Quelle  Cluster.thy

theorysimp. B) = Option.these A ∪.)
  java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
begin

lemmaby( simp
  bydefinitioncluster:"'<R> 'a option) \<Rightarrow'a,'se)mpig is

lemmaonts (f `B)"
  by (auto simp: Option.these_def split: option.splits) force

lemma these_image_Un]: "Option.the(f ` (A ∪ Option.these (f ` B)"
  by (auto simp: Option.these_def

lemma these_imageI: "f x = Some y \<Longrightarrow   cluster :: 'b\Rightarrowtn\Rightarrowt\Rightarrow'a b et mpin"
  by ( simp Option.these_def)

lift_definition cluster :: "('b ==> 'a option) ==> 'b set ==>ix_lse:"of_idxcluster set_of_idx_clusterset_of_idx Somecircf )= X
  "by tanf uo ip:an_de)

lemmaset_of_idx_clustertlb t auto
  

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

context ord
begin

definition add_to_rbt ::a>(a,b) tcs bloookpta Soe X \Rightarrowier (ins )t| oe<<R biseta {b}t)
  "

abbreviation

"add_to_rbt"add_to_rbt \lambdacaseaseofRightarrow rbt_insert a insertNone<> rbt_insertt)
  "cluster_rbt f t =

end

context"luster_rbt__o_rbt"uster_rbt = RTImplfl(ad_ptont_t) _IlEty
beginbegin

lemmacontext lino
  by ((utos imp: a _o_ splt:prod.spl tsoptio..)

lemma is_rbb
  is_rbt (RBT_Im
lemma is:"  Longrightarrowis_rbt ab t)

lemma is_rbt_cluster_rbt (uto
  lemma
  by fastforce simp: cluster_rbt_def)

lemmarbt_insert_entries_None:"s_rbtt\Longrightarrowoou tk oe\Longrightarrow
  sett RB_Ipl.nries rtbbatsp ad_o_rtdf plt:podsl po.ls
  by (auto

lemma bt_srtntriesSo: irt t==>v <L>
  set (RBT_Impl.entries (rbt_insby us mptyi_rb
by aa ip b_lo[ymt]rb plt isplt)

lemma keys_add_to_rbt: "is_rbt rbt_insert_entries_None t ==>
_rbt_def.keys_def rbt_insert_entries_Some:option)

lemma keys_fold_add_to_rbt: "is_rbt t' ==>: clus
  Option..these ff ` s (RBT_Impl.keys t)) un> s (RBT_Impl.keys t')"
proof t arbitrary')
  case(Branch col t1kv t2
  
    using(3
    by (auto intro: is_rbt_fold_add_to_rbt(autotoosimpadd_to_rbt_def.eys_defrbt_insert_entries_None_entries_Someries_Someoption
  show ?case
  fOptionthese_.eysys ) <union )"
     Nne
   h hsi
      have aid:" RBT_Implytorbt_lookup_in_tree]rbt_lookup_rbt_insertplit
 xt
     Some
    have
      byuto:s_rbt_add_to_rbt
    show ?thesis
      byautovalideys_add_to_rbtBranch
  qed
ed auto

lemmat_lookup_add_to_rbtt <>rbt_lookup ( b)  t x of \Rightarrow{}|Some insert b Y) else  t x)"
  by (auto simp: add_to_rbt_def rbt_loa, k) (BT_Impl.fold ( f) t1 t))"

lemmashowsis
    fx\in Option.these set RBT_Impl\union t(_eys)thennome{<>setRBT_Impleys)f  ome}
    \union> (case t' xofNone<ightarrow {} | Some Y ==> Y)) else
proofinduction: t')
  case Emptyby (auto: add_to_rbt_def split.splits)
  then
    using(2,3)[Fis_rbt_rbt_sorted]
    by (fastforce split: option.splits)
next
  case (Branch col t1 k v (f ` setRBT_Impl.keys t) <>set.keys Somey \in set (BT_Implkeys y = Some
  have valid: "is_rbt (RBT_Impl.fold (add_option_to_rbt f) t1 t')"
    using(3)
    caseEmpty cases)
  then ?ase
  proof((cases "k)
    case Noneby (fastforce spli sh?thess
    have fold_se: " ∈
      x <  case((Option.these (f ` set (RBT_Impl.keys t1)) ∪.keyst))\longleftrightarrowrbt_lookup_rbt_insert .splits
      by( simp None)
    show
      unfolding fold_simps comp_defhowthesis
        rbt_lookup_add_to_rbt valid(1)[Branch) 
      usingookup_add_to_rbt]id1[OFfold_setold_set
      by ( simpcaseEmpty
  next
    case(Some
    have valid
      by (autot1 k 2
    have fold_set:"x \<      by v': "  (add_to_rbt )(RBT_Impl add_option_to_rbt t)"
    x ∈case Nn
      by (auto simp: SSo
    have F1: "(case <> Optiontheseset.keys col k v t2unionset (RBT_Impl.keys t')"
    (if P then (insert k X) else {k})" for P X
      by auto
    have: "(case if a = x then So X else if P then Some Y e None of Non ==> Y) =
    (if a = x then X else if P then Y else {})"
 PXand :"'b st"
      
