(* Author: Tobias Nipkow *) section "Deterministic List Update" theory Move_to_Front imports Swaps On_Off Competitive_Analysis begin declare Let_def[simp] subsection "Function ‹mtf›" definition mtf :: "'a ⇒ 'a list ⇒ 'a list" where "mtf x xs = (if x ∈ set xs then x # (take (index xs x) xs) @ drop (index xs x + 1) xs else xs)" lemma mtf_id[simp]: "x ∉ set xs ⟹ mtf x xs = xs" by(simp add: mtf_def) lemma mtf0[simp]: "x ∈ set xs ⟹ mtf x xs ! 0 = x" by(auto simp: mtf_def) lemma before_in_mtf: assumes "z ∈ set xs" shows "x < y in mtf z xs ⟷ (y ≠ z ∧ (if x=z then y ∈ set xs else x < y in xs))" proof- have 0: "index xs z < size xs" by (metis assms index_less_size_conv) let ?xs = "take (index xs z) xs @ xs ! index xs z # drop (Suc (index xs z)) xs" have "x < y in mtf z xs = (y ≠ z ∧ (if x=z then y ∈ set ?xs else x < y in ?xs))" using assms by(auto simp add: mtf_def before_in_def index_append) (metis add_lessD1 index_less_size_conv length_take less_Suc_eq not_less_eq) with id_take_nth_drop[OF 0, symmetric] show ?thesis by(simp) qed lemma Inv_mtf: "set xs = set ys ⟹ z : set ys ⟹ Inv xs (mtf z ys) = Inv xs ys ∪ {(x,z)|x. x < z in xs ∧ x < z in ys} - {(z,x)|x. z < x in xs ∧ x < z in ys}" by(auto simp add: Inv_def before_in_mtf not_before_in dest: before_in_setD1) lemma set_mtf[simp]: "set(mtf x xs) = set xs" by(simp add: mtf_def) (metis append_take_drop_id Cons_nth_drop_Suc index_less le_refl Un_insert_right nth_index set_append set_simps(2)) lemma length_mtf[simp]: "size (mtf x xs) = size xs" by (auto simp add: mtf_def min_def) (metis index_less_size_conv leD) lemma distinct_mtf[simp]: "distinct (mtf x xs) = distinct xs" by (metis length_mtf set_mtf card_distinct distinct_card) subsection "Function ‹mtf2›" definition mtf2 :: "nat ⇒ 'a ⇒ 'a list ⇒ 'a list" where "mtf2 n x xs = (if x : set xs then swaps [index xs x - n..<index xs x] xs else xs)" lemma mtf_eq_mtf2: "mtf x xs = mtf2 (length xs - 1) x xs" proof - have "x : set xs ⟹ index xs x - (size xs - Suc 0) = 0" by (auto simp: less_Suc_eq_le[symmetric]) thus ?thesis by(auto simp: mtf_def mtf2_def swaps_eq_nth_take_drop) qed lemma mtf20[simp]: "mtf2 0 x xs = xs" by(auto simp add: mtf2_def) lemma length_mtf2[simp]: "length (mtf2 n x xs) = length xs" by (auto simp: mtf2_def index_less_size_conv[symmetric] simp del:index_conv_size_if_notin) lemma set_mtf2[simp]: "set(mtf2 n x xs) = set xs" by (auto simp: mtf2_def index_less_size_conv[symmetric] simp del:index_conv_size_if_notin) lemma distinct_mtf2[simp]: "distinct (mtf2 n x xs) = distinct xs" by (metis length_mtf2 set_mtf2 card_distinct distinct_card) lemma card_Inv_mtf2: "xs!j = ys!0 ⟹ j < length xs ⟹ dist_perm xs ys ⟹ card (Inv (swaps [i..<j] xs) ys) = card (Inv xs ys) - int(j-i)" proof(induction j arbitrary: xs) case (Suc j) show ?case proof cases assume "i > j" thus ?thesis by simp next assume [arith]: "¬ i > j" have 0: "Suc j < length ys" by (metis Suc.prems(2,3) distinct_card) have 1: "(ys ! 0, xs ! j) : Inv ys xs" proof (auto simp: Inv_def) show "ys ! 0 < xs ! j in ys" using Suc.prems by (metis Suc_lessD n_not_Suc_n not_before0 not_before_in nth_eq_iff_index_eq nth_mem) show "xs ! j < ys ! 0 in xs" using Suc.prems by (metis Suc_lessD before_id lessI) qed have 2: "card(Inv ys xs) ≠ 0" using 1 by auto have "int(card (Inv (swaps [i..<Suc j] xs) ys)) = card (Inv (swap j xs) ys) - int (j-i)" using Suc by simp also have "… = card (Inv ys (swap j xs)) - int (j-i)" by(simp add: card_Inv_sym) also have "… = card (Inv ys xs - {(ys ! 0, xs ! j)}) - int (j - i)" using Suc.prems 0 by(simp add: Inv_swap) also have "… = int(card (Inv ys xs) - 1) - (j - i)" using 1 by(simp add: card_Diff_singleton) also have "… = card (Inv ys xs) - int (Suc j - i)" using 2 by arith also have "… = card (Inv xs ys) - int (Suc j - i)" by(simp add: card_Inv_sym) finally show ?thesis . qed qed simp subsection "Function Lxy" definition Lxy :: "'a list ⇒ 'a set ⇒ 'a list" where "Lxy xs S = filter (λz. z∈S) xs" thm inter_set_filter lemma Lxy_length_cons: "length (Lxy xs S) ≤ length (Lxy (x#xs) S)" unfolding Lxy_def by(simp) lemma Lxy_empty[simp]: "Lxy [] S = []" unfolding Lxy_def by simp lemma Lxy_set_filter: "set (Lxy xs S) = S ∩ set xs" by (simp add: Lxy_def inter_set_filter) lemma Lxy_distinct: "distinct xs ⟹ distinct (Lxy xs S)" by (simp add: Lxy_def) lemma Lxy_append: "Lxy (xs@ys) S = Lxy xs S @ Lxy ys S" by(simp add: Lxy_def) lemma Lxy_snoc: "Lxy (xs@[x]) S = (if x∈S then Lxy xs S @ [x] else Lxy xs S)" by(simp add: Lxy_def) lemma Lxy_not: "S ∩ set xs = {} ⟹ Lxy xs S = []" unfolding Lxy_def apply(induct xs) by simp_all lemma Lxy_notin: "set xs ∩ S = {} ⟹ Lxy xs S = []" apply(induct xs) by(simp_all add: Lxy_def) lemma Lxy_in: "x∈S ⟹ Lxy [x] S = [x]" by(simp add: Lxy_def) lemma Lxy_project: assumes "x≠y" "x ∈ set xs" "y∈set xs" "distinct xs" and "x < y in xs" shows "Lxy xs {x,y} = [x,y]" proof - from assms have ij: "index xs x < index xs y" and xinxs: "index xs x < length xs" and yinxs: "index xs y < length xs" unfolding before_in_def by auto from xinxs obtain a as where dec1: "a @ [xs!index xs x] @ as = xs" and "a = take (index xs x) xs" and "as = drop (Suc (index xs x)) xs" and length_a: "length a = index xs x" and length_as: "length as = length xs - index xs x- 1" using id_take_nth_drop by fastforce have "index xs y≥length (a @ [xs!index xs x])" using length_a ij by auto then have "((a @ [xs!index xs x]) @ as) ! index xs y = as ! (index xs y-length (a @ [xs ! index xs x]))" using nth_append[where xs="a @ [xs!index xs x]" and ys="as"] by(simp) then have xsj: "xs ! index xs y = as ! (index xs y-index xs x-1)" using dec1 length_a by auto have las: "(index xs y-index xs x-1) < length as" using length_as yinxs ij by simp obtain b c where dec2: "b @ [xs!index xs y] @ c = as" and "b = take (index xs y-index xs x-1) as" "c=drop (Suc (index xs y-index xs x-1)) as" and length_b: "length b = index xs y-index xs x-1" using id_take_nth_drop[OF las] xsj by force have xs_dec: "a @ [xs!index xs x] @ b @ [xs!index xs y] @ c = xs" using dec1 dec2 by auto from xs_dec assms(4) have "distinct ((a @ [xs!index xs x] @ b @ [xs!index xs y]) @ c)" by simp then have c_empty: "set c ∩ {x,y} = {}" and b_empty: "set b ∩ {x,y} = {}"and a_empty: "set a ∩ {x,y} = {}" by(auto simp add: assms(2,3)) have "Lxy (a @ [xs!index xs x] @ b @ [xs!index xs y] @ c) {x,y} = [x,y]" apply(simp only: Lxy_append) apply(simp add: assms(2,3)) using a_empty b_empty c_empty by(simp add: Lxy_notin Lxy_in) with xs_dec show ?thesis by auto qed lemma Lxy_mono: "{x,y} ⊆ set xs ⟹ distinct xs ⟹ x < y in xs = x < y in Lxy xs {x,y}" apply(cases "x=y") apply(simp add: before_in_irefl) proof - assume xyset: "{x,y} ⊆ set xs" assume dxs: "distinct xs" assume xy: "x≠y" { fix x y assume 1: "{x,y} ⊆ set xs" assume xny: "x≠y" assume 3: "x < y in xs" have "Lxy xs {x,y} = [x,y]" apply(rule Lxy_project) using xny 1 3 dxs by(auto) then have "x < y in Lxy xs {x,y}" using xny by(simp add: before_in_def) } note aha=this have a: "x < y in xs ⟹ x < y in Lxy xs {x,y}" apply(subst Lxy_project) using xy xyset dxs by(simp_all add: before_in_def) have t: "{x,y}={y,x}" by(auto) have f: "~ x < y in xs ⟹ y < x in Lxy xs {x,y}" unfolding t apply(rule aha) using xyset apply(simp) using xy apply(simp) using xy xyset by(simp add: not_before_in) have b: "~ x < y in xs ⟹ ~ x < y in Lxy xs {x,y}" proof - assume "~ x < y in xs" then have "y < x in Lxy xs {x,y}" using f by auto then have "~ x < y in Lxy xs {x,y}" using xy by(simp add: not_before_in) then show ?thesis . qed from a b show ?thesis by metis qed subsection "List Update as Online/Offline Algorithm" type_synonym 'a state = "'a list" type_synonym answer = "nat * nat list" definition step :: "'a state ⇒ 'a ⇒ answer ⇒ 'a state" where "step s r a = (let (k,sws) = a in mtf2 k r (swaps sws s))" definition t :: "'a state ⇒ 'a ⇒ answer ⇒ nat" where "t s r a = (let (mf,sws) = a in index (swaps sws s) r + 1 + size sws)" definition static where "static s rs = (set rs ⊆ set s)" interpretation On_Off step t static . type_synonym 'a alg_off = "'a state ⇒ 'a list ⇒ answer list" type_synonym ('a,'is) alg_on = "('a state,'is,'a,answer) alg_on" lemma T_ge_len: "length as = length rs ⟹ T s rs as ≥ length rs" by(induction arbitrary: s rule: list_induct2) (auto simp: t_def trans_le_add2) lemma T_off_neq0: "(⋀rs s0. size(alg s0 rs) = length rs) ⟹ rs ≠ [] ⟹ T_off alg s0 rs ≠ 0" apply(erule_tac x=rs in meta_allE) apply(erule_tac x=s0 in meta_allE) apply (auto simp: neq_Nil_conv length_Suc_conv t_def) done lemma length_step[simp]: "length (step s r as) = length s" by(simp add: step_def split_def) lemma step_Nil_iff[simp]: "step xs r act = [] ⟷ xs = []" by(auto simp add: step_def mtf2_def split: prod.splits) lemma set_step2: "set(step s r (mf,sws)) = set s" by(auto simp add: step_def) lemma set_step: "set(step s r act) = set s" by(cases act)(simp add: set_step2) lemma distinct_step: "distinct(step s r as) = distinct s" by (auto simp: step_def split_def) subsection "Online Algorithm Move-to-Front is 2-Competitive" definition MTF :: "('a,unit) alg_on" where "MTF = (λ_. (), λs r. ((size (fst s) - 1,[]), ()))" text‹It was first proved by Sleator and Tarjan~\cite{SleatorT-CACM85} that the Move-to-Front algorithm is 2-competitive.