1(* 2Author: Tobias Nipkow, Daniel St��we 3*) 4 5section \<open>AA Tree Implementation of Sets\<close> 6 7theory AA_Set 8imports 9 Isin2 10 Cmp 11begin 12 13type_synonym 'a aa_tree = "('a,nat) tree" 14 15definition empty :: "'a aa_tree" where 16"empty = Leaf" 17 18fun lvl :: "'a aa_tree \<Rightarrow> nat" where 19"lvl Leaf = 0" | 20"lvl (Node _ _ lv _) = lv" 21 22fun 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 28fun 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 33fun 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 40hide_const (open) insert 41 42fun 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 50fun 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 55definition 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 75text\<open>In the paper, the last case of @{const adjust} is expressed with the help of an 76incorrect auxiliary function \texttt{nlvl}. 77 78Function @{text split_max} below is called \texttt{dellrg} in the paper. 79The latter is incorrect for two reasons: \texttt{dellrg} is meant to delete the largest 80element but recurses on the left instead of the right subtree; the invariant 81is not restored.\<close> 82 83fun 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 87fun 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 96fun 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 101declare pre_adjust.simps [simp del] 102 103subsection "Auxiliary Proofs" 104 105lemma 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)" 111by(auto split: tree.split) 112 113lemma 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)" 117by(auto split: tree.split) 118 119lemma lvl_0_iff: "invar t \<Longrightarrow> lvl t = 0 \<longleftrightarrow> t = Leaf" 120by(cases t) auto 121 122lemma lvl_Suc_iff: "lvl t = Suc n \<longleftrightarrow> (\<exists> l a r. t = Node l a (Suc n) r)" 123by(cases t) auto 124 125lemma lvl_skew: "lvl (skew t) = lvl t" 126by(cases t rule: skew.cases) auto 127 128lemma lvl_split: "lvl (split t) = lvl t \<or> lvl (split t) = lvl t + 1 \<and> sngl (split t)" 129by(cases t rule: split.cases) auto 130 131lemma 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)))" 134by simp 135 136lemma invar_NodeLeaf[simp]: 137 "invar (Node l x lv Leaf) = (invar l \<and> lv = Suc (lvl l) \<and> lv = Suc 0)" 138by simp 139 140lemma sngl_if_invar: "invar (Node l a n r) \<Longrightarrow> n = lvl r \<Longrightarrow> sngl r" 141by(cases r rule: sngl.cases) clarsimp+ 142 143 144subsection "Invariance" 145 146subsubsection "Proofs for insert" 147 148lemma lvl_insert_aux: 149 "lvl (insert x t) = lvl t \<or> lvl (insert x t) = lvl t + 1 \<and> sngl (insert x t)" 150apply(induction t) 151apply (auto simp: lvl_skew) 152apply (metis Suc_eq_plus1 lvl.simps(2) lvl_split lvl_skew)+ 153done 154 155lemma lvl_insert: obtains 156 (Same) "lvl (insert x t) = lvl t" | 157 (Incr) "lvl (insert x t) = lvl t + 1" "sngl (insert x t)" 158using lvl_insert_aux by blast 159 160lemma lvl_insert_sngl: "invar t \<Longrightarrow> sngl t \<Longrightarrow> lvl(insert x t) = lvl t" 161proof (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 177qed simp 178 179lemma skew_invar: "invar t \<Longrightarrow> skew t = t" 180by(cases t rule: skew.cases) auto 181 182lemma split_invar: "invar t \<Longrightarrow> split t = t" 183by(cases t rule: split.cases) clarsimp+ 184 185lemma invar_NodeL: 186 "\<lbrakk> invar(Node l x n r); invar l'; lvl l' = lvl l \<rbrakk> \<Longrightarrow> invar(Node l' x n r)" 187by(auto) 188 189lemma 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')" 191by(auto) 192 193lemma 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')" 195by(cases r' rule: sngl.cases) clarsimp+ 196 197 198lemma 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)" 200apply(cases t) 201apply(auto simp add: skew_case split_case split: if_splits) 202apply(auto split: tree.splits if_splits) 203done 204 205lemma invar_insert: "invar t \<Longrightarrow> invar(insert a t)" 206proof(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 283qed simp 284 285 286subsubsection "Proofs for delete" 287 288lemma invarL: "ASSUMPTION(invar \<langle>l, a, lv, r\<rangle>) \<Longrightarrow> invar l" 289by(simp add: ASSUMPTION_def) 290 291lemma invarR: "ASSUMPTION(invar \<langle>lv, l, a, r\<rangle>) \<Longrightarrow> invar r" 292by(simp add: ASSUMPTION_def) 293 294lemma sngl_NodeI: 295 "sngl (Node l a lv