author | wenzelm |
Mon, 02 May 2011 16:33:21 +0200 | |
changeset 42616 | 92715b528e78 |
parent 41166 | 4b2a457b17e8 |
child 44306 | 33572a766836 |
permissions | -rw-r--r-- |
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(* Author : Jacques D. Fleuriot |
12224 | 2 |
Copyright : 2001 University of Edinburgh |
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3 |
Conversion to Isar and new proofs by Lawrence C Paulson, 2004 |
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Conversion of Mac Laurin to Isar by Lukas Bulwahn and Bernhard Häupler, 2005 |
12224 | 5 |
*) |
6 |
||
15944 | 7 |
header{*MacLaurin Series*} |
8 |
||
15131 | 9 |
theory MacLaurin |
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imports Transcendental |
15131 | 11 |
begin |
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12 |
|
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subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*} |
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14 |
|
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text{*This is a very long, messy proof even now that it's been broken down |
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16 |
into lemmas.*} |
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|
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lemma Maclaurin_lemma: |
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"0 < h ==> |
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\<exists>B. f h = (\<Sum>m=0..<n. (j m / real (fact m)) * (h^m)) + |
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(B * ((h^n) / real(fact n)))" |
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by (rule exI[where x = "(f h - (\<Sum>m=0..<n. (j m / real (fact m)) * h^m)) * |
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real(fact n) / (h^n)"]) simp |
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24 |
|
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lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))" |
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26 |
by arith |
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|
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lemma fact_diff_Suc [rule_format]: |
29 |
"n < Suc m ==> fact (Suc m - n) = (Suc m - n) * fact (m - n)" |
|
30 |
by (subst fact_reduce_nat, auto) |
|
31 |
||
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lemma Maclaurin_lemma2: |
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fixes B |
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assumes DERIV : "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t" |
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and INIT : "n = Suc k" |
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defines "difg \<equiv> (\<lambda>m t. diff m t - ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) + |
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B * (t ^ (n - m) / real (fact (n - m)))))" (is "difg \<equiv> (\<lambda>m t. diff m t - ?difg m t)") |
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38 |
shows "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t" |
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proof (rule allI impI)+ |
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40 |
fix m t assume INIT2: "m < n & 0 \<le> t & t \<le> h" |
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41 |
have "DERIV (difg m) t :> diff (Suc m) t - |
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((\<Sum>x = 0..<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / real (fact x)) + |
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real (n - m) * t ^ (n - Suc m) * B / real (fact (n - m)))" unfolding difg_def |
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44 |
by (auto intro!: DERIV_intros DERIV[rule_format, OF INIT2]) |
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bulwahn
parents:
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45 |
moreover |
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46 |
from INIT2 have intvl: "{..<n - m} = insert 0 (Suc ` {..<n - Suc m})" and "0 < n - m" |
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47 |
unfolding atLeast0LessThan[symmetric] by auto |
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48 |
have "(\<Sum>x = 0..<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / real (fact x)) = |
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(\<Sum>x = 0..<n - Suc m. real (Suc x) * t ^ x * diff (Suc m + x) 0 / real (fact (Suc x)))" |
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50 |
unfolding intvl atLeast0LessThan by (subst setsum.insert) (auto simp: setsum.reindex) |
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51 |
moreover |
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52 |
have fact_neq_0: "\<And>x::nat. real (fact x) + real x * real (fact x) \<noteq> 0" |
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53 |
by (metis fact_gt_zero_nat not_add_less1 real_of_nat_add real_of_nat_mult real_of_nat_zero_iff) |
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54 |
have "\<And>x. real (Suc x) * t ^ x * diff (Suc m + x) 0 / real (fact (Suc x)) = |
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55 |
diff (Suc m + x) 0 * t^x / real (fact x)" |
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56 |
by (auto simp: field_simps real_of_nat_Suc fact_neq_0 intro!: nonzero_divide_eq_eq[THEN iffD2]) |
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57 |
moreover |
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58 |
have "real (n - m) * t ^ (n - Suc m) * B / real (fact (n - m)) = |
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59 |
B * (t ^ (n - Suc m) / real (fact (n - Suc m)))" |
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60 |
using `0 < n - m` by (simp add: fact_reduce_nat) |
