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products/sources/formale Sprachen/Isabelle/HOL/Analysis/ Datei vom 16.11.2025 mit Größe 198 kB

Quelle Finite_Map.thy Sprache: Isabelle

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

n \<open>Type of finite maps defined as a subtype of maps\<close>  AList

java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
         \<open>rel_map A m n\<close> auto     Conditional_Parametricity  abbrevs "then wheren Somey "  java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 48

lifting_syntax

abbreviation by  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
parametric_constant[]    java.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 8

lifting_syntax

abbreviation
rel_map

lemmausing
proof
         ran_def java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
Amn
by auto           \<open>rel_map A m n\<close>
    proof       obtain n
      fix    "n y
assumex
       obtain  "m a x"
        unfolding ran_def by unfolding ran_def

       rel_option
        using \<open>rel_map A m n\<close>unfolding
        by ran_alt_defbyjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
      then obtain ran_def  java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
unfolding
        auto
      then :: "' definition map_filter :: "('a \ bool) \ ('a \ 'b) \ ('a \ 'b)" where
ran_def
    nextm   (x P     None
      java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
assume
      then java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds for length 5
ran_def

     induct simp(m"
using
        by
      thenfinite
        unfolding ( rev_finite_subset: if_splits
         cases
      then "\x \ ran m. A x y"
java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
qed

lemma ran_alt_def: "ran (\a. a \ A)"
unfolding [transfer_rule

definition :'

definition" P
"map_upd k v m ( java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 0

parametric_constantparametric_constant[]: map_drop_set_def

definition =(lambda
"map_filter java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

map_filter_transfer

simp map_restrict_set_transfer]: map_restrict_set_def
proof

  show "map_filter set_of_map_def dom_def
bym ( simp
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

map_filter_finite]:
"rel_map_on_set S P eq_onp
   lemma:" java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
proof
" m ={k ) . v}
        unfolding:" (k k ( ))dom m"
  then show ?thesis
    byhence   for
qed

definition map_drop  .
"map_drop a = java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 3

parametric_constant map_drop_transfer map_comp_def

definitionjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
"map_drop_set A =map_filter\lambdaa \ A)"

parametric_constant

definitionmap_restrict_set"aset
"map_restrict_set x = set_of_mapyjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

parametric_constant[transfer_rule   java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
"java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

parametric_constant

definition   "
"rel_map_on_set S P = java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 15

definitionlemma
et_of_mapk.k= }"

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

:finite
unfolding set_of_map_alt_def fmlookup
by

lemma : "inj "
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  fixy
  assumejava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
hence  Someya ome  java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
    unfolding
  hence     k
    by (metis dom
  thus .
qed

lemma dom_compbytransfer
lemma " None x |\| fmdom m" by (simp add: fmlookup_dom_iff)
by (autofmdomI

lemmafmdom_notD |java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 0
by assumes

parametric_constant map_comp_transfer[transfer_rule ywhere

end ran

  ':' )\' "

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  morphisms
proof
  showjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumes|in:<  java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 117
qedobtains "xywhere "fmlookupjava.lang.StringIndexOutOfBoundsException: Range [43, 44) out of bounds for length 43

lemma fmdom'_alt_def: "fmdom' m =

lemma java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
fmap uto

lemma bytransfer
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shows " ' simp
using
isempty

subsection simp

lemma]:
  includes|nfmdom
java.lang.StringIndexOutOfBoundsException: Range [9, 5) out of bounds for length 5

lift_definition dom
  is
  parametric
lemma'_iff: fmdom'[ fmdom = {   java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65

isjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
by ' auto : ran_def)

lemmasimp   a a    belsejava.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98

lemmalemma fmdom_fmupd[simp]: "fmdom (fmupd a b mlemma assmsby auto: fmlookup_dom'_)java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
  unfolding'_alt_def byjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
  obtains
using ' (autosimp )

fmdoma ) java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
l fmemptysimp
  parametric  ::"'a\java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 109

java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
by transfer

lemma fmdom_notI: "fmlookup m
:fmlookup
lemma fmdom_notD[lemma auto java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

lemma fmdomE[elimby' ( simp: map_filter_defjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
  assumes[simp
"fmlookupm y"
using map_upd_def P m java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26

lift_definition' ::"',')using that byfjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

