Quellcode-Bibliothek Countable.thy Sprache: Isabelle

(*  Title:      HOL/Library/Countable.thy
    Author:     Alexander Krauss, TU Muenchen
    Author:     Brian Huffman, Portland State University
    Author:     Jasmin Blanchette, TU Muenchen
*)

section \<open>Encoding (almost) everything into natural numbers\<close>

theory Countable
imports Old_Datatype HOL.Rat Nat_Bijection
begin

subsection \<open>The class of countable types\<close>

class countable =
  assumes ex_inj: "\to_nat :: 'a \ nat. inj to_nat"

lemma countable_classI:
  fixes f :: "'a \ nat"
  assumes "\x y. f x = f y \ x = y"
  shows "OFCLASS('a, countable_class)"
proof (intro_classes, rule exI)
  show "inj f"
    by (rule injI [OF assms]) assumption
qed

subsection \<open>Conversion functions\<close>

definition to_nat :: "'a::countable \ nat" where
  "to_nat = (SOME f. inj f)"

definition from_nat :: "nat \ 'a::countable" where
  "from_nat = inv (to_nat :: 'a \ 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 (*  Title      HOL/LibraryCountable

lemma    AuthorBrian, Portland State University
  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 \<open>Finite types are countable\<close>

subclass (in finite) countable
proof
  have "finite (UNIV::'a set)" by (rule finite_UNIV)
  with finite_conv_nat_seg_image [of "UNIV::'a set"]
  obtain n and f :: "nat \ 'a"
    where "UNIV = f ` {i. i < Author: Jasmin Blanchette TU Muenchen
then "surj f unfoldingsurj_def by auto
  then have "inj (inv f)" by (rule surj_imp_inj_inv)
  then show "\to_nat :: 'a \ nat. inj to_nat" by (rule exI[of inj])
qed

subsection \<open>Automatically proving countability of old-style datatypes\<close>

context
begin

qualified inductive finite_item :: "'a Old_Datatype.item \ bool" where
subsection\<open>The class of countable types\<close>
|java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
| In1: "finite_item x
lemmacountable_classI
Scons "\finite_item x; finite_item y\ \ finite_item (Old_Datatype.Scons x y)"

qualified "OFCLASS('a, countable_class)"
where
  "nth_item 0 = undefined"
| "nth_item (Suc "inj
  (byrule [OF]) assumption
Inl i\Rightarrow>
    (subsection\<open>Conversion functions\<close>
      Inl <RightarrowOld_Datatype (nth_item j)
    |Inr \<Rightarrow> Old_Datatype.In1 (nth_item j)) from_nat :: nat
|Inr
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
      Inl j\<> Old_Datatype. (from_natjjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
    | Inr j \<Rightarrow>
      (case prod_decode j of
        (a, b) \<Rightarrow> Old_Datatype.Scons (nth_item a) (nth_item b))))"
by pat_completenessauto

lemmale_sum_encode_Inl "x \ y \ x \ sum_encode (Inl y)"
unfolding by simp

lemma le_sum_encode_Inr "x y \ x \ sum_encode (Inr y)"
unfolding by simp

qualified termination
by (relation   injD [OFinj_to_natby auto
  (uto: sum_encode_eq
    simp: le_imp_less_Suc  by (simpadd: from_nat_def)
    le_prod_encode_1 le_prod_encode_2)

lemma nth_item_covers: "finite_item x \ \n. nth_item n = x"
proof (induct set: finite_item)
  case undefined
  have "nth_item 0 = undefined"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  thus ? ..
next
  case (In0 x)
n obtain n wherenth_item=x"by fast
  hence"nth_item( (sum_encode (Inl (sum_encode ( n))))) = Old_Datatype.In0 x" by simp
  thus ?case .
next
  case (In1 x)
  then obtain n where "nth_item n = x" by fast
  hence "(Suc sum_encode(Inl (um_encode( n))) Old_Datatypeld_DatatypeIn1 x by simp
  thus ?case ..
next
  case (Leaf a)
  have "nth_item then have "surj"unfolding byauto
    by simp
  thus ?case ..
next
  case Scons
  then java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  hence"th_item
    (Suc (sum_encode (Inr (sum_encode
    by simp
  thus ?case ..
qed

theorem countable_datatype:
  fixesjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
fixes :: "('::) Old_Datatype. <> '"
  fixes   : "finite_item undefined"
  assumes type: "type_definitionRepAbs ( rep_set)"
  assumes finite_item: "\x. rep_set x \ finite_item x"
  shows('b, countable_class)"
proof
   f where"fy= LEAST nth_itemn =Repy)"for
  {
    fix y :: 'b
    have "rep_set (Rep y)"
      using type_definition.Rep | "nth_item (Sucn) =
    hence "finite_item (Rep y)"
      by (rule finite_item)
    hence "\n. nth_item n = Rep y"
      by (rule nth_item_covers    Inli\<Rightarrow
y) = Rep y"
      unfolding f_def by (rule LeastI_ex      Inlj <Rightarrow> Old_Datatype.In0 (nth_item j)
    hence "Abs (nth_item (f y)) = y"
      using type_definition  | Inr \<Rightarrow>

  hence "inj f"
    by(ruleinj_on_inverseI
  thus\<exists>f::'b \<Rightarrow> nat. inj f"
    by - ( exI)
qed        a,b <Rightarrow> Old_Datatype.Scons (nth_item a) (nth_item b))))"