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61 |
ultimately show "DERIV (difg m) t :> difg (Suc m) t" |
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62 |
unfolding difg_def by simp |
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adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
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63 |
qed |
32038 | 64 |
|
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parents:
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65 |
lemma Maclaurin: |
29187 | 66 |
assumes h: "0 < h" |
67 |
assumes n: "0 < n" |
|
68 |
assumes diff_0: "diff 0 = f" |
|
69 |
assumes diff_Suc: |
|
70 |
"\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t" |
|
71 |
shows |
|
72 |
"\<exists>t. 0 < t & t < h & |
|
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parents:
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73 |
f h = |
15539 | 74 |
setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} + |
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75 |
(diff n t / real (fact n)) * h ^ n" |
29187 | 76 |
proof - |
77 |
from n obtain m where m: "n = Suc m" |
|
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78 |
by (cases n) (simp add: n) |
29187 | 79 |
|
80 |
obtain B where f_h: "f h = |
|
81 |
(\<Sum>m = 0..<n. diff m (0\<Colon>real) / real (fact m) * h ^ m) + |
|
82 |
B * (h ^ n / real (fact n))" |
|
83 |
using Maclaurin_lemma [OF h] .. |
|
84 |
||
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85 |
def g \<equiv> "(\<lambda>t. f t - |
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86 |
(setsum (\<lambda>m. (diff m 0 / real(fact m)) * t^m) {0..<n} |
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87 |
+ (B * (t^n / real(fact n)))))" |
29187 | 88 |
|
89 |
have g2: "g 0 = 0 & g h = 0" |
|
90 |
apply (simp add: m f_h g_def del: setsum_op_ivl_Suc) |
|
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91 |
apply (cut_tac n = m and k = "Suc 0" in sumr_offset2) |
29187 | 92 |
apply (simp add: eq_diff_eq' diff_0 del: setsum_op_ivl_Suc) |
93 |
done |
|
94 |
||
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95 |
def difg \<equiv> "(%m t. diff m t - |
29187 | 96 |
(setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {0..<n-m} |
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97 |
+ (B * ((t ^ (n - m)) / real (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" |
|
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104 |
using diff_Suc m unfolding difg_def by (rule Maclaurin_lemma2) |
29187 | 105 |
|
106 |
have difg_eq_0: "\<forall>m. m < n --> difg m 0 = 0" |
|
107 |
apply clarify |
|
108 |
apply (simp add: m difg_def) |
|
109 |
apply (frule less_iff_Suc_add [THEN iffD1], clarify) |
|
110 |
apply (simp del: setsum_op_ivl_Suc) |
|
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111 |
apply (insert sumr_offset4 [of "Suc 0"]) |
32047 | 112 |
apply (simp del: setsum_op_ivl_Suc fact_Suc) |
29187 | 113 |
done |
114 |
||
115 |
have isCont_difg: "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> isCont (difg m) x" |
|
116 |
by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp |
|
117 |
||
118 |
have differentiable_difg: |
|
119 |
"\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> difg m differentiable x" |
|
120 |
by (rule differentiableI [OF difg_Suc [rule_format]]) simp |
|
121 |
||
122 |
have difg_Suc_eq_0: "\<And>m t. \<lbrakk>m < n; 0 \<le> t; t \<le> h; DERIV (difg m) t :> 0\<rbrakk> |
|
123 |
\<Longrightarrow> difg (Suc m) t = 0" |
|
124 |
by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp |
|
125 |
||
126 |
have "m < n" using m by simp |
|
127 |
||
128 |
have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m) t :> 0" |
|
129 |
using `m < n` |
|
130 |
proof (induct m) |
|
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131 |
case 0 |
29187 | 132 |
show ?case |
133 |
proof (rule Rolle) |
|
134 |
show "0 < h" by fact |
|
135 |
show "difg 0 0 = difg 0 h" by (simp add: difg_0 g2) |
|
136 |
show "\<forall>x. 0 \<le> x \<and> x \<le> h \<longrightarrow> isCont (difg (0\<Colon>nat)) x" |
|
137 |
by (simp add: isCont_difg n) |
|
138 |
show "\<forall>x. 0 < x \<and> x < h \<longrightarrow> difg (0\<Colon>nat) differentiable x" |
|
139 |
by (simp add: differentiable_difg n) |
|
140 |
qed |
|
141 |
next |
|
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142 |
case (Suc m') |
29187 | 143 |
hence "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m') t :> 0" by simp |
144 |
then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0" by fast |
|
145 |
have "\<exists>t'. 0 < t' \<and> t' < t \<and> DERIV (difg (Suc m')) t' :> 0" |
|
146 |
proof (rule Rolle) |
|
147 |
show "0 < t" by fact |
|
148 |
show "difg (Suc m') 0 = difg (Suc m') t" |
|
149 |
using t `Suc m' < n` by (simp add: difg_Suc_eq_0 difg_eq_0) |
|
150 |
show "\<forall>x. 0 \<le> x \<and> x \<le> t \<longrightarrow> isCont (difg (Suc m')) x" |
|
151 |
using `t < h` `Suc m' < n` by (simp add: isCont_difg) |
|
152 |
show "\<forall>x. 0 < x \<and> x < t \<longrightarrow> difg (Suc m') differentiable x" |
|
153 |
using `t < h` `Suc m' < n` by (simp add: differentiable_difg) |
|
154 |
qed |
|
155 |
thus ?case |
|
156 |
using `t < h` by auto |
|
157 |
qed |
|
158 |
||
159 |
then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0" by fast |
|
160 |
||
161 |
hence "difg (Suc m) t = 0" |
|
162 |
using `m < n` by (simp add: difg_Suc_eq_0) |
|
163 |
||
164 |
show ?thesis |
|
165 |
proof (intro exI conjI) |
|
166 |
show "0 < t" by fact |
|
167 |
show "t < h" by fact |
|
168 |
show "f h = |
|
169 |
(\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) + |
|
170 |
diff n t / real (fact n) * h ^ n" |
|
171 |
using `difg (Suc m) t = 0` |
|
32047 | 172 |
by (simp add: m f_h difg_def del: fact_Suc) |
29187 | 173 |
qed |
174 |
qed |
