# HG changeset patch
# User wenzelm
# Date 1015433288 -3600
# Node ID 5765aa72afac56248b07014bb3a928562c710d75
# Parent  84e4ba7fb033a9a47b43bd1dc6e31295f06460d4
moved to Hyperreal-ex;

diff -r 84e4ba7fb033 -r 5765aa72afac src/HOL/Real/ex/ROOT.ML
--- a/src/HOL/Real/ex/ROOT.ML	Wed Mar 06 17:47:51 2002 +0100
+++ b/src/HOL/Real/ex/ROOT.ML	Wed Mar 06 17:48:08 2002 +0100
@@ -4,7 +4,4 @@
 Miscellaneous HOL-Real Examples.
 *)
 
-no_document use_thy "Primes";
-time_use_thy "Sqrt";
-time_use_thy "Sqrt_Script";
 time_use_thy "BinEx";
diff -r 84e4ba7fb033 -r 5765aa72afac src/HOL/Real/ex/Sqrt.thy
--- a/src/HOL/Real/ex/Sqrt.thy	Wed Mar 06 17:47:51 2002 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,152 +0,0 @@
-(*  Title:      HOL/Real/ex/Sqrt.thy
-    ID:         $Id$
-    Author:     Markus Wenzel, TU Muenchen
-    License:    GPL (GNU GENERAL PUBLIC LICENSE)
-*)
-
-header {*  Square roots of primes are irrational *}
-
-theory Sqrt = Primes + Real:
-
-subsection {* The set of rational numbers *}
-
-constdefs
-  rationals :: "real set"    ("\<rat>")
-  "\<rat> == {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
-
-theorem rationals_rep: "x \<in> \<rat> ==>
-  \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real m / real n \<and> gcd (m, n) = 1"
-proof -
-  assume "x \<in> \<rat>"
-  then obtain m n :: nat where n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n"
-    by (unfold rationals_def) blast
-  let ?gcd = "gcd (m, n)"
-  from n have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero)
-  let ?k = "m div ?gcd"
-  let ?l = "n div ?gcd"
-  let ?gcd' = "gcd (?k, ?l)"
-  have "?gcd dvd m" .. hence gcd_k: "?gcd * ?k = m"
-    by (rule dvd_mult_div_cancel)
-  have "?gcd dvd n" .. hence gcd_l: "?gcd * ?l = n"
-    by (rule dvd_mult_div_cancel)
-
-  from n gcd_l have "?l \<noteq> 0"
-    by (auto iff del: neq0_conv)
-  moreover
-  have "\<bar>x\<bar> = real ?k / real ?l"
-  proof -
-    from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)"
-      by (simp add: real_mult_div_cancel1)
-    also from gcd_k gcd_l have "... = real m / real n" by simp
-    also from x_rat have "... = \<bar>x\<bar>" ..
-    finally show ?thesis ..
-  qed
-  moreover
-  have "?gcd' = 1"
-  proof -
-    have "?gcd * ?gcd' = gcd (?gcd * ?k, ?gcd * ?l)"
-      by (rule gcd_mult_distrib2)
-    with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
-    with gcd show ?thesis by simp
-  qed
-  ultimately show ?thesis by blast
-qed
-
-lemma [elim?]: "r \<in> \<rat> ==>
-  (!!m n. n \<noteq> 0 ==> \<bar>r\<bar> = real m / real n ==> gcd (m, n) = 1 ==> C)
-    ==> C"
-  by (insert rationals_rep) blast
-
-
-subsection {* Main theorem *}
-
-text {*
-  The square root of any prime number (including @{text 2}) is
-  irrational.
-*}
-
-theorem sqrt_prime_irrational: "x\<twosuperior> = real p ==> p \<in> prime ==> x \<notin> \<rat>"
-proof
-  assume x_sqrt: "x\<twosuperior> = real p"
-  assume p_prime: "p \<in> prime"
-  hence p: "1 < p" by (simp add: prime_def)
-  assume "x \<in> \<rat>"
-  then obtain m n where
-    n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n" and gcd: "gcd (m, n) = 1" ..
