src/HOL/Library/Sublist.thy
author nipkow
Thu May 26 09:05:00 2016 +0200 (2016-05-26)
changeset 63155 ea8540c71581
parent 63149 f5dbab18c404
child 63173 3413b1cf30cd
permissions -rw-r--r--
added function "prefixes" and some lemmas
     1 (*  Title:      HOL/Library/Sublist.thy
     2     Author:     Tobias Nipkow and Markus Wenzel, TU Muenchen
     3     Author:     Christian Sternagel, JAIST
     4 *)
     5 
     6 section \<open>List prefixes, suffixes, and homeomorphic embedding\<close>
     7 
     8 theory Sublist
     9 imports Main
    10 begin
    11 
    12 subsection \<open>Prefix order on lists\<close>
    13 
    14 definition prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
    15   where "prefix xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"
    16 
    17 definition strict_prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
    18   where "strict_prefix xs ys \<longleftrightarrow> prefix xs ys \<and> xs \<noteq> ys"
    19 
    20 interpretation prefix_order: order prefix strict_prefix
    21   by standard (auto simp: prefix_def strict_prefix_def)
    22 
    23 interpretation prefix_bot: order_bot Nil prefix strict_prefix
    24   by standard (simp add: prefix_def)
    25 
    26 lemma prefixI [intro?]: "ys = xs @ zs \<Longrightarrow> prefix xs ys"
    27   unfolding prefix_def by blast
    28 
    29 lemma prefixE [elim?]:
    30   assumes "prefix xs ys"
    31   obtains zs where "ys = xs @ zs"
    32   using assms unfolding prefix_def by blast
    33 
    34 lemma strict_prefixI' [intro?]: "ys = xs @ z # zs \<Longrightarrow> strict_prefix xs ys"
    35   unfolding strict_prefix_def prefix_def by blast
    36 
    37 lemma strict_prefixE' [elim?]:
    38   assumes "strict_prefix xs ys"
    39   obtains z zs where "ys = xs @ z # zs"
    40 proof -
    41   from \<open>strict_prefix xs ys\<close> obtain us where "ys = xs @ us" and "xs \<noteq> ys"
    42     unfolding strict_prefix_def prefix_def by blast
    43   with that show ?thesis by (auto simp add: neq_Nil_conv)
    44 qed
    45 
    46 (* FIXME rm *)
    47 lemma strict_prefixI [intro?]: "prefix xs ys \<Longrightarrow> xs \<noteq> ys \<Longrightarrow> strict_prefix xs ys"
    48 by(fact prefix_order.le_neq_trans)
    49 
    50 lemma strict_prefixE [elim?]:
    51   fixes xs ys :: "'a list"
    52   assumes "strict_prefix xs ys"
    53   obtains "prefix xs ys" and "xs \<noteq> ys"
    54   using assms unfolding strict_prefix_def by blast
    55 
    56 
    57 subsection \<open>Basic properties of prefixes\<close>
    58 
    59 (* FIXME rm *)
    60 theorem Nil_prefix [iff]: "prefix [] xs"
    61 by(fact prefix_bot.bot_least)
    62 
    63 (* FIXME rm *)
    64 theorem prefix_Nil [simp]: "(prefix xs []) = (xs = [])"
    65 by(fact prefix_bot.bot_unique)
    66 
    67 lemma prefix_snoc [simp]: "prefix xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefix xs ys"
    68 proof
    69   assume "prefix xs (ys @ [y])"
    70   then obtain zs where zs: "ys @ [y] = xs @ zs" ..
