src/HOL/Library/Countable_Set.thy
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(*  Title:      HOL/Library/Countable_Set.thy
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    Author:     Johannes Hölzl
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    Author:     Andrei Popescu
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*)
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header {* Countable sets *}
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theory Countable_Set
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imports Countable Infinite_Set
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begin
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subsection {* Predicate for countable sets *}
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definition countable :: "'a set \<Rightarrow> bool" where
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  "countable S \<longleftrightarrow> (\<exists>f::'a \<Rightarrow> nat. inj_on f S)"
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lemma countableE:
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  assumes S: "countable S" obtains f :: "'a \<Rightarrow> nat" where "inj_on f S"
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  using S by (auto simp: countable_def)
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lemma countableI: "inj_on (f::'a \<Rightarrow> nat) S \<Longrightarrow> countable S"
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  by (auto simp: countable_def)
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lemma countableI': "inj_on (f::'a \<Rightarrow> 'b::countable) S \<Longrightarrow> countable S"
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  using comp_inj_on[of f S to_nat] by (auto intro: countableI)
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lemma countableE_bij:
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  assumes S: "countable S" obtains f :: "nat \<Rightarrow> 'a" and C :: "nat set" where "bij_betw f C S"
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  using S by (blast elim: countableE dest: inj_on_imp_bij_betw bij_betw_inv)
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lemma countableI_bij: "bij_betw f (C::nat set) S \<Longrightarrow> countable S"
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  by (blast intro: countableI bij_betw_inv_into bij_betw_imp_inj_on)
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lemma countable_finite: "finite S \<Longrightarrow> countable S"
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  by (blast dest: finite_imp_inj_to_nat_seg countableI)
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lemma countableI_bij1: "bij_betw f A B \<Longrightarrow> countable A \<Longrightarrow> countable B"
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  by (blast elim: countableE_bij intro: bij_betw_trans countableI_bij)
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lemma countableI_bij2: "bij_betw f B A \<Longrightarrow> countable A \<Longrightarrow> countable B"
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  by (blast elim: countableE_bij intro: bij_betw_trans bij_betw_inv_into countableI_bij)
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lemma countable_iff_bij[simp]: "bij_betw f A B \<Longrightarrow> countable A \<longleftrightarrow> countable B"
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  by (blast intro: countableI_bij1 countableI_bij2)
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lemma countable_subset: "A \<subseteq> B \<Longrightarrow> countable B \<Longrightarrow> countable A"
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  by (auto simp: countable_def intro: subset_inj_on)
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lemma countableI_type[intro, simp]: "countable (A:: 'a :: countable set)"
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  using countableI[of to_nat A] by auto
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subsection {* Enumerate a countable set *}
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lemma countableE_infinite:
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  assumes "countable S" "infinite S"
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  obtains e :: "'a \<Rightarrow> nat" where "bij_betw e S UNIV"
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proof -
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  from `countable S`[THEN countableE] guess f .
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  then have "bij_betw f S (f`S)"
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    unfolding bij_betw_def by simp
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  moreover
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  from `inj_on f S` `infinite S` have inf_fS: "infinite (f`S)"
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    by (auto dest: finite_imageD)
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  then have "bij_betw (the_inv_into UNIV (enumerate (f`S))) (f`S) UNIV"
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    by (intro bij_betw_the_inv_into bij_enumerate)
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  ultimately have "bij_betw (the_inv_into UNIV (enumerate (f`S)) \<circ> f) S UNIV"
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    by (rule bij_betw_trans)
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  then show thesis ..