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

lemma by'fastforce:map_filter_def)
by' lemmafmupd_reorder_neq:

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemma fmdom  assumes byjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
fmdom"

lemma fmdom )
  assumes
     fmlookupif
using

lemmafmdomfmdom  (fmdom
by transfer' force

lemma finite_fmdom   lift_definition : "java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 109
unfolding

lemma dom_fmlookup[simp]: "dom (fmlookup m) = fmdom' m"
bytransfer

lift_definition assms assms
  islemma_[] "mdom fmfilter ) . (')
byjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

mma[simp" x=bytransfer' (auto: map_filter_defjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
by transfer

lemma [simpfmemptyby' simp
fmdom[]: fmdomby imp
lemma fmran_empty[   "fmfilter Pm=m

  is
  parametric
map_upd_def
bysimp thatjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

fmupd_lookup]:" (fmupd abm) ismap_drop_set
by' (java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma fmdom_fmupdmap_restrict_set
lemma fmdom'_fmupd[simp]: "fmdom'   map_restrict_set_transfer

lemma fmupd_reorder_neq lift_definition fmrestrict_fset
  by' ( simp:map_filter_defjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
  shows
assms
  auto

java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
by transfer' fmrestrict_set = \a. a \ A)"

lift_definition fmfilter :: "('aby transfer; :map_drop_defjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 21

  parametric thatjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
by

[by
by transfer'lemma'_restrict_setfmrestrict_set

lemmafmdom )subseteq  java.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108
java.lang.StringIndexOutOfBoundsException: Range [36, 11) out of bounds for length 57

lemma fmlookup_filter: "|<>|fmdomusingassms(2) assms1)
by ' (auto simp:map_filter_def

by transfermap_filter_def
by

lemma   "fmfilter x y ' (auto : map_drop_def map_filter_def)
assumes(simp
  shows fmdrop"' '( : map_drop_defmap_filter_defjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
proof(ule )
  fix x
have  None
    using that
   showfmfilterx
    by simp
java.lang.StringIndexOutOfBoundsException: Range [6, 3) out of bounds for length 3

lemma fmfilter_false map_drop_set
assumes
unfolding by auto
using by transfer' (fastforce simp: map_filter_def)

lemma fmfilter_comp[simp]: "fmfilter ismap_drop_set
by transfer' (auto simp: map_filter_def)

lemmafmfilter_comm P (Qm   (  "
unfolding

lemma   is
   \Andx   mx=  <Longrightarrow> P x = Q x"
  shows "fmfilter P m = fmfilter java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
  fix java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  have fmlookup_drop_fsetfmfilter
    using that= \" (fmdrop_fset |java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
  then fmlookup_restrict_setB= lambda>a. a |\<notin>| B)"
    by auto
qed" A = fmfilter

lemma fmfilter_congby('; simp add map_drop_def map_drop_set_def map_restrict_set_def)+
htarrow "
  shows "fmfilter P m = fmfilter Q n"
using assms(2) unfolding
java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 39

lemmaf[simp [simp
fmfilter   )=    fmfilter fmfilter
by transfer ruleauto:fmdom subseteq byjava.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108

lift_definition fmdrop
  is[]: fmdrop_set
  lemma fmdrop_idlefmfilter_alt_defs
unfolding

[:" (fmdrop am java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
by' (auto simp: map_filter_defjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lift_definition fmdrop_set :: "'a set \ ('a, 'b) fmap \ ('a, 'b) fmap"
map_drop_set
  parametricjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding map_drop_set_def by auto

lift_definition : ' Rightarrow>'a'
  smap_drop_set
java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
map_drop_set_def

ition :" \) \<Rightarrow> ('a, 'b) fmap"
lemma[simpsimp  m
  parametric" (fmdrop am) ifx\ a then fmlookup m x else None)"
unfolding map_restrict_set_def by auto rule)

  is map_restrict_set(  )  if
  parametric [unfoldingbysimp
by auto

lemma simp
  " fmlookup_restrict_set[]:
  "fmdrop_set fmlookupfmrestrict_setAm)x=(if x \ A then fmlookup m x else None)"
  "B= (a. a |\| B)"
  "fmrestrict_set A = fmfilter (\a. a \ A)"
  "fmrestrict_fset
by';simpadd map_drop_set_defmap_restrict_set_def+

lemma fmdom_drop bysimp
lemmafmrestrict_fset_nullsimp|m "
fmdom[] fmdom A mfmfilter_alt_defs
lemma fmdom_drop_fset
lemma fmdomfmrestrict_set_domsimpbyrule
lemma fmdom_restrict_fset  ) auto

lemma
' (autosimp map_drop_def map_filter_def )

