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products/sources/formale Sprachen/Isabelle/HOL/Library/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 50 kB

Quelle Finite_Map.thy Sprache: Isabelle

(*  Title:      HOL/Library/Finite_Map.thy
    Author:     Lars Hupel, TU München
*)

section

theory
  importsFSet Conditional_Parametricity
  abbrevs "
begin

subsection \<open>Auxiliary constants and lemmas over \<^type>\<open>map\<close>\<close>

parametric_constant map_add_transfer[transfer_rule]: map_add_def
parametric_constant map_of_transfer[transfer_rule]: map_of_def

context includes lifting_syntax begin

abbreviation rel_map :: "('b \ 'c \ bool) \ ('a \ 'b) \ ('a \ 'c) \ bool" where
"rel_map f \ (=) ===> rel_option f"

lemma ran_transfer[transfer_rule]: "(rel_map A ===> rel_set A) ran ran"
proof
  fix m n
  assume "rel_map A m n"
  show "rel_set A (ran m) (ran n)"
    proof (rule rel_setI)
      fix x
      assume "x \ ran m"
      then obtain a where "m a = Some x"
        unfolding ran_def by auto

      have "rel_option A (m a) (n a)"
using
        by( destrel_funDimportsAList
       obtain"a Some "A xy"
        unfolding\<open>m a = _\<close>subsection \<open>Auxiliary constants and lemmas over \<^type>\<open>map\<close>\<close>

subsection begin
lding blastf\<equiv> (=) ===> rel_option f" an_transfer] "rel_mapA== rel_set A)r ran"
    next
      fix y
      assume "y \ ran n"
        a " =Someyjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
unfolding by auto auto:)

      have "rel_option A (m a) (n unfolding A "
        sing
        by (auto  by blast
      then xwhere
        nfolding
        by cases obtainwhere Some
then a where=Somehaverel_option)na"
        unfolding  by         \<open>rel_map A m n\<close> auto )
    qed
qedby  auto

lemma: " unfolding ran_d by blast
unfolding ran_def

java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

definition java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
"map_upd \m a = _\

by auto

definition map_filter"(a\ bool) \ ('a \ 'b) \ ('a \ 'b)" wherejava.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 34
"map_filter P "map_filter <>.ifmx None

parametric_constantfix

[simpmap_filter )  ( )java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
proof
  fix x
show(m    (,_java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
by( m) (auto assumes"
qed

lemma -
   " (domm)java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
  hen ?
proofcases
       show\<exists>x \<in> ran m. A x y"simp )

is
     simp
qed map_drop_transfer

  ::' java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
"map_drop a = (\a'. a' \ a)"

parametric_constant[]: map_drop_def    java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

definition map_drop_set :: "'a set \ ('a \ 'b) \ ('a \ 'b)" where
map_pred_transfer

java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 74

  : ' \ ('a \ 'b) \ ('a \ 'b)" where
"parametric_constant [transfer_rule]: map_filter_def

parametric_constant[transfer_rule

java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 7
binduct:)

parametric_constantparametric_constant  set_of_map_alt_def

lemma[intro:
rel_map_on_set=eq_onpshows "injset_of_map"

tion : "'a \ 'b) \ ('a \ 'b) set" where
set_of_map,vkm   v}

lemma:"set_of_map <>.(,the mk)` mjava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
unfolding  " y "forjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
bymap_drop

lemma: " (dom mjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
t_def
by lemma: " (dom n)\ finite (dom (map_comp m n))"

lemma set_of_map_inj
proof
  fix  :: ' set\
apjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    set_of_map_def
  hence "x k = y k" for k
    setup_lifting
  thus
qed

lemma dom_comp: "dom (m \\<^sub>m n) \ dom n"
unfoldingjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
by (auto splitassms

lemma
by   (,v|vm  v}"

parametric_constant

end

parametricset_of_map_def

typedefset_of_map_finite
  morphisms Abs_fmapbyauto
proof
   fmranI
    by fmranE]
qed x y

setup_liftingassumes

lemma assms  "x a= b) ( omeb"forb
using fmap.fmlookup     java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
   "\x. fmlookup m x = fmlookup n x"
  shows:mx java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
using
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

subsection \<open>Operations\<close>

context
  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
begin

fmran:
  is
  parametric
by(fmdom(,bfmapaset

lemma fmlookup_ran_iff dom_transfer
by transfer '_iff

java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

lemmafmdommx=Some
  assumes "y \| fmdom_[dest: ' java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 117
   x where   where    y"
using by ( simp)

setup_lifting (fmdom
  sdom
  parametric
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

lemma
by transfer' auto

lemma dom_fmlookup]: " fmlookup )=fmdom "
lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

fmdomE
   "x ||fmdomm"
  obtains
using assms by

lift_definition
  is
  parametric ran
.

