src/HOL/Library/Sublist.thy
author Andreas Lochbihler
Fri Sep 20 10:09:16 2013 +0200 (2013-09-20)
changeset 53745 788730ab7da4
parent 53015 a1119cf551e8
child 54483 9f24325c2550
permissions -rw-r--r--
prefer Code.abort over code_abort
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(*  Title:      HOL/Library/Sublist.thy
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    Author:     Tobias Nipkow and Markus Wenzel, TU Muenchen
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    Author:     Christian Sternagel, JAIST
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*)
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header {* List prefixes, suffixes, and homeomorphic embedding *}
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theory Sublist
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imports Main
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begin
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subsection {* Prefix order on lists *}
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definition prefixeq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
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  where "prefixeq xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"
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definition prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
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  where "prefix xs ys \<longleftrightarrow> prefixeq xs ys \<and> xs \<noteq> ys"
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interpretation prefix_order: order prefixeq prefix
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  by default (auto simp: prefixeq_def prefix_def)
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interpretation prefix_bot: order_bot Nil prefixeq prefix
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  by default (simp add: prefixeq_def)
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lemma prefixeqI [intro?]: "ys = xs @ zs \<Longrightarrow> prefixeq xs ys"
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  unfolding prefixeq_def by blast
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lemma prefixeqE [elim?]:
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  assumes "prefixeq xs ys"
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  obtains zs where "ys = xs @ zs"
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  using assms unfolding prefixeq_def by blast
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lemma prefixI' [intro?]: "ys = xs @ z # zs \<Longrightarrow> prefix xs ys"
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  unfolding prefix_def prefixeq_def by blast
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lemma prefixE' [elim?]:
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  assumes "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 `prefix xs ys` obtain us where "ys = xs @ us" and "xs \<noteq> ys"
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    unfolding prefix_def prefixeq_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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lemma prefixI [intro?]: "prefixeq xs ys \<Longrightarrow> xs \<noteq> ys \<Longrightarrow> prefix xs ys"
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  unfolding prefix_def by blast
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lemma prefixE [elim?]:
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  fixes xs ys :: "'a list"
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  assumes "prefix xs ys"
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  obtains "prefixeq xs ys" and "xs \<noteq> ys"
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  using assms unfolding prefix_def by blast
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subsection {* Basic properties of prefixes *}
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theorem Nil_prefixeq [iff]: "prefixeq [] xs"
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  by (simp add: prefixeq_def)
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theorem prefixeq_Nil [simp]: "(prefixeq xs []) = (xs = [])"
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  by (induct xs) (simp_all add: prefixeq_def)
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lemma prefixeq_snoc [simp]: "prefixeq xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefixeq xs ys"
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proof
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  assume "prefixeq 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> prefixeq xs ys"
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    by (metis append_Nil2 butlast_append butlast_snoc prefixeqI zs)
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next
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  assume "xs = ys @ [y] \<or> prefixeq xs ys"
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  then show "prefixeq xs (ys @ [y])"
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    by (metis prefix_order.eq_iff prefix_order.order_trans prefixeqI)
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qed
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lemma Cons_prefixeq_Cons [simp]: "prefixeq (x # xs) (y # ys) = (x = y \<and> prefixeq xs ys)"
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  by (auto simp add: prefixeq_def)
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lemma prefixeq_code [code]:
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  "prefixeq [] xs \<longleftrightarrow> True"
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  "prefixeq (x # xs) [] \<longleftrightarrow> False"
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  "prefixeq (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefixeq xs ys"
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  by simp_all
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lemma same_prefixeq_prefixeq [simp]: "prefixeq (xs @ ys) (xs @ zs) = prefixeq ys zs"
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  by (induct xs) simp_all
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lemma same_prefixeq_nil [iff]: "prefixeq (xs @ ys) xs = (ys = [])"
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  by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixeqI)
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lemma prefixeq_prefixeq [simp]: "prefixeq xs ys \<Longrightarrow> prefixeq xs (ys @ zs)"
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  by (metis prefix_order.le_less_trans prefixeqI prefixE prefixI)
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lemma append_prefixeqD: "prefixeq (xs @ ys) zs \<Longrightarrow> prefixeq xs zs"
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  by (auto simp add: prefixeq_def)
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theorem prefixeq_Cons: "prefixeq xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefixeq zs ys))"
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  by (cases xs) (auto simp add: prefixeq_def)
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theorem prefixeq_append:
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  "prefixeq xs (ys @ zs) = (prefixeq xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefixeq us zs))"
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  apply (induct zs rule: rev_induct)
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   apply force
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  apply (simp del: append_assoc add: append_assoc [symmetric])
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  apply (metis append_eq_appendI)
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  done
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lemma append_one_prefixeq:
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  "prefixeq xs ys \<Longrightarrow> length xs < length ys \<Longrightarrow> prefixeq (xs @ [ys ! length xs]) ys"
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  unfolding prefixeq_def
