author | wenzelm |
Sun, 02 Nov 2014 18:21:45 +0100 | |
changeset 58889 | 5b7a9633cfa8 |
parent 58834 | 773b378d9313 |
child 58981 | 11b6c099f5f3 |
permissions | -rw-r--r-- |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
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parents:
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|
1 |
(* Title: HOL/Transcendental.thy |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
2 |
Author: Jacques D. Fleuriot, University of Cambridge, University of Edinburgh |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3 |
Author: Lawrence C Paulson |
51527 | 4 |
Author: Jeremy Avigad |
12196 | 5 |
*) |
6 |
||
58889 | 7 |
section{*Power Series, Transcendental Functions etc.*} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
8 |
|
15131 | 9 |
theory Transcendental |
25600 | 10 |
imports Fact Series Deriv NthRoot |
15131 | 11 |
begin |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
12 |
|
57025 | 13 |
lemma root_test_convergence: |
14 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
|
15 |
assumes f: "(\<lambda>n. root n (norm (f n))) ----> x" -- "could be weakened to lim sup" |
|
16 |
assumes "x < 1" |
|
17 |
shows "summable f" |
|
18 |
proof - |
|
19 |
have "0 \<le> x" |
|
20 |
by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1]) |
|
21 |
from `x < 1` obtain z where z: "x < z" "z < 1" |
|
22 |
by (metis dense) |
|
23 |
from f `x < z` |
|
24 |
have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially" |
|
25 |
by (rule order_tendstoD) |
|
26 |
then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially" |
|
27 |
using eventually_ge_at_top |
|
28 |
proof eventually_elim |
|
29 |
fix n assume less: "root n (norm (f n)) < z" and n: "1 \<le> n" |
|
30 |
from power_strict_mono[OF less, of n] n |
|
31 |
show "norm (f n) \<le> z ^ n" |
|
32 |
by simp |
|
33 |
qed |
|
34 |
then show "summable f" |
|
35 |
unfolding eventually_sequentially |
|
36 |
using z `0 \<le> x` by (auto intro!: summable_comparison_test[OF _ summable_geometric]) |
|
37 |
qed |
|
38 |
||
29164 | 39 |
subsection {* Properties of Power Series *} |
15077
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converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
40 |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
41 |
lemma lemma_realpow_diff: |
31017 | 42 |
fixes y :: "'a::monoid_mult" |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
43 |
shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
44 |
proof - |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
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diff
changeset
|
45 |
assume "p \<le> n" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
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diff
changeset
|
46 |
hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le) |
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents:
30082
diff
changeset
|
47 |
thus ?thesis by (simp add: power_commutes) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
48 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
49 |
|
15229 | 50 |
lemma lemma_realpow_diff_sumr2: |
53079 | 51 |
fixes y :: "'a::{comm_ring,monoid_mult}" |
52 |
shows |
|
53 |
"x ^ (Suc n) - y ^ (Suc n) = |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
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diff
changeset
|
54 |
(x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))" |
54573 | 55 |
proof (induct n) |
56 |
case (Suc n) |
|
57 |
have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x ^ n) - y * (y * y ^ n)" |
|
58 |
by simp |
|
59 |
also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x ^ n)" |
|
60 |
by (simp add: algebra_simps) |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
61 |
also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)" |
54573 | 62 |
by (simp only: Suc) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
63 |
also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
64 |
by (simp only: mult.left_commute) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
65 |
also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
66 |
by (simp add: field_simps Suc_diff_le setsum_left_distrib setsum_right_distrib) |
54573 | 67 |
finally show ?case . |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
68 |
qed simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
69 |
|
55832 | 70 |
corollary power_diff_sumr2: --{* @{text COMPLEX_POLYFUN} in HOL Light *} |
55734 | 71 |
fixes x :: "'a::{comm_ring,monoid_mult}" |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
72 |
shows "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)" |
55734 | 73 |
using lemma_realpow_diff_sumr2[of x "n - 1" y] |
74 |
by (cases "n = 0") (simp_all add: field_simps) |
|
75 |
||
15229 | 76 |
lemma lemma_realpow_rev_sumr: |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
77 |
"(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) = |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
78 |
(\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))" |
57129
7edb7550663e
introduce more powerful reindexing rules for big operators
hoelzl
parents:
57025
diff
changeset
|
79 |
by (subst nat_diff_setsum_reindex[symmetric]) simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
80 |
|
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
81 |
lemma power_diff_1_eq: |
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
82 |
fixes x :: "'a::{comm_ring,monoid_mult}" |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
83 |
shows "n \<noteq> 0 \<Longrightarrow> x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))" |
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
84 |
using lemma_realpow_diff_sumr2 [of x _ 1] |
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
85 |
by (cases n) auto |
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
86 |
|
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
87 |
lemma one_diff_power_eq': |
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
88 |
fixes x :: "'a::{comm_ring,monoid_mult}" |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
89 |
shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))" |
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