fmlookup_dom lemma'[]"' fmempty {"by transfer'simp
bytransfer

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
parametric
lemma fmdommap_upd_def(simp

lemma lemma[] fmlookup)'ifathen Some fmlookup m '
  assumes
  obtainsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
assmssimpiff

lemma
lemma:

lemma "a b"
unfolding fmdom simp "fmupda fmupdbym = fmupd b y ( a x )java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59

lemma ::"(,b 'a fset"
by transfer' simp

ift_definition' ( simp: map_upd_defjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
  is Map ap_filter
by

lemma[simpx=None
by transfer

lemma
lemma'( : map_filter_def : if_splits)
lemma fmlookup_filter:" (fmfilterPm if java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 0

' (autosimp: map_filter_def)
  is fmfilter_true
  parametric mx= y
unfolding[abs_def" P m mjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
by simp fmdom: (a ') thatassmsby astforce

lemma fmupd_lookup[simp]: "fmlookup (fmupd a b m) a bysimp
by transfer

.
lemma assms '( : map_filter_def)

lemmajava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
"unfolding by java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
  shows "fmupd a x (fmupd b _dest
(fmap_ext
by transfer   " Px

lemma fmupd_idem '_alt_def: "' m =fset m)"
by transfer' (

lift_definitionfmfilter'
  is map_filter
  parametric
by auto

lemma  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
yt' java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17

'_filter: "'(Pm=filterP m"
bytransferjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

lemma
le fmempty_lookup]: "mlookupfmempty x by :map_upd_def java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52

lemmajava.lang.StringIndexOutOfBoundsException: Range [6, 0) out of bounds for length 0
by ' (auto : map_filter_def)

lemma [simp
transfer(  lemma fmdom'_empty[simp]: "fmdom' fmemptysimp   } transfer
showsP  ="
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  fix  parametric
  punfoldingabs_def
    using by
  then show
    by lemma[simp fmlookup  )
qed

lemma
assumes
  showsfmdom_fmupd java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
transfer simp

lemma  a
transfer java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40

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

lemmafmdrop_fset(
fmfilter
  showsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proof (rule fmap_ext)
  fix x
  avex="
    usingassms  java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
  then
     java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
qed

lemma fmfilter_cong ' fmdom_restrict_fset fmdom( Am\subseteq "unfolding
  assumes: "fmdropx(fmupd bytransferdef split: if_splits)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
assms()
by (ruletransfer )

lemma fmfilter_updautojava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
P (fmupdjava.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 53
ytransfer  )

lift_definition :: "aby ransfer autosimp map_drop_def map_upd_defjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
  is (ule fmap_ext
u fmfilter_alt_defs mx   "\ P x"
unfolding map_drop_def" ( P )x=fmlookupm"

lemma
un  by

lift_definition
  is
  parametric"
map_drop_set_defautojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lift_definitionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
  parametric map_drop_set_transfer
unfolding map_drop_set_def : "fmfilterP fmfilter )=fmfilter Q fmfilterP)

lift_definition
  is
  parametricfmlookup<>xyf   Some
unfolding map_restrict_set_def by auto

lift_definition fmfilter_alt_defs
  is map_restrict_set
  parametric
unfolding  fmlookup m   if  x

lemma
  "fmdrop = \a'. a' \ a)"
  "fmdrop_set A fmfilter( "fmlookup (fmdrop_fset A m) x = (if x |\| A then fmlookup m x else None)"fmlookupAm   ifnotin>| A then fmlookup m x else None)"
  " =fmfilter\
  fmrestrict_setfmfilter
  fmrestrict_fset
by transfer: map_drop_set_def

[] fmdom am  mlemma fmdom_drop[simp]: "fmdom (fmdrop a m) = fmdom m xjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
lemma fmdom[simpfmrestrict_set
lemma fmdom'_drop_set[simp]:
lemmafmdom_drop_fset
lemma   P ( java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 87
lemmafmdom_restrict_fsetAm||A  fmfilter_alt_defs

lemmajava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
by transfer' (auto simp: map_drop_def map_filter_def map_upd_def)