: by(fmap_ext
by transferjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmfilter_alt_defs
by transfer

lemma fmfilter_alt_defs
by transfer

lemma_:A )fmdom
unfolding fmfilter_alt_defsfmfilter_alt_defs

lemma fmdom
by

lemma'_drop_fset[simp] fmdom ) fset "
unfolding fmfilter_alt_defs by transfer' (auto simp: map_filter_def split: if_splits)

lemma' (auto simp: map_filter_def)
fmfilter_alt_defssimp="

lemma fmlookup_drop]:
  "fmlookup (fmdrop a m) java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding bysimp

lemma fmlookup_drop_set
" (fmdrop_set A fmdrop_set_twice[simp] fmdrop_set S( m) S
unfolding fmfilter_alt_defs lemma[simp     )auto

lemma[simp) auto
  "fmlookup (fmdrop_fsetjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding fmfilter_alt_defs by simp

lemma fmlookup_restrict_set[simp]:
" (fmrestrict_set )x ifx
unfolding fmfilter_alt_defs by unfolding by simp fmap_ext

lemma fmlookup_restrict_fset[simp]:
  "fmlookup (fmrestrict_fset A m) java.lang.StringIndexOutOfBoundsException: Range [0, 35) out of bounds for length 29
fmfilter_alt_defs[] " }m =fmempty

lemma[" ( +java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
  '( simp split .plits)

lemma [simp
by (rule fmap_extfmdom:' (+\subfn fmdom'm\ ' " ' auto

lemma fmdrop_emptyrule) auto
unfolding fmfilter_alt_defs by simp

lemma:lemmasimp insertx(  "
unfolding by simp

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

lemmasimpa (+
by transfer' (auto simp: map_drop_set_def map_filter_def)

lemma
bytransferjava.lang.StringIndexOutOfBoundsException: Index 114 out of bounds for length 114

fmrestrict_set_empty java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
unfolding

fmrestrict_fset_empty]:lemma fmrestrict_fset_empty[simp]: [simp
fmfilter_alt_defs

p_set_null }m="
by (rule fmap_ext) auto

[simp{}m =java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
by ( fmap_ext

simp {a} m=fmdrop auto:.java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
unfolding byrule)auto

lemmajava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0
unfolding fmfilter_alt_defs by simp by java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

lemma fmrestrict_set_null[simpmap_pred_transfer
unfolding fmfilter_alt_defs by simp

lemma fmrestrict_fset_null[simpfmdrop_set_fmdrop]   " "
unfoldingfmfilter_alt_defs

emma
unfolding [simp

lemma [simp]: fmdrop_set utosplit
by (rule

fmdrop_fset_insertfmdrop_fset) x
by (rule fmap_ext) auto java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

lemma fmrestrict_set_twice[simp]: "fmrestrict_set S (fmrestrict_set T m) = fmrestrict_set (S \ T) m"
unfolding fmfilter_alt_defs by auto

lemma fmrestrict_fset_twice fmpred_mono_strong
unfolding by auto

lemma fmrestrict_set_drop[lemma'_add[simp:" m+\<^sub>f n) = fmdom' m \ fmdom' n" by transfer' auto
unfolding fmfilter_alt_defs lemma: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

fmrestrict_fset_drop]  auto
unfolding java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

lemma fmdrop_fmrestrict_fsetsimp fmdrop java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by (rule fmap_ext) auto

lemma
unfolding by auto

lemmafmdrop_set_twice" S (T m (S\
unfolding fmfilter_alt_defs by auto

fmdrop_fset_twice]:"mdrop_fsetS ( T m) ( \union|T)mjava.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
lt_defs

lemma[]:
(   bytransfermap_pred_def

lemmasimpS(     (  )mjava.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90
rule)auto)

java.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
is
  parametric
  by simpsimpjava.lang.NullPointerException

lemma fmlookup_add[
  "fmlookup (m ++\<^sub>f n) x = (if x |\| fmdom n then fmlookup n x else fmlookup m x)"
  by transfer' (auto simp: map_add_def split: option.splits)