    71   show "xs = ys @ [y] \<or> prefix xs ys"
    72     by (metis append_Nil2 butlast_append butlast_snoc prefixI zs)
    73 next
    74   assume "xs = ys @ [y] \<or> prefix xs ys"
    75   then show "prefix xs (ys @ [y])"
    76     by (metis prefix_order.eq_iff prefix_order.order_trans prefixI)
    77 qed
    78 
    79 lemma Cons_prefix_Cons [simp]: "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)"
    80   by (auto simp add: prefix_def)
    81 
    82 lemma prefix_code [code]:
    83   "prefix [] xs \<longleftrightarrow> True"
    84   "prefix (x # xs) [] \<longleftrightarrow> False"
    85   "prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys"
    86   by simp_all
    87 
    88 lemma same_prefix_prefix [simp]: "prefix (xs @ ys) (xs @ zs) = prefix ys zs"
    89   by (induct xs) simp_all
    90 
    91 lemma same_prefix_nil [iff]: "prefix (xs @ ys) xs = (ys = [])"
    92   by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixI)
    93 
    94 lemma prefix_prefix [simp]: "prefix xs ys \<Longrightarrow> prefix xs (ys @ zs)"
    95   by (metis prefix_order.le_less_trans prefixI strict_prefixE strict_prefixI)
    96 
    97 lemma append_prefixD: "prefix (xs @ ys) zs \<Longrightarrow> prefix xs zs"
    98   by (auto simp add: prefix_def)
    99 
   100 theorem prefix_Cons: "prefix xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefix zs ys))"
   101   by (cases xs) (auto simp add: prefix_def)
   102 
   103 theorem prefix_append:
   104   "prefix xs (ys @ zs) = (prefix xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefix us zs))"
   105   apply (induct zs rule: rev_induct)
   106    apply force
   107   apply (simp del: append_assoc add: append_assoc [symmetric])
   108   apply (metis append_eq_appendI)
   109   done
   110 
   111 lemma append_one_prefix:
   112   "prefix xs ys \<Longrightarrow> length xs < length ys \<Longrightarrow> prefix (xs @ [ys ! length xs]) ys"
   113   proof (unfold prefix_def)
   114     assume a1: "\<exists>zs. ys = xs @ zs"
   115     then obtain sk :: "'a list" where sk: "ys = xs @ sk" by fastforce
   116     assume a2: "length xs < length ys"
   117     have f1: "\<And>v. ([]::'a list) @ v = v" using append_Nil2 by simp
   118     have "[] \<noteq> sk" using a1 a2 sk less_not_refl by force
   119     hence "\<exists>v. xs @ hd sk # v = ys" using sk by (metis hd_Cons_tl)
   120     thus "\<exists>zs. ys = (xs @ [ys ! length xs]) @ zs" using f1 by fastforce
   121   qed
   122 
   123 theorem prefix_length_le: "prefix xs ys \<Longrightarrow> length xs \<le> length ys"
   124   by (auto simp add: prefix_def)
   125 
   126 lemma prefix_same_cases:
   127   "prefix (xs\<^sub>1::'a list) ys \<Longrightarrow> prefix xs\<^sub>2 ys \<Longrightarrow> prefix xs\<^sub>1 xs\<^sub>2 \<or> prefix xs\<^sub>2 xs\<^sub>1"
   128   unfolding prefix_def by (force simp: append_eq_append_conv2)
   129 
   130 lemma set_mono_prefix: "prefix xs ys \<Longrightarrow> set xs \<subseteq> set ys"
   131   by (auto simp add: prefix_def)
   132 
   133 lemma take_is_prefix: "prefix (take n xs) xs"
   134   unfolding prefix_def by (metis append_take_drop_id)
   135 
   136 lemma prefixeq_butlast: "prefix (butlast xs) xs"
   137 by (simp add: butlast_conv_take take_is_prefix)
   138 
   139 lemma map_prefixI: "prefix xs ys \<Longrightarrow> prefix (map f xs) (map f ys)"
   140   by (auto simp: prefix_def)
   141 
   142 lemma prefix_length_less: "strict_prefix xs ys \<Longrightarrow> length xs < length ys"
   143   by (auto simp: strict_prefix_def prefix_def)
   144 
   145 lemma prefix_snocD: "prefix (xs@[x]) ys \<Longrightarrow> strict_prefix xs ys"
   146   by (simp add: strict_prefixI' prefix_order.dual_order.strict_trans1)
   147 
   148 lemma strict_prefix_simps [simp, code]:
   149   "strict_prefix xs [] \<longleftrightarrow> False"
   150   "strict_prefix [] (x # xs) \<longleftrightarrow> True"
   151   "strict_prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> strict_prefix xs ys"
   152   by (simp_all add: strict_prefix_def cong: conj_cong)
   153 
   154 lemma take_strict_prefix: "strict_prefix xs ys \<Longrightarrow> strict_prefix (take n xs) ys"
   155   apply (induct n arbitrary: xs ys)
   156    apply (case_tac ys; simp)
   157   apply (metis prefix_order.less_trans strict_prefixI take_is_prefix)
   158   done
   159 
   160 lemma not_prefix_cases:
   161   assumes pfx: "\<not> prefix ps ls"
   162   obtains
