(* Title: HOL/Library/Sublist.thy
Author: Tobias Nipkow and Markus Wenzel, TU Muenchen
Author: Christian Sternagel, JAIST
*)
section \List prefixes, suffixes, and homeomorphic embedding\
theory Sublist
imports Main
begin
subsection \Prefix order on lists\
definition prefixeq :: "'a list \ 'a list \ bool"
where "prefixeq xs ys \ (\zs. ys = xs @ zs)"
definition prefix :: "'a list \ 'a list \ bool"
where "prefix xs ys \ prefixeq xs ys \ xs \ ys"
interpretation prefix_order: order prefixeq prefix
by standard (auto simp: prefixeq_def prefix_def)
interpretation prefix_bot: order_bot Nil prefixeq prefix
by standard (simp add: prefixeq_def)
lemma prefixeqI [intro?]: "ys = xs @ zs \ 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 \ 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 \ 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 \ xs \ ys \ 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 \ 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]) \ xs = ys @ [y] \ prefixeq xs ys"
proof
assume "prefixeq xs (ys @ [y])"
then obtain zs where zs: "ys @ [y] = xs @ zs" ..
show "xs = ys @ [y] \ prefixeq xs ys"
by (metis append_Nil2 butlast_append butlast_snoc prefixeqI zs)
next
assume "xs = ys @ [y] \ 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 \ prefixeq xs ys)"
by (auto simp add: prefixeq_def)
lemma prefixeq_code [code]:
"prefixeq [] xs \ True"
"prefixeq (x # xs) [] \ False"
"prefixeq (x # xs) (y # ys) \ x = y \ 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 \ prefixeq xs (ys @ zs)"
by (metis prefix_order.le_less_trans prefixeqI prefixE prefixI)
lemma append_prefixeqD: "prefixeq (xs @ ys) zs \ prefixeq xs zs"
by (auto simp add: prefixeq_def)
theorem prefixeq_Cons: "prefixeq xs (y # ys) = (xs = [] \ (\zs. xs = y # zs \ prefixeq zs ys))"
by (cases xs) (auto simp add: prefixeq_def)
theorem prefixeq_append:
"prefixeq xs (ys @ zs) = (prefixeq xs ys \ (\us. xs = ys @ us \ 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 \ length xs < length ys \ prefixeq (xs @ [ys ! length xs]) ys"
proof (unfold prefixeq_def)
assume a1: "\zs. ys = xs @ zs"
then obtain sk :: "'a list" where sk: "ys = xs @ sk" by fastforce
assume a2: "length xs < length ys"
have f1: "\v. ([]::'a list) @ v = v" using append_Nil2 by simp
have "[] \ sk" using a1 a2 sk less_not_refl by force
hence "\v. xs @ hd sk # v = ys" using sk by (metis hd_Cons_tl)
thus "\zs. ys = (xs @ [ys ! length xs]) @ zs" using f1 by fastforce
qed
theorem prefixeq_length_le: "prefixeq xs ys \ length xs \ length ys"
by (auto simp add: prefixeq_def)
lemma prefixeq_same_cases:
"prefixeq (xs\<^sub>1::'a list) ys \ prefixeq xs\<^sub>2 ys \ prefixeq xs\<^sub>1 xs\<^sub>2 \ prefixeq xs\<^sub>2 xs\<^sub>1"
unfolding prefixeq_def by (force simp: append_eq_append_conv2)
