Quelle Tree23_Set.thy Sprache: Isabelle

theory java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  text\<open>\\<^sub>i (Of l a r) = height l"
begin

declare  full  auto up

definition empty :: "'
"empty Leaf"

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

  (case
inductive_cases full_elims
     EQ" njava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
     GT
"isin (Node3 l a m b r) x =
  inductive_casesfull_Suc_elim" Sucn)"
     LT
     EQ \<Rightarrow> True |
      full_Suc_elimfull tjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
( elim intro)
           \<Rightarrow> isin m x |
Rightarrow
          GT \<Rightarrow> isin r x))"(   .introsbyautofull_elims:full)

java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0

java.lang.NullPointerException
"\<^sub>i (Eq\<^sub>i t) = t" |
"\<^sub>i (Oflar =java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0

fun b autoSuc pmq)\<longleftrightarrow> full n l \<and> full n m \<and> full n r"( elim
"ins x Leaf = Of Leaf x Leaf"  byinduct , )
( =
   (casejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 22
        ll"t (\n. full n t)"
           Eq\<^sub>i l' => Eq\<^sub>i (Node2 l' a r) |
Of  full_imp_complete

      GT \<Rightarrow>
        (case ins x r of
           Eq
           Of <open>The \<^const>\<open>insert\<close> function either preserves the height of the \<^term>\<open>Of l p r\<close> indicates an increase in height.\<close>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
form
   (
      LT \<Rightarrow>full \<open>The \<^const>\<open>insert\<close> operation preserves completeance.\<close>
        ( complete_iff_full
           Eqerule
(: .)(:<ub
      EQ
      GT introjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
        ( (druled(java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
            \<Rightarrow>
              rjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
                \^>ir >
                Of<
           ( l a 'rule: .

           LT\Rightarrowjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
(case
                Eq\<^sub>i m' => Eq\<^sub>i (Node3 l a m' b r) |
java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68

hide_const\<lbrakk> complete (tree\<^sub>d l'); complete m; complete r; h\<^sub>d l' = height r; height m = height r \<rbrakk><complete

definition insert  <>complete (tree\<^sub>d (node33 l a m b r'))"
"java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
by l mbrrule
datatypejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

fun tree\<^sub>d :: "'a up\<^sub>d \<Rightarrow> 'a tree23" where
"byinductl amb'rule.induct
"tree\<^sub>d (Uf t) = t"

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

fun<d_node31
"node21 (Eq\<^sub>d t1) a t2 = Eq\<^sub>d(Node2 t1 a t2)" |
"node21 (Uf t1) a: .)()
"node21 (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

fun node22 :
" '_node31:
"node22 (Node2 "height m > 0 \java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
"node22 (Node3 t1 heightr> \ h\<^sub>d(node32 l a m b r) =

funnode31 br :induct: java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
"" m max)(java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
Eq
"node31 (Uf t1) a (Node3java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

:"atree23
"node32 t1 a (Eq\<^sub>d t2) b t3 = Eq\<^sub>d(Node3 t1 a t2 b t3)" |
"node32 t1 max (heightl java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
"node33 t1 a t2 b (Eq\<^sub>d t3) = Eq\<^sub>d(Node3 t1 a t2 b t3)" |
( t : x java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
node33heights prod)

fun
byinduct "t \ h\<^sub>d(del x t) = height t"
"(autosimp: heights split prod)
" (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
split_min am  =let x'=split_min in(x,ode31 m r"

textjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
in which caseby t arbitraryrule

fun  auto:heights split.
"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  Node2
  (ifjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
  q
      x  then Leaf
     elsesubsection
"del x (Node2 l a r) =
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
emptyandjava.lang.StringIndexOutOfBoundsException: Range [35, 32) out of bounds for length 75

     EQ
" x ( l a m )=
  (case cmp x a of
     LT inorder  and  complete
      \<Rightarrow> let (a',m') = split_min m in node32 l a' m' b r |
     GTjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
       (casecmp x
LTjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
          EQ
case)

definition
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

subsection "Functional Correctness"

subsubsection "Proofs for isin"

lemma isin_set: "sorted(qed(imp add: empty_def)+
by (induction

subsubsectionjava.lang.StringIndexOutOfBoundsException: Index 0 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) (auto simp: ins_list_simps split: up\<^sub>i.splits)

lemma inorder_insert:
  "sorted(inorder t) \ inorder(insert a t) = ins_list a (inorder t)"
by(simp add: insert_def inorder_ins)