   163     (c1) "ps \<noteq> []" and "ls = []"
   164   | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> prefix as xs"
   165   | (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a"
   166 proof (cases ps)
   167   case Nil
   168   then show ?thesis using pfx by simp
   169 next
   170   case (Cons a as)
   171   note c = \<open>ps = a#as\<close>
   172   show ?thesis
   173   proof (cases ls)
   174     case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefix_nil)
   175   next
   176     case (Cons x xs)
   177     show ?thesis
   178     proof (cases "x = a")
   179       case True
   180       have "\<not> prefix as xs" using pfx c Cons True by simp
   181       with c Cons True show ?thesis by (rule c2)
   182     next
   183       case False
   184       with c Cons show ?thesis by (rule c3)
   185     qed
   186   qed
   187 qed
   188 
   189 lemma not_prefix_induct [consumes 1, case_names Nil Neq Eq]:
   190   assumes np: "\<not> prefix ps ls"
   191     and base: "\<And>x xs. P (x#xs) []"
   192     and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)"
   193     and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefix xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"
   194   shows "P ps ls" using np
   195 proof (induct ls arbitrary: ps)
   196   case Nil then show ?case
   197     by (auto simp: neq_Nil_conv elim!: not_prefix_cases intro!: base)
   198 next
   199   case (Cons y ys)
   200   then have npfx: "\<not> prefix ps (y # ys)" by simp
   201   then obtain x xs where pv: "ps = x # xs"
   202     by (rule not_prefix_cases) auto
   203   show ?case by (metis Cons.hyps Cons_prefix_Cons npfx pv r1 r2)
   204 qed
   205 
   206 
   207 subsection \<open>Prefixes\<close>
   208 
   209 fun prefixes where
   210 "prefixes [] = [[]]" |
   211 "prefixes (x#xs) = [] # map (op # x) (prefixes xs)"
   212 
   213 lemma in_set_prefixes[simp]: "xs \<in> set (prefixes ys) \<longleftrightarrow> prefix xs ys"
   214 by (induction "xs" arbitrary: "ys"; rename_tac "ys", case_tac "ys"; auto)
   215 
   216 lemma length_prefixes[simp]: "length (prefixes xs) = length xs+1"
   217 by (induction xs) auto
   218 
   219 lemma prefixes_snoc[simp]:
   220   "prefixes (xs@[x]) = prefixes xs @ [xs@[x]]"
   221 by (induction xs) auto
   222 
   223 lemma prefixes_eq_Snoc:
   224   "prefixes ys = xs @ [x] \<longleftrightarrow>
   225   (ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = zs@[z] \<and> xs = prefixes zs)) \<and> x = ys"
   226 by (cases ys rule: rev_cases) auto
   227 
   228 
   229 subsection \<open>Parallel lists\<close>
   230 
   231 definition parallel :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "\<parallel>" 50)
   232   where "(xs \<parallel> ys) = (\<not> prefix xs ys \<and> \<not> prefix ys xs)"
   233 
   234 lemma parallelI [intro]: "\<not> prefix xs ys \<Longrightarrow> \<not> prefix ys xs \<Longrightarrow> xs \<parallel> ys"
   235   unfolding parallel_def by blast
   236 
   237 lemma parallelE [elim]:
   238   assumes "xs \<parallel> ys"
   239   obtains "\<not> prefix xs ys \<and> \<not> prefix ys xs"
   240   using assms unfolding parallel_def by blast
   241 
   242 theorem prefix_cases:
   243   obtains "prefix xs ys" | "strict_prefix ys xs" | "xs \<parallel> ys"
   244   unfolding parallel_def strict_prefix_def by blast
   245 
   246 theorem parallel_decomp:
   247   "xs \<parallel> ys \<Longrightarrow> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
   248 proof (induct xs rule: rev_induct)
   249   case Nil
   250   then have False by auto
   251   then show ?case ..
   252 next
   253   case (snoc x xs)
   254   show ?case
   255   proof (rule prefix_cases)
   256     assume le: "prefix xs ys"
   257     then obtain ys' where ys: "ys = xs @ ys'" ..
   258     show ?thesis
   259     proof (cases ys')
   260       assume "ys' = []"
   261       then show ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys)
   262     next
   263       fix c cs assume ys': "ys' = c # cs"
   264       have "x \<noteq> c" using snoc.prems ys ys' by fastforce
   265       thus "\<exists>as b bs c cs. b \<noteq> c \<and> xs @ [x] = as @ b # bs \<and> ys = as @ c # cs"
   266         using ys ys' by blast
   267     qed
   268   next
   269     assume "strict_prefix ys xs"
   270     then have "prefix ys (xs @ [x])" by (simp add: strict_prefix_def)
   271     with snoc have False by blast
   272     then show ?thesis ..
