Quellcode-Bibliothek Cluster.thy

theory Cluster
  imports Mapping_Code
begin

lemma these_Un[simp]: "Option.these (A ∪ B) = Option.these A ∪ Option.these B"
  by (auto simp: Option.these_def)

lemma these_insert[simp]: "Option.these (insert x A) = (case x of Some a ==> insert a | None ==> id) (Option.these A)"
  by (auto simp: Optionbyuto_ef

lemma these_image_Un]: "Option.these (f ` (A ∪ ) ∪ (f ` B)"
  by (auto (auto: Option.these_defsplit: option) force

lemma these_imageI" x = ome\<>x
  by (force sim (auto simp: Option.these_def)

lift_definitioncluster :: "(' <> 'aoption<Rightarrow 'b set ==>" is
  "λf Y x. if Some x ∈  :: "('b ==> 'b set ==>map is

lemma set_of_idx_cluster: "set_of_idx (cluster f) X) = X"
  by transfer (auto simp: ran_def)

lemma lookup_cluster': "Mapping.lookup (clustertransfer
  by transfer

context ord transfer auto ord
begin add_to_rbt : "a \times'b \Rightarrow ('a, 'b set) rbt ==>

definition add_to_rbt :: "'<>b==>> ('a, 'b set) rbt ==> ('a, 'b set) rbt" where
  "add_to_rbt(a, b) t. case rbt_lookup t aof X  <Rightarrowrbt_insert a (insert b X) t | None ==> rbt_insert a{}t)

abbreviation add_option_to_rbt (λea \ add_to_rbt (a, b) t | None ==> t)"

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

end

context linorder
begin

lemma is_rbt_add_to_rbt: "is_rbt t ==> is_rbt (add_to_rbt
  by (auto simp: add_to_rbt_def split: prod.splits option.splits)

lemma is_rbt_fold_add_to_rbt"cluster_rbt f t = f t = RBT_Impl.fold (add_option_to_rbt f) t RBT_Impl.Empty"
  is_rbt ((RBT_Impl.fold add_option_to_rbt ')
  by (induction t arbitrary

lemma is_rbt_cluster_rbt: (luster_rbt t
  using is_rbt_fold_add_to_rbt (auto simp
  by ( simp cluster_rbt_def

lemma rbt_insert_entries_None: "is_rbt t ==> rbt_lookup t k = None ==>induction t arbitrary: t') (auto 0 0 simp: is_rbt_add_to_rbt s: option.splits)
   is_rbt_fold_add_to Empty_is_rbt
  by ( simp: rbt_lookup_in_tree[symmetric] rbt_lookup_rbtinsert split: if_splits)

lemma rbt_ins (RBT_Impl.entri (rbt_insert k v t)) = insert (k, v) (set (RBT_Impentries t))"
  setRBT_Impl (rbt_insert) insert) (set.entries{(k v''))"
  by (auto simp:

lemma set (.entries (rbt_insert k v t)) = insert(, v) (s (RBT_Impl.entries t) - {(k, v')})
  to imp: add__to_rbt__def RBT_Impl.keys_def rbt_insert_entries_None rbt_insert_entries_Some sit: oio.splits)

lemma eys_fld_add_to_rbt: "is_rbt 🚫: ')
  .these ` setRBT_Implkeys )) <union (RBT_Impl t')"
proof (induction t arbitrary: t')
  case(Bran col t1 k v t2)
  have valid: " (RBT_Implfold (dd_option_to_rbt t')"
    using Branch(3)
    autointro: is_rbt_fold_addd_to_rb
  show ?se
  rooff (c k"java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
    case None
    show ?hesis
      by (auto simp: None Branch(2)[OF valid]  by (auto add_to_rbt_def_insertption)
  nextt
        if<ion BT_Impl set (RBT_Impl.keys t') then Some(\in  BT_ImplSome
    have valid': "is_rbt (add_to_rbt (a, k) (RBT_Impl.fold (aptirbt f) tt11 "
       auto: is_rbt_add_to_rbt valid
     Empty
      by (auto simp: using(2,3OF]
  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 ==> {b} | Some Y ==>: " (RBT_Impl (add_option_to_rbt t')"
  by (auto simp: add_to_rbt_def rbt_lookup_rbt_insert split: optop.splits)

