author | wenzelm |
Tue, 03 Jul 2007 22:27:11 +0200 | |
changeset 23555 | 16e5fd18905c |
parent 23069 | cdfff0241c12 |
child 25134 | 3d4953e88449 |
permissions | -rw-r--r-- |
17634 | 1 |
(* Title: HOL/Hyperreal/Taylor.thy |
2 |
ID: $Id$ |
|
3 |
Author: Lukas Bulwahn, Bernhard Haeupler, Technische Universitaet Muenchen |
|
4 |
*) |
|
5 |
||
6 |
header {* Taylor series *} |
|
7 |
||
8 |
theory Taylor |
|
9 |
imports MacLaurin |
|
10 |
begin |
|
11 |
||
12 |
text {* |
|
13 |
We use MacLaurin and the translation of the expansion point @{text c} to @{text 0} |
|
14 |
to prove Taylor's theorem. |
|
15 |
*} |
|
16 |
||
17 |
lemma taylor_up: |
|
18 |
assumes INIT: "0 < n" "diff 0 = f" |
|
19 |
and DERIV: "(\<forall> m t. m < n & a \<le> t & t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t))" |
|
20 |
and INTERV: "a \<le> c" "c < b" |
|
21 |
shows "\<exists> t. c < t & t < b & |
|
22 |
f b = setsum (%m. (diff m c / real (fact m)) * (b - c)^m) {0..<n} + |
|
23 |
(diff n t / real (fact n)) * (b - c)^n" |
|
24 |
proof - |
|
25 |
from INTERV have "0 < b-c" by arith |
|
26 |
moreover |
|
27 |
from INIT have "0<n" "((\<lambda>m x. diff m (x + c)) 0) = (\<lambda>x. f (x + c))" by auto |
|
28 |
moreover |
|
29 |
have "ALL m t. m < n & 0 <= t & t <= b - c --> DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)" |
|
30 |
proof (intro strip) |
|
31 |
fix m t |
|
32 |
assume "m < n & 0 <= t & t <= b - c" |
|
33 |
with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)" by auto |
|
34 |
moreover |
|
23069
cdfff0241c12
rename lemmas LIM_ident, isCont_ident, DERIV_ident
huffman
parents:
17634
diff
changeset
|
35 |
from DERIV_ident and DERIV_const have "DERIV (%x. x + c) t :> 1+0" by (rule DERIV_add) |
17634 | 36 |
ultimately have "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c) * (1+0)" |
37 |
by (rule DERIV_chain2) |
|
38 |
thus "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)" by simp |
|
39 |
qed |
|
40 |
ultimately |
|
41 |
have EX:"EX t>0. t < b - c & |
|
42 |
f (b - c + c) = (SUM m = 0..<n. diff m (0 + c) / real (fact m) * (b - c) ^ m) + |
|
43 |
diff n (t + c) / real (fact n) * (b - c) ^ n" |
|
44 |
by (rule Maclaurin) |
|
45 |
show ?thesis |
|
46 |
proof - |
|
47 |
from EX obtain x where |
|
48 |
X: "0 < x & x < b - c & |
|
49 |
f (b - c + c) = (\<Sum>m = 0..<n. diff m (0 + c) / real (fact m) * (b - c) ^ m) + |
|
50 |
diff n (x + c) / real (fact n) * (b - c) ^ n" .. |
|
51 |
let ?H = "x + c" |
|
52 |
from X have "c<?H & ?H<b \<and> f b = (\<Sum>m = 0..<n. diff m c / real (fact m) * (b - c) ^ m) + |
|
53 |
diff n ?H / real (fact n) * (b - c) ^ n" |
|
54 |
by fastsimp |
|
55 |
thus ?thesis by fastsimp |
|
56 |
qed |
|
57 |
qed |
|
58 |
||
59 |
lemma taylor_down: |
|
60 |
assumes INIT: "0 < n" "diff 0 = f" |
|
61 |
and DERIV: "(\<forall> m t. m < n & a \<le> t & t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t))" |
|
62 |
and INTERV: "a < c" "c \<le> b" |
|
63 |
shows "\<exists> t. a < t & t < c & |
|
64 |
f a = setsum (% m. (diff m c / real (fact m)) * (a - c)^m) {0..<n} + |
|
65 |
(diff n t / real (fact n)) * (a - c)^n" |
|
66 |
proof - |
|
67 |
from INTERV have "a-c < 0" by arith |
|
68 |
moreover |
|
69 |
from INIT have "0<n" "((\<lambda>m x. diff m (x + c)) 0) = (\<lambda>x. f (x + c))" by auto |
