section ‹The Euler--MacLaurin summation formula› theory Euler_MacLaurin imports "HOL-Complex_Analysis.Complex_Analysis" Bernoulli.Periodic_Bernpoly Bernoulli.Bernoulli_FPS begin subsection ‹Auxiliary facts› (* TODO Move? *) lemma pbernpoly_of_int [simp]: "pbernpoly n (of_int a) = bernoulli n" by (simp add: pbernpoly_def) lemma continuous_on_bernpoly' [continuous_intros]: assumes "continuous_on A f" shows "continuous_on A (λx. bernpoly n (f x) :: 'a :: real_normed_algebra_1)" using continuous_on_compose2[OF continuous_on_bernpoly assms, of UNIV n] by auto lemma sum_atLeastAtMost_int_last: assumes "a < (b :: int)" shows "sum f {a..b} = sum f {a..<b} + f b" proof - from assms have "{a..b} = insert b {a..<b}" by auto also have "sum f … = sum f {a..<b} + f b" by (subst sum.insert) (auto simp: add_ac) finally show ?thesis . qed lemma sum_atLeastAtMost_int_head: assumes "a < (b :: int)" shows "sum f {a..b} = f a + sum f {a<..b}" proof - from assms have "{a..b} = insert a {a<..b}" by auto also have "sum f … = f a + sum f {a<..b}" by (subst sum.insert) auto finally show ?thesis . qed lemma not_in_nonpos_Reals_imp_add_nonzero: assumes "z ∉ ℝ⇩_{≤}⇩_{0}" "x ≥ 0" shows "z + of_real x ≠ 0" using assms by (auto simp: add_eq_0_iff2) (* END TODO *) lemma negligible_atLeastAtMostI: "b ≤ a ⟹ negligible {a..(b::real)}" by (cases "b < a") auto lemma integrable_on_negligible: "negligible A ⟹ (f :: 'n :: euclidean_space ⇒ 'a :: banach) integrable_on A" by (subst integrable_spike_set_eq[of _ "{}"]) (simp_all add: integrable_on_empty) lemma Union_atLeastAtMost_real_of_int: assumes "a < b" shows "(⋃n∈{a..<b}. {real_of_int n..real_of_int (n + 1)}) = {real_of_int a..real_of_int b}" proof (intro equalityI subsetI) fix x assume x: "x ∈ {real_of_int a..real_of_int b}" thus "x ∈ (⋃n∈{a..<b}. {real_of_int n..real_of_int (n + 1)})" proof (cases "x = real_of_int b") case True with assms show ?thesis by (auto intro!: bexI[of _ "b - 1"]) next case False with x have x: "x ≥ real_of_int a" "x < real_of_int b" by simp_all hence "x ≥ of_int ⌊x⌋" "x ≤ of_int ⌊x⌋ + 1" by linarith+ moreover from x have "⌊x⌋ ≥ a" "⌊x⌋ < b" by linarith+ ultimately have "∃n∈{a..<b}. x ∈ {of_int n..of_int (n + 1)}" by (intro bexI[of _ "⌊x⌋"]) simp_all thus ?thesis by blast qed qed auto subsection ‹The remainder terms› text ‹ The following describes the remainder term in the classical version of the Euler--MacLaurin formula. › definition EM_remainder' :: "nat ⇒ (real ⇒ 'a :: banach) ⇒ real ⇒ real ⇒ 'a" where "EM_remainder' n f a b = ((-1) ^ Suc n / fact n) *⇩_{R}integral {a..b} (λt. pbernpoly n t *⇩_{R}f t)" text ‹ Next, we define the remainder term that occurs when one lets the right bound of summation in the Euler--MacLaurin formula tend to infinity. › definition EM_remainder_converges :: "nat ⇒ (real ⇒ 'a :: banach) ⇒ int ⇒ bool" where "EM_remainder_converges n f a ⟷ (∃L. ((λx. EM_remainder' n f a (of_int x)) ⤏ L) at_top)" definition EM_remainder :: "nat ⇒ (real ⇒ 'a :: banach) ⇒ int ⇒ 'a" where "EM_remainder n f a = (if EM_remainder_converges n f a then Lim at_top (λx. EM_remainder' n f a (of_int x)) else 0)" text ‹ The following lemmas are fairly obvious -- but tedious to prove -- properties of the remainder terms. › lemma EM_remainder_eqI: fixes L assumes "((λx. EM_remainder' n f b (of_int x)) ⤏ L) at_top" shows "EM_remainder n f b = L" using assms by (auto simp: EM_remainder_def EM_remainder_converges_def intro!: tendsto_Lim) lemma integrable_EM_remainder'_int: fixes a b :: int and f :: "real ⇒ 'a :: banach" assumes "continuous_on {of_int a..of_int b} f" shows "(λt. pbernpoly n t *⇩_{R}f t) integrable_on {a..b}" proof - have [continuous_intros]: "continuous_on A f" if "A ⊆ {of_int a..of_int b}" for A using continuous_on_subset[OF assms that] . consider "a > b" | "a = b" | "a < b" "n = 1" | "a < b" "n ≠ 1" by (cases a b rule: linorder_cases) auto thus ?thesis proof cases assume "a < b" and "n ≠ 1" thus ?thesis by (intro integrable_continuous_real continuous_intros) auto next assume ab: "a < b" and [simp]: "n = 1" let ?A = "(λn. {real_of_int n..real_of_int (n+1)}) ` {a..<b}" show ?thesis proof (rule integrable_combine_division; (intro ballI)?) show "?A division_of {of_int a..of_int b}" using Union_atLeastAtMost_real_of_int[OF ab] by (simp add: division_of_def) next fix I assume "I ∈ ?A" then obtain i where i: "i ∈ {a..<b}" "I = {of_int i..of_int (i + 1)}" by auto show "(λt. pbernpoly n t *⇩_{R}f t) integrable_on I" proof (rule integrable_spike) show "(λt. (t - of_int i - 1/2) *⇩_{R}f t) integrable_on I" using i by (auto intro!: integrable_continuous_real continuous_intros) next fix x assume "x ∈ I - {of_int (i + 1)}" with i have "of_int i ≤ x" "x < of_int i + 1" by simp_all hence "floor x = i" by linarith thus "pbernpoly n x *⇩_{R}f x = (x - of_int i - 1 / 2) *⇩_{R}f x" by (simp add: pbernpoly_def bernpoly_def frac_def) qed simp_all qed qed (simp_all add: integrable_on_negligible) qed lemma integrable_EM_remainder': fixes a b :: real and f :: "real ⇒ 'a :: banach" assumes "continuous_on {a..b} f" shows "(λt. pbernpoly n t *⇩_{R}f t) integrable_on {a..b}" proof (cases "⌈a⌉ ≤ ⌊b⌋") case True define a' b' where "a' = ⌈a⌉" and "b' = ⌊b⌋" from True have *: "a' ≤ b'" "a' ≥ a" "b' ≤ b" by (auto simp: a'_def b'_def) from * have A: "(λt. pbernpoly n t *⇩_{R}f t) integrable_on ({a'..b'})" by (intro integrable_EM_remainder'_int continuous_on_subset[OF assms]) auto have B: "(λt. pbernpoly n t *⇩_{R}f t) integrable_on ({a..a'})" proof (rule integrable_spike) show "pbernpoly n x *⇩_{R}f x = bernpoly n (x - of_int (floor a)) *⇩_{R}f x" if "x ∈ {a..real_of_int a'} - {real_of_int a'}"for x proof - have "x ≥ a" "x <real_of_int a'" using that by auto with True have "floor x = floor a" unfolding