(* Title : Series.thy Author : Jacques D. Fleuriot Copyright : 1998 University of Cambridge Converted to Isar and polished by lcp Converted to setsum and polished yet more by TNN Additional contributions by Jeremy Avigad *) header{*Finite Summation and Infinite Series*} theory Series imports SEQ begin definition sums :: "(nat => 'a::real_normed_vector) => 'a => bool" (infixr "sums" 80) where "f sums s = (%n. setsum f {0..<n}) ----> s" definition summable :: "(nat => 'a::real_normed_vector) => bool" where "summable f = (∃s. f sums s)" definition suminf :: "(nat => 'a::real_normed_vector) => 'a" where "suminf f = (THE s. f sums s)" syntax "_suminf" :: "idt => 'a => 'a" ("∑_. _" [0, 10] 10) translations "∑i. b" == "CONST suminf (%i. b)" lemma sumr_diff_mult_const: "setsum f {0..<n} - (real n*r) = setsum (%i. f i - r) {0..<n::nat}" by (simp add: diff_minus setsum_addf real_of_nat_def) lemma real_setsum_nat_ivl_bounded: "(!!p. p < n ==> f(p) ≤ K) ==> setsum f {0..<n::nat} ≤ real n * K" using setsum_bounded[where A = "{0..<n}"] by (auto simp:real_of_nat_def) (* Generalize from real to some algebraic structure? *) lemma sumr_minus_one_realpow_zero [simp]: "(∑i=0..<2*n. (-1) ^ Suc i) = (0::real)" by (induct "n", auto) (* FIXME this is an awful lemma! *) lemma sumr_one_lb_realpow_zero [simp]: "(∑n=Suc 0..<n. f(n) * (0::real) ^ n) = 0" by (rule setsum_0', simp) lemma sumr_group: "(∑m=0..<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {0 ..< n * k}" apply (subgoal_tac "k = 0 | 0 < k", auto) apply (induct "n") apply (simp_all add: setsum_add_nat_ivl add_commute) done lemma sumr_offset3: "setsum f {0::nat..<n+k} = (∑m=0..<n. f (m+k)) + setsum f {0..<k}" apply (subst setsum_shift_bounds_nat_ivl [symmetric]) apply (simp add: setsum_add_nat_ivl add_commute) done lemma sumr_offset: fixes f :: "nat => 'a::ab_group_add" shows "(∑m=0..<n. f(m+k)) = setsum f {0..<n+k} - setsum f {0..<k}" by (simp add: sumr_offset3) lemma sumr_offset2: "∀f. (∑m=0..<n::nat. f(m+k)::real) = setsum f {0..<n+k} - setsum f {0..<k}" by (simp add: sumr_offset) lemma sumr_offset4: "∀n f. setsum f {0::nat..<n+k} = (∑m=0..<n. f (m+k)::real) + setsum f {0..<k}" by (clarify, rule sumr_offset3) (* lemma sumr_from_1_from_0: "0 < n ==> (∑n=Suc 0 ..< Suc n. if even(n) then 0 else ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n = (∑n=0..<Suc n. if even(n) then 0 else ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n" by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto) *) subsection{* Infinite Sums, by the Properties of Limits*} (*---------------------- suminf is the sum ---------------------*) lemma sums_summable: "f sums l ==> summable f" by (simp add: sums_def summable_def, blast) lemma summable_sums: "summable f ==> f sums (suminf f)" apply (simp add: summable_def suminf_def sums_def) apply (blast intro: theI LIMSEQ_unique) done lemma summable_sumr_LIMSEQ_suminf: "summable f ==> (%n. setsum f {0..<n}) ----> (suminf f)" by (rule summable_sums [unfolded sums_def]) (*------------------- sum is unique ------------------*) lemma sums_unique: "f sums s ==> (s = suminf f)" apply (frule sums_summable [THEN summable_sums]) apply (auto intro!: LIMSEQ_unique simp add: sums_def) done lemma sums_split_initial_segment: "f sums s ==> (%n. f(n + k)) sums (s - (SUM i = 0..