author | wenzelm |
Tue, 31 Mar 2015 22:31:05 +0200 | |
changeset 59886 | e0dc738eb08c |
parent 59730 | b7c394c7a619 |
child 60017 | b785d6d06430 |
permissions | -rw-r--r-- |
28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
27239
diff
changeset
|
1 |
(* Author : Jacques D. Fleuriot |
12224 | 2 |
Copyright : 2001 University of Edinburgh |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
3 |
Conversion to Isar and new proofs by Lawrence C Paulson, 2004 |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
4 |
Conversion of Mac Laurin to Isar by Lukas Bulwahn and Bernhard Häupler, 2005 |
12224 | 5 |
*) |
6 |
||
58889 | 7 |
section{*MacLaurin Series*} |
15944 | 8 |
|
15131 | 9 |
theory MacLaurin |
29811
026b0f9f579f
fixed Proofs and dependencies ; Theory Dense_Linear_Order moved to Library
chaieb@chaieb-laptop
parents:
29803
diff
changeset
|
10 |
imports Transcendental |
15131 | 11 |
begin |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
12 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
13 |
subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*} |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
14 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
15 |
text{*This is a very long, messy proof even now that it's been broken down |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
16 |
into lemmas.*} |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
17 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
18 |
lemma Maclaurin_lemma: |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
19 |
"0 < h ==> |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
20 |
\<exists>B::real. f h = (\<Sum>m<n. (j m / (fact m)) * (h^m)) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
21 |
(B * ((h^n) /(fact n)))" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
22 |
by (rule exI[where x = "(f h - (\<Sum>m<n. (j m / (fact m)) * h^m)) * (fact n) / (h^n)"]) simp |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
23 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
24 |
lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
25 |
by arith |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
26 |
|
32038 | 27 |
lemma fact_diff_Suc [rule_format]: |
28 |
"n < Suc m ==> fact (Suc m - n) = (Suc m - n) * fact (m - n)" |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
29 |
by (subst fact_reduce, auto) |
32038 | 30 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
31 |
lemma Maclaurin_lemma2: |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
32 |
fixes B |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
33 |
assumes DERIV : "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
34 |
and INIT : "n = Suc k" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
35 |
defines "difg \<equiv> |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
36 |
(\<lambda>m t::real. diff m t - |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
37 |
((\<Sum>p<n - m. diff (m + p) 0 / (fact p) * t ^ p) + B * (t ^ (n - m) / (fact (n - m)))))" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
38 |
(is "difg \<equiv> (\<lambda>m t. diff m t - ?difg m t)") |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
39 |
shows "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t" |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
40 |
proof (rule allI impI)+ |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
41 |
fix m and t::real |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
42 |
assume INIT2: "m < n & 0 \<le> t & t \<le> h" |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
43 |
have "DERIV (difg m) t :> diff (Suc m) t - |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
44 |
((\<Sum>x<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / (fact x)) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
45 |
real (n - m) * t ^ (n - Suc m) * B / (fact (n - m)))" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
46 |
unfolding difg_def |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56238
diff
changeset
|
47 |
by (auto intro!: derivative_eq_intros DERIV[rule_format, OF INIT2] |
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56238
diff
changeset
|
48 |
simp: real_of_nat_def[symmetric]) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
49 |
moreover |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
50 |
from INIT2 have intvl: "{..<n - m} = insert 0 (Suc ` {..<n - Suc m})" and "0 < n - m" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
51 |
unfolding atLeast0LessThan[symmetric] by auto |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
52 |
have "(\<Sum>x<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / (fact x)) = |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
53 |
(\<Sum>x<n - Suc m. real (Suc x) * t ^ x * diff (Suc m + x) 0 / (fact (Suc x)))" |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
54 |
unfolding intvl atLeast0LessThan by (subst setsum.insert) (auto simp: setsum.reindex) |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
55 |
moreover |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
56 |
have fact_neq_0: "\<And>x. (fact x) + real x * (fact x) \<noteq> 0" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
57 |
by (metis add_pos_pos fact_gt_zero less_add_same_cancel1 less_add_same_cancel2 less_numeral_extra(3) mult_less_0_iff not_real_of_nat_less_zero) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
58 |
have "\<And>x. (Suc x) * t ^ x * diff (Suc m + x) 0 / (fact (Suc x)) = |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
59 |
diff (Suc m + x) 0 * t^x / (fact x)" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
60 |
by (rule nonzero_divide_eq_eq[THEN iffD2]) auto |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
61 |
moreover |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
62 |
have "(n - m) * t ^ (n - Suc m) * B / (fact (n - m)) = |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
63 |
B * (t ^ (n - Suc m) / (fact (n - Suc m)))" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
64 |
