists>qs. ?P qs \<and> ?Q qs")
+proof -
+ from assms have "all_prime (ms @ ns)"
+ by (simp add: all_prime_append)
+ moreover from assms have "(PROD m\<Colon>nat:#multiset_of (ms @ ns). m) =
+ (PROD m\<Colon>nat:#multiset_of ms. m) * (PROD m\<Colon>nat:#multiset_of ns. m)"
+ by (simp add: msetprod_Un)
+ ultimately have "?P (ms @ ns) \<and> ?Q (ms @ ns)" ..
+ then show ?thesis ..
+qed
+
+lemma all_prime_nempty_g_one:
+ assumes "all_prime ps" and "ps \<noteq> []"
+ shows "Suc 0 < (PROD m\<Colon>nat:#multiset_of ps. m)"
+ using `ps \<noteq> []` `all_prime ps` unfolding One_nat_def [symmetric] by (induct ps rule: list_nonempty_induct)
+ (simp_all add: all_prime_simps msetprod_singleton msetprod_Un prime_gt_1_nat less_1_mult del: One_nat_def)
+
+lemma factor_exists: "Suc 0 < n \<Longrightarrow> (\<exists>ps. all_prime ps \<and> (PROD m\<Colon>nat:#multiset_of ps. m) = n)"
+proof (induct n rule: nat_wf_ind)
+ case (1 n)
+ from `Suc 0 < n`
+ have "(\<exists>m k. Suc 0 < m \<and> Suc 0 < k \<and> m < n \<and> k < n \<and> n = m * k) \<or> prime n"
+ by (rule not_prime_ex_mk)
+ then show ?case
+ proof
+ assume "\<exists>m k. Suc 0 < m \<and> Suc 0 < k \<and> m < n \<and> k < n \<and> n = m * k"
+ then obtain m k where m: "Suc 0 < m" and k: "Suc 0 < k" and mn: "m < n"
+ and kn: "k < n" and nmk: "n = m * k" by iprover
+ from mn and m have "\<exists>ps. all_prime ps \<and> (PROD m\<Colon>nat:#multiset_of ps. m) = m" by (rule 1)
+ then obtain ps1 where "all_prime ps1" and prod_ps1_m: "(PROD m\<Colon>nat:#multiset_of ps1. m) = m"
+ by iprover
+ from kn and k have "\<exists>ps. all_prime ps \<and> (PROD m\<Colon>nat:#multiset_of ps. m) = k" by (rule 1)
+ then obtain ps2 where "all_prime ps2" and prod_ps2_k: "(PROD m\<Colon>nat:#multiset_of ps2. m) = k"
+ by iprover
+ from `all_prime ps1` `all_prime ps2`
+ have "\<exists>ps. all_prime ps \<and> (PROD m\<Colon>nat:#multiset_of ps. m) =
+ (PROD m\<Colon>nat:#multiset_of ps1. m) * (PROD m\<Colon>nat:#multiset_of ps2. m)"
+ by (rule split_all_prime)
+ with prod_ps1_m prod_ps2_k nmk show ?thesis by simp
+ next
+ assume "prime n" then have "all_prime [n]" by (simp add: all_prime_simps)
+ moreover have "(PROD m\<Colon>nat:#multiset_of [n]. m) = n" by (simp add: msetprod_singleton)
+ ultimately have "all_prime [n] \<and> (PROD m\<Colon>nat:#multiset_of [n]. m) = n" ..
+ then show ?thesis ..
+ qed
+qed
+
+lemma prime_factor_exists:
+ assumes N: "(1::nat) < n"
+ shows "\<exists>p. prime p \<and> p dvd n"
+proof -
+ from N obtain ps where "all_prime ps"
+ and prod_ps: "n = (PROD m\<Colon>nat:#multiset_of ps. m)" using factor_exists
+ by simp iprover
+ with N have "ps \<noteq> []"
+ by (auto simp add: all_prime_nempty_g_one msetprod_empty)
+ then obtain p qs where ps: "ps = p # qs" by (cases ps) simp
+ with `all_prime ps` have "prime p" by (simp add: all_prime_simps)
+ moreover from `all_prime ps` ps prod_ps
+ have "p dvd n" by (simp only: dvd_prod)
+ ultimately show ?thesis by iprover
+qed
+
+text {*
+Euclid's theorem: there are infinitely many primes.
+*}
+
+lemma Euclid: "\<exists>p::nat. prime p \<and> n < p"
+proof -
+ let ?k = "fact n + 1"
+ have "1 < fact n + 1" by simp
+ then obtain p where prime: "prime p" and dvd: "p dvd ?k" using prime_factor_exists by iprover
+ have "n < p"
+ proof -
+ have "\<not> p \<le> n"
+ proof
+ assume pn: "p \<le> n"
+ from `prime p` have "0 < p" by (rule prime_gt_0_nat)
+ then have "p dvd fact n" using pn by (rule dvd_factorial)
+ with dvd have "p dvd ?k - fact n" by (rule dvd_diff_nat)
+ then have "p dvd 1" by simp
+ with prime show False by auto
+ qed
+ then show ?thesis by simp
+ qed
+ with prime show ?thesis by iprover
+qed
+
+extract Euclid
+
+text {*
+The program extracted from the proof of Euclid's theorem looks as follows.
+@{thm [display] Euclid_def}
+The program corresponding to the proof of the factorization theorem is
+@{thm [display] factor_exists_def}
+*}
+
+instantiation nat :: default
+begin
+
+definition "default = (0::nat)"
+
+instance ..
+
+end
+
+instantiation list :: (type) default
+begin
+
+definition "default = []"
+
+instance ..
+
+end
+
+primrec iterate :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a list" where
+ "iterate 0 f x = []"
+ | "iterate (Suc n) f x = (let y = f x in y # iterate n f y)"
+
+lemma "factor_exists 1007 = [53, 19]" by eval
+lemma "factor_exists 567 = [7, 3, 3, 3, 3]" by eval
+lemma "factor_exists 345 = [23, 5, 3]" by eval
+lemma "factor_exists 999 = [37, 3, 3, 3]" by eval
+lemma "factor_exists 876 = [73, 3, 2, 2]" by eval
+
+lemma "iterate 4 Euclid 0 = [2, 3, 7, 71]" by eval
+
+consts_code
+ default ("(error \"default\")")
+
+lemma "factor_exists 1007 = [53, 19]" by evaluation
+lemma "factor_exists 567 = [7, 3, 3, 3, 3]" by evaluation
+lemma "factor_exists 345 = [23, 5, 3]" by evaluation
+lemma "factor_exists 999 = [37, 3, 3, 3]" by evaluation
+lemma "factor_exists 876 = [73, 3, 2, 2]" by evaluation
+
+lemma "iterate 4 Euclid 0 = [2, 3, 7, 71]" by evaluation
+
+end