(* Author: Tobias Nipkow, TU Muenchen *) section ‹Sum and product over lists› theory Groups_List imports List begin locale monoid_list = monoid begin definition F :: "'a list ⇒ 'a" where eq_foldr [code]: "F xs = foldr f xs ❙1" lemma Nil [simp]: "F [] = ❙1" by (simp add: eq_foldr) lemma Cons [simp]: "F (x # xs) = x ❙*F xs" by (simp add: eq_foldr) lemma append [simp]: "F (xs @ ys) = F xs ❙*F ys" by (induct xs) (simp_all add: assoc) end locale comm_monoid_list = comm_monoid + monoid_list begin lemma rev [simp]: "F (rev xs) = F xs" by (simp add: eq_foldr foldr_fold fold_rev fun_eq_iff assoc left_commute) end locale comm_monoid_list_set = list: comm_monoid_list + set: comm_monoid_set begin lemma distinct_set_conv_list: "distinct xs ⟹ set.F g (set xs) = list.F (map g xs)" by (induct xs) simp_all lemma set_conv_list [code]: "set.F g (set xs) = list.F (map g (remdups xs))" by (simp add: distinct_set_conv_list [symmetric]) lemma list_conv_set_nth: "list.F xs = set.F (λi. xs ! i) {0..<length xs}" proof - have "xs = map (λi. xs ! i) [0..<length xs]" by (simp add: map_nth) also have "list.F … = set.F (λi. xs ! i) {0..<length xs}" by (subst distinct_set_conv_list [symmetric]) auto finally show ?thesis . qed end subsection ‹List summation› context monoid_add begin sublocale sum_list: monoid_list plus 0 defines sum_list = sum_list.F .. end context comm_monoid_add begin sublocale sum_list: comm_monoid_list plus 0 rewrites "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 simp qed sublocale sum: comm_monoid_list_set plus 0 rewrites "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) qed end text ‹Some syntactic sugar for summing a function over a list:› syntax (ASCII) "_sum_list" :: "pttrn => 'a list => 'b => 'b" ("(3SUM _<-_. _)" [0, 51, 10] 10) syntax "_sum_list" :: "pttrn => 'a list => 'b => 'b" ("(3∑_←_. _)" [0, 51, 10] 10) translations ― ‹Beware of argument permutation!› "∑x←xs. b" == "CONST sum_list (CONST map (λx. b) xs)" context includes lifting_syntax begin lemma 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_prover end text ‹TODO duplicates› lemmas sum_list_simps = sum_list.Nil sum_list.Cons lemmas sum_list_append = sum_list.append lemmas sum_list_rev = sum_list.rev lemma (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 "… = sum_list (rev xs) + x" by simp finally show "fold plus xs x = sum_list (rev xs) + x" . qed lemma (in comm_monoid_add) sum_list_map_remove1: "x ∈ set xs ⟹ 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) auto lemma (in monoid_add) length_concat: "length (concat xss) = sum_list (map length xss)" by (induct xss) simp_all lemma (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 simp lemma (in monoid_add) sum_list_map_filter: assumes "⋀x. x ∈ set xs ⟹ ¬ P x ⟹ f x = 0" shows "sum_list (map f (filter P xs)) = sum_list (map f xs)" using assms by (induct xs) auto lemma sum_list_filter_le_nat: fixes f :: "'a ⇒ nat" shows "sum_list (map f (filter P xs)) ≤ sum_list (map f xs)" by(induction xs; simp) lemma (in comm_monoid_add) distinct_sum_list_conv_Sum: "distinct xs ⟹ sum_list xs = Sum (set xs)" by (induct xs) simp_all lemma sum_list_upt[simp]: "m ≤ n ⟹ sum_list [m..<n] = ∑ {m..