src/HOL/Library/Sublist.thy
author Christian Sternagel
Wed, 29 Aug 2012 16:25:35 +0900
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permissions -rw-r--r--
introduced "sub" as abbreviation for "emb (op =)"; Sublist_Order is now based on Sublist.sub; simplified and moved most lemmas on sub from Sublist_Order to Sublist; Sublist_Order merely contains ord and order instances for sub plus some lemmas on the strict part of the order
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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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*)
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header {* List prefixes, suffixes, and embedding*}
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theory Sublist
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imports List Main
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begin
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subsection {* Prefix order on lists *}
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definition prefixeq :: "'a list => 'a list => bool" where
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  "prefixeq xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"
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definition prefix :: "'a list => 'a list => bool" where
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  "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: bot prefixeq prefix Nil
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  by default (simp add: prefixeq_def)
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lemma prefixeqI [intro?]: "ys = xs @ zs ==> 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 ==> 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 ==> xs \<noteq> ys ==> 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 ==> 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 ==> length xs < length ys ==> 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 ==> 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\<^isub>1::'a list) ys \<Longrightarrow> prefixeq xs\<^isub>2 ys \<Longrightarrow> prefixeq xs\<^isub>1 xs\<^isub>2 \<or> prefixeq xs\<^isub>2 xs\<^isub>1"
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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 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
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  parallel :: "'a list => 'a list => bool"  (infixl "\<parallel>" 50) where
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  "(xs \<parallel> ys) = (\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs)"
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lemma parallelI [intro]: "\<not> prefixeq xs ys ==> \<not> prefixeq ys xs ==> 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 ==> \<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" 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
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  suffixeq :: "'a list => 'a list => bool" where
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  "suffixeq xs ys = (\<exists>zs. ys = zs @ xs)"
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definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
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  "suffix xs ys \<equiv> \<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 ==> 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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parents: 45236
diff changeset
   280
lemma suffix_trans:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   281
  "suffix xs ys \<Longrightarrow> suffix ys zs \<Longrightarrow> suffix xs zs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   282
  by (auto simp: suffix_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   283
lemma suffixeq_trans: "\<lbrakk>suffixeq xs ys; suffixeq ys zs\<rbrakk> \<Longrightarrow> suffixeq xs zs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   284
  by (auto simp add: suffixeq_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   285
lemma suffixeq_antisym: "\<lbrakk>suffixeq xs ys; suffixeq ys xs\<rbrakk> \<Longrightarrow> xs = ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   286
  by (auto simp add: suffixeq_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   287
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   288
lemma suffixeq_tl [simp]: "suffixeq (tl xs) xs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   289
  by (induct xs) (auto simp: suffixeq_def)
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   290
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   291
lemma suffix_tl [simp]: "xs \<noteq> [] \<Longrightarrow> suffix (tl xs) xs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   292
  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
   293
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   294
lemma Nil_suffixeq [iff]: "suffixeq [] xs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   295
  by (simp add: suffixeq_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   296
lemma suffixeq_Nil [simp]: "(suffixeq xs []) = (xs = [])"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   297
  by (auto simp add: suffixeq_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   298
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   299
lemma suffixeq_ConsI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (y#ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   300
  by (auto simp add: suffixeq_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   301
lemma suffixeq_ConsD: "suffixeq (x#xs) ys \<Longrightarrow> suffixeq xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   302
  by (auto simp add: suffixeq_def)
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   303
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   304
lemma suffixeq_appendI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (zs @ ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   305
  by (auto simp add: suffixeq_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   306
lemma suffixeq_appendD: "suffixeq (zs @ xs) ys \<Longrightarrow> suffixeq xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   307
  by (auto simp add: suffixeq_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   308
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   309
lemma suffix_set_subset:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   310
  "suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffix_def)
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   311
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   312
lemma suffixeq_set_subset:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   313
  "suffixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffixeq_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   314
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   315
lemma suffixeq_ConsD2: "suffixeq (x#xs) (y#ys) ==> suffixeq xs ys"
21305
d41eddfd2b66 tuned proofs;
wenzelm
parents: 19086
diff changeset
   316
proof -
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   317
  assume "suffixeq (x#xs) (y#ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   318
  then obtain zs where "y#ys = zs @ x#xs" ..
