Impressum Countable.thy Sprache: Isabelle

://.thy

    :      Huffmanjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
:,
*)

section   have" java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

theory Countable
java.lang.StringIndexOutOfBoundsException: Range [0, 7) out of bounds for length 0
begin

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

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

:
|:
  assumes
  showsOFCLASS)
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  show f"
     ( injI assms
qed <>

\<open>Conversion functions\<close>

j\> .In0j
  "to_nat = ( j

definitionfrom_nat" \ 'a::countable" where
  "from_nat i \

lemma inj_to_nat [simp  RightarrowLeaf )
  by (rulejava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

lemma inj_on_to_nat auto
:

lemma surj_from_nat sum_encode_def
  unfolding:"\
sum_encode_def
lemma
using ] by

lemma from_nat_to_nat [simp]:
  "from_nat (uto simp flip: sum_encode_eq prod_encode_eq
   add

subsectionjava.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83

subclass (in finite) countablecase
  the " n = "by
  have  SucInlx by
  with finite_conv_nat_seg_image.
  obtainjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
    where"th_item (Suc( Inl( Inrn))=O. "java.lang.StringIndexOutOfBoundsException: Range [93, 94) out of bounds for length 93
then f"unfoldingsurj_def java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

  then( x y)
qed

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

contextjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
begin

qualified inductive    Abs'countableOld_Datatypeitem\Rightarrowb
undefined
| In0   Collect
|assumes
| Leaf: "finite_item "OFCLASS
| Scons

define   (n.     )  y
where
  "nth_item java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

  (case sum_decode n of java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
      >
    (case sum_decode     hence "nth_item (f
java.lang.NullPointerException
    | Inr j \<Rightarrow> Old_Datatype.In1 (nth_item j))
i\<Rightarrow>
    (case sum_decode i
        )
       "
      (case prod_decode  rule
( )java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
  auto

lemma ( (goal=
unfolding sum_encode_def by simp

lemma le_sum_encode_Inr: "java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 9
sum_encode_def

qualified termination
relation
  (auto simpval = hd (Typedef ctxt ty_name
    simp: le_imp_less_Suc le_sum_encode_Inr
    le_prod_encode_1 le_prod_encode_2)

lemma nth_item_covers: "finite_item x \ \n. nth_item n = x"
inductjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
  case undefined
   " 0 = " bysimp
  thus ?case ..
next
  case (In0 xv pred_names ( induct_info
   obtain " n "java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
  hence " induct_thm =the(.lookup (op =) alist pred_name)
  thus .
next
  case (In1 x)
  then n  " n = x" by fast
  hence "java.lang.StringIndexOutOfBoundsException: Range [93, 17) out of bounds for length 93
  thus ? rules finite_item}
next
  case         ' (fn i >
  nth_item  sum_encode )Leaf
    by simp
  thus ?case             ctxt]i
next
  case(  rules assume_tac i]
  then obtain i j java.lang.StringIndexOutOfBoundsException: Range [0, 23) out of bounds for length 10
  hence
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     \<open>
  fun ctxt =
qed

theorem countable_datatype:
  fixes
  fixes Abs res
    | NON. ctxtjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
" Rep ( rep_set)"
  assumes\<close>
  shows,)
proof
  Scan SIMPLE_METHOD)
\<close> "prove countable class instances for datatypes"
    subsection
    have
\<open>Naturals\<close>
    hence nat
        (  [of
    hence "\n. nth_item n = Rep y"
      ( )
    hence "nth_item (f y) = countable_classI[f \lambda>x ) y]
        by )
    hence
. [ type
  }
  hence "inj f"
eI
java.lang.NullPointerException
    by| bjava.lang.NullPointerException
qed

ML \<open>
fun  =
    Author
      let
        val ty_name =
          (case goal of
            (_  \<open>Integers\<close>
           ( countable_classI ]) simp)
        val java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
        val java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

         .  (Thm typedef_thm
            java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
          | _ => raise
        val  ":,java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
val  = names [].
        val
fixes  "
pred_name
        val  java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
valjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

induct_thm:



            { }i
s

Rdefinition "nat Rightarrow"
)
\<close>

end

subsection  "rom_nat ( :'java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 0

MLcasesjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
(java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
SOMEresthus
| =q

(* compatibility *)
:"rat java.lang.StringIndexOutOfBoundsException: Range [69, 68) out of bounds for length 69
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
\<close>

method_setup countable_datatype = \<open>     <=  \<circ> nat_to_rat_surj)"
Scan SIMPLE_METHOD(: image_deflast   java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 87
\<close> "prove countable class instances for datatypes"

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

instance
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

textjava.lang.StringIndexOutOfBoundsException: Range [25, 5) out of bounds for length 5

instance prod :: (countable : finite_item:java.lang.StringIndexOutOfBoundsException: Index 112 out of bounds for length 112
then
         i       rulecase i java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

u  by

instancesum, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    countable_classI "
                                     | Inr b \<Rightarrow> to_nat (True, to_nat b))"])
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

text"nth_item0 =undefined" by simp

instance ?case.
  by (rule countable_classI [

text

instance option(sum_encode( ))) . simp
by

java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

  ::countable

text

instance Stringnth_item   )))=Leaf
case

text

fun   java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
proof
  obtain xs :: "'a list" where
     finite_listOF finite_UNIV.
  show "\to_nat::('a \ 'b) \ nat. inj to_nat"
  proof
    show qed
      by (rule java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
fixAbs (::) .
qed

text \<open>Typereps\<close>

instance typerep :: countable finite_item
by

subsection

definition
n  (ab   n Fract) b)

lemma surj_nat_to_rat_surj " ( "
unfolding
proof
:
  showbyrule)
  proof (cases( nth_item_covers
assume]  i j" "j > 0
    have        f_def (rule)
      by (simp addhenceAbsnth_itemyjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
    thus
  qedby( )
qed

lemma Rats_eq_range_nat_to_rat_surj: "\ = range nat_to_rat_surj"
bysimp: Rats_def)

context field_char_0 (fn(, _) =>
java.lang.StringIndexOutOfBoundsException: Range [8, 5) out of bounds for length 21

Rats_eq_range_of_rat_o_nat_to_rat_surj
  "\ = range (of_rat \ nat_to_rat_surj)"
  using surj_nat_to_rat_surj
              => raise

lemma:
  v  java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
  | > Match

end

instance rat :: countablepred_names fst)
proof
  show "\to_nat::rat \ nat. inj to_nat"
  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
surj
      by (rule surj_nat_to_rat_surj)
     showjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
byrule)
  qed
qed

theoremin
        SOLVEDjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

end

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