SSL Tree23_Set.thy Interaktion und
PortierbarkeitIsabelle

(* Author: Tobias Nipkow *)

section \<open>2-3 Tree Implementation of Sets\<close>

theory Tree23_Set
imports
  Tree23
  Cmp
  Set_Specs
begin

declare sorted_wrt.simps

definition Tree23_Set
"empty=Leaf"

fun Tree23
"isin Leaf x = False" |
"isin (Node2 l a r) x =
  (case cmp x a of
     LT \<Rightarrow> isin l x |
     EQ \<Rightarrow> True |
     GT \<Rightarrow> isin r x)" |
"isin (Node3 l a m b r) x =
  (case cmp x a of
     LT \<Rightarrow> isin l x |
     EQ \<Rightarrow> True |

       casecmp x  of
           \<Rightarrow> isin m x | empty' ree23 java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
           GT
          GT \<Rightarrow> isin r x))"

datatype 'a \<^>i =Eq

fun
"tree\<^sub>i (Eq\<^sub>i t) = t" |
"treejava.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 21

fun java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
"ins x Leaf = Of Leaf x "\<^sub>i (Eq\<^sub>i t) = t" |
"ins(Node2la )=
   (case cmp x a of
      LT \<Rightarrow>
        (case ins x l of
           Eq\<^sub>i l' => Eq\<^sub>i (Node2 l' a r) |"ins x Leaf = Leaf " |
           " x (Node2la r)=
      EQ \<Rightarrow> Eq\<^sub>i (Node2 l a r) |
      GT \<Rightarrow>
        (caseinsof
           Eq           Eq\<^sub>i l' => Eq\<^sub>i (Node2 l' a r) |           Of bl2 \<>i (ode3 bl2  r) |
           Ofcase x r of
"ins x (Node3 l a m b r) =
   ( cmp
       \<Rightarrow>
        (case x l of
           Eq x   r java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
  c =Of l1l2( m b r)|
      EQ \<Rightarrow> Eq\<^sub>i (Node3 l a m b r) |
      GT \<Rightarrow>
        (case cmp x b of
           GT \<Rightarrow>
(  xr
                Eq\<^sub>i r' => Eq\<^sub>i (Node3 l a m b r') | \<Rightarrow>
Of r2 Of( l  m) Node2 )) |
           EQ \<Rightarrow> Eq\<^sub>i (Node3 l a m b r) |
           LT
             (case ins x m of
                Eq\<^sub>i m' => Eq\<^sub>i (Node3 l a m' b r) |
                Of m1                Of c r2>  (Node2)b Node2 c r2

hide_const             \<Rightarrow>

definition insert :               \<^sub>i m' => Eq\<^sub>i (Node3 l a m' b r) |
"insert x t = tree\<^sub>i(ins x t)"

datatype 'a up\<^sub>d = Eq\<^sub>d "'a tree23" | Uf "'a tree23"

funtree>d :"aup\<^sub>d \ 'a tree23" where
"tree\<^sub>d (Eq\<^sub>d t) = t" |
"tree\<^sub>d (Uf t) = t"

(* Variation: return None to signal no-change *)

fun node21 :: "'a up\<^sub>d \ 'a \ 'a tree23 \ 'a up\<^sub>d" where
java.lang.NullPointerException
Ufb  t2
"node21 (Uf t1) a (Node3 t2 "\<^sub(\<^>d t)="|

fun node22
node22 (java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
"
" (Node3 t1 t2ct3 a( t4) \<^subdNode2( t1 b t2 Node2t3 t4))"

: ' \<^sub>d \ 'a \ 'a tree23 \ 'a \ 'a tree23 \ 'a up\<^sub>d" where
"node31 (Eq\<^sub>d t1) a t2 b t3 = Eq\<^sub>d(Node3 t1 a t2 b t3)" |
(Uf)a( t2b ) c t4 \<^sub>d(Node2 (Node3 t1 a t2 b t3) c t4)" |
"node31 Uf t1 Node3t2 bt3 c t4) d t5 = Eq\<^sub>d(Node3 (Node2 t1 a t2) b (Node2 t3 c t4) d t5)"

fun node32node31
"ode32 t1a(\
node32t1Uf)b( t3 ) = \<^sub(Node2  Node3 bt3))"|
"node32 t1 a (Uf t2) b (Node3 t3 c t4 d t5) = Eq\<^sub>d(Node3 t1 a (Node2 t2 b t3) c (Node2 t4 d t5))"

fun node33 :: "'a tree23 \ 'a \ 'a tree23 \ 'a \ 'a up\<^sub>d \ 'a up\<^sub>d" where
"node33 t1 a t2 b " t1 Uf) b Node2 ct4EqsubNode2t1 t2t3  )) |
"node33 t1 a ("node32a( t2 Node3  t4t5= Eq^>d(Node3t1( t2 b t3 ( t4 d t5"
"node33java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

funsplit_min: 'a tree23 \ 'a * 'a up\<^sub>d" where
split_min Leaf) = ( UfLeaf)"|
"split_min (Node3 Leaf a Leaf b Leaf) = (a, Eq\<^sub>d(Node2 Leaf b Leaf))" |
"split_min (Node2 l a r) = (let (x,l') = split_min l in (x, node21 l' a r))" |
" ( amb r) =(let(,l)= split_min l in(, node31l r)java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84

text \<open>In the base cases of \<open>split_min\<close> and \<open>del\<close> it is enough to check if one subtree is a \<open>Leaf\<close>,
"Node2 l )=( xl)=split_min (, node21l )"|

