src/HOL/Library/Sublist.thy
author wenzelm
Wed, 08 Mar 2017 10:50:59 +0100
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child 65869 a6ed757b8585
permissions -rw-r--r--
tuned proofs;
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(*  Title:      HOL/Library/Sublist.thy
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    Author:     Tobias Nipkow and Markus Wenzel, TU Muenchen
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    Author:     Christian Sternagel, JAIST
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*)
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section \<open>List prefixes, suffixes, and homeomorphic embedding\<close>
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theory Sublist
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imports Main
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begin
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subsection \<open>Prefix order on lists\<close>
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definition prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
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  where "prefix xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"
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definition strict_prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
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  where "strict_prefix xs ys \<longleftrightarrow> prefix xs ys \<and> xs \<noteq> ys"
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interpretation prefix_order: order prefix strict_prefix
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  by standard (auto simp: prefix_def strict_prefix_def)
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interpretation prefix_bot: order_bot Nil prefix strict_prefix
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  by standard (simp add: prefix_def)
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lemma prefixI [intro?]: "ys = xs @ zs \<Longrightarrow> prefix xs ys"
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  unfolding prefix_def by blast
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lemma prefixE [elim?]:
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  assumes "prefix xs ys"
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  obtains zs where "ys = xs @ zs"
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  using assms unfolding prefix_def by blast
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lemma strict_prefixI' [intro?]: "ys = xs @ z # zs \<Longrightarrow> strict_prefix xs ys"
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  unfolding strict_prefix_def prefix_def by blast
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lemma strict_prefixE' [elim?]:
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  assumes "strict_prefix xs ys"
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  obtains z zs where "ys = xs @ z # zs"
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proof -
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  from \<open>strict_prefix xs ys\<close> obtain us where "ys = xs @ us" and "xs \<noteq> ys"
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    unfolding strict_prefix_def prefix_def by blast
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  with that show ?thesis by (auto simp add: neq_Nil_conv)
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qed
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(* FIXME rm *)
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lemma strict_prefixI [intro?]: "prefix xs ys \<Longrightarrow> xs \<noteq> ys \<Longrightarrow> strict_prefix xs ys"
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by(fact prefix_order.le_neq_trans)
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lemma strict_prefixE [elim?]:
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  fixes xs ys :: "'a list"
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  assumes "strict_prefix xs ys"
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  obtains "prefix xs ys" and "xs \<noteq> ys"
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  using assms unfolding strict_prefix_def by blast
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subsection \<open>Basic properties of prefixes\<close>
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(* FIXME rm *)
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theorem Nil_prefix [iff]: "prefix [] xs"
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by(fact prefix_bot.bot_least)
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(* FIXME rm *)
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theorem prefix_Nil [simp]: "(prefix xs []) = (xs = [])"
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by(fact prefix_bot.bot_unique)
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lemma prefix_snoc [simp]: "prefix xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefix xs ys"
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proof
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  assume "prefix xs (ys @ [y])"
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  then obtain zs where zs: "ys @ [y] = xs @ zs" ..
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  show "xs = ys @ [y] \<or> prefix xs ys"
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    by (metis append_Nil2 butlast_append butlast_snoc prefixI zs)
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next
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  assume "xs = ys @ [y] \<or> prefix xs ys"
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  then show "prefix xs (ys @ [y])"
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    by (metis prefix_order.eq_iff prefix_order.order_trans prefixI)
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qed
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lemma Cons_prefix_Cons [simp]: "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)"
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  by (auto simp add: prefix_def)
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lemma prefix_code [code]:
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  "prefix [] xs \<longleftrightarrow> True"
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  "prefix (x # xs) [] \<longleftrightarrow> False"
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  "prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys"
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  by simp_all
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lemma same_prefix_prefix [simp]: "prefix (xs @ ys) (xs @ zs) = prefix ys zs"
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  by (induct xs) simp_all
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lemma same_prefix_nil [iff]: "prefix (xs @ ys) xs = (ys = [])"
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  by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixI)
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lemma prefix_prefix [simp]: "prefix xs ys \<Longrightarrow> prefix xs (ys @ zs)"
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  unfolding prefix_def by fastforce
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lemma append_prefixD: "prefix (xs @ ys) zs \<Longrightarrow> prefix xs zs"
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  by (auto simp add: prefix_def)
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theorem prefix_Cons: "prefix xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefix zs ys))"
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  by (cases xs) (auto simp add: prefix_def)
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theorem prefix_append:
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  "prefix xs (ys @ zs) = (prefix xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefix us zs))"
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  apply (induct zs rule: rev_induct)
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   apply force
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  apply (simp del: append_assoc add: append_assoc [symmetric])
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  apply (metis append_eq_appendI)
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  done
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lemma append_one_prefix:
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  "prefix xs ys \<Longrightarrow> length xs < length ys \<Longrightarrow> prefix (xs @ [ys ! length xs]) ys"
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  proof (unfold prefix_def)
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    assume a1: "\<exists>zs. ys = xs @ zs"
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    then obtain sk :: "'a list" where sk: "ys = xs @ sk" by fastforce
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    assume a2: "length xs < length ys"
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    have f1: "\<And>v. ([]::'a list) @ v = v" using append_Nil2 by simp
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    have "[] \<noteq> sk" using a1 a2 sk less_not_refl by force
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    hence "\<exists>v. xs @ hd sk # v = ys" using sk by (metis hd_Cons_tl)
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    thus "\<exists>zs. ys = (xs @ [ys ! length xs]) @ zs" using f1 by fastforce
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  qed
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theorem prefix_length_le: "prefix xs ys \<Longrightarrow> length xs \<le> length ys"
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  by (auto simp add: prefix_def)
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lemma prefix_same_cases:
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  "prefix (xs\<^sub>1::'a list) ys \<Longrightarrow> prefix xs\<^sub>2 ys \<Longrightarrow> prefix xs\<^sub>1 xs\<^sub>2 \<or> prefix xs\<^sub>2 xs\<^sub>1"
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  unfolding prefix_def by (force simp: append_eq_append_conv2)
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lemma prefix_length_prefix:
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  "prefix ps xs \<Longrightarrow> prefix qs xs \<Longrightarrow> length ps \<le> length qs \<Longrightarrow> prefix ps qs"
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by (auto simp: prefix_def) (metis append_Nil2 append_eq_append_conv_if)
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lemma set_mono_prefix: "prefix xs ys \<Longrightarrow> set xs \<subseteq> set ys"
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  by (auto simp add: prefix_def)
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lemma take_is_prefix: "prefix (take n xs) xs"
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   138