using `0 < n - m` |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
65 |
by (simp add: divide_simps fact_reduce) |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
66 |
ultimately show "DERIV (difg m) t :> difg (Suc m) t" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
67 |
unfolding difg_def by simp |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
68 |
qed |
32038 | 69 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
70 |
lemma Maclaurin: |
29187 | 71 |
assumes h: "0 < h" |
72 |
assumes n: "0 < n" |
|
73 |
assumes diff_0: "diff 0 = f" |
|
74 |
assumes diff_Suc: |
|
75 |
"\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t" |
|
76 |
shows |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
77 |
"\<exists>t::real. 0 < t & t < h & |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
78 |
f h = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
79 |
setsum (%m. (diff m 0 / (fact m)) * h ^ m) {..<n} + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
80 |
(diff n t / (fact n)) * h ^ n" |
29187 | 81 |
proof - |
82 |
from n obtain m where m: "n = Suc m" |
|
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
83 |
by (cases n) (simp add: n) |
29187 | 84 |
|
85 |
obtain B where f_h: "f h = |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
86 |
(\<Sum>m<n. diff m (0\<Colon>real) / (fact m) * h ^ m) + B * (h ^ n / (fact n))" |
29187 | 87 |
using Maclaurin_lemma [OF h] .. |
88 |
||
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
89 |
def g \<equiv> "(\<lambda>t. f t - |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
90 |
(setsum (\<lambda>m. (diff m 0 / (fact m)) * t^m) {..<n} + (B * (t^n / (fact n)))))" |
29187 | 91 |
|
92 |
have g2: "g 0 = 0 & g h = 0" |
|
57418 | 93 |
by (simp add: m f_h g_def lessThan_Suc_eq_insert_0 image_iff diff_0 setsum.reindex) |
29187 | 94 |
|
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
95 |
def difg \<equiv> "(%m t. diff m t - |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
96 |
(setsum (%p. (diff (m + p) 0 / (fact p)) * (t ^ p)) {..<n-m} |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
97 |
+ (B * ((t ^ (n - m)) / (fact (n - m))))))" |
29187 | 98 |
|
99 |
have difg_0: "difg 0 = g" |
|
100 |
unfolding difg_def g_def by (simp add: diff_0) |
|
101 |
||
102 |
have difg_Suc: "\<forall>(m\<Colon>nat) t\<Colon>real. |
|
103 |
m < n \<and> (0\<Colon>real) \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t" |
|
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
104 |
using diff_Suc m unfolding difg_def by (rule Maclaurin_lemma2) |
29187 | 105 |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
106 |
have difg_eq_0: "\<forall>m<n. difg m 0 = 0" |
57418 | 107 |
by (auto simp: difg_def m Suc_diff_le lessThan_Suc_eq_insert_0 image_iff setsum.reindex) |
29187 | 108 |
|
109 |
have isCont_difg: "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> isCont (difg m) x" |
|
110 |
by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp |
|
111 |
||
112 |
have differentiable_difg: |
|
56181
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
51489
diff
changeset
|
113 |
"\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> difg m differentiable (at x)" |
29187 | 114 |
by (rule differentiableI [OF difg_Suc [rule_format]]) simp |
115 |
||
116 |
have difg_Suc_eq_0: "\<And>m t. \<lbrakk>m < n; 0 \<le> t; t \<le> h; DERIV (difg m) t :> 0\<rbrakk> |
|
117 |
\<Longrightarrow> difg (Suc m) t = 0" |
|
118 |
by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp |
|
119 |
||
120 |
have "m < n" using m by simp |
|
121 |
||
122 |
have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m) t :> 0" |
|
123 |
using `m < n` |
|
124 |
proof (induct m) |
|
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
125 |
case 0 |
29187 | 126 |
show ?case |
127 |
proof (rule Rolle) |
|
128 |
show "0 < h" by fact |
|
129 |
show "difg 0 0 = difg 0 h" by (simp add: difg_0 g2) |
|
130 |
show "\<forall>x. 0 \<le> x \<and> x \<le> h \<longrightarrow> isCont (difg (0\<Colon>nat)) x" |
|
131 |
by (simp add: isCont_difg n) |
|
56181
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
51489
diff
changeset
|
132 |
show "\<forall>x. 0 < x \<and> x < h \<longrightarrow> difg (0\<Colon>nat) differentiable (at x)" |
29187 | 133 |
by (simp add: differentiable_difg n) |
134 |
qed |
|
135 |
next |
|
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
136 |
case (Suc m') |
29187 | 137 |
hence "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m') t :> 0" by simp |
138 |
then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0" by fast |
|
139 |
have "\<exists>t'. 0 < t' \<and> t' < t \<and> DERIV (difg (Suc m')) t' :> 0" |
|
140 |
proof (rule Rolle) |
|
141 |
show "0 < t" by fact |
|
142 |
show "difg (Suc m') 0 = difg (Suc m') t" |
|
143 |
using t `Suc m' < n` by (simp add: difg_Suc_eq_0 difg_eq_0) |
|
144 |
show "\<forall>x. 0 \<le> x \<and> x \<le> t \<longrightarrow> isCont (difg (Suc m')) x" |
|
145 |
using `t < h` `Suc m' < n` by (simp add: isCont_difg) |
|
56181
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
51489
diff
changeset
|
146 |
show "\<forall>x. 0 < x \<and> x < t \<longrightarrow> difg (Suc m') differentiable (at x)" |
29187 | 147 |
using `t < h` `Suc m' < n` by (simp add: differentiable_difg) |
148 |
qed |
|
149 |
thus ?case |
|
150 |
using `t < h` by auto |
|
151 |
qed |
|
152 |
then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0" by fast |
|
153 |
||
154 |
hence "difg (Suc m) t = 0" |
|
155 |
using `m < n` by (simp add: difg_Suc_eq_0) |
|
156 |
||
157 |
show ?thesis |
|
158 |
proof (intro exI conjI) |
|
159 |
show "0 < t" by fact |
|
160 |
show "t < h" by fact |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
161 |
show "f h = (\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + diff n t / (fact n) * h ^ n" |
29187 | 162 |
using `difg (Suc m) t = 0` |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
163 |
by (simp add: m f_h difg_def) |
29187 | 164 |
qed |
165 |
qed |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
