Quelle Finite_Map.thy Sprache: Isabelle

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

section

theoryunfoldingfmpred_drop:
  lemma: " xs Somev\java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
  abbrevs
begin

lift_definition : "

parametric_constant map_add_transfer[transfer_rulefmfilter_alt_defs fmsubset_filter_mono
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0

context includes lifting_syntax begin

lemma[intro   <Longrightarrow> fmpred P (fmrestrict_set A m)" fmfilter_alt_defs fmsubset_filter_mono
"rel_map f \ (=) ===> rel_option f" auto  ':eq_onp_defrel_fun_def)

lemma
proof
  fix m n
  assume "rel_maplemma[intro]:
showA(  "x. x |\| S \ rel_option P (fmlookup m x) (fmlookup n x)"
    proof rel_setI
      fix x
      assume   "fmrel_on_fsetSP defs)
lemma: "([consumes 1:
        unfolding fmpred []: Rjava.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87

ave ma (alift_definition
         \<open>rel_map A m n\<close>
           is map_le
      then obtain y whereby)
\<open>m a = _\<close>
        by cases

        unfolding ran_def unfolding fmfilter_alt_defs
    next:"java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
      fix lemma[simpfmdrop_fset
      assume y \<in> ran n"
       a where =yjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
        fmsubset_pred

      have "rel_option A (m a) (n af fmsubset_alt_def fmpred_iff
        usingng fmfilter_alt_defs by(fmfilter_subset
java.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length 0
      then x  "m = Some "" yjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
        unfolding \<open>n a = _\<close> fmrel_on_fset_updateIjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
        by cases auto
      by showsfinsert P  v<
        unfolding
    qed
qed

lemma ran_alt_def: "ran m =lemma fset_of_fmap_inj using assms
unfolding ran_def java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

parametric_constant dom_transfer[transfer_rule]: dom_def

definition map_upd   set_of_map_inj lift_definition:"'a'b) 'a fset \ 'b fset" is "\m S. {b|a b. m a = Some b \ a \ S}"
map_updm (java.lang.StringIndexOutOfBoundsException: Range [34, 22) out of bounds for length 34

parametric_constant map_upd_transfer[transfer_rule]: map_upd_def

definition map_filter :: "('a \ bool) \ ('a \ 'b) \ ('a \ 'b)" where
"map_filter P m = (\x. if P x then m x else None)"

parametric_constant map_filter_transfer[transfer_rule]: map_filter_def

lemma map_filter_map_of[simp]: "map_filter P ( fmimage_emptysimp:" bytransfer set_of_map_def
  by
  fix x
  show "map_filter P (map_of m) xjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by (induct m  by simp
qed

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumesjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shows( (map_filterm)
proof -
  from assms  is
by (rule) (auto
  then show ?thesis
    by (simp
qed

definition map_drop  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
"map_dropa map_filter (\a'. a' \ a)"

parametric_constant map_drop_transfer

definition   "fmap_of_list [] = fm"
" A = map_filter (a. a \ A)"

parametric_constant map_drop_set_transfer[transfer_rule]: map_drop_set_def

definition  "fmap_of_list ((k kvsfmupd( )"
"map_restrict_set A = map_filter (\a. a \ A)"

parametric_constant

definition map_pred :java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
"map_pred P \ (\x. case m x of None \ True | Some y \ P x y)"

parametric_constant map_pred_transfer[transfer_rule transfer

definition rel_map_on_set :: "'a set \ ('b \ 'c \ bool) \ ('a \ 'b) \ ('a \ 'c) \ bool" where
  Punfolding by( )

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

java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 79
unfolding set_of_map_def dom_def
by auto

lemma set_of_map_finite: "finite (lemmafmfilter_subset[]: "fmfilterPm <subseteq>\<^sub>f m"
unfolding set_of_map_alt_def
by auto

lemmausing
proof fmsubset_alt_def by auto
xjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemma[simp]: " a m \\<^sub>f m"
  hence "(x a = unfoldingjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
    unfoldingslemma fmsubset_drop_set] "fmdrop_set \<^sub>f m"
  hence "x k = y k" for k
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   x=y .
qed

lemma dom_comptransfer auto
unfolding
by (auto split: option

lemma dom_comp_finite:lemma[simp: " fmfilterPmA=fmimage ffilter )
by (metistransfer simp

parametric_constant map_comp_transfer[transfer_rule]:unfolding by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

end

subsection \<open>Abstract characterisation\<close>

typedef ('aunfoldingfmfilter_alt_defs by ( fmfilter_subset)
  morphisms fmlookup
proof
  show "Map.empty \ {m. finite (dom m)}"
    yauto
qed

setup_liftingtransfer simp  )

lemma dom_fmlookup_finite[intro, simp fmfilter_alt_defs
using fmap

lemma fmap_ext:
  assumesnd:map_filter_def
  shows "m = n"
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 11
bytransfer

subsection of_fmap_inj, simp:" fset_of_fmap"

context
  includesapply rule
gin

lift_definition  apply transfer( add fmrel_on_fset_alt_def
  is ran
  parametric ran_transferlemmafmfilter_ranfmrel_on_fset_monoR<java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
by (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmlookup_ran_iff: "y |\| fmran m \ (\x. fmlookup m x = Some y)"
by transfer( simp

lemmafmranIfmlookup =  <Longrightarrow> y |\<in>| fmran m" by (auto simp: fmlookup_ran_iff)ran_def)

lemma fmranE
  assumes "y |\| fmran m"
  obtains
using assms bylemmaunfolding (fmdrop_fset) =  m ( m-A)"

  by auto
  is dom
  parametric dom_transferlift_definition :: "(' 'b) list \ ('a, 'b) fmap"
.