  unfolding prefix_def by (metis append_take_drop_id)
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   139
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lemma prefixeq_butlast: "prefix (butlast xs) xs"
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   141
by (simp add: butlast_conv_take take_is_prefix)
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   142
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   143
lemma map_prefixI: "prefix xs ys \<Longrightarrow> prefix (map f xs) (map f ys)"
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   144
  by (auto simp: prefix_def)
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   145
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lemma prefix_length_less: "strict_prefix xs ys \<Longrightarrow> length xs < length ys"
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   147
  by (auto simp: strict_prefix_def prefix_def)
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   148
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lemma prefix_snocD: "prefix (xs@[x]) ys \<Longrightarrow> strict_prefix xs ys"
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  by (simp add: strict_prefixI' prefix_order.dual_order.strict_trans1)
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   151
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lemma strict_prefix_simps [simp, code]:
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  "strict_prefix xs [] \<longleftrightarrow> False"
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   154
  "strict_prefix [] (x # xs) \<longleftrightarrow> True"
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  "strict_prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> strict_prefix xs ys"
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   156
  by (simp_all add: strict_prefix_def cong: conj_cong)
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   157
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   158
lemma take_strict_prefix: "strict_prefix xs ys \<Longrightarrow> strict_prefix (take n xs) ys"
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proof (induct n arbitrary: xs ys)
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   160
  case 0
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  then show ?case by (cases ys) simp_all
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   162
next
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  case (Suc n)
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  then show ?case by (metis prefix_order.less_trans strict_prefixI take_is_prefix)
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   165
qed
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   166
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lemma not_prefix_cases:
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  assumes pfx: "\<not> prefix ps ls"
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   169
  obtains
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    (c1) "ps \<noteq> []" and "ls = []"
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  | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> prefix as xs"
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  | (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a"
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   173
proof (cases ps)
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  case Nil
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  then show ?thesis using pfx by simp
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   176
next
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  case (Cons a as)
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  note c = \<open>ps = a#as\<close>
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  show ?thesis
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  proof (cases ls)
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    case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefix_nil)
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   182
  next
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    case (Cons x xs)
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    show ?thesis
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   185
    proof (cases "x = a")
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      case True
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      have "\<not> prefix as xs" using pfx c Cons True by simp
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   188
      with c Cons True show ?thesis by (rule c2)
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   189
    next
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      case False
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      with c Cons show ?thesis by (rule c3)
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   192
    qed
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  qed
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   194
qed
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   195
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   196
lemma not_prefix_induct [consumes 1, case_names Nil Neq Eq]:
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  assumes np: "\<not> prefix ps ls"
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   198
    and base: "\<And>x xs. P (x#xs) []"
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   199
    and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)"
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   200
    and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefix xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"
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   201
  shows "P ps ls" using np
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   202
proof (induct ls arbitrary: ps)
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  case Nil
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   204
  then show ?case
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   205
    by (auto simp: neq_Nil_conv elim!: not_prefix_cases intro!: base)
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   206
next
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   207
  case (Cons y ys)
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  then have npfx: "\<not> prefix ps (y # ys)" by simp
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   209
  then obtain x xs where pv: "ps = x # xs"
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   210
    by (rule not_prefix_cases) auto
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   211
  show ?case by (metis Cons.hyps Cons_prefix_Cons npfx pv r1 r2)
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   212
qed
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   213
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subsection \<open>Prefixes\<close>
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   216
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fun prefixes where
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   218
"prefixes [] = [[]]" |
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"prefixes (x#xs) = [] # map (op # x) (prefixes xs)"
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   220
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   221
lemma in_set_prefixes[simp]: "xs \<in> set (prefixes ys) \<longleftrightarrow> prefix xs ys"
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   222
proof (induct xs arbitrary: ys)
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   223
  case Nil
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   224
  then show ?case by (cases ys) auto
e690d6f2185b tuned proofs;
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   225
next
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   226
  case (Cons a xs)
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   227
  then show ?case by (cases ys) auto
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   228
qed
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   229
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   230
lemma length_prefixes[simp]: "length (prefixes xs) = length xs+1"
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   231
by (induction xs) auto
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   232
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   233
lemma prefixes_snoc[simp]:
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   234
  "prefixes (xs@[x]) = prefixes xs @ [xs@[x]]"
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   235
by (induction xs) auto
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   236
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   237
lemma prefixes_eq_Snoc:
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   238
  "prefixes ys = xs @ [x] \<longleftrightarrow>
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   239
  (ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = zs@[z] \<and> xs = prefixes zs)) \<and> x = ys"
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   240
by (cases ys rule: rev_cases) auto
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   241
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   242
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   243
subsection \<open>Longest Common Prefix\<close>
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   244
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   245
definition Longest_common_prefix :: "'a list set \<Rightarrow> 'a list" where
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   246
"Longest_common_prefix L = (GREATEST ps WRT length. \<forall>xs \<in> L. prefix ps xs)"
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   247
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   248
lemma Longest_common_prefix_ex: "L \<noteq> {} \<Longrightarrow>
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   249
  \<exists>ps. (\<forall>xs \<in> L. prefix ps xs) \<and> (\<forall>qs. (\<forall>xs \<in> L. prefix qs xs) \<longrightarrow> size qs \<le> size ps)"
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   250
  (is "_ \<Longrightarrow> \<exists>ps. ?P L ps")
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   251
proof(induction "LEAST n. \<exists>xs \<in>L. n = length xs" arbitrary: L)
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   252
  case 0
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   253
  have "[] : L" using "0.hyps" LeastI[of "\<lambda>n. \<exists>xs\<in>L. n = length xs"] \<open>L \<noteq> {}\<close>
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   254
    by auto
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   255
  hence "?P L []" by(auto)
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   256
  thus ?case ..
