Quelle Finite_Map.thy Sprache: Isabelle

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

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
theoryby: )
FSet
  abbrevs         ="A"
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

cludesbegin

ran_def
"rel_mapf\ (=) ===> rel_option f"

lemmar[transfer_rule]: (  =>rel_set an
proof
  fix map_of_transfer[ransfer_rulejava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
  assume "rel_mapthenobtaina where na=Some "
  show "rel_set A (ran m) (ran n)"
    proof (rule rel_setI)
      fix x
      assume "x \ ran m"
      then
        includes begin

      " f \ (=) ===> rel_option f"
        using \<open>rel_map A m n\<close>
        by( dest rel_funD
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding"
         cases
      thenshowusing
        unfoldingran_defjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
    ext
      fix y
      assume "unfolding \n a = _\
      then a where a= "
        unfolding        " \ ran m"

       "rel_option A (m a)(n a)java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
        using
        by( dest: rel_funD
      thenhave" A (m a) (n a)"
        unfoldingqed
        bycases
      then show
        efjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
    qed
qed

lemma ran_alt_def: "ran m = (the \ m) ` dom m"
unfolding dom_def byforce

parametric_constant dom_transfer[transfer_rule]: dom_def

definition map_upd          \<open>m a = _\<close>cases
java.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 34

parametric_constant map_upd_transfer[transfer_rule]: map_upd_def

definition         unfolding by blast
P m=\lambda  xthenelse)"

parametric_constant y

lemmamap_filter_map_of]: "map_filter P (map_ofm =map_of[k,_) m. P k]"
proof
  fix x
  show "map_filter P assume "y \<in> ran n"
       "map_filter P (map_of ) x=map_of[k _ m. P k] x"
qed

lemma         unfolding by auto
  "finite (om )"
  shows "finite (dom \rel_map A m n\
proof
  assumes( m"
    byrule) (auto split)
then showthesis       bycases auto
    by ( add:map_filter_def
qed

        unfolding
"map_drop a by(simp add: map_filter_def)

parametric_constant    qed

definitiondefinitionmap_drop "a\ ('a \ 'b) \ ('a \ 'b)" where
lter

parametric_constant map_drop_set_transfermap_dropmap_filter

definition map_drop_transfertransfer_rule
"map_restrict_setA=map_filter(a. a \ A)"

parametric_constantmap_restrict_set_transfer]: map_restrict_set_def

map_pred: "(a \ 'b \ bool) \ ('a \ 'b) \ bool" where
map_pred m\<longleftrightarrow> (\<forall>x. case m x of None \<Rightarrow> True | Some y \<Rightarrow> P x y)"=m(kjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

parametric_constant[transfer_rule

definition map_drop_set_transfer[transfer_rulemap_drop_set_def
"rel_map_on_set SP eq_onp \<>x. x \ S) ===> rel_option P"

definition definitionmap_restrict_set:"a set\ ('a \ 'b) \ ('a \ 'b)" where
"set_of_map lemmamap_filter_map_of[]:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemmadefinitiondefinition map_pred
unfoldingdom_def
by auto

lemma     ( m) (auto map_filter_def
unfolding

set_of_map_inj
proof
  fix  set_of_map:(
  assume -
  set_of_map (|v k=Some
unfolding set_of_map_alt_def m=(lambda m)  mjava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
   x k yk for k
    by (metisqed
  thus "x =y"..
qed

lemma set_of_map_finitefinite) \<Longrightarrow> finite (set_of_map m)"
unfolding dom_def
by map_drop_set

dom_comp_finitefinite n
by (metismap_drop_set  (<>.a\<notin> A)"

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

end

subsection \<open>Abstract characterisation\<close>

typedef ap y"
  morphisms fmlookup Abs_fmap
proof
parametric_constant map_restrict_set_transfer]: map_restrict_set_defunfolding by auto
    by auto
qed

setup_lifting type_definition_fmap

lemma dom_fmlookup_finitejava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
using fmap

lemma fmap_ext
assumes<And>x. fmlookup m x = fmlookup n x"
  shows "m = n"
using assms
by

subsectionm={(, )| v  Some

context
  includes
begin

lift_definition
  is
   ran_transfer  dom_def
by (rule lemma " (domm)\ finite (set_of_map m)"

lemma auto
by transferset_of_map_inj set_of_map

mma:proof

lemma[elim]java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
   "y |\| fmran m"
  obtains
using by ( hence( a=Some =(  =S ) forab

lift_definition "x k= y k"for
  is
  parametric "x = y".
.

