Quellcode-Bibliothek RBT_Set.thy Sprache: Isabelle

(*  Title:      HOL/Library/RBT_Set.thy
    Author:     Ondrej Kuncar
*)

section \<open>Implementation of sets using RBT trees\<close>

theory RBT_Set
imports RBT Product_Lexorder
begin

(*
  Users should be aware that by including this file all code equations
  outside of List.thy using 'a list as an implementation of sets cannot be
  used for code generation. If such equations are not needed, they can be
  deleted from the code generator. Otherwise, a user has to provide their
  own equations using RBT trees.
*)

section \<open>Definition of code datatype constructors\<close>

definition Set :: "('a::linorder, unit) rbt \ 'a set"
  where "Set t = {x . RBT.lookup t x = Some ()}"

definition Coset :: "('a::linorder, unit) rbt \ 'a set"
  where [simp]: "Coset t = - Set t"

section \<open>Lemmas\<close>

subsection \<open>Auxiliary lemmas\<close>

lemma [simp]: "x \ Some () \ x = None"
by (auto simp: not_Some_eq[THEN iffD1])

lemma Set_set_keys: "Set x = dom (RBT.lookup x)"
by (auto simp: Set_def)

lemma finite_Set [simp, intro!]: "finite (Set x)"
by (simp add: Set_set_keys)

lemma set_keys: "Set t = set(RBT.keys t)"
by (simp add: Set_set_keys lookup_keys)

subsection \<open>fold and filter\<close>

lemma finite_fold_rbt_fold_eq:
  assumes "comp_fun_commute f"
  shows "Finite_Set.fold f A (set (RBT.entries t)) = RBT.fold (curry f) t A"
proof -
  interpret comp_fun_commute: comp_fun_commute f
    by (fact assms)
  have *: "remdups (RBT.entries t) = RBT.entries t"
    using distinct_entries distinct_map by (auto intro: distinct_remdups_id)
  show ?thesis using assms by (auto simp: fold_def_alt comp_fun_commute.fold_set_fold_remdups *)
qed

definition fold_keys :: "('a :: linorder \ 'b \ 'b) \ ('a, _) rbt \ 'b \ 'b"
  where [code_unfold]:"fold_keys f t A = RBT.fold (\k _ t. f k t) t A"

lemma fold_keys_def_alt:
  "fold_keys f t s = List.fold f (RBT.keys t) s"
by (auto simp: fold_map o_def split_def fold_def_alt keys_def_alt fold_keys_def)

lemma finite_fold_fold_keys:
  assumes "comp_fun_commute f"
  shows "Finite_Set.fold f A (Set t) = fold_keys f t A"
using assms
proof -
  interpret comp_fun_commute f by fact
  have "set (RBT.keys t) = fst ` (set (RBT.entries t))" by (auto simp: fst_eq_Domain keys_entries)
  moreover have "inj_on fst (set (RBT.entries t))" using distinct_entries distinct_map by auto
  ultimately show ?thesis
    by (auto simp add: set_keys fold_keys_def curry_def fold_image finite_fold_rbt_fold_eq
      comp_comp_fun_commute)
qed

definition rbt_filter :: "('a :: linorder \ bool) \ ('a, 'b) rbt \ 'a set" where
  "rbt_filter P t = RBTfold \

lemma Set_filter_rbt_filter:
  "Set.filter P (Set t) = rbt_filter P t"
  by (substSet_filter_fold)
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

subsection \<open>foldi and Ball\<close>

lemma Ball_False: "RBT_Impl.fold (\k v s. s \ P k) t False = False"
by (induction t) auto

lemma rbt_foldi_fold_conj:
  "RBT_Impl.foldi (\s. s = True) (\k v s. s \ P k) t val = RBT_Impl.fold (\k v s. s \ P k) t val"
proof   deleted from the code generator. Otherwise, a user  own equations using RBT trees
  case(Branch) then ?case
    by (cases "RBT_Impl.fold (\k v s. s \ P k) t1 True") (simp_all add: Ball_False)
qed simp

