src/HOL/Library/Sublist.thy
author Andreas Lochbihler
Wed Feb 27 10:33:30 2013 +0100 (2013-02-27)
changeset 51288 be7e9a675ec9
parent 50516 ed6b40d15d1c
child 52729 412c9e0381a1
permissions -rw-r--r--
add wellorder instance for Numeral_Type (suggested by Jesus Aransay)
     1 (*  Title:      HOL/Library/Sublist.thy
     2     Author:     Tobias Nipkow and Markus Wenzel, TU Muenchen
     3     Author:     Christian Sternagel, JAIST
     4 *)
     5 
     6 header {* List prefixes, suffixes, and homeomorphic embedding *}
     7 
     8 theory Sublist
     9 imports Main
    10 begin
    11 
    12 subsection {* Prefix order on lists *}
    13 
    14 definition prefixeq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
    15   where "prefixeq xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"
    16 
    17 definition prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
    18   where "prefix xs ys \<longleftrightarrow> prefixeq xs ys \<and> xs \<noteq> ys"
    19 
    20 interpretation prefix_order: order prefixeq prefix
    21   by default (auto simp: prefixeq_def prefix_def)
    22 
    23 interpretation prefix_bot: bot prefixeq prefix Nil
    24   by default (simp add: prefixeq_def)
    25 
    26 lemma prefixeqI [intro?]: "ys = xs @ zs \<Longrightarrow> prefixeq xs ys"
    27   unfolding prefixeq_def by blast
    28 
    29 lemma prefixeqE [elim?]:
    30   assumes "prefixeq xs ys"
    31   obtains zs where "ys = xs @ zs"
    32   using assms unfolding prefixeq_def by blast
    33 
    34 lemma prefixI' [intro?]: "ys = xs @ z # zs \<Longrightarrow> prefix xs ys"
    35   unfolding prefix_def prefixeq_def by blast
    36 
    37 lemma prefixE' [elim?]:
    38   assumes "prefix xs ys"
    39   obtains z zs where "ys = xs @ z # zs"
    40 proof -
    41   from `prefix xs ys` obtain us where "ys = xs @ us" and "xs \<noteq> ys"
    42     unfolding prefix_def prefixeq_def by blast
    43   with that show ?thesis by (auto simp add: neq_Nil_conv)
    44 qed
    45 
    46 lemma prefixI [intro?]: "prefixeq xs ys \<Longrightarrow> xs \<noteq> ys \<Longrightarrow> prefix xs ys"
    47   unfolding prefix_def by blast
    48 
    49 lemma prefixE [elim?]:
    50   fixes xs ys :: "'a list"
    51   assumes "prefix xs ys"
    52   obtains "prefixeq xs ys" and "xs \<noteq> ys"
    53   using assms unfolding prefix_def by blast
    54 
    55 
    56 subsection {* Basic properties of prefixes *}
    57 
    58 theorem Nil_prefixeq [iff]: "prefixeq [] xs"
    59   by (simp add: prefixeq_def)
    60 
    61 theorem prefixeq_Nil [simp]: "(prefixeq xs []) = (xs = [])"
    62   by (induct xs) (simp_all add: prefixeq_def)
    63 
    64 lemma prefixeq_snoc [simp]: "prefixeq xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefixeq xs ys"
    65 proof
    66   assume "prefixeq xs (ys @ [y])"
    67   then obtain zs where zs: "ys @ [y] = xs @ zs" ..
