(* Title: HOL/Proofs/Extraction/Euclid.thy
Author: Markus Wenzel, TU Muenchen
Author: Freek Wiedijk, Radboud University Nijmegen
Author: Stefan Berghofer, TU Muenchen
*)
section \<open>Euclid's theorem\<close>
theory Euclid
imports
"~~/src/HOL/Number_Theory/UniqueFactorization"
Util
"~~/src/HOL/Library/Code_Target_Numeral"
begin
text \<open>
A constructive version of the proof of Euclid's theorem by
Markus Wenzel and Freek Wiedijk @{cite "Wenzel-Wiedijk-JAR2002"}.
\<close>
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_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: ac_simps)
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 \<open>m \<le> Suc n\<close> show ?case
proof (rule le_SucE)
assume "m \<le> n"
with \<open>0 < m\<close> 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: ac_simps)
then show ?thesis by (simp add: mult.commute)
qed
qed
lemma dvd_prod [iff]: "n dvd (\<Prod>m::nat \<in># mset (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::nat \<in># mset qs. m) =
(\<Prod>m::nat \<in># mset ms. m) * (\<Prod>m::nat \<in># mset 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::nat \<in># mset (ms @ ns). m) =
(\<Prod>m::nat \<in># mset ms. m) * (\<Prod>m::nat \<in># mset 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::nat \<in># mset ps. m)"
using \<open>ps \<noteq> []\<close> \<open>all_prime ps\<close> 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::nat \<in># mset ps. m) = n)"
proof (induct n rule: nat_wf_ind)
case (1 n)
from \<open>Suc 0 < n\<close>
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::nat \<in># mset ps. m) = m" by (rule 1)
then obtain ps1 where "all_prime ps1" and prod_ps1_m: "(\<Prod>m::nat \<in># mset ps1. m) = m"
by iprover
from kn and k have "\<exists>ps. all_prime ps \<and> (\<Prod>m::nat \<in># mset ps. m) = k" by (rule 1)
then obtain ps2 where "all_prime ps2" and prod_ps2_k: "(\<Prod>m::nat \<in># mset ps2. m) = k"
by iprover
from \<open>all_prime ps1\<close> \<open>all_prime ps2\<close>
have "\<exists>ps. all_prime ps \<and> (\<Prod>m::nat \<in># mset ps. m) =
(\<Prod>m::nat \<in># mset ps1. m) * (\<Prod>m::nat \<in># mset 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::nat \<in># mset [n]. m) = n" by (simp add: msetprod_singleton)
ultimately have "all_prime [n] \<and> (\<Prod>m::nat \<in># mset [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::nat \<in># mset 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 \<open>all_prime ps\<close> have "prime p" by (simp add: all_prime_simps)
moreover from \<open>all_prime ps\<close> ps prod_ps
have "p dvd n" by (simp only: dvd_prod)
ultimately show ?thesis by iprover
qed
text \<open>
Euclid's theorem: there are infinitely many primes.
\<close>
lemma Euclid: "\<exists>p::nat. prime p \<and> n < p"
proof -
let ?k = "fact n + (1::nat)"
have "1 < ?k" 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 \<open>prime p\<close> 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 \<open>
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}
\<close>
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
end