src/HOL/Data_Structures/AA_Set.thy
 author nipkow Wed Jun 13 15:24:20 2018 +0200 (10 months ago) changeset 68440 6826718f732d parent 68431 b294e095f64c child 69505 cc2d676d5395 permissions -rw-r--r--
qualify interpretations to avoid clashes
```     1 (*
```
```     2 Author: Tobias Nipkow, Daniel StÃ¼we
```
```     3 *)
```
```     4
```
```     5 section \<open>AA Tree Implementation of Sets\<close>
```
```     6
```
```     7 theory AA_Set
```
```     8 imports
```
```     9   Isin2
```
```    10   Cmp
```
```    11 begin
```
```    12
```
```    13 type_synonym 'a aa_tree = "('a,nat) tree"
```
```    14
```
```    15 definition empty :: "'a aa_tree" where
```
```    16 "empty = Leaf"
```
```    17
```
```    18 fun lvl :: "'a aa_tree \<Rightarrow> nat" where
```
```    19 "lvl Leaf = 0" |
```
```    20 "lvl (Node _ _ lv _) = lv"
```
```    21
```
```    22 fun invar :: "'a aa_tree \<Rightarrow> bool" where
```
```    23 "invar Leaf = True" |
```
```    24 "invar (Node l a h r) =
```
```    25  (invar l \<and> invar r \<and>
```
```    26   h = lvl l + 1 \<and> (h = lvl r + 1 \<or> (\<exists>lr b rr. r = Node lr b h rr \<and> h = lvl rr + 1)))"
```
```    27
```
```    28 fun skew :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
```
```    29 "skew (Node (Node t1 b lvb t2) a lva t3) =
```
```    30   (if lva = lvb then Node t1 b lvb (Node t2 a lva t3) else Node (Node t1 b lvb t2) a lva t3)" |
```
```    31 "skew t = t"
```
```    32
```
```    33 fun split :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
```
```    34 "split (Node t1 a lva (Node t2 b lvb (Node t3 c lvc t4))) =
```
```    35    (if lva = lvb \<and> lvb = lvc \<comment> \<open>\<open>lva = lvc\<close> suffices\<close>
```
```    36     then Node (Node t1 a lva t2) b (lva+1) (Node t3 c lva t4)
```
```    37     else Node t1 a lva (Node t2 b lvb (Node t3 c lvc t4)))" |
```
```    38 "split t = t"
```
```    39
```
```    40 hide_const (open) insert
```
```    41
```
```    42 fun insert :: "'a::linorder \<Rightarrow> 'a aa_tree \<Rightarrow> 'a aa_tree" where
```
```    43 "insert x Leaf = Node Leaf x 1 Leaf" |
```
```    44 "insert x (Node t1 a lv t2) =
```
```    45   (case cmp x a of
```
```    46      LT \<Rightarrow> split (skew (Node (insert x t1) a lv t2)) |
```
```    47      GT \<Rightarrow> split (skew (Node t1 a lv (insert x t2))) |
```
```    48      EQ \<Rightarrow> Node t1 x lv t2)"
```
```    49
```
```    50 fun sngl :: "'a aa_tree \<Rightarrow> bool" where
```
```    51 "sngl Leaf = False" |
```
```    52 "sngl (Node _ _ _ Leaf) = True" |
```
```    53 "sngl (Node _ _ lva (Node _ _ lvb _)) = (lva > lvb)"
```
```    54
```
```    55 definition adjust :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
```
```    56 "adjust t =
```
```    57  (case t of
```
```    58   Node l x lv r \<Rightarrow>
```
```    59    (if lvl l >= lv-1 \<and> lvl r >= lv-1 then t else
```
```    60     if lvl r < lv-1 \<and> sngl l then skew (Node l x (lv-1) r) else
```
```    61     if lvl r < lv-1
```
```    62     then case l of
```
```    63            Node t1 a lva (Node t2 b lvb t3)
```
```    64              \<Rightarrow> Node (Node t1 a lva t2) b (lvb+1) (Node t3 x (lv-1) r)
```
```    65     else
```
```    66     if lvl r < lv then split (Node l x (lv-1) r)
```
```    67     else
```
```    68       case r of
```
```    69         Node t1 b lvb t4 \<Rightarrow>
```
```    70           (case t1 of
```
```    71              Node t2 a lva t3
```
```    72                \<Rightarrow> Node (Node l x (lv-1) t2) a (lva+1)
```
```    73                     (split (Node t3 b (if sngl t1 then lva else lva+1) t4)))))"
```
```    74
```
```    75 text\<open>In the paper, the last case of @{const adjust} is expressed with the help of an
```
```    76 incorrect auxiliary function \texttt{nlvl}.