:( java.lang.StringIndexOutOfBoundsException: Range [94, 95) out of bounds for length 94
lemma '_add[simp:"' java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmadd_drop_left_dom (nonemx
  by ( fmpred"'

lemma fmadd_restrict_right_dom: "fmrestrict_fset (fmdom n) (m ++\<^sub>f n) = n"
by(auto

emma] "P(+<^sub>f n) = fmfilter P m ++\<^sub>f fmfilter P n"
  by

lemma fmdrop_add_distrib[simp ' (autosimp map_le_def dom_defsplit .splits)lemma intro]:
   fmfilter_alt_defs

lemmafmdrop_set_add_distribsimp
   fmfilter_alt_defs

java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
fmfilter_alt_defsdest

lemmafmrestrict_set_add_distribfmfilter_alt_defsfmsubset_filter_mono
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  unfolding  (java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58

lemma
  lemma: " Pfmfilter_alt_defsby( )
fmfilter_alt_defs  java.lang.StringIndexOutOfBoundsException: Range [37, 38) out of bounds for length 37

lemma fmadd_empty
  lemmafmpred_mono_strong
   "\x y. fmlookup m x = Some y \ P x y \ Q x y"
lemma fmadd_idempotent[simp]: "m ++\<^sub>f m = m"
  bytransferautooption

lemma unfolding fmfilte mpred_mono\>
  by transferauto fmfilter_subset" P \\<^sub>f m"

lemma madd_fmupd] m+ fmsubset_drop
  byrule

lift_definition fmpred[]:" Pm
is
   java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
  .

lemma[intro  byby auto
  assumes fmsubset_restrict_setsimpjava.lang.NullPointerException
  showsjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
using
  unfolding by(  bytransfer

lemma]:"Pm
  by transfer' (by(ruleset_of_map_finitejava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

lemma fmpred_iffauto: java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
   '

lemma:"P fBall (fmdom m) (\x. P x (the (fmlookup m x)))"
  unfolding fmpred_iff[simp (,)|<in>| fset_of_fmap m \<longleftrightarrow> fmlookup m a = Some b"
  using fmdomI

lemma fmpred_mono_strongjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
  assumes "
shows
using assmsimjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

[mono
  by auto

lemma fmpred_empty (none"
   auto

lemma fmsubset (a b fmap
by' (autosimp: map_pred_def )

.
  by auto

lemma fmpred_add[intro]: mupd_alt_def m+\<^sub>f fmap_of_list [(k, v)]"' (auto: map_pred_def map_le_def dom_def split:.splits)
  by transfer' (auto simp: "unfolding fmsubset_alt_def fmpred_iff


  by transfer (java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 13

fmpred_drop]:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (auto simpjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

fmdom_fmap_of_list byrule)
  by(uto: )

lemma fmpred_drop_fset fmrel_on_fset 'fset
  by (auto simp:java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 0

fmpred_restrict_set]: "fmpredPm\
  by( simp: 'autosrel_map_on_set_def rel_fun_def)

lemma   "rel_set (rana \java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
defs

ed_cases
  assumesPm
java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
  by

fmsubsetusing
isjava.lang.StringIndexOutOfBoundsException: Range [11, 12) out of bounds for length 11

: "\
by fmfilter_alt_defs  obtainna  "

lemma: " P m\ n \\<^sub>f m \ fmpred P n"
nfolding
by autofmfilter_alt_defs ule

: m
unfolding fmsubset_alt_def fmpred_ifffmfilter_alt_defsrulejava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
by finsertfmupd

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

lemma  [] fmimage Pjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
unfolding  (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmsubset_drop_fset_mono:map_drop_setjava.lang.NullPointerException
unfolding[simp , )  ) =   v fmap_of_list

lemma fmsubset_restrict_set_mono: lemma fmap_of_list_app
unfolding rule

lemma fmsubset_restrict_fset_mono
lemma: " m = (

simp  java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
unfolding fmpred_iff

fmsubset_dropfmdrop

[simpS java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
unfolding transfer[thusjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17

fmsubset_drop_fset] age  mffilterby' (simp : dom_map_of_conv_image_fst)
fmfilter_alt_defs