lemma foldi_fold_conj: "RBT.foldi (\s. s = True) (\k v s. s \ P k) t val = fold_keys (\k s. s \ P k) t val"
unfolding fold_keys_def including rbt.lifting by transfer (rule rbt_foldi_fold_conj)

subsection\<open>foldi and Bex\<close>where]: "Coset t = -Settjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

lemma Bex_True: "RBT_Impl.fold (java.lang.StringIndexOutOfBoundsException: Range [0, 75) out of bounds for length 0
by ( t) auto

lemmaby (simp add: Set_set_keys)
  "RBT_Impl.foldi (\s. s = False) (\k v s. s \ P k) t val = RBT_Impl.fold (\k v s. s \ P k) t val"
proofinduction: val
  case (Branch c t1) then show ?case
v s. \<or> P k) t1 False") (simp_all add: Bex_True)
qed simp

lemma foldi_fold_disj finite_fold_rbt_fold_eq "comp_fun_commute f
unfolding including.liftingtransfer ( rbt_foldi_fold_disj

subsection

subsubsection <open>concrete\<close>

text \<open>The concrete part is here because it's probably not general enough to be moved to \<open>RBT_Impl\<close>\<close>

definition assmshave:" (RBT.entriest) RBT. t"
  wheref t= List tl ) hdt)"

paragraph \<open>minimum\<close>

definition rbt_minthesis  auto comp_fun_commute *)
  where

:(Branch  vrt
by  (auto simp: rbt_greater_propwhere[]:"fold_keys ftA RBT.fold (\k _ t. f k t) t A"

left_le_key(  kvrt
by (auto( simp  o_defsplit_def keys_def_alt)

lemma:
  fixes k: _::linorder
  shows "Finite_Set. f A (Set t) = f t A"
-

lemma " (RBT. t) = fst ` (setRBTentriest) by autosimp keys_entriesjava.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98
  "is_rbt (Branch c RBT_Implultimately show thesis
  : fold_min_triv  simpjava.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104

=\> .  kSet ) }
(Branch.Empty) =k
  "rbt_min_opt (Branch c ( SetfilterPSett Ptjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

lemma rbt_min_opt_Branch \<open>foldi and Ball\<close> Ball_False(java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
" \ rbt.Empty \ rbt_min_opt (Branch c t1 k () t2) = rbt_min_opt t1"
by cases) auto

lemma [case_names left_non_empty
      ( "RBT_Impl.fold \lambdakvs s\ P k True" simp_all: Ball_False
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   "
  assumes \Andcolorb.
  shows "P t"
  using assms
proof (induct t)
  case Empty
  then
next
  case Bex_True fold
  then by( t) auto
qed"BT_Implfoldi(<>.s=False)\lambdak .s\ P k) t val = RBT_Impl.fold (\k v s. s \ P k) t val"

lemma rbt_min_opt_in_set:
  fixesinduction : val
  assumes( c t1)then?case
  showscasesfold
using assmsqedsimp

lemma rbt_min_opt_is_min:
  fixes t ::java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumes "rbt_sorted t"unfoldingfold_keys_def rbt by transfer )
  subsection
  shows\Andy y \<in> set (RBT_Impl.keys t) \<Longrightarrow> y \<ge> rbt_min_opt t"
using \<open>concrete\<close>
proofinduction: rbt_min_opt_induct
  case empty  whereft=Listtl) ((. t)"
  then ?case bysimp
next
  caseleft_empty
then  ?case (uto:key_le_right delrbt_sorted)
next
  case (by  (auto simp: rbt_greater_prop)
then "y "|" set (RBT_Impl.keys t1)" | "y \ set (RBT_Impl.keys t2)"
    by auto
  then show ?case
  proof cases
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    with left_non_empty "(\x\set xs. k \ x) \ List.fold min xs k = k"
by simp:  introleft_le_key)
  next
    case 2
    with left_non_empty:
      by (autoBranchEmpty)java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
  nextjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    case y: java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
    have "rbt_min_opt t1 \ k"
      using left_non_empty by ( add )
    moreover "k \ y"
      using y by ( add key_le_right
    ultimately show ?thesis cases) auto
      using left_non_empty y by (simp add: rbt_min_opt_Branch)
  qed
qed "P .Empty"