a'_def using ceiling_diff_floor_le_1[of a] by (intro floor_unique; linarith) thus ?thesis by (simp add: pbernpoly_def frac_def) qed qed (insert *, auto intro!: continuous_intros integrable_continuous_real continuous_on_subset[OF assms]) have C: "(λt. pbernpoly n t *⇩_{R}f t) integrable_on ({b'..b})" proof (rule integrable_spike) show "pbernpoly n x *⇩_{R}f x = bernpoly n (x - of_int b') *⇩_{R}f x" if "x ∈ {real_of_int b'..b} - {real_of_int b'}" for x proof - have "x ≤ b" "x > real_of_int b'" using that by auto with True have "floor x = b'" unfolding b'_def by (intro floor_unique; linarith) thus ?thesis by (simp add: pbernpoly_def frac_def) qed qed (insert *, auto intro!: continuous_intros integrable_continuous_real continuous_on_subset[OF assms]) have "(λt. pbernpoly n t *⇩_{R}f t) integrable_on ({a..a'} ∪ {a'..b'} ∪ {b'..b})" using * A B C by (intro integrable_Un; (subst ivl_disj_un)?) (auto simp: ivl_disj_un max_def min_def) also have "{a..a'} ∪ {a'..b'} ∪ {b'..b} = {a..b}" using * by auto finally show ?thesis . next assume *: "¬ceiling a ≤ floor b" show ?thesis proof (rule integrable_spike) show "(λt. bernpoly n (t - floor a) *⇩_{R}f t) integrable_on {a..b}" using * by (auto intro!: integrable_continuous_real continuous_intros assms) next show "pbernpoly n x *⇩_{R}f x = bernpoly n (x - floor a) *⇩_{R}f x" if "x ∈ {a..b} - {}" for x proof - from * have **: "b < floor a + 1" unfolding ceiling_altdef by (auto split: if_splits simp: le_floor_iff) from that have x: "x ≥ a" "x ≤ b" by simp_all with * ** have "floor x = floor a" by linarith thus ?thesis by (simp add: pbernpoly_def frac_def) qed qed simp_all qed lemma EM_remainder'_bounded_linear: assumes "bounded_linear h" assumes "continuous_on {a..b} f" shows "EM_remainder' n (λx. h (f x)) a b = h (EM_remainder' n f a b)" proof - have "integral {a..b} (λt. pbernpoly n t *⇩_{R}h (f t)) = integral {a..b} (λt. h (pbernpoly n t *⇩_{R}f t))" using assms by (simp add: linear_simps) also have "… = h (integral {a..b} (λt. pbernpoly n t *⇩_{R}f t))" by (subst integral_linear [OF _ assms(1), symmetric]) (auto intro!: integrable_EM_remainder' assms(2) simp: o_def) finally show ?thesis using assms(1) by (simp add: EM_remainder'_def linear_simps) qed lemma EM_remainder_converges_of_real: assumes "EM_remainder_converges n f a" "continuous_on {of_int a..} f" shows "EM_remainder_converges n (λx. of_real (f x)) a" proof - from assms obtain L where L: "((λb. EM_remainder' n f (real_of_int a) (real_of_int b)) ⤏ L) at_top" by (auto simp: EM_remainder_converges_def) have "((λb. EM_remainder' n (λx. of_real (f x)) (of_int a) (of_int b)) ⤏ of_real L) at_top" proof (rule Lim_transform_eventually) show "eventually (λb. of_real (EM_remainder' n f (of_int a) (of_int b)) = EM_remainder' n (λx. of_real (f x)) (of_int a) (of_int b)) at_top" using eventually_ge_at_top[of a] by eventually_elim (intro EM_remainder'_bounded_linear [OF bounded_linear_of_real, symmetric] continuous_on_subset[OF assms(2)], auto) qed (intro tendsto_intros L) thus ?thesis unfolding EM_remainder_converges_def .. qed lemma EM_remainder_converges_of_real_iff: fixes f :: "real ⇒ real" assumes "continuous_on {of_int a..} f" shows "EM_remainder_converges n (λx. of_real (f x) :: 'a :: {banach, real_normed_algebra_1, real_inner}) a ⟷ EM_remainder_converges n f a" proof assume "EM_remainder_converges n (λx. of_real (f x) :: 'a) a" then obtain L :: 'a where L: "((λb. EM_remainder' n (λx. of_real (f x)) (of_int a) (of_int b)) ⤏ L) at_top" by (auto simp: EM_remainder_converges_def) have "((λb. EM_remainder' n f (of_int a) (of_int b)) ⤏ L ∙ 1) at_top" proof (rule Lim_transform_eventually) show "eventually (λb. EM_remainder' n (λx. of_real (f x) :: 'a) (of_int a) (of_int b) ∙ 1 = EM_remainder' n f (of_int a) (of_int b)) at_top" using eventually_ge_at_top[of a] by eventually_elim (subst EM_remainder'_bounded_linear [OF bounded_linear_of_real], auto intro!: continuous_on_subset[OF assms]) qed (intro tendsto_intros L) thus "EM_remainder_converges n f a" unfolding EM_remainder_converges_def .. qed (intro EM_remainder_converges_of_real assms) lemma EM_remainder_of_real: assumes "continuous_on {a..} f" shows "EM_remainder n (λx. of_real (f x) :: 'a :: {banach, real_normed_algebra_1, real_inner}) a = of_real (EM_remainder n f a)" proof - have eq: "EM_remainder' n (λx. of_real (f x) :: 'a) (real_of_int a) = (λx::int. of_real (EM_remainder' n f a x))" by (intro ext EM_remainder'_bounded_linear[OF bounded_linear_of_real] continuous_on_subset[OF assms]) auto show ?thesis proof (cases "EM_remainder_converges n f a") case False with EM_remainder_converges_of_real_iff[OF assms, of n] show ?thesis by (auto simp: EM_remainder_def) next case True then obtain L where L: "((λx. EM_remainder' n f a (real_of_int x)) ⤏ L) at_top" by (auto simp: EM_remainder_converges_def) have L': "((λx. EM_remainder' n (λx. of_real (f x) :: 'a) a (real_of_int x)) ⤏ of_real L) at_top" unfolding eq by (intro tendsto_of_real L) from L L' tendsto_Lim[OF _ L] tendsto_Lim[OF _ L'] show ?thesis by (auto simp: EM_remainder_def EM_remainder_converges_def) qed qed lemma EM_remainder'_cong: assumes "⋀x. x ∈ {a..b} ⟹ f x = g x" "n = n'" "a = a'" "b = b'" shows "EM_remainder' n f a b = EM_remainder' n' g a' b'" proof - have "integral {a..b} (λt. pbernpoly n t *⇩_{R}f t) = integral {a'..b'} (λt. pbernpoly n' t *⇩_{R}g t)" unfolding assms using assms by (intro integral_cong) auto with assms show ?thesis by (simp add: EM_remainder'_def) qed lemma EM_remainder_converges_cong: assumes "⋀x. x ≥ of_int a ⟹ f x = g x" "n = n'" "a = a'" shows "EM_remainder_converges n f a = EM_remainder_converges n' g a'" unfolding EM_remainder_converges_def by (subst EM_remainder'_cong[OF _ refl refl refl, of _ _ f g]) (use assms in auto) lemma EM_remainder_cong: assumes "⋀x. x ≥ of_int a ⟹ f x = g x" "n = n'" "a = a'" shows "EM_remainder n f a = EM_remainder n' g a'" proof - have *: "EM_remainder_converges n f a = EM_remainder_converges n' g a'" using assms by (intro EM_remainder_converges_cong) auto show ?thesis unfolding EM_remainder_def by (subst EM_remainder'_cong[OF _ refl refl refl, of _ _ f g]) (use assms * in auto) qed lemma EM_remainder_converges_cnj: assumes "continuous_on {a..