(* Title: HOL/Library/Sublist.thy Author: Tobias Nipkow and Markus Wenzel, TU München Author: Christian Sternagel, JAIST Author: Manuel Eberl, TU München *) section ‹List prefixes, suffixes, and homeomorphic embedding› theory Sublist imports Main begin subsection ‹Prefix order on lists› definition prefix :: "'a list ⇒ 'a list ⇒ bool" where "prefix xs ys ⟷ (∃zs. ys = xs @ zs)" definition strict_prefix :: "'a list ⇒ 'a list ⇒ bool" where "strict_prefix xs ys ⟷ prefix xs ys ∧ xs ≠ ys" interpretation prefix_order: order prefix strict_prefix by standard (auto simp: prefix_def strict_prefix_def) interpretation prefix_bot: order_bot Nil prefix strict_prefix by standard (simp add: prefix_def) lemma prefixI [intro?]: "ys = xs @ zs ⟹ prefix xs ys" unfolding prefix_def by blast lemma prefixE [elim?]: assumes "prefix xs ys" obtains zs where "ys = xs @ zs" using assms unfolding prefix_def by blast lemma strict_prefixI' [intro?]: "ys = xs @ z # zs ⟹ strict_prefix xs ys" unfolding strict_prefix_def prefix_def by blast lemma strict_prefixE' [elim?]: assumes "strict_prefix xs ys" obtains z zs where "ys = xs @ z # zs" proof - from ‹strict_prefix xs ys› obtain us where "ys = xs @ us" and "xs ≠ ys" unfolding strict_prefix_def prefix_def by blast with that show ?thesis by (auto simp add: neq_Nil_conv) qed (* FIXME rm *) lemma strict_prefixI [intro?]: "prefix xs ys ⟹ xs ≠ ys ⟹ strict_prefix xs ys" by(fact prefix_order.le_neq_trans) lemma strict_prefixE [elim?]: fixes xs ys :: "'a list" assumes "strict_prefix xs ys" obtains "prefix xs ys" and "xs ≠ ys" using assms unfolding strict_prefix_def by blast subsection ‹Basic properties of prefixes› (* FIXME rm *) theorem Nil_prefix [simp]: "prefix [] xs" by (fact prefix_bot.bot_least) (* FIXME rm *) theorem prefix_Nil [simp]: "(prefix xs []) = (xs = [])" by (fact prefix_bot.bot_unique) lemma prefix_snoc [simp]: "prefix xs (ys @ [y]) ⟷ xs = ys @ [y] ∨ prefix xs ys" proof assume "prefix xs (ys @ [y])" then obtain zs where zs: "ys @ [y] = xs @ zs" .. show "xs = ys @ [y] ∨ prefix xs ys" by (metis append_Nil2 butlast_append butlast_snoc prefixI zs) next assume "xs = ys @ [y] ∨ prefix xs ys" then show "prefix xs (ys @ [y])" by (metis prefix_order.eq_iff prefix_order.order_trans prefixI) qed lemma Cons_prefix_Cons [simp]: "prefix (x # xs) (y # ys) = (x = y ∧ prefix xs ys)" by (auto simp add: prefix_def) lemma prefix_code [code]: "prefix [] xs ⟷ True" "prefix (x # xs) [] ⟷ False" "prefix (x # xs) (y # ys) ⟷ x = y ∧ prefix xs ys" by simp_all lemma same_prefix_prefix [simp]: "prefix (xs @ ys) (xs @ zs) = prefix ys zs" by (induct xs) simp_all lemma same_prefix_nil [simp]: "prefix (xs @ ys) xs = (ys = [])" by (metis append_Nil2 append_self_conv prefix_order.eq_iff prefixI) lemma prefix_prefix [simp]: "prefix xs ys ⟹ prefix xs (ys @ zs)" unfolding prefix_def by fastforce lemma append_prefixD: "prefix (xs @ ys) zs ⟹ prefix xs zs" by (auto simp add: prefix_def) theorem prefix_Cons: "prefix xs (y # ys) = (xs = [] ∨ (∃zs. xs = y # zs ∧ prefix zs ys))" by (cases xs) (auto simp add: prefix_def) theorem prefix_append: "prefix xs (ys @ zs) = (prefix xs ys ∨ (∃us. xs = ys @ us ∧ prefix us zs))" apply (induct zs rule: rev_induct) apply force apply (simp flip: append_assoc) apply (metis append_eq_appendI) done lemma append_one_prefix: "prefix xs ys ⟹ length xs < length ys ⟹ prefix (xs @ [ys ! length xs]) ys" proof (unfold prefix_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 prefix_length_le: "prefix xs ys ⟹ length xs ≤ length ys" by (auto simp add: prefix_def) lemma prefix_same_cases: "prefix (xs⇩_{1}::'a list) ys ⟹ prefix xs⇩_{2}ys ⟹ prefix xs⇩_{1}xs⇩_{2}∨ prefix xs⇩_{2}xs⇩_{1}" unfolding prefix_def by (force simp: append_eq_append_conv2) lemma prefix_length_prefix: "prefix ps xs ⟹ prefix qs xs ⟹ length ps ≤ length qs ⟹ prefix ps qs" by (auto simp: prefix_def) (metis append_Nil2 append_eq_append_conv_if) lemma set_mono_prefix: "prefix xs ys ⟹ set xs ⊆ set ys" by (auto simp add: prefix_def) lemma take_is_prefix: "prefix (take n xs) xs" unfolding prefix_def by (metis append_take_drop_id) lemma prefixeq_butlast: "prefix (butlast xs) xs" by (simp add: butlast_conv_take take_is_prefix) lemma prefix_map_rightE: assumes "prefix xs (map f ys)" shows "∃xs'. prefix xs' ys ∧ xs = map f xs'" proof - define n where "n = length xs" have "xs = take n (map f ys)" using assms by (auto simp: prefix_def n_def) thus ?thesis by (intro exI[of _ "take n ys"]) (auto simp: take_map take_is_prefix) qed lemma map_mono_prefix: "prefix xs ys ⟹ prefix (map f xs) (map f ys)" by (auto simp: prefix_def) lemma filter_mono_prefix: "prefix xs ys ⟹ prefix (filter P xs) (filter P ys)" by (auto simp: prefix_def) lemma sorted_antimono_prefix: "prefix xs ys ⟹ sorted ys ⟹ sorted xs" by (metis sorted_append prefix_def) lemma prefix_length_less: "strict_prefix xs ys ⟹ length xs < length ys" by (auto simp: strict_prefix_def prefix_def) lemma prefix_snocD: "prefix (xs@[x]) ys ⟹ strict_prefix xs ys" by (simp add: strict_prefixI' prefix_order.dual_order.strict_trans1) lemma