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src/HOL/Probability/Discrete_Topology.thy

author | wenzelm |

Thu, 18 Apr 2013 17:07:01 +0200 | |

changeset 51717 | 9e7d1c139569 |

parent 51343 | b61b32f62c78 |

child 61808 | fc1556774cfe |

permissions | -rw-r--r-- |

simplifier uses proper Proof.context instead of historic type simpset;

(* Title: HOL/Probability/Discrete_Topology.thy Author: Fabian Immler, TU München *) theory Discrete_Topology imports "~~/src/HOL/Multivariate_Analysis/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) 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_def) 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