166 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
167 |
lemma Maclaurin_objl: |
25162 | 168 |
"0 < h & n>0 & diff 0 = f & |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
169 |
(\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
170 |
--> (\<exists>t::real. 0 < t & t < h & |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
171 |
f h = (\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
172 |
diff n t / (fact n) * h ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
173 |
by (blast intro: Maclaurin) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
174 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
175 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
176 |
lemma Maclaurin2: |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
177 |
assumes INIT1: "0 < h " and INIT2: "diff 0 = f" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
178 |
and DERIV: "\<forall>m t::real. |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
179 |
m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t" |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
180 |
shows "\<exists>t. 0 < t \<and> t \<le> h \<and> f h = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
181 |
(\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
182 |
diff n t / (fact n) * h ^ n" |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
183 |
proof (cases "n") |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44319
diff
changeset
|
184 |
case 0 with INIT1 INIT2 show ?thesis by fastforce |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
185 |
next |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
186 |
case Suc |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
187 |
hence "n > 0" by simp |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
188 |
from INIT1 this INIT2 DERIV have "\<exists>t>0. t < h \<and> |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
189 |
f h = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
190 |
(\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + diff n t / (fact n) * h ^ n" |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
191 |
by (rule Maclaurin) |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44319
diff
changeset
|
192 |
thus ?thesis by fastforce |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
193 |
qed |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
194 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
195 |
lemma Maclaurin2_objl: |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
196 |
"0 < h & diff 0 = f & |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
197 |
(\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
198 |
--> (\<exists>t::real. 0 < t & |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
199 |
t \<le> h & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
200 |
f h = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
201 |
(\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
202 |
diff n t / (fact n) * h ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
203 |
by (blast intro: Maclaurin2) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
204 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
205 |
lemma Maclaurin_minus: |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
206 |
fixes h::real |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
207 |
assumes "h < 0" "0 < n" "diff 0 = f" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
208 |
and DERIV: "\<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t" |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
209 |
shows "\<exists>t. h < t & t < 0 & |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
210 |
f h = (\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
211 |
diff n t / (fact n) * h ^ n" |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
212 |
proof - |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56238
diff
changeset
|
213 |
txt "Transform @{text ABL'} into @{text derivative_intros} format." |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
214 |
note DERIV' = DERIV_chain'[OF _ DERIV[rule_format], THEN DERIV_cong] |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
215 |
from assms |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
216 |
have "\<exists>t>0. t < - h \<and> |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
217 |
f (- (- h)) = |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
218 |
(\<Sum>m<n. |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
219 |
(- 1) ^ m * diff m (- 0) / (fact m) * (- h) ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
220 |
(- 1) ^ n * diff n (- t) / (fact n) * (- h) ^ n" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56238
diff
changeset
|
221 |
by (intro Maclaurin) (auto intro!: derivative_eq_intros DERIV') |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
222 |
then guess t .. |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
223 |
moreover |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
224 |
have "(- 1) ^ n * diff n (- t) * (- h) ^ n / (fact n) = diff n (- t) * h ^ n / (fact n)" |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
225 |
by (auto simp add: power_mult_distrib[symmetric]) |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
226 |
moreover |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
227 |
have "(SUM m<n. (- 1) ^ m * diff m 0 * (- h) ^ m / (fact m)) = (SUM m<n. diff m 0 * h ^ m / (fact m))" |
57418 | 228 |
by (auto intro: setsum.cong simp add: power_mult_distrib[symmetric]) |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
229 |
ultimately have " h < - t \<and> |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
230 |
- t < 0 \<and> |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
231 |
f h = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
232 |
(\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + diff n (- t) / (fact n) * h ^ n" |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
233 |
by auto |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
234 |
