src/HOL/ex/Sublist.thy
changeset 30689 b14b2cc4e25e
parent 30688 2d1d426e00e4
child 30690 55ef8e045931
     1.1 --- a/src/HOL/ex/Sublist.thy	Mon Mar 23 19:01:17 2009 +0100
     1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.3 @@ -1,505 +0,0 @@
     1.4 -(* $Id$ *)
     1.5 -
     1.6 -header {* Slices of lists *}
     1.7 -
     1.8 -theory Sublist
     1.9 -imports Multiset
    1.10 -begin
    1.11 -
    1.12 -
    1.13 -lemma sublist_split: "i \<le> j \<and> j \<le> k \<Longrightarrow> sublist xs {i..<j} @ sublist xs {j..<k} = sublist xs {i..<k}" 
    1.14 -apply (induct xs arbitrary: i j k)
    1.15 -apply simp
    1.16 -apply (simp only: sublist_Cons)
    1.17 -apply simp
    1.18 -apply safe
    1.19 -apply simp
    1.20 -apply (erule_tac x="0" in meta_allE)
    1.21 -apply (erule_tac x="j - 1" in meta_allE)
    1.22 -apply (erule_tac x="k - 1" in meta_allE)
    1.23 -apply (subgoal_tac "0 \<le> j - 1 \<and> j - 1 \<le> k - 1")
    1.24 -apply simp
    1.25 -apply (subgoal_tac "{ja. Suc ja < j} = {0..<j - Suc 0}")
    1.26 -apply (subgoal_tac "{ja. j \<le> Suc ja \<and> Suc ja < k} = {j - Suc 0..<k - Suc 0}")
    1.27 -apply (subgoal_tac "{j. Suc j < k} = {0..<k - Suc 0}")
    1.28 -apply simp
    1.29 -apply fastsimp
    1.30 -apply fastsimp
    1.31 -apply fastsimp
    1.32 -apply fastsimp
    1.33 -apply (erule_tac x="i - 1" in meta_allE)
    1.34 -apply (erule_tac x="j - 1" in meta_allE)
    1.35 -apply (erule_tac x="k - 1" in meta_allE)
    1.36 -apply (subgoal_tac " {ja. i \<le> Suc ja \<and> Suc ja < j} = {i - 1 ..<j - 1}")
    1.37 -apply (subgoal_tac " {ja. j \<le> Suc ja \<and> Suc ja < k} = {j - 1..<k - 1}")
    1.38 -apply (subgoal_tac "{j. i \<le> Suc j \<and> Suc j < k} = {i - 1..<k - 1}")
    1.39 -apply (subgoal_tac " i - 1 \<le> j - 1 \<and> j - 1 \<le> k - 1")
    1.40 -apply simp
    1.41 -apply fastsimp
    1.42 -apply fastsimp
    1.43 -apply fastsimp
    1.44 -apply fastsimp
    1.45 -done
    1.46 -
    1.47 -lemma sublist_update1: "i \<notin> inds \<Longrightarrow> sublist (xs[i := v]) inds = sublist xs inds"
    1.48 -apply (induct xs arbitrary: i inds)
    1.49 -apply simp
    1.50 -apply (case_tac i)
    1.51 -apply (simp add: sublist_Cons)
    1.52 -apply (simp add: sublist_Cons)
    1.53 -done
    1.54 -
    1.55 -lemma sublist_update2: "i \<in> inds \<Longrightarrow> sublist (xs[i := v]) inds = (sublist xs inds)[(card {k \<in> inds. k < i}):= v]"
    1.56 -proof (induct xs arbitrary: i inds)
    1.57 -  case Nil thus ?case by simp
    1.58 -next
    1.59 -  case (Cons x xs)
    1.60 -  thus ?case
    1.61 -  proof (cases i)
    1.62 -    case 0 with Cons show ?thesis by (simp add: sublist_Cons)
    1.63 -  next
    1.64 -    case (Suc i')
    1.65 -    with Cons show ?thesis
    1.66 -      apply simp
    1.67 -      apply (simp add: sublist_Cons)
    1.68 -      apply auto