lemma set_mono_prefixeq: "prefixeq xs ys \ set xs \ 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 \ prefixeq (map f xs) (map f ys)"
by (auto simp: prefixeq_def)
lemma prefixeq_length_less: "prefix xs ys \ length xs < length ys"
by (auto simp: prefix_def prefixeq_def)
lemma prefix_simps [simp, code]:
"prefix xs [] \ False"
"prefix [] (x # xs) \ True"
"prefix (x # xs) (y # ys) \ x = y \ prefix xs ys"
by (simp_all add: prefix_def cong: conj_cong)
lemma take_prefix: "prefix xs ys \ prefix (take n xs) ys"
apply (induct n arbitrary: xs ys)
apply (case_tac ys; simp)
apply (metis prefix_order.less_trans prefixI take_is_prefixeq)
done
lemma not_prefixeq_cases:
assumes pfx: "\ prefixeq ps ls"
obtains
(c1) "ps \ []" and "ls = []"
| (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\ prefixeq as xs"
| (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \ 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 "\ 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: "\ prefixeq ps ls"
and base: "\x xs. P (x#xs) []"
and r1: "\x xs y ys. x \ y \ P (x#xs) (y#ys)"
and r2: "\x xs y ys. \ x = y; \ prefixeq xs ys; P xs ys \ \ 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: "\ 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 \ 'a list \ bool" (infixl "\" 50)
where "(xs \ ys) = (\ prefixeq xs ys \ \ prefixeq ys xs)"
lemma parallelI [intro]: "\ prefixeq xs ys \ \ prefixeq ys xs \ xs \ ys"
unfolding parallel_def by blast
lemma parallelE [elim]:
assumes "xs \ ys"
obtains "\ prefixeq xs ys \ \ prefixeq ys xs"
using assms unfolding parallel_def by blast
theorem prefixeq_cases:
obtains "prefixeq xs ys" | "prefix ys xs" | "xs \ ys"
unfolding parallel_def prefix_def by blast
theorem parallel_decomp:
"xs \ ys \ \as b bs c cs. b \ c \ xs = as @ b # bs \ ys = as @ c # cs"
proof (induct xs rule: rev_induct)
case Nil
then have False by auto
then show ?case ..
next
case (snoc x xs)
show ?case
proof (rule prefixeq_cases)
assume le: "prefixeq xs ys"
then obtain ys' where ys: "ys = xs @ ys'" ..
show ?thesis
proof (cases ys')
assume "ys' = []"
then show ?thesis by (metis append_Nil2 parallelE prefixeqI snoc.prems ys)
next
fix c cs assume ys': "ys' = c # cs"
have "x \ c" using snoc.prems ys ys' by fastforce
thus "\as b bs c cs. b \ c \ xs @ [x] = as @ b # bs \ ys = as @ c # cs"
using ys ys' by blast
qed
next
assume "prefix ys xs"
then have "prefixeq ys (xs @ [x])" by (simp add: prefix_def)
with snoc have False by blast
then show ?thesis ..
next
assume "xs \ ys"
with snoc obtain as b bs c cs where neq: "(b::'a) \ c"
and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
by blast
from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
with neq ys show ?thesis by blast
qed
qed
lemma parallel_append: "a \ b \ a @ c \ b @ d"
apply (rule parallelI)
