src/HOL/Library/Sublist.thy
author nipkow
Thu, 23 Apr 2020 09:57:41 +0200
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added lemmas
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(*  Title:      HOL/Library/Sublist.thy
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    Author:     Tobias Nipkow and Markus Wenzel, TU München
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    Author:     Christian Sternagel, JAIST
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    Author:     Manuel Eberl, TU München
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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 [simp]: "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 [simp]: "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 flip: append_assoc)
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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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  unfolding prefix_def by (metis append_take_drop_id)
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lemma prefixeq_butlast: "prefix (butlast xs) xs"
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by (simp add: butlast_conv_take take_is_prefix)
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lemma prefix_map_rightE:
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  assumes "prefix xs (map f ys)"
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  shows   "\<exists>xs'. prefix xs' ys \<and> xs = map f xs'"
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proof -
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  define n where "n = length xs"
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  have "xs = take n (map f ys)"
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    using assms by (auto simp: prefix_def n_def)
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  thus ?thesis
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    by (intro exI[of _ "take n ys"]) (auto simp: take_map take_is_prefix)
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qed
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lemma map_mono_prefix: "prefix xs ys \<Longrightarrow> prefix (map f xs) (map f ys)"
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by (auto simp: prefix_def)
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lemma filter_mono_prefix: "prefix xs ys \<Longrightarrow> prefix (filter P xs) (filter P ys)"
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by (auto simp: prefix_def)
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lemma sorted_antimono_prefix: "prefix xs ys \<Longrightarrow> sorted ys \<Longrightarrow> sorted xs"
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by (metis sorted_append prefix_def)
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lemma prefix_length_less: "strict_prefix xs ys \<Longrightarrow> length xs < length ys"
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  by (auto simp: strict_prefix_def prefix_def)
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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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lemma strict_prefix_simps [simp, code]:
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  "strict_prefix xs [] \<longleftrightarrow> False"
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  "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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  by (simp_all add: strict_prefix_def cong: conj_cong)
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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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  case 0
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  then show ?case by (cases ys) simp_all
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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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qed
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lemma prefix_takeWhile:
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  assumes "prefix xs ys"
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  shows   "prefix (takeWhile P xs) (takeWhile P ys)"
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proof -
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  from assms obtain zs where ys: "ys = xs @ zs"
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    by (auto simp: prefix_def)
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  have "prefix (takeWhile P xs) (takeWhile P (xs @ zs))"
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    by (induction xs) auto
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  thus ?thesis by (simp add: ys)
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qed
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lemma prefix_dropWhile:
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  assumes "prefix xs ys"
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  shows   "prefix (dropWhile P xs) (dropWhile P ys)"
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   199
proof -
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  from assms obtain zs where ys: "ys = xs @ zs"
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    by (auto simp: prefix_def)
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  have "prefix (dropWhile P xs) (dropWhile P (xs @ zs))"
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    by (induction xs) auto
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  thus ?thesis by (simp add: ys)
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qed
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lemma prefix_remdups_adj:
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  assumes "prefix xs ys"
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  shows   "prefix (remdups_adj xs) (remdups_adj ys)"
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  using assms
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proof (induction "length xs" arbitrary: xs ys rule: less_induct)
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  case (less xs)
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  show ?case
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  proof (cases xs)
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    case [simp]: (Cons x xs')
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    then obtain y ys' where [simp]: "ys = y # ys'"
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      using \<open>prefix xs ys\<close> by (cases ys) auto
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    from less show ?thesis
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      by (auto simp: remdups_adj_Cons' less_Suc_eq_le length_dropWhile_le
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               intro!: less prefix_dropWhile)
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  qed auto
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qed
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lemma not_prefix_cases:
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  assumes pfx: "\<not> prefix ps ls"
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  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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proof (cases ps)
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  case Nil
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  then show ?thesis using pfx by simp
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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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  next
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    case (Cons x xs)
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    show ?thesis
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    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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      with c Cons True show ?thesis by (rule c2)
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    next
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      case False
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      with c Cons show ?thesis by (rule c3)
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    qed
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  qed
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qed
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   252
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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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    and base: "\<And>x xs. P (x#xs) []"
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    and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)"
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   257
    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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  shows "P ps ls" using np
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   259
proof (induct ls arbitrary: ps)
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  case Nil
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   261
  then show ?case
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    by (auto simp: neq_Nil_conv elim!: not_prefix_cases intro!: base)
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   263
next
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   264
  case (Cons y ys)
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   265
  then have npfx: "\<not> prefix ps (y # ys)" by simp
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  then obtain x xs where pv: "ps = x # xs"
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    by (rule not_prefix_cases) auto
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  show ?case by (metis Cons.hyps Cons_prefix_Cons npfx pv r1 r2)
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qed
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   270
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subsection \<open>Prefixes\<close>
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primrec prefixes where
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"prefixes [] = [[]]" |
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"prefixes (x#xs) = [] # map ((#) x) (prefixes xs)"
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lemma in_set_prefixes[simp]: "xs \<in> set (prefixes ys) \<longleftrightarrow> prefix xs ys"
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   279
proof (induct xs arbitrary: ys)
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  case Nil
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   281
  then show ?case by (cases ys) auto
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   282
next
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   283
  case (Cons a xs)
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   284
  then show ?case by (cases ys) auto
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   285
qed
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   286
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lemma length_prefixes[simp]: "length (prefixes xs) = length xs+1"
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   288
  by (induction xs) auto
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   289
    
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   290
lemma distinct_prefixes [intro]: "distinct (prefixes xs)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   291
  by (induction xs) (auto simp: distinct_map)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   292
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   293
lemma prefixes_snoc [simp]: "prefixes (xs@[x]) = prefixes xs @ [xs@[x]]"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   294
  by (induction xs) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   295
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   296
lemma prefixes_not_Nil [simp]: "prefixes xs \<noteq> []"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   297
  by (cases xs) auto
63155
ea8540c71581 added function "prefixes" and some lemmas
nipkow
parents: 63149
diff changeset
   298
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   299
lemma hd_prefixes [simp]: "hd (prefixes xs) = []"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   300
  by (cases xs) simp_all
63155
ea8540c71581 added function "prefixes" and some lemmas
nipkow
parents: 63149
diff changeset
   301
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   302
lemma last_prefixes [simp]: "last (prefixes xs) = xs"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   303
  by (induction xs) (simp_all add: last_map)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   304
    