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qed
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lemma countable_enum_cases:
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  assumes "countable S"
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  obtains (finite) f :: "'a \<Rightarrow> nat" where "finite S" "bij_betw f S {..<card S}"
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        | (infinite) f :: "'a \<Rightarrow> nat" where "infinite S" "bij_betw f S UNIV"
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  using ex_bij_betw_finite_nat[of S] countableE_infinite `countable S`
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  by (cases "finite S") (auto simp add: atLeast0LessThan)
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definition to_nat_on :: "'a set \<Rightarrow> 'a \<Rightarrow> nat" where
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  "to_nat_on S = (SOME f. if finite S then bij_betw f S {..< card S} else bij_betw f S UNIV)"
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definition from_nat_into :: "'a set \<Rightarrow> nat \<Rightarrow> 'a" where
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  "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)"
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lemma to_nat_on_finite: "finite S \<Longrightarrow> bij_betw (to_nat_on S) S {..< card S}"
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  using ex_bij_betw_finite_nat unfolding to_nat_on_def
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  by (intro someI2_ex[where Q="\<lambda>f. bij_betw f S {..<card S}"]) (auto simp add: atLeast0LessThan)
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lemma to_nat_on_infinite: "countable S \<Longrightarrow> infinite S \<Longrightarrow> bij_betw (to_nat_on S) S UNIV"
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  using countableE_infinite unfolding to_nat_on_def
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  by (intro someI2_ex[where Q="\<lambda>f. bij_betw f S UNIV"]) auto
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lemma bij_betw_from_nat_into_finite: "finite S \<Longrightarrow> bij_betw (from_nat_into S) {..< card S} S"
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  unfolding from_nat_into_def[abs_def]
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  using to_nat_on_finite[of S]
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  apply (subst bij_betw_cong)
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  apply (split split_if)
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  apply (simp add: bij_betw_def)
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  apply (auto cong: bij_betw_cong
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              intro: bij_betw_inv_into to_nat_on_finite)
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  done
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lemma bij_betw_from_nat_into: "countable S \<Longrightarrow> infinite S \<Longrightarrow> bij_betw (from_nat_into S) UNIV S"
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  unfolding from_nat_into_def[abs_def]
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  using to_nat_on_infinite[of S, unfolded bij_betw_def]
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  by (auto cong: bij_betw_cong intro: bij_betw_inv_into to_nat_on_infinite)
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lemma inj_on_to_nat_on[intro]: "countable A \<Longrightarrow> inj_on (to_nat_on A) A"
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  using to_nat_on_infinite[of A] to_nat_on_finite[of A]
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  by (cases "finite A") (auto simp: bij_betw_def)
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lemma to_nat_on_inj[simp]:
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  "countable A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> to_nat_on A a = to_nat_on A b \<longleftrightarrow> a = b"
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  using inj_on_to_nat_on[of A] by (auto dest: inj_onD)
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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"
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  by (auto simp: from_nat_into_def intro!: inv_into_f_f)
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lemma subset_range_from_nat_into: "countable A \<Longrightarrow> A \<subseteq> range (from_nat_into A)"
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  by (auto intro: from_nat_into_to_nat_on[symmetric])
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lemma from_nat_into: "A \<noteq> {} \<Longrightarrow> from_nat_into A n \<in> A"
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  unfolding from_nat_into_def by (metis equals0I inv_into_into someI_ex)
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lemma range_from_nat_into_subset: "A \<noteq> {} \<Longrightarrow> range (from_nat_into A) \<subseteq> A"
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  using from_nat_into[of A] by auto
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lemma range_from_nat_into[simp]: "A \<noteq> {} \<Longrightarrow> countable A \<Longrightarrow> range (from_nat_into A) = A"
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  by (metis equalityI range_from_nat_into_subset subset_range_from_nat_into)
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lemma image_to_nat_on: "countable A \<Longrightarrow> infinite A \<Longrightarrow> to_nat_on A ` A = UNIV"
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  using to_nat_on_infinite[of A] by (simp add: bij_betw_def)
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lemma to_nat_on_surj: "countable A \<Longrightarrow> infinite A \<Longrightarrow> \<exists>a\<in>A. to_nat_on A a = n"
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  by (metis (no_types) image_iff iso_tuple_UNIV_I image_to_nat_on)
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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"
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  by (simp add: f_inv_into_f from_nat_into_def)
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lemma to_nat_on_from_nat_into_infinite[simp]:
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  "countable A \<Longrightarrow> infinite A \<Longrightarrow> to_nat_on A (from_nat_into A n) = n"
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  by (metis image_iff to_nat_on_surj to_nat_on_from_nat_into)
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lemma from_nat_into_inj:
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  "countable A \<Longrightarrow> m \<in> to_nat_on A ` A \<Longrightarrow> n \<in> to_nat_on A ` A \<Longrightarrow>
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    from_nat_into A m = from_nat_into A n \<longleftrightarrow> m = n"
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  by (subst to_nat_on_inj[symmetric, of A]) auto
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lemma from_nat_into_inj_infinite[simp]:
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  "countable A \<Longrightarrow> infinite A \<Longrightarrow> from_nat_into A m = from_nat_into A n \<longleftrightarrow> m = n"
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  using image_to_nat_on[of A] from_nat_into_inj[of A m n] by simp
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lemma eq_from_nat_into_iff:
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  "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"
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  by auto
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lemma from_nat_into_surj: "countable A \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>n. from_nat_into A n = a"
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  by (rule exI[of _ "to_nat_on A a"]) simp
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lemma from_nat_into_inject[simp]:
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  "A \<noteq> {} \<Longrightarrow> countable A \<Longrightarrow> B \<noteq> {} \<Longrightarrow> countable B \<Longrightarrow> from_nat_into A = from_nat_into B \<longleftrightarrow> A = B"