strict_prefix_simps [simp, code]: "strict_prefix xs [] ⟷ False" "strict_prefix [] (x # xs) ⟷ True" "strict_prefix (x # xs) (y # ys) ⟷ x = y ∧ strict_prefix xs ys" by (simp_all add: strict_prefix_def cong: conj_cong) lemma take_strict_prefix: "strict_prefix xs ys ⟹ strict_prefix (take n xs) ys" proof (induct n arbitrary: xs ys) case 0 then show ?case by (cases ys) simp_all next case (Suc n) then show ?case by (metis prefix_order.less_trans strict_prefixI take_is_prefix) qed lemma prefix_takeWhile: assumes "prefix xs ys" shows "prefix (takeWhile P xs) (takeWhile P ys)" proof - from assms obtain zs where ys: "ys = xs @ zs" by (auto simp: prefix_def) have "prefix (takeWhile P xs) (takeWhile P (xs @ zs))" by (induction xs) auto thus ?thesis by (simp add: ys) qed lemma prefix_dropWhile: assumes "prefix xs ys" shows "prefix (dropWhile P xs) (dropWhile P ys)" proof - from assms obtain zs where ys: "ys = xs @ zs" by (auto simp: prefix_def) have "prefix (dropWhile P xs) (dropWhile P (xs @ zs))" by (induction xs) auto thus ?thesis by (simp add: ys) qed lemma prefix_remdups_adj: assumes "prefix xs ys" shows "prefix (remdups_adj xs) (remdups_adj ys)" using assms proof (induction "length xs" arbitrary: xs ys rule: less_induct) case (less xs) show ?case proof (cases xs) case [simp]: (Cons x xs') then obtain y ys' where [simp]: "ys = y # ys'" using ‹prefix xs ys› by (cases ys) auto from less show ?thesis by (auto simp: remdups_adj_Cons' less_Suc_eq_le length_dropWhile_le intro!: less prefix_dropWhile) qed auto qed lemma not_prefix_cases: assumes pfx: "¬ prefix ps ls" obtains (c1) "ps ≠ []" and "ls = []" | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "¬ prefix 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_prefix_nil) next case (Cons x xs) show ?thesis proof (cases "x = a") case True have "¬ prefix 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_prefix_induct [consumes 1, case_names Nil Neq Eq]: assumes np: "¬ prefix 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; ¬ prefix 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_prefix_cases intro!: base) next case (Cons y ys) then have npfx: "¬ prefix ps (y # ys)" by simp then obtain x xs where pv: "ps = x # xs" by (rule not_prefix_cases) auto show ?case by (metis Cons.hyps Cons_prefix_Cons npfx pv r1 r2) qed subsection ‹Prefixes› primrec prefixes where "prefixes [] = [[]]" | "prefixes (x#xs) = [] # map ((#) x) (prefixes xs)" lemma in_set_prefixes[simp]: "xs ∈ set (prefixes ys) ⟷ prefix xs ys" proof (induct xs arbitrary: ys) case Nil then show ?case by (cases ys) auto next case (Cons a xs) then show ?case by (cases ys) auto qed lemma length_prefixes[simp]: "length (prefixes xs) = length xs+1" by (induction xs) auto lemma distinct_prefixes [intro]: "distinct (prefixes xs)" by (induction xs) (auto simp: distinct_map) lemma prefixes_snoc [simp]: "prefixes (xs@[x]) = prefixes xs @ [xs@[x]]" by (induction xs) auto lemma prefixes_not_Nil [simp]: "prefixes xs ≠ []" by (cases xs) auto lemma hd_prefixes [simp]: "hd (prefixes xs) = []" by (cases xs) simp_all lemma last_prefixes [simp]: "last (prefixes xs) = xs" by (induction xs) (simp_all add: last_map) lemma prefixes_append: "prefixes (xs @ ys) = prefixes xs @ map (λys'. xs @ ys') (tl (prefixes ys))" proof (induction xs) case Nil thus ?case by (cases ys) auto qed simp_all lemma prefixes_eq_snoc: "prefixes ys = xs @ [x] ⟷ (ys = [] ∧ xs = [] ∨ (∃z zs. ys = zs@[z] ∧ xs = prefixes zs)) ∧ x = ys" by (cases ys rule: rev_cases) auto lemma prefixes_tailrec [code]: "prefixes xs = rev (snd (foldl (λ(acc1, acc2) x. (x#acc1, rev (x#acc1)#acc2)) ([],[[]]) xs))" proof - have "foldl (λ(acc1, acc2) x. (x#acc1, rev (x#acc1)#acc2)) (ys, rev ys # zs) xs = (rev xs @ ys, rev (map (λas. rev ys @ as) (prefixes xs)) @ zs)" for ys zs proof (induction xs arbitrary: ys zs) case (Cons x xs ys zs) from Cons.IH[of "x # ys" "rev ys # zs"] show ?case by (simp add: o_def) qed simp_all from this [of "[]" "[]"] show ?thesis by simp qed lemma set_prefixes_eq: "set (prefixes xs) = {ys. prefix ys xs}" by auto lemma card_set_prefixes [simp]: "card (set (prefixes xs)) = Suc (length xs)" by (subst distinct_card) auto lemma set_prefixes_append: "set (prefixes (xs @ ys)) = set (prefixes xs) ∪ {xs @ ys' |ys'. ys' ∈ set (prefixes ys)}" by (subst prefixes_append, cases ys) auto subsection ‹Longest Common Prefix› definition Longest_common_prefix :: "'a list set ⇒ 'a list" where "Longest_common_prefix L = (ARG_MAX length ps. ∀xs ∈ L. prefix ps xs)" lemma Longest_common_prefix_ex: "L ≠ {} ⟹ ∃ps. (∀xs ∈ L. prefix ps xs) ∧ (∀qs. (∀xs ∈ L. prefix qs xs) ⟶ size qs ≤ size ps)" (is "_ ⟹ ∃ps. ?P L ps") proof(induction "LEAST n. ∃xs ∈L. n = length xs" arbitrary: L) case 0 have "[] ∈ L" using "0.hyps" LeastI[of "λn. ∃xs∈L. n = length xs"] ‹L ≠ {}› by auto hence "?P L []" by(auto) thus ?case .. next case (Suc n) let ?EX = "λn. ∃xs∈L. n = length xs" obtain x xs where xxs: "x#xs ∈ L" "size xs = n" using Suc.prems Suc.hyps(2) by(metis