wenzelm@10330: (*  Title:      HOL/Library/List_Prefix.thy
wenzelm@10330:     ID:         $Id$
wenzelm@10330:     Author:     Tobias Nipkow and Markus Wenzel, TU Muenchen
wenzelm@11780:     License:    GPL (GNU GENERAL PUBLIC LICENSE)
wenzelm@10330: *)
wenzelm@10330: 
wenzelm@10330: header {*
wenzelm@10330:   \title{List prefixes}
wenzelm@10330:   \author{Tobias Nipkow and Markus Wenzel}
wenzelm@10330: *}
wenzelm@10330: 
wenzelm@10330: theory List_Prefix = Main:
wenzelm@10330: 
wenzelm@10330: subsection {* Prefix order on lists *}
wenzelm@10330: 
wenzelm@10330: instance list :: ("term") ord ..
wenzelm@10330: 
wenzelm@10330: defs (overloaded)
wenzelm@10389:   prefix_def: "xs \<le> ys == \<exists>zs. ys = xs @ zs"
wenzelm@10389:   strict_prefix_def: "xs < ys == xs \<le> ys \<and> xs \<noteq> (ys::'a list)"
wenzelm@10330: 
wenzelm@10330: instance list :: ("term") order
wenzelm@10389:   by intro_classes (auto simp add: prefix_def strict_prefix_def)
wenzelm@10330: 
wenzelm@10389: lemma prefixI [intro?]: "ys = xs @ zs ==> xs \<le> ys"
wenzelm@10389:   by (unfold prefix_def) blast
wenzelm@10330: 
wenzelm@10389: lemma prefixE [elim?]: "xs \<le> ys ==> (!!zs. ys = xs @ zs ==> C) ==> C"
wenzelm@10389:   by (unfold prefix_def) blast
wenzelm@10330: 
wenzelm@10870: lemma strict_prefixI' [intro?]: "ys = xs @ z # zs ==> xs < ys"
wenzelm@10870:   by (unfold strict_prefix_def prefix_def) blast
wenzelm@10870: 
wenzelm@10870: lemma strict_prefixE' [elim?]:
wenzelm@10870:     "xs < ys ==> (!!z zs. ys = xs @ z # zs ==> C) ==> C"
wenzelm@10870: proof -
wenzelm@10870:   assume r: "!!z zs. ys = xs @ z # zs ==> C"
wenzelm@10870:   assume "xs < ys"
wenzelm@10870:   then obtain us where "ys = xs @ us" and "xs \<noteq> ys"
wenzelm@10870:     by (unfold strict_prefix_def prefix_def) blast
wenzelm@10870:   with r show ?thesis by (auto simp add: neq_Nil_conv)
wenzelm@10870: qed
wenzelm@10870: 
wenzelm@10389: lemma strict_prefixI [intro?]: "xs \<le> ys ==> xs \<noteq> ys ==> xs < (ys::'a list)"
wenzelm@10389:   by (unfold strict_prefix_def) blast
wenzelm@10330: 
wenzelm@10389: lemma strict_prefixE [elim?]:
wenzelm@10389:     "xs < ys ==> (xs \<le> ys ==> xs \<noteq> (ys::'a list) ==> C) ==> C"
wenzelm@10389:   by (unfold strict_prefix_def) blast
wenzelm@10330: 
wenzelm@10330: 
wenzelm@10389: subsection {* Basic properties of prefixes *}
wenzelm@10330: 
wenzelm@10330: theorem Nil_prefix [iff]: "[] \<le> xs"
wenzelm@10389:   by (simp add: prefix_def)
wenzelm@10330: 
wenzelm@10330: theorem prefix_Nil [simp]: "(xs \<le> []) = (xs = [])"
wenzelm@10389:   by (induct xs) (simp_all add: prefix_def)
wenzelm@10330: 
wenzelm@10330: lemma prefix_snoc [simp]: "(xs \<le> ys @ [y]) = (xs = ys @ [y] \<or> xs \<le> ys)"
wenzelm@10389: proof
wenzelm@10389:   assume "xs \<le> ys @ [y]"
wenzelm@10389:   then obtain zs where zs: "ys @ [y] = xs @ zs" ..