    68   show "xs = ys @ [y] \<or> prefixeq xs ys"
    69     by (metis append_Nil2 butlast_append butlast_snoc prefixeqI zs)
    70 next
    71   assume "xs = ys @ [y] \<or> prefixeq xs ys"
    72   then show "prefixeq xs (ys @ [y])"
    73     by (metis prefix_order.eq_iff prefix_order.order_trans prefixeqI)
    74 qed
    75 
    76 lemma Cons_prefixeq_Cons [simp]: "prefixeq (x # xs) (y # ys) = (x = y \<and> prefixeq xs ys)"
    77   by (auto simp add: prefixeq_def)
    78 
    79 lemma prefixeq_code [code]:
    80   "prefixeq [] xs \<longleftrightarrow> True"
    81   "prefixeq (x # xs) [] \<longleftrightarrow> False"
    82   "prefixeq (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefixeq xs ys"
    83   by simp_all
    84 
    85 lemma same_prefixeq_prefixeq [simp]: "prefixeq (xs @ ys) (xs @ zs) = prefixeq ys zs"
    86   by (induct xs) simp_all
    87 
    88 lemma same_prefixeq_nil [iff]: "prefixeq (xs @ ys) xs = (ys = [])"
    89   by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixeqI)
    90 
    91 lemma prefixeq_prefixeq [simp]: "prefixeq xs ys \<Longrightarrow> prefixeq xs (ys @ zs)"
    92   by (metis prefix_order.le_less_trans prefixeqI prefixE prefixI)
    93 
    94 lemma append_prefixeqD: "prefixeq (xs @ ys) zs \<Longrightarrow> prefixeq xs zs"
    95   by (auto simp add: prefixeq_def)
    96 
    97 theorem prefixeq_Cons: "prefixeq xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefixeq zs ys))"
    98   by (cases xs) (auto simp add: prefixeq_def)
    99 
   100 theorem prefixeq_append:
   101   "prefixeq xs (ys @ zs) = (prefixeq xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefixeq us zs))"
   102   apply (induct zs rule: rev_induct)
   103    apply force
   104   apply (simp del: append_assoc add: append_assoc [symmetric])
   105   apply (metis append_eq_appendI)
   106   done
   107 
   108 lemma append_one_prefixeq:
   109   "prefixeq xs ys \<Longrightarrow> length xs < length ys \<Longrightarrow> prefixeq (xs @ [ys ! length xs]) ys"
   110   unfolding prefixeq_def
   111   by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj
   112     eq_Nil_appendI nth_drop')
   113 
   114 theorem prefixeq_length_le: "prefixeq xs ys \<Longrightarrow> length xs \<le> length ys"
   115   by (auto simp add: prefixeq_def)
   116 
   117 lemma prefixeq_same_cases:
   118   "prefixeq (xs\<^isub>1::'a list) ys \<Longrightarrow> prefixeq xs\<^isub>2 ys \<Longrightarrow> prefixeq xs\<^isub>1 xs\<^isub>2 \<or> prefixeq xs\<^isub>2 xs\<^isub>1"
   119   unfolding prefixeq_def by (metis append_eq_append_conv2)
   120 
   121 lemma set_mono_prefixeq: "prefixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys"
   122   by (auto simp add: prefixeq_def)
   123 
   124 lemma take_is_prefixeq: "prefixeq (take n xs) xs"
   125   unfolding prefixeq_def by (metis append_take_drop_id)
   126 
   127 lemma map_prefixeqI: "prefixeq xs ys \<Longrightarrow> prefixeq (map f xs) (map f ys)"
   128   by (auto simp: prefixeq_def)
   129 
   130 lemma prefixeq_length_less: "prefix xs ys \<Longrightarrow> length xs < length ys"
   131   by (auto simp: prefix_def prefixeq_def)
   132 
   133 lemma prefix_simps [simp, code]:
   134   "prefix xs [] \<longleftrightarrow> False"
   135   "prefix [] (x # xs) \<longleftrightarrow> True"
   136   "prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys"
   137   by (simp_all add: prefix_def cong: conj_cong)
   138 
   139 lemma take_prefix: "prefix xs ys \<Longrightarrow> prefix (take n xs) ys"
   140   apply (induct n arbitrary: xs ys)
   141    apply (case_tac ys, simp_all)[1]
   142   apply (metis prefix_order.less_trans prefixI take_is_prefixeq)
