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_Int_Bij
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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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primrec sum :: "nat \<Rightarrow> nat"
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where
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  "sum 0 = 0"
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| "sum (Suc n) = Suc n + sum n"
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lemma sum_arith: "sum n = n * Suc n div 2"
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  by (induct n) auto
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lemma sum_mono: "n \<ge> m \<Longrightarrow> sum n \<ge> sum m"
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  by (induct n m rule: diff_induct) auto
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definition
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  "pair_encode = (\<lambda>(m, n). sum (m + n) + m)"
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lemma inj_pair_cencode: "inj pair_encode"
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  unfolding pair_encode_def
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proof (rule injI, simp only: split_paired_all split_conv)
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  fix a b c d
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  assume eq: "sum (a + b) + a = sum (c + d) + c"
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  have "a + b = c + d \<or> a + b \<ge> Suc (c + d) \<or> c + d \<ge> Suc (a + b)" by arith
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  then
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  show "(a, b) = (c, d)"
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  proof (elim disjE)
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    assume sumeq: "a + b = c + d"
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    then have "a = c" using eq by auto
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    moreover from sumeq this have "b = d" by auto
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    ultimately show ?thesis by simp
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  next
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    assume "a + b \<ge> Suc (c + d)"
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    from sum_mono[OF this] eq
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    show ?thesis by auto
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  next
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    assume "c + d \<ge> Suc (a + b)"
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    from sum_mono[OF this] eq
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    show ?thesis by auto
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  qed
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qed
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instance "*" :: (countable, countable) countable
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by (rule countable_classI [of "\<lambda>(x, y). pair_encode (to_nat x, to_nat y)"])
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  (auto dest: injD [OF inj_pair_cencode] injD [OF inj_to_nat])
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text {* Sums *}
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instance "+":: (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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    (auto split:sum.splits)
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text {* Integers *}
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lemma int_cases: "(i::int) = 0 \<or> i < 0 \<or> i > 0"
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by presburger
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lemma int_pos_neg_zero:
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  obtains (zero) "(z::int) = 0" "sgn z = 0" "abs z = 0"
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  | (pos) n where "z = of_nat n" "sgn z = 1" "abs z = of_nat n"
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  | (neg) n where "z = - (of_nat n)" "sgn z = -1" "abs z = of_nat n"
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apply atomize_elim
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apply (insert int_cases[of z])
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apply (auto simp:zsgn_def)
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apply (rule_tac x="nat (-z)" in exI, simp)
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apply (rule_tac x="nat z" in exI, simp)
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done
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instance int :: countable
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proof (rule countable_classI [of "(\<lambda>i. to_nat (nat (sgn i + 1), nat (abs i)))"], 
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    auto dest: injD [OF inj_to_nat])
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  fix x y 
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  assume a: "nat (sgn x + 1) = nat (sgn y + 1)" "nat (abs x) = nat (abs y)"
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  show "x = y"
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  proof (cases rule: int_pos_neg_zero[of x])
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    case zero 
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    with a show "x = y" by (cases rule: int_pos_neg_zero[of y]) auto
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  next
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    case (pos n)
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    with a show "x = y" by (cases rule: int_pos_neg_zero[of y]) auto
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  next
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    case (neg n)
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    with a show "x = y" by (cases rule: int_pos_neg_zero[of y]) auto
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  qed
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qed
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text {* Options *}
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instance option :: (countable) countable
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by (rule countable_classI[of "\<lambda>x. case x of None \<Rightarrow> 0
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                                     | Some y \<Rightarrow> Suc (to_nat y)"])
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 (auto split:option.splits)
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text {* Lists *}
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lemma from_nat_to_nat_map [simp]: "map from_nat (map to_nat xs) = xs"
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  by (simp add: comp_def)
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primrec
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  list_encode :: "'a\<Colon>countable list \<Rightarrow> nat"
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where
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  "list_encode [] = 0"
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| "list_encode (x#xs) = Suc (to_nat (x, list_encode xs))"
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instance list :: (countable) countable
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proof (rule countable_classI [of "list_encode"])
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  fix xs ys :: "'a list"
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  assume cenc: "list_encode xs = list_encode ys"
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  then show "xs = ys"
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  proof (induct xs arbitrary: ys)
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    case (Nil ys)
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    with cenc show ?case by (cases ys, auto)
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  next
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    case (Cons x xs' ys)
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    thus ?case by (cases ys) auto
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  qed
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qed
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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 expand_fun_eq)
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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) = nat_to_nat2 n
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                      in Fract (nat_to_int_bij a) (nat_to_int_bij 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 = inv nat_to_int_bij i; n = inv nat_to_int_bij j
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               in nat_to_rat_surj(nat2_to_nat (m,n)))"
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      using nat2_to_nat_inj surj_f_inv_f[OF surj_nat_to_int_bij]
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      by(simp add:Let_def nat_to_rat_surj_def nat_to_nat2_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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end
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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