```
```    77
```
```    78 Function @{text split_max} below is called \texttt{dellrg} in the paper.
```
```    79 The latter is incorrect for two reasons: \texttt{dellrg} is meant to delete the largest
```
```    80 element but recurses on the left instead of the right subtree; the invariant
```
```    81 is not restored.\<close>
```
```    82
```
```    83 fun split_max :: "'a aa_tree \<Rightarrow> 'a aa_tree * 'a" where
```
```    84 "split_max (Node l a lv Leaf) = (l,a)" |
```
```    85 "split_max (Node l a lv r) = (let (r',b) = split_max r in (adjust(Node l a lv r'), b))"
```
```    86
```
```    87 fun delete :: "'a::linorder \<Rightarrow> 'a aa_tree \<Rightarrow> 'a aa_tree" where
```
```    88 "delete _ Leaf = Leaf" |
```
```    89 "delete x (Node l a lv r) =
```
```    90   (case cmp x a of
```
```    91      LT \<Rightarrow> adjust (Node (delete x l) a lv r) |
```
```    92      GT \<Rightarrow> adjust (Node l a lv (delete x r)) |
```
```    93      EQ \<Rightarrow> (if l = Leaf then r
```
```    94             else let (l',b) = split_max l in adjust (Node l' b lv r)))"
```
```    95
```
```    96 fun pre_adjust where
```
```    97 "pre_adjust (Node l a lv r) = (invar l \<and> invar r \<and>
```
```    98     ((lv = lvl l + 1 \<and> (lv = lvl r + 1 \<or> lv = lvl r + 2 \<or> lv = lvl r \<and> sngl r)) \<or>
```
```    99      (lv = lvl l + 2 \<and> (lv = lvl r + 1 \<or> lv = lvl r \<and> sngl r))))"
```
```   100
```
```   101 declare pre_adjust.simps [simp del]
```
```   102
```
```   103 subsection "Auxiliary Proofs"
```
```   104
```
```   105 lemma split_case: "split t = (case t of
```
```   106   Node t1 x lvx (Node t2 y lvy (Node t3 z lvz t4)) \<Rightarrow>
```
```   107    (if lvx = lvy \<and> lvy = lvz
```
```   108     then Node (Node t1 x lvx t2) y (lvx+1) (Node t3 z lvx t4)
```
```   109     else t)
```
```   110   | t \<Rightarrow> t)"
```
```   111 by(auto split: tree.split)
```
```   112
```
```   113 lemma skew_case: "skew t = (case t of
```
```   114   Node (Node t1 y lvy t2) x lvx t3 \<Rightarrow>
```
```   115   (if lvx = lvy then Node t1 y lvx (Node t2 x lvx t3) else t)
```
```   116  | t \<Rightarrow> t)"
```
```   117 by(auto split: tree.split)
```
```   118
```
```   119 lemma lvl_0_iff: "invar t \<Longrightarrow> lvl t = 0 \<longleftrightarrow> t = Leaf"
```
```   120 by(cases t) auto
```
```   121
```
```   122 lemma lvl_Suc_iff: "lvl t = Suc n \<longleftrightarrow> (\<exists> l a r. t = Node l a (Suc n) r)"
```
```   123 by(cases t) auto
```
```   124
```
```   125 lemma lvl_skew: "lvl (skew t) = lvl t"
```
```   126 by(cases t rule: skew.cases) auto
```
```   127
```
```   128 lemma lvl_split: "lvl (split t) = lvl t \<or> lvl (split t) = lvl t + 1 \<and> sngl (split t)"
```
```   129 by(cases t rule: split.cases) auto