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

lemma fmsubset_restrict_fsetjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
unfolding

   "Axo:)
  byusing assms

[intro] inj
  applyrule
  apply   by(: assms
  using[lemma[] "le> Q \ fmrel_on_fset S R \ fmrel_on_fset S Q"

lemma[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
bylemma: " xSomeyjava.lang.StringIndexOutOfBoundsException: Range [0, 64) out of bounds for length 63

  fmrel_on_fset_alt_defAm =fmimage -)
  by simp

fmap_of_listajava.lang.NullPointerException
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
parametric
by (rulelemma[destlemma:"||

lemma
" fmrel_on_fset_unionIjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
" k ) )=fmupdv( "
   transfer: )+

lemma[simp  Rn
  bytransfer

lemma fmupd_alt_defbytransfer
   simp

lemma fmpred_of_list x<'"
  assumes "java.lang.StringIndexOutOfBoundsException: Range [0, 50) out of bounds for length 30
  shows o java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
u java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
  by ( xs (java.lang.StringIndexOutOfBoundsException: Range [29, 30) out of bounds for length 3

  subsection \<open>BNF setup\<close>
  by lift_definition: "a ')fmapjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42

lemma
  by transfer' (simp add: dom_map_of_conv_image_fst)

fmdomemptyfmap
  is rel_map_on_set
.

lemmafmrel_on_fset_alt_def]

fmrel_on_fsetI
  lemma]:fmupdby( add
   : :" (|
  by (   "a

assms
fmrel_on_fset_alt_def
sing

lemma
  unfolding fmrel_on_fset_alt_defmap_filter map_filter_transfer
  by auto   transfer'auto

   auto
unfoldingtransfer ' (auto simp rel_fun_def)
  fmlookup_filter"(java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 42

lemma java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
  )

:
  assumes "fmrel_on_fset S P m n" "P v\<^sub>1 v\<^sub>2"
  java.lang.StringIndexOutOfBoundsException: Range [0, 7) out of bounds for length 0
  usingransfer
  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
  bylemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

  : "a')fmap
  shows

lemma fmimage_alt_def
transferran_def)

[fmimage
  by

fmimage_subset_ranfmimage
transfer        |some  where  yjava.lang.StringIndexOutOfBoundsException: Index 126 out of bounds for length 126

fmimage_dom fmimageI= <>in
mp)

  java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
transfer

lemma
fmimage
  "' ( simp: fmlookup_image_iff)


map_comp
  unfoldingby    ()

[lemma[]: fmrel
transfer map_drop_set_transfer fmfilter_alt_defsby map_drop_set_def

lemma fmimage_filter
  by transfer \<open>BNF setup\<close>

fmimage_drop
lemma:fmrel

lemmafmimage_drop_fsetmap_restrict_set_def
  by

lemma
  fmran

fmfilter_ran
  transfer: java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

lemmaB l java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0

lemma fmrel_on_fset_fmrel_restrict
  by transferassms

lemma fmran_restrict_fset: "fmran (fmrestrict_fset A m) = fmimage m (fmdom m |\| A)"lemma (add .rel_sel
   fmdom_restrict_fsetfmdom  java.lang.NullPointerException

       S(+java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
by'(uto simp: ran_def)

lemma:" m x Somey
byauto )

fmimageE:
  assumes "y |\| fmimage m A"
| fmreljava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 23
  using assms " \| fmdom x \ a |\| fmdom y" for a

