src/HOL/Probability/Discrete_Topology.thy
author hoelzl
Thu Jan 31 11:31:30 2013 +0100 (2013-01-31)
changeset 51000 c9adb50f74ad
parent 50881 ae630bab13da
child 51343 b61b32f62c78
permissions -rw-r--r--
use order topology for extended reals
     1 (*  Title:      HOL/Probability/Discrete_Topology.thy
     2     Author:     Fabian Immler, TU M√ľnchen
     3 *)
     4 
     5 theory Discrete_Topology
     6 imports "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
     7 begin
     8 
     9 text {* Copy of discrete types with discrete topology. This space is polish. *}
    10 
    11 typedef 'a discrete = "UNIV::'a set"
    12 morphisms of_discrete discrete
    13 ..
    14 
    15 instantiation discrete :: (type) topological_space
    16 begin
    17 
    18 definition open_discrete::"'a discrete set \<Rightarrow> bool"
    19   where "open_discrete s = True"
    20 
    21 instance proof qed (auto simp: open_discrete_def)
    22 end
    23 
    24 instantiation discrete :: (type) metric_space
    25 begin
    26 
    27 definition dist_discrete::"'a discrete \<Rightarrow> 'a discrete \<Rightarrow> real"
    28   where "dist_discrete n m = (if n = m then 0 else 1)"
    29 
    30 instance proof qed (auto simp: open_discrete_def dist_discrete_def intro: exI[where x=1])
    31 end
    32 
    33 instance discrete :: (type) complete_space
    34 proof
    35   fix X::"nat\<Rightarrow>'a discrete" assume "Cauchy X"
    36   hence "\<exists>n. \<forall>m\<ge>n. X n = X m"
    37     by (force simp: dist_discrete_def Cauchy_def split: split_if_asm dest:spec[where x=1])
    38   then guess n ..
    39   thus "convergent X"
    40     by (intro convergentI[where L="X n"] tendstoI eventually_sequentiallyI[of n])
    41        (simp add: dist_discrete_def)
    42 qed
    43 
    44 instance discrete :: (countable) countable
    45 proof
    46   have "inj (\<lambda>c::'a discrete. to_nat (of_discrete c))"
    47     by (simp add: inj_on_def of_discrete_inject)
    48   thus "\<exists>f::'a discrete \<Rightarrow> nat. inj f" by blast
    49 qed
    50 
    51 instance discrete :: (countable) second_countable_topology
    52 proof
    53   let ?B = "(range (\<lambda>n::nat. {from_nat n::'a discrete}))"
    54   have "topological_basis ?B"
    55   proof (intro topological_basisI)
    56     fix x::"'a discrete" and O' assume "open O'" "x \<in> O'"
    57     thus "\<exists>B'\<in>range (\<lambda>n. {from_nat n}). x \<in> B' \<and> B' \<subseteq> O'"
    58       by (auto intro: exI[where x="to_nat x"])
    59   qed (simp add: open_discrete_def)
    60   moreover have "countable ?B" by simp
    61   ultimately show "\<exists>B::'a discrete set set. countable B \<and> topological_basis B" by blast
    62 qed
    63 
    64 instance discrete :: (countable) polish_space ..
    65 
    66 end