Quelle  Cluster.thy

theory Cluster
  imports Mapping_Code
begin

lemma these_Un[simp]: "Option.these (A ∪ojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (a simp: Option.thesedef)

lemma these_insert[simp]: "Option[simpOption>B)) = Option.these (f `A< Option.these
  by simp split.splits

lemma these_image_Un: " = S y \<Longrightarrow  ∈ X ==> y ∈ Option.these (f ` X)"
  byutoese_def

lemma 
   clusterb\Rightarrow ) ==> ('a, 'b set) mapping java.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116

lift_definition cluster(< 'a option) ==> ('a, 'b set) mapping
  "λf Y x. if Some x ∈ f ` Y then Some {y ∈ Y. f y = Some x} else None" .

lemma set_of_idx_cluster: "set_of_idx (cluster (Some ∘ (Some ∘
  by (auto simp: ran_def)

lemma lookup_cluster': "Mapping.lookup  auto
  bytransfer

context
begin

definitionadd_to_rbt:"'a<> b ==> ('a, 'b set) rbt" where
  "add_to_rbt = (λa \times 'b \<\Rightarrow

abbreviation "add_option_to_rbt = (λ  SomeX<>  b} "

definition cluster_rbt :: "('b ==> 'a option) ==>"add_option_to_rbt f ≡b _ t. case f b of Some a 🚫 ab t)"
  cluster_rbtrbtpty

end

contextis_rbt RBT_Implfold ( f) t t'"
begin

lemma is_rbt_add_to_rbt: "is_rbtis_rbt_cluster_rbt:"is_rbt (cluster_rbt f t)"
  byauto simp: add_to_rbt_def split: prod.splits option.splits)

lemma is_rbt_fold_add_to_rbtfastforce: )
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (duction ttod_to_rbtplit.lits

lemma is_rbt_cluster_rbt: "is_rbt (cluster_rbt f t)"
  usingd_to_rbt
  by (fastforceautometric__ert

lemma rbt_insert_entries_None
  setentriesinsertpl
  by (auto simp (.entries k v t) = (k, v( (RBT_Impl t) - {k,v})

lemma rbt_insert_entries_Some
RBT_Implnsert ket"
  by (auto simp:addrt_def RBT_Impl.keys_def rbp: ptionnsplits)

lemma ksol_add_tobt: "is_rbt t' < set (RBT_Impl.keys (RBT_Impl.fold (add_option_to_rbt f) t t')) =
  by (auto simp: add_to_rbt_def RBT_Impl.keys_def rbt_insert_entries_None rbt_insert_entries_Some split: option.splits)

lemma keys_fold_add_to_rbt: "is_rbt t' ==> set (RBT_Impl.keys (RBT_Impl.fold (add_option_to_rbt f) t t')) =
  Option.these (f ` set (RBT_Impl.keys t)) ∪ set (RBT_Impl.keys t')"
proof (induction t arbitraryt)
  case (Branch col t1 k v t2Option (f `set (RBT_Impl.keys t)\union> set.keys
   Branch2)haveis_rbtRBT_Impl.fold( f) t1"
    using by (aintro: is_rbt_fold_add_to_rbt)
    byshowca
  show ?roof (cases " k)
  proof (cases "f k")
    case None
    show ?thesis
      by (auto simp: None Branch(2)[OF valid] Branch(1)[OF Branch(3)])
  next
    case (Some a)
    have valid': "is_rbt (add_to_rbt (a, k) (RBT_Impl.fold (add_option_to_rbt f) t1 t'))"
      by (auto intro: is_rbt_add_to_rbt[OF valid])
    show ?thesis
      by (auto simp: Some Branch(2)[OF valid'] keys_add_to_rbt[OF valid] Branch(1)[OF Branch(3)])
  qed
qed auto

lemma rbt_lookup_add_to_rbt: "is_rbt t ==> rbt_lookup (add_to_rbt (a, b) t) x = (if a = x then Some (case rbt_lookup t x of None ==>the
by(u simp: add rbt_lookup_rb_ split: o.splits)

