isabelle: src/HOL/Probability/Discrete_Topology.thy@ae630bab13da (annotated)

50089 1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	1	(* Title: HOL/Probability/Discrete_Topology.thy
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	2	Author: Fabian Immler, TU München
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	3	*)
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	4
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	5	theory Discrete_Topology
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	6	imports Multivariate_Analysis
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	7	begin
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	8
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	9	text {* Copy of discrete types with discrete topology. This space is polish. *}
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	10
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	11	typedef 'a discrete = "UNIV::'a set"
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	12	morphisms of_discrete discrete
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	13	..
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	14
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	15	instantiation discrete :: (type) topological_space
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	16	begin
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	17
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	18	definition open_discrete::"'a discrete set \<Rightarrow> bool"
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	19	where "open_discrete s = True"
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	20
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	21	instance proof qed (auto simp: open_discrete_def)
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	22	end
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	23
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	24	instantiation discrete :: (type) metric_space
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	25	begin
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	26
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	27	definition dist_discrete::"'a discrete \<Rightarrow> 'a discrete \<Rightarrow> real"
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	28	where "dist_discrete n m = (if n = m then 0 else 1)"
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	29
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	30	instance proof qed (auto simp: open_discrete_def dist_discrete_def intro: exI[where x=1])
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	31	end
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	32
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	33	instance discrete :: (type) complete_space
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	34	proof
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	35	fix X::"nat\<Rightarrow>'a discrete" assume "Cauchy X"
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	36	hence "\<exists>n. \<forall>m\<ge>n. X n = X m"
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	37	by (force simp: dist_discrete_def Cauchy_def split: split_if_asm dest:spec[where x=1])
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	38	then guess n ..
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	39	thus "convergent X"
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	40	by (intro convergentI[where L="X n"] tendstoI eventually_sequentiallyI[of n])
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	41	(simp add: dist_discrete_def)
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	42	qed
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	43
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	44	instance discrete :: (countable) countable
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	45	proof
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	46	have "inj (\<lambda>c::'a discrete. to_nat (of_discrete c))"
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	47	by (simp add: inj_on_def of_discrete_inject)
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	48	thus "\<exists>f::'a discrete \<Rightarrow> nat. inj f" by blast
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	49	qed
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	50
50881 ae630bab13da renamed countable_basis_space to second_countable_topology hoelzl parents: 50245 diff changeset	51	instance discrete :: (countable) second_countable_topology
50089 1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	52	proof
50245 dea9363887a6 based countable topological basis on Countable_Set immler parents: 50089 diff changeset	53	let ?B = "(range (\<lambda>n::nat. {from_nat n::'a discrete}))"
dea9363887a6 based countable topological basis on Countable_Set immler parents: 50089 diff changeset	54	have "topological_basis ?B"
50089 1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	55	proof (intro topological_basisI)
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	56	fix x::"'a discrete" and O' assume "open O'" "x \<in> O'"
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	57	thus "\<exists>B'\<in>range (\<lambda>n. {from_nat n}). x \<in> B' \<and> B' \<subseteq> O'"
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	58	by (auto intro: exI[where x="to_nat x"])
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	59	qed (simp add: open_discrete_def)
50245 dea9363887a6 based countable topological basis on Countable_Set immler parents: 50089 diff changeset	60	moreover have "countable ?B" by simp
dea9363887a6 based countable topological basis on Countable_Set immler parents: 50089 diff changeset	61	ultimately show "\<exists>B::'a discrete set set. countable B \<and> topological_basis B" by blast
50089 1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	62	qed
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	63
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	64	instance discrete :: (countable) polish_space ..
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	65
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	66	end

author	hoelzl
	Mon, 14 Jan 2013 17:29:04 +0100
changeset 50881	ae630bab13da
parent 50245	dea9363887a6
child 51000	c9adb50f74ad
permissions	-rw-r--r--