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   305
lemma prefixes_append: 
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   306
  "prefixes (xs @ ys) = prefixes xs @ map (\<lambda>ys'. xs @ ys') (tl (prefixes ys))"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   307
proof (induction xs)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   308
  case Nil
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   309
  thus ?case by (cases ys) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   310
qed simp_all
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   311
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   312
lemma prefixes_eq_snoc:
63155
ea8540c71581 added function "prefixes" and some lemmas
nipkow
parents: 63149
diff changeset
   313
  "prefixes ys = xs @ [x] \<longleftrightarrow>
ea8540c71581 added function "prefixes" and some lemmas
nipkow
parents: 63149
diff changeset
   314
  (ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = zs@[z] \<and> xs = prefixes zs)) \<and> x = ys"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   315
  by (cases ys rule: rev_cases) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   316
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   317
lemma prefixes_tailrec [code]: 
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   318
  "prefixes xs = rev (snd (foldl (\<lambda>(acc1, acc2) x. (x#acc1, rev (x#acc1)#acc2)) ([],[[]]) xs))"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   319
proof -
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   320
  have "foldl (\<lambda>(acc1, acc2) x. (x#acc1, rev (x#acc1)#acc2)) (ys, rev ys # zs) xs =
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   321
          (rev xs @ ys, rev (map (\<lambda>as. rev ys @ as) (prefixes xs)) @ zs)" for ys zs
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   322
  proof (induction xs arbitrary: ys zs)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   323
    case (Cons x xs ys zs)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   324
    from Cons.IH[of "x # ys" "rev ys # zs"]
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   325
      show ?case by (simp add: o_def)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   326
  qed simp_all
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   327
  from this [of "[]" "[]"] show ?thesis by simp
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   328
qed
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   329
  