< k. f i))" apply (unfold sums_def); apply (simp add: sumr_offset); apply (rule LIMSEQ_diff_const) apply (rule LIMSEQ_ignore_initial_segment) apply assumption done lemma summable_ignore_initial_segment: "summable f ==> summable (%n. f(n + k))" apply (unfold summable_def) apply (auto intro: sums_split_initial_segment) done lemma suminf_minus_initial_segment: "summable f ==> suminf f = s ==> suminf (%n. f(n + k)) = s - (SUM i = 0..< k. f i)" apply (frule summable_ignore_initial_segment) apply (rule sums_unique [THEN sym]) apply (frule summable_sums) apply (rule sums_split_initial_segment) apply auto done lemma suminf_split_initial_segment: "summable f ==> suminf f = (SUM i = 0..< k. f i) + suminf (%n. f(n + k))" by (auto simp add: suminf_minus_initial_segment) lemma series_zero: "(∀m. n ≤ m --> f(m) = 0) ==> f sums (setsum f {0..<n})" apply (simp add: sums_def LIMSEQ_def diff_minus[symmetric], safe) apply (rule_tac x = n in exI) apply (clarsimp simp add:setsum_diff[symmetric] cong:setsum_ivl_cong) done lemma sums_zero: "(λn. 0) sums 0" unfolding sums_def by (simp add: LIMSEQ_const) lemma summable_zero: "summable (λn. 0)" by (rule sums_zero [THEN sums_summable]) lemma suminf_zero: "suminf (λn. 0) = 0" by (rule sums_zero [THEN sums_unique, symmetric]) lemma (in bounded_linear) sums: "(λn. X n) sums a ==> (λn. f (X n)) sums (f a)" unfolding sums_def by (drule LIMSEQ, simp only: setsum) lemma (in bounded_linear) summable: "summable (λn. X n) ==> summable (λn. f (X n))" unfolding summable_def by (auto intro: sums) lemma (in bounded_linear) suminf: "summable (λn. X n) ==> f (∑n. X n) = (∑n. f (X n))" by (intro sums_unique sums summable_sums) lemma sums_mult: fixes c :: "'a::real_normed_algebra" shows "f sums a ==> (λn. c * f n) sums (c * a)" by (rule mult_right.sums) lemma summable_mult: fixes c :: "'a::real_normed_algebra" shows "summable f ==> summable (%n. c * f n)" by (rule mult_right.summable) lemma suminf_mult: fixes c :: "'a::real_normed_algebra" shows "summable f ==> suminf (λn. c * f n) = c * suminf f"; by (rule mult_right.suminf [symmetric]) lemma sums_mult2: fixes c :: "'a::real_normed_algebra" shows "f sums a ==> (λn. f n * c) sums (a * c)" by (rule mult_left.sums) lemma summable_mult2: fixes c :: "'a::real_normed_algebra" shows "summable f ==> summable (λn. f n * c)" by (rule mult_left.summable) lemma suminf_mult2: fixes c :: "'a::real_normed_algebra" shows "summable f ==> suminf f * c = (∑n. f n * c)" by (rule mult_left.suminf) lemma sums_divide: fixes c :: "'a::real_normed_field" shows "f sums a ==> (λn. f n / c) sums (a / c)" by (rule divide.sums) lemma summable_divide: fixes c :: "'a::real_normed_field" shows "summable f ==> summable (λn. f n / c)" by (rule divide.summable) lemma suminf_divide: fixes c :: "'a::real_normed_field" shows "summable f ==> suminf (λn. f n / c) = suminf f / c" by (rule divide.suminf [symmetric]) lemma sums_add: "[|X sums a; Y sums b|] ==> (λn. X n + Y n) sums (a + b)" unfolding sums_def by (simp add: setsum_addf LIMSEQ_add) lemma summable_add: "[|summable X; summable Y|] ==> summable (λn. X n + Y n)" unfolding summable_def by (auto intro: sums_add) lemma suminf_add: "[|summable X; summable Y|] ==> suminf X + suminf Y = (∑n. X n + Y n)" by (intro sums_unique sums_add summable_sums) lemma sums_diff: "[|X sums a; Y sums b|] ==> (λn. X n - Y n) sums (a - b)" unfolding sums_def