   143   done
   144 
   145 lemma not_prefixeq_cases:
   146   assumes pfx: "\<not> prefixeq ps ls"
   147   obtains
   148     (c1) "ps \<noteq> []" and "ls = []"
   149   | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> prefixeq as xs"
   150   | (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a"
   151 proof (cases ps)
   152   case Nil
   153   then show ?thesis using pfx by simp
   154 next
   155   case (Cons a as)
   156   note c = `ps = a#as`
   157   show ?thesis
   158   proof (cases ls)
   159     case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefixeq_nil)
   160   next
   161     case (Cons x xs)
   162     show ?thesis
   163     proof (cases "x = a")
   164       case True
   165       have "\<not> prefixeq as xs" using pfx c Cons True by simp
   166       with c Cons True show ?thesis by (rule c2)
   167     next
   168       case False
   169       with c Cons show ?thesis by (rule c3)
   170     qed
   171   qed
   172 qed
   173 
   174 lemma not_prefixeq_induct [consumes 1, case_names Nil Neq Eq]:
   175   assumes np: "\<not> prefixeq ps ls"
   176     and base: "\<And>x xs. P (x#xs) []"
   177     and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)"
   178     and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefixeq xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"
   179   shows "P ps ls" using np
   180 proof (induct ls arbitrary: ps)
   181   case Nil then show ?case
   182     by (auto simp: neq_Nil_conv elim!: not_prefixeq_cases intro!: base)
   183 next
   184   case (Cons y ys)
   185   then have npfx: "\<not> prefixeq ps (y # ys)" by simp
   186   then obtain x xs where pv: "ps = x # xs"
   187     by (rule not_prefixeq_cases) auto
   188   show ?case by (metis Cons.hyps Cons_prefixeq_Cons npfx pv r1 r2)
   189 qed
   190 
   191 
   192 subsection {* Parallel lists *}
   193 
   194 definition parallel :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "\<parallel>" 50)
   195   where "(xs \<parallel> ys) = (\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs)"
   196 
   197 lemma parallelI [intro]: "\<not> prefixeq xs ys \<Longrightarrow> \<not> prefixeq ys xs \<Longrightarrow> xs \<parallel> ys"
   198   unfolding parallel_def by blast
   199 
   200 lemma parallelE [elim]:
   201   assumes "xs \<parallel> ys"
   202   obtains "\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs"
   203   using assms unfolding parallel_def by blast
   204 
   205 theorem prefixeq_cases:
   206   obtains "prefixeq xs ys" | "prefix ys xs" | "xs \<parallel> ys"
   207   unfolding parallel_def prefix_def by blast
   208 
   209 theorem parallel_decomp:
   210   "xs \<parallel> ys \<Longrightarrow> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
   211 proof (induct xs rule: rev_induct)
   212   case Nil
   213   then have False by auto
   214   then show ?case ..
   215 next
   216   case (snoc x xs)
   217   show ?case
   218   proof (rule prefixeq_cases)
   219     assume le: "prefixeq xs ys"
   220     then obtain ys' where ys: "ys = xs @ ys'" ..
   221     show ?thesis
   222     proof (cases ys')
   223       assume "ys' = []"
   224       then show ?thesis by (metis append_Nil2 parallelE prefixeqI snoc.prems ys)
   225     next
   226       fix c cs assume ys': "ys' = c # cs"
   227       then show ?thesis
   228         by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixeqI
   229           same_prefixeq_prefixeq snoc.prems ys)
   230     qed
   231   next
   232     assume "prefix ys xs"
   233     then have "prefixeq ys (xs @ [x])" by (simp add: prefix_def)
   234     with snoc have False by blast
   235     then show ?thesis ..