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  by (metis range_from_nat_into)
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lemma inj_on_from_nat_into: "inj_on from_nat_into ({A. A \<noteq> {} \<and> countable A})"
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  unfolding inj_on_def by auto
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subsection {* Closure properties of countability *}
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lemma countable_SIGMA[intro, simp]:
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  "countable I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (A i)) \<Longrightarrow> countable (SIGMA i : I. A i)"
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  by (intro countableI'[of "\<lambda>(i, a). (to_nat_on I i, to_nat_on (A i) a)"]) (auto simp: inj_on_def)
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lemma countable_image[intro, simp]: assumes A: "countable A" shows "countable (f`A)"
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proof -
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  from A guess g by (rule countableE)
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  moreover have "inj_on (inv_into A f) (f`A)" "inv_into A f ` f ` A \<subseteq> A"
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    by (auto intro: inj_on_inv_into inv_into_into)
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  ultimately show ?thesis
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    by (blast dest: comp_inj_on subset_inj_on intro: countableI)
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qed
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lemma countable_UN[intro, simp]:
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  fixes I :: "'i set" and A :: "'i => 'a set"
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  assumes I: "countable I"
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  assumes A: "\<And>i. i \<in> I \<Longrightarrow> countable (A i)"
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  shows "countable (\<Union>i\<in>I. A i)"
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proof -
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  have "(\<Union>i\<in>I. A i) = snd ` (SIGMA i : I. A i)" by (auto simp: image_iff)
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  then show ?thesis by (simp add: assms)
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qed
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lemma countable_Un[intro]: "countable A \<Longrightarrow> countable B \<Longrightarrow> countable (A \<union> B)"
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  by (rule countable_UN[of "{True, False}" "\<lambda>True \<Rightarrow> A | False \<Rightarrow> B", simplified])
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     (simp split: bool.split)
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lemma countable_Un_iff[simp]: "countable (A \<union> B) \<longleftrightarrow> countable A \<and> countable B"
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  by (metis countable_Un countable_subset inf_sup_ord(3,4))
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lemma countable_Plus[intro, simp]:
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  "countable A \<Longrightarrow> countable B \<Longrightarrow> countable (A <+> B)"
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  by (simp add: Plus_def)
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lemma countable_empty[intro, simp]: "countable {}"
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  by (blast intro: countable_finite)
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lemma countable_insert[intro, simp]: "countable A \<Longrightarrow> countable (insert a A)"
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  using countable_Un[of "{a}" A] by (auto simp: countable_finite)
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lemma countable_Int1[intro, simp]: "countable A \<Longrightarrow> countable (A \<inter> B)"
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  by (force intro: countable_subset)
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lemma countable_Int2[intro, simp]: "countable B \<Longrightarrow> countable (A \<inter> B)"
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  by (blast intro: countable_subset)
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lemma countable_INT[intro, simp]: "i \<in> I \<Longrightarrow> countable (A i) \<Longrightarrow> countable (\<Inter>i\<in>I. A i)"
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  by (blast intro: countable_subset)
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lemma countable_Diff[intro, simp]: "countable A \<Longrightarrow> countable (A - B)"
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  by (blast intro: countable_subset)
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lemma countable_vimage: "B \<subseteq> range f \<Longrightarrow> countable (f -` B) \<Longrightarrow> countable B"
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  by (metis Int_absorb2 assms countable_image image_vimage_eq)
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lemma surj_countable_vimage: "surj f \<Longrightarrow> countable (f -` B) \<Longrightarrow> countable B"
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  by (metis countable_vimage top_greatest)
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lemma countable_Collect[simp]: "countable A \<Longrightarrow> countable {a \<in> A. \<phi> a}"
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  by (metis Collect_conj_eq Int_absorb Int_commute Int_def countable_Int1)
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lemma countable_lists[intro, simp]:
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  assumes A: "countable A" shows "countable (lists A)"
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   231
proof -
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   232
  have "countable (lists (range (from_nat_into A)))"
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   233
    by (auto simp: lists_image)
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   234
  with A show ?thesis
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   235
    by (auto dest: subset_range_from_nat_into countable_subset lists_mono)
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qed
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lemma Collect_finite_eq_lists: "Collect finite = set ` lists UNIV"
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  using finite_list by auto
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lemma countable_Collect_finite: "countable (Collect (finite::'a::countable set\<Rightarrow>bool))"
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  by (simp add: Collect_finite_eq_lists)
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50936
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lemma countable_rat: "countable \<rat>"
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  unfolding Rats_def by auto
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lemma Collect_finite_subset_eq_lists: "{A. finite A \<and> A \<subseteq> T} = set ` lists T"
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  using finite_list by (auto simp: lists_eq_set)
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lemma countable_Collect_finite_subset:
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  "countable T \<Longrightarrow> countable {A. finite A \<and> A \<subseteq> T}"
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  unfolding Collect_finite_subset_eq_lists by auto
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subsection {* Misc lemmas *}
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lemma countable_all:
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  assumes S: "countable S"
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   258
  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))"
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   259
  using S[THEN subset_range_from_nat_into] by auto
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end