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 München
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> {}"
```