src/HOL/Library/Sublist.thy
author blanchet
Wed Nov 20 18:58:00 2013 +0100 (2013-11-20)
changeset 54538 ba7392b52a7c
parent 54483 9f24325c2550
child 55579 207538943038
permissions -rw-r--r--
factor 'List_Prefix' out of 'Sublist' and move to 'Main' (needed for codatatypes)
     1 (*  Title:      HOL/Library/Sublist.thy
     2     Author:     Tobias Nipkow and Markus Wenzel, TU Muenchen
     3     Author:     Christian Sternagel, JAIST
     4 *)
     5 
     6 header {* Parallel lists, list suffixes, and homeomorphic embedding *}
     7 
     8 theory Sublist
     9 imports Main
    10 begin
    11 
    12 subsection {* Parallel lists *}
    13 
    14 definition parallel :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "\<parallel>" 50)
    15   where "(xs \<parallel> ys) = (\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs)"
    16 
    17 lemma parallelI [intro]: "\<not> prefixeq xs ys \<Longrightarrow> \<not> prefixeq ys xs \<Longrightarrow> xs \<parallel> ys"
    18   unfolding parallel_def by blast
    19 
    20 lemma parallelE [elim]:
    21   assumes "xs \<parallel> ys"
    22   obtains "\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs"
    23   using assms unfolding parallel_def by blast
    24 
    25 theorem prefixeq_cases:
    26   obtains "prefixeq xs ys" | "prefix ys xs" | "xs \<parallel> ys"
    27   unfolding parallel_def prefix_def by blast
    28 
    29 theorem parallel_decomp:
    30   "xs \<parallel> ys \<Longrightarrow> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
    31 proof (induct xs rule: rev_induct)
    32   case Nil
    33   then have False by auto
    34   then show ?case ..
    35 next
    36   case (snoc x xs)
    37   show ?case
    38   proof (rule prefixeq_cases)
    39     assume le: "prefixeq xs ys"
    40     then obtain ys' where ys: "ys = xs @ ys'" ..
    41     show ?thesis
    42     proof (cases ys')
    43       assume "ys' = []"
    44       then show ?thesis by (metis append_Nil2 parallelE prefixeqI snoc.prems ys)
    45     next
    46       fix c cs assume ys': "ys' = c # cs"
    47       have "x \<noteq> c" using snoc.prems ys ys' by fastforce
    48       thus "\<exists>as b bs c cs. b \<noteq> c \<and> xs @ [x] = as @ b # bs \<and> ys = as @ c # cs"
    49         using ys ys' by blast
    50     qed
    51   next
    52     assume "prefix ys xs"
    53     then have "prefixeq ys (xs @ [x])" by (simp add: prefix_def)
    54     with snoc have False by blast
    55     then show ?thesis ..
    56   next
    57     assume "xs \<parallel> ys"
    58     with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c"
    59       and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
    60       by blast
    61     from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
    62     with neq ys show ?thesis by blast
    63   qed
    64 qed
    65 
    66 lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
    67   apply (rule parallelI)
    68     apply (erule parallelE, erule conjE,
    69       induct rule: not_prefixeq_induct, simp+)+
    70   done
    71 
    72 lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y"
    73   by (simp add: parallel_append)
    74 
    75 lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"
    76   unfolding parallel_def by auto
    77 
    78 
    79 subsection {* Suffix order on lists *}
    80 
    81 definition suffixeq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
