--- a/src/HOL/MacLaurin.thy Tue Mar 18 14:32:23 2014 +0100
+++ b/src/HOL/MacLaurin.thy Tue Mar 18 15:53:48 2014 +0100
@@ -17,10 +17,9 @@
lemma Maclaurin_lemma:
"0 < h ==>
- \<exists>B. f h = (\<Sum>m=0..<n. (j m / real (fact m)) * (h^m)) +
+ \<exists>B. f h = (\<Sum>m<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
+by (rule exI[where x = "(f h - (\<Sum>m<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
@@ -33,20 +32,20 @@
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) +
+ defines "difg \<equiv> (\<lambda>m t. diff m t - ((\<Sum>p<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)) +
+ ((\<Sum>x<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
+ 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)))"
+ have "(\<Sum>x<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / real (fact x)) =
+ (\<Sum>x<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"
@@ -71,29 +70,26 @@
shows
"\<exists>t. 0 < t & t < h &
f h =
- setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
+ setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {..<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) +
+ (\<Sum>m<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}
+ (setsum (\<lambda>m. (diff m 0 / real(fact m)) * t^m) {..<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
+ 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 / real (fact p)) * (t ^ p)) {0..<n-m}
+ (setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {..<n-m}
+ (B * ((t ^ (n - m)) / real (fact (n - m))))))"
have difg_0: "difg 0 = g"
@@ -103,14 +99,8 @@
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 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
@@ -166,7 +156,7 @@
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) +
+ (\<Sum>m<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)
@@ -177,7 +167,7 @@
"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) +
+ f h = (\<Sum>m<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n)"
by (blast intro: Maclaurin)
@@ -187,7 +177,7 @@
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) +
+ (\<Sum>m<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 fastforce
@@ -196,7 +186,7 @@
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"
+ (\<Sum>m<n. diff m 0 / real (fact m) * h ^ m) + diff n t / real (fact n) * h ^ n"
by (rule Maclaurin)
thus ?thesis by fastforce
qed
@@ -208,7 +198,7 @@
--> (\<exists>t. 0 < t &
t \<le> h &
f h =
- (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
+ (\<Sum>m<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n)"
by (blast intro: Maclaurin2)
@@ -216,7 +206,7 @@
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) +
+ f h = (\<Sum>m<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."
@@ -224,7 +214,7 @@
from assms
have "\<exists>t>0. t < - h \<and>
f (- (- h)) =
- (\<Sum>m = 0..<n.
+ (\<Sum>m<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')
@@ -233,12 +223,12 @@
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))"
+ have "(SUM m<n. -1 ^ m * diff m 0 * (- h) ^ m / real (fact m)) = (SUM m<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"
+ (\<Sum>m<n. diff m 0 / real (fact m) * h ^ m) + diff n (- t) / real (fact n) * h ^ n"
by auto
thus ?thesis ..
qed
@@ -250,7 +240,7 @@
--> (\<exists>t. h < t &
t < 0 &
f h =
- (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
+ (\<Sum>m<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n)"
by (blast intro: Maclaurin_minus)
@@ -262,7 +252,7 @@
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)) +
+ (\<Sum>m<n. diff m 0 * 0 ^ m / real (fact m)) +
diff n 0 * 0 ^ n / real (fact n)"
by (induct "n") auto
@@ -271,7 +261,7 @@
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) +
+ (\<Sum>m<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
@@ -303,7 +293,7 @@
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) +
+ (\<Sum>m<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"..
@@ -327,14 +317,14 @@
(\<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) +
+ f x = (\<Sum>m<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) =
+ (\<Sum>m<n. (diff m (0::real) / real (fact m)) * x ^ m) =
diff 0 0"
by (induct n, auto)
@@ -343,7 +333,7 @@
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) +
+ (\<Sum>m<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
@@ -352,7 +342,7 @@
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"
+ with `n \<noteq> 0` have "(\<Sum>m<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 ..
@@ -369,7 +359,7 @@
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) +
+ f x = (\<Sum>m<n. (diff m 0 / real (fact m)) * x ^ m) +
(diff n t / real (fact n)) * x ^ n)"
by (blast intro: Maclaurin_all_le)
@@ -379,14 +369,14 @@
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 x = (\<Sum>m<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 x = (\<Sum>m<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)
@@ -420,7 +410,7 @@
lemma Maclaurin_sin_expansion2:
"\<exists>t. abs t \<le> abs x &
sin x =
- (\<Sum>m=0..<n. sin_coeff m * x ^ m)
+ (\<Sum>m<n. sin_coeff 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)
@@ -440,7 +430,7 @@
lemma Maclaurin_sin_expansion:
"\<exists>t. sin x =
- (\<Sum>m=0..<n. sin_coeff m * x ^ m)
+ (\<Sum>m<n. sin_coeff 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:)
@@ -450,7 +440,7 @@
"[| n > 0; 0 < x |] ==>
\<exists>t. 0 < t & t < x &
sin x =
- (\<Sum>m=0..<n. sin_coeff m * x ^ m)
+ (\<Sum>m<n. sin_coeff 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
@@ -467,7 +457,7 @@
"0 < x ==>
\<exists>t. 0 < t & t \<le> x &
sin x =
- (\<Sum>m=0..<n. sin_coeff m * x ^ m)
+ (\<Sum>m<n. sin_coeff 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
@@ -484,7 +474,7 @@
subsection{*Maclaurin Expansion for Cosine Function*}
lemma sumr_cos_zero_one [simp]:
- "(\<Sum>m=0..<(Suc n). cos_coeff m * 0 ^ m) = 1"
+ "(\<Sum>m<(Suc n). cos_coeff m * 0 ^ m) = 1"
by (induct "n", auto)
lemma cos_expansion_lemma:
@@ -494,14 +484,14 @@
lemma Maclaurin_cos_expansion:
"\<exists>t. abs t \<le> abs x &
cos x =
- (\<Sum>m=0..<n. cos_coeff m * x ^ m)
+ (\<Sum>m<n. cos_coeff m * 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 (simp del: setsum_lessThan_Suc)
apply (rule ccontr, simp)
apply (drule_tac x = x in spec, simp)
apply (erule ssubst)
@@ -514,7 +504,7 @@
"[| 0 < x; n > 0 |] ==>
\<exists>t. 0 < t & t < x &
cos x =
- (\<Sum>m=0..<n. cos_coeff m * x ^ m)
+ (\<Sum>m<n. cos_coeff m * 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
@@ -530,7 +520,7 @@
"[| x < 0; n > 0 |] ==>
\<exists>t. x < t & t < 0 &
cos x =
- (\<Sum>m=0..<n. cos_coeff m * x ^ m)
+ (\<Sum>m<n. cos_coeff m * 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
@@ -551,7 +541,7 @@
by auto
lemma Maclaurin_sin_bound:
- "abs(sin x - (\<Sum>m=0..<n. sin_coeff m * x ^ m))
+ "abs(sin x - (\<Sum>m<n. sin_coeff 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"
@@ -567,7 +557,7 @@
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) +
+ t2: "sin x = (\<Sum>m<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