(* Title: HOL/MacLaurin.thy Author: Jacques D. Fleuriot, 2001 University of Edinburgh Author: Lawrence C Paulson, 2004 Author: Lukas Bulwahn and Bernhard HÃ¤upler, 2005 *) section ‹MacLaurin and Taylor Series› theory MacLaurin imports Transcendental begin subsection ‹Maclaurin's Theorem with Lagrange Form of Remainder› text ‹This is a very long, messy proof even now that it's been broken down into lemmas.› lemma Maclaurin_lemma: "0 < h ⟹ ∃B::real. f h = (∑m<n. (j m / (fact m)) * (h^m)) + (B * ((h^n) /(fact n)))" by (rule exI[where x = "(f h - (∑m<n. (j m / (fact m)) * h^m)) * (fact n) / (h^n)"]) simp lemma eq_diff_eq': "x = y - z ⟷ y = x + z" for x y z :: real by arith lemma fact_diff_Suc: "n < Suc m ⟹ fact (Suc m - n) = (Suc m - n) * fact (m - n)" by (subst fact_reduce) auto lemma Maclaurin_lemma2: fixes B assumes DERIV: "∀m t. m < n ∧ 0≤t ∧ t≤h ⟶ DERIV (diff m) t :> diff (Suc m) t" and INIT: "n = Suc k" defines "difg ≡ (λm t::real. diff m t - ((∑p<n - m. diff (m + p) 0 / fact p * t ^ p) + B * (t ^ (n - m) / fact (n - m))))" (is "difg ≡ (λm t. diff m t - ?difg m t)") shows "∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h ⟶ DERIV (difg m) t :> difg (Suc m) t" proof (rule allI impI)+ fix m t assume INIT2: "m < n ∧ 0 ≤ t ∧ t ≤ h" have "DERIV (difg m) t :> diff (Suc m) t - ((∑x<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / fact x) + real (n - m) * t ^ (n - Suc m) * B / fact (n - m))" by (auto simp: difg_def intro!: derivative_eq_intros DERIV[rule_format, OF INIT2]) moreover from INIT2 have intvl: "{..<n - m} = insert 0 (Suc ` {..<n - Suc m})" and "0 < n - m" unfolding atLeast0LessThan[symmetric] by auto have "(∑x<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / fact x) = (∑x<n - Suc m. real (Suc x) * t ^ x * diff (Suc m + x) 0 / fact (Suc x))" unfolding intvl by (subst sum.insert) (auto simp: sum.reindex) moreover have fact_neq_0: "⋀x. (fact x) + real x * (fact x) ≠ 0" by (metis add_pos_pos fact_gt_zero less_add_same_cancel1 less_add_same_cancel2 less_numeral_extra(3) mult_less_0_iff of_nat_less_0_iff) have "⋀x. (Suc x) * t ^ x * diff (Suc m + x) 0 / fact (Suc x) = diff (Suc m + x) 0 * t^x / fact x" by (rule nonzero_divide_eq_eq[THEN iffD2]) auto moreover have "(n - m) * t ^ (n - Suc m) * B / fact (n - m) = B * (t ^ (n - Suc m) / fact (n - Suc m))" using ‹0 < n - m› by (simp add: divide_simps fact_reduce) ultimately show "DERIV (difg m) t :> difg (Suc m) t" unfolding difg_def by (simp add: mult.commute) qed lemma Maclaurin: assumes h: "0 < h" and n: "0 < n" and diff_0: "diff 0 = f" and diff_Suc: "∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h ⟶ DERIV (diff m) t :> diff (Suc m) t" shows "∃t::real. 0 < t ∧ t < h ∧ f h = sum (λm. (diff m 0 / fact m) * h ^ m) {..