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