--- a/src/HOL/Proofs/Extraction/Euclid.thy Mon Dec 28 18:03:26 2015 +0100
+++ b/src/HOL/Proofs/Extraction/Euclid.thy Mon Dec 28 19:23:15 2015 +0100
@@ -123,7 +123,7 @@
qed
qed
-lemma dvd_prod [iff]: "n dvd (PROD m::nat:#mset (n # ns). m)"
+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
@@ -140,13 +140,13 @@
lemma split_all_prime:
assumes "all_prime ms" and "all_prime ns"
- shows "\<exists>qs. all_prime qs \<and> (PROD m::nat:#mset qs. m) =
- (PROD m::nat:#mset ms. m) * (PROD m::nat:#mset ns. m)" (is "\<exists>qs. ?P qs \<and> ?Q qs")
+ 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:#mset (ms @ ns). m) =
- (PROD m::nat:#mset ms. m) * (PROD m::nat:#mset ns. m)"
+ 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 ..
@@ -154,11 +154,11 @@
lemma all_prime_nempty_g_one:
assumes "all_prime ps" and "ps \<noteq> []"
- shows "Suc 0 < (PROD m::nat:#mset ps. m)"
+ shows "Suc 0 < (\<Prod>m::nat \<in># mset 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::nat:#mset ps. m) = n)"
+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 `Suc 0 < n`
@@ -169,21 +169,21 @@
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:#mset ps. m) = m" by (rule 1)
- then obtain ps1 where "all_prime ps1" and prod_ps1_m: "(PROD m::nat:#mset ps1. m) = m"
+ 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:#mset ps. m) = k" by (rule 1)
- then obtain ps2 where "all_prime ps2" and prod_ps2_k: "(PROD m::nat:#mset ps2. m) = k"
+ 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 `all_prime ps1` `all_prime ps2`
- have "\<exists>ps. all_prime ps \<and> (PROD m::nat:#mset ps. m) =
- (PROD m::nat:#mset ps1. m) * (PROD m::nat:#mset ps2. m)"
+ 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:#mset [n]. m) = n" by (simp add: msetprod_singleton)
- ultimately have "all_prime [n] \<and> (PROD m::nat:#mset [n]. m) = n" ..
+ 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
@@ -193,7 +193,7 @@
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:#mset ps. m)" using factor_exists
+ 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)