From 417a4ed168b8982f7f8db417e2deb23693beedc7 Mon Sep 17 00:00:00 2001
From: David Aspinall
Date: Tue, 3 Aug 2010 12:48:09 +0000
Subject: Move distribution examples into subdir

---
 isar/ex/Sqrt_Script.thy | 70 +++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 70 insertions(+)
 create mode 100644 isar/ex/Sqrt_Script.thy

(limited to 'isar/ex/Sqrt_Script.thy')

diff --git a/isar/ex/Sqrt_Script.thy b/isar/ex/Sqrt_Script.thy
new file mode 100644
index 00000000..08634ea7
--- /dev/null
+++ b/isar/ex/Sqrt_Script.thy
@@ -0,0 +1,70 @@
+(*  Title:      HOL/ex/Sqrt_Script.thy
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   2001  University of Cambridge
+*)
+
+header {* Square roots of primes are irrational (script version) *}
+
+theory Sqrt_Script
+imports Complex_Main "~~/src/HOL/Number_Theory/Primes"
+begin
+
+text {*
+  \medskip Contrast this linear Isabelle/Isar script with Markus
+  Wenzel's more mathematical version.
+*}
+
+subsection {* Preliminaries *}
+
+lemma prime_nonzero:  "prime (p::nat) \<Longrightarrow> p \<noteq> 0"
+  by (force simp add: prime_nat_def)
+
+lemma prime_dvd_other_side:
+    "(n::nat) * n = p * (k * k) \<Longrightarrow> prime p \<Longrightarrow> p dvd n"
+  apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult_nat)
+  apply auto
+  done
+
+lemma reduction: "prime (p::nat) \<Longrightarrow>
+    0 < k \<Longrightarrow> k * k = p * (j * j) \<Longrightarrow> k < p * j \<and> 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_nat_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:
+    "prime (p::nat) \<Longrightarrow> (\<And>k. 0 < k \<Longrightarrow> m * m \<noteq> p * (k * k))"
+  apply (induct 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 {* Main theorem *}
+
+text {*
+  The square root of any prime number (including @{text 2}) is
+  irrational.
+*}
+
+theorem prime_sqrt_irrational:
+    "prime (p::nat) \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>"
+  apply (rule notI)
+  apply (erule Rats_abs_nat_div_natE)
+  apply (simp del: real_of_nat_mult
+              add: abs_if divide_eq_eq prime_not_square real_of_nat_mult [symmetric])
+  done
+
+lemmas two_sqrt_irrational =
+  prime_sqrt_irrational [OF two_is_prime_nat]
+
+end
-- 
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