src/HOL/Library/List_Prefix.thy
author wenzelm
Thu Nov 08 20:09:17 2007 +0100 (2007-11-08)
changeset 25355 69c0a39ba028
parent 25322 e2eac0c30ff5
child 25356 059c03630d6e
permissions -rw-r--r--
avoid implicit use of prems;
tuned proofs;
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(*  Title:      HOL/Library/List_Prefix.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow and Markus Wenzel, TU Muenchen
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*)
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header {* List prefixes and postfixes *}
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theory List_Prefix
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imports Main
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begin
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subsection {* Prefix order on lists *}
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instance list :: (type) ord ..
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defs (overloaded)
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  prefix_def: "xs \<le> ys == \<exists>zs. ys = xs @ zs"
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  strict_prefix_def: "xs < ys == xs \<le> ys \<and> xs \<noteq> (ys::'a list)"
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instance list :: (type) order
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  by intro_classes (auto simp add: prefix_def strict_prefix_def)
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lemma prefixI [intro?]: "ys = xs @ zs ==> xs \<le> ys"
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  unfolding prefix_def by blast
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lemma prefixE [elim?]:
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  assumes "xs \<le> ys"
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  obtains zs where "ys = xs @ zs"
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  using assms unfolding prefix_def by blast
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lemma strict_prefixI' [intro?]: "ys = xs @ z # zs ==> xs < ys"
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  unfolding strict_prefix_def prefix_def by blast
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lemma strict_prefixE' [elim?]:
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  assumes "xs < ys"
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  obtains z zs where "ys = xs @ z # zs"
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proof -
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  from `xs < ys` obtain us where "ys = xs @ us" and "xs \<noteq> ys"
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    unfolding strict_prefix_def prefix_def by blast
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  with that show ?thesis by (auto simp add: neq_Nil_conv)
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qed
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lemma strict_prefixI [intro?]: "xs \<le> ys ==> xs \<noteq> ys ==> xs < (ys::'a list)"
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  unfolding strict_prefix_def by blast
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lemma strict_prefixE [elim?]:
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  fixes xs ys :: "'a list"
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  assumes "xs < ys"
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  obtains "xs \<le> ys" and "xs \<noteq> ys"
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  using assms unfolding strict_prefix_def by blast
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subsection {* Basic properties of prefixes *}
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theorem Nil_prefix [iff]: "[] \<le> xs"
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  by (simp add: prefix_def)
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theorem prefix_Nil [simp]: "(xs \<le> []) = (xs = [])"
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  by (induct xs) (simp_all add: prefix_def)
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lemma prefix_snoc [simp]: "(xs \<le> ys @ [y]) = (xs = ys @ [y] \<or> xs \<le> ys)"
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proof
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  assume "xs \<le> ys @ [y]"
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  then obtain zs where zs: "ys @ [y] = xs @ zs" ..
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  show "xs = ys @ [y] \<or> xs \<le> ys"
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  proof (cases zs rule: rev_cases)
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    assume "zs = []"
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    with zs have "xs = ys @ [y]" by simp
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    then show ?thesis ..
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  next
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    fix z zs' assume "zs = zs' @ [z]"
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    with zs have "ys = xs @ zs'" by simp
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    then have "xs \<le> ys" ..
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    then show ?thesis ..
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  qed
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next
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  assume "xs = ys @ [y] \<or> xs \<le> ys"
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  then show "xs \<le> ys @ [y]"
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  proof
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    assume "xs = ys @ [y]"
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    then show ?thesis by simp
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  next
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    assume "xs \<le> ys"
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    then obtain zs where "ys = xs @ zs" ..
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    then have "ys @ [y] = xs @ (zs @ [y])" by simp
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    then show ?thesis ..
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  qed
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qed
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lemma Cons_prefix_Cons [simp]: "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)"
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  by (auto simp add: prefix_def)
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lemma same_prefix_prefix [simp]: "(xs @ ys \<le> xs @ zs) = (ys \<le> zs)"
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  by (induct xs) simp_all
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lemma same_prefix_nil [iff]: "(xs @ ys \<le> xs) = (ys = [])"
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proof -
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  have "(xs @ ys \<le> xs @ []) = (ys \<le> [])" by (rule same_prefix_prefix)
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  then show ?thesis by simp
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qed
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lemma prefix_prefix [simp]: "xs \<le> ys ==> xs \<le> ys @ zs"
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proof -
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  assume "xs \<le> ys"
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  then obtain us where "ys = xs @ us" ..
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  then have "ys @ zs = xs @ (us @ zs)" by simp
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  then show ?thesis ..