   273   next
   274     assume "xs \<parallel> ys"
   275     with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c"
   276       and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
   277       by blast
   278     from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
   279     with neq ys show ?thesis by blast
   280   qed
   281 qed
   282 
   283 lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
   284   apply (rule parallelI)
   285     apply (erule parallelE, erule conjE,
   286       induct rule: not_prefix_induct, simp+)+
   287   done
   288 
   289 lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y"
   290   by (simp add: parallel_append)
   291 
   292 lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"
   293   unfolding parallel_def by auto
   294 
   295 
   296 subsection \<open>Suffix order on lists\<close>
   297 
   298 definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
   299   where "suffix xs ys = (\<exists>zs. ys = zs @ xs)"
   300 
   301 definition strict_suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
   302   where "strict_suffix xs ys \<longleftrightarrow> (\<exists>us. ys = us @ xs \<and> us \<noteq> [])"
   303 
   304 lemma strict_suffix_imp_suffix:
   305   "strict_suffix xs ys \<Longrightarrow> suffix xs ys"
   306   by (auto simp: suffix_def strict_suffix_def)
   307 
   308 lemma suffixI [intro?]: "ys = zs @ xs \<Longrightarrow> suffix xs ys"
   309   unfolding suffix_def by blast
   310 
   311 lemma suffixE [elim?]:
   312   assumes "suffix xs ys"
   313   obtains zs where "ys = zs @ xs"
   314   using assms unfolding suffix_def by blast
   315 
   316 lemma suffix_refl [iff]: "suffix xs xs"
   317   by (auto simp add: suffix_def)
   318 
   319 lemma suffix_trans:
   320   "suffix xs ys \<Longrightarrow> suffix ys zs \<Longrightarrow> suffix xs zs"
   321   by (auto simp: suffix_def)
   322 
   323 lemma strict_suffix_trans:
   324   "\<lbrakk>strict_suffix xs ys; strict_suffix ys zs\<rbrakk> \<Longrightarrow> strict_suffix xs zs"
   325 by (auto simp add: strict_suffix_def)
   326 
   327 lemma suffix_antisym: "\<lbrakk>suffix xs ys; suffix ys xs\<rbrakk> \<Longrightarrow> xs = ys"
   328   by (auto simp add: suffix_def)
   329 
   330 lemma suffix_tl [simp]: "suffix (tl xs) xs"
   331   by (induct xs) (auto simp: suffix_def)
   332 
   333 lemma strict_suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> strict_suffix (tl xs) xs"
   334   by (induct xs) (auto simp: strict_suffix_def)
   335 
   336 lemma Nil_suffix [iff]: "suffix [] xs"
   337   by (simp add: suffix_def)
   338 
   339 lemma suffix_Nil [simp]: "(suffix xs []) = (xs = [])"
   340   by (auto simp add: suffix_def)
   341 
   342 lemma suffix_ConsI: "suffix xs ys \<Longrightarrow> suffix xs (y # ys)"
   343   by (auto simp add: suffix_def)
   344 
   345 lemma suffix_ConsD: "suffix (x # xs) ys \<Longrightarrow> suffix xs ys"
   346   by (auto simp add: suffix_def)
   347 
   348 lemma suffix_appendI: "suffix xs ys \<Longrightarrow> suffix xs (zs @ ys)"
   349   by (auto simp add: suffix_def)
   350 
   351 lemma suffix_appendD: "suffix (zs @ xs) ys \<Longrightarrow> suffix xs ys"
   352   by (auto simp add: suffix_def)
   353 
   354 lemma strict_suffix_set_subset: "strict_suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys"
   355 by (auto simp: strict_suffix_def)
   356 
   357 lemma suffix_set_subset: "suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys"
   358 by (auto simp: suffix_def)
   359 
   360 lemma suffix_ConsD2: "suffix (x # xs) (y # ys) \<Longrightarrow> suffix xs ys"
   361 proof -
   362   assume "suffix (x # xs) (y # ys)"
   363   then obtain zs where "y # ys = zs @ x # xs" ..
   364   then show ?thesis
   365     by (induct zs) (auto intro!: suffix_appendI suffix_ConsI)
   366 qed
   367 
   368 lemma suffix_to_prefix [code]: "suffix xs ys \<longleftrightarrow> prefix (rev xs) (rev ys)"
   369 proof
   370   assume "suffix xs ys"
   371   then obtain zs where "ys = zs @ xs" ..