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

fmdom_notIfmlookup= <Longrightarrow> x |\<notin>| fmdom m" by (simp add: fmlookup_dom_iff)
lemma fmdomI assms
lemma [dest]:"x \

lemma
  java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 7
  obtains  "fmlookup mbegin
using efinition :java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

lift_definition : "a ' 'aset"
  typedef
  parametricdom_transfer
.

lemmafmlookup_dom: "x \ fmdom' m \ (\a. fmlookup m x = Some a)"
byjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

lemma fmdom
lemma 'I: "fmlookup x = Some y \ x \ fmdom' m" by (simp add: fmlookup_dom'_iff)
lemma'notD[]:x\notin>fmdomm fmlookup m x = None" by (simp add: fmlookup_dom'_iff)

lemma fmdom'E[elim]:
  assumes qed
  obtainswhere mx=Some
using assms by (autousing assms(auto: fmlookup_ran_iff

fset m)"
by transferi dom

lemma finite_fmdom.
unfolding fmdomusing.fmlookup byajava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27

lemmadom_fmlookup[simp"dom(m 'm"
bytransfer

lift_definition assms
   Map.java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
bysimp

[elim
byassumes|i>|  mjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

lemma fmdom_empty[simp
lemma'emptysimp:"java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 65
lemma fmran_empty transfer' auto

lift_definition
  is map_upd
  parametric map_upd_transfer
unfolding map_upd_deftransferautoran_def
by simp

fmupd_lookup:" (fmupd abm '=( ='then else a'"
by transfer "x \ fmdom' m"

java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 104
lemma by( simp_)

fmupd_reorder_neq
umesjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
  shows  x(  y ) fmupd(fmupdm"
using assms
by transfer :map_upd_def

lemma fmupd_idemdefinitionfmdom "(,')fmap\java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
by transferauto)

lift_definitionfmfilter ('a bool) \ ('a, 'b) fmap \ ('a, 'b) fmap"
  ismap_filter
  parametric
byauto fmempty_lookup]: "fmlookup fmempty = None"

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

lemma fmdom'_filter[lemmafmdomI " m x = Some y\ x |\| fmdom m" by (simp add: fmlookup_dom_iff)
by transfersimpsplit

lemma[simp] fmlookup  ) x=( Pxthen
by transferauto)

lemma fmfilter_empty[simp
bytransfer simp

lemmafmfilter_true]:
  assumes   obtains y where   Some
  showsfmfilter ="
proof (rule fmap_ext)
  fix x
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   usingassms
  then show "fmlookup (fmfilter Pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
qed

lemma fmfilter_false[simp]:
  assumes "\x y. fmlookup m x = Some y \ \ P x"
  shows
usingassmsby transfersimp)

lemma transfer fmupd_reorder_neq
by transfer'

lemma fmfilter_comm: "fmfilter P (fmfilter Q m) = fmfilter Q (fmfilter P m)"
fmfilter_compmeson

lemma fmfilter_cong[cong]:
  assumes "\x y. fmlookup m x = Some y \ P x = Q x"
lemma'_notD[]:"\ fmdom' m \ fmlookup m x = None" by (simp add: fmlookup_dom'_iff)
proofrule
  fix x
have"mx =None "P  \<noteq> Q x"
    using java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
then : "a\ bool) \ ('a, 'b) fmap \ ('a, 'b) fmap"
    by auto map_filter_transfer
qed

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
b ransfer'simp
  shows
using(2) unfolding(1)
fmdomsimpmdom )Set fmdom

lemmaby transfer simp
  "fmfilter P (fmupd x y m) = (if P xjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
transfer simp)

lift_definition fmdrop :: "'a \ ('a, 'b) fmap \ ('a, 'b) fmap"
  is
  parametricfmdom_empty]: "fmdom = {||}"  transfer
unfolding map_drop_deftransfersimp

lemma fmdrop_lookup fmfilter_true]:
by transfer' auto simp:map_drop_deflemma'_empty: "'fmempty={" 'sjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65

lift_definition map_upd
  is map_upd_transfer
  arametric[]
unfoldingby simp    using assms fastforce

lift_definition fmdrop_fset java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
is
  parametric map_drop_set_transfer transfer
unfolding map_drop_set_def

   "\x y. fmlookup m x = Some y \ \ P x"
  isjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
  parametric
unfolding using assms by' (fastforcesimp: map_filter_def)

::"' fset\ ('a, 'b) fmap \ ('a, 'b) fmap"
  is map_restrict_setauto)
  parametric map_restrict_set_transfer
unfolding map_restrict_set_def by auto

lemmausing
  "fmdrop a bytransfer' (auto simp: map_upd_def)
  "fmdrop_set em
  " B = fmfilter (\a. a |\| B)"
  "fmrestrict_setA fmfilter(\a. a \ A)"
  "fmrestrict_fset Bjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(' simpadd java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 21

h "fmlookup m =None"
lemma    using assms byfastforce
lemma fmdom' auto
lemma lemmafmdom_filterauto
fmdom: "fmdom' (fmrestrict_set A m)java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
lemma:" (fmrestrict_fset m)||A unfolding fmfilter_alt_defs by auto

lemma fmdrop_fmupd x fmupdjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
by

lemma fmdrop_idlex \notin   assmsunfoldingjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
by' ( simp: map_drop_def )

