src/HOL/Library/List_Prefix.thy
author wenzelm
Mon, 06 Nov 2000 22:56:07 +0100
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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 {*
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  \title{List prefixes}
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  \author{Tobias Nipkow and Markus Wenzel}
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*}
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theory List_Prefix = Main:
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subsection {* Prefix order on lists *}
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instance list :: ("term") 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 :: ("term") 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?]: "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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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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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 \<le> xs ==> C) ==>
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    (xs \<parallel> ys ==> C) ==> C"
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  by (unfold parallel_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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  (is "PROP ?P xs" concl is "?E xs")
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proof (induct xs rule: rev_induct)
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  assume "[] \<parallel> ys" hence False by auto
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  thus "?E []" ..
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next
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  fix x xs
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  assume hyp: "PROP ?P xs"
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  assume asm: "xs @ [x] \<parallel> ys"
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  show "?E (xs @ [x])"
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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 asm have "[x] \<parallel> []" by auto
d8b3613158b1 improved: 'induct' handle non-atomic goals;
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   156
      hence False by blast
10389
c7d8901ab269 proper setup of "parallel";
wenzelm
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   157
      thus ?thesis ..
c7d8901ab269 proper setup of "parallel";
wenzelm
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   158
    next
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   159
      fix c cs assume ys': "ys' = c # cs"
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   160
      with asm ys have "xs @ [x] \<parallel> xs @ c # cs" by (simp only:)
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   161
      hence "x \<noteq> c" by auto
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   162
      moreover have "xs @ [x] = xs @ x # []" by simp
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   163
      moreover from ys ys' have "ys = xs @ c # cs" by (simp only:)
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   164
      ultimately show ?thesis by blast
10389
c7d8901ab269 proper setup of "parallel";
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   165
    qed
10408
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   166
  next
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   167
    assume "ys \<le> xs" hence "ys \<le> xs @ [x]" by simp
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   168
    with asm have False by blast
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   169
    thus ?thesis ..
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   170
  next
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   171
    assume "xs \<parallel> ys"
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   172
    with hyp obtain as b bs c cs where neq: "(b::'a) \<noteq> c"
d8b3613158b1 improved: 'induct' handle non-atomic goals;
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   173
      and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
d8b3613158b1 improved: 'induct' handle non-atomic goals;
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   174
      by blast
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   175
    from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
d8b3613158b1 improved: 'induct' handle non-atomic goals;
wenzelm
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   176
    with neq ys show ?thesis by blast
10389
c7d8901ab269 proper setup of "parallel";
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   177
  qed
c7d8901ab269 proper setup of "parallel";
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   178
qed
10330
4362e906b745 "List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
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   179
4362e906b745 "List prefixes" library theory (replaces old Lex/Prefix);
wenzelm
parents:
diff changeset
   180
end