Quelle  Cluster.thy

theory Cluster
  imports Cluster
begin

lemma these_Un[]: "Optionthese (A ∪ Option.these B"
  by (auto simp: Optionthese_def

lemmaimports
  by (auto: Optionthese_defsplit:option) force

lemma
  (auto simp:Option.these_def)

lemmaby auto: Option.these_def)
  lemmathese_insert] ""Option.thesex ofRightarrow insert a | None ==>

_ cluster :: "(b \Rightarrow<> 'b set ==> (','bset)t apn"
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma set_of_d_cutr set_of_ (c set_of_idx_cluster cl(Some ∘
   ras (at sm: r rnde)

lemma se_et
   uo

context ord
begin

definitionadd_to_rbt : "\times
  > ) asebt_lookuprt insertsertXtone\Rightarrow>_ert {bt)"

abbreviation

definition d  add_to_rbt =(\lambda>(a, b) t. ca rbt_lookup t a of Some X ==> ==>
 uster_rbt =RTmll a_pobftBmy

end

contextnorder(to m:adtospltposlis ins
lemmas_rbt_add_to_rbt it ==>

java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
 by (utm adt_rbdf pltprplitsoinpt

lemma is_rbt_fold_add_to_rbt: " utoo
  bt_insert_entries_Someme_tt\Longrightarrowrbt_lookup t k = Some 'Longrightarrow
  by is_rbt_fold_add_to_rbt

lemmay autoimpookup_in_treemetricrbt_lookup_rbt_insertlit
  usingis_rbtrbt_lookup ==>RBT_Impl RBT_Impl rbt_insert_entries_None split.splits
  by (fastforce simp

lemma rbt_insert_entries_None(RBT_Impl ( sets )nion set')
  set Branch col t1 k v)
  by (auto simp: rbt_lookup_in_tree

lemma rbt_insert_entries_Some: "is_rbt t ==> (auauto :: aRBker rbt_insert_entries_Some split: o.splits)
  set (RBT_Impl.entries (rbt_insert k v t)) = insert (k,proo (t (f ` set (RBT_Ikeys t) ∪set (RBT_Impl.keys t'"
 simptric split

lemma keys_add_to_rbt:singext
  Some

lemma      utojava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
  Option
proof bt_lookup_add_to_rbt t \Longrightarrow  (add_to_rbtb) )x  if x  Somecase rbt_lookupof <ightarrow>b 
      by(
  show
    if>Option (f `set(RBT_ImplunionBT_Impl ' enSome( in (RBT_Impleys) f=ome
    by (auto<unioncase rbt_lookup  None 🚫
  show (induction : Branch
  proof "f k"
    caseusing (cases")
 ?ei
      by (auto simp: None Branch(2)[OF va
  next
    case (Some a)
     (r col t1 k v t)
      by au intro: is[OF valid])
    show ?th
      by (auto simp: Some Branch(2) by (auto in: ii)
  qed
qed aut

lemma rbt_lookup_add_tha fold_set:" > Option.these `set.keys))<>  set (RBT_Impl ') ⟷
  by (auto simp: add_to_rbt_def splitoptionsplits

lemma rbt_lookup_fold_add_to_rbt: "is_rbt l ∈
    (sh ?
    \<uniondingOB3)]fold_
proofusinglo[OF vai] Branch(1[OF Branch(3)] fol
   Empty
  then show ?case
    usingne
    by (fastforce splicase (S a)
next
  case (Branch col t1 k vk v t)
  have valid: "is_rbt fold_set x< (auto simpNone: optionsplits) (auto dest)
     is_rbt.foldt')"
    by (auto intro: is_rbt_fold_add_to_rbt)
  show ?case
  proof (cases "f k")
    case o
     fo: "x∈case
      x \   Y set
      by (auto simp
    showthesisauto
      unfolding comp_def(1  Branch(]fold_set F2
        rbt_lookup_add_to_rbt]Branch Branch
      using rbt_lookup_iff_keys P X andY: "b s: option.splits(auto t)
      by(au sim: Nonesplit option.splits) d: )
   co Some opt.case(2) Bran(2[ valid]k[OF valid] keOF ()
     (Some )
    have valid': usir(2,3)[OF is_rb[OF Branch(3)]]
      by (auto intro: is_rbt_add_to_rbt[OF valid])
    have fol:"<Option (.keysunion ((set \unionRBT_Impl )java.lang.StringIndexOutOfBoundsException: Index 187 out of bounds for length 187
    x ∈ 'b ==>a bset ('a, 'b set) rbt" were
      by (a : Some)
    have F1: "(case if P then Some X  X==>
    (if P then (insert  {k})" forP
      by auto
    have F2: "casefa=xthenomeeXelseifeYelse==> {} SomeY <> 
    (if x thenXelseifP then Y  {)java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
      for P X and
       
