src/HOL/Library/Countable_Set.thy
 author Christian Sternagel Thu Dec 13 13:11:38 2012 +0100 (2012-12-13) changeset 50516 ed6b40d15d1c parent 50245 dea9363887a6 child 50936 b28f258ebc1a permissions -rw-r--r--
renamed "emb" to "list_hembeq";
make "list_hembeq" reflexive independent of the base order;
renamed "sub" to "sublisteq";
dropped "transp_on" (state transitivity explicitly instead);
no need to hide "sub" after renaming;
replaced some ASCII symbols by proper Isabelle symbols;
NEWS
```     1 (*  Title:      HOL/Library/Countable_Set.thy
```
```     2     Author:     Johannes Hölzl
```
```     3     Author:     Andrei Popescu
```
```     4 *)
```
```     5
```
```     6 header {* Countable sets *}
```
```     7
```
```     8 theory Countable_Set
```
```     9   imports "~~/src/HOL/Library/Countable" "~~/src/HOL/Library/Infinite_Set"
```
```    10 begin
```
```    11
```
```    12 subsection {* Predicate for countable sets *}
```
```    13
```
```    14 definition countable :: "'a set \<Rightarrow> bool" where
```
```    15   "countable S \<longleftrightarrow> (\<exists>f::'a \<Rightarrow> nat. inj_on f S)"
```
```    16
```
```    17 lemma countableE:
```
```    18   assumes S: "countable S" obtains f :: "'a \<Rightarrow> nat" where "inj_on f S"
```
```    19   using S by (auto simp: countable_def)
```
```    20
```
```    21 lemma countableI: "inj_on (f::'a \<Rightarrow> nat) S \<Longrightarrow> countable S"
```
```    22   by (auto simp: countable_def)
```
```    23
```
```    24 lemma countableI': "inj_on (f::'a \<Rightarrow> 'b::countable) S \<Longrightarrow> countable S"
```
```    25   using comp_inj_on[of f S to_nat] by (auto intro: countableI)
```
```    26
```
```    27 lemma countableE_bij:
```
```    28   assumes S: "countable S" obtains f :: "nat \<Rightarrow> 'a" and C :: "nat set" where "bij_betw f C S"
```
```    29   using S by (blast elim: countableE dest: inj_on_imp_bij_betw bij_betw_inv)
```
```    30
```
```    31 lemma countableI_bij: "bij_betw f (C::nat set) S \<Longrightarrow> countable S"
```
```    32   by (blast intro: countableI bij_betw_inv_into bij_betw_imp_inj_on)
```
```    33
```
```    34 lemma countable_finite: "finite S \<Longrightarrow> countable S"
```
```    35   by (blast dest: finite_imp_inj_to_nat_seg countableI)
```
```    36
```
```    37 lemma countableI_bij1: "bij_betw f A B \<Longrightarrow> countable A \<Longrightarrow> countable B"
```
```    38   by (blast elim: countableE_bij intro: bij_betw_trans countableI_bij)
```
```    39
```
```    40 lemma countableI_bij2: "bij_betw f B A \<Longrightarrow> countable A \<Longrightarrow> countable B"
```
```    41   by (blast elim: countableE_bij intro: bij_betw_trans bij_betw_inv_into countableI_bij)
```
```    42
```
```    43 lemma countable_iff_bij[simp]: "bij_betw f A B \<Longrightarrow> countable A \<longleftrightarrow> countable B"
```
```    44   by (blast intro: countableI_bij1 countableI_bij2)
```
```    45
```
```    46 lemma countable_subset: "A \<subseteq> B \<Longrightarrow> countable B \<Longrightarrow> countable A"
```
```    47   by (auto simp: countable_def intro: subset_inj_on)
```
```    48
```
```    49 lemma countableI_type[intro, simp]: "countable (A:: 'a :: countable set)"
```
```    50   using countableI[of to_nat A] by auto
```
```    51
```
```    52 subsection {* Enumerate a countable set *}
```
```    53
```
```    54 lemma countableE_infinite:
```
```    55   assumes "countable S" "infinite S"
```
```    56   obtains e :: "'a \<Rightarrow> nat" where "bij_betw e S UNIV"
```
```    57 proof -
```
```    58   from `countable S`[THEN countableE] guess f .
```
```    59   then have "bij_betw f S (f`S)"
```
```    60     unfolding bij_betw_def by simp
```
```    61   moreover
```
```    62   from `inj_on f S` `infinite S` have inf_fS: "infinite (f`S)"
```
```    63     by (auto dest: finite_imageD)
```
```    64   then have "bij_betw (the_inv_into UNIV (enumerate (f`S))) (f`S) UNIV"
```
```    65     by (intro bij_betw_the_inv_into bij_enumerate)
```
```    66   ultimately have "bij_betw (the_inv_into UNIV (enumerate (f`S)) \<circ> f) S UNIV"
```
```    67     by (rule bij_betw_trans)
```
```    68   then show thesis ..
