(* 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 \<open>MacLaurin and Taylor Series\<close>
theory MacLaurin
imports Transcendental
begin
subsection \<open>Maclaurin's Theorem with Lagrange Form of Remainder\<close>
text \<open>This is a very long, messy proof even now that it's been broken down
into lemmas.\<close>
lemma Maclaurin_lemma:
"0 < h \<Longrightarrow>
\<exists>B::real. f h = (\<Sum>m<n. (j m / (fact m)) * (h^m)) + (B * ((h^n) /(fact n)))"
by (rule exI[where x = "(f h - (\<Sum>m<n. (j m / (fact m)) * h^m)) * (fact n) / (h^n)"]) simp
lemma eq_diff_eq': "x = y - z \<longleftrightarrow> y = x + z"
for x y z :: real
by arith
lemma fact_diff_Suc: "n < Suc m \<Longrightarrow> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
by (subst fact_reduce) auto
lemma Maclaurin_lemma2:
fixes B
assumes DERIV: "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
and INIT: "n = Suc k"
defines "difg \<equiv>
(\<lambda>m t::real. diff m t -
((\<Sum>p<n - m. diff (m + p) 0 / fact p * t ^ p) + B * (t ^ (n - m) / fact (n - m))))"
(is "difg \<equiv> (\<lambda>m t. diff m t - ?difg m t)")
shows "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t"
proof (rule allI impI)+
fix m t
assume INIT2: "m < n \<and> 0 \<le> t \<and> t \<le> h"
have "DERIV (difg m) t :> diff (Suc m) t -
((\<Sum>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 "(\<Sum>x<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / fact x) =
(\<Sum>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 add: sum.reindex)
moreover
have fact_neq_0: "\<And>x. (fact x) + real x * (fact x) \<noteq> 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 "\<And>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 \<open>0 < n - m\<close> 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: "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
shows
"\<exists>t::real. 0 < t \<and> t < h \<and>
f h = sum (\<lambda>m. (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 = (\<Sum>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 (\<lambda>m. (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 (\<lambda>p. (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: "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> 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: "\<forall>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: "\<And>m x. m < n \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> h \<Longrightarrow> isCont (difg m) x"
by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp
have differentiable_difg: "\<And>m x. m < n \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> h \<Longrightarrow> difg m differentiable (at x)"
by (rule differentiableI [OF difg_Suc [rule_format]]) simp
have difg_Suc_eq_0:
"\<And>m t. m < n \<Longrightarrow> 0 \<le> t \<Longrightarrow> t \<le> h \<Longrightarrow> DERIV (difg m) t :> 0 \<Longrightarrow> difg (Suc m) t = 0"
by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp
have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m) t :> 0"
using \<open>m < n\<close>
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 "\<forall>x. 0 \<le> x \<and> x \<le> h \<longrightarrow> isCont (difg (0::nat)) x"
by (simp add: isCont_difg n)
show "\<forall>x. 0 < x \<and> x < h \<longrightarrow> difg (0::nat) differentiable (at x)"
by (simp add: differentiable_difg n)
qed
next
case (Suc m')
then have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m') t :> 0"
by simp
then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0"
by fast
have "\<exists>t'. 0 < t' \<and> t' < t \<and> 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 \<open>Suc m' < n\<close> by (simp add: difg_Suc_eq_0 difg_eq_0)
show "\<forall>x. 0 \<le> x \<and> x \<le> t \<longrightarrow> isCont (difg (Suc m')) x"
using \<open>t < h\<close> \<open>Suc m' < n\<close> by (simp add: isCont_difg)
show "\<forall>x. 0 < x \<and> x < t \<longrightarrow> difg (Suc m') differentiable (at x)"
using \<open>t < h\<close> \<open>Suc m' < n\<close> by (simp add: differentiable_difg)
qed
with \<open>t < h\<close> show ?case
by auto
qed
then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0"
by fast
with \<open>m < n\<close> 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 = (\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + diff n t / (fact n) * h ^ n"
using \<open>difg (Suc m) t = 0\<close> by (simp add: m f_h difg_def)
qed
qed
lemma Maclaurin_objl:
"0 < h \<and> n > 0 \<and> diff 0 = f \<and>
(\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t) \<longrightarrow>
(\<exists>t. 0 < t \<and> t < h \<and> f h = (\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + diff n t / fact n * h ^ n)"
for n :: nat and h :: real
by (blast intro: Maclaurin)
lemma Maclaurin2:
fixes n :: nat
and h :: real
assumes INIT1: "0 < h"
and INIT2: "diff 0 = f"
and DERIV: "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
shows "\<exists>t. 0 < t \<and> t \<le> h \<and> f h = (\<Sum>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 "\<exists>t>0. t < h \<and> f h = (\<Sum>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 Maclaurin2_objl:
"0 < h \<and> diff 0 = f \<and>
(\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t) \<longrightarrow>
(\<exists>t. 0 < t \<and> t \<le> h \<and> f h = (\<Sum>m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n)"
for n :: nat and h :: real
by (blast intro: Maclaurin2)
lemma Maclaurin_minus:
fixes n :: nat and h :: real
assumes "h < 0" "0 < n" "diff 0 = f"
and DERIV: "\<forall>m t. m < n \<and> h \<le> t \<and> t \<le> 0 \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
shows "\<exists>t. h < t \<and> t < 0 \<and> f h = (\<Sum>m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n"
proof -
txt \<open>Transform \<open>ABL'\<close> into \<open>derivative_intros\<close> format.\<close>
note DERIV' = DERIV_chain'[OF _ DERIV[rule_format], THEN DERIV_cong]
let ?sum = "\<lambda>t.