LeastI_ex[of ?EX] Suc_length_conv ex_in_conv) hence "[] ∉ L" using Suc.hyps(2) by auto show ?case proof (cases "∀xs ∈ L. ∃ys. xs = x#ys") case True let ?L = "{ys. x#ys ∈ L}" have 1: "(LEAST n. ∃xs ∈ ?L. n = length xs) = n" using xxs Suc.prems Suc.hyps(2) Least_le[of "?EX"] by - (rule Least_equality, fastforce+) have 2: "?L ≠ {}" using ‹x # xs ∈ L› by auto from Suc.hyps(1)[OF 1[symmetric] 2] obtain ps where IH: "?P ?L ps" .. { fix qs assume "∀qs. (∀xa. x # xa ∈ L ⟶ prefix qs xa) ⟶ length qs ≤ length ps" and "∀xs∈L. prefix qs xs" hence "length (tl qs) ≤ length ps" by (metis Cons_prefix_Cons hd_Cons_tl list.sel(2) Nil_prefix) hence "length qs ≤ Suc (length ps)" by auto } hence "?P L (x#ps)" using True IH by auto thus ?thesis .. next case False then obtain y ys where yys: "x≠y" "y#ys ∈ L" using ‹[] ∉ L› by (auto) (metis list.exhaust) have "∀qs. (∀xs∈L. prefix qs xs) ⟶ qs = []" using yys ‹x#xs ∈ L› by auto (metis Cons_prefix_Cons prefix_Cons) hence "?P L []" by auto thus ?thesis .. qed qed lemma Longest_common_prefix_unique: "L ≠ {} ⟹ ∃! ps. (∀xs ∈ L. prefix ps xs) ∧ (∀qs. (∀xs ∈ L. prefix qs xs) ⟶ size qs ≤ size ps)" by(rule ex_ex1I[OF Longest_common_prefix_ex]; meson equals0I prefix_length_prefix prefix_order.antisym) lemma Longest_common_prefix_eq: "⟦ L ≠ {}; ∀xs ∈ L. prefix ps xs; ∀qs. (∀xs ∈ L. prefix qs xs) ⟶ size qs ≤ size ps ⟧ ⟹ Longest_common_prefix L = ps" unfolding Longest_common_prefix_def arg_max_def is_arg_max_linorder by(rule some1_equality[OF Longest_common_prefix_unique]) auto lemma Longest_common_prefix_prefix: "xs ∈ L ⟹ prefix (Longest_common_prefix L) xs" unfolding Longest_common_prefix_def arg_max_def is_arg_max_linorder by(rule someI2_ex[OF Longest_common_prefix_ex]) auto lemma Longest_common_prefix_longest: "L ≠ {} ⟹ ∀xs∈L. prefix ps xs ⟹ length ps ≤ length(Longest_common_prefix L)" unfolding Longest_common_prefix_def arg_max_def is_arg_max_linorder by(rule someI2_ex[OF Longest_common_prefix_ex]) auto lemma Longest_common_prefix_max_prefix: "L ≠ {} ⟹ ∀xs∈L. prefix ps xs ⟹ prefix ps (Longest_common_prefix L)" by(metis Longest_common_prefix_prefix Longest_common_prefix_longest prefix_length_prefix ex_in_conv) lemma Longest_common_prefix_Nil: "[] ∈ L ⟹ Longest_common_prefix L = []" using Longest_common_prefix_prefix prefix_Nil by blast lemma Longest_common_prefix_image_Cons: "L ≠ {} ⟹ Longest_common_prefix ((#) x ` L) = x # Longest_common_prefix L" apply(rule Longest_common_prefix_eq) apply(simp) apply (simp add: Longest_common_prefix_prefix) apply simp by(metis Longest_common_prefix_longest[of L] Cons_prefix_Cons Nitpick.size_list_simp(2) Suc_le_mono hd_Cons_tl order.strict_implies_order zero_less_Suc) lemma Longest_common_prefix_eq_Cons: assumes "L ≠ {}" "[] ∉ L" "∀xs∈L. hd xs = x" shows "Longest_common_prefix L = x # Longest_common_prefix {ys. x#ys ∈ L}" proof - have "L = (#) x ` {ys. x#ys ∈ L}" using assms(2,3) by (auto simp: image_def)(metis hd_Cons_tl) thus ?thesis by (metis Longest_common_prefix_image_Cons image_is_empty assms(1)) qed lemma Longest_common_prefix_eq_Nil: "⟦x#ys ∈ L; y#zs ∈ L; x ≠ y ⟧ ⟹ Longest_common_prefix L = []" by (metis Longest_common_prefix_prefix list.inject prefix_Cons) fun longest_common_prefix :: "'a list ⇒ 'a list ⇒ 'a list" where "longest_common_prefix (x#xs) (y#ys) = (if x=y then x # longest_common_prefix xs ys else [])" | "longest_common_prefix _ _ = []" lemma longest_common_prefix_prefix1: "prefix (longest_common_prefix xs ys) xs" by(induction xs ys rule: longest_common_prefix.induct) auto lemma longest_common_prefix_prefix2: "prefix (longest_common_prefix xs ys) ys" by(induction xs ys rule: longest_common_prefix.induct) auto lemma longest_common_prefix_max_prefix: "⟦ prefix ps xs; prefix ps ys ⟧ ⟹ prefix ps (longest_common_prefix xs ys)" by(induction xs ys arbitrary: ps rule: longest_common_prefix.induct) (auto simp: prefix_Cons) subsection ‹Parallel lists› definition parallel :: "'a list ⇒ 'a list ⇒ bool" (infixl "∥" 50) where "(xs ∥ ys) = (¬ prefix xs ys ∧ ¬ prefix ys xs)" lemma parallelI [intro]: "¬ prefix xs ys ⟹ ¬ prefix ys xs ⟹ xs ∥ ys" unfolding parallel_def by blast lemma parallelE [elim]: assumes "xs ∥ ys" obtains "¬ prefix xs ys ∧ ¬ prefix ys xs" using assms unfolding parallel_def by blast theorem prefix_cases: obtains "prefix xs ys" | "strict_prefix ys xs" | "xs ∥ ys" unfolding parallel_def strict_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 prefix_cases) assume le: "prefix 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 prefixI 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 "strict_prefix ys xs" then have "prefix ys (xs @ [x])" by (simp add: strict_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_prefix_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 suffix :: "'a list ⇒ 'a list ⇒ bool" where "suffix xs ys = (∃zs. ys = zs @ xs)" definition strict_suffix :: "'a list ⇒ 'a list ⇒ bool" where "strict_suffix xs ys ⟷ suffix xs ys ∧ xs ≠ ys" interpretation suffix_order: order suffix strict_suffix by standard (auto simp: suffix_def strict_suffix_def) interpretation suffix_bot: order_bot Nil suffix strict_suffix by standard (simp add: suffix_def) lemma suffixI [intro?]: "ys = zs @ xs ⟹ suffix xs ys" unfolding suffix_def by blast lemma suffixE [elim?]: assumes "suffix xs ys" obtains zs where "ys = zs @ xs" using assms unfolding suffix_def by blast lemma suffix_tl [simp]: "suffix (tl xs) xs" by (induct xs) (auto simp: suffix_def) lemma strict_suffix_tl [simp]: "xs ≠ [] ⟹ strict_suffix (tl xs) xs" by (induct xs) (auto simp: strict_suffix_def suffix_def) lemma Nil_suffix [simp]: "suffix [] xs" by (simp add: suffix_def) lemma suffix_Nil [simp]: "(suffix xs []) = (xs = [])" by (auto simp add: suffix_def) lemma suffix_ConsI: "suffix xs ys ⟹ suffix xs (y # ys)" by (auto simp add: suffix_def) lemma suffix_ConsD: "suffix (x # xs) ys ⟹ suffix xs ys" by (auto simp add: suffix_def) lemma suffix_appendI: "suffix xs ys ⟹ suffix xs (zs @ ys)" by (auto simp add: suffix_def) lemma suffix_appendD: "suffix (zs @ xs) ys ⟹ suffix xs ys" by (auto simp add: suffix_def) lemma strict_suffix_set_subset: "strict_suffix xs ys ⟹ set xs ⊆ set ys" by (auto simp: strict_suffix_def suffix_def) lemma set_mono_suffix: "suffix xs ys ⟹ set xs ⊆ set ys" by (auto simp: suffix_def) lemma sorted_antimono_suffix: "suffix xs ys ⟹ sorted ys ⟹ sorted xs" by (metis sorted_append suffix_def) lemma suffix_ConsD2: "suffix (x # xs) (y # ys) ⟹ suffix xs ys" proof - assume "suffix (x # xs) (y # ys)" then obtain zs where "y # ys = zs @ x # xs" .. then show ?thesis by (induct zs) (auto intro!: suffix_appendI suffix_ConsI) qed lemma suffix_to_prefix [code]: "suffix xs ys ⟷ prefix (rev xs) (rev ys)" proof assume "suffix xs ys" then obtain zs where "ys = zs @ xs" .. then have "rev ys = rev xs @ rev zs" by simp then show "prefix (rev xs) (rev ys)" .. next assume "prefix (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 "suffix xs ys" .. qed lemma strict_suffix_to_prefix [code]: "strict_suffix xs ys ⟷ strict_prefix (rev xs) (rev ys)" by (auto simp: suffix_to_prefix strict_suffix_def strict_prefix_def) lemma distinct_suffix: "distinct ys ⟹ suffix xs ys ⟹ distinct xs" by (clarsimp elim!: suffixE) lemma map_mono_suffix: "suffix xs ys ⟹ suffix (map f xs) (map f ys)" by (auto elim!: suffixE intro: suffixI) lemma filter_mono_suffix: "suffix xs ys ⟹ suffix (filter P xs) (filter P ys)" by (auto simp: suffix_def) lemma suffix_drop: "suffix (drop n as) as" unfolding suffix_def by (rule exI [where x = "take n as"]) simp lemma suffix_take: "suffix xs ys ⟹ ys = take (length ys - length xs) ys @ xs" by (auto elim!: suffixE) lemma strict_suffix_reflclp_conv: "strict_suffix⇧^{=}⇧^{=}= suffix" by (intro ext) (auto simp: suffix_def strict_suffix_def) lemma suffix_lists: "suffix xs ys ⟹ ys ∈ lists A ⟹ xs ∈ lists A" unfolding suffix_def by auto lemma suffix_snoc [simp]: "suffix xs (ys @ [y]) ⟷ xs = [] ∨ (∃zs. xs = zs @ [y] ∧ suffix zs ys)" by (cases xs rule: rev_cases) (auto simp: suffix_def) lemma snoc_suffix_snoc [simp]: "suffix (xs @ [x]) (ys @ [y]) = (x = y ∧ suffix xs ys)" by (auto simp add: suffix_def) lemma same_suffix_suffix [simp]: "suffix (ys @ xs) (zs @ xs) = suffix ys zs" by (simp add: suffix_to_prefix) lemma same_suffix_nil [simp]: "suffix (ys @ xs) xs = (ys = [])" by (simp add: suffix_to_prefix) theorem suffix_Cons: "suffix xs (y # ys) ⟷ xs = y # ys ∨ suffix xs ys" unfolding suffix_def by (auto simp: Cons_eq_append_conv) theorem suffix_append: "suffix xs (ys @ zs) ⟷ suffix xs zs ∨ (∃xs'. xs = xs' @ zs ∧ suffix xs' ys)" by (auto simp: suffix_def append_eq_append_conv2) theorem suffix_length_le: "suffix xs ys ⟹ length xs ≤ length ys" by (auto simp add: suffix_def) lemma suffix_same_cases: "suffix (xs⇩_{1}::'a list) ys ⟹ suffix xs⇩_{2}ys ⟹ suffix xs⇩_{1}xs⇩_{2}∨ suffix xs⇩_{2}xs⇩_{1}" unfolding suffix_def by (force simp: append_eq_append_conv2) lemma suffix_length_suffix: "suffix ps xs ⟹ suffix qs xs ⟹ length ps ≤ length qs ⟹ suffix ps qs" by (auto simp: suffix_to_prefix intro: prefix_length_prefix) lemma suffix_length_less: "strict_suffix xs ys ⟹ length xs < length ys" by (auto simp: strict_suffix_def suffix_def) lemma suffix_ConsD': "suffix (x#xs) ys ⟹ strict_suffix xs ys" by (auto simp: strict_suffix_def suffix_def) lemma drop_strict_suffix: "strict_suffix xs ys ⟹ strict_suffix (drop n xs) ys" proof (induct n arbitrary: xs ys) case 0 then show ?case by (cases ys) simp_all next case (Suc n) then show ?case by (cases xs) (auto intro: Suc dest: suffix_ConsD' suffix_order.less_imp_le) qed lemma suffix_map_rightE: assumes "suffix xs (map f ys)" shows "∃xs'. suffix xs' ys ∧ xs = map f xs'" proof - from assms obtain xs' where xs': "map f ys = xs' @ xs" by (auto simp: suffix_def) define n where "n = length xs'" have "xs = drop n (map f ys)" by (simp add: xs' n_def) thus ?thesis by (intro exI[of _ "drop n ys"]) (auto simp: drop_map suffix_drop) qed lemma suffix_remdups_adj: "suffix xs ys ⟹ suffix (remdups_adj xs) (remdups_adj ys)" using prefix_remdups_adj[of "rev xs" "rev ys"] by (simp add: suffix_to_prefix) lemma not_suffix_cases: assumes pfx: "¬ suffix ps ls" obtains (c1) "ps ≠ []" and "ls = []" | (c2) a as x xs where "ps = as@[a]" and "ls = xs@[x]" and "x = a" and "¬ suffix as xs" | (c3) a as x xs where "ps = as@[a]" and "ls = xs@[x]" and "x ≠ a" proof (cases ps rule: rev_cases) case Nil then show ?thesis using pfx by simp next case (snoc as a) note c = ‹ps = as@[a]› show ?thesis proof (cases ls rule: rev_cases) case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_suffix_nil) next case (snoc xs x) show ?thesis proof (cases "x = a") case True have "¬ suffix as xs" using pfx c snoc True by simp with c snoc True show ?thesis by (rule c2) next case False with c snoc show ?thesis by (rule c3) qed qed qed lemma not_suffix_induct [consumes 1, case_names Nil Neq Eq]: assumes np: "¬ suffix ps ls" and base: "⋀x xs. P (xs@[x]) []" and r1: "⋀x xs y ys. x ≠ y ⟹ P (xs@[x]) (ys@[y])" and r2: "⋀x xs y ys. ⟦ x = y; ¬ suffix xs ys; P xs ys ⟧ ⟹ P (xs@[x]) (ys@[y])" shows "P ps ls" using np proof (induct ls arbitrary: ps rule: rev_induct) case Nil then show ?case by (cases ps rule: rev_cases) (auto intro: base) next case (snoc y ys ps) then have npfx: "¬ suffix ps (ys @ [y])" by simp then obtain x xs where pv: "ps = xs @ [x]" by (rule not_suffix_cases) auto show ?case by (metis snoc.hyps snoc_suffix_snoc npfx pv r1 r2) qed lemma parallelD1: "x ∥ y ⟹ ¬ prefix x y" by blast lemma parallelD2: "x ∥ y ⟹ ¬ prefix 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_prefix_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 subsection ‹Suffixes› primrec suffixes where "suffixes [] = [[]]" | "suffixes (x#xs) = suffixes xs @ [x # xs]" lemma in_set_suffixes [simp]: "xs ∈ set (suffixes ys) ⟷ suffix xs ys" by (induction ys) (auto simp: suffix_def Cons_eq_append_conv) lemma distinct_suffixes [intro]: "distinct (suffixes xs)" by (induction xs) (auto simp: suffix_def) lemma length_suffixes [simp]: "length (suffixes xs) = Suc (length xs)" by (induction xs) auto lemma suffixes_snoc [simp]: "suffixes (xs @ [x]) = [] # map (λys. ys @ [x]) (suffixes xs)" by (induction xs) auto lemma suffixes_not_Nil [simp]: "suffixes xs ≠ []" by (cases xs) auto lemma hd_suffixes [simp]: "hd (suffixes xs) = []" by (induction xs) simp_all lemma last_suffixes [simp]: "last (suffixes xs) = xs" by (cases xs) simp_all lemma suffixes_append: "suffixes (xs @ ys) = suffixes ys @ map (λxs'. xs' @ ys) (tl (suffixes xs))" proof (induction ys rule: rev_induct) case Nil thus ?case by (cases xs rule: rev_cases) auto next case (snoc y ys) show ?case by (simp only: append.assoc [symmetric] suffixes_snoc snoc.IH) simp qed lemma suffixes_eq_snoc: "suffixes ys = xs @ [x] ⟷ (ys = [] ∧ xs = [] ∨ (∃z zs. ys = z#zs ∧ xs = suffixes zs)) ∧ x = ys" by (cases ys) auto lemma suffixes_tailrec [code]: "suffixes xs = rev (snd (foldl (λ(acc1, acc2) x. (x#acc1, (x#acc1)#acc2)) ([],[[]]) (rev xs)))" proof - have "foldl (λ(acc1, acc2) x. (x#acc1, (x#acc1)#acc2)) (ys, ys # zs) (rev xs) = (xs @ ys, rev (map (λas. as @ ys) (suffixes xs)) @ zs)" for ys zs proof (induction xs arbitrary: ys zs) case (Cons x xs ys zs) from Cons.IH[of ys zs] show ?case by (simp add: o_def case_prod_unfold) qed simp_all from this [of "[]" "[]"] show ?thesis by simp qed lemma set_suffixes_eq: "set (suffixes xs) = {ys. suffix ys xs}" by auto lemma card_set_suffixes [simp]: "card (set (suffixes xs)) = Suc (length xs)" by (subst distinct_card) auto lemma set_suffixes_append: "set (suffixes (xs @ ys)) = set (suffixes ys) ∪ {xs' @ ys |xs'. xs' ∈ set (suffixes xs)}" by (subst suffixes_append, cases xs rule: rev_cases) auto lemma suffixes_conv_prefixes: "suffixes xs = map rev (prefixes (rev xs))" by (induction xs) auto lemma prefixes_conv_suffixes: "prefixes xs = map rev (suffixes (rev xs))" by (induction xs) auto lemma prefixes_rev: "prefixes (rev xs) = map rev (suffixes xs)" by (induction xs) auto lemma suffixes_rev: "suffixes (rev xs) = map rev (prefixes xs)" by (induction xs) 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_strict_suffix: assumes "list_emb P xs ys" and "strict_suffix ys zs" shows "list_emb P xs zs" using assms(2) and list_emb_append2 [OF assms(1)] by (auto simp: strict_suffix_def suffix_def) lemma list_emb_suffix: assumes "list_emb P xs ys" and "suffix ys zs" shows "list_emb P xs zs" using assms and list_emb_strict_suffix unfolding strict_suffix_reflclp_conv[symmetric] 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 lemma list_emb_Cons_iff1 [simp]: assumes "P x y" shows "list_emb P (x#xs) (y#ys) ⟷ list_emb P xs ys" using assms by (subst list_emb.simps) (auto dest: list_emb_ConsD) lemma list_emb_Cons_iff2 [simp]: assumes "¬P x y" shows "list_emb P (x#xs) (y#ys) ⟷ list_emb P (x#xs) ys" using assms by (subst list_emb.simps) auto lemma list_emb_code [code]: "list_emb P [] ys ⟷ True" "list_emb P (x#xs) [] ⟷ False" "list_emb P (x#xs) (y#ys) ⟷ (if P x y then list_emb P xs ys else list_emb P (x#xs) ys)" by simp_all subsection ‹Subsequences (special case of homeomorphic embedding)› abbreviation