› (* The core idea with upper bounds: *) lemma potential: fixes t :: "nat ⇒ 'a::linordered_ab_group_add" and p :: "nat ⇒ 'a" assumes p0: "p 0 = 0" and ppos: "⋀n. p n ≥ 0" and ub: "⋀n. t n + p(n+1) - p n ≤ u n" shows "(∑i<n. t i) ≤ (∑i<n. u i)" proof- let ?a = "λn. t n + p(n+1) - p n" have 1: "(∑i<n. t i) = (∑i<n. ?a i) - p(n)" by(induction n) (simp_all add: p0) thus ?thesis by (metis (erased, lifting) add.commute diff_add_cancel le_add_same_cancel2 order.trans ppos sum_mono ub) qed lemma potential2: fixes t :: "nat ⇒ 'a::linordered_ab_group_add" and p :: "nat ⇒ 'a" assumes p0: "p 0 = 0" and ppos: "⋀n. p n ≥ 0" and ub: "⋀m. m<n ⟹ t m + p(m+1) - p m ≤ u m" shows "(∑i<n. t i) ≤ (∑i<n. u i)" proof- let ?a = "λn. t n + p(n+1) - p n" have "(∑i<n. t i) = (∑i<n. ?a i) - p(n)" by(induction n) (simp_all add: p0) also have "… ≤ (∑i<n. ?a i)" using ppos by auto also have "… ≤ (∑i<n. u i)" apply(rule sum_mono) apply(rule ub) by auto finally show ?thesis . qed abbreviation "before x xs ≡ {y. y < x in xs}" abbreviation "after x xs ≡ {y. x < y in xs}" lemma finite_before[simp]: "finite (before x xs)" apply(rule finite_subset[where B = "set xs"]) apply (auto dest: before_in_setD1) done lemma finite_after[simp]: "finite (after x xs)" apply(rule finite_subset[where B = "set xs"]) apply (auto dest: before_in_setD2) done lemma before_conv_take: "x : set xs ⟹ before x xs = set(take (index xs x) xs)" by (auto simp add: before_in_def set_take_if_index index_le_size) (metis index_take leI) lemma card_before: "distinct xs ⟹ x : set xs ⟹ card (before x xs) = index xs x" using index_le_size[of xs x] by(simp add: before_conv_take distinct_card[OF distinct_take] min_def) lemma before_Un: "set xs = set ys ⟹ x : set xs ⟹ before x ys = before x xs ∩ before x ys Un after x xs ∩ before x ys" by(auto)(metis before_in_setD1 not_before_in) lemma phi_diff_aux: "card (Inv xs ys ∪ {(y, x) |y. y < x in xs ∧ y < x in ys} - {(x, y) |y. x < y in xs ∧ y < x in ys}) = card(Inv xs ys) + card(before x xs ∩ before x ys) - int(card(after x xs ∩ before x ys))" (is "card(?I ∪ ?B - ?A) = card ?I + card ?b - int(card ?a)") proof- have 1: "?I ∩ ?B = {}" by(auto simp: Inv_def) (metis no_before_inI) have 2: "?A ⊆ ?I ∪ ?B" by(auto simp: Inv_def) have 3: "?A ⊆ ?I" by(auto simp: Inv_def) have "int(card(?I ∪ ?B - ?A)) = int(card ?I + card ?B) - int(card ?A)" using card_mono[OF _ 3] by(simp add: card_Un_disjoint[OF _ _ 1] card_Diff_subset[OF _ 2]) also have "card ?B = card (fst ` ?B)" by(auto simp: card_image inj_on_def) also have "fst ` ?B = ?b" by force also have "card ?A = card (snd ` ?A)" by(auto simp: card_image inj_on_def) also have "snd ` ?A = ?a" by force finally show ?thesis . qed lemma not_before_Cons[simp]: "¬ x < y in y # xs" by (simp add: before_in_def) lemma before_Cons[simp]: "y ∈ set xs ⟹ y ≠ x ⟹ before y (x#xs) = insert x (before y xs)" by(auto simp: before_in_def) lemma card_before_le_index: "card (before x xs) ≤ index xs x" apply(cases "x ∈ set xs") prefer 2 apply (simp add: before_in_def) apply(induction xs) apply (simp add: before_in_def) apply (auto simp: card_insert_if) done lemma config_config_length: "length (fst (config A init qs)) = length init" apply (induct rule: config_induct) by (simp_all) lemma config_config_distinct: shows " distinct (fst (config A init qs)) = distinct init" apply (induct rule: config_induct) by (simp_all add: distinct_step) lemma config_config_set: shows "set (fst (config A init qs)) = set init" apply(induct rule: config_induct) by(simp_all add: set_step) lemma config_config: "set (fst (config A init qs)) = set init ∧ distinct (fst (config A init qs)) = distinct init ∧ length (fst (config A init qs)) = length init" using config_config_distinct config_config_set config_config_length by metis lemma config_dist_perm: "distinct init ⟹ dist_perm (fst (config A init qs)) init" using config_config_distinct config_config_set by metis lemma config_rand_length: "∀x∈set_pmf (config_rand A init qs). length (fst x) = length init" apply (induct rule: config_rand_induct) by (simp_all) lemma config_rand_distinct: shows "∀x ∈ (config_rand