# Theory MacLaurin

theory MacLaurin
imports Transcendental
```(*  Title:      HOL/MacLaurin.thy
Author:     Jacques D. Fleuriot, 2001 University of Edinburgh
Author:     Lawrence C Paulson, 2004
Author:     Lukas Bulwahn and Bernhard Häupler, 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"
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)"
using difg_Suc real_differentiable_def by auto
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"
show "continuous_on {0..h} (difg 0)"
by (simp add: continuous_at_imp_continuous_on isCont_difg n)
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)
have "⋀x. 0 ≤ x ∧ x ≤ t ⟹ isCont (difg (Suc m')) x"
using ‹t < h› ‹Suc m' < n› by (simp add: isCont_difg)
then show "continuous_on {0..t} (difg (Suc m'))"
qed (use ‹t < h› ‹Suc m' < n› in ‹simp add: differentiable_difg›)
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"
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)"

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)"

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"
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"
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
```