    1.69 -      apply (auto simp add: nat.split)
    1.70 -      apply (simp add: card_less_Suc[symmetric])
    1.71 -      apply (simp add: card_less_Suc2)
    1.72 -      done
    1.73 -  qed
    1.74 -qed
    1.75 -
    1.76 -lemma sublist_update: "sublist (xs[i := v]) inds = (if i \<in> inds then (sublist xs inds)[(card {k \<in> inds. k < i}) := v] else sublist xs inds)"
    1.77 -by (simp add: sublist_update1 sublist_update2)
    1.78 -
    1.79 -lemma sublist_take: "sublist xs {j. j < m} = take m xs"
    1.80 -apply (induct xs arbitrary: m)
    1.81 -apply simp
    1.82 -apply (case_tac m)
    1.83 -apply simp
    1.84 -apply (simp add: sublist_Cons)
    1.85 -done
    1.86 -
    1.87 -lemma sublist_take': "sublist xs {0..<m} = take m xs"
    1.88 -apply (induct xs arbitrary: m)
    1.89 -apply simp
    1.90 -apply (case_tac m)
    1.91 -apply simp
    1.92 -apply (simp add: sublist_Cons sublist_take)
    1.93 -done
    1.94 -
    1.95 -lemma sublist_all[simp]: "sublist xs {j. j < length xs} = xs"
    1.96 -apply (induct xs)
    1.97 -apply simp
    1.98 -apply (simp add: sublist_Cons)
    1.99 -done
   1.100 -
   1.101 -lemma sublist_all'[simp]: "sublist xs {0..<length xs} = xs"
   1.102 -apply (induct xs)
   1.103 -apply simp
   1.104 -apply (simp add: sublist_Cons)
   1.105 -done
   1.106 -
   1.107 -lemma sublist_single: "a < length xs \<Longrightarrow> sublist xs {a} = [xs ! a]"
   1.108 -apply (induct xs arbitrary: a)
   1.109 -apply simp
   1.110 -apply(case_tac aa)
   1.111 -apply simp
   1.112 -apply (simp add: sublist_Cons)
   1.113 -apply simp
   1.114 -apply (simp add: sublist_Cons)
   1.115 -done
   1.116 -
   1.117 -lemma sublist_is_Nil: "\<forall>i \<in> inds. i \<ge> length xs \<Longrightarrow> sublist xs inds = []" 
   1.118 -apply (induct xs arbitrary: inds)
   1.119 -apply simp
   1.120 -apply (simp add: sublist_Cons)
   1.121 -apply auto
   1.122 -apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
   1.123 -apply auto
   1.124 -done
   1.125 -
   1.126 -lemma sublist_Nil': "sublist xs inds = [] \<Longrightarrow> \<forall>i \<in> inds. i \<ge> length xs"
   1.127 -apply (induct xs arbitrary: inds)
   1.128 -apply simp
   1.129 -apply (simp add: sublist_Cons)
   1.130 -apply (auto split: if_splits)
   1.131 -apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
   1.132 -apply (case_tac x, auto)
   1.133 -done
   1.134 -
   1.135 -lemma sublist_Nil[simp]: "(sublist xs inds = []) = (\<forall>i \<in> inds. i \<ge> length xs)"
   1.136 -apply (induct xs arbitrary: inds)
   1.137 -apply simp
   1.138 -apply (simp add: sublist_Cons)
   1.139 -apply auto
   1.140 -apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
   1.141 -apply (case_tac x, auto)
   1.142 -done
   1.143 -
   1.144 -lemma sublist_eq_subseteq: " \<lbrakk> inds' \<subseteq> inds; sublist xs inds = sublist ys inds \<rbrakk> \<Longrightarrow> sublist xs inds' = sublist ys inds'"
