Theory Discrete_Topology

(*  Title:      HOL/Probability/Discrete_Topology.thy
    Author:     Fabian Immler, TU München
*)

theory Discrete_Topology
imports "HOL-Analysis.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) metric_space
begin

definition dist_discrete :: "'a discrete ⇒ 'a discrete ⇒ real"
  where "dist_discrete n m = (if n = m then 0 else 1)"

definition uniformity_discrete :: "('a discrete × 'a discrete) filter" where
  "(uniformity::('a discrete × 'a discrete) filter) = (INF e∈{0 <..}. principal {(x, y). dist x y < e})"

definition "open_discrete" :: "'a discrete set ⇒ bool" where
  "(open::'a discrete set ⇒ bool) U ⟷ (∀x∈U. eventually (λ(x', y). x' = x ⟶ y ∈ 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⇒'a discrete"
  assume "Cauchy X"
  then obtain n where "∀m≥n. X n = X m"
    by (force simp: dist_discrete_def Cauchy_def split: if_split_asm dest:spec[where x=1])
  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 (λc::'a discrete. to_nat (of_discrete c))"
    by (simp add: inj_on_def of_discrete_inject)
  thus "∃f::'a discrete ⇒ nat. inj f" by blast
qed

instance discrete :: (countable) second_countable_topology
proof
  let ?B = "range (λn::'a discrete. {n})"
  have "⋀S. generate_topology ?B (⋃x∈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 "∃B::'a discrete set set. countable B ∧ open = generate_topology B" by blast
qed

instance discrete :: (countable) polish_space ..

end