author | wenzelm |
Fri, 18 Aug 2017 20:47:47 +0200 | |
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child 73811 | f143d0a4cb6a |
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(* Title: HOL/ex/Sqrt_Script.thy |
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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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section \<open>Square roots of primes are irrational (script version)\<close> |
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theory Sqrt_Script |
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session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
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imports Complex_Main "HOL-Computational_Algebra.Primes" |
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begin |
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text \<open> |
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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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\<close> |
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subsection \<open>Preliminaries\<close> |
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lemma prime_nonzero: "prime (p::nat) \<Longrightarrow> p \<noteq> 0" |
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by (force simp add: prime_nat_iff) |
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lemma prime_dvd_other_side: |
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"(n::nat) * 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_nat) |
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apply auto |
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done |
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lemma reduction: "prime (p::nat) \<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_nat_iff) |
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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: ac_simps) |
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lemma prime_not_square: |
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"prime (p::nat) \<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 \<open>Main theorem\<close> |
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text \<open> |
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The square root of any prime number (including \<open>2\<close>) is |
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irrational. |
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\<close> |
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theorem prime_sqrt_irrational: |
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"prime (p::nat) \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>" |
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apply (rule notI) |
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apply (erule Rats_abs_nat_div_natE) |
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apply (simp del: of_nat_mult |
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add: abs_if divide_eq_eq prime_not_square 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_nat] |
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end |