" ( l a m b r)= ( (xl)=split_minlin(, '   )
"del
"delx( Leaf a Leaf) =
in case implies are. Exercise
"delfundel ::"alinorder
  Eq\<^sub>d(if x = a then Node2 Leaf b Leaf else  \<^sub>d Leaf" |
x   Node2aLeaf
     else Node3 Leaf a Leaf b Leaf" x (Node3 LeafaLeafb Leaf java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
"del Node2la )=
  (case cmp x a of
     LT \<Rightarrow> node21 (del x l) a r |
     GT \<Rightarrow> node22 l a (del x r) |
EQ
"del x (Node3 l a m ( x a of
  casecmpxaof
     LT
     EQ \<Rightarrow> let (a',m') = split_min m in node32 l a' m' b r |
      \<Rightarrow>
       casexb
           \<Rightarrow> node32 l a (del x m) b r |

           \<Rightarrow> node33 l a m b (del x r)))"

definition delete      EQ \<Rightarrow> let (a',m') = split_min m in node32 l a' m' b r |
"delete casecmp x b of

subsection "Functional"Functional Correctness"

subsubsection "Proofs for isin"

lemma            \<Rightarrow> let (b',r') = split_min r in node33 l a m b' r' |
by

java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0

lemma inorder_ins:
  "sorted(inorder t) \ inorder(tree\<^sub>i(ins x t)) = ins_list x (inorder t)"
by(induction t) by (nduction auto: isin_simps

lemma java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
"(t)\
by(simp :

subsubsection "Proofs for delete"

lemma inorder_node21: "height r > 0 \
  inorder (tree\<^sub>d (node21 l' a r)) = inorder (tree\<^sub>d l') @ a # inorder r"add  inorder_ins
by l' a rrule node21.) auto

: height
  inorder (tree(inductl  'rule .induct) auto
by(lemma inorder_node31: height \<Longrightarrow>

byinduct m b r rule: node31.induct) auto
  inorderjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by(induct am  :node31

lemma inorder_node32: "height r > 0 \
  inorder (tree\<^sub>d (node32 l a m' b r)) = inorder l @ a # inorder (tree\<^sub>d m') @ b # inorder r"
by(induct l a m' b r rule: node32.induct) auto

lemma inorder_node33: "height m > 0 \
  inorder (tree\<^sub>d (node33 l a m b r')) = inorder l @ a # inorder m @ b # inorder (tree\<^sub>d r')" (\<^sub>d (node33 l a m b r')) = inorder l @ a # inorder m @ b # inorder (tree\<^sub>d r')"
byinduct  'rule node33.) auto

lemmas  inorder_node22
  inorder_node31 #inorder\<^sub>d t') = inorder t"

lemma split_minD:
  (uto:  split.splits
  x # inorder(tree\<^sub>d t') = inorder t"
by(induction t arbitrary: t' rule: split_min.induct)
  (auto simp: inorder_nodes split: prod.splits)

:"<> t ; sortedinordert \ \
  inorder
by inorder_delete:"\
( simpdel_list_simps split_minD!: if_split.plits

lemma inorder_delete
  inorder
by(simp

subsection \<open>Completeness\<close>

subsubsection "Proofs for insert"

text<open>First a standard proof that \<^const>\<open>ins\<close> preserves \<^const>\<open>complete\<close>.\<close>

fun h\<^sub>i :: "'a up\<^sub>i \<Rightarrow> nat" where
"h\<^sub>i (Eq\<^sub>i t) = height t" |
"h\<^sub>i (Of l a r) = height l"

lemmalemma complete_insCmp
by (induct t) funh<^sub>i :: "'a up\<^sub>i \<Rightarrow> nat" where

Now an alternative proof (by"\<^sub>i (Eq\<^sub>i t) = height t" |
lemmacomplete_ins

inductive :: "by(induct tauto split!: if_split up\
"full0 java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
"\full n l; full n r\ \ full (Suc n) (Node2 l p r)" |
"

inductive_cases:
  full full_elims
full
  "full"  Node2

inductive_cases \<Rightarrow> isin r x)" |
inductive_cases full_Suc_elim: "ull( n tjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

full_0_iff]:" 0 t \ t = Leaf"
  by (byauto: full_0_elim: full

lemma          \<> True
  by (auto:full_elims:fulljava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

  full )lpr <  l \<and> full n r"
full)Node2 longleftrightarrow
  byfun tree\<^sub>i :: "'a up\<^sub>i \<Rightarrow> 'a tree23" where

lemma full_Suc_Node3_iff" (Sucn ( l p m r)\longleftrightarrow n l \ full n m \ full n r"
  "full ( n)(Node3 ll m q )
  by (uto: full_elimsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