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   319
  then show ?thesis
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   320
    by (induct zs) (auto intro!: suffixeq_appendI suffixeq_ConsI)
21305
d41eddfd2b66 tuned proofs;
wenzelm
parents: 19086
diff changeset
   321
qed
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   322
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   323
lemma suffixeq_to_prefixeq [code]: "suffixeq xs ys \<longleftrightarrow> prefixeq (rev xs) (rev ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   324
proof
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   325
  assume "suffixeq xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   326
  then obtain zs where "ys = zs @ xs" ..
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   327
  then have "rev ys = rev xs @ rev zs" by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   328
  then show "prefixeq (rev xs) (rev ys)" ..
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   329
next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   330
  assume "prefixeq (rev xs) (rev ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   331
  then obtain zs where "rev ys = rev xs @ zs" ..
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   332
  then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   333
  then have "ys = rev zs @ xs" by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   334
  then show "suffixeq xs ys" ..
21305
d41eddfd2b66 tuned proofs;
wenzelm
parents: 19086
diff changeset
   335
qed
14538
1d9d75a8efae removed o2l and fold_rel; moved postfix to Library/List_Prefix.thy
oheimb
parents: 14300
diff changeset
   336
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   337
lemma distinct_suffixeq: "distinct ys \<Longrightarrow> suffixeq xs ys \<Longrightarrow> distinct xs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   338
  by (clarsimp elim!: suffixeqE)
17201
3bdf1dfcdee4 reactivate postfix by change of syntax;
wenzelm
parents: 15355
diff changeset
   339
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   340
lemma suffixeq_map: "suffixeq xs ys \<Longrightarrow> suffixeq (map f xs) (map f ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   341
  by (auto elim!: suffixeqE intro: suffixeqI)
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   342
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   343
lemma suffixeq_drop: "suffixeq (drop n as) as"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   344
  unfolding suffixeq_def
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   345
  apply (rule exI [where x = "take n as"])
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   346
  apply simp
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   347
  done
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   348
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   349
lemma suffixeq_take: "suffixeq xs ys \<Longrightarrow> ys = take (length ys - length xs) ys @ xs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   350
  by (clarsimp elim!: suffixeqE)
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   351
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   352
lemma suffixeq_suffix_reflclp_conv:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   353
  "suffixeq = suffix\<^sup>=\<^sup>="
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   354
proof (intro ext iffI)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   355
  fix xs ys :: "'a list"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   356
  assume "suffixeq xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   357
  show "suffix\<^sup>=\<^sup>= xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   358
  proof
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   359
    assume "xs \<noteq> ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   360
    with `suffixeq xs ys` show "suffix xs ys" by (auto simp: suffixeq_def suffix_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   361
  qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   362
next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   363
  fix xs ys :: "'a list"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   364
  assume "suffix\<^sup>=\<^sup>= xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   365
  thus "suffixeq xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   366
  proof
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   367
    assume "suffix xs ys" thus "suffixeq xs ys" by (rule suffix_imp_suffixeq)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   368
  next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   369
    assume "xs = ys" thus "suffixeq xs ys" by (auto simp: suffixeq_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   370
  qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   371
qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   372
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   373
lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefixeq x y"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   374
  by blast
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   375
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   376
lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefixeq y x"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   377
  by blast
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   378
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   379
lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   380
  unfolding parallel_def by simp
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   381
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   382
lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   383
  unfolding parallel_def by simp
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   384
25564
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25356
diff changeset
   385
lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
25692
eda4958ab0d2 tuned proofs, document;
wenzelm
parents: 25665
diff changeset
   386
  by auto
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   387
25564
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25356
diff changeset
   388
lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   389
  by (metis Cons_prefixeq_Cons parallelE parallelI)
25665
faabc08af882 removed legacy proofs
nipkow
parents: 25595
diff changeset
   390
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   391
lemma not_equal_is_parallel:
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   392
  assumes neq: "xs \<noteq> ys"
25356
059c03630d6e tuned presentation;
wenzelm
parents: 25355
diff changeset
   393
