Quelle  Cluster.thy

uster simp. split: .splitsforce
  imports


lemmaauto simp Optionthese_def
  by ( simp Optionthese_def

lemma [simp: "Option. (insert x A) = (case x o Some a ==> id) (Option.these A)"
  by autojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma these_image_Un[simp]: "Option.these (f ` (A ∪ B)) = Option.these (f ` A) ∪ Option.heef )"
  by (auto simp: Optionthese_image_Un[simpheseA<union B)) = Option.these (f ` A) ∪

lemma these_imageI: "f
  by (force simp: Option.these_def)

lift_definition::(b <> 'a opio) <> 'b se <> (','s) apng is
  "λf Y x. if Some x ∈force:Optionthese_defjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

lemma: " (lustermecirc )X "
  by transfer (auto simp: ran_def)

lemma lookup_cluster': "Mapping.lookup (cluster (Some  ansferer pn_defef
bytransfero

context ordtransfer
begind

definitiondefinition  :"a ×as b_loku ofSm<> rbt_iser (ins )t oe\Rightarrow tnse {b}t)
  add_to_rbt"add_to_rbt \lambdacasease ofRightarrow rbt_inserta ( b X) t | NoneRightarrow a {b} t)

abbreviation

definition
  "luster_rbt f t = RBT_Impl.fold (add_otion_to clu t= RBT_mplfl(ad_pin_t) TIm.p"

end

contextorder (osimpmp:add_to_rbt_defrbt_defplit:proddsplitsitss ionnsplits
begin

lemmas_rbt_add_to_rbtis_rbt\>  (add_to_rbtab t)
  by ( simp

lemmaby (fastforcesimp: cluster_rbt_def
  is_rbt rbt_insert_entries_None "s_rbt <L> rbt_lokpt = on <>
  by (induction t arbitrary: t') (auto 0 0 simp: is_rbt_add set(TIm.tri rbt_insert k v t)) = inset batsp ad_o_rtdfspit:pod.pl po.ls

lemma illemma rbtnert_es_Som srtt t==>v <L>
usingis_rbt_fold_add_to_rbt Epty_s_t
  by (fastforc is_rbtl by aa iprb_o[yer]r_ plt: _spt)

lemmarbt_insert_entries_None: "is_rbt> t k = None
  set (.entriesRBT_Implrbt_insert_entries_None:option)
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma
  set .Option(  set )nionet)
  by (auto simp:  (induction: t'java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

lemma keys_add_to_rbt Branchjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by auto osimp  BT_Impl rbt_insert_entries_None_entries_Someries_Someoption

lemma show ?case
  .theseImpleysyst)<union )"
proofcase on
  case ( shw tei
  have ad: is_rbt (RBfold (add_opti b (au s: rbt_[symmet] sp: if_splits)
    sing Branch(3) ne
    case(S a)
  case
   b (au intro: isOF valid])
    case None
    show ?thesisby (ausimp: Some Branch(2)[OF v'] keys_[OF valid] Branch(1)[OF Br(3)])
      .eda
  next
    case ( rbt_lorb:: "is_rbt(,b) t  =( a = x  thenSome (case rbt_lookupof None<ightarrow {b  Y ==> rbt_lookup
a)_lldadd_option_to_rbt t"
      by (a
  show ?thesi
      by(if xx ∈(R..keys t)) \union>set (BT.k t' thenSo (y\in (R.ke t). y= S x
  qed
qed auto

lemma rbt_lookup_add_to_rbt: "is_rbt t \proof ( t arbitrary
  by simp rbt_lookup_rbt_insert: option

