src/HOL/Library/Sublist.thy
author Christian Sternagel
Wed Aug 29 12:23:14 2012 +0900 (2012-08-29)
changeset 49083 01081bca31b6
parent 45236 src/HOL/Library/List_Prefix.thy@ac4a2a66707d
permissions -rw-r--r--
dropped ord and bot instance for list prefixes (use locale interpretation instead, which allows users to decide what order to use on lists)
     1 (*  Title:      HOL/Library/Sublist.thy
     2     Author:     Tobias Nipkow and Markus Wenzel, TU Muenchen
     3 *)
     4 
     5 header {* List prefixes, suffixes, and embedding*}
     6 
     7 theory Sublist
     8 imports List Main
     9 begin
    10 
    11 subsection {* Prefix order on lists *}
    12 
    13 definition prefixeq :: "'a list => 'a list => bool" where
    14   "prefixeq xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"
    15 
    16 definition prefix :: "'a list => 'a list => bool" where
    17   "prefix xs ys \<longleftrightarrow> prefixeq xs ys \<and> xs \<noteq> ys"
    18 
    19 interpretation prefix_order: order prefixeq prefix
    20   by default (auto simp: prefixeq_def prefix_def)
    21 
    22 interpretation prefix_bot: bot prefixeq prefix Nil
    23   by default (simp add: prefixeq_def)
    24 
    25 lemma prefixeqI [intro?]: "ys = xs @ zs ==> prefixeq xs ys"
    26   unfolding prefixeq_def by blast
    27 
    28 lemma prefixeqE [elim?]:
    29   assumes "prefixeq xs ys"
    30   obtains zs where "ys = xs @ zs"
    31   using assms unfolding prefixeq_def by blast
    32 
    33 lemma prefixI' [intro?]: "ys = xs @ z # zs ==> prefix xs ys"
    34   unfolding prefix_def prefixeq_def by blast
    35 
    36 lemma prefixE' [elim?]:
    37   assumes "prefix xs ys"
    38   obtains z zs where "ys = xs @ z # zs"
    39 proof -
    40   from `prefix xs ys` obtain us where "ys = xs @ us" and "xs \<noteq> ys"
    41     unfolding prefix_def prefixeq_def by blast
    42   with that show ?thesis by (auto simp add: neq_Nil_conv)
    43 qed
    44 
    45 lemma prefixI [intro?]: "prefixeq xs ys ==> xs \<noteq> ys ==> prefix xs ys"
    46   unfolding prefix_def by blast
    47 
    48 lemma prefixE [elim?]:
    49   fixes xs ys :: "'a list"
    50   assumes "prefix xs ys"
    51   obtains "prefixeq xs ys" and "xs \<noteq> ys"
    52   using assms unfolding prefix_def by blast
    53 
    54 
    55 subsection {* Basic properties of prefixes *}
    56 
    57 theorem Nil_prefixeq [iff]: "prefixeq [] xs"
    58   by (simp add: prefixeq_def)
    59 
    60 theorem prefixeq_Nil [simp]: "(prefixeq xs []) = (xs = [])"
    61   by (induct xs) (simp_all add: prefixeq_def)
    62 
    63 lemma prefixeq_snoc [simp]: "prefixeq xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefixeq xs ys"
    64 proof
    65   assume "prefixeq xs (ys @ [y])"
    66   then obtain zs where zs: "ys @ [y] = xs @ zs" ..