|
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175 |
|
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176 |
lemma Maclaurin_objl: |
25162 | 177 |
"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
|
178 |
(\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
179 |
--> (\<exists>t. 0 < t & t < h & |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
180 |
f h = (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) + |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
181 |
diff n t / real (fact n) * h ^ n)" |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
182 |
by (blast intro: Maclaurin) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
183 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
184 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
185 |
lemma Maclaurin2: |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
186 |
assumes INIT1: "0 < h " and INIT2: "diff 0 = f" |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
187 |
and DERIV: "\<forall>m t. |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
188 |
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
|
189 |
shows "\<exists>t. 0 < t \<and> t \<le> h \<and> f h = |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
190 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) + |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
191 |
diff n t / real (fact n) * h ^ n" |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
192 |
proof (cases "n") |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
193 |
case 0 with INIT1 INIT2 show ?thesis by fastsimp |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
194 |
next |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
195 |
case Suc |
41120
74e41b2d48ea
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bulwahn
parents:
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diff
changeset
|
196 |
hence "n > 0" by simp |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
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diff
changeset
|
197 |
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
|
198 |
f h = |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
199 |
(\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) + diff n t / real (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
|
200 |
by (rule Maclaurin) |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
201 |
thus ?thesis by fastsimp |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
202 |
qed |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
203 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
204 |
lemma Maclaurin2_objl: |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
205 |
"0 < h & diff 0 = f & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
206 |
(\<forall>m t. |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
207 |
m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
208 |
--> (\<exists>t. 0 < t & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
209 |
t \<le> h & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
210 |
f h = |
15539 | 211 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) + |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
212 |
diff n t / real (fact n) * h ^ n)" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
213 |
by (blast intro: Maclaurin2) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
214 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
215 |
lemma Maclaurin_minus: |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
216 |
assumes "h < 0" "0 < n" "diff 0 = f" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
217 |
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
|
218 |
shows "\<exists>t. h < t & t < 0 & |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
219 |
f h = (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) + |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
220 |
diff n t / real (fact n) * h ^ n" |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
221 |
proof - |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
222 |
txt "Transform @{text ABL'} into @{text DERIV_intros} format." |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
223 |
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
|
224 |
from assms |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
225 |
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
|
226 |
f (- (- h)) = |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
227 |
(\<Sum>m = 0..<n. |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
228 |
(- 1) ^ m * diff m (- 0) / real (fact m) * (- h) ^ m) + |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
229 |
(- 1) ^ n * diff n (- t) / real (fact n) * (- h) ^ n" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
230 |
by (intro Maclaurin) (auto intro!: DERIV_intros DERIV') |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
231 |
then guess t .. |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
232 |
moreover |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
233 |
have "-1 ^ n * diff n (- t) * (- h) ^ n / real (fact n) = diff n (- t) * h ^ n / real (fact n)" |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
234 |
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
|
235 |
moreover |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
236 |
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))" |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
237 |
by (auto intro: setsum_cong 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
|
238 |
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
|
239 |
- t < 0 \<and> |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
240 |
f h = |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
241 |
(\<Sum>m = 0..<n. diff m 0 / real (fact m) * h ^ m) + diff n (- t) / real (fact n) * h ^ n" |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