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"
by(induct l' a r rule: node21.induct) auto

lemma inorder_node22: "height l > 0 \
  inorder (tree\<^sub>d (node22 l a r')) = inorder l @ a # inorder (tree\<^sub>d r')"
by(induct l a r' rule: node22.induct) auto

lemma inorder_node31: "height m > 0 \
  inorder (tree\<^sub>d (node31 l' a m b r)) = inorder (tree\<^sub>d l') @ a # inorder m @ b # inorder r"
by(induct l' a m b r rule: node31.induct) auto

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')"
by(induct l a m b r' rule: node33.induct) auto

lemmas inorder_nodes = inorder_node21 inorder_node22
  inorder_node31 inorder_node32 inorder_node33

lemma split_minD:
  "split_min t = (x,t') \ complete t \ height t > 0 \
  x # inorder(tree\<^sub>d t') = inorder t"
by(induction t arbitrary: t' rule: split_min.induct)
  (auto simp: inorder_nodes split: prod.splits)

lemma inorder_del: "\ complete t ; sorted(inorder t) \ \
  inorder(tree\<^sub>d (del x t)) = del_list x (inorder t)"
by(induction t rule: del.induct)
  (auto simp: del_list_simps inorder_nodes split_minD split!: if_split prod.splits)

lemma inorder_delete: "\ complete t ; sorted(inorder t) \ \
  inorder(delete x t) = del_list x (inorder t)"
by(simp add: delete_def inorder_del)

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"

lemma complete_ins: "complete t \ complete (tree\<^sub>i(ins a t)) \ h\<^sub>i(ins a t) = height t"
by (induct t) (auto split!: if_split up\<^sub>i.split) (* 15 secs in 2015 *)

text\<open>Now an alternative proof (by Brian Huffman) that runs faster because
two properties (completeness and height) are combined in one predicate.\<close>

inductive full :: "nat \ 'a tree23 \ bool" where
"full 0 Leaf" |
"\full n l; full n r\ \ full (Suc n) (Node2 l p r)" |
"\full n l; full n m; full n r\ \ full (Suc n) (Node3 l p m q r)"

inductive_cases full_elims:
  "full n Leaf"
  "full n (Node2 l p r)"
  "full n (Node3 l p m q r)"

inductive_cases full_0_elim: "full 0 t"
inductive_cases full_Suc_elim: "full (Suc n) t"

lemma full_0_iff [simp]: "full 0 t \ t = Leaf"
  by (auto elim: full_0_elim intro: full.intros)

lemma full_Leaf_iff [simp]: "full n Leaf \ n = 0"
  by (auto elim: full_elims intro: full.intros)

lemma full_Suc_Node2_iff [simp]:
  "full (Suc n) (Node2 l p r) \ full n l \ full n r"
  by (auto elim: full_elims intro: full.intros)

lemma full_Suc_Node3_iff [simp]:
  "full (Suc n) (Node3 l p m q r) \ full n l \ full n m \ full n r"
  by (auto elim: full_elims intro: full.intros)

lemma full_imp_height: "full n t \ height t = n"
  by (induct set: full, simp_all)

lemma full_imp_complete: "full n t \ complete t"
  by (induct set: full, auto dest: full_imp_height)

lemma complete_imp_full: "complete t \ full (height t) t"
  by (induct t, simp_all)

lemma complete_iff_full: "complete t \ (\n. full n t)"
  by (auto elim!: complete_imp_full full_imp_complete)

text \<open>The \<^const>\<open>insert\<close> function either preserves the height of the
tree, or increases it by one. The constructor returned by the \<^term>\<open>insert\<close> function determines which: A return value of the form \<^term>\<open>Eq\<^sub>i t\<close> indicates that the height will be the same. A value of the
form \<^term>\<open>Of l p r\<close> indicates an increase in height.\<close>

fun full\<^sub>i :: "nat \<Rightarrow> 'a up\<^sub>i \<Rightarrow> bool" where
"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.induct) (auto split: up\<^sub>i.split)

text \<open>The \<^const>\<open>insert\<close> operation preserves completeance.\<close>

lemma complete_insert: "complete t \ complete (insert a t)"
unfolding complete_iff_full insert_def
apply (erule exE)
apply (drule full\<^sub>i_ins [of _ _ a])
apply (cases "ins a t")
apply (auto intro: full.intros)
done

subsection "Proofs for delete"

fun h\<^sub>d :: "'a up\<^sub>d \<Rightarrow> nat" where
"h\<^sub>d (Eq\<^sub>d t) = height t" |
"h\<^sub>d (Uf t) = height t + 1"