   236   next
   237     assume "xs \<parallel> ys"
   238     with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c"
   239       and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
   240       by blast
   241     from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
   242     with neq ys show ?thesis by blast
   243   qed
   244 qed
   245 
   246 lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
   247   apply (rule parallelI)
   248     apply (erule parallelE, erule conjE,
   249       induct rule: not_prefixeq_induct, simp+)+
   250   done
   251 
   252 lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y"
   253   by (simp add: parallel_append)
   254 
   255 lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"
   256   unfolding parallel_def by auto
   257 
   258 
   259 subsection {* Suffix order on lists *}
   260 
   261 definition suffixeq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
   262   where "suffixeq xs ys = (\<exists>zs. ys = zs @ xs)"
   263 
   264 definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
   265   where "suffix xs ys \<longleftrightarrow> (\<exists>us. ys = us @ xs \<and> us \<noteq> [])"
   266 
   267 lemma suffix_imp_suffixeq:
   268   "suffix xs ys \<Longrightarrow> suffixeq xs ys"
   269   by (auto simp: suffixeq_def suffix_def)
   270 
   271 lemma suffixeqI [intro?]: "ys = zs @ xs \<Longrightarrow> suffixeq xs ys"
   272   unfolding suffixeq_def by blast
   273 
   274 lemma suffixeqE [elim?]:
   275   assumes "suffixeq xs ys"
   276   obtains zs where "ys = zs @ xs"
   277   using assms unfolding suffixeq_def by blast
   278 
   279 lemma suffixeq_refl [iff]: "suffixeq xs xs"
   280   by (auto simp add: suffixeq_def)
   281 lemma suffix_trans:
   282   "suffix xs ys \<Longrightarrow> suffix ys zs \<Longrightarrow> suffix xs zs"
   283   by (auto simp: suffix_def)
   284 lemma suffixeq_trans: "\<lbrakk>suffixeq xs ys; suffixeq ys zs\<rbrakk> \<Longrightarrow> suffixeq xs zs"
   285   by (auto simp add: suffixeq_def)
   286 lemma suffixeq_antisym: "\<lbrakk>suffixeq xs ys; suffixeq ys xs\<rbrakk> \<Longrightarrow> xs = ys"
   287   by (auto simp add: suffixeq_def)
   288 
   289 lemma suffixeq_tl [simp]: "suffixeq (tl xs) xs"
   290   by (induct xs) (auto simp: suffixeq_def)
   291 
   292 lemma suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> suffix (tl xs) xs"
   293   by (induct xs) (auto simp: suffix_def)
   294 
   295 lemma Nil_suffixeq [iff]: "suffixeq [] xs"
   296   by (simp add: suffixeq_def)
   297 lemma suffixeq_Nil [simp]: "(suffixeq xs []) = (xs = [])"
   298   by (auto simp add: suffixeq_def)
   299 
   300 lemma suffixeq_ConsI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (y # ys)"
   301   by (auto simp add: suffixeq_def)
   302 lemma suffixeq_ConsD: "suffixeq (x # xs) ys \<Longrightarrow> suffixeq xs ys"
   303   by (auto simp add: suffixeq_def)
   304 
   305 lemma suffixeq_appendI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (zs @ ys)"
   306   by (auto simp add: suffixeq_def)
   307 lemma suffixeq_appendD: "suffixeq (zs @ xs) ys \<Longrightarrow> suffixeq xs ys"
   308   by (auto simp add: suffixeq_def)
   309 
   310 lemma suffix_set_subset:
   311   "suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffix_def)
   312 
   313 lemma suffixeq_set_subset:
   314   "suffixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffixeq_def)
   315 
   316 lemma suffixeq_ConsD2: "suffixeq (x # xs) (y # ys) \<Longrightarrow> suffixeq xs ys"
   317 proof -
   318   assume "suffixeq (x # xs) (y # ys)"
   319   then obtain zs where "y # ys = zs @ x # xs" ..
   320   then show ?thesis
   321     by (induct zs) (auto intro!: suffixeq_appendI suffixeq_ConsI)
   322 qed
   323 
   324 lemma suffixeq_to_prefixeq [code]: "suffixeq xs ys \<longleftrightarrow> prefixeq (rev xs) (rev ys)"
   325 proof
   326   assume "suffixeq xs ys"
   327   then obtain zs where "ys = zs @ xs" ..
   328   then have "rev ys = rev xs @ rev zs" by simp
   329   then show "prefixeq (rev xs) (rev ys)" ..
   330 next
   331   assume "prefixeq (rev xs) (rev ys)"
   332   then obtain zs where "rev ys = rev xs @ zs" ..
   333   then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp
   334   then have "ys = rev zs @ xs" by simp
   335   then show "suffixeq xs ys" ..