242 |
by auto |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
243 |
thus ?thesis .. |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
244 |
qed |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
245 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
246 |
lemma Maclaurin_minus_objl: |
25162 | 247 |
"(h < 0 & n > 0 & diff 0 = f & |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
248 |
(\<forall>m t. |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
249 |
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
|
250 |
--> (\<exists>t. h < t & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
251 |
t < 0 & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
252 |
f h = |
15539 | 253 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) + |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
254 |
diff n t / real (fact n) * h ^ n)" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
255 |
by (blast intro: Maclaurin_minus) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
256 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
257 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
258 |
subsection{*More Convenient "Bidirectional" Version.*} |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
259 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
260 |
(* not good for PVS sin_approx, cos_approx *) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
261 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
262 |
lemma Maclaurin_bi_le_lemma [rule_format]: |
25162 | 263 |
"n>0 \<longrightarrow> |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
264 |
diff 0 0 = |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
265 |
(\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) + |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
266 |
diff n 0 * 0 ^ n / real (fact n)" |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
267 |
by (induct "n") auto |
14738 | 268 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
269 |
lemma Maclaurin_bi_le: |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
270 |
assumes "diff 0 = f" |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
271 |
and DERIV : "\<forall>m t. m < n & abs t \<le> abs x --> 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
|
272 |
shows "\<exists>t. abs t \<le> abs x & |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
273 |
f x = |
15539 | 274 |
(\<Sum>m=0..<n. diff m 0 / real (fact m) * x ^ m) + |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
275 |
diff n t / real (fact n) * x ^ n" (is "\<exists>t. _ \<and> f x = ?f x t") |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
276 |
proof cases |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
277 |
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
|
278 |
next |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
279 |
assume "n \<noteq> 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
280 |
show ?thesis |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
281 |
proof (cases rule: linorder_cases) |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
282 |
assume "x = 0" with `n \<noteq> 0` `diff 0 = f` DERIV |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
283 |
have "\<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0" by (force simp add: Maclaurin_bi_le_lemma) |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
284 |
thus ?thesis .. |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
285 |
next |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
286 |
assume "x < 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
287 |
with `n \<noteq> 0` DERIV |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
288 |
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
|
289 |
then guess t .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
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|
290 |
with `x < 0` `diff 0 = f` have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp |
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|
291 |
thus ?thesis .. |
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changeset
|
292 |
next |
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beautify MacLaurin proofs; make better use of DERIV_intros
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changeset
|
293 |
assume "x > 0" |
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|
294 |
with `n \<noteq> 0` `diff 0 = f` DERIV |
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|
295 |
have "\<exists>t>0. t < x \<and> diff 0 x = ?f x t" by (intro Maclaurin) auto |
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changeset
|
296 |
then guess t .. |
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parents:
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diff
changeset
|
297 |
with `x > 0` `diff 0 = f` have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp |
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parents:
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changeset
|
298 |
thus ?thesis .. |
41120
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|
299 |
qed |
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adding an Isar version of the MacLaurin theorem from some students' work in 2005
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|
300 |
qed |
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adding an Isar version of the MacLaurin theorem from some students' work in 2005
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parents:
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diff
changeset
|
301 |
|
15079
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conversion of Hyperreal/MacLaurin_lemmas to Isar script
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parents:
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diff
changeset
|
302 |
lemma Maclaurin_all_lt: |
41120
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|
303 |
assumes INIT1: "diff 0 = f" and INIT2: "0 < n" and INIT3: "x \<noteq> 0" |
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|
304 |
and DERIV: "\<forall>m x. DERIV (diff m) x :> diff(Suc m) x" |
41166
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diff
changeset
|
305 |
shows "\<exists>t. 0 < abs t & abs t < abs x & f x = |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
306 |
(\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) + |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
307 |
(diff n t / real (fact n)) * x ^ n" (is "\<exists>t. _ \<and> _ \<and> f x = ?f x t") |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
308 |
proof (cases rule: linorder_cases) |
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beautify MacLaurin proofs; make better use of DERIV_intros
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parents:
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diff
changeset
|
309 |
assume "x = 0" with INIT3 show "?thesis".. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
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diff
changeset
|
310 |
next |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
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parents:
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diff
changeset
|
311 |
assume "x < 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
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diff
changeset
|
312 |
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
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parents:
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diff
changeset
|
313 |
then guess t .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
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diff
changeset
|
314 |
with `x < 0` have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t" by simp |
4b2a457b17e8
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hoelzl
parents:
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diff
changeset
|
315 |
thus ?thesis .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
316 |
next |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
317 |
assume "x > 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
318 |
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
|
319 |
then guess t .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
320 |
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
|
321 |
thus ?thesis .. |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
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parents:
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diff
changeset
|
322 |
qed |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
323 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
324 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
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diff
changeset
|
325 |
lemma Maclaurin_all_lt_objl: |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
326 |
"diff 0 = f & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
327 |
(\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) & |
25162 | 328 |
x ~= 0 & n > 0 |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
329 |
--> (\<exists>t. 0 < abs t & abs t < abs x & |
15539 | 330 |
f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) + |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
331 |
(diff n t / real (fact n)) * x ^ n)" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
332 |
by (blast intro: Maclaurin_all_lt) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
333 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
334 |
lemma Maclaurin_zero [rule_format]: |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
335 |
"x = (0::real) |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
336 |
==> n \<noteq> 0 --> |
15539 | 337 |
(\<Sum>m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) = |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
338 |
diff 0 0" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
339 |
by (induct n, auto) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
340 |
|
41120
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adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
341 |
|
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
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parents:
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diff
changeset
|
342 |
lemma Maclaurin_all_le: |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
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parents:
36974
diff
changeset
|
343 |
assumes INIT: "diff 0 = f" |
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parents:
36974
diff
changeset
|
344 |
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
|
345 |
shows "\<exists>t. abs t \<le> abs x & f x = |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
346 |
(\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) + |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
347 |
(diff n t / real (fact n)) * x ^ n" (is "\<exists>t. _ \<and> f x = ?f x t") |
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 "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
|
350 |
next |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
351 |
assume "n \<noteq> 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
352 |
show ?thesis |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
353 |
proof cases |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
354 |
assume "x = 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
355 |
with `n \<noteq> 0` have "(\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) = diff 0 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
