src/HOL/Library/Countable.thy

more formal class Markup_Tree.Elements;

(*  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

ML {*
  fun countable_tac ctxt =
    SUBGOAL (fn (goal, i) =>
      let
        val ty_name =
          (case goal of
            (_ $ Const (@{const_name TYPE}, Type (@{type_name 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 vars = rev (Term.add_vars (Thm.prop_of induct_thm) [])
        val thy = Proof_Context.theory_of ctxt
        val insts = vars |> map (fn (_, T) => try (Thm.cterm_of thy)
          (Const (@{const_name Countable.finite_item}, T)))
        val induct_thm' = Drule.instantiate' [] insts induct_thm
        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)
*}

method_setup countable_datatype = {*
  Scan.succeed (fn ctxt => SIMPLE_METHOD' (countable_tac ctxt))
*} "prove countable class instances for datatypes"

hide_const (open) finite_item nth_item


subsection {* Countable datatypes *}

instance typerep :: countable
  by countable_datatype

end