```
```    69 qed
```
```    70
```
```    71 lemma countable_enum_cases:
```
```    72   assumes "countable S"
```
```    73   obtains (finite) f :: "'a \<Rightarrow> nat" where "finite S" "bij_betw f S {..<card S}"
```
```    74         | (infinite) f :: "'a \<Rightarrow> nat" where "infinite S" "bij_betw f S UNIV"
```
```    75   using ex_bij_betw_finite_nat[of S] countableE_infinite `countable S`
```
```    76   by (cases "finite S") (auto simp add: atLeast0LessThan)
```
```    77
```
```    78 definition to_nat_on :: "'a set \<Rightarrow> 'a \<Rightarrow> nat" where
```
```    79   "to_nat_on S = (SOME f. if finite S then bij_betw f S {..< card S} else bij_betw f S UNIV)"
```
```    80
```
```    81 definition from_nat_into :: "'a set \<Rightarrow> nat \<Rightarrow> 'a" where
```
```    82   "from_nat_into S n = (if n \<in> to_nat_on S ` S then inv_into S (to_nat_on S) n else SOME s. s\<in>S)"
```
```    83
```
```    84 lemma to_nat_on_finite: "finite S \<Longrightarrow> bij_betw (to_nat_on S) S {..< card S}"
```
```    85   using ex_bij_betw_finite_nat unfolding to_nat_on_def
```
```    86   by (intro someI2_ex[where Q="\<lambda>f. bij_betw f S {..<card S}"]) (auto simp add: atLeast0LessThan)
```
```    87
```
```    88 lemma to_nat_on_infinite: "countable S \<Longrightarrow> infinite S \<Longrightarrow> bij_betw (to_nat_on S) S UNIV"
```
```    89   using countableE_infinite unfolding to_nat_on_def
```
```    90   by (intro someI2_ex[where Q="\<lambda>f. bij_betw f S UNIV"]) auto
```
```    91
```
```    92 lemma bij_betw_from_nat_into_finite: "finite S \<Longrightarrow> bij_betw (from_nat_into S) {..< card S} S"
```
```    93   unfolding from_nat_into_def[abs_def]
```
```    94   using to_nat_on_finite[of S]
```
```    95   apply (subst bij_betw_cong)
```
```    96   apply (split split_if)
```
```    97   apply (simp add: bij_betw_def)
```
```    98   apply (auto cong: bij_betw_cong
```
```    99               intro: bij_betw_inv_into to_nat_on_finite)
```
```   100   done
```
```   101
```
```   102 lemma bij_betw_from_nat_into: "countable S \<Longrightarrow> infinite S \<Longrightarrow> bij_betw (from_nat_into S) UNIV S"
```
```   103   unfolding from_nat_into_def[abs_def]
```
```   104   using to_nat_on_infinite[of S, unfolded bij_betw_def]
```
```   105   by (auto cong: bij_betw_cong intro: bij_betw_inv_into to_nat_on_infinite)
```
```   106
```
```   107 lemma inj_on_to_nat_on[intro]: "countable A \<Longrightarrow> inj_on (to_nat_on A) A"
```
```   108   using to_nat_on_infinite[of A] to_nat_on_finite[of A]
```
```   109   by (cases "finite A") (auto simp: bij_betw_def)
```
```   110
```
```   111 lemma to_nat_on_inj[simp]:
```
```   112   "countable A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> to_nat_on A a = to_nat_on A b \<longleftrightarrow> a = b"
```
```   113   using inj_on_to_nat_on[of A] by (auto dest: inj_onD)
```
```   114
```
```   115 lemma from_nat_into_to_nat_on[simp]: "countable A \<Longrightarrow> a \<in> A \<Longrightarrow> from_nat_into A (to_nat_on A a) = a"
```
```   116   by (auto simp: from_nat_into_def intro!: inv_into_f_f)
```
```   117
```
```   118 lemma subset_range_from_nat_into: "countable A \<Longrightarrow> A \<subseteq> range (from_nat_into A)"
```
```   119   by (auto intro: from_nat_into_to_nat_on[symmetric])
```
```   120
```
```   121 lemma from_nat_into: "A \<noteq> {} \<Longrightarrow> from_nat_into A n \<in> A"
```
```   122   unfolding from_nat_into_def by (metis equals0I inv_into_into someI_ex)
```
```   123
```
```   124 lemma range_from_nat_into_subset: "A \<noteq> {} \<Longrightarrow> range (from_nat_into A) \<subseteq> A"
```
```   125   using from_nat_into[of A] by auto
```
```   126
```