thus ?thesis .. |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
235 |
qed |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
236 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
237 |
lemma Maclaurin_minus_objl: |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
238 |
fixes h::real |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
239 |
shows |
25162 | 240 |
"(h < 0 & n > 0 & diff 0 = f & |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
241 |
(\<forall>m t. |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
242 |
m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t)) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
243 |
--> (\<exists>t. h < t & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
244 |
t < 0 & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
245 |
f h = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
246 |
(\<Sum>m<n. diff m 0 / (fact m) * h ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
247 |
diff n t / (fact n) * h ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
248 |
by (blast intro: Maclaurin_minus) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
249 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
250 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
251 |
subsection{*More Convenient "Bidirectional" Version.*} |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
252 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
253 |
(* not good for PVS sin_approx, cos_approx *) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
254 |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
255 |
lemma Maclaurin_bi_le_lemma: |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
256 |
"n>0 \<Longrightarrow> |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
257 |
diff 0 0 = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
258 |
(\<Sum>m<n. diff m 0 * 0 ^ m / (fact m)) + diff n 0 * 0 ^ n / (fact n :: real)" |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
259 |
by (induct "n") auto |
14738 | 260 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
261 |
lemma Maclaurin_bi_le: |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
262 |
assumes "diff 0 = f" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
263 |
and DERIV : "\<forall>m t::real. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t" |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
264 |
shows "\<exists>t. abs t \<le> abs x & |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
265 |
f x = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
266 |
(\<Sum>m<n. diff m 0 / (fact m) * x ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
267 |
diff n t / (fact n) * x ^ n" (is "\<exists>t. _ \<and> f x = ?f x t") |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
268 |
proof cases |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
269 |
assume "n = 0" with `diff 0 = f` show ?thesis by force |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
270 |
next |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
271 |
assume "n \<noteq> 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
272 |
show ?thesis |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
273 |
proof (cases rule: linorder_cases) |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
274 |
assume "x = 0" with `n \<noteq> 0` `diff 0 = f` DERIV |
56238
5d147e1e18d1
a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents:
56193
diff
changeset
|
275 |
have "\<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0" by (auto simp add: Maclaurin_bi_le_lemma) |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
276 |
thus ?thesis .. |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
277 |
next |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
278 |
assume "x < 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
279 |
with `n \<noteq> 0` DERIV |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
280 |
have "\<exists>t>x. t < 0 \<and> diff 0 x = ?f x t" by (intro Maclaurin_minus) auto |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
281 |
then guess t .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
282 |
with `x < 0` `diff 0 = f` have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
283 |
thus ?thesis .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
284 |
next |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
285 |
assume "x > 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
286 |
with `n \<noteq> 0` `diff 0 = f` DERIV |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
287 |
have "\<exists>t>0. t < x \<and> diff 0 x = ?f x t" by (intro Maclaurin) auto |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
288 |
then guess t .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
289 |
with `x > 0` `diff 0 = f` have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
290 |
thus ?thesis .. |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
291 |
qed |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
292 |
qed |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
293 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
294 |
lemma Maclaurin_all_lt: |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
295 |
fixes x::real |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
296 |
assumes INIT1: "diff 0 = f" and INIT2: "0 < n" and INIT3: "x \<noteq> 0" |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
297 |
and DERIV: "\<forall>m x. DERIV (diff m) x :> diff(Suc m) x" |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
298 |
shows "\<exists>t. 0 < abs t & abs t < abs x & f x = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
299 |
(\<Sum>m<n. (diff m 0 / (fact m)) * x ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
300 |
(diff n t / (fact n)) * x ^ n" (is "\<exists>t. _ \<and> _ \<and> f x = ?f x t") |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
301 |
proof (cases rule: linorder_cases) |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
302 |
assume "x = 0" with INIT3 show "?thesis".. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