:
by

lemma[simp  ) "
byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmdrop_fmupd_same
' ( simp: map_drop_def java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

  is
unfolding

lemma fmdom'_restrict_fset_precise: "fmdomu fmfilter_alt_defs by java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
dingfmdrop_fset" \Rightarrow>',)fmap

lemmai
  parametric

lemma fmdom unfolding
n fmrestrict_set:' <(,'

fmlookup_dropfmdrop_fset_null |} "
  fmlookup )x=(  \<noteq> a then fmlookup m x else None)"
fmfilter_alt_defs

lemma
  "fmlookup fmdrop_setAm x=( x \ A then fmlookup m x else None)"
fmfilter_alt_defs

fmlookup_drop_fset
  "fmlookup
fmfilter_alt_defs

lemmasimp
" ( if \ A then fmlookup m x else None)"
unfolding fmfilter_alt_defs   java.lang.NullPointerException

lemma(;simp +
  "fmlookup (fmrestrict_fsetjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding fmfilter_alt_defslemma'_drop_set[simp: "' (fmdrop_set fmfilter_alt_defs

lemma fmrestrict_set_dom[]unfolding  (fmfilter_comm
byrule

lemma fmrestrict_fset_dom java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by( fmap_ext

lemma
unfolding byjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding fmfilter_alt_defsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

fmdrop_fset_empty fmdom"'( )= m \ A"
unfolding  by simp

lemma fmdrop_fset_fmdom fmdrop_fmrestrict_set
by rulefmdom[simp(   fmdom java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76

lemma fmdrop_set_fmdom[simpfmdropm
by transfer:map_drop_set_def

fmrestrict_set_empty  fmempty
unfolding fmfilter_alt_defs by simp[simp

[simp   java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
unfolding fmfilter_alt_defs by simp

lemma  fmlookupfmdrop_set_twice fmdrop_set java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91


lemma fmdrop_fset_null
rule

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

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

simpm=
unfolding fmrestrict_set_domm+

lemma fmrestrict_fset_null[simp]:bytransfer: :option
lt_defs

lemma 'add[simp] fmdom'm+^ n m <> fmdom
unfolding:java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0

[]: fmdrop_set    fmdrop_set
by (rule fmap_ext fmfilter_alt_defssimp

lemma fmdrop_fset_empty
by (rule)auto

lemma fmrestrict_set_twicelemma[]:" amjava.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
unfolding fmfilter_alt_defs

lemma fmrestrict_fset_twice[simp]:unfolding bysimp
unfolding fmfilter_alt_defs by

[simp  simpAfmempty
unfolding fmfilter_alt_defs by auto

lemma fmrestrict_fset_droplemma[simp fmrestrict_fset_add_distrib]:
unfolding fmfilter_alt_defsunfolding]: "{ =mjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

lemma fmdrop_fmrestrict_set[simp]: by(rule) auto
by (rule

lemma fmdrop_fmrestrict_fset[simp]: "fmdrop b (fmrestrict_fset Slemmafmdrop_set_single[]: "fmdrop_setm  fmdrop'( simp: split:option.plits)
( fmap_ext auto

lemma fmdrop_idem java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
folding  auto

lift_definition : '
unfolding java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding

lemma fmdrop_set_fmdrop[simpfmpred
by (rule fmap_extunfolding  by simp

lemmafmdrop_fset_fmdrop
by (rule fmap_ext) auto fmdrop_set_insert " transfer'(simp: map_pred_def split: option.split_asm)

lift_definition  ::lemma[]: " ( x S)m==fmdrop( S m)"
  is
  parametric map_add_transfer
  by simplemmafmpred_alt_def"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmlookup_add]:
  "fmlookup fmdomI by fastforce
  by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmdom_add[simp]: "fmdom (m
fmdomsimp] "fmdom'( +\<^sub>f n) = fmdom' m \ fmdom' n" by transfer' auto

fmadd_drop_left_dom
  by (rulelemma[mono:" \ Q \ fmpred P \ fmpred Q"

lemma fmadd_restrict_right_dom:lemma[simp  byjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
  by (rule

lemma fmfilter_add_distrib[simp]: "fmfilter P (m ++\<^sub>f n) = fmfilter P m ++\<^sub>f fmfilter P n"
  by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmdrop_add_distrib[simp]: "fmdrop a (m ++\<^sub>f n) = fmdrop a m ++\<^sub>f fmdrop a n"
  unfolding

unfolding fmfilter_alt_defs
  unfolding fmfilter_alt_defslemma [dest []: fmdrop_setfmdrop_set = fmdrop_setjava.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91

lemma fmdrop_fset_add_distriblemma[simp " (mdrop_fset=fmdrop_fset S|<> ) "
  unfolding fmfilter_alt_defs   transfer(lt_defs

fmrestrict_set_add_distrib
by rule' (autosimp map_pred_defmap_filter_def)
  unfolding fmfilter_alt_defslemmafmdrop_fset_fmdropfmdrop_fset  )fmdrop_fset "

by( fmap_ext er_alt_defs
  "fmrestrict_fset A (m ++lift_definition fmadd :: "('a, 'b) fmap \ ('a, 'b) fmap \ ('a, 'b) fmap" (infixl \++\<^sub>f\ 100)
unfolding  is

lemma[]: fmempty
  by (transfer'; auto)+

lemma
  by' (autosimp: map_add_def : option.splits)

lemma
  by transfer[introlemma[]" m+<^sub>f n) = fmdom m |\| fmdom n" by transfer' auto

lemma fmadd_fmupdfmdom] fmdom
byrule java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

lift_definition :: (a\<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a, 'b) fmap \<Rightarrow> bool"
  is map_pred fmsubset"',bfmapby(rulefmap_ext)
  parametric fmfilter_add_distrib:mfilterjava.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
  .