   1.145 -apply (induct xs arbitrary: ys inds inds')
   1.146 -apply simp
   1.147 -apply (drule sym, rule sym)
   1.148 -apply (simp add: sublist_Nil, fastsimp)
   1.149 -apply (case_tac ys)
   1.150 -apply (simp add: sublist_Nil, fastsimp)
   1.151 -apply (auto simp add: sublist_Cons)
   1.152 -apply (erule_tac x="list" in meta_allE)
   1.153 -apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
   1.154 -apply (erule_tac x="{j. Suc j \<in> inds'}" in meta_allE)
   1.155 -apply fastsimp
   1.156 -apply (erule_tac x="list" in meta_allE)
   1.157 -apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
   1.158 -apply (erule_tac x="{j. Suc j \<in> inds'}" in meta_allE)
   1.159 -apply fastsimp
   1.160 -done
   1.161 -
   1.162 -lemma sublist_eq: "\<lbrakk> \<forall>i \<in> inds. ((i < length xs) \<and> (i < length ys)) \<or> ((i \<ge> length xs ) \<and> (i \<ge> length ys)); \<forall>i \<in> inds. xs ! i = ys ! i \<rbrakk> \<Longrightarrow> sublist xs inds = sublist ys inds"
   1.163 -apply (induct xs arbitrary: ys inds)
   1.164 -apply simp
   1.165 -apply (rule sym, simp add: sublist_Nil)
   1.166 -apply (case_tac ys)
   1.167 -apply (simp add: sublist_Nil)
   1.168 -apply (auto simp add: sublist_Cons)
   1.169 -apply (erule_tac x="list" in meta_allE)
   1.170 -apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
   1.171 -apply fastsimp
   1.172 -apply (erule_tac x="list" in meta_allE)
   1.173 -apply (erule_tac x="{j. Suc j \<in> inds}" in meta_allE)
   1.174 -apply fastsimp
   1.175 -done
   1.176 -
   1.177 -lemma sublist_eq_samelength: "\<lbrakk> length xs = length ys; \<forall>i \<in> inds. xs ! i = ys ! i \<rbrakk> \<Longrightarrow> sublist xs inds = sublist ys inds"
   1.178 -by (rule sublist_eq, auto)
   1.179 -
   1.180 -lemma sublist_eq_samelength_iff: "length xs = length ys \<Longrightarrow> (sublist xs inds = sublist ys inds) = (\<forall>i \<in> inds. xs ! i = ys ! i)"
   1.181 -apply (induct xs arbitrary: ys inds)
   1.182 -apply simp
   1.183 -apply (rule sym, simp add: sublist_Nil)
   1.184 -apply (case_tac ys)
   1.185 -apply (simp add: sublist_Nil)
   1.186 -apply (auto simp add: sublist_Cons)
   1.187 -apply (case_tac i)
   1.188 -apply auto
   1.189 -apply (case_tac i)
   1.190 -apply auto
   1.191 -done
   1.192 -
   1.193 -section {* Another sublist function *}
   1.194 -
   1.195 -function sublist' :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
   1.196 -where
   1.197 -  "sublist' n m [] = []"
   1.198 -| "sublist' n 0 xs = []"
   1.199 -| "sublist' 0 (Suc m) (x#xs) = (x#sublist' 0 m xs)"
   1.200 -| "sublist' (Suc n) (Suc m) (x#xs) = sublist' n m xs"
   1.201 -by pat_completeness auto
   1.202 -termination by lexicographic_order
   1.203 -
   1.204 -subsection {* Proving equivalence to the other sublist command *}
   1.205 -
   1.206 -lemma sublist'_sublist: "sublist' n m xs = sublist xs {j. n \<le> j \<and> j < m}"