} f" and "EM_remainder_converges n f a" shows "EM_remainder_converges n (λx. cnj (f x)) a" proof - interpret bounded_linear cnj by (rule bounded_linear_cnj) obtain L where L: "((λx. EM_remainder' n f (real_of_int a) (real_of_int x)) ⤏ L) at_top" using assms unfolding EM_remainder_converges_def by blast note tendsto_cnj [OF this] also have "(λx. cnj (EM_remainder' n f (real_of_int a) (real_of_int x))) = (λx. EM_remainder' n (λx. cnj (f x)) (real_of_int a) (real_of_int x))" by (subst EM_remainder'_bounded_linear [OF bounded_linear_cnj]) (rule continuous_on_subset [OF assms(1)], auto) finally have L': "(… ⤏ cnj L) at_top" . thus "EM_remainder_converges n (λx. cnj (f x)) a" by (auto simp: EM_remainder_converges_def) qed lemma EM_remainder_converges_cnj_iff: assumes "continuous_on {of_int a..} f" shows "EM_remainder_converges n (λx. cnj (f x)) a ⟷ EM_remainder_converges n f a" proof assume "EM_remainder_converges n (λx. cnj (f x)) a" hence "EM_remainder_converges n (λx. cnj (cnj (f x))) a" by (rule EM_remainder_converges_cnj [rotated]) (auto intro: continuous_intros assms) thus "EM_remainder_converges n f a" by simp qed (intro EM_remainder_converges_cnj assms) lemma EM_remainder_cnj: assumes "continuous_on {a..} f" shows "EM_remainder n (λx. cnj (f x)) a = cnj (EM_remainder n f a)" proof (cases "EM_remainder_converges n f a") case False hence "¬EM_remainder_converges n (λx. cnj (f x)) a" by (subst EM_remainder_converges_cnj_iff [OF assms]) with False show ?thesis by (simp add: EM_remainder_def) next case True then obtain L where L: "((λx. EM_remainder' n f (real_of_int a) (real_of_int x)) ⤏ L) at_top" unfolding EM_remainder_converges_def by blast note tendsto_cnj [OF this] also have "(λx. cnj (EM_remainder' n f (real_of_int a) (real_of_int x))) = (λx. EM_remainder' n (λx. cnj (f x)) (real_of_int a) (real_of_int x))" by (subst EM_remainder'_bounded_linear [OF bounded_linear_cnj]) (rule continuous_on_subset [OF assms(1)], auto) finally have L': "(… ⤏ cnj L) at_top" . moreover from assms and L have "EM_remainder n f a = L" by (intro EM_remainder_eqI) ultimately show "EM_remainder n (λx. cnj (f x)) a = cnj (EM_remainder n f a)" using L' by (intro EM_remainder_eqI) simp_all qed lemma EM_remainder'_combine: fixes f :: "real ⇒ 'a :: banach" assumes [continuous_intros]: "continuous_on {a..c} f" assumes "a ≤ b" "b ≤ c" shows "EM_remainder' n f a b + EM_remainder' n f b c = EM_remainder' n f a c" proof - have "integral {a..b} (λt. pbernpoly n t *⇩_{R}f t) + integral {b..c} (λt. pbernpoly n t *⇩_{R}f t) = integral {a..c} (λt. pbernpoly n t *⇩_{R}f t)" by (intro Henstock_Kurzweil_Integration.integral_combine assms integrable_EM_remainder') from this [symmetric] show ?thesis by (simp add: EM_remainder'_def algebra_simps) qed lemma uniformly_convergent_EM_remainder': fixes f :: "'a ⇒ real ⇒ 'b :: {banach,real_normed_algebra}" assumes deriv: "⋀y. a ≤ y ⟹ (G has_real_derivative g y) (at y within {a..})" assumes integrable: "⋀a' b y. y ∈ A ⟹ a ≤ a' ⟹ a' ≤ b ⟹ (λt. pbernpoly n t *⇩_{R}f y t) integrable_on {a'..b}" assumes conv: "convergent (λy. G (real y))" assumes bound: "eventually (λx. ∀y∈A. norm (f y x) ≤ g x) at_top" shows "uniformly_convergent_on A (λb s. EM_remainder' n (f s) a b)" proof - interpret bounded_linear "λx::'b. ((- 1) ^ Suc n / fact n) *⇩_{R}x" by (rule bounded_linear_scaleR_right) from bounded_pbernpoly obtain C where C: "⋀x. norm (pbernpoly n x) ≤ C" by auto from C[of 0] have [simp]: "C ≥ 0" by simp show ?thesis unfolding EM_remainder'_def proof (intro uniformly_convergent_on uniformly_convergent_improper_integral') fix x assume "x ≥ a" thus "((λx. C * G x) has_real_derivative C * g x) (at x within {a..})" by (intro DERIV_cmult deriv) next fix y a' b assume "y ∈ A" "a ≤ a'" "a' ≤ b" thus "(λt. pbernpoly n t *⇩_{R}f y t) integrable_on {a'..b}" by (rule integrable) next from conv obtain L where "(λy. G (real y)) ⇢ L" by (auto simp: convergent_def) from tendsto_mult[OF tendsto_const[of C] this] show "convergent (λy. C * G (real y))" by (auto simp: convergent_def) next show "∀⇩_{F}x in at_top. ∀y∈A. norm (pbernpoly n x *⇩_{R}f y x) ≤ C * g x" using C unfolding norm_scaleR by (intro eventually_mono[OF bound] ballI mult_mono) auto qed qed lemma uniform_limit_EM_remainder: fixes f :: "'a ⇒ real ⇒ 'b :: {banach,real_normed_algebra}" assumes deriv: "⋀y. a ≤ y ⟹ (G has_real_derivative g y) (at y within {a..})" assumes integrable: "⋀a' b y. y ∈ A ⟹ a ≤ a' ⟹ a' ≤ b ⟹ (λt. pbernpoly n t *⇩_{R}f y t) integrable_on {a'..b}" assumes conv: "convergent (λy. G (real y))" assumes bound: "eventually (λx. ∀y∈A. norm (f y x) ≤ g x) at_top" shows "uniform_limit A (λb s. EM_remainder' n (f s) a b) (λs. EM_remainder n (f s) a) sequentially" proof - have *: "uniformly_convergent_on A (λb s. EM_remainder' n (f s) a b)" by (rule uniformly_convergent_EM_remainder'[OF assms]) also have "?this ⟷ ?thesis" unfolding uniformly_convergent_uniform_limit_iff proof (intro uniform_limit_cong refl always_eventually allI ballI) fix s assume "s ∈ A" with * have **: "convergent (λb. EM_remainder' n (f s) a b)" by (rule uniformly_convergent_imp_convergent) show "lim (λb. EM_remainder' n (f s) a b) = EM_remainder n (f s) a" proof (rule sym, rule EM_remainder_eqI) have "((λx. EM_remainder' n (f s) (real_of_int a) (real x)) ⤏ lim (λx. EM_remainder' n (f s) (real_of_int a) (real x))) at_top" (is "(_ ⤏ ?L) _") using ** unfolding convergent_LIMSEQ_iff by blast hence "((λx. EM_remainder' n (f s) (real_of_int a) (real (nat x))) ⤏ ?L) at_top" by (rule filterlim_compose) (fact filterlim_nat_sequentially) thus "((λx. EM_remainder' n (f s) (real_of_int a) (real_of_int x)) ⤏ ?L) at_top" by (rule Lim_transform_eventually) (auto intro: eventually_mono[OF eventually_ge_at_top[of 0]]) qed qed finally show … . qed lemma tendsto_EM_remainder: fixes f :: "real ⇒ 'b :: {banach,real_normed_algebra}" assumes deriv: "⋀y. a ≤ y ⟹ (G has_real_derivative g y) (at y within {a..