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   330
lemma set_prefixes_eq: "set (prefixes xs) = {ys. prefix ys xs}"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   331
  by auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   332
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   333
lemma card_set_prefixes [simp]: "card (set (prefixes xs)) = Suc (length xs)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   334
  by (subst distinct_card) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   335
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   336
lemma set_prefixes_append: 
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   337
  "set (prefixes (xs @ ys)) = set (prefixes xs) \<union> {xs @ ys' |ys'. ys' \<in> set (prefixes ys)}"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   338
  by (subst prefixes_append, cases ys) auto
63155
ea8540c71581 added function "prefixes" and some lemmas
nipkow
parents: 63149
diff changeset
   339
ea8540c71581 added function "prefixes" and some lemmas
nipkow
parents: 63149
diff changeset
   340
63173
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   341
subsection \<open>Longest Common Prefix\<close>
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   342
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   343
definition Longest_common_prefix :: "'a list set \<Rightarrow> 'a list" where
65954
431024edc9cf introduced arg_max
nipkow
parents: 65869
diff changeset
   344
"Longest_common_prefix L = (ARG_MAX length ps. \<forall>xs \<in> L. prefix ps xs)"
63173
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   345
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   346
lemma Longest_common_prefix_ex: "L \<noteq> {} \<Longrightarrow>
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   347
  \<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
   348
  (is "_ \<Longrightarrow> \<exists>ps. ?P L ps")
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   349
proof(induction "LEAST n. \<exists>xs \<in>L. n = length xs" arbitrary: L)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   350
  case 0
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67606
diff changeset
   351
  have "[] \<in> L" using "0.hyps" LeastI[of "\<lambda>n. \<exists>xs\<in>L. n = length xs"] \<open>L \<noteq> {}\<close>
63173
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   352
    by auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   353
  hence "?P L []" by(auto)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   354
  thus ?case ..
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   355
next
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   356
  case (Suc n)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   357
  let ?EX = "\<lambda>n. \<exists>xs\<in>L. n = length xs"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   358
  obtain x xs where xxs: "x#xs \<in> L" "size xs = n" using Suc.prems Suc.hyps(2)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   359
    by(metis LeastI_ex[of ?EX] Suc_length_conv ex_in_conv)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   360
  hence "[] \<notin> L" using Suc.hyps(2) by auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   361
  show ?case
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   362
  proof (cases "\<forall>xs \<in> L. \<exists>ys. xs = x#ys")
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   363
    case True
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   364
    let ?L = "{ys. x#ys \<in> L}"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   365
    have 1: "(LEAST n. \<exists>xs \<in> ?L. n = length xs) = n"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   366
      using xxs Suc.prems Suc.hyps(2) Least_le[of "?EX"]
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   367
      by - (rule Least_equality, fastforce+)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   368
    have 2: "?L \<noteq> {}" using \<open>x # xs \<in> L\<close> by auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   369
    from Suc.hyps(1)[OF 1[symmetric] 2] obtain ps where IH: "?P ?L ps" ..
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   370
    { fix qs
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   371
      assume "\<forall>qs. (\<forall>xa. x # xa \<in> L \<longrightarrow> prefix qs xa) \<longrightarrow> length qs \<le> length ps"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   372
      and "\<forall>xs\<in>L. prefix qs xs"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   373
      hence "length (tl qs) \<le> length ps"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   374
        by (metis Cons_prefix_Cons hd_Cons_tl list.sel(2) Nil_prefix) 
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   375
      hence "length qs \<le> Suc (length ps)" by auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   376
    }
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   377
    hence "?P L (x#ps)" using True IH by auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   378
    thus ?thesis ..
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   379
  next
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   380
    case False
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   381
    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
   382
      by (auto) (metis list.exhaust)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   383
    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
   384
      by auto (metis Cons_prefix_Cons prefix_Cons)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   385
    hence "?P L []" by auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   386
    thus ?thesis ..
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   387
  qed
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   388
qed
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   389
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   390
lemma Longest_common_prefix_unique: "L \<noteq> {} \<Longrightarrow>
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   391
  \<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
   392
by(rule ex_ex1I[OF Longest_common_prefix_ex];
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   393
   meson equals0I prefix_length_prefix prefix_order.antisym)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   394
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   395
lemma Longest_common_prefix_eq:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   396
 "\<lbrakk> L \<noteq> {};  \<forall>xs \<in> L. prefix ps xs;
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   397
    \<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
   398
  \<Longrightarrow> Longest_common_prefix L = ps"
65954
431024edc9cf introduced arg_max
nipkow
parents: 65869
diff changeset
   399
unfolding Longest_common_prefix_def arg_max_def is_arg_max_linorder
63173
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   400
by(rule some1_equality[OF Longest_common_prefix_unique]) auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   401
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   402
lemma Longest_common_prefix_prefix:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   403
  "xs \<in> L \<Longrightarrow> prefix (Longest_common_prefix L) xs"
65954
431024edc9cf introduced arg_max
nipkow
parents: 65869
diff changeset
   404
unfolding Longest_common_prefix_def arg_max_def is_arg_max_linorder
63173
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   405
by(rule someI2_ex[OF Longest_common_prefix_ex]) auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   406
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   407
lemma Longest_common_prefix_longest:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   408
  "L \<noteq> {} \<Longrightarrow> \<forall>xs\<in>L. prefix ps xs \<Longrightarrow> length ps \<le> length(Longest_common_prefix L)"
65954
431024edc9cf introduced arg_max
nipkow
parents: 65869
diff changeset
   409
unfolding Longest_common_prefix_def arg_max_def is_arg_max_linorder
63173
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   410
by(rule someI2_ex[OF Longest_common_prefix_ex]) auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   411
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   412
lemma Longest_common_prefix_max_prefix:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   413
  "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
   414
by(metis Longest_common_prefix_prefix Longest_common_prefix_longest
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   415
     prefix_length_prefix ex_in_conv)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   416
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   417
lemma Longest_common_prefix_Nil: "[] \<in> L \<Longrightarrow> Longest_common_prefix L = []"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   418
using Longest_common_prefix_prefix prefix_Nil by blast
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   419
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   420
lemma Longest_common_prefix_image_Cons: "L \<noteq> {} \<Longrightarrow>
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
   421
  Longest_common_prefix ((#) x ` L) = x # Longest_common_prefix L"
63173
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   422
apply(rule Longest_common_prefix_eq)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   423
  apply(simp)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   424
 apply (simp add: Longest_common_prefix_prefix)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   425
apply simp
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   426
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
   427
     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
   428
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   429
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
   430
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
   431
proof -
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
   432
  have "L = (#) x ` {ys. x#ys \<in> L}" using assms(2,3)
63173
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   433
    by (auto simp: image_def)(metis hd_Cons_tl)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   434
  thus ?thesis
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   435
    by (metis Longest_common_prefix_image_Cons image_is_empty assms(1))
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   436
qed
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   437
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   438
lemma Longest_common_prefix_eq_Nil:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   439
  "\<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
   440
by (metis Longest_common_prefix_prefix list.inject prefix_Cons)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   441
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   442
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
   443
"longest_common_prefix (x#xs) (y#ys) =
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   444
  (if x=y then x # longest_common_prefix xs ys else [])" |
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   445
"longest_common_prefix _ _ = []"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   446
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   447
lemma longest_common_prefix_prefix1:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   448
  "prefix (longest_common_prefix xs ys) xs"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   449
by(induction xs ys rule: longest_common_prefix.induct) auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   450
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   451
lemma longest_common_prefix_prefix2:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   452
  "prefix (longest_common_prefix xs ys) ys"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   453
by(induction xs ys rule: longest_common_prefix.induct) auto
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   454
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   455
lemma longest_common_prefix_max_prefix:
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   456
  "\<lbrakk> prefix ps xs; prefix ps ys \<rbrakk>
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   457
   \<Longrightarrow> prefix ps (longest_common_prefix xs ys)"
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   458
by(induction xs ys arbitrary: ps rule: longest_common_prefix.induct)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   459
  (auto simp: prefix_Cons)
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   460
3413b1cf30cd added subtheory of longest common prefix
nipkow
parents: 63155
diff changeset
   461
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   462
subsection \<open>Parallel lists\<close>
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   463
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   464
definition parallel :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "\<parallel>" 50)
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   465
  where "(xs \<parallel> ys) = (\<not> prefix xs ys \<and> \<not> prefix ys xs)"
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   466
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   467
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
   468
  unfolding parallel_def by blast
10330
4362e906b745 "List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff changeset
   469
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   470
lemma parallelE [elim]:
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   471
  assumes "xs \<parallel> ys"
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   472
  obtains "\<not> prefix xs ys \<and> \<not> prefix ys xs"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   473
  using assms unfolding parallel_def by blast
10330
4362e906b745 "List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff changeset
   474
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   475
theorem prefix_cases:
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   476
  obtains "prefix xs ys" | "strict_prefix ys xs" | "xs \<parallel> ys"
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   477
  unfolding parallel_def strict_prefix_def by blast
10330
4362e906b745 "List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff changeset
   478
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   479
theorem parallel_decomp:
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
   480
  "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
   481
proof (induct xs rule: rev_induct)
11987
bf31b35949ce tuned induct proofs;
wenzelm
parents: 11780
diff changeset
   482
  case Nil
23254
99644a53f16d tuned proofs;
wenzelm
parents: 22178
diff changeset
   483
  then have False by auto
99644a53f16d tuned proofs;
wenzelm
parents: 22178
diff changeset
   484
  then show ?case ..
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   485
next
11987
bf31b35949ce tuned induct proofs;
wenzelm
parents: 11780
diff changeset
   486
  case (snoc x xs)
bf31b35949ce tuned induct proofs;
wenzelm
parents: 11780
diff changeset
   487
  show ?case
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   488
  proof (rule prefix_cases)
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   489
    assume le: "prefix xs ys"
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   490
    then obtain ys' where ys: "ys = xs @ ys'" ..
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   491
    show ?thesis
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   492
    proof (cases ys')
25564
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25356
diff changeset
   493
      assume "ys' = []"
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   494
      then show ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys)
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   495
    next
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   496
      fix c cs assume ys': "ys' = c # cs"
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   497
      have "x \<noteq> c" using snoc.prems ys ys' by fastforce
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   498
      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
   499
        using ys ys' by blast
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   500
    qed
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   501
  next
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   502
    assume "strict_prefix ys xs"
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   503
    then have "prefix ys (xs @ [x])" by (simp add: strict_prefix_def)
11987
bf31b35949ce tuned induct proofs;
wenzelm
parents: 11780
diff changeset
   504
    with snoc have False by blast
23254
99644a53f16d tuned proofs;
wenzelm
parents: 22178
diff changeset
   505
    then show ?thesis ..
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   506
  next
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   507
    assume "xs \<parallel> ys"
11987
bf31b35949ce tuned induct proofs;
wenzelm
parents: 11780
diff changeset
   508
    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
   509
      and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   510
      by blast
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   511
    from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
parents: 10389
diff changeset
   512
    with neq ys show ?thesis by blast
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   513
  qed
c7d8901ab269 proper setup of "parallel";
wenzelm
parents: 10330
diff changeset
   514
qed
10330
4362e906b745 "List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff changeset
   515
25564
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25356
diff changeset
   516
lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   517
  apply (rule parallelI)
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   518
    apply (erule parallelE, erule conjE,
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   519
      induct rule: not_prefix_induct, simp+)+
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   520
  done
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   521
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   522
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
   523
  by (simp add: parallel_append)
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   524
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   525
lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   526
  unfolding parallel_def by auto
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   527
25356
059c03630d6e tuned presentation;
wenzelm
parents: 25355
diff changeset
   528
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   529
subsection \<open>Suffix order on lists\<close>
17201
3bdf1dfcdee4 reactivate postfix by change of syntax;
wenzelm
parents: 15355
diff changeset
   530
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   531
definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   532
  where "suffix xs ys = (\<exists>zs. ys = zs @ xs)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   533
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   534
definition strict_suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   535
  where "strict_suffix xs ys \<longleftrightarrow> suffix xs ys \<and> xs \<noteq> ys"
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   536
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   537
interpretation suffix_order: order suffix strict_suffix
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   538
  by standard (auto simp: suffix_def strict_suffix_def)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   539
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   540
interpretation suffix_bot: order_bot Nil suffix strict_suffix
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   541
  by standard (simp add: suffix_def)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   542
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   543
lemma suffixI [intro?]: "ys = zs @ xs \<Longrightarrow> suffix xs ys"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   544
  unfolding suffix_def by blast
21305
d41eddfd2b66 tuned proofs;
wenzelm
parents: 19086
diff changeset
   545
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   546
lemma suffixE [elim?]:
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   547
  assumes "suffix xs ys"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   548
  obtains zs where "ys = zs @ xs"
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   549
  using assms unfolding suffix_def by blast
65957
558ba6b37f5c Tuned Library/Sublist.thy
eberlm <eberlm@in.tum.de>
parents: 65956
diff changeset
   550
    