<n} + (diff n t / fact n) * h ^ n" proof - from n obtain m where m: "n = Suc m" by (cases n) (simp add: n) from m have "m < n" by simp obtain B where f_h: "f h = (∑m<n. diff m 0 / fact m * h ^ m) + B * (h ^ n / fact n)" using Maclaurin_lemma [OF h] .. define g where [abs_def]: "g t = f t - (sum (λm. (diff m 0 / fact m) * t^m) {..<n} + B * (t^n / fact n))" for t have g2: "g 0 = 0" "g h = 0" by (simp_all add: m f_h g_def lessThan_Suc_eq_insert_0 image_iff diff_0 sum.reindex) define difg where [abs_def]: "difg m t = diff m t - (sum (λp. (diff (m + p) 0 / fact p) * (t ^ p)) {..<n-m} + B * ((t ^ (n - m)) / fact (n - m)))" for m t have difg_0: "difg 0 = g" by (simp add: difg_def g_def diff_0) have difg_Suc: "∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h ⟶ DERIV (difg m) t :> difg (Suc m) t" using diff_Suc m unfolding difg_def [abs_def] by (rule Maclaurin_lemma2) have difg_eq_0: "∀m<n. difg m 0 = 0" by (auto simp: difg_def m Suc_diff_le lessThan_Suc_eq_insert_0 image_iff sum.reindex) have isCont_difg: "⋀m x. m < n ⟹ 0 ≤ x ⟹ x ≤ h ⟹ isCont (difg m) x" by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp have differentiable_difg: "⋀m x. m < n ⟹ 0 ≤ x ⟹ x ≤ h ⟹ difg m differentiable (at x)" by (rule differentiableI [OF difg_Suc [rule_format]]) simp have difg_Suc_eq_0: "⋀m t. m < n ⟹ 0 ≤ t ⟹ t ≤ h ⟹ DERIV (difg m) t :> 0 ⟹ difg (Suc m) t = 0" by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp have "∃t. 0 < t ∧ t < h ∧ DERIV (difg m) t :> 0" using ‹m < n› proof (induct m) case 0 show ?case proof (rule Rolle) show "0 < h" by fact show "difg 0 0 = difg 0 h" by (simp add: difg_0 g2) show "∀x. 0 ≤ x ∧ x ≤ h ⟶ isCont (difg (0::nat)) x" by (simp add: isCont_difg n) show "∀x. 0 < x ∧ x < h ⟶ difg (0::nat) differentiable (at x)" by (simp add: differentiable_difg n) qed next case (Suc m') then have "∃t. 0 < t ∧ t < h ∧ DERIV (difg m') t :> 0" by simp then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0" by fast have "∃t'. 0 < t' ∧ t' < t ∧ DERIV (difg (Suc m')) t' :> 0" proof (rule Rolle) show "0 < t" by fact show "difg (Suc m') 0 = difg (Suc m') t" using t ‹Suc m' < n› by (simp add: difg_Suc_eq_0 difg_eq_0) show "∀x. 0 ≤ x ∧ x ≤ t ⟶ isCont (difg (Suc m')) x" using ‹t < h› ‹Suc m' < n› by (simp add: isCont_difg) show "∀x. 0 < x ∧ x < t ⟶ difg (Suc m') differentiable (at x)" using ‹t < h› ‹Suc m' < n› by (simp add: differentiable_difg) qed with ‹t < h› show ?case by auto qed then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0" by fast with ‹m < n› have "difg (Suc m) t = 0" by (simp add: difg_Suc_eq_0) show ?thesis proof (intro exI conjI) show "0 < t" by fact show "t < h" by fact show "f h = (∑m<n. diff m 0 / (fact m) * h ^ m) + diff n t / (fact n) * h ^ n" using ‹difg (Suc m) t = 0› by (simp add: m f_h difg_def) qed qed lemma Maclaurin2: fixes n :: nat and h :: real assumes INIT1: "0 < h" and INIT2: "diff 0 = f" and DERIV: "∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h ⟶ DERIV (diff m) t :> diff (Suc m) t" shows "∃t. 0 < t ∧ t ≤ h ∧ f h = (∑m<n. diff m 0 / (fact m) * h ^ m) + diff n t / fact n * h ^ n" proof (cases n) case 0 with INIT1 INIT2 show ?thesis by fastforce next case Suc then have "n > 0" by simp from INIT1 this INIT2 DERIV have "∃t>0. t < h ∧ f h = (∑m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n" by (rule Maclaurin) then show ?thesis by fastforce qed lemma Maclaurin_minus: fixes n :: nat and h :: real assumes "h < 0" "0 < n" "diff 0 = f" and DERIV: "∀m t. m < n ∧ h ≤ t ∧ t ≤ 0 ⟶ DERIV (diff m) t :> diff (Suc m) t" shows "∃t. h < t ∧ t < 0 ∧ f h = (∑m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n" proof - txt ‹Transform ‹ABL'› into ‹derivative_intros› format.› note DERIV' = DERIV_chain'[OF _ DERIV[rule_format], THEN DERIV_cong] let ?sum = "λt. (∑m<n. (- 1) ^ m * diff m (- 0) / (fact m) * (- h) ^ m) + (- 1) ^ n * diff n (- t) / (fact n) * (- h) ^ n" from assms have "∃t>0. t < - h ∧ f (- (- h)) = ?sum t" by (intro Maclaurin) (auto intro!: derivative_eq_intros DERIV') then obtain t where "0 < t" "t < - h" "f (- (- h)) = ?sum t" by blast moreover have "(- 1) ^ n * diff n (- t) * (- h) ^ n / fact n = diff n (- t) * h ^ n / fact n" by (auto simp: power_mult_distrib[symmetric]) moreover have "(∑m<n. (- 1) ^ m * diff m 0 * (- h) ^ m / fact m) = (∑m<n. diff m 0 * h ^ m / fact m)" by (auto intro: sum.cong simp add: power_mult_distrib[symmetric]) ultimately have "h < - t ∧ - t < 0 ∧ f h = (∑m<n. diff m 0 / (fact m) * h ^ m) + diff n (- t) / (fact n) * h ^ n" by auto then show ?thesis .. qed subsection ‹More Convenient "Bidirectional" Version.› lemma Maclaurin_bi_le: fixes n :: nat and x :: real assumes "diff 0 = f" and DERIV : "∀m t. m < n ∧ ¦t¦ ≤ ¦x¦ ⟶ DERIV (diff m) t :> diff (Suc m) t" shows "∃t. ¦t¦ ≤ ¦x¦ ∧ f x = (∑m<n. diff m 0 / (fact m) * x ^ m) + diff n t / (fact n) * x ^ n" (is "∃t. _ ∧ f x = ?f x t") proof (cases "n = 0") case True with ‹diff 0 = f› show ?thesis by force next case False show ?thesis proof (cases rule: linorder_cases) assume "x = 0" with ‹n ≠ 0› ‹diff 0 = f› DERIV have "¦0¦ ≤ ¦x¦ ∧ f x = ?f x 0" by auto then show ?thesis .. next assume "x < 0" with ‹n ≠ 0› DERIV have "∃t>x. t < 0 ∧ diff 0 x = ?f x t" by (intro Maclaurin_minus) auto then obtain t where "x < t" "t < 0" "diff 0 x = (∑m<n. diff m 0 / fact m * x ^ m) + diff n t / fact n * x ^ n" by blast with ‹x < 0› ‹diff 0 = f› have "¦t¦ ≤ ¦x¦ ∧ f x = ?f x t" by simp then show ?thesis .. next assume "x > 0" with ‹n ≠ 0› ‹diff 0 = f› DERIV have "∃t>0. t < x ∧ diff 0 x = ?f x