   336 qed
   337 
   338 lemma distinct_suffixeq: "distinct ys \<Longrightarrow> suffixeq xs ys \<Longrightarrow> distinct xs"
   339   by (clarsimp elim!: suffixeqE)
   340 
   341 lemma suffixeq_map: "suffixeq xs ys \<Longrightarrow> suffixeq (map f xs) (map f ys)"
   342   by (auto elim!: suffixeqE intro: suffixeqI)
   343 
   344 lemma suffixeq_drop: "suffixeq (drop n as) as"
   345   unfolding suffixeq_def
   346   apply (rule exI [where x = "take n as"])
   347   apply simp
   348   done
   349 
   350 lemma suffixeq_take: "suffixeq xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs"
   351   by (auto elim!: suffixeqE)
   352 
   353 lemma suffixeq_suffix_reflclp_conv: "suffixeq = suffix\<^sup>=\<^sup>="
   354 proof (intro ext iffI)
   355   fix xs ys :: "'a list"
   356   assume "suffixeq xs ys"
   357   show "suffix\<^sup>=\<^sup>= xs ys"
   358   proof
   359     assume "xs \<noteq> ys"
   360     with `suffixeq xs ys` show "suffix xs ys"
   361       by (auto simp: suffixeq_def suffix_def)
   362   qed
   363 next
   364   fix xs ys :: "'a list"
   365   assume "suffix\<^sup>=\<^sup>= xs ys"
   366   then show "suffixeq xs ys"
   367   proof
   368     assume "suffix xs ys" then show "suffixeq xs ys"
   369       by (rule suffix_imp_suffixeq)
   370   next
   371     assume "xs = ys" then show "suffixeq xs ys"
   372       by (auto simp: suffixeq_def)
   373   qed
   374 qed
   375 
   376 lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefixeq x y"
   377   by blast
   378 
   379 lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefixeq y x"
   380   by blast
   381 
   382 lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
   383   unfolding parallel_def by simp
   384 
   385 lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
   386   unfolding parallel_def by simp
   387 
   388 lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
   389   by auto
   390 
   391 lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
   392   by (metis Cons_prefixeq_Cons parallelE parallelI)
   393 
   394 lemma not_equal_is_parallel:
   395   assumes neq: "xs \<noteq> ys"
   396     and len: "length xs = length ys"
   397   shows "xs \<parallel> ys"
   398   using len neq
   399 proof (induct rule: list_induct2)
   400   case Nil
   401   then show ?case by simp
   402 next
   403   case (Cons a as b bs)
   404   have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact
   405   show ?case
   406   proof (cases "a = b")
   407     case True
   408     then have "as \<noteq> bs" using Cons by simp
   409     then show ?thesis by (rule Cons_parallelI2 [OF True ih])
   410   next
   411     case False
   412     then show ?thesis by (rule Cons_parallelI1)
   413   qed
   414 qed
   415 
   416 lemma suffix_reflclp_conv: "suffix\<^sup>=\<^sup>= = suffixeq"
   417   by (intro ext) (auto simp: suffixeq_def suffix_def)
   418 
   419 lemma suffix_lists: "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A"
   420   unfolding suffix_def by auto
   421 
   422 
   423 subsection {* Homeomorphic embedding on lists *}
   424 
   425 inductive list_hembeq :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
   426   for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
   427 where
   428   list_hembeq_Nil [intro, simp]: "list_hembeq P [] ys"
   429 | list_hembeq_Cons [intro] : "list_hembeq P xs ys \<Longrightarrow> list_hembeq P xs (y#ys)"
   430 | list_hembeq_Cons2 [intro]: "P\<^sup>=\<^sup>= x y \<Longrightarrow> list_hembeq P xs ys \<Longrightarrow> list_hembeq P (x#xs) (y#ys)"
   431 
   432 lemma list_hembeq_Nil2 [simp]:
   433   assumes "list_hembeq P xs []" shows "xs = []"
   434   using assms by (cases rule: list_hembeq.cases) auto