   1.207 -apply (induct xs arbitrary: n m)
   1.208 -apply simp
   1.209 -apply (case_tac n)
   1.210 -apply (case_tac m)
   1.211 -apply simp
   1.212 -apply (simp add: sublist_Cons)
   1.213 -apply (case_tac m)
   1.214 -apply simp
   1.215 -apply (simp add: sublist_Cons)
   1.216 -done
   1.217 -
   1.218 -
   1.219 -lemma "sublist' n m xs = sublist xs {n..<m}"
   1.220 -apply (induct xs arbitrary: n m)
   1.221 -apply simp
   1.222 -apply (case_tac n, case_tac m)
   1.223 -apply simp
   1.224 -apply simp
   1.225 -apply (simp add: sublist_take')
   1.226 -apply (case_tac m)
   1.227 -apply simp
   1.228 -apply (simp add: sublist_Cons sublist'_sublist)
   1.229 -done
   1.230 -
   1.231 -
   1.232 -subsection {* Showing equivalence to use of drop and take for definition *}
   1.233 -
   1.234 -lemma "sublist' n m xs = take (m - n) (drop n xs)"
   1.235 -apply (induct xs arbitrary: n m)
   1.236 -apply simp
   1.237 -apply (case_tac m)
   1.238 -apply simp
   1.239 -apply (case_tac n)
   1.240 -apply simp
   1.241 -apply simp
   1.242 -done
   1.243 -
   1.244 -subsection {* General lemma about sublist *}
   1.245 -
   1.246 -lemma sublist'_Nil[simp]: "sublist' i j [] = []"
   1.247 -by simp
   1.248 -
   1.249 -lemma sublist'_Cons[simp]: "sublist' i (Suc j) (x#xs) = (case i of 0 \<Rightarrow> (x # sublist' 0 j xs) | Suc i' \<Rightarrow>  sublist' i' j xs)"
   1.250 -by (cases i) auto
   1.251 -
   1.252 -lemma sublist'_Cons2[simp]: "sublist' i j (x#xs) = (if (j = 0) then [] else ((if (i = 0) then [x] else []) @ sublist' (i - 1) (j - 1) xs))"
   1.253 -apply (cases j)
   1.254 -apply auto
   1.255 -apply (cases i)
   1.256 -apply auto
   1.257 -done
   1.258 -
   1.259 -lemma sublist_n_0: "sublist' n 0 xs = []"
   1.260 -by (induct xs, auto)
   1.261 -
   1.262 -lemma sublist'_Nil': "n \<ge> m \<Longrightarrow> sublist' n m xs = []"
   1.263 -apply (induct xs arbitrary: n m)
   1.264 -apply simp
   1.265 -apply (case_tac m)
   1.266 -apply simp
   1.267 -apply (case_tac n)
   1.268 -apply simp
   1.269 -apply simp
   1.270 -done
   1.271 -
   1.272 -lemma sublist'_Nil2: "n \<ge> length xs \<Longrightarrow> sublist' n m xs = []"
   1.273 -apply (induct xs arbitrary: n m)
   1.274 -apply simp
   1.275 -apply (case_tac m)
   1.276 -apply simp
   1.277 -apply (case_tac n)
   1.278 -apply simp
   1.279 -apply simp
   1.280 -done
   1.281 -
   1.282 -lemma sublist'_Nil3: "(sublist' n m xs = []) = ((n \<ge> m) \<or> (n \<ge> length xs))"
   1.283 -apply (induct xs arbitrary: n m)
   1.284 -apply simp
   1.285 -apply (case_tac m)
   1.286 -apply simp
   1.287 -apply (case_tac n)
   1.288 -apply simp
   1.289 -apply simp
   1.290 -done
   1.291 -
   1.292 -lemma sublist'_notNil: "\<lbrakk> n < length xs; n < m \<rbrakk> \<Longrightarrow> sublist' n m xs \<noteq> []"
   1.293 -apply (induct xs arbitrary: n m)