t" by (intro Maclaurin) auto then obtain t where "0 < t" "t < x" "diff 0 x = (∑m<n. diff m 0 / fact m * x ^ m) + diff n t / fact n * x ^ n" by blast with ‹x > 0› ‹diff 0 = f› have "¦t¦ ≤ ¦x¦ ∧ f x = ?f x t" by simp then show ?thesis .. qed qed lemma Maclaurin_all_lt: fixes x :: real assumes INIT1: "diff 0 = f" and INIT2: "0 < n" and INIT3: "x ≠ 0" and DERIV: "∀m x. DERIV (diff m) x :> diff(Suc m) x" shows "∃t. 0 < ¦t¦ ∧ ¦t¦ < ¦x¦ ∧ f x = (∑m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n" (is "∃t. _ ∧ _ ∧ f x = ?f x t") proof (cases rule: linorder_cases) assume "x = 0" with INIT3 show ?thesis .. next assume "x < 0" with assms have "∃t>x. t < 0 ∧ f x = ?f x t" by (intro Maclaurin_minus) auto then obtain t where "t > x" "t < 0" "f x = ?f x t" by blast with ‹x < 0› have "0 < ¦t¦ ∧ ¦t¦ < ¦x¦ ∧ f x = ?f x t" by simp then show ?thesis .. next assume "x > 0" with assms have "∃t>0. t < x ∧ f x = ?f x t" by (intro Maclaurin) auto then obtain t where "t > 0" "t < x" "f x = ?f x t" by blast with ‹x > 0› have "0 < ¦t¦ ∧ ¦t¦ < ¦x¦ ∧ f x = ?f x t" by simp then show ?thesis .. qed lemma Maclaurin_zero: "x = 0 ⟹ n ≠ 0 ⟹ (∑m<n. (diff m 0 / fact m) * x ^ m) = diff 0 0" for x :: real and n :: nat by simp lemma Maclaurin_all_le: fixes x :: real and n :: nat assumes INIT: "diff 0 = f" and DERIV: "∀m x. DERIV (diff m) x :> diff (Suc m) x" shows "∃t. ¦t¦ ≤ ¦x¦ ∧ f x = (∑m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n" (is "∃t. _ ∧ f x = ?f x t") proof (cases "n = 0") case True with INIT show ?thesis by force next case False show ?thesis proof (cases "x = 0") case True with ‹n ≠ 0› have "(∑m<n. diff m 0 / (fact m) * x ^ m) = diff 0 0" by (intro Maclaurin_zero) auto with INIT ‹x = 0› ‹n ≠ 0› have " ¦0¦ ≤ ¦x¦ ∧ f x = ?f x 0" by force then show ?thesis .. next case False with INIT ‹n ≠ 0› DERIV have "∃t. 0 < ¦t¦ ∧ ¦t¦ < ¦x¦ ∧ f x = ?f x t" by (intro Maclaurin_all_lt) auto then obtain t where "0 < ¦t¦ ∧ ¦t¦ < ¦x¦ ∧ f x = ?f x t" .. then have "¦t¦ ≤ ¦x¦ ∧ f x = ?f x t" by simp then show ?thesis .. qed qed lemma Maclaurin_all_le_objl: "diff 0 = f ∧ (∀m x. DERIV (diff m) x :> diff (Suc m) x) ⟶ (∃t::real. ¦t¦ ≤ ¦x¦ ∧ f x = (∑m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n)" for x :: real and n :: nat by (blast intro: Maclaurin_all_le) subsection ‹Version for Exponential Function› lemma Maclaurin_exp_lt: fixes x :: real and n :: nat shows "x ≠ 0 ⟹ n > 0 ⟹ (∃t. 0 < ¦t¦ ∧ ¦t¦ < ¦x¦ ∧ exp x = (∑m<n. (x ^ m) / fact m) + (exp t / fact n) * x ^ n)" using Maclaurin_all_lt [where diff = "λn. exp" and f = exp and x = x and n = n] by auto lemma Maclaurin_exp_le: fixes x :: real and n :: nat shows "∃t. ¦t¦ ≤ ¦x¦ ∧ exp x = (∑m<n. (x ^ m) / fact m) + (exp t / fact n) * x ^ n" using Maclaurin_all_le_objl [where diff = "λn. exp" and f = exp and x = x and n = n] by auto corollary exp_lower_taylor_quadratic: "0 ≤ x ⟹ 1 + x + x⇧^{2}/ 2 ≤ exp x" for x :: real using Maclaurin_exp_le [of x 3] by (auto simp: numeral_3_eq_3 power2_eq_square) corollary ln_2_less_1: "ln 2 < (1::real)" proof - have "2 < 5/(2::real)" by simp also have "5/2 ≤ exp (1::real)" using exp_lower_taylor_quadratic[of 1, simplified] by simp finally have "exp (ln 2) < exp (1::real)" by simp thus "ln 2 < (1::real)" by (subst (asm) exp_less_cancel_iff) simp qed subsection ‹Version for Sine Function› lemma mod_exhaust_less_4: "m mod 4 = 0 ∨ m mod 4 = 1 ∨ m mod 4 = 2 ∨ m mod 4 = 3" for m :: nat by auto text ‹It is unclear why so many variant results are needed.› lemma sin_expansion_lemma: "sin (x + real (Suc m) * pi / 2) = cos (x + real m * pi / 2)" by (auto simp: cos_add sin_add add_divide_distrib distrib_right) lemma Maclaurin_sin_expansion2: "∃t. ¦t¦ ≤ ¦x¦ ∧ sin x = (∑m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n" proof (cases "n = 0 ∨ x = 0") case False let ?diff = "λn x. sin (x + 1/2 * real n * pi)" have "∃t. 0 < ¦t¦ ∧ ¦t¦ < ¦x¦ ∧ sin x = (∑m<n. (?diff m 0 / fact m) * x ^ m) + (?diff n t / fact n) * x ^ n" proof (rule Maclaurin_all_lt) show "∀m x. ((λt. sin (t + 1/2 * real m * pi)) has_real_derivative sin (x + 1/2 * real (Suc m) * pi)) (at x)" by (rule allI derivative_eq_intros | use sin_expansion_lemma in force)+ qed (use False in auto) then show ?thesis apply (rule ex_forward, simp) apply (rule sum.cong[OF refl]) apply (auto simp: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc) done qed auto lemma Maclaurin_sin_expansion: "∃t. sin x = (∑m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n" using Maclaurin_sin_expansion2 [of x n] by blast lemma Maclaurin_sin_expansion3: assumes "n > 0" "x > 0" shows "∃t. 0 < t ∧ t < x ∧ sin x = (∑m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n" proof - let ?diff = "λn x. sin (x + 1/2 * real n * pi)" have "∃t. 0 < t ∧ t < x ∧ sin x = (∑m<n. ?diff m 0 / (fact m) * x ^ m) + ?diff n t / fact n * x ^ n" proof (rule Maclaurin) show "∀m t. m < n ∧ 0 ≤ t ∧ t ≤ x ⟶ ((λu. sin (u + 1/2 * real m * pi)) has_real_derivative sin (t + 1/2 * real (Suc m) * pi)) (at t)" apply (simp add: sin_expansion_lemma del: of_nat_Suc) apply (force intro!: derivative_eq_intros) done qed (use assms in auto) then show ?thesis apply (rule ex_forward, simp) apply (rule sum.cong[OF refl]) apply (auto simp: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc) done qed lemma Maclaurin_sin_expansion4: assumes "0 < x" shows "∃t. 0 < t ∧ t ≤ x ∧ sin x = (∑m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n" proof - let ?diff = "λn x. sin (x + 1/2 * real n * pi)" have "∃t. 0 < t ∧ t ≤ x ∧ sin x = (∑m<n. ?diff m 0 / (fact