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

lemma fmdrop_fmupd_same: "fmdrop x by ' (auto simp: map_upd_defmap_filter_def)
byt auto )

lemma fmdom'_restrict_set_precise: "fmdom' (fmrestrict_set A m) = fmdom' m \ A"
nfolding   "fmlookup m = None"if

lemmathen fmlookupPm     java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
foldingfmfilter_alt_defs auto

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  parametric   "\x y. fmlookup m x = Some y \ \ P x"

lemma fmdom assms
unfolding

lemma map_restrict_set
  "assumes"And.mlookup yjava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
unfolding by simp

lemma fmlookup_drop_setmap_restrict_set_transfer
" (fmdrop_set Am)x =( x \ A then fmlookup m x else None)"
unfolding fmfilter_alt_defs

lemmafmlookup_drop_fset a=(\<lambda>a'. a' \<noteq> a)"
fmlookup  )x=( x \<
unfolding fmfilter_alt_defs by simp

lemma[simpfmdrop_fset(<
  "fmlookup (fmrestrict_set A m) x = (if x \ A then fmlookup m x else None)"
unfolding fmfilter_alt_defs

lemma fmlookup_restrict_fset"fmrestrict_fsetjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  "fmlookup (fmrestrict_fset Alemmafmdom_dropsimp]:" (fmdrop ) =fmdom> P x = Q"
unfolding fmfilter_alt_defs by simp

lemma fmrestrict_set_dom]: "java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 39
by (rule mfilter_upd]:

  " (mupd xym) = if P x then fmupdxy(fmfilter P m) else fmfilter P m)"
by( fmap_ext) auto :" (fmrestrict_fset )\ "unfolding auto

lemma fmdrop_empty[simp]: "fmdropjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding fmfilter_alt_defs by simp

lemma fmdrop_set_emptysimp " A fmempty = fmempty"
unfolding bysimp

lemma
unfolding fmfilter_alt_defslemmafmdrop_lookup] fmlookupfmdropa=None

lemma fmdrop_fset_fmdom[simp]: "fmdrop_fset (fmdom A) A = fmempty"
byby transferautomap_drop_def

lemma fmdrop_set_fmdom[simp]:bytransferauto java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0
by transfer' (auto simp: map_drop_set_def map_filter_def)

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

lemma : "afset<>'a b \<Rightarrow> ('a, 'b) fmap"
unfolding unfolding map_drop_set_transfer

lemmafmdrop_set_null byauto
byunfoldingfmrestrict_set : 'set\',bfmap

lemma []: "fmdrop_fset{||} m=java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
by( fmap_ext autounfolding by simp

lemmafmdrop_set_single  by simp
lemmaet_def

lemma fmdrop_fset_singleunfoldingby java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
unfolding byfmdrop_fsetfmfilterjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

lemmaby transfer :map_drop_def)java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
unfolding fmfilter_alt_defs simp

lemma fmrestrict_fset_null[]: "fmrestrict_fset {|} =fmemptyjava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
unfolding by simp

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

lemmaby( map_ext
by (rule

lemma bytransfer :map_drop_defmap_upd_def
by (rulelemmafmdrop_idle ule) auto

lemma fmfilter_alt_defsby simp
unfolding fmfilter_alt_defs by auto

fmrestrict_fset_twice
unfolding by auto

lemma
unfolding by auto

lemmalemma[simp'_restrict_set_precise:fmdom fmrestrict_setm ' \ A"
unfolding fmfilter_alt_defs by auto

lemma by auto
by( fmap_ext fmdomdrop_fset:"'(fmdrop_fsetAm='m-fsetAjava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76

lemma fmdrop_fmrestrict_fset]: "fmdrop b (fmrestrict_fset S java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by unfolding bylemma[]: "fmrestrict_setA fmempty fmemptyjava.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70

lemmalemmafmrestrict_fset_empty]: "fmrestrict_fset Afmempty=fmempty"
unfolding fmfilter_alt_defs by auto fmfilter_alt_defs java.lang.StringIndexOutOfBoundsException: Range [35, 36) out of bounds for length 35

lemmafmdrop_set_twice:" fmdrop_setT)=fmdrop_set(S\T)m"
unfolding fmfilter_alt_defs by auto

fmdrop_fset_twice]: "fmdrop_fset S(fmdrop_fsetTmby(rulemap_ext) java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
unfoldinglemma fmlookup_drop_fsetby( fmap_ext

lemma fmdrop_set_fmdrop[simp]: "fmdrop_set S (fmdrop b m) = fmdrop_set (insert b S) m"
bylemma fmdrop_set_singlesimp"lemma fmdrop_set_single[simp]: "fmdrop_set

  fmlookupAm  =( x \<in> A then fmlookup m x else None)"
by (rule) auto

lift_definitionfmadd:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  is map_add
  parametric map_add_transfer
  by simp

lemmaunfoldinglemmafmrestrict_set_null:"mrestrict_set{ "
  fmlookup+<^sub>f n) x = (if x |\<in>| fmdom n then fmlookup n x else fmlookup m x)"
y (utomap_add_def.plits

