(* Author : Jacques D. Fleuriot
Copyright : 2001 University of Edinburgh
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
Conversion of Mac Laurin to Isar by Lukas Bulwahn and Bernhard Häupler, 2005
*)
section\<open>MacLaurin 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 ==>
\<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) = (y = x + (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 & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t"
proof (rule allI impI)+
fix m and t::real
assume INIT2: "m < n & 0 \<le> t & 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)))"
unfolding difg_def
by (auto 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 atLeast0LessThan by (subst setsum.insert) (auto simp: setsum.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"
assumes n: "0 < n"
assumes diff_0: "diff 0 = f"
assumes diff_Suc:
"\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t"
shows
"\<exists>t::real. 0 < t & t < h &
f h =
setsum (%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)
obtain B where f_h: "f h =
(\<Sum>m<n. diff m (0::real) / (fact m) * h ^ m) + B * (h ^ n / (fact n))"
using Maclaurin_lemma [OF h] ..
def g \<equiv> "(\<lambda>t. f t -
(setsum (\<lambda>m. (diff m 0 / (fact m)) * t^m) {..<n} + (B * (t^n / (fact n)))))"
have g2: "g 0 = 0 & g h = 0"
by (simp add: m f_h g_def lessThan_Suc_eq_insert_0 image_iff diff_0 setsum.reindex)
def difg \<equiv> "(%m t. diff m t -
(setsum (%p. (diff (m + p) 0 / (fact p)) * (t ^ p)) {..<n-m}
+ (B * ((t ^ (n - m)) / (fact (n - m))))))"
have difg_0: "difg 0 = g"
unfolding difg_def g_def by (simp add: diff_0)
have difg_Suc: "\<forall>(m::nat) t::real.
m < n \<and> (0::real) \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t"
using diff_Suc m unfolding difg_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 setsum.reindex)
have isCont_difg: "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> isCont (difg m) x"
by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp
have differentiable_difg:
"\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> difg m differentiable (at x)"
by (rule differentiableI [OF difg_Suc [rule_format]]) simp
have difg_Suc_eq_0: "\<And>m t. \<lbrakk>m < n; 0 \<le> t; t \<le> h; DERIV (difg m) t :> 0\<rbrakk>
\<Longrightarrow> difg (Suc m) t = 0"
by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp
have "m < n" using m by 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')
hence "\<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
thus ?case
using \<open>t < h\<close> by auto
qed
then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0" by fast
hence "difg (Suc m) t = 0"
using \<open>m < n\<close> 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 & n>0 & diff 0 = f &
(\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
--> (\<exists>t::real. 0 < t & t < h &
f h = (\<Sum>m<n. diff m 0 / (fact m) * h ^ m) +
diff n t / (fact n) * h ^ n)"
by (blast intro: Maclaurin)
lemma Maclaurin2:
assumes INIT1: "0 < h " and INIT2: "diff 0 = f"
and DERIV: "\<forall>m t::real.
m < n & 0 \<le> t & t \<le> h --> 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
hence "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)
thus ?thesis by fastforce
qed
lemma Maclaurin2_objl:
"0 < h & diff 0 = f &
(\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
--> (\<exists>t::real. 0 < t &
t \<le> h &
f h =
(\<Sum>m<n. diff m 0 / (fact m) * h ^ m) +
diff n t / (fact n) * h ^ n)"
by (blast intro: Maclaurin2)
lemma Maclaurin_minus:
fixes h::real
assumes "h < 0" "0 < n" "diff 0 = f"
and DERIV: "\<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t"
shows "\<exists>t. h < t & t < 0 &
f h = (\<Sum>m<n. diff m 0 / (fact m) * h ^ m) +
diff n t / (fact n) * h ^ n"
proof -
txt "Transform \<open>ABL'\<close> into \<open>derivative_intros\<close> format."
note DERIV' = DERIV_chain'[OF _ DERIV[rule_format], THEN DERIV_cong]
from assms
have "\<exists>t>0. t < - h \<and>
f (- (- h)) =
(\<Sum>m<n.