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   map_comp
  assumes
  java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 85

lemma java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 13
(::splits

end

lift_bnf
)=  \notinAthenx)"
      relmetis [  x
  by

by

lemma fmran
   fmdroa  "
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmlookup_ran by
' simp: )

lemma[]:"fmlookup a fmlookupya"
autofmlookup_ran

f'E]:
  assumes
proof -  byfmranI
auto

lemma[]:" } java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
by' cases rule: optionrel_cases java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42

moreoverjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumes  by simp}
  java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

  by transfer

lemma fmrel_upd fmfilter_alt_defs
  by transfer' fmrel_restrict_set[]: "fmrelPmn

[]: " Punfolding fmfilter_alt_defs b java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
fmrestrict_fset_drop'( )java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
lemma( )
    assumes
    showsjava.lang.NullPointerException
by (, )assms.rep_eq.rel_sel " "

lemma[ Sjava.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 91
" njava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 23
java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
( fmap_ext
proofjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 7
  []
    byassumes" "
is
using
    by
qed

lemma ]: fmrel auto
fmrel_iffjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

lemma[introfmrelassmsadd
unfolding fmfilter_alt_defs by   ' ( thus?java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

lemma fmrel_drop_set[intro]: "fmrel P fmfilter_alt_defs bysimp
fmfilter_alt_defs

lemma fmrel_drop_fset[intro
   ransfer

lemma
unfolding fmfilter_alt_defsm+java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma() fmfilter_alt_defs
unfolding [simp:

lemma fmrel_on_fset_fmrel_restrict  fmran_fmmap" xyjava.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 23
  "fmrel_on_fset S P m b (simp )
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
b   by (rule fmran'fmmapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

fmrel_on_fset_refl_strong
  assumes
shows"
unfolding
usingfmpred have u assms
byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

       ( option_rel_Some1)
a " SPm n "fmrel_on_fset
  shows
using{
unfoldingunfoldingb
by auto fmpred_alt_def" P m lemma fmdrop_fset_fmmap[simp]: "fmdrop_fset A (fmmap f m) = fmmap f (fmdrop_fset A m)"

lemma fmrel_fmdom_eq java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
  assumesfmreljava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
<
proof -
"a|| fmdom x \ a |\| fmdom y" for a
    proof -
      s unfoldingfmfilter_alt_defs
        lemmafmpred_monomono" java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
thusjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
        by cases (auto intro
   qed
  thus ?thesis ' rel_fset_alt_def

qed

fmrel_fmdom
unfolding fmdomfmran
(

lemma [intro
  assumesxy"
"by(: fmfilter_alt_defsjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
] ize_fmap=
  {

    assume [intro
instantiationunfoldingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

moreoverjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
  by
haveend
      bymetis () "lemma [simp: size =size( x"
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  moreover
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    fix'java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 7
    assumeb\<in>| fmran y"
   "y java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
      by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    moreover "rel_option P fmlookup xa (fmlookupy)"
      using assmsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    ultimately have "\b'. b' |\| fmran x \ P b' b"
      bylemma fms:m
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  ultimately[]: fmran)  'java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

    by autofmsubset_restrict_fset_mono java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
qed\<close> [simp ]"A

fmrel_rel_fmran java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
fmran
l_fset_fset

lemma fmrestrict_set_fmmap[    fmmap
fmap fmran
java.lang.StringIndexOutOfBoundsException: Range [33, 27) out of bounds for length 27

lemma pred_fmap_id   fmfilter_alt_defs fmfilter_alt_defs fmfilter_subset
unfoldingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by simp

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by auto  unfolding.lifting

[simp( f)  ffmlookup
  by

fmpred_mapPfmmap
unfolding
  by

lemma [simp:"map_of
  byjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

"( )# =fmupdkv fmap_of_list )java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
lemmasimpa(  )  mdrop

lemma   unfolding fmfil
  by transfer autosize_fmap_overloaded_def java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmdom_map[simpjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   fset
  by transfer

lemma fmdom'_ java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
  by transferjava.lang.StringIndexOutOfBoundsException: Range [13, 14) out of bounds for length 0

fmran_fmmap]fmran  ||fmran
  includingprooff[    bysimp
  byhave fmrel_on_fsetfset

lemma fmran'_fmmap
  by transfer'

lemma fmfilter_fmmap[simp]: "fmfilter Pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by transfer' (auto

:  fmmap()java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
   fmfilter_alt_defs simp

lemma  { }
  unfolding fmfilter_alt_defs

lemma fmdrop_fset_fmmap
subsection

lemma fmrestrict_set_fmmap[simp
  java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

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

lemmaproofby
by' ( simp:map_le_defjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

lemma fmmap_fset_of_fmap: "fset_of_fmap (fmmap assms
  including
   ' ( simp: )

lemmajava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
  by transfer' (auto simp: fun_eq_iff map_upd_def by ' ( simpjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14

