# HG changeset patch
# User paulson
# Date 1004964948 -3600
# Node ID 650f854b73107ef9370a745be099df3f870f8366
# Parent  6934c005aec4f145978f1bd3394ebe369ae27cc3
new Sqrt example

diff -r 6934c005aec4 -r 650f854b7310 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Sun Nov 04 21:12:03 2001 +0100
+++ b/src/HOL/IsaMakefile	Mon Nov 05 13:55:48 2001 +0100
@@ -154,7 +154,8 @@
 HOL-Real-ex: HOL-Real $(LOG)/HOL-Real-ex.gz
 
 $(LOG)/HOL-Real-ex.gz: $(OUT)/HOL-Real Real/ex/ROOT.ML \
-  Real/ex/BinEx.thy Real/ex/document/root.tex Real/ex/Sqrt_Irrational.thy
+  Real/ex/BinEx.thy Real/ex/document/root.tex Real/ex/Sqrt_Irrational.thy\
+      Real/ex/Sqrt_Script.thy
 	@cd Real; $(ISATOOL) usedir $(OUT)/HOL-Real ex
 
 
diff -r 6934c005aec4 -r 650f854b7310 src/HOL/Real/ex/Sqrt_Script.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Real/ex/Sqrt_Script.thy	Mon Nov 05 13:55:48 2001 +0100
@@ -0,0 +1,81 @@
+(*  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 *}
+
+text {*
+  \medskip Contrast this linear Isar script with Markus Wenzel's 
+           more mathematical version.
+*}
+
+theory Sqrt_Script = Primes + Real:
+
+section {* 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
+
+
+section {* The set of rational numbers [Borrowed from Markus Wenzel] *}
+
+constdefs
+  rationals :: "real set"    ("\<rat>")
+  "\<rat> \<equiv> {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
+
+
+section {* Main theorem *}
+
+text {*
+  \tweakskip 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 [THEN sym])
+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