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  by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj
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    eq_Nil_appendI nth_drop')
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theorem prefixeq_length_le: "prefixeq xs ys \<Longrightarrow> length xs \<le> length ys"
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  by (auto simp add: prefixeq_def)
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lemma prefixeq_same_cases:
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  "prefixeq (xs\<^sub>1::'a list) ys \<Longrightarrow> prefixeq xs\<^sub>2 ys \<Longrightarrow> prefixeq xs\<^sub>1 xs\<^sub>2 \<or> prefixeq xs\<^sub>2 xs\<^sub>1"
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  unfolding prefixeq_def by (metis append_eq_append_conv2)
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lemma set_mono_prefixeq: "prefixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys"
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  by (auto simp add: prefixeq_def)
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lemma take_is_prefixeq: "prefixeq (take n xs) xs"
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  unfolding prefixeq_def by (metis append_take_drop_id)
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lemma map_prefixeqI: "prefixeq xs ys \<Longrightarrow> prefixeq (map f xs) (map f ys)"
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  by (auto simp: prefixeq_def)
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lemma prefixeq_length_less: "prefix xs ys \<Longrightarrow> length xs < length ys"
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  by (auto simp: prefix_def prefixeq_def)
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lemma prefix_simps [simp, code]:
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  "prefix xs [] \<longleftrightarrow> False"
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  "prefix [] (x # xs) \<longleftrightarrow> True"
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  "prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys"
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  by (simp_all add: prefix_def cong: conj_cong)
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lemma take_prefix: "prefix xs ys \<Longrightarrow> prefix (take n xs) ys"
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  apply (induct n arbitrary: xs ys)
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   apply (case_tac ys, simp_all)[1]
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  apply (metis prefix_order.less_trans prefixI take_is_prefixeq)
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  done
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lemma not_prefixeq_cases:
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  assumes pfx: "\<not> prefixeq 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> prefixeq 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 = `ps = a#as`
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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_prefixeq_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> prefixeq 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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lemma not_prefixeq_induct [consumes 1, case_names Nil Neq Eq]:
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  assumes np: "\<not> prefixeq 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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    and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefixeq 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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proof (induct ls arbitrary: ps)
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  case Nil then show ?case
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    by (auto simp: neq_Nil_conv elim!: not_prefixeq_cases intro!: base)
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next
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  case (Cons y ys)
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  then have npfx: "\<not> prefixeq 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_prefixeq_cases) auto
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  show ?case by (metis Cons.hyps Cons_prefixeq_Cons npfx pv r1 r2)
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qed
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subsection {* Parallel lists *}
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definition parallel :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "\<parallel>" 50)
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  where "(xs \<parallel> ys) = (\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs)"
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lemma parallelI [intro]: "\<not> prefixeq xs ys \<Longrightarrow> \<not> prefixeq ys xs \<Longrightarrow> xs \<parallel> ys"
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  unfolding parallel_def by blast
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lemma parallelE [elim]:
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  assumes "xs \<parallel> ys"
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  obtains "\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs"
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  using assms unfolding parallel_def by blast
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theorem prefixeq_cases:
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  obtains "prefixeq xs ys" | "prefix ys xs" | "xs \<parallel> ys"
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  unfolding parallel_def prefix_def by blast
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theorem parallel_decomp:
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  "xs \<parallel> ys \<Longrightarrow> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
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proof (induct xs rule: rev_induct)
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  case Nil
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  then have False by auto
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  then show ?case ..
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next
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  case (snoc x xs)
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  show ?case
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  proof (rule prefixeq_cases)
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    assume le: "prefixeq xs ys"
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    then obtain ys' where ys: "ys = xs @ ys'" ..
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    show ?thesis
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    proof (cases ys')
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      assume "ys' = []"
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      then show ?thesis by (metis append_Nil2 parallelE prefixeqI snoc.prems ys)
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    next
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      fix c cs assume ys': "ys' = c # cs"
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      then show ?thesis
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        by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixeqI
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          same_prefixeq_prefixeq snoc.prems ys)
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    qed
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  next
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    assume "prefix ys xs"
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    then have "prefixeq ys (xs @ [x])" by (simp add: prefix_def)
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    with snoc have False by blast
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    then show ?thesis ..