(\<Sum>m<n. (- 1) ^ m * diff m (- 0) / (fact m) * (- h) ^ m) +
(- 1) ^ n * diff n (- t) / (fact n) * (- h) ^ n"
from assms have "\<exists>t>0. t < - h \<and> 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 "(\<Sum>m<n. (- 1) ^ m * diff m 0 * (- h) ^ m / fact m) = (\<Sum>m<n. diff m 0 * h ^ m / fact m)"
by (auto intro: sum.cong simp add: power_mult_distrib[symmetric])
ultimately have "h < - t \<and> - t < 0 \<and>
f h = (\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + diff n (- t) / (fact n) * h ^ n"
by auto
then show ?thesis ..
qed
lemma Maclaurin_minus_objl:
fixes n :: nat and h :: real
shows
"h < 0 \<and> n > 0 \<and> diff 0 = f \<and>
(\<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t) \<longrightarrow>
(\<exists>t. h < t \<and> t < 0 \<and> f h = (\<Sum>m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n)"
by (blast intro: Maclaurin_minus)
subsection \<open>More Convenient "Bidirectional" Version.\<close>
(* not good for PVS sin_approx, cos_approx *)
lemma Maclaurin_bi_le_lemma:
"n > 0 \<Longrightarrow>
diff 0 0 = (\<Sum>m<n. diff m 0 * 0 ^ m / (fact m)) + diff n 0 * 0 ^ n / (fact n :: real)"
by (induct n) auto
lemma Maclaurin_bi_le:
fixes n :: nat and x :: real
assumes "diff 0 = f"
and DERIV : "\<forall>m t. m < n \<and> \<bar>t\<bar> \<le> \<bar>x\<bar> \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
shows "\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = (\<Sum>m<n. diff m 0 / (fact m) * x ^ m) + diff n t / (fact n) * x ^ n"
(is "\<exists>t. _ \<and> f x = ?f x t")
proof (cases "n = 0")
case True
with \<open>diff 0 = f\<close> show ?thesis by force
next
case False
show ?thesis
proof (cases rule: linorder_cases)
assume "x = 0"
with \<open>n \<noteq> 0\<close> \<open>diff 0 = f\<close> DERIV have "\<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0"
by (auto simp add: Maclaurin_bi_le_lemma)
then show ?thesis ..
next
assume "x < 0"
with \<open>n \<noteq> 0\<close> DERIV have "\<exists>t>x. t < 0 \<and> diff 0 x = ?f x t"
by (intro Maclaurin_minus) auto
then obtain t where "x < t" "t < 0"
"diff 0 x = (\<Sum>m<n. diff m 0 / fact m * x ^ m) + diff n t / fact n * x ^ n"
by blast
with \<open>x < 0\<close> \<open>diff 0 = f\<close> have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t"
by simp
then show ?thesis ..
next
assume "x > 0"
with \<open>n \<noteq> 0\<close> \<open>diff 0 = f\<close> DERIV have "\<exists>t>0. t < x \<and> diff 0 x = ?f x t"
by (intro Maclaurin) auto
then obtain t where "0 < t" "t < x"
"diff 0 x = (\<Sum>m<n. diff m 0 / fact m * x ^ m) + diff n t / fact n * x ^ n"
by blast
with \<open>x > 0\<close> \<open>diff 0 = f\<close> have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> 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 \<noteq> 0"
and DERIV: "\<forall>m x. DERIV (diff m) x :> diff(Suc m) x"
shows "\<exists>t. 0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x =
(\<Sum>m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n"
(is "\<exists>t. _ \<and> _ \<and> f x = ?f x t")
proof (cases rule: linorder_cases)
assume "x = 0"
with INIT3 show ?thesis ..
