isabelle: src/HOL/Probability/Discrete_Topology.thy@34bd10a9a2ad (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
51000 c9adb50f74ad use order topology for extended reals hoelzl parents: 50881 diff changeset	6	imports "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
50089 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
51343 b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis hoelzl parents: 51000 diff changeset	53	let ?B = "range (\<lambda>n::'a discrete. {n})"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis hoelzl parents: 51000 diff changeset	54	have "\<And>S. generate_topology ?B (\<Union>x\<in>S. {x})"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis hoelzl parents: 51000 diff changeset	55	by (intro generate_topology_Union) (auto intro: generate_topology.intros)
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis hoelzl parents: 51000 diff changeset	56	then have "open = generate_topology ?B"
b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis hoelzl parents: 51000 diff changeset	57	by (auto intro!: ext simp: open_discrete_def)
50245 dea9363887a6 based countable topological basis on Countable_Set immler parents: 50089 diff changeset	58	moreover have "countable ?B" by simp
51343 b61b32f62c78 use generate_topology for second countable topologies, does not require intersection stable basis hoelzl parents: 51000 diff changeset	59	ultimately show "\<exists>B::'a discrete set set. countable B \<and> open = generate_topology B" by blast
50089 1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	60	qed
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	61
1badf63e5d97 generalized to copy of countable types instead of instantiation of nat for discrete topology immler parents: diff changeset	62	instance discrete :: (countable) polish_space ..
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	end

author	blanchet
	Thu, 13 Mar 2014 13:18:14 +0100
changeset 56090	34bd10a9a2ad
parent 51343	b61b32f62c78
child 61808	fc1556774cfe
permissions	-rw-r--r--