```
```   130
```
```   131 lemma invar_2Nodes:"invar (Node l x lv (Node rl rx rlv rr)) =
```
```   132      (invar l \<and> invar \<langle>rl, rx, rlv, rr\<rangle> \<and> lv = Suc (lvl l) \<and>
```
```   133      (lv = Suc rlv \<or> rlv = lv \<and> lv = Suc (lvl rr)))"
```
```   134 by simp
```
```   135
```
```   136 lemma invar_NodeLeaf[simp]:
```
```   137   "invar (Node l x lv Leaf) = (invar l \<and> lv = Suc (lvl l) \<and> lv = Suc 0)"
```
```   138 by simp
```
```   139
```
```   140 lemma sngl_if_invar: "invar (Node l a n r) \<Longrightarrow> n = lvl r \<Longrightarrow> sngl r"
```
```   141 by(cases r rule: sngl.cases) clarsimp+
```
```   142
```
```   143
```
```   144 subsection "Invariance"
```
```   145
```
```   146 subsubsection "Proofs for insert"
```
```   147
```
```   148 lemma lvl_insert_aux:
```
```   149   "lvl (insert x t) = lvl t \<or> lvl (insert x t) = lvl t + 1 \<and> sngl (insert x t)"
```
```   150 apply(induction t)
```
```   151 apply (auto simp: lvl_skew)
```
```   152 apply (metis Suc_eq_plus1 lvl.simps(2) lvl_split lvl_skew)+
```
```   153 done
```
```   154
```
```   155 lemma lvl_insert: obtains
```
```   156   (Same) "lvl (insert x t) = lvl t" |
```
```   157   (Incr) "lvl (insert x t) = lvl t + 1" "sngl (insert x t)"
```
```   158 using lvl_insert_aux by blast
```
```   159
```
```   160 lemma lvl_insert_sngl: "invar t \<Longrightarrow> sngl t \<Longrightarrow> lvl(insert x t) = lvl t"
```
```   161 proof (induction t rule: insert.induct)
```
```   162   case (2 x t1 a lv t2)
```
```   163   consider (LT) "x < a" | (GT) "x > a" | (EQ) "x = a"
```
```   164     using less_linear by blast
```
```   165   thus ?case proof cases
```
```   166     case LT
```
```   167     thus ?thesis using 2 by (auto simp add: skew_case split_case split: tree.splits)
```
```   168   next
```
```   169     case GT
```
```   170     thus ?thesis using 2 proof (cases t1)
```
```   171       case Node
```
```   172       thus ?thesis using 2 GT
```
```   173         apply (auto simp add: skew_case split_case split: tree.splits)
```
```   174         by (metis less_not_refl2 lvl.simps(2) lvl_insert_aux n_not_Suc_n sngl.simps(3))+
```
```   175     qed (auto simp add: lvl_0_iff)
```
```   176   qed simp
```
```   177 qed simp
```
```   178
```
```   179 lemma skew_invar: "invar t \<Longrightarrow> skew t = t"
```
```   180 by(cases t rule: skew.cases) auto
```
```   181
```
```   182 lemma split_invar: "invar t \<Longrightarrow> split t = t"
```
```   183 by(cases t rule: split.cases) clarsimp+
```
```   184
```
```   185 lemma invar_NodeL:
```
```   186   "\<lbrakk> invar(Node l x n r); invar l'; lvl l' = lvl l \<rbrakk> \<Longrightarrow> invar(Node l' x n r)"
```
```   187 by(auto)
```
```   188
```