90 |
using lemma_realpow_diff_sumr2 [of 1 _ x] |
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
91 |
by (cases n) auto |
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
92 |
|
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
93 |
lemma one_diff_power_eq: |
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
94 |
fixes x :: "'a::{comm_ring,monoid_mult}" |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
95 |
shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^i)" |
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
96 |
by (metis one_diff_power_eq' [of n x] nat_diff_setsum_reindex) |
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
97 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
98 |
text{*Power series has a `circle` of convergence, i.e. if it sums for @{term |
53079 | 99 |
x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
100 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
101 |
lemma powser_insidea: |
53599 | 102 |
fixes x z :: "'a::real_normed_div_algebra" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
103 |
assumes 1: "summable (\<lambda>n. f n * x ^ n)" |
53079 | 104 |
and 2: "norm z < norm x" |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
105 |
shows "summable (\<lambda>n. norm (f n * z ^ n))" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
106 |
proof - |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
107 |
from 2 have x_neq_0: "x \<noteq> 0" by clarsimp |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
108 |
from 1 have "(\<lambda>n. f n * x ^ n) ----> 0" |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
109 |
by (rule summable_LIMSEQ_zero) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
110 |
hence "convergent (\<lambda>n. f n * x ^ n)" |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
111 |
by (rule convergentI) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
112 |
hence "Cauchy (\<lambda>n. f n * x ^ n)" |
44726 | 113 |
by (rule convergent_Cauchy) |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
114 |
hence "Bseq (\<lambda>n. f n * x ^ n)" |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
115 |
by (rule Cauchy_Bseq) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
116 |
then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
117 |
by (simp add: Bseq_def, safe) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
118 |
have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le> |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
119 |
K * norm (z ^ n) * inverse (norm (x ^ n))" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
120 |
proof (intro exI allI impI) |
53079 | 121 |
fix n::nat |
122 |
assume "0 \<le> n" |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
123 |
have "norm (norm (f n * z ^ n)) * norm (x ^ n) = |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
124 |
norm (f n * x ^ n) * norm (z ^ n)" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
125 |
by (simp add: norm_mult abs_mult) |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
126 |
also have "\<dots> \<le> K * norm (z ^ n)" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
127 |
by (simp only: mult_right_mono 4 norm_ge_zero) |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
128 |
also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
129 |
by (simp add: x_neq_0) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
130 |
also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
131 |
by (simp only: mult.assoc) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
132 |
finally show "norm (norm (f n * z ^ n)) \<le> |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
133 |
K * norm (z ^ n) * inverse (norm (x ^ n))" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
134 |
by (simp add: mult_le_cancel_right x_neq_0) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
135 |
qed |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
136 |
moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
137 |
proof - |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
138 |
from 2 have "norm (norm (z * inverse x)) < 1" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
139 |
using x_neq_0 |
53599 | 140 |
by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric]) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
141 |
hence "summable (\<lambda>n. norm (z * inverse x) ^ n)" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
142 |
by (rule summable_geometric) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
143 |
hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
144 |
by (rule summable_mult) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
145 |
thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
146 |
using x_neq_0 |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
147 |
by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
148 |
power_inverse norm_power mult.assoc) |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
149 |
qed |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
150 |
ultimately show "summable (\<lambda>n. norm (f n * z ^ n))" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
151 |
by (rule summable_comparison_test) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
152 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
153 |
|
15229 | 154 |
lemma powser_inside: |
53599 | 155 |
fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}" |
53079 | 156 |
shows |
157 |
"summable (\<lambda>n. f n * (x ^ n)) \<Longrightarrow> norm z < norm x \<Longrightarrow> |
|
158 |
summable (\<lambda>n. f n * (z ^ n))" |
|
159 |
by (rule powser_insidea [THEN summable_norm_cancel]) |
|
160 |
||
161 |
lemma sum_split_even_odd: |
|
162 |
fixes f :: "nat \<Rightarrow> real" |
|
163 |
shows |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
164 |
"(\<Sum>i<2 * n. if even i then f i else g i) = |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
165 |
(\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
166 |
proof (induct n) |
53079 | 167 |
case 0 |
168 |
then show ?case by simp |
|
169 |
next |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
170 |
case (Suc n) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