A init qs). distinct (fst x) = distinct init" apply (induct rule: config_rand_induct) by (simp_all add: distinct_step) lemma config_rand_set: shows " ∀x ∈ (config_rand A init qs). set (fst x) = set init" apply(induct rule: config_rand_induct) by(simp_all add: set_step) lemma config_rand: "∀x ∈ (config_rand A init qs). set (fst x) = set init ∧ distinct (fst x) = distinct init ∧ length (fst x) = length init" using config_rand_distinct config_rand_set config_rand_length by metis lemma config_rand_dist_perm: "distinct init ⟹ ∀x ∈ (config_rand A init qs). dist_perm (fst x) init" using config_rand_distinct config_rand_set by metis (*fixme start from Inv*) lemma amor_mtf_ub: assumes "x : set ys" "set xs = set ys" shows "int(card(before x xs Int before x ys)) - card(after x xs Int before x ys) ≤ 2 * int(index xs x) - card (before x ys)" (is "?m - ?n ≤ 2 * ?j - ?k") proof- have xxs: "x ∈ set xs" using assms(1,2) by simp let ?bxxs = "before x xs" let ?bxys = "before x ys" let ?axxs = "after x xs" have 0: "?bxxs ∩ ?axxs = {}" by (auto simp: before_in_def) hence 1: "(?bxxs ∩ ?bxys) ∩ (?axxs ∩ ?bxys) = {}" by blast have "(?bxxs ∩ ?bxys) ∪ (?axxs ∩ ?bxys) = ?bxys" using assms(2) before_Un xxs by fastforce hence "?m + ?n = ?k" using card_Un_disjoint[OF _ _ 1] by simp hence "?m - ?n = 2 * ?m - ?k" by arith also have "?m ≤ ?j" using card_before_le_index[of x xs] card_mono[of ?bxxs, OF _ Int_lower1] by(auto intro: order_trans) finally show ?thesis by auto qed locale MTF_Off = fixes as :: "answer list" fixes rs :: "'a list" fixes s0 :: "'a list" assumes dist_s0[simp]: "distinct s0" assumes len_as: "length as = length rs" begin definition mtf_A :: "nat list" where "mtf_A = map fst as" definition sw_A :: "nat list list" where "sw_A = map snd as" fun s_A :: "nat ⇒ 'a list" where "s_A 0 = s0" | "s_A(Suc n) = step (s_A n) (rs!n) (mtf_A!n, sw_A!n)" lemma length_s_A[simp]: "length(s_A n) = length s0" by (induction n) simp_all lemma dist_s_A[simp]: "distinct(s_A n)" by(induction n) (simp_all add: step_def) lemma set_s_A[simp]: "set(s_A n) = set s0" by(induction n) (simp_all add: step_def) fun s_mtf :: "nat ⇒ 'a list" where "s_mtf 0 = s0" | "s_mtf (Suc n) = mtf (rs!n) (s_mtf n)" definition t_mtf :: "nat ⇒ int" where "t_mtf n = index (s_mtf n) (rs!n) + 1" definition T_mtf :: "nat ⇒ int" where "T_mtf n = (∑i<n. t_mtf i)" definition c_A :: "nat ⇒ int" where "c_A n = index (swaps (sw_A!n) (s_A n)) (rs!n) + 1" definition f_A :: "nat ⇒ int" where "f_A n = min (mtf_A!n) (index (swaps (sw_A!n) (s_A n)) (rs!n))" definition p_A :: "nat ⇒ int" where "p_A n = size(sw_A!n)" definition t_A :: "nat ⇒ int" where "t_A n = c_A n + p_A n" definition T_A :: "nat ⇒ int" where "T_A n = (∑i<n. t_A i)" lemma length_s_mtf[simp]: "length(s_mtf n) = length s0" by (induction n) simp_all lemma dist_s_mtf[simp]: "distinct(s_mtf n)" apply(induction n) apply (simp) apply (auto simp: mtf_def index_take set_drop_if_index) apply (metis set_drop_if_index index_take less_Suc_eq_le linear) done lemma set_s_mtf[simp]: "set (s_mtf n) = set s0" by (induction n) (simp_all) lemma dperm_inv: "dist_perm (s_A n) (s_mtf n)" by (metis dist_s_mtf dist_s_A set_s_mtf set_s_A) definition Phi :: "nat ⇒ int" ("Φ") where "Phi n = card(Inv (s_A n) (s_mtf n))" lemma phi0: "Phi 0 = 0" by(simp add: Phi_def) lemma phi_pos: "Phi n ≥ 0" by(simp add: Phi_def) lemma mtf_ub: "t_mtf n + Phi (n+1) - Phi n ≤ 2 * c_A n - 1 + p_A n - f_A n" proof - let ?xs = "s_A n" let ?ys = "s_mtf n" let ?x = "rs!n" let ?xs' = "swaps (sw_A!n) ?xs" let ?ys' = "mtf ?x ?ys" show ?thesis proof cases assume xin: "?x ∈ set ?ys" let ?bb = "before ?x ?xs ∩ before ?x ?ys" let ?ab = "after ?x ?xs ∩ before ?x ?ys" have phi_mtf: "card(Inv ?xs' ?ys') - int(card (Inv ?xs' ?ys)) ≤ 2 * int(index ?xs' ?x) - int(card (before ?x ?ys))" using xin by(simp add: Inv_mtf phi_diff_aux amor_mtf_ub) have phi_sw: "card(Inv ?xs' ?ys) ≤ Phi n + length(sw_A!n)" proof - have "int(card (Inv ?xs' ?ys)) ≤ card(Inv ?xs' ?xs) + int(card(Inv ?xs ?ys))" using card_Inv_tri_ineq[of ?xs' ?xs ?ys] xin by (simp) also have "card(Inv ?xs' ?xs) = card(Inv ?xs ?xs')" by (rule card_Inv_sym) also have "card(Inv ?xs ?xs') ≤ size(sw_A!n)" by (metis card_Inv_swaps_le dist_s_A) finally show ?thesis by(fastforce simp: Phi_def) qed have phi_free: "card(Inv ?xs' ?ys') - Phi (Suc n) = f_A n" using xin by(simp add: Phi_def mtf2_def step_def card_Inv_mtf2 index_less_size_conv f_A_def) show ?thesis using xin phi_sw