m) * x ^ m) + ?diff n t / fact n * x ^ n" proof (rule Maclaurin2) show "∀m t. m < n ∧ 0 ≤ t ∧ t ≤ x ⟶ ((λu. sin (u + 1/2 * real m * pi)) has_real_derivative sin (t + 1/2 * real (Suc m) * pi)) (at t)" apply (simp add: sin_expansion_lemma del: of_nat_Suc) apply (force intro!: derivative_eq_intros) done qed (use assms in auto) then show ?thesis apply (rule ex_forward, simp) apply (rule sum.cong[OF refl]) apply (auto simp: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc) done qed subsection ‹Maclaurin Expansion for Cosine Function› lemma sumr_cos_zero_one [simp]: "(∑m<Suc n. cos_coeff m * 0 ^ m) = 1" by (induct n) auto lemma cos_expansion_lemma: "cos (x + real (Suc m) * pi / 2) = - sin (x + real m * pi / 2)" by (auto simp: cos_add sin_add distrib_right add_divide_distrib) lemma Maclaurin_cos_expansion: "∃t::real. ¦t¦ ≤ ¦x¦ ∧ cos x = (∑m<n. cos_coeff m * x ^ m) + (cos(t + 1/2 * real n * pi) / fact n) * x ^ n" proof (cases "n = 0 ∨ x = 0") case False let ?diff = "λn x. cos (x + 1/2 * real n * pi)" have "∃t. 0 < ¦t¦ ∧ ¦t¦ < ¦x¦ ∧ cos x = (∑m<n. (?diff m 0 / fact m) * x ^ m) + (?diff n t / fact n) * x ^ n" proof (rule Maclaurin_all_lt) show "∀m x. ((λt. cos (t + 1/2 * real m * pi)) has_real_derivative cos (x + 1/2 * real (Suc m) * pi)) (at x)" apply (rule allI derivative_eq_intros | simp)+ using cos_expansion_lemma by force qed (use False in auto) then show ?thesis apply (rule ex_forward, simp) apply (rule sum.cong[OF refl]) apply (auto simp: cos_coeff_def cos_zero_iff elim: evenE simp del: of_nat_Suc) done qed auto lemma Maclaurin_cos_expansion2: assumes "x > 0" "n > 0" shows "∃t. 0 < t ∧ t < x ∧ cos x = (∑m<n. cos_coeff m * x ^ m) + (cos (t + 1/2 * real n * pi) / fact n) * x ^ n" proof - let ?diff = "λn x. cos (x + 1/2 * real n * pi)" have "∃t. 0 < t ∧ t < x ∧ cos x = (∑m<n. ?diff m 0 / (fact m) * x ^ m) + ?diff n t / fact n * x ^ n" proof (rule Maclaurin) show "∀m t. m < n ∧ 0 ≤ t ∧ t ≤ x ⟶ ((λu. cos (u + 1 / 2 * real m * pi)) has_real_derivative cos (t + 1 / 2 * real (Suc m) * pi)) (at t)" by (simp add: cos_expansion_lemma del: of_nat_Suc) qed (use assms in auto) then show ?thesis apply (rule ex_forward, simp) apply (rule sum.cong[OF refl]) apply (auto simp: cos_coeff_def cos_zero_iff elim: evenE) done qed lemma Maclaurin_minus_cos_expansion: assumes "n > 0" "x < 0" shows "∃t. x < t ∧ t < 0 ∧ cos x = (∑m<n. cos_coeff m * x ^ m) + ((cos (t + 1/2 * real n * pi) / fact n) * x ^ n)" proof - let ?diff = "λn x. cos (x + 1/2 * real n * pi)" have "∃t. x < t ∧ t < 0 ∧ cos x = (∑m<n. ?diff m 0 / (fact m) * x ^ m) + ?diff n t / fact n * x ^ n" proof (rule Maclaurin_minus) show "∀m t. m < n ∧ x ≤ t ∧ t ≤ 0 ⟶ ((λu. cos (u + 1 / 2 * real m * pi)) has_real_derivative cos (t + 1 / 2 * real (Suc m) * pi)) (at