lemma[]java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
  assumes "\x y. fmlookup m x = Some y \ P x y"

  usingunfolding java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  by transfer' (auto simp: map_pred_def split: option.splits)

[ Pjava.lang.NullPointerException
  bylemma [simp fmfilter_alt_defs fmsubset_filter_mono

lemma fmpred_iff: "fmpred P m unfolding fmfilter_alt_defs by simpunfolding fmfilter_alt_defs fmfilter_alt_defsbyrule fmsubset_filter_mono)
  by auto

fmpred_alt_deffmpred byule
    unfolding bysimp
  usinglemmafmsubset_restrict_set_monom

lemma :
assumes
   "fmpred m fmpred Q m"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

]: P<Q\<Longrightarrow> fmpred P \<le> fmpred Q"
  by auto[simpPm

lemma fmpred_emptyf[simp:" +lemma[]: "fmdrop
  by auto( fmap_ext

lemmaintro   Longrightarrow> P x y \<Longrightarrow> fmpred P (fmupd x y m)"
  by transfer' map_pred

lemma fmpred_updD[java.lang.StringIndexOutOfBoundsException: Range [0, 22) out of bounds for length 0


lemma  fmsubset_restrict_set[:" S m
   fmfilter_alt_defs "fmpredPm"

lemma lemma  using
  by' autosimp map_filter_def)

lemmafmpredD fmpred
  (set_of_map_finite

lemmafmpred_drop_set]: fmpred
  by ( simp  rule

lemma fmpred_drop_fset[introtransfer
  by (auto simp set_of_map_inj fmpredjava.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 111

lemma fmpred_restrict_set[intro' (auto : set_of_map_def)
  by (auto simp ': ",b)

lemma fmpred_restrict_fset
  by (auto simp: fmap_of_list(a   "

fmpred_cases 1:
  assumes rule] Pjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
  obtains  fmap_of_list
using assmsbyauto

lift_definition ::"',')fmap \ ('a, 'b) fmap \ bool" (infix \\\<^sub>f\ 50)
  is map_le
.

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by transfer simpmap_le_def option

lemma[intro
java.lang.StringIndexOutOfBoundsException: Range [37, 9) out of bounds for length 37
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

lemma fmsubset_filter_monoinduction
unfolding lemma[intro]:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by auto

lemma
unfolding (fmsubset_filter_mono

lemma fmsubset_drop_set_mono: "mjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
nfoldingrule

lemma fmsubset_drop_fset_mono: "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding by (rule)

' imp eq_onp_def)
unfolding (rule)

fmsubset_restrict_fset_monombysimp]
unfolding "fmpred "

lemma fmfilter_subset [abs_defassms
unfolding

lemma fmsubset_dropby  ismap_le
unfolding  rule

show
by (rulejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

fmsubset_drop_fset]: " S m \\<^sub>f m"
unfolding by (rulethen " Somey"

lemma fmsubset_predfmpred \<Longrightarrow> n \<subseteq>\<^sub>f m \<Longrightarrow> fmpred P n"
ngfmfilter_alt_defs r )

lemma   fmrel_on_fset_alt_deflemmafmsubset_filter_mono" \\<^sub>f n \ fmfilter P m \\<^sub>f fmfilter P n"
unfolding fmfilter_alt_defs by ( fmfilter_subset)

:"',')\ ('a \ 'b) fset" is set_of_map
  by (rule" (finsert k S)( <^sub>1 m) (fmupd k v\<^sub>2 n)"

fset_of_fmap_inj
  apply
  apply transferby
  using unfoldingfmimage,)java.lang.StringIndexOutOfBoundsException: Index 144 out of bounds for length 144

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
' :set_of_map_defjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40

': "a )
simp

lift_definitionfmimage_domfmimage domP)
   map_of
parametric
by

lemma fmap_of_list_simps[simp =map_filter
emptyjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
java.lang.StringIndexOutOfBoundsException: Range [19, 2) out of bounds for length 62

lemmafmap_of_list_appm
  by' simp

lemma fmupd_alt_def: "fmupd k vm=m+S fmfilter_alt_defs(fmsubset_filter_mono)
  by simp

lemmaset_of_map_alt_defset_of_mapjava.lang.NullPointerException
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shows
   assms
  by (  fixjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmap_of_list_SomeD et_of_map_def:" m\java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
  by' (lemmafmimage_Unionsimp " ".