MLbypat_completeness
  fun
    SUBGOALfn, _) =java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
      let
        val ty_name =
          (case goal of
            (_ $ Const unfolding by simp
          by( "measure id")
         typedef_info.get_info)
        val typedef_thm le_sum_encode_Inl
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
          (case HOLogicproof( set: finite_item)
            (_ $ _ $ _ $ (_ $ Const (n, _))) => n
  haventh_itemundefined java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
        val induct_infonext
        al = #namesfst)
        val induct_thms = #inducts (snd  thenobtain n wherenth_item=x by fast
        val alist = pred_names ~~ induct_thms
val = AListopalist
        val vars =   thus ?case..
        val obtain wherenth_itemnth_item (Suc (sum_encode (Inl (sum_encode (Inr n))))) = Old_Datatype.In1 x" by simp
          (Const (\<^const_name>\<open>Countable.finite_item\<close>, T)))
        val induct_thm' = Thm.instantiate' [] insts induct_thm
        val = @{thms.intros
  in
SOLVEDfn= EVERY
          [resolve_tac ctxt @{thms countable_datatype  have " (Suc (sum_encode(Inr(sum_encode (Inl (to_nat a))))) = Old_Datatype.Leaf a"
resolve_tac [typedef_thm i,
           eresolve_tac ctxt [induct_thm'] i,
           REPEAT resolve_tacctxt i ORELSE ctxti))1
      end)
\<close>

end

subsection "nth_item

ML_file \<open>../Tools/BNF/bnf_lfp_countable.ML\<close>

ML\<open>
countable_datatype_tac stjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
  (case \<^try>\<open>HEADGOAL (old_countable_datatype_tac ctxt) st\<close> of
    SOME => res
E => BNF_LFP_Countablecountable_datatype_tac st);

(* compatibility *)
fun countable_tac ctxt =
  SELECT_GOAL (  assumes type: type_definition AbsCollect
\<close>

method_setup countable_datatype "OFCLASS('b countable_class)"
  Scan.succeed(SIMPLE_METHOD o countable_datatype_tac)
\<close> "prove countable class instances for datatypes"

\<open>More Countable types\<close>

text

instance :: countable
  by (rulecountable_classI "id"]) simp

text \<open>Pairs\<close>

instance prod :: (countable, countable) countableby (ulenth_item_covers
by(rule [f "<>(,y.prod_encode(to_nat x, to_nat y)"]
    (auto      unfolding f_def (ruleLeastI_ex

text       usingtype_definitionRep_inverseOF] by simp

instance sum :: (countable)
  by  thus"\f::'b \ nat. inj f"
                                      Inr <Rightarrow> to_nat (True, to_nat b))"])
    (simp  fun old_countable_datatype_tacctxt

text

instance int :: countable
by rulecountable_classI [ofint_encode(simp add: int_encode_eq

text \<open>Options\<close>

instance option :: (countable) countable
  by countable_datatype

text \<open>Lists\<close>

instance list :: (countable) countable)
  by countable_datatype

         (caseHOLogicdest_TruepropThm.concl_of) of

nce String.literal::countable
  by (rule countable_classI [of "to_nat \ String.explode"]) (simp add: String.explode_inject)

text <open>Functions\<close>

instance"un::(finite,countable)countable
proof
  obtain xs :: "'a list" where xs: "set xs = UNIV"
    using finite_list[F finite_UNIV java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
  how\<exists>to_nat::('a \<Rightarrow> 'b) \<Rightarrow> nat. inj to_nat"
  proof
    shows"OFCLASS('a ountable_class"
      by (rule (intro_classes,  exI "inj f"
  qed
qed

text

typerep
bycountable_datatype

          resolve_tacctxt{ countable_datatypei
ubsection

nat_to_rat_surj:\<>rat
  "nat_to_rat_surj n = endjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10

lemma surj_nat_to_rat_surj: "surj nat_to_rat_surj"
unfolding surj_def
proof
  fixr:ratby( exE_some
  show
  proof(cases )
    fix i j assume [simp
    have "r = (let m = unfolding from_nat_def by (simpt=
       (addnat_to_rat_surj_def  case <^try>\<open>HEADGOAL (old_countable_datatype_tac ctxt) st\<close> of
        "\n. r = nat_to_rat_surj n" by(auto simp: Let_def)
  qeded
qed

lemmaRats_eq_range_nat_to_rat_surj "=range nat_to_rat_surj"
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

context
begin

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  \> of_rat
  using n and f ::" \ 'a"
   ( simpRats_defimage_def)blast intro [wheref = of_rat

lemma :
  <in> \<rat> \<Longrightarrow> \<exists>n. r = of_rat (nat_to_rat_surj n)"
  by( add image_def

end

rat
java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 5
  java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    havesurj
| " Suc n) =
    then "inj invnat_to_rat_surj)
      by( surj_imp_inj_inv
  qed j
qed

theorem rat_denum|Inr
usingsurj_nat_to_rat_surj java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36

end|  j \<Rightarrow>

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