explicit type class for discrete linordered semidoms
(* Author: Tobias Nipkow, TU Muenchen *)section \<open>Sum and product over lists\<close>theory Groups_Listimports Listbeginlocale monoid_list = monoidbegindefinition F :: "'a list \<Rightarrow> 'a"where eq_foldr [code]: "F xs = foldr f xs \<^bold>1"lemma Nil [simp]: "F [] = \<^bold>1" by (simp add: eq_foldr)lemma Cons [simp]: "F (x # xs) = x \<^bold>* F xs" by (simp add: eq_foldr)lemma append [simp]: "F (xs @ ys) = F xs \<^bold>* F ys" by (induct xs) (simp_all add: assoc)endlocale comm_monoid_list = comm_monoid + monoid_listbeginlemma rev [simp]: "F (rev xs) = F xs" by (simp add: eq_foldr foldr_fold fold_rev fun_eq_iff assoc left_commute)endlocale comm_monoid_list_set = list: comm_monoid_list + set: comm_monoid_setbeginlemma distinct_set_conv_list: "distinct xs \<Longrightarrow> set.F g (set xs) = list.F (map g xs)" by (induct xs) simp_alllemma set_conv_list [code]: "set.F g (set xs) = list.F (map g (remdups xs))" by (simp add: distinct_set_conv_list [symmetric])endsubsection \<open>List summation\<close>context monoid_addbeginsublocale sum_list: monoid_list plus 0defines sum_list = sum_list.F ..endcontext comm_monoid_addbeginsublocale sum_list: comm_monoid_list plus 0rewrites "monoid_list.F plus 0 = sum_list"proof - show "comm_monoid_list plus 0" .. then interpret sum_list: comm_monoid_list plus 0 . from sum_list_def show "monoid_list.F plus 0 = sum_list" by simpqedsublocale sum: comm_monoid_list_set plus 0rewrites "monoid_list.F plus 0 = sum_list" and "comm_monoid_set.F plus 0 = sum"proof - show "comm_monoid_list_set plus 0" .. then interpret sum: comm_monoid_list_set plus 0 . from sum_list_def show "monoid_list.F plus 0 = sum_list" by simp from sum_def show "comm_monoid_set.F plus 0 = sum" by (auto intro: sym)qedendtext \<open>Some syntactic sugar for summing a function over a list:\<close>syntax (ASCII) "_sum_list" :: "pttrn => 'a list => 'b => 'b" ("(3SUM _<-_. _)" [0, 51, 10] 10)syntax "_sum_list" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)translations \<comment> \<open>Beware of argument permutation!\<close> "\<Sum>x\<leftarrow>xs. b" == "CONST sum_list (CONST map (\<lambda>x. b) xs)"context includes lifting_syntaxbeginlemma sum_list_transfer [transfer_rule]: "(list_all2 A ===> A) sum_list sum_list" if [transfer_rule]: "A 0 0" "(A ===> A ===> A) (+) (+)" unfolding sum_list.eq_foldr [abs_def] by transfer_proverendtext \<open>TODO duplicates\<close>lemmas sum_list_simps = sum_list.Nil sum_list.Conslemmas sum_list_append = sum_list.appendlemmas sum_list_rev = sum_list.revlemma (in monoid_add) fold_plus_sum_list_rev: "fold plus xs = plus (sum_list (rev xs))"proof fix x have "fold plus xs x = sum_list (rev xs @ [x])" by (simp add: foldr_conv_fold sum_list.eq_foldr) also have "\<dots> = sum_list (rev xs) + x" by simp finally show "fold plus xs x = sum_list (rev xs) + x" .qedlemma (in comm_monoid_add) sum_list_map_remove1: "x \<in> set xs \<Longrightarrow> sum_list (map f xs) = f x + sum_list (map f (remove1 x xs))" by (induct xs) (auto simp add: ac_simps)lemma (in monoid_add) size_list_conv_sum_list: "size_list f xs = sum_list (map f xs) + size