lemma "\color t1 a b t2. P t1 \ P t2 \ t1 \ rbt.Empty \ P (Branch color t1 a b t2)"
  assumes "t \ RBT_Impl.Empty"
  assumesshows Ptjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
  shows Empty show? bysimp
proof
   assms hd. )#tl.keys .t  ( t) simp_all
  with assms   showcasebycases=rbtsimp_all
    by (simp add
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [5, 3) out of bounds for length 60

<openjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32

:(a:,unit
  where " "

lemma:
  fixes assms
  shows "(\x\set xs. x \ k) \ List.fold max xs k = k"
by( xs (uto add)

lemma fold_max_rev_eqcasejava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  fixes java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 4
  assumes" \ []"
  showsnext
add.set_eq_fold])

lemma rbt_max_simpsconsiderk  y
  assumesb java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  proofjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13

  have       auto:  intro )
    java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
   show by( simp: rbt_max_defrbt_fold1_keys_def)
qed

  n
  " "rbt_min_opt \<le> k"
  "rbt_max_opt (Branchc lt k v(Branch rc rlc rk rv rrt)) = rbt_max_opt(Branch rcrlcrk rvrrt)java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99

lemma rbt_max_opt_Branch:
  "t2 \ rbt.Empty \ rbt_max_opt (Branch c t1 k () t2) = rbt_max_opt t2"
by (cases t2) auto

lemma       using left_non_empty ( add: key_le_right
  fixes:"' :linorder unit) RBT_Impl.rbt"
  assumes "P rbt.Empty"
  assumes "\color t1 a b t2. P t1 \ P t2 \ t2 = rbt.Empty \ P (Branch color t1 a b t2)"
assumes" P t2\ t2 \ rbt.Empty \ P (Branch color t1 a b t2)"
  shows "P t"
  using assms
proof (induct t)
  case Empty
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
next
  ssumes
shows tjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
qed

lemma rbt_max_opt_in_set:
  fixes:"' :linorder, unit) RBT_Impl.rbt"
\<noteq> rbt.Empty"
  shows  <  RBT_Impl
using

java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
t : (a: , RBT_Impl
  assumes "rbt_sorted t"
assumes
  shows "\y. y \ set (RBT_Impl.keys t) \ y \ rbt_max_opt t"

proofshows
  case empty( xs( simp: max_def
  then? by simp
next
  fixes :: (a :linorder
then?ase intro simp: rbt_sorted)
next
  case right_non_empty k vt2
then " "|"\java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
    by auto
t showcase
  proofshows( c    RBT_Impl
       ".fold max (tl(revRBT_Impl.keys lt@[]) hd (rev(RBT_Impl. lt @ [k])) kjava.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
    with show ?
      by (auto ?thesis  add rbt_fold1_keys_def)
  next
    case java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    with
      by  rbt_max_opt   (    rrtBranch)
  next
    case y: 3
lemma:
      using
    moreover " \ k"
      using rbt_max_opt_induct empty right_non_empty
     ?java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
      usingright_non_empty add)
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
qed inductjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16

lemma
assumes \<noteq> RBT_Impl.Empty"
    then show?caseby casesrbt
  showsjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 3
proof -
  from   assumes  <noteq> rbt.Empty"
  with assms show ?thesis
    by (simp add: rbt_max_def    "rbt_max_opt t \ set (RBT_Impl.keys t)"
      .set_eq_foldsymmetric rbt_max_opt_in_set
qed

\<open>abstract\<close>

context   "\y. y \ set (RBT_Impl.keys t) \ y \ rbt_max_opt t"
lift_definition fold1_keys (induction rule)
is .