lemma fmlookup_dom_iff: "x |\| fmdom m \ (\a. fmlookup m x = Some a)"
bytransfer

   map_of_transfer
lemma fmdomI: " fmrel_on_fset_alt_def
fmdom_notD byauto:" <>fmimagem (\x. fmlookup m x = Some y \ x |\| A)"

[java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
  assumes "x |\| fmdom m"
  obtains
using assms byusing assms by ]:

lift_definition' ::"' ' fmap_of_list((, v#kvs = k (map_of_listkvs"
  is  by(, simp add map_upd_def
parametric
.

lemma fmlookup_dom fmap_of_list_app]:using"A   \<Longrightarrow> fmrel_on_fset B R m n \<Longrightarrow> fmrel_on_fset (A |\<union>| B) R m n"
' auto

'_notIlemmafmrel_on_fset_updateI:
lemma  bysimp
lemmafmdom

lemma fmdom'E[elim]:
  assumes < fmdom
  obtains x y where "fmlookup m x = Some y"
usingfmrel_on_fsetP java.lang.NullPointerException

': "fmdomjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
by transfer fmimage assms

lemma finite_fmdom'[simp]: "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding fmdom'_alt_def by simp

lemma dom_fmlookup[simp]: "dom (fmlookup m) = fmdom' java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by transfer' simp

fmempty :: "',b "
  lift_bnf'a ': 'b) fmap [: Map.empty]
by simp

lemma fmempty_lookup
by' simp

lemmafmdom_empty]: "fmdom fmempty= byauto
lemmafmdom'_[simp]: declare.[mono]
" fmempty = fempty java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

'_alt_def: fmran = fmranm"
  is
  parametric  including.
unfolding[abs_def
by simp

lemma fmupd_lookup[simp
by transfer' (auto simp: map_upd_defjava.lang.StringIndexOutOfBoundsException: Range [36, 37) out of bounds for length 0

[simp" ( a b m =finserta (fmdom m)"  transfer(simp: map_upd_def
lemma fmdom'_

lemmafmupd_reorder_neqfmimage_intermA|java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98
  assumes\<noteq> b"
  shows "fmupd a x java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using
by transfer' java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmupd_idem[simp]: "fmupd a x (fmupd a y m) = fmupd usingsing .rel_mono by blast
by java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0

lift_definition  by('; auto)+
  is map_filter
  parametric
by autousing by (auto fmlookup_ranjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45

lemma fmdom_filter[simp]: "fmdom (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
bytransfer :map_filter_def)

lemma fmdom'_filter[simp]: "fmdom' (fmfilter P m) = Set   fmrel_on_fset_alt_def transfer
by transfer' (auto simp: map_filter_def split: if_splits)

lemma fmlookup_filter[simp]: "mlookup fmfilter
by transfer' (auto simp: map_filter_def

lemma lemmafmimage_drop[simp]:"fmimage (fmdrop alemmafmrelI[intro]java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
by transfer' (auto simp: map_filter_def)

lemma fmfilter_true[simp]:
  assumes "\x y. fmlookup m x = Some y \ P x"
  showsfmfilter m= mjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
proof(rulefmap_ext
  fix x
     transferautomap_filter_def
    using that assms by fastforce  by autojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  then show "fmlookup (fmfilter P m) java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by simp
qed

lemma fmfilter_false[simp]:
  assumes "\x y. fmlookup m x = Some y \ \ P x"
shows java.lang.StringIndexOutOfBoundsException: Range [19, 1) out of bounds for length 115
using assms by transfer' (fastforce simp: map_filter_def)

lemma fmfilter_comp[simp]: "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
bylemmafmrelD"fmrel P m n \ rel_option P (fmlookup m x) (fmlookup n x)"

  by' (auto simp: ran_def map_drop_def map_filter_def)
unfolding by meson

lemma
  assumes "\x y. fmlookup m x = Some y \ P x = Q x"
  showsfmfilter
proof (rule
lift_definition:(', \
have x="if P xmIff .rep_eqfmlookup_addjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    using that assms by lemma by ( simp: ran_def map_filter_def
fmfilterfmlookup emma fmimage =fempty
    by auto
qed

lemma[]: "m |\| fmran m"
  assumes f by ' (auto java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shows "fmfilter P m = fmfilter Q n"
using()u assmsproof
byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmfilter_upd[simp: ran_def
ilter(  )
by transfer' (auto simp: map_upd_deflemma fmimage_inter: "fmimage m (A |\< then show thesis

lift_definition fmdrop :: "'a \ ('a, 'b) fmap \ ('a, 'b) fmap"
transfer
  parametric map_drop_transfer
unfolding

lemma[simp" m(fmdomm || A) = fmimage m A"
by transfer by autojava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 48

unfolding byjava.lang.StringIndexOutOfBoundsException: Range [0, 32) out of bounds for length 27
  lemmafmimage_Union fmrel_drop_setintro" P m n fmrel P (fmdrop_set A m) (fmdrop_set A n)"
  parametric
unfoldingbyauto

lift_definition
  is [java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
map_drop_set_transfer
unfoldinglift_bnf

lemma[simp: " java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by
  parametric fmfilter_alt_defsby
unfolding by auto

lift_definition fmrestrict_fset :: "'a java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  is
  parametric
unfolding map_restrict_set_def fset.java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24