    showthesis
      unfolding fold_simpsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        rbt_lookup_add_to_rbt[OFlemmacluster_rbt_comprbt_comp = ord
      using rbt_lookup_iff_keys
      by (autojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  qed
qed

end

context
  fixes c :: "'a comparator"
begin

definition add_to_rbt_comp :: "'a ×"cluster_rbt_comp ccomp"
  add_to_rbt_comp= (\\amba(,b)t aert_oplopctao o <> rbt_oop_isrc a{}t
  | Some X ==>b fatoc

abbreviation "add_option_to_rbt_comp f ≡ 'a :: ccompare option" and t :: "('b, unit) mapping_rbt"

definition cluster_rbt_comp :: "('b ==>: ccompare'')(λ(RBT_set))
  "cluster_rbt_comp f t = RBT_Impl.fold (add_option_to_rbt_comp f) t RBT."

contextc'< (RBT_Mapping (mapping_of_cluster f (RBT_Mapping2.impl_of t)))))"
  assumes c: "comparator c"
begin

lemma add_to_rbt_comp: "add_to_rbt_comp = ord.add_to_rbt (lt_of_comp-
  unfolding add_to_rbt_comp_def c : optionID = Some c :: 'b comparatoroption"
   m

lemma cluster_rbt_comp: "cluster_rbt_comp = ord.cluster_rbt
   cluster_rbt_comp_deford.cluster_rbt_def add_to_rbt_comp
  by simp

end

end have c "comparator (ccomp :: 'a comparator)"

lift_definition mapping_of_cluster :: "('b ==><R> rbt_comp_insert c a using ID_cco[OF assms(1)]
  
  using li.i sg _oeOFasss)]
  by ( tsp: 'de)

lemmaluster_code[co[ce]
  fixes f :: "'b :: ccompare ==>lookup ssmp
  shows "cluster f (RBT_set t) = (case ID C
   ode.ao T''ute coa Nn'') lbda>_clter(RBT_t )
    | m ud rd.ad_rtdfbcpopF c] rt_cmietF c
    Code.abort (STR ''clusthyi
    | Some c' ==>
proof -
 
    fix c c'
    ssumeass ID ccompare = (Some c : 'acomoption)"" cc = (S c' : 'bb c o)"
    have
       1)
      by have
    have cuto
       assms
      by auto assms'_]
omp: 'a comparator"
      using ID_ccompare'[OF assms(1)]
      by (auto simp: c using norei_btl_dd_rtOF cmaao.iodrO Dcopr'] r.mt_i_b]
    haveb by (t i: pinn.spli)
      usingI_cmae
      by (auto simp: c'_def)
    e class opaar.liorde[O ]
    note c'_class = comparator.linorder[OF c']
    haveve rblook_clutr:"rdupclessess (cluster_rbt_comp)=
      (λ (case ID CCOMPARE('b) of None ==>
       ord cless (t : ',unit) ∨ : b  option" f t
    proof -
      have is_rbt_t: "ord -
        using assms thatfix c cc'
        byto
      show ?thesis
        unfolding cluster_rbt_comp[OF c] ord.cluster_rbt_def linorder.rbt_lookup_fold_add_to_rbtusing()
        by (autoc'_def  ==ccomp
    qed
    have dom_ord_rbt_lookup: "ord.is_rbt cless t ==>
      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