subseq :: "'a list ⇒ 'a list ⇒ bool" where "subseq xs ys ≡ list_emb (=) xs ys" definition strict_subseq where "strict_subseq xs ys ⟷ xs ≠ ys ∧ subseq xs ys" lemma subseq_Cons2: "subseq xs ys ⟹ subseq (x#xs) (x#ys)" by auto lemma subseq_same_length: assumes "subseq xs ys" and "length xs = length ys" shows "xs = ys" using assms by (induct) (auto dest: list_emb_length) lemma not_subseq_length [simp]: "length ys < length xs ⟹ ¬ subseq xs ys" by (metis list_emb_length linorder_not_less) lemma subseq_Cons': "subseq (x#xs) ys ⟹ subseq xs ys" by (induct xs, simp, blast dest: list_emb_ConsD) lemma subseq_Cons2': assumes "subseq (x#xs) (x#ys)" shows "subseq xs ys" using assms by (cases) (rule subseq_Cons') lemma subseq_Cons2_neq: assumes "subseq (x#xs) (y#ys)" shows "x ≠ y ⟹ subseq (x#xs) ys" using assms by (cases) auto lemma subseq_Cons2_iff [simp]: "subseq (x#xs) (y#ys) = (if x = y then subseq xs ys else subseq (x#xs) ys)" by simp lemma subseq_append': "subseq (zs @ xs) (zs @ ys) ⟷ subseq xs ys" by (induct zs) simp_all interpretation subseq_order: order subseq strict_subseq proof fix xs ys :: "'a list" { assume "subseq xs ys" and "subseq ys xs" thus "xs = ys" 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 subseq_Cons' by fastforce thus ?case .. qed } thus "strict_subseq xs ys ⟷ (subseq xs ys ∧ ¬subseq ys xs)" by (auto simp: strict_subseq_def) qed (auto simp: list_emb_refl intro: list_emb_trans) lemma in_set_subseqs [simp]: "xs ∈ set (subseqs ys) ⟷ subseq xs ys" proof assume "xs ∈ set (subseqs ys)" thus "subseq xs ys" by (induction ys arbitrary: xs) (auto simp: Let_def) next have [simp]: "[] ∈ set (subseqs ys)" for ys :: "'a list" by (induction ys) (auto simp: Let_def) assume "subseq xs ys" thus "xs ∈ set (subseqs ys)" by (induction xs ys rule: list_emb.induct) (auto simp: Let_def) qed lemma set_subseqs_eq: "set (subseqs ys) = {xs. subseq xs ys}" by auto lemma subseq_append_le_same_iff: "subseq (xs @ ys) ys ⟷ xs = []" by (auto dest: list_emb_length) lemma subseq_singleton_left: "subseq [x] ys ⟷ x ∈ set ys" by (fastforce dest: list_emb_ConsD split_list_last) lemma list_emb_append_mono: "⟦ list_emb P xs xs'; list_emb P ys ys' ⟧ ⟹ list_emb P (xs@ys) (xs'@ys')" by (induct rule: list_emb.induct) auto lemma prefix_imp_subseq [intro]: "prefix xs ys ⟹ subseq xs ys" by (auto simp: prefix_def) lemma suffix_imp_subseq [intro]: "suffix xs ys ⟹ subseq xs ys" by (auto simp: suffix_def) subsection ‹Appending elements› lemma subseq_append [simp]: "subseq (xs @ zs) (ys @ zs) ⟷ subseq xs ys" (is "?l = ?r") proof { fix xs' ys' xs ys zs :: "'a list" assume "subseq xs' ys'" then have "xs' = xs @ zs ∧ ys' = ys @ zs ⟶ subseq 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 ‹(=) 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 subseq_order.order_refl) qed lemma subseq_append_iff: "subseq xs (ys @ zs) ⟷ (∃xs1 xs2. xs = xs1 @ xs2 ∧ subseq xs1 ys ∧ subseq xs2 zs)" (is "?lhs = ?rhs") proof assume ?lhs thus ?rhs proof (induction xs "ys @ zs" arbitrary: ys zs rule: list_emb.induct) case (list_emb_Cons xs ws y ys zs) from list_emb_Cons(2)[of "tl ys" zs] and list_emb_Cons(2)[of "[]" "tl zs"] and list_emb_Cons(1,3) show ?case by (cases ys) auto next case (list_emb_Cons2 x y xs ws ys zs) from list_emb_Cons2(3)[of "tl ys" zs] and list_emb_Cons2(3)[of "[]" "tl zs"] and list_emb_Cons2(1,2,4) show ?case by (cases ys) (auto simp: Cons_eq_append_conv) qed auto qed (auto intro: list_emb_append_mono) lemma subseq_appendE [case_names append]: assumes "subseq xs (ys @ zs)" obtains xs1 xs2 where "xs = xs1 @ xs2" "subseq xs1 ys" "subseq xs2 zs" using assms by (subst (asm) subseq_append_iff) auto lemma subseq_drop_many: "subseq xs ys ⟹ subseq xs (zs @ ys)" by (induct zs) auto lemma subseq_rev_drop_many: "subseq xs ys ⟹ subseq xs (ys @ zs)" by (metis append_Nil2 list_emb_Nil list_emb_append_mono) subsection ‹Relation to standard list operations› lemma subseq_map: assumes "subseq xs ys" shows "subseq (map f xs) (map f ys)" using assms by (induct) auto lemma subseq_filter_left [simp]: "subseq (filter P xs) xs" by (induct xs) auto lemma subseq_filter [simp]: assumes "subseq xs ys" shows "subseq (filter P xs) (filter P ys)" using assms by induct auto lemma subseq_conv_nths: "subseq xs ys ⟷ (∃N. xs = nths ys N)" (is "?L = ?R") proof assume ?L then show ?R proof (induct) case list_emb_Nil show ?case by (metis nths_empty) next case (list_emb_Cons xs ys x) then obtain N where "xs = nths ys N" by blast then have "xs = nths (x#ys) (Suc ` N)" by (clarsimp simp add: nths_Cons inj_image_mem_iff) then show ?case by blast next case (list_emb_Cons2 x y xs ys) then obtain N where "xs = nths ys N" by blast then have "x#xs = nths (x#ys) (insert 0 (Suc ` N))" by (clarsimp simp add: nths_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 = nths ys N" .. moreover have "subseq (nths ys N) ys" proof (induct ys arbitrary: N) case Nil show ?case by simp next case Cons then show ?case by (auto simp: nths_Cons) qed ultimately show ?L by simp qed subsection ‹Contiguous sublists› subsubsection ‹‹sublist›› definition sublist :: "'a list ⇒ 'a list ⇒ bool" where "sublist xs ys = (∃ps ss. ys = ps @ xs @ ss)" definition strict_sublist :: "'a list ⇒ 'a list ⇒ bool" where "strict_sublist xs ys ⟷ sublist xs ys ∧ xs ≠ ys" interpretation sublist_order: order sublist strict_sublist proof fix xs ys zs :: "'a list" assume "sublist xs ys" "sublist ys zs" then obtain xs1 xs2 ys1 ys2 where "ys = xs1 @ xs @ xs2" "zs = ys1 @ ys @ ys2" by (auto simp: sublist_def) hence "zs = (ys1 @ xs1) @ xs @ (xs2 @ ys2)" by simp thus "sublist xs zs" unfolding sublist_def by blast next fix xs ys :: "'a list" { assume "sublist xs ys" "sublist ys xs" then obtain as bs cs ds where xs: "xs = as @ ys @ bs" and ys: "ys = cs @ xs @ ds" by (auto simp: sublist_def) have "xs = as @ cs @ xs @ ds @ bs" by (subst xs, subst ys) auto also have "length … = length as + length cs + length xs + length bs + length ds" by simp finally have "as = []" "bs = []" by simp_all with xs show "xs = ys" by simp } thus "strict_sublist xs ys ⟷ (sublist xs ys ∧ ¬sublist ys xs)" by (auto simp: strict_sublist_def) qed (auto simp: strict_sublist_def sublist_def intro: exI[of _ "[]"]) lemma sublist_Nil_left [simp, intro]: "sublist [] ys" by (auto simp: sublist_def) lemma sublist_Cons_Nil [simp]: "¬sublist (x#xs) []" by (auto simp: sublist_def) lemma sublist_Nil_right [simp]: "sublist xs [] ⟷ xs = []" by (cases xs) auto lemma sublist_appendI [simp, intro]: "sublist xs (ps @ xs @ ss)" by (auto simp: sublist_def) lemma sublist_append_leftI [simp, intro]: "sublist xs (ps @ xs)" by (auto simp: sublist_def intro: exI[of _ "[]"]) lemma sublist_append_rightI [simp, intro]: "sublist xs (xs @ ss)" by (auto simp: sublist_def intro: exI[of _ "[]"]) lemma sublist_altdef: "sublist xs ys ⟷ (∃ys'. prefix ys' ys ∧ suffix xs ys')" proof safe assume "sublist xs ys" then obtain ps ss where "ys = ps @ xs @ ss" by (auto simp: sublist_def) thus "∃ys'. prefix ys' ys ∧ suffix xs ys'" by (intro exI[of _ "ps @ xs"] conjI suffix_appendI) auto next fix ys' assume "prefix ys' ys" "suffix xs ys'" thus "sublist xs ys" by (auto simp: prefix_def suffix_def) qed lemma sublist_altdef': "sublist xs ys ⟷ (∃ys'. suffix ys' ys ∧ prefix xs ys')" proof safe assume "sublist xs ys" then obtain ps ss where "ys = ps @ xs @ ss" by (auto simp: sublist_def) thus "∃ys'. suffix ys' ys ∧ prefix xs ys'" by (intro exI[of _ "xs @ ss"] conjI suffixI) auto next fix ys' assume "suffix ys' ys" "prefix xs ys'" thus "sublist xs ys" by (auto simp: prefix_def suffix_def) qed lemma sublist_Cons_right: "sublist xs (y # ys) ⟷ prefix xs (y # ys) ∨ sublist xs ys" by (auto simp: sublist_def prefix_def Cons_eq_append_conv) lemma sublist_code [code]: "sublist [] ys ⟷ True" "sublist (x # xs) [] ⟷ False" "sublist (x # xs) (y # ys) ⟷ prefix (x # xs) (y # ys) ∨ sublist (x # xs) ys" by (simp_all add: sublist_Cons_right) lemma sublist_append: "sublist xs (ys @ zs) ⟷ sublist xs ys ∨ sublist xs zs ∨ (∃xs1 xs2. xs = xs1 @ xs2 ∧ suffix xs1 ys ∧ prefix xs2 zs)" by (auto simp: sublist_altdef prefix_append suffix_append) lemma map_mono_sublist: assumes "sublist xs ys" shows "sublist (map f xs) (map f ys)" proof - from assms obtain xs1 xs2 where ys: "ys = xs1 @ xs @ xs2" by (auto simp: sublist_def) have "map f ys = map f xs1 @ map f xs @ map f xs2" by (auto simp: ys) thus ?thesis by (auto simp: sublist_def) qed lemma sublist_length_le: "sublist xs ys ⟹ length xs ≤ length ys" by (auto simp add: sublist_def) lemma set_mono_sublist: "sublist xs ys ⟹ set xs ⊆ set ys" by (auto simp add: sublist_def) lemma prefix_imp_sublist [simp, intro]: "prefix xs ys ⟹ sublist xs ys" by (auto simp: sublist_def prefix_def intro: exI[of _ "[]"]) lemma suffix_imp_sublist [simp, intro]: "suffix xs ys ⟹ sublist xs ys" by (auto simp: sublist_def suffix_def intro: exI[of _ "[]"]) lemma sublist_take [simp, intro]: "sublist (take n xs) xs" by (rule prefix_imp_sublist) (simp_all add: take_is_prefix) lemma sublist_drop [simp, intro]: "sublist (drop n xs) xs" by (rule suffix_imp_sublist) (simp_all add: suffix_drop) lemma sublist_tl [simp, intro]: "sublist (tl xs) xs" by (rule suffix_imp_sublist) (simp_all add: suffix_drop) lemma sublist_butlast [simp, intro]: "sublist (butlast xs) xs" by (rule prefix_imp_sublist) (simp_all add: prefixeq_butlast) lemma sublist_rev [simp]: "sublist (rev xs) (rev ys) = sublist xs ys" proof assume "sublist (rev xs) (rev ys)" then obtain as bs where "rev ys = as @ rev xs @ bs" by (auto simp: sublist_def) also have "rev … = rev bs @ xs @ rev as" by simp finally show "sublist xs ys" by simp next assume "sublist xs ys" then obtain as bs where "ys = as @ xs @ bs" by (auto simp: sublist_def) also have "rev … = rev bs @ rev xs @ rev as" by simp finally show "sublist (rev xs) (rev ys)" by simp qed lemma sublist_rev_left: "sublist (rev xs) ys = sublist xs (rev ys)" by (subst sublist_rev [symmetric]) (simp only: rev_rev_ident) lemma sublist_rev_right: "sublist xs (rev ys) = sublist (rev xs) ys" by (subst sublist_rev [symmetric]) (simp