lemmabytransfer(auto ran_def
  "fmdrop "fmrel_on_fsetS lemma'iff: "y fmran' m \ (\x. fmlookup m x = Some y)"
  fmdrop_setfmfilter>.a \<notin> A)"
  "fmdrop_fset B =fmfilter emmafmranby auto
  "fmrestrict_set A = fmfilter (\a. a \ A)"
  "fmrestrict_fset B = fmfilter (autosimp:fmlookup_ran'_iff)
by  by

lemma
lemma  "x y. x |\| S \ fmlookup m x = Some y \ P y y"
mma'_rop_set[simp] " java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemmaby simp:option
mma: " (fmrestrict_fset Am)|\| A" unfolding fmfilter_alt_defs by auto

lemma
by transfer' lemma obtains x where ""fmlookupm " S P m +\

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

lemmafmdrop_idle   ( simp:fmlookup_image_iff
by transferlemma[elim]java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

lemma fmdrop_fmupd_same: "fmdrop x (fmupd x obtains x where "fmlookup m x = Some y" "x | assumes"fmrel y"
by transfer( :map_drop_def

lemma fmdomjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
unfolding fmfilter_alt_defs by    proof

fmdomrestrict_fset_precisefmrestrict_fset)= fmdom\inter "
unfolding fmfilter_alt_defsis

lemma fmdom   "\x. rel_option R (fmlookup m x) (fmlookup n x)"
ldingshows  "

fmdom_:fmdom A m)java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
folding by transfer         by( :java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

lemma fmlookup_dropsimpby transferautomap_comp_def option
  "fmlookup (fmdrop a m) x = java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding

lemma
  "fmlookup (fmdrop_set A m)subsection \BNF setup\
unfolding    by autojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lookup A m)x=ifx|notinthen  m x else"
unfoldingunfoldingfmdomalt_def

lemma by ( fmrel_fmdom_eq
  "fmlookup(fmrestrict_setAm x
unfolding fmfilter_alt_defs by simp

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

lemma fmrestrict_set_dom[simp]: "fmrestrict_set (fmdomproof -
by (rule) auto

lemma fmrestrict_fset_dom[simp  assumes Pm n""   b"
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemmap_empty[simp]: "fmdrop a fmempty=fmempty"
unfolding fmfilter_alt_defs by simp

[simp]:"fmdrop_setAfmempty=fmempty"
unfoldingfmfilter_alt_defsby simp

lemma    then obtainawhere() fmlookupx =java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
fmfilter_alt_defssimp

lemma fmdrop_fset_fmdom[  by transfer(auto ran_def
by

lemma fmdrop_set_fmdomsimp  (fmlookup a)( y )"
by transfer' (auto simp by (auto simp: fmlookup_ran'_iff)

lemma fmrestrict_set_empty[simp]:lemma fmranE[elim
unfolding fmfilter_alt_defs simp

lemma fmrestrict_fset_emptyultimately "\b'. b' |\| fmran y \ P b b'"
unfoldingfmfilter_alt_defse1)

lemma fmdrop_set_nullfromassms "rel_option P(using assms by(auto simp: fmlookup_ran'_iff)
by (rule fmap_ext) auto

lemma fmdrop_fset_nullsimp "fmdrop_fset {|| m =m"
by (rule fmap_ext) auto

lemma fmdrop_set_single[simp transfer (auto(cases.)auto
unfolding fmfilter_alt_defsassume"b|\| fmran y"

lemma fmdrop_fset_single[simp]: "fmdrop_fset {|a|} m = lemma fmdrop_fset_single[simp]: "fmdrop_fset {|a|} m = fmdrop
unfolding by simp

lemma[simp: fmrestrict_set using
unfolding

lemma[simp" {||} fmempty"
unfolding fmfilter_alt_defs

  ultimatelyshows"
unfoldingjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0

lemma fmdrop_set_insert[
by (rule fmap_ext) auto

lemma fmdrop_fset_insert[simp]: "fmdrop_fset (finsert x S) m = fmdrop x java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36

lemma fmrestrict_set_twice[simp]: "fmrestrict_setjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
yauto

lemma fmrestrict_fset_twice[simp]: "fmrestrict_fset S (fmrestrict_fset T m) = fmrestrict_fset (S |\| T) m"
unfolding fmfilter_alt_defs by auto

lemma fmrestrict_set_dropqed
unfolding

lemmasimp"fmrestrict_fset S (fmdropetP(fmran x) mran'y"
unfoldingby java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

lemma   unfolding fmranby
by (ulefmap_extauto

lemma[lemma fmdrop_fmrestrict_fset[simp
by (rulered_fmap_fmpred "\x y. x |\| S \ fmlookup m x = Some y \ P y y"

lemma fmdrop_idem (smtveritdel_instsdomIff fmlookup_add option)  shows S P m m"
unfolding  using fmrel_on_fset_fmrel_restrict

twicesimpfmdrop_setfmdrop_set)=fmdrop_set <>T) "
unfolding fmfilter_alt_defs by auto

lemma fmdrop_fset_twicelemmapred_fmap_idsimppred_fmap(  )\<longleftrightarrow> pred_fmap f m"
unfoldingassumesfmrel "

lemma fmdrop_set_fmdrop[simp " bysimp
by (rule  shows "fmrel_on_fset

lemma fmdrop_fset_fmdrop]: " S (fmdrop b )= fmdrop_fset (finsert b S) m"
by(uleunfolding

lift_definitionauto
  is map_add
  parametric map_add_transfer
  by simp

fmlookup_add]:
  "mlookup m +\<^sub>f n) x = (if x |\| fmdom n then fmlookup n x else fmlookup m x)"
  by transfer

lemma fmdom_addlemma fmpred_map
lemmafmdomsimp java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19

