--- a/src/HOL/Library/Sublist.thy Wed Nov 20 18:32:25 2013 +0100
+++ b/src/HOL/Library/Sublist.thy Wed Nov 20 18:58:00 2013 +0100
@@ -3,198 +3,12 @@
Author: Christian Sternagel, JAIST
*)
-header {* List prefixes, suffixes, and homeomorphic embedding *}
+header {* Parallel lists, list suffixes, and homeomorphic embedding *}
theory Sublist
imports Main
begin
-subsection {* Prefix order on lists *}
-
-definition prefixeq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
- where "prefixeq xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"
-
-definition prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
- where "prefix xs ys \<longleftrightarrow> prefixeq xs ys \<and> xs \<noteq> ys"
-
-interpretation prefix_order: order prefixeq prefix
- by default (auto simp: prefixeq_def prefix_def)
-
-interpretation prefix_bot: order_bot Nil prefixeq prefix
- by default (simp add: prefixeq_def)
-
-lemma prefixeqI [intro?]: "ys = xs @ zs \<Longrightarrow> prefixeq xs ys"
- unfolding prefixeq_def by blast
-
-lemma prefixeqE [elim?]:
- assumes "prefixeq xs ys"
- obtains zs where "ys = xs @ zs"
- using assms unfolding prefixeq_def by blast
-
-lemma prefixI' [intro?]: "ys = xs @ z # zs \<Longrightarrow> prefix xs ys"
- unfolding prefix_def prefixeq_def by blast
-
-lemma prefixE' [elim?]:
- assumes "prefix xs ys"
- obtains z zs where "ys = xs @ z # zs"
-proof -
- from `prefix xs ys` obtain us where "ys = xs @ us" and "xs \<noteq> ys"
- unfolding prefix_def prefixeq_def by blast
- with that show ?thesis by (auto simp add: neq_Nil_conv)
-qed
-
-lemma prefixI [intro?]: "prefixeq xs ys \<Longrightarrow> xs \<noteq> ys \<Longrightarrow> prefix xs ys"
- unfolding prefix_def by blast
-
-lemma prefixE [elim?]:
- fixes xs ys :: "'a list"
- assumes "prefix xs ys"
- obtains "prefixeq xs ys" and "xs \<noteq> ys"
- using assms unfolding prefix_def by blast
-
-
-subsection {* Basic properties of prefixes *}
-
-theorem Nil_prefixeq [iff]: "prefixeq [] xs"
- by (simp add: prefixeq_def)
-
-theorem prefixeq_Nil [simp]: "(prefixeq xs []) = (xs = [])"
- by (induct xs) (simp_all add: prefixeq_def)
-
-lemma prefixeq_snoc [simp]: "prefixeq xs (ys @ [y]) \<longleftrightarrow> xs = ys @ [y] \<or> prefixeq xs ys"
-proof
- assume "prefixeq xs (ys @ [y])"
- then obtain zs where zs: "ys @ [y] = xs @ zs" ..
- show "xs = ys @ [y] \<or> prefixeq xs ys"
- by (metis append_Nil2 butlast_append butlast_snoc prefixeqI zs)
-next
- assume "xs = ys @ [y] \<or> prefixeq xs ys"
- then show "prefixeq xs (ys @ [y])"
- by (metis prefix_order.eq_iff prefix_order.order_trans prefixeqI)
-qed
-
-lemma Cons_prefixeq_Cons [simp]: "prefixeq (x # xs) (y # ys) = (x = y \<and> prefixeq xs ys)"
- by (auto simp add: prefixeq_def)
-
-lemma prefixeq_code [code]:
- "prefixeq [] xs \<longleftrightarrow> True"
- "prefixeq (x # xs) [] \<longleftrightarrow> False"
- "prefixeq (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefixeq xs ys"
- by simp_all
-
-lemma same_prefixeq_prefixeq [simp]: "prefixeq (xs @ ys) (xs @ zs) = prefixeq ys zs"
- by (induct xs) simp_all
-
-lemma same_prefixeq_nil [iff]: "prefixeq (xs @ ys) xs = (ys = [])"
- by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixeqI)
-
-lemma prefixeq_prefixeq [simp]: "prefixeq xs ys \<Longrightarrow> prefixeq xs (ys @ zs)"
- by (metis prefix_order.le_less_trans prefixeqI prefixE prefixI)
-
-lemma append_prefixeqD: "prefixeq (xs @ ys) zs \<Longrightarrow> prefixeq xs zs"
- by (auto simp add: prefixeq_def)
-
-theorem prefixeq_Cons: "prefixeq xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefixeq zs ys))"
- by (cases xs) (auto simp add: prefixeq_def)
-
-theorem prefixeq_append:
- "prefixeq xs (ys @ zs) = (prefixeq xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefixeq us zs))"