})" assumes integrable: "⋀a' b . a ≤ a' ⟹ a' ≤ b ⟹ (λt. pbernpoly n t *⇩_{R}f t) integrable_on {a'..b}" assumes conv: "convergent (λy. G (real y))" assumes bound: "eventually (λx. norm (f x) ≤ g x) at_top" shows "filterlim (λb. EM_remainder' n f a b) (nhds (EM_remainder n f a)) sequentially" proof - have "uniform_limit {()} (λb s. EM_remainder' n f a b) (λs. EM_remainder n f a) sequentially" using assms by (intro uniform_limit_EM_remainder[where G = G and g = g]) auto moreover have "() ∈ {()}" by simp ultimately show ?thesis by (rule tendsto_uniform_limitI) qed lemma EM_remainder_0 [simp]: "EM_remainder n (λx. 0) a = 0" by (rule EM_remainder_eqI) (simp add: EM_remainder'_def) lemma holomorphic_EM_remainder': assumes deriv: "⋀z t. z ∈ U ⟹ t ∈ {a..x} ⟹ ((λz. f z t) has_field_derivative f' z t) (at z within U)" assumes int: "⋀b c z e. a ≤ b ⟹ c ≤ x ⟹ z ∈ U ⟹ (λt. of_real (bernpoly n (t - e)) * f z t) integrable_on {b..c}" assumes cont: "continuous_on (U × {a..x}) (λ(z, t). f' z t)" assumes "convex U" shows "(λs. EM_remainder' n (f s) a x) holomorphic_on U" unfolding EM_remainder'_def scaleR_conv_of_real proof (intro holomorphic_intros) have holo: "(λz. integral (cbox b c) (λt. of_real (bernpoly n (t - e)) * f z t)) holomorphic_on U" if "b ≥ a" "c ≤ x" for b c e :: real proof (rule leibniz_rule_holomorphic) fix z t assume "z ∈ U" "t ∈ cbox b c" thus "((λz. complex_of_real (bernpoly n (t - e)) * f z t) has_field_derivative complex_of_real (bernpoly n (t - e)) * f' z t) (at z within U)" using that by (intro DERIV_cmult deriv) auto next fix z assume "z ∈ U" thus "(λt. complex_of_real (bernpoly n (t - e)) * f z t) integrable_on cbox b c" using that int[of b c z] by auto next have "continuous_on (U × {b..c}) (λ(z, t). f' z t)" using cont by (rule continuous_on_subset) (insert that, auto) thus "continuous_on (U × cbox b c) (λ(z, t). complex_of_real (bernpoly n (t - e)) * f' z t)" by (auto simp: case_prod_unfold intro!: continuous_intros) qed fact+ consider "a > x" | "a ≤ x" "floor x ≤ a" | "a ≤ x" "floor x > a" by force hence "(λz. integral (cbox a x) (λt. of_real (pbernpoly n t) * f z t)) holomorphic_on U" (is "?f a x holomorphic_on _") proof cases case 2 have "(λz. integral (cbox a x) (λt. of_real (bernpoly n (t - of_int ⌊x⌋)) * f z t)) holomorphic_on U" by (intro holo) auto also have "(λz. integral (cbox a x) (λt. of_real (bernpoly n (t - of_int ⌊x⌋)) * f z t)) = ?f a x" proof (intro ext integral_cong, goal_cases) case (1 z t) hence "t ≥ a" "t ≤ x" by auto hence "floor t = floor x" using 2 by linarith thus ?case by (simp add: pbernpoly_def frac_def) qed finally show ?thesis . next case 3 define N :: "int set" where "N = {⌈a⌉..<⌊x⌋}" define A where "A = insert {a..of_int ⌈a⌉} (insert {of_int ⌊x⌋..x} ((λn. {of_int n..of_int n + 1}) ` N))" { fix X assume "X ∈ A" then consider "X = {a..of_int ⌈a⌉}" | "X = {of_int ⌊x⌋..x}" | n where "X = {of_int n..of_int n + 1}" "n ∈ N" by (auto simp: A_def) } note A_cases = this have division: "A division_of {a..x}" proof (rule division_ofI) show "finite A" by (auto simp: A_def N_def) fix K assume K: "K ∈ A" from 3 have "of_int ⌈a⌉ ≤ x" using ceiling_le[of a "floor x"] by linarith moreover from 3 have "of_int ⌊x⌋ ≥ a" by linarith ultimately show "K ⊆ {a..x}" using K 3 by (auto simp: A_def N_def) linarith+ from K show "K ≠ {}" and "∃a b. K = cbox a b" by (auto simp: A_def) next fix K1 K2 assume K: "K1 ∈ A" "K2 ∈ A" "K1 ≠ K2" have F1: "interior {a..⌈a⌉} ∩ interior {⌊x⌋..x} = {}" using 3 ceiling_le[of a "floor x"] by (auto simp: min_def max_def) hence F2: "interior {⌊x⌋..x} ∩ interior {a..⌈a⌉} = {}" by simp have F3: "interior {a..⌈a⌉} ∩ interior {of_int n..of_int n+1} = {}" "interior {⌊x⌋..x} ∩ interior {of_int n..of_int n+1} = {}" "interior {of_int n..of_int n+1} ∩ interior {a..⌈a⌉} = {}" "interior {of_int n..of_int n+1} ∩ interior {⌊x⌋..x} = {}"if "n ∈ N" for n using 3 ceiling_le[of a "floor x"] that by (auto simp: min_def max_def N_def) have F4: "interior {real_of_int n..of_int n+1} ∩ interior {of_int m..of_int m+1} = {}" if "{real_of_int n..of_int n+1} ≠ {of_int m..of_int m+1}" for m n proof - from that have "n ≠ m" by auto thus ?thesis by simp qed from F1 F2 F3 F4 K show "interior K1 ∩ interior K2 = {}" by (elim A_cases) (simp_all only: not_False_eq_True) next show "⋃A = {a..x}" proof (cases "⌈a⌉ = ⌊x⌋") case True thus ?thesis using 3 by (auto simp: A_def N_def intro: order.trans) linarith+ next case False with 3 have *: "⌈a⌉ < ⌊x⌋" by linarith have "⋃A = {a..of_int ⌈a⌉} ∪ (⋃n∈N. {of_int n..of_int (n + 1)}) ∪ {of_int ⌊x⌋..x}" by (simp add: A_def Un_ac) also have "(⋃n∈N. {of_int n..of_int (n + 1)}) = {of_int ⌈a⌉..real_of_int ⌊x⌋}" using * unfolding N_def by (intro Union_atLeastAtMost_real_of_int) also have "{a..of_int ⌈a⌉} ∪ … = {a..real_of_int ⌊x⌋}" using 3 * by (intro ivl_disj_un) auto also have "… ∪ {of_int ⌊x⌋..x} = {a..x}" using 3 * by (intro ivl_disj_un) auto finally show ?thesis . qed qed have "(λz. ∑X∈A. integral X (λt. of_real (bernpoly n (t - ⌊Inf X⌋)) * f z t)) holomorphic_on U" proof (intro holomorphic_on_sum holo, goal_cases) case (1 X) from 1 and division have subset: "X ⊆ {a..x}" by (auto simp: division_of_def) from 1 obtain b c where [simp]: "X = cbox b c" "b ≤ c" by (auto simp: A_def) from subset have "b ≥ a" "c ≤ x" by auto hence "(λx. integral (cbox b c) (λt. of_real (bernpoly n (t - ⌊Inf {b..c}⌋)) * f x t)) holomorphic_on U" by (intro holo) auto thus ?case by simp qed also have "?this ⟷ (λz. integral {a..x} (λt. of_real (pbernpoly n t) * f z t)) holomorphic_on U" proof (intro holomorphic_cong refl, goal_cases) case (1 z) have "((λt. of_real (pbernpoly n t) * f z t) has_integral (∑X∈A. integral X (λt. of_real (bernpoly n (t - ⌊Inf X⌋)) * f z t))) {a..x}" using division proof (rule has_integral_combine_division) fix X assume X: "X ∈ A" then obtain b c where X': "X = {b..c}" "b ≤ c" by (elim A_cases) auto from X and division have "X ⊆ {a..x}" by (auto simp: division_of_def) with X' have bc: "b ≥ a" "c ≤ x" by auto have "((λt. of_real (bernpoly n (t - of_int ⌊Inf X⌋)) * f z t) has_integral integral X (λt. of_real (bernpoly n (t - of_int ⌊Inf X⌋)) * f z t)) X" unfolding X' using ‹z ∈ U› bc by (intro integrable_integral int) also have "?this ⟷ ((λt. of_real (pbernpoly n t) * f z t) has_integral integral X (λt. of_real (bernpoly n (t - of_int ⌊Inf X⌋)) * f z t)) X" proof (rule has_integral_spike_eq[of "{Sup X}"], goal_cases) case (2 t) note t = this from ‹X ∈ A› have "⌊t⌋ = ⌊Inf X⌋" proof (cases rule: A_cases [consumes 1]) case 1 with t show ?thesis by (intro floor_unique) (auto simp: ceiling_altdef split: if_splits, (linarith+)?) next case 2 with t show ?thesis by (intro floor_unique) (auto simp: ceiling_altdef split: if_splits, (linarith+)?) next case 3 with t show ?thesis by (intro floor_unique) (auto simp: ceiling_altdef N_def split: if_splits) qed thus ?case by (simp add: pbernpoly_def frac_def) qed auto finally show … . qed thus ?case by (simp add: has_integral_iff) qed finally show ?thesis by simp qed auto thus "(λz. integral {a..x} (λt. of_real (pbernpoly n t) * f z t)) holomorphic_on U" by simp qed lemma assumes deriv: "⋀y. a ≤ y ⟹ (G has_real_derivative g y) (at y within {a..})" assumes deriv': "⋀z t x. z ∈ U ⟹ x ≥ a ⟹ t ∈ {a..x} ⟹ ((λz. f z t) has_field_derivative f' z t) (at z within U)" assumes cont: "continuous_on (U × {of_int a..}) (λ(z, t). f' z t)" assumes int: "⋀b c z e. a ≤ b ⟹ z ∈ U ⟹ (λt. of_real (bernpoly n (t - e)) * f z t) integrable_on {b..c}" assumes int': "⋀a' b y. y ∈ U ⟹ a ≤ a' ⟹ a' ≤ b ⟹ (λt. pbernpoly n t *⇩_{R}f y t) integrable_on {a'..b}" assumes conv: "convergent (λy. G (real y))" assumes bound: "eventually (λx. ∀y∈U. norm (f y x) ≤ g x) at_top" assumes "open U" shows analytic_EM_remainder: "(λs::complex. EM_remainder n (f s) a) analytic_on U" and holomorphic_EM_remainder: "(λs::complex. EM_remainder n (f s) a) holomorphic_on U" proof - show "(λs::complex. EM_remainder n (f s) a) analytic_on U" unfolding analytic_on_def proof fix z assume "z ∈ U" from ‹z ∈ U› and ‹open U› obtain ε where ε: "ε > 0" "ball z ε ⊆ U" by (auto simp: open_contains_ball) have "(λs. EM_remainder n (f s) a) holomorphic_on ball z ε" proof (rule holomorphic_uniform_sequence) fix x :: nat show "(λs. EM_remainder' n (f s) a x) holomorphic_on ball z ε" proof (rule holomorphic_EM_remainder', goal_cases) fix s t assume "s ∈ ball z ε" "t ∈ {real_of_int a..real x}" thus "((λz. f z t) has_field_derivative f' s t) (at s within ball z ε)" using ε by (intro DERIV_subset[OF deriv'[of _ x]]) auto next case (2 b c s e) with ε have "s ∈ U" by blast with 2 show ?case using ε int[of b s e c] by (cases "a ≤ x") auto next from cont show "continuous_on (ball z ε × {real_of_int a..real x}) (λ(z, t). f' z t)" by (rule continuous_on_subset) (insert ε, auto) qed (auto) next fix s assume s: "s ∈ ball z ε" have "open (ball z ε)" by simp with s obtain δ where δ: "δ > 0" "cball s δ ⊆ ball z ε" unfolding open_contains_cball by blast moreover have bound': "eventually (λx. ∀y∈cball s δ. norm (f y x) ≤ g x) at_top" by (intro eventually_mono [OF bound]) (insert δ ε, auto) have "uniform_limit (cball s δ) (λx s. EM_remainder' n (f s) (real_of_int a) (real x)) (λs. EM_remainder n (f s) a) sequentially" by (rule uniform_limit_EM_remainder[OF deriv int' conv bound']) (insert δ ε s, auto) ultimately show "∃δ>0. cball s δ ⊆ ball z ε ∧ uniform_limit (cball s δ) (λx s. EM_remainder' n (f s) (real_of_int a) (real x)) (λs. EM_remainder n (f s) a) sequentially" by blast qed auto with ε show "∃ε>0. (λs. EM_remainder n (f s) a) holomorphic_on ball z ε" by blast qed thus "(λs::complex. EM_remainder n (f s) a) holomorphic_on U" by (rule analytic_imp_holomorphic) qed text ‹ The following lemma is the first step in the proof of the Euler--MacLaurin formula: We show the relationship between the first-order remainder term and the difference of the integral and the sum. › context fixes f f' :: "real ⇒ 'a :: banach" fixes a b :: int and I S :: 'a fixes Y :: "real set" assumes "a ≤ b" assumes fin: "finite Y" assumes cont: "continuous_on {real_of_int a..real_of_int b} f" assumes deriv [derivative_intros]: "⋀x::real. x ∈ {a..b} - Y ⟹ (f has_vector_derivative f' x) (at x)" defines S_def: "S ≡ (∑i∈{a<..b}. f i)" and I_def: "I ≡ integral {a..b} f" begin lemma diff_sum_integral_has_integral_int: "((λt. (frac t - 1/2) *⇩_{R}f' t) has_integral (S - I - (f b - f a) /⇩_{R}2)) {a..b}" proof (cases "a = b") case True thus ?thesis by (simp_all add: S_def I_def has_integral_refl) next case False with ‹a ≤ b› have ab: "a < b" by simp let ?A = "(λn. {real_of_int n..real_of_int (n+1)}) ` {a..