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diff changeset
   257
next
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   258
  case (Suc n)
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   259
  let ?EX = "\<lambda>n. \<exists>xs\<in>L. n = length xs"
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   260
  obtain x xs where xxs: "x#xs \<in> L" "size xs = n" using Suc.prems Suc.hyps(2)
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   261
    by(metis LeastI_ex[of ?EX] Suc_length_conv ex_in_conv)
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   262
  hence "[] \<notin> L" using Suc.hyps(2) by auto
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   263
  show ?case
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   264
  proof (cases "\<forall>xs \<in> L. \<exists>ys. xs = x#ys")
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   265
    case True
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   266
    let ?L = "{ys. x#ys \<in> L}"
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   267
    have 1: "(LEAST n. \<exists>xs \<in> ?L. n = length xs) = n"
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   268
      using xxs Suc.prems Suc.hyps(2) Least_le[of "?EX"]
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   269
      by - (rule Least_equality, fastforce+)
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   270
    have 2: "?L \<noteq> {}" using \<open>x # xs \<in> L\<close> by auto
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diff changeset
   271
    from Suc.hyps(1)[OF 1[symmetric] 2] obtain ps where IH: "?P ?L ps" ..
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   272
    { fix qs
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   273
      assume "\<forall>qs. (\<forall>xa. x # xa \<in> L \<longrightarrow> prefix qs xa) \<longrightarrow> length qs \<le> length ps"
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   274
      and "\<forall>xs\<in>L. prefix qs xs"
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   275
      hence "length (tl qs) \<le> length ps"
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   276
        by (metis Cons_prefix_Cons hd_Cons_tl list.sel(2) Nil_prefix) 
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   277
      hence "length qs \<le> Suc (length ps)" by auto
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diff changeset
   278
    }
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   279
    hence "?P L (x#ps)" using True IH by auto
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   280
    thus ?thesis ..
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   281
  next
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   282
    case False
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   283
    then obtain y ys where yys: "x\<noteq>y" "y#ys \<in> L" using \<open>[] \<notin> L\<close>
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   284
      by (auto) (metis list.exhaust)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   285
    have "\<forall>qs. (\<forall>xs\<in>L. prefix qs xs) \<longrightarrow> qs = []" using yys \<open>x#xs \<in> L\<close>
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   286
      by auto (metis Cons_prefix_Cons prefix_Cons)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   287
    hence "?P L []" by auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   288
    thus ?thesis ..
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   289
  qed
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   290
qed
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   291
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   292
lemma Longest_common_prefix_unique: "L \<noteq> {} \<Longrightarrow>
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   293
  \<exists>! ps. (\<forall>xs \<in> L. prefix ps xs) \<and> (\<forall>qs. (\<forall>xs \<in> L. prefix qs xs) \<longrightarrow> size qs \<le> size ps)"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   294
by(rule ex_ex1I[OF Longest_common_prefix_ex];
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   295
   meson equals0I prefix_length_prefix prefix_order.antisym)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   296
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   297
lemma Longest_common_prefix_eq:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   298
 "\<lbrakk> L \<noteq> {};  \<forall>xs \<in> L. prefix ps xs;
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   299
    \<forall>qs. (\<forall>xs \<in> L. prefix qs xs) \<longrightarrow> size qs \<le> size ps \<rbrakk>
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   300
  \<Longrightarrow> Longest_common_prefix L = ps"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   301
unfolding Longest_common_prefix_def GreatestM_def
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   302
by(rule some1_equality[OF Longest_common_prefix_unique]) auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   303
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   304
lemma Longest_common_prefix_prefix:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   305
  "xs \<in> L \<Longrightarrow> prefix (Longest_common_prefix L) xs"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   306
unfolding Longest_common_prefix_def GreatestM_def
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   307
by(rule someI2_ex[OF Longest_common_prefix_ex]) auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   308
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   309
lemma Longest_common_prefix_longest:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   310
  "L \<noteq> {} \<Longrightarrow> \<forall>xs\<in>L. prefix ps xs \<Longrightarrow> length ps \<le> length(Longest_common_prefix L)"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   311
unfolding Longest_common_prefix_def GreatestM_def
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   312
by(rule someI2_ex[OF Longest_common_prefix_ex]) auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   313
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   314
lemma Longest_common_prefix_max_prefix:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   315
  "L \<noteq> {} \<Longrightarrow> \<forall>xs\<in>L. prefix ps xs \<Longrightarrow> prefix ps (Longest_common_prefix L)"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   316
by(metis Longest_common_prefix_prefix Longest_common_prefix_longest
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   317
     prefix_length_prefix ex_in_conv)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   318
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   319
lemma Longest_common_prefix_Nil: "[] \<in> L \<Longrightarrow> Longest_common_prefix L = []"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   320
using Longest_common_prefix_prefix prefix_Nil by blast
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   321
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   322
lemma Longest_common_prefix_image_Cons: "L \<noteq> {} \<Longrightarrow>
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   323
  Longest_common_prefix (op # x ` L) = x # Longest_common_prefix L"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   324
apply(rule Longest_common_prefix_eq)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   325
  apply(simp)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   326
 apply (simp add: Longest_common_prefix_prefix)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   327
apply simp
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   328
by(metis Longest_common_prefix_longest[of L] Cons_prefix_Cons Nitpick.size_list_simp(2)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   329