171 |
have "(\<Sum>i<2 * Suc n. if even i then f i else g i) = |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
172 |
(\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))" |
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset
|
173 |
using Suc.hyps unfolding One_nat_def by auto |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
174 |
also have "\<dots> = (\<Sum>i<Suc n. f (2 * i)) + (\<Sum>i<Suc n. g (2 * i + 1))" |
53079 | 175 |
by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
176 |
finally show ?case . |
53079 | 177 |
qed |
178 |
||
179 |
lemma sums_if': |
|
180 |
fixes g :: "nat \<Rightarrow> real" |
|
181 |
assumes "g sums x" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
182 |
shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
183 |
unfolding sums_def |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
184 |
proof (rule LIMSEQ_I) |
53079 | 185 |
fix r :: real |
186 |
assume "0 < r" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
187 |
from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this] |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
188 |
obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g {..<n} - x) < r)" by blast |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
189 |
|
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
190 |
let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)" |
53079 | 191 |
{ |
192 |
fix m |
|
193 |
assume "m \<ge> 2 * no" |
|
194 |
hence "m div 2 \<ge> no" by auto |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
195 |
have sum_eq: "?SUM (2 * (m div 2)) = setsum g {..< m div 2}" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
196 |
using sum_split_even_odd by auto |
53079 | 197 |
hence "(norm (?SUM (2 * (m div 2)) - x) < r)" |
198 |
using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
199 |
moreover |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
200 |
have "?SUM (2 * (m div 2)) = ?SUM m" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
201 |
proof (cases "even m") |
53079 | 202 |
case True |
58710
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58709
diff
changeset
|
203 |
then show ?thesis by (auto simp add: even_two_times_div_two) |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
204 |
next |
53079 | 205 |
case False |
58834 | 206 |
then have eq: "Suc (2 * (m div 2)) = m" by simp |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
207 |
hence "even (2 * (m div 2))" using `odd m` by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
208 |
have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq .. |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
209 |
also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
210 |
finally show ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
211 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
212 |
ultimately have "(norm (?SUM m - x) < r)" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
213 |
} |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
214 |
thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
215 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
216 |
|
53079 | 217 |
lemma sums_if: |
218 |
fixes g :: "nat \<Rightarrow> real" |
|
219 |
assumes "g sums x" and "f sums y" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
220 |
shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
221 |
proof - |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
222 |
let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)" |
53079 | 223 |
{ |
224 |
fix B T E |
|
225 |
have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)" |
|
226 |
by (cases B) auto |
|
227 |
} note if_sum = this |
|
228 |
have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" |
|
229 |
using sums_if'[OF `g sums x`] . |
|
41970 | 230 |
{ |
41550 | 231 |
have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
232 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
233 |
have "?s sums y" using sums_if'[OF `f sums y`] . |
41970 | 234 |
from this[unfolded sums_def, THEN LIMSEQ_Suc] |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
235 |
have "(\<lambda> n. if even n then f (n div 2) else 0) sums y" |
57418 | 236 |
by (simp add: lessThan_Suc_eq_insert_0 image_iff setsum.reindex if_eq sums_def cong del: if_cong) |
53079 | 237 |
} |
238 |
from sums_add[OF g_sums this] show ?thesis unfolding if_sum . |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
239 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
240 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
241 |
subsection {* Alternating series test / Leibniz formula *} |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
242 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
243 |
lemma sums_alternating_upper_lower: |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
244 |
fixes a :: "nat \<Rightarrow> real" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
245 |
assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0" |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
246 |
shows "\<exists>l. ((\<forall>n. (\<Sum>i<2*n. (- 1)^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i<2*n. (- 1)^i*a i) ----> l) \<and> |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
247 |
((\<forall>n. l \<le> (\<Sum>i<2*n + 1. (- 1)^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. (- 1)^i*a i) ----> l)" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
248 |
(is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)") |
53079 | 249 |
proof (rule nested_sequence_unique) |
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset
|
250 |
have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
251 |
|
53079 | 252 |
show "\<forall>n. ?f n \<le> ?f (Suc n)" |
253 |
proof |
|
254 |
fix n |
|
255 |
show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto |
|
256 |
qed |
|
257 |
show "\<forall>n. ?g (Suc n) \<le> ?g n" |
|
258 |
proof |
|
259 |
fix n |
|
260 |
show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"] |
|
261 |
unfolding One_nat_def by auto |
|
262 |
qed |
|
263 |
show "\<forall>n. ?f n \<le> ?g n" |
|
264 |
proof |
|
265 |
fix n |
|
266 |
show "?f n \<le> ?g n" using fg_diff a_pos |
|