    82   where "suffixeq xs ys = (\<exists>zs. ys = zs @ xs)"
    83 
    84 definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
    85   where "suffix xs ys \<longleftrightarrow> (\<exists>us. ys = us @ xs \<and> us \<noteq> [])"
    86 
    87 lemma suffix_imp_suffixeq:
    88   "suffix xs ys \<Longrightarrow> suffixeq xs ys"
    89   by (auto simp: suffixeq_def suffix_def)
    90 
    91 lemma suffixeqI [intro?]: "ys = zs @ xs \<Longrightarrow> suffixeq xs ys"
    92   unfolding suffixeq_def by blast
    93 
    94 lemma suffixeqE [elim?]:
    95   assumes "suffixeq xs ys"
    96   obtains zs where "ys = zs @ xs"
    97   using assms unfolding suffixeq_def by blast
    98 
    99 lemma suffixeq_refl [iff]: "suffixeq xs xs"
   100   by (auto simp add: suffixeq_def)
   101 lemma suffix_trans:
   102   "suffix xs ys \<Longrightarrow> suffix ys zs \<Longrightarrow> suffix xs zs"
   103   by (auto simp: suffix_def)
   104 lemma suffixeq_trans: "\<lbrakk>suffixeq xs ys; suffixeq ys zs\<rbrakk> \<Longrightarrow> suffixeq xs zs"
   105   by (auto simp add: suffixeq_def)
   106 lemma suffixeq_antisym: "\<lbrakk>suffixeq xs ys; suffixeq ys xs\<rbrakk> \<Longrightarrow> xs = ys"
   107   by (auto simp add: suffixeq_def)
   108 
   109 lemma suffixeq_tl [simp]: "suffixeq (tl xs) xs"
   110   by (induct xs) (auto simp: suffixeq_def)
   111 
   112 lemma suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> suffix (tl xs) xs"
   113   by (induct xs) (auto simp: suffix_def)
   114 
   115 lemma Nil_suffixeq [iff]: "suffixeq [] xs"
   116   by (simp add: suffixeq_def)
   117 lemma suffixeq_Nil [simp]: "(suffixeq xs []) = (xs = [])"
   118   by (auto simp add: suffixeq_def)
   119 
   120 lemma suffixeq_ConsI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (y # ys)"
   121   by (auto simp add: suffixeq_def)
   122 lemma suffixeq_ConsD: "suffixeq (x # xs) ys \<Longrightarrow> suffixeq xs ys"
   123   by (auto simp add: suffixeq_def)
   124 
   125 lemma suffixeq_appendI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (zs @ ys)"
   126   by (auto simp add: suffixeq_def)
   127 lemma suffixeq_appendD: "suffixeq (zs @ xs) ys \<Longrightarrow> suffixeq xs ys"
   128   by (auto simp add: suffixeq_def)
   129 
   130 lemma suffix_set_subset:
   131   "suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffix_def)
   132 
   133 lemma suffixeq_set_subset:
   134   "suffixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffixeq_def)
   135 
   136 lemma suffixeq_ConsD2: "suffixeq (x # xs) (y # ys) \<Longrightarrow> suffixeq xs ys"
   137 proof -
   138   assume "suffixeq (x # xs) (y # ys)"
   139   then obtain zs where "y # ys = zs @ x # xs" ..
   140   then show ?thesis
   141     by (induct zs) (auto intro!: suffixeq_appendI suffixeq_ConsI)
   142 qed
   143 
   144 lemma suffixeq_to_prefixeq [code]: "suffixeq xs ys \<longleftrightarrow> prefixeq (rev xs) (rev ys)"
   145 proof
   146   assume "suffixeq xs ys"
   147   then obtain zs where "ys = zs @ xs" ..
   148   then have "rev ys = rev xs @ rev zs" by simp
   149   then show "prefixeq (rev xs) (rev ys)" ..
   150 next
   151   assume "prefixeq (rev xs) (rev ys)"
   152   then obtain zs where "rev ys = rev xs @ zs" ..
   153   then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp
   154   then have "ys = rev zs @ xs" by simp
   155   then show "suffixeq xs ys" ..