lemma fold1_keys_def_alt
  "fold1_keys f t = List.fold f (tl (RBT.keys t)) (hd (RBT.keys t))"
  by transfer

_fold1_keys
  assumes fjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
  assumes "\ RBT.is_empty t"   consider  "|\java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
  shows    cases
proof -     right_non_empty?
  from      auto: intro rbt_max_opt_in_set
  casejava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
    by auto:fold1_keys_def_alt fold_def_alt set_eq_fold])
qed

paragraph

lift_definitionhave"bt_max_opt\

lift_definition  "y\le>java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

lemma r_min_alt_def
qed

:
java.lang.NullPointerException
   "rbt_max t t"
using  by ( intro)

fold_keys_min_top_eq
fixest:"':linorder,bounded_lattice_top},unit) rbt"

  showsMax [] Max_eqI)
proof
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      List fold1_keys"'\Rightarrow a
    ( add[symmetric
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    add
  show  (simp:)
    using assms
    unfolding fold1_keys_def_alt
    apply "not>RBT.is_empty "
     case_tac

    apply (subst
     apply   show t usingassmsjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
    apply (substqed
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    done
qed

paragraph \<open>maximum\<close>

lift_definition r_max_opt: "'a : linorder, unit)rbt \ 'a" is rbt_min_opt .

lift_definition r_max_opt r_min_alt_def:"r_min t =fold1_keysmin t"

lemma r_max_alt_def: "r_max t = fold1_keys max t"
by transfer (simp addtransfer addrbt_min_defjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

lemma "r_mint t"
  assumes
  shows "r_max t = r_max_opt t"
using unfolding bytransfer intro)

lemma fold_keys_max_bot_eq:
  fixes t :   "\ (RBT.is_empty t)"
  assumes
  shows " -
proof
have:"t RBT_Impl. t \ [] \ List.fold max (RBT_Impl.keys t) bot =
.fold hd.keyst  (RBT_Impl t) "
    by (simp   * ".min( )top=Listfoldminxs x" x :: ' ndxs
  have **: "List.fold max (x # tric])
    bysimp:sup_max])
  show ?thesis
    using assms
     is_empty_empty
    apply transfer
   apply( t)
     apply applycase_tac)java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
     subst
     apply simp
    apply (subst *
    apply simp
    done
qed

end

section \<open>Code equations\<close>

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

declare list.set[code] (* needed? *)

lemma  [code
  "Set.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by (auto simp: Set_def)

lemma UNIV_Coset [code]:
  "UNIV = Coset RBT.using unfolding is_empty_empty transfer( intro: rbt_max_eq_rbt_max_optjava.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
by (auto simp t :"':{,bounded_lattice_bot} unit) rbt"

lemma is_empty_Set [code]:
  "Set.s "fold_keys t  = fold1_keys java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
  using non_empty_keys [of t] by       ist. max((RBT_Impl t) #tl.keys"

lemma compl_code [code]:
  " =Coset "
  "- Coset xs = Set xs"
by (simp_all  (imp: sup_max[symmetric

lemma [code
       fold_keys_def_alt
  "x \ (Coset t) = (RBT.lookup t x = None)"
by (    apply( t)

lemmaapply(subst
  "Set applysimp
     ( **)
by      simp

lemma remove_code [code]:

  SetremoveCoset =Coset.insert )"
by (auto simp: code_datatype Set

lemmadeclarelistset[code (* needed? *)
"A \ Set t = rbt_filter (\k. k \ A) t"
simp Set_filter_rbt_filter inter_Set_filter

lemmaunion_Set_Set]:
  "Set t1 \ Set t2 = Set (RBT.union t1 t2)"
by (auto UNIV_Coset [code:

lemmaunion_Setjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
ld_keys.insertA"
proof
  interpret comp_fun_idem  ".is_empty ( t) = RBTis_empty t"
    by fact)
  from  xs xs
? by ( simp: union_fold_insert
qed

