(* 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_tac x = "(f h - (\<Sum>m=0..<n. (j m / real (fact m)) * h^m)) *
real(fact n) / (h^n)"
in exI, 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:
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"
and DIFG_DEF: "difg = (\<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)))))"
shows "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t"
proof (rule allI)+
fix m
fix t
show "m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t"
proof
assume INIT2: "m < n & 0 \<le> t & t \<le> h"
hence INTERV: "0 \<le> t & t \<le> h" ..
from INIT2 and INIT have mtok: "m < Suc k" by arith
have "DERIV (\<lambda>t. diff m t -
((\<Sum>p = 0..<Suc k - m. diff (m + p) 0 / real (fact p) * t ^ p) +
B * (t ^ (Suc k - m) / real (fact (Suc k - m)))))
t :> diff (Suc m) t -
((\<Sum>p = 0..<Suc k - Suc m. diff (Suc m + p) 0 / real (fact p) * t ^ p) +
B * (t ^ (Suc k - Suc m) / real (fact (Suc k - Suc m))))"
proof -
from DERIV and INIT2 have "DERIV (diff m) t :> diff (Suc m) t" by simp
moreover
have " DERIV (\<lambda>x. (\<Sum>p = 0..<Suc k - m. diff (m + p) 0 / real (fact p) * x ^ p) +
B * (x ^ (Suc k - m) / real (fact (Suc k - m))))
t :> (\<Sum>p = 0..<Suc k - Suc m. diff (Suc m + p) 0 / real (fact p) * t ^ p) +
B * (t ^ (Suc k - Suc m) / real (fact (Suc k - Suc m)))"
proof -
have "DERIV (\<lambda>x. \<Sum>p = 0..<Suc k - m. diff (m + p) 0 / real (fact p) * x ^ p) t
:> (\<Sum>p = 0..<Suc k - Suc m. diff (Suc m + p) 0 / real (fact p) * t ^ p)"
proof -
have "\<exists> d. k = m + d"
proof -
from INIT2 have "m < n" ..
hence "\<exists> d. n = m + d + Suc 0" by arith
with INIT show ?thesis by (simp del: setsum_op_ivl_Suc)
qed
from this obtain d where kmd: "k = m + d" ..
have "DERIV (\<lambda>x. (\<Sum>ma = 0..<d. diff (Suc (m + ma)) 0 * x ^ Suc ma / real (fact (Suc ma))) +
diff m 0)
t :> (\<Sum>p = 0..<d. diff (Suc (m + p)) 0 * t ^ p / real (fact p))"
proof -
have "DERIV (\<lambda>x. (\<Sum>ma = 0..<d. diff (Suc (m + ma)) 0 * x ^ Suc ma / real (fact (Suc ma))) + diff m 0) t :> (\<Sum>r = 0..<d. diff (Suc (m + r)) 0 * t ^ r / real (fact r)) + 0"
proof -
from DERIV and INTERV have "DERIV (\<lambda>x. (\<Sum>ma = 0..<d. diff (Suc (m + ma)) 0 * x ^ Suc ma / real (fact (Suc ma)))) t :> (\<Sum>r = 0..<d. diff (Suc (m + r)) 0 * t ^ r / real (fact r))"
proof -
have "\<forall>r. 0 \<le> r \<and> r < 0 + d \<longrightarrow>
DERIV (\<lambda>x. diff (Suc (m + r)) 0 * x ^ Suc r / real (fact (Suc r))) t
:> diff (Suc (m + r)) 0 * t ^ r / real (fact r)"
proof (rule allI)
fix r
show " 0 \<le> r \<and> r < 0 + d \<longrightarrow>
DERIV (\<lambda>x. diff (Suc (m + r)) 0 * x ^ Suc r / real (fact (Suc r))) t
:> diff (Suc (m + r)) 0 * t ^ r / real (fact r)"
proof
assume "0 \<le> r & r < 0 + d"
have "DERIV (\<lambda>x. diff (Suc (m + r)) 0 *
(x ^ Suc r * inverse (real (fact (Suc r)))))
t :> diff (Suc (m + r)) 0 * (t ^ r * inverse (real (fact r)))"
proof -
have "(1 + real r) * real (fact r) \<noteq> 0" by auto
from this have "real (fact r) + real r * real (fact r) \<noteq> 0"