by (simp add: setsum_subtractf LIMSEQ_diff) lemma summable_diff: "[|summable X; summable Y|] ==> summable (λn. X n - Y n)" unfolding summable_def by (auto intro: sums_diff) lemma suminf_diff: "[|summable X; summable Y|] ==> suminf X - suminf Y = (∑n. X n - Y n)" by (intro sums_unique sums_diff summable_sums) lemma sums_minus: "X sums a ==> (λn. - X n) sums (- a)" unfolding sums_def by (simp add: setsum_negf LIMSEQ_minus) lemma summable_minus: "summable X ==> summable (λn. - X n)" unfolding summable_def by (auto intro: sums_minus) lemma suminf_minus: "summable X ==> (∑n. - X n) = - (∑n. X n)" by (intro sums_unique [symmetric] sums_minus summable_sums) lemma sums_group: "[|summable f; 0 < k |] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)" apply (drule summable_sums) apply (simp only: sums_def sumr_group) apply (unfold LIMSEQ_def, safe) apply (drule_tac x="r" in spec, safe) apply (rule_tac x="no" in exI, safe) apply (drule_tac x="n*k" in spec) apply (erule mp) apply (erule order_trans) apply simp done text{*A summable series of positive terms has limit that is at least as great as any partial sum.*} lemma series_pos_le: fixes f :: "nat => real" shows "[|summable f; ∀m≥n. 0 ≤ f m|] ==> setsum f {0..<n} ≤ suminf f" apply (drule summable_sums) apply (simp add: sums_def) apply (cut_tac k = "setsum f {0..<n}" in LIMSEQ_const) apply (erule LIMSEQ_le, blast) apply (rule_tac x="n" in exI, clarify) apply (rule setsum_mono2) apply auto done lemma series_pos_less: fixes f :: "nat => real" shows "[|summable f; ∀m≥n. 0 < f m|] ==> setsum f {0..<n} < suminf f" apply (rule_tac y="setsum f {0..<Suc n}" in order_less_le_trans) apply simp apply (erule series_pos_le) apply (simp add: order_less_imp_le) done lemma suminf_gt_zero: fixes f :: "nat => real" shows "[|summable f; ∀n. 0 < f n|] ==> 0 < suminf f" by (drule_tac n="0" in series_pos_less, simp_all) lemma suminf_ge_zero: fixes f :: "nat => real" shows "[|summable f; ∀n. 0 ≤ f n|] ==> 0 ≤ suminf f" by (drule_tac n="0" in series_pos_le, simp_all) lemma sumr_pos_lt_pair: fixes f :: "nat => real" shows "[|summable f; ∀d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))|] ==> setsum f {0..<k} < suminf f" apply (subst suminf_split_initial_segment [where k="k"]) apply assumption apply simp apply (drule_tac k="k" in summable_ignore_initial_segment) apply (drule_tac k="Suc (Suc 0)" in sums_group, simp) apply simp apply (frule sums_unique) apply (drule sums_summable) apply simp apply (erule suminf_gt_zero) apply (simp add: add_ac) done text{*Sum of a geometric progression.*} lemmas sumr_geometric = geometric_sum [where 'a = real] lemma geometric_sums: fixes x :: "'a::{real_normed_field,recpower}" shows "norm x < 1 ==> (λn. x ^ n) sums (1 / (1 - x))" proof - assume less_1: "norm x < 1" hence neq_1: "x ≠ 1" by auto hence neq_0: "x - 1 ≠ 0" by simp from less_1 have lim_0: "(λn. x ^ n) ----> 0" by (rule LIMSEQ_power_zero) hence "(λn. x ^ n / (x - 1) - 1 / (x - 1)) ----> 0 / (x - 1) - 1 / (x - 1)" using neq_0 by (intro LIMSEQ_divide LIMSEQ_diff LIMSEQ_const) hence "(λn. (x ^ n - 1) / (x - 1)) ----> 1 / (1 - x)" by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib) thus "(λn. x ^ n) sums (1 / (1 - x))" by (simp add: sums_def geometric_sum neq_1) qed lemma summable_geometric: fixes x :: "'a::{real_normed_field,recpower}" shows "norm x < 1 ==> summable (λn. x ^ n)" by (rule geometric_sums [THEN sums_summable]) text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*} lemma summable_convergent_sumr_iff: "summable f = convergent (%n. setsum f {0..