356 |
by (intro Maclaurin_zero) auto |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
357 |
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
|
358 |
thus ?thesis .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
359 |
next |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
360 |
assume "x \<noteq> 0" |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
361 |
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
|
362 |
by (intro Maclaurin_all_lt) auto |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
363 |
then guess t .. |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
364 |
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
|
365 |
thus ?thesis .. |
41120
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
366 |
qed |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
367 |
qed |
74e41b2d48ea
adding an Isar version of the MacLaurin theorem from some students' work in 2005
bulwahn
parents:
36974
diff
changeset
|
368 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
369 |
lemma Maclaurin_all_le_objl: "diff 0 = f & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
370 |
(\<forall>m x. DERIV (diff m) x :> diff (Suc m) x) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
371 |
--> (\<exists>t. abs t \<le> abs x & |
15539 | 372 |
f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) + |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
373 |
(diff n t / real (fact n)) * x ^ n)" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
374 |
by (blast intro: Maclaurin_all_le) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
375 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
376 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
377 |
subsection{*Version for Exponential Function*} |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
378 |
|
25162 | 379 |
lemma Maclaurin_exp_lt: "[| x ~= 0; n > 0 |] |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
380 |
==> (\<exists>t. 0 < abs t & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
381 |
abs t < abs x & |
15539 | 382 |
exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) + |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
383 |
(exp t / real (fact n)) * x ^ n)" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
384 |
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
|
385 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
386 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
387 |
lemma Maclaurin_exp_le: |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
388 |
"\<exists>t. abs t \<le> abs x & |
15539 | 389 |
exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) + |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
390 |
(exp t / real (fact n)) * x ^ n" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
391 |
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
|
392 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
393 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
394 |
subsection{*Version for Sine Function*} |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
395 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
396 |
lemma mod_exhaust_less_4: |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
397 |
"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
|
398 |
by auto |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
399 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
400 |
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
|
401 |
"n\<noteq>0 --> Suc (Suc (2 * n - 2)) = 2*n" |
15251 | 402 |
by (induct "n", auto) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
403 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
404 |
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
|
405 |
"n\<noteq>0 --> Suc (Suc (4*n - 2)) = 4*n" |
15251 | 406 |
by (induct "n", auto) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
407 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
408 |
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
|
409 |
"n\<noteq>0 --> Suc (2 * n - 1) = 2*n" |
15251 | 410 |
by (induct "n", auto) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
411 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
412 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
413 |
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
|
414 |
|
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
|
415 |
lemma sin_expansion_lemma: |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
416 |
"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
|
417 |
cos (x + real (m) * pi / 2)" |
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 |
by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto) |
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
|
419 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
420 |
lemma Maclaurin_sin_expansion2: |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
421 |
"\<exists>t. abs t \<le> abs x & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
422 |
sin x = |
15539 | 423 |
(\<Sum>m=0..<n. (if even m then 0 |
23177 | 424 |
else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) * |
15539 | 425 |
x ^ m) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
426 |
+ ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
427 |
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
|
428 |
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
|
429 |
apply safe |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
430 |
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
|
431 |
apply (simp (no_asm) add: sin_expansion_lemma) |
23242 | 432 |
apply (case_tac "n", clarify, simp, simp add: lemma_STAR_sin) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
433 |
apply (rule ccontr, simp) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