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  next
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    assume "xs \<parallel> ys"
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    with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> c"
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      and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
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      by blast
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    from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
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    with neq ys show ?thesis by blast
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  qed
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qed
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lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
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  apply (rule parallelI)
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    apply (erule parallelE, erule conjE,
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      induct rule: not_prefixeq_induct, simp+)+
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  done
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lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y"
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  by (simp add: parallel_append)
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lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"
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  unfolding parallel_def by auto
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subsection {* Suffix order on lists *}
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definition suffixeq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
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  where "suffixeq xs ys = (\<exists>zs. ys = zs @ xs)"
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definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
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  where "suffix xs ys \<longleftrightarrow> (\<exists>us. ys = us @ xs \<and> us \<noteq> [])"
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lemma suffix_imp_suffixeq:
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  "suffix xs ys \<Longrightarrow> suffixeq xs ys"
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  by (auto simp: suffixeq_def suffix_def)
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lemma suffixeqI [intro?]: "ys = zs @ xs \<Longrightarrow> suffixeq xs ys"
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  unfolding suffixeq_def by blast
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lemma suffixeqE [elim?]:
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  assumes "suffixeq xs ys"
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  obtains zs where "ys = zs @ xs"
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  using assms unfolding suffixeq_def by blast
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lemma suffixeq_refl [iff]: "suffixeq xs xs"
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  by (auto simp add: suffixeq_def)
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lemma suffix_trans:
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  "suffix xs ys \<Longrightarrow> suffix ys zs \<Longrightarrow> suffix xs zs"
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  by (auto simp: suffix_def)
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lemma suffixeq_trans: "\<lbrakk>suffixeq xs ys; suffixeq ys zs\<rbrakk> \<Longrightarrow> suffixeq xs zs"
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  by (auto simp add: suffixeq_def)
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lemma suffixeq_antisym: "\<lbrakk>suffixeq xs ys; suffixeq ys xs\<rbrakk> \<Longrightarrow> xs = ys"
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  by (auto simp add: suffixeq_def)
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lemma suffixeq_tl [simp]: "suffixeq (tl xs) xs"
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  by (induct xs) (auto simp: suffixeq_def)
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lemma suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> suffix (tl xs) xs"
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  by (induct xs) (auto simp: suffix_def)
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lemma Nil_suffixeq [iff]: "suffixeq [] xs"
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  by (simp add: suffixeq_def)
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lemma suffixeq_Nil [simp]: "(suffixeq xs []) = (xs = [])"
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  by (auto simp add: suffixeq_def)
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lemma suffixeq_ConsI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (y # ys)"
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  by (auto simp add: suffixeq_def)
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lemma suffixeq_ConsD: "suffixeq (x # xs) ys \<Longrightarrow> suffixeq xs ys"
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  by (auto simp add: suffixeq_def)
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lemma suffixeq_appendI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (zs @ ys)"
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  by (auto simp add: suffixeq_def)
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lemma suffixeq_appendD: "suffixeq (zs @ xs) ys \<Longrightarrow> suffixeq xs ys"
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  by (auto simp add: suffixeq_def)
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lemma suffix_set_subset:
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  "suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffix_def)
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lemma suffixeq_set_subset:
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  "suffixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffixeq_def)
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lemma suffixeq_ConsD2: "suffixeq (x # xs) (y # ys) \<Longrightarrow> suffixeq xs ys"
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proof -
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  assume "suffixeq (x # xs) (y # ys)"
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  then obtain zs where "y # ys = zs @ x # xs" ..
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  then show ?thesis
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   321
    by (induct zs) (auto intro!: suffixeq_appendI suffixeq_ConsI)
wenzelm@21305
   322
qed
oheimb@14538
   323
Christian@49087
   324
lemma suffixeq_to_prefixeq [code]: "suffixeq xs ys \<longleftrightarrow> prefixeq (rev xs) (rev ys)"
Christian@49087
   325
proof
Christian@49087
   326
  assume "suffixeq xs ys"
Christian@49087
   327
  then obtain zs where "ys = zs @ xs" ..
Christian@49087
   328
  then have "rev ys = rev xs @ rev zs" by simp
Christian@49087
   329
  then show "prefixeq (rev xs) (rev ys)" ..
Christian@49087
   330
next
Christian@49087
   331
  assume "prefixeq (rev xs) (rev ys)"
Christian@49087
   332
  then obtain zs where "rev ys = rev xs @ zs" ..