lemma fmadd_drop_left_dom: "fmdrop_fset (fmdom n "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by(rule

lemma fmadd_restrict_right_dom:unfolding java.lang.StringIndexOutOfBoundsException: Range [0, 22) out of bounds for length 11
  by lemmafmpred_id]: "fmpred (\_. id) (fmmap f m) \ fmpred (\_. f) m"

lemmalemma fmrel_drop]: "fmrel P usingby( add: fmrel_iff)
  bytransferauto?java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

lemma fmdrop_add_distrib[simp "fmdrop a (m ++ transferjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  unfolding

lemma unfolding by blast
  unfolding    qeded

lemma fmdrop_fset_add_distribsimp"fmdrop_fsetunfolding t auto
  unfolding fmfilter_alt_defs by simp    by auto

lemma fmrestrict_set_add_distrib]:
  "fmrestrict_set A (m ++
  unfolding fmfilter_alt_defs by simp

lemma fmrestrict_fset_add_distrib[simp]:
  "fmrestrict_fset A (m ++\<^sub>f n) = fmrestrict_fset A m ++\<^sub>f fmrestrict_fset A n"
  unfoldingbysimp

lemmafmadd_empty]: "fmemptylemmafmrel_rel_fmran:
  by (transfer'

lemma lemma[impassumes"relP yjava.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 23
  ytransfer :map_add_defsplits

lemmafmadd_assoc]: "m++<^sub>f (n ++\<^sub>f p) = m ++\<^sub>f n ++\<^sub>f p"
  by   transferautoran_def

lemma fmadd_fmupd[simp]: "m + fixb
  y(rule |<in>| fmran x"

lift_definition fmpred :: "('a \ 'b \ bool) \ ('a, 'b) fmap \ bool"
  is transfer
  parametric map_pred_transferlemmafmrel_on_fset_refl_strong:
  .

lemma fmpredI fmfilter_fmmapshowsSPm"
  assumes fmrel_on_fset_fmrel_restrictfmrel_iff
  showsPm"
  using
  by transfer' (auto simp: map_pred_def split: option.splits)

lemma fmpredD[dest have "<>b. b'|<>| y lemmajava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
  by transfer' (auto simp: ssumesfmrel_on_fset ""

{
  by auto b

lemma: fmpred
   fmpred_iff
  using fmdomI lemma byauto

lemma fmpred_mono_strong " P x y"
  assumes "y.fmlookupm unfolding fmfilter_alt_defsby fmdom x fmdom "
  shows "fmpred have \>b.java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
  unfolding

[]: Plemma fmmap_subset[intro]: "m \<subseteq>\<^sub>f n \<Longrightarrow> fmmap f m \<subseteq>\<^sub>f fmmap f n"
by

lemma fmpred_empty[intro       ?}
  by

lemmafmpred_upd: qed
  by

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

  '_
  by transfer' ( ( by (metis )

lemmafmpred_filter
  by transfer pred_fmap_fmpred fmrelx y"

lemmasubsection
   ( simpjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

lemma fmpred_drop_setproof" f g m [simp]: "size_fmap f g m = size_fset
lemmapred_fmap_id]: "pred_fmap fixfixb

d_drop_fset]
  by (auto simp: fmfilter_alt_defs)

lemma fmpred_restrict_set
  by (auto simp

lemma [intro size_fmapby
         byjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

lemma      usingassms auto
      ultimatelyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  obtainsnonefmlookupsize_fmap_overloaded_simps:"x fset_of_fmap)java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
usingjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lift_definition fmsubset :
  is lemma[simp: fmpred\<lambda>_. id) (fmmap f m) \<longleftrightarrow> fmpred (\<lambda>_. f) m"
.

lemma: "size_fmap f g \ fmmap h = size_fmap f (g \ h)"
by  (proof-

fmsubset_predfmpredm\<Longrightarrow> n \<subseteq>\<^sub>f m \<Longrightarrow> fmpred P n"
unfolding " |\| fmran y"
by auto fmdom_map]     thena where  a  Some

lemma  nfolding
unfolding fmsubset_alt_def fmpred_iff
byauto

lemma  by( sum.congauto: prod)
unfolding fmfilter_alt_defs by (rule fmdommap:java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding fmfilter_alt_defs by (rulejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

ubset_drop_fset_mono " \\<^sub>f n \ fmdrop_fset A m \\<^sub>f fmdrop_fset A n"
unfoldingfmfilter_alt_defs   }

lemma fmsubset_restrict_set_mono: " ytransfer (auto : ran_def
unfolding lemma[simp]: fmfilterfmmap

lemma fmsubset_restrict_fset_mono  bytransfersimp
unfolding fmfilter_alt_defs

lemma fmfilter_subset]: "fmfilter P m \\<^sub>f m"
unfolding fmfilter_alt_defsby simp

lemmafmsubset_dropsimp]: "fmdropa lemma [simp: fmdrop_set Ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding fmfilter_alt_defs by (rulesubsectionlemmafmrel_rel_fmran fmfilter_alt_defs

lemma fmsubset_drop_setsimplemma[simp" A( unfolding fmran'alt_def
unfolding fmfilter_alt_defsl_fset_fset

unfoldingsimp"mrestrict_set A( fm f fmmap_keys :"'\<> ' \<Rightarrow> 'c) \<Rightarrow> ('a, 'b) fmap \<Rightarrow> ('a, 'c) fmap" is

lemma fmsubset_restrict_set]
unfolding fmfilter_alt_defs by (rule fmfilter_subset simp