<b}" have division: "?A division_of {of_int a..of_int b}" using Union_atLeastAtMost_real_of_int[OF ab] by (simp add: division_of_def) have cont' [continuous_intros]: "continuous_on A f" if "A ⊆ {of_int a..of_int b}" for A using continuous_on_subset[OF cont that] . define d where "d = (λx. (f x + f (x + 1)) /⇩_{R}2 - integral {x..x+1} f)" have "((λt. (frac t - 1/2) *⇩_{R}f' t) has_integral d i) {of_int i..of_int (i+1)}" if i: "i ∈ {a..<b}" for i proof (rule has_integral_spike) show "(frac x - 1 / 2) *⇩_{R}f' x = (x - of_int i - 1 / 2) *⇩_{R}f' x" if "x ∈ {of_int i..of_int (i + 1)} - {of_int (i + 1)}" for x proof - have "x ≥ of_int i" "x < of_int (i + 1)" using that by auto hence "floor x = of_int i" by (subst floor_unique) auto thus ?thesis by (simp add: frac_def) qed next define h where "h = (λx::real. (x - of_int i - 1 / 2) *⇩_{R}f' x)" define g where "g = (λx::real. (x - of_int i - 1/2) *⇩_{R}f x - integral {of_int i..x} f)" have *: "((λx. integral {real_of_int i..x} f) has_vector_derivative f x) (at x within {i..i+1})" if "x ∈ {of_int i<..<of_int i + 1}" for x using that i by (intro integral_has_vector_derivative cont') auto have "((λx. integral {real_of_int i..x} f) has_vector_derivative f x) (at x)" if "x ∈ {of_int i<..<of_int i + 1}" for x using that i at_within_interior[of x "{of_int i..of_int (i + 1)}"] *[of x] by simp hence "(h has_integral g (of_int (i + 1)) - g (of_int i)) {of_int i..of_int (i+1)}" unfolding g_def h_def using that by (intro fundamental_theorem_of_calculus_interior_strong[OF fin]) (auto intro!: derivative_eq_intros continuous_intros indefinite_integral_continuous_1 integrable_continuous_real) also have "g (of_int (i + 1)) - g (of_int i) = d i" by (simp add: g_def scaleR_add_right [symmetric] d_def) finally show "(h has_integral d i) {of_int i..of_int (i + 1)}" . qed simp_all hence *: "⋀I. I∈?A ⟹ ((λx. (frac x - 1 / 2) *⇩_{R}f' x) has_integral d (⌊Inf I⌋)) I" by (auto simp: add_ac) have "((λx::real. (frac x - 1 / 2) *⇩_{R}f' x) has_integral (∑I∈?A. d (⌊Inf I⌋))) (⋃?A)" by (intro has_integral_Union * finite_imageI) (force intro!: negligible_atLeastAtMostI pairwiseI)+ also have "⋃?A = {of_int a..of_int b}" by (intro Union_atLeastAtMost_real_of_int ab) also have "(∑I∈?A. d (⌊Inf I⌋)) = (∑i=a..<b. d i)" by (subst sum.reindex) (auto simp: inj_on_def) also have "… = (1 / 2) *⇩_{R}((∑i = a..<b. f (real_of_int i)) + (∑i = a..<b. f (real_of_int (i + 1)))) - (∑i = a..<b. integral {real_of_int i..1 + real_of_int i} f)" (is "_ = _ *⇩_{R}(?S1 + ?S2) - ?S3") by (simp add: d_def algebra_simps sum.distrib sum_subtractf scaleR_sum_right) also have "?S1 = (∑i = a..b. f (real_of_int i)) - f b" unfolding S_def using ab by (subst sum_atLeastAtMost_int_last) auto also have "(∑i = a..b. f (real_of_int i)) = S + f a" unfolding S_def using ab by (subst sum_atLeastAtMost_int_head) auto also have "?S2 = S" unfolding S_def by (intro sum.reindex_bij_witness[of _ "λi. i-1" "λi. i+1"]) auto also have "(1 / 2) *⇩_{R}(S + f a - f b + S) = (1/2) *⇩_{R}S + (1/2) *⇩_{R}S - (f b - f a) /⇩_{R}2" by (simp add: algebra_simps) also have "(1/2) *⇩_{R}S + (1/2) *⇩_{R}S = S" by (simp add: scaleR_add_right [symmetric]) also have "?S3 = (∑I∈?A. integral I f)" by (subst sum.reindex) (auto simp: inj_on_def add_ac) also have "… = I" unfolding I_def by (intro integral_combine_division_topdown [symmetric] division integrable_continuous_real continuous_intros) simp_all finally show ?thesis by (simp add: algebra_simps) qed lemma diff_sum_integral_has_integral_int': "((λt. pbernpoly 1 t *⇩_{R}f' t) has_integral (S - I - (f b - f a) /⇩_{R}2 )) {a..b}" using diff_sum_integral_has_integral_int by (simp add: pbernpoly_def bernpoly_def) lemma EM_remainder'_Suc_0: "EM_remainder' (Suc 0) f' a b = S - I - (f b - f a) /⇩_{R}2" using diff_sum_integral_has_integral_int' by (simp add: has_integral_iff EM_remainder'_def) end text ‹ Next, we show that the $n$-th-order remainder can be expressed in terms of the $n+1$-th-order remainder term. Iterating this essentially yields the Euler--MacLaurin formula. › context fixes f f' :: "real ⇒ 'a :: banach" and a b :: int and n :: nat and A :: "real set" assumes ab: "a ≤ b" and n: "n > 0" assumes fin: "finite A" assumes cont: "continuous_on {of_int a..of_int b} f" assumes cont': "continuous_on {of_int a..of_int b} f'" assumes deriv: "⋀x. x ∈ {of_int a<..<of_int b} - A ⟹ (f has_vector_derivative f' x) (at x)" begin lemma EM_remainder'_integral_conv_Suc: shows "integral {a..b} (λt. pbernpoly n t *⇩_{R}f t) = (bernoulli (Suc n) / real (Suc n)) *⇩_{R}(f b - f a) - integral {a..b} (λt. pbernpoly (Suc n) t *⇩_{R}f' t) /⇩_{R}real (Suc n)" unfolding EM_remainder'_def proof - let ?h = "λi. (pbernpoly (Suc n) (real_of_int i) / real (Suc n)) *⇩_{R}f (real_of_int i)" define T where "T = integral {a..b} (λt. (pbernpoly (Suc n) t / real (Suc n)) *⇩_{R}f' t)" note [derivative_intros] = has_field_derivative_pbernpoly_Suc' let ?A = "real_of_int ` {a..b} ∪ A" have "((λt. pbernpoly n t *⇩_{R}f t) has_integral (-T + (?h b - ?h a))) {a..b}" proof (rule integration_by_parts_interior_strong[OF bounded_bilinear_scaleR]) from fin show "finite ?A" by simp from ‹n > 0› show "continuous_on {of_int a..of_int b} (λt. pbernpoly (Suc n) t / real (Suc n))" by (intro continuous_intros) auto show "continuous_on {of_int a..of_int b} f" by fact show "(f has_vector_derivative f' t) (at t)" if "t ∈ {of_int a<..