lemma fmdom_add by simp
lemma_add:fmdom +<> )= union n bytransfer

lemma fmadd_drop_left_dom "fmdrop_fset
  by (rule fmap_extjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

lemma fmadd_restrict_right_domfmdrop_set_insert:"fmdrop_set( x S)m =fmdropx (fmdrop_setSm)java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
  by (rule fmap_ext) auto

lemma fmfilter_add_distrib fmap_ext auto
  by

fmdrop_add_distrib fmdrop( +\<^sub>f n) = fmdrop a m ++\<^sub>f fmdrop a n"
  unfolding fmfilter_alt_defs by simp

lemmaby ' (auto simp: map_drop_set_def map_filter_def)
   fmfilter_alt_defs simp

lemma fmdrop_fset_add_distrib[simp]: "fmdrop_fset A (m ++\<^sub>f n) = fmdrop_fset A m ++\<^sub>f fmdrop_fset A n"
  unfolding fmfilter_alt_defs simp

lemma fmrestrict_set_add_distrib[simplemmafmrestrict_set_dropsimp]: lemma[]: "fmrestrict_set fmempty= fmempty"
  "fmrestrict_set A (m ++\<^sub>f n) = fmrestrict_set A m ++\<^sub>f fmrestrict_set A n"
  unfolding

lemmafmrestrict_fset_add_distribsimp
  unfolding by simp
  unfolding[simpfmdrop_set=mjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

lemma fmadd_empty[simp]: lemmafmdrop_fset_null]: "fmdrop_fset || m = m"
  by

lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by transfermap_add_def.)

lemma fmadd_assoc[simp]: "m ++\<^sub>f (n ++\<^sub>f p) = m ++\<^sub>f n ++\<^sub>f p"
  by transfer' simp

lemma fmadd_fmupd[simp]: "m ++\<^sub>f fmupd a b n = fmupd a b (m ++\<^sub>f n)"
  foldingfmfilter_alt_defsauto

fmpred::"(a \ 'b \ bool) \ ('a, 'b) fmap \ bool"
  is map_pred
  parametric
  .

lemma fmpredI[intro]:
  assumes "\x y. fmlookup m x = Some y \ P x y"
  showsPmjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
  using assms
  by transfer'emma fmdrop_comm: "

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by( :map_pred_def

lemmalift_definitionfmadd simp finsert   fmdrop_fset
  byauto

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

lemma:
fmfilter_alt_defs
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
using assms unfolding

fmpred_mono] P\<le> Q \<Longrightarrow> fmpred P \<le> fmpred Q"


lemma fmpred_empty[intro!, simp
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

fmpred_updDsimp  ) ( <union> T) m"
  by auto

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by' auto by auto

lemma fmpred_filtersimp
transfer : )

lemma []: "fmdrop_fset S (mdropbm= (insertb S mjava.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90
r_alt_defs

java.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
   unfolding  map_add

lemma fmpred_drop_fsetsimp fmadd_empty"fmempty ++<^sub>f m = m" "m ++\<^sub>f fmempty = m"
  by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmpred_restrict_set transfer simpsplit)
  by

lemma fmpred_restrict_fset]: " fmdom_addsimp] fmdom (m +<^sub>f n) = fmdom m |\| fmdom n" by transfer' auto
  by (auto simp: fmfilter_alt_defs)

lemma fmpred_cases[   ( fmap_ext)simp
  assumes
  obtains) "fmlookup m java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
usingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lift_definition :: "(a ') ( ))
  is
lemma[simp" m +<^sub>f n) = fmfilter P m ++\<^sub>f fmfilter P n"

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
bytransfer :map_pred_def :optionfmpredI

  unfolding by simp
unfolding fmsubset_alt_def fmdrop_set_add_distrib[
byauto

lemma fmsubset_filter_mono: "m \\<^sub>f n \ fmfilter P m \\<^sub>f fmfilter P n"
unfolding fmsubset_alt_def fmpred_iff
by auto

lemma  unfoldingfmfilter_alt_defslemmafmpredD]:"fmpred m fmlookup m x = Some y \ P x y"
unfolding by (rule)

lemma  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding rulejava.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58

lemma fmsubset_drop_fset_mono: "m \\<^sub>f n \ fmdrop_fset A m \\<^sub>f fmdrop_fset A n"
unfolding by (fmsubset_filter_mono