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qed
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lemma append_prefixD: "xs @ ys \<le> zs \<Longrightarrow> xs \<le> zs"
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  by (auto simp add: prefix_def)
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theorem prefix_Cons: "(xs \<le> y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> zs \<le> ys))"
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  by (cases xs) (auto simp add: prefix_def)
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theorem prefix_append:
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    "(xs \<le> ys @ zs) = (xs \<le> ys \<or> (\<exists>us. xs = ys @ us \<and> us \<le> 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 simp
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  apply blast
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  done
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lemma append_one_prefix:
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    "xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys"
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  apply (unfold prefix_def)
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  apply (auto simp add: nth_append)
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  apply (case_tac zs)
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   apply auto
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  done
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theorem prefix_length_le: "xs \<le> ys ==> length xs \<le> length ys"
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  by (auto simp add: prefix_def)
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lemma prefix_same_cases:
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    "(xs\<^isub>1::'a list) \<le> ys \<Longrightarrow> xs\<^isub>2 \<le> ys \<Longrightarrow> xs\<^isub>1 \<le> xs\<^isub>2 \<or> xs\<^isub>2 \<le> xs\<^isub>1"
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  apply (simp add: prefix_def)
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  apply (erule exE)+
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  apply (simp add: append_eq_append_conv_if split: if_splits)
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   apply (rule disjI2)
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   apply (rule_tac x = "drop (size xs\<^isub>2) xs\<^isub>1" in exI)
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   apply clarify
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   apply (drule sym)
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   apply (insert append_take_drop_id [of "length xs\<^isub>2" xs\<^isub>1])
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   apply simp
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  apply (rule disjI1)
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  apply (rule_tac x = "drop (size xs\<^isub>1) xs\<^isub>2" in exI)
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  apply clarify
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  apply (insert append_take_drop_id [of "length xs\<^isub>1" xs\<^isub>2])
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  apply simp
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  done
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lemma set_mono_prefix:
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    "xs \<le> ys \<Longrightarrow> set xs \<subseteq> set ys"
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  by (auto simp add: prefix_def)
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lemma take_is_prefix:
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  "take n xs \<le> xs"
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  apply (simp add: prefix_def)
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  apply (rule_tac x="drop n xs" in exI)
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  apply simp
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  done
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lemma map_prefixI:
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  "xs \<le> ys \<Longrightarrow> map f xs \<le> map f ys"
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  by (clarsimp simp: prefix_def)
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lemma prefix_length_less:
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  "xs < ys \<Longrightarrow> length xs < length ys"
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  apply (clarsimp simp: strict_prefix_def)
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  apply (frule prefix_length_le)
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  apply (rule ccontr, simp)
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  apply (clarsimp simp: prefix_def)
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  done
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lemma strict_prefix_simps [simp]:
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  "xs < [] = False"
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  "[] < (x # xs) = True"
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  "(x # xs) < (y # ys) = (x = y \<and> xs < ys)"
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  by (simp_all add: strict_prefix_def cong: conj_cong)
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lemma take_strict_prefix:
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  "xs < ys \<Longrightarrow> 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 (case_tac xs, simp)
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  apply (case_tac ys, simp_all)
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  done
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lemma not_prefix_cases:
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  assumes pfx: "\<not> ps \<le> ls"
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  and c1: "\<lbrakk> ps \<noteq> []; ls = [] \<rbrakk> \<Longrightarrow> R"
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  and c2: "\<And>a as x xs. \<lbrakk> ps = a#as; ls = x#xs; x = a; \<not> as \<le> xs\<rbrakk> \<Longrightarrow> R"
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  and c3: "\<And>a as x xs. \<lbrakk> ps = a#as; ls = x#xs; x \<noteq> a\<rbrakk> \<Longrightarrow> R"
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  shows "R"
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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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  then have c: "ps = a#as" .
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  show ?thesis
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  proof (cases ls)
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    case Nil
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    have "ps \<noteq> []" by (simp add: Nil Cons)
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    from this and Nil show ?thesis by (rule c1)
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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> as \<le> 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_prefix_induct [consumes 1, case_names Nil Neq Eq]:
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  assumes np: "\<not> ps \<le> 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> xs \<le> ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"
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  shows "P ps ls"
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  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_prefix_cases intro!: base)
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next
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  case (Cons y ys)
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  then have npfx: "\<not> ps \<le> (y # ys)" by simp
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  then obtain x xs where pv: "ps = x # xs"
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    by (rule not_prefix_cases) auto
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  from Cons
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  have ih: "\<And>ps. \<not>ps \<le> ys \<Longrightarrow> P ps ys" by simp
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  show ?case using npfx
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    by (simp only: pv) (erule not_prefix_cases, auto intro: r1 r2 ih)
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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> xs \<le> ys \<and> \<not> ys \<le> xs)"
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lemma parallelI [intro]: "\<not> xs \<le> ys ==> \<not> ys \<le> 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> xs \<le> ys \<and> \<not> ys \<le> xs"
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  using assms unfolding parallel_def by blast
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theorem prefix_cases:
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  obtains "xs \<le> ys" | "ys < xs" | "xs \<parallel> ys"
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  unfolding parallel_def strict_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 prefix_cases)
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    assume le: "xs \<le> 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' = []" with ys have "xs = ys" by simp
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      with snoc have "[x] \<parallel> []" by auto
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      then have False by blast
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      then show ?thesis ..