   435 
   436 lemma list_hembeq_refl [simp, intro!]:
   437   "list_hembeq P xs xs"
   438   by (induct xs) auto
   439 
   440 lemma list_hembeq_Cons_Nil [simp]: "list_hembeq P (x#xs) [] = False"
   441 proof -
   442   { assume "list_hembeq P (x#xs) []"
   443     from list_hembeq_Nil2 [OF this] have False by simp
   444   } moreover {
   445     assume False
   446     then have "list_hembeq P (x#xs) []" by simp
   447   } ultimately show ?thesis by blast
   448 qed
   449 
   450 lemma list_hembeq_append2 [intro]: "list_hembeq P xs ys \<Longrightarrow> list_hembeq P xs (zs @ ys)"
   451   by (induct zs) auto
   452 
   453 lemma list_hembeq_prefix [intro]:
   454   assumes "list_hembeq P xs ys" shows "list_hembeq P xs (ys @ zs)"
   455   using assms
   456   by (induct arbitrary: zs) auto
   457 
   458 lemma list_hembeq_ConsD:
   459   assumes "list_hembeq P (x#xs) ys"
   460   shows "\<exists>us v vs. ys = us @ v # vs \<and> P\<^sup>=\<^sup>= x v \<and> list_hembeq P xs vs"
   461 using assms
   462 proof (induct x \<equiv> "x # xs" ys arbitrary: x xs)
   463   case list_hembeq_Cons
   464   then show ?case by (metis append_Cons)
   465 next
   466   case (list_hembeq_Cons2 x y xs ys)
   467   then show ?case by (cases xs) (auto, blast+)
   468 qed
   469 
   470 lemma list_hembeq_appendD:
   471   assumes "list_hembeq P (xs @ ys) zs"
   472   shows "\<exists>us vs. zs = us @ vs \<and> list_hembeq P xs us \<and> list_hembeq P ys vs"
   473 using assms
   474 proof (induction xs arbitrary: ys zs)
   475   case Nil then show ?case by auto
   476 next
   477   case (Cons x xs)
   478   then obtain us v vs where "zs = us @ v # vs"
   479     and "P\<^sup>=\<^sup>= x v" and "list_hembeq P (xs @ ys) vs" by (auto dest: list_hembeq_ConsD)
   480   with Cons show ?case by (metis append_Cons append_assoc list_hembeq_Cons2 list_hembeq_append2)
   481 qed
   482 
   483 lemma list_hembeq_suffix:
   484   assumes "list_hembeq P xs ys" and "suffix ys zs"
   485   shows "list_hembeq P xs zs"
   486   using assms(2) and list_hembeq_append2 [OF assms(1)] by (auto simp: suffix_def)
   487 
   488 lemma list_hembeq_suffixeq:
   489   assumes "list_hembeq P xs ys" and "suffixeq ys zs"
   490   shows "list_hembeq P xs zs"
   491   using assms and list_hembeq_suffix unfolding suffixeq_suffix_reflclp_conv by auto
   492 
   493 lemma list_hembeq_length: "list_hembeq P xs ys \<Longrightarrow> length xs \<le> length ys"
   494   by (induct rule: list_hembeq.induct) auto
   495 
   496 lemma list_hembeq_trans:
   497   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"
   498   shows "\<And>xs ys zs. \<lbrakk>xs \<in> lists A; ys \<in> lists A; zs \<in> lists A;
   499     list_hembeq P xs ys; list_hembeq P ys zs\<rbrakk> \<Longrightarrow> list_hembeq P xs zs"
   500 proof -
   501   fix xs ys zs
   502   assume "list_hembeq P xs ys" and "list_hembeq P ys zs"
   503     and "xs \<in> lists A" and "ys \<in> lists A" and "zs \<in> lists A"
   504   then show "list_hembeq P xs zs"
   505   proof (induction arbitrary: zs)
   506     case list_hembeq_Nil show ?case by blast
   507   next
   508     case (list_hembeq_Cons xs ys y)
   509     from list_hembeq_ConsD [OF `list_hembeq P (y#ys) zs`] obtain us v vs
   510       where zs: "zs = us @ v # vs" and "P\<^sup>=\<^sup>= y v" and "list_hembeq P ys vs" by blast
   511     then have "list_hembeq P ys (v#vs)" by blast
   512     then have "list_hembeq P ys zs" unfolding zs by (rule list_hembeq_append2)
   513     from list_hembeq_Cons.IH [OF this] and list_hembeq_Cons.prems show ?case by simp
   514   next
   515     case (list_hembeq_Cons2 x y xs ys)
   516     from list_hembeq_ConsD [OF `list_hembeq P (y#ys) zs`] obtain us v vs