- apply (induct zs rule: rev_induct)
- apply force
- apply (simp del: append_assoc add: append_assoc [symmetric])
- apply (metis append_eq_appendI)
- done
-
-lemma append_one_prefixeq:
- "prefixeq xs ys \<Longrightarrow> length xs < length ys \<Longrightarrow> prefixeq (xs @ [ys ! length xs]) ys"
- proof (unfold prefixeq_def)
- assume a1: "\<exists>zs. ys = xs @ zs"
- then obtain sk :: "'a list" where sk: "ys = xs @ sk" by fastforce
- assume a2: "length xs < length ys"
- have f1: "\<And>v. ([]\<Colon>'a list) @ v = v" using append_Nil2 by simp
- have "[] \<noteq> sk" using a1 a2 sk less_not_refl by force
- hence "\<exists>v. xs @ hd sk # v = ys" using sk by (metis hd_Cons_tl)
- thus "\<exists>zs. ys = (xs @ [ys ! length xs]) @ zs" using f1 by fastforce
- qed
-
-theorem prefixeq_length_le: "prefixeq xs ys \<Longrightarrow> length xs \<le> length ys"
- by (auto simp add: prefixeq_def)
-
-lemma prefixeq_same_cases:
- "prefixeq (xs\<^sub>1::'a list) ys \<Longrightarrow> prefixeq xs\<^sub>2 ys \<Longrightarrow> prefixeq xs\<^sub>1 xs\<^sub>2 \<or> prefixeq xs\<^sub>2 xs\<^sub>1"
- unfolding prefixeq_def by (force simp: append_eq_append_conv2)
-
-lemma set_mono_prefixeq: "prefixeq xs ys \<Longrightarrow> set xs \<subseteq> set ys"
- by (auto simp add: prefixeq_def)
-
-lemma take_is_prefixeq: "prefixeq (take n xs) xs"
- unfolding prefixeq_def by (metis append_take_drop_id)
-
-lemma map_prefixeqI: "prefixeq xs ys \<Longrightarrow> prefixeq (map f xs) (map f ys)"
- by (auto simp: prefixeq_def)
-
-lemma prefixeq_length_less: "prefix xs ys \<Longrightarrow> length xs < length ys"
- by (auto simp: prefix_def prefixeq_def)
-
-lemma prefix_simps [simp, code]:
- "prefix xs [] \<longleftrightarrow> False"
- "prefix [] (x # xs) \<longleftrightarrow> True"
- "prefix (x # xs) (y # ys) \<longleftrightarrow> x = y \<and> prefix xs ys"
- by (simp_all add: prefix_def cong: conj_cong)
-
-lemma take_prefix: "prefix xs ys \<Longrightarrow> prefix (take n xs) ys"
- apply (induct n arbitrary: xs ys)
- apply (case_tac ys, simp_all)[1]
- apply (metis prefix_order.less_trans prefixI take_is_prefixeq)
- done
-
-lemma not_prefixeq_cases:
- assumes pfx: "\<not> prefixeq ps ls"
- obtains
- (c1) "ps \<noteq> []" and "ls = []"
- | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> prefixeq as xs"
- | (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a"
-proof (cases ps)
- case Nil
- then show ?thesis using pfx by simp
-next
- case (Cons a as)
- note c = `ps = a#as`
- show ?thesis
- proof (cases ls)
- case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefixeq_nil)
- next
- case (Cons x xs)
- show ?thesis
- proof (cases "x = a")
- case True
- have "\<not> prefixeq as xs" using pfx c Cons True by simp
- with c Cons True show ?thesis by (rule c2)
- next
- case False
- with c Cons show ?thesis by (rule c3)
- qed
- qed
-qed
-
-lemma not_prefixeq_induct [consumes 1, case_names Nil Neq Eq]:
- assumes np: "\<not> prefixeq ps ls"
- and base: "\<And>x xs. P (x#xs) []"
- and r1: "\<And>x xs y ys. x \<noteq> y \<Longrightarrow> P (x#xs) (y#ys)"
- and r2: "\<And>x xs y ys. \<lbrakk> x = y; \<not> prefixeq xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys)"
- shows "P ps ls" using np
-proof (induct ls arbitrary: ps)
- case Nil then show ?case
- by (auto simp: neq_Nil_conv elim!: not_prefixeq_cases intro!: base)
-next
- case (Cons y ys)
- then have npfx: "\<not> prefixeq ps (y # ys)" by simp
- then obtain x xs where pv: "ps = x # xs"
- by (rule not_prefixeq_cases) auto
- show ?case by (metis Cons.hyps Cons_prefixeq_Cons npfx pv r1 r2)
-qed
-
-
subsection {* Parallel lists *}
definition parallel :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "\<parallel>" 50)