```   189 lemma invar_NodeR:
```
```   190   "\<lbrakk> invar(Node l x n r); n = lvl r + 1; invar r'; lvl r' = lvl r \<rbrakk> \<Longrightarrow> invar(Node l x n r')"
```
```   191 by(auto)
```
```   192
```
```   193 lemma invar_NodeR2:
```
```   194   "\<lbrakk> invar(Node l x n r); sngl r'; n = lvl r + 1; invar r'; lvl r' = n \<rbrakk> \<Longrightarrow> invar(Node l x n r')"
```
```   195 by(cases r' rule: sngl.cases) clarsimp+
```
```   196
```
```   197
```
```   198 lemma lvl_insert_incr_iff: "(lvl(insert a t) = lvl t + 1) \<longleftrightarrow>
```
```   199   (\<exists>l x r. insert a t = Node l x (lvl t + 1) r \<and> lvl l = lvl r)"
```
```   200 apply(cases t)
```
```   201 apply(auto simp add: skew_case split_case split: if_splits)
```
```   202 apply(auto split: tree.splits if_splits)
```
```   203 done
```
```   204
```
```   205 lemma invar_insert: "invar t \<Longrightarrow> invar(insert a t)"
```
```   206 proof(induction t)
```
```   207   case N: (Node l x n r)
```
```   208   hence il: "invar l" and ir: "invar r" by auto
```
```   209   note iil = N.IH(1)[OF il]
```
```   210   note iir = N.IH(2)[OF ir]
```
```   211   let ?t = "Node l x n r"
```
```   212   have "a < x \<or> a = x \<or> x < a" by auto
```
```   213   moreover
```
```   214   have ?case if "a < x"
```
```   215   proof (cases rule: lvl_insert[of a l])
```
```   216     case (Same) thus ?thesis
```
```   217       using \<open>a<x\<close> invar_NodeL[OF N.prems iil Same]
```
```   218       by (simp add: skew_invar split_invar del: invar.simps)
```
```   219   next
```
```   220     case (Incr)
```
```   221     then obtain t1 w t2 where ial[simp]: "insert a l = Node t1 w n t2"
```
```   222       using N.prems by (auto simp: lvl_Suc_iff)
```
```   223     have l12: "lvl t1 = lvl t2"
```
```   224       by (metis Incr(1) ial lvl_insert_incr_iff tree.inject)
```
```   225     have "insert a ?t = split(skew(Node (insert a l) x n r))"
```
```   226       by(simp add: \<open>a<x\<close>)
```
```   227     also have "skew(Node (insert a l) x n r) = Node t1 w n (Node t2 x n r)"
```
```   228       by(simp)
```
```   229     also have "invar(split \<dots>)"
```
```   230     proof (cases r)
```
```   231       case Leaf
```
```   232       hence "l = Leaf" using N.prems by(auto simp: lvl_0_iff)
```
```   233       thus ?thesis using Leaf ial by simp
```
```   234     next
```
```   235       case [simp]: (Node t3 y m t4)
```
```   236       show ?thesis (*using N(3) iil l12 by(auto)*)
```
```   237       proof cases
```
```   238         assume "m = n" thus ?thesis using N(3) iil by(auto)
```
```   239       next
```
```   240         assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
```
```   241       qed
```
```   242     qed
```
```   243     finally show ?thesis .