phi_mtf phi_free card_before[of "s_mtf n"] by(simp add: t_mtf_def c_A_def p_A_def) next assume notin: "?x ∉ set ?ys" have "int (card (Inv ?xs' ?ys)) - card (Inv ?xs ?ys) ≤ card(Inv ?xs ?xs')" using card_Inv_tri_ineq[OF _ dperm_inv, of ?xs' n] swaps_inv[of "sw_A!n" "s_A n"] by(simp add: card_Inv_sym) also have "… ≤ size(sw_A!n)" by(simp add: card_Inv_swaps_le dperm_inv) finally show ?thesis using notin by(simp add: t_mtf_def step_def c_A_def p_A_def f_A_def Phi_def mtf2_def) qed qed theorem Sleator_Tarjan: "T_mtf n ≤ (∑i<n. 2*c_A i + p_A i - f_A i) - n" proof- have "(∑i<n. t_mtf i) ≤ (∑i<n. 2*c_A i - 1 + p_A i - f_A i)" by(rule potential[where p=Phi,OF phi0 phi_pos mtf_ub]) also have "… = (∑i<n. (2*c_A i + p_A i - f_A i) - 1)" by (simp add: algebra_simps) also have "… = (∑i<n. 2*c_A i + p_A i - f_A i) - n" by(simp add: sumr_diff_mult_const2[symmetric]) finally show ?thesis by(simp add: T_mtf_def) qed corollary Sleator_Tarjan': "T_mtf n ≤ 2*T_A n - n" proof - have "T_mtf n ≤ (∑i<n. 2*c_A i + p_A i - f_A i) - n" by (fact Sleator_Tarjan) also have "(∑i<n. 2*c_A i + p_A i - f_A i) ≤ (∑i<n. 2*(c_A i + p_A i))" by(intro sum_mono) (simp add: p_A_def f_A_def) also have "… ≤ 2* T_A n" by (simp add: sum_distrib_left T_A_def t_A_def) finally show "T_mtf n ≤ 2* T_A n - n" by auto qed lemma T_A_nneg: "0 ≤ T_A n" by(auto simp add: sum_nonneg T_A_def t_A_def c_A_def p_A_def) lemma T_mtf_ub: "∀i<n. rs!i ∈ set s0 ⟹ T_mtf n ≤ n * size s0" proof(induction n) case 0 show ?case by(simp add: T_mtf_def) next case (Suc n) thus ?case using index_less_size_conv[of "s_mtf n" "rs!n"] by(simp add: T_mtf_def t_mtf_def less_Suc_eq del: index_less) qed corollary T_mtf_competitive: assumes "s0 ≠ []" and "∀i<n. rs!i ∈ set s0" shows "T_mtf n ≤ (2 - 1/(size s0)) * T_A n" proof cases assume 0: "real_of_int(T_A n) ≤ n * (size s0)" have "T_mtf n ≤ 2 * T_A n - n" proof - have "T_mtf n ≤ (∑i<n. 2*c_A i + p_A i - f_A i) - n" by(rule Sleator_Tarjan) also have "(∑i<n. 2*c_A i + p_A i - f_A i) ≤ (∑i<n. 2*(c_A i + p_A i))" by(intro sum_mono) (simp add: p_A_def f_A_def) also have "… ≤ 2 * T_A n" by (simp add: sum_distrib_left T_A_def t_A_def) finally show ?thesis by simp qed hence "real_of_int(T_mtf n) ≤ 2 * of_int(T_A n) - n" by simp also have "… = 2 * of_int(T_A n) - (n * size s0) / size s0" using assms(1) by simp also have "… ≤ 2 * real_of_int(T_A n) - T_A n / size s0" by(rule diff_left_mono[OF divide_right_mono[OF 0]]) simp also have "… = (2 - 1 / size s0) * T_A n" by algebra finally show ?thesis . next assume 0: "¬ real_of_int(T_A n) ≤ n * (size s0)" have "2 - 1 / size s0 ≥ 1" using assms(1) by (auto simp add: field_simps neq_Nil_conv) have "real_of_int (T_mtf n) ≤ n * size s0" using T_mtf_ub[OF assms(2)] by linarith also have "… < of_int(T_A n)" using 0 by simp also have "… ≤ (2 - 1 / size s0) * T_A n" using assms(1) T_A_nneg[of n] by(auto simp add: mult_le_cancel_right1 field_simps neq_Nil_conv) finally show ?thesis by linarith qed lemma t_A_t: "n < length rs ⟹ t_A n = int (t (s_A n) (rs ! n) (as ! n))" by(simp add: t_A_def t_def c_A_def p_A_def sw_A_def len_as split: prod.split) lemma T_A_eq_lem: "(∑i=0..<length rs. t_A i) = T (s_A 0) (drop 0 rs) (drop 0 as)" proof(induction rule: zero_induct[of _ "size rs"]) case 1 thus ?case by (simp add: len_as) next case (2 n) show ?case proof cases assume "n < length rs" thus ?case using 2 by(simp add: Cons_nth_drop_Suc[symmetric,where i=n] len_as sum.atLeast_Suc_lessThan t_A_t mtf_A_def sw_A_def) next assume "¬ n < length rs" thus ?case by (simp add: len_as) qed qed lemma T_A_eq: "T_A (length rs) = T s0 rs as" using T_A_eq_lem by(simp add: T_A_def atLeast0LessThan) lemma nth_off_MTF: "n < length rs ⟹ off2 MTF s rs ! n = (size(fst s) - 1,[])" by(induction rs arbitrary: s n)(auto simp add: MTF_def nth_Cons' Step_def) lemma t_mtf_MTF: "n < length rs ⟹ t_mtf n = int (t (s_mtf n) (rs ! n) (off MTF s rs ! n))" by(simp add: t_mtf_def t_def nth_off_MTF split: prod.split) lemma mtf_MTF: "n < length rs ⟹ length s = length s0 ⟹ mtf (rs ! n) s = step s (rs ! n) (off MTF s0 rs ! n)" by(auto simp add: nth_off_MTF step_def mtf_eq_mtf2) lemma T_mtf_eq_lem: "(∑i=0..