```   127 lemma range_from_nat_into[simp]: "A \<noteq> {} \<Longrightarrow> countable A \<Longrightarrow> range (from_nat_into A) = A"
```
```   128   by (metis equalityI range_from_nat_into_subset subset_range_from_nat_into)
```
```   129
```
```   130 lemma image_to_nat_on: "countable A \<Longrightarrow> infinite A \<Longrightarrow> to_nat_on A ` A = UNIV"
```
```   131   using to_nat_on_infinite[of A] by (simp add: bij_betw_def)
```
```   132
```
```   133 lemma to_nat_on_surj: "countable A \<Longrightarrow> infinite A \<Longrightarrow> \<exists>a\<in>A. to_nat_on A a = n"
```
```   134   by (metis (no_types) image_iff iso_tuple_UNIV_I image_to_nat_on)
```
```   135
```
```   136 lemma to_nat_on_from_nat_into[simp]: "n \<in> to_nat_on A ` A \<Longrightarrow> to_nat_on A (from_nat_into A n) = n"
```
```   137   by (simp add: f_inv_into_f from_nat_into_def)
```
```   138
```
```   139 lemma to_nat_on_from_nat_into_infinite[simp]:
```
```   140   "countable A \<Longrightarrow> infinite A \<Longrightarrow> to_nat_on A (from_nat_into A n) = n"
```
```   141   by (metis image_iff to_nat_on_surj to_nat_on_from_nat_into)
```
```   142
```
```   143 lemma from_nat_into_inj:
```
```   144   "countable A \<Longrightarrow> m \<in> to_nat_on A ` A \<Longrightarrow> n \<in> to_nat_on A ` A \<Longrightarrow>
```
```   145     from_nat_into A m = from_nat_into A n \<longleftrightarrow> m = n"
```
```   146   by (subst to_nat_on_inj[symmetric, of A]) auto
```
```   147
```
```   148 lemma from_nat_into_inj_infinite[simp]:
```
```   149   "countable A \<Longrightarrow> infinite A \<Longrightarrow> from_nat_into A m = from_nat_into A n \<longleftrightarrow> m = n"
```
```   150   using image_to_nat_on[of A] from_nat_into_inj[of A m n] by simp
```
```   151
```
```   152 lemma eq_from_nat_into_iff:
```
```   153   "countable A \<Longrightarrow> x \<in> A \<Longrightarrow> i \<in> to_nat_on A ` A \<Longrightarrow> x = from_nat_into A i \<longleftrightarrow> i = to_nat_on A x"
```
```   154   by auto
```
```   155
```
```   156 lemma from_nat_into_surj: "countable A \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>n. from_nat_into A n = a"
```
```   157   by (rule exI[of _ "to_nat_on A a"]) simp
```
```   158
```
```   159 lemma from_nat_into_inject[simp]:
```
```   160   "A \<noteq> {} \<Longrightarrow> countable A \<Longrightarrow> B \<noteq> {} \<Longrightarrow> countable B \<Longrightarrow> from_nat_into A = from_nat_into B \<longleftrightarrow> A = B"
```
```   161   by (metis range_from_nat_into)
```
```   162
```
```   163 lemma inj_on_from_nat_into: "inj_on from_nat_into ({A. A \<noteq> {} \<and> countable A})"
```
```   164   unfolding inj_on_def by auto
```
```   165
```
```   166 subsection {* Closure properties of countability *}
```
```   167
```
```   168 lemma countable_SIGMA[intro, simp]:
```
```   169   "countable I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (A i)) \<Longrightarrow> countable (SIGMA i : I. A i)"
```
```   170   by (intro countableI'[of "\<lambda>(i, a). (to_nat_on I i, to_nat_on (A i) a)"]) (auto simp: inj_on_def)
```
```   171
```
```   172 lemma countable_image[intro, simp]: assumes A: "countable A" shows "countable (f`A)"
```
```   173 proof -
```
```   174   from A guess g by (rule countableE)
```
```   175   moreover have "inj_on (inv_into A f) (f`A)" "inv_into A f ` f ` A \<subseteq> A"
```
```   176     by (auto intro: inj_on_inv_into inv_into_into)
```
```   177   ultimately show ?thesis
```
```   178     by (blast dest: comp_inj_on subset_inj_on intro: countableI)
```
```   179 qed
```
```   180
```
```   181 lemma countable_UN[intro, simp]:
```
```   182   fixes I :: "'i set" and A :: "'i => 'a set"
```
```   183   assumes I: "countable I"
```
```   184   assumes A: "\<And>i. i \<in> I \<Longrightarrow> countable (A i)"