t)" by (simp add: cos_expansion_lemma del: of_nat_Suc) qed (use assms in auto) then show ?thesis apply (rule ex_forward, simp) apply (rule sum.cong[OF refl]) apply (auto simp: cos_coeff_def cos_zero_iff elim: evenE) done qed (* Version for ln(1 +/- x). Where is it?? *) lemma sin_bound_lemma: "x = y ⟹ ¦u¦ ≤ v ⟹ ¦(x + u) - y¦ ≤ v" for x y u v :: real by auto lemma Maclaurin_sin_bound: "¦sin x - (∑m<n. sin_coeff m * x ^ m)¦ ≤ inverse (fact n) * ¦x¦ ^ n" proof - have est: "x ≤ 1 ⟹ 0 ≤ y ⟹ x * y ≤ 1 * y" for x y :: real by (rule mult_right_mono) simp_all let ?diff = "λ(n::nat) (x::real). if n mod 4 = 0 then sin x else if n mod 4 = 1 then cos x else if n mod 4 = 2 then - sin x else - cos x" have diff_0: "?diff 0 = sin" by simp have "DERIV (?diff m) x :> ?diff (Suc m) x" for m and x using mod_exhaust_less_4 [of m] by (auto simp: mod_Suc intro!: derivative_eq_intros) then have DERIV_diff: "∀m x. DERIV (?diff m) x :> ?diff (Suc m) x" by blast from Maclaurin_all_le [OF diff_0 DERIV_diff] obtain t where t1: "¦t¦ ≤ ¦x¦" and t2: "sin x = (∑m<n. ?diff m 0 / (fact m) * x ^ m) + ?diff n t / (fact n) * x ^ n" by fast have diff_m_0: "?diff m 0 = (if even m then 0 else (- 1) ^ ((m - Suc 0) div 2))" for m using mod_exhaust_less_4 [of m] by (auto simp: minus_one_power_iff even_even_mod_4_iff [of m] dest: even_mod_4_div_2 odd_mod_4_div_2) show ?thesis unfolding sin_coeff_def apply (subst t2) apply (rule sin_bound_lemma) apply (rule sum.cong[OF refl]) apply (subst diff_m_0, simp) using est apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono simp: ac_simps divide_inverse power_abs [symmetric] abs_mult) done qed section ‹Taylor series› text ‹ We use MacLaurin and the translation of the expansion point ‹c› to ‹0› to prove Taylor's theorem. › lemma taylor_up: assumes INIT: "n > 0" "diff 0 = f" and DERIV: "∀m t. m < n ∧ a ≤ t ∧ t ≤ b ⟶ DERIV (diff m) t :> (diff (Suc m) t)" and INTERV: "a ≤ c" "c < b" shows "∃t::real. c < t ∧ t < b ∧ f b = (∑m<n. (diff m c / fact m) * (b - c)^m) + (diff n t / fact n) * (b - c)^n" proof - from INTERV have "0 < b - c" by arith moreover from INIT have "n > 0" "(λm x. diff m (x + c)) 0 = (λx. f (x + c))" by auto moreover have "∀m t. m < n ∧ 0 ≤ t ∧ t ≤ b - c ⟶ DERIV (λx. diff m (x + c)) t :> diff (Suc m) (t + c)" proof (intro strip) fix m t assume "m < n ∧ 0 ≤ t ∧ t ≤ b - c" with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)" by auto moreover from DERIV_ident and DERIV_const have "DERIV (λx. x + c) t :> 1 + 0" by (rule DERIV_add) ultimately have "DERIV (λx. diff m (x + c)) t :> diff (Suc m) (t + c) * (1 + 0)" by (rule DERIV_chain2) then show "DERIV (λx. diff m (x + c)) t :> diff (Suc m) (t + c)" by simp qed ultimately obtain x where "0 < x ∧ x < b - c ∧ f (b - c + c) = (∑m<n. diff