Christian@49087
   333
  then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp
Christian@49087
   334
  then have "ys = rev zs @ xs" by simp
Christian@49087
   335
  then show "suffixeq xs ys" ..
wenzelm@21305
   336
qed
oheimb@14538
   337
Christian@49087
   338
lemma distinct_suffixeq: "distinct ys \<Longrightarrow> suffixeq xs ys \<Longrightarrow> distinct xs"
Christian@49087
   339
  by (clarsimp elim!: suffixeqE)
wenzelm@17201
   340
Christian@49087
   341
lemma suffixeq_map: "suffixeq xs ys \<Longrightarrow> suffixeq (map f xs) (map f ys)"
Christian@49087
   342
  by (auto elim!: suffixeqE intro: suffixeqI)
kleing@25299
   343
Christian@49087
   344
lemma suffixeq_drop: "suffixeq (drop n as) as"
Christian@49087
   345
  unfolding suffixeq_def
wenzelm@25692
   346
  apply (rule exI [where x = "take n as"])
wenzelm@25692
   347
  apply simp
wenzelm@25692
   348
  done
kleing@25299
   349
Christian@49087
   350
lemma suffixeq_take: "suffixeq xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs"
wenzelm@49107
   351
  by (auto elim!: suffixeqE)
kleing@25299
   352
wenzelm@49107
   353
lemma suffixeq_suffix_reflclp_conv: "suffixeq = suffix\<^sup>=\<^sup>="
Christian@49087
   354
proof (intro ext iffI)
Christian@49087
   355
  fix xs ys :: "'a list"
Christian@49087
   356
  assume "suffixeq xs ys"
Christian@49087
   357
  show "suffix\<^sup>=\<^sup>= xs ys"
Christian@49087
   358
  proof
Christian@49087
   359
    assume "xs \<noteq> ys"
wenzelm@49107
   360
    with `suffixeq xs ys` show "suffix xs ys"
wenzelm@49107
   361
      by (auto simp: suffixeq_def suffix_def)
Christian@49087
   362
  qed
Christian@49087
   363
next
Christian@49087
   364
  fix xs ys :: "'a list"
Christian@49087
   365
  assume "suffix\<^sup>=\<^sup>= xs ys"
wenzelm@49107
   366
  then show "suffixeq xs ys"
Christian@49087
   367
  proof
wenzelm@49107
   368
    assume "suffix xs ys" then show "suffixeq xs ys"
wenzelm@49107
   369
      by (rule suffix_imp_suffixeq)
Christian@49087
   370
  next
wenzelm@49107
   371
    assume "xs = ys" then show "suffixeq xs ys"
wenzelm@49107
   372
      by (auto simp: suffixeq_def)
Christian@49087
   373
  qed
Christian@49087
   374
qed
Christian@49087
   375
Christian@49087
   376
lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefixeq x y"
wenzelm@25692
   377
  by blast
kleing@25299
   378
Christian@49087
   379
lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefixeq y x"
wenzelm@25692
   380
  by blast
wenzelm@25355
   381
wenzelm@25355
   382
lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
wenzelm@25692
   383
  unfolding parallel_def by simp
wenzelm@25355
   384
kleing@25299
   385
lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
wenzelm@25692
   386
  unfolding parallel_def by simp
kleing@25299
   387
nipkow@25564
   388
lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
wenzelm@25692
   389
  by auto
kleing@25299
   390
nipkow@25564
   391
lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
Christian@49087
   392
  by (metis Cons_prefixeq_Cons parallelE parallelI)
nipkow@25665
   393
kleing@25299
   394
lemma not_equal_is_parallel:
kleing@25299
   395
  assumes neq: "xs \<noteq> ys"
wenzelm@25356
   396
    and len: "length xs = length ys"
wenzelm@25356
   397
  shows "xs \<parallel> ys"
kleing@25299
   398
  using len neq
wenzelm@25355
   399
proof (induct rule: list_induct2)
haftmann@26445
   400
  case Nil
wenzelm@25356
   401
  then show ?case by simp
kleing@25299
   402
next
haftmann@26445
   403
  case (Cons a as b bs)
wenzelm@25355
   404
  have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact
kleing@25299
   405
  show ?case
kleing@25299
   406
  proof (cases "a = b")
wenzelm@25355
   407
    case True
haftmann@26445
   408
    then have "as \<noteq> bs" using Cons by simp
wenzelm@25355
   409
    then show ?thesis by (rule Cons_parallelI2 [OF True ih])
kleing@25299
   410
  next
kleing@25299
   411
    case False
wenzelm@25355
   412
    then show ?thesis by (rule Cons_parallelI1)
kleing@25299
   413
  qed
kleing@25299
   414
qed
haftmann@22178
   415
wenzelm@49107
   416
lemma suffix_reflclp_conv: "suffix\<^sup>=\<^sup>= = suffixeq"
Christian@49087
   417
  by (intro ext) (auto simp: suffixeq_def suffix_def)
Christian@49087
   418
wenzelm@49107
   419
lemma suffix_lists: "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A"
Christian@49087
   420
  unfolding suffix_def by auto