<n})" by (simp add: summable_def sums_def convergent_def) lemma summable_LIMSEQ_zero: "summable f ==> f ----> 0" apply (drule summable_convergent_sumr_iff [THEN iffD1]) apply (drule convergent_Cauchy) apply (simp only: Cauchy_def LIMSEQ_def, safe) apply (drule_tac x="r" in spec, safe) apply (rule_tac x="M" in exI, safe) apply (drule_tac x="Suc n" in spec, simp) apply (drule_tac x="n" in spec, simp) done lemma summable_Cauchy: "summable (f::nat => 'a::banach) = (∀e > 0. ∃N. ∀m ≥ N. ∀n. norm (setsum f {m..<n}) < e)" apply (simp only: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def, safe) apply (drule spec, drule (1) mp) apply (erule exE, rule_tac x="M" in exI, clarify) apply (rule_tac x="m" and y="n" in linorder_le_cases) apply (frule (1) order_trans) apply (drule_tac x="n" in spec, drule (1) mp) apply (drule_tac x="m" in spec, drule (1) mp) apply (simp add: setsum_diff [symmetric]) apply simp apply (drule spec, drule (1) mp) apply (erule exE, rule_tac x="N" in exI, clarify) apply (rule_tac x="m" and y="n" in linorder_le_cases) apply (subst norm_minus_commute) apply (simp add: setsum_diff [symmetric]) apply (simp add: setsum_diff [symmetric]) done text{*Comparison test*} lemma norm_setsum: fixes f :: "'a => 'b::real_normed_vector" shows "norm (setsum f A) ≤ (∑i∈A. norm (f i))" apply (case_tac "finite A") apply (erule finite_induct) apply simp apply simp apply (erule order_trans [OF norm_triangle_ineq add_left_mono]) apply simp done lemma summable_comparison_test: fixes f :: "nat => 'a::banach" shows "[|∃N. ∀n≥N. norm (f n) ≤ g n; summable g|] ==> summable f" apply (simp add: summable_Cauchy, safe) apply (drule_tac x="e" in spec, safe) apply (rule_tac x = "N + Na" in exI, safe) apply (rotate_tac 2) apply (drule_tac x = m in spec) apply (auto, rotate_tac 2, drule_tac x = n in spec) apply (rule_tac y = "∑k=m..<n. norm (f k)" in order_le_less_trans) apply (rule norm_setsum) apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans) apply (auto intro: setsum_mono simp add: abs_less_iff) done lemma summable_norm_comparison_test: fixes f :: "nat => 'a::banach" shows "[|∃N. ∀n≥N. norm (f n) ≤ g n; summable g|] ==> summable (λn. norm (f n))" apply (rule summable_comparison_test) apply (auto) done lemma summable_rabs_comparison_test: fixes f :: "nat => real" shows "[|∃N. ∀n≥N. ¦f n¦ ≤ g n; summable g|] ==> summable (λn. ¦f n¦)" apply (rule summable_comparison_test) apply (auto) done text{*Summability of geometric series for real algebras*} lemma complete_algebra_summable_geometric: fixes x :: "'a::{real_normed_algebra_1,banach,recpower}" shows "norm x < 1 ==> summable (λn. x ^ n)" proof (rule summable_comparison_test) show "∃N. ∀n≥N. norm (x ^ n) ≤ norm x ^ n" by (simp add: norm_power_ineq) show "norm x < 1 ==> summable (λn. norm x ^ n)" by (simp add: summable_geometric) qed text{*Limit comparison property for series (c.f. jrh)*} lemma summable_le: fixes f g :: "nat => real" shows "[|∀n. f n ≤ g n; summable f; summable g|] ==> suminf f ≤ suminf g" apply (drule summable_sums)+ apply (simp only: sums_def, erule (1) LIMSEQ_le) apply (rule exI) apply (auto