by (metis add_nonneg_eq_0_iff mult_nonneg_nonneg real_of_nat_fact_not_zero real_of_nat_ge_zero)
from this have "DERIV (\<lambda>x. x ^ Suc r * inverse (real (fact (Suc r)))) t :> real (Suc r) * t ^ (Suc r - Suc 0) * inverse (real (fact (Suc r))) +
0 * t ^ Suc r"
apply - by ( rule DERIV_intros | rule refl)+ auto
moreover
have "real (Suc r) * t ^ (Suc r - Suc 0) * inverse (real (fact (Suc r))) +
0 * t ^ Suc r =
t ^ r * inverse (real (fact r))"
proof -
have " real (Suc r) * t ^ (Suc r - Suc 0) *
inverse (real (Suc r) * real (fact r)) +
0 * t ^ Suc r =
t ^ r * inverse (real (fact r))" by (simp add: mult_ac)
hence "real (Suc r) * t ^ (Suc r - Suc 0) * inverse (real (Suc r * fact r)) +
0 * t ^ Suc r =
t ^ r * inverse (real (fact r))" by (subst real_of_nat_mult)
thus ?thesis by (subst fact_Suc)
qed
ultimately have " DERIV (\<lambda>x. x ^ Suc r * inverse (real (fact (Suc r)))) t
:> t ^ r * inverse (real (fact r))" by (rule lemma_DERIV_subst)
thus ?thesis by (rule DERIV_cmult)
qed
thus "DERIV (\<lambda>x. diff (Suc (m + r)) 0 * x ^ Suc r /
real (fact (Suc r)))
t :> diff (Suc (m + r)) 0 * t ^ r / real (fact r)" by (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc power_Suc)
qed
qed
thus ?thesis by (rule DERIV_sumr)
qed
moreover
from DERIV_const have "DERIV (\<lambda>x. diff m 0) t :> 0" .
ultimately show ?thesis by (rule DERIV_add)
qed
moreover
have " (\<Sum>r = 0..<d. diff (Suc (m + r)) 0 * t ^ r / real (fact r)) + 0 = (\<Sum>p = 0..<d. diff (Suc (m + p)) 0 * t ^ p / real (fact p))" by simp
ultimately show ?thesis by (rule lemma_DERIV_subst)
qed
with kmd and sumr_offset4 [of 1] show ?thesis by (simp del: setsum_op_ivl_Suc fact_Suc power_Suc)
qed
moreover
have " DERIV (\<lambda>x. B * (x ^ (Suc k - m) / real (fact (Suc k - m)))) t
:> B * (t ^ (Suc k - Suc m) / real (fact (Suc k - Suc m)))"
proof -
have " DERIV (\<lambda>x. x ^ (Suc k - m) / real (fact (Suc k - m))) t
:> t ^ (Suc k - Suc m) / real (fact (Suc k - Suc m))"
proof -
have "DERIV (\<lambda>x. x ^ (Suc k - m)) t :> real (Suc k - m) * t ^ (Suc k - m - Suc 0)" by (rule DERIV_pow)
moreover
have "DERIV (\<lambda>x. real (fact (Suc k - m))) t :> 0" by (rule DERIV_const)
moreover
have "(\<lambda>x. real (fact (Suc k - m))) t \<noteq> 0" by simp
ultimately have " DERIV (\<lambda>y. y ^ (Suc k - m) / real (fact (Suc k - m))) t
:> ( real (Suc k - m) * t ^ (Suc k - m - Suc 0) * real (fact (Suc k - m)) + - (0 * t ^ (Suc k - m))) /
real (fact (Suc k - m)) ^ Suc (Suc 0)"
apply -
apply (rule DERIV_cong) by (rule DERIV_intros | rule refl)+ auto
moreover
from mtok and INIT have "( real (Suc k - m) * t ^ (Suc k - m - Suc 0) * real (fact (Suc k - m)) + - (0 * t ^ (Suc k - m))) /
real (fact (Suc k - m)) ^ Suc (Suc 0) = t ^ (Suc k - Suc m) / real (fact (Suc k - Suc m))" by (simp add: fact_diff_Suc)
ultimately show ?thesis by (rule lemma_DERIV_subst)
qed
moreover
thus ?thesis by (rule DERIV_cmult)
qed
ultimately show ?thesis by (rule DERIV_add)
qed
ultimately show ?thesis by (rule DERIV_diff)
qed
from INIT and this and DIFG_DEF show "DERIV (difg m) t :> difg (Suc m) t" by clarify
qed
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] ..