(- 1) ^ m * diff m (- 0) / (fact m) * (- h) ^ m) +
(- 1) ^ n * diff n (- t) / (fact n) * (- h) ^ n"
by (intro Maclaurin) (auto intro!: derivative_eq_intros DERIV')
then guess t ..
moreover
have "(- 1) ^ n * diff n (- t) * (- h) ^ n / (fact n) = diff n (- t) * h ^ n / (fact n)"
by (auto simp add: 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: setsum.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
thus ?thesis ..
qed
lemma Maclaurin_minus_objl:
fixes h::real
shows
"(h < 0 & n > 0 & diff 0 = f &
(\<forall>m t.
m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
--> (\<exists>t. h < t &
t < 0 &
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:
assumes "diff 0 = f"
and DERIV : "\<forall>m t::real. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t"
shows "\<exists>t. abs t \<le> abs x &
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
assume "n = 0" with \<open>diff 0 = f\<close> show ?thesis by force
next
assume "n \<noteq> 0"
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)
thus ?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 guess t ..
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
thus ?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 guess t ..
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
thus ?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 < abs t & abs t < abs x & 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 guess t ..
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
thus ?thesis ..
next
assume "x > 0"
with assms have "\<exists>t>0. t < x \<and> f x = ?f x t " by (intro Maclaurin) auto
then guess t ..
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
thus ?thesis ..
qed
lemma Maclaurin_all_lt_objl:
fixes x::real
shows
"diff 0 = f &
(\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
x ~= 0 & n > 0
--> (\<exists>t. 0 < abs t & abs t < abs x &
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 [rule_format]:
"x = (0::real)
==> n \<noteq> 0 -->
(\<Sum>m<n. (diff m (0::real) / (fact m)) * x ^ m) =
diff 0 0"
by (induct n, auto)
lemma Maclaurin_all_le:
assumes INIT: "diff 0 = f"
and DERIV: "\<forall>m x::real. DERIV (diff m) x :> diff (Suc m) x"
shows "\<exists>t. abs t \<le> abs x & 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
assume "n = 0" with INIT show ?thesis by force
next
assume "n \<noteq> 0"
show ?thesis
proof cases
assume "x = 0"
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
thus ?thesis ..
next
assume "x \<noteq> 0"
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 guess t ..
hence "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp
thus ?thesis ..
qed
qed
lemma Maclaurin_all_le_objl:
"diff 0 = f &
(\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
--> (\<exists>t::real. abs t \<le> abs x &
f x = (\<Sum>m<n. (diff m 0 / (fact m)) * x ^ m) +
(diff n t / (fact n)) * x ^ n)"
by (blast intro: Maclaurin_all_le)
subsection\<open>Version for Exponential Function\<close>
lemma Maclaurin_exp_lt:
fixes x::real
shows
"[| x ~= 0; n > 0 |]
==> (\<exists>t. 0 < abs t &
abs t < abs x &
exp x = (\<Sum>m<n. (x ^ m) / (fact m)) +
(exp t / (fact n)) * x ^ n)"
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)
lemma Maclaurin_exp_le:
"\<exists>t::real. abs t \<le> abs x &
exp x = (\<Sum>m<n. (x ^ m) / (fact m)) +
(exp t / (fact n)) * x ^ n"
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
lemma exp_lower_taylor_quadratic:
fixes x::real
shows "0 \<le> x \<Longrightarrow> 1 + x + x\<^sup>2 / 2 \<le> exp x"
using Maclaurin_exp_le [of x 3]
by (auto simp: numeral_3_eq_3 power2_eq_square power_Suc)
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::nat)"
by auto
lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
"n\<noteq>0 --> Suc (Suc (2 * n - 2)) = 2*n"
by (induct "n", auto)
lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
"n\<noteq>0 --> Suc (Suc (4*n - 2)) = 4*n"
by (induct "n", auto)
lemma Suc_mult_two_diff_one [rule_format, simp]:
"n\<noteq>0 --> 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 (simp only: cos_add sin_add of_nat_Suc add_divide_distrib distrib_right, auto)