    67   show "xs = ys @ [y] \<or> prefixeq xs ys"
    68     by (metis append_Nil2 butlast_append butlast_snoc prefixeqI zs)
    69 next
    70   assume "xs = ys @ [y] \<or> prefixeq xs ys"
    71   then show "prefixeq xs (ys @ [y])"
    72     by (metis prefix_order.eq_iff prefix_order.order_trans prefixeqI)
    73 qed
    74 
    75 lemma Cons_prefixeq_Cons [simp]: "prefixeq (x # xs) (y # ys) = (x = y \<and> prefixeq xs ys)"
    76   by (auto simp add: prefixeq_def)
    77 
    78 lemma prefixeq_code [code]:
    79   "prefixeq [] xs \<longleftrightarrow> True"
    80   "prefixeq (x # xs) [] \<longleftrightarrow> False"
    81   "prefixeq (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefixeq xs ys"
    82   by simp_all
    83 
    84 lemma same_prefixeq_prefixeq [simp]: "prefixeq (xs @ ys) (xs @ zs) = prefixeq ys zs"
    85   by (induct xs) simp_all
    86 
    87 lemma same_prefixeq_nil [iff]: "prefixeq (xs @ ys) xs = (ys = [])"
    88   by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixeqI)
    89 
    90 lemma prefixeq_prefixeq [simp]: "prefixeq xs ys ==> prefixeq xs (ys @ zs)"
    91   by (metis prefix_order.le_less_trans prefixeqI prefixE prefixI)
    92 
    93 lemma append_prefixeqD: "prefixeq (xs @ ys) zs \<Longrightarrow> prefixeq xs zs"
    94   by (auto simp add: prefixeq_def)
    95 
    96 theorem prefixeq_Cons: "prefixeq xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefixeq zs ys))"
    97   by (cases xs) (auto simp add: prefixeq_def)
    98 
    99 theorem prefixeq_append:
   100   "prefixeq xs (ys @ zs) = (prefixeq xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefixeq us zs))"
   101   apply (induct zs rule: rev_induct)
   102    apply force
   103   apply (simp del: append_assoc add: append_assoc [symmetric])
   104   apply (metis append_eq_appendI)
   105   done
   106 
   107 lemma append_one_prefixeq:
   108   "prefixeq xs ys ==> length xs < length ys ==> prefixeq (xs @ [ys ! length xs]) ys"
   109   unfolding prefixeq_def
   110   by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj
   111     eq_Nil_appendI nth_drop')
   112 
   113 theorem prefixeq_length_le: "prefixeq xs ys ==> length xs \<le> length ys"
   114   by (auto simp add: prefixeq_def)
   115 
   116 lemma prefixeq_same_cases:
   117   "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"
   118   unfolding prefixeq_def by (metis append_eq_append_conv2)
   119 
   120 lemma set_mono_prefixeq: "prefixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys"
   121   by (auto simp add: prefixeq_def)
   122 
   123 lemma take_is_prefixeq: "prefixeq (take n xs) xs"
   124   unfolding prefixeq_def by (metis append_take_drop_id)
   125 
   126 lemma map_prefixeqI: "prefixeq xs ys \<Longrightarrow> prefixeq (map f xs) (map f ys)"
   127   by (auto simp: prefixeq_def)
   128 
   129 lemma prefixeq_length_less: "prefix xs ys \<Longrightarrow> length xs < length ys"
   130   by (auto simp: prefix_def prefixeq_def)
   131 
   132 lemma prefix_simps [simp, code]:
   133   "prefix xs [] \<longleftrightarrow> False"
   134   "prefix [] (x # xs) \<longleftrightarrow> True"
   135   "prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys"
   136   by (simp_all add: prefix_def cong: conj_cong)
   137 
   138 lemma take_prefix: "prefix xs ys \<Longrightarrow> prefix (take n xs) ys"
   139   apply (induct n arbitrary: xs ys)
   140    apply (case_tac ys, simp_all)[1]
   141   apply (metis prefix_order.less_trans prefixI take_is_prefixeq)
   142   done
   143 
   144 lemma not_prefixeq_cases:
   145   assumes pfx: "\<not> prefixeq ps ls"
   146   obtains
   147     (c1) "ps \<noteq> []" and "ls = []"
   148   | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> prefixeq as xs"
   149   | (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a"
   150 proof (cases ps)
   151   case Nil then show ?thesis using pfx by simp
   152 next
   153   case (Cons a as)
   154   note c = `ps = a#as`
   155   show ?thesis
   156   proof (cases ls)
   157     case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefixeq_nil)
   158   next
   159     case (Cons x xs)