   156 qed
   157 
   158 lemma distinct_suffixeq: "distinct ys \<Longrightarrow> suffixeq xs ys \<Longrightarrow> distinct xs"
   159   by (clarsimp elim!: suffixeqE)
   160 
   161 lemma suffixeq_map: "suffixeq xs ys \<Longrightarrow> suffixeq (map f xs) (map f ys)"
   162   by (auto elim!: suffixeqE intro: suffixeqI)
   163 
   164 lemma suffixeq_drop: "suffixeq (drop n as) as"
   165   unfolding suffixeq_def
   166   apply (rule exI [where x = "take n as"])
   167   apply simp
   168   done
   169 
   170 lemma suffixeq_take: "suffixeq xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs"
   171   by (auto elim!: suffixeqE)
   172 
   173 lemma suffixeq_suffix_reflclp_conv: "suffixeq = suffix\<^sup>=\<^sup>="
   174 proof (intro ext iffI)
   175   fix xs ys :: "'a list"
   176   assume "suffixeq xs ys"
   177   show "suffix\<^sup>=\<^sup>= xs ys"
   178   proof
   179     assume "xs \<noteq> ys"
   180     with `suffixeq xs ys` show "suffix xs ys"
   181       by (auto simp: suffixeq_def suffix_def)
   182   qed
   183 next
   184   fix xs ys :: "'a list"
   185   assume "suffix\<^sup>=\<^sup>= xs ys"
   186   then show "suffixeq xs ys"
   187   proof
   188     assume "suffix xs ys" then show "suffixeq xs ys"
   189       by (rule suffix_imp_suffixeq)
   190   next
   191     assume "xs = ys" then show "suffixeq xs ys"
   192       by (auto simp: suffixeq_def)
   193   qed
   194 qed
   195 
   196 lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefixeq x y"
   197   by blast
   198 
   199 lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefixeq y x"
   200   by blast
   201 
   202 lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
   203   unfolding parallel_def by simp
   204 
   205 lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
   206   unfolding parallel_def by simp
   207 
   208 lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
   209   by auto
   210 
   211 lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
   212   by (metis Cons_prefixeq_Cons parallelE parallelI)
   213 
   214 lemma not_equal_is_parallel:
   215   assumes neq: "xs \<noteq> ys"
   216     and len: "length xs = length ys"
   217   shows "xs \<parallel> ys"
   218   using len neq
   219 proof (induct rule: list_induct2)
   220   case Nil
   221   then show ?case by simp
   222 next
   223   case (Cons a as b bs)
   224   have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact
   225   show ?case
   226   proof (cases "a = b")
   227     case True
   228     then have "as \<noteq> bs" using Cons by simp
   229     then show ?thesis by (rule Cons_parallelI2 [OF True ih])
   230   next
   231     case False
   232     then show ?thesis by (rule Cons_parallelI1)
   233   qed
   234 qed
   235 
   236 lemma suffix_reflclp_conv: "suffix\<^sup>=\<^sup>= = suffixeq"
   237   by (intro ext) (auto simp: suffixeq_def suffix_def)
   238 
   239 lemma suffix_lists: "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A"
   240   unfolding suffix_def by auto
   241 
   242 
   243 subsection {* Homeomorphic embedding on lists *}
   244 
   245 inductive list_hembeq :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
   246   for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
   247 where
   248   list_hembeq_Nil [intro, simp]: "list_hembeq P [] ys"
   249 | list_hembeq_Cons [intro] : "list_hembeq P xs ys \<Longrightarrow> list_hembeq P xs (y#ys)"
   250 | list_hembeq_Cons2 [intro]: "P\<^sup>=\<^sup>= x y \<Longrightarrow> list_hembeq P xs ys \<Longrightarrow> list_hembeq P (x#xs) (y#ys)"
   251 
   252 lemma list_hembeq_Nil2 [simp]:
   253   assumes "list_hembeq P xs []" shows "xs = []"