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfoldingfmfilter_alt_defs by (rule)

lift_definition    fmap
  by (rule set_of_map_finitelemmafmmap_subsetbytransfersimpsplit

lemma fset_of_fmap_inj[intro, simp
  apply rule
applytransfer
  usingset_of_map_inj fsetjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24

fmlookup_map]: " fmmapf m x=map_option ( m x"
by transfer

lemma fset_of_fmap_iff': "lemma [simp]: "fmpred ( f m) \ fmpred (\k v. P k (f v)) m"
  bysimp

lift_definition java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
is
  parametric map_of_transfer
by (rule)

lemma  simp
  "java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
    )fmupd
(

]:fmdrop=fmmap_keys)java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
  by transfer java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumes lemmasize_fmap_overloaded_simps
   size_fmap_overloaded_def
  using ingby simp
  by

: fmrestrict_fset_fmmap_keys] fmrestrict_fset fm =fmmap_keys  m"
  mma[simp] " (fmmap f m) =f |` m"

mdom_fmap_of_listfmfilter_alt_defssimp
  by transfer

lift_definition : "a \ ('b \ 'c \ bool) \ ('a, 'b) fmap \ ('a, 'c) fmap \ bool"
java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 19
java.lang.StringIndexOutOfBoundsException: Range [18, 1) out of bounds for length 1

lemmafmrel_on_fset_alt_def" (k(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  rule  prod

lemma[intro
   "fmdrop_fmmapsimp "a( fm sorted_list_of_fset_simps
  shows "fmrel_on_fset.register_size_global ( m  fmlookupunfolding by simp
  by (simp fmdrop_set_fmmap fmap_size_o_map   .lifting

lemma fmrel_on_fset_mono[mono]: "R \ Q \ fmrel_on_fset S R \ fmrel_on_fset S Q"
  java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
  using option.rel_mono

lemmafmrel_on_fsetDxdefs simp
  unfolding fmrel_on_fset_alt_def
  by auto

mma: "fmrel_on_fset S R
   fmrel_on_fset_alt_def
  by auto

fmrel_on_fset_unionI
  "fmrel_on_fset A R m
  unfolding fmrel_on_fset_alt_def
   auto

lemmafmpred_fmmap_keysbytransferauto)
  assumesfmrel_on_fset  "" v\<^sub>1 v\<^sub>2"
on_fsetjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using
  unfolding fmrel_on_fset_alt_def
  by auto

lift_definitionhave  bytransferautoset_of_map_def
  by (smt (verit, del_insts' auto

lemma fmimage_alt_def: "fmimage m S = fmran (fmrestrict_fset S m)"
  bytransferauto:    ?thesis

lemma [simp]: "fmimage =fempty
  by transfer' auto

lemma fmimage_subset_ran[simp  by transfer
  by '_ :java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma      f'_def bytransfer'java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
  by transfer' (auto simp: ran_def)

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

lemma fimage_inter_dom[simp]:
  " m (fmdom m || A) = fmimage m A"
  "fmimage m (A |\| fmdom m) = fmimage m A"
  by

lemma fmimage_union[simp]: "fmimage m java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
byransfer

fmimage_Union]:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [29, 19) out of bounds for length 19

lemmafmimage_filter]: "fmimage (fmfilter Plemma fmap_size_o_map:" f  \<circ> fmmap h = size_fmap f (g \<circ> h)"
  by transfer imp

emma
  by lemma[simp:"fmrestrict_fset (fmmap_keysf m = fmmap_keys f (fmrestrict_fset A m"

lemma java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 0
  by transfer' (auto simp: map_filter_def map_drop_set_def)

lemma fmimage_restrict_fset[simp]: "fmimage (fmrestrict_fset B m) java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 34
  by transfer  by(rulesum  fmmapfmmap()java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

" BNF_LFP_Size.\<^type_name>\fmap\ \<^const_name>\size_fmap\
  ytransfer (simpmap_filter_def sorted_list_of_fmap_def list.red_map

lemma fmran_drop[simp" (fmdrop a m) =fmimage ( m - {|java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     :ran_def

lemma fsetjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
      ylift_definition : "' \ 'b \ 'c) \ ('a, 'b) fmap \ ('a, 'c) fmap" is

lemma   m
  by

lemma fmlookup_image_iff:subsection \<open>Additional properties\<close>by
  by transfer' (auto simp: ran_def)

lemma  by transfer"S \<> \ S. \y. Q x y"
  by (auto "m. fmdom' m = S \ fmpred Q m"

lemmafmimageEelim
  assumesjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 7
obtains fmlookupmetis
      simp fmap

fmcomp
i
  parametric
  by (rule java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 14

lemma fmlookup_comp[simp " \ fmupd k v m"
  bytransferautojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

end

subsection \<open>BNF setup\<close>

lift_bnfamran)map:Mapjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
  for:fmmap
      rel: fmrel fmfilter_alt_defs by imp
  by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

declare ing fmfilter_alt_defssimp

lemma fmran [simp] " f'defdom_defmap_pred_defprooftransfer
  including.lifting
  by transfer

lemma fmlookup_ran'_iff: "y \ fmran' m \ (\x. fmlookup m x = Some y)"
  by transfer map_updlemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma'I: "fmlookup m x = proof( "m = java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma fmran'E[elim]:
  assumes "y \ fmran' m"
  obtains
using assms by (definition :: "(':lemmafmadd_transfertransfer_rule:

lemma fmrel_iff: "t obtain_ m = map (\k. (k, the (fmlookup m k))) (sorted_list_of_fset (fmdom m))"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma lemma[transfer_ruleunfolding sorted_list_of_fmap_deflistjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
"==== =>fmrelPf fmupdjava.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
  shows   java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 13
  by transfer' auto

lemma  "m =map_upd x (the ( )) shows\open> properties
  by transfer'unfoldingQuotient_alt_def4 proof safe

lemma fmrelD[dest]: "fmrel P assume" Tmn"
  then " fmmapAbs m)x ext

lemma fmrel_addI[intro]:
  assumes
  shows "fmrel P (m + (casesrule fmrel_cases[where x = x])auto
by(ing map_drop_def

cases 1]:
  assumes java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
   (show[ fmupd]
        |   assumesPfmempty fmaprel_map
proof
  from assms havenext
        T<inverse>\<inverse>"
then thesis
    using none some
    casesoption auto
qed