   517       where zs: "zs = us @ v # vs" and "P\<^sup>=\<^sup>= y v" and "list_hembeq P ys vs" by blast
   518     with list_hembeq_Cons2 have "list_hembeq P xs vs" by simp
   519     moreover have "P\<^sup>=\<^sup>= x v"
   520     proof -
   521       from zs and `zs \<in> lists A` have "v \<in> A" by auto
   522       moreover have "x \<in> A" and "y \<in> A" using list_hembeq_Cons2 by simp_all
   523       ultimately show ?thesis
   524         using `P\<^sup>=\<^sup>= x y` and `P\<^sup>=\<^sup>= y v` and assms
   525         by blast
   526     qed
   527     ultimately have "list_hembeq P (x#xs) (v#vs)" by blast
   528     then show ?case unfolding zs by (rule list_hembeq_append2)
   529   qed
   530 qed
   531 
   532 
   533 subsection {* Sublists (special case of homeomorphic embedding) *}
   534 
   535 abbreviation sublisteq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
   536   where "sublisteq xs ys \<equiv> list_hembeq (op =) xs ys"
   537 
   538 lemma sublisteq_Cons2: "sublisteq xs ys \<Longrightarrow> sublisteq (x#xs) (x#ys)" by auto
   539 
   540 lemma sublisteq_same_length:
   541   assumes "sublisteq xs ys" and "length xs = length ys" shows "xs = ys"
   542   using assms by (induct) (auto dest: list_hembeq_length)
   543 
   544 lemma not_sublisteq_length [simp]: "length ys < length xs \<Longrightarrow> \<not> sublisteq xs ys"
   545   by (metis list_hembeq_length linorder_not_less)
   546 
   547 lemma [code]:
   548   "list_hembeq P [] ys \<longleftrightarrow> True"
   549   "list_hembeq P (x#xs) [] \<longleftrightarrow> False"
   550   by (simp_all)
   551 
   552 lemma sublisteq_Cons': "sublisteq (x#xs) ys \<Longrightarrow> sublisteq xs ys"
   553   by (induct xs) (auto dest: list_hembeq_ConsD)
   554 
   555 lemma sublisteq_Cons2':
   556   assumes "sublisteq (x#xs) (x#ys)" shows "sublisteq xs ys"
   557   using assms by (cases) (rule sublisteq_Cons')
   558 
   559 lemma sublisteq_Cons2_neq:
   560   assumes "sublisteq (x#xs) (y#ys)"
   561   shows "x \<noteq> y \<Longrightarrow> sublisteq (x#xs) ys"
   562   using assms by (cases) auto
   563 
   564 lemma sublisteq_Cons2_iff [simp, code]:
   565   "sublisteq (x#xs) (y#ys) = (if x = y then sublisteq xs ys else sublisteq (x#xs) ys)"
   566   by (metis list_hembeq_Cons sublisteq_Cons2 sublisteq_Cons2' sublisteq_Cons2_neq)
   567 
   568 lemma sublisteq_append': "sublisteq (zs @ xs) (zs @ ys) \<longleftrightarrow> sublisteq xs ys"
   569   by (induct zs) simp_all
   570 
   571 lemma sublisteq_refl [simp, intro!]: "sublisteq xs xs" by (induct xs) simp_all
   572 
   573 lemma sublisteq_antisym:
   574   assumes "sublisteq xs ys" and "sublisteq ys xs"
   575   shows "xs = ys"
   576 using assms
   577 proof (induct)
   578   case list_hembeq_Nil
   579   from list_hembeq_Nil2 [OF this] show ?case by simp
   580 next
   581   case list_hembeq_Cons2
   582   then show ?case by simp
   583 next
   584   case list_hembeq_Cons
   585   then show ?case
   586     by (metis sublisteq_Cons' list_hembeq_length Suc_length_conv Suc_n_not_le_n)
   587 qed
   588 
   589 lemma sublisteq_trans: "sublisteq xs ys \<Longrightarrow> sublisteq ys zs \<Longrightarrow> sublisteq xs zs"
   590   by (rule list_hembeq_trans [of UNIV "op ="]) auto
   591 
   592 lemma sublisteq_append_le_same_iff: "sublisteq (xs @ ys) ys \<longleftrightarrow> xs = []"
   593   by (auto dest: list_hembeq_length)
   594 
   595 lemma list_hembeq_append_mono:
   596   "\<lbrakk> list_hembeq P xs xs'; list_hembeq P ys ys' \<rbrakk> \<Longrightarrow> list_hembeq P (xs@ys) (xs'@ys')"
   597   apply (induct rule: list_hembeq.induct)
   598     apply (metis eq_Nil_appendI list_hembeq_append2)