: "m \\<^sub>f n \ fmrestrict_set A m \\<^sub>f fmrestrict_set A n"
unfoldingshowsPjava.lang.NullPointerException

lemma fmsubset_restrict_fset_monoby transfer' ( simp: map_add_def split: option.splits)
r_alt_defslemma[mono" le \ fmpred P \ fmpred Q"

lemmafmfilter_subset]: fmfilter \<subseteq>\<^sub>f m"
unfolding

lemmasimp am \<subseteq>\<^sub>f m"
unfolding rule) simp

lemma fmpred_upd fmpredjava.lang.StringIndexOutOfBoundsException: Range [102, 53) out of bounds for length 102
unfolding fmfilter_alt_defsparametricmap_pred_transfer

lemma fmsubset_drop_fset
unfolding fmfilter_alt_defs by (rule fmfilter_subset fmpredI]:

lemmafmsubset_restrict_set[] fmrestrict_set \<subseteq>\<^sub>f m"
unfolding by  shows

fmsubset_restrict_fset assms
unfolding fmfilter_alt_defs (rule '( :map_pred_defmap_filter_def)

lift_definition fset_of_fmap [dest" m \ fmlookup m x = Some y \ P x y"
   ( )

lemma fset_of_fmap_inj fmpred_drop_set[intro" P m \ fmpred P (fmdrop_set A m)"
applyjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  applyjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
  using fmpred_alt_def  m\<

lemma fset_of_fmap_iffsimp]:"(a, b \java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
by transfersimp

lemmafset_of_fmap_iff(a b) \<in> fset (fset_of_fmap m) \<longleftrightarrow> fmlookup m a = Some b":
  by simp

lift_definition :: "'fmpredPm\ fmpred Q m"
  s ap_of
  parametric map_of_transferlemma[consumes1]:
by( lemma fmpred_mono:" Q \ fmpred P \ fmpred Q"

lemma fmap_of_list_simps[simp]:
  " [] = fmemptyjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
  "fmap_of_list ((k, v) # kvs) = fmupd k v (fmap_of_list kvs)java.lang.StringIndexOutOfBoundsException: Range [61, 62) out of bounds for length 0
  by (transfer, simp add:   transfer simpmap_upd_def

lemma fmap_of_list_app[simp
lemma

lemmaf: "fmupd k v m= ++\<^sub>f fmap_of_list [(k, v)]"
  by simp

lemma fmpred_of_list]:
assumesjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
  shows "lemmabyauto
java.lang.StringIndexOutOfBoundsException: Range [20, 13) out of bounds for length 13
  by (inductionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

fmap_of_list_SomeD"mlookup(fmap_of_list )k= (k, v) \ set xs"
  by transfer' (auto dest: map_of_SomeD)

lemma[simp fmfilter_alt_defs fmsubset_filter_mono)
  by  by auto simpfmfilter_alt_defs

lift_definitionfmrel_on_fset:a \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('a, 'b) fmap \<Rightarrow> ('a, 'c) fmap \<Rightarrow> bool"]:  by (rule)
  is rel_map_on_set
.

lemma fmrel_on_fset_alt_def
bytransferimp eq_onp_def

fmrel_on_fsetI:
ssumesjava.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
  shows  Pm_defs
   (ed_cases]:

lemma fmrel_on_fset_monomono"R e" | (some) y where "fmlookup m x = Some y" "P x y"
  unfoldingfmrel_on_fset_alt_defusing auto
  using "rel_option A ( ) (n ) java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmfilter_alt_defs( fmfilter_subset
          unfoldingjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
        then"\y \ ran n. A x y"

lemma fmrel_on_fsubset ".
  unfoldinglemmafmsubset_alt_def" subseteq>\<^sub>f n \ fmpred (\k v. fmlookup n k = Some v) m"
  by      "

lemma fmrel_on_fset_unionI        unfoldingjava.lang.StringIndexOutOfBoundsException: Range [0, 18) out of bounds for length 0

  unfolding
  by auto obtain wherea=x Axy"

lemmafmrel_on_fset_updateI:
  assumes "lift_definition fset_of_fmap: ('a 'b fmap \ ('a \ 'b) fset" is set_of_map
   fmrel_on_fsetk S)  kjava.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 83
  using
  unfoldingapplyrule
  by

  : (a bfmap
  by (smt (verit" k v m = mk \ v)"

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

lemma[] fmimage'(auto simp set_of_map_def)
transfer' auto

lemmafset_of_fmap_iff(,b \<in> fset (fset_of_fmap m) \<longleftrightarrow> fmlookup m a = Some b"
  by

lemma[imp: "fmimage shows "finitemap_filter m"
       rev_finite_subset splitmap_of_transfer

:"fmimage mAjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

age_inter_dom,v#k kvs
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  "fmimagejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  byfmupd_alt_def      rel_map_on_setfmfilter_alt_defsfmsubset_filter_mono

set_of_map_alt_def <lambda>k. (k, the (m k))) ` dom m"
  by"Someunfoldingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

simp .
  by transfer'auto

(  )   (PA)
  by ' (autosimp: map_filter_def)

lemma[simp: "fmimage lemmafmsubset_restrict_setsimp: fmrestrict_set
  by( add

lemma
  by     java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

lemma
   map_restrict_set_def

lemma] le> Q \<Longrightarrow> fmrel_on_fset S R \<le> fmrel_on_fset S Q"
   transferfset_of_fmap_iff

lemma fmran_drop' auto: ran_def)
transfer simp java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63

an  )fmdom)java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
transfersimp

lemma fmran_restrict_fset: "fmran (fmrestrict_fset A java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
parametric

fmlookup_image_iffinA
  by transfer' lemmafmdomEelim]:

lemma
  by   emma:

elim
  assumes fmdom (a,b  (k v  =  k ()"
  obtains x where " parametric dom_transfer
  using  fmrel_on_fsetm \<Longrightarrow> fmrel_on_fset B R m n \<Longrightarrow> fmrel_on_fset (A |\<union>| B) R m n"

lemma fmdom java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
  is"\in> fmdom mjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
  parametric assmsshows finsert)P fmupd<
  by (rulelemma fmdom_alt_deft_def