434 |
apply (drule_tac x = x in spec, simp) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
435 |
apply (erule ssubst) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
436 |
apply (rule_tac x = t in exI, simp) |
15536 | 437 |
apply (rule setsum_cong[OF refl]) |
15539 | 438 |
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
439 |
done |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
440 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
441 |
lemma Maclaurin_sin_expansion: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
442 |
"\<exists>t. sin x = |
15539 | 443 |
(\<Sum>m=0..<n. (if even m then 0 |
23177 | 444 |
else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) * |
15539 | 445 |
x ^ m) |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
446 |
+ ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)" |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
447 |
apply (insert Maclaurin_sin_expansion2 [of x n]) |
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
448 |
apply (blast intro: elim:) |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
449 |
done |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
450 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
451 |
lemma Maclaurin_sin_expansion3: |
25162 | 452 |
"[| n > 0; 0 < x |] ==> |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
453 |
\<exists>t. 0 < t & t < x & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
454 |
sin x = |
15539 | 455 |
(\<Sum>m=0..<n. (if even m then 0 |
23177 | 456 |
else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) * |
15539 | 457 |
x ^ m) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
458 |
+ ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
459 |
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
|
460 |
apply safe |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
461 |
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
|
462 |
apply (simp (no_asm) add: sin_expansion_lemma) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
463 |
apply (erule ssubst) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
464 |
apply (rule_tac x = t in exI, simp) |
15536 | 465 |
apply (rule setsum_cong[OF refl]) |
15539 | 466 |
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
467 |
done |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
468 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
469 |
lemma Maclaurin_sin_expansion4: |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
470 |
"0 < x ==> |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
471 |
\<exists>t. 0 < t & t \<le> x & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
472 |
sin x = |
15539 | 473 |
(\<Sum>m=0..<n. (if even m then 0 |
23177 | 474 |
else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) * |
15539 | 475 |
x ^ m) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
476 |
+ ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
477 |
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
|
478 |
apply safe |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
479 |
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
|
480 |
apply (simp (no_asm) add: sin_expansion_lemma) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
481 |
apply (erule ssubst) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
482 |
apply (rule_tac x = t in exI, simp) |
15536 | 483 |
apply (rule setsum_cong[OF refl]) |
15539 | 484 |
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
485 |
done |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
486 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
487 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
488 |
subsection{*Maclaurin Expansion for Cosine Function*} |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
489 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
490 |
lemma sumr_cos_zero_one [simp]: |
15539 | 491 |
"(\<Sum>m=0..<(Suc n). |
23177 | 492 |
(if even m then -1 ^ (m div 2)/(real (fact m)) else 0) * 0 ^ m) = 1" |
15251 | 493 |
by (induct "n", auto) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
494 |
|
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
|
495 |
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
|
496 |
"cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)" |
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
|
497 |
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto) |
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
|
498 |
|
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
499 |
lemma Maclaurin_cos_expansion: |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
500 |
"\<exists>t. abs t \<le> abs x & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
501 |
cos x = |
15539 | 502 |
(\<Sum>m=0..<n. (if even m |
23177 | 503 |
then -1 ^ (m div 2)/(real (fact m)) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
504 |
else 0) * |
15539 | 505 |
x ^ m) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
506 |
+ ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
507 |
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
|
508 |
apply safe |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
509 |
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
|
510 |
apply (simp (no_asm) add: cos_expansion_lemma) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
511 |
apply (case_tac "n", simp) |
15561 | 512 |
apply (simp del: setsum_op_ivl_Suc) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
513 |
apply (rule ccontr, simp) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
514 |
apply (drule_tac x = x in spec, simp) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
515 |
apply (erule ssubst) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
516 |
apply (rule_tac x = t in exI, simp) |
15536 | 517 |