lemma minus_Set [code]:
  "A - Set t = fold_keys Set.remove t A"
proof -
  interpret comp_fun_idem( add)
    lemma [code
  from finite_fold_fold_keys[OF]
  show ?thesisinsert )=Cosetxt"
qed

lemma inter_Coset_Coset [code]:
byauto: Set_defjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
by (auto simp add: lookup_union map_add_Some_iff Set_def)

lemma inter_Coset [code]:
Ajava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
by (imp add: Diff_eq symmetric)

lemma inter_Set]:
  A\inter Set(<lambda>k. k \<in> A) t"
proof
  have *: "lemma [code]:
  show ?thesis by (simp( simp: lookup_union Set_def
qed

lemma minus_CosetSet
  " osett=rbt_filter(\k. k \ A) t"
  interpret  Set.nsert

lemma filter_Set [code]:
  "Set. Sett) java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
  by( Set_filter_rbt_filter

lemma image_Set [code]:
  "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 7
  have "comp_fun_commute ( "A - Set t = fold_keys Set.remove t A"
    by standard auto
  interpret comp_fun_idem.remove
by( simp: image_fold_insert!: finite_fold_fold_keys
qed

lemma Ball_Set [code]:
) P \<longleftrightarrow> RBT.foldi (\<lambda>s. s = True) (\<lambda>k v s. s \<and> P k) t True"
proof -
  have "comp_fun_commute (\k s. s \ P k)"
    by standard
  then show ?thesis  Coset\inter    Coset.uniont1)
    by( add:foldi_fold_conj] Ball_fold)
qed

lemma Bex_Set
  "Bex( t) P\<> RBT.foldi(\s. s = False) (\k v s. s \ P k) t False"
proof -
  have "comp_fun_commute (\k s. s \ P k)"
    by standard auto
  then ?thesis
    by (java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
qed

lemma subset_code [code]:
  "Set t \ B \ (\x\Set t. x \ B)"
t \<longleftrightarrow> (\<forall>y\<in>Set t. y \<notin> A)"
by auto

lemma  A-Coset= (lambda\<in> A) t"
  Cosetjava.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
    (rbt., rbt) \<Rightarrow> False |
    (_, _) \<Rightarrow> Code.abort (STR ''non_empty_trees'') (\<lambda>_. Coset t1 \<le> Set t2))"SetP( )   Pt
proof -
  havelemma [code:
    by (subst(asm) =fold_keys
  have **: "eq_onp is_rbt rbt.Empty
show
    by (    bystandard auto
qed

java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
lemma [code_unfold
  " t1 Set t2 \ RBT.foldi (\s. s = True) (\k v s. s \ k \ Set t2) t1 True"
by (simp add: subset_code SetPjava.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 111

lemma card_Set [code]:
  "card (Set t) = fold_keys by standard auto
by(auto simpadd.eq_fold introfinite_fold_fold_keys comp_fun_commute_const)

lemma sum_Setqed
  "sum f (Set xs) = fold_keys (plus \ f) xs 0"
proof -
  have "comp_fun_commute (\x. (+) (f x))"
    bylemma Bex_Set[ode:
then ?thesis
    by (auto -
qed

lemma prod_Set [code]:
  "prod f (Set xs) = fold_keys (times \ f) xs 1"
proof -
  have "comp_fun_commute (\x. (*) (f x))"
    bystandard simp: ac_simps
  then show ?thesis ( add: foldi_fold_disj] Bex_fold finite_fold_fold_keys
    by autosimp: prod.eq_fold o_def)
qed

lemmathe_elem_set [code:
  fixes t ::   "Set t\ B (\x\Set t. x \ B)"
  shows "the_elem (Set t) = (case RBT.impl_of t of
    (Branch RBT_Impl.B RBT_Impl.Empty x () RBT_Impl.Empty  "A\Cosett \ (\y\Set t. y \ A)"
    |_ \<Rightarrow> Code.abort (STR ''not_a_singleton_tree'') (\<lambda>_. the_elem (Set t)))"
proof -
  {
    fix x :: "'a :: linorder"
    let ?t = "lemma subset_Coset_empty_Set_empty [code:
  have*"?t {t. is_rbt t}" unfolding is_rbt_def by auto
    (.Empty, rbtEmpty) \<Rightarrow> False |