     Suc_le_mono hd_Cons_tl order.strict_implies_order zero_less_Suc)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   330
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   331
lemma Longest_common_prefix_eq_Cons: assumes "L \<noteq> {}" "[] \<notin> L"  "\<forall>xs\<in>L. hd xs = x"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   332
shows "Longest_common_prefix L = x # Longest_common_prefix {ys. x#ys \<in> L}"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   333
proof -
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   334
  have "L = op # x ` {ys. x#ys \<in> L}" using assms(2,3)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   335
    by (auto simp: image_def)(metis hd_Cons_tl)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   336
  thus ?thesis
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   337
    by (metis Longest_common_prefix_image_Cons image_is_empty assms(1))
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   338
qed
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   339
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   340
lemma Longest_common_prefix_eq_Nil:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   341
  "\<lbrakk>x#ys \<in> L; y#zs \<in> L; x \<noteq> y \<rbrakk> \<Longrightarrow> Longest_common_prefix L = []"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   342
by (metis Longest_common_prefix_prefix list.inject prefix_Cons)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   343
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   344
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   345
fun longest_common_prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   346
"longest_common_prefix (x#xs) (y#ys) =
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   347
  (if x=y then x # longest_common_prefix xs ys else [])" |
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   348
"longest_common_prefix _ _ = []"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   349
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   350
lemma longest_common_prefix_prefix1:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   351
  "prefix (longest_common_prefix xs ys) xs"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   352
by(induction xs ys rule: longest_common_prefix.induct) auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   353
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   354
lemma longest_common_prefix_prefix2:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   355
  "prefix (longest_common_prefix xs ys) ys"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   356
by(induction xs ys rule: longest_common_prefix.induct) auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   357
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   358
lemma longest_common_prefix_max_prefix:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   359
  "\<lbrakk> prefix ps xs; prefix ps ys \<rbrakk>
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   360
   \<Longrightarrow> prefix ps (longest_common_prefix xs ys)"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   361
by(induction xs ys arbitrary: ps rule: longest_common_prefix.induct)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   362
  (auto simp: prefix_Cons)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   363
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   364
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   365
subsection \<open>Parallel lists\<close>
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   366
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   367
definition parallel :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "\<parallel>" 50)
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   368
  where "(xs \<parallel> ys) = (\<not> prefix xs ys \<and> \<not> prefix ys xs)"
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   369
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   370
lemma parallelI [intro]: "\<not> prefix xs ys \<Longrightarrow> \<not> prefix ys xs \<Longrightarrow> xs \<parallel> ys"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   371
  unfolding parallel_def by blast
10330
4362e906b745 "List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff changeset
   372
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   373
lemma parallelE [elim]:
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   374
  assumes "xs \<parallel> ys"
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   375
  obtains "\<not> prefix xs ys \<and> \<not> prefix ys xs"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   376
  using assms unfolding parallel_def by blast
10330
4362e906b745 "List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff changeset
   377
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   378
theorem prefix_cases:
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   379
  obtains "prefix xs ys" | "strict_prefix ys xs" | "xs \<parallel> ys"
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   380
  unfolding parallel_def strict_prefix_def by blast
10330
4362e906b745 "List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff changeset
   381
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   382
theorem parallel_decomp:
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   383
  "xs \<parallel> ys \<Longrightarrow> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   384
proof (induct xs rule: rev_induct)
11987
bf31b35949ce tuned induct proofs;
wenzelm
parents: 11780
diff changeset
   385
  case Nil
23254
99644a53f16d tuned proofs;
wenzelm
parents: 22178
diff changeset
   386
  then have False by auto
99644a53f16d tuned proofs;
wenzelm
parents: 22178
diff changeset
   387
  then show ?case ..
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   388
next
11987
bf31b35949ce tuned induct proofs;
wenzelm
parents: 11780
diff changeset
   389
  case (snoc x xs)
bf31b35949ce tuned induct proofs;
wenzelm
parents: 11780
diff changeset
   390
  show ?case
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   391
  proof (rule prefix_cases)
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   392
    assume le: "prefix xs ys"
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   393
    then obtain ys' where ys: "ys = xs @ ys'" ..
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   394
    show ?thesis
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   395
    proof (cases ys')
25564
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25356
diff changeset
   396
      assume "ys' = []"
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   397
      then show ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys)
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   398
    next
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   399
      fix c cs assume ys': "ys' = c # cs"
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   400
      have "x \<noteq> c" using snoc.prems ys ys' by fastforce
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   401
      thus "\<exists>as b bs c cs. b \<noteq> c \<and> xs @ [x] = as @ b # bs \<and> ys = as @ c # cs"
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   402
        using ys ys' by blast
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   403
    qed
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   404
  next
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   405
    assume "strict_prefix ys xs"
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   406
    then have "prefix ys (xs @ [x])" by (simp add: strict_prefix_def)
11987
bf31b35949ce tuned induct proofs;
wenzelm
parents: 11780
diff changeset
   407
    with snoc have False by blast
23254
99644a53f16d tuned proofs;
wenzelm
parents: 22178
diff changeset
   408
    then show ?thesis ..