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   551
lemma suffix_tl [simp]: "suffix (tl xs) xs"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   552
  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
   553
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   554
lemma strict_suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> strict_suffix (tl xs) xs"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   555
  by (induct xs) (auto simp: strict_suffix_def suffix_def)
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   556
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   557
lemma Nil_suffix [simp]: "suffix [] xs"
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   558
  by (simp add: suffix_def)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   559
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   560
lemma suffix_Nil [simp]: "(suffix xs []) = (xs = [])"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   561
  by (auto simp add: suffix_def)
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   562
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   563
lemma suffix_ConsI: "suffix xs ys \<Longrightarrow> suffix xs (y # ys)"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   564
  by (auto simp add: suffix_def)
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   565
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   566
lemma suffix_ConsD: "suffix (x # xs) ys \<Longrightarrow> suffix xs ys"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   567
  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
   568
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   569
lemma suffix_appendI: "suffix xs ys \<Longrightarrow> suffix xs (zs @ ys)"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   570
  by (auto simp add: suffix_def)
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   571
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   572
lemma suffix_appendD: "suffix (zs @ xs) ys \<Longrightarrow> suffix xs ys"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   573
  by (auto simp add: suffix_def)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   574
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   575
lemma strict_suffix_set_subset: "strict_suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   576
  by (auto simp: strict_suffix_def suffix_def)
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   577
67606
3b3188ae63da added lemmas
nipkow
parents: 67399
diff changeset
   578
lemma set_mono_suffix: "suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys"
3b3188ae63da added lemmas
nipkow
parents: 67399
diff changeset
   579
by (auto simp: suffix_def)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   580
67612
e4e57da0583a New theory ex/Radix_Sort.thy
nipkow
parents: 67606
diff changeset
   581
lemma sorted_antimono_suffix: "suffix xs ys \<Longrightarrow> sorted ys \<Longrightarrow> sorted xs"
e4e57da0583a New theory ex/Radix_Sort.thy
nipkow
parents: 67606
diff changeset
   582
by (metis sorted_append suffix_def)
e4e57da0583a New theory ex/Radix_Sort.thy
nipkow
parents: 67606
diff changeset
   583
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   584
lemma suffix_ConsD2: "suffix (x # xs) (y # ys) \<Longrightarrow> suffix xs ys"
21305
d41eddfd2b66 tuned proofs;
wenzelm
parents: 19086
diff changeset
   585
proof -
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   586
  assume "suffix (x # xs) (y # ys)"
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   587
  then obtain zs where "y # ys = zs @ x # xs" ..
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   588
  then show ?thesis
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   589
    by (induct zs) (auto intro!: suffix_appendI suffix_ConsI)
21305
d41eddfd2b66 tuned proofs;
wenzelm
parents: 19086
diff changeset
   590
qed
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   591
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   592
lemma suffix_to_prefix [code]: "suffix xs ys \<longleftrightarrow> prefix (rev xs) (rev ys)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   593
proof
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   594
  assume "suffix xs ys"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   595
  then obtain zs where "ys = zs @ xs" ..
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   596
  then have "rev ys = rev xs @ rev zs" by simp
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   597
  then show "prefix (rev xs) (rev ys)" ..
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   598
next
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   599
  assume "prefix (rev xs) (rev ys)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   600
  then obtain zs where "rev ys = rev xs @ zs" ..
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   601
  then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   602
  then have "ys = rev zs @ xs" by simp
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   603
  then show "suffix xs ys" ..
21305
d41eddfd2b66 tuned proofs;
wenzelm
parents: 19086
diff changeset
   604
qed
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   605
  