```
```   185   shows "countable (\<Union>i\<in>I. A i)"
```
```   186 proof -
```
```   187   have "(\<Union>i\<in>I. A i) = snd ` (SIGMA i : I. A i)" by (auto simp: image_iff)
```
```   188   then show ?thesis by (simp add: assms)
```
```   189 qed
```
```   190
```
```   191 lemma countable_Un[intro]: "countable A \<Longrightarrow> countable B \<Longrightarrow> countable (A \<union> B)"
```
```   192   by (rule countable_UN[of "{True, False}" "\<lambda>True \<Rightarrow> A | False \<Rightarrow> B", simplified])
```
```   193      (simp split: bool.split)
```
```   194
```
```   195 lemma countable_Un_iff[simp]: "countable (A \<union> B) \<longleftrightarrow> countable A \<and> countable B"
```
```   196   by (metis countable_Un countable_subset inf_sup_ord(3,4))
```
```   197
```
```   198 lemma countable_Plus[intro, simp]:
```
```   199   "countable A \<Longrightarrow> countable B \<Longrightarrow> countable (A <+> B)"
```
```   200   by (simp add: Plus_def)
```
```   201
```
```   202 lemma countable_empty[intro, simp]: "countable {}"
```
```   203   by (blast intro: countable_finite)
```
```   204
```
```   205 lemma countable_insert[intro, simp]: "countable A \<Longrightarrow> countable (insert a A)"
```
```   206   using countable_Un[of "{a}" A] by (auto simp: countable_finite)
```
```   207
```
```   208 lemma countable_Int1[intro, simp]: "countable A \<Longrightarrow> countable (A \<inter> B)"
```
```   209   by (force intro: countable_subset)
```
```   210
```
```   211 lemma countable_Int2[intro, simp]: "countable B \<Longrightarrow> countable (A \<inter> B)"
```
```   212   by (blast intro: countable_subset)
```
```   213
```
```   214 lemma countable_INT[intro, simp]: "i \<in> I \<Longrightarrow> countable (A i) \<Longrightarrow> countable (\<Inter>i\<in>I. A i)"
```
```   215   by (blast intro: countable_subset)
```
```   216
```
```   217 lemma countable_Diff[intro, simp]: "countable A \<Longrightarrow> countable (A - B)"
```
```   218   by (blast intro: countable_subset)
```
```   219
```
```   220 lemma countable_vimage: "B \<subseteq> range f \<Longrightarrow> countable (f -` B) \<Longrightarrow> countable B"
```
```   221   by (metis Int_absorb2 assms countable_image image_vimage_eq)
```
```   222
```
```   223 lemma surj_countable_vimage: "surj f \<Longrightarrow> countable (f -` B) \<Longrightarrow> countable B"
```
```   224   by (metis countable_vimage top_greatest)
```
```   225
```
```   226 lemma countable_Collect[simp]: "countable A \<Longrightarrow> countable {a \<in> A. \<phi> a}"
```
```   227   by (metis Collect_conj_eq Int_absorb Int_commute Int_def countable_Int1)
```
```   228
```
```   229 lemma countable_lists[intro, simp]:
```
```   230   assumes A: "countable A" shows "countable (lists A)"
```
```   231 proof -
```
```   232   have "countable (lists (range (from_nat_into A)))"
```
```   233     by (auto simp: lists_image)
```
```   234   with A show ?thesis
```
```   235     by (auto dest: subset_range_from_nat_into countable_subset lists_mono)
```
```   236 qed
```
```   237
```
```   238 lemma Collect_finite_eq_lists: "Collect finite = set ` lists UNIV"
```
```   239   using finite_list by auto
```
```   240
```
```   241 lemma countable_Collect_finite: "countable (Collect (finite::'a::countable set\<Rightarrow>bool))"
```
```   242   by (simp add: Collect_finite_eq_lists)
```
```   243
```
```   244 subsection {* Misc lemmas *}
```
```   245
```
```   246 lemma countable_all:
```
```   247   assumes S: "countable S"
```
```   248   shows "(\<forall>s\<in>S. P s) \<longleftrightarrow> (\<forall>n::nat. from_nat_into S n \<in> S \<longrightarrow> P (from_nat_into S n))"
```
```   249   using S[THEN subset_range_from_nat_into] by auto
```
```   250
```
```   251 end
```