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   409
  next
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   410
    assume "xs \<parallel> ys"
11987
bf31b35949ce tuned induct proofs;
wenzelm
parents: 11780
diff changeset
   411
    with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c"
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   412
      and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   413
      by blast
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   414
    from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   415
    with neq ys show ?thesis by blast
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   416
  qed
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   417
qed
10330
4362e906b745 "List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff changeset
   418
25564
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25356
diff changeset
   419
lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   420
  apply (rule parallelI)
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   421
    apply (erule parallelE, erule conjE,
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   422
      induct rule: not_prefix_induct, simp+)+
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   423
  done
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   424
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   425
lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y"
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   426
  by (simp add: parallel_append)
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   427
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   428
lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   429
  unfolding parallel_def by auto
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   430
25356
059c03630d6e tuned presentation;
wenzelm
parents: 25355
diff changeset
   431
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   432
subsection \<open>Suffix order on lists\<close>
17201
3bdf1dfcdee4 reactivate postfix by change of syntax;
wenzelm
parents: 15355
diff changeset
   433
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   434
definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   435
  where "suffix xs ys = (\<exists>zs. ys = zs @ xs)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   436
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   437
definition strict_suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   438
  where "strict_suffix xs ys \<longleftrightarrow> (\<exists>us. ys = us @ xs \<and> us \<noteq> [])"
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   439
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   440
lemma strict_suffix_imp_suffix:
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   441
  "strict_suffix xs ys \<Longrightarrow> suffix xs ys"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   442
  by (auto simp: suffix_def strict_suffix_def)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   443
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   444
lemma suffixI [intro?]: "ys = zs @ xs \<Longrightarrow> suffix xs ys"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   445
  unfolding suffix_def by blast
21305
d41eddfd2b66 tuned proofs;
wenzelm
parents: 19086
diff changeset
   446
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   447
lemma suffixE [elim?]:
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   448
  assumes "suffix xs ys"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   449
  obtains zs where "ys = zs @ xs"
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   450
  using assms unfolding suffix_def by blast
21305
d41eddfd2b66 tuned proofs;
wenzelm
parents: 19086
diff changeset
   451
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   452
lemma suffix_refl [iff]: "suffix xs xs"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   453
  by (auto simp add: suffix_def)
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   454
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   455
lemma suffix_trans:
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   456
  "suffix xs ys \<Longrightarrow> suffix ys zs \<Longrightarrow> suffix xs zs"
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   457
  by (auto simp: suffix_def)
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   458
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   459
lemma strict_suffix_trans:
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   460
  "\<lbrakk>strict_suffix xs ys; strict_suffix ys zs\<rbrakk> \<Longrightarrow> strict_suffix xs zs"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   461
by (auto simp add: strict_suffix_def)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   462
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   463
lemma suffix_antisym: "\<lbrakk>suffix xs ys; suffix ys xs\<rbrakk> \<Longrightarrow> xs = ys"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   464
  by (auto simp add: suffix_def)
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   465
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   466
lemma suffix_tl [simp]: "suffix (tl xs) xs"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   467
  by (induct xs) (auto simp: suffix_def)
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   468
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   469
lemma strict_suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> strict_suffix (tl xs) xs"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   470
  by (induct xs) (auto simp: strict_suffix_def)
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   471
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   472
lemma Nil_suffix [iff]: "suffix [] xs"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   473
  by (simp add: suffix_def)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   474
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   475
lemma suffix_Nil [simp]: "(suffix xs []) = (xs = [])"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   476
  by (auto simp add: suffix_def)
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   477
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   478
lemma suffix_ConsI: "suffix xs ys \<Longrightarrow> suffix xs (y # ys)"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   479
  by (auto simp add: suffix_def)
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   480
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   481
lemma suffix_ConsD: "suffix (x # xs) ys \<Longrightarrow> suffix xs ys"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   482
  by (auto simp add: suffix_def)
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   483
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   484
lemma suffix_appendI: "suffix xs ys \<Longrightarrow> suffix xs (zs @ ys)"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   485
  by (auto simp add: suffix_def)
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   486
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   487
lemma suffix_appendD: "suffix (zs @ xs) ys \<Longrightarrow> suffix xs ys"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   488
  by (auto simp add: suffix_def)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   489
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   490
lemma strict_suffix_set_subset: "strict_suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   491
by (auto simp: strict_suffix_def)
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   492
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   493
lemma suffix_set_subset: "suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   494
by (auto simp: suffix_def)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   495
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   496
lemma suffix_ConsD2: "suffix (x # xs) (y # ys) \<Longrightarrow> suffix xs ys"
21305
d41eddfd2b66 tuned proofs;
wenzelm
parents: 19086
diff changeset
   497
proof -
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   498
  assume "suffix (x # xs) (y # ys)"
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   499
  then obtain zs where "y # ys = zs @ x # xs" ..
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   500
  then show ?thesis
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   501
    by (induct zs) (auto intro!: suffix_appendI suffix_ConsI)
21305
d41eddfd2b66 tuned proofs;
wenzelm
parents: 19086
diff changeset
   502
qed
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   503
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   504
lemma suffix_to_prefix [code]: "suffix xs ys \<longleftrightarrow> prefix (rev xs) (rev ys)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   505
proof
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   506
  assume "suffix xs ys"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   507
  then obtain zs where "ys = zs @ xs" ..
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   508
  then have "rev ys = rev xs @ rev zs" by simp
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   509
  then show "prefix (rev xs) (rev ys)" ..
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   510
next
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   511
  assume "prefix (rev xs) (rev ys)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   512
  then obtain zs where "rev ys = rev xs @ zs" ..
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   513
  then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   514
  then have "ys = rev zs @ xs" by simp
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   515
  then show "suffix xs ys" ..