<of_int b} - ?A" for t using deriv[of t] that by auto have "(λt. pbernpoly (Suc n) t *⇩_{R}f' t) integrable_on {a..b}" by (intro integrable_EM_remainder' cont') hence "(λt. (1 / real (Suc n)) *⇩_{R}pbernpoly (Suc n) t *⇩_{R}f' t) integrable_on {a..b}" by (rule integrable_cmul) also have "(λt. (1 / real (Suc n)) *⇩_{R}pbernpoly (Suc n) t *⇩_{R}f' t) = (λt. (pbernpoly (Suc n) t / real (Suc n)) *⇩_{R}f' t)" by (rule ext) (simp add: algebra_simps) finally show "((λt. (pbernpoly (Suc n) t / real (Suc n)) *⇩_{R}f' t) has_integral ?h b - ?h a - (- T + (?h b - ?h a))) {a..b}" using integrable_EM_remainder'[of a b f' "Suc n"] by (simp add: has_integral_iff T_def) qed (insert ab n, auto intro!: derivative_eq_intros simp: has_real_derivative_iff_has_vector_derivative [symmetric] not_le elim!: Ints_cases) also have "?h b - ?h a = (bernoulli (Suc n) / real (Suc n)) *⇩_{R}(f b - f a)" using n by (simp add: algebra_simps bernoulli'_def) finally have "integral {a..b} (λt. pbernpoly n t *⇩_{R}f t) = … - T" by (simp add: has_integral_iff) also have "T = integral {a..b} (λt. (1 / real (Suc n)) *⇩_{R}(pbernpoly (Suc n) t) *⇩_{R}f' t)" by (simp add: T_def) also have "… = integral {a..b} (λt. pbernpoly (Suc n) t *⇩_{R}f' t) /⇩_{R}real (Suc n)" by (subst integral_cmul) (simp_all add: divide_simps) finally show ?thesis . qed lemma EM_remainder'_conv_Suc: "EM_remainder' n f a b = ((-1) ^ Suc n * bernoulli (Suc n) / fact (Suc n)) *⇩_{R}(f b - f a) + EM_remainder' (Suc n) f' a b" by (simp add: EM_remainder'_def EM_remainder'_integral_conv_Suc scaleR_diff_right scaleR_add_right field_simps del: of_nat_Suc) end context fixes f f' :: "real ⇒ 'a :: banach" and a :: int and n :: nat and A :: "real set" and C assumes n: "n > 0" assumes fin: "finite A" assumes cont: "continuous_on {of_int a..} f" assumes cont': "continuous_on {of_int a..} f'" assumes lim: "(f ⤏ C) at_top" assumes deriv: "⋀x. x ∈ {of_int a<..} - A ⟹ (f has_vector_derivative f' x) (at x)" begin lemma shows EM_remainder_converges_iff_Suc_converges: "EM_remainder_converges n f a ⟷ EM_remainder_converges (Suc n) f' a" and EM_remainder_conv_Suc: "EM_remainder_converges n f a ⟹ EM_remainder n f a = ((-1) ^ Suc n * bernoulli (Suc n) / fact (Suc n)) *⇩_{R}(C - f a) + EM_remainder (Suc n) f' a" proof (rule iffI) define g where "g = (λx. ((-1) ^ Suc n * bernoulli (Suc n) / fact (Suc n)) *⇩_{R}(f x - f a))" define G where "G = ((-1) ^ Suc n * bernoulli (Suc n) / fact (Suc n)) *⇩_{R}(C - f a)" have limit_g: "(g ⤏ G) at_top" unfolding g_def G_def by (intro tendsto_intros lim) have *: "eventually (λx. EM_remainder' n f (real_of_int a) (real_of_int x) = g x + EM_remainder' (Suc n) f' (real_of_int a) (real_of_int x)) at_top" using eventually_ge_at_top[of a] proof eventually_elim case (elim b) thus ?case using EM_remainder'_conv_Suc[OF elim n fin continuous_on_subset[OF cont] continuous_on_subset[OF cont'] deriv] by (auto simp: g_def) qed { assume "EM_remainder_converges n f a" then obtain L where L: "((λb. EM_remainder' n f (real_of_int a) (real_of_int b)) ⤏ L) at_top" by (auto simp: EM_remainder_converges_def) have *: "((λb. EM_remainder' (Suc n) f' (real_of_int a) (real_of_int b)) ⤏ L - G) at_top" proof (rule Lim_transform_eventually) show "∀⇩_{F}x in at_top. EM_remainder' n f (real_of_int a) (real_of_int x) - g x = EM_remainder' (Suc n) f' (real_of_int a) (real_of_int x)" using * by (simp add: algebra_simps) show "((λx. EM_remainder' n f (real_of_int a) (real_of_int x) - g x) ⤏ L - G) at_top" by (intro tendsto_intros filterlim_compose[OF limit_g] L) qed from * show "EM_remainder_converges (Suc n) f' a" unfolding EM_remainder_converges_def .. from * have "EM_remainder (Suc n) f' a = L - G" by (rule EM_remainder_eqI) moreover from L have "EM_remainder n f a = L" by (rule EM_remainder_eqI) ultimately show "EM_remainder n f a = G + EM_remainder (Suc n) f' a" by (simp add: G_def) } { assume "EM_remainder_converges (Suc n) f' a" then obtain L where L: "((λb. EM_remainder' (Suc n) f' (real_of_int a) (real_of_int b)) ⤏ L) at_top" by (auto simp: EM_remainder_converges_def) have *: "((λb. EM_remainder' n f (real_of_int a) (real_of_int b)) ⤏ G + L) at_top" proof (rule Lim_transform_eventually) show "∀⇩_{F}x in at_top. g x + EM_remainder' (Suc n) f' (real_of_int a) (real_of_int x) = EM_remainder' n f (real_of_int a) (real_of_int x)" using * by (subst eq_commute) show "((λx. g x + EM_remainder' (Suc n) f' (real_of_int a) (real_of_int x)) ⤏ G + L) at_top" by (intro tendsto_intros filterlim_compose[OF limit_g] L) qed thus "EM_remainder_converges n f a" unfolding EM_remainder_converges_def .. } qed end subsection ‹The conventional version of the Euler--MacLaurin formula› text ‹ The following theorems are the classic Euler--MacLaurin formula that can be found, with slight variations, in many sources (e.\,g.\ \<^cite>‹"AS_HMF" and "apostol99" and "GKP_CM"›). › context fixes f :: "real ⇒ 'a :: banach" fixes fs :: "nat ⇒ real ⇒ 'a" fixes a b :: int assumes ab: "a ≤ b" fixes N :: nat assumes N: "N > 0" fixes Y :: "real set" assumes fin: "finite Y" assumes fs_0 [simp]: "fs 0 = f" assumes fs_cont [continuous_intros]: "⋀k. k ≤ N ⟹ continuous_on {real_of_int a..real_of_int b} (fs k)" assumes fs_deriv [derivative_intros]: "⋀k x. k < N ⟹ x ∈ {a..b} - Y ⟹ (fs k has_vector_derivative fs (Suc k) x) (at x)" begin theorem euler_maclaurin_raw_strong_int: defines "S ≡ (∑i∈{a<..b}. f (of_int i))" defines "I ≡ integral {of_int a..of_int b} f" defines "c' ≡ λk. (bernoulli' (Suc k) / fact (Suc k)) *⇩_{R}(fs k b - fs k a)" shows "S - I = (∑k<N. c' k) + EM_remainder' N (fs N) a b" proof - define c :: "nat ⇒ 'a" where "c = (λk. ((-1) ^ (Suc k) * bernoulli (Suc k) / fact (Suc k)) *⇩_{R}(fs k b - fs k a))" have "S - I = (∑k<m. c k) + EM_remainder' m (fs m) a b" if "m ≥ 1" "m ≤ N" for m using that proof (induction m rule: dec_induct) case base with ab fin fs_cont[of 0] show ?case using fs_deriv[of 0] N unfolding One_nat_def by (subst EM_remainder'_Suc_0[of _ _ Y f]) (simp_all add: algebra_simps S_def I_def c_def) next case (step n) from step.prems have "S - I = (∑k<n. c k) + EM_remainder' n (fs n) a b" by (intro step.IH) simp_all also have "(∑k<n. c k) = (∑k<Suc n. c k) + (((-1) ^ n * bernoulli (Suc n) / fact (Suc n)) *⇩_{R}(fs n b - fs n a))" (is "_ = _ + ?c") by (simp add: EM_remainder'_Suc_0 c_def) also have "… + EM_remainder' n (fs n) a b = (∑k<Suc n. c k) + (?c + EM_remainder' n (fs n) a b)" by (simp add: add.assoc) also from step.prems step.hyps ab fin have "?c + EM_remainder' n (fs n) a b = EM_remainder' (Suc n) (fs (Suc n)) a b" by (subst EM_remainder'_conv_Suc [where A = Y]) (auto intro!: fs_deriv fs_cont) finally show ?case . qed from this[of N] and N have "S - I = sum c {..