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   606
lemma strict_suffix_to_prefix [code]: "strict_suffix xs ys \<longleftrightarrow> strict_prefix (rev xs) (rev ys)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   607
  by (auto simp: suffix_to_prefix strict_suffix_def strict_prefix_def)
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   608
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   609
lemma distinct_suffix: "distinct ys \<Longrightarrow> suffix xs ys \<Longrightarrow> distinct xs"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   610
  by (clarsimp elim!: suffixE)
17201
3bdf1dfcdee4 reactivate postfix by change of syntax;
wenzelm
parents: 15355
diff changeset
   611
67606
3b3188ae63da added lemmas
nipkow
parents: 67399
diff changeset
   612
lemma map_mono_suffix: "suffix xs ys \<Longrightarrow> suffix (map f xs) (map f ys)"
3b3188ae63da added lemmas
nipkow
parents: 67399
diff changeset
   613
by (auto elim!: suffixE intro: suffixI)
3b3188ae63da added lemmas
nipkow
parents: 67399
diff changeset
   614
3b3188ae63da added lemmas
nipkow
parents: 67399
diff changeset
   615
lemma filter_mono_suffix: "suffix xs ys \<Longrightarrow> suffix (filter P xs) (filter P ys)"
3b3188ae63da added lemmas
nipkow
parents: 67399
diff changeset
   616
by (auto simp: suffix_def)
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   617
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   618
lemma suffix_drop: "suffix (drop n as) as"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   619
  unfolding suffix_def by (rule exI [where x = "take n as"]) simp
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   620
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   621
lemma suffix_take: "suffix xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   622
  by (auto elim!: suffixE)
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   623
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   624
lemma strict_suffix_reflclp_conv: "strict_suffix\<^sup>=\<^sup>= = suffix"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   625
  by (intro ext) (auto simp: suffix_def strict_suffix_def)
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   626
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   627
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
   628
  unfolding suffix_def by auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   629
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   630
lemma suffix_snoc [simp]: "suffix xs (ys @ [y]) \<longleftrightarrow> xs = [] \<or> (\<exists>zs. xs = zs @ [y] \<and> suffix zs ys)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   631
  by (cases xs rule: rev_cases) (auto simp: suffix_def)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   632
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   633
lemma snoc_suffix_snoc [simp]: "suffix (xs @ [x]) (ys @ [y]) = (x = y \<and> suffix xs ys)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   634
  by (auto simp add: suffix_def)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   635
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   636
lemma same_suffix_suffix [simp]: "suffix (ys @ xs) (zs @ xs) = suffix ys zs"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   637
  by (simp add: suffix_to_prefix)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   638
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   639
lemma same_suffix_nil [simp]: "suffix (ys @ xs) xs = (ys = [])"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   640
  by (simp add: suffix_to_prefix)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   641
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   642
theorem suffix_Cons: "suffix xs (y # ys) \<longleftrightarrow> xs = y # ys \<or> suffix xs ys"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   643
  unfolding suffix_def by (auto simp: Cons_eq_append_conv)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   644
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   645
theorem suffix_append: 
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   646
  "suffix xs (ys @ zs) \<longleftrightarrow> suffix xs zs \<or> (\<exists>xs'. xs = xs' @ zs \<and> suffix xs' ys)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   647
  by (auto simp: suffix_def append_eq_append_conv2)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   648
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   649
theorem suffix_length_le: "suffix xs ys \<Longrightarrow> length xs \<le> length ys"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   650
  by (auto simp add: suffix_def)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   651
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   652
lemma suffix_same_cases:
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   653
  "suffix (xs\<^sub>1::'a list) ys \<Longrightarrow> suffix xs\<^sub>2 ys \<Longrightarrow> suffix xs\<^sub>1 xs\<^sub>2 \<or> suffix xs\<^sub>2 xs\<^sub>1"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   654
  unfolding suffix_def by (force simp: append_eq_append_conv2)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   655
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   656
lemma suffix_length_suffix:
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   657
  "suffix ps xs \<Longrightarrow> suffix qs xs \<Longrightarrow> length ps \<le> length qs \<Longrightarrow> suffix ps qs"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   658
  by (auto simp: suffix_to_prefix intro: prefix_length_prefix)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   659
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   660
lemma suffix_length_less: "strict_suffix xs ys \<Longrightarrow> length xs < length ys"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   661
  by (auto simp: strict_suffix_def suffix_def)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   662
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   663
lemma suffix_ConsD': "suffix (x#xs) ys \<Longrightarrow> strict_suffix xs ys"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   664
  by (auto simp: strict_suffix_def suffix_def)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   665
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   666
lemma drop_strict_suffix: "strict_suffix xs ys \<Longrightarrow> strict_suffix (drop n xs) ys"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   667
proof (induct n arbitrary: xs ys)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   668
  case 0
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   669
  then show ?case by (cases ys) simp_all
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   670
next
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   671
  case (Suc n)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   672
  then show ?case 
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   673
    by (cases xs) (auto intro: Suc dest: suffix_ConsD' suffix_order.less_imp_le)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   674
qed
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   675
71789
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   676
lemma suffix_map_rightE:
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   677
  assumes "suffix xs (map f ys)"
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   678
  shows   "\<exists>xs'. suffix xs' ys \<and> xs = map f xs'"
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   679
proof -
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   680
  from assms obtain xs' where xs': "map f ys = xs' @ xs"
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   681
    by (auto simp: suffix_def)
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   682
  define n where "n = length xs'"
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   683
  have "xs = drop n (map f ys)"
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   684
    by (simp add: xs' n_def)
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   685
  thus ?thesis
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   686
    by (intro exI[of _ "drop n ys"]) (auto simp: drop_map suffix_drop)
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   687
qed
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   688
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   689
lemma suffix_remdups_adj: "suffix xs ys \<Longrightarrow> suffix (remdups_adj xs) (remdups_adj ys)"
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   690
  using prefix_remdups_adj[of "rev xs" "rev ys"]
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   691
  by (simp add: suffix_to_prefix)
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   692
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   693
lemma not_suffix_cases:
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   694
  assumes pfx: "\<not> suffix ps ls"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   695
  obtains
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   696
    (c1) "ps \<noteq> []" and "ls = []"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   697
  | (c2) a as x xs where "ps = as@[a]" and "ls = xs@[x]" and "x = a" and "\<not> suffix as xs"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   698
  | (c3) a as x xs where "ps = as@[a]" and "ls = xs@[x]" and "x \<noteq> a"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   699
proof (cases ps rule: rev_cases)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   700
  case Nil
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   701
  then show ?thesis using pfx by simp
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   702
next
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   703
  case (snoc as a)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   704
  note c = \<open>ps = as@[a]\<close>
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   705
  show ?thesis
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   706
  proof (cases ls rule: rev_cases)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   707
    case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_suffix_nil)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   708
  next
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   709
    case (snoc xs x)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   710
    show ?thesis
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   711
    proof (cases "x = a")
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   712
      case True
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   713
      have "\<not> suffix as xs" using pfx c snoc True by simp
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   714
      with c snoc True show ?thesis by (rule c2)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   715
    next
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   716
      case False
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   717
      with c snoc show ?thesis by (rule c3)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   718
    qed
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   719
  qed
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   720
qed
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   721
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   722
lemma not_suffix_induct [consumes 1, case_names Nil Neq Eq]:
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   723
  assumes np: "\<not> suffix ps ls"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   724
    and base: "\<And>x xs. P (xs@[x]) []"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   725
    and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (xs@[x]) (ys@[y])"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   726
    and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> suffix xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (xs@[x]) (ys@[y])"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   727
  shows "P ps ls" using np
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   728
proof (induct ls arbitrary: ps rule: rev_induct)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   729
  case Nil
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   730
  then show ?case by (cases ps rule: rev_cases) (auto intro: base)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   731
next
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   732
  case (snoc y ys ps)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   733
  then have npfx: "\<not> suffix ps (ys @ [y])" by simp
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   734
  then obtain x xs where pv: "ps = xs @ [x]"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   735
    by (rule not_suffix_cases) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   736
  show ?case by (metis snoc.hyps snoc_suffix_snoc npfx pv r1 r2)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   737
qed
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   738
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   739
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   740
lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefix x y"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   741
  by blast
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   742
63117
acb6d72fc42e renamed prefix* in Library/Sublist
nipkow
parents: 61076
diff changeset
   743
lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefix y x"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   744
  by blast
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   745
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   746
lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   747
  unfolding parallel_def by simp
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   748
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   749
lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   750
  unfolding parallel_def by simp
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   751
25564
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25356
diff changeset
   752
lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   753
  by auto
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   754
25564
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25356
diff changeset
   755
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
   756
  by (metis Cons_prefix_Cons parallelE parallelI)
25665
faabc08af882 removed legacy proofs
nipkow
parents: 25595
diff changeset
   757
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   758
lemma not_equal_is_parallel:
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   759
  assumes neq: "xs \<noteq> ys"
25356
059c03630d6e tuned presentation;
wenzelm
parents: 25355
diff changeset
   760
    and len: "length xs = length ys"
059c03630d6e tuned presentation;
wenzelm
parents: 25355
diff changeset
   761
  shows "xs \<parallel> ys"
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   762
  using len neq
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   763
proof (induct rule: list_induct2)
26445
17223cf843d8 explicit case names for rule list_induct2
haftmann
parents: 25764
diff changeset
   764
  case Nil
25356
059c03630d6e tuned presentation;
wenzelm
parents: 25355
diff changeset
   765
  then show ?case by simp
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   766
next
26445
17223cf843d8 explicit case names for rule list_induct2
haftmann
parents: 25764
diff changeset
   767
  case (Cons a as b bs)
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   768
  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
   769
  show ?case
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   770
  proof (cases "a = b")
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   771
    case True
26445
17223cf843d8 explicit case names for rule list_induct2
haftmann
parents: 25764
diff changeset
   772
    then have "as \<noteq> bs" using Cons by simp
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   773
    then show ?thesis by (rule Cons_parallelI2 [OF True ih])
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   774
  next
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   775
    case False
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   776
    then show ?thesis by (rule Cons_parallelI1)
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   777
  qed
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   778
qed
22178
29b95968272b made executable
haftmann
parents: 21404
diff changeset
   779
71789
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
   780
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   781
subsection \<open>Suffixes\<close>
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   782
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
   783
primrec suffixes where
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   784
  "suffixes [] = [[]]"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   785
| "suffixes (x#xs) = suffixes xs @ [x # xs]"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   786
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   787
lemma in_set_suffixes [simp]: "xs \<in> set (suffixes ys) \<longleftrightarrow> suffix xs ys"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   788
  by (induction ys) (auto simp: suffix_def Cons_eq_append_conv)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   789
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   790
lemma distinct_suffixes [intro]: "distinct (suffixes xs)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   791
  by (induction xs) (auto simp: suffix_def)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   792
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   793
lemma length_suffixes [simp]: "length (suffixes xs) = Suc (length xs)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   794
  by (induction xs) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   795
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   796
lemma suffixes_snoc [simp]: "suffixes (xs @ [x]) = [] # map (\<lambda>ys. ys @ [x]) (suffixes xs)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   797
  by (induction xs) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   798
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   799
lemma suffixes_not_Nil [simp]: "suffixes xs \<noteq> []"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   800
  by (cases xs) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   801
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   802
lemma hd_suffixes [simp]: "hd (suffixes xs) = []"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   803
  by (induction xs) simp_all
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   804
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   805
lemma last_suffixes [simp]: "last (suffixes xs) = xs"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   806
  by (cases xs) simp_all
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   807
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   808
lemma suffixes_append: 
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   809
  "suffixes (xs @ ys) = suffixes ys @ map (\<lambda>xs'. xs' @ ys) (tl (suffixes xs))"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   810
proof (induction ys rule: rev_induct)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   811
  case Nil
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   812
  thus ?case by (cases xs rule: rev_cases) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   813
next
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   814
  case (snoc y ys)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   815
  show ?case
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   816
    by (simp only: append.assoc [symmetric] suffixes_snoc snoc.IH) simp
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   817
qed
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   818
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   819
lemma suffixes_eq_snoc:
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   820
  "suffixes ys = xs @ [x] \<longleftrightarrow>
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   821
     (ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = z#zs \<and> xs = suffixes zs)) \<and> x = ys"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   822
  by (cases ys) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   823
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   824
lemma suffixes_tailrec [code]: 
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   825
  "suffixes xs = rev (snd (foldl (\<lambda>(acc1, acc2) x. (x#acc1, (x#acc1)#acc2)) ([],[[]]) (rev xs)))"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   826
proof -
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   827
  have "foldl (\<lambda>(acc1, acc2) x. (x#acc1, (x#acc1)#acc2)) (ys, ys # zs) (rev xs) =
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   828
          (xs @ ys, rev (map (\<lambda>as. as @ ys) (suffixes xs)) @ zs)" for ys zs
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   829
  proof (induction xs arbitrary: ys zs)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   830
    case (Cons x xs ys zs)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   831
    from Cons.IH[of ys zs]
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   832
      show ?case by (simp add: o_def case_prod_unfold)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   833
  qed simp_all
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   834
  from this [of "[]" "[]"] show ?thesis by simp
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   835
qed
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   836
  