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    next
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      fix c cs assume ys': "ys' = c # cs"
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      with snoc ys have "xs @ [x] \<parallel> xs @ c # cs" by (simp only:)
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      then have "x \<noteq> c" by auto
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      moreover have "xs @ [x] = xs @ x # []" by simp
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      moreover from ys ys' have "ys = xs @ c # cs" by (simp only:)
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      ultimately show ?thesis by blast
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    qed
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  next
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    assume "ys < xs" then have "ys \<le> xs @ [x]" by (simp add: strict_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:
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  "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
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  by (rule parallelI)
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     (erule parallelE, erule conjE,
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            induct rule: not_prefix_induct, simp+)+
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lemma parallel_appendI:
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  "\<lbrakk> xs \<parallel> ys; x = xs @ xs' ; y = ys @ ys' \<rbrakk> \<Longrightarrow> x \<parallel> y"
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  by simp (rule parallel_append)
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lemma parallel_commute:
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  "(a \<parallel> b) = (b \<parallel> a)"
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  unfolding parallel_def by auto
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subsection {* Postfix order on lists *}
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definition
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  postfix :: "'a list => 'a list => bool"  ("(_/ >>= _)" [51, 50] 50) where
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  "(xs >>= ys) = (\<exists>zs. xs = zs @ ys)"
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lemma postfixI [intro?]: "xs = zs @ ys ==> xs >>= ys"
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  unfolding postfix_def by blast
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lemma postfixE [elim?]:
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  assumes "xs >>= ys"
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  obtains zs where "xs = zs @ ys"
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  using assms unfolding postfix_def by blast
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lemma postfix_refl [iff]: "xs >>= xs"
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  by (auto simp add: postfix_def)
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lemma postfix_trans: "\<lbrakk>xs >>= ys; ys >>= zs\<rbrakk> \<Longrightarrow> xs >>= zs"
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  by (auto simp add: postfix_def)
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lemma postfix_antisym: "\<lbrakk>xs >>= ys; ys >>= xs\<rbrakk> \<Longrightarrow> xs = ys"
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  by (auto simp add: postfix_def)
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lemma Nil_postfix [iff]: "xs >>= []"
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  by (simp add: postfix_def)
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lemma postfix_Nil [simp]: "([] >>= xs) = (xs = [])"
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  by (auto simp add: postfix_def)
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lemma postfix_ConsI: "xs >>= ys \<Longrightarrow> x#xs >>= ys"
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  by (auto simp add: postfix_def)
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lemma postfix_ConsD: "xs >>= y#ys \<Longrightarrow> xs >>= ys"
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  by (auto simp add: postfix_def)
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lemma postfix_appendI: "xs >>= ys \<Longrightarrow> zs @ xs >>= ys"
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  by (auto simp add: postfix_def)
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lemma postfix_appendD: "xs >>= zs @ ys \<Longrightarrow> xs >>= ys"
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  by (auto simp add: postfix_def)
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lemma postfix_is_subset: "xs >>= ys ==> set ys \<subseteq> set xs"
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proof -
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  assume "xs >>= ys"
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  then obtain zs where "xs = zs @ ys" ..
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  then show ?thesis by (induct zs) auto
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qed
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lemma postfix_ConsD2: "x#xs >>= y#ys ==> xs >>= ys"
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proof -
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  assume "x#xs >>= y#ys"
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  then obtain zs where "x#xs = zs @ y#ys" ..
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  then show ?thesis
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    by (induct zs) (auto intro!: postfix_appendI postfix_ConsI)
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qed
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lemma postfix_to_prefix: "xs >>= ys \<longleftrightarrow> rev ys \<le> rev xs"
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proof
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  assume "xs >>= ys"
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  then obtain zs where "xs = zs @ ys" ..
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  then have "rev xs = rev ys @ rev zs" by simp
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  then show "rev ys <= rev xs" ..
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next
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  assume "rev ys <= rev xs"
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  then obtain zs where "rev xs = rev ys @ zs" ..
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   377
  then have "rev (rev xs) = rev zs @ rev (rev ys)" by simp
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   378
  then have "xs = rev zs @ ys" by simp
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   379
  then show "xs >>= ys" ..