   1.294 -apply simp
   1.295 -apply (case_tac m)
   1.296 -apply simp
   1.297 -apply (case_tac n)
   1.298 -apply simp
   1.299 -apply simp
   1.300 -done
   1.301 -
   1.302 -lemma sublist'_single: "n < length xs \<Longrightarrow> sublist' n (Suc n) xs = [xs ! n]"
   1.303 -apply (induct xs arbitrary: n)
   1.304 -apply simp
   1.305 -apply simp
   1.306 -apply (case_tac n)
   1.307 -apply (simp add: sublist_n_0)
   1.308 -apply simp
   1.309 -done
   1.310 -
   1.311 -lemma sublist'_update1: "i \<ge> m \<Longrightarrow> sublist' n m (xs[i:=v]) = sublist' n m xs"
   1.312 -apply (induct xs arbitrary: n m i)
   1.313 -apply simp
   1.314 -apply simp
   1.315 -apply (case_tac i)
   1.316 -apply simp
   1.317 -apply simp
   1.318 -done
   1.319 -
   1.320 -lemma sublist'_update2: "i < n \<Longrightarrow> sublist' n m (xs[i:=v]) = sublist' n m xs"
   1.321 -apply (induct xs arbitrary: n m i)
   1.322 -apply simp
   1.323 -apply simp
   1.324 -apply (case_tac i)
   1.325 -apply simp
   1.326 -apply simp
   1.327 -done
   1.328 -
   1.329 -lemma sublist'_update3: "\<lbrakk>n \<le> i; i < m\<rbrakk> \<Longrightarrow> sublist' n m (xs[i := v]) = (sublist' n m xs)[i - n := v]"
   1.330 -proof (induct xs arbitrary: n m i)
   1.331 -  case Nil thus ?case by auto
   1.332 -next
   1.333 -  case (Cons x xs)
   1.334 -  thus ?case
   1.335 -    apply -
   1.336 -    apply auto
   1.337 -    apply (cases i)
   1.338 -    apply auto
   1.339 -    apply (cases i)
   1.340 -    apply auto
   1.341 -    done
   1.342 -qed
   1.343 -
   1.344 -lemma "\<lbrakk> sublist' i j xs = sublist' i j ys; n \<ge> i; m \<le> j \<rbrakk> \<Longrightarrow> sublist' n m xs = sublist' n m ys"
   1.345 -proof (induct xs arbitrary: i j ys n m)
   1.346 -  case Nil
   1.347 -  thus ?case
   1.348 -    apply -
   1.349 -    apply (rule sym, drule sym)
   1.350 -    apply (simp add: sublist'_Nil)
   1.351 -    apply (simp add: sublist'_Nil3)
   1.352 -    apply arith
   1.353 -    done
   1.354 -next
   1.355 -  case (Cons x xs i j ys n m)
   1.356 -  note c = this
   1.357 -  thus ?case
   1.358 -  proof (cases m)
   1.359 -    case 0 thus ?thesis by (simp add: sublist_n_0)
   1.360 -  next
   1.361 -    case (Suc m')
   1.362 -    note a = this
   1.363 -    thus ?thesis
   1.364 -    proof (cases n)
   1.365 -      case 0 note b = this
   1.366 -      show ?thesis
   1.367 -      proof (cases ys)
   1.368 -	case Nil  with a b Cons.prems show ?thesis by (simp add: sublist'_Nil3)
   1.369 -      next
   1.370 -	case (Cons y ys)
   1.371 -	show ?thesis
   1.372 -	proof (cases j)
   1.373 -	  case 0 with a b Cons.prems show ?thesis by simp
   1.374 -	next
   1.375 -	  case (Suc j') with a b Cons.prems Cons show ?thesis 
   1.376 -	    apply -
   1.377 -	    apply (simp, rule Cons.hyps [of "0" "j'" "ys" "0" "m'"], auto)
   1.378 -	    done
   1.379 -	qed
   1.380 -      qed