next
assume "x < 0"
with assms have "\<exists>t>x. t < 0 \<and> 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 \<open>x < 0\<close> have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t"
by simp
then show ?thesis ..
next
assume "x > 0"
with assms have "\<exists>t>0. t < x \<and> 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 \<open>x > 0\<close> have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t"
by simp
then show ?thesis ..
qed
lemma Maclaurin_all_lt_objl:
fixes x :: real
shows
"diff 0 = f \<and> (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x) \<and> x \<noteq> 0 \<and> n > 0 \<longrightarrow>
(\<exists>t. 0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and>
f x = (\<Sum>m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n)"
by (blast intro: Maclaurin_all_lt)
lemma Maclaurin_zero: "x = 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> (\<Sum>m<n. (diff m 0 / fact m) * x ^ m) = diff 0 0"
for x :: real and n :: nat
by (induct n) auto
lemma Maclaurin_all_le:
fixes x :: real and n :: nat
assumes INIT: "diff 0 = f"
and DERIV: "\<forall>m x. DERIV (diff m) x :> diff (Suc m) x"
shows "\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = (\<Sum>m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n"
(is "\<exists>t. _ \<and> 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 \<open>n \<noteq> 0\<close> have "(\<Sum>m<n. diff m 0 / (fact m) * x ^ m) = diff 0 0"
by (intro Maclaurin_zero) auto
with INIT \<open>x = 0\<close> \<open>n \<noteq> 0\<close> have " \<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0"
by force
then show ?thesis ..
next
case False
with INIT \<open>n \<noteq> 0\<close> DERIV have "\<exists>t. 0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t"
by (intro Maclaurin_all_lt) auto
then obtain t where "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t" ..
then have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t"
by simp
then show ?thesis ..
qed
qed
lemma Maclaurin_all_le_objl:
"diff 0 = f \<and> (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x) \<longrightarrow>
(\<exists>t::real. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = (\<Sum>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 \<open>Version for Exponential Function\<close>
lemma Maclaurin_exp_lt:
fixes x :: real and n :: nat
shows
"x \<noteq> 0 \<Longrightarrow> n > 0 \<Longrightarrow>
(\<exists>t. 0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> exp x = (\<Sum>m<n. (x ^ m) / fact m) + (exp t / fact n) * x ^ n)"
using Maclaurin_all_lt_objl [where diff = "\<lambda>n. exp" and f = exp and x = x and n = n] by auto
lemma Maclaurin_exp_le:
fixes x :: real and n :: nat
shows "\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> exp x = (\<Sum>m<n. (x ^ m) / fact m) + (exp t / fact n) * x ^ n"
using Maclaurin_all_le_objl [where diff = "\<lambda>n. exp" and f = exp and x = x and n = n] by auto
lemma exp_lower_taylor_quadratic: "0 \<le> x \<Longrightarrow> 1 + x + x\<^sup>2 / 2 \<le> exp x"
for x :: real
using Maclaurin_exp_le [of x 3] by (auto simp: numeral_3_eq_3 power2_eq_square)
subsection \<open>Version for Sine Function\<close>
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
lemma Suc_Suc_mult_two_diff_two [simp]: "n \<noteq> 0 \<Longrightarrow> Suc (Suc (2 * n - 2)) = 2 * n"
by (induct n) auto
lemma lemma_Suc_Suc_4n_diff_2 [simp]: "n \<noteq> 0 \<Longrightarrow> Suc (Suc (4 * n - 2)) = 4 * n"
by (induct n) auto
lemma Suc_mult_two_diff_one [simp]: "n \<noteq> 0 \<Longrightarrow> Suc (2 * n - 1) = 2 * n"
by (induct n) auto
text \<open>It is unclear why so many variant results are needed.\<close>
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:
"\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and>
sin x = (\<Sum>m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n"
using Maclaurin_all_lt_objl
[where f = sin and n = n and x = x and diff = "\<lambda>n x. sin (x + 1/2 * real n * pi)"]
apply safe
apply simp
apply (simp add: sin_expansion_lemma del: of_nat_Suc)