   254   using assms by (cases rule: list_hembeq.cases) auto
   255 
   256 lemma list_hembeq_refl [simp, intro!]:
   257   "list_hembeq P xs xs"
   258   by (induct xs) auto
   259 
   260 lemma list_hembeq_Cons_Nil [simp]: "list_hembeq P (x#xs) [] = False"
   261 proof -
   262   { assume "list_hembeq P (x#xs) []"
   263     from list_hembeq_Nil2 [OF this] have False by simp
   264   } moreover {
   265     assume False
   266     then have "list_hembeq P (x#xs) []" by simp
   267   } ultimately show ?thesis by blast
   268 qed
   269 
   270 lemma list_hembeq_append2 [intro]: "list_hembeq P xs ys \<Longrightarrow> list_hembeq P xs (zs @ ys)"
   271   by (induct zs) auto
   272 
   273 lemma list_hembeq_prefix [intro]:
   274   assumes "list_hembeq P xs ys" shows "list_hembeq P xs (ys @ zs)"
   275   using assms
   276   by (induct arbitrary: zs) auto
   277 
   278 lemma list_hembeq_ConsD:
   279   assumes "list_hembeq P (x#xs) ys"
   280   shows "\<exists>us v vs. ys = us @ v # vs \<and> P\<^sup>=\<^sup>= x v \<and> list_hembeq P xs vs"
   281 using assms
   282 proof (induct x \<equiv> "x # xs" ys arbitrary: x xs)
   283   case list_hembeq_Cons
   284   then show ?case by (metis append_Cons)
   285 next
   286   case (list_hembeq_Cons2 x y xs ys)
   287   then show ?case by blast
   288 qed
   289 
   290 lemma list_hembeq_appendD:
   291   assumes "list_hembeq P (xs @ ys) zs"
   292   shows "\<exists>us vs. zs = us @ vs \<and> list_hembeq P xs us \<and> list_hembeq P ys vs"
   293 using assms
   294 proof (induction xs arbitrary: ys zs)
   295   case Nil then show ?case by auto
   296 next
   297   case (Cons x xs)
   298   then obtain us v vs where
   299     zs: "zs = us @ v # vs" and p: "P\<^sup>=\<^sup>= x v" and lh: "list_hembeq P (xs @ ys) vs"
   300     by (auto dest: list_hembeq_ConsD)
   301   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
   302     sk: "\<forall>x\<^sub>0 x\<^sub>1. \<not> list_hembeq 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_hembeq P xs (sk\<^sub>0 x\<^sub>0 x\<^sub>1) \<and> list_hembeq P x\<^sub>0 (sk\<^sub>1 x\<^sub>0 x\<^sub>1)"
   303     using Cons(1) by (metis (no_types))
   304   hence "\<forall>x\<^sub>2. list_hembeq P (x # xs) (x\<^sub>2 @ v # sk\<^sub>0 ys vs)" using p lh by auto
   305   thus ?case using lh zs sk by (metis (no_types) append_Cons append_assoc)
   306 qed
   307 
   308 lemma list_hembeq_suffix:
   309   assumes "list_hembeq P xs ys" and "suffix ys zs"
   310   shows "list_hembeq P xs zs"
   311   using assms(2) and list_hembeq_append2 [OF assms(1)] by (auto simp: suffix_def)
   312 
   313 lemma list_hembeq_suffixeq:
   314   assumes "list_hembeq P xs ys" and "suffixeq ys zs"
   315   shows "list_hembeq P xs zs"
   316   using assms and list_hembeq_suffix unfolding suffixeq_suffix_reflclp_conv by auto
   317 
   318 lemma list_hembeq_length: "list_hembeq P xs ys \<Longrightarrow> length xs \<le> length ys"
   319   by (induct rule: list_hembeq.induct) auto
   320 
   321 lemma list_hembeq_trans:
   322   assumes "\<And>x y z. \<lbrakk>x \<in> A; y \<in> A; z \<in> A; P x y; P y z\<rbrakk> \<Longrightarrow> P x z"
   323   shows "\<And>xs ys zs. \<lbrakk>xs \<in> lists A; ys \<in> lists A; zs \<in> lists A;
   324     list_hembeq P xs ys; list_hembeq P ys zs\<rbrakk> \<Longrightarrow> list_hembeq P xs zs"
   325 proof -
   326   fix xs ys zs
   327   assume "list_hembeq P xs ys" and "list_hembeq P ys zs"
   328     and "xs \<in> lists A" and "ys \<in> lists A" and "zs \<in> lists A"
   329   then show "list_hembeq P xs zs"
   330   proof (induction arbitrary: zs)
   331     case list_hembeq_Nil show ?case by blast