```
```   244   qed
```
```   245   moreover
```
```   246   have ?case if "x < a"
```
```   247   proof -
```
```   248     from \<open>invar ?t\<close> have "n = lvl r \<or> n = lvl r + 1" by auto
```
```   249     thus ?case
```
```   250     proof
```
```   251       assume 0: "n = lvl r"
```
```   252       have "insert a ?t = split(skew(Node l x n (insert a r)))"
```
```   253         using \<open>a>x\<close> by(auto)
```
```   254       also have "skew(Node l x n (insert a r)) = Node l x n (insert a r)"
```
```   255         using N.prems by(simp add: skew_case split: tree.split)
```
```   256       also have "invar(split \<dots>)"
```
```   257       proof -
```
```   258         from lvl_insert_sngl[OF ir sngl_if_invar[OF \<open>invar ?t\<close> 0], of a]
```
```   259         obtain t1 y t2 where iar: "insert a r = Node t1 y n t2"
```
```   260           using N.prems 0 by (auto simp: lvl_Suc_iff)
```
```   261         from N.prems iar 0 iir
```
```   262         show ?thesis by (auto simp: split_case split: tree.splits)
```
```   263       qed
```
```   264       finally show ?thesis .
```
```   265     next
```
```   266       assume 1: "n = lvl r + 1"
```
```   267       hence "sngl ?t" by(cases r) auto
```
```   268       show ?thesis
```
```   269       proof (cases rule: lvl_insert[of a r])
```
```   270         case (Same)
```
```   271         show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF N.prems 1 iir Same]
```
```   272           by (auto simp add: skew_invar split_invar)
```
```   273       next
```
```   274         case (Incr)
```
```   275         thus ?thesis using invar_NodeR2[OF \<open>invar ?t\<close> Incr(2) 1 iir] 1 \<open>x < a\<close>
```
```   276           by (auto simp add: skew_invar split_invar split: if_splits)
```
```   277       qed
```
```   278     qed
```
```   279   qed
```
```   280   moreover
```
```   281   have "a = x \<Longrightarrow> ?case" using N.prems by auto
```
```   282   ultimately show ?case by blast
```
```   283 qed simp
```
```   284
```
```   285
```
```   286 subsubsection "Proofs for delete"
```
```   287
```
```   288 lemma invarL: "ASSUMPTION(invar \<langle>l, a, lv, r\<rangle>) \<Longrightarrow> invar l"
```
```   289 by(simp add: ASSUMPTION_def)
```
```   290
```
```   291 lemma invarR: "ASSUMPTION(invar \<langle>lv, l, a, r\<rangle>) \<Longrightarrow> invar r"
```
```   292 by(simp add: ASSUMPTION_def)
```
```   293
```
```   294 lemma sngl_NodeI:
```
```   295   "sngl (Node l a lv r) \<Longrightarrow> sngl (Node l' a' lv r)"
```
```   296 by(cases r) (simp_all)
```
```   297
```
```   298
```
```   299 declare invarL[simp] invarR[simp]
```
```   300
```
```   301 lemma pre_cases:
```
```   302 assumes "pre_adjust (Node l x lv r)"
```
```   303 obtains
```
```   304  (tSngl) "invar l \<and> invar r \<and>
```
```   305     lv = Suc (lvl r) \<and> lvl l = lvl r" |
```
```   306  (tDouble) "invar l \<and> invar r \<and>
```
```   307     lv = lvl r \<and> Suc (lvl l) = lvl r \<and> sngl r " |
```
```   308  (rDown) "invar l \<and> invar r \<and>
```
```   309     lv = Suc (Suc (lvl r)) \<and>  lv = Suc (lvl l)" |
```
```   310  (lDown_tSngl) "invar l \<and> invar r \<and>
```
```   311     lv = Suc (lvl r) \<and> lv = Suc (Suc (lvl l))" |
```
```   312  (lDown_tDouble) "invar l \<and> invar r \<and>
```
```   313     lv = lvl r \<and> lv = Suc (Suc (lvl l)) \<and> sngl r"
