(* Title: HOL/List_Prefix.thy
Author: Tobias Nipkow and Markus Wenzel, TU Muenchen
Author: Christian Sternagel, JAIST
*)
header {* Parallel lists, list suffixes, and homeomorphic embedding *}
theory List_Prefix
imports List
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
end