<length rs. t_mtf i) = T (s_mtf 0) (drop 0 rs) (drop 0 (off MTF s0 rs))" proof(induction rule: zero_induct[of _ "size rs"]) case 1 thus ?case by (simp add: len_as) next case (2 n) show ?case proof cases assume "n < length rs" thus ?case using 2 by(simp add: Cons_nth_drop_Suc[symmetric,where i=n] len_as sum.atLeast_Suc_lessThan t_mtf_MTF[where s=s0] mtf_A_def sw_A_def mtf_MTF) next assume "¬ n < length rs" thus ?case by (simp add: len_as) qed qed lemma T_mtf_eq: "T_mtf (length rs) = T_on MTF s0 rs" using T_mtf_eq_lem by(simp add: T_mtf_def atLeast0LessThan) corollary MTF_competitive2: "s0 ≠ [] ⟹ ∀i<length rs. rs!i ∈ set s0 ⟹ T_on MTF s0 rs ≤ (2 - 1/(size s0)) * T s0 rs as" by (metis T_mtf_competitive T_A_eq T_mtf_eq of_int_of_nat_eq) corollary MTF_competitive': "T_on MTF s0 rs ≤ 2 * T s0 rs as" using Sleator_Tarjan'[of "length rs"] T_A_eq T_mtf_eq by auto end theorem compet_MTF: assumes "s0 ≠ []" "distinct s0" "set rs ⊆ set s0" shows "T_on MTF s0 rs ≤ (2 - 1/(size s0)) * T_opt s0 rs" proof- from assms(3) have 1: "∀i < length rs. rs!i ∈ set s0" by auto { fix as :: "answer list" assume len: "length as = length rs" interpret MTF_Off as rs s0 proof qed (auto simp: assms(2) len) from MTF_competitive2[OF assms(1) 1] assms(1) have "T_on MTF s0 rs / (2 - 1 / (length s0)) ≤ of_int(T s0 rs as)" by(simp add: field_simps length_greater_0_conv[symmetric] del: length_greater_0_conv) } hence "T_on MTF s0 rs / (2 - 1/(size s0)) ≤ T_opt s0 rs" apply(simp add: T_opt_def Inf_nat_def) apply(rule LeastI2_wellorder) using length_replicate[of "length rs" undefined] apply fastforce apply auto done thus ?thesis using assms by(simp add: field_simps length_greater_0_conv[symmetric] del: length_greater_0_conv) qed theorem compet_MTF': assumes "distinct s0" shows "T_on MTF s0 rs ≤ (2::real) * T_opt s0 rs" proof- { fix as :: "answer list" assume len: "length as = length rs" interpret MTF_Off as rs s0 proof qed (auto simp: assms(1) len) from MTF_competitive' have "T_on MTF s0 rs / 2 ≤ of_int(T s0 rs as)" by(simp add: field_simps length_greater_0_conv[symmetric] del: length_greater_0_conv) } hence "T_on MTF s0 rs / 2 ≤ T_opt s0 rs" apply(simp add: T_opt_def Inf_nat_def) apply(rule LeastI2_wellorder) using length_replicate[of "length rs" undefined] apply fastforce apply auto done thus ?thesis using assms by(simp add: field_simps length_greater_0_conv[symmetric] del: length_greater_0_conv) qed theorem MTF_is_2_competitive: "compet MTF 2 {s . distinct s}" unfolding compet_def using compet_MTF' by fastforce subsection "Lower Bound for Competitiveness" text‹This result is independent of MTF but is based on the list update problem defined in this theory.› lemma rat_fun_lem: fixes l c :: real assumes [simp]: "F ≠ bot" assumes "0 < l" assumes ev: "eventually (λn. l ≤ f n / g n) F" "eventually (λn. (f n + c) / (g n + d) ≤ u) F" and g: "LIM n F. g n :> at_top" shows "l ≤ u" proof (rule dense_le_bounded[OF ‹0 < l›]) fix x assume x: "0 < x" "x < l" define m where "m = (x - l) / 2" define k where "k = l / (x - m)" have "x = l / k + m" "1 < k" "m < 0" unfolding k_def m_def using x by (auto simp: divide_simps) from ‹1 < k› have "LIM n F. (k - 1) * g n :> at_top" by (intro filterlim_tendsto_pos_mult_at_top[OF tendsto_const _ g]) (simp add: field_simps) then have "eventually (λn. d ≤ (k - 1) * g n) F" by (simp add: filterlim_at_top) moreover have "eventually (λn. 1 ≤ g n) F" "eventually (λn. 1 - d ≤ g n) F" "eventually (λn. c / m - d ≤ g n) F" using g by (auto simp add: filterlim_at_top) ultimately have "eventually (λn. x ≤ u) F" using ev proof eventually_elim fix n assume d: "d ≤ (k - 1) * g n" "1 ≤ g n" "1 - d ≤ g n" "c / m - d ≤ g n" and l: "l ≤ f n / g n" and u: "(f n + c) / (g n + d) ≤ u" from d have "g n + d ≤ k * g n" by (simp add: field_simps) from d have g: "0 < g n" "0 < g n + d" by (auto simp: field_simps) with ‹0 < l› l have "0 < f n" by (auto simp: field_simps intro: mult_pos_pos less_le_trans) note ‹x = l / k + m› also have "l / k ≤ f n / (k * g n)" using l ‹1 < k› by (simp add: field_simps) also have "… ≤ f n / (g n + d)" using d ‹1 < k› ‹0 < f n› by (intro divide_left_mono mult_pos_pos) (auto simp: field_simps) also have "m ≤ c / (g n + d)" using ‹c / m - d ≤ g n› ‹0 < g n› ‹0 < g n + d› ‹m < 0› by (simp add: field_simps) also have "f n / (g n + d) + c / (g n + d) = (f n + c) / (g n + d)" using ‹0 < g n + d› by (auto simp: add_divide_distrib) also note u finally show "x ≤ u" by simp qed then show "x ≤ u" by auto qed lemma compet_lb0: fixes a Aon Aoff cruel defines "f s0 rs == real(T_on Aon s0 rs)" defines "g s0 rs == real(T_off Aoff s0 rs)" assumes "⋀rs s0. size(Aoff s0 rs) = length rs" and "⋀n. cruel n ≠ []" assumes "compet Aon c S0" and "c≥0" and "s0 ∈ S0" and l: "eventually (λn. f s0 (cruel n) / (g s0 (cruel n) + a) ≥ l) sequentially" and g: "LIM n sequentially. g s0 (cruel n) :> at_top" and "l > 0" and "⋀n. static s0 (cruel n)" shows "l ≤ c" proof- let ?h = "λb s0 rs. (f s0 rs - b) / g s0 rs" have g': "LIM n sequentially. g s0 (cruel n) + a :> at_top" using filterlim_tendsto_add_at_top[OF tendsto_const g] by (simp add: ac_simps) from competE[OF assms(5) ‹c≥0› _ ‹s0 ∈ S0›] assms(3) obtain b where "∀rs. static s0 rs ∧ rs ≠ [] ⟶ ?h b s0 rs ≤ c " by (fastforce simp del: neq0_conv simp: neq0_conv[symmetric] field_simps f_def g_def T_off_neq0[of Aoff, OF assms(3)]) hence "∀n. (?h b s0 o cruel) n ≤ c" using assms(4,11) by simp with rat_fun_lem[OF sequentially_bot ‹l>0› _ _ g', of "f s0 o cruel" "-b" "- a" c] assms(7) l show "l ≤ c" by (auto) qed text ‹Sorting› fun ins_sws where "ins_sws k x [] = []" | "ins_sws k x (y#ys) = (if k x ≤ k y then [] else map Suc (ins_sws k x ys) @ [0])" fun sort_sws where "sort_sws k [] = []" | "sort_sws k (x#xs) = ins_sws k x (sort_key k xs) @ map Suc (sort_sws k xs)" lemma length_ins_sws: "length(ins_sws k x xs) ≤ length xs" by(induction xs) auto lemma length_sort_sws_le: "length(sort_sws k xs) ≤ length xs ^ 2" proof(induction xs) case (Cons x xs) thus ?case using length_ins_sws[of k x "sort_key k xs"] by (simp add: numeral_eq_Suc) qed simp lemma swaps_ins_sws: "swaps (ins_sws k x xs) (x#xs) = insort_key k x xs" by(induction xs)(auto simp: swap_def[of 0]) lemma swaps_sort_sws[simp]: "swaps (sort_sws k xs) xs = sort_key k xs" by(induction xs)(auto simp: swaps_ins_sws) text‹The cruel adversary:› fun cruel :: "('a,'is) alg_on ⇒ 'a state * 'is ⇒ nat ⇒ 'a list" where "cruel A s 0 = []" | "cruel A s (Suc n) = last (fst s) # cruel A (Step A s (last (fst s))) n" definition adv :: "('a,'is) alg_on ⇒ ('a::linorder) alg_off" where "adv A s rs = (if rs=[] then [] else let crs = cruel A (Step A (s, fst A s) (last s)) (size rs - 1) in (0,sort_sws (λx. size rs - 1 - count_list crs x) s) # replicate (size rs - 1) (0,[]))" lemma set_cruel: "s ≠ [] ⟹ set(cruel A (s,is) n) ⊆ set s" apply(induction n arbitrary: s "is") apply(auto simp: step_def Step_def split: prod.split) by (metis empty_iff swaps_inv last_in_set list.set(1) rev_subsetD set_mtf2) lemma static_cruel: "s ≠ [] ⟹ static s (cruel A (s,is) n)" by(simp add: set_cruel static_def) (* Do not convert into structured proof - eta conversion screws it up! *) lemma T_cruel: "s ≠ [] ⟹ distinct s ⟹ T s (cruel A (s,is) n) (off2 A (s,is) (cruel A (s,is) n)) ≥ n*(length s)" apply(induction n arbitrary: s "is") apply(simp) apply(erule_tac x = "fst(Step A (s, is) (last s))" in meta_allE) apply(erule_tac x = "snd(Step A (s, is) (last s))" in meta_allE) apply(frule_tac sws = "snd(fst(snd A (s,is) (last s)))" in index_swaps_last_size) apply(simp add: distinct_step t_def split_def Step_def length_greater_0_conv[symmetric] del: length_greater_0_conv) done lemma length_cruel[simp]: "length (cruel A s n) = n" by (induction n arbitrary: s) (auto) lemma t_sort_sws: "t s r (mf, sort_sws k s) ≤ size s ^ 2 + size s + 1" using length_sort_sws_le[of k s] index_le_size[of "sort_key k s" r] by (simp add: t_def add_mono index_le_size algebra_simps) lemma T_noop: "n = length rs ⟹ T s rs (replicate n (0, [])) = (∑r←rs. index s r + 1)" by(induction rs arbitrary: s n)(auto simp: t_def step_def) lemma sorted_asc: "j≤i ⟹ i<size ss ⟹ ∀x ∈ set ss. ∀y ∈ set ss. k(x) ≤ k(y) ⟶ f y ≤ f x ⟹ sorted (map k ss) ⟹ f (ss ! i) ≤ f (ss ! j)" by (auto simp: sorted_iff_nth_mono) lemma sorted_weighted_gauss_Ico_div2: fixes f :: "nat ⇒ nat" assumes "⋀i j. i ≤ j ⟹ j < n ⟹ f i ≥ f j" shows "(∑i=0..<n. (i + 1) * f i) ≤ (n + 1) * sum f {0..<n} div 2" proof (cases n) case 0 then show ?thesis by simp next case (Suc n) with assms have "Suc n * (∑i=0..<Suc n. Suc i * f i) ≤ (∑i=0..<Suc n. Suc i) * sum f {0..<Suc n}" by (intro Chebyshev_sum_upper_nat [of "Suc n" Suc f]) auto then have "Suc n * (2 * (∑i=0..n. Suc i * f i)) ≤ 2 * (∑i=0..n. Suc i) * sum f {0..n}" by (simp add: atLeastLessThanSuc_atLeastAtMost) also have "2 * (∑i=0..n. Suc i) = Suc n * (n + 2)" using arith_series_nat [of 1 1 n] by simp finally have "2 * (∑i=0..n. Suc i * f i) ≤ (n + 2) * sum f {0..n}" by (simp only: ac_simps Suc_mult_le_cancel1) with Suc show ?thesis by (simp only: atLeastLessThanSuc_atLeastAtMost) simp qed lemma T_adv: assumes "l ≠ 0" shows "T_off (adv A) [0..<l] (cruel A ([0..<l],fst A [0..