base Sublist_Order on Sublist (using a simplified form of embedding as sublist relation)
(* Title: HOL/Library/Sublist.thy
Author: Tobias Nipkow and Markus Wenzel, TU Muenchen
*)
header {* List prefixes, suffixes, and embedding*}
theory Sublist
imports List Main
begin
subsection {* Prefix order on lists *}
definition prefixeq :: "'a list => 'a list => bool" where
"prefixeq xs ys \<longleftrightarrow> (\<exists>zs. ys = xs @ zs)"
definition prefix :: "'a list => 'a list => 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: bot prefixeq prefix Nil
by default (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 \<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 ==> xs \<noteq> 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 \<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 ==> 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 ==> length xs < length ys ==> prefixeq (xs @ [ys ! length xs]) ys"
unfolding prefixeq_def
by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj
eq_Nil_appendI nth_drop')
theorem prefixeq_length_le: "prefixeq xs ys ==> length xs \<le> length ys"
by (auto simp add: prefixeq_def)
lemma prefixeq_same_cases:
"prefixeq (xs\<^isub>1::'a list) ys \<Longrightarrow> prefixeq xs\<^isub>2 ys \<Longrightarrow> prefixeq xs\<^isub>1 xs\<^isub>2 \<or> prefixeq xs\<^isub>2 xs\<^isub>1"
unfolding prefixeq_def by (metis 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 => 'a list => bool" (infixl "\<parallel>" 50) where
"(xs \<parallel> ys) = (\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs)"
lemma parallelI [intro]: "\<not> prefixeq xs ys ==> \<not> prefixeq ys xs ==> xs \<parallel> ys"
unfolding parallel_def by blast
lemma parallelE [elim]:
assumes "xs \<parallel> ys"
obtains "\<not> prefixeq xs ys \<and> \<not> prefixeq ys xs"
using assms unfolding parallel_def by blast
theorem prefixeq_cases:
obtains "prefixeq xs ys" | "prefix ys xs" | "xs \<parallel> ys"
unfolding parallel_def prefix_def by blast
theorem parallel_decomp:
"xs \<parallel> ys ==> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> 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"
then show ?thesis
by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixeqI
same_prefixeq_prefixeq snoc.prems ys)
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 \<parallel> ys"
with snoc obtain as b bs c cs where neq: "(b::'a) \<noteq> 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 \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
apply (rule parallelI)
apply (erule parallelE, erule conjE,
induct rule: not_prefixeq_induct, simp+)+
done
lemma parallel_appendI: "xs \<parallel> ys \<Longrightarrow> x = xs @ xs' \<Longrightarrow> y = ys @ ys' \<Longrightarrow> x \<parallel> y"
by (simp add: parallel_append)
lemma parallel_commute: "a \<parallel> b \<longleftrightarrow> b \<parallel> a"
unfolding parallel_def by auto
subsection {* Suffix order on lists *}
definition
suffixeq :: "'a list => 'a list => bool" where
"suffixeq xs ys = (\<exists>zs. ys = zs @ xs)"
definition suffix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
"suffix xs ys \<equiv> \<exists>us. ys = us @ xs \<and> us \<noteq> []"
lemma suffix_imp_suffixeq:
"suffix xs ys \<Longrightarrow> 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 \<Longrightarrow> suffix ys zs \<Longrightarrow> suffix xs zs"
by (auto simp: suffix_def)
lemma suffixeq_trans: "\<lbrakk>suffixeq xs ys; suffixeq ys zs\<rbrakk> \<Longrightarrow> suffixeq xs zs"
by (auto simp add: suffixeq_def)
lemma suffixeq_antisym: "\<lbrakk>suffixeq xs ys; suffixeq ys xs\<rbrakk> \<Longrightarrow> 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 \<noteq> [] \<Longrightarrow> 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 \<Longrightarrow> suffixeq xs (y#ys)"
by (auto simp add: suffixeq_def)
lemma suffixeq_ConsD: "suffixeq (x#xs) ys \<Longrightarrow> suffixeq xs ys"
by (auto simp add: suffixeq_def)