wenzelm@10389:   show "xs = ys @ [y] \<or> xs \<le> ys"
wenzelm@10389:   proof (cases zs rule: rev_cases)
wenzelm@10389:     assume "zs = []"
wenzelm@10389:     with zs have "xs = ys @ [y]" by simp
wenzelm@10389:     thus ?thesis ..
wenzelm@10389:   next
wenzelm@10389:     fix z zs' assume "zs = zs' @ [z]"
wenzelm@10389:     with zs have "ys = xs @ zs'" by simp
wenzelm@10389:     hence "xs \<le> ys" ..
wenzelm@10389:     thus ?thesis ..
wenzelm@10389:   qed
wenzelm@10389: next
wenzelm@10389:   assume "xs = ys @ [y] \<or> xs \<le> ys"
wenzelm@10389:   thus "xs \<le> ys @ [y]"
wenzelm@10389:   proof
wenzelm@10389:     assume "xs = ys @ [y]"
wenzelm@10389:     thus ?thesis by simp
wenzelm@10389:   next
wenzelm@10389:     assume "xs \<le> ys"
wenzelm@10389:     then obtain zs where "ys = xs @ zs" ..
wenzelm@10389:     hence "ys @ [y] = xs @ (zs @ [y])" by simp
wenzelm@10389:     thus ?thesis ..
wenzelm@10389:   qed
wenzelm@10389: qed
wenzelm@10330: 
wenzelm@10330: lemma Cons_prefix_Cons [simp]: "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)"
wenzelm@10389:   by (auto simp add: prefix_def)
wenzelm@10330: 
wenzelm@10330: lemma same_prefix_prefix [simp]: "(xs @ ys \<le> xs @ zs) = (ys \<le> zs)"
wenzelm@10389:   by (induct xs) simp_all
wenzelm@10330: 
wenzelm@10389: lemma same_prefix_nil [iff]: "(xs @ ys \<le> xs) = (ys = [])"
wenzelm@10389: proof -
wenzelm@10389:   have "(xs @ ys \<le> xs @ []) = (ys \<le> [])" by (rule same_prefix_prefix)
wenzelm@10389:   thus ?thesis by simp
wenzelm@10389: qed
wenzelm@10330: 
wenzelm@10330: lemma prefix_prefix [simp]: "xs \<le> ys ==> xs \<le> ys @ zs"
wenzelm@10389: proof -
wenzelm@10389:   assume "xs \<le> ys"
wenzelm@10389:   then obtain us where "ys = xs @ us" ..
wenzelm@10389:   hence "ys @ zs = xs @ (us @ zs)" by simp
wenzelm@10389:   thus ?thesis ..