obtain g where g_def: "g = (%t. f t -
(setsum (%m. (diff m 0 / real(fact m)) * t^m) {0..<n}
+ (B * (t^n / real(fact n)))))" by blast
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
obtain difg where difg_def: "difg = (%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))))))" by blast
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 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 INTERV : "h < 0" and
INIT : "0 < n" "diff 0 = f" and
ABL : "\<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 -
from INTERV have "0 < -h" by simp
moreover
from INIT have "0 < n" by simp
moreover
from INIT have "(% x. ( - 1) ^ 0 * diff 0 (- x)) = (% x. f (- x))" by simp
moreover
have "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> - h \<longrightarrow>
DERIV (\<lambda>x. (- 1) ^ m * diff m (- x)) t :> (- 1) ^ Suc m * diff (Suc m) (- t)"
proof (rule allI impI)+
fix m t
assume tINTERV:" m < n \<and> 0 \<le> t \<and> t \<le> - h"
with ABL show "DERIV (\<lambda>x. (- 1) ^ m * diff m (- x)) t :> (- 1) ^ Suc m * diff (Suc m) (- t)"
proof -
from ABL and tINTERV have "DERIV (\<lambda>x. diff m (- x)) t :> - diff (Suc m) (- t)" (is ?tABL)
proof -
from ABL and tINTERV have "DERIV (diff m) (- t) :> diff (Suc m) (- t)" by force
moreover
from DERIV_ident[of t] have "DERIV uminus t :> (- 1)" by (rule DERIV_minus)
ultimately have "DERIV (\<lambda>x. diff m (- x)) t :> diff (Suc m) (- t) * - 1" by (rule DERIV_chain2)
thus ?thesis by simp
qed
thus ?thesis
proof -
assume ?tABL hence "DERIV (\<lambda>x. -1 ^ m * diff m (- x)) t :> -1 ^ m * - diff (Suc m) (- t)" by (rule DERIV_cmult)
hence "DERIV (\<lambda>x. -1 ^ m * diff m (- x)) t :> - (-1 ^ m * diff (Suc m) (- t))" by (subst minus_mult_right)
thus ?thesis by simp
qed
qed
qed
ultimately have t_exists: "\<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" (is "\<exists> t. ?P t") by (rule Maclaurin)
from this obtain t where t_def: "?P 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 INIT : "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"
proof (cases "n = 0")
case True from INIT True show ?thesis by force
next
case False
from this have n_not_zero:"n \<noteq> 0" .
from False INIT DERIV show ?thesis
proof (cases "x = 0")
case True show ?thesis
proof -
from n_not_zero True INIT DERIV have "\<bar>0\<bar> \<le> \<bar>x\<bar> \<and>
f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n 0 / real (fact n) * x ^ n" by(force simp add: Maclaurin_bi_le_lemma)
thus ?thesis ..
qed
next
case False
note linorder_less_linear [of "x" "0"]
thus ?thesis
proof (elim disjE)
assume "x = 0" with False show ?thesis ..