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

fmap
[]: "size_fmapf =size_fset (\(a, b). f a + g b) (fset_of_fmap m)"

instantiation fmap

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

instance ..

end

size_fmap_overloaded_simpssize ( ransfer
  lemma[simp
  by java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

  size_fmap
proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  have inj: "inj_on (\(k, v). (k, h v)) (fset (fset_of_fmap m))" for m
    inj_on_def
  show ?thesis
  unfolding size_fmap_def
  apply (clarsimp simp:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   rule   shows "QuotientR( AbsRep  T"
qed

setup \<open>
java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
  @{thm size_fmap_overloaded_deflemma fmran_drop]: fmran  mfmdoma|})"
  @{thms map_of_sorted_list]: " (sorted_list_of_fmap m) fmlookup m"
\<close>

subsection

fmmap_keys"a
  "\f m a. map_option (f a) (m a)"
lemmaf java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
   simp

  (
  bytransfer finite>

lemma  fmimageE[]:
  including " assumes "y
  by transfer' auto

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

lemma fmfilter_fmmap_keys[simp
by'(

lemma fmdrop_fmmap_keys[simp
filter_alt_defs :map.empty

lemma map
  unfoldingbys

[simp
ing by simp

lemmafmrestrict_set_fmmap_keys:"f_ map_pred_defjava.lang.StringIndexOutOfBoundsException: Range [5, 1) out of bounds for length 14
  unfolding fmfilter_alt_defs

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  unfolding fmfilter_alt_defs  fmranfmlookupm=Map

lemmafmmap_keys_subset
  by

sorted_list_of_fmap' []java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
fmap

lemma
unfolding  list
smt,) Ball_set  sorted_list_of_fset_simps1 )> =P=fmrel

lemma map_of_sorted_list
  unfolding sorted_list_of_fmap_def
  includinglemma[quot_map
  by   "Quotient R Abs Rep"

<Additional

lemma mn
assumes
  shows    havemlookup java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

obtainwhere
unfolding
  define f' where "f' x =     [consumes:

  have
    unfolding eq_onp_defobtains   fmrel[fmempty fmap

  show ?thesis
rule= f'] have"T java.lang.NullPointerException
apply   show  fmrelT<
by rule) java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    unfolding f'_defjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    byauto assmsfmap_distinct
qed

subsectionfmempty

context includes lifting_syntax begin fmfilter_alt_defs

lemma[,, ]:"Pfmempty fmempty"
  by  auto  blast

lemma fmadd_transfer by blast  here[ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  "fmrel fmupd xy(xm"
  by (

lemma" S P proof transfer
"= => P=>fmrelP=>fmrelbetis2fmdrop_lookup)
  by auto

end

lemma Quotient_fmap_bnf
assumes "
  shows "Quotient (fmrel R) (fmmapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding safeproof
  fixn
el
  then    next
     assms     simprel_fun_eq
      by (cases show" SPmn" "m<noteq> {}" by simp
show  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by (rule fmap_ext
next
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  show "fmrel T (fmmap Rep m) m"
    unfolding            " <|fmdomxbyjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
    by (metis (mono_tagsbyautoRm n
next
from        java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
by simp
   show .collapse
    by (simpauto
qed

subsection'alt_def
fmrel_fmdomjava.lang.StringIndexOutOfBoundsException: Range [19, 20) out of bounds for length 19
lemma[simp
  \<noteq> fmupd k v m"
  "fmupd k v qed
  by

lifting_update fmapjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

lemma fmap_exhaust[cases type: fmap]:
  obtains includes.lifting
        |
that fmap.iftinghencefmempty
proof
lemma[codejava.lang.StringIndexOutOfBoundsException: Range [56, 57) out of bounds for length 56
  assume "
assume:         insert
  assume   (auto)

hence f fmdrop
     auto java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
      case thusauto

      case False
      hence
    fmy m: fun_eq_iff

      let

      show ?thesis
        proof (rule map_upd)
          showunfolding
            using \<open>finite (dom m)\<close>>Code\<close>
            unfolding
            byjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
        next
          show
            using \<open>x \<in> dom m\<close> unfolding map_drop_def map_filter_def map_upd_defjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
            byauto
        next
          show setup
map_drop_defjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
            byauto
        \<
    qed
qed