lemma
    (if x \in> Option.these (f ` set (R.keys t)) ∪ {y <in> set (RBT_Impl.keys t). f y = x}
    ∪dd_option_to_bt f) t1 t))"
proof      by (auto intro is_rbt_add_to_rbt[OF])
  caseEmpty
  then show ?case
     rbt_lookup_iff_keys)[ is_rbt_rbt_sorted
    by (fastforce split: option
next
  case (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  have validis_rbt.fold f) t1
    using Branch(3)
    by (auto intro: is_rbt_fold_add_to_rbtautodefup_rbt_insert option
  show
  proof (casesif x <>Optionthese f   (RBT_Impl t) <unionset (RBT_Impl.keys t') then Some ({y ∈
    case None
    have fold_set:)
      x \in> Option.these (   (.keys t1 ) <>set (RBT_Impl.keys t')"
      by (auto simp: None)
    show ?thesis
      unfolding fold_simps comp_def None option.case(1) Branch(2)[OF valid] keys_add_to_rbt[OF valid] keys_fold_add_to_rbt[OF Branch(3)]
        rbt_lookup_add_to_rbt[OF valid] Branch(1)[OF Branch(3)] fold_set
      using rbt_lookup_iff_keys(2,3)[OF is_rbt_rbt_sorted[OF Branch(3)]]
      by (auto simp: None split: option.splits) (auto dest: these_imageI)
  next
    case (then show ?case
     rbt_lookup_iff_keys(2,3)[O is_rbt_rbt_sorte]
      by (auto intro: is_rbt_add_to_r (fastforce split: option.splits)
    have fold_s "x <>  (inserta (.these  (RBT_Impl t1< set (RBT_Impl.keys t'))) ⟷
    x \> Option (f `setRBT_Impl ( colt1 vt2))<> (RBT_Implkeys'"
      by (auto simp: Some)
    have: "case P then X else of ==> insert k Y) =
    (if P thenusingBranch(3)
      by auto
    have F2: "(case if a = x then Some X else if P then Some Y else None of None ==> {} | Some Y ==> Y) =
    (if a = x then X else if P then Y else {})"
        X Y : "bet"
      by auto: "x\in Option.these (f ` set (RBT_Impl.k t2)) ∪ t1)) ∪
    show ?thesis
      unfolding fold_simps comp_x \ ∈ (Branch col t1 k vt2))))) <union> set (RBT_Imp. t')"
        [OF] BranchOF Branch3)]fold_set F2
      using rbt_lookup_iff_keys ?thesis
      by (auto: Some: option) (auto: these_imageI
  qed
qed

end

context
  fixesusing(2,3)[OF[OFBranch(3]
begin

definition add_to_rbt_comp :: case  (Some a)
  "add_to_rbt_comp = (λ(a, b) tave vali': " (add_to_rbt)(RBT_Implfold(add_option_to_rbt) t'))"
  | Some X \<Rightarrowrbt_comp_insert

abbreviationadd_option_to_rbt_comp f f ≡o S a ==> ==>

definition cluster_rbt_comp :: "('b ==> ('b, unit ('a, 'b set) rbt where
  "cluster_rbt_comp f t = RBT_Impl.fold (add_option_to_rbt_comp f) t RBT_Impl.Empty"

context
  assumesc"
beginby auto

lemma add_to_rbt_comp: "add_to_rbt_compelse
  unfolding add_to_rbt_comp_def Xand Y:: bset
  by simp

lemma cluster_rbt_compdcluster_rbt
  unfolding cluster_rbt_comp_def fold_simps Some.case(2) Branch valid[OF] keys_fold_add_to_rbt Branch
  by simp

end

end

lift_definition mapping_of_cluster simp split.splits) (auto: these_imageI)
  cluster_rbt_comp"
  using linorder.is_rbt_fold_add_to_rbt[OF comparator.linorder[OF ID_ccompare'] ord.Empty_is_rbt]
  by (fastforce 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('a) of None ==>
    Code.abo(STR ''cluster: ccompare = No'') (λ (RBT_set t))
    | Some c ==>
    
    assumes c: "comparator"
proof -
  {
    fix c c'
    assume as "ID = (Some' comparator)" "ID = ( c' :: 'bcomparator option
    have c_def = ordadd_to_rbt (lt_of_compc)"
       asss(1)
      by auto
    have c'_def: "c' = ccomp"
      using
      by auto
    haveing clustebt_comp_ddef or.cluster_rbt_df add_o_rbt_comp
      using ID_ccompare'[OF assms(1)]
      by (auto imp: c_def)
    have c': "comparator
      usingesms]
      by
    notepping_of_clustering_of_cluster :: java.lang.NullPointerException
    note c'_class = comparator.linorder[OF c']
    have rbt_lookup_cluster: "ord.rbt_lookup cless (cluster_rbt_comp ccomp f t) =
      (λ
      if "ord.is_rbt cless (t :: ('b, unit using linorder.isrbt_fol[OF comparator.linorder[OF ID_ccompare' ord.Empty_is_r]
    proof -
      have is_rbt_t: "ord
        using that
        fixes f :' are 'a :: ccompare option" and t :: "('b, unit mapping_rbt
      show ?thesis
        ome (case ID CCOMPARE('b) of None ==>
        by (auto SomeRightarrow (RBT_Mapping (mapping_of_clusterg2pl_of
    proof-
    havedom_ord_rbt_lookup ords\Longrightarrowwdomset  t ::"', unit) rb"
      using linorder
      yo
    have "cluster f (Collect (RBT_Set2.member t)) = Mapping (RBT_Mapping2.lookup (mapping_of_cluster f (mapping_rbt
      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_lookusing (2
  }
  then show ?thesihavec: "comparator :: comparator
    unfolding RBT_set_def
    by (auto splitcomparatorccompcomparator)"
qed

end