(* 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⇩_{1}::'a list) ys ⟹ prefixeq xs⇩_{2}ys ⟹ prefixeq xs⇩_{1}xs⇩_{2}∨ prefixeq xs⇩_{2}xs⇩_{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⇧^{=}⇧^{=}" 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 ∥ 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⇧^{=}⇧^{=}= 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⇩_{0}:: "'a list ⇒ 'a list ⇒ 'a list" and sk⇩_{1}:: "'a list ⇒ 'a list ⇒ 'a list" where sk: "∀x⇩_{0}x⇩_{1}. ¬ list_emb P (xs @ x⇩_{0}) x⇩_{1}∨ sk⇩_{0}x⇩_{0}x⇩_{1}@ sk⇩_{1}x⇩_{0}x⇩_{1}= x⇩_{1}∧ list_emb P xs (sk⇩_{0}x⇩_{0}x⇩_{1}) ∧ list_emb P x⇩_{0}(sk⇩_{1}x⇩_{0}x⇩_{1})" using Cons(1) by (metis (no_types)) hence "∀x⇩_{2}. list_emb P (x # xs) (x⇩_{2}@ v # sk⇩_{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⇧^{=}⇧^{=}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 ⇒ bool" where "sublisteq xs ys ≡ list_emb (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_emb_length) lemma not_sublisteq_length [simp]: "length ys < length xs ⟹ ¬ sublisteq xs ys" by (metis list_emb_length linorder_not_less) lemma [code]: "list_emb P [] ys ⟷ True" "list_emb P (x#xs) [] ⟷ False" by (simp_all) lemma sublisteq_Cons': "sublisteq (x#xs) ys ⟹ sublisteq xs ys" by (induct xs, simp, blast dest: list_emb_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_emb_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_emb_Nil from list_emb_Nil2 [OF this] show ?case by simp next case list_emb_Cons2 thus ?case by simp next case list_emb_Cons hence False using sublisteq_Cons' by fastforce thus ?case .. qed lemma sublisteq_trans: "sublisteq xs ys ⟹ sublisteq ys zs ⟹ sublisteq xs zs" by (rule list_emb_trans [of _ _ _ "op ="]) auto lemma sublisteq_append_le_same_iff: "sublisteq (xs @ ys) ys ⟷ xs = []" by (auto dest: list_emb_length) lemma list_emb_append_mono: "⟦ list_emb P xs xs'; list_emb P ys ys' ⟧ ⟹ list_emb P (xs@ys) (xs'@ys')" apply (induct rule: list_emb.induct) apply (metis eq_Nil_appendI list_emb_append2) apply (metis append_Cons list_emb_Cons) apply (metis append_Cons list_emb_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_emb_Nil show ?case by simp next case (list_emb_Cons xs' ys' x) { assume "ys=[]" then have ?case using list_emb_Cons(1) by auto } moreover { fix us assume "ys = x#us" then have ?case using list_emb_Cons(2) by(simp add: list_emb.list_emb_Cons) } ultimately show ?case by (auto simp:Cons_eq_append_conv) next case (list_emb_Cons2 x y xs' ys') { assume "xs=[]" then have ?case using list_emb_Cons2(1) by auto } moreover { fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using list_emb_Cons2 by auto} moreover { fix us assume "xs=x#us" "ys=[]" then have ?case using list_emb_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_emb_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_emb_Nil list_emb_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_emb_Nil show ?case by (metis sublist_empty) next case (list_emb_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_emb_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_emb_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