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   837
lemma set_suffixes_eq: "set (suffixes xs) = {ys. suffix ys xs}"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   838
  by auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   839
    
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   840
lemma card_set_suffixes [simp]: "card (set (suffixes xs)) = Suc (length xs)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   841
  by (subst distinct_card) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   842
  
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   843
lemma set_suffixes_append: 
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   844
  "set (suffixes (xs @ ys)) = set (suffixes ys) \<union> {xs' @ ys |xs'. xs' \<in> set (suffixes xs)}"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   845
  by (subst suffixes_append, cases xs rule: rev_cases) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   846
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   847
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   848
lemma suffixes_conv_prefixes: "suffixes xs = map rev (prefixes (rev xs))"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   849
  by (induction xs) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   850
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   851
lemma prefixes_conv_suffixes: "prefixes xs = map rev (suffixes (rev xs))"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   852
  by (induction xs) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   853
    
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   854
lemma prefixes_rev: "prefixes (rev xs) = map rev (suffixes xs)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   855
  by (induction xs) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   856
    
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   857
lemma suffixes_rev: "suffixes (rev xs) = map rev (prefixes xs)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   858
  by (induction xs) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   859
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   860
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   861
subsection \<open>Homeomorphic embedding on lists\<close>
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   862
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   863
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
   864
  for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   865
where
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   866
  list_emb_Nil [intro, simp]: "list_emb P [] ys"
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   867
| 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
   868
| 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
   869
57499
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   870
lemma list_emb_mono:                         
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   871
  assumes "\<And>x y. P x y \<longrightarrow> Q x y"
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   872
  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
   873
proof                                        
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   874
  assume "list_emb P xs ys"                    
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   875
  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
   876
qed 
7e22776f2d32 added monotonicity lemma for list embedding
Christian Sternagel
parents: 57498
diff changeset
   877
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   878
lemma list_emb_Nil2 [simp]:
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   879
  assumes "list_emb P xs []" shows "xs = []"
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   880
  using assms by (cases rule: list_emb.cases) auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   881
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
   882
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
   883
  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
   884
  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
   885
  using assms by (induct xs) auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   886
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   887
lemma list_emb_Cons_Nil [simp]: "list_emb P (x#xs) [] = False"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   888
proof -
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   889
  { assume "list_emb P (x#xs) []"
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   890
    from list_emb_Nil2 [OF this] have False by simp
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   891
  } moreover {
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   892
    assume False
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   893
    then have "list_emb P (x#xs) []" by simp
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   894
  } ultimately show ?thesis by blast
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   895
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   896
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   897
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
   898
  by (induct zs) auto
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   899
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   900
lemma list_emb_prefix [intro]:
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   901
  assumes "list_emb P xs ys" shows "list_emb P xs (ys @ zs)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   902
  using assms
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   903
  by (induct arbitrary: zs) auto
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   904
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   905
lemma list_emb_ConsD:
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   906
  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
   907
  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
   908
using assms
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   909
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
   910
  case list_emb_Cons
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   911
  then show ?case by (metis append_Cons)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   912
next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   913
  case (list_emb_Cons2 x y xs ys)
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   914
  then show ?case by blast
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   915
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   916
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   917
lemma list_emb_appendD:
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   918
  assumes "list_emb P (xs @ ys) zs"
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   919
  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
   920
using assms
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   921
proof (induction xs arbitrary: ys zs)
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
   922
  case Nil then show ?case by auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   923
next
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   924
  case (Cons x xs)
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   925
  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
   926
    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
   927
    by (auto dest: list_emb_ConsD)
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
   928
  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
   929
    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
   930
    using Cons(1) by (metis (no_types))
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   931
  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
   932
  thus ?case using lh zs sk by (metis (no_types) append_Cons append_assoc)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   933
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   934
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   935
lemma list_emb_strict_suffix:
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   936
  assumes "list_emb P xs ys" and "strict_suffix ys zs"
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   937
  shows "list_emb P xs zs"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   938
  using assms(2) and list_emb_append2 [OF assms(1)] by (auto simp: strict_suffix_def suffix_def)
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   939
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   940
lemma list_emb_suffix:
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   941
  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
   942
  shows "list_emb P xs zs"
63149
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   943
using assms and list_emb_strict_suffix
f5dbab18c404 renamed suffix(eq)
nipkow
parents: 63117
diff changeset
   944
unfolding strict_suffix_reflclp_conv[symmetric] by auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   945
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   946
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
   947
  by (induct rule: list_emb.induct) auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   948
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   949
lemma list_emb_trans:
57500
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   950
  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
   951
  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
   952
proof -
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   953
  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
   954
  then show "list_emb P xs zs" using assms
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   955
  proof (induction arbitrary: zs)
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   956
    case list_emb_Nil show ?case by blast
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   957
  next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   958
    case (list_emb_Cons xs ys y)
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   959
    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
   960
      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
   961
    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
   962
    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
   963
    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
   964
  next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   965
    case (list_emb_Cons2 x y xs ys)
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   966
    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
   967
      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
   968
    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
   969
    moreover have "P x v"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   970
    proof -
57500
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   971
      from zs have "v \<in> set zs" by auto
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   972
      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
   973
      ultimately show ?thesis
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
   974
        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
   975
        by blast
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   976
    qed
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
   977
    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
   978
    then show ?case unfolding zs by (rule list_emb_append2)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   979
  qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   980
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
   981
57500
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   982
lemma list_emb_set:
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   983
  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
   984
  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
   985
  using assms by (induct) auto
5a8b3e9d82a4 weaker assumption for "list_emb_trans"; added lemma
Christian Sternagel
parents: 57499
diff changeset
   986
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   987
lemma list_emb_Cons_iff1 [simp]:
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   988
  assumes "P x y"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   989
  shows   "list_emb P (x#xs) (y#ys) \<longleftrightarrow> list_emb P xs ys"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   990
  using assms by (subst list_emb.simps) (auto dest: list_emb_ConsD)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   991
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   992
lemma list_emb_Cons_iff2 [simp]:
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   993
  assumes "\<not>P x y"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   994
  shows   "list_emb P (x#xs) (y#ys) \<longleftrightarrow> list_emb P (x#xs) ys"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   995
  using assms by (subst list_emb.simps) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   996
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   997
lemma list_emb_code [code]:
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   998
  "list_emb P [] ys \<longleftrightarrow> True"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
   999
  "list_emb P (x#xs) [] \<longleftrightarrow> False"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1000
  "list_emb P (x#xs) (y#ys) \<longleftrightarrow> (if P x y then list_emb P xs ys else list_emb P (x#xs) ys)"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1001
  by simp_all
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1002
    