intro!: setsum_mono) done lemma summable_le2: fixes f g :: "nat => real" shows "[|∀n. ¦f n¦ ≤ g n; summable g|] ==> summable f ∧ suminf f ≤ suminf g" apply (subgoal_tac "summable f") apply (auto intro!: summable_le) apply (simp add: abs_le_iff) apply (rule_tac g="g" in summable_comparison_test, simp_all) done (* specialisation for the common 0 case *) lemma suminf_0_le: fixes f::"nat=>real" assumes gt0: "∀n. 0 ≤ f n" and sm: "summable f" shows "0 ≤ suminf f" proof - let ?g = "(λn. (0::real))" from gt0 have "∀n. ?g n ≤ f n" by simp moreover have "summable ?g" by (rule summable_zero) moreover from sm have "summable f" . ultimately have "suminf ?g ≤ suminf f" by (rule summable_le) then show "0 ≤ suminf f" by (simp add: suminf_zero) qed text{*Absolute convergence imples normal convergence*} lemma summable_norm_cancel: fixes f :: "nat => 'a::banach" shows "summable (λn. norm (f n)) ==> summable f" apply (simp only: summable_Cauchy, safe) apply (drule_tac x="e" in spec, safe) apply (rule_tac x="N" in exI, safe) apply (drule_tac x="m" in spec, safe) apply (rule order_le_less_trans [OF norm_setsum]) apply (rule order_le_less_trans [OF abs_ge_self]) apply simp done lemma summable_rabs_cancel: fixes f :: "nat => real" shows "summable (λn. ¦f n¦) ==> summable f" by (rule summable_norm_cancel, simp) text{*Absolute convergence of series*} lemma summable_norm: fixes f :: "nat => 'a::banach" shows "summable (λn. norm (f n)) ==> norm (suminf f) ≤ (∑n. norm (f n))" by (auto intro: LIMSEQ_le LIMSEQ_norm summable_norm_cancel summable_sumr_LIMSEQ_suminf norm_setsum) lemma summable_rabs: fixes f :: "nat => real" shows "summable (λn. ¦f n¦) ==> ¦suminf f¦ ≤ (∑n. ¦f n¦)" by (fold real_norm_def, rule summable_norm) subsection{* The Ratio Test*} lemma norm_ratiotest_lemma: fixes x y :: "'a::real_normed_vector" shows "[|c ≤ 0; norm x ≤ c * norm y|] ==> x = 0" apply (subgoal_tac "norm x ≤ 0", simp) apply (erule order_trans) apply (simp add: mult_le_0_iff) done lemma rabs_ratiotest_lemma: "[| c ≤ 0; abs x ≤ c * abs y |] ==> x = (0::real)" by (erule norm_ratiotest_lemma, simp) lemma le_Suc_ex: "(k::nat) ≤ l ==> (∃n. l = k + n)" apply (drule le_imp_less_or_eq) apply (auto dest: less_imp_Suc_add) done lemma le_Suc_ex_iff: "((k::nat) ≤ l) = (∃n. l = k + n)" by (auto simp add: le_Suc_ex) (*All this trouble just to get 0<c *) lemma ratio_test_lemma2: fixes f :: "nat => 'a::banach" shows "[|∀n≥N. norm (f (Suc n)) ≤ c * norm (f n)|] ==> 0 < c ∨ summable f" apply (simp (no_asm) add: linorder_not_le [symmetric]) apply (simp add: summable_Cauchy) apply (safe, subgoal_tac "∀n. N < n --> f (n) = 0") prefer 2 apply clarify apply(erule_tac x = "n - 1" in allE) apply (simp add:diff_Suc split:nat.splits) apply (blast intro: norm_ratiotest_lemma) apply (rule_tac x = "Suc N" in exI, clarify) apply(simp cong:setsum_ivl_cong) done lemma ratio_test: fixes f :: "nat => 'a::banach" shows "[|c < 1; ∀n≥N. norm (f (Suc n)) ≤ c * norm (f n)|] ==> summable f" apply (frule ratio_test_lemma2, auto) apply (rule_tac g = "%n. (norm (f N) / (c ^ N))*c ^ n" in summable_comparison_test) apply (rule_tac x = N in exI, safe) apply (drule le_Suc_ex_iff [THEN iffD1]) apply (auto simp add: power_add field_power_not_zero) apply (induct_tac "na", auto) apply (rule_tac y = "c * norm (f (N + n))" in order_trans) apply (auto intro: mult_right_mono simp add: summable_def) apply (simp add: mult_ac) apply (rule_tac x = "norm (f N) * (1/ (1 - c)) / (c ^ N)" in exI) apply (rule sums_divide) apply (rule sums_mult) apply (auto intro!: geometric_sums) done subsection {* Cauchy Product Formula *} (* Proof based on Analysis WebNotes: Chapter 07, Class 41 http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm *) lemma setsum_triangle_reindex: fixes n :: nat shows "(∑(i,j)∈{(i,j). i+j < n}. f i j) = (∑k=0..