only: rev_rev_ident) lemma snoc_sublist_snoc: "sublist (xs @ [x]) (ys @ [y]) ⟷ (x = y ∧ suffix xs ys ∨ sublist (xs @ [x]) ys) " by (subst (1 2) sublist_rev [symmetric]) (simp del: sublist_rev add: sublist_Cons_right suffix_to_prefix) lemma sublist_snoc: "sublist xs (ys @ [y]) ⟷ suffix xs (ys @ [y]) ∨ sublist xs ys" by (subst (1 2) sublist_rev [symmetric]) (simp del: sublist_rev add: sublist_Cons_right suffix_to_prefix) lemma sublist_imp_subseq [intro]: "sublist xs ys ⟹ subseq xs ys" by (auto simp: sublist_def) lemma sublist_map_rightE: assumes "sublist xs (map f ys)" shows "∃xs'. sublist xs' ys ∧ xs = map f xs'" proof - note takedrop = sublist_take sublist_drop define n where "n = (length ys - length xs)" from assms obtain xs1 xs2 where xs12: "map f ys = xs1 @ xs @ xs2" by (auto simp: sublist_def) define n where "n = length xs1" have "xs = take (length xs) (drop n (map f ys))" by (simp add: xs12 n_def) thus ?thesis by (intro exI[of _ "take (length xs) (drop n ys)"]) (auto simp: take_map drop_map intro!: takedrop[THEN sublist_order.order.trans]) qed lemma sublist_remdups_adj: assumes "sublist xs ys" shows "sublist (remdups_adj xs) (remdups_adj ys)" proof - from assms obtain xs1 xs2 where ys: "ys = xs1 @ xs @ xs2" by (auto simp: sublist_def) have "suffix (remdups_adj (xs @ xs2)) (remdups_adj (xs1 @ xs @ xs2))" by (rule suffix_remdups_adj, rule suffix_appendI) auto then obtain zs1 where zs1: "remdups_adj (xs1 @ xs @ xs2) = zs1 @ remdups_adj (xs @ xs2)" by (auto simp: suffix_def) have "prefix (remdups_adj xs) (remdups_adj (xs @ xs2))" by (intro prefix_remdups_adj) auto then obtain zs2 where zs2: "remdups_adj (xs @ xs2) = remdups_adj xs @ zs2" by (auto simp: prefix_def) show ?thesis by (simp add: ys zs1 zs2) qed subsubsection ‹‹sublists›› primrec sublists :: "'a list ⇒ 'a list list" where "sublists [] = [[]]" | "sublists (x # xs) = sublists xs @ map ((#) x) (prefixes xs)" lemma in_set_sublists [simp]: "xs ∈ set (sublists ys) ⟷ sublist xs ys" by (induction ys arbitrary: xs) (auto simp: sublist_Cons_right prefix_Cons) lemma set_sublists_eq: "set (sublists xs) = {ys. sublist ys xs}" by auto lemma length_sublists [simp]: "length (sublists xs) = Suc (length xs * Suc (length xs) div 2)" by (induction xs) simp_all subsection ‹Parametricity› context includes lifting_syntax begin private lemma prefix_primrec: "prefix = rec_list (λxs. True) (λx xs xsa ys. case ys of [] ⇒ False | y # ys ⇒ x = y ∧ xsa ys)" proof (intro ext, goal_cases) case (1 xs ys) show ?case by (induction xs arbitrary: ys) (auto simp: prefix_Cons split: list.splits) qed private lemma sublist_primrec: "sublist = (λxs ys. rec_list (λxs. xs = []) (λy ys ysa xs. prefix xs (y # ys) ∨ ysa xs) ys xs)" proof (intro ext, goal_cases) case (1 xs ys) show ?case by (induction ys) (auto simp: sublist_Cons_right) qed private lemma list_emb_primrec: "list_emb = (λuu uua uuaa. rec_list (λP xs. List.null xs) (λy ys ysa P xs. case xs of [] ⇒ True | x # xs ⇒ if P x y then ysa P xs else ysa P (x # xs)) uuaa uu uua)" proof (intro ext, goal_cases) case (1 P xs ys) show ?case by (induction ys arbitrary: xs) (auto simp: list_emb_code List.null_def split: list.splits) qed lemma prefix_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 A ===> (=)) prefix prefix" unfolding prefix_primrec by transfer_prover lemma suffix_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 A ===> (=)) suffix suffix" unfolding suffix_to_prefix [abs_def] by transfer_prover lemma sublist_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 A ===> (=)) sublist sublist" unfolding sublist_primrec by transfer_prover lemma parallel_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 A ===> (=)) parallel parallel" unfolding parallel_def by transfer_prover lemma list_emb_transfer [transfer_rule]: "((A ===> A ===> (=)) ===> list_all2 A ===> list_all2 A ===> (=)) list_emb list_emb" unfolding list_emb_primrec by transfer_prover lemma strict_prefix_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 A ===> (=)) strict_prefix strict_prefix" unfolding strict_prefix_def by transfer_prover lemma strict_suffix_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 A ===> (=)) strict_suffix strict_suffix" unfolding strict_suffix_def by transfer_prover lemma strict_subseq_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 A ===> (=)) strict_subseq strict_subseq" unfolding strict_subseq_def by transfer_prover lemma strict_sublist_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 A ===> (=)) strict_sublist strict_sublist" unfolding strict_sublist_def by transfer_prover lemma prefixes_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 (list_all2 A)) prefixes prefixes" unfolding prefixes_def by transfer_prover lemma suffixes_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 (list_all2 A)) suffixes suffixes" unfolding suffixes_def by transfer_prover lemma sublists_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 (list_all2 A)) sublists sublists" unfolding sublists_def by transfer_prover end end