   332   next
   333     case (list_hembeq_Cons xs ys y)
   334     from list_hembeq_ConsD [OF `list_hembeq P (y#ys) zs`] obtain us v vs
   335       where zs: "zs = us @ v # vs" and "P\<^sup>=\<^sup>= y v" and "list_hembeq P ys vs" by blast
   336     then have "list_hembeq P ys (v#vs)" by blast
   337     then have "list_hembeq P ys zs" unfolding zs by (rule list_hembeq_append2)
   338     from list_hembeq_Cons.IH [OF this] and list_hembeq_Cons.prems show ?case by simp
   339   next
   340     case (list_hembeq_Cons2 x y xs ys)
   341     from list_hembeq_ConsD [OF `list_hembeq P (y#ys) zs`] obtain us v vs
   342       where zs: "zs = us @ v # vs" and "P\<^sup>=\<^sup>= y v" and "list_hembeq P ys vs" by blast
   343     with list_hembeq_Cons2 have "list_hembeq P xs vs" by simp
   344     moreover have "P\<^sup>=\<^sup>= x v"
   345     proof -
   346       from zs and `zs \<in> lists A` have "v \<in> A" by auto
   347       moreover have "x \<in> A" and "y \<in> A" using list_hembeq_Cons2 by simp_all
   348       ultimately show ?thesis
   349         using `P\<^sup>=\<^sup>= x y` and `P\<^sup>=\<^sup>= y v` and assms
   350         by blast
   351     qed
   352     ultimately have "list_hembeq P (x#xs) (v#vs)" by blast
   353     then show ?case unfolding zs by (rule list_hembeq_append2)
   354   qed
   355 qed
   356 
   357 
   358 subsection {* Sublists (special case of homeomorphic embedding) *}
   359 
   360 abbreviation sublisteq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
   361   where "sublisteq xs ys \<equiv> list_hembeq (op =) xs ys"
   362 
   363 lemma sublisteq_Cons2: "sublisteq xs ys \<Longrightarrow> sublisteq (x#xs) (x#ys)" by auto
   364 
   365 lemma sublisteq_same_length:
   366   assumes "sublisteq xs ys" and "length xs = length ys" shows "xs = ys"
   367   using assms by (induct) (auto dest: list_hembeq_length)
   368 
   369 lemma not_sublisteq_length [simp]: "length ys < length xs \<Longrightarrow> \<not> sublisteq xs ys"
   370   by (metis list_hembeq_length linorder_not_less)
   371 
   372 lemma [code]:
   373   "list_hembeq P [] ys \<longleftrightarrow> True"
   374   "list_hembeq P (x#xs) [] \<longleftrightarrow> False"
   375   by (simp_all)
   376 
   377 lemma sublisteq_Cons': "sublisteq (x#xs) ys \<Longrightarrow> sublisteq xs ys"
   378   by (induct xs, simp, blast dest: list_hembeq_ConsD)
   379 
   380 lemma sublisteq_Cons2':
   381   assumes "sublisteq (x#xs) (x#ys)" shows "sublisteq xs ys"
   382   using assms by (cases) (rule sublisteq_Cons')
   383 
   384 lemma sublisteq_Cons2_neq:
   385   assumes "sublisteq (x#xs) (y#ys)"
   386   shows "x \<noteq> y \<Longrightarrow> sublisteq (x#xs) ys"
   387   using assms by (cases) auto
   388 
   389 lemma sublisteq_Cons2_iff [simp, code]:
   390   "sublisteq (x#xs) (y#ys) = (if x = y then sublisteq xs ys else sublisteq (x#xs) ys)"
   391   by (metis list_hembeq_Cons sublisteq_Cons2 sublisteq_Cons2' sublisteq_Cons2_neq)
   392 
   393 lemma sublisteq_append': "sublisteq (zs @ xs) (zs @ ys) \<longleftrightarrow> sublisteq xs ys"
   394   by (induct zs) simp_all
   395 
   396 lemma sublisteq_refl [simp, intro!]: "sublisteq xs xs" by (induct xs) simp_all
   397 
   398 lemma sublisteq_antisym:
   399   assumes "sublisteq xs ys" and "sublisteq ys xs"
   400   shows "xs = ys"
   401 using assms
   402 proof (induct)
   403   case list_hembeq_Nil
   404   from list_hembeq_Nil2 [OF this] show ?case by simp
   405 next
   406   case list_hembeq_Cons2
   407   thus ?case by simp
   408 next
   409   case list_hembeq_Cons
   410   hence False using sublisteq_Cons' by fastforce
   411   thus ?case ..