303 |
next |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
304 |
assume "x < 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
305 |
with assms have "\<exists>t>x. t < 0 \<and> f x = ?f x t" by (intro Maclaurin_minus) auto |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
306 |
then guess t .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
307 |
with `x < 0` have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t" by simp |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
308 |
thus ?thesis .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
309 |
next |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
310 |
assume "x > 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
311 |
with assms have "\<exists>t>0. t < x \<and> f x = ?f x t " by (intro Maclaurin) auto |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
312 |
then guess t .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
313 |
with `x > 0` have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t" by simp |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
314 |
thus ?thesis .. |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
315 |
qed |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
316 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
317 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
318 |
lemma Maclaurin_all_lt_objl: |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
319 |
fixes x::real |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
320 |
shows |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
321 |
"diff 0 = f & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
322 |
(\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) & |
25162 | 323 |
x ~= 0 & n > 0 |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
324 |
--> (\<exists>t. 0 < abs t & abs t < abs x & |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
325 |
f x = (\<Sum>m<n. (diff m 0 / (fact m)) * x ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
326 |
(diff n t / (fact n)) * x ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
327 |
by (blast intro: Maclaurin_all_lt) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
328 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
329 |
lemma Maclaurin_zero [rule_format]: |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
330 |
"x = (0::real) |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
331 |
==> n \<noteq> 0 --> |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
332 |
(\<Sum>m<n. (diff m (0::real) / (fact m)) * x ^ m) = |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
333 |
diff 0 0" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
334 |
by (induct n, auto) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
335 |
|
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
336 |
|
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
337 |
lemma Maclaurin_all_le: |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
338 |
assumes INIT: "diff 0 = f" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
339 |
and DERIV: "\<forall>m x::real. DERIV (diff m) x :> diff (Suc m) x" |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
340 |
shows "\<exists>t. abs t \<le> abs x & f x = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
341 |
(\<Sum>m<n. (diff m 0 / (fact m)) * x ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
342 |
(diff n t / (fact n)) * x ^ n" (is "\<exists>t. _ \<and> f x = ?f x t") |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
343 |
proof cases |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
344 |
assume "n = 0" with INIT show ?thesis by force |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
345 |
next |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
346 |
assume "n \<noteq> 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
347 |
show ?thesis |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
348 |
proof cases |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
349 |
assume "x = 0" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
350 |
with `n \<noteq> 0` have "(\<Sum>m<n. diff m 0 / (fact m) * x ^ m) = diff 0 0" |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
351 |
by (intro Maclaurin_zero) auto |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
352 |
with INIT `x = 0` `n \<noteq> 0` have " \<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0" by force |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
353 |
thus ?thesis .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
354 |
next |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
355 |
assume "x \<noteq> 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
356 |
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" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
357 |
by (intro Maclaurin_all_lt) auto |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
358 |
then guess t .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
359 |
hence "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
360 |
thus ?thesis .. |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
361 |
qed |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
362 |
qed |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
363 |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
364 |
lemma Maclaurin_all_le_objl: |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
365 |
"diff 0 = f & |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
366 |
(\<forall>m x. DERIV (diff m) x :> diff (Suc m) x) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
367 |
--> (\<exists>t::real. abs t \<le> abs x & |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
368 |
f x = (\<Sum>m<n. (diff m 0 / (fact m)) * x ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
369 |
(diff n t / (fact n)) * x ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
370 |
by (blast intro: Maclaurin_all_le) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
371 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
