src/HOL/Probability/Regularity.thy
changeset 50089 1badf63e5d97
parent 50087 635d73673b5e
child 50125 4319691be975
     1.1 --- a/src/HOL/Probability/Regularity.thy	Thu Nov 15 11:16:58 2012 +0100
     1.2 +++ b/src/HOL/Probability/Regularity.thy	Thu Nov 15 15:50:01 2012 +0100
     1.3 @@ -2,53 +2,12 @@
     1.4      Author:     Fabian Immler, TU M√ľnchen
     1.5  *)
     1.6  
     1.7 +header {* Regularity of Measures *}
     1.8 +
     1.9  theory Regularity
    1.10  imports Measure_Space Borel_Space
    1.11  begin
    1.12  
    1.13 -instantiation nat::topological_space
    1.14 -begin
    1.15 -
    1.16 -definition open_nat::"nat set \<Rightarrow> bool"
    1.17 -  where "open_nat s = True"
    1.18 -
    1.19 -instance proof qed (auto simp: open_nat_def)
    1.20 -end
    1.21 -
    1.22 -instantiation nat::metric_space
    1.23 -begin
    1.24 -
    1.25 -definition dist_nat::"nat \<Rightarrow> nat \<Rightarrow> real"
    1.26 -  where "dist_nat n m = (if n = m then 0 else 1)"
    1.27 -
    1.28 -instance proof qed (auto simp: open_nat_def dist_nat_def intro: exI[where x=1])
    1.29 -end
    1.30 -
    1.31 -instance nat::complete_space
    1.32 -proof
    1.33 -  fix X::"nat\<Rightarrow>nat" assume "Cauchy X"
    1.34 -  hence "\<exists>n. \<forall>m\<ge>n. X m = X n"
    1.35 -    by (force simp: dist_nat_def Cauchy_def split: split_if_asm dest:spec[where x=1])
    1.36 -  then guess n ..
    1.37 -  thus "convergent X"
    1.38 -    apply (intro convergentI[where L="X n"] tendstoI)
    1.39 -    unfolding eventually_sequentially dist_nat_def
    1.40 -    apply (intro exI[where x=n])
    1.41 -    apply (intro allI)
    1.42 -    apply (drule_tac x=na in spec)
    1.43 -    apply simp
    1.44 -    done
    1.45 -qed
    1.46 -
    1.47 -instance nat::enumerable_basis
    1.48 -proof
    1.49 -  have "topological_basis (range (\<lambda>n::nat. {n}))"
    1.50 -    by (intro topological_basisI) (auto simp: open_nat_def)
    1.51 -  thus "\<exists>f::nat\<Rightarrow>nat set. topological_basis (range f)" by blast
    1.52 -qed
    1.53 -
    1.54 -subsection {* Regularity of Measures *}
    1.55 -
    1.56  lemma ereal_approx_SUP:
    1.57    fixes x::ereal
    1.58    assumes A_notempty: "A \<noteq> {}"