```
```   314 using assms unfolding pre_adjust.simps
```
```   315 by auto
```
```   316
```
```   317 declare invar.simps(2)[simp del] invar_2Nodes[simp add]
```
```   318
```
```   319 lemma invar_adjust:
```
```   320   assumes pre: "pre_adjust (Node l a lv r)"
```
```   321   shows  "invar(adjust (Node l a lv r))"
```
```   322 using pre proof (cases rule: pre_cases)
```
```   323   case (tDouble) thus ?thesis unfolding adjust_def by (cases r) (auto simp: invar.simps(2))
```
```   324 next
```
```   325   case (rDown)
```
```   326   from rDown obtain llv ll la lr where l: "l = Node ll la llv lr" by (cases l) auto
```
```   327   from rDown show ?thesis unfolding adjust_def by (auto simp: l invar.simps(2) split: tree.splits)
```
```   328 next
```
```   329   case (lDown_tDouble)
```
```   330   from lDown_tDouble obtain rlv rr ra rl where r: "r = Node rl ra rlv rr" by (cases r) auto
```
```   331   from lDown_tDouble and r obtain rrlv rrr rra rrl where
```
```   332     rr :"rr = Node rrr rra rrlv rrl" by (cases rr) auto
```
```   333   from  lDown_tDouble show ?thesis unfolding adjust_def r rr
```
```   334     apply (cases rl) apply (auto simp add: invar.simps(2) split!: if_split)
```
```   335     using lDown_tDouble by (auto simp: split_case lvl_0_iff  elim:lvl.elims split: tree.split)
```
```   336 qed (auto simp: split_case invar.simps(2) adjust_def split: tree.splits)
```
```   337
```
```   338 lemma lvl_adjust:
```
```   339   assumes "pre_adjust (Node l a lv r)"
```
```   340   shows "lv = lvl (adjust(Node l a lv r)) \<or> lv = lvl (adjust(Node l a lv r)) + 1"
```
```   341 using assms(1) proof(cases rule: pre_cases)
```
```   342   case lDown_tSngl thus ?thesis
```
```   343     using lvl_split[of "\<langle>l, a, lvl r, r\<rangle>"] by (auto simp: adjust_def)
```
```   344 next
```
```   345   case lDown_tDouble thus ?thesis
```
```   346     by (auto simp: adjust_def invar.simps(2) split: tree.split)
```
```   347 qed (auto simp: adjust_def split: tree.splits)
```
```   348
```
```   349 lemma sngl_adjust: assumes "pre_adjust (Node l a lv r)"
```
```   350   "sngl \<langle>l, a, lv, r\<rangle>" "lv = lvl (adjust \<langle>l, a, lv, r\<rangle>)"
```
```   351   shows "sngl (adjust \<langle>l, a, lv, r\<rangle>)"
```
```   352 using assms proof (cases rule: pre_cases)
```
```   353   case rDown
```
```   354   thus ?thesis using assms(2,3) unfolding adjust_def
```
```   355     by (auto simp add: skew_case) (auto split: tree.split)
```
```   356 qed (auto simp: adjust_def skew_case split_case split: tree.split)
```
```   357
```
```   358 definition "post_del t t' ==
```
```   359   invar t' \<and>
```
```   360   (lvl t' = lvl t \<or> lvl t' + 1 = lvl t) \<and>
```
```   361   (lvl t' = lvl t \<and> sngl t \<longrightarrow> sngl t')"
```
```   362
```
```   363 lemma pre_adj_if_postR:
```
```   364   "invar\<langle>lv, l, a, r\<rangle> \<Longrightarrow> post_del r r' \<Longrightarrow> pre_adjust \<langle>lv, l, a, r'\<rangle>"
```
```   365 by(cases "sngl r")
```
```   366   (auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims)
```
```   367
```
```   368 lemma pre_adj_if_postL:
```
```   369   "invar\<langle>l, a, lv, r\<rangle> \<Longrightarrow> post_del l l' \<Longrightarrow> pre_adjust \<langle>l', b, lv, r\<rangle>"
```
```   370 by(cases "sngl r")
```