Christian@49087
   421
Christian@49087
   422
Christian@50516
   423
subsection {* Homeomorphic embedding on lists *}
Christian@49087
   424
Christian@50516
   425
inductive list_hembeq :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
Christian@49087
   426
  for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
Christian@49087
   427
where
Christian@50516
   428
  list_hembeq_Nil [intro, simp]: "list_hembeq P [] ys"
Christian@50516
   429
| list_hembeq_Cons [intro] : "list_hembeq P xs ys \<Longrightarrow> list_hembeq P xs (y#ys)"
Christian@50516
   430
| list_hembeq_Cons2 [intro]: "P\<^sup>=\<^sup>= x y \<Longrightarrow> list_hembeq P xs ys \<Longrightarrow> list_hembeq P (x#xs) (y#ys)"
Christian@50516
   431
Christian@50516
   432
lemma list_hembeq_Nil2 [simp]:
Christian@50516
   433
  assumes "list_hembeq P xs []" shows "xs = []"
Christian@50516
   434
  using assms by (cases rule: list_hembeq.cases) auto
Christian@49087
   435
Christian@50516
   436
lemma list_hembeq_refl [simp, intro!]:
Christian@50516
   437
  "list_hembeq P xs xs"
Christian@50516
   438
  by (induct xs) auto
Christian@49087
   439
Christian@50516
   440
lemma list_hembeq_Cons_Nil [simp]: "list_hembeq P (x#xs) [] = False"
Christian@49087
   441
proof -
Christian@50516
   442
  { assume "list_hembeq P (x#xs) []"
Christian@50516
   443
    from list_hembeq_Nil2 [OF this] have False by simp
Christian@49087
   444
  } moreover {
Christian@49087
   445
    assume False
Christian@50516
   446
    then have "list_hembeq P (x#xs) []" by simp
Christian@49087
   447
  } ultimately show ?thesis by blast
Christian@49087
   448
qed
Christian@49087
   449
Christian@50516
   450
lemma list_hembeq_append2 [intro]: "list_hembeq P xs ys \<Longrightarrow> list_hembeq P xs (zs @ ys)"
Christian@49087
   451
  by (induct zs) auto
Christian@49087
   452
Christian@50516
   453
lemma list_hembeq_prefix [intro]:
Christian@50516
   454
  assumes "list_hembeq P xs ys" shows "list_hembeq P xs (ys @ zs)"
Christian@49087
   455
  using assms
Christian@49087
   456
  by (induct arbitrary: zs) auto
Christian@49087
   457
Christian@50516
   458
lemma list_hembeq_ConsD:
Christian@50516
   459
  assumes "list_hembeq P (x#xs) ys"
Christian@50516
   460
  shows "\<exists>us v vs. ys = us @ v # vs \<and> P\<^sup>=\<^sup>= x v \<and> list_hembeq P xs vs"
Christian@49087
   461
using assms
wenzelm@49107
   462
proof (induct x \<equiv> "x # xs" ys arbitrary: x xs)
Christian@50516
   463
  case list_hembeq_Cons
wenzelm@49107
   464
  then show ?case by (metis append_Cons)
Christian@49087
   465
next
Christian@50516
   466
  case (list_hembeq_Cons2 x y xs ys)
wenzelm@49107
   467
  then show ?case by (cases xs) (auto, blast+)
Christian@49087
   468
qed
Christian@49087
   469
Christian@50516
   470
lemma list_hembeq_appendD:
Christian@50516
   471
  assumes "list_hembeq P (xs @ ys) zs"
Christian@50516
   472
  shows "\<exists>us vs. zs = us @ vs \<and> list_hembeq P xs us \<and> list_hembeq P ys vs"
Christian@49087
   473
using assms
Christian@49087
   474
proof (induction xs arbitrary: ys zs)
wenzelm@49107
   475
  case Nil then show ?case by auto
Christian@49087
   476
next
Christian@49087
   477
  case (Cons x xs)
Christian@49087
   478
  then obtain us v vs where "zs = us @ v # vs"
Christian@50516
   479
    and "P\<^sup>=\<^sup>= x v" and "list_hembeq P (xs @ ys) vs" by (auto dest: list_hembeq_ConsD)
Christian@50516
   480
  with Cons show ?case by (metis append_Cons append_assoc list_hembeq_Cons2 list_hembeq_append2)
Christian@49087
   481
qed
Christian@49087
   482
Christian@50516
   483
lemma list_hembeq_suffix:
Christian@50516
   484
  assumes "list_hembeq P xs ys" and "suffix ys zs"
Christian@50516
   485
  shows "list_hembeq P xs zs"
Christian@50516
   486
  using assms(2) and list_hembeq_append2 [OF assms(1)] by (auto simp: suffix_def)
Christian@49087
   487
Christian@50516
   488
lemma list_hembeq_suffixeq:
Christian@50516
   489
  assumes "list_hembeq P xs ys" and "suffixeq ys zs"
Christian@50516
   490
  shows "list_hembeq P xs zs"
Christian@50516
   491
  using assms and list_hembeq_suffix unfolding suffixeq_suffix_reflclp_conv by auto
Christian@49087
   492
Christian@50516
   493
lemma list_hembeq_length: "list_hembeq P xs ys \<Longrightarrow> length xs \<le> length ys"
Christian@50516
   494
  by (induct rule: list_hembeq.induct) auto
Christian@49087
   495
Christian@50516
   496
lemma list_hembeq_trans:
Christian@50516
   497
  assumes "\<And>x y z. \<lbrakk>x \<in> A; y \<in> A; z \<in> A; P x y; P y z\<rbrakk> \<Longrightarrow> P x z"
Christian@50516
   498
  shows "\<And>xs ys zs. \<lbrakk>xs \<in> lists A; ys \<in> lists A; zs \<in> lists A;
Christian@50516
   499
    list_hembeq P xs ys; list_hembeq P ys zs\<rbrakk> \<Longrightarrow> list_hembeq P xs zs"
Christian@50516
   500
proof -
Christian@49087
   501
  fix xs ys zs
Christian@50516
   502
  assume "list_hembeq P xs ys" and "list_hembeq P ys zs"
Christian@49087
   503
    and "xs \<in> lists A" and "ys \<in> lists A" and "zs \<in> lists A"
Christian@50516
   504
  then show "list_hembeq P xs zs"
Christian@49087
   505
  proof (induction arbitrary: zs)
Christian@50516
   506
    case list_hembeq_Nil show ?case by blast
Christian@49087
   507
  next
Christian@50516
   508
    case (list_hembeq_Cons xs ys y)
Christian@50516
   509
    from list_hembeq_ConsD [OF `list_hembeq P (y#ys) zs`] obtain us v vs
Christian@50516
   510
      where zs: "zs = us @ v # vs" and "P\<^sup>=\<^sup>= y v" and "list_hembeq P ys vs" by blast
Christian@50516
   511
    then have "list_hembeq P ys (v#vs)" by blast
Christian@50516
   512
    then have "list_hembeq P ys zs" unfolding zs by (rule list_hembeq_append2)
Christian@50516
   513
    from list_hembeq_Cons.IH [OF this] and list_hembeq_Cons.prems show ?case by simp
Christian@49087
   514
  next
Christian@50516
   515
    case (list_hembeq_Cons2 x y xs ys)
Christian@50516
   516
    from list_hembeq_ConsD [OF `list_hembeq P (y#ys) zs`] obtain us v vs
Christian@50516
   517
      where zs: "zs = us @ v # vs" and "P\<^sup>=\<^sup>= y v" and "list_hembeq P ys vs" by blast
Christian@50516
   518
    with list_hembeq_Cons2 have "list_hembeq P xs vs" by simp
Christian@50516
   519
    moreover have "P\<^sup>=\<^sup>= x v"
Christian@49087
   520
    proof -
Christian@49087
   521
      from zs and `zs \<in> lists A` have "v \<in> A" by auto
Christian@50516
   522
      moreover have "x \<in> A" and "y \<in> A" using list_hembeq_Cons2 by simp_all
Christian@50516
   523
      ultimately show ?thesis
Christian@50516
   524
        using `P\<^sup>=\<^sup>= x y` and `P\<^sup>=\<^sup>= y v` and assms
Christian@50516
   525
        by blast
Christian@49087
   526
    qed
Christian@50516
   527
    ultimately have "list_hembeq P (x#xs) (v#vs)" by blast
Christian@50516
   528
    then show ?case unfolding zs by (rule list_hembeq_append2)
Christian@49087
   529
  qed
Christian@49087
   530
qed
Christian@49087
   531
Christian@49087
   532
Christian@50516
   533
subsection {* Sublists (special case of homeomorphic embedding) *}
Christian@49087
   534
Christian@50516
   535
abbreviation sublisteq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
Christian@50516
   536
  where "sublisteq xs ys \<equiv> list_hembeq (op =) xs ys"
Christian@49087
   537
Christian@50516
   538
lemma sublisteq_Cons2: "sublisteq xs ys \<Longrightarrow> sublisteq (x#xs) (x#ys)" by auto
Christian@49087
   539
Christian@50516
   540
lemma sublisteq_same_length:
Christian@50516
   541
  assumes "sublisteq xs ys" and "length xs = length ys" shows "xs = ys"
Christian@50516
   542
  using assms by (induct) (auto dest: list_hembeq_length)
Christian@49087
   543
Christian@50516
   544
lemma not_sublisteq_length [simp]: "length ys < length xs \<Longrightarrow> \<not> sublisteq xs ys"
Christian@50516
   545
  by (metis list_hembeq_length linorder_not_less)
Christian@49087
   546
Christian@49087
   547
lemma [code]:
Christian@50516
   548
  "list_hembeq P [] ys \<longleftrightarrow> True"
Christian@50516
   549
  "list_hembeq P (x#xs) [] \<longleftrightarrow> False"
Christian@49087
   550
  by (simp_all)
Christian@49087
   551
Christian@50516
   552
lemma sublisteq_Cons': "sublisteq (x#xs) ys \<Longrightarrow> sublisteq xs ys"
Christian@50516
   553
  by (induct xs) (auto dest: list_hembeq_ConsD)
Christian@49087
   554
Christian@50516
   555
lemma sublisteq_Cons2':
Christian@50516
   556
  assumes "sublisteq (x#xs) (x#ys)" shows "sublisteq xs ys"
Christian@50516
   557
  using assms by (cases) (rule sublisteq_Cons')
Christian@49087
   558
Christian@50516
   559
lemma sublisteq_Cons2_neq:
Christian@50516
   560
  assumes "sublisteq (x#xs) (y#ys)"
Christian@50516
   561
  shows "x \<noteq> y \<Longrightarrow> sublisteq (x#xs) ys"
Christian@49087
   562
  using assms by (cases) auto
Christian@49087
   563