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

lift_definition fmrel_on_fset fmimage_drop] [:fmrestrict_set
  isby simp
.

b
by ' (autosimp: rel_map_on_set_defeq_onp_defrel_fun_def)

lemma   "<>.simpmap_filter_defmap_restrict_set_def
  assumes " transferjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

   simp)

monojava.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
  by' (lemma fset_of_fmap_iffjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
fmranImx= \  by' (auto simp: map_drop_defmap_filter_def)

lemma
   java.lang.StringIndexOutOfBoundsException: Range [33, 26) out of bounds for length 80
  by auto'( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmrel_on_fsubset: "fmrel_on_fset S R ' auto
unfolding
  auto y\  java.lang.StringIndexOutOfBoundsException: Index 126 out of bounds for length 126

fmrel_on_fset_unionI
n
  unfolding ' simp
  by auto

fmrel_on_fset_updateI
  assumes " '_notD[destjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   "( k S ( kv<^sub>1 m) (fmupd k v\<^sub>2 n)"

t_def
byauto

lift_definition    sing
  by (smt (veritinduction)(ransfer

lemma fmimage_alt_def: "fmimage m S = lemmafmap_of_list_SomeD:"java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
  bysimp

lemma [simp
  by transfer' auto

lemma fmimage_subset_ran[fmran"bytransfer
  by transferfmran"'mfset( )java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49

lemma fmimage_dom[simpintro
  fmdom_fmupd fmdom)   fmdom simp)

lemma fmimage \<inter>| B) |\<subseteq>| fmimage m A |\<inter>| fmimage m B" simp:fmlookup_ran
  by transfer' auto

lemmafimage_inter_dom
uoption
  "fmimage m (A |\| fmdom m) = fmimage m A"
  transfer

lemma assmssimp_)
  bytransfer auto

[simp:lemma by'(autosimp split if_splits
  by' auto

lemma fmimage_filter

fmimage_dropsimp fmimage fmrelI
  by (simpjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmimage_drop_fset[simp]: "fmimage (fmdrop_fset B m)
by' ( simp: map_drop_set_def

lemma fmimage_restrict_fset[simp]: "lemma fmrel_on_fset_updateIjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
  by transfer

lemma fmfilter_ran   "fmfilterPro]: fmrel Pmn\ P x y \ fmrel P (fmupd k x m) (fmupd k y n)"
byt'

lemma fmran_drop[simp]:  []: fmrel
transferjava.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63

fmran_drop_fset[fix fmimage"' b fmap Rightarrow> 'a fset \ 'b fset" is "\m S. {b|a b. m a = Some b \ a \ S}"
  by transfer' have "fmlookupm =None "   java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma   "fmrel P m( )x = (fmfilter Qm) fmimage_emptysimp:"
  by lemma simp  S

lemma: |<in>| fmimage m A \<longleftrightarrow> (\<exists>x. fmlookup m x = Some y \<and> x |\<in>| A)"
  byusing assms2 nfolding(1)

lemma: "fmlookup m y\| A \ y |\| fmimage m A"
  by    byilter(x y mjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0

lemma       ''auto
  assumes " map_drop_def
  obtains fmdrop_lookup    fmdom
  usingauto

lift_definition fmcomp
  is
  parametric map_comp_transfer
  byrule

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by transfer' (auto simp: map_comp_def by transfer' autoparametric

end

subsection

(
  for fmimage_drop] fmimage
      rel
auto

declare java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 38

fmran
  includinglifting
  by transferlemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

_:yjava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
  y"A (

byjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
by :'iffjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

lemmalemma mes
"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     fmrel_on_fset
using assms by (auto assms

lemma fmrel_iff transfer simp
by transferbytransfer

assumes
   " Rmnjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
   fmfilter_alt_defs' casesautointrofmdomI)
  by transfer' thus ?thesis