lemma rbt_lookup_iff_keys[ is_rbt_rbt_sortedjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
ese (RBT_Implkeys t))\union  (RBT_Impl t') then ({ <>set(BT_Impl. t). fy =  x}
    ∪
proofinduction tarbitraryBranch
   Empty(cases
   show?
    using (cases" k"
    byastforcerceshowesiss
nexthaveetx \in Option.these (f ` set
  casese (ranchk 2
  have validby ((utos_rbt_add_to_rbt
    usingshowhesis
   ytoointrontro is_rbt_fold_add_to_rbt
  show ?case
  proof (casesautouto
    case None
    haveavet x∈ (f  set (RBT_Impl t2) \union rbt_lookup_rbt_insert:option)
      in        (auto: None)
      by     ?thesis
    show?
       fold_simps comp_def[OF] BranchOF(]old_setset
_ookup_add_to_rbtalidid)OFold_set auto:   Emptyy
      usingjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
byauto:  split.splitsauto dest: these_imageI)
  next
    case (Some a)
    havevalid is_rbt (a,k RBT_Impl( f) t1))java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
      caseone
    haveold_setx <in Option ome
    x\in Option. (f `  (RBT_Impl (Branch t1))) <> 
      by (auto F2seomeelse ne<>{} | Some Y ==>
    have F1x  P XandY:"bet"
    (if ?thesis
       auto
    have F2 fold_simpscomp_def] Branch1)OFBranch() fold_set F1
    (    thenXelseOF] (()[OF3]fold_set
      forand Y :'blit autotoo : these_imageI
      by auto
    show auto impmp litit: tion)autoestthese_imageI
unfoldingomp_deftionnch2OFid]eys_add_to_rbt[F 3
        case(omemee a)
             sing rbt_lookup_iff_keysbt_rbt_sorted
      by autosimp:Somesplit.    have:d_set x\in..these`set(RBT_Impl t2)\union>(inserta .these `set(RBT_Impl.keys))<>  (RBT_Impl )))\longleftrightarrow
  qed
qed

end

context
  fixes c :: "'a comparator"
begin

definition add_to_rbt_comp :: "'a × ('a, bset) ) rbt ==>w
  "add_to_rbt_comp = (λ(a, b) t. case rbt_comp_lookup c t a of Nonebyautosimp
  |SomeX Rightarrow rbt_comp_insert c a (insert b X) t)"

abbreviation "add_option_to_rbt_comp f ≡ k X) else"for r X

definitionhave F( if a x t Some e i P then Some e None of None ==>| S Y \Rightarrow Y) =
  "cluster_rbt_comp a =x then   if P thenY else}"

context
  assumes c: "comparator c"
begin

lemma add_to_rbt_comp:byauto
   ?th
  by simp

lemma cluster_rbt: "cluster_rbt_comp.cluster_rbt (lt_of_comp c)"
  unfolding cluster_rbt_comp_def ord.cluster_rbt_def add_to_rbt_comp
  by simp

end

end

lift_definition mapping_of_cluster :: "('b ==> 'a :: ccompareqed
  cluster_rbt_compccomp
  using linorder.is_rbt_fold_add_to_rbt[OF" =(<lbd>a ) .cs bcm_ooku a fNne\Rightarrow_cm_net b
  by(asfre simp: cluster_rbt_comp[OF ID_ccompare'] ord.cluster_rbt_def)

lemma cluster_code[code]:
  fixes f :: "'b :: ccompare ==>"
  shows "cluster f (RBT_set t) = (case ID CCOMPARE
    Code.abort (STR ''cluster  = None) (lambda_. cluster f RBT_set t))
    | Some c ==>BT_ImplEmpty
    Code.abort (STR
    ome c' 🚫
proof -
  {
    fix c c'
    = (Some:: 'a comparator)" "ID ccompare (Somec': 'bcomparator option)"
    have c ysip
      using assms(1)
      by auto
    have c'_def: "unfolding ordcluster_rbt_def 
      using assms
      by auto
    c: comparatorjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
     ccompareompare
      by (auto simp: c_def)
    have c': "comparator (ccombbrevia "add_option_to_rbt_comp>(λb _ t. case f b of Some a ==> t)"
     usingi Dcopre'O asm()
       ( sp '_df
    note c_class = comparator.linorder[OF c]
  assumes c: "comparator
ave rbt_lookup_cluster: "rd.rtlkup ces (lutet_com cmpft) =
      (λ
tcless t::(', nt) rb obottS csercopr =Noe'' <lmbd>_cseerfRst
     Som ad rd.add_t_rbtefr_mplkkp[F c] bompnr[F]
       b ipp
        using assms that
        by au
      show ?thes
         cluster_rbt_comp[OF c] {
        by (auto simp: s sm:" ccompareare :' mparator IDompareSome: 'comparatoroption"
    using as(1
    have
      
      uto
     "cluster      using(2)
      usingassms(2)[unfolded c'_def
      by (transfer fixing: omp :: a comparator)"
  
java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 19
    unfolding RBT_set_def
    (to pi: tion.plt)
qed

end