   372   then have "rev ys = rev xs @ rev zs" by simp
   373   then show "prefix (rev xs) (rev ys)" ..
   374 next
   375   assume "prefix (rev xs) (rev ys)"
   376   then obtain zs where "rev ys = rev xs @ zs" ..
   377   then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp
   378   then have "ys = rev zs @ xs" by simp
   379   then show "suffix xs ys" ..
   380 qed
   381 
   382 lemma distinct_suffix: "distinct ys \<Longrightarrow> suffix xs ys \<Longrightarrow> distinct xs"
   383   by (clarsimp elim!: suffixE)
   384 
   385 lemma suffix_map: "suffix xs ys \<Longrightarrow> suffix (map f xs) (map f ys)"
   386   by (auto elim!: suffixE intro: suffixI)
   387 
   388 lemma suffix_drop: "suffix (drop n as) as"
   389   unfolding suffix_def
   390   apply (rule exI [where x = "take n as"])
   391   apply simp
   392   done
   393 
   394 lemma suffix_take: "suffix xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs"
   395   by (auto elim!: suffixE)
   396 
   397 lemma strict_suffix_reflclp_conv: "strict_suffix\<^sup>=\<^sup>= = suffix"
   398 by (intro ext) (auto simp: suffix_def strict_suffix_def)
   399 
   400 lemma suffix_lists: "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A"
   401   unfolding suffix_def by auto
   402 
   403 lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefix x y"
   404   by blast
   405 
   406 lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefix y x"
   407   by blast
   408 
   409 lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
   410   unfolding parallel_def by simp
   411 
   412 lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
   413   unfolding parallel_def by simp
   414 
   415 lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
   416   by auto
   417 
   418 lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
   419   by (metis Cons_prefix_Cons parallelE parallelI)
   420 
   421 lemma not_equal_is_parallel:
   422   assumes neq: "xs \<noteq> ys"
   423     and len: "length xs = length ys"
   424   shows "xs \<parallel> ys"
   425   using len neq
   426 proof (induct rule: list_induct2)
   427   case Nil
   428   then show ?case by simp
   429 next
   430   case (Cons a as b bs)
   431   have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact
   432   show ?case
   433   proof (cases "a = b")
   434     case True
   435     then have "as \<noteq> bs" using Cons by simp
   436     then show ?thesis by (rule Cons_parallelI2 [OF True ih])
   437   next
   438     case False
   439     then show ?thesis by (rule Cons_parallelI1)
   440   qed
   441 qed
   442 
   443 
   444 subsection \<open>Homeomorphic embedding on lists\<close>
   445 
   446 inductive list_emb :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
   447   for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
   448 where
   449   list_emb_Nil [intro, simp]: "list_emb P [] ys"
   450 | list_emb_Cons [intro] : "list_emb P xs ys \<Longrightarrow> list_emb P xs (y#ys)"
   451 | list_emb_Cons2 [intro]: "P x y \<Longrightarrow> list_emb P xs ys \<Longrightarrow> list_emb P (x#xs) (y#ys)"
   452 
   453 lemma list_emb_mono:                         
   454   assumes "\<And>x y. P x y \<longrightarrow> Q x y"
   455   shows "list_emb P xs ys \<longrightarrow> list_emb Q xs ys"
   456 proof                                        
   457   assume "list_emb P xs ys"                    
   458   then show "list_emb Q xs ys" by (induct) (auto simp: assms)
   459 qed 
   460 
   461 lemma list_emb_Nil2 [simp]:
   462   assumes "list_emb P xs []" shows "xs = []"
   463   using assms by (cases rule: list_emb.cases) auto
