(* 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
*)
header{*MacLaurin 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 ==>
\<exists>B. f h = (\<Sum>m=0..<n. (j m / real (fact m)) * (h^m)) +
(B * ((h^n) / real(fact n)))"
by (rule exI[where x = "(f h - (\<Sum>m=0..<n. (j m / real (fact m)) * h^m)) *
real(fact n) / (h^n)"]) simp
lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
by arith
lemma fact_diff_Suc [rule_format]:
"n < Suc m ==> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
by (subst fact_reduce_nat, 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. diff m t - ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
B * (t ^ (n - m) / real (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 t assume INIT2: "m < n & 0 \<le> t & t \<le> h"
have "DERIV (difg m) t :> diff (Suc m) t -
((\<Sum>x = 0..<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / real (fact x)) +
real (n - m) * t ^ (n - Suc m) * B / real (fact (n - m)))" unfolding difg_def
by (auto intro!: DERIV_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 = 0..<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / real (fact x)) =
(\<Sum>x = 0..<n - Suc m. real (Suc x) * t ^ x * diff (Suc m + x) 0 / real (fact (Suc x)))"
unfolding intvl atLeast0LessThan by (subst setsum.insert) (auto simp: setsum.reindex)
moreover
have fact_neq_0: "\<And>x::nat. real (fact x) + real x * real (fact x) \<noteq> 0"
by (metis fact_gt_zero_nat not_add_less1 real_of_nat_add real_of_nat_mult real_of_nat_zero_iff)
have "\<And>x. real (Suc x) * t ^ x * diff (Suc m + x) 0 / real (fact (Suc x)) =
diff (Suc m + x) 0 * t^x / real (fact x)"
by (auto simp: field_simps real_of_nat_Suc fact_neq_0 intro!: nonzero_divide_eq_eq[THEN iffD2])
moreover
have "real (n - m) * t ^ (n - Suc m) * B / real (fact (n - m)) =
B * (t ^ (n - Suc m) / real (fact (n - Suc m)))"
using `0 < n - m` by (simp add: fact_reduce_nat)
ultimately show "DERIV (difg m) t :> difg (Suc m) t"
unfolding difg_def by simp
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. 0 < t & t < h &
f h =
setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
(diff n t / real (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 = 0..<n. diff m (0\<Colon>real) / real (fact m) * h ^ m) +
B * (h ^ n / real (fact n))"
using Maclaurin_lemma [OF h] ..
def g \<equiv> "(\<lambda>t. f t -
(setsum (\<lambda>m. (diff m 0 / real(fact m)) * t^m) {0..<n}
+ (B * (t^n / real(fact n)))))"
have g2: "g 0 = 0 & g h = 0"
apply (simp add: m f_h g_def del: setsum_op_ivl_Suc)
apply (cut_tac n = m and k = "Suc 0" in sumr_offset2)
apply (simp add: eq_diff_eq' diff_0 del: setsum_op_ivl_Suc)
done
def difg \<equiv> "(%m t. diff m t -
(setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {0..<n-m}
+ (B * ((t ^ (n - m)) / real (fact (n - m))))))"
have difg_0: "difg 0 = g"
unfolding difg_def g_def by (simp add: diff_0)
have difg_Suc: "\<forall>(m\<Colon>nat) t\<Colon>real.
m < n \<and> (0\<Colon>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. m < n --> difg m 0 = 0"
apply clarify
apply (simp add: m difg_def)
apply (frule less_iff_Suc_add [THEN iffD1], clarify)
apply (simp del: setsum_op_ivl_Suc)
apply (insert sumr_offset4 [of "Suc 0"])
apply (simp del: setsum_op_ivl_Suc fact_Suc)
done
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 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 `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 "\<forall>x. 0 \<le> x \<and> x \<le> h \<longrightarrow> isCont (difg (0\<Colon>nat)) x"
by (simp add: isCont_difg n)
show "\<forall>x. 0 < x \<and> x < h \<longrightarrow> difg (0\<Colon>nat) differentiable 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 `Suc m' < n` 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 `t < h` `Suc m' < n` by (simp add: isCont_difg)
show "\<forall>x. 0 < x \<and> x < t \<longrightarrow> difg (Suc m') differentiable x"
using `t < h` `Suc m' < n` by (simp add: differentiable_difg)
qed
thus ?case
using `t < h` 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 `m < n` 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 = 0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n"
using `difg (Suc m) t = 0`
by (simp add: m f_h difg_def del: fact_Suc)
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. 0 < t & t < h &
f h = (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n)"
by (blast intro: Maclaurin)
lemma Maclaurin2:
assumes INIT1: "0 < h " and INIT2: "diff 0 = f"
and DERIV: "\<forall>m t.
