src/HOL/Library/List_Prefix.thy
author wenzelm
Fri Nov 25 19:20:56 2005 +0100 (2005-11-25)
changeset 18258 836491e9b7d5
parent 17201 3bdf1dfcdee4
child 18730 843da46f89ac
permissions -rw-r--r--
tuned induct 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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  by (unfold prefix_def) blast
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lemma prefixE [elim?]: "xs \<le> ys ==> (!!zs. ys = xs @ zs ==> C) ==> C"
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  by (unfold prefix_def) blast
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lemma strict_prefixI' [intro?]: "ys = xs @ z # zs ==> xs < ys"
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  by (unfold strict_prefix_def prefix_def) blast
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lemma strict_prefixE' [elim?]:
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  assumes lt: "xs < ys"
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    and r: "!!z zs. ys = xs @ z # zs ==> C"
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  shows C
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proof -
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  from lt obtain us where "ys = xs @ us" and "xs \<noteq> ys"
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    by (unfold strict_prefix_def prefix_def) blast
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  with r 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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  by (unfold strict_prefix_def) blast
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lemma strict_prefixE [elim?]:
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    "xs < ys ==> (xs \<le> ys ==> xs \<noteq> (ys::'a list) ==> C) ==> C"
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  by (unfold strict_prefix_def) 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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    thus ?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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    hence "xs \<le> ys" ..
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    thus ?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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  thus "xs \<le> ys @ [y]"
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  proof
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    assume "xs = ys @ [y]"
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    thus ?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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    hence "ys @ [y] = xs @ (zs @ [y])" by simp
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    thus ?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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  thus ?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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  hence "ys @ zs = xs @ (us @ zs)" by simp
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  thus ?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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subsection {* Parallel lists *}
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constdefs
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  parallel :: "'a list => 'a list => bool"    (infixl "\<parallel>" 50)
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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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  by (unfold parallel_def) blast
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lemma parallelE [elim]:
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    "xs \<parallel> ys ==> (\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> C) ==> C"
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  by (unfold parallel_def) blast
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theorem prefix_cases:
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  "(xs \<le> ys ==> C) ==>
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    (ys < xs ==> C) ==>
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    (xs \<parallel> ys ==> C) ==> C"
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  by (unfold parallel_def strict_prefix_def) 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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  hence False by auto
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  thus ?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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      hence False by blast
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      thus ?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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      hence "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" hence "ys \<le> xs @ [x]" by (simp add: strict_prefix_def)
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    with snoc have False by blast
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    thus ?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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subsection {* Postfix order on lists *}
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constdefs
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  postfix :: "'a list => 'a list => bool"  ("(_/ >>= _)" [51, 50] 50)
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  "xs >>= ys == \<exists>zs. xs = zs @ ys"
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lemma postfix_refl [simp, intro!]: "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_lemma: "xs = zs @ ys \<Longrightarrow> set ys \<subseteq> set xs"
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  by (induct zs) auto
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lemma postfix_is_subset: "xs >>= ys \<Longrightarrow> set ys \<subseteq> set xs"
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  by (unfold postfix_def, erule exE, erule postfix_is_subset_lemma)
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lemma postfix_ConsD2_lemma: "x#xs = zs @ y#ys \<Longrightarrow> xs >>= ys"
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  by (induct zs) (auto intro!: postfix_appendI postfix_ConsI)
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lemma postfix_ConsD2: "x#xs >>= y#ys \<Longrightarrow> xs >>= ys"
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  by (auto simp add: postfix_def dest!: postfix_ConsD2_lemma)
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lemma postfix2prefix: "(xs >>= ys) = (rev ys <= rev xs)"
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  apply (unfold prefix_def postfix_def, safe)
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   apply (rule_tac x = "rev zs" in exI, simp)
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  apply (rule_tac x = "rev zs" in exI)
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  apply (rule rev_is_rev_conv [THEN iffD1], simp)
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  done
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end