src/HOL/Probability/Discrete_Topology.thy
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Thu, 18 Apr 2013 17:07:01 +0200
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simplifier uses proper Proof.context instead of historic type simpset;
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(*  Title:      HOL/Probability/Discrete_Topology.thy
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    Author:     Fabian Immler, TU M√ľnchen
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*)
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theory Discrete_Topology
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imports "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
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begin
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text {* Copy of discrete types with discrete topology. This space is polish. *}
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typedef 'a discrete = "UNIV::'a set"
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morphisms of_discrete discrete
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..
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instantiation discrete :: (type) topological_space
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begin
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definition open_discrete::"'a discrete set \<Rightarrow> bool"
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  where "open_discrete s = True"
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instance proof qed (auto simp: open_discrete_def)
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end
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instantiation discrete :: (type) metric_space
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begin
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definition dist_discrete::"'a discrete \<Rightarrow> 'a discrete \<Rightarrow> real"
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  where "dist_discrete n m = (if n = m then 0 else 1)"
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instance proof qed (auto simp: open_discrete_def dist_discrete_def intro: exI[where x=1])
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end
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instance discrete :: (type) complete_space
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proof
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  fix X::"nat\<Rightarrow>'a discrete" assume "Cauchy X"
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  hence "\<exists>n. \<forall>m\<ge>n. X n = X m"
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    by (force simp: dist_discrete_def Cauchy_def split: split_if_asm dest:spec[where x=1])
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  then guess n ..
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  thus "convergent X"
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    by (intro convergentI[where L="X n"] tendstoI eventually_sequentiallyI[of n])
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       (simp add: dist_discrete_def)
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qed
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instance discrete :: (countable) countable
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proof
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  have "inj (\<lambda>c::'a discrete. to_nat (of_discrete c))"
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    by (simp add: inj_on_def of_discrete_inject)
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  thus "\<exists>f::'a discrete \<Rightarrow> nat. inj f" by blast
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qed
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instance discrete :: (countable) second_countable_topology
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proof
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  let ?B = "range (\<lambda>n::'a discrete. {n})"
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  have "\<And>S. generate_topology ?B (\<Union>x\<in>S. {x})"
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    by (intro generate_topology_Union) (auto intro: generate_topology.intros)
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  then have "open = generate_topology ?B"
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    by (auto intro!: ext simp: open_discrete_def)
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  moreover have "countable ?B" by simp
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  ultimately show "\<exists>B::'a discrete set set. countable B \<and> open = generate_topology B" by blast
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qed
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instance discrete :: (countable) polish_space ..
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end