372 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
373 |
subsection{*Version for Exponential Function*} |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
374 |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
375 |
lemma Maclaurin_exp_lt: |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
376 |
fixes x::real |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
377 |
shows |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
378 |
"[| x ~= 0; n > 0 |] |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
379 |
==> (\<exists>t. 0 < abs t & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
380 |
abs t < abs x & |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
381 |
exp x = (\<Sum>m<n. (x ^ m) / (fact m)) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
382 |
(exp t / (fact n)) * x ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
383 |
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
384 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
385 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
386 |
lemma Maclaurin_exp_le: |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
387 |
"\<exists>t::real. abs t \<le> abs x & |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
388 |
exp x = (\<Sum>m<n. (x ^ m) / (fact m)) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
389 |
(exp t / (fact n)) * x ^ n" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
390 |
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
391 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
392 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
393 |
subsection{*Version for Sine Function*} |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
394 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
395 |
lemma mod_exhaust_less_4: |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
396 |
"m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)" |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19765
diff
changeset
|
397 |
by auto |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
398 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
399 |
lemma Suc_Suc_mult_two_diff_two [rule_format, simp]: |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
400 |
"n\<noteq>0 --> Suc (Suc (2 * n - 2)) = 2*n" |
15251 | 401 |
by (induct "n", auto) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
402 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
403 |
lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]: |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
404 |
"n\<noteq>0 --> Suc (Suc (4*n - 2)) = 4*n" |
15251 | 405 |
by (induct "n", auto) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
406 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
407 |
lemma Suc_mult_two_diff_one [rule_format, simp]: |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
408 |
"n\<noteq>0 --> Suc (2 * n - 1) = 2*n" |
15251 | 409 |
by (induct "n", auto) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
410 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
411 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
412 |
text{*It is unclear why so many variant results are needed.*} |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
413 |
|
36974
b877866b5b00
remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents:
32047
diff
changeset
|
414 |
lemma sin_expansion_lemma: |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
415 |
"sin (x + real (Suc m) * pi / 2) = |
36974
b877866b5b00
remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents:
32047
diff
changeset
|
416 |
cos (x + real (m) * pi / 2)" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44890
diff
changeset
|
417 |
by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib distrib_right, auto) |
36974
b877866b5b00
remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents:
32047
diff
changeset
|
418 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
419 |
lemma Maclaurin_sin_expansion2: |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
420 |
"\<exists>t. abs t \<le> abs x & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
421 |
sin x = |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
422 |
(\<Sum>m<n. sin_coeff m * x ^ m) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
423 |
+ ((sin(t + 1/2 * real (n) *pi) / (fact n)) * x ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
424 |
apply (cut_tac f = sin and n = n and x = x |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
425 |
and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
426 |
apply safe |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
427 |
apply (simp (no_asm)) |
36974
b877866b5b00
remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents:
32047
diff
changeset
|
428 |
apply (simp (no_asm) add: sin_expansion_lemma) |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56238
diff
changeset
|
429 |
apply (force intro!: derivative_eq_intros) |
57492
74bf65a1910a
Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
57418
diff
changeset
|
430 |
apply (subst (asm) setsum.neutral, auto)[1] |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
431 |
apply (rule ccontr, simp) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
432 |
apply (drule_tac x = x in spec, simp) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
433 |
apply (erule ssubst) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
434 |
apply (rule_tac x = t in exI, simp) |
57418 | 435 |
apply (rule setsum.cong[OF refl]) |
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58410
diff
changeset
|
436 |
apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
437 |
done |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
438 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
439 |
lemma Maclaurin_sin_expansion: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
440 |
"\<exists>t. sin x = |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
441 |
(\<Sum>m<n. sin_coeff m * x ^ m) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
442 |