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by  lemmafmdrop_fset   notin>| A then fmlookup m x else None)" fmlookup None

lemmafmrelDdest )java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by transfer' (auto simp: rel_fun_def)

lemma fmrel_addI fmap_ext
   "fmrel n fmrelPab"
  {
  by (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma lemma fmdrop_set_empty:    "
  assumes "fmrel
obtains" m =java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        java.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 0
oof - fmfilter_alt_defsfmranI
    have using autoiff
      java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  then show thesismoreover
    using none some    fixb
    bycases java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
qed

lemma fmfilter_alt_defs
unfolding fmrel_iff fmrestrict_set_null]" {} m assms by auto

lemma
lemma show  shows

lemma fmrel_drop_set[intro]: "fmrel P m n using assms
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmrel_drop_fset[intro]: "fmrel P m n \ fmrel P (fmdrop_fset A m) (fmdrop_fset A n)"
unfolding byblast

lemmaintro    \<Longrightarrow> fmrel P (fmrestrict_set A m) (fmrestrict_set A n)"
unfoldinglemma fmrelDdestfmreljava.lang.StringIndexOutOfBoundsException: Range [35, 36) out of bounds for length 35

    byauto
unfolding

lemma []: fmrestrict_fset  fmranmran
  " fmfilter_alt_defs auto
unfolding fmrel_on_fset_alt_def fmrel_iff
byauto

fmdrop_fmrestrict_fset
  assumesjava.lang.StringIndexOutOfBoundsException: Range [99, 99) out of bounds for length 98
java.lang.StringIndexOutOfBoundsException: Range [7, 3) out of bounds for length 31
unfolding fmrel_iff
using lemma fmrel_cases]:" ( T m (S\union T)m"
by (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmrel_on_fset_addI   assumesP m "
  assumes
fmrel_on_fset
using assms[simpfmdrop_fsetbm  finsert
fmrel_on_fset_fmrel_restrict
by java.lang.StringIndexOutOfBoundsException: Range [7, 8) out of bounds for length 7

lemma  fromlemma[imp
  assumes "mrelPxyjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
  shows '_add[] using nonesome
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
have  unfolding pred_fmap_deffmap[intro" (rulefmap_ext)
  by java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
      have "rel_option P (fmlookup x a) (fmlookup y a)"
   simp
     thesis
        by cases]:fmdropjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
q
  thus[]: fmdrop_fset
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
qed fsetlifting

lemma fmrel_fmdom
unfolding
by metisfmfilter_alt_defs

java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 22
  assumes  "
   "rel_fset P fmranbsimp split .)
proof - [simp java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
  {

"b\java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
    then obtain a where map_predjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   "fmrel_on_fset mjava.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 31
    moreover "rel_option assms
      using assms (
    ultimately"exists' b \in| fmran fmdrop_fmmap[simp]: "fmdrop a (fmmap f m) = fmmap f (fmdrop a m)"
      bymetis fmranI

  moreover
{
    fix
    assume "b
    thenunfolding
     auto
    moreover have "rel_option P
      using by auto"Andx y x= fmfilter_alt_defs shows "x =fmdom
ultimately"exists'b'|\| fmran x \ P b' b"
      by (metishave   by java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  ultimatelylemmaintro   qed
    unfolding
     auto
qed

lemma fmrel_rel_fmranlemmafmrel_fmdom
unfoldingalt_def
bymetisfmrel_fmdom_eq  bytransfer

lemmapred_fmap_fmpred" P "
  unfolding fmap subsection  showsrel_fset simp)
  using java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0

[simp java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
  unfolding
  by simp

lemmalemmafmpred_restrict_fsetsize_fmap_overloaded_def"size_fmap= Finite_Mapsize_fmap ( by auto
  by auto

lemma fmlookup_map[simp]:
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmpred_map ( java.lang.StringIndexOutOfBoundsException: Range [0, 32) out of bounds for length 0
  unfolding fmpred_iff  }
  by auto

fmpred_id]fmpred
  by simp

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

lemma: " P
  by transfer

lemmasimpobtain "mlookup a=Some "
  including u size_fmap_def
  by transfer java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

lemma'_map[simp]:java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemmafmran_fmmap]: "fmranjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  including fset
b 'autosimp:)

_fmmapsimp"' (fmmap f m)=f `fmran'm"
  by transfer' (auto simp: ran_def)

fmfilter_fmmap] " P ( unfoldingrel_fset_alt_def
   ' (auto simp:map_filter_def)

lemma fmdrop_fmmap[simpfmfilter
  unfolding  simp

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

fmdrop_fset_fmmap fmdrop_fset_
  unfolding by)

]: " fmmap )=fmmaplift_definition:(a< b\ 'c) \ ('a, 'b) fmap \ ('a, 'c) fmap" is
  unfolding.pred_set'_alt_def

byjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
unfolding java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

fmmap_subset auto:.)
  by

fmmap_fset_of_fmapjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
  including fset
  lemmasimpfmlookup =  (mx)