next
assume x_less_zero: "x < 0" moreover
from n_not_zero have "0 < n" by simp moreover
have "diff 0 = diff 0" by simp moreover
have "\<forall>m t. m < n \<and> x \<le> t \<and> t \<le> 0 \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
proof (rule allI, rule allI, rule impI)
fix m t
assume "m < n & x \<le> t & t \<le> 0"
with DERIV show " DERIV (diff m) t :> diff (Suc m) t" by (fastsimp simp add: abs_if)
qed
ultimately have t_exists:"\<exists>t>x. t < 0 \<and>
diff 0 x =
(\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" (is "\<exists> t. ?P t") by (rule Maclaurin_minus)
from this obtain t where t_def: "?P t" ..
from t_def x_less_zero INIT have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and>
f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n"
by (simp add: abs_if order_less_le)
thus ?thesis by (rule exI)
next
assume x_greater_zero: "x > 0" moreover
from n_not_zero have "0 < n" by simp moreover
have "diff 0 = diff 0" by simp moreover
have "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> x \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
proof (rule allI, rule allI, rule impI)
fix m t
assume "m < n & 0 \<le> t & t \<le> x"
with DERIV show " DERIV (diff m) t :> diff (Suc m) t" by fastsimp
qed
ultimately have t_exists:"\<exists>t>0. t < x \<and>
diff 0 x =
(\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" (is "\<exists> t. ?P t") by (rule Maclaurin)
from this obtain t where t_def: "?P t" ..
from t_def x_greater_zero INIT have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and>
f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n"
by fastsimp
thus ?thesis ..
qed
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"
proof -
have "(x = 0) \<Longrightarrow> ?thesis"
proof -
assume "x = 0"
with INIT3 show "(x = 0) \<Longrightarrow> ?thesis"..
qed
moreover have "(x < 0) \<Longrightarrow> ?thesis"
proof -
assume x_less_zero: "x < 0"
from DERIV have "\<forall>m t. m < n \<and> x \<le> t \<and> t \<le> 0 \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t" by simp
with x_less_zero INIT2 INIT1 have "\<exists>t>x. t < 0 \<and> 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. ?P t") by (rule Maclaurin_minus)
from this obtain t where "?P t" ..
with x_less_zero have "0 < \<bar>t\<bar> \<and>
\<bar>t\<bar> < \<bar>x\<bar> \<and>
f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" by simp
thus ?thesis ..
qed
moreover have "(x > 0) \<Longrightarrow> ?thesis"
proof -
assume x_greater_zero: "x > 0"
from DERIV have "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> x \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t" by simp
with x_greater_zero INIT2 INIT1 have "\<exists>t>0. t < x \<and> 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. ?P t") by (rule Maclaurin)
from this obtain t where "?P t" ..
with x_greater_zero have "0 < \<bar>t\<bar> \<and>
\<bar>t\<bar> < \<bar>x\<bar> \<and>
f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" by fastsimp
thus ?thesis ..
qed
ultimately show ?thesis by (fastsimp)
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"
proof -
note linorder_le_less_linear [of n 0]
thus ?thesis
proof
assume "n\<le> 0" with INIT show ?thesis by force
next
assume n_greater_zero: "n > 0" show ?thesis
proof (cases "x = 0")
case True
from n_greater_zero have "n \<noteq> 0" by auto
from True this have f_0:"(\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) = diff 0 0" by (rule Maclaurin_zero)
from n_greater_zero have "n \<noteq> 0" by (rule gr_implies_not0)
hence "\<exists> m. n = Suc m" by (rule not0_implies_Suc)
with f_0 True INIT have " \<bar>0\<bar> \<le> \<bar>x\<bar> \<and>
f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n 0 / real (fact n) * x ^ n"
by force
thus ?thesis ..
next
case False
from INIT n_greater_zero this DERIV have "\<exists>t. 0 < \<bar>t\<bar> \<and>
\<bar>t\<bar> < \<bar>x\<bar> \<and> 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. ?P t") by (rule Maclaurin_all_lt)
from this obtain t where "?P t" ..
hence "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and>
f x = (\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" by (simp add: order_less_le)
thus ?thesis ..
qed
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