generalized to copy of countable types instead of instantiation of nat for discrete topology

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Discrete_Topology.thy	Thu Nov 15 15:50:01 2012 +0100
@@ -0,0 +1,64 @@
+(*  Title:      HOL/Probability/Discrete_Topology.thy
+    Author:     Fabian Immler, TU München
+*)
+
+theory Discrete_Topology
+imports Multivariate_Analysis
+begin
+
+text {* Copy of discrete types with discrete topology. This space is polish. *}
+
+typedef 'a discrete = "UNIV::'a set"
+morphisms of_discrete discrete
+..
+
+instantiation discrete :: (type) topological_space
+begin
+
+definition open_discrete::"'a discrete set \<Rightarrow> bool"
+  where "open_discrete s = True"
+
+instance proof qed (auto simp: open_discrete_def)
+end
+
+instantiation discrete :: (type) metric_space
+begin
+
+definition dist_discrete::"'a discrete \<Rightarrow> 'a discrete \<Rightarrow> real"
+  where "dist_discrete n m = (if n = m then 0 else 1)"
+
+instance proof qed (auto simp: open_discrete_def dist_discrete_def intro: exI[where x=1])
+end
+
+instance discrete :: (type) complete_space
+proof
+  fix X::"nat\<Rightarrow>'a discrete" assume "Cauchy X"
+  hence "\<exists>n. \<forall>m\<ge>n. X n = X m"
+    by (force simp: dist_discrete_def Cauchy_def split: split_if_asm dest:spec[where x=1])
+  then guess n ..
+  thus "convergent X"
+    by (intro convergentI[where L="X n"] tendstoI eventually_sequentiallyI[of n])
+       (simp add: dist_discrete_def)
+qed
+
+instance discrete :: (countable) countable
+proof
+  have "inj (\<lambda>c::'a discrete. to_nat (of_discrete c))"
+    by (simp add: inj_on_def of_discrete_inject)
+  thus "\<exists>f::'a discrete \<Rightarrow> nat. inj f" by blast
+qed
+
+instance discrete :: (countable) enumerable_basis
+proof
+  have "topological_basis (range (\<lambda>n::nat. {from_nat n::'a discrete}))"
+  proof (intro topological_basisI)
+    fix x::"'a discrete" and O' assume "open O'" "x \<in> O'"
+    thus "\<exists>B'\<in>range (\<lambda>n. {from_nat n}). x \<in> B' \<and> B' \<subseteq> O'"
+      by (auto intro: exI[where x="to_nat x"])
+  qed (simp add: open_discrete_def)
+  thus "\<exists>f::nat\<Rightarrow>'a discrete set. topological_basis (range f)" by blast
+qed
+
+instance discrete :: (countable) polish_space ..
+
+end

--- a/src/HOL/Probability/Probability.thy	Thu Nov 15 11:16:58 2012 +0100
+++ b/src/HOL/Probability/Probability.thy	Thu Nov 15 15:50:01 2012 +0100
@@ -1,5 +1,6 @@
 theory Probability
 imports
+  Discrete_Topology
   Complete_Measure
   Probability_Measure
   Infinite_Product_Measure

--- a/src/HOL/Probability/Regularity.thy	Thu Nov 15 11:16:58 2012 +0100
+++ b/src/HOL/Probability/Regularity.thy	Thu Nov 15 15:50:01 2012 +0100
@@ -2,53 +2,12 @@
     Author:     Fabian Immler, TU München
 *)
 
+header {* Regularity of Measures *}
+
 theory Regularity
 imports Measure_Space Borel_Space
 begin
 
-instantiation nat::topological_space
-begin
-
-definition open_nat::"nat set \<Rightarrow> bool"
-  where "open_nat s = True"
-
-instance proof qed (auto simp: open_nat_def)
-end
-
-instantiation nat::metric_space
-begin
-
-definition dist_nat::"nat \<Rightarrow> nat \<Rightarrow> real"
-  where "dist_nat n m = (if n = m then 0 else 1)"
-
-instance proof qed (auto simp: open_nat_def dist_nat_def intro: exI[where x=1])
-end
-
-instance nat::complete_space
-proof
-  fix X::"nat\<Rightarrow>nat" assume "Cauchy X"
-  hence "\<exists>n. \<forall>m\<ge>n. X m = X n"
-    by (force simp: dist_nat_def Cauchy_def split: split_if_asm dest:spec[where x=1])
-  then guess n ..
-  thus "convergent X"
-    apply (intro convergentI[where L="X n"] tendstoI)
-    unfolding eventually_sequentially dist_nat_def
-    apply (intro exI[where x=n])
-    apply (intro allI)
-    apply (drule_tac x=na in spec)
-    apply simp
-    done
-qed
-
-instance nat::enumerable_basis
-proof
-  have "topological_basis (range (\<lambda>n::nat. {n}))"
-    by (intro topological_basisI) (auto simp: open_nat_def)
-  thus "\<exists>f::nat\<Rightarrow>nat set. topological_basis (range f)" by blast
-qed
-
-subsection {* Regularity of Measures *}
-
 lemma ereal_approx_SUP:
   fixes x::ereal
   assumes A_notempty: "A \<noteq> {}"