```   371   (auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims)
```
```   372
```
```   373 lemma post_del_adjL:
```
```   374   "\<lbrakk> invar\<langle>l, a, lv, r\<rangle>; pre_adjust \<langle>l', b, lv, r\<rangle> \<rbrakk>
```
```   375   \<Longrightarrow> post_del \<langle>l, a, lv, r\<rangle> (adjust \<langle>l', b, lv, r\<rangle>)"
```
```   376 unfolding post_del_def
```
```   377 by (metis invar_adjust lvl_adjust sngl_NodeI sngl_adjust lvl.simps(2))
```
```   378
```
```   379 lemma post_del_adjR:
```
```   380 assumes "invar\<langle>lv, l, a, r\<rangle>" "pre_adjust \<langle>lv, l, a, r'\<rangle>" "post_del r r'"
```
```   381 shows "post_del \<langle>lv, l, a, r\<rangle> (adjust \<langle>lv, l, a, r'\<rangle>)"
```
```   382 proof(unfold post_del_def, safe del: disjCI)
```
```   383   let ?t = "\<langle>lv, l, a, r\<rangle>"
```
```   384   let ?t' = "adjust \<langle>lv, l, a, r'\<rangle>"
```
```   385   show "invar ?t'" by(rule invar_adjust[OF assms(2)])
```
```   386   show "lvl ?t' = lvl ?t \<or> lvl ?t' + 1 = lvl ?t"
```
```   387     using lvl_adjust[OF assms(2)] by auto
```
```   388   show "sngl ?t'" if as: "lvl ?t' = lvl ?t" "sngl ?t"
```
```   389   proof -
```
```   390     have s: "sngl \<langle>lv, l, a, r'\<rangle>"
```
```   391     proof(cases r')
```
```   392       case Leaf thus ?thesis by simp
```
```   393     next
```
```   394       case Node thus ?thesis using as(2) assms(1,3)
```
```   395       by (cases r) (auto simp: post_del_def)
```
```   396     qed
```
```   397     show ?thesis using as(1) sngl_adjust[OF assms(2) s] by simp
```
```   398   qed
```
```   399 qed
```
```   400
```
```   401 declare prod.splits[split]
```
```   402
```
```   403 theorem post_split_max:
```
```   404  "\<lbrakk> invar t; (t', x) = split_max t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> post_del t t'"
```
```   405 proof (induction t arbitrary: t' rule: split_max.induct)
```
```   406   case (2 lv l a lvr rl ra rr)
```
```   407   let ?r =  "\<langle>lvr, rl, ra, rr\<rangle>"
```
```   408   let ?t = "\<langle>lv, l, a, ?r\<rangle>"
```
```   409   from "2.prems"(2) obtain r' where r': "(r', x) = split_max ?r"
```
```   410     and [simp]: "t' = adjust \<langle>lv, l, a, r'\<rangle>" by auto
```
```   411   from  "2.IH"[OF _ r'] \<open>invar ?t\<close> have post: "post_del ?r r'" by simp
```
```   412   note preR = pre_adj_if_postR[OF \<open>invar ?t\<close> post]
```
```   413   show ?case by (simp add: post_del_adjR[OF "2.prems"(1) preR post])
```
```   414 qed (auto simp: post_del_def)
```
```   415
```
```   416 theorem post_delete: "invar t \<Longrightarrow> post_del t (delete x t)"
```
```   417 proof (induction t)
```
```   418   case (Node l a lv r)
```
```   419
```
```   420   let ?l' = "delete x l" and ?r' = "delete x r"
```
```   421   let ?t = "Node l a lv r" let ?t' = "delete x ?t"
```
```   422
```
```   423   from Node.prems have inv_l: "invar l" and inv_r: "invar r" by (auto)
```
```   424
```
```   425   note post_l' = Node.IH(1)[OF inv_l]
```
```   426   note preL = pre_adj_if_postL[OF Node.prems post_l']
```
```   427
```
```   428   note post_r' = Node.IH(2)[OF inv_r]
```
```   429   note preR = pre_adj_if_postR[OF Node.prems post_r']
```
```   430
```
```   431   show ?case
```
```   432   proof (cases rule: linorder_cases[of x a])
```
```   433     case less
```