21305
d41eddfd2b66 tuned proofs;
wenzelm
parents: 19086
diff changeset
   516
qed
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   517
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   518
lemma distinct_suffix: "distinct ys \<Longrightarrow> suffix xs ys \<Longrightarrow> distinct xs"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   519
  by (clarsimp elim!: suffixE)
17201
3bdf1dfcdee4 reactivate postfix by change of syntax;
wenzelm
parents: 15355
diff changeset
   520
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   521
lemma suffix_map: "suffix xs ys \<Longrightarrow> suffix (map f xs) (map f ys)"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   522
  by (auto elim!: suffixE intro: suffixI)
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   523
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   524
lemma suffix_drop: "suffix (drop n as) as"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   525
  unfolding suffix_def
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   526
  apply (rule exI [where x = "take n as"])
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   527
  apply simp
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   528
  done
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   529
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   530
lemma suffix_take: "suffix xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   531
  by (auto elim!: suffixE)
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   532
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   533
lemma strict_suffix_reflclp_conv: "strict_suffix\<^sup>=\<^sup>= = suffix"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   534
by (intro ext) (auto simp: suffix_def strict_suffix_def)
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   535
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   536
lemma suffix_lists: "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   537
  unfolding suffix_def by auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   538
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   539
lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefix x y"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   540
  by blast
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   541
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   542
lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefix y x"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   543
  by blast
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   544
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   545
lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   546
  unfolding parallel_def by simp
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   547
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   548
lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   549
  unfolding parallel_def by simp
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   550
25564
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25356
diff changeset
   551
lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   552
  by auto
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   553
25564
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25356
diff changeset
   554
lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   555
  by (metis Cons_prefix_Cons parallelE parallelI)
25665
faabc08af882 removed legacy proofs
nipkow
parents: 25595
diff changeset
   556
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   557
lemma not_equal_is_parallel:
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   558
  assumes neq: "xs \<noteq> ys"
25356
059c03630d6e tuned presentation;
wenzelm
parents: 25355
diff changeset
   559
    and len: "length xs = length ys"
059c03630d6e tuned presentation;
wenzelm
parents: 25355
diff changeset
   560
  shows "xs \<parallel> ys"
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   561
  using len neq
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   562
proof (induct rule: list_induct2)
26445
17223cf843d8 explicit case names for rule list_induct2
haftmann
parents: 25764
diff changeset
   563
  case Nil
25356
059c03630d6e tuned presentation;
wenzelm
parents: 25355
diff changeset
   564
  then show ?case by simp
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   565
next
26445
17223cf843d8 explicit case names for rule list_induct2
haftmann
parents: 25764
diff changeset
   566
  case (Cons a as b bs)
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   567
  have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   568
  show ?case
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   569
  proof (cases "a = b")
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   570
    case True
26445
17223cf843d8 explicit case names for rule list_induct2
haftmann
parents: 25764
diff changeset
   571
    then have "as \<noteq> bs" using Cons by simp
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   572
    then show ?thesis by (rule Cons_parallelI2 [OF True ih])
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   573
  next
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   574
    case False
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   575
    then show ?thesis by (rule Cons_parallelI1)
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   576
  qed
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   577
qed
22178
29b95968272b made executable
haftmann
parents: 21404
diff changeset
   578
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   579
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   580
subsection \<open>Homeomorphic embedding on lists\<close>
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   581
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   582
inductive list_emb :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   583
  for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   584
where
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   585
  list_emb_Nil [intro, simp]: "list_emb P [] ys"
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   586
| list_emb_Cons [intro] : "list_emb P xs ys \<Longrightarrow> list_emb P xs (y#ys)"
57498
ea44ec62a574 no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents: 57497
diff changeset
   587
| list_emb_Cons2 [intro]: "P x y \<Longrightarrow> list_emb P xs ys \<Longrightarrow> list_emb P (x#xs) (y#ys)"
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   588
57499
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   589
lemma list_emb_mono:                         
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   590
  assumes "\<And>x y. P x y \<longrightarrow> Q x y"
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   591
  shows "list_emb P xs ys \<longrightarrow> list_emb Q xs ys"
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   592
proof                                        
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   593
  assume "list_emb P xs ys"                    
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   594
  then show "list_emb Q xs ys" by (induct) (auto simp: assms)
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   595
qed 
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   596
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   597
lemma list_emb_Nil2 [simp]:
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   598
  assumes "list_emb P xs []" shows "xs = []"
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   599
  using assms by (cases rule: list_emb.cases) auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   600
57498
ea44ec62a574 no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents: 57497
diff changeset
   601
lemma list_emb_refl:
ea44ec62a574 no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents: 57497
diff changeset
   602
  assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x x"
ea44ec62a574 no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents: 57497
diff changeset
   603
  shows "list_emb P xs xs"
ea44ec62a574 no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents: 57497
diff changeset
   604
  using assms by (induct xs) auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   605
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   606
lemma list_emb_Cons_Nil [simp]: "list_emb P (x#xs) [] = False"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   607
proof -
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   608
  { assume "list_emb P (x#xs) []"
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   609
    from list_emb_Nil2 [OF this] have False by simp
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   610
  } moreover {
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   611
    assume False
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   612
    then have "list_emb P (x#xs) []" by simp
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   613
  } ultimately show ?thesis by blast
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   614
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   615
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   616
lemma list_emb_append2 [intro]: "list_emb P xs ys \<Longrightarrow> list_emb P xs (zs @ ys)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   617
  by (induct zs) auto
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   618
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   619
lemma list_emb_prefix [intro]:
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   620
  assumes "list_emb P xs ys" shows "list_emb P xs (ys @ zs)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   621
  using assms
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   622
  by (induct arbitrary: zs) auto
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   623
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   624
lemma list_emb_ConsD:
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   625
  assumes "list_emb P (x#xs) ys"
57498
ea44ec62a574 no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents: 57497
diff changeset
   626
  shows "\<exists>us v vs. ys = us @ v # vs \<and> P x v \<and> list_emb P xs vs"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   627
using assms
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   628
proof (induct x \<equiv> "x # xs" ys arbitrary: x xs)
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   629
  case list_emb_Cons
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   630
  then show ?case by (metis append_Cons)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   631
next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   632
  case (list_emb_Cons2 x y xs ys)
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   633
  then show ?case by blast
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   634
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   635
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   636
lemma list_emb_appendD:
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   637
  assumes "list_emb P (xs @ ys) zs"
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   638
  shows "\<exists>us vs. zs = us @ vs \<and> list_emb P xs us \<and> list_emb P ys vs"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   639
using assms
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   640
proof (induction xs arbitrary: ys zs)
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   641
  case Nil then show ?case by auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   642
next
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   643
  case (Cons x xs)
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   644
  then obtain us v vs where