   160     show ?thesis
   161     proof (cases "x = a")
   162       case True
   163       have "\<not> prefixeq as xs" using pfx c Cons True by simp
   164       with c Cons True show ?thesis by (rule c2)
   165     next
   166       case False
   167       with c Cons show ?thesis by (rule c3)
   168     qed
   169   qed
   170 qed
   171 
   172 lemma not_prefixeq_induct [consumes 1, case_names Nil Neq Eq]:
   173   assumes np: "\<not> prefixeq ps ls"
   174     and base: "\<And>x xs. P (x#xs) []"
   175     and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)"
   176     and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefixeq xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"
   177   shows "P ps ls" using np
   178 proof (induct ls arbitrary: ps)
   179   case Nil then show ?case
   180     by (auto simp: neq_Nil_conv elim!: not_prefixeq_cases intro!: base)
   181 next
   182   case (Cons y ys)
   183   then have npfx: "\<not> prefixeq ps (y # ys)" by simp
   184   then obtain x xs where pv: "ps = x # xs"
   185     by (rule not_prefixeq_cases) auto
   186   show ?case by (metis Cons.hyps Cons_prefixeq_Cons npfx pv r1 r2)
   187 qed
   188 
   189 
   190 subsection {* Parallel lists *}
   191 
   192 definition
   193   parallel :: "'a list => 'a list => bool"  (infixl "\<parallel>" 50) where
   194   "(xs \<parallel> ys) = (\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs)"
   195 
   196 lemma parallelI [intro]: "\<not> prefixeq xs ys ==> \<not> prefixeq ys xs ==> xs \<parallel> ys"
   197   unfolding parallel_def by blast
   198 
   199 lemma parallelE [elim]:
   200   assumes "xs \<parallel> ys"
   201   obtains "\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs"
   202   using assms unfolding parallel_def by blast
   203 
   204 theorem prefixeq_cases:
   205   obtains "prefixeq xs ys" | "prefix ys xs" | "xs \<parallel> ys"
   206   unfolding parallel_def prefix_def by blast
   207 
   208 theorem parallel_decomp:
   209   "xs \<parallel> ys ==> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
   210 proof (induct xs rule: rev_induct)
   211   case Nil
   212   then have False by auto
   213   then show ?case ..
   214 next
   215   case (snoc x xs)
   216   show ?case
   217   proof (rule prefixeq_cases)
   218     assume le: "prefixeq xs ys"
   219     then obtain ys' where ys: "ys = xs @ ys'" ..
   220     show ?thesis
   221     proof (cases ys')
   222       assume "ys' = []"
   223       then show ?thesis by (metis append_Nil2 parallelE prefixeqI snoc.prems ys)
   224     next
   225       fix c cs assume ys': "ys' = c # cs"
   226       then show ?thesis
   227         by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixeqI
   228           same_prefixeq_prefixeq snoc.prems ys)
   229     qed
   230   next
   231     assume "prefix ys xs" then have "prefixeq ys (xs @ [x])" by (simp add: prefix_def)
   232     with snoc have False by blast
   233     then show ?thesis ..
   234   next
   235     assume "xs \<parallel> ys"
   236     with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c"
   237       and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
   238       by blast
   239     from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
   240     with neq ys show ?thesis by blast
   241   qed
   242 qed
   243 
   244 lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
   245   apply (rule parallelI)
   246     apply (erule parallelE, erule conjE,
   247       induct rule: not_prefixeq_induct, simp+)+
   248   done
   249 
   250 lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y"
   251   by (simp add: parallel_append)
   252 
   253 lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"
   254   unfolding parallel_def by auto
   255 
   256 
   257 subsection {* Suffix order on lists *}
   258 
   259 definition
   260   suffixeq :: "'a list => 'a list => bool" where
   261   "suffixeq xs ys = (\<exists>zs. ys = zs @ xs)"
   262 
   263 definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
   264   "suffix xs ys \<equiv> \<exists>us. ys = us @ xs \<and> us \<noteq> []"
