author | wenzelm |
Fri, 17 Nov 2006 02:20:03 +0100 | |
changeset 21404 | eb85850d3eb7 |
parent 19736 | d8d0f8f51d69 |
child 27651 | 16a26996c30e |
permissions | -rw-r--r-- |
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(* Title: HOL/Hyperreal/ex/Sqrt_Script.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 2001 University of Cambridge |
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*) |
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header {* Square roots of primes are irrational (script version) *} |
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theory Sqrt_Script |
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imports Primes Complex_Main |
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begin |
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text {* |
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\medskip Contrast this linear Isabelle/Isar script with Markus |
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Wenzel's more mathematical version. |
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*} |
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subsection {* Preliminaries *} |
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lemma prime_nonzero: "prime p \<Longrightarrow> p \<noteq> 0" |
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by (force simp add: prime_def) |
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lemma prime_dvd_other_side: |
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"n * n = p * (k * k) \<Longrightarrow> prime p \<Longrightarrow> p dvd n" |
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apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult) |
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apply (rule_tac j = "k * k" in dvd_mult_left, simp) |
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done |
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lemma reduction: "prime p \<Longrightarrow> |
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0 < k \<Longrightarrow> k * k = p * (j * j) \<Longrightarrow> k < p * j \<and> 0 < j" |
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apply (rule ccontr) |
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apply (simp add: linorder_not_less) |
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apply (erule disjE) |
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apply (frule mult_le_mono, assumption) |
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apply auto |
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apply (force simp add: prime_def) |
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done |
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lemma rearrange: "(j::nat) * (p * j) = k * k \<Longrightarrow> k * k = p * (j * j)" |
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by (simp add: mult_ac) |
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lemma prime_not_square: |
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"prime p \<Longrightarrow> (\<And>k. 0 < k \<Longrightarrow> m * m \<noteq> p * (k * k))" |
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apply (induct m rule: nat_less_induct) |
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apply clarify |
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apply (frule prime_dvd_other_side, assumption) |
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apply (erule dvdE) |
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apply (simp add: nat_mult_eq_cancel_disj prime_nonzero) |
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apply (blast dest: rearrange reduction) |
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done |
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subsection {* The set of rational numbers *} |
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definition |
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rationals :: "real set" ("\<rat>") where |
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"\<rat> = {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}" |
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subsection {* Main theorem *} |
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text {* |
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The square root of any prime number (including @{text 2}) is |
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irrational. |
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*} |
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theorem prime_sqrt_irrational: |
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"prime p \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>" |
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apply (simp add: rationals_def real_abs_def) |
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apply clarify |
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apply (erule_tac P = "real m / real n * ?x = ?y" in rev_mp) |
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apply (simp del: real_of_nat_mult |
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add: divide_eq_eq prime_not_square real_of_nat_mult [symmetric]) |
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done |
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lemmas two_sqrt_irrational = |
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prime_sqrt_irrational [OF two_is_prime] |
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end |