-  have eq: "m\<twosuperior> = p * n\<twosuperior>"
-  proof -
-    from n x_rat have "real m = \<bar>x\<bar> * real n" by simp
-    hence "real (m\<twosuperior>) = x\<twosuperior> * real (n\<twosuperior>)"
-      by (simp add: power_two real_power_two split: abs_split)
-    also from x_sqrt have "... = real (p * n\<twosuperior>)" by simp
-    finally show ?thesis ..
-  qed
-  have "p dvd m \<and> p dvd n"
-  proof
-    from eq have "p dvd m\<twosuperior>" ..
-    with p_prime show "p dvd m" by (rule prime_dvd_power_two)
-    then obtain k where "m = p * k" ..
-    with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power_two mult_ac)
-    with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power_two)
-    hence "p dvd n\<twosuperior>" ..
-    with p_prime show "p dvd n" by (rule prime_dvd_power_two)
-  qed
-  hence "p dvd gcd (m, n)" ..
-  with gcd have "p dvd 1" by simp
-  hence "p \<le> 1" by (simp add: dvd_imp_le)
-  with p show False by simp
-qed
-
-
-subsection {* Variations *}
-
-text {*
-  Just for the record: we instantiate the main theorem for the
-  specific prime number @{text 2} (real mathematicians would never do
-  this formally :-).
-*}
-
-theorem "x\<twosuperior> = real (2::nat) ==> x \<notin> \<rat>"
-proof (rule sqrt_prime_irrational)
-  {
-    fix m :: nat assume dvd: "m dvd 2"
-    hence "m \<le> 2" by (simp add: dvd_imp_le)
-    moreover from dvd have "m \<noteq> 0" by (auto iff del: neq0_conv)
-    ultimately have "m = 1 \<or> m = 2" by arith
-  }
-  thus "2 \<in> prime" by (simp add: prime_def)
-qed
-
-text {*
-  \medskip An alternative version of the main proof, using mostly
-  linear forward-reasoning.  While this results in less top-down
-  structure, it is probably closer to proofs seen in mathematics.
-*}
-
-theorem "x\<twosuperior> = real p ==> p \<in> prime ==> x \<notin> \<rat>"
-proof
-  assume x_sqrt: "x\<twosuperior> = real p"
-  assume p_prime: "p \<in> prime"
-  hence p: "1 < p" by (simp add: prime_def)
-  assume "x \<in> \<rat>"
-  then obtain m n where
-    n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n" and gcd: "gcd (m, n) = 1" ..
-  from n x_rat have "real m = \<bar>x\<bar> * real n" by simp
-  hence "real (m\<twosuperior>) = x\<twosuperior> * real (n\<twosuperior>)"
-    by (simp add: power_two real_power_two split: abs_split)
-  also from x_sqrt have "... = real (p * n\<twosuperior>)" by simp
-  finally have eq: "m\<twosuperior> = p * n\<twosuperior>" ..
-  hence "p dvd m\<twosuperior>" ..
-  with p_prime have dvd_m: "p dvd m" by (rule prime_dvd_power_two)
-  then obtain k where "m = p * k" ..
-  with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power_two mult_ac)
-  with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power_two)
-  hence "p dvd n\<twosuperior>" ..
-  with p_prime have "p dvd n" by (rule prime_dvd_power_two)
-  with dvd_m have "p dvd gcd (m, n)" by (rule gcd_greatest)
-  with gcd have "p dvd 1" by simp
-  hence "p \<le> 1" by (simp add: dvd_imp_le)
-  with p show False by simp
-qed
-
-end
diff -r 84e4ba7fb033 -r 5765aa72afac src/HOL/Real/ex/Sqrt_Script.thy
--- a/src/HOL/Real/ex/Sqrt_Script.thy	Wed Mar 06 17:47:51 2002 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,81 +0,0 @@
-(*  Title:      HOL/Real/ex/Sqrt_Script.thy
-    ID:         $Id$
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   2001  University of Cambridge
-*)
-
-header {*  Square roots of primes are irrational *}
-
-theory Sqrt_Script = Primes + Real:
-
-text {*
-  \medskip Contrast this linear Isar script with Markus Wenzel's more
-  mathematical version.