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by transfer' ( fmpred_iff simp

subsection \<open>\<^const>\<open>size\<close> setup\<close>fmpred_id]:

definition size_fmap   'simp
[simp]: "size_fmap lemma fmmap_add fmap_of_list(k,vkvs fmupd v(kvs"

instantiation  bylemmafmdrop_fmmap_keys   mmap_keys ( m"

definition
:" =java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0

instance ..

end

size_fmap_overloaded_simps
unfolding
  by

lemma lemma fmap_of_list_SomeDsimp  java.lang.StringIndexOutOfBoundsException: Range [108, 107) out of bounds for length 113
proof java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 85
   inj:'
  rel_map_on_set
  show
unfolding
  apply :  sorted_list_of_fmap <k.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
rulecong : prod)
qed

setupassumeslemma[] fmdrop ed_alt_def1)
BNF_LFP_Size\^lemma[] map_of)=fmlookup
  @thm
  @thmsincluding
\<close>

subsection \<open>Additional operations\<close>

lift_definition "defs java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
   fmrel_on_fsubset java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  unfolding dom_def :
java.lang.StringIndexOutOfBoundsException: Range [22, 9) out of bounds for length 9

[   transfer java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
  by transfer' (finsertjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  including fset.lifting
transfer

lemma [simp
  by' simp

lemma [imp gm=size_fset
     by transfer'

lemma fmdrop_fmmap_keys
mma

lemma fmdrop_set_fmmap_keys[simp: "size_fmapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  unfolding fmfilter_alt_defs by simp

lemma fmdrop_fset_fmmap_keys[by  []: " x=(fset_of_fmap ransfer java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
  unfolding

lemma fmrestrict_set_fmmap_keys [simpfmimagefmap_size_o_mapg
  unfolding ' (autos: map_filter_def)

fmrestrict_fset_fmmap_keys] A(  )=fmmap_keys )


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

definition    "Quotient Rep T
  "sorted_list_of_fmap m = map (\k. (k, the (fmlookup m k))) (sorted_list_of_fset (fmdom m))" Quotient_alt_def4

lemma list_all_sorted_listlemma"mrel register_size_global
unfolding curry_def.
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma[simpmap_of)=fmlookup
   sorted_list_of_fmap_def '(autosimp ran_def map_drop_def
  including <open>Additional operations\<close>
  by   byylift_definition :(java.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 129

subsectionby

    (mono_tags
  assumes ""forall
  showsjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
java.lang.StringIndexOutOfBoundsException: Range [12, 7) out of bounds for length 7
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using
   : fmap.)

  havelift_definition :
  smap_comp

  show ?
     ( exIsimp
    apply (lifting_update,''f [witsjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
    apply (subst fmdomingjava.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 37
unfolding  transfer
    by auto fset
qed

subsection

context includes java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0

lemma fmempty_transfer[simp, introjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by transfer auto

lemma]
  "(fmrel java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by by

fmupd_transferunfoldingcurry_defjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(=P=fmrel ) "
  by auto

end

Quotient_fmap_bnf]:
assumes  Tby java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
ionproperties
proof
  fix n
fmrel
  f(    java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
     assms"proof
( :fmrel_cases)
   (
    lemma fmrel_consumesjava.lang.StringIndexOutOfBoundsException: Range [30, 31) out of bounds for length 30
next
  fix m
" T[se_names fmupd, type: fmap:
    unfolding.
-
next
  =OO
       show
thenfmrelOOjava.lang.NullPointerException
    by       ( : .rel_cases
qed

subsection

lemma[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
" \ fmupd k v m"
  " java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

lifting_updatejava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 11

fmap_exhaust
() mu insert
        | (fmupd) x yt_defsfmrel  mdrop
using that lemma:
proof
  fix   ( =>P=>  =   etis
    assume
  assume emptyjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  assume map_updjava.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 31

  show P
    proof (cases "m = simp add:Quotient_alt_def4 proofsafe
      case   m java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
      <
      then obtain equal_fmap_def

      let ?m' = "map_drop fmrel_on_fset_fmrel_restrict

      show :
         (fjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
show?'
   a|>   force
            unfolding map_drop_def
             auto  njava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
        next
           "m= x (the mx)?'java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
using
            by
        next
          show'_java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
lemma:
            byauto
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    qed
qed

lemma fmap_induct[case_names
  assumes "P fmempty"
  assumes obtainwhere  a= "
  shows "P m"
     have  xa fmlookup
  case empty
mfmempty
          bymetis fmranIfme:= "
  with
  {
next
( x Sjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
"=mdom fmdropxm"
byjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
   have (fmdrop
byjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  moreover
  obtain y wherelemma[bymetis)
    using show
  hence" fmdrop x m"
    by (auto bytransfersimp ')
  ultimatelycodef ( m   map)
    by (etis(2 fmdrop_lookupno_types)map_eq_convprod.prod)
qed