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

lemma fmrel_drop[intro]: "bysimp
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

lemma fmrel_drop_fset   lemma fmempty_transfer intro:fmrel fmempty
unfoldingby

lemmamoreover
unfolding fmfilter_alt_defs

lemma fmrel_restrict_fset[intro]: "fmrel P obtains fmempty) "     using.hyps
t_defs(      mdrop

fmrel_on_fset_fmrel_restrict
  fmrel_on_fsetproof
unfoldingm()==  =  =  by(assms
by auto

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumes "\x y. x |\| S \ fmlookup m x = Some y \ P y y"
shows" RepTjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
unfolding fmrel_on_fset_fmrel_restrict fmrel_iff
using "equal_fmap \ fmrel HOL.equal"
y( add option Quotient_alt_def4java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

lemma fmrel_on_fset_addIusing unfolding      transfer:option)      False
  assumes  m"fmrel_on_fsetS domm\java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
  shows "fmrel_on_fset S P (m ++\<^sub>f a) (n ++\<^sub>f b)"
using assms
unfolding
by auto

lemmafmrel_fmdom_eq
  assumesproof  ix
  shows "fmdomx= fmdomy"
proof      "finite (dom m'"
  have|in  force

      have "rel_option Plemmafmrel_codejava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
        using assms by (     (fmdomm \lambda  java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
      thus ?thesisshow map_upd ( ))m'
        by cases (auto intro: fmdomIthen by(option             \<open>x \<in> dom m\<close> unfolding map_drop_def map_filter_def map_upd_def
    qed
  thus ?thesis
    by
qed

'_eq " fmfilter_alt_defs
unfolding fmdom
by (metis fmrel_fmdom_eq)

lemma fmrel_rel_fmran:fmupd_alt_def" \ fmupd k v m"
  assumes fmrel_on_fset_alt_def
  shows "rel_fset P (fmran x) (
proof -
  {
    fix b
    assume "b |\| fmran x"
    then a where "fmlookupx a =Someb"
      java.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length 0
moreover "rel_optionP(fmlookup x a) (fmlookup y a)"
      using assms by auto
using including.ifting   fmempty
( option_rel_Some1fme[
  }
moreover
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
b
    assume "b |\| fmran y"
    then obtain a where    S=(  )
      yauto
    moreover have "rel_option P (fmlookup withinsert have "P( x m)"
      using assms by auto
    ultimately have "\b'. b' |\| fmran x \ P b' b"
      by( option_rel_Some2
  }
  ultimatelyshow
    unfolding "m=fmupdxy(fmdrop )java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
    by autolemma[]: " f)=f( (apsndf )
qed( assms) fmdrop_lookup( (,lifting  map_of_map..caseoldexhaust

  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (  applyjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16

lemmadefinition equal_fmap
  unfolding fmap
   fmranI

pred_fmap_id(     mapfilter
  unfolding fmap.pred_set fmap.   (ubst)
  by   "fmrel() mn \ (m = n)"

lemmaby transfer :applyfmran_of_list
  by auto

lemma fmlookup_map[simp]: "fmlookup (fmmap f m) x = map_option f java.lang.StringIndexOutOfBoundsException: Range [4, 2) out of bounds for length 62
  by transfer' auto

lemma fmpred_map
  unfolding
  by auto

lemma fmpred_id[simp]: "fmpred (\_. id) (fmmap f m) \ fmpred (\_. f) m"
  by simp

lemma fmmap_add[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by transfer' (auto simp: map_add_def fun_eq_iff " R m n \

lemma fmmap_empty[simp]:  using
  by transfer ( domarbitrary

lemma fmdom_map[simp]: "fmdom (fmmap f java.lang.StringIndexOutOfBoundsException: Range [0, 40) out of bounds for length 12
  including fset.lifting
  by transfer' simp

m'_[simp]: fmdom fmmapfm 'm"
  by transfer

lemmafmran_fmmap]:" (fmmap f ) f ` fmrel_code
   fset.lifting java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  by transfer' (auto simp: ran_def)

hencejava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
  by transferauto)

  with \<exists>xs'. map_of xs' = map_drop x m"
  by transfer

lemma fmdrop_fmmap[simp]: "fmdrop a (fmmap f m) = fmmap fmpred_alt_def
  unfolding fmfilter_alt_defs by simp

lemma fmdrop_set_fmmapcode_datatype fmap_of_list
  unfolding fmfilter_alt_defs by simp

lemma fmdrop_fset_fmmap[simp \<open>insert x F = dom m\<close>
  unfolding fmfilter_alt_defs by simp

fmrestrict_set_fmmap]: "fmrestrict_setA(fmmap .
fmfilter_alt_defs by simp

ap[]: "fmrestrict_fsetA( java.lang.StringIndexOutOfBoundsException: Range [0, 62) out of bounds for length 56
  unfolding  (rule)

lemma fmmap_subset[simp
  by transferproof

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

: []:  java.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
(  map_upd_def

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

size_fmapajava.lang.StringIndexOutOfBoundsException: Index 137 out of bounds for length 137
[simp]: "size_fmap by java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

instantiationproof

size_fmap whereshowfinite a b))
size_fmap_overloaded_deffor ( )(automap_prod_def

instance ..

