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src/HOL/Library/Countable.thy

author | haftmann |

Thu, 26 Jun 2008 10:07:01 +0200 | |

changeset 27368 | 9f90ac19e32b |

parent 26585 | 3bf2ebb7148e |

child 27487 | c8a6ce181805 |

permissions | -rw-r--r-- |

established Plain theory and image

(* Title: HOL/Library/Countable.thy ID: $Id$ Author: Alexander Krauss, TU Muenchen *) header {* Encoding (almost) everything into natural numbers *} theory Countable imports Plain List Hilbert_Choice begin subsection {* The class of countable types *} class countable = itself + assumes ex_inj: "\<exists>to_nat \<Colon> 'a \<Rightarrow> nat. inj to_nat" lemma countable_classI: fixes f :: "'a \<Rightarrow> nat" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "OFCLASS('a, countable_class)" proof (intro_classes, rule exI) show "inj f" by (rule injI [OF assms]) assumption qed subsection {* Conversion functions *} definition to_nat :: "'a\<Colon>countable \<Rightarrow> nat" where "to_nat = (SOME f. inj f)" definition from_nat :: "nat \<Rightarrow> 'a\<Colon>countable" where "from_nat = inv (to_nat \<Colon> 'a \<Rightarrow> nat)" lemma inj_to_nat [simp]: "inj to_nat" by (rule exE_some [OF ex_inj]) (simp add: to_nat_def) lemma to_nat_split [simp]: "to_nat x = to_nat y \<longleftrightarrow> x = y" using injD [OF inj_to_nat] by auto lemma from_nat_to_nat [simp]: "from_nat (to_nat x) = x" by (simp add: from_nat_def) subsection {* Countable types *} instance nat :: countable by (rule countable_classI [of "id"]) simp subclass (in finite) countable proof unfold_locales have "finite (UNIV\<Colon>'a set)" by (rule finite_UNIV) with finite_conv_nat_seg_image [of UNIV] obtain n and f :: "nat \<Rightarrow> 'a" where "UNIV = f ` {i. i < n}" by auto then have "surj f" unfolding surj_def by auto then have "inj (inv f)" by (rule surj_imp_inj_inv) then show "\<exists>to_nat \<Colon> 'a \<Rightarrow> nat. inj to_nat" by (rule exI[of inj]) qed text {* Pairs *} primrec sum :: "nat \<Rightarrow> nat" where "sum 0 = 0" | "sum (Suc n) = Suc n + sum n" lemma sum_arith: "sum n = n * Suc n div 2" by (induct n) auto lemma sum_mono: "n \<ge> m \<Longrightarrow> sum n \<ge> sum m" by (induct n m rule: diff_induct) auto definition "pair_encode = (\<lambda>(m, n). sum (m + n) + m)" lemma inj_pair_cencode: "inj pair_encode" unfolding pair_encode_def proof (rule injI, simp only: split_paired_all split_conv) fix a b c d assume eq: "sum (a + b) + a = sum (c + d) + c" have "a + b = c + d \<or> a + b \<ge> Suc (c + d) \<or> c + d \<ge> Suc (a + b)" by arith then show "(a, b) = (c, d)" proof (elim disjE) assume sumeq: "a + b = c + d" then have "a = c" using eq by auto moreover from sumeq this have "b = d" by auto ultimately show ?thesis by simp next assume "a + b \<ge> Suc (c + d)" from sum_mono[OF this] eq show ?thesis by auto next assume "c + d \<ge> Suc (a + b)" from sum_mono[OF this] eq show ?thesis by auto qed qed instance "*" :: (countable, countable) countable by (rule countable_classI [of "\<lambda>(x, y). pair_encode (to_nat x, to_nat y)"]) (auto dest: injD [OF inj_pair_cencode] injD [OF inj_to_nat]) text {* Sums *} instance "+":: (countable, countable) countable by (rule countable_classI [of "(\<lambda>x. case x of Inl a \<Rightarrow> to_nat (False, to_nat a) | Inr b \<Rightarrow> to_nat (True, to_nat b))"]) (auto split:sum.splits) text {* Integers *} lemma int_cases: "(i::int) = 0 \<or> i < 0 \<or> i > 0" by presburger lemma int_pos_neg_zero: obtains (zero) "(z::int) = 0" "sgn z = 0" "abs z = 0" | (pos) n where "z = of_nat n" "sgn z = 1" "abs z = of_nat n" | (neg) n where "z = - (of_nat n)" "sgn z = -1" "abs z = of_nat n" apply atomize_elim apply (insert int_cases[of z]) apply (auto simp:zsgn_def) apply (rule_tac x="nat (-z)" in exI, simp) apply (rule_tac x="nat z" in exI, simp) done instance int :: countable proof (rule countable_classI [of "(\<lambda>i. to_nat (nat (sgn i + 1), nat (abs i)))"], auto dest: injD [OF inj_to_nat]) fix x y assume a: "nat (sgn x + 1) = nat (sgn y + 1)" "nat (abs x) = nat (abs y)" show "x = y" proof (cases rule: int_pos_neg_zero[of x]) case zero with a show "x = y" by (cases rule: int_pos_neg_zero[of y]) auto next case (pos n) with a show "x = y" by (cases rule: int_pos_neg_zero[of y]) auto next case (neg n) with a show "x = y" by (cases rule: int_pos_neg_zero[of y]) auto qed qed text {* Options *} instance option :: (countable) countable by (rule countable_classI[of "\<lambda>x. case x of None \<Rightarrow> 0 | Some y \<Rightarrow> Suc (to_nat y)"]) (auto split:option.splits) text {* Lists *} lemma from_nat_to_nat_map [simp]: "map from_nat (map to_nat xs) = xs" by (simp add: comp_def map_compose [symmetric]) primrec list_encode :: "'a\<Colon>countable list \<Rightarrow> nat" where "list_encode [] = 0" | "list_encode (x#xs) = Suc (to_nat (x, list_encode xs))" instance list :: (countable) countable proof (rule countable_classI [of "list_encode"]) fix xs ys :: "'a list" assume cenc: "list_encode xs = list_encode ys" then show "xs = ys" proof (induct xs arbitrary: ys) case (Nil ys) with cenc show ?case by (cases ys, auto) next case (Cons x xs' ys) thus ?case by (cases ys) auto qed qed text {* Functions *} instance "fun" :: (finite, countable) countable proof obtain xs :: "'a list" where xs: "set xs = UNIV" using finite_list [OF finite_UNIV] .. show "\<exists>to_nat::('a \<Rightarrow> 'b) \<Rightarrow> nat. inj to_nat" proof show "inj (\<lambda>f. to_nat (map f xs))" by (rule injI, simp add: xs expand_fun_eq) qed qed end