apply (force intro!: derivative_eq_intros)
apply (subst (asm) sum.neutral; auto)
apply (rule ccontr)
apply simp
apply (drule_tac x = x in spec)
apply simp
apply (erule ssubst)
apply (rule_tac x = t in exI)
apply simp
apply (rule sum.cong[OF refl])
apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc)
done
lemma Maclaurin_sin_expansion:
"\<exists>t. sin x = (\<Sum>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:
"n > 0 \<Longrightarrow> 0 < x \<Longrightarrow>
\<exists>t. 0 < t \<and> t < x \<and>
sin x = (\<Sum>m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n"
using Maclaurin_objl
[where f = sin and n = n and h = x and diff = "\<lambda>n x. sin (x + 1/2 * real n * pi)"]
apply safe
apply simp
apply (simp (no_asm) add: sin_expansion_lemma del: of_nat_Suc)
apply (force intro!: derivative_eq_intros)
apply (erule ssubst)
apply (rule_tac x = t in exI)
apply simp
apply (rule sum.cong[OF refl])
apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc)
done
lemma Maclaurin_sin_expansion4:
"0 < x \<Longrightarrow>
\<exists>t. 0 < t \<and> t \<le> x \<and>
sin x = (\<Sum>m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n"
using Maclaurin2_objl
[where f = sin and n = n and h = x and diff = "\<lambda>n x. sin (x + 1/2 * real n * pi)"]
apply safe
apply simp
apply (simp (no_asm) add: sin_expansion_lemma del: of_nat_Suc)
apply (force intro!: derivative_eq_intros)
apply (erule ssubst)
apply (rule_tac x = t in exI)
apply simp
apply (rule sum.cong[OF refl])
apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc)
done
subsection \<open>Maclaurin Expansion for Cosine Function\<close>
lemma sumr_cos_zero_one [simp]: "(\<Sum>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:
"\<exists>t::real. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and>
cos x = (\<Sum>m<n. cos_coeff m * x ^ m) + (cos(t + 1/2 * real n * pi) / fact n) * x ^ n"
using Maclaurin_all_lt_objl
[where f = cos and n = n and x = x and diff = "\<lambda>n x. cos (x + 1/2 * real n * pi)"]
apply safe
apply simp
apply (simp add: cos_expansion_lemma del: of_nat_Suc)
apply (cases n)
apply simp
apply (simp del: sum_lessThan_Suc)
apply (rule ccontr)
apply simp
apply (drule_tac x = x in spec)
apply simp
apply (erule ssubst)
apply (rule_tac x = t in exI)
apply simp
apply (rule sum.cong[OF refl])
apply (auto simp add: cos_coeff_def cos_zero_iff elim: evenE)
done
lemma Maclaurin_cos_expansion2:
"0 < x \<Longrightarrow> n > 0 \<Longrightarrow>
\<exists>t. 0 < t \<and> t < x \<and>
cos x = (\<Sum>m<n. cos_coeff m * x ^ m) + (cos (t + 1/2 * real n * pi) / fact n) * x ^ n"
using Maclaurin_objl
[where f = cos and n = n and h = x and diff = "\<lambda>n x. cos (x + 1/2 * real n * pi)"]
apply safe
apply simp
apply (simp add: cos_expansion_lemma del: of_nat_Suc)
apply (erule ssubst)
apply (rule_tac x = t in exI)
apply simp
apply (rule sum.cong[OF refl])
apply (auto simp: cos_coeff_def cos_zero_iff elim: evenE)
done
lemma Maclaurin_minus_cos_expansion:
"x < 0 \<Longrightarrow> n > 0 \<Longrightarrow>
\<exists>t. x < t \<and> t < 0 \<and>
cos x = (\<Sum>m<n. cos_coeff m * x ^ m) + ((cos (t + 1/2 * real n * pi) / fact n) * x ^ n)"
using Maclaurin_minus_objl
[where f = cos and n = n and h = x and diff = "\<lambda>n x. cos (x + 1/2 * real n *pi)"]
apply safe
apply simp
apply (simp add: cos_expansion_lemma del: of_nat_Suc)
apply (erule ssubst)
apply (rule_tac x = t in exI)
apply simp
apply (rule sum.cong[OF refl])
apply (auto simp: cos_coeff_def cos_zero_iff elim: evenE)
done
(* Version for ln(1 +/- x). Where is it?? *)
lemma sin_bound_lemma: "x = y \<Longrightarrow> \<bar>u\<bar> \<le> v \<Longrightarrow> \<bar>(x + u) - y\<bar> \<le> v"
for x y u v :: real
by auto
lemma Maclaurin_sin_bound: "\<bar>sin x - (\<Sum>m<n. sin_coeff m * x ^ m)\<bar> \<le> inverse (fact n) * \<bar>x\<bar> ^ n"
proof -
have est: "x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y" for x y :: real
by (rule mult_right_mono) simp_all
let ?diff = "\<lambda>(n::nat) x.
