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src/HOL/Induct/Sigma_Algebra.thy

author | haftmann |

Fri, 17 Jun 2005 16:12:49 +0200 | |

changeset 16417 | 9bc16273c2d4 |

parent 14981 | e73f8140af78 |

child 23746 | a455e69c31cc |

permissions | -rw-r--r-- |

migrated theory headers to new format

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