apply (erule parallelE, erule conjE,
induct rule: not_prefixeq_induct, simp+)+
done
lemma parallel_appendI: "xs \ ys \ x = xs @ xs' \ y = ys @ ys' \ x \ y"
by (simp add: parallel_append)
lemma parallel_commute: "a \ b \ b \ a"
unfolding parallel_def by auto
subsection \Suffix order on lists\
definition suffixeq :: "'a list \ 'a list \ bool"
where "suffixeq xs ys = (\zs. ys = zs @ xs)"
definition suffix :: "'a list \ 'a list \ bool"
where "suffix xs ys \ (\us. ys = us @ xs \ us \ [])"
lemma suffix_imp_suffixeq:
"suffix xs ys \ suffixeq xs ys"
by (auto simp: suffixeq_def suffix_def)
lemma suffixeqI [intro?]: "ys = zs @ xs \ suffixeq xs ys"
unfolding suffixeq_def by blast
lemma suffixeqE [elim?]:
assumes "suffixeq xs ys"
obtains zs where "ys = zs @ xs"
using assms unfolding suffixeq_def by blast
lemma suffixeq_refl [iff]: "suffixeq xs xs"
by (auto simp add: suffixeq_def)
lemma suffix_trans:
"suffix xs ys \ suffix ys zs \ suffix xs zs"
by (auto simp: suffix_def)
lemma suffixeq_trans: "\suffixeq xs ys; suffixeq ys zs\ \ suffixeq xs zs"
by (auto simp add: suffixeq_def)
lemma suffixeq_antisym: "\suffixeq xs ys; suffixeq ys xs\ \ xs = ys"
by (auto simp add: suffixeq_def)
lemma suffixeq_tl [simp]: "suffixeq (tl xs) xs"
by (induct xs) (auto simp: suffixeq_def)
lemma suffix_tl [simp]: "xs \ [] \ suffix (tl xs) xs"
by (induct xs) (auto simp: suffix_def)
lemma Nil_suffixeq [iff]: "suffixeq [] xs"
by (simp add: suffixeq_def)
lemma suffixeq_Nil [simp]: "(suffixeq xs []) = (xs = [])"
by (auto simp add: suffixeq_def)
lemma suffixeq_ConsI: "suffixeq xs ys \ suffixeq xs (y # ys)"
by (auto simp add: suffixeq_def)
lemma suffixeq_ConsD: "suffixeq (x # xs) ys \ suffixeq xs ys"
by (auto simp add: suffixeq_def)
lemma suffixeq_appendI: "suffixeq xs ys \ suffixeq xs (zs @ ys)"
by (auto simp add: suffixeq_def)
lemma suffixeq_appendD: "suffixeq (zs @ xs) ys \ suffixeq xs ys"
by (auto simp add: suffixeq_def)
lemma suffix_set_subset:
"suffix xs ys \ set xs \ set ys" by (auto simp: suffix_def)
lemma suffixeq_set_subset:
"suffixeq xs ys \ set xs \ set ys" by (auto simp: suffixeq_def)
lemma suffixeq_ConsD2: "suffixeq (x # xs) (y # ys) \ suffixeq xs ys"
proof -
assume "suffixeq (x # xs) (y # ys)"
then obtain zs where "y # ys = zs @ x # xs" ..
then show ?thesis
by (induct zs) (auto intro!: suffixeq_appendI suffixeq_ConsI)
qed
lemma suffixeq_to_prefixeq [code]: "suffixeq xs ys \ prefixeq (rev xs) (rev ys)"
proof
assume "suffixeq xs ys"
then obtain zs where "ys = zs @ xs" ..
then have "rev ys = rev xs @ rev zs" by simp
then show "prefixeq (rev xs) (rev ys)" ..
next
assume "prefixeq (rev xs) (rev ys)"
then obtain zs where "rev ys = rev xs @ zs" ..
then have "rev (rev ys) = rev zs @ rev (rev xs)" by simp
then have "ys = rev zs @ xs" by simp
then show "suffixeq xs ys" ..