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

(,fmranwits
  for
      rel:  transfer
  byjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

pred_mono

lemmaalt_def= fmran
   fset.lifting : map_upd_def
  by by

lemma[]:
  byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmran
  by (auto '_iff)

lemma
  assumes  unfolding [
  obtainsjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
by :'iff

lemma lemma fmimage_Union]  ': :if_splits
bytransfer :java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

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

  byhave  java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 33

" \ P x y \ fmrel P (fmupd k x m) (fmupd k y n)"
   '

destjava.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
  byunfolding fmfilter_comp

lemma fmrel_addI[intro " P m = fmfilter Q m"
lemma
"fmrel P (m++\<^sub>f a) (n ++\<^sub>f b)"
fmdom

   'auto map_restrict_set_def
assumespPm  =fmfilter]" m fempty "
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
         () a b "fmlookup_image_iff"java.lang.StringIndexOutOfBoundsException: Index 126 out of bounds for length 126
-
lemma[simpfmimageIxSome>|in
P mupd
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
by

  m  java.lang.NullPointerException
  assms)

lemma fmrel_drop
fmfilter_alt_defs dom_comp_finite

fmrel_drop_set mjava.lang.NullPointerException
unfolding   java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34

  parametric
unfolding

       fmrel
unfolding   [simp by

unfolding fmfilter_alt_defs   map_restrict_set  bylemma   map_restrict_set_transfer

' auto simp: ran_def map_filter_def)
  fmlookup_ran
unfolding fmrel_on_fset_alt_defb = (lambda
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

lemma
java.lang.NullPointerException
   fmdom_rop_setassumes
unfolding fmrel_iff
using java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  add)

lemma

shows <^sub>f a) (n ++\<^sub>f b)"
using
unfolding
   (simp

lemma fmrel_fmdom_eq fmimageI fmlookup =  \<Longrightarrow> x |\<in>| A \<Longrightarrow> y |\<in>| fmimage m A"lemma ': "x java.lang.StringIndexOutOfBoundsException: Range [0, 59) out of bounds for length 36
" P x"
  shows' autosimp map_filter_def map_upd_def)
proof -
  havea|
proof
lemma '_restrict_fset_precise: "fmdom ( A m m |<>|Ajava.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
  byfmrel
      lemma'restrict_fset: "' (fmrestrict_fset <subseteq> fset A"
   )
    qed
thesis[]  t' auto simp map_comp_def split option.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by
qed

(A)( <>|  fmlookup
'_
)

lemma
  assumes
    by
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

      by
    then     none
      by  java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
( xa  java.lang.StringIndexOutOfBoundsException: Range [63, 62) out of bounds for length 62
by
     have
java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

    then
      by
     have "
using
     fmfilter_alt_defs
      by fmrestrict_fset_null]: fmrestrict_fset m=fmempty
  }
ultimately "fmrel Rmnjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

fmran(yjava.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
unfolding

assumes
  smt   fmdom fmrel_ifffmrel_on_fset
using java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

[]: " id fmmapfm \ pred_fmap f m"
  unfolding


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

lemma fmlookup_map
"( +

java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
fmpred_iff fmapfmrel_filterby)
auto

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

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

lemma fmmap_empty[simp]: "fmmap java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by auto

lemma fmdom_map[simp[simp
  including.
  by transferjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

[imp " "
  including fsetshowsP(   ' (autosimp split:optionsplitsjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
by'( simp:ran_defjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

lemma'_fmmap[simp]: "fmran ( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by' java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma[simpP java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
  by transfer   "fmpred java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

  unfoldingbymetisfmranI

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


java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
  unfolding     auto

lemma assms
unfoldingshows =yjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27

lemma   have java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
  unfolding simp

java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
     auto

lemma auto
  including [] qed
  bytransfer

lemmajava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
   ' (autojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

\<open>\<^const>\<open>size\<close> setup\<close>auto

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

fmap

definition
:  .      java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13

instance

end

]  java.lang.StringIndexOutOfBoundsException: Range [70, 68) out of bounds for length 72
  unfolding fmpred_id" (
    {

fmap_size_o_map


    using auto
  [:    obtainmlookupb
size_fmap_def
auto
rule) ( split.splits
qed have P(  )(  a"

setup \<open>
[simp
  @{thm size_fmap_overloaded_def} @{  byjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  @{thmsby
\<close>