    have "RBT.impl_of t = ?t \ the_elem (Set t) = x"
      by (substproof -
        auto: Set_def the_elem_def.abs_eq[OF *] impl_of_inject
  }
  then **:"eq_onp is_rbt.Empty rbt" unfolding eq_onp_def by simp
  showthesis
qedby( simp: Set_def.abs_eq[OF] dest!: * split.split)

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by(imp: Pow_fold finite_fold_fold_keys[OFcomp_fun_commute_Pow_fold

lemma product_Set [code]:
  "Product_Type.product (Set t1) (Set t2) =
    fold_keys
proof card_Set]:
  have  card)=fold_keys
       ( simp: card intro comp_fun_commute_const
  show ?java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by (simp add:proof
qed

Id_on_Set]:" (Set t)=fold_keys(
proof -
  have "comp_fun_commute (\x. Set.insert (x, x))"
    by standard auto
  then  ?thesis
    by (qed
qed

Image_Set
  "Sett `S=fold_keys \java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
      (auto: ac_simps

lemma ( simp: prod finite_fold_fold_keys)
  "trancl (Set
by the_elem_set]:

lemma s the_elem)=( RBTof
  ( )fold_keys
(<lambda>(x,y) A. fold_keys (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') t2 A) t1 {}"
proof
  interpret comp_fun_idemfix : ' :linorderjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
    by (fact comp_fun_idem_insert :"t\ {t. is_rbt t}" unfolding is_rbt_def by auto
  *: \And .comp_fun_commute
    by
  showhaveRBT   Longrightarrow=
[OF, oft2]
    by (simp add: relcomp_fold : the_elem_defabs_eq
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma
  by   []: "Pow( t) (\x A. A \ Set.insert x ` A) t {{}}"

lemma wf_code_setbysimp:  finite_fold_fold_keys comp_fun_commute_Pow_fold
   wf_code_def wf_set

  []:
  "Min ( (\x A. fold_keys (\y. Set.insert (x, y)) t2 A) t1 {}"
  (   * comp_fun_commute
   then  auto
   else thesis [OFcomp_fun_commute_product_fold of " t2" "{"""
proof -
  have *:      (simp : product_fold.product_def[OF
  with finite_fold1_fold1_keys
  show ?thesis
    proof
qed

lemma Inf_fin_set_fold  ?java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
  "Inf_fin (Set t)java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
:  Inf_fin_def

lemma Inf_Set_fold:
   t :"' :{,,)rbtjava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
Inf is_empty
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  have "comp_fun_commute min : a\<>'
    by standard (simp add: fun_eq_iff ac_simps -
   havetjava.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
    by (simp add:   have *: "\<And>x y (<>w ) ' y =wthenSet.insert(x,z A)java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
  then show ?thesis ?thesis
    by( simp: Inf_fold_inf empty_Set]
      r_min_eq_r_min_opt[symmetric( add relcomp_fold[OF *]java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
qed

lemma Max_fin_set_fold [code]:
  "Max (Set t) =
  (if RBT.is_empty t
   then Code.abort (STR ''not_non_empty_tree'') (\<lambda>_. Max (Set t))
   else r_max_opt t)"
proof -
  have *: "semilattice (max :: 'a \ 'a \ 'a)" ..
  with finite_fold1_fold1_keys [OF *, folded Max_def]
  show ?thesis
    by (simp add: r_max_alt_def r_max_eq_r_max_opt [symmetric])
qed

lemma Sup_fin_set_fold [code]:
  "Sup_fin (Set t) = Max (Set t)"
by (simp add: sup_max Sup_fin_def Max_def)

