(*  Title:      HOL/Library/Sublist.thy

    Author:     Tobias Nipkow and Markus Wenzel, TU Muenchen

*)

header {* List prefixes, suffixes, and embedding*}

theory Sublist

imports List Main

begin

subsection {* Prefix order on lists *}

definition prefixeq :: "'a list => 'a list => bool" where

  "prefixeq xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"

definition prefix :: "'a list => 'a list => bool" where

  "prefix xs ys \<longleftrightarrow> prefixeq xs ys \<and> xs \<noteq> ys"

interpretation prefix_order: order prefixeq prefix

  by default (auto simp: prefixeq_def prefix_def)

interpretation prefix_bot: bot prefixeq prefix Nil

  by default (simp add: prefixeq_def)

lemma prefixeqI [intro?]: "ys = xs @ zs ==> prefixeq xs ys"

  unfolding prefixeq_def by blast

lemma prefixeqE [elim?]:

  assumes "prefixeq xs ys"

  obtains zs where "ys = xs @ zs"

  using assms unfolding prefixeq_def by blast

lemma prefixI' [intro?]: "ys = xs @ z # zs ==> prefix xs ys"

  unfolding prefix_def prefixeq_def by blast

lemma prefixE' [elim?]:

  assumes "prefix xs ys"

  obtains z zs where "ys = xs @ z # zs"

proof -

  from `prefix xs ys` obtain us where "ys = xs @ us" and "xs \<noteq> ys"

    unfolding prefix_def prefixeq_def by blast

  with that show ?thesis by (auto simp add: neq_Nil_conv)

qed

lemma prefixI [intro?]: "prefixeq xs ys ==> xs \<noteq> ys ==> prefix xs ys"

  unfolding prefix_def by blast

lemma prefixE [elim?]:

  fixes xs ys :: "'a list"

  assumes "prefix xs ys"

  obtains "prefixeq xs ys" and "xs \<noteq> ys"

  using assms unfolding prefix_def by blast

subsection {* Basic properties of prefixes *}

theorem Nil_prefixeq [iff]: "prefixeq [] xs"

  by (simp add: prefixeq_def)

theorem prefixeq_Nil [simp]: "(prefixeq xs []) = (xs = [])"

  by (induct xs) (simp_all add: prefixeq_def)

lemma prefixeq_snoc [simp]: "prefixeq xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefixeq xs ys"

proof

  assume "prefixeq xs (ys @ [y])"

  then obtain zs where zs: "ys @ [y] = xs @ zs" ..

  show "xs = ys @ [y] \<or> prefixeq xs ys"

    by (metis append_Nil2 butlast_append butlast_snoc prefixeqI zs)

next

  assume "xs = ys @ [y] \<or> prefixeq xs ys"

  then show "prefixeq xs (ys @ [y])"

    by (metis prefix_order.eq_iff prefix_order.order_trans prefixeqI)

qed

lemma Cons_prefixeq_Cons [simp]: "prefixeq (x # xs) (y # ys) = (x = y \<and> prefixeq xs ys)"

  by (auto simp add: prefixeq_def)

lemma prefixeq_code [code]:

  "prefixeq [] xs \<longleftrightarrow> True"

  "prefixeq (x # xs) [] \<longleftrightarrow> False"

  "prefixeq (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefixeq xs ys"

  by simp_all

lemma same_prefixeq_prefixeq [simp]: "prefixeq (xs @ ys) (xs @ zs) = prefixeq ys zs"

  by (induct xs) simp_all

lemma same_prefixeq_nil [iff]: "prefixeq (xs @ ys) xs = (ys = [])"

  by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixeqI)

lemma prefixeq_prefixeq [simp]: "prefixeq xs ys ==> prefixeq xs (ys @ zs)"

  by (metis prefix_order.le_less_trans prefixeqI prefixE prefixI)

lemma append_prefixeqD: "prefixeq (xs @ ys) zs \<Longrightarrow> prefixeq xs zs"

  by (auto simp add: prefixeq_def)

theorem prefixeq_Cons: "prefixeq xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefixeq zs ys))"

  by (cases xs) (auto simp add: prefixeq_def)

theorem prefixeq_append:

  "prefixeq xs (ys @ zs) = (prefixeq xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefixeq us zs))"

  apply (induct zs rule: rev_induct)

   apply force

  apply (simp del: append_assoc add: append_assoc [symmetric])

  apply (metis append_eq_appendI)

  done

lemma append_one_prefixeq:

  "prefixeq xs ys ==> length xs < length ys ==> prefixeq (xs @ [ys ! length xs]) ys"

  unfolding prefixeq_def

  by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj

    eq_Nil_appendI nth_drop')

theorem prefixeq_length_le: "prefixeq xs ys ==> length xs \<le> length ys"

  by (auto simp add: prefixeq_def)

lemma prefixeq_same_cases:

  "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"

  unfolding prefixeq_def by (metis append_eq_append_conv2)

lemma set_mono_prefixeq: "prefixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys"

  by (auto simp add: prefixeq_def)

lemma take_is_prefixeq: "prefixeq (take n xs) xs"

  unfolding prefixeq_def by (metis append_take_drop_id)

lemma map_prefixeqI: "prefixeq xs ys \<Longrightarrow> prefixeq (map f xs) (map f ys)"

  by (auto simp: prefixeq_def)

lemma prefixeq_length_less: "prefix xs ys \<Longrightarrow> length xs < length ys"

  by (auto simp: prefix_def prefixeq_def)

lemma prefix_simps [simp, code]:

  "prefix xs [] \<longleftrightarrow> False"

  "prefix [] (x # xs) \<longleftrightarrow> True"

  "prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys"

  by (simp_all add: prefix_def cong: conj_cong)

lemma take_prefix: "prefix xs ys \<Longrightarrow> prefix (take n xs) ys"

  apply (induct n arbitrary: xs ys)

   apply (case_tac ys, simp_all)[1]

  apply (metis prefix_order.less_trans prefixI take_is_prefixeq)

  done

lemma not_prefixeq_cases:

  assumes pfx: "\<not> prefixeq ps ls"

  obtains

    (c1) "ps \<noteq> []" and "ls = []"

  | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> prefixeq as xs"

  | (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a"

proof (cases ps)

  case Nil then show ?thesis using pfx by simp

next

  case (Cons a as)

  note c = `ps = a#as`

  show ?thesis

  proof (cases ls)

    case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefixeq_nil)

  next

    case (Cons x xs)

    show ?thesis

    proof (cases "x = a")

      case True

      have "\<not> prefixeq as xs" using pfx c Cons True by simp

      with c Cons True show ?thesis by (rule c2)

    next

      case False

      with c Cons show ?thesis by (rule c3)

    qed

  qed

qed

lemma not_prefixeq_induct [consumes 1, case_names Nil Neq Eq]:

  assumes np: "\<not> prefixeq ps ls"

    and base: "\<And>x xs. P (x#xs) []"

    and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)"

    and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefixeq xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"

  shows "P ps ls" using np

proof (induct ls arbitrary: ps)

  case Nil then show ?case

    by (auto simp: neq_Nil_conv elim!: not_prefixeq_cases intro!: base)

next

  case (Cons y ys)

  then have npfx: "\<not> prefixeq ps (y # ys)" by simp

  then obtain x xs where pv: "ps = x # xs"

    by (rule not_prefixeq_cases) auto

  show ?case by (metis Cons.hyps Cons_prefixeq_Cons npfx pv r1 r2)

qed

subsection {* Parallel lists *}

definition

  parallel :: "'a list => 'a list => bool"  (infixl "\<parallel>" 50) where

  "(xs \<parallel> ys) = (\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs)"

lemma parallelI [intro]: "\<not> prefixeq xs ys ==> \<not> prefixeq ys xs ==> xs \<parallel> ys"

  unfolding parallel_def by blast

lemma parallelE [elim]:

  assumes "xs \<parallel> ys"

  obtains "\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs"

  using assms unfolding parallel_def by blast

theorem prefixeq_cases:

  obtains "prefixeq xs ys" | "prefix ys xs" | "xs \<parallel> ys"

  unfolding parallel_def prefix_def by blast

theorem parallel_decomp:

  "xs \<parallel> ys ==> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"

proof (induct xs rule: rev_induct)

  case Nil

  then have False by auto

  then show ?case ..

next

  case (snoc x xs)

  show ?case

  proof (rule prefixeq_cases)

    assume le: "prefixeq xs ys"

    then obtain ys' where ys: "ys = xs @ ys'" ..

    show ?thesis

    proof (cases ys')

      assume "ys' = []"

      then show ?thesis by (metis append_Nil2 parallelE prefixeqI snoc.prems ys)

    next

      fix c cs assume ys': "ys' = c # cs"

      then show ?thesis

        by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixeqI

          same_prefixeq_prefixeq snoc.prems ys)

    qed

  next

    assume "prefix ys xs" then have "prefixeq ys (xs @ [x])" by (simp add: prefix_def)

    with snoc have False by blast

    then show ?thesis ..

  next

    assume "xs \<parallel> ys"

    with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c"

      and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"

      by blast

    from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp

    with neq ys show ?thesis by blast

  qed

qed

lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"

  apply (rule parallelI)

    apply (erule parallelE, erule conjE,

      induct rule: not_prefixeq_induct, simp+)+

  done

lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y"

  by (simp add: parallel_append)

lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"

  unfolding parallel_def by auto

subsection {* Suffix order on lists *}

definition

  suffixeq :: "'a list => 'a list => bool" where

  "suffixeq xs ys = (\<exists>zs. ys = zs @ xs)"

definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where

  "suffix xs ys \<equiv> \<exists>us. ys = us @ xs \<and> us \<noteq> []"

lemma suffix_imp_suffixeq:

  "suffix xs ys \<Longrightarrow> suffixeq xs ys"

  by (auto simp: suffixeq_def suffix_def)

lemma suffixeqI [intro?]: "ys = zs @ xs ==> suffixeq xs ys"

  unfolding suffixeq_def by blast

lemma suffixeqE [elim?]:

  assumes "suffixeq xs ys"

  obtains zs where "ys = zs @ xs"

  using assms unfolding suffixeq_def by blast

lemma suffixeq_refl [iff]: "suffixeq xs xs"

  by (auto simp add: suffixeq_def)

lemma suffix_trans:

  "suffix xs ys \<Longrightarrow> suffix ys zs \<Longrightarrow> suffix xs zs"

  by (auto simp: suffix_def)

lemma suffixeq_trans: "\<lbrakk>suffixeq xs ys; suffixeq ys zs\<rbrakk> \<Longrightarrow> suffixeq xs zs"

  by (auto simp add: suffixeq_def)

lemma suffixeq_antisym: "\<lbrakk>suffixeq xs ys; suffixeq ys xs\<rbrakk> \<Longrightarrow> xs = ys"

  by (auto simp add: suffixeq_def)

lemma suffixeq_tl [simp]: "suffixeq (tl xs) xs"

  by (induct xs) (auto simp: suffixeq_def)

lemma suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> suffix (tl xs) xs"

  by (induct xs) (auto simp: suffix_def)

lemma Nil_suffixeq [iff]: "suffixeq [] xs"

  by (simp add: suffixeq_def)

lemma suffixeq_Nil [simp]: "(suffixeq xs []) = (xs = [])"

  by (auto simp add: suffixeq_def)

lemma suffixeq_ConsI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (y#ys)"

  by (auto simp add: suffixeq_def)

lemma suffixeq_ConsD: "suffixeq (x#xs) ys \<Longrightarrow> suffixeq xs ys"

  by (auto simp add: suffixeq_def)

lemma suffixeq_appendI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (zs @ ys)"

  by (auto simp add: suffixeq_def)

lemma suffixeq_appendD: "suffixeq (zs @ xs) ys \<Longrightarrow> suffixeq xs ys"

  by (auto simp add: suffixeq_def)

lemma suffix_set_subset:

  "suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffix_def)

lemma suffixeq_set_subset:

  "suffixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffixeq_def)

lemma suffixeq_ConsD2: "suffixeq (x#xs) (y#ys) ==> suffixeq xs ys"

proof -

  assume "suffixeq (x#xs) (y#ys)"

  then obtain zs where "y#ys = zs @ x#xs" ..

  then show ?thesis

    by (induct zs) (auto intro!: suffixeq_appendI suffixeq_ConsI)

qed

lemma suffixeq_to_prefixeq [code]: "suffixeq xs ys \<longleftrightarrow> prefixeq (rev xs) (rev ys)"

proof

  assume "suffixeq xs ys"

  then obtain zs where "ys = zs @ xs" ..

  then have "rev ys = rev xs @ rev zs" by simp

  then show "prefixeq (rev xs) (rev ys)" ..

next

  assume "prefixeq (rev xs) (rev ys)"

  then obtain zs where "rev ys = rev xs @ zs" ..

  then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp

  then have "ys = rev zs @ xs" by simp

  then show "suffixeq xs ys" ..

qed

lemma distinct_suffixeq: "distinct ys \<Longrightarrow> suffixeq xs ys \<Longrightarrow> distinct xs"

  by (clarsimp elim!: suffixeqE)

lemma suffixeq_map: "suffixeq xs ys \<Longrightarrow> suffixeq (map f xs) (map f ys)"

  by (auto elim!: suffixeqE intro: suffixeqI)

lemma suffixeq_drop: "suffixeq (drop n as) as"

  unfolding suffixeq_def

  apply (rule exI [where x = "take n as"])

  apply simp

  done

lemma suffixeq_take: "suffixeq xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs"

  by (clarsimp elim!: suffixeqE)

lemma suffixeq_suffix_reflclp_conv:

  "suffixeq = suffix\<^sup>=\<^sup>="

proof (intro ext iffI)

  fix xs ys :: "'a list"

  assume "suffixeq xs ys"

  show "suffix\<^sup>=\<^sup>= xs ys"

  proof

    assume "xs \<noteq> ys"

    with `suffixeq xs ys` show "suffix xs ys" by (auto simp: suffixeq_def suffix_def)

  qed

next

  fix xs ys :: "'a list"

  assume "suffix\<^sup>=\<^sup>= xs ys"

  thus "suffixeq xs ys"

  proof

    assume "suffix xs ys" thus "suffixeq xs ys" by (rule suffix_imp_suffixeq)

  next

    assume "xs = ys" thus "suffixeq xs ys" by (auto simp: suffixeq_def)

  qed

qed

lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefixeq x y"

  by blast

lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefixeq y x"

  by blast

lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"

  unfolding parallel_def by simp

lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"

  unfolding parallel_def by simp

lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"

  by auto

lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"

  by (metis Cons_prefixeq_Cons parallelE parallelI)

lemma not_equal_is_parallel:

  assumes neq: "xs \<noteq> ys"

    and len: "length xs = length ys"

  shows "xs \<parallel> ys"

  using len neq

proof (induct rule: list_induct2)

  case Nil

  then show ?case by simp

next

  case (Cons a as b bs)

  have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact

  show ?case

  proof (cases "a = b")

    case True

    then have "as \<noteq> bs" using Cons by simp

    then show ?thesis by (rule Cons_parallelI2 [OF True ih])

  next

    case False

    then show ?thesis by (rule Cons_parallelI1)

  qed

qed

lemma suffix_reflclp_conv:

  "suffix\<^sup>=\<^sup>= = suffixeq"

  by (intro ext) (auto simp: suffixeq_def suffix_def)

lemma suffix_lists:

  "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A"

  unfolding suffix_def by auto

subsection {* Embedding on lists *}

inductive

  emb :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"

  for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"

where

  emb_Nil [intro, simp]: "emb P [] ys"

| emb_Cons [intro] : "emb P xs ys \<Longrightarrow> emb P xs (y#ys)"

| emb_Cons2 [intro]: "P x y \<Longrightarrow> emb P xs ys \<Longrightarrow> emb P (x#xs) (y#ys)"

lemma emb_Nil2 [simp]:

  assumes "emb P xs []" shows "xs = []"

  using assms by (cases rule: emb.cases) auto

lemma emb_Cons_Nil [simp]:

  "emb P (x#xs) [] = False"

proof -

  { assume "emb P (x#xs) []"

    from emb_Nil2 [OF this] have False by simp

  } moreover {

    assume False

    hence "emb P (x#xs) []" by simp

  } ultimately show ?thesis by blast

qed

lemma emb_append2 [intro]:

  "emb P xs ys \<Longrightarrow> emb P xs (zs @ ys)"

  by (induct zs) auto

lemma emb_prefix [intro]:

  assumes "emb P xs ys" shows "emb P xs (ys @ zs)"

  using assms

  by (induct arbitrary: zs) auto

lemma emb_ConsD:

  assumes "emb P (x#xs) ys"

  shows "\<exists>us v vs. ys = us @ v # vs \<and> P x v \<and> emb P xs vs"

using assms

proof (induct x\<equiv>"x#xs" y\<equiv>"ys" arbitrary: x xs ys)

  case emb_Cons thus ?case by (metis append_Cons)

next

  case (emb_Cons2 x y xs ys)

  thus ?case by (cases xs) (auto, blast+)

qed

lemma emb_appendD:

  assumes "emb P (xs @ ys) zs"

  shows "\<exists>us vs. zs = us @ vs \<and> emb P xs us \<and> emb P ys vs"

using assms

proof (induction xs arbitrary: ys zs)

  case Nil thus ?case by auto

next

  case (Cons x xs)

  then obtain us v vs where "zs = us @ v # vs"

    and "P x v" and "emb P (xs @ ys) vs" by (auto dest: emb_ConsD)

  with Cons show ?case by (metis append_Cons append_assoc emb_Cons2 emb_append2)

qed

lemma emb_suffix:

  assumes "emb P xs ys" and "suffix ys zs"

  shows "emb P xs zs"

  using assms(2) and emb_append2 [OF assms(1)] by (auto simp: suffix_def)

lemma emb_suffixeq:

  assumes "emb P xs ys" and "suffixeq ys zs"

  shows "emb P xs zs"

  using assms and emb_suffix unfolding suffixeq_suffix_reflclp_conv by auto

lemma emb_length: "emb P xs ys \<Longrightarrow> length xs \<le> length ys"

  by (induct rule: emb.induct) auto

(*FIXME: move*)

definition transp_on :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where

  "transp_on P A \<equiv> \<forall>a\<in>A. \<forall>b\<in>A. \<forall>c\<in>A. P a b \<and> P b c \<longrightarrow> P a c"

lemma transp_onI [Pure.intro]:

  "(\<And>a b c. \<lbrakk>a \<in> A; b \<in> A; c \<in> A; P a b; P b c\<rbrakk> \<Longrightarrow> P a c) \<Longrightarrow> transp_on P A"

  unfolding transp_on_def by blast

lemma transp_on_emb:

  assumes "transp_on P A"

  shows "transp_on (emb P) (lists A)"

proof

  fix xs ys zs

  assume "emb P xs ys" and "emb P ys zs"

    and "xs \<in> lists A" and "ys \<in> lists A" and "zs \<in> lists A"

  thus "emb P xs zs"

  proof (induction arbitrary: zs)

    case emb_Nil show ?case by blast

  next

    case (emb_Cons xs ys y)

    from emb_ConsD [OF `emb P (y#ys) zs`] obtain us v vs

      where zs: "zs = us @ v # vs" and "P y v" and "emb P ys vs" by blast

    hence "emb P ys (v#vs)" by blast

    hence "emb P ys zs" unfolding zs by (rule emb_append2)

    from emb_Cons.IH [OF this] and emb_Cons.prems show ?case by simp

  next

    case (emb_Cons2 x y xs ys)

    from emb_ConsD [OF `emb P (y#ys) zs`] obtain us v vs

      where zs: "zs = us @ v # vs" and "P y v" and "emb P ys vs" by blast

    with emb_Cons2 have "emb P xs vs" by simp

    moreover have "P x v"

    proof -

      from zs and `zs \<in> lists A` have "v \<in> A" by auto

      moreover have "x \<in> A" and "y \<in> A" using emb_Cons2 by simp_all

      ultimately show ?thesis using `P x y` and `P y v` and assms

        unfolding transp_on_def by blast

    qed

    ultimately have "emb P (x#xs) (v#vs)" by blast

    thus ?case unfolding zs by (rule emb_append2)

  qed

qed

subsection {* Sublists (special case of embedding) *}

abbreviation sub :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where

  "sub xs ys \<equiv> emb (op =) xs ys"

lemma sub_Cons2: "sub xs ys \<Longrightarrow> sub (x#xs) (x#ys)" by auto

lemma sub_same_length:

  assumes "sub xs ys" and "length xs = length ys" shows "xs = ys"

  using assms by (induct) (auto dest: emb_length)

lemma not_sub_length [simp]: "length ys < length xs \<Longrightarrow> \<not> sub xs ys"

  by (metis emb_length linorder_not_less)

lemma [code]:

  "emb P [] ys \<longleftrightarrow> True"

  "emb P (x#xs) [] \<longleftrightarrow> False"

  by (simp_all)

lemma sub_Cons': "sub (x#xs) ys \<Longrightarrow> sub xs ys"

  by (induct xs) (auto dest: emb_ConsD)

lemma sub_Cons2':

  assumes "sub (x#xs) (x#ys)" shows "sub xs ys"

  using assms by (cases) (rule sub_Cons')

lemma sub_Cons2_neq:

  assumes "sub (x#xs) (y#ys)"

  shows "x \<noteq> y \<Longrightarrow> sub (x#xs) ys"

  using assms by (cases) auto

lemma sub_Cons2_iff [simp, code]:

  "sub (x#xs) (y#ys) = (if x = y then sub xs ys else sub (x#xs) ys)"

  by (metis emb_Cons emb_Cons2 [of "op =", OF refl] sub_Cons2' sub_Cons2_neq)

lemma sub_append': "sub (zs @ xs) (zs @ ys) \<longleftrightarrow> sub xs ys"

  by (induct zs) simp_all

lemma sub_refl [simp, intro!]: "sub xs xs" by (induct xs) simp_all

lemma sub_antisym:

  assumes "sub xs ys" and "sub ys xs"

  shows "xs = ys"

using assms

proof (induct)

  case emb_Nil

  from emb_Nil2 [OF this] show ?case by simp

next

  case emb_Cons2 thus ?case by simp

next

  case emb_Cons thus ?case

    by (metis sub_Cons' emb_length Suc_length_conv Suc_n_not_le_n)

qed

lemma transp_on_sub: "transp_on sub UNIV"

proof -

  have "transp_on (op =) UNIV" by (simp add: transp_on_def)

  from transp_on_emb [OF this] show ?thesis by simp

qed

lemma sub_trans: "sub xs ys \<Longrightarrow> sub ys zs \<Longrightarrow> sub xs zs"

  using transp_on_sub [unfolded transp_on_def] by blast

lemma sub_append_le_same_iff: "sub (xs @ ys) ys \<longleftrightarrow> xs = []"

  by (auto dest: emb_length)

lemma emb_append_mono:

  "\<lbrakk> emb P xs xs'; emb P ys ys' \<rbrakk> \<Longrightarrow> emb P (xs@ys) (xs'@ys')"

apply (induct rule: emb.induct)

  apply (metis eq_Nil_appendI emb_append2)

 apply (metis append_Cons emb_Cons)

by (metis append_Cons emb_Cons2)

subsection {* Appending elements *}

lemma sub_append [simp]:

  "sub (xs @ zs) (ys @ zs) \<longleftrightarrow> sub xs ys" (is "?l = ?r")

proof

  { fix xs' ys' xs ys zs :: "'a list" assume "sub xs' ys'"

    hence "xs' = xs @ zs & ys' = ys @ zs \<longrightarrow> sub xs ys"

    proof (induct arbitrary: xs ys zs)

      case emb_Nil show ?case by simp

    next

      case (emb_Cons xs' ys' x)

      { assume "ys=[]" hence ?case using emb_Cons(1) by auto }

      moreover

      { fix us assume "ys = x#us"

        hence ?case using emb_Cons(2) by(simp add: emb.emb_Cons) }

      ultimately show ?case by (auto simp:Cons_eq_append_conv)

    next

      case (emb_Cons2 x y xs' ys')

      { assume "xs=[]" hence ?case using emb_Cons2(1) by auto }

      moreover

      { fix us vs assume "xs=x#us" "ys=x#vs" hence ?case using emb_Cons2 by auto}

      moreover

      { fix us assume "xs=x#us" "ys=[]" hence ?case using emb_Cons2(2) by bestsimp }

      ultimately show ?case using `x = y` by (auto simp: Cons_eq_append_conv)

    qed }

  moreover assume ?l

  ultimately show ?r by blast

next

  assume ?r thus ?l by (metis emb_append_mono sub_refl)

qed

lemma sub_drop_many: "sub xs ys \<Longrightarrow> sub xs (zs @ ys)"

  by (induct zs) auto

lemma sub_rev_drop_many: "sub xs ys \<Longrightarrow> sub xs (ys @ zs)"

  by (metis append_Nil2 emb_Nil emb_append_mono)

subsection {* Relation to standard list operations *}

lemma sub_map:

  assumes "sub xs ys" shows "sub (map f xs) (map f ys)"

  using assms by (induct) auto

lemma sub_filter_left [simp]: "sub (filter P xs) xs"

  by (induct xs) auto

lemma sub_filter [simp]:

  assumes "sub xs ys" shows "sub (filter P xs) (filter P ys)"

  using assms by (induct) auto

lemma "sub xs ys \<longleftrightarrow> (\<exists> N. xs = sublist ys N)" (is "?L = ?R")

proof

  assume ?L

  thus ?R

  proof (induct)

    case emb_Nil show ?case by (metis sublist_empty)

  next

    case (emb_Cons xs ys x)

    then obtain N where "xs = sublist ys N" by blast

    hence "xs = sublist (x#ys) (Suc ` N)"

      by (clarsimp simp add:sublist_Cons inj_image_mem_iff)

    thus ?case by blast

  next

    case (emb_Cons2 x y xs ys)

    then obtain N where "xs = sublist ys N" by blast

    hence "x#xs = sublist (x#ys) (insert 0 (Suc ` N))"

      by (clarsimp simp add:sublist_Cons inj_image_mem_iff)

    thus ?case unfolding `x = y` by blast

  qed

next

  assume ?R

  then obtain N where "xs = sublist ys N" ..

  moreover have "sub (sublist ys N) ys"

  proof (induct ys arbitrary:N)

    case Nil show ?case by simp

  next

    case Cons thus ?case by (auto simp: sublist_Cons)

  qed

  ultimately show ?L by simp

qed

end