lemma rbt_lookup_fold_add_to_rbt: " ?case
    (if\in Option.these( `set.keys) <>  set (RBT_Impl.keys t). f y = Some x}
    ∪
prooft arbitrary t')
  case Empty∈ (ff `set(RBT_Impl (Branch col k vt2)) <nion keysBranch
  hen?ase
    using3Fs_rbt_rbt_sortedd
    yastforcets
nextfold_setinOption.these (f ` set (RBT_Impl.keys t2)) ∪ aOption (f ` set.keys)) ∪
  case (Branch<in.these ` set (.keysBranch t1 k t2)) \union setRBT_Impl.keys t)"
  have hF1" if Some None None {k} | Some Y ==>
     Branch3java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
    by (hen
  show ?case
  proof (cases "f k")
    case NoneforPX and:"'b set"
    have fold_set <>OptionBT_Impleys ((Option.these (f ` set (RBT_Impl.keys \union set (RBT_Impl.keys t'))) ⟷
      x \>Option.these (f ` set (RBT_Impl.keyst1v 2 unionBT_Impllkeys
      by (auto simp: None)rbt_lookup_add_to_rbt valid(1)[OFBranch(3]fold_set F1
    showthesis
      unfolding fold_simps simp split.splitsauto dest)
        java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
      using rbt_lookup_iff_keysOF is_rbt_rbt_sorted Branch(]
      by (auto simp: None
  next
    case(Some ajava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
    d'is_rbt (a, k) RBT_Impl.fold (add_option_to_rbt f t1
      by (auto intro> rbt_comp_insert c a (insert b X) t)"
     "add_option_to_rbt_comp (λb _ t. case f b fome add_to_rbt_comp (a, b) t | None t)"
    x ∈ 'a option) ==>) rbt ==>"where
      by (
    have F1
    (if c: "comparator c"
      byo
    have F2: "
    (if a = x then X if P then Y else {})"
      for P and : "'b s "
      by auto
    show ?thesis: "cluster_rbt_comp = ord.clu (lt_of_comp c)"
      unfolding comp_def option(2 (2)[OF'] keys_add_to_rbt valid[OF(3)]
        rbt_lookup_add_to_rbt[OF
      java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      by (auto: Some: option.splits ((auto destthese_imageI
  qed
qed

end

context
  fixes c :: "'a comparator"
begin

definition add_to_rbt_comp :: "'a × 'b ==> ('a " ccomp"
  "add_to_rbt_comp = (λareter_rbt_def
  | Some 

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

definition cluster_rbt_comp :    abortTRne_. cluster f(T_set
  "cluster_rbt_comp f t = RBT_Impl.fold (add_option_to_rbt_comp f) t RBT_Impl.Empty"

context
   ccomparator cjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
beginssms ccompare c :: 'a comparator option ccompare(Someb  comparator)"

lemmabt_comp: "add_to_rbt_comp.add_to_rbtlt_of_comp c)"
  unfolding add_to_rbt_comp_def ord.add_to_rbt_def rbt_comp_lusingassms1)java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
  by simp

lemmauto
  ster_rbt_comp_def o_rbt_comp_defd.cluster_rbt_defmp
  osmp_def)

end

end ID_ccompare'[OF assms(2)

lift_definition mappi<Rightarrow> 'a :: ccompare option) ==> ('b, unit) rbt ==> ('a, 'b set) mapping_rbt" is
  "cluster_rbt_comp ccomp"
  usinginorderr_bt_fold_add_to_rbt _mpare] rd_bt
  by (fastforce simp: cluster_rbt_comp[OF ID_ccompare'] ordproof

lemma cluster_code[codeusing assms
  fixes : "':: ccomp ==>) ma"
  shows "cluster f (RBT_set t) = (case ID CCOMPARE('a) of None ==>
    Code.abort (STR ''cluster: ccompare = None'') (λ_. cluster f (RBT_set t))
    | Some c c ==>
    Code.abort (STR ''cluster: ccompare = None'') (λ_. cluster f (RBT_set t))
    |S c' ==> f (RBT_Mappin.impl_of t)))))"
proof 
  h dom_ord_rbt_lookup:"ord.is_rbt cless t <Longrightarro> d(ord.rbt_lookup cless t) = set (RBT_Impl.keys t)"for:"((b, unit"
    
    assume assms: "ID ccompare = (Some c :: 'a coy auauto
    
      using assms(1)
      by auto
    have c'_def: "c' = ccomp"
      assms2)
      by auto
     c: " (ccomp:'a)"
      using ID_ccompare'[OF assms(1)]
      by (auto simp: c_def)
    have c': "comparator ( :: 'b comparator)"
      using ID_ccompare'[OF assms(2)]
      by
    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) ∨ ID ccompare = (None :: 'b comparator option)" for t
    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 dom_ord_rbt_lookup: "ord.is_rbt cless t ==> 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