    and len: "length xs = length ys"
059c03630d6e tuned presentation;
wenzelm
parents: 25355
diff changeset
   394
  shows "xs \<parallel> ys"
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   395
  using len neq
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   396
proof (induct rule: list_induct2)
26445
17223cf843d8 explicit case names for rule list_induct2
haftmann
parents: 25764
diff changeset
   397
  case Nil
25356
059c03630d6e tuned presentation;
wenzelm
parents: 25355
diff changeset
   398
  then show ?case by simp
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   399
next
26445
17223cf843d8 explicit case names for rule list_induct2
haftmann
parents: 25764
diff changeset
   400
  case (Cons a as b bs)
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   401
  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
   402
  show ?case
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   403
  proof (cases "a = b")
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   404
    case True
26445
17223cf843d8 explicit case names for rule list_induct2
haftmann
parents: 25764
diff changeset
   405
    then have "as \<noteq> bs" using Cons by simp
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   406
    then show ?thesis by (rule Cons_parallelI2 [OF True ih])
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   407
  next
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   408
    case False
25355
69c0a39ba028 avoid implicit use of prems;
wenzelm
parents: 25322
diff changeset
   409
    then show ?thesis by (rule Cons_parallelI1)
25299
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   410
  qed
c3542f70b0fd misc lemmas about prefix, postfix, and parallel
kleing
parents: 23394
diff changeset
   411
qed
22178
29b95968272b made executable
haftmann
parents: 21404
diff changeset
   412
49085
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   413
lemma suffix_reflclp_conv:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   414
  "suffix\<^sup>=\<^sup>= = suffixeq"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   415
  by (intro ext) (auto simp: suffixeq_def suffix_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   416
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   417
lemma suffix_lists:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   418
  "suffix xs ys \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   419
  unfolding suffix_def by auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   420
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   421
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   422
subsection {* Embedding on lists *}
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   423
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   424
inductive
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   425
  emb :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   426
  for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   427
where
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   428
  emb_Nil [intro, simp]: "emb P [] ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   429
| emb_Cons [intro] : "emb P xs ys \<Longrightarrow> emb P xs (y#ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   430
| emb_Cons2 [intro]: "P x y \<Longrightarrow> emb P xs ys \<Longrightarrow> emb P (x#xs) (y#ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   431
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   432
lemma emb_Nil2 [simp]:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   433
  assumes "emb P xs []" shows "xs = []"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   434
  using assms by (cases rule: emb.cases) auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   435
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   436
lemma emb_Cons_Nil [simp]:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   437
  "emb P (x#xs) [] = False"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   438
proof -
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   439
  { assume "emb P (x#xs) []"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   440
    from emb_Nil2 [OF this] have False by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   441
  } moreover {
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   442
    assume False
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   443
    hence "emb P (x#xs) []" by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   444
  } ultimately show ?thesis by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   445
qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   446
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   447
lemma emb_append2 [intro]:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   448
  "emb P xs ys \<Longrightarrow> emb P xs (zs @ ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   449
  by (induct zs) auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   450
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   451
lemma emb_prefix [intro]:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   452
  assumes "emb P xs ys" shows "emb P xs (ys @ zs)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   453
  using assms
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   454
  by (induct arbitrary: zs) auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   455
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   456
lemma emb_ConsD:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   457
  assumes "emb P (x#xs) ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   458
  shows "\<exists>us v vs. ys = us @ v # vs \<and> P x v \<and> emb P xs vs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   459
using assms
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   460
proof (induct x\<equiv>"x#xs" y\<equiv>"ys" arbitrary: x xs ys)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   461
  case emb_Cons thus ?case by (metis append_Cons)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   462
next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   463
  case (emb_Cons2 x y xs ys)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   464
  thus ?case by (cases xs) (auto, blast+)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   465
qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   466
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   467
lemma emb_appendD:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   468
  assumes "emb P (xs @ ys) zs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   469
  shows "\<exists>us vs. zs = us @ vs \<and> emb P xs us \<and> emb P ys vs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   470
using assms
