src/HOL/Probability/Discrete_Topology.thy
 author hoelzl Fri Feb 19 13:40:50 2016 +0100 (2016-02-19) changeset 62378 85ed00c1fe7c parent 62101 26c0a70f78a3 child 62390 842917225d56 permissions -rw-r--r--
generalize more theorems to support enat and ennreal
```     1 (*  Title:      HOL/Probability/Discrete_Topology.thy
```
```     2     Author:     Fabian Immler, TU München
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```     3 *)
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```     4
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```     5 theory Discrete_Topology
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```     6 imports "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
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```     7 begin
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```     8
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```     9 text \<open>Copy of discrete types with discrete topology. This space is polish.\<close>
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```    10
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```    11 typedef 'a discrete = "UNIV::'a set"
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```    12 morphisms of_discrete discrete
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```    13 ..
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```    14
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```    15 instantiation discrete :: (type) metric_space
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```    16 begin
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```    17
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```    18 definition dist_discrete :: "'a discrete \<Rightarrow> 'a discrete \<Rightarrow> real"
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```    19   where "dist_discrete n m = (if n = m then 0 else 1)"
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```    20
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```    21 definition uniformity_discrete :: "('a discrete \<times> 'a discrete) filter" where
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```    22   "(uniformity::('a discrete \<times> 'a discrete) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"
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```    23
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```    24 definition "open_discrete" :: "'a discrete set \<Rightarrow> bool" where
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```    25   "(open::'a discrete set \<Rightarrow> bool) U \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
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```    26
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```    27 instance proof qed (auto simp: uniformity_discrete_def open_discrete_def dist_discrete_def intro: exI[where x=1])
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```    28 end
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```    29
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```    30 lemma open_discrete: "open (S :: 'a discrete set)"
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```    31   unfolding open_dist dist_discrete_def by (auto intro!: exI[of _ "1 / 2"])
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```    32
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```    33 instance discrete :: (type) complete_space
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```    34 proof
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```    35   fix X::"nat\<Rightarrow>'a discrete" assume "Cauchy X"
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```    36   hence "\<exists>n. \<forall>m\<ge>n. X n = X m"
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```    37     by (force simp: dist_discrete_def Cauchy_def split: split_if_asm dest:spec[where x=1])
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```    38   then guess n ..
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```    39   thus "convergent X"
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```    40     by (intro convergentI[where L="X n"] tendstoI eventually_sequentiallyI[of n])
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```    41        (simp add: dist_discrete_def)
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```    42 qed
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```    43
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```    44 instance discrete :: (countable) countable
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```    45 proof
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```    46   have "inj (\<lambda>c::'a discrete. to_nat (of_discrete c))"
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```    47     by (simp add: inj_on_def of_discrete_inject)
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```    48   thus "\<exists>f::'a discrete \<Rightarrow> nat. inj f" by blast
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```    49 qed
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```    50
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```    51 instance discrete :: (countable) second_countable_topology
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```    52 proof
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```    53   let ?B = "range (\<lambda>n::'a discrete. {n})"
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```    54   have "\<And>S. generate_topology ?B (\<Union>x\<in>S. {x})"
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```    55     by (intro generate_topology_Union) (auto intro: generate_topology.intros)
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```    56   then have "open = generate_topology ?B"
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```    57     by (auto intro!: ext simp: open_discrete)
```
```    58   moreover have "countable ?B" by simp
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```    59   ultimately show "\<exists>B::'a discrete set set. countable B \<and> open = generate_topology B" by blast
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```    60 qed
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```    61
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```    62 instance discrete :: (countable) polish_space ..
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```    63
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```    64 end
```