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1003
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1004
subsection \<open>Subsequences (special case of homeomorphic embedding)\<close>
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1005
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1006
abbreviation subseq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
  1007
  where "subseq xs ys \<equiv> list_emb (=) xs ys"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1008
  
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1009
definition strict_subseq where "strict_subseq xs ys \<longleftrightarrow> xs \<noteq> ys \<and> subseq xs ys"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1010
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1011
lemma subseq_Cons2: "subseq xs ys \<Longrightarrow> subseq (x#xs) (x#ys)" by auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1012
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1013
lemma subseq_same_length:
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1014
  assumes "subseq 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
  1015
  using assms by (induct) (auto dest: list_emb_length)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1016
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1017
lemma not_subseq_length [simp]: "length ys < length xs \<Longrightarrow> \<not> subseq xs ys"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1018
  by (metis list_emb_length linorder_not_less)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1019
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1020
lemma subseq_Cons': "subseq (x#xs) ys \<Longrightarrow> subseq xs ys"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1021
  by (induct xs, simp, blast dest: list_emb_ConsD)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1022
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1023
lemma subseq_Cons2':
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1024
  assumes "subseq (x#xs) (x#ys)" shows "subseq xs ys"
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1025
  using assms by (cases) (rule subseq_Cons')
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1026
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1027
lemma subseq_Cons2_neq:
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1028
  assumes "subseq (x#xs) (y#ys)"
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1029
  shows "x \<noteq> y \<Longrightarrow> subseq (x#xs) ys"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1030
  using assms by (cases) auto
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1031
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1032
lemma subseq_Cons2_iff [simp]:
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1033
  "subseq (x#xs) (y#ys) = (if x = y then subseq xs ys else subseq (x#xs) ys)"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1034
  by simp
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1035
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1036
lemma subseq_append': "subseq (zs @ xs) (zs @ ys) \<longleftrightarrow> subseq xs ys"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1037
  by (induct zs) simp_all
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1038
    