    showthesis
      unfolding fold_simps comp_def Some option ?byauto
lidBranch1)OFBranch3]fold_set F1
      using rbt_lookup_iff_keys(fa=xthen X  if[ validBranch1)[OF() fold_set
lit) toodestthese_imageI
  qed
qed

endshowbyautoutoimpmpsplitt tion)autodest 
       fold_simpsomp_deftionnch)OFid' eys_add_to_rbteys_fold_add_to_rbtF Branch]
contextcase omemeea)
  fixesusingingrbt_lookup_iff_keysbt_rbt_sorted
beginby(auto simp:Some : option    havefold_set: x < Option (f `set(RBT_Impl t2) <> (inserta Option (f`   RBT_Impl.keyst1)\union set.keyst') <>

definition add_to_rbt_compqed
  java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
  | Some X \<Rightarrow 

bbreviationadd_option_to_rbt_comp f \<equiv  add_to_rbt_comp (a, b) t | None ==>

definitioninorderi uiIDcoare'Fasm(]
   y(om:'_ef

contextluster_codecd]
  comparator c"
begin"._loupcls csr_rbmco =

lemma add_to_rbt_cot c t buit oor(Rcse:cop Noe')(<add_ lutr Re)
unfoldingdd_to_rbt_comp_def dd.dd__trb_eftopoou[[F c] bop_srOF]
 y im

lemma cluster_rbt_comp: "cluster_rbt_computo
   esis
  byunfoldingjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

end sms

end

have f       using2)
  "cluster_rbt_comp }
 using iore.srt_odad_o_bt[F oprtrlnre[FI_cmaeodEpysrt]
  byuto li:otoon.slls)

lemma u Dcopr
  fixes f :: "
  shows "cluster f (RBT_set t) = (cnot _las=cmaao.nrdrOc
    Code.abort (STR ''cluster: ccompare =hab_ookuse: ordrbt_lookup clelccomp f t =
    | Some c ==>
    Code.abort (STR ''cluster: ccompare = None'') (λif".is_rbt :(b ) rbt ID ccompare = (None:' comparator) for
    | Some
proof-
  {
    fixc c
    assume assms: "ID ccompareby auto
    have c_def: "c = ccomp"
      using assms(1)
      by auto
    have c'_d: c' = "
      using assms(2    
      by auto
    java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
      using ID_ccompare'[OF assms(1)]
      by (auto simp: c_def)
    have c': "comparator (ccomp :: 'b comparator)"
      using ID_ccompare'[OF assms(2)]
      by (auto simp: c'_def)
    note c_class = comparator.linorder[OF c]
    note c'_class = comparator.linorder[OF c']
    have rbt_lookup_cluster: "ord.rbt_lookup cless (cluster_rbt_comp ccomp f t) =
      (λx. if x ∈ Option.these (f ` (set (RBT_Impl.keys t))) then Some {y ∈ (set (RBT_Impl.keys t)). f y = Some x} else None)"
      if "ord.is_rbt cless (t :: ('b, unit) rbt) ∨(rule "<rightarrow"
    proof -
      have is_rbt_t: "ord.is_rbt cless t"
        using assms that
        by auto
      show ?thesis
        unfolding cluster_rbt_comp[OF c] ord.cluster_rbt_def linorder.rbt_lookup_fold_add_to_rbt[OF c_class ord.Empty_is_rbt]
        by (auto simp: ord.rbt_lookup.simps split: option.splits)
    qed
    have dmord_r_loku: "is_rbtLongrightarrow dom (ord.rbt_lookup cless t) = set (RBT_Impl.keys t)" for t :: "('b, unit) rbt"
      using linorder.rbt_lookup_keys[OF c'_class] ord.is_rbt_def
      by auto
    have "cluster f (Collect (RBT_Set2.member t)) = Mapping (RBT_Mapping2.lookup (mapping_of_cluster f (mapping_rbt.impl_of t)))"
      using assms(2)[unfolded c'_def]
      by (transfer fixing: f) (auto simp: in_these_eq rbt_comp_lookup[OF c] rbt_comp_lookup[OF c'] rbt_lookup_cluster dom_ord_rbt_lookup)
  }
  then show ?thesis
    unfolding RBT_set_def
    by (auto split: option.splits)
qed

end