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: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x"
apply clarify
apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
apply (cut_tac m=m in mod_exhaust_less_4)
apply safe
apply (auto intro!: derivative_eq_intros)
done
from Maclaurin_all_le [OF diff_0 DERIV_diff]
obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>"
and t2: "sin x = (\<Sum>m<n. ?diff m 0 / (fact m) * x ^ m) + ?diff n t / (fact n) * x ^ n"
by fast
have diff_m_0: "\<And>m. ?diff m 0 = (if even m then 0 else (- 1) ^ ((m - Suc 0) div 2))"
apply (subst even_even_mod_4_iff)
apply (cut_tac m=m in mod_exhaust_less_4)
apply (elim disjE)
apply simp_all
apply (safe dest!: mod_eqD)
apply simp_all
done
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 \<open>Taylor series\<close>
text \<open>
We use MacLaurin and the translation of the expansion point \<open>c\<close> to \<open>0\<close>
to prove Taylor's theorem.
\<close>
lemma taylor_up:
assumes INIT: "n > 0" "diff 0 = f"
and DERIV: "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t)"
and INTERV: "a \<le> c" "c < b"
shows "\<exists>t::real. c < t \<and> t < b \<and>
f b = (\<Sum>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" "(\<lambda>m x. diff m (x + c)) 0 = (\<lambda>x. f (x + c))"
by auto
moreover
have "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> b - c \<longrightarrow> DERIV (\<lambda>x. diff m (x + c)) t :> diff (Suc m) (t + c)"
proof (intro strip)
fix m t
assume "m < n \<and> 0 \<le> t \<and> t \<le> 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 (\<lambda>x. x + c) t :> 1 + 0"
by (rule DERIV_add)
ultimately have "DERIV (\<lambda>x. diff m (x + c)) t :> diff (Suc m) (t + c) * (1 + 0)"
by (rule DERIV_chain2)
then show "DERIV (\<lambda>x. diff m (x + c)) t :> diff (Suc m) (t + c)"
by simp
qed
ultimately obtain x where
"0 < x \<and> x < b - c \<and>
f (b - c + c) =
(\<Sum>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 \<and> x + c < b \<and> f b =
(\<Sum>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: "(\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> b \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t)"
and INTERV: "a < c" "c \<le> b"
shows "\<exists>t. a < t \<and> t < c \<and>
f a = (\<Sum>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" "(\<lambda>m x. diff m (x + c)) 0 = (\<lambda>x. f (x + c))"
by auto
moreover
have "\<forall>m t. m < n \<and> a - c \<le> t \<and> t \<le> 0 \<longrightarrow> DERIV (\<lambda>x. diff m (x + c)) t :> diff (Suc m) (t + c)"
proof (rule allI impI)+
fix m t
assume "m < n \<and> a - c \<le> t \<and> t \<le> 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 (\<lambda>x. x + c) t :> 1 + 0"
by (rule DERIV_add)
ultimately have "DERIV (\<lambda>x. diff m (x + c)) t :> diff (Suc m) (t + c) * (1 + 0)"
by (rule DERIV_chain2)
then show "DERIV (\<lambda>x. diff m (x + c)) t :> diff (Suc m) (t + c)"
by simp
qed
ultimately obtain x where
"a - c < x \<and> x < 0 \<and>
f (a - c + c) =
(\<Sum>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 \<and> x + c < c \<and>
f a = (\<Sum>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: "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> b \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
and INTERV: "a \<le> c " "c \<le> b" "a \<le> x" "x \<le> b" "x \<noteq> c"
shows "\<exists>t.
(if x < c then x < t \<and> t < c else c < t \<and> t < x) \<and>
f x = (\<Sum>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 "\<forall>m t. m < n \<and> x \<le> t \<and> t \<le> b \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
using DERIV and INTERV by fastforce
moreover note True
moreover from INTERV have "c \<le> b"
by simp
ultimately have "\<exists>t>x. t < c \<and> f x =
(\<Sum>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 "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> x \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
using DERIV and INTERV by fastforce
moreover from INTERV have "a \<le> c"
by arith
moreover from False and INTERV have "c < x"
by arith
ultimately have "\<exists>t>c. t < x \<and> f x =
(\<Sum>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