   464 
   465 lemma list_emb_refl:
   466   assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x x"
   467   shows "list_emb P xs xs"
   468   using assms by (induct xs) auto
   469 
   470 lemma list_emb_Cons_Nil [simp]: "list_emb P (x#xs) [] = False"
   471 proof -
   472   { assume "list_emb P (x#xs) []"
   473     from list_emb_Nil2 [OF this] have False by simp
   474   } moreover {
   475     assume False
   476     then have "list_emb P (x#xs) []" by simp
   477   } ultimately show ?thesis by blast
   478 qed
   479 
   480 lemma list_emb_append2 [intro]: "list_emb P xs ys \<Longrightarrow> list_emb P xs (zs @ ys)"
   481   by (induct zs) auto
   482 
   483 lemma list_emb_prefix [intro]:
   484   assumes "list_emb P xs ys" shows "list_emb P xs (ys @ zs)"
   485   using assms
   486   by (induct arbitrary: zs) auto
   487 
   488 lemma list_emb_ConsD:
   489   assumes "list_emb P (x#xs) ys"
   490   shows "\<exists>us v vs. ys = us @ v # vs \<and> P x v \<and> list_emb P xs vs"
   491 using assms
   492 proof (induct x \<equiv> "x # xs" ys arbitrary: x xs)
   493   case list_emb_Cons
   494   then show ?case by (metis append_Cons)
   495 next
   496   case (list_emb_Cons2 x y xs ys)
   497   then show ?case by blast
   498 qed
   499 
   500 lemma list_emb_appendD:
   501   assumes "list_emb P (xs @ ys) zs"
   502   shows "\<exists>us vs. zs = us @ vs \<and> list_emb P xs us \<and> list_emb P ys vs"
   503 using assms
   504 proof (induction xs arbitrary: ys zs)
   505   case Nil then show ?case by auto
   506 next
   507   case (Cons x xs)
   508   then obtain us v vs where
   509     zs: "zs = us @ v # vs" and p: "P x v" and lh: "list_emb P (xs @ ys) vs"
   510     by (auto dest: list_emb_ConsD)
   511   obtain sk\<^sub>0 :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" and sk\<^sub>1 :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   512     sk: "\<forall>x\<^sub>0 x\<^sub>1. \<not> list_emb P (xs @ x\<^sub>0) x\<^sub>1 \<or> sk\<^sub>0 x\<^sub>0 x\<^sub>1 @ sk\<^sub>1 x\<^sub>0 x\<^sub>1 = x\<^sub>1 \<and> list_emb P xs (sk\<^sub>0 x\<^sub>0 x\<^sub>1) \<and> list_emb P x\<^sub>0 (sk\<^sub>1 x\<^sub>0 x\<^sub>1)"
   513     using Cons(1) by (metis (no_types))
   514   hence "\<forall>x\<^sub>2. list_emb P (x # xs) (x\<^sub>2 @ v # sk\<^sub>0 ys vs)" using p lh by auto
   515   thus ?case using lh zs sk by (metis (no_types) append_Cons append_assoc)
   516 qed
   517 
   518 lemma list_emb_strict_suffix:
   519   assumes "list_emb P xs ys" and "strict_suffix ys zs"
   520   shows "list_emb P xs zs"
   521   using assms(2) and list_emb_append2 [OF assms(1)] by (auto simp: strict_suffix_def)
   522 
   523 lemma list_emb_suffix:
   524   assumes "list_emb P xs ys" and "suffix ys zs"
   525   shows "list_emb P xs zs"
   526 using assms and list_emb_strict_suffix
   527 unfolding strict_suffix_reflclp_conv[symmetric] by auto
   528 
   529 lemma list_emb_length: "list_emb P xs ys \<Longrightarrow> length xs \<le> length ys"
   530   by (induct rule: list_emb.induct) auto
   531 
   532 lemma list_emb_trans:
   533   assumes "\<And>x y z. \<lbrakk>x \<in> set xs; y \<in> set ys; z \<in> set zs; P x y; P y z\<rbrakk> \<Longrightarrow> P x z"
   534   shows "\<lbrakk>list_emb P xs ys; list_emb P ys zs\<rbrakk> \<Longrightarrow> list_emb P xs zs"
   535 proof -
   536   assume "list_emb P xs ys" and "list_emb P ys zs"
   537   then show "list_emb P xs zs" using assms
   538   proof (induction arbitrary: zs)
   539     case list_emb_Nil show ?case by blast
   540   next
   541     case (list_emb_Cons xs ys y)
   542     from list_emb_ConsD [OF \<open>list_emb P (y#ys) zs\<close>] obtain us v vs