xs" by (induct xs) autolemma (in monoid_add) length_concat: "length (concat xss) = sum_list (map length xss)" by (induct xss) simp_alllemma (in monoid_add) length_product_lists: "length (product_lists xss) = foldr (*) (map length xss) 1"proof (induct xss) case (Cons xs xss) then show ?case by (induct xs) (auto simp: length_concat o_def)qed simplemma (in monoid_add) sum_list_map_filter: assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0" shows "sum_list (map f (filter P xs)) = sum_list (map f xs)" using assms by (induct xs) autolemma sum_list_filter_le_nat: fixes f :: "'a \<Rightarrow> nat" shows "sum_list (map f (filter P xs)) \<le> sum_list (map f xs)"by(induction xs; simp)lemma (in comm_monoid_add) distinct_sum_list_conv_Sum: "distinct xs \<Longrightarrow> sum_list xs = Sum (set xs)" by (induct xs) simp_alllemma sum_list_upt[simp]: "m \<le> n \<Longrightarrow> sum_list [m..<n] = \<Sum> {m..<n}"by(simp add: distinct_sum_list_conv_Sum)context ordered_comm_monoid_addbeginlemma sum_list_nonneg: "(\<And>x. x \<in> set xs \<Longrightarrow> 0 \<le> x) \<Longrightarrow> 0 \<le> sum_list xs"by (induction xs) autolemma sum_list_nonpos: "(\<And>x. x \<in> set xs \<Longrightarrow> x \<le> 0) \<Longrightarrow> sum_list xs \<le> 0"by (induction xs) (auto simp: add_nonpos_nonpos)lemma sum_list_nonneg_eq_0_iff: "(\<And>x. x \<in> set xs \<Longrightarrow> 0 \<le> x) \<Longrightarrow> sum_list xs = 0 \<longleftrightarrow> (\<forall>x\<in> set xs. x = 0)"by (induction xs) (simp_all add: add_nonneg_eq_0_iff sum_list_nonneg)endcontext canonically_ordered_monoid_addbeginlemma sum_list_eq_0_iff [simp]: "sum_list ns = 0 \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"by (simp add: sum_list_nonneg_eq_0_iff)lemma member_le_sum_list: "x \<in> set xs \<Longrightarrow> x \<le> sum_list xs"by (induction xs) (auto simp: add_increasing add_increasing2)lemma elem_le_sum_list: "k < size ns \<Longrightarrow> ns ! k \<le> sum_list (ns)"by (rule member_le_sum_list) simpendlemma (in ordered_cancel_comm_monoid_diff) sum_list_update: "k < size xs \<Longrightarrow> sum_list (xs[k := x]) = sum_list xs + x - xs ! k"apply(induction xs arbitrary:k) apply (auto simp: add_ac split: nat.split)apply(drule elem_le_sum_list)by (simp add: local.add_diff_assoc local.add_increasing)lemma (in monoid_add) sum_list_triv: "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r" by (induct xs) (simp_all add: distrib_right)lemma (in monoid_add) sum_list_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0" by (induct xs) (simp_all add: distrib_right)text\<open>For non-Abelian groups \<open>xs\<close> needs to be reversed on one side:\<close>lemma (in ab_group_add) uminus_sum_list_map: "- sum_list (map f xs) = sum_list (map (uminus \<circ> f) xs)" by (induct xs) simp_alllemma (in comm_monoid_add) sum_list_addf: "(\<Sum>x\<leftarrow>xs. f x + g x) = sum_list (map f xs) + sum_list (map g xs)" by (induct xs) (simp_all add: algebra_simps)lemma (in ab_group_add) sum_list_subtractf: "(\<Sum>x\<leftarrow>xs. f x - g x) = sum_list (map f xs) - sum_list (map g xs)" by (induct xs) (simp_all add: algebra_simps)lemma (in semiring_0) sum_list_const_mult: "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)" by (induct xs) (simp_all add: algebra_simps)lemma (in semiring_0) sum_list_mult_const: "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c" by (induct xs) (simp_all add: algebra_simps)lemma (in ordered_ab_group_add_abs) sum_list_abs: "\<bar>sum_list xs\<bar> \<le> sum_list (map abs xs)" by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])lemma sum_list_mono: fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}" shows "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sum>x\<leftarrow>xs. f x) \<le> (\<Sum>x\<leftarrow>xs. g x)"by (induct xs) (simp, simp add: add_mono)lemma sum_list_strict_mono: fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, strict_ordered_ab_semigroup_add}" shows "\<lbrakk> xs \<noteq> []; \<And>x. x \<in> set xs \<Longrightarrow> f x < g x \<rbrakk> \<Longrightarrow> sum_list (map f xs) < sum_list (map g xs)"proof (induction xs) case Nil thus ?case by simpnext case C: (Cons _ xs) show ?case proof (cases xs) case Nil thus ?thesis using C.prems by simp next case Cons thus ?thesis using C by(simp add: add_strict_mono) qedqedtext \<open>A much more general version of this monotonicity lemmacan be formulated with multisets and the multiset order\<close>lemma sum_list_mono2: fixes xs :: "'a ::ordered_comm_monoid_add list"shows "\<lbrakk> length xs = length ys; \<And>i. i < length xs \<longrightarrow> xs!i \<le> ys!i \<rbrakk> \<Longrightarrow> sum_list xs \<le> sum_list ys"apply(induction xs ys rule: list_induct2)by(auto simp: nth_Cons' less_Suc_eq_0_disj imp_ex add_mono)lemma (in monoid_add) sum_list_distinct_conv_sum_set: "distinct xs \<Longrightarrow> sum_list (map f xs) = sum f (set xs)" by (induct xs) simp_alllemma (in monoid_add) interv_sum_list_conv_sum_set_nat: "sum_list (map f [m..<n]) = sum f (set [m..<n])" by (simp add: sum_list_distinct_conv_sum_set)lemma (in monoid_add) interv_sum_list_conv_sum_set_int: "sum_list (map f [k..l]) = sum f (set [k..l])" by (simp add: sum_list_distinct_conv_sum_set)text \<open>General equivalence between \<^const>\<open>sum_list\<close> and \<^const>\<open>sum\<close>\<close>lemma (in monoid_add) sum_list_sum_nth: "sum_list xs = (\<Sum> i = 0 ..< length xs. xs ! i)" using interv_sum_list_conv_sum_set_nat [of "(!) xs" 0 "length xs"] by (simp add: map_nth)lemma sum_list_map_eq_sum_count: "sum_list (map f xs) = sum (\<lambda>x. count_list xs x * f x) (set xs)"proof(induction xs) case (Cons x xs) show ?case (is "?l = ?r") proof cases assume "x \<in> set xs" have "?l = f x + (\<Sum>x\<in>set xs. count_list xs x * f x)" by (simp add: Cons.IH) also have "set xs = insert x (set xs - {x})" using \<open>x \<in> set xs\<close>by blast also have "f x + (\<Sum>x\<in>insert x (set xs - {x}). count_list xs x * f x) = ?r" by (simp add: sum.insert_remove eq_commute) finally show ?thesis . next assume "x \<notin> set xs" hence "\<And>xa. xa \<in> set xs \<Longrightarrow> x \<noteq> xa" by blast thus ?thesis by (simp add: Cons.IH \<open>x \<notin> set xs\<close>) qedqed simplemma sum_list_map_eq_sum_count2:assumes "set xs \<subseteq> X" "finite X"shows "sum_list (map f xs) = sum (\<lambda>x. count_list xs x * f