   265 
   266 lemma suffix_imp_suffixeq:
   267   "suffix xs ys \<Longrightarrow> suffixeq xs ys"
   268   by (auto simp: suffixeq_def suffix_def)
   269 
   270 lemma suffixeqI [intro?]: "ys = zs @ xs ==> suffixeq xs ys"
   271   unfolding suffixeq_def by blast
   272 
   273 lemma suffixeqE [elim?]:
   274   assumes "suffixeq xs ys"
   275   obtains zs where "ys = zs @ xs"
   276   using assms unfolding suffixeq_def by blast
   277 
   278 lemma suffixeq_refl [iff]: "suffixeq xs xs"
   279   by (auto simp add: suffixeq_def)
   280 lemma suffix_trans:
   281   "suffix xs ys \<Longrightarrow> suffix ys zs \<Longrightarrow> suffix xs zs"
   282   by (auto simp: suffix_def)
   283 lemma suffixeq_trans: "\<lbrakk>suffixeq xs ys; suffixeq ys zs\<rbrakk> \<Longrightarrow> suffixeq xs zs"
   284   by (auto simp add: suffixeq_def)
   285 lemma suffixeq_antisym: "\<lbrakk>suffixeq xs ys; suffixeq ys xs\<rbrakk> \<Longrightarrow> xs = ys"
   286   by (auto simp add: suffixeq_def)
   287 
   288 lemma suffixeq_tl [simp]: "suffixeq (tl xs) xs"
   289   by (induct xs) (auto simp: suffixeq_def)
   290 
   291 lemma suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> suffix (tl xs) xs"
   292   by (induct xs) (auto simp: suffix_def)
   293 
   294 lemma Nil_suffixeq [iff]: "suffixeq [] xs"
   295   by (simp add: suffixeq_def)
   296 lemma suffixeq_Nil [simp]: "(suffixeq xs []) = (xs = [])"
   297   by (auto simp add: suffixeq_def)
   298 
   299 lemma suffixeq_ConsI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (y#ys)"
   300   by (auto simp add: suffixeq_def)
   301 lemma suffixeq_ConsD: "suffixeq (x#xs) ys \<Longrightarrow> suffixeq xs ys"
   302   by (auto simp add: suffixeq_def)
   303 
   304 lemma suffixeq_appendI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (zs @ ys)"
   305   by (auto simp add: suffixeq_def)
   306 lemma suffixeq_appendD: "suffixeq (zs @ xs) ys \<Longrightarrow> suffixeq xs ys"
   307   by (auto simp add: suffixeq_def)
   308 
   309 lemma suffix_set_subset:
   310   "suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffix_def)
   311 
   312 lemma suffixeq_set_subset:
   313   "suffixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffixeq_def)
   314 
   315 lemma suffixeq_ConsD2: "suffixeq (x#xs) (y#ys) ==> suffixeq xs ys"
   316 proof -
   317   assume "suffixeq (x#xs) (y#ys)"
   318   then obtain zs where "y#ys = zs @ x#xs" ..
   319   then show ?thesis
   320     by (induct zs) (auto intro!: suffixeq_appendI suffixeq_ConsI)
   321 qed
   322 
   323 lemma suffixeq_to_prefixeq [code]: "suffixeq xs ys \<longleftrightarrow> prefixeq (rev xs) (rev ys)"
   324 proof
   325   assume "suffixeq xs ys"
   326   then obtain zs where "ys = zs @ xs" ..
   327   then have "rev ys = rev xs @ rev zs" by simp
   328   then show "prefixeq (rev xs) (rev ys)" ..
   329 next
   330   assume "prefixeq (rev xs) (rev ys)"
   331   then obtain zs where "rev ys = rev xs @ zs" ..
   332   then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp
   333   then have "ys = rev zs @ xs" by simp
   334   then show "suffixeq xs ys" ..
   335 qed
   336 
   337 lemma distinct_suffixeq: "distinct ys \<Longrightarrow> suffixeq xs ys \<Longrightarrow> distinct xs"
   338   by (clarsimp elim!: suffixeqE)
   339 
   340 lemma suffixeq_map: "suffixeq xs ys \<Longrightarrow> suffixeq (map f xs) (map f ys)"
   341   by (auto elim!: suffixeqE intro: suffixeqI)
   342 
   343 lemma suffixeq_drop: "suffixeq (drop n as) as"
   344   unfolding suffixeq_def
   345   apply (rule exI [where x = "take n as"])
   346   apply simp
   347   done
   348 
   349 lemma suffixeq_take: "suffixeq xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs"
   350   by (clarsimp elim!: suffixeqE)
   351 
   352 lemma suffixeq_suffix_reflclp_conv:
   353   "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" by (auto simp: suffixeq_def suffix_def)
   361   qed
   362 next
   363   fix xs ys :: "'a list"
   364   assume "suffix\<^sup>=\<^sup>= xs ys"
   365   thus "suffixeq xs ys"
   366   proof
   367     assume "suffix xs ys" thus "suffixeq xs ys" by (rule suffix_imp_suffixeq)
