Theory Sigma_Algebra

(*  Title:      HOL/Induct/Sigma_Algebra.thy
    Author:     Markus Wenzel, TU Muenchen
*)

section ‹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 ‹σ›-algebra over a given set of sets.
›

inductive_set σ_algebra :: "'a set set ⇒ 'a set set" for A :: "'a set set"
where
  basic: "a ∈ σ_algebra A" if "a ∈ A" for a
| UNIV: "UNIV ∈ σ_algebra A"
| complement: "- a ∈ σ_algebra A" if "a ∈ σ_algebra A" for a
| Union: "(⋃i. a i) ∈ σ_algebra A" if "⋀i::nat. a i ∈ σ_algebra A" for a

text ‹
  The following basic facts are consequences of the closure properties
  of any ‹σ›-algebra, merely using the introduction rules, but
  no induction nor cases.
›

theorem sigma_algebra_empty: "{} ∈ σ_algebra A"
proof -
  have "UNIV ∈ σ_algebra A" by (rule σ_algebra.UNIV)
  then have "-UNIV ∈ σ_algebra A" by (rule σ_algebra.complement)
  also have "-UNIV = {}" by simp
  finally show ?thesis .
qed

theorem sigma_algebra_Inter:
  "(⋀i::nat. a i ∈ σ_algebra A) ⟹ (⋂i. a i) ∈ σ_algebra A"
proof -
  assume "⋀i::nat. a i ∈ σ_algebra A"
  then have "⋀i::nat. -(a i) ∈ σ_algebra A" by (rule σ_algebra.complement)
  then have "(⋃i. -(a i)) ∈ σ_algebra A" by (rule σ_algebra.Union)
  then have "-(⋃i. -(a i)) ∈ σ_algebra A" by (rule σ_algebra.complement)
  also have "-(⋃i. -(a i)) = (⋂i. a i)" by simp
  finally show ?thesis .
qed

end