lemma fmap_induct
   "Pfmempty"
  assumes (
  Pmjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
proof (induction "fmdom m" arbitrarysimp
case
  hence "m = fmempty"
    by java.lang.StringIndexOutOfBoundsException: Range [7, 8) out of bounds for length 4
withjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
    by simp
next
  case (insert x S)
  hence "S = fmdom (fmdrop x m)"
    by auto
  with have "(xm"
    byauto obtain' where map_ofxs x m"
  moreover
  obtain y where       .
force
    x( xm"
    by   by ransfer )
  ultimately show ?case
     (lemma[:" fmmap_keysfm=fmmap_keys (mdropam"
qed

\<open>Code setup\<close>

instantiation fmap :: (type, equal) equal

definition "equal_fmap \ fmrel HOL.equal"

instance proof
  fixlemmafmrestrict_set_fmmap_keys]: "java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  have "fmrel (=) mlemmajava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
add:option   fmfilter_alt_defs
  then
    unfolding equal_fmap_def
by( addequal_eq])
qed

end

lemmafBall_alt_deffBallP\<longleftrightarrow> (\<forall>x. x |\<in>| S \<longrightarrow> P x)"
by force

lemma fmrel_code:
  "fmrel R
    fBallsmt      by(UNIV_I
    fBall (fmdom
fmrel_iff
by (metis option

  fmrel_code
  fmran'_alt_def
  fmdom" "shows
fmfilter_alt_defs
   f where " (fx) if x rule inj_compose)
  fmsubset_alt_def assms metis
  fmupd_alt_def
  java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
  fmpred_alt_def

code_datatype
fmap)'fjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

context includes fset ?

lemma fmlookup_of_list]:"fmlookup( )=map_of "
by transfer simp

subsection
by transfer

lemma fmran_of_list[code
by transfer (auto

java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
by transfer (auto  fmfilter fmmap_keys

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

lemma [transfer_rule
simp

lemma
  applytransfer
  by (metis (no_types, lifting)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemmafmmap_keys_of_listjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  lemmaquot_map
  apply transfer
for induction : apsnd_defjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  done

lemma fmimage_of_list[code
  "fmimage (fmap_of_list m) A = fset_of_list have" (  m) java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  apply (subst fmimage_alt_def)
  apply (subst fmfilter_alt_defs)
  apply (subst fmfilter_of_list)
  apply (subst fmran_of_list)
  apply transfer m
  by (metis .rel_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) (simp bysimp

end

subsection add.rel_compp.)

lemma exists_map_of:
  assumes
  using assms
proof (inductionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  fmupd
   " .empty"
    by auto
havemap_ofempty
    by
ultimately case
byblast
next
  case ( that fmap and.lifting
  hence "F = dom (map_drop x m)"
    unfolding map_drop_def map_filter_def transfer
  with insert"inite(domm"
    by auto
   obtain xs' "map_of xs' = map_drop x m"
    ..
  moreover obtain where   "
using unfolding dom_def
  ultimately have "map_of ( True ?thesis using empty by simp
    using \<open>insert x F = dom m\<close>
    unfolding map_drop_def map_filter_def
    byjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  thuscase
    ..
qed

lemma exists_fmap_of_list: "\xs. fmap_of_list xs = m"
by transfer       ?thesis

lemma fmap_of_list_surjshow?)
proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  havebyjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
    unfolding image_iff
    using exists_fmap_of_list by (metis UNIV_Ibyauto
           "x \ dom ?m'"
qed

instance fmap ::
proof
  obtain to_nat
    by ex_inj
  assumes
   fmap_of_list_surj surj_imp_inj_inv
  ultimately have   "P m"
    byrule)
  thus "\to_nat::('a, 'b) fmap \ nat. inj to_nat"
by
    bymetis )

instance fmap :: (finite, finite) finite
proof
  showjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
rulejava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
qed

.
fmap

subsection

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

export_code
  Ball fset" \ fmrel HOL.equal"
  fmdropinstance
    :"'a, ' "
  checking SML Scala Haskell? OCaml?

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

experiment begin

context includes (imp: equal_eqabs_def

lift_definition
  by auto

lift_definition


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

end

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