Christian@50516
   564
lemma sublisteq_Cons2_iff [simp, code]:
Christian@50516
   565
  "sublisteq (x#xs) (y#ys) = (if x = y then sublisteq xs ys else sublisteq (x#xs) ys)"
Christian@50516
   566
  by (metis list_hembeq_Cons sublisteq_Cons2 sublisteq_Cons2' sublisteq_Cons2_neq)
Christian@49087
   567
Christian@50516
   568
lemma sublisteq_append': "sublisteq (zs @ xs) (zs @ ys) \<longleftrightarrow> sublisteq xs ys"
Christian@49087
   569
  by (induct zs) simp_all
Christian@49087
   570
Christian@50516
   571
lemma sublisteq_refl [simp, intro!]: "sublisteq xs xs" by (induct xs) simp_all
Christian@49087
   572
Christian@50516
   573
lemma sublisteq_antisym:
Christian@50516
   574
  assumes "sublisteq xs ys" and "sublisteq ys xs"
Christian@49087
   575
  shows "xs = ys"
Christian@49087
   576
using assms
Christian@49087
   577
proof (induct)
Christian@50516
   578
  case list_hembeq_Nil
Christian@50516
   579
  from list_hembeq_Nil2 [OF this] show ?case by simp
Christian@49087
   580
next
Christian@50516
   581
  case list_hembeq_Cons2
wenzelm@49107
   582
  then show ?case by simp
Christian@49087
   583
next
Christian@50516
   584
  case list_hembeq_Cons
wenzelm@49107
   585
  then show ?case
Christian@50516
   586
    by (metis sublisteq_Cons' list_hembeq_length Suc_length_conv Suc_n_not_le_n)
Christian@49087
   587
qed
Christian@49087
   588
Christian@50516
   589
lemma sublisteq_trans: "sublisteq xs ys \<Longrightarrow> sublisteq ys zs \<Longrightarrow> sublisteq xs zs"
Christian@50516
   590
  by (rule list_hembeq_trans [of UNIV "op ="]) auto
Christian@49087
   591
Christian@50516
   592
lemma sublisteq_append_le_same_iff: "sublisteq (xs @ ys) ys \<longleftrightarrow> xs = []"
Christian@50516
   593
  by (auto dest: list_hembeq_length)
Christian@49087
   594
Christian@50516
   595
lemma list_hembeq_append_mono:
Christian@50516
   596
  "\<lbrakk> list_hembeq P xs xs'; list_hembeq P ys ys' \<rbrakk> \<Longrightarrow> list_hembeq P (xs@ys) (xs'@ys')"
Christian@50516
   597
  apply (induct rule: list_hembeq.induct)
Christian@50516
   598
    apply (metis eq_Nil_appendI list_hembeq_append2)
Christian@50516
   599
   apply (metis append_Cons list_hembeq_Cons)
Christian@50516
   600
  apply (metis append_Cons list_hembeq_Cons2)
wenzelm@49107
   601
  done
Christian@49087
   602
Christian@49087
   603
Christian@49087
   604
subsection {* Appending elements *}
Christian@49087
   605
Christian@50516
   606
lemma sublisteq_append [simp]:
Christian@50516
   607
  "sublisteq (xs @ zs) (ys @ zs) \<longleftrightarrow> sublisteq xs ys" (is "?l = ?r")
Christian@49087
   608
proof
Christian@50516
   609
  { fix xs' ys' xs ys zs :: "'a list" assume "sublisteq xs' ys'"
Christian@50516
   610
    then have "xs' = xs @ zs & ys' = ys @ zs \<longrightarrow> sublisteq xs ys"
Christian@49087
   611
    proof (induct arbitrary: xs ys zs)
Christian@50516
   612
      case list_hembeq_Nil show ?case by simp
Christian@49087
   613
    next
Christian@50516
   614
      case (list_hembeq_Cons xs' ys' x)
Christian@50516
   615
      { assume "ys=[]" then have ?case using list_hembeq_Cons(1) by auto }
Christian@49087
   616
      moreover
Christian@49087
   617
      { fix us assume "ys = x#us"
Christian@50516
   618
        then have ?case using list_hembeq_Cons(2) by(simp add: list_hembeq.list_hembeq_Cons) }
Christian@49087
   619
      ultimately show ?case by (auto simp:Cons_eq_append_conv)
Christian@49087
   620
    next
Christian@50516
   621
      case (list_hembeq_Cons2 x y xs' ys')
Christian@50516
   622
      { assume "xs=[]" then have ?case using list_hembeq_Cons2(1) by auto }
Christian@49087
   623
      moreover
Christian@50516
   624
      { fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using list_hembeq_Cons2 by auto}
Christian@49087
   625
      moreover
Christian@50516
   626
      { fix us assume "xs=x#us" "ys=[]" then have ?case using list_hembeq_Cons2(2) by bestsimp }
Christian@50516
   627
      ultimately show ?case using `op =\<^sup>=\<^sup>= x y` by (auto simp: Cons_eq_append_conv)
Christian@49087
   628
    qed }
Christian@49087
   629
  moreover assume ?l
Christian@49087
   630
  ultimately show ?r by blast
Christian@49087
   631
next
Christian@50516
   632
  assume ?r then show ?l by (metis list_hembeq_append_mono sublisteq_refl)
Christian@49087
   633
qed
Christian@49087
   634
Christian@50516
   635
lemma sublisteq_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (zs @ ys)"