r) \<Longrightarrow> sngl (Node l' a' lv r)" 296by(cases r) (simp_all) 297 298 299declare invarL[simp] invarR[simp] 300 301lemma pre_cases: 302assumes "pre_adjust (Node l x lv r)" 303obtains 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" 314using assms unfolding pre_adjust.simps 315by auto 316 317declare invar.simps(2)[simp del] invar_2Nodes[simp add] 318 319lemma invar_adjust: 320 assumes pre: "pre_adjust (Node l a lv r)" 321 shows "invar(adjust (Node l a lv r))" 322using pre proof (cases rule: pre_cases) 323 case (tDouble) thus ?thesis unfolding adjust_def by (cases r) (auto simp: invar.simps(2)) 324next 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) 328next 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) 336qed (auto simp: split_case invar.simps(2) adjust_def split: tree.splits) 337 338lemma 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" 341using 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) 344next 345 case lDown_tDouble thus ?thesis 346 by (auto simp: adjust_def invar.simps(2) split: tree.split) 347qed (auto simp: adjust_def split: tree.splits) 348 349lemma 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>)" 352using 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) 356qed (auto simp: adjust_def skew_case split_case split: tree.split) 357 358definition "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 363lemma 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>" 365by(cases "sngl r") 366 (auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims) 367 368lemma 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>" 370by(cases "sngl r") 371 (auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims) 372 373lemma 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>)" 376unfolding post_del_def 377by (metis invar_adjust lvl_adjust sngl_NodeI sngl_adjust lvl.simps(2)) 378 379lemma post_del_adjR: 380assumes "invar\<langle>lv, l, a, r\<rangle>" "pre_adjust \<langle>lv, l, a, r'\<rangle>" "post_del r r'" 381shows "post_del \<langle>lv, l, a, r\<rangle> (adjust \<langle>lv, l, a, r'\<rangle>)" 382proof(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 399qed 400 401declare prod.splits[split] 402 403theorem post_split_max: 404 "\<lbrakk> invar t; (t', x) = split_max t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> post_del t t'" 405proof (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]) 414qed (auto simp: post_del_def) 415 416theorem post_delete: "invar t \<Longrightarrow> post_del t (delete x t)" 417proof (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 449qed (simp add: post_del_def) 450 451declare invar_2Nodes[simp del] 452 453 454subsection "Functional Correctness" 455 456 457subsubsection "Proofs for insert" 458 459lemma inorder_split: "inorder(split t) = inorder t" 460by(cases t rule: split.cases) (auto) 461 462lemma inorder_skew: "inorder(skew t) = inorder t" 463by(cases t rule: skew.cases) (auto) 464 465lemma inorder_insert: 466 "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)" 467by(induction t) (auto simp: ins_list_simps inorder_split inorder_skew) 468 469 470subsubsection "Proofs for delete" 471 472lemma inorder_adjust: "t \<noteq> Leaf \<Longrightarrow> pre_adjust t \<Longrightarrow> inorder(adjust t) = inorder t" 473by(cases t) 474 (auto simp: adjust_def inorder_skew inorder_split invar.simps(2) pre_adjust.simps 475 split: tree.splits) 476 477lemma split_maxD: 478 "\<lbrakk> split_max t = (t',x); t \<noteq> Leaf; invar t \<rbrakk> \<Longrightarrow> inorder t' @ [x] = inorder t" 479by(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 482lemma inorder_delete: 483 "invar t \<Longrightarrow> sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)" 484by(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 488interpretation S: Set_by_Ordered 489where empty = empty and isin = isin and insert = insert and delete = delete 490and inorder = inorder and inv = invar 491proof (standard, goal_cases) 492 case 1 show ?case by (simp add: empty_def) 493next 494 case 2 thus ?case by(simp add: isin_set_inorder) 495next 496 case 3 thus ?case by(simp add: inorder_insert) 497next 498 case 4 thus ?case by(simp add: inorder_delete) 499next 500 case 5 thus ?case by(simp add: empty_def) 501next 502 case 6 thus ?case by(simp add: invar_insert) 503next 504 case 7 thus ?case using post_delete by(auto simp: post_del_def) 505qed 506 507end 508