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   471
proof (induction xs arbitrary: ys zs)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   472
  case Nil thus ?case by auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   473
next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   474
  case (Cons x xs)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   475
  then obtain us v vs where "zs = us @ v # vs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   476
    and "P x v" and "emb P (xs @ ys) vs" by (auto dest: emb_ConsD)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   477
  with Cons show ?case by (metis append_Cons append_assoc emb_Cons2 emb_append2)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   478
qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   479
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   480
lemma emb_suffix:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   481
  assumes "emb P xs ys" and "suffix ys zs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   482
  shows "emb P xs zs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   483
  using assms(2) and emb_append2 [OF assms(1)] by (auto simp: suffix_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   484
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   485
lemma emb_suffixeq:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   486
  assumes "emb P xs ys" and "suffixeq ys zs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   487
  shows "emb P xs zs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   488
  using assms and emb_suffix unfolding suffixeq_suffix_reflclp_conv by auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   489
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   490
lemma emb_length: "emb P xs ys \<Longrightarrow> length xs \<le> length ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   491
  by (induct rule: emb.induct) auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   492
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   493
(*FIXME: move*)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   494
definition transp_on :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   495
  "transp_on P A \<equiv> \<forall>a\<in>A. \<forall>b\<in>A. \<forall>c\<in>A. P a b \<and> P b c \<longrightarrow> P a c"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   496
lemma transp_onI [Pure.intro]:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   497
  "(\<And>a b c. \<lbrakk>a \<in> A; b \<in> A; c \<in> A; P a b; P b c\<rbrakk> \<Longrightarrow> P a c) \<Longrightarrow> transp_on P A"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   498
  unfolding transp_on_def by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   499
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   500
lemma transp_on_emb:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   501
  assumes "transp_on P A"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   502
  shows "transp_on (emb P) (lists A)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   503
proof
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   504
  fix xs ys zs
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   505
  assume "emb P xs ys" and "emb P ys zs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   506
    and "xs \<in> lists A" and "ys \<in> lists A" and "zs \<in> lists A"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   507
  thus "emb P xs zs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   508
  proof (induction arbitrary: zs)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   509
    case emb_Nil show ?case by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   510
  next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   511
    case (emb_Cons xs ys y)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   512
    from emb_ConsD [OF `emb P (y#ys) zs`] obtain us v vs
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   513
      where zs: "zs = us @ v # vs" and "P y v" and "emb P ys vs" by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   514
    hence "emb P ys (v#vs)" by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   515
    hence "emb P ys zs" unfolding zs by (rule emb_append2)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   516
    from emb_Cons.IH [OF this] and emb_Cons.prems show ?case by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   517
  next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   518
    case (emb_Cons2 x y xs ys)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   519
    from emb_ConsD [OF `emb P (y#ys) zs`] obtain us v vs
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   520
      where zs: "zs = us @ v # vs" and "P y v" and "emb P ys vs" by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   521
    with emb_Cons2 have "emb P xs vs" by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   522
    moreover have "P x v"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   523
    proof -
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   524
      from zs and `zs \<in> lists A` have "v \<in> A" by auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   525
      moreover have "x \<in> A" and "y \<in> A" using emb_Cons2 by simp_all
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   526
      ultimately show ?thesis using `P x y` and `P y v` and assms
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   527
        unfolding transp_on_def by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   528
    qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   529
    ultimately have "emb P (x#xs) (v#vs)" by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   530
    thus ?case unfolding zs by (rule emb_append2)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   531
  qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   532
qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   533
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   534
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   535
subsection {* Sublists (special case of embedding) *}
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   536
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   537
abbreviation sub :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   538
  "sub xs ys \<equiv> emb (op =) xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   539
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   540
lemma sub_Cons2: "sub xs ys \<Longrightarrow> sub (x#xs) (x#ys)" by auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   541
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   542
