(* Title: HOL/Library/Sublist.thy

Author: Tobias Nipkow and Markus Wenzel, TU Muenchen

Author: Christian Sternagel, JAIST

*)

header {* 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 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 ==> 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"

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 ≤ length ys"

by (auto simp add: prefixeq_def)

lemma prefixeq_same_cases:

"prefixeq (xs⇩_{1}::'a list) ys ==> prefixeq xs⇩_{2}ys ==> prefixeq xs⇩_{1}xs⇩_{2}∨ prefixeq xs⇩_{2}xs⇩_{1}"

unfolding prefixeq_def by (metis 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_all)[1]

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 "\<parallel>" 50)

where "(xs \<parallel> ys) = (¬ prefixeq xs ys ∧ ¬ prefixeq ys xs)"

lemma parallelI [intro]: "¬ prefixeq xs ys ==> ¬ prefixeq ys xs ==> xs \<parallel> ys"

unfolding parallel_def by blast

lemma parallelE [elim]:

assumes "xs \<parallel> 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 \<parallel> ys"

unfolding parallel_def prefix_def by blast

theorem parallel_decomp:

"xs \<parallel> 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"

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) ≠ 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 ==> 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 ==> x = xs @ xs' ==> y = ys @ ys' ==> x \<parallel> y"

by (simp add: parallel_append)

lemma parallel_commute: "a \<parallel> b <-> b \<parallel> 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⇧^{=}⇧^{=}"

proof (intro ext iffI)

fix xs ys :: "'a list"

assume "suffixeq xs ys"

show "suffix⇧^{=}⇧^{=}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⇧^{=}⇧^{=}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 \<parallel> y ==> ¬ prefixeq x y"

by blast

lemma parallelD2: "x \<parallel> y ==> ¬ prefixeq y x"

by blast

lemma parallel_Nil1 [simp]: "¬ x \<parallel> []"

unfolding parallel_def by simp

lemma parallel_Nil2 [simp]: "¬ [] \<parallel> x"

unfolding parallel_def by simp

lemma Cons_parallelI1: "a ≠ b ==> a # as \<parallel> b # bs"

by auto

lemma Cons_parallelI2: "[| a = b; as \<parallel> bs |] ==> a # as \<parallel> 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 \<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 ≠ bs ==> as \<parallel> 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⇧^{=}⇧^{=}= 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_hembeq :: "('a => 'a => bool) => 'a list => 'a list => bool"

for P :: "('a => 'a => bool)"

where

list_hembeq_Nil [intro, simp]: "list_hembeq P [] ys"

| list_hembeq_Cons [intro] : "list_hembeq P xs ys ==> list_hembeq P xs (y#ys)"

| list_hembeq_Cons2 [intro]: "P⇧^{=}⇧^{=}x y ==> list_hembeq P xs ys ==> list_hembeq P (x#xs) (y#ys)"

lemma list_hembeq_Nil2 [simp]:

assumes "list_hembeq P xs []" shows "xs = []"

using assms by (cases rule: list_hembeq.cases) auto

lemma list_hembeq_refl [simp, intro!]:

"list_hembeq P xs xs"

by (induct xs) auto

lemma list_hembeq_Cons_Nil [simp]: "list_hembeq P (x#xs) [] = False"

proof -

{ assume "list_hembeq P (x#xs) []"

from list_hembeq_Nil2 [OF this] have False by simp

} moreover {

assume False

then have "list_hembeq P (x#xs) []" by simp

} ultimately show ?thesis by blast

qed

lemma list_hembeq_append2 [intro]: "list_hembeq P xs ys ==> list_hembeq P xs (zs @ ys)"

by (induct zs) auto

lemma list_hembeq_prefix [intro]:

assumes "list_hembeq P xs ys" shows "list_hembeq P xs (ys @ zs)"

using assms

by (induct arbitrary: zs) auto

lemma list_hembeq_ConsD:

assumes "list_hembeq P (x#xs) ys"

shows "∃us v vs. ys = us @ v # vs ∧ P⇧^{=}⇧^{=}x v ∧ list_hembeq P xs vs"

using assms

proof (induct x ≡ "x # xs" ys arbitrary: x xs)

case list_hembeq_Cons

then show ?case by (metis append_Cons)

next

case (list_hembeq_Cons2 x y xs ys)

then show ?case by (cases xs) (auto, blast+)

qed

lemma list_hembeq_appendD:

assumes "list_hembeq P (xs @ ys) zs"

shows "∃us vs. zs = us @ vs ∧ list_hembeq P xs us ∧ list_hembeq 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 = us @ v # vs"

and "P⇧^{=}⇧^{=}x v" and "list_hembeq P (xs @ ys) vs" by (auto dest: list_hembeq_ConsD)