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=0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n"
proof (cases "n")
case 0 with INIT1 INIT2 show ?thesis by fastsimp
next
case Suc
hence "n > 0" by simp
from INIT1 this INIT2 DERIV have "\<exists>t>0. t < h \<and>
f h =
(\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) + diff n t / real (fact n) * h ^ n"
by (rule Maclaurin)
thus ?thesis by fastsimp
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. 0 < t &
t \<le> h &
f h =
(\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n)"
by (blast intro: Maclaurin2)
lemma Maclaurin_minus:
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=0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n"
proof -
txt "Transform @{text ABL'} into @{text DERIV_intros} 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 = 0..<n.
(- 1) ^ m * diff m (- 0) / real (fact m) * (- h) ^ m) +
(- 1) ^ n * diff n (- t) / real (fact n) * (- h) ^ n"
by (intro Maclaurin) (auto intro!: DERIV_intros DERIV')
then guess t ..
moreover
have "-1 ^ n * diff n (- t) * (- h) ^ n / real (fact n) = diff n (- t) * h ^ n / real (fact n)"
by (auto simp add: power_mult_distrib[symmetric])
moreover
have "(SUM m = 0..<n. -1 ^ m * diff m 0 * (- h) ^ m / real (fact m)) = (SUM m = 0..<n. diff m 0 * h ^ m / real (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 = 0..<n. diff m 0 / real (fact m) * h ^ m) + diff n (- t) / real (fact n) * h ^ n"
by auto
thus ?thesis ..
qed
lemma Maclaurin_minus_objl:
"(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=0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n)"
by (blast intro: Maclaurin_minus)
subsection{*More Convenient "Bidirectional" Version.*}
(* not good for PVS sin_approx, cos_approx *)
lemma Maclaurin_bi_le_lemma [rule_format]:
"n>0 \<longrightarrow>
diff 0 0 =
(\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
diff n 0 * 0 ^ n / real (fact n)"
by (induct "n") auto
lemma Maclaurin_bi_le:
assumes "diff 0 = f"
and DERIV : "\<forall>m t. 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=0..<n. diff m 0 / real (fact m) * x ^ m) +
diff n t / real (fact n) * x ^ n" (is "\<exists>t. _ \<and> f x = ?f x t")
proof cases
assume "n = 0" with `diff 0 = f` show ?thesis by force
next
assume "n \<noteq> 0"
show ?thesis
proof (cases rule: linorder_cases)
assume "x = 0" with `n \<noteq> 0` `diff 0 = f` DERIV
have "\<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0" by (force simp add: Maclaurin_bi_le_lemma)
thus ?thesis ..
next
assume "x < 0"
with `n \<noteq> 0` DERIV
have "\<exists>t>x. t < 0 \<and> diff 0 x = ?f x t" by (intro Maclaurin_minus) auto
then guess t ..
with `x < 0` `diff 0 = f` have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp
thus ?thesis ..
next
assume "x > 0"
with `n \<noteq> 0` `diff 0 = f` DERIV
have "\<exists>t>0. t < x \<and> diff 0 x = ?f x t" by (intro Maclaurin) auto
then guess t ..
with `x > 0` `diff 0 = f` have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp
thus ?thesis ..
