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(* Title: HOL/Induct/Sigma_Algebra.thy
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ID: $Id$
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Author: Markus Wenzel, TU Muenchen
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License: GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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theory Sigma_Algebra = Main:
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text {*
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This is just a tiny example demonstrating the use of inductive
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definitions in classical mathematics. We define the least @{text
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\<sigma>}-algebra over a given set of sets.
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*}
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consts
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sigma_algebra :: "'a set set => 'a set set" ("\<sigma>'_algebra")
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inductive "\<sigma>_algebra A"
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intros
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basic: "a \<in> A ==> a \<in> \<sigma>_algebra A"
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UNIV: "UNIV \<in> \<sigma>_algebra A"
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complement: "a \<in> \<sigma>_algebra A ==> -a \<in> \<sigma>_algebra A"
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Union: "(!!i::nat. a i \<in> \<sigma>_algebra A) ==> (\<Union>i. a i) \<in> \<sigma>_algebra A"
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text {*
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The following basic facts are consequences of the closure properties
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of any @{text \<sigma>}-algebra, merely using the introduction rules, but
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no induction nor cases.
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*}
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theorem sigma_algebra_empty: "{} \<in> \<sigma>_algebra A"
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proof -
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have "UNIV \<in> \<sigma>_algebra A" by (rule sigma_algebra.UNIV)
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hence "-UNIV \<in> \<sigma>_algebra A" by (rule sigma_algebra.complement)
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also have "-UNIV = {}" by simp
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finally show ?thesis .
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qed
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theorem sigma_algebra_Inter:
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"(!!i::nat. a i \<in> \<sigma>_algebra A) ==> (\<Inter>i. a i) \<in> \<sigma>_algebra A"
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proof -
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assume "!!i::nat. a i \<in> \<sigma>_algebra A"
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hence "!!i::nat. -(a i) \<in> \<sigma>_algebra A" by (rule sigma_algebra.complement)
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hence "(\<Union>i. -(a i)) \<in> \<sigma>_algebra A" by (rule sigma_algebra.Union)
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hence "-(\<Union>i. -(a i)) \<in> \<sigma>_algebra A" by (rule sigma_algebra.complement)
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also have "-(\<Union>i. -(a i)) = (\<Inter>i. a i)" by simp
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finally show ?thesis .
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qed
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end
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