lemma sub_same_length:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   543
  assumes "sub xs ys" and "length xs = length ys" shows "xs = ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   544
  using assms by (induct) (auto dest: emb_length)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   545
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   546
lemma not_sub_length [simp]: "length ys < length xs \<Longrightarrow> \<not> sub xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   547
  by (metis emb_length linorder_not_less)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   548
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   549
lemma [code]:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   550
  "emb P [] ys \<longleftrightarrow> True"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   551
  "emb P (x#xs) [] \<longleftrightarrow> False"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   552
  by (simp_all)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   553
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   554
lemma sub_Cons': "sub (x#xs) ys \<Longrightarrow> sub xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   555
  by (induct xs) (auto dest: emb_ConsD)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   556
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   557
lemma sub_Cons2':
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   558
  assumes "sub (x#xs) (x#ys)" shows "sub xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   559
  using assms by (cases) (rule sub_Cons')
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   560
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   561
lemma sub_Cons2_neq:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   562
  assumes "sub (x#xs) (y#ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   563
  shows "x \<noteq> y \<Longrightarrow> sub (x#xs) ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   564
  using assms by (cases) auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   565
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   566
lemma sub_Cons2_iff [simp, code]:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   567
  "sub (x#xs) (y#ys) = (if x = y then sub xs ys else sub (x#xs) ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   568
  by (metis emb_Cons emb_Cons2 [of "op =", OF refl] sub_Cons2' sub_Cons2_neq)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   569
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   570
lemma sub_append': "sub (zs @ xs) (zs @ ys) \<longleftrightarrow> sub xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   571
  by (induct zs) simp_all
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   572
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   573
lemma sub_refl [simp, intro!]: "sub xs xs" by (induct xs) simp_all
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   574
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   575
lemma sub_antisym:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   576
  assumes "sub xs ys" and "sub ys xs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   577
  shows "xs = ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   578
using assms
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   579
proof (induct)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   580
  case emb_Nil
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   581
  from emb_Nil2 [OF this] show ?case by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   582
next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   583
  case emb_Cons2 thus ?case by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   584
next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   585
  case emb_Cons thus ?case
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   586
    by (metis sub_Cons' emb_length Suc_length_conv Suc_n_not_le_n)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   587
qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   588
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   589
lemma transp_on_sub: "transp_on sub UNIV"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   590
proof -
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   591
  have "transp_on (op =) UNIV" by (simp add: transp_on_def)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   592
  from transp_on_emb [OF this] show ?thesis by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   593
qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   594
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   595
lemma sub_trans: "sub xs ys \<Longrightarrow> sub ys zs \<Longrightarrow> sub xs zs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   596
  using transp_on_sub [unfolded transp_on_def] by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   597
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   598
lemma sub_append_le_same_iff: "sub (xs @ ys) ys \<longleftrightarrow> xs = []"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   599
  by (auto dest: emb_length)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   600
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   601
lemma emb_append_mono:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   602
  "\<lbrakk> emb P xs xs'; emb P ys ys' \<rbrakk> \<Longrightarrow> emb P (xs@ys) (xs'@ys')"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   603
apply (induct rule: emb.induct)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   604
  apply (metis eq_Nil_appendI emb_append2)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   605
 apply (metis append_Cons emb_Cons)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   606
by (metis append_Cons emb_Cons2)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   607
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   608
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   609
subsection {* Appending elements *}
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   610
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   611
lemma sub_append [simp]:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   612
  "sub (xs @ zs) (ys @ zs) \<longleftrightarrow> sub xs ys" (is "?l = ?r")
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   613
proof
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   614
  { fix xs' ys' xs ys zs :: "'a list" assume "sub xs' ys'"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   615
    hence "xs' = xs @ zs & ys' = ys @ zs \<longrightarrow> sub xs ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   616
    proof (induct arbitrary: xs ys zs)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   617
