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
```