+ ((sin(t + 1/2 * real (n) *pi) / (fact n)) * x ^ n)" |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
443 |
apply (insert Maclaurin_sin_expansion2 [of x n]) |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
444 |
apply (blast intro: elim:) |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
445 |
done |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
446 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
447 |
lemma Maclaurin_sin_expansion3: |
25162 | 448 |
"[| n > 0; 0 < x |] ==> |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
449 |
\<exists>t. 0 < t & t < x & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
450 |
sin x = |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
451 |
(\<Sum>m<n. sin_coeff m * x ^ m) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
452 |
+ ((sin(t + 1/2 * real(n) *pi) / (fact n)) * x ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
453 |
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) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
454 |
apply safe |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
455 |
apply simp |
36974
b877866b5b00
remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents:
32047
diff
changeset
|
456 |
apply (simp (no_asm) add: sin_expansion_lemma) |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56238
diff
changeset
|
457 |
apply (force intro!: derivative_eq_intros) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
458 |
apply (erule ssubst) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
459 |
apply (rule_tac x = t in exI, simp) |
57418 | 460 |
apply (rule setsum.cong[OF refl]) |
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58410
diff
changeset
|
461 |
apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
462 |
done |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
463 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
464 |
lemma Maclaurin_sin_expansion4: |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
465 |
"0 < x ==> |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
466 |
\<exists>t. 0 < t & t \<le> x & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
467 |
sin x = |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
468 |
(\<Sum>m<n. sin_coeff m * x ^ m) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
469 |
+ ((sin(t + 1/2 * real (n) *pi) / (fact n)) * x ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
470 |
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) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
471 |
apply safe |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
472 |
apply simp |
36974
b877866b5b00
remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents:
32047
diff
changeset
|
473 |
apply (simp (no_asm) add: sin_expansion_lemma) |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56238
diff
changeset
|
474 |
apply (force intro!: derivative_eq_intros) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
475 |
apply (erule ssubst) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
476 |
apply (rule_tac x = t in exI, simp) |
57418 | 477 |
apply (rule setsum.cong[OF refl]) |
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58410
diff
changeset
|
478 |
apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
479 |
done |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
480 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
481 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
482 |
subsection{*Maclaurin Expansion for Cosine Function*} |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
483 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
484 |
lemma sumr_cos_zero_one [simp]: |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
485 |
"(\<Sum>m<(Suc n). cos_coeff m * 0 ^ m) = 1" |
15251 | 486 |
by (induct "n", auto) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
487 |
|
36974
b877866b5b00
remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents:
32047
diff
changeset
|
488 |
lemma cos_expansion_lemma: |
b877866b5b00
remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents:
32047
diff
changeset
|
489 |
"cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
44890
diff
changeset
|
490 |
by (simp only: cos_add sin_add real_of_nat_Suc distrib_right add_divide_distrib, auto) |
36974
b877866b5b00
remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents:
32047
diff
changeset
|
491 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
492 |
lemma Maclaurin_cos_expansion: |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
493 |
"\<exists>t::real. abs t \<le> abs x & |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
494 |
cos x = |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
495 |
(\<Sum>m<n. cos_coeff m * x ^ m) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
496 |
+ ((cos(t + 1/2 * real (n) *pi) / (fact n)) * x ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
497 |
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) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
498 |
apply safe |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
499 |
apply (simp (no_asm)) |
36974
b877866b5b00
remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents:
32047
diff
changeset
|
500 |
apply (simp (no_asm) add: cos_expansion_lemma) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
501 |
apply (case_tac "n", simp) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
502 |
apply (simp del: setsum_lessThan_Suc) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
503 |
apply (rule ccontr, simp) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
504 |
apply (drule_tac x = x in spec, simp) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
505 |
apply (erule ssubst) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
506 |
apply (rule_tac x = t in exI, simp) |
57418 | 507 |
apply (rule setsum.cong[OF refl]) |
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58410
diff
changeset
|
508 |
apply (auto simp add: cos_coeff_def cos_zero_iff elim: evenE) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
509 |