-*}
-
-subsection {* Preliminaries *}
-
-lemma prime_nonzero:  "p \<in> prime \<Longrightarrow> p\<noteq>0"
-by (force simp add: prime_def)
-
-lemma prime_dvd_other_side: "\<lbrakk>n*n = p*(k*k); p \<in> prime\<rbrakk> \<Longrightarrow> p dvd n"
-apply (subgoal_tac "p dvd n*n", blast dest: prime_dvd_mult)
-apply (rule_tac j="k*k" in dvd_mult_left, simp)
-done
-
-lemma reduction: "\<lbrakk>p \<in> prime; 0 < k; k*k = p*(j*j)\<rbrakk> \<Longrightarrow> k < p*j & 0 < j"
-apply (rule ccontr)
-apply (simp add: linorder_not_less)
-apply (erule disjE)
- apply (frule mult_le_mono, assumption)
- apply auto
-apply (force simp add: prime_def)
-done
-
-lemma rearrange: "(j::nat) * (p*j) = k*k \<Longrightarrow> k*k = p*(j*j)"
-by (simp add: mult_ac)
-
-lemma prime_not_square [rule_format]:
-     "p \<in> prime \<Longrightarrow> \<forall>k. 0<k \<longrightarrow> m*m \<noteq> p*(k*k)"
-apply (induct_tac m rule: nat_less_induct)
-apply clarify 
-apply (frule prime_dvd_other_side, assumption)
-apply (erule dvdE)
-apply (simp add: nat_mult_eq_cancel_disj prime_nonzero)
-apply (blast dest: rearrange reduction)
-done
-
-
-subsection {* The set of rational numbers *}
-
-constdefs
-  rationals :: "real set"    ("\<rat>")
-  "\<rat> \<equiv> {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
-
-
-subsection {* Main theorem *}
-
-text {*
-  The square root of any prime number (including @{text 2}) is
-  irrational.
-*}
-
-theorem prime_sqrt_irrational: "\<lbrakk>p \<in> prime; x*x = real p; 0 \<le> x\<rbrakk> \<Longrightarrow> x \<notin> \<rat>"
-apply (simp add: rationals_def real_abs_def)
-apply clarify
-apply (erule_tac P="real m / real n * ?x = ?y" in rev_mp) 
-apply (simp del: real_of_nat_mult
-	    add: real_divide_eq_eq prime_not_square
-                 real_of_nat_mult [symmetric])
-done
-
-lemma two_is_prime: "2 \<in> prime"
-apply (auto simp add: prime_def)
-apply (case_tac m)
-apply (auto dest!: dvd_imp_le)
-apply arith
-done
-
-lemmas two_sqrt_irrational = prime_sqrt_irrational [OF two_is_prime]
-
-end
diff -r 84e4ba7fb033 -r 5765aa72afac src/HOL/Real/ex/document/root.tex
--- a/src/HOL/Real/ex/document/root.tex	Wed Mar 06 17:47:51 2002 +0100
+++ b/src/HOL/Real/ex/document/root.tex	Wed Mar 06 17:48:08 2002 +0100
@@ -1,7 +1,8 @@
 
 \documentclass[11pt,a4paper]{article}
-\usepackage{isabelle,isabellesym,pdfsetup}
 \usepackage[latin1]{inputenc}
+\usepackage{isabelle,isabellesym}
+\usepackage{pdfsetup}
 
 \urlstyle{rm}
 \isabellestyle{it}