x) X"proof- let ?F = "\<lambda>x. count_list xs x * f x" have "sum ?F X = sum ?F (set xs \<union> (X - set xs))" using Un_absorb1[OF assms(1)] by(simp) also have "\<dots> = sum ?F (set xs)" using assms(2) by(simp add: sum.union_disjoint[OF _ _ Diff_disjoint] del: Un_Diff_cancel) finally show ?thesis by(simp add:sum_list_map_eq_sum_count)qedlemma sum_list_replicate: "sum_list (replicate n c) = of_nat n * c"by(induction n)(auto simp add: distrib_right)lemma sum_list_nonneg: "(\<And>x. x \<in> set xs \<Longrightarrow> (x :: 'a :: ordered_comm_monoid_add) \<ge> 0) \<Longrightarrow> sum_list xs \<ge> 0" by (induction xs) simp_alllemma sum_list_Suc: "sum_list (map (\<lambda>x. Suc(f x)) xs) = sum_list (map f xs) + length xs"by(induction xs; simp)lemma (in monoid_add) sum_list_map_filter': "sum_list (map f (filter P xs)) = sum_list (map (\<lambda>x. if P x then f x else 0) xs)" by (induction xs) simp_alltext \<open>Summation of a strictly ascending sequence with length \<open>n\<close> can be upper-bounded by summation over \<open>{0..<n}\<close>.\<close>lemma sorted_wrt_less_sum_mono_lowerbound: fixes f :: "nat \<Rightarrow> ('b::ordered_comm_monoid_add)" assumes mono: "\<And>x y. x\<le>y \<Longrightarrow> f x \<le> f y" shows "sorted_wrt (<) ns \<Longrightarrow> (\<Sum>i\<in>{0..<length ns}. f i) \<le> (\<Sum>i\<leftarrow>ns. f i)"proof (induction ns rule: rev_induct) case Nil then show ?case by simpnext case (snoc n ns) have "sum f {0..<length (ns @ [n])} = sum f {0..<length ns} + f (length ns)" by simp also have "sum f {0..<length ns} \<le> sum_list (map f ns)" using snoc by (auto simp: sorted_wrt_append) also have "length ns \<le> n" using sorted_wrt_less_idx[OF snoc.prems(1), of "length ns"] by auto finally have "sum f {0..<length (ns @ [n])} \<le> sum_list (map f ns) + f n" using mono add_mono by blast thus ?case by simpqedsubsection \<open>Horner sums\<close>context comm_semiring_0begindefinition horner_sum :: \<open>('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'a\<close> where horner_sum_foldr: \<open>horner_sum f a xs = foldr (\<lambda>x b. f x + a * b) xs 0\<close>lemma horner_sum_simps [simp]: \<open>horner_sum f a [] = 0\<close> \<open>horner_sum f a (x # xs) = f x + a * horner_sum f a xs\<close> by (simp_all add: horner_sum_foldr)lemma horner_sum_eq_sum_funpow: \<open>horner_sum f a xs = (\<Sum>n = 0..<length xs. ((*) a ^^ n) (f (xs ! n)))\<close>proof (induction xs) case Nil then show ?case by simpnext case (Cons x xs) then show ?case by (simp add: sum.atLeast0_lessThan_Suc_shift sum_distrib_left del: sum.op_ivl_Suc)qedendcontext includes lifting_syntaxbeginlemma horner_sum_transfer [transfer_rule]: \<open>((B ===> A) ===> A ===> list_all2 B ===> A) horner_sum horner_sum\<close> if [transfer_rule]: \<open>A 0 0\<close> and [transfer_rule]: \<open>(A ===> A ===> A) (+) (+)\<close> and [transfer_rule]: \<open>(A ===> A ===> A) (*) (*)\<close> by (unfold horner_sum_foldr) transfer_proverendcontext comm_semiring_1beginlemma horner_sum_eq_sum: \<open>horner_sum f a xs = (\<Sum>n = 0..