src/HOL/Library/Countable.thy
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(*  Title:      HOL/Library/Countable.thy
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    Author:     Alexander Krauss, TU Muenchen
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*)
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header {* Encoding (almost) everything into natural numbers *}
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theory Countable
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imports Main Rat Nat_Bijection
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begin
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subsection {* The class of countable types *}
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class countable =
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  assumes ex_inj: "\<exists>to_nat \<Colon> 'a \<Rightarrow> nat. inj to_nat"
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lemma countable_classI:
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  fixes f :: "'a \<Rightarrow> nat"
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  assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
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  shows "OFCLASS('a, countable_class)"
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proof (intro_classes, rule exI)
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  show "inj f"
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    by (rule injI [OF assms]) assumption
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qed
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subsection {* Conversion functions *}
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definition to_nat :: "'a\<Colon>countable \<Rightarrow> nat" where
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  "to_nat = (SOME f. inj f)"
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definition from_nat :: "nat \<Rightarrow> 'a\<Colon>countable" where
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  "from_nat = inv (to_nat \<Colon> 'a \<Rightarrow> nat)"
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lemma inj_to_nat [simp]: "inj to_nat"
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  by (rule exE_some [OF ex_inj]) (simp add: to_nat_def)
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lemma surj_from_nat [simp]: "surj from_nat"
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  unfolding from_nat_def by (simp add: inj_imp_surj_inv)
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lemma to_nat_split [simp]: "to_nat x = to_nat y \<longleftrightarrow> x = y"
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  using injD [OF inj_to_nat] by auto
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lemma from_nat_to_nat [simp]:
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  "from_nat (to_nat x) = x"
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  by (simp add: from_nat_def)
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subsection {* Countable types *}
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instance nat :: countable
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  by (rule countable_classI [of "id"]) simp
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subclass (in finite) countable
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proof
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  have "finite (UNIV\<Colon>'a set)" by (rule finite_UNIV)
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  with finite_conv_nat_seg_image [of "UNIV::'a set"]
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  obtain n and f :: "nat \<Rightarrow> 'a" 
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    where "UNIV = f ` {i. i < n}" by auto
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  then have "surj f" unfolding surj_def by auto
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  then have "inj (inv f)" by (rule surj_imp_inj_inv)
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  then show "\<exists>to_nat \<Colon> 'a \<Rightarrow> nat. inj to_nat" by (rule exI[of inj])
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qed
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text {* Pairs *}
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instance prod :: (countable, countable) countable
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  by (rule countable_classI [of "\<lambda>(x, y). prod_encode (to_nat x, to_nat y)"])
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    (auto simp add: prod_encode_eq)
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text {* Sums *}
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instance sum :: (countable, countable) countable
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  by (rule countable_classI [of "(\<lambda>x. case x of Inl a \<Rightarrow> to_nat (False, to_nat a)
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                                     | Inr b \<Rightarrow> to_nat (True, to_nat b))"])
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    (simp split: sum.split_asm)
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text {* Integers *}
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instance int :: countable
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  by (rule countable_classI [of "int_encode"])
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    (simp add: int_encode_eq)
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text {* Options *}
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instance option :: (countable) countable
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  by (rule countable_classI [of "option_case 0 (Suc \<circ> to_nat)"])
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    (simp split: option.split_asm)
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text {* Lists *}
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instance list :: (countable) countable
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  by (rule countable_classI [of "list_encode \<circ> map to_nat"])
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    (simp add: list_encode_eq)
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text {* Further *}
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instance String.literal :: countable
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  by (rule countable_classI [of "String.literal_case to_nat"])
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   (auto split: String.literal.splits)
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instantiation typerep :: countable
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begin
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fun to_nat_typerep :: "typerep \<Rightarrow> nat" where
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  "to_nat_typerep (Typerep.Typerep c ts) = to_nat (to_nat c, to_nat (map to_nat_typerep ts))"
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instance proof (rule countable_classI)
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  fix t t' :: typerep and ts ts' :: "typerep list"
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  assume "to_nat_typerep t = to_nat_typerep t'"
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  moreover have "to_nat_typerep t = to_nat_typerep t' \<Longrightarrow> t = t'"
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    and "map to_nat_typerep ts = map to_nat_typerep ts' \<Longrightarrow> ts = ts'"
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  proof (induct t and ts arbitrary: t' and ts' rule: typerep.inducts)
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    case (Typerep c ts t')
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    then obtain c' ts' where t': "t' = Typerep.Typerep c' ts'" by (cases t') auto
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    with Typerep have "c = c'" and "ts = ts'" by simp_all
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    with t' show "Typerep.Typerep c ts = t'" by simp
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  next
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    case Nil_typerep then show ?case by simp
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  next
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    case (Cons_typerep t ts) then show ?case by auto
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  qed
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  ultimately show "t = t'" by simp
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qed
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end
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text {* Functions *}
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instance "fun" :: (finite, countable) countable
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proof
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  obtain xs :: "'a list" where xs: "set xs = UNIV"
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    using finite_list [OF finite_UNIV] ..
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  show "\<exists>to_nat::('a \<Rightarrow> 'b) \<Rightarrow> nat. inj to_nat"
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  proof
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    show "inj (\<lambda>f. to_nat (map f xs))"
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      by (rule injI, simp add: xs ext_iff)
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  qed
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qed
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subsection {* The Rationals are Countably Infinite *}
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definition nat_to_rat_surj :: "nat \<Rightarrow> rat" where
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"nat_to_rat_surj n = (let (a,b) = prod_decode n
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                      in Fract (int_decode a) (int_decode b))"
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lemma surj_nat_to_rat_surj: "surj nat_to_rat_surj"
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unfolding surj_def
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proof
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  fix r::rat
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  show "\<exists>n. r = nat_to_rat_surj n"
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  proof (cases r)
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    fix i j assume [simp]: "r = Fract i j" and "j > 0"
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    have "r = (let m = int_encode i; n = int_encode j
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               in nat_to_rat_surj(prod_encode (m,n)))"
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      by (simp add: Let_def nat_to_rat_surj_def)
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    thus "\<exists>n. r = nat_to_rat_surj n" by(auto simp:Let_def)
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  qed
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qed
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lemma Rats_eq_range_nat_to_rat_surj: "\<rat> = range nat_to_rat_surj"
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by (simp add: Rats_def surj_nat_to_rat_surj surj_range)
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context field_char_0
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begin
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lemma Rats_eq_range_of_rat_o_nat_to_rat_surj:
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  "\<rat> = range (of_rat o nat_to_rat_surj)"
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using surj_nat_to_rat_surj
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by (auto simp: Rats_def image_def surj_def)
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   (blast intro: arg_cong[where f = of_rat])
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lemma surj_of_rat_nat_to_rat_surj:
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  "r\<in>\<rat> \<Longrightarrow> \<exists>n. r = of_rat(nat_to_rat_surj n)"
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by(simp add: Rats_eq_range_of_rat_o_nat_to_rat_surj image_def)
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26169
73027318f9ba added theory for countable types
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end
29880
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instance rat :: countable
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proof
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  show "\<exists>to_nat::rat \<Rightarrow> nat. inj to_nat"
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  proof
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    have "surj nat_to_rat_surj"
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      by (rule surj_nat_to_rat_surj)
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    then show "inj (inv nat_to_rat_surj)"
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      by (rule surj_imp_inj_inv)
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  qed
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qed
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end