<N} + EM_remainder' N (fs N) (real_of_int a) (real_of_int b)" by simp also have "sum c {..<N} = sum c' {..<N}" proof (intro sum.cong refl) fix k :: nat show "c k = c' k" by (cases "even k") (auto simp: c_def c'_def bernoulli'_def algebra_simps bernoulli_odd_eq_0) qed finally show ?thesis . qed end theorem euler_maclaurin_strong_raw_nat: assumes "a ≤ b" "0 < N" "finite Y" "fs 0 = f" "(⋀k. k ≤ N ⟹ continuous_on {real a..real b} (fs k))" "(⋀k x. k < N ⟹ x ∈ {real a..real b} - Y ⟹ (fs k has_vector_derivative fs (Suc k) x) (at x))" shows "(∑i∈{a<..b}. f (real i)) - integral {real a..real b} f = (∑k<N. (bernoulli' (Suc k) / fact (Suc k)) *⇩_{R}(fs k (real b) - fs k (real a))) + EM_remainder' N (fs N) (real a) (real b)" proof - have "(∑i∈{int a<..int b}. f (real_of_int i)) - integral {real_of_int (int a)..real_of_int (int b)} f = (∑k<N. (bernoulli' (Suc k) / fact (Suc k)) *⇩_{R}(fs k (real_of_int (int b)) - fs k (real_of_int (int a)))) + EM_remainder' N (fs N) (real_of_int (int a)) (real_of_int (int b))" using assms by (intro euler_maclaurin_raw_strong_int[where Y = Y] assms) simp_all also have "(∑i∈{int a<..int b}. f (real_of_int i)) = (∑i∈{a<..b}. f (real i))" by (intro sum.reindex_bij_witness[of _ int nat]) auto finally show ?thesis by simp qed subsection ‹The ``Concrete Mathematics'' version of the Euler--MacLaurin formula› text ‹ As explained in \textit{Concrete Mathematics}~\<^cite>‹"GKP_CM"›, the above form of the formula has some drawbacks: When applying it to determine the asymptotics of some concrete function, one is usually left with several different unwieldy constant terms that are difficult to get rid of. There is no general way to determine what these constant terms are, but in concrete applications, they can often be determined or estimated by other means. We can therefore simply group all the constant terms into a single constant and have the user provide a proof of what it is. › locale euler_maclaurin_int = fixes F f :: "real ⇒ 'a :: banach" fixes fs :: "nat ⇒ real ⇒ 'a" fixes a :: int fixes N :: nat assumes N: "N > 0" fixes C :: 'a fixes Y :: "real set" assumes fin: "finite Y" assumes fs_0 [simp]: "fs 0 = f" assumes fs_cont [continuous_intros]: "⋀k. k ≤ N ⟹ continuous_on {real_of_int a..} (fs k)" assumes fs_deriv [derivative_intros]: "⋀k x. k < N ⟹ x ∈ {of_int a..} - Y ⟹ (fs k has_vector_derivative fs (Suc k) x) (at x)" assumes F_cont [continuous_intros]: "continuous_on {of_int a..} F" assumes F_deriv [derivative_intros]: "⋀x. x ∈ {of_int a..} - Y ⟹ (F has_vector_derivative f x) (at x)" assumes limit: "((λb. (∑k=a..b. f k) - F (of_int b) - (∑i<N. (bernoulli' (Suc i) / fact (Suc i)) *⇩_{R}fs i (of_int b))) ⤏ C) at_top" begin context fixes C' T defines "C' ≡ -f a + F a + C + (∑k<N. (bernoulli' (Suc k) / fact (Suc k)) *⇩_{R}(fs k (of_int a)))" and "T ≡ (λx. ∑i<N. (bernoulli' (Suc i) / fact (Suc i)) *⇩_{R}fs i x)" begin lemma euler_maclaurin_strong_int_aux: assumes ab: "a ≤ b" defines "S ≡ (∑k=a..b. f (of_int k))" shows "S - F (of_int b) - T (of_int b) = EM_remainder' N (fs N) (of_int a) (of_int b) + (C - C')" proof (cases "a = b") case True thus ?thesis unfolding C'_def by (simp add: S_def EM_remainder'_def T_def) next case False with assms have ab: "a < b" by simp define T' where "T' = (∑k<N. (bernoulli' (Suc k) / fact (Suc k)) *⇩_{R}(fs k (of_int a)))" have "(∑i∈{a<..b}. f (of_int i)) - integral {of_int a..of_int b} f = (∑k<N. (bernoulli' (Suc k) / fact (Suc k)) *⇩_{R}(fs k (of_int b) - fs k (of_int a))) + EM_remainder' N (fs N) (of_int a) (of_int b)" using ab by (intro euler_maclaurin_raw_strong_int [where Y = Y] N fin fs_0 continuous_on_subset[OF fs_cont] fs_deriv) auto also have "(f has_integral (F b - F a)) {of_int a..of_int b}" using ab by (intro fundamental_theorem_of_calculus_strong[OF fin]) (auto intro!: continuous_on_subset[OF F_cont] derivative_intros) hence "integral {of_int a..of_int b} f = F (of_int b) - F (of_int a)" by (simp add: has_integral_iff) also have "(∑k<N. (bernoulli' (Suc k) / fact (Suc k)) *⇩_{R}(fs k (of_int b) - fs k (of_int a))) = T (of_int b) - T'" by (simp add: T_def T'_def algebra_simps sum_subtractf) also have "(∑i∈{a<..b}. f (of_int i)) = S - f (of_int a)" unfolding S_def using ab by (subst sum_atLeastAtMost_int_head) auto finally show ?thesis by (simp add: algebra_simps C'_def T'_def) qed lemma EM_remainder_limit: assumes ab: "a ≤ b" defines "D ≡ EM_remainder' N (fs N) (of_int a) (of_int b)" shows "EM_remainder N (fs N) b = C' - D" and EM_remainder_converges: "EM_remainder_converges N (fs N) b" proof - note limit also have "((λb. (∑k = a..b. f (of_int k)) - F (of_int b) - (∑i<N. (bernoulli' (Suc i) / fact (Suc i)) *⇩_{R}fs i (of_int b))) ⤏ C) at_top = ((λb. (∑k = a..b. f (of_int k)) - F (of_int b) - T (of_int b)) ⤏ C) at_top" unfolding T_def .. also have "eventually (λx. (∑k=a..x. f k) - F (of_int x) - T (of_int x) = EM_remainder' N (fs N) (of_int a) (of_int x) + (C - C')) at_top" (is "eventually (λx. ?f x = ?g x) _") using eventually_gt_at_top[of b] by eventually_elim (rule euler_maclaurin_strong_int_aux, insert ab, simp_all) hence "(?f ⤏ C) at_top ⟷ (?g ⤏ C) at_top" by (intro filterlim_cong refl) finally have "((λx. ?g x - (C - C')) ⤏ (C - (C - C'))) at_top" by (rule tendsto_diff[OF _ tendsto_const]) hence *: "((λx. EM_remainder' N (fs N) (of_int a) (of_int x)) ⤏ C') at_top" by simp