57498
ea44ec62a574 no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents: 57497
diff changeset
   645
    zs: "zs = us @ v # vs" and p: "P x v" and lh: "list_emb P (xs @ ys) vs"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   646
    by (auto dest: list_emb_ConsD)
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   647
  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
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   648
    sk: "\<forall>x\<^sub>0 x\<^sub>1. \<not> list_emb 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_emb P xs (sk\<^sub>0 x\<^sub>0 x\<^sub>1) \<and> list_emb P x\<^sub>0 (sk\<^sub>1 x\<^sub>0 x\<^sub>1)"
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   649
    using Cons(1) by (metis (no_types))
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   650
  hence "\<forall>x\<^sub>2. list_emb P (x # xs) (x\<^sub>2 @ v # sk\<^sub>0 ys vs)" using p lh by auto
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   651
  thus ?case using lh zs sk by (metis (no_types) append_Cons append_assoc)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   652
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   653
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   654
lemma list_emb_strict_suffix:
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   655
  assumes "list_emb P xs ys" and "strict_suffix ys zs"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   656
  shows "list_emb P xs zs"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   657
  using assms(2) and list_emb_append2 [OF assms(1)] by (auto simp: strict_suffix_def)
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   658
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   659
lemma list_emb_suffix:
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   660
  assumes "list_emb P xs ys" and "suffix ys zs"
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   661
  shows "list_emb P xs zs"
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   662
using assms and list_emb_strict_suffix
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   663
unfolding strict_suffix_reflclp_conv[symmetric] by auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   664
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   665
lemma list_emb_length: "list_emb P xs ys \<Longrightarrow> length xs \<le> length ys"
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   666
  by (induct rule: list_emb.induct) auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   667
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   668
lemma list_emb_trans:
57500
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   669
  assumes "\<And>x y z. \<lbrakk>x \<in> set xs; y \<in> set ys; z \<in> set zs; P x y; P y z\<rbrakk> \<Longrightarrow> P x z"
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   670
  shows "\<lbrakk>list_emb P xs ys; list_emb P ys zs\<rbrakk> \<Longrightarrow> list_emb P xs zs"
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   671
proof -
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   672
  assume "list_emb P xs ys" and "list_emb P ys zs"
57500
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   673
  then show "list_emb P xs zs" using assms
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   674
  proof (induction arbitrary: zs)
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   675
    case list_emb_Nil show ?case by blast
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   676
  next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   677
    case (list_emb_Cons xs ys y)
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   678
    from list_emb_ConsD [OF \<open>list_emb P (y#ys) zs\<close>] obtain us v vs
57500
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   679
      where zs: "zs = us @ v # vs" and "P\<^sup>=\<^sup>= y v" and "list_emb P ys vs" by blast
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   680
    then have "list_emb P ys (v#vs)" by blast
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   681
    then have "list_emb P ys zs" unfolding zs by (rule list_emb_append2)
57500
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   682
    from list_emb_Cons.IH [OF this] and list_emb_Cons.prems show ?case by auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   683
  next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   684
    case (list_emb_Cons2 x y xs ys)
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   685
    from list_emb_ConsD [OF \<open>list_emb P (y#ys) zs\<close>] obtain us v vs
57498
ea44ec62a574 no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents: 57497
diff changeset
   686
      where zs: "zs = us @ v # vs" and "P y v" and "list_emb P ys vs" by blast
57500
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   687
    with list_emb_Cons2 have "list_emb P xs vs" by auto
57498
ea44ec62a574 no built-in reflexivity of list embedding (which is more standard; now embedding is reflexive whenever the base-order is)
Christian Sternagel
parents: 57497
diff changeset
   688
    moreover have "P x v"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   689
    proof -
57500
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   690
      from zs have "v \<in> set zs" by auto
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   691
      moreover have "x \<in> set (x#xs)" and "y \<in> set (y#ys)" by simp_all
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   692
      ultimately show ?thesis
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   693
        using \<open>P x y\<close> and \<open>P y v\<close> and list_emb_Cons2
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   694
        by blast
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   695
    qed
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   696
    ultimately have "list_emb P (x#xs) (v#vs)" by blast
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   697
    then show ?case unfolding zs by (rule list_emb_append2)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   698
  qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   699
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   700
57500
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   701
lemma list_emb_set:
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   702
  assumes "list_emb P xs ys" and "x \<in> set xs"
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   703
  obtains y where "y \<in> set ys" and "P x y"
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   704
  using assms by (induct) auto
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   705
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   706
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   707
subsection \<open>Sublists (special case of homeomorphic embedding)\<close>
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   708
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   709
abbreviation sublisteq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   710
  where "sublisteq xs ys \<equiv> list_emb (op =) xs ys"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   711
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   712
lemma sublisteq_Cons2: "sublisteq xs ys \<Longrightarrow> sublisteq (x#xs) (x#ys)" by auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   713
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   714
lemma sublisteq_same_length:
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   715
  assumes "sublisteq xs ys" and "length xs = length ys" shows "xs = ys"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   716
  using assms by (induct) (auto dest: list_emb_length)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   717
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   718
lemma not_sublisteq_length [simp]: "length ys < length xs \<Longrightarrow> \<not> sublisteq xs ys"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   719
  by (metis list_emb_length linorder_not_less)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   720
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   721
lemma [code]:
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   722
  "list_emb P [] ys \<longleftrightarrow> True"
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   723
  "list_emb P (x#xs) [] \<longleftrightarrow> False"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   724
  by (simp_all)
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   725
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   726
lemma sublisteq_Cons': "sublisteq (x#xs) ys \<Longrightarrow> sublisteq xs ys"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   727
  by (induct xs, simp, blast dest: list_emb_ConsD)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   728
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   729
lemma sublisteq_Cons2':
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   730
  assumes "sublisteq (x#xs) (x#ys)" shows "sublisteq xs ys"
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   731
  using assms by (cases) (rule sublisteq_Cons')
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   732
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   733
lemma sublisteq_Cons2_neq:
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   734
  assumes "sublisteq (x#xs) (y#ys)"
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   735
  shows "x \<noteq> y \<Longrightarrow> sublisteq (x#xs) ys"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   736
  using assms by (cases) auto
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   737
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   738
lemma sublisteq_Cons2_iff [simp, code]:
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   739
  "sublisteq (x#xs) (y#ys) = (if x = y then sublisteq xs ys else sublisteq (x#xs) ys)"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   740
  by (metis list_emb_Cons sublisteq_Cons2 sublisteq_Cons2' sublisteq_Cons2_neq)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   741
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   742
lemma sublisteq_append': "sublisteq (zs @ xs) (zs @ ys) \<longleftrightarrow> sublisteq xs ys"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   743
  by (induct zs) simp_all
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   744
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   745
lemma sublisteq_refl [simp, intro!]: "sublisteq xs xs" by (induct xs) simp_all
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   746
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   747
lemma sublisteq_antisym:
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   748
  assumes "sublisteq xs ys" and "sublisteq ys xs"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   749
  shows "xs = ys"
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   750
using assms
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   751
proof (induct)
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   752
  case list_emb_Nil
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   753
  from list_emb_Nil2 [OF this] show ?case by simp
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   754
next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   755
  case list_emb_Cons2
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   756
  thus ?case by simp
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   757
next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   758
  case list_emb_Cons
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   759
  hence False using sublisteq_Cons' by fastforce
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   760
  thus ?case ..