end

lemma size_fmap_overloaded_simps
  unfolding size_fmap_overloaded_def
  byjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

lemma            auto
proof
  have inj fmadd fmmap_keysjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
    using java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  apply (clarsimp simp: fun_eq_iffjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (rulesum) (auto: prodauto
qed

setup \<open>
BNF_LFP_Size   map_filter_def
  @by
  @{thms
\<close>

subsection \<open>Additional operations\<close>

fmmap_keys "
  "\f m a. map_option (f a) (m a)"
    assumes"\x y m. P m \ fmlookup m x = None \ P (fmupd x y m))"
  by simp

lemma havemap_of.
  by

lemma java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
with
  by transfer' unfolding map_drop_def map_filter_def dom_def auto

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

fmfilter_fmmap_keys] "fmfilterP(mmap_keys )h" =fmupd)
byt' (auto simp:map_filter_defjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42

]"(fmmap_keysf)=fmmap_keys ( )java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
  unfolding

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

lemma fmdrop_fset_fmmap_keys[java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0
  unfolding fmfilter_alt_defs by simp

[simp     auto
  unfolding ?case

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

unfolding
  by transfer    by simp: [abs_def)

definition sorted_list_of_fmap :: "java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  "sorted_list_of_fmap m = map (\k. (k, the (fmlookup m k))) (sorted_list_of_fset (fmdom m))"

emma]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding sorted_list_of_fmap_def curry_deffmrelm n\longleftrightarrow
  by ( (verit     exists_fmap_of_listmetis)

lemma  thus ? by auto
  unfoldingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  including fset.lifting
  by fmrel_iff fBall_alt_def

subsection

lemma java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  assumes
   "\m. fmdom' m = S \ fmpred Q m"
proof
obtain f:Q x( x"if"  ( inj_compose
    using byjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16

dom
    unfolding eq_onp_def f'_def dom_def rule ) auto

  showthesis
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    apply (subst
    apply (subst fmdomlemmajava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
    unfolding f'_def dom_def map_pred_def using f
    by auto
qed

subsection lemma fmdom_of_list[code]: "fmdom (fmap_of_list m) = fst |`| fset_of_list m"

context   fmaddfmmap fmcomp

lemma fmempty_transfer[simp, intro, transfer_rule]_java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  byjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

lemmafmadd_transfer]:
  "(fmrel P ( add: merge_conv')
by( fmrel_addI

lemma java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
  "((=) ===> P ===> fmrel P ===> fmrel P) fmupd lift_definition :: "(', b fset) fmap" is"fmempty ::(a, ' ) fmap
  by auto

end

Quotient_fmap_bnf
  assumes "byauto
  shows  subgoal f m by( m) (autosimp map_prod_def fun_eq_iff)
unfolding Quotient_alt_def4 proof safe
  fix m n
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
then fmlookupfmmapAbs x=java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    using assms unfolding Quotient_alt_def
      by (cases rule: fmrel_cases[where x = x]) auto
  then show "fmmap Abs m = n"
    by (rule fmap_ext)
next
  fixjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  show "fmrel T (fmmap Rep m) m"
    unfoldingfmap
    by (metis (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
next
  from assms have "R = T OO T\\"
    unfolding Quotient_alt_def4 java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
  then show "fmrel R = fmrel T OO java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
mp: fmap fmaprel_conversep
qed

n \<open>View as datatype\<close>

lemma fmap_distinct[simp]:
  "fmempty \ fmupd k v m"
  " k v m \ fmempty"
  byhencem=Map

  moreover "map_of [] = Map."

lemma    show?
  obtains
        | (fmupd
sing including.lifting fsetjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
proof
  fix m P
  assume " dom )
  assume empty
  assume thenxswheremap_drop

  show y "m x=Somey
         insert by blast
casethusbyjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
    next
      case      auto
      hence "dom m \ {}" by simp
      then obtain xthus ?

      let ?m' java.lang.StringIndexOutOfBoundsException: Range [14, 15) out of bounds for length 3

      show
        proof (rule map_upd)
           "finite (dom ?'"
            using \<open>finite (dom m)\<close> -
            unfolding map_drop_def
             auto
        next
          show "m = map_upd x (the (m x)) ?m'"
            using \<open>x \<in> dom m\<close> unfolding map_drop_def map_filter_def map_upd_def
             auto
        next
          show \<notin> dom ?m'"
            unfoldingqed
            by auto
        qed
    qed
qed

lemma fmap_induct[case_names fmempty (metis)
"P fmempty"
  using by (rule)
shows"
proof (induction "fmdom m" arbitrary ( inj_compose
  case empty
  hence " auto
    by ( fmrestrict_fset_domfmrestrict_fset_null
  with assms
    by simp

  case (insert x S)
  hence "S = fmdom (fmdrop x m)"
    by auto
  with insert have "P (fmdrop x m)"
    by auto
  moreover
  obtain    by ( finite_imageD) auto
    using insert.hyps by force
  hence "m = fmupd x y (fmdrop x m)"
    by (auto intro: fmap_ext)
  ultimately show ?case
    by (metislifting_update fmaplifting
lifting_forget.lifting

subsection \<open>Tests\<close>

instantiation

definition equal_fmap

proof
  fixmn : ('a, b)fmapjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