apply (rule setsum_cong[OF refl]) |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
518 |
apply (auto simp add: cos_zero_iff even_mult_two_ex) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
519 |
done |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
520 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
521 |
lemma Maclaurin_cos_expansion2: |
25162 | 522 |
"[| 0 < x; n > 0 |] ==> |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
523 |
\<exists>t. 0 < t & t < x & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
524 |
cos x = |
15539 | 525 |
(\<Sum>m=0..<n. (if even m |
23177 | 526 |
then -1 ^ (m div 2)/(real (fact m)) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
527 |
else 0) * |
15539 | 528 |
x ^ m) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
529 |
+ ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
530 |
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
|
531 |
apply safe |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
532 |
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
|
533 |
apply (simp (no_asm) add: cos_expansion_lemma) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
534 |
apply (erule ssubst) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
535 |
apply (rule_tac x = t in exI, simp) |
15536 | 536 |
apply (rule setsum_cong[OF refl]) |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
537 |
apply (auto simp add: cos_zero_iff even_mult_two_ex) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
538 |
done |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
539 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
540 |
lemma Maclaurin_minus_cos_expansion: |
25162 | 541 |
"[| x < 0; n > 0 |] ==> |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
542 |
\<exists>t. x < t & t < 0 & |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
543 |
cos x = |
15539 | 544 |
(\<Sum>m=0..<n. (if even m |
23177 | 545 |
then -1 ^ (m div 2)/(real (fact m)) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
546 |
else 0) * |
15539 | 547 |
x ^ m) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
548 |
+ ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)" |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
549 |
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
|
550 |
apply safe |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
551 |
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
|
552 |
apply (simp (no_asm) add: cos_expansion_lemma) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
553 |
apply (erule ssubst) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
554 |
apply (rule_tac x = t in exI, simp) |
15536 | 555 |
apply (rule setsum_cong[OF refl]) |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
556 |
apply (auto simp add: cos_zero_iff even_mult_two_ex) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
557 |
done |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
558 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
559 |
(* ------------------------------------------------------------------------- *) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
560 |
(* Version for ln(1 +/- x). Where is it?? *) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
561 |
(* ------------------------------------------------------------------------- *) |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
562 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
563 |
lemma sin_bound_lemma: |
15081 | 564 |
"[|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
|
565 |
by auto |
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
566 |
|
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
567 |
lemma Maclaurin_sin_bound: |
23177 | 568 |
"abs(sin x - (\<Sum>m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) * |
15081 | 569 |
x ^ m)) \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n" |
14738 | 570 |
proof - |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
571 |
have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y" |
14738 | 572 |
by (rule_tac mult_right_mono,simp_all) |
573 |
note est = this[simplified] |
|
22985 | 574 |
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)" |
575 |
have diff_0: "?diff 0 = sin" by simp |
|
576 |
have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x" |
|
577 |
apply (clarify) |
|
578 |
apply (subst (1 2 3) mod_Suc_eq_Suc_mod) |
|
579 |
apply (cut_tac m=m in mod_exhaust_less_4) |
|
31881 | 580 |
apply (safe, auto intro!: DERIV_intros) |
22985 | 581 |
done |
582 |
from Maclaurin_all_le [OF diff_0 DERIV_diff] |
|
583 |
obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>" and |
|
584 |
t2: "sin x = (\<Sum>m = 0..<n. ?diff m 0 / real (fact m) * x ^ m) + |
|
585 |
?diff n t / real (fact n) * x ^ n" by fast |
|
586 |
have diff_m_0: |
|
587 |
"\<And>m. ?diff m 0 = (if even m then 0 |
|
23177 | 588 |
else -1 ^ ((m - Suc 0) div 2))" |
22985 | 589 |
apply (subst even_even_mod_4_iff) |
590 |
apply (cut_tac m=m in mod_exhaust_less_4) |
|
591 |
apply (elim disjE, simp_all) |
|
592 |
apply (safe dest!: mod_eqD, simp_all) |
|
593 |
done |
|
14738 | 594 |
show ?thesis |
22985 | 595 |
apply (subst t2) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
596 |
apply (rule sin_bound_lemma) |
15536 | 597 |
apply (rule setsum_cong[OF refl]) |
22985 | 598 |
apply (subst diff_m_0, simp) |
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
599 |
apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono |
41166
4b2a457b17e8
beautify MacLaurin proofs; make better use of DERIV_intros
hoelzl
parents:
41120
diff
changeset
|
600 |
simp add: est mult_nonneg_nonneg mult_ac divide_inverse |
16924 | 601 |
power_abs [symmetric] abs_mult) |
14738 | 602 |
done |
603 |
qed |
|
604 |
||
15079
2ef899e4526d
conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents:
14738
diff
changeset
|
605 |
end |