src/HOL/Complex/ex/Sqrt_Script.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ex/Sqrt_Script.thy	Mon May 05 18:22:31 2003 +0200
@@ -0,0 +1,78 @@
+(*  Title:      HOL/Hyperreal/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 (script version) *}
+
+theory Sqrt_Script = Primes + Hyperreal:
+
+text {*
+  \medskip Contrast this linear Isabelle/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:
+    "n * n = p * (k * k) \<Longrightarrow> p \<in> prime \<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: "p \<in> prime \<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_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:
+    "p \<in> prime \<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 {* 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:
+    "p \<in> prime \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<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
+
+lemmas two_sqrt_irrational =
+  prime_sqrt_irrational [OF two_is_prime]
+
+end