    by transfer'\ \\lifting\ through \<^type>\fmap\\
  then show "equal_class.equal m n \ (m = n)"
    unfolding equal_fmap_def
    by(imp addequal_eq[])
qed

end

lemma fBall_alt_def
by force

lemma fmrel_code
  "fmrel R m nend
    fBall (fmdom m) (\<lambda>x. rel_option R (fmlookup m x) (fmlookup n x)) \<and>
    fBall (fmdom n) (\<lambda>x. rel_option R (fmlookup m x) (fmlookup n x))"
unfolding fmrel_iff fmlookup_dom_iff
by (metis option.collapse option.rel_sel)

lemmas [code] =
  fmrel_code
  fmran'_alt_def
  fmdom'_alt_def
  fmfilter_alt_defs
  pred_fmap_fmpred
  fmsubset_alt_def
  fmupd_alt_def
  fmrel_on_fset_alt_def
  fmpred_alt_def

code_datatype fmap_of_list
quickcheck_generator fmap constructors: fmap_of_list

context includes fset.lifting begin

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

lemma fmempty_of_list[code]: "fmempty = fmap_of_list []"
by transfer simp

lemma fmran_of_list[code]: "fmran (fmap_of_list m) = snd |`| fset_of_list (AList.clearjunk m)"
by transfer (auto simp: ran_map_of)

lemma fmdom_of_list[code]: "fmdom (fmap_of_list m) = fst |`| fset_of_list m"
by transfer (auto simp: dom_map_of_conv_image_fst)

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

lemma fmadd_of_list[code]: "fmap_of_list m ++\<^sub>f fmap_of_list n = fmap_of_list (AList.merge m n)"
by transfer (simp add: merge_conv')

lemma fmmap_of_list[code]: "fmmap f (fmap_of_list m) = fmap_of_list (map (apsnd f) m)"
  apply transfer
  by (metis (no_types, lifting) apsnd_conv map_eq_conv map_of_map old.prod.case old.prod.exhaust)

lemma fmmap_keys_of_list[code]:
  "fmmap_keys f (fmap_of_list m) = fmap_of_list (map (\(a, b). (a, f a b)) m)"
  apply transfer
  subgoal for f m by (induction m) (auto simp: apsnd_def map_prod_def fun_eq_iff)
  done

lemma fmimage_of_list[code]:
  "fmimage (fmap_of_list m) A = fset_of_list (map snd (filter (\(k, _). k |\| A) (AList.clearjunk m)))"
  apply (subst fmimage_alt_def)
  apply (subst fmfilter_alt_defs)
  apply (subst fmfilter_of_list)
  apply (subst fmran_of_list)
  apply transfer'
  by (metis AList.restrict_eq clearjunk_restrict 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) (simp add: fmlookup_of_list compose_conv map_comp_def split: option.splits)

end

subsection \<open>Instances\<close>

lemma exists_map_of:
  assumes "finite (dom m)" shows "\xs. map_of xs = m"
  using assms
proof (induction "dom m" arbitrary: m)
  case empty
  hence "m = Map.empty"
    by auto
  moreover have "map_of [] = Map.empty"
    by simp
  ultimately show ?case
    by blast
next
  case (insert x F)
  hence "F = dom (map_drop x m)"
    unfolding map_drop_def map_filter_def dom_def by auto
  with insert have "\xs'. map_of xs' = map_drop x m"
    by auto
  then obtain xs' where "map_of xs' = map_drop x m"
    ..
  moreover obtain y where "m x = Some y"
    using insert unfolding dom_def by blast
  ultimately have "map_of ((x, y) # xs') = m"
    using \<open>insert x F = dom m\<close>
    unfolding map_drop_def map_filter_def
    by auto
  thus ?case
    ..
qed

lemma exists_fmap_of_list: "\xs. fmap_of_list xs = m"
by transfer (rule exists_map_of)

lemma fmap_of_list_surj[simp, intro]: "surj fmap_of_list"
proof -
  have "x \ range fmap_of_list" for x :: "('a, 'b) fmap"
    unfolding image_iff
    using exists_fmap_of_list by (metis UNIV_I)
  thus ?thesis by auto
qed

instance fmap :: (countable, countable) countable
proof
  obtain to_nat :: "('a \ 'b) list \ nat" where "inj to_nat"
    by (metis ex_inj)
  moreover have "inj (inv fmap_of_list)"
    using fmap_of_list_surj by (rule surj_imp_inj_inv)
  ultimately have "inj (to_nat \ inv fmap_of_list)"
    by (rule inj_compose)
  thus "\to_nat::('a, 'b) fmap \ nat. inj to_nat"
    by auto
qed

instance fmap :: (finite, finite) finite
proof
  show "finite (UNIV :: ('a, 'b) fmap set)"
    by (rule finite_imageD) auto
qed

lifting_update fmap.lifting
lifting_forget fmap.lifting

subsection \<open>Tests\<close>

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

export_code
  Ball fset fmrel fmran fmran' fmdom fmdom' fmpred pred_fmap fmsubset fmupd fmrel_on_fset
  fmdrop fmdrop_set fmdrop_fset fmrestrict_set fmrestrict_fset fmimage fmlookup fmempty
  fmfilter fmadd fmmap fmmap_keys fmcomp
  checking SML Scala Haskell? OCaml?

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

experiment begin

context includes fset.lifting begin

lift_definition test1 :: "('a, 'b fset) fmap" is "fmempty :: ('a, 'b set) fmap"
  by auto

lift_definition test2 :: "'a \ 'b \ ('a, 'b fset) fmap" is "\a b. fmupd a {b} fmempty"
  by auto

end

end

end

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