   1.381 -    next
   1.382 -      case (Suc n')
   1.383 -      show ?thesis
   1.384 -      proof (cases ys)
   1.385 -	case Nil with Suc a Cons.prems show ?thesis by (auto simp add: sublist'_Nil3)
   1.386 -      next
   1.387 -	case (Cons y ys) with Suc a Cons.prems show ?thesis
   1.388 -	  apply -
   1.389 -	  apply simp
   1.390 -	  apply (cases j)
   1.391 -	  apply simp
   1.392 -	  apply (cases i)
   1.393 -	  apply simp
   1.394 -	  apply (rule_tac j="nat" in Cons.hyps [of "0" _ "ys" "n'" "m'"])
   1.395 -	  apply simp
   1.396 -	  apply simp
   1.397 -	  apply simp
   1.398 -	  apply simp
   1.399 -	  apply (rule_tac i="nata" and j="nat" in Cons.hyps [of _ _ "ys" "n'" "m'"])
   1.400 -	  apply simp
   1.401 -	  apply simp
   1.402 -	  apply simp
   1.403 -	  done
   1.404 -      qed
   1.405 -    qed
   1.406 -  qed
   1.407 -qed
   1.408 -
   1.409 -lemma length_sublist': "j \<le> length xs \<Longrightarrow> length (sublist' i j xs) = j - i"
   1.410 -by (induct xs arbitrary: i j, auto)
   1.411 -
   1.412 -lemma sublist'_front: "\<lbrakk> i < j; i < length xs \<rbrakk> \<Longrightarrow> sublist' i j xs = xs ! i # sublist' (Suc i) j xs"
   1.413 -apply (induct xs arbitrary: a i j)
   1.414 -apply simp
   1.415 -apply (case_tac j)
   1.416 -apply simp
   1.417 -apply (case_tac i)
   1.418 -apply simp
   1.419 -apply simp
   1.420 -done
   1.421 -
   1.422 -lemma sublist'_back: "\<lbrakk> i < j; j \<le> length xs \<rbrakk> \<Longrightarrow> sublist' i j xs = sublist' i (j - 1) xs @ [xs ! (j - 1)]"
   1.423 -apply (induct xs arbitrary: a i j)
   1.424 -apply simp
   1.425 -apply simp
   1.426 -apply (case_tac j)
   1.427 -apply simp
   1.428 -apply auto
   1.429 -apply (case_tac nat)
   1.430 -apply auto
   1.431 -done
   1.432 -
   1.433 -(* suffices that j \<le> length xs and length ys *) 
   1.434 -lemma sublist'_eq_samelength_iff: "length xs = length ys \<Longrightarrow> (sublist' i j xs  = sublist' i j ys) = (\<forall>i'. i \<le> i' \<and> i' < j \<longrightarrow> xs ! i' = ys ! i')"
   1.435 -proof (induct xs arbitrary: ys i j)
   1.436 -  case Nil thus ?case by simp
   1.437 -next
   1.438 -  case (Cons x xs)
   1.439 -  thus ?case
   1.440 -    apply -
   1.441 -    apply (cases ys)
   1.442 -    apply simp
   1.443 -    apply simp
   1.444 -    apply auto
   1.445 -    apply (case_tac i', auto)
   1.446 -    apply (erule_tac x="Suc i'" in allE, auto)
   1.447 -    apply (erule_tac x="i' - 1" in allE, auto)
   1.448 -    apply (case_tac i', auto)
   1.449 -    apply (erule_tac x="Suc i'" in allE, auto)
   1.450 -    done
   1.451 -qed
   1.452 -
   1.453 -lemma sublist'_all[simp]: "sublist' 0 (length xs) xs = xs"
   1.454 -by (induct xs, auto)
   1.455 -
   1.456 -lemma sublist'_sublist': "sublist' n m (sublist' i j xs) = sublist' (i + n) (min (i + m) j) xs" 
   1.457 -by (induct xs arbitrary: i j n m) (auto simp add: min_diff)