<length xs. f (xs ! n) * a ^ n)\<close>proof - have \<open>(*) a ^^ n = (*) (a ^ n)\<close> for n by (induction n) (simp_all add: ac_simps) then show ?thesis by (simp add: horner_sum_eq_sum_funpow ac_simps)qedlemma horner_sum_append: \<open>horner_sum f a (xs @ ys) = horner_sum f a xs + a ^ length xs * horner_sum f a ys\<close> using sum.atLeastLessThan_shift_bounds [of _ 0 \<open>length xs\<close> \<open>length ys\<close>] atLeastLessThan_add_Un [of 0 \<open>length xs\<close> \<open>length ys\<close>] by (simp add: horner_sum_eq_sum sum_distrib_left sum.union_disjoint ac_simps nth_append power_add)endcontext linordered_semidombeginlemma horner_sum_nonnegative: \<open>0 \<le> horner_sum of_bool 2 bs\<close> by (induction bs) simp_allendcontext discrete_linordered_semidombeginlemma horner_sum_bound: \<open>horner_sum of_bool 2 bs < 2 ^ length bs\<close>proof (induction bs) case Nil then show ?case by simpnext case (Cons b bs) moreover define a where \<open>a = 2 ^ length bs - horner_sum of_bool 2 bs\<close> ultimately have *: \<open>2 ^ length bs = horner_sum of_bool 2 bs + a\<close> by simp have \<open>0 < a\<close> using Cons * by simp moreover have \<open>1 \<le> a\<close> using \<open>0 < a\<close> by (simp add: less_eq_iff_succ_less) ultimately have \<open>0 + 1 < a + a\<close> by (rule add_less_le_mono) then have \<open>1 < a * 2\<close> by (simp add: mult_2_right) with Cons show ?case by (simp add: * algebra_simps)qedendlemma nat_horner_sum [simp]: \<open>nat (horner_sum of_bool 2 bs) = horner_sum of_bool 2 bs\<close> by (induction bs) (auto simp add: nat_add_distrib horner_sum_nonnegative)context discrete_linordered_semidombeginlemma horner_sum_less_eq_iff_lexordp_eq: \<open>horner_sum of_bool 2 bs \<le> horner_sum of_bool 2 cs \<longleftrightarrow> lexordp_eq (rev bs) (rev cs)\<close> if \<open>length bs = length cs\<close>proof - have \<open>horner_sum of_bool 2 (rev bs) \<le> horner_sum of_bool 2 (rev cs) \<longleftrightarrow> lexordp_eq bs cs\<close> if \<open>length bs = length cs\<close> for bs cs using that proof (induction bs cs rule: list_induct2) case Nil then show ?case by simp next case (Cons b bs c cs) with horner_sum_nonnegative [of \<open>rev bs\<close>] horner_sum_nonnegative [of \<open>rev cs\<close>] horner_sum_bound [of \<open>rev bs\<close>] horner_sum_bound [of \<open>rev cs\<close>] show ?case by (auto simp add: horner_sum_append not_le Cons intro: add_strict_increasing2 add_increasing) qed from that this [of \<open>rev bs\<close> \<open>rev cs\<close>] show ?thesis by simpqedlemma horner_sum_less_iff_lexordp: \<open>horner_sum of_bool 2 bs < horner_sum of_bool 2 cs \<longleftrightarrow> ord_class.lexordp (rev bs) (rev cs)\<close> if \<open>length bs = length cs\<close>proof - have \<open>horner_sum of_bool 2 (rev bs) < horner_sum of_bool 2 (rev cs) \<longleftrightarrow> ord_class.lexordp bs cs\<close> if \<open>length bs = length cs\<close> for bs cs using that proof (induction bs cs rule: list_induct2) case Nil then show ?case by simp next case (Cons b bs c cs) with horner_sum_nonnegative [of \<open>rev bs\<close>] horner_sum_nonnegative [of \<open>rev cs\<close>] horner_sum_bound [of \<open>rev bs\<close>] horner_sum_bound [of \<open>rev cs\<close>] show ?case by (auto simp add: horner_sum_append