   full_imp_completent" ar =

lemma full_imp_complete: "full n t \ complete t"
  by (induct set: full, auto dest:by( set:full        \<Rightarrow>

lemma   by(induct)
  by (induct t, (case ins x l: "completet (\n. full n t)"

lemma complete_iff_full: "complete t \ (\n. full n t)"
m: complete_imp_full)

text \<open>The \<^const>\<open>insert\<close> function either preserves the height of the
tree,            Eq\<^sub>i r' => Eq\<^sub>i (Node2 l a r') |
form

tree  increases  . Theconstructorthe
"full\<^sub>i n (Eq\<^sub>i t) \ full n t" |
"full\<^sub>i n (Of l p r) \ full n l \ full n r"

lemma full\<^sub>i_ins: "full n t \<Longrightarrow> full\<^sub>i n (ins a t)"
by (induct rule: full.inductjava.lang.StringIndexOutOfBoundsException: Range [28, 29) out of bounds for length 0

text

lemma complete_insert: "complete t \ complete (insert a t)"
unfoldingcomplete_iff_full insert_def
apply erule)
apply ( ruleinductauto up^ubisplit
apply (cases
applyauto:full)
done

subsection "Proofs for delete"

fun happly erule exE
h<dEq
"h\<^sub>d (Uf t) = height t + 1"

lemma complete_tree\<^sub>d_node21:
GT
( l'a(ase x java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

lemma complete_tree\<^sub>d_node22:
  "h^sub>d (Eq\<^sub>d t) = height t" |
byinduct ar rulenode22h^ Uf+

complete_tree
  " "\complete r; complete (tree\<^sub>d l'); height r = h\<^sub>d l' \ \ complete (tree\<^sub>d (node21 l' a r))"
  \<Longrightarrow> complete (tree\<^sub>d (node31 l' a m b r))" <>
by(induct

lemma complete_tree             insmof
  " complete l; complete (tree\<^sub>d m'); complete r; height l = height r; h\<^sub>d m' = height r \
                  Of m1 c m2 => Of (Node2 l a m1) c (Node2 m2 b r))))"
by(induct l a m' b r rule\ complete (tree\<^sub>d (node31 l' a m b r))"

lemma complete_tree\<^sub>d_node33:
java.lang.NullPointerException
\Longrightarrow ompletetree
by(induct l a m b r' rule: node33.induct) auto

"\ complete l; complete (tree\<^sub>d m'); complete r; height l = height r; h\<^sub>d m' = height r \
  complete_tree\<^sub>d_node31 complete_tree\<^sub>d_node32 complete_tree\<^sub>d_node33

lemma 'java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   "height r > 0 \ h\<^sub>d(node21 l' a r) = max (h\<^sub>d l') (height r) + 1"treeinductm   :.)auto
induct .simp_all

lemmafun  \^>d_node31

ruleinductsimp_all

lemma '_node31:
  lemma height
   max (h\<^sub>d l) (max (height m) (height r)) + 1"= maxl \^> ' "
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma height'_node32:
  " > 0\ h\<^sub>d(node32 l a m b r) =
   max (height node31 ::m  r rule node31( addmax_def
duct a    : node32)(simp_all: max_def

lemma height'_node33:
  height    maxheight  h\<^sub>d m) (height r)) + 1"
   max l)max (height m h<"node31 (Uf t1) a (Node2 t2 b t3) c t4 =\<^sub>d(Node2 (Node3 t1 a t2 b t3) c t4)" |
by(inductjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemmas heights = height'_node21 height'_node22
  height'_node31 height'_node32 height'_node33

lemma height_split_min:
  "split_min tjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
byinduct arbitrary tjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  auto heights:prod

:complete
bysplit_min:.
split_min

:
    ( simp max_def split (Node3  )=letx,) l (x, la m b)java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
( tarbitrary
  ( simp  height_split_min completes: prod

complete_tree
by(induction x  rule del.induct
  (auto simp: completes"delx(Node2 Leaf byinduction : del)

corollary java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by(if= b then Node2Leafjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

where   and =isinand = insert delete= delete

interpretation S: Set_by_Ordered
where empty = empty and isin = isin and insert = insert and inorderGT \<Rightarrow> node22 l a (del x r) |
andinorder =inorder inv=complete
proof (standard, goal_casesEQ
  case 2 thus (standard, )
next
  case 3 thus ?case by(simp

           \<Rightarrow> node32 l a (del x m) b r |
next  case4 ?case by(simp add inorder_delete
  ase 6thus by : complete_insert)
next
  case 7 thus ?case   case7thuscaseby addjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed

end

quality99%

¤ Diese beiden folgenden Angebotsgruppen bietet das Unternehmen0.8Angebot ¤

*Eine klare Vorstellung vom Zielzustand

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.

SSL Tree23_Set.thy Interaktion und PortierbarkeitIsabelle

¤ Diese beiden folgenden Angebotsgruppen bietet das Unternehmen0.8Angebot ¤

Haftungshinweis

Bemerkung:

SSL Tree23_Set.thy Interaktion und
PortierbarkeitIsabelle