  ' by
[]: ]:   java.lang.StringIndexOutOfBoundsException: Range [11, 2) out of bounds for length 26
Rightarrow
  "\f m a. map_option (f a) (m a)"
  unfolding[simp
   java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   '( simp: map_pred_def :optionsplitsjava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

lemma: " '
  including
  by transfer' auto

lemma fmlookup_fmmap_keys[   simp finite_dom_map_of
  bytransfer

lemma fmfilter_fmmap_keys[simp]: "fmfilter P (java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 0
  by transfer' by (transfer, simpadd

lemma[simp " a(mmap_keys f m fmmap_keysf (mdropa )java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
ter_alt_defs

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

including
  unfolding'

s size_fmap_overloaded_def
ingbysimp

lemma[:" A(fmmap_keys ) fmmap_keys f(fmrestrict_fsetAm)java.lang.StringIndexOutOfBoundsException: Index 113 out of bounds for length 113
unfolding simp

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

definition  unfolding size_fmap_def
"m =map\lambda>k.kjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma list_all_sorted_list  by( sum.) (autosplit.splits
unfolding sorted_list_of_fmap_def curry_def fmrel_on_fsetI
)

map_of_sorted_listsorted_list_of_fmap m"
  unfolding sorted_list_of_fmap_def@{ size_fmap_overloaded_def
  fset
  by transfer   unfolding fmrel_on_fset_alt_def

subsection :_byjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

lemma fmchoice
  lemma:
  shows
proofbyjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
  obtain f where " S Pmn P\<^sub>1 v\<^sub>2"
    using assms by metis
  define

  havetransfer:java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    unfolding eq_onp_def f'_def

show
    apply (rule exIfmimage_empty] "fmimagemfempty=fempty"
    apply java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
    apply (subst fmdom'. bytransfer fmap:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfoldingdom_def
    by
qed

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

context includes   fmimagejava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 3
  by  by simp  sizet'auto

lemma fmadd_transfersimp fmfilter_alt_defs
  "(fmrel P ===> fmrel Pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (intro -

lemma fmupd_transfer[ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  "((=) using by force
  by autofmimage_drop_fset

end

assumesRAbs"
(fmrelR)(fmmap) ( Rep) fmrel
unfolding proof safe
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
umeTBNF_LFP_Size
b transfer' (uto simp: ran_def map_filter_def)
    usingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      by (cases
  then showunfolding\<close>  bytransfer  java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 0
bjava.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 129
next
  fix
  show "fmrel T (fmmapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    unfolding fmap
    bymetisjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
next
  from assms have "R java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    unfolding java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 19
   x where" m assms by
   by(add.rel_compprel_conversep
qed

subsection \<open>View as datatype\<close> map_comp_transfer

lemma fmap_distinct[simpapply(ulebytransfer
fmempty
' ( simp: map_comp_defjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (transfer'; by transfer (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lifting_update (' ' b map Map]

lemmafor java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
  obtains (fmempty) "m = fmempty"
        |lemmafmdrop_fset_fmmap_keys]: "fmdrop_fsetjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using that java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
transfer
  fix m P
  assume "finite (dom m)"
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumemap_updfmrestrict_fset_fmmap_keys

  show P
cases java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
      case True thus ?thesis [intro
    next
      case False
      hence "dom m \ {}" by simp
      henfmap

      let ?m' = java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

      show   by( (verit best comp_deffmpred_alt_def(1 ((=  =  P=>)mupd
        proof (rule map_upd "fmrelRmn"
          showend
            using \<open>finite (dom m)\<close>
            unfolding map_drop_def
            byauto
        next
          show =map_upd(x)    open>Additional properties\<close>\
            using \<open>x \<in> dom m\<close> unfolding map_drop_def map_filter_def map_upd_def
            by auto
        java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
          show    using unfolding "java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 7
            unfoldingmap_filter_def
java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of bounds for length 30
        qed  assumes "fmrel P m n"
    qed
qed

ctfmemptyinduct
assumes " "
  assumes "(\x y m. P m \ fmlookup m x = None \ P (fmupd x y m))"
  shows "P( exI[where x "Abs_fmap") omassms" <inverse>\<inverse>"
proof (induction "fmdom m" arbitrary     (  then " R= fmrel T OO (fmrel T)inverse>\"
  case
  hence "m = fmempty"
    by (metis \<open>View as datatype\<close>
  with show fmap_distinct

next
  casesubsection  fmempty
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    byauto
  with [imptransfer_rule  fmempty
    by autotransfer fmfilter_alt_defs
  moreover
  obtainyw "fmlookup lemma [casestype
using. by force
=fmupd(  )java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
    by (auto intro: fmap_ext)
  ultimately show ?case
  y( (2 )
qed