wenzelm@10389: qed
wenzelm@10330: 
wenzelm@10330: theorem prefix_Cons: "(xs \<le> y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> zs \<le> ys))"
wenzelm@10389:   by (cases xs) (auto simp add: prefix_def)
wenzelm@10330: 
wenzelm@10330: theorem prefix_append:
wenzelm@10330:     "(xs \<le> ys @ zs) = (xs \<le> ys \<or> (\<exists>us. xs = ys @ us \<and> us \<le> zs))"
wenzelm@10330:   apply (induct zs rule: rev_induct)
wenzelm@10330:    apply force
wenzelm@10330:   apply (simp del: append_assoc add: append_assoc [symmetric])
wenzelm@10330:   apply simp
wenzelm@10330:   apply blast
wenzelm@10330:   done
wenzelm@10330: 
wenzelm@10330: lemma append_one_prefix:
wenzelm@10330:     "xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys"
wenzelm@10330:   apply (unfold prefix_def)
wenzelm@10330:   apply (auto simp add: nth_append)
wenzelm@10389:   apply (case_tac zs)
wenzelm@10330:    apply auto
wenzelm@10330:   done
wenzelm@10330: 
wenzelm@10330: theorem prefix_length_le: "xs \<le> ys ==> length xs \<le> length ys"
wenzelm@10389:   by (auto simp add: prefix_def)
wenzelm@10330: 
wenzelm@10330: 
wenzelm@10389: subsection {* Parallel lists *}
wenzelm@10389: 
wenzelm@10389: constdefs
wenzelm@10389:   parallel :: "'a list => 'a list => bool"    (infixl "\<parallel>" 50)
wenzelm@10389:   "xs \<parallel> ys == \<not> xs \<le> ys \<and> \<not> ys \<le> xs"
wenzelm@10389: 
wenzelm@10389: lemma parallelI [intro]: "\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> xs \<parallel> ys"
wenzelm@10389:   by (unfold parallel_def) blast
wenzelm@10330: 
wenzelm@10389: lemma parallelE [elim]:
wenzelm@10389:     "xs \<parallel> ys ==> (\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> C) ==> C"
wenzelm@10389:   by (unfold parallel_def) blast
wenzelm@10330: 
wenzelm@10389: theorem prefix_cases:
wenzelm@10389:   "(xs \<le> ys ==> C) ==>
wenzelm@10512:     (ys < xs ==> C) ==>
wenzelm@10389:     (xs \<parallel> ys ==> C) ==> C"
wenzelm@10512:   by (unfold parallel_def strict_prefix_def) blast
wenzelm@10330: 
wenzelm@10389: theorem parallel_decomp:
wenzelm@10389:   "xs \<parallel> ys ==> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
wenzelm@10408:   (is "PROP ?P xs" concl is "?E xs")
wenzelm@10408: proof (induct xs rule: rev_induct)
wenzelm@10408:   assume "[] \<parallel> ys" hence False by auto
wenzelm@10408:   thus "?E []" ..
wenzelm@10408: next
wenzelm@10408:   fix x xs
wenzelm@10408:   assume hyp: "PROP ?P xs"
wenzelm@10408:   assume asm: "xs @ [x] \<parallel> ys"
wenzelm@10408:   show "?E (xs @ [x])"
wenzelm@10408:   proof (rule prefix_cases)
wenzelm@10408:     assume le: "xs \<le> ys"
wenzelm@10408:     then obtain ys' where ys: "ys = xs @ ys'" ..
wenzelm@10408:     show ?thesis
wenzelm@10408:     proof (cases ys')
wenzelm@10408:       assume "ys' = []" with ys have "xs = ys" by simp
wenzelm@10408:       with asm have "[x] \<parallel> []" by auto
wenzelm@10408:       hence False by blast
wenzelm@10389:       thus ?thesis ..
wenzelm@10389:     next
wenzelm@10408:       fix c cs assume ys': "ys' = c # cs"
wenzelm@10408:       with asm ys have "xs @ [x] \<parallel> xs @ c # cs" by (simp only:)
wenzelm@10408:       hence "x \<noteq> c" by auto
wenzelm@10408:       moreover have "xs @ [x] = xs @ x # []" by simp
wenzelm@10408:       moreover from ys ys' have "ys = xs @ c # cs" by (simp only:)
wenzelm@10408:       ultimately show ?thesis by blast
wenzelm@10389:     qed
wenzelm@10408:   next
wenzelm@10512:     assume "ys < xs" hence "ys \<le> xs @ [x]" by (simp add: strict_prefix_def)
wenzelm@10408:     with asm have False by blast
wenzelm@10408:     thus ?thesis ..
wenzelm@10408:   next
wenzelm@10408:     assume "xs \<parallel> ys"
wenzelm@10408:     with hyp obtain as b bs c cs where neq: "(b::'a) \<noteq> c"
wenzelm@10408:       and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
wenzelm@10408:       by blast
wenzelm@10408:     from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
wenzelm@10408:     with neq ys show ?thesis by blast
wenzelm@10389:   qed
wenzelm@10389: qed
wenzelm@10330: 
wenzelm@10330: end