--- a/src/HOL/Extraction/Greatest_Common_Divisor.thy Tue Sep 07 11:51:53 2010 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,96 +0,0 @@
-(* Title: HOL/Extraction/Greatest_Common_Divisor.thy
- Author: Stefan Berghofer, TU Muenchen
- Helmut Schwichtenberg, LMU Muenchen
-*)
-
-header {* Greatest common divisor *}
-
-theory Greatest_Common_Divisor
-imports QuotRem
-begin
-
-theorem greatest_common_divisor:
- "\<And>n::nat. Suc m < n \<Longrightarrow> \<exists>k n1 m1. k * n1 = n \<and> k * m1 = Suc m \<and>
- (\<forall>l l1 l2. l * l1 = n \<longrightarrow> l * l2 = Suc m \<longrightarrow> l \<le> k)"
-proof (induct m rule: nat_wf_ind)
- case (1 m n)
- from division obtain r q where h1: "n = Suc m * q + r" and h2: "r \<le> m"
- by iprover
- show ?case
- proof (cases r)
- case 0
- with h1 have "Suc m * q = n" by simp
- moreover have "Suc m * 1 = Suc m" by simp
- moreover {
- fix l2 have "\<And>l l1. l * l1 = n \<Longrightarrow> l * l2 = Suc m \<Longrightarrow> l \<le> Suc m"
- by (cases l2) simp_all }
- ultimately show ?thesis by iprover
- next
- case (Suc nat)
- with h2 have h: "nat < m" by simp
- moreover from h have "Suc nat < Suc m" by simp
- ultimately have "\<exists>k m1 r1. k * m1 = Suc m \<and> k * r1 = Suc nat \<and>
- (\<forall>l l1 l2. l * l1 = Suc m \<longrightarrow> l * l2 = Suc nat \<longrightarrow> l \<le> k)"
- by (rule 1)
- then obtain k m1 r1 where
- h1': "k * m1 = Suc m"
- and h2': "k * r1 = Suc nat"
- and h3': "\<And>l l1 l2. l * l1 = Suc m \<Longrightarrow> l * l2 = Suc nat \<Longrightarrow> l \<le> k"
- by iprover
- have mn: "Suc m < n" by (rule 1)
- from h1 h1' h2' Suc have "k * (m1 * q + r1) = n"
- by (simp add: add_mult_distrib2 nat_mult_assoc [symmetric])
- moreover have "\<And>l l1 l2. l * l1 = n \<Longrightarrow> l * l2 = Suc m \<Longrightarrow> l \<le> k"
- proof -
- fix l l1 l2
- assume ll1n: "l * l1 = n"
- assume ll2m: "l * l2 = Suc m"
- moreover have "l * (l1 - l2 * q) = Suc nat"
- by (simp add: diff_mult_distrib2 h1 Suc [symmetric] mn ll1n ll2m [symmetric])
- ultimately show "l \<le> k" by (rule h3')
- qed
- ultimately show ?thesis using h1' by iprover
- qed
-qed
-
-extract greatest_common_divisor
-
-text {*
-The extracted program for computing the greatest common divisor is
-@{thm [display] greatest_common_divisor_def}
-*}
-
-instantiation nat :: default
-begin
-
-definition "default = (0::nat)"
-
-instance ..
-
-end
-
-instantiation prod :: (default, default) default
-begin
-
-definition "default = (default, default)"
-
-instance ..
-
-end
-
-instantiation "fun" :: (type, default) default
-begin
-
-definition "default = (\<lambda>x. default)"
-
-instance ..
-
-end
-
-consts_code
- default ("(error \"default\")")
-
-lemma "greatest_common_divisor 7 12 = (4, 3, 2)" by evaluation
-lemma "greatest_common_divisor 7 12 = (4, 3, 2)" by eval
-
-end