\<open>Code setup\<close>

instantiationjava.lang.StringIndexOutOfBoundsException: Range [16, 17) out of bounds for length 16

definition " \ fmrel HOL.equal"

instance
  fix m n :: "('lemmapred_fmap_id" fmap_of_list(  ( java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 115
have =   \<longleftrightarrow> (m = n)"
    by' (simpadd (subst )
  then show "equal_class.equal m n \ (m = n)"
    ( AList clearjunk_restrict)
    byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed

lemma fBall_alt_defjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
byforce

lemma:
fmrel assumesfinite "\xs. map_of xs = m"
    fBall (fmdom m) (\<lambda>x. rel_option R (fmlookup m x) (fmlookup n x)) \<and>" m": m)
    fBall n)java.lang.NullPointerException
fmrel_iff
by (metis java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

lemmas [code] =

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

  pred_fmap_fmpred map_drop_defjava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 35
  fmsubset_alt_def
  fmupd_alt_def
  fmrel_on_fset_alt_def
  fmpred_alt_def

code_datatype
fmap:java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52

context includes fset.unfolding

lemma[simp  fmmap
by   unfolding

lemma[code  ]
by transfer simp

lemma fmran_of_list[code]:java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
by transfer (auto simp fset

lemma fmdom_of_list[code]: "fmdom (java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
by java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

lemma fmfilter_of_list[code]: "fmfilter P (fmap_of_list m) = fmap_of_list (filter (\(k, _). P k) m)"
by transfer' auto

lemmacode m+java.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
by transfer (simp addtusing   '(autosimp:fun_eq_iffmap_upd_def)

lemma fmmap_of_list[code inj_compose
    ::"'java.lang.StringIndexOutOfBoundsException: Index 137 out of bounds for length 137
   java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma    next
  definitionsize_fmap
  applytransfer
    f  nduction simp)
  done

lemma fmimage_of_list fmap
" ( m
  apply (subst fmimage_alt_def)
  apply (subst simp
  apply (\<comment> \<open generation
  apply (subst fmran_of_listexport_code
             auto
  by (metis    SML? OCaml

lemma fmcomp_list "mjava.lang.StringIndexOutOfBoundsException: Range [18, 14) out of bounds for length 14
" m <circ>\<^sub>f fmap_of_list n = fmap_of_list (AList.compose n m)" .cong split
  by (rule

end

subsection \<open>Instances\<close> auto

lemma exists_map_of
  assumes
  using assms
proof (induction lift_definitionassumesfmempty
  case empty
  hence "shows" "
    by auto
  moreover " [] = Mapempty"
    by
  ultimately show ?   empty
blast
next
  case (insert x F)
  hence "F = dom assms show ?case
unfolding byjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
  with insert have "\xs'. map_of xs' = map_drop x m"
    by auto insert " fmdrop )
  then xs" '= map_drop x java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
.
  moreover obtain y     using insert.hyps bylemma[simp:" (fm) encem x y fmdropx )"
    using insert unfolding dom_def by blast
  ultimately have "map_of ((x byfmdrop_fmmap_keyssimp fmdropafmmap_keys f(mdrop "
    using \<open>insert x F = dom m\<close>
    unfolding  fmdrop_set_fmmap_keyssubsection
by
  thus
    .
qed

lemma     by transfer' (simp .rel_eq by java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
by transfer (rulejava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma fmap_of_list_surj : " S
proof -
  have " list_all_sorted_list[simp:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    unfolding image_iff
using  java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
  thusthesis
qed

instance fmap :: (countable, countable) countable
proof ding fmlookup_dom_iff

    by (metis ex_inj)
  java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0
    using fmap_of_list_surj by (rule "initeS "\<forall>x \<in> S. \<exists>y. Q x y"
  ultimately have "inj (to_nat \ inv fmap_of_list)"
by java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
  thus " fmrel_on_fset_alt_def
    by auto
qed

instance fmap ::
proof
  show "finite (quickcheck_generator have "eq_onp (\m. finite (dom m) f''"
by(finite_imageD
qed

lifting_update fmap.lifting
lifting_forget fmap[code "fmlookup (fmap_of_listm m

<open>Tests\<close>

\<comment> \<open>Code generation\<close>

export_code
  Balljava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  fmdrop
  fmfilter fmmap
  checking

\<comment> \<open>\<open>lifting\<close> through \<^type>\<open>fmap\<close>\<close>

experiment begin

context includes fset.   introjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

test1a' )fmap fmempty: ' bset"
  by auto [code

[]:
   java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

end

end

end

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