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1039
interpretation subseq_order: order subseq strict_subseq
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1040
proof
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1041
  fix xs ys :: "'a list"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1042
  {
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1043
    assume "subseq xs ys" and "subseq ys xs"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1044
    thus "xs = ys"
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1045
    proof (induct)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1046
      case list_emb_Nil
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1047
      from list_emb_Nil2 [OF this] show ?case by simp
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1048
    next
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1049
      case list_emb_Cons2
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1050
      thus ?case by simp
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1051
    next
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1052
      case list_emb_Cons
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1053
      hence False using subseq_Cons' by fastforce
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1054
      thus ?case ..
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1055
    qed
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1056
  }
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1057
  thus "strict_subseq xs ys \<longleftrightarrow> (subseq xs ys \<and> \<not>subseq ys xs)"
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1058
    by (auto simp: strict_subseq_def)
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1059
qed (auto simp: list_emb_refl intro: list_emb_trans)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1060
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1061
lemma in_set_subseqs [simp]: "xs \<in> set (subseqs ys) \<longleftrightarrow> subseq xs ys"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1062
proof
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1063
  assume "xs \<in> set (subseqs ys)"
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1064
  thus "subseq xs ys"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1065
    by (induction ys arbitrary: xs) (auto simp: Let_def)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1066
next
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1067
  have [simp]: "[] \<in> set (subseqs ys)" for ys :: "'a list" 
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1068
    by (induction ys) (auto simp: Let_def)
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1069
  assume "subseq xs ys"
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1070
  thus "xs \<in> set (subseqs ys)"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1071
    by (induction xs ys rule: list_emb.induct) (auto simp: Let_def)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1072
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1073
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1074
lemma set_subseqs_eq: "set (subseqs ys) = {xs. subseq xs ys}"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1075
  by auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1076
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1077
lemma subseq_append_le_same_iff: "subseq (xs @ ys) ys \<longleftrightarrow> xs = []"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1078
  by (auto dest: list_emb_length)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1079
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1080
lemma subseq_singleton_left: "subseq [x] ys \<longleftrightarrow> x \<in> set ys"
64886
cea327ecb8e3 added lemma
blanchet
parents: 63649
diff changeset
  1081
  by (fastforce dest: list_emb_ConsD split_list_last)
cea327ecb8e3 added lemma
blanchet
parents: 63649
diff changeset
  1082
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1083
lemma list_emb_append_mono:
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1084
  "\<lbrakk> list_emb P xs xs'; list_emb P ys ys' \<rbrakk> \<Longrightarrow> list_emb P (xs@ys) (xs'@ys')"
65957
558ba6b37f5c Tuned Library/Sublist.thy
eberlm <eberlm@in.tum.de>
parents: 65956
diff changeset
  1085
  by (induct rule: list_emb.induct) auto
558ba6b37f5c Tuned Library/Sublist.thy
eberlm <eberlm@in.tum.de>
parents: 65956
diff changeset
  1086
558ba6b37f5c Tuned Library/Sublist.thy
eberlm <eberlm@in.tum.de>
parents: 65956
diff changeset
  1087
lemma prefix_imp_subseq [intro]: "prefix xs ys \<Longrightarrow> subseq xs ys"
558ba6b37f5c Tuned Library/Sublist.thy
eberlm <eberlm@in.tum.de>
parents: 65956
diff changeset
  1088
  by (auto simp: prefix_def)
558ba6b37f5c Tuned Library/Sublist.thy
eberlm <eberlm@in.tum.de>
parents: 65956
diff changeset
  1089
558ba6b37f5c Tuned Library/Sublist.thy
eberlm <eberlm@in.tum.de>
parents: 65956
diff changeset
  1090
lemma suffix_imp_subseq [intro]: "suffix xs ys \<Longrightarrow> subseq xs ys"
558ba6b37f5c Tuned Library/Sublist.thy
eberlm <eberlm@in.tum.de>
parents: 65956
diff changeset
  1091
  by (auto simp: suffix_def)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1092
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1093
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
  1094
subsection \<open>Appending elements\<close>
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1095
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1096
lemma subseq_append [simp]:
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1097
  "subseq (xs @ zs) (ys @ zs) \<longleftrightarrow> subseq xs ys" (is "?l = ?r")
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1098
proof
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1099
  { fix xs' ys' xs ys zs :: "'a list" assume "subseq xs' ys'"
67091
1393c2340eec more symbols;
wenzelm
parents: 65957
diff changeset
  1100
    then have "xs' = xs @ zs \<and> ys' = ys @ zs \<longrightarrow> subseq xs ys"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1101
    proof (induct arbitrary: xs ys zs)
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1102
      case list_emb_Nil show ?case by simp
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1103
    next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1104
      case (list_emb_Cons xs' ys' x)
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1105
      { assume "ys=[]" then have ?case using list_emb_Cons(1) by auto }
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1106
      moreover
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1107
      { fix us assume "ys = x#us"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1108
        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
  1109
      ultimately show ?case by (auto simp:Cons_eq_append_conv)
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1110
    next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1111
      case (list_emb_Cons2 x y xs' ys')
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1112
      { assume "xs=[]" then have ?case using list_emb_Cons2(1) by auto }
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1113
      moreover
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1114
      { 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
  1115
      moreover
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1116
      { fix us assume "xs=x#us" "ys=[]" then have ?case using list_emb_Cons2(2) by bestsimp }
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
  1117
      ultimately show ?case using \<open>(=) x y\<close> by (auto simp: Cons_eq_append_conv)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1118
    qed }
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1119
  moreover assume ?l
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1120
  ultimately show ?r by blast
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1121
next
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1122
  assume ?r then show ?l by (metis list_emb_append_mono subseq_order.order_refl)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1123
qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1124
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1125
lemma subseq_append_iff: 
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1126
  "subseq xs (ys @ zs) \<longleftrightarrow> (\<exists>xs1 xs2. xs = xs1 @ xs2 \<and> subseq xs1 ys \<and> subseq xs2 zs)"
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1127
  (is "?lhs = ?rhs")
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1128
proof
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1129
  assume ?lhs thus ?rhs
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1130
  proof (induction xs "ys @ zs" arbitrary: ys zs rule: list_emb.induct)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1131
    case (list_emb_Cons xs ws y ys zs)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1132
    from list_emb_Cons(2)[of "tl ys" zs] and list_emb_Cons(2)[of "[]" "tl zs"] and list_emb_Cons(1,3)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1133
      show ?case by (cases ys) auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1134
  next
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1135
    case (list_emb_Cons2 x y xs ws ys zs)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1136
    from list_emb_Cons2(3)[of "tl ys" zs] and list_emb_Cons2(3)[of "[]" "tl zs"]
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1137
       and list_emb_Cons2(1,2,4)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1138
    show ?case by (cases ys) (auto simp: Cons_eq_append_conv)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1139
  qed auto
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1140
qed (auto intro: list_emb_append_mono)
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1141
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1142
lemma subseq_appendE [case_names append]: 
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1143
  assumes "subseq xs (ys @ zs)"
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1144
  obtains xs1 xs2 where "xs = xs1 @ xs2" "subseq xs1 ys" "subseq xs2 zs"
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1145
  using assms by (subst (asm) subseq_append_iff) auto
65869
a6ed757b8585 more on sublists
eberlm <eberlm@in.tum.de>
parents: 64886
diff changeset
  1146
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1147
lemma subseq_drop_many: "subseq xs ys \<Longrightarrow> subseq xs (zs @ ys)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1148
  by (induct zs) auto
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1149
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1150
lemma subseq_rev_drop_many: "subseq xs ys \<Longrightarrow> subseq xs (ys @ zs)"
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1151
  by (metis append_Nil2 list_emb_Nil list_emb_append_mono)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1152
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1153
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59997
diff changeset
  1154
subsection \<open>Relation to standard list operations\<close>
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1155
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1156
lemma subseq_map:
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1157
  assumes "subseq xs ys" shows "subseq (map f xs) (map f ys)"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1158
  using assms by (induct) auto
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1159
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1160
lemma subseq_filter_left [simp]: "subseq (filter P xs) xs"
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1161
  by (induct xs) auto
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1162
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1163
lemma subseq_filter [simp]:
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1164
  assumes "subseq xs ys" shows "subseq (filter P xs) (filter P ys)"
54483
9f24325c2550 optimized more bad apples
blanchet
parents: 53015
diff changeset
  1165
  using assms by induct auto
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1166
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1167
lemma subseq_conv_nths: 
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1168
  "subseq xs ys \<longleftrightarrow> (\<exists>N. xs = nths ys N)" (is "?L = ?R")
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1169
proof
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1170
  assume ?L
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
  1171
  then show ?R
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1172
  proof (induct)
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1173
    case list_emb_Nil show ?case by (metis nths_empty)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1174
  next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1175
    case (list_emb_Cons xs ys x)
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1176
    then obtain N where "xs = nths ys N" by blast
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1177
    then have "xs = nths (x#ys) (Suc ` N)"
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1178
      by (clarsimp simp add: nths_Cons inj_image_mem_iff)
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
  1179
    then show ?case by blast
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1180
  next
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1181
    case (list_emb_Cons2 x y xs ys)
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1182
    then obtain N where "xs = nths ys N" by blast
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1183
    then have "x#xs = nths (x#ys) (insert 0 (Suc ` N))"
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1184
      by (clarsimp simp add: nths_Cons inj_image_mem_iff)
57497
4106a2bc066a renamed "list_hembeq" into slightly shorter "list_emb"
Christian Sternagel
parents: 55579
diff changeset
  1185
    moreover from list_emb_Cons2 have "x = y" by simp
50516
ed6b40d15d1c renamed "emb" to "list_hembeq";
Christian Sternagel
parents: 49107
diff changeset
  1186
    ultimately show ?case by blast
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1187
  qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1188
next
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1189
  assume ?R
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1190
  then obtain N where "xs = nths ys N" ..
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1191
  moreover have "subseq (nths ys N) ys"
49107
ec34e9df0514 misc tuning;
wenzelm
parents: 49087
diff changeset
  1192
  proof (induct ys arbitrary: N)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1193
    case Nil show ?case by simp
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1194
  next
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1195
    case Cons then show ?case by (auto simp: nths_Cons)
49087
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1196
  qed
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1197
  ultimately show ?L by simp
7a17ba4bc997 added author
Christian Sternagel
parents: 45236
diff changeset
  1198
qed
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1199
  
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1200
  
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1201
subsection \<open>Contiguous sublists\<close>
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1202
71789
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
  1203
subsubsection \<open>\<open>sublist\<close>\<close>
3b6547bdf6e2 added lemmas
nipkow
parents: 68406
diff changeset
  1204
65956
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1205
definition sublist :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where 
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1206
  "sublist xs ys = (\<exists>ps ss. ys = ps @ xs @ ss)"
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1207
  
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1208
definition strict_sublist :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where 
639eb3617a86 reorganised material on sublists
eberlm <eberlm@in.tum.de>
parents: 65954
diff changeset
  1209
  "strict_sublist xs ys \<longleftrightarrow> sublist xs ys \<and> xs \<noteq