```   434     thus ?thesis using Node.prems by (simp add: post_del_adjL preL)
```
```   435   next
```
```   436     case greater
```
```   437     thus ?thesis using Node.prems by (simp add: post_del_adjR preR post_r')
```
```   438   next
```
```   439     case equal
```
```   440     show ?thesis
```
```   441     proof cases
```
```   442       assume "l = Leaf" thus ?thesis using equal Node.prems
```
```   443         by(auto simp: post_del_def invar.simps(2))
```
```   444     next
```
```   445       assume "l \<noteq> Leaf" thus ?thesis using equal
```
```   446         by simp (metis Node.prems inv_l post_del_adjL post_split_max pre_adj_if_postL)
```
```   447     qed
```
```   448   qed
```
```   449 qed (simp add: post_del_def)
```
```   450
```
```   451 declare invar_2Nodes[simp del]
```
```   452
```
```   453
```
```   454 subsection "Functional Correctness"
```
```   455
```
```   456
```
```   457 subsubsection "Proofs for insert"
```
```   458
```
```   459 lemma inorder_split: "inorder(split t) = inorder t"
```
```   460 by(cases t rule: split.cases) (auto)
```
```   461
```
```   462 lemma inorder_skew: "inorder(skew t) = inorder t"
```
```   463 by(cases t rule: skew.cases) (auto)
```
```   464
```
```   465 lemma inorder_insert:
```
```   466   "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
```
```   467 by(induction t) (auto simp: ins_list_simps inorder_split inorder_skew)
```
```   468
```
```   469
```
```   470 subsubsection "Proofs for delete"
```
```   471
```
```   472 lemma inorder_adjust: "t \<noteq> Leaf \<Longrightarrow> pre_adjust t \<Longrightarrow> inorder(adjust t) = inorder t"
```
```   473 by(cases t)
```
```   474   (auto simp: adjust_def inorder_skew inorder_split invar.simps(2) pre_adjust.simps
```
```   475      split: tree.splits)
```
```   476
```
```   477 lemma split_maxD:
```
```   478   "\<lbrakk> split_max t = (t',x); t \<noteq> Leaf; invar t \<rbrakk> \<Longrightarrow> inorder t' @ [x] = inorder t"
```
```   479 by(induction t arbitrary: t' rule: split_max.induct)
```
```   480   (auto simp: sorted_lems inorder_adjust pre_adj_if_postR post_split_max split: prod.splits)
```
```   481
```
```   482 lemma inorder_delete:
```
```   483   "invar t \<Longrightarrow> sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
```
```   484 by(induction t)
```
```   485   (auto simp: del_list_simps inorder_adjust pre_adj_if_postL pre_adj_if_postR
```
```   486               post_split_max post_delete split_maxD split: prod.splits)
```
```   487
```
```   488 interpretation S: Set_by_Ordered
```
```   489 where empty = empty and isin = isin and insert = insert and delete = delete
```
```   490 and inorder = inorder and inv = invar
```
```   491 proof (standard, goal_cases)
```
```   492   case 1 show ?case by (simp add: empty_def)
```
```   493 next
```
```   494   case 2 thus ?case by(simp add: isin_set_inorder)
```
```   495 next
```
```   496   case 3 thus ?case by(simp add: inorder_insert)
```
```   497 next
```
```   498   case 4 thus ?case by(simp add: inorder_delete)
```
```   499 next
```
```   500   case 5 thus ?case by(simp add: empty_def)
```
```   501 next
```
```   502   case 6 thus ?case by(simp add: invar_insert)
```
```   503 next
```
```   504   case 7 thus ?case using post_delete by(auto simp: post_del_def)
```
```   505 qed
```
```   506
```
```   507 end
```