   1.458 -
   1.459 -lemma sublist'_append: "\<lbrakk> i \<le> j; j \<le> k \<rbrakk> \<Longrightarrow>(sublist' i j xs) @ (sublist' j k xs) = sublist' i k xs"
   1.460 -by (induct xs arbitrary: i j k) auto
   1.461 -
   1.462 -lemma nth_sublist': "\<lbrakk> k < j - i; j \<le> length xs \<rbrakk> \<Longrightarrow> (sublist' i j xs) ! k = xs ! (i + k)"
   1.463 -apply (induct xs arbitrary: i j k)
   1.464 -apply auto
   1.465 -apply (case_tac k)
   1.466 -apply auto
   1.467 -apply (case_tac i)
   1.468 -apply auto
   1.469 -done
   1.470 -
   1.471 -lemma set_sublist': "set (sublist' i j xs) = {x. \<exists>k. i \<le> k \<and> k < j \<and> k < List.length xs \<and> x = xs ! k}"
   1.472 -apply (simp add: sublist'_sublist)
   1.473 -apply (simp add: set_sublist)
   1.474 -apply auto
   1.475 -done
   1.476 -
   1.477 -lemma all_in_set_sublist'_conv: "(\<forall>j. j \<in> set (sublist' l r xs) \<longrightarrow> P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < List.length xs \<longrightarrow> P (xs ! k))"
   1.478 -unfolding set_sublist' by blast
   1.479 -
   1.480 -lemma ball_in_set_sublist'_conv: "(\<forall>j \<in> set (sublist' l r xs). P j) = (\<forall>k. l \<le> k \<and> k < r \<and> k < List.length xs \<longrightarrow> P (xs ! k))"
   1.481 -unfolding set_sublist' by blast
   1.482 -
   1.483 -
   1.484 -lemma multiset_of_sublist:
   1.485 -assumes l_r: "l \<le> r \<and> r \<le> List.length xs"
   1.486 -assumes left: "\<forall> i. i < l \<longrightarrow> (xs::'a list) ! i = ys ! i"
   1.487 -assumes right: "\<forall> i. i \<ge> r \<longrightarrow> (xs::'a list) ! i = ys ! i"
   1.488 -assumes multiset: "multiset_of xs = multiset_of ys"
   1.489 -  shows "multiset_of (sublist' l r xs) = multiset_of (sublist' l r ys)"
   1.490 -proof -
   1.491 -  from l_r have xs_def: "xs = (sublist' 0 l xs) @ (sublist' l r xs) @ (sublist' r (List.length xs) xs)" (is "_ = ?xs_long") 
   1.492 -    by (simp add: sublist'_append)
   1.493 -  from multiset have length_eq: "List.length xs = List.length ys" by (rule multiset_of_eq_length)
   1.494 -  with l_r have ys_def: "ys = (sublist' 0 l ys) @ (sublist' l r ys) @ (sublist' r (List.length ys) ys)" (is "_ = ?ys_long") 
   1.495 -    by (simp add: sublist'_append)
   1.496 -  from xs_def ys_def multiset have "multiset_of ?xs_long = multiset_of ?ys_long" by simp
   1.497 -  moreover
   1.498 -  from left l_r length_eq have "sublist' 0 l xs = sublist' 0 l ys"
   1.499 -    by (auto simp add: length_sublist' nth_sublist' intro!: nth_equalityI)
   1.500 -  moreover
   1.501 -  from right l_r length_eq have "sublist' r (List.length xs) xs = sublist' r (List.length ys) ys"
   1.502 -    by (auto simp add: length_sublist' nth_sublist' intro!: nth_equalityI)
   1.503 -  moreover
   1.504 -  ultimately show ?thesis by (simp add: multiset_of_append)
   1.505 -qed
   1.506 -
   1.507 -
   1.508 -end