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   761
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   762
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   763
lemma sublisteq_trans: "sublisteq xs ys \<Longrightarrow> sublisteq ys zs \<Longrightarrow> sublisteq xs zs"
57500
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   764
  by (rule list_emb_trans [of _ _ _ "op ="]) auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   765
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   766
lemma sublisteq_append_le_same_iff: "sublisteq (xs @ ys) ys \<longleftrightarrow> xs = []"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   767
  by (auto dest: list_emb_length)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   768
64886
cea327ecb8e3 added lemma
blanchet
parents: 63649
diff changeset
   769
lemma sublisteq_singleton_left: "sublisteq [x] ys \<longleftrightarrow> x \<in> set ys"
cea327ecb8e3 added lemma
blanchet
parents: 63649
diff changeset
   770
  by (fastforce dest: list_emb_ConsD split_list_last)
cea327ecb8e3 added lemma
blanchet
parents: 63649
diff changeset
   771
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   772
lemma list_emb_append_mono:
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   773
  "\<lbrakk> list_emb P xs xs'; list_emb P ys ys' \<rbrakk> \<Longrightarrow> list_emb P (xs@ys) (xs'@ys')"
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   774
  apply (induct rule: list_emb.induct)
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   775
    apply (metis eq_Nil_appendI list_emb_append2)
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   776
   apply (metis append_Cons list_emb_Cons)
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   777
  apply (metis append_Cons list_emb_Cons2)
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   778
  done
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   779
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   780
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   781
subsection \<open>Appending elements\<close>
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   782
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   783
lemma sublisteq_append [simp]:
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   784
  "sublisteq (xs @ zs) (ys @ zs) \<longleftrightarrow> sublisteq xs ys" (is "?l = ?r")
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   785
proof
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   786
  { fix xs' ys' xs ys zs :: "'a list" assume "sublisteq xs' ys'"
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   787
    then have "xs' = xs @ zs & ys' = ys @ zs \<longrightarrow> sublisteq xs ys"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   788
    proof (induct arbitrary: xs ys zs)
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   789
      case list_emb_Nil show ?case by simp
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   790
    next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   791
      case (list_emb_Cons xs' ys' x)
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   792
      { assume "ys=[]" then have ?case using list_emb_Cons(1) by auto }
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   793
      moreover
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   794
      { fix us assume "ys = x#us"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   795
        then have ?case using list_emb_Cons(2) by(simp add: list_emb.list_emb_Cons) }
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   796
      ultimately show ?case by (auto simp:Cons_eq_append_conv)
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   797
    next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   798
      case (list_emb_Cons2 x y xs' ys')
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   799
      { assume "xs=[]" then have ?case using list_emb_Cons2(1) by auto }
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   800
      moreover
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   801
      { fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using list_emb_Cons2 by auto}
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   802
      moreover
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   803
      { fix us assume "xs=x#us" "ys=[]" then have ?case using list_emb_Cons2(2) by bestsimp }
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   804
      ultimately show ?case using \<open>op = x y\<close> by (auto simp: Cons_eq_append_conv)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   805
    qed }
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   806
  moreover assume ?l
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   807
  ultimately show ?r by blast
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   808
next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   809
  assume ?r then show ?l by (metis list_emb_append_mono sublisteq_refl)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   810
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   811
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   812
lemma sublisteq_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (zs @ ys)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   813
  by (induct zs) auto
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   814
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   815
lemma sublisteq_rev_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (ys @ zs)"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   816
  by (metis append_Nil2 list_emb_Nil list_emb_append_mono)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   817
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   818
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   819
subsection \<open>Relation to standard list operations\<close>
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   820
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   821
lemma sublisteq_map:
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   822
  assumes "sublisteq xs ys" shows "sublisteq (map f xs) (map f ys)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   823
  using assms by (induct) auto
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   824
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   825
lemma sublisteq_filter_left [simp]: "sublisteq (filter P xs) xs"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   826
  by (induct xs) auto
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   827
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   828
lemma sublisteq_filter [simp]:
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   829
  assumes "sublisteq xs ys" shows "sublisteq (filter P xs) (filter P ys)"
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   830
  using assms by induct auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   831
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   832
lemma "sublisteq xs ys \<longleftrightarrow> (\<exists>N. xs = sublist ys N)" (is "?L = ?R")
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   833
proof
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   834
  assume ?L
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   835
  then show ?R
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   836
  proof (induct)
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   837
    case list_emb_Nil show ?case by (metis sublist_empty)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   838
  next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   839
    case (list_emb_Cons xs ys x)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   840
    then obtain N where "xs = sublist ys N" by blast
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   841
    then have "xs = sublist (x#ys) (Suc ` N)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   842
      by (clarsimp simp add:sublist_Cons inj_image_mem_iff)
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   843
    then show ?case by blast
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   844
  next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   845
    case (list_emb_Cons2 x y xs ys)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   846
    then obtain N where "xs = sublist ys N" by blast
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   847
    then have "x#xs = sublist (x#ys) (insert 0 (Suc ` N))"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   848
      by (clarsimp simp add:sublist_Cons inj_image_mem_iff)
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   849
    moreover from list_emb_Cons2 have "x = y" by simp
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   850
    ultimately show ?case by blast
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   851
  qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   852
next
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   853
  assume ?R
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   854
  then obtain N where "xs = sublist ys N" ..
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   855
  moreover have "sublisteq (sublist ys N) ys"
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   856
  proof (induct ys arbitrary: N)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   857
    case Nil show ?case by simp
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   858
  next
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   859
    case Cons then show ?case by (auto simp: sublist_Cons)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   860
  qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   861
  ultimately show ?L by simp
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   862
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   863
10330
4362e906b745 "List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff changeset
   864
end