<n}" by(simp add: distinct_sum_list_conv_Sum) context ordered_comm_monoid_add begin lemma sum_list_nonneg: "(⋀x. x ∈ set xs ⟹ 0 ≤ x) ⟹ 0 ≤ sum_list xs" by (induction xs) auto lemma sum_list_nonpos: "(⋀x. x ∈ set xs ⟹ x ≤ 0) ⟹ sum_list xs ≤ 0" by (induction xs) (auto simp: add_nonpos_nonpos) lemma sum_list_nonneg_eq_0_iff: "(⋀x. x ∈ set xs ⟹ 0 ≤ x) ⟹ sum_list xs = 0 ⟷ (∀x∈ set xs. x = 0)" by (induction xs) (simp_all add: add_nonneg_eq_0_iff sum_list_nonneg) end context canonically_ordered_monoid_add begin lemma sum_list_eq_0_iff [simp]: "sum_list ns = 0 ⟷ (∀n ∈ set ns. n = 0)" by (simp add: sum_list_nonneg_eq_0_iff) lemma member_le_sum_list: "x ∈ set xs ⟹ x ≤ sum_list xs" by (induction xs) (auto simp: add_increasing add_increasing2) lemma elem_le_sum_list: "k < size ns ⟹ ns ! k ≤ sum_list (ns)" by (rule member_le_sum_list) simp end lemma (in ordered_cancel_comm_monoid_diff) sum_list_update: "k < size xs ⟹ 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: "(∑x←xs. r) = of_nat (length xs) * r" by (induct xs) (simp_all add: distrib_right) lemma (in monoid_add) sum_list_0 [simp]: "(∑x←xs. 0) = 0" by (induct xs) (simp_all add: distrib_right) text‹For non-Abelian groups ‹xs› needs to be reversed on one side:› lemma (in ab_group_add) uminus_sum_list_map: "- sum_list (map f xs) = sum_list (map (uminus ∘ f) xs)" by (induct xs) simp_all lemma (in comm_monoid_add) sum_list_addf: "(∑x←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: "(∑x←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: "(∑x←xs. c * f x) = c * (∑x←xs. f x)" by (induct xs) (simp_all add: algebra_simps) lemma (in semiring_0) sum_list_mult_const: "(∑x←xs. f x * c) = (∑x←xs. f x) * c" by (induct xs) (simp_all add: algebra_simps) lemma (in ordered_ab_group_add_abs) sum_list_abs: "¦sum_list xs¦ ≤ 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 ⇒ 'b::{monoid_add, ordered_ab_semigroup_add}" shows "(⋀x. x ∈ set xs ⟹ f x ≤ g x) ⟹ (∑x←xs. f x) ≤ (∑x←xs. g x)" by (induct xs) (simp, simp add: add_mono) lemma sum_list_strict_mono: fixes f g :: "'a ⇒ 'b::{monoid_add, strict_ordered_ab_semigroup_add}" shows "⟦ xs ≠ []; ⋀x. x ∈ set xs ⟹ f x < g x ⟧ ⟹ sum_list (map f xs) < sum_list (map g xs)" proof (induction xs) case Nil thus ?case by simp next 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) qed qed text ‹A much more general version of this monotonicity lemma can be formulated with multisets and the multiset order› lemma sum_list_mono2: fixes xs :: "'a ::ordered_comm_monoid_add list" shows "⟦ length xs = length ys; ⋀i. i < length xs ⟶ xs!i ≤ ys!i ⟧ ⟹ sum_list xs ≤ 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 ⟹ sum_list (map f xs) = sum f (set xs)" by (induct xs) simp_all lemma (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 ‹General equivalence between \<^const>‹sum_list› and \<^const>‹sum›› lemma (in monoid_add) sum_list_sum_nth: "sum_list xs = (∑ 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 (λx. count_list xs x * f x) (set xs)" proof(induction xs) case (Cons x xs) show ?case (is "?l = ?r") proof cases assume "x ∈ set xs" have "?l = f x + (∑x∈set xs. count_list xs x * f x)" by (simp add: Cons.IH) also have "set xs = insert x (set xs - {x})" using ‹x ∈ set xs›by blast also have "f x + (∑x∈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 ∉ set xs" hence "⋀xa. xa ∈ set xs ⟹ x ≠ xa" by blast thus ?thesis by (simp add: Cons.IH ‹x ∉ set xs›) qed qed simp lemma sum_list_map_eq_sum_count2: assumes "set xs ⊆ X" "finite X" shows "sum_list (map f xs) = sum (λx. count_list xs x * f x) X" proof- let ?F = "λx. count_list xs x * f x" have "sum ?F X = sum ?F (set xs ∪ (X - set xs))" using Un_absorb1[OF assms(1)] by(simp) also have "… = 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) qed lemma sum_list_replicate: "sum_list (replicate n c) = of_nat n * c" by(induction n)(auto simp add: distrib_right) lemma sum_list_nonneg: "(⋀x. x ∈ set xs ⟹ (x :: 'a :: ordered_comm_monoid_add) ≥ 0) ⟹ sum_list xs ≥ 0" by (induction xs) simp_all lemma sum_list_Suc: "sum_list (map (λx. 