lemma complete_tree\<^sub>d_node21:
  "\complete r; complete (tree\<^sub>d l'); height r = h\<^sub>d l' \ \ complete (tree\<^sub>d (node21 l' a r))"
by(induct l' a r rule: node21.induct) auto

lemma complete_tree\<^sub>d_node22:
  "\complete(tree\<^sub>d r'); complete l; h\<^sub>d r' = height l \ \ complete (tree\<^sub>d (node22 l a r'))"
by(induct l a r' rule: node22.induct) auto

lemma complete_tree\<^sub>d_node31:
  "\<lbrakk> complete (tree\<^sub>d l'); complete m; complete r; h\<^sub>d l' = height r; height m = height r \<rbrakk>
  \<Longrightarrow> complete (tree\<^sub>d (node31 l' a m b r))"
by(induct l' a m b r rule: node31.induct) auto

lemma complete_tree\<^sub>d_node32:
  "\ complete l; complete (tree\<^sub>d m'); complete r; height l = height r; h\<^sub>d m' = height r \
  \<Longrightarrow> complete (tree\<^sub>d (node32 l a m' b r))"
by(induct l a m' b r rule: node32.induct) auto

lemma complete_tree\<^sub>d_node33:
  "\ complete l; complete m; complete(tree\<^sub>d r'); height l = h\<^sub>d r'; height m = h\<^sub>d r' \
  \<Longrightarrow> complete (tree\<^sub>d (node33 l a m b r'))"
by(induct l a m b r' rule: node33.induct) auto

lemmas completes = complete_tree\<^sub>d_node21 complete_tree\<^sub>d_node22
  complete_tree\<^sub>d_node31 complete_tree\<^sub>d_node32 complete_tree\<^sub>d_node33

lemma height'_node21:
   "height r > 0 \ h\<^sub>d(node21 l' a r) = max (h\<^sub>d l') (height r) + 1"
by(induct l' a r rule: node21.induct)(simp_all)

lemma height'_node22:
   "height l > 0 \ h\<^sub>d(node22 l a r') = max (height l) (h\<^sub>d r') + 1"
by(induct l a r' rule: node22.induct)(simp_all)

lemma height'_node31:
  "height m > 0 \ h\<^sub>d(node31 l a m b r) =
   max (h\<^sub>d l) (max (height m) (height r)) + 1"
by(induct l a m b r rule: node31.induct)(simp_all add: max_def)

lemma height'_node32:
  "height r > 0 \ h\<^sub>d(node32 l a m b r) =
   max (height l) (max (h\<^sub>d m) (height r)) + 1"
by(induct l a m b r rule: node32.induct)(simp_all add: max_def)

lemma height'_node33:
  "height m > 0 \ h\<^sub>d(node33 l a m b r) =
   max (height l) (max (height m) (h\<^sub>d r)) + 1"
by(induct l a m b r rule: node33.induct)(simp_all add: max_def)

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

lemma height_split_min:
  "split_min t = (x, t') \ height t > 0 \ complete t \ h\<^sub>d t' = height t"
by(induct t arbitrary: x t' rule: split_min.induct)
  (auto simp: heights split: prod.splits)

lemma height_del: "complete t \ h\<^sub>d(del x t) = height t"
by(induction x t rule: del.induct)
  (auto simp: heights max_def height_split_min split: prod.splits)

lemma complete_split_min:
  "\ split_min t = (x, t'); complete t; height t > 0 \ \ complete (tree\<^sub>d t')"
by(induct t arbitrary: x t' rule: split_min.induct)
  (auto simp: heights height_split_min completes split: prod.splits)

lemma complete_tree\<^sub>d_del: "complete t \<Longrightarrow> complete(tree\<^sub>d(del x t))"
by(induction x t rule: del.induct)
  (auto simp: completes complete_split_min height_del height_split_min split: prod.splits)

corollary complete_delete: "complete t \ complete(delete x t)"
by(simp add: delete_def complete_tree\<^sub>d_del)

subsection \<open>Overall Correctness\<close>

interpretation S: Set_by_Ordered
where empty = empty and isin = isin and insert = insert and delete = delete
and inorder = inorder and inv = complete
proof (standard, goal_cases)
  case 2 thus ?case by(simp add: isin_set)
next
  case 3 thus ?case by(simp add: inorder_insert)
next
  case 4 thus ?case by(simp add: inorder_delete)
next
  case 6 thus ?case by(simp add: complete_insert)
next
  case 7 thus ?case by(simp add: complete_delete)
qed (simp add: empty_def)+

end

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