<n. ∑i=0..k. f i (k - i))" proof - have "(∑(i, j)∈{(i, j). i + j < n}. f i j) = (∑(k, i)∈(SIGMA k:{0..<n}. {0..k}). f i (k - i))" proof (rule setsum_reindex_cong) show "inj_on (λ(k,i). (i, k - i)) (SIGMA k:{0..<n}. {0..k})" by (rule inj_on_inverseI [where g="λ(i,j). (i+j, i)"], auto) show "{(i,j). i + j < n} = (λ(k,i). (i, k - i)) ` (SIGMA k:{0..<n}. {0..k})" by (safe, rule_tac x="(a+b,a)" in image_eqI, auto) show "!!a. (λ(k, i). f i (k - i)) a = split f ((λ(k, i). (i, k - i)) a)" by clarify qed thus ?thesis by (simp add: setsum_Sigma) qed lemma Cauchy_product_sums: fixes a b :: "nat => 'a::{real_normed_algebra,banach}" assumes a: "summable (λk. norm (a k))" assumes b: "summable (λk. norm (b k))" shows "(λk. ∑i=0..k. a i * b (k - i)) sums ((∑k. a k) * (∑k. b k))" proof - let ?S1 = "λn::nat. {0..<n} × {0..<n}" let ?S2 = "λn::nat. {(i,j). i + j < n}" have S1_mono: "!!m n. m ≤ n ==> ?S1 m ⊆ ?S1 n" by auto have S2_le_S1: "!!n. ?S2 n ⊆ ?S1 n" by auto have S1_le_S2: "!!n. ?S1 (n div 2) ⊆ ?S2 n" by auto have finite_S1: "!!n. finite (?S1 n)" by simp with S2_le_S1 have finite_S2: "!!n. finite (?S2 n)" by (rule finite_subset) let ?g = "λ(i,j). a i * b j" let ?f = "λ(i,j). norm (a i) * norm (b j)" have f_nonneg: "!!x. 0 ≤ ?f x" by (auto simp add: mult_nonneg_nonneg) hence norm_setsum_f: "!!A. norm (setsum ?f A) = setsum ?f A" unfolding real_norm_def by (simp only: abs_of_nonneg setsum_nonneg [rule_format]) have "(λn. (∑k=0..<n. a k) * (∑k=0..<n. b k)) ----> (∑k. a k) * (∑k. b k)" by (intro LIMSEQ_mult summable_sumr_LIMSEQ_suminf summable_norm_cancel [OF a] summable_norm_cancel [OF b]) hence 1: "(λn. setsum ?g (?S1 n)) ----> (∑k. a k) * (∑k. b k)" by (simp only: setsum_product setsum_Sigma [rule_format] finite_atLeastLessThan) have "(λn. (∑k=0..<n. norm (a k)) * (∑k=0..<n. norm (b k))) ----> (∑k. norm (a k)) * (∑k. norm (b k))" using a b by (intro LIMSEQ_mult summable_sumr_LIMSEQ_suminf) hence "(λn. setsum ?f (?S1 n)) ----> (∑k. norm (a k)) * (∑k. norm (b k))" by (simp only: setsum_product setsum_Sigma [rule_format] finite_atLeastLessThan) hence "convergent (λn. setsum ?f (?S1 n))" by (rule convergentI) hence Cauchy: "Cauchy (λn. setsum ?f (?S1 n))" by (rule convergent_Cauchy) have "Zseq (λn. setsum ?f (?S1 n - ?S2 n))" proof (rule ZseqI, simp only: norm_setsum_f) fix r :: real assume r: "0 < r" from CauchyD [OF Cauchy r] obtain N where "∀m≥N. ∀n≥N. norm (setsum ?f (?S1 m) - setsum ?f (?S1 n)) < r" .. hence "!!m n. [|N ≤ n; n ≤ m|] ==> norm (setsum ?f (?S1 m - ?S1 n)) < r" by (simp only: setsum_diff finite_S1 S1_mono) hence N: "!!m n. [|N ≤ n; n ≤ m|] ==> setsum ?f (?S1 m - ?S1 n) < r" by (simp only: norm_setsum_f) show "∃N. ∀n≥N. setsum ?f (?S1 n - ?S2 n) < r" proof (intro exI allI impI) fix n assume "2 * N ≤ n" hence n: "N ≤ n div 2" by simp have "setsum ?f (?S1 n - ?S2 n) ≤ setsum ?f (?S1 n - ?S1 (n div 