      case emb_Nil show ?case by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   618
    next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   619
      case (emb_Cons xs' ys' x)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   620
      { assume "ys=[]" hence ?case using emb_Cons(1) by auto }
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   621
      moreover
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   622
      { fix us assume "ys = x#us"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   623
        hence ?case using emb_Cons(2) by(simp add: emb.emb_Cons) }
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   624
      ultimately show ?case by (auto simp:Cons_eq_append_conv)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   625
    next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   626
      case (emb_Cons2 x y xs' ys')
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   627
      { assume "xs=[]" hence ?case using emb_Cons2(1) by auto }
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   628
      moreover
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   629
      { fix us vs assume "xs=x#us" "ys=x#vs" hence ?case using emb_Cons2 by auto}
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   630
      moreover
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   631
      { fix us assume "xs=x#us" "ys=[]" hence ?case using emb_Cons2(2) by bestsimp }
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   632
      ultimately show ?case using `x = y` by (auto simp: Cons_eq_append_conv)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   633
    qed }
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   634
  moreover assume ?l
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   635
  ultimately show ?r by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   636
next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   637
  assume ?r thus ?l by (metis emb_append_mono sub_refl)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   638
qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   639
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   640
lemma sub_drop_many: "sub xs ys \<Longrightarrow> sub xs (zs @ ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   641
  by (induct zs) auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   642
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   643
lemma sub_rev_drop_many: "sub xs ys \<Longrightarrow> sub xs (ys @ zs)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   644
  by (metis append_Nil2 emb_Nil emb_append_mono)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   645
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   646
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   647
subsection {* Relation to standard list operations *}
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   648
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   649
lemma sub_map:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   650
  assumes "sub xs ys" shows "sub (map f xs) (map f ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   651
  using assms by (induct) auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   652
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   653
lemma sub_filter_left [simp]: "sub (filter P xs) xs"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   654
  by (induct xs) auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   655
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   656
lemma sub_filter [simp]:
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   657
  assumes "sub xs ys" shows "sub (filter P xs) (filter P ys)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   658
  using assms by (induct) auto
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   659
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   660
lemma "sub xs ys \<longleftrightarrow> (\<exists> N. xs = sublist ys N)" (is "?L = ?R")
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   661
proof
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   662
  assume ?L
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   663
  thus ?R
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   664
  proof (induct)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   665
    case emb_Nil show ?case by (metis sublist_empty)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   666
  next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   667
    case (emb_Cons xs ys x)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   668
    then obtain N where "xs = sublist ys N" by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   669
    hence "xs = sublist (x#ys) (Suc ` N)"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   670
      by (clarsimp simp add:sublist_Cons inj_image_mem_iff)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   671
    thus ?case by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   672
  next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   673
    case (emb_Cons2 x y xs ys)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   674
    then obtain N where "xs = sublist ys N" by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   675
    hence "x#xs = sublist (x#ys) (insert 0 (Suc ` N))"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   676
      by (clarsimp simp add:sublist_Cons inj_image_mem_iff)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   677
    thus ?case unfolding `x = y` by blast
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   678
  qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   679
next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   680
  assume ?R
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   681
  then obtain N where "xs = sublist ys N" ..
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   682
  moreover have "sub (sublist ys N) ys"
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   683
  proof (induct ys arbitrary:N)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   684
    case Nil show ?case by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   685
  next
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   686
    case Cons thus ?case by (auto simp: sublist_Cons)
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   687
  qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   688
  ultimately show ?L by simp
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   689
qed
4eef5c2ff5ad introduced "sub" as abbreviation for "emb (op =)";
Christian Sternagel
parents: 45236
diff changeset
   690
10330
4362e906b745 "List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff changeset
   691
end