(*  Title:      HOL/Library/Countable.thy

    Author:     Alexander Krauss, TU Muenchen

    Author:     Brian Huffman, Portland State University

*)

header {* Encoding (almost) everything into natural numbers *}

theory Countable

imports Main Rat Nat_Bijection

begin

subsection {* The class of countable types *}

class countable =

  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 inj_on_to_nat[simp, intro]: "inj_on to_nat S"

  using inj_to_nat by (auto simp: inj_on_def)

lemma surj_from_nat [simp]: "surj from_nat"

  unfolding from_nat_def by (simp add: inj_imp_surj_inv)

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

  have "finite (UNIV\<Colon>'a set)" by (rule finite_UNIV)

  with finite_conv_nat_seg_image [of "UNIV::'a set"]

  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 *}

instance prod :: (countable, countable) countable

  by (rule countable_classI [of "\<lambda>(x, y). prod_encode (to_nat x, to_nat y)"])

    (auto simp add: prod_encode_eq)

text {* Sums *}

instance sum :: (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))"])

    (simp split: sum.split_asm)

text {* Integers *}

instance int :: countable

  by (rule countable_classI [of "int_encode"])

    (simp add: int_encode_eq)

text {* Options *}

instance option :: (countable) countable

  by (rule countable_classI [of "option_case 0 (Suc \<circ> to_nat)"])

    (simp split: option.split_asm)

text {* Lists *}

instance list :: (countable) countable

  by (rule countable_classI [of "list_encode \<circ> map to_nat"])

    (simp add: list_encode_eq)

text {* Further *}

instance String.literal :: countable

  by (rule countable_classI [of "to_nat o explode"])

    (auto simp add: explode_inject)

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 fun_eq_iff)

  qed

qed

subsection {* The Rationals are Countably Infinite *}

definition nat_to_rat_surj :: "nat \<Rightarrow> rat" where

"nat_to_rat_surj n = (let (a,b) = prod_decode n

                      in Fract (int_decode a) (int_decode b))"

lemma surj_nat_to_rat_surj: "surj nat_to_rat_surj"

unfolding surj_def

proof

  fix r::rat

  show "\<exists>n. r = nat_to_rat_surj n"

  proof (cases r)

    fix i j assume [simp]: "r = Fract i j" and "j > 0"

    have "r = (let m = int_encode i; n = int_encode j

               in nat_to_rat_surj(prod_encode (m,n)))"

      by (simp add: Let_def nat_to_rat_surj_def)

    thus "\<exists>n. r = nat_to_rat_surj n" by(auto simp:Let_def)

  qed

qed

lemma Rats_eq_range_nat_to_rat_surj: "\<rat> = range nat_to_rat_surj"

by (simp add: Rats_def surj_nat_to_rat_surj)

context field_char_0

begin

lemma Rats_eq_range_of_rat_o_nat_to_rat_surj:

  "\<rat> = range (of_rat o nat_to_rat_surj)"

using surj_nat_to_rat_surj

by (auto simp: Rats_def image_def surj_def)

   (blast intro: arg_cong[where f = of_rat])

lemma surj_of_rat_nat_to_rat_surj:

  "r\<in>\<rat> \<Longrightarrow> \<exists>n. r = of_rat(nat_to_rat_surj n)"

by(simp add: Rats_eq_range_of_rat_o_nat_to_rat_surj image_def)

end

instance rat :: countable

proof

  show "\<exists>to_nat::rat \<Rightarrow> nat. inj to_nat"

  proof

    have "surj nat_to_rat_surj"

      by (rule surj_nat_to_rat_surj)

    then show "inj (inv nat_to_rat_surj)"

      by (rule surj_imp_inj_inv)

  qed

qed

subsection {* Automatically proving countability of datatypes *}

inductive finite_item :: "'a Datatype.item \<Rightarrow> bool" where

  undefined: "finite_item undefined"

| In0: "finite_item x \<Longrightarrow> finite_item (Datatype.In0 x)"

| In1: "finite_item x \<Longrightarrow> finite_item (Datatype.In1 x)"

| Leaf: "finite_item (Datatype.Leaf a)"

| Scons: "\<lbrakk>finite_item x; finite_item y\<rbrakk> \<Longrightarrow> finite_item (Datatype.Scons x y)"

function

  nth_item :: "nat \<Rightarrow> ('a::countable) Datatype.item"

where

  "nth_item 0 = undefined"

| "nth_item (Suc n) =

  (case sum_decode n of

    Inl i \<Rightarrow>

    (case sum_decode i of

      Inl j \<Rightarrow> Datatype.In0 (nth_item j)

    | Inr j \<Rightarrow> Datatype.In1 (nth_item j))

  | Inr i \<Rightarrow>

    (case sum_decode i of

      Inl j \<Rightarrow> Datatype.Leaf (from_nat j)

    | Inr j \<Rightarrow>

      (case prod_decode j of

        (a, b) \<Rightarrow> Datatype.Scons (nth_item a) (nth_item b))))"

by pat_completeness auto

lemma le_sum_encode_Inl: "x \<le> y \<Longrightarrow> x \<le> sum_encode (Inl y)"

unfolding sum_encode_def by simp

lemma le_sum_encode_Inr: "x \<le> y \<Longrightarrow> x \<le> sum_encode (Inr y)"

unfolding sum_encode_def by simp

termination

by (relation "measure id")