2))" by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg Diff_mono subset_refl S1_le_S2) also have "… < r" using n div_le_dividend by (rule N) finally show "setsum ?f (?S1 n - ?S2 n) < r" . qed qed hence "Zseq (λn. setsum ?g (?S1 n - ?S2 n))" apply (rule Zseq_le [rule_format]) apply (simp only: norm_setsum_f) apply (rule order_trans [OF norm_setsum setsum_mono]) apply (auto simp add: norm_mult_ineq) done hence 2: "(λn. setsum ?g (?S1 n) - setsum ?g (?S2 n)) ----> 0" by (simp only: LIMSEQ_Zseq_iff setsum_diff finite_S1 S2_le_S1 diff_0_right) with 1 have "(λn. setsum ?g (?S2 n)) ----> (∑k. a k) * (∑k. b k)" by (rule LIMSEQ_diff_approach_zero2) thus ?thesis by (simp only: sums_def setsum_triangle_reindex) qed lemma Cauchy_product: fixes a b :: "nat => 'a::{real_normed_algebra,banach}" assumes a: "summable (λk. norm (a k))" assumes b: "summable (λk. norm (b k))" shows "(∑k. a k) * (∑k. b k) = (∑k. ∑i=0..k. a i * b (k - i))" using a b by (rule Cauchy_product_sums [THEN sums_unique]) end
lemma sumr_diff_mult_const:
setsum f {0..<n} - real n * r = (∑i = 0..<n. f i - r)
lemma real_setsum_nat_ivl_bounded:
(!!p. p < n ==> f p ≤ K) ==> setsum f {0..<n} ≤ real n * K
lemma sumr_minus_one_realpow_zero:
(∑i = 0..<2 * n. -1 ^ Suc i) = 0
lemma sumr_one_lb_realpow_zero:
(∑n = Suc 0..<n. f n * 0 ^ n) = 0
lemma sumr_group:
(∑m = 0..<n. setsum f {m * k..<m * k + k}) = setsum f {0..<n * k}
lemma sumr_offset3:
setsum f {0..<n + k} = (∑m = 0..<n. f (m + k)) + setsum f {0..<k}
lemma sumr_offset:
(∑m = 0..<n. f (m + k)) = setsum f {0..<n + k} - setsum f {0..<k}
lemma sumr_offset2:
∀f. (∑m = 0..<n. f (m + k)) = setsum f {0..<n + k} - setsum f {0..<k}
lemma sumr_offset4:
∀n f. setsum f {0..<n + k} = (∑m = 0..<n. f (m + k)) + setsum f {0..<k}
lemma sums_summable:
f sums l ==> summable f
lemma summable_sums:
summable f ==> f sums suminf f
lemma summable_sumr_LIMSEQ_suminf:
summable f ==> (λn. setsum f {0..<n}) ----> suminf f
lemma sums_unique:
f sums s ==> s = suminf f
lemma sums_split_initial_segment:
f sums s ==> (λn. f (n + k)) sums (s - setsum f {0..<k})
lemma summable_ignore_initial_segment:
summable f ==> summable (λn. f (n + k))
lemma suminf_minus_initial_segment:
[| summable f; suminf f = s |] ==> (∑n. f (n + k)) = s - setsum f {0..<k}
lemma suminf_split_initial_segment:
summable f ==> suminf f = setsum f {0..<k} + (∑n. f (n + k))
lemma series_zero:
∀m≥n. f m = (0::'a) ==> f sums setsum f {0..<n}
lemma sums_zero:
(λn. 0::'a) sums (0::'a)
lemma summable_zero:
summable (λn. 0::'a)
lemma suminf_zero:
(∑n. (0::'a)) = (0::'a)
lemma sums:
X sums a ==> (λn. f (X n)) sums f a
lemma summable:
summable X ==> summable (λn. f (X n))
lemma suminf:
summable X ==> f (suminf X) = (∑n. f (X n))
lemma sums_mult:
f sums a ==> (λn. c * f n) sums (c * a)
lemma summable_mult:
summable f ==> summable (λn. c * f n)
lemma suminf_mult:
summable f ==> (∑n. c * f n) = c * suminf f
lemma sums_mult2:
f sums a ==> (λn. f n * c) sums (a * c)
lemma summable_mult2:
summable f ==> summable (λn. f n * c)
lemma suminf_mult2:
summable f ==> suminf f * c = (∑n. f n * c)
lemma sums_divide:
f sums a ==> (λn. f n / c) sums (a / c)
lemma summable_divide:
summable f ==> summable (λn. f n / c)
lemma suminf_divide:
summable f ==> (∑n. f n / c) = suminf f / c
lemma sums_add:
[| X sums a; Y sums b |] ==> (λn. X n + Y n) sums (a + b)
lemma summable_add:
[| summable X; summable Y |] ==> summable (λn. X n + Y n)
lemma suminf_add:
[| summable X; summable Y |] ==> suminf X + suminf Y = (∑n. X n + Y n)
lemma sums_diff:
[| X sums a; Y sums b |] ==> (λn. X n - Y n) sums (a - b)
lemma summable_diff:
[| summable X; summable Y |] ==> summable (λn. X n - Y n)
lemma suminf_diff:
[| summable X; summable Y |] ==> suminf X - suminf Y = (∑n. X n - Y n)
lemma sums_minus:
X sums a ==> (λn. - X n) sums - a
lemma summable_minus:
summable X ==> summable (λn. - X n)
lemma suminf_minus:
summable X ==> (∑n. - X n) = - suminf X
lemma sums_group:
[| summable f; 0 < k |] ==> (λn. setsum f {n * k..<n * k + k}) sums suminf f
lemma series_pos_le:
[| summable f; ∀m≥n. 0 ≤ f m |] ==> setsum f {0..<n} ≤ suminf f
lemma series_pos_less:
[| summable f; ∀m≥n. 0 < f m |] ==> setsum f {0..<n} < suminf f
lemma suminf_gt_zero:
[| summable f; ∀n. 0 < f n |] ==> 0 < suminf f
lemma suminf_ge_zero:
[| summable f; ∀n. 0 ≤ f n |] ==> 0 ≤ suminf f
lemma sumr_pos_lt_pair:
[| summable f;
∀d. 0 < f (k + Suc (Suc 0) * d) + f (k + (Suc (Suc 0) * d + 1)) |]
==> setsum f {0..<k} < suminf f
lemma sumr_geometric:
x ≠ 1 ==> setsum (op ^ x) {0..<n} = (x ^ n - 1) / (x - 1)
lemma geometric_sums:
norm x < 1 ==> op ^ x sums ((1::'a) / ((1::'a) - x))
lemma summable_geometric:
norm x < 1 ==> summable (op ^ x)
lemma summable_convergent_sumr_iff:
summable f = convergent (λn. setsum f {0..<n})
lemma summable_LIMSEQ_zero:
summable f ==> f ----> (0::'a)
lemma summable_Cauchy:
summable f = (∀e>0. ∃N. ∀m≥N. ∀n. norm (setsum f {m..<n}) < e)
lemma norm_setsum:
norm (setsum f A) ≤ (∑i∈A. norm (f i))
lemma summable_comparison_test:
[| ∃N. ∀n≥N. norm (f n) ≤ g n; summable g |] ==> summable f
lemma summable_norm_comparison_test:
[| ∃N. ∀n≥N. norm (f n) ≤ g n; summable g |] ==> summable (λn. norm (f n))
lemma summable_rabs_comparison_test:
[| ∃N. ∀n≥N. ¦f n¦ ≤ g n; summable g |] ==> summable (λn. ¦f n¦)
lemma complete_algebra_summable_geometric:
norm x < 1 ==> summable (op ^ x)
lemma summable_le:
[| ∀n. f n ≤ g n; summable f; summable g |] ==> suminf f ≤ suminf g
lemma summable_le2:
[| ∀n. ¦f n¦ ≤ g n; summable g |] ==> summable f ∧ suminf f ≤ suminf g
lemma suminf_0_le:
[| ∀n. 0 ≤ f n; summable f |] ==> 0 ≤ suminf f
lemma summable_norm_cancel:
summable (λn. norm (f n)) ==> summable f
lemma summable_rabs_cancel:
summable (λn. ¦f n¦) ==> summable f
lemma summable_norm:
summable (λn. norm (f n)) ==> norm (suminf f) ≤ (∑n. norm (f n))
lemma summable_rabs:
summable (λn. ¦f n¦) ==> ¦suminf f¦ ≤ (∑n. ¦f n¦)
lemma norm_ratiotest_lemma:
[| c ≤ 0; norm x ≤ c * norm y |] ==> x = (0::'a)
lemma rabs_ratiotest_lemma:
[| c ≤ 0; ¦x¦ ≤ c * ¦y¦ |] ==> x = 0
lemma le_Suc_ex:
k ≤ l ==> ∃n. l = k + n
lemma le_Suc_ex_iff:
(k ≤ l) = (∃n. l = k + n)
lemma ratio_test_lemma2:
∀n≥N. norm (f (Suc n)) ≤ c * norm (f n) ==> 0 < c ∨ summable f
lemma ratio_test:
[| c < 1; ∀n≥N. norm (f (Suc n)) ≤ c * norm (f n) |] ==> summable f
lemma setsum_triangle_reindex:
(∑(i, j)∈{(i, j). i + j < n}. f i j) = (∑k = 0..<n. ∑i = 0..k. f i (k - i))
lemma Cauchy_product_sums:
[| summable (λk. norm (a k)); summable (λk. norm (b k)) |]
==> (λk. ∑i = 0..k. a i * b (k - i)) sums (suminf a * suminf b)
lemma Cauchy_product:
[| summable (λk. norm (a k)); summable (λk. norm (b k)) |]
==> suminf a * suminf b = (∑k. ∑i = 0..k. a i * b (k - i))