not_less Cons intro: add_strict_increasing2 add_increasing) qed from that this [of \<open>rev bs\<close> \<open>rev cs\<close>] show ?thesis by simpqedendsubsection \<open>Further facts about \<^const>\<open>List.n_lists\<close>\<close>lemma length_n_lists: "length (List.n_lists n xs) = length xs ^ n" by (induct n) (auto simp add: comp_def length_concat sum_list_triv)lemma distinct_n_lists: assumes "distinct xs" shows "distinct (List.n_lists n xs)"proof (rule card_distinct) from assms have card_length: "card (set xs) = length xs" by (rule distinct_card) have "card (set (List.n_lists n xs)) = card (set xs) ^ n" proof (induct n) case 0 then show ?case by simp next case (Suc n) moreover have "card (\<Union>ys\<in>set (List.n_lists n xs). (\<lambda>y. y # ys) ` set xs) = (\<Sum>ys\<in>set (List.n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))" by (rule card_UN_disjoint) auto moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)" by (rule card_image) (simp add: inj_on_def) ultimately show ?case by auto qed also have "\<dots> = length xs ^ n" by (simp add: card_length) finally show "card (set (List.n_lists n xs)) = length (List.n_lists n xs)" by (simp add: length_n_lists)qedsubsection \<open>Tools setup\<close>lemmas sum_code = sum.set_conv_listlemma sum_set_upto_conv_sum_list_int [code_unfold]: "sum f (set [i..j::int]) = sum_list (map f [i..j])" by (simp add: interv_sum_list_conv_sum_set_int)lemma sum_set_upt_conv_sum_list_nat [code_unfold]: "sum f (set [m..<n]) = sum_list (map f [m..<n])" by (simp add: interv_sum_list_conv_sum_set_nat)subsection \<open>List product\<close>context monoid_multbeginsublocale prod_list: monoid_list times 1defines prod_list = prod_list.F ..endcontext comm_monoid_multbeginsublocale prod_list: comm_monoid_list times 1rewrites "monoid_list.F times 1 = prod_list"proof - show "comm_monoid_list times 1" .. then interpret prod_list: comm_monoid_list times 1 . from prod_list_def show "monoid_list.F times 1 = prod_list" by simpqedsublocale prod: comm_monoid_list_set times 1rewrites "monoid_list.F times 1 = prod_list" and "comm_monoid_set.F times 1 = prod"proof - show "comm_monoid_list_set times 1" .. then interpret prod: comm_monoid_list_set times 1 . from prod_list_def show "monoid_list.F times 1 = prod_list" by simp from prod_def show "comm_monoid_set.F times 1 = prod" by (auto intro: sym)qedendtext \<open>Some syntactic sugar:\<close>syntax (ASCII) "_prod_list" :: "pttrn => 'a list => 'b => 'b" ("(3PROD _<-_. _)" [0, 51, 10] 10)syntax "_prod_list" :: "pttrn => 'a list => 'b => 'b" ("(3\<Prod>_\<leftarrow>_. _)" [0, 51, 10] 10)translations \<comment> \<open>Beware of argument permutation!\<close> "\<Prod>x\<leftarrow>xs. b" \<rightleftharpoons> "CONST prod_list (CONST map (\<lambda>x. b) xs)"context includes lifting_syntaxbeginlemma prod_list_transfer [transfer_rule]: "(list_all2 A ===> A) prod_list prod_list" if [transfer_rule]: "A 1 1" "(A ===> A ===> A) (*) (*)" unfolding prod_list.eq_foldr [abs_def] by transfer_proverendlemma prod_list_zero_iff: "prod_list xs = 0 \<longleftrightarrow> (0 :: 'a :: {semiring_no_zero_divisors, semiring_1}) \<in> set xs" by (induction xs) simp_allend