src/HOL/Library/Countable.thy
changeset 26169 73027318f9ba
child 26243 69592314f977
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Countable.thy	Wed Feb 27 21:41:07 2008 +0100
@@ -0,0 +1,183 @@
+(*  Title:      HOL/Library/Countable.thy
+    ID:         $Id$
+    Author:     Tobias Nipkow
+*)
+
+header {* Encoding (almost) everything into natural numbers *}
+
+theory Countable
+imports Finite_Set 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 {* Converion 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 elim_to_cases
+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
+
+end