subsection \<open>Code setup\<close>

   " QuotientRAbsRepTjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32

definitionequal_fmap

instance java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
  fix m n ::  assume "fmr T m n"
  have "fmrel (=) m n
by' ( add: option.rel_eq rel_fun_eq) caseFalse
  then   fmrel_on_fset  " hencedom {}" by simp
    unfolding
      then "fmmap Absm java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed

end

lemmashows" yjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
by

:
  "fmrel m n
fBall )(<>x.rel_option
    fBall   assmsjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
unfolding     unfolding Quotient_alt_def4
metis option

lemmas [code] =
  fmrel_code
  fmranalt_def
  fmdom_
fmfilter_alt_defs
  pred_fmap_fmpred
  fmsubset_alt_def fmap_distinct
  fmupd_alt_deffmempty
  fmrel_on_fset_alt_def
  fmpred_alt_def

code_datatype fmap_of_list
quickcheck_generator fmap constructors: fmap_of_listjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

contextincludes fset begin

lemma fmlookup_of_list[code]: "fmlookup (fmap_of_list m) = map_of m"
by transfer  "m= fmempty"

mpty_of_list]:"fmempty =fmap_of_list]java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
by transfer simp

lemma fmran_of_list[code]   empty    fix caseinsert)
bytransfer simp: ran_map_of

bjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
by transfer True

fmfilter_of_list metis fmranI
by}

lemma java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
by transfer ( add:merge_conv

fmmap_of_list fmmap(fmap_of_listfmap_of_list )m)
  apply transfer
  by metis lifting apsnd_convmap_of_map oldcase old.exhaust

lemma fmmap_keys_of_list[code]:
  "fmmap_keys f (fmap_of_list m) = fmap_of_listsubsectionjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
  apply transfer
  subgoal for f m by (induction m) (auto simp: apsnd_def map_prod_def fun_eq_iff)
  done

lemma fmimage_of_listusing proof
  fmimage m)A= fset_of_listsnd(<lambda>(k, _). k |\<in>| A) (AList.clearjunk m)))"
  apply( fmimage_alt_def
  apply (subst fmfilter_alt_defs)
  apply (subst fmfilter_of_list)
subst
  apply transfer'
metis.restrict_eq list.set_map

lemma fmcomp_list[code]:
  "fmap_of_list m \\<^sub>f fmap_of_list n = fmap_of_list (AList.compose n m)"
  by (rule fmap_ext) (simpunfolding

end

subsection force

lemma exists_map_oflemma fmrel_code
  assumes " (dom m)" shows
using assms
proofinduction arbitrary
  case empty (fmdom)(<lambda>x. rel_option R (fmlookup m x) (fmlookup n x))"
  hence " hence "mfmrel_iff fmlookup_dom_iff fBall_alt_def
    byauto
  moreovermap"'( )= fmdom "
    by simp
  ultimately [simp fmranm =f||
    byblast
next
  case (insert x F)
    fmfilter_alt_defs
    unfolding map_filter_def dom_def' ( simp: ran_def)
insert have"
    by auto
  then obtain xs'java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    ..
  moreover obtain y where "m x
    using insert unfolding dom_def byquickcheck_generator constructors fmap_of_list
  ultimately have "map_of ((x, y) # xs') = m"
    using
     map_drop_defmap_filter_def
    by auto
  thus ?case
    .
qed

lemma exists_fmap_of_list[imp  fmmap fmempty_of_list]: "fmempty =fmap_of_list ["
by transfer exists_map_of

lemma fmap_of_list_surj, intro]: "surj fmap_of_list"
proof -
  have "x \ range fmap_of_list" for x :: "('a, 'b) fmap"
    unfolding image_iff
    using   including.lifting
  thus ?thesis by auto
qed

instance
of
  obtain to_nat :: "('a \ 'b) list \ nat" where "inj to_nat"
    by (metis ex_injlemmafmmap_fmupdfmadd_of_list:"fmap_of_list m +<^sub>f fmap_of_list n = fmap_of_list (AList.merge m n)"
  moreover have "inj (inv fmap_of_list)"
    fmap_of_list_surjbytransferauto map_upd_def
  ultimately have "inj (to_nat \ inv fmap_of_list)"
    by (rule)
  thus "\to_nat::('a, 'b) fmap \ nat. inj to_nat"  finition:: (' Rightarrow> nat) \ ('b \ nat) \ ('a, 'b) fmap \ nat" where
    by auto
qed

instance
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
   " (UNIV :: ('a,'b) fmap set"
    by (rule finite_imageD:subgoal b( m (auto apsnd_def map_prod_deffun_eq_iff
qed

lifting_update
lifting_forget fmap  fmimagefmap_of_list

subsection \<open>Tests\<close>

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

export_code
  Ball fset fmrel fmap_size_o_mapauto
  fmdrop fmdrop_set fmdrop_fset fmrestrict_set -
  fmfilter fmmapfmmap_keys fmcomp
checking Scala HaskellOCaml?

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

experiment begin

context includes fset  fmap_of_list\<

lift_definition
  by auto

lift_definition            unfolding
  by auto

end

end

end

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