qed
lemma distinct_suffixeq: "distinct ys \ suffixeq xs ys \ distinct xs"
by (clarsimp elim!: suffixeqE)
lemma suffixeq_map: "suffixeq xs ys \ suffixeq (map f xs) (map f ys)"
by (auto elim!: suffixeqE intro: suffixeqI)
lemma suffixeq_drop: "suffixeq (drop n as) as"
unfolding suffixeq_def
apply (rule exI [where x = "take n as"])
apply simp
done
lemma suffixeq_take: "suffixeq xs ys \ ys = take (length ys - length xs) ys @ xs"
by (auto elim!: suffixeqE)
lemma suffixeq_suffix_reflclp_conv: "suffixeq = suffix\<^sup>=\<^sup>="
proof (intro ext iffI)
fix xs ys :: "'a list"
assume "suffixeq xs ys"
show "suffix\<^sup>=\<^sup>= xs ys"
proof
assume "xs \ ys"
with \suffixeq xs ys\ show "suffix xs ys"
by (auto simp: suffixeq_def suffix_def)
qed
next
fix xs ys :: "'a list"
assume "suffix\<^sup>=\<^sup>= xs ys"
then show "suffixeq xs ys"
proof
assume "suffix xs ys" then show "suffixeq xs ys"
by (rule suffix_imp_suffixeq)
next
assume "xs = ys" then show "suffixeq xs ys"
by (auto simp: suffixeq_def)
qed
qed
lemma parallelD1: "x \ y \ \ prefixeq x y"
by blast
lemma parallelD2: "x \ y \ \ prefixeq y x"
by blast
lemma parallel_Nil1 [simp]: "\ x \ []"
unfolding parallel_def by simp
lemma parallel_Nil2 [simp]: "\ [] \ x"
unfolding parallel_def by simp
lemma Cons_parallelI1: "a \ b \ a # as \ b # bs"
by auto
lemma Cons_parallelI2: "\ a = b; as \ bs \ \ a # as \ b # bs"
by (metis Cons_prefixeq_Cons parallelE parallelI)
lemma not_equal_is_parallel:
assumes neq: "xs \ ys"
and len: "length xs = length ys"
shows "xs \ ys"
using len neq
proof (induct rule: list_induct2)
case Nil
then show ?case by simp
next
case (Cons a as b bs)
have ih: "as \ bs \ as \ bs" by fact
show ?case
proof (cases "a = b")
case True
then have "as \ bs" using Cons by simp
then show ?thesis by (rule Cons_parallelI2 [OF True ih])
next
case False
then show ?thesis by (rule Cons_parallelI1)
qed
qed
lemma suffix_reflclp_conv: "suffix\<^sup>=\<^sup>= = suffixeq"
by (intro ext) (auto simp: suffixeq_def suffix_def)
lemma suffix_lists: "suffix xs ys \ ys \ lists A \ xs \ lists A"
unfolding suffix_def by auto
subsection \Homeomorphic embedding on lists\
inductive list_emb :: "('a \ 'a \ bool) \ 'a list \ 'a list \ bool"
for P :: "('a \ 'a \ bool)"
where
list_emb_Nil [intro, simp]: "list_emb P [] ys"
| list_emb_Cons [intro] : "list_emb P xs ys \ list_emb P xs (y#ys)"
| list_emb_Cons2 [intro]: "P x y \ list_emb P xs ys \ list_emb P (x#xs) (y#ys)"
lemma list_emb_mono:
assumes "\x y. P x y \ Q x y"
shows "list_emb P xs ys \ list_emb Q xs ys"
proof
assume "list_emb P xs ys"
then show "list_emb Q xs ys" by (induct) (auto simp: assms)
qed
lemma list_emb_Nil2 [simp]:
assumes "list_emb P xs []" shows "xs = []"
using assms by (cases rule: list_emb.cases) auto
lemma list_emb_refl:
assumes "\x. x \ set xs \ P x x"
shows "list_emb P xs xs"
using assms by (induct xs) auto
lemma list_emb_Cons_Nil [simp]: "list_emb P (x#xs) [] = False"
proof -
{ assume "list_emb P (x#xs) []"
from list_emb_Nil2 [OF this] have False by simp
} moreover {
assume False
then have "list_emb P (x#xs) []" by simp
} ultimately show ?thesis by blast
qed
lemma list_emb_append2 [intro]: "list_emb P xs ys \ list_emb P xs (zs @ ys)"
by (induct zs) auto
lemma list_emb_prefix [intro]:
assumes "list_emb P xs ys" shows "list_emb P xs (ys @ zs)"
using assms
by (induct arbitrary: zs) auto
lemma list_emb_ConsD:
assumes "list_emb P (x#xs) ys"
shows "\us v vs. ys = us @ v # vs \ P x v \ list_emb P xs vs"
using assms
proof (induct x \ "x # xs" ys arbitrary: x xs)
case list_emb_Cons
then show ?case by (metis append_Cons)
next
case (list_emb_Cons2 x y xs ys)
then show ?case by blast
qed
lemma list_emb_appendD:
assumes "list_emb P (xs @ ys) zs"
shows "\us vs. zs = us @ vs \ list_emb P xs us \ list_emb P ys vs"
using assms
proof (induction xs arbitrary: ys zs)
case Nil then show ?case by auto
next
case (Cons x xs)
then obtain us v vs where
zs: "zs = us @ v # vs" and p: "P x v" and lh: "list_emb P (xs @ ys) vs"
by (auto dest: list_emb_ConsD)
obtain sk\<^sub>0 :: "'a list \ 'a list \ 'a list" and sk\<^sub>1 :: "'a list \ 'a list \ 'a list" where
sk: "\x\<^sub>0 x\<^sub>1. \ list_emb P (xs @ x\<^sub>0) x\<^sub>1 \ sk\<^sub>0 x\<^sub>0 x\<^sub>1 @ sk\<^sub>1 x\<^sub>0 x\<^sub>1 = x\<^sub>1 \ list_emb P xs (sk\<^sub>0 x\<^sub>0 x\<^sub>1) \ list_emb P x\<^sub>0 (sk\<^sub>1 x\<^sub>0 x\<^sub>1)"
using Cons(1) by (metis (no_types))
hence "\x\<^sub>2. list_emb P (x # xs) (x\<^sub>2 @ v # sk\<^sub>0 ys vs)" using p lh by auto
thus ?case using lh zs sk by (metis (no_types) append_Cons append_assoc)
qed
lemma list_emb_suffix:
assumes "list_emb P xs ys" and "suffix ys zs"
shows "list_emb P xs zs"
using assms(2) and list_emb_append2 [OF assms(1)] by (auto simp: suffix_def)
lemma list_emb_suffixeq:
assumes "list_emb P xs ys" and "suffixeq ys zs"
shows "list_emb P xs zs"
using assms and list_emb_suffix unfolding suffixeq_suffix_reflclp_conv by auto
lemma list_emb_length: "list_emb P xs ys \ length xs \ length ys"
by (induct rule: list_emb.induct) auto
lemma list_emb_trans:
assumes "\x y z. \x \ set xs; y \ set ys; z \ set zs; P x y; P y z\ \ P x z"
shows "\list_emb P xs ys; list_emb P ys zs\ \ list_emb P xs zs"
proof -
assume "list_emb P xs ys" and "list_emb P ys zs"
then show "list_emb P xs zs" using assms
proof (induction arbitrary: zs)
case list_emb_Nil show ?case by blast
next
case (list_emb_Cons xs ys y)
from list_emb_ConsD [OF \list_emb P (y#ys) zs\] obtain us v vs
where zs: "zs = us @ v # vs" and "P\<^sup>=\<^sup>= y v" and "list_emb P ys vs" by blast
then have "list_emb P ys (v#vs)" by blast
then have "list_emb P ys zs" unfolding zs by (rule list_emb_append2)
from list_emb_Cons.IH [OF this] and list_emb_Cons.prems show ?case by auto
next
case (list_emb_Cons2 x y xs ys)
from list_emb_ConsD [OF \list_emb P (y#ys) zs\] obtain us v vs
where zs: "zs = us @ v # vs" and "P y v" and "list_emb P ys vs" by blast
with list_emb_Cons2 have "list_emb P xs vs" by auto
moreover have "P x v"
proof -
from zs have "v \ set zs" by auto
moreover have "x \ set (x#xs)" and "y \ set (y#ys)" by simp_all
ultimately show ?thesis
using \P x y\ and \P y v\ and list_emb_Cons2
by blast
qed
ultimately have "list_emb P (x#xs) (v#vs)" by blast
then show ?case unfolding zs by (rule list_emb_append2)
qed
qed
lemma list_emb_set:
assumes "list_emb P xs ys" and "x \ set xs"
obtains y where "y \ set ys" and "P x y"
using assms by (induct) auto
subsection \Sublists (special case of homeomorphic embedding)\
abbreviation sublisteq :: "'a list \ 'a list \