   543       where zs: "zs = us @ v # vs" and "P\<^sup>=\<^sup>= y v" and "list_emb P ys vs" by blast
   544     then have "list_emb P ys (v#vs)" by blast
   545     then have "list_emb P ys zs" unfolding zs by (rule list_emb_append2)
   546     from list_emb_Cons.IH [OF this] and list_emb_Cons.prems show ?case by auto
   547   next
   548     case (list_emb_Cons2 x y xs ys)
   549     from list_emb_ConsD [OF \<open>list_emb P (y#ys) zs\<close>] obtain us v vs
   550       where zs: "zs = us @ v # vs" and "P y v" and "list_emb P ys vs" by blast
   551     with list_emb_Cons2 have "list_emb P xs vs" by auto
   552     moreover have "P x v"
   553     proof -
   554       from zs have "v \<in> set zs" by auto
   555       moreover have "x \<in> set (x#xs)" and "y \<in> set (y#ys)" by simp_all
   556       ultimately show ?thesis
   557         using \<open>P x y\<close> and \<open>P y v\<close> and list_emb_Cons2
   558         by blast
   559     qed
   560     ultimately have "list_emb P (x#xs) (v#vs)" by blast
   561     then show ?case unfolding zs by (rule list_emb_append2)
   562   qed
   563 qed
   564 
   565 lemma list_emb_set:
   566   assumes "list_emb P xs ys" and "x \<in> set xs"
   567   obtains y where "y \<in> set ys" and "P x y"
   568   using assms by (induct) auto
   569 
   570 
   571 subsection \<open>Sublists (special case of homeomorphic embedding)\<close>
   572 
   573 abbreviation sublisteq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
   574   where "sublisteq xs ys \<equiv> list_emb (op =) xs ys"
   575 
   576 lemma sublisteq_Cons2: "sublisteq xs ys \<Longrightarrow> sublisteq (x#xs) (x#ys)" by auto
   577 
   578 lemma sublisteq_same_length:
   579   assumes "sublisteq xs ys" and "length xs = length ys" shows "xs = ys"
   580   using assms by (induct) (auto dest: list_emb_length)
   581 
   582 lemma not_sublisteq_length [simp]: "length ys < length xs \<Longrightarrow> \<not> sublisteq xs ys"
   583   by (metis list_emb_length linorder_not_less)
   584 
   585 lemma [code]:
   586   "list_emb P [] ys \<longleftrightarrow> True"
   587   "list_emb P (x#xs) [] \<longleftrightarrow> False"
   588   by (simp_all)
   589 
   590 lemma sublisteq_Cons': "sublisteq (x#xs) ys \<Longrightarrow> sublisteq xs ys"
   591   by (induct xs, simp, blast dest: list_emb_ConsD)
   592 
   593 lemma sublisteq_Cons2':
   594   assumes "sublisteq (x#xs) (x#ys)" shows "sublisteq xs ys"
   595   using assms by (cases) (rule sublisteq_Cons')
   596 
   597 lemma sublisteq_Cons2_neq:
   598   assumes "sublisteq (x#xs) (y#ys)"
   599   shows "x \<noteq> y \<Longrightarrow> sublisteq (x#xs) ys"
   600   using assms by (cases) auto
   601 
   602 lemma sublisteq_Cons2_iff [simp, code]:
   603   "sublisteq (x#xs) (y#ys) = (if x = y then sublisteq xs ys else sublisteq (x#xs) ys)"
   604   by (metis list_emb_Cons sublisteq_Cons2 sublisteq_Cons2' sublisteq_Cons2_neq)
   605 
   606 lemma sublisteq_append': "sublisteq (zs @ xs) (zs @ ys) \<longleftrightarrow> sublisteq xs ys"
   607   by (induct zs) simp_all
   608 
   609 lemma sublisteq_refl [simp, intro!]: "sublisteq xs xs" by (induct xs) simp_all
   610 
   611 lemma sublisteq_antisym:
   612   assumes "sublisteq xs ys" and "sublisteq ys xs"
   613   shows "xs = ys"
   614 using assms
   615 proof (induct)
   616   case list_emb_Nil
   617   from list_emb_Nil2 [OF this] show ?case by simp
   618 next
   619   case list_emb_Cons2
   620   thus ?case by simp
   621 next
   622   case list_emb_Cons
   623   hence False using sublisteq_Cons' by fastforce
   624   thus ?case ..