done |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
510 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
511 |
lemma Maclaurin_cos_expansion2: |
25162 | 512 |
"[| 0 < x; n > 0 |] ==> |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
513 |
\<exists>t. 0 < t & t < x & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
514 |
cos x = |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
515 |
(\<Sum>m<n. cos_coeff m * x ^ m) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
516 |
+ ((cos(t + 1/2 * real (n) *pi) / (fact n)) * x ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
517 |
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) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
518 |
apply safe |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
519 |
apply simp |
36974
b877866b5b00
remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents:
32047
diff
changeset
|
520 |
apply (simp (no_asm) add: cos_expansion_lemma) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
521 |
apply (erule ssubst) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
522 |
apply (rule_tac x = t in exI, simp) |
57418 | 523 |
apply (rule setsum.cong[OF refl]) |
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58410
diff
changeset
|
524 |
apply (auto simp add: cos_coeff_def cos_zero_iff elim: evenE) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
525 |
done |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
526 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
527 |
lemma Maclaurin_minus_cos_expansion: |
25162 | 528 |
"[| x < 0; n > 0 |] ==> |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
529 |
\<exists>t. x < t & t < 0 & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
530 |
cos x = |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
531 |
(\<Sum>m<n. cos_coeff m * x ^ m) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
532 |
+ ((cos(t + 1/2 * real (n) *pi) / (fact n)) * x ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
533 |
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) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
534 |
apply safe |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
535 |
apply simp |
36974
b877866b5b00
remove some unnamed simp rules from Transcendental.thy; move the needed ones to MacLaurin.thy where they are used
huffman
parents:
32047
diff
changeset
|
536 |
apply (simp (no_asm) add: cos_expansion_lemma) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
537 |
apply (erule ssubst) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
538 |
apply (rule_tac x = t in exI, simp) |
57418 | 539 |
apply (rule setsum.cong[OF refl]) |
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58410
diff
changeset
|
540 |
apply (auto simp add: cos_coeff_def cos_zero_iff elim: evenE) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
541 |
done |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
542 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
543 |
(* ------------------------------------------------------------------------- *) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
544 |
(* Version for ln(1 +/- x). Where is it?? *) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
545 |
(* ------------------------------------------------------------------------- *) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
546 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
547 |
lemma sin_bound_lemma: |
15081 | 548 |
"[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
549 |
by auto |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
550 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
551 |
lemma Maclaurin_sin_bound: |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
552 |
"abs(sin x - (\<Sum>m<n. sin_coeff m * x ^ m)) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
553 |
\<le> inverse((fact n)) * \<bar>x\<bar> ^ n" |
14738 | 554 |
proof - |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
555 |
have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y" |
14738 | 556 |
by (rule_tac mult_right_mono,simp_all) |
557 |
note est = this[simplified] |
|
22985 | 558 |
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)" |
559 |
have diff_0: "?diff 0 = sin" by simp |
|
560 |
have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x" |
|
561 |
apply (clarify) |
|
562 |
apply (subst (1 2 3) mod_Suc_eq_Suc_mod) |
|
563 |
apply (cut_tac m=m in mod_exhaust_less_4) |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56238
diff
changeset
|
564 |
apply (safe, auto intro!: derivative_eq_intros) |
22985 | 565 |
done |
566 |
from Maclaurin_all_le [OF diff_0 DERIV_diff] |
|
567 |
obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>" and |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
568 |
t2: "sin x = (\<Sum>m<n. ?diff m 0 / (fact m) * x ^ m) + |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
569 |
?diff n t / (fact n) * x ^ n" by fast |
22985 | 570 |
have diff_m_0: |
571 |
"\<And>m. ?diff m 0 = (if even m then 0 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
572 |
else (- 1) ^ ((m - Suc 0) div 2))" |
22985 | 573 |
apply (subst even_even_mod_4_iff) |
574 |
apply (cut_tac m=m in mod_exhaust_less_4) |
|
575 |
apply (elim disjE, simp_all) |
|
576 |
apply (safe dest!: mod_eqD, simp_all) |
|
577 |
done |
|
14738 | 578 |
show ?thesis |
44306
33572a766836
fold definitions of sin_coeff and cos_coeff in Maclaurin lemmas
huffman
parents:
41166
diff
changeset
|
579 |
unfolding sin_coeff_def |
22985 | 580 |
apply (subst t2) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
581 |
apply (rule sin_bound_lemma) |
57418 | 582 |
apply (rule setsum.cong[OF refl]) |
22985 | 583 |
apply (subst diff_m_0, simp) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
584 |
apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57492
diff
changeset
|
585 |
simp add: est ac_simps divide_inverse power_abs [symmetric] abs_mult) |
14738 | 586 |
done |
587 |
qed |
|
588 |
||
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
589 |
end |