m (0 + c) / fact m * (b - c) ^ m) + diff n (x + c) / fact n * (b - c) ^ n" by (rule Maclaurin [THEN exE]) then have "c < x + c ∧ x + c < b ∧ f b = (∑m<n. diff m c / fact m * (b - c) ^ m) + diff n (x + c) / fact n * (b - c) ^ n" by fastforce then show ?thesis by fastforce qed lemma taylor_down: fixes a :: real and n :: nat assumes INIT: "n > 0" "diff 0 = f" and DERIV: "(∀m t. m < n ∧ a ≤ t ∧ t ≤ b ⟶ DERIV (diff m) t :> diff (Suc m) t)" and INTERV: "a < c" "c ≤ b" shows "∃t. a < t ∧ t < c ∧ f a = (∑m<n. (diff m c / fact m) * (a - c)^m) + (diff n t / fact n) * (a - c)^n" proof - from INTERV have "a-c < 0" by arith moreover from INIT have "n > 0" "(λm x. diff m (x + c)) 0 = (λx. f (x + c))" by auto moreover have "∀m t. m < n ∧ a - c ≤ t ∧ t ≤ 0 ⟶ DERIV (λx. diff m (x + c)) t :> diff (Suc m) (t + c)" proof (rule allI impI)+ fix m t assume "m < n ∧ a - c ≤ t ∧ t ≤ 0" with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)" by auto moreover from DERIV_ident and DERIV_const have "DERIV (λx. x + c) t :> 1 + 0" by (rule DERIV_add) ultimately have "DERIV (λx. diff m (x + c)) t :> diff (Suc m) (t + c) * (1 + 0)" by (rule DERIV_chain2) then show "DERIV (λx. diff m (x + c)) t :> diff (Suc m) (t + c)" by simp qed ultimately obtain x where "a - c < x ∧ x < 0 ∧ f (a - c + c) = (∑m<n. diff m (0 + c) / fact m * (a - c) ^ m) + diff n (x + c) / fact n * (a - c) ^ n" by (rule Maclaurin_minus [THEN exE]) then have "a < x + c ∧ x + c < c ∧ f a = (∑m<n. diff m c / fact m * (a - c) ^ m) + diff n (x + c) / fact n * (a - c) ^ n" by fastforce then show ?thesis by fastforce qed theorem taylor: fixes a :: real and n :: nat assumes INIT: "n > 0" "diff 0 = f" and DERIV: "∀m t. m < n ∧ a ≤ t ∧ t ≤ b ⟶ DERIV (diff m) t :> diff (Suc m) t" and INTERV: "a ≤ c " "c ≤ b" "a ≤ x" "x ≤ b" "x ≠ c" shows "∃t. (if x < c then x < t ∧ t < c else c < t ∧ t < x) ∧ f x = (∑m<n. (diff m c / fact m) * (x - c)^m) + (diff n t / fact n) * (x - c)^n" proof (cases "x < c") case True note INIT moreover have "∀m t. m < n ∧ x ≤ t ∧ t ≤ b ⟶ DERIV (diff m) t :> diff (Suc m) t" using DERIV and INTERV by fastforce moreover note True moreover from INTERV have "c ≤ b" by simp ultimately have "∃t>x. t < c ∧ f x = (∑m<n. diff m c / (fact m) * (x - c) ^ m) + diff n t / (fact n) * (x - c) ^ n" by (rule taylor_down) with True show ?thesis by simp next case False note INIT moreover have "∀m t. m < n ∧ a ≤ t ∧ t ≤ x ⟶ DERIV (diff m) t :> diff (Suc m) t" using DERIV and INTERV by fastforce moreover from INTERV have "a ≤ c" by arith moreover from False and INTERV have "c < x" by arith ultimately have "∃t>c. t < x ∧ f x = (∑m<n. diff m c / (fact m) * (x - c) ^ m) + diff n t / (fact n) * (x - c) ^ n" by (rule taylor_up) with False show ?thesis by simp qed end