<l]) (Suc n)) ≤ l⇧^{2}+ l + 1 + (l + 1) * n div 2" (is "?l ≤ ?r") proof- let ?s = "[0..<l]" let ?r = "last ?s" let ?S' = "Step A (?s,fst A ?s) ?r" let ?s' = "fst ?S'" let ?cr = "cruel A ?S' n" let ?c = "count_list ?cr" let ?k = "λx. n - ?c x" let ?sort = "sort_key ?k ?s" have 1: "set ?s' = {0..<l}" by(simp add: set_step Step_def split: prod.split) have 3: "⋀x. x < l ⟹ ?c x ≤ n" by(simp) (metis count_le_length length_cruel) have "?l = t ?s (last ?s) (0, sort_sws ?k ?s) + (∑x∈set ?s'. ?c x * (index ?sort x + 1))" using assms apply(simp add: adv_def T_noop sum_list_map_eq_sum_count2[OF set_cruel] Step_def split: prod.split) apply(subst (3) step_def) apply(simp) done also have "(∑x∈set ?s'. ?c x * (index ?sort x + 1)) = (∑x∈{0..<l}. ?c x * (index ?sort x + 1))" by (simp add: 1) also have "… = (∑x∈{0..<l}. ?c (?sort ! x) * (index ?sort (?sort ! x) + 1))" by(rule sum.reindex_bij_betw[where ?h = "nth ?sort", symmetric]) (simp add: bij_betw_imageI inj_on_nth nth_image) also have "… = (∑x∈{0..<l}. ?c (?sort ! x) * (x+1))" by(simp add: index_nth_id) also have "… ≤ (∑x∈{0..<l}. (x+1) * ?c (?sort ! x))" by (simp add: algebra_simps) also(ord_eq_le_subst) have "… ≤ (l+1) * (∑x∈{0..<l}. ?c (?sort ! x)) div 2" apply(rule sorted_weighted_gauss_Ico_div2) apply(erule sorted_asc[where k = "λx. n - count_list (cruel A ?S' n) x"]) apply(auto simp add: index_nth_id dest!: 3) using assms [[linarith_split_limit = 20]] by simp also have "(∑x∈{0..<l}. ?c (?sort ! x)) = (∑x∈{0..<l}. ?c (?sort ! (index ?sort x)))" by(rule sum.reindex_bij_betw[where ?h = "index ?sort", symmetric]) (simp add: bij_betw_imageI inj_on_index2 index_image) also have "… = (∑x∈{0..<l}. ?c x)" by(simp) also have "… = length ?cr" using set_cruel[of ?s' A _ n] assms 1 by(auto simp add: sum_count_set Step_def split: prod.split) also have "… = n" by simp also have "t ?s (last ?s) (0, sort_sws ?k ?s) ≤ (length ?s)^2 + length ?s + 1" by(rule t_sort_sws) also have "… = l^2 + l + 1" by simp finally show "?l ≤ l⇧^{2}+ l + 1 + (l + 1) * n div 2" by auto qed text ‹The main theorem:› theorem compet_lb2: assumes "compet A c {xs::nat list. size xs = l}" and "l ≠ 0" and "c ≥ 0" shows "c ≥ 2*l/(l+1)" proof (rule compet_lb0[OF _ _ assms(1) ‹c≥0›]) let ?S0 = "{xs::nat list. size xs = l}" let ?s0 = "[0..<l]" let ?cruel = "cruel A (?s0,fst A ?s0) o Suc" let ?on = "λn. T_on A ?s0 (?cruel n)" let ?off = "λn. T_off (adv A) ?s0 (?cruel n)" show "⋀s0 rs. length (adv A s0 rs) = length rs" by(simp add: adv_def) show "⋀n. ?cruel n ≠ []" by auto show "?s0 ∈ ?S0" by simp { fix Z::real and n::nat assume "n ≥ nat(ceiling Z)" have "?off n ≥ length(?cruel n)" by(rule T_ge_len) (simp add: adv_def) hence "?off n > n" by simp hence "Z ≤ ?off n" using ‹n ≥ nat(ceiling Z)› by linarith } thus "LIM n sequentially. real (?off n) :> at_top" by(auto simp only: filterlim_at_top eventually_sequentially) let ?a = "- (l^2 + l + 1)" { fix n assume "n ≥ l^2 + l + 1" have "2*l/(l+1) = 2*l*(n+1) / ((l+1)*(n+1))" by (simp del: One_nat_def) also have "… = 2*real(l*(n+1)) / ((l+1)*(n+1))" by simp also have "l * (n+1) ≤ ?on n" using T_cruel[of ?s0 "Suc n"] ‹l ≠ 0› by (simp add: ac_simps) also have "2*real(?on n) / ((l+1)*(n+1)) ≤ 2*real(?on n)/(2*(?off n + ?a))" proof - have 0: "2*real(?on n) ≥ 0" by simp have 1: "0 < real ((l + 1) * (n + 1))" by (simp del: of_nat_Suc) have "?off n ≥ length(?cruel n)" by(rule T_ge_len) (simp add: adv_def) hence "?off n > n" by simp hence "?off n + ?a > 0" using ‹n ≥ l^2 + l + 1› by linarith hence 2: "real_of_int(2*(?off n + ?a)) > 0" by(simp only: of_int_0_less_iff zero_less_mult_iff zero_less_numeral simp_thms) have "?off n + ?a ≤ (l+1)*(n) div 2" using T_adv[OF ‹l≠0›, of A n] by (simp only: o_apply of_nat_add of_nat_le_iff) also have "… ≤ (l+1)*(n+1) div 2" by (simp) finally have "2*(?off n + ?a) ≤ (l+1)*(n+1)" by (simp add: zdiv_int) hence "of_int(2*(?off n + ?a)) ≤ real((l+1)*(n+1))" by (simp only: of_int_le_iff) from divide_left_mono[OF this 0 mult_pos_pos[OF 1 2]] show ?thesis . qed also have "… = ?on n / (?off n + ?a)" by (simp del: distrib_left_numeral One_nat_def cruel.simps) finally have "2*l/(l+1) ≤ ?on n / (real (?off n) + ?a)" by (auto simp: divide_right_mono) } thus "eventually (λn. (2 * l) / (l + 1) ≤ ?on n / (real(?off n) + ?a)) sequentially" by(auto simp add: filterlim_at_top eventually_sequentially) show "0 < 2*l / (l+1)" using ‹l ≠ 0› by(simp) show "⋀n. static ?s0 (?cruel n)" using ‹l ≠ 0› by(simp add: static_cruel del: cruel.simps) qed end