lemma suffixeq_appendI: "suffixeq xs ys \<Longrightarrow> suffixeq xs (zs @ ys)"
by (auto simp add: suffixeq_def)
lemma suffixeq_appendD: "suffixeq (zs @ xs) ys \<Longrightarrow> suffixeq xs ys"
by (auto simp add: suffixeq_def)
lemma suffix_set_subset:
"suffix xs ys \<Longrightarrow> set xs \<subseteq> set ys" by (auto simp: suffix_def)
lemma suffixeq_set_subset:
"suffixeq xs ys \<Longrightarrow> set xs \<subseteq> 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 \<longleftrightarrow> 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 \<Longrightarrow> suffixeq xs ys \<Longrightarrow> distinct xs"
by (clarsimp elim!: suffixeqE)
lemma suffixeq_map: "suffixeq xs ys \<Longrightarrow> 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 \<Longrightarrow> ys = take (length ys - length xs) ys @ xs"
by (clarsimp 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 \<noteq> 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"
thus "suffixeq xs ys"
proof
assume "suffix xs ys" thus "suffixeq xs ys" by (rule suffix_imp_suffixeq)
next
assume "xs = ys" thus "suffixeq xs ys" by (auto simp: suffixeq_def)
qed
qed
lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> prefixeq x y"
by blast
lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> prefixeq y x"
by blast
lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
unfolding parallel_def by simp
lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
unfolding parallel_def by simp
lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
by auto
lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
by (metis Cons_prefixeq_Cons parallelE parallelI)
lemma not_equal_is_parallel:
assumes neq: "xs \<noteq> ys"
and len: "length xs = length ys"
shows "xs \<parallel> 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 \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact
show ?case
proof (cases "a = b")
case True
then have "as \<noteq> 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 \<Longrightarrow> ys \<in> lists A \<Longrightarrow> xs \<in> lists A"
unfolding suffix_def by auto
subsection {* Embedding on lists *}
inductive
emb :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
for P :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
where
emb_Nil [intro, simp]: "emb P [] ys"
| emb_Cons [intro] : "emb P xs ys \<Longrightarrow> emb P xs (y#ys)"
| emb_Cons2 [intro]: "P x y \<Longrightarrow> emb P xs ys \<Longrightarrow> emb P (x#xs) (y#ys)"
lemma emb_Nil2 [simp]:
assumes "emb P xs []" shows "xs = []"
using assms by (cases rule: emb.cases) auto
lemma emb_append2 [intro]:
"emb P xs ys \<Longrightarrow> emb P xs (zs @ ys)"
by (induct zs) auto
lemma emb_prefix [intro]:
assumes "emb P xs ys" shows "emb P xs (ys @ zs)"
using assms
by (induct arbitrary: zs) auto
lemma emb_ConsD:
assumes "emb P (x#xs) ys"
shows "\<exists>us v vs. ys = us @ v # vs \<and> P x v \<and> emb P xs vs"
using assms
proof (induct x\<equiv>"x#xs" y\<equiv>"ys" arbitrary: x xs ys)
case emb_Cons thus ?case by (metis append_Cons)
next
case (emb_Cons2 x y xs ys)
thus ?case by (cases xs) (auto, blast+)
qed
lemma emb_appendD:
assumes "emb P (xs @ ys) zs"
shows "\<exists>us vs. zs = us @ vs \<and> emb P xs us \<and> emb P ys vs"
using assms
proof (induction xs arbitrary: ys zs)
case Nil thus ?case by auto
next
case (Cons x xs)
then obtain us v vs where "zs = us @ v # vs"
and "P x v" and "emb P (xs @ ys) vs" by (auto dest: emb_ConsD)
with Cons show ?case by (metis append_Cons append_assoc emb_Cons2 emb_append2)
qed
lemma emb_suffix:
assumes "emb P xs ys" and "suffix ys zs"
shows "emb P xs zs"
using assms(2) and emb_append2 [OF assms(1)] by (auto simp: suffix_def)
lemma emb_suffixeq:
assumes "emb P xs ys" and "suffixeq ys zs"
shows "emb P xs zs"
using assms and emb_suffix unfolding suffixeq_suffix_reflclp_conv by auto
lemma emb_length: "emb P xs ys \<Longrightarrow> length xs \<le> length ys"
by (induct rule: emb.induct) auto
end