qed
qed
lemma Maclaurin_all_lt:
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=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
(diff n t / real (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 `x < 0` 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 `x > 0` 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:
"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=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
(diff n t / real (fact n)) * x ^ n)"
by (blast intro: Maclaurin_all_lt)
lemma Maclaurin_zero [rule_format]:
"x = (0::real)
==> n \<noteq> 0 -->
(\<Sum>m=0..<n. (diff m (0::real) / 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. DERIV (diff m) x :> diff (Suc m) x"
shows "\<exists>t. abs t \<le> abs x & f x =
(\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
(diff n t / real (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 `n \<noteq> 0` have "(\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) = diff 0 0"
by (intro Maclaurin_zero) auto
with INIT `x = 0` `n \<noteq> 0` 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 `n \<noteq> 0` 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. abs t \<le> abs x &
f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
(diff n t / real (fact n)) * x ^ n)"
by (blast intro: Maclaurin_all_le)
subsection{*Version for Exponential Function*}
lemma Maclaurin_exp_lt: "[| x ~= 0; n > 0 |]
==> (\<exists>t. 0 < abs t &
abs t < abs x &
exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
(exp t / real (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. abs t \<le> abs x &
exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
(exp t / real (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)
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::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{*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 (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)
lemma Maclaurin_sin_expansion2:
"\<exists>t. abs t \<le> abs x &
sin x =
(\<Sum>m=0..<n. (if even m then 0
else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
x ^ m)
+ ((sin(t + 1/2 * real (n) *pi) / real (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 (no_asm))
apply (simp (no_asm) add: sin_expansion_lemma)
apply (case_tac "n", clarify, simp, simp add: lemma_STAR_sin)
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_zero_iff odd_Suc_mult_two_ex)
done
lemma Maclaurin_sin_expansion:
"\<exists>t. sin x =
(\<Sum>m=0..<n. (if even m then 0
else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
x ^ m)
+ ((sin(t + 1/2 * real (n) *pi) / real (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=0..<n. (if even m then 0
else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
x ^ m)
+ ((sin(t + 1/2 * real(n) *pi) / real (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)
apply (erule ssubst)
apply (rule_tac x = t in exI, simp)
apply (rule setsum_cong[OF refl])
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
done
lemma Maclaurin_sin_expansion4:
"0 < x ==>
\<exists>t. 0 < t & t \<le> x &
sin x =
(\<Sum>m=0..<n. (if even m then 0
else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
x ^ m)
+ ((sin(t + 1/2 * real (n) *pi) / real (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)
apply (erule ssubst)
apply (rule_tac x = t in exI, simp)
apply (rule setsum_cong[OF refl])
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
done
subsection{*Maclaurin Expansion for Cosine Function*}
lemma sumr_cos_zero_one [simp]:
"(\<Sum>m=0..<(Suc n).
(if even m then -1 ^ (m div 2)/(real (fact m)) else 0) * 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 real_of_nat_Suc left_distrib add_divide_distrib, auto)
lemma Maclaurin_cos_expansion:
"\<exists>t. abs t \<le> abs x &
cos x =
(\<Sum>m=0..<n. (if even m
then -1 ^ (m div 2)/(real (fact m))
else 0) *
x ^ m)
+ ((cos(t + 1/2 * real (n) *pi) / real (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)
apply (case_tac "n", simp)
apply (simp del: setsum_op_ivl_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_zero_iff even_mult_two_ex)
done
lemma Maclaurin_cos_expansion2:
"[| 0 < x; n > 0 |] ==>
\<exists>t. 0 < t & t < x &
cos x =
(\<Sum>m=0..<n. (if even m
then -1 ^ (m div 2)/(real (fact m))
else 0) *
x ^ m)
+ ((cos(t + 1/2 * real (n) *pi) / real (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)
apply (erule ssubst)
apply (rule_tac x = t in exI, simp)
apply (rule setsum_cong[OF refl])
apply (auto simp add: cos_zero_iff even_mult_two_ex)
done
lemma Maclaurin_minus_cos_expansion:
"[| x < 0; n > 0 |] ==>
\<exists>t. x < t & t < 0 &
cos x =
(\<Sum>m=0..<n. (if even m
then -1 ^ (m div 2)/(real (fact m))
else 0) *
x ^ m)
+ ((cos(t + 1/2 * real (n) *pi) / real (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)
apply (erule ssubst)
apply (rule_tac x = t in exI, simp)
apply (rule setsum_cong[OF refl])
apply (auto simp add: cos_zero_iff even_mult_two_ex)
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=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
x ^ m)) \<le> inverse(real (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!: DERIV_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 = 0..<n. ?diff m 0 / real (fact m) * x ^ m) +
?diff n t / real (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
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 mult_nonneg_nonneg mult_ac divide_inverse
power_abs [symmetric] abs_mult)
done
qed
end