   368   next
   369     assume "xs = ys" thus "suffixeq xs ys" by (auto simp: suffixeq_def)
   370   qed
   371 qed
   372 
   373 lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefixeq x y"
   374   by blast
   375 
   376 lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefixeq y x"
   377   by blast
   378 
   379 lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
   380   unfolding parallel_def by simp
   381 
   382 lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
   383   unfolding parallel_def by simp
   384 
   385 lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
   386   by auto
   387 
   388 lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
   389   by (metis Cons_prefixeq_Cons parallelE parallelI)
   390 
   391 lemma not_equal_is_parallel:
   392   assumes neq: "xs \<noteq> ys"
   393     and len: "length xs = length ys"
   394   shows "xs \<parallel> ys"
   395   using len neq
   396 proof (induct rule: list_induct2)
   397   case Nil
   398   then show ?case by simp
   399 next
   400   case (Cons a as b bs)
   401   have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact
   402   show ?case
   403   proof (cases "a = b")
   404     case True
   405     then have "as \<noteq> bs" using Cons by simp
   406     then show ?thesis by (rule Cons_parallelI2 [OF True ih])
   407   next
   408     case False
   409     then show ?thesis by (rule Cons_parallelI1)
   410   qed
   411 qed
   412 
   413 lemma suffix_reflclp_conv:
   414   "suffix\<^sup>=\<^sup>= = suffixeq"
   415   by (intro ext) (auto simp: suffixeq_def suffix_def)
   416 
   417 lemma suffix_lists:
   418   "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A"
   419   unfolding suffix_def by auto
   420 
   421 
   422 subsection {* Embedding on lists *}
   423 
   424 inductive
   425   emb :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
   426   for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
   427 where
   428   emb_Nil [intro, simp]: "emb P [] ys"
   429 | emb_Cons [intro] : "emb P xs ys \<Longrightarrow> emb P xs (y#ys)"
   430 | emb_Cons2 [intro]: "P x y \<Longrightarrow> emb P xs ys \<Longrightarrow> emb P (x#xs) (y#ys)"
   431 
   432 lemma emb_Nil2 [simp]:
   433   assumes "emb P xs []" shows "xs = []"
   434   using assms by (cases rule: emb.cases) auto
   435 
   436 lemma emb_append2 [intro]:
   437   "emb P xs ys \<Longrightarrow> emb P xs (zs @ ys)"
   438   by (induct zs) auto
   439 
   440 lemma emb_prefix [intro]:
   441   assumes "emb P xs ys" shows "emb P xs (ys @ zs)"
   442   using assms
   443   by (induct arbitrary: zs) auto
   444 
   445 lemma emb_ConsD:
   446   assumes "emb P (x#xs) ys"
   447   shows "\<exists>us v vs. ys = us @ v # vs \<and> P x v \<and> emb P xs vs"
   448 using assms
   449 proof (induct x\<equiv>"x#xs" y\<equiv>"ys" arbitrary: x xs ys)
   450   case emb_Cons thus ?case by (metis append_Cons)
   451 next
   452   case (emb_Cons2 x y xs ys)
   453   thus ?case by (cases xs) (auto, blast+)
   454 qed
   455 
   456 lemma emb_appendD:
   457   assumes "emb P (xs @ ys) zs"
   458   shows "\<exists>us vs. zs = us @ vs \<and> emb P xs us \<and> emb P ys vs"
   459 using assms
   460 proof (induction xs arbitrary: ys zs)
   461   case Nil thus ?case by auto
   462 next
   463   case (Cons x xs)
   464   then obtain us v vs where "zs = us @ v # vs"
   465     and "P x v" and "emb P (xs @ ys) vs" by (auto dest: emb_ConsD)
   466   with Cons show ?case by (metis append_Cons append_assoc emb_Cons2 emb_append2)
   467 qed
   468 
   469 lemma emb_suffix:
   470   assumes "emb P xs ys" and "suffix ys zs"
   471   shows "emb P xs zs"
   472   using assms(2) and emb_append2 [OF assms(1)] by (auto simp: suffix_def)
   473 
   474 lemma emb_suffixeq:
   475   assumes "emb P xs ys" and "suffixeq ys zs"
   476   shows "emb P xs zs"
   477   using assms and emb_suffix unfolding suffixeq_suffix_reflclp_conv by auto
   478 
   479 lemma emb_length: "emb P xs ys \<Longrightarrow> length xs \<le> length ys"
   480   by (induct rule: emb.induct) auto
   481 
   482 end