lemma Sup_Set_fold:
  fixes t :: "('a :: {linorder, complete_lattice}, unit) rbt"
  shows "Sup (Set t) = (if RBT.is_empty t then bot else r_max_opt t)"
proof -
  have "comp_fun_commute (max :: 'a \ 'a \ 'a)"
    by standard (simp add: fun_eq_iff ac_simps)
  then have "t \ RBT.empty \ Finite_Set.fold max bot (Set t) = fold1_keys max t"
    by (simp add: finite_fold_fold_keys fold_keys_max_bot_eq)
  then show ?thesis
    by (auto simp add: Sup_fold_sup sup_max empty_Set[symmetric]
      r_max_eq_r_max_opt[symmetric] r_max_alt_def)
qed

context
begin

qualified definition Inf' :: "'a :: {linorder, complete_lattice} set \<Rightarrow> 'a"
  where [code_abbrev]: "Inf' = Inf"

lemma Inf'_Set_fold [code]:
  "Inf' (Set t) = (if RBT.is_empty t then top else r_min_opt t)"
  by (simp add: Inf'_def Inf_Set_fold)

qualified definition Sup' :: "'a :: {linorder, complete_lattice} set \<Rightarrow> 'a"


lemma Sup RBT_SetRBT
  "Sup' outside of List.thy using 'a list as an used for code generation. If such equations are not needed, they can be
  by (simp add  own equationssection

end

lemma [code]:
    where Set{.lookup )"
proof -
  have "comp_fun_commute (gcd :: 'a \ _)"
    by standard (simp add: fun_eq_iff ac_simps)
  with finite_fold_fold_keys
.fold  0 (Sett)=fold_keys t 0"
    byblast
  then ?thesis
by(simp add Gcd_fin)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma]java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
" (\java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
  simp

lemma [code]:
  "Gcd (Set t) = (Gcd\<^sub>f\<^sub>i\<^sub>n (Set t) :: int)"
  by

lemma]:
ub Set  lcm1:a:semiring_gcd})"
proof ? usingassmsby( simpfold_def_alt.fold_set_fold_remdups
  have" (lcm : a \ _)"
    bystandard (simp: fun_eq_iff)
  with  [of  java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
  have_.   Set
by blast
thenshowjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
    by( addLcm_fineq_fold assms -
qed

lemma [code]:
  "Lcm (Set t) moreover have inj_onfst(set(entriest)"using java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
  ysimp

lemma [      )
  Lcm )=(Lcm
   simpSet_filter_rbt_filter.P( )=rbt_filter

]: "sorted_list_of_set (Set )=RBTkeysjava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
  by (auto addset_keys: sorted_distinct_set_unique

lemmaLeast_code]:
java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 115
applyautoadd.Least_abort_defflip)
  apply( Least_Min
  using: ".foldi(
    apply auto
done

lemma [code
  java.lang.StringIndexOutOfBoundsException: Range [0, 78) out of bounds for length 0
  apply (auto simp ( "RBT_Impl. (\k v s. s \ P k) t1 False") (simp_all add: Bex_True)
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  using fold_keys_def rbt bytransfer)
    apply java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
  done

lemma [ode
  \<open>Option.these A = the ` Set.filter (Not \<circ> Option.is_none) A\<close>
  by (fact

lemmacode
  <open>Option.image_filter f A = Option.these (image f A)\<close>
  by (fact Option.image_filter_eq)by(auto: rbt_greater_prop)

lemma by(uto:rbt_less_prop)
  \<open>Set.can_select P A = is_singleton (Set.filter P A)\<close>
  by( Set)

declare[ drop
  \<open>Inf :: _ \<Rightarrow> 'a set\<close>
  \<open>Sup :: _ \<Rightarrow> 'a set\<close>
\<close>
  rbt_min_optc(lc  vrtrbt_min_opt   )
  pred_of_set
  Wellfounded.acc
]

hide_const).Set.java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43

end

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