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 (λx. if P x then f x else 0) xs)" by (induction xs) simp_all text ‹Summation of a strictly ascending sequence with length ‹n› can be upper-bounded by summation over ‹{0..<n}›.› lemma sorted_wrt_less_sum_mono_lowerbound: fixes f :: "nat ⇒ ('b::ordered_comm_monoid_add)" assumes mono: "⋀x y. x≤y ⟹ f x ≤ f y" shows "sorted_wrt (<) ns ⟹ (∑i∈{0..<length ns}. f i) ≤ (∑i←ns. f i)" proof (induction ns rule: rev_induct) case Nil then show ?case by simp next 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} ≤ sum_list (map f ns)" using snoc by (auto simp: sorted_wrt_append) also have "length ns ≤ n" using sorted_wrt_less_idx[OF snoc.prems(1), of "length ns"] by auto finally have "sum f {0..<length (ns @ [n])} ≤ sum_list (map f ns) + f n" using mono add_mono by blast thus ?case by simp qed subsection ‹Horner sums› context comm_semiring_0 begin definition horner_sum :: ‹('b ⇒ 'a) ⇒ 'a ⇒ 'b list ⇒ 'a› where horner_sum_foldr: ‹horner_sum f a xs = foldr (λx b. f x + a * b) xs 0› lemma horner_sum_simps [simp]: ‹horner_sum f a [] = 0› ‹horner_sum f a (x # xs) = f x + a * horner_sum f a xs› by (simp_all add: horner_sum_foldr) lemma horner_sum_eq_sum_funpow: ‹horner_sum f a xs = (∑n = 0..<length xs. ((*) a ^^ n) (f (xs ! n)))› proof (induction xs) case Nil then show ?case by simp next case (Cons x xs) then show ?case by (simp add: sum.atLeast0_lessThan_Suc_shift sum_distrib_left del: sum.op_ivl_Suc) qed end context includes lifting_syntax begin lemma horner_sum_transfer [transfer_rule]: ‹((B ===> A) ===> A ===> list_all2 B ===> A) horner_sum horner_sum› if [transfer_rule]: ‹A 0 0› and [transfer_rule]: ‹(A ===> A ===> A) (+) (+)› and [transfer_rule]: ‹(A ===> A ===> A) (*) (*)› by (unfold horner_sum_foldr) transfer_prover end context comm_semiring_1 begin lemma horner_sum_eq_sum: ‹horner_sum f a xs = (∑n = 0..<length xs. f (xs ! n) * a ^ n)› proof - have ‹(*) a ^^ n = (*) (a ^ n)› for n by (induction n) (simp_all add: ac_simps) then show ?thesis by (simp add: horner_sum_eq_sum_funpow ac_simps) qed lemma horner_sum_append: ‹horner_sum f a (xs @ ys) = horner_sum f a xs + a ^ length xs * horner_sum f a ys› using sum.atLeastLessThan_shift_bounds [of _ 0 ‹length xs› ‹length ys›] atLeastLessThan_add_Un [of 0 ‹length xs› ‹length ys›] by (simp add: horner_sum_eq_sum sum_distrib_left sum.union_disjoint ac_simps nth_append power_add) end context linordered_semidom begin lemma horner_sum_nonnegative: ‹0 ≤ horner_sum of_bool 2 bs› by (induction bs) simp_all end context discrete_linordered_semidom begin lemma horner_sum_bound: ‹horner_sum of_bool 2 bs < 2 ^ length bs› proof (induction bs) case Nil then show ?case by simp next case (Cons b bs) moreover define a where ‹a = 2 ^ length bs - horner_sum of_bool 2 bs› ultimately have *: ‹2 ^ length bs = horner_sum of_bool 2 bs + a› by simp have ‹0 < a› using Cons * by simp moreover have ‹1 ≤ a› using ‹0 < a› by (simp add: less_eq_iff_succ_less) ultimately have ‹0 + 1 < a + a› by (rule add_less_le_mono) then have ‹1 < a * 2› by (simp add: mult_2_right) with Cons show ?case by (simp add: * algebra_simps) qed lemma horner_sum_of_bool_2_less: ‹(horner_sum of_bool 2 bs) < 2 ^ length bs› by (fact horner_sum_bound) end lemma nat_horner_sum [simp]: ‹nat (horner_sum of_bool 2 bs) = horner_sum of_bool 2 bs› by (induction bs) (auto simp add: nat_add_distrib horner_sum_nonnegative) context discrete_linordered_semidom begin lemma horner_sum_less_eq_iff_lexordp_eq: ‹horner_sum of_bool 2 bs ≤ horner_sum of_bool 2 cs ⟷ lexordp_eq (rev bs) (rev cs)› if ‹length bs = length cs› proof - have ‹horner_sum of_bool 2 (rev bs) ≤ horner_sum of_bool 2 (rev cs) ⟷ lexordp_eq bs cs› if ‹length bs = length cs› 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 ‹rev bs›] horner_sum_nonnegative [of ‹rev cs›] horner_sum_bound [of ‹rev bs›] horner_sum_bound [of ‹rev cs›] show ?case by (auto simp add: horner_sum_append not_le Cons intro: add_strict_increasing2 add_increasing) qed from that this [of ‹rev bs› ‹rev cs›] show ?thesis by simp qed lemma horner_sum_less_iff_lexordp: ‹horner_sum of_bool 2 bs < horner_sum of_bool 2 cs ⟷ ord_class.lexordp (rev bs) (rev cs)› if ‹length bs = length cs› proof - have ‹horner_sum of_bool 2 (rev bs) < horner_sum of_bool 2 (rev cs) ⟷ ord_class.lexordp bs cs› if ‹length bs = length cs› 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 ‹rev bs›] horner_sum_nonnegative [of ‹rev cs›] horner_sum_bound [of ‹rev bs›] horner_sum_bound [of ‹rev cs›] show ?case by (auto simp add: horner_sum_append not_less Cons intro: add_strict_increasing2 add_increasing) qed from that this [of ‹rev bs› ‹rev cs›] show ?thesis by simp qed end subsection ‹Further facts about \<^const>‹List.n_lists›› 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 (⋃ys∈set (List.n_lists n xs). (λy. y # ys) ` set xs) = (∑ys∈set (List.n_lists n xs). card ((λy. y # ys) ` set xs))" by (rule card_UN_disjoint) auto moreover have "⋀ys. card ((λy. y # ys) ` set xs) = card (set xs)" by (rule card_image) (simp add: inj_on_def) ultimately show ?case by auto qed also have "… = 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) qed subsection ‹Tools setup› lemmas sum_code = sum.set_conv_list lemma 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 ‹List product› context monoid_mult begin sublocale prod_list: monoid_list times 1 defines prod_list = prod_list.F .. end context comm_monoid_mult begin sublocale prod_list: comm_monoid_list times 1 rewrites "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 simp qed sublocale prod: comm_monoid_list_set times 1 rewrites "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) qed end text ‹Some syntactic sugar:› syntax (ASCII) "_prod_list" :: "pttrn => 'a list => 'b => 'b" ("(3PROD _<-_. _)" [0, 51, 10] 10) syntax "_prod_list" :: "pttrn => 'a list => 'b => 'b" ("(3∏_←_. _)" [0, 51, 10] 10) translations ― ‹Beware of argument permutation!› "∏x←xs. b" ⇌ "CONST prod_list (CONST map (λx. b) xs)" context includes lifting_syntax begin lemma 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_prover end lemma prod_list_zero_iff: "prod_list xs = 0 ⟷ (0 :: 'a :: {semiring_no_zero_divisors, semiring_1}) ∈ set xs" by (induction xs) simp_all end