   599    apply (metis append_Cons list_hembeq_Cons)
   600   apply (metis append_Cons list_hembeq_Cons2)
   601   done
   602 
   603 
   604 subsection {* Appending elements *}
   605 
   606 lemma sublisteq_append [simp]:
   607   "sublisteq (xs @ zs) (ys @ zs) \<longleftrightarrow> sublisteq xs ys" (is "?l = ?r")
   608 proof
   609   { fix xs' ys' xs ys zs :: "'a list" assume "sublisteq xs' ys'"
   610     then have "xs' = xs @ zs & ys' = ys @ zs \<longrightarrow> sublisteq xs ys"
   611     proof (induct arbitrary: xs ys zs)
   612       case list_hembeq_Nil show ?case by simp
   613     next
   614       case (list_hembeq_Cons xs' ys' x)
   615       { assume "ys=[]" then have ?case using list_hembeq_Cons(1) by auto }
   616       moreover
   617       { fix us assume "ys = x#us"
   618         then have ?case using list_hembeq_Cons(2) by(simp add: list_hembeq.list_hembeq_Cons) }
   619       ultimately show ?case by (auto simp:Cons_eq_append_conv)
   620     next
   621       case (list_hembeq_Cons2 x y xs' ys')
   622       { assume "xs=[]" then have ?case using list_hembeq_Cons2(1) by auto }
   623       moreover
   624       { fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using list_hembeq_Cons2 by auto}
   625       moreover
   626       { fix us assume "xs=x#us" "ys=[]" then have ?case using list_hembeq_Cons2(2) by bestsimp }
   627       ultimately show ?case using `op =\<^sup>=\<^sup>= x y` by (auto simp: Cons_eq_append_conv)
   628     qed }
   629   moreover assume ?l
   630   ultimately show ?r by blast
   631 next
   632   assume ?r then show ?l by (metis list_hembeq_append_mono sublisteq_refl)
   633 qed
   634 
   635 lemma sublisteq_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (zs @ ys)"
   636   by (induct zs) auto
   637 
   638 lemma sublisteq_rev_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (ys @ zs)"
   639   by (metis append_Nil2 list_hembeq_Nil list_hembeq_append_mono)
   640 
   641 
   642 subsection {* Relation to standard list operations *}
   643 
   644 lemma sublisteq_map:
   645   assumes "sublisteq xs ys" shows "sublisteq (map f xs) (map f ys)"
   646   using assms by (induct) auto
   647 
   648 lemma sublisteq_filter_left [simp]: "sublisteq (filter P xs) xs"
   649   by (induct xs) auto
   650 
   651 lemma sublisteq_filter [simp]:
   652   assumes "sublisteq xs ys" shows "sublisteq (filter P xs) (filter P ys)"
   653   using assms by (induct) auto
   654 
   655 lemma "sublisteq xs ys \<longleftrightarrow> (\<exists>N. xs = sublist ys N)" (is "?L = ?R")
   656 proof
   657   assume ?L
   658   then show ?R
   659   proof (induct)
   660     case list_hembeq_Nil show ?case by (metis sublist_empty)
   661   next
   662     case (list_hembeq_Cons xs ys x)
   663     then obtain N where "xs = sublist ys N" by blast
   664     then have "xs = sublist (x#ys) (Suc ` N)"
   665       by (clarsimp simp add:sublist_Cons inj_image_mem_iff)
   666     then show ?case by blast
   667   next
   668     case (list_hembeq_Cons2 x y xs ys)
   669     then obtain N where "xs = sublist ys N" by blast
   670     then have "x#xs = sublist (x#ys) (insert 0 (Suc ` N))"
   671       by (clarsimp simp add:sublist_Cons inj_image_mem_iff)
   672     moreover from list_hembeq_Cons2 have "x = y" by simp
   673     ultimately show ?case by blast
   674   qed
   675 next
   676   assume ?R
   677   then obtain N where "xs = sublist ys N" ..
   678   moreover have "sublisteq (sublist ys N) ys"
   679   proof (induct ys arbitrary: N)
   680     case Nil show ?case by simp
   681   next
   682     case Cons then show ?case by (auto simp: sublist_Cons)
   683   qed
   684   ultimately show ?L by simp
   685 qed
   686 
   687 end