|
70 |
moreover |
|
71 |
have "ALL m t. m < n & a-c <= t & t <= 0 --> DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)" |
|
72 |
proof (rule allI impI)+ |
|
73 |
fix m t |
|
74 |
assume "m < n & a-c <= t & t <= 0" |
|
75 |
with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)" by auto |
|
76 |
moreover |
|
23069
cdfff0241c12
rename lemmas LIM_ident, isCont_ident, DERIV_ident
huffman
parents:
17634
diff
changeset
|
77 |
from DERIV_ident and DERIV_const have "DERIV (%x. x + c) t :> 1+0" by (rule DERIV_add) |
17634 | 78 |
ultimately have "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c) * (1+0)" by (rule DERIV_chain2) |
79 |
thus "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)" by simp |
|
80 |
qed |
|
81 |
ultimately |
|
82 |
have EX: "EX t>a - c. t < 0 & |
|
83 |
f (a - c + c) = (SUM m = 0..<n. diff m (0 + c) / real (fact m) * (a - c) ^ m) + |
|
84 |
diff n (t + c) / real (fact n) * (a - c) ^ n" |
|
85 |
by (rule Maclaurin_minus) |
|
86 |
show ?thesis |
|
87 |
proof - |
|
88 |
from EX obtain x where X: "a - c < x & x < 0 & |
|
89 |
f (a - c + c) = (SUM m = 0..<n. diff m (0 + c) / real (fact m) * (a - c) ^ m) + |
|
90 |
diff n (x + c) / real (fact n) * (a - c) ^ n" .. |
|
91 |
let ?H = "x + c" |
|
92 |
from X have "a<?H & ?H<c \<and> f a = (\<Sum>m = 0..<n. diff m c / real (fact m) * (a - c) ^ m) + |
|
93 |
diff n ?H / real (fact n) * (a - c) ^ n" |
|
94 |
by fastsimp |
|
95 |
thus ?thesis by fastsimp |
|
96 |
qed |
|
97 |
qed |
|
98 |
||
99 |
lemma taylor: |
|
100 |
assumes INIT: "0 < n" "diff 0 = f" |
|
101 |
and DERIV: "(\<forall> m t. m < n & a \<le> t & t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t))" |
|
102 |
and INTERV: "a \<le> c " "c \<le> b" "a \<le> x" "x \<le> b" "x \<noteq> c" |
|
103 |
shows "\<exists> t. (if x<c then (x < t & t < c) else (c < t & t < x)) & |
|
104 |
f x = setsum (% m. (diff m c / real (fact m)) * (x - c)^m) {0..<n} + |
|
105 |
(diff n t / real (fact n)) * (x - c)^n" |
|
106 |
proof (cases "x<c") |
|
107 |
case True |
|
108 |
note INIT |
|
109 |
moreover from DERIV and INTERV |
|
110 |
have "\<forall>m t. m < n \<and> x \<le> t \<and> t \<le> b \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t" |
|
111 |
by fastsimp |
|
112 |
moreover note True |
|
113 |
moreover from INTERV have "c \<le> b" by simp |
|
114 |
ultimately have EX: "\<exists>t>x. t < c \<and> f x = |
|
115 |
(\<Sum>m = 0..<n. diff m c / real (fact m) * (x - c) ^ m) + |
|
116 |
diff n t / real (fact n) * (x - c) ^ n" |
|
117 |
by (rule taylor_down) |
|
118 |
with True show ?thesis by simp |
|
119 |
next |
|
120 |
case False |
|
121 |
note INIT |
|
122 |
moreover from DERIV and INTERV |
|
123 |
have "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> x \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t" |
|
124 |
by fastsimp |
|
125 |
moreover from INTERV have "a \<le> c" by arith |
|
126 |
moreover from False and INTERV have "c < x" by arith |
|
127 |
ultimately have EX: "\<exists>t>c. t < x \<and> f x = |
|
128 |
(\<Sum>m = 0..<n. diff m c / real (fact m) * (x - c) ^ m) + |
|
129 |
diff n t / real (fact n) * (x - c) ^ n" |
|
130 |
by (rule taylor_up) |
|
131 |
with False show ?thesis by simp |
|
132 |
qed |
|
133 |
||
134 |
end |