with Cons show ?case by (metis append_Cons append_assoc list_hembeq_Cons2 list_hembeq_append2)

qed

lemma list_hembeq_suffix:

assumes "list_hembeq P xs ys" and "suffix ys zs"

shows "list_hembeq P xs zs"

using assms(2) and list_hembeq_append2 [OF assms(1)] by (auto simp: suffix_def)

lemma list_hembeq_suffixeq:

assumes "list_hembeq P xs ys" and "suffixeq ys zs"

shows "list_hembeq P xs zs"

using assms and list_hembeq_suffix unfolding suffixeq_suffix_reflclp_conv by auto

lemma list_hembeq_length: "list_hembeq P xs ys ==> length xs ≤ length ys"

by (induct rule: list_hembeq.induct) auto

lemma list_hembeq_trans:

assumes "!!x y z. [|x ∈ A; y ∈ A; z ∈ A; P x y; P y z|] ==> P x z"

shows "!!xs ys zs. [|xs ∈ lists A; ys ∈ lists A; zs ∈ lists A;

list_hembeq P xs ys; list_hembeq P ys zs|] ==> list_hembeq P xs zs"

proof -

fix xs ys zs

assume "list_hembeq P xs ys" and "list_hembeq P ys zs"

and "xs ∈ lists A" and "ys ∈ lists A" and "zs ∈ lists A"

then show "list_hembeq P xs zs"

proof (induction arbitrary: zs)

case list_hembeq_Nil show ?case by blast

next

case (list_hembeq_Cons xs ys y)

from list_hembeq_ConsD [OF `list_hembeq P (y#ys) zs`] obtain us v vs

where zs: "zs = us @ v # vs" and "P⇧^{=}⇧^{=}y v" and "list_hembeq P ys vs" by blast

then have "list_hembeq P ys (v#vs)" by blast

then have "list_hembeq P ys zs" unfolding zs by (rule list_hembeq_append2)

from list_hembeq_Cons.IH [OF this] and list_hembeq_Cons.prems show ?case by simp

next

case (list_hembeq_Cons2 x y xs ys)

from list_hembeq_ConsD [OF `list_hembeq P (y#ys) zs`] obtain us v vs

where zs: "zs = us @ v # vs" and "P⇧^{=}⇧^{=}y v" and "list_hembeq P ys vs" by blast

with list_hembeq_Cons2 have "list_hembeq P xs vs" by simp

moreover have "P⇧^{=}⇧^{=}x v"

proof -

from zs and `zs ∈ lists A` have "v ∈ A" by auto

moreover have "x ∈ A" and "y ∈ A" using list_hembeq_Cons2 by simp_all

ultimately show ?thesis

using `P⇧^{=}⇧^{=}x y` and `P⇧^{=}⇧^{=}y v` and assms

by blast

qed

ultimately have "list_hembeq P (x#xs) (v#vs)" by blast

then show ?case unfolding zs by (rule list_hembeq_append2)

qed

qed

subsection {* Sublists (special case of homeomorphic embedding) *}

abbreviation sublisteq :: "'a list => 'a list => bool"

where "sublisteq xs ys ≡ list_hembeq (op =) xs ys"

lemma sublisteq_Cons2: "sublisteq xs ys ==> sublisteq (x#xs) (x#ys)" by auto

lemma sublisteq_same_length:

assumes "sublisteq xs ys" and "length xs = length ys" shows "xs = ys"

using assms by (induct) (auto dest: list_hembeq_length)

lemma not_sublisteq_length [simp]: "length ys < length xs ==> ¬ sublisteq xs ys"

by (metis list_hembeq_length linorder_not_less)

lemma [code]:

"list_hembeq P [] ys <-> True"

"list_hembeq P (x#xs) [] <-> False"

by (simp_all)

lemma sublisteq_Cons': "sublisteq (x#xs) ys ==> sublisteq xs ys"

by (induct xs) (auto dest: list_hembeq_ConsD)

lemma sublisteq_Cons2':

assumes "sublisteq (x#xs) (x#ys)" shows "sublisteq xs ys"

using assms by (cases) (rule sublisteq_Cons')

lemma sublisteq_Cons2_neq:

assumes "sublisteq (x#xs) (y#ys)"

shows "x ≠ y ==> sublisteq (x#xs) ys"

using assms by (cases) auto

lemma sublisteq_Cons2_iff [simp, code]:

"sublisteq (x#xs) (y#ys) = (if x = y then sublisteq xs ys else sublisteq (x#xs) ys)"

by (metis list_hembeq_Cons sublisteq_Cons2 sublisteq_Cons2' sublisteq_Cons2_neq)

lemma sublisteq_append': "sublisteq (zs @ xs) (zs @ ys) <-> sublisteq xs ys"

by (induct zs) simp_all

lemma sublisteq_refl [simp, intro!]: "sublisteq xs xs" by (induct xs) simp_all

lemma sublisteq_antisym:

assumes "sublisteq xs ys" and "sublisteq ys xs"

shows "xs = ys"

using assms

proof (induct)

case list_hembeq_Nil

from list_hembeq_Nil2 [OF this] show ?case by simp

next

case list_hembeq_Cons2

then show ?case by simp

next

case list_hembeq_Cons

then show ?case

by (metis sublisteq_Cons' list_hembeq_length Suc_length_conv Suc_n_not_le_n)

qed

lemma sublisteq_trans: "sublisteq xs ys ==> sublisteq ys zs ==> sublisteq xs zs"

by (rule list_hembeq_trans [of UNIV "op ="]) auto

lemma sublisteq_append_le_same_iff: "sublisteq (xs @ ys) ys <-> xs = []"

by (auto dest: list_hembeq_length)

lemma list_hembeq_append_mono:

"[| list_hembeq P xs xs'; list_hembeq P ys ys' |] ==> list_hembeq P (xs@ys) (xs'@ys')"

apply (induct rule: list_hembeq.induct)

apply (metis eq_Nil_appendI list_hembeq_append2)

apply (metis append_Cons list_hembeq_Cons)

apply (metis append_Cons list_hembeq_Cons2)

done

subsection {* Appending elements *}

lemma sublisteq_append [simp]:

"sublisteq (xs @ zs) (ys @ zs) <-> sublisteq xs ys" (is "?l = ?r")

proof

{ fix xs' ys' xs ys zs :: "'a list" assume "sublisteq xs' ys'"

then have "xs' = xs @ zs & ys' = ys @ zs --> sublisteq xs ys"

proof (induct arbitrary: xs ys zs)

case list_hembeq_Nil show ?case by simp

next

case (list_hembeq_Cons xs' ys' x)

{ assume "ys=[]" then have ?case using list_hembeq_Cons(1) by auto }

moreover

{ fix us assume "ys = x#us"

then have ?case using list_hembeq_Cons(2) by(simp add: list_hembeq.list_hembeq_Cons) }

ultimately show ?case by (auto simp:Cons_eq_append_conv)

next

case (list_hembeq_Cons2 x y xs' ys')

{ assume "xs=[]" then have ?case using list_hembeq_Cons2(1) by auto }

moreover

{ fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using list_hembeq_Cons2 by auto}

moreover

{ fix us assume "xs=x#us" "ys=[]" then have ?case using list_hembeq_Cons2(2) by bestsimp }

ultimately show ?case using `op =⇧^{=}⇧^{=}x y` by (auto simp: Cons_eq_append_conv)

qed }

moreover assume ?l

ultimately show ?r by blast

next

assume ?r then show ?l by (metis list_hembeq_append_mono sublisteq_refl)

qed

lemma sublisteq_drop_many: "sublisteq xs ys ==> sublisteq xs (zs @ ys)"

by (induct zs) auto

lemma sublisteq_rev_drop_many: "sublisteq xs ys ==> sublisteq xs (ys @ zs)"

by (metis append_Nil2 list_hembeq_Nil list_hembeq_append_mono)

subsection {* Relation to standard list operations *}

lemma sublisteq_map:

assumes "sublisteq xs ys" shows "sublisteq (map f xs) (map f ys)"

using assms by (induct) auto

lemma sublisteq_filter_left [simp]: "sublisteq (filter P xs) xs"

by (induct xs) auto

lemma sublisteq_filter [simp]:

assumes "sublisteq xs ys" shows "sublisteq (filter P xs) (filter P ys)"

using assms by (induct) auto

lemma "sublisteq xs ys <-> (∃N. xs = sublist ys N)" (is "?L = ?R")

proof

assume ?L

then show ?R

proof (induct)

case list_hembeq_Nil show ?case by (metis sublist_empty)

next

case (list_hembeq_Cons xs ys x)

then obtain N where "xs = sublist ys N" by blast

then have "xs = sublist (x#ys) (Suc ` N)"

by (clarsimp simp add:sublist_Cons inj_image_mem_iff)

then show ?case by blast

next

case (list_hembeq_Cons2 x y xs ys)

then obtain N where "xs = sublist ys N" by blast

then have "x#xs = sublist (x#ys) (insert 0 (Suc ` N))"

by (clarsimp simp add:sublist_Cons inj_image_mem_iff)

moreover from list_hembeq_Cons2 have "x = y" by simp

ultimately show ?case by blast

qed

next

assume ?R

then obtain N where "xs = sublist ys N" ..

moreover have "sublisteq (sublist ys N) ys"

proof (induct ys arbitrary: N)

case Nil show ?case by simp

next

case Cons then show ?case by (auto simp: sublist_Cons)

qed

ultimately show ?L by simp

qed

end