   625 qed
   626 
   627 lemma sublisteq_trans: "sublisteq xs ys \<Longrightarrow> sublisteq ys zs \<Longrightarrow> sublisteq xs zs"
   628   by (rule list_emb_trans [of _ _ _ "op ="]) auto
   629 
   630 lemma sublisteq_append_le_same_iff: "sublisteq (xs @ ys) ys \<longleftrightarrow> xs = []"
   631   by (auto dest: list_emb_length)
   632 
   633 lemma list_emb_append_mono:
   634   "\<lbrakk> list_emb P xs xs'; list_emb P ys ys' \<rbrakk> \<Longrightarrow> list_emb P (xs@ys) (xs'@ys')"
   635   apply (induct rule: list_emb.induct)
   636     apply (metis eq_Nil_appendI list_emb_append2)
   637    apply (metis append_Cons list_emb_Cons)
   638   apply (metis append_Cons list_emb_Cons2)
   639   done
   640 
   641 
   642 subsection \<open>Appending elements\<close>
   643 
   644 lemma sublisteq_append [simp]:
   645   "sublisteq (xs @ zs) (ys @ zs) \<longleftrightarrow> sublisteq xs ys" (is "?l = ?r")
   646 proof
   647   { fix xs' ys' xs ys zs :: "'a list" assume "sublisteq xs' ys'"
   648     then have "xs' = xs @ zs & ys' = ys @ zs \<longrightarrow> sublisteq xs ys"
   649     proof (induct arbitrary: xs ys zs)
   650       case list_emb_Nil show ?case by simp
   651     next
   652       case (list_emb_Cons xs' ys' x)
   653       { assume "ys=[]" then have ?case using list_emb_Cons(1) by auto }
   654       moreover
   655       { fix us assume "ys = x#us"
   656         then have ?case using list_emb_Cons(2) by(simp add: list_emb.list_emb_Cons) }
   657       ultimately show ?case by (auto simp:Cons_eq_append_conv)
   658     next
   659       case (list_emb_Cons2 x y xs' ys')
   660       { assume "xs=[]" then have ?case using list_emb_Cons2(1) by auto }
   661       moreover
   662       { fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using list_emb_Cons2 by auto}
   663       moreover
   664       { fix us assume "xs=x#us" "ys=[]" then have ?case using list_emb_Cons2(2) by bestsimp }
   665       ultimately show ?case using \<open>op = x y\<close> by (auto simp: Cons_eq_append_conv)
   666     qed }
   667   moreover assume ?l
   668   ultimately show ?r by blast
   669 next
   670   assume ?r then show ?l by (metis list_emb_append_mono sublisteq_refl)
   671 qed
   672 
   673 lemma sublisteq_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (zs @ ys)"
   674   by (induct zs) auto
   675 
   676 lemma sublisteq_rev_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (ys @ zs)"
   677   by (metis append_Nil2 list_emb_Nil list_emb_append_mono)
   678 
   679 
   680 subsection \<open>Relation to standard list operations\<close>
   681 
   682 lemma sublisteq_map:
   683   assumes "sublisteq xs ys" shows "sublisteq (map f xs) (map f ys)"
   684   using assms by (induct) auto
   685 
   686 lemma sublisteq_filter_left [simp]: "sublisteq (filter P xs) xs"
   687   by (induct xs) auto
   688 
   689 lemma sublisteq_filter [simp]:
   690   assumes "sublisteq xs ys" shows "sublisteq (filter P xs) (filter P ys)"
   691   using assms by induct auto
   692 
   693 lemma "sublisteq xs ys \<longleftrightarrow> (\<exists>N. xs = sublist ys N)" (is "?L = ?R")
   694 proof
   695   assume ?L
   696   then show ?R
   697   proof (induct)
   698     case list_emb_Nil show ?case by (metis sublist_empty)
   699   next
   700     case (list_emb_Cons xs ys x)
   701     then obtain N where "xs = sublist ys N" by blast
   702     then have "xs = sublist (x#ys) (Suc ` N)"
   703       by (clarsimp simp add:sublist_Cons inj_image_mem_iff)
   704     then show ?case by blast
   705   next
   706     case (list_emb_Cons2 x y xs ys)
   707     then obtain N where "xs = sublist ys N" by blast
   708     then have "x#xs = sublist (x#ys) (insert 0 (Suc ` N))"
   709       by (clarsimp simp add:sublist_Cons inj_image_mem_iff)
   710     moreover from list_emb_Cons2 have "x = y" by simp
   711     ultimately show ?case by blast
   712   qed
   713 next
   714   assume ?R
   715   then obtain N where "xs = sublist ys N" ..
   716   moreover have "sublisteq (sublist ys N) ys"
   717   proof (induct ys arbitrary: N)
   718     case Nil show ?case by simp
   719   next
   720     case Cons then show ?case by (auto simp: sublist_Cons)
   721   qed
   722   ultimately show ?L by simp
   723 qed
   724 
   725 end