   412 qed
   413 
   414 lemma sublisteq_trans: "sublisteq xs ys \<Longrightarrow> sublisteq ys zs \<Longrightarrow> sublisteq xs zs"
   415   by (rule list_hembeq_trans [of UNIV "op ="]) auto
   416 
   417 lemma sublisteq_append_le_same_iff: "sublisteq (xs @ ys) ys \<longleftrightarrow> xs = []"
   418   by (auto dest: list_hembeq_length)
   419 
   420 lemma list_hembeq_append_mono:
   421   "\<lbrakk> list_hembeq P xs xs'; list_hembeq P ys ys' \<rbrakk> \<Longrightarrow> list_hembeq P (xs@ys) (xs'@ys')"
   422   apply (induct rule: list_hembeq.induct)
   423     apply (metis eq_Nil_appendI list_hembeq_append2)
   424    apply (metis append_Cons list_hembeq_Cons)
   425   apply (metis append_Cons list_hembeq_Cons2)
   426   done
   427 
   428 
   429 subsection {* Appending elements *}
   430 
   431 lemma sublisteq_append [simp]:
   432   "sublisteq (xs @ zs) (ys @ zs) \<longleftrightarrow> sublisteq xs ys" (is "?l = ?r")
   433 proof
   434   { fix xs' ys' xs ys zs :: "'a list" assume "sublisteq xs' ys'"
   435     then have "xs' = xs @ zs & ys' = ys @ zs \<longrightarrow> sublisteq xs ys"
   436     proof (induct arbitrary: xs ys zs)
   437       case list_hembeq_Nil show ?case by simp
   438     next
   439       case (list_hembeq_Cons xs' ys' x)
   440       { assume "ys=[]" then have ?case using list_hembeq_Cons(1) by auto }
   441       moreover
   442       { fix us assume "ys = x#us"
   443         then have ?case using list_hembeq_Cons(2) by(simp add: list_hembeq.list_hembeq_Cons) }
   444       ultimately show ?case by (auto simp:Cons_eq_append_conv)
   445     next
   446       case (list_hembeq_Cons2 x y xs' ys')
   447       { assume "xs=[]" then have ?case using list_hembeq_Cons2(1) by auto }
   448       moreover
   449       { fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using list_hembeq_Cons2 by auto}
   450       moreover
   451       { fix us assume "xs=x#us" "ys=[]" then have ?case using list_hembeq_Cons2(2) by bestsimp }
   452       ultimately show ?case using `op =\<^sup>=\<^sup>= x y` by (auto simp: Cons_eq_append_conv)
   453     qed }
   454   moreover assume ?l
   455   ultimately show ?r by blast
   456 next
   457   assume ?r then show ?l by (metis list_hembeq_append_mono sublisteq_refl)
   458 qed
   459 
   460 lemma sublisteq_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (zs @ ys)"
   461   by (induct zs) auto
   462 
   463 lemma sublisteq_rev_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (ys @ zs)"
   464   by (metis append_Nil2 list_hembeq_Nil list_hembeq_append_mono)
   465 
   466 
   467 subsection {* Relation to standard list operations *}
   468 
   469 lemma sublisteq_map:
   470   assumes "sublisteq xs ys" shows "sublisteq (map f xs) (map f ys)"
   471   using assms by (induct) auto
   472 
   473 lemma sublisteq_filter_left [simp]: "sublisteq (filter P xs) xs"
   474   by (induct xs) auto
   475 
   476 lemma sublisteq_filter [simp]:
   477   assumes "sublisteq xs ys" shows "sublisteq (filter P xs) (filter P ys)"
   478   using assms by induct auto
   479 
   480 lemma "sublisteq xs ys \<longleftrightarrow> (\<exists>N. xs = sublist ys N)" (is "?L = ?R")
   481 proof
   482   assume ?L
   483   then show ?R
   484   proof (induct)
   485     case list_hembeq_Nil show ?case by (metis sublist_empty)
   486   next
   487     case (list_hembeq_Cons xs ys x)
   488     then obtain N where "xs = sublist ys N" by blast
   489     then have "xs = sublist (x#ys) (Suc ` N)"
   490       by (clarsimp simp add:sublist_Cons inj_image_mem_iff)
   491     then show ?case by blast
   492   next
   493     case (list_hembeq_Cons2 x y xs ys)
   494     then obtain N where "xs = sublist ys N" by blast
   495     then have "x#xs = sublist (x#ys) (insert 0 (Suc ` N))"
   496       by (clarsimp simp add:sublist_Cons inj_image_mem_iff)
   497     moreover from list_hembeq_Cons2 have "x = y" by simp
   498     ultimately show ?case by blast
   499   qed
   500 next
   501   assume ?R
   502   then obtain N where "xs = sublist ys N" ..
   503   moreover have "sublisteq (sublist ys N) ys"
   504   proof (induct ys arbitrary: N)
   505     case Nil show ?case by simp
   506   next
   507     case Cons then show ?case by (auto simp: sublist_Cons)
   508   qed
   509   ultimately show ?L by simp
   510 qed
   511 
   512 end