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src/HOL/Induct/Sigma_Algebra.thy

author | wenzelm |

Wed, 14 Mar 2012 00:34:56 +0100 | |

changeset 46914 | c2ca2c3d23a6 |

parent 36862 | 952b2b102a0a |

child 58889 | 5b7a9633cfa8 |

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

misc tuning;

(* Title: HOL/Induct/Sigma_Algebra.thy 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. *} inductive_set \<sigma>_algebra :: "'a set set => 'a set set" for A :: "'a set set" where 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) then have "-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" then have "!!i::nat. -(a i) \<in> \<sigma>_algebra A" by (rule \<sigma>_algebra.complement) then have "(\<Union>i. -(a i)) \<in> \<sigma>_algebra A" by (rule \<sigma>_algebra.Union) then have "-(\<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