lemma Maclaurin_sin_expansion2:
"\<exists>t. abs t \<le> abs x &
sin x =
(\<Sum>m<n. sin_coeff m * x ^ m)
+ ((sin(t + 1/2 * real (n) *pi) / (fact n)) * x ^ n)"
apply (cut_tac f = sin and n = n and x = x
and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
apply safe
apply (simp)
apply (simp add: sin_expansion_lemma del: of_nat_Suc)
apply (force intro!: derivative_eq_intros)
apply (subst (asm) setsum.neutral, auto)[1]
apply (rule ccontr, simp)
apply (drule_tac x = x in spec, simp)
apply (erule ssubst)
apply (rule_tac x = t in exI, simp)
apply (rule setsum.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)"
apply (insert Maclaurin_sin_expansion2 [of x n])
apply (blast intro: elim:)
done
lemma Maclaurin_sin_expansion3:
"[| n > 0; 0 < x |] ==>
\<exists>t. 0 < t & t < x &
sin x =
(\<Sum>m<n. sin_coeff m * x ^ m)
+ ((sin(t + 1/2 * real(n) *pi) / (fact n)) * x ^ n)"
apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin_objl)
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, simp)
apply (rule setsum.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 ==>
\<exists>t. 0 < t & t \<le> x &
sin x =
(\<Sum>m<n. sin_coeff m * x ^ m)
+ ((sin(t + 1/2 * real (n) *pi) / (fact n)) * x ^ n)"
apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin2_objl)
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, simp)
apply (rule setsum.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 (simp only: cos_add sin_add of_nat_Suc distrib_right add_divide_distrib, auto)
lemma Maclaurin_cos_expansion:
"\<exists>t::real. abs t \<le> abs x &
cos x =
(\<Sum>m<n. cos_coeff m * x ^ m)
+ ((cos(t + 1/2 * real (n) *pi) / (fact n)) * x ^ n)"
apply (cut_tac f = cos and n = n and x = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_all_lt_objl)
apply safe
apply (simp (no_asm))
apply (simp (no_asm) add: cos_expansion_lemma del: of_nat_Suc)
apply (case_tac "n", simp)
apply (simp del: setsum_lessThan_Suc)
apply (rule ccontr, simp)
apply (drule_tac x = x in spec, simp)
apply (erule ssubst)
apply (rule_tac x = t in exI, simp)
apply (rule setsum.cong[OF refl])
apply (auto simp add: cos_coeff_def cos_zero_iff elim: evenE)
done
lemma Maclaurin_cos_expansion2:
"[| 0 < x; n > 0 |] ==>
\<exists>t. 0 < t & t < x &
cos x =
(\<Sum>m<n. cos_coeff m * x ^ m)
+ ((cos(t + 1/2 * real (n) *pi) / (fact n)) * x ^ n)"
apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_objl)
apply safe
apply simp
apply (simp (no_asm) add: cos_expansion_lemma del: of_nat_Suc)
apply (erule ssubst)
apply (rule_tac x = t in exI, simp)
apply (rule setsum.cong[OF refl])
apply (auto simp add: cos_coeff_def cos_zero_iff elim: evenE)
done
lemma Maclaurin_minus_cos_expansion:
"[| x < 0; n > 0 |] ==>
\<exists>t. x < t & t < 0 &
cos x =
(\<Sum>m<n. cos_coeff m * x ^ m)
+ ((cos(t + 1/2 * real (n) *pi) / (fact n)) * x ^ n)"
apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_minus_objl)
apply safe
apply simp
apply (simp (no_asm) add: cos_expansion_lemma del: of_nat_Suc)
apply (erule ssubst)
apply (rule_tac x = t in exI, simp)
apply (rule setsum.cong[OF refl])
apply (auto simp add: cos_coeff_def cos_zero_iff elim: evenE)
done
(* ------------------------------------------------------------------------- *)
(* Version for ln(1 +/- x). Where is it?? *)
(* ------------------------------------------------------------------------- *)
lemma sin_bound_lemma:
"[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
by auto
lemma Maclaurin_sin_bound:
"abs(sin x - (\<Sum>m<n. sin_coeff m * x ^ m))
\<le> inverse((fact n)) * \<bar>x\<bar> ^ n"
proof -
have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
by (rule_tac mult_right_mono,simp_all)
note est = this[simplified]
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, 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, simp_all)
apply (safe dest!: mod_eqD, simp_all)
done
show ?thesis
unfolding sin_coeff_def
apply (subst t2)
apply (rule sin_bound_lemma)
apply (rule setsum.cong[OF refl])
apply (subst diff_m_0, simp)
apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
simp add: est ac_simps divide_inverse power_abs [symmetric] abs_mult)
done
qed
end