  (auto simp add: sum_encode_eq [symmetric] prod_encode_eq [symmetric]

    le_imp_less_Suc le_sum_encode_Inl le_sum_encode_Inr

    le_prod_encode_1 le_prod_encode_2)

lemma nth_item_covers: "finite_item x \<Longrightarrow> \<exists>n. nth_item n = x"

proof (induct set: finite_item)

  case undefined

  have "nth_item 0 = undefined" by simp

  thus ?case ..

next

  case (In0 x)

  then obtain n where "nth_item n = x" by fast

  hence "nth_item (Suc (sum_encode (Inl (sum_encode (Inl n)))))

    = Datatype.In0 x" by simp

  thus ?case ..

next

  case (In1 x)

  then obtain n where "nth_item n = x" by fast

  hence "nth_item (Suc (sum_encode (Inl (sum_encode (Inr n)))))

    = Datatype.In1 x" by simp

  thus ?case ..

next

  case (Leaf a)

  have "nth_item (Suc (sum_encode (Inr (sum_encode (Inl (to_nat a))))))

    = Datatype.Leaf a" by simp

  thus ?case ..

next

  case (Scons x y)

  then obtain i j where "nth_item i = x" and "nth_item j = y" by fast

  hence "nth_item

    (Suc (sum_encode (Inr (sum_encode (Inr (prod_encode (i, j)))))))

      = Datatype.Scons x y" by simp

  thus ?case ..

qed

theorem countable_datatype:

  fixes Rep :: "'b \<Rightarrow> ('a::countable) Datatype.item"

  fixes Abs :: "('a::countable) Datatype.item \<Rightarrow> 'b"

  fixes rep_set :: "('a::countable) Datatype.item \<Rightarrow> bool"

  assumes type: "type_definition Rep Abs (Collect rep_set)"

  assumes finite_item: "\<And>x. rep_set x \<Longrightarrow> finite_item x"

  shows "OFCLASS('b, countable_class)"

proof

  def f \<equiv> "\<lambda>y. LEAST n. nth_item n = Rep y"

  {

    fix y :: 'b

    have "rep_set (Rep y)"

      using type_definition.Rep [OF type] by simp

    hence "finite_item (Rep y)"

      by (rule finite_item)

    hence "\<exists>n. nth_item n = Rep y"

      by (rule nth_item_covers)

    hence "nth_item (f y) = Rep y"

      unfolding f_def by (rule LeastI_ex)

    hence "Abs (nth_item (f y)) = y"

      using type_definition.Rep_inverse [OF type] by simp

  }

  hence "inj f"

    by (rule inj_on_inverseI)

  thus "\<exists>f::'b \<Rightarrow> nat. inj f"

    by - (rule exI)

qed

method_setup countable_datatype = {*

let

  fun countable_tac ctxt =

    SUBGOAL (fn (goal, i) =>

      let

        val ty_name =

          (case goal of

            (_ $ Const ("TYPE", Type ("itself", [Type (n, _)]))) => n

          | _ => raise Match)

        val typedef_info = hd (Typedef.get_info ctxt ty_name)

        val typedef_thm = #type_definition (snd typedef_info)

        val pred_name =

          (case HOLogic.dest_Trueprop (concl_of typedef_thm) of

            (typedef $ rep $ abs $ (collect $ Const (n, _))) => n

          | _ => raise Match)

        val induct_info = Inductive.the_inductive ctxt pred_name

        val pred_names = #names (fst induct_info)

        val induct_thms = #inducts (snd induct_info)

        val alist = pred_names ~~ induct_thms

        val induct_thm = the (AList.lookup (op =) alist pred_name)

        val rules = @{thms finite_item.intros}

      in

        SOLVED' (fn i => EVERY

          [rtac @{thm countable_datatype} i,

           rtac typedef_thm i,

           etac induct_thm i,

           REPEAT (resolve_tac rules i ORELSE atac i)]) 1

      end)

in

  Scan.succeed (fn ctxt => SIMPLE_METHOD' (countable_tac ctxt))

end

*} "prove countable class instances for datatypes"

hide_const (open) finite_item nth_item

subsection {* Countable datatypes *}

instance typerep :: countable

  by countable_datatype

end