Christian@49087
   636
  by (induct zs) auto
Christian@49087
   637
Christian@50516
   638
lemma sublisteq_rev_drop_many: "sublisteq xs ys \<Longrightarrow> sublisteq xs (ys @ zs)"
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  by (metis append_Nil2 list_hembeq_Nil list_hembeq_append_mono)
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   640
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   641
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   642
subsection {* Relation to standard list operations *}
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   644
lemma sublisteq_map:
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  assumes "sublisteq xs ys" shows "sublisteq (map f xs) (map f ys)"
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   646
  using assms by (induct) auto
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   647
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   648
lemma sublisteq_filter_left [simp]: "sublisteq (filter P xs) xs"
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   649
  by (induct xs) auto
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   650
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   651
lemma sublisteq_filter [simp]:
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  assumes "sublisteq xs ys" shows "sublisteq (filter P xs) (filter P ys)"
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   653
  using assms by (induct) auto
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   654
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   655
lemma "sublisteq xs ys \<longleftrightarrow> (\<exists>N. xs = sublist ys N)" (is "?L = ?R")
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proof
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  assume ?L
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   658
  then show ?R
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   659
  proof (induct)
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    case list_hembeq_Nil show ?case by (metis sublist_empty)
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   661
  next
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    case (list_hembeq_Cons xs ys x)
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   663
    then obtain N where "xs = sublist ys N" by blast
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   664
    then have "xs = sublist (x#ys) (Suc ` N)"
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   665
      by (clarsimp simp add:sublist_Cons inj_image_mem_iff)
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   666
    then show ?case by blast
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   667
  next
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   668
    case (list_hembeq_Cons2 x y xs ys)
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   669
    then obtain N where "xs = sublist ys N" by blast
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   670
    then have "x#xs = sublist (x#ys) (insert 0 (Suc ` N))"
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   671
      by (clarsimp simp add:sublist_Cons inj_image_mem_iff)
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    moreover from list_hembeq_Cons2 have "x = y" by simp
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    ultimately show ?case by blast
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   674
  qed
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   675
next
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  assume ?R
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   677
  then obtain N where "xs = sublist ys N" ..
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   678
  moreover have "sublisteq (sublist ys N) ys"
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  proof (induct ys arbitrary: N)
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    case Nil show ?case by simp
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  next
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    case Cons then show ?case by (auto simp: sublist_Cons)
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  qed
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  ultimately show ?L by simp
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   685
qed
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wenzelm@10330
   687
end