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   380
qed
wenzelm@17201
   381
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   382
lemma distinct_postfix:
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   383
  assumes dx: "distinct xs"
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   384
  and     pf: "xs >>= ys"
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   385
  shows   "distinct ys"
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   386
  using dx pf by (clarsimp elim!: postfixE)
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   387
kleing@25299
   388
lemma postfix_map:
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   389
  assumes pf: "xs >>= ys"
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   390
  shows   "map f xs >>= map f ys"
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   391
  using pf by (auto elim!: postfixE intro: postfixI)
kleing@25299
   392
kleing@25299
   393
lemma postfix_drop:
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   394
  "as >>= drop n as"
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   395
  unfolding postfix_def
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   396
  by (rule exI [where x = "take n as"]) simp
kleing@25299
   397
kleing@25299
   398
lemma postfix_take:
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   399
  "xs >>= ys \<Longrightarrow> xs = take (length xs - length ys) xs @ ys"
kleing@25299
   400
  by (clarsimp elim!: postfixE)
kleing@25299
   401
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   402
lemma parallelD1:
kleing@25299
   403
  "x \<parallel> y \<Longrightarrow> \<not> x \<le> y" by blast
kleing@25299
   404
wenzelm@25355
   405
lemma parallelD2:
kleing@25299
   406
  "x \<parallel> y \<Longrightarrow> \<not> y \<le> x" by blast
wenzelm@25355
   407
wenzelm@25355
   408
lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
kleing@25299
   409
  unfolding parallel_def by simp
wenzelm@25355
   410
kleing@25299
   411
lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
kleing@25299
   412
  unfolding parallel_def by simp
kleing@25299
   413
kleing@25299
   414
lemma Cons_parallelI1:
kleing@25299
   415
  "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs" by auto
kleing@25299
   416
kleing@25299
   417
lemma Cons_parallelI2:
wenzelm@25355
   418
  "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
kleing@25299
   419
  apply simp
kleing@25299
   420
  apply (rule parallelI)
kleing@25299
   421
   apply simp
kleing@25299
   422
   apply (erule parallelD1)
kleing@25299
   423
  apply simp
kleing@25299
   424
  apply (erule parallelD2)
kleing@25299
   425
 done
kleing@25299
   426
kleing@25299
   427
lemma not_equal_is_parallel:
kleing@25299
   428
  assumes neq: "xs \<noteq> ys"
kleing@25299
   429
  and     len: "length xs = length ys"
kleing@25299
   430
  shows   "xs \<parallel> ys"
kleing@25299
   431
  using len neq
wenzelm@25355
   432
proof (induct rule: list_induct2)
wenzelm@25355
   433
  case 1 then show ?case by simp
kleing@25299
   434
next
kleing@25299
   435
  case (2 a as b bs)
wenzelm@25355
   436
  have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact
kleing@25299
   437
kleing@25299
   438
  show ?case
kleing@25299
   439
  proof (cases "a = b")
wenzelm@25355
   440
    case True
wenzelm@25355
   441
    then have "as \<noteq> bs" using 2 by simp
wenzelm@25355
   442
    then show ?thesis by (rule Cons_parallelI2 [OF True ih])
kleing@25299
   443
  next
kleing@25299
   444
    case False
wenzelm@25355
   445
    then show ?thesis by (rule Cons_parallelI1)
kleing@25299
   446
  qed
kleing@25299
   447
qed
haftmann@22178
   448
wenzelm@25355
   449
haftmann@22178
   450
subsection {* Exeuctable code *}
haftmann@22178
   451
haftmann@22178
   452
lemma less_eq_code [code func]:
haftmann@22178
   453
  "([]\<Colon>'a\<Colon>{eq, ord} list) \<le> xs \<longleftrightarrow> True"
haftmann@22178
   454
  "(x\<Colon>'a\<Colon>{eq, ord}) # xs \<le> [] \<longleftrightarrow> False"
haftmann@22178
   455
  "(x\<Colon>'a\<Colon>{eq, ord}) # xs \<le> y # ys \<longleftrightarrow> x = y \<and> xs \<le> ys"
haftmann@22178
   456
  by simp_all
haftmann@22178
   457
haftmann@22178
   458
lemma less_code [code func]:
haftmann@22178
   459
  "xs < ([]\<Colon>'a\<Colon>{eq, ord} list) \<longleftrightarrow> False"
haftmann@22178
   460
  "[] < (x\<Colon>'a\<Colon>{eq, ord})# xs \<longleftrightarrow> True"
haftmann@22178
   461
  "(x\<Colon>'a\<Colon>{eq, ord}) # xs < y # ys \<longleftrightarrow> x = y \<and> xs < ys"
haftmann@22178
   462
  unfolding strict_prefix_def by auto
haftmann@22178
   463
haftmann@22178
   464
lemmas [code func] = postfix_to_prefix
haftmann@22178
   465
wenzelm@10330
   466
end