(* Title: HOL/Probability/Discrete_Topology.thy
Author: Fabian Immler, TU München
*)
theory Discrete_Topology
imports "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
begin
text \<open>Copy of discrete types with discrete topology. This space is polish.\<close>
typedef 'a discrete = "UNIV::'a set"
morphisms of_discrete discrete
..
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)"
definition uniformity_discrete :: "('a discrete \<times> 'a discrete) filter" where
"(uniformity::('a discrete \<times> 'a discrete) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"
definition "open_discrete" :: "'a discrete set \<Rightarrow> bool" where
"(open::'a discrete set \<Rightarrow> bool) U \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
instance proof qed (auto simp: uniformity_discrete_def open_discrete_def dist_discrete_def intro: exI[where x=1])
end
lemma open_discrete: "open (S :: 'a discrete set)"
unfolding open_dist dist_discrete_def by (auto intro!: exI[of _ "1 / 2"])
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) second_countable_topology
proof
let ?B = "range (\<lambda>n::'a discrete. {n})"
have "\<And>S. generate_topology ?B (\<Union>x\<in>S. {x})"
by (intro generate_topology_Union) (auto intro: generate_topology.intros)
then have "open = generate_topology ?B"
by (auto intro!: ext simp: open_discrete)
moreover have "countable ?B" by simp
ultimately show "\<exists>B::'a discrete set set. countable B \<and> open = generate_topology B" by blast
qed
instance discrete :: (countable) polish_space ..
end