267 |
unfolding One_nat_def by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
268 |
qed |
53079 | 269 |
show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff |
270 |
proof (rule LIMSEQ_I) |
|
271 |
fix r :: real |
|
272 |
assume "0 < r" |
|
273 |
with `a ----> 0`[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" |
|
274 |
by auto |
|
275 |
hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto |
|
276 |
thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto |
|
277 |
qed |
|
41970 | 278 |
qed |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
279 |
|
53079 | 280 |
lemma summable_Leibniz': |
281 |
fixes a :: "nat \<Rightarrow> real" |
|
282 |
assumes a_zero: "a ----> 0" |
|
283 |
and a_pos: "\<And> n. 0 \<le> a n" |
|
284 |
and a_monotone: "\<And> n. a (Suc n) \<le> a n" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
285 |
shows summable: "summable (\<lambda> n. (-1)^n * a n)" |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
286 |
and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
287 |
and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
288 |
and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i<2*n+1. (-1)^i*a i)" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
289 |
and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
290 |
proof - |
53079 | 291 |
let ?S = "\<lambda>n. (-1)^n * a n" |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
292 |
let ?P = "\<lambda>n. \<Sum>i<n. ?S i" |
53079 | 293 |
let ?f = "\<lambda>n. ?P (2 * n)" |
294 |
let ?g = "\<lambda>n. ?P (2 * n + 1)" |
|
295 |
obtain l :: real |
|
296 |
where below_l: "\<forall> n. ?f n \<le> l" |
|
297 |
and "?f ----> l" |
|
298 |
and above_l: "\<forall> n. l \<le> ?g n" |
|
299 |
and "?g ----> l" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
300 |
using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast |
41970 | 301 |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
302 |
let ?Sa = "\<lambda>m. \<Sum>n<m. ?S n" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
303 |
have "?Sa ----> l" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
304 |
proof (rule LIMSEQ_I) |
53079 | 305 |
fix r :: real |
306 |
assume "0 < r" |
|
41970 | 307 |
with `?f ----> l`[THEN LIMSEQ_D] |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
308 |
obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
309 |
|
41970 | 310 |
from `0 < r` `?g ----> l`[THEN LIMSEQ_D] |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
311 |
obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
312 |
|
53079 | 313 |
{ |
314 |
fix n :: nat |
|
315 |
assume "n \<ge> (max (2 * f_no) (2 * g_no))" |
|
316 |
hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
317 |
have "norm (?Sa n - l) < r" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
318 |
proof (cases "even n") |
53079 | 319 |
case True |
58710
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58709
diff
changeset
|
320 |
then have n_eq: "2 * (n div 2) = n" by (simp add: even_two_times_div_two) |
53079 | 321 |
with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" |
322 |
by auto |
|
323 |
from f[OF this] show ?thesis |
|
324 |
unfolding n_eq atLeastLessThanSuc_atLeastAtMost . |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
325 |
next |
53079 | 326 |
case False |
327 |
hence "even (n - 1)" by simp |
|
58710
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58709
diff
changeset
|
328 |
then have n_eq: "2 * ((n - 1) div 2) = n - 1" |
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58709
diff
changeset
|
329 |
by (simp add: even_two_times_div_two) |
53079 | 330 |
hence range_eq: "n - 1 + 1 = n" |
331 |
using odd_pos[OF False] by auto |
|
332 |
||
333 |
from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" |
|
334 |
by auto |
|
335 |
from g[OF this] show ?thesis |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
336 |
unfolding n_eq range_eq . |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
337 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
338 |
} |
53079 | 339 |
thus "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
340 |
qed |
53079 | 341 |
hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l" |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
342 |
unfolding sums_def . |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
343 |
thus "summable ?S" using summable_def by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
344 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
345 |
have "l = suminf ?S" using sums_unique[OF sums_l] . |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
346 |
|
53079 | 347 |
fix n |
348 |
show "suminf ?S \<le> ?g n" |
|
349 |
unfolding sums_unique[OF sums_l, symmetric] using above_l by auto |
|
350 |
show "?f n \<le> suminf ?S" |
|
351 |
unfolding sums_unique[OF sums_l, symmetric] using below_l by auto |
|
352 |
show "?g ----> suminf ?S" |
|
353 |
using `?g ----> l` `l = suminf ?S` by auto |
|
354 |
show "?f ----> suminf ?S" |
|
355 |
using `?f ----> l` `l = suminf ?S` by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
356 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
357 |
|
53079 | 358 |
theorem summable_Leibniz: |
359 |
fixes a :: "nat \<Rightarrow> real" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
360 |
assumes a_zero: "a ----> 0" and "monoseq a" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
361 |
shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable") |
53079 | 362 |
and "0 < a 0 \<longrightarrow> |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
363 |
(\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n. (- 1)^i * a i .. \<Sum>i<2*n+1. (- 1)^i * a i})" (is "?pos") |
53079 | 364 |
and "a 0 < 0 \<longrightarrow> |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
365 |
(\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n+1. (- 1)^i * a i .. \<Sum>i<2*n. (- 1)^i * a i})" (is "?neg") |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
366 |
and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) ----> (\<Sum>i. (- 1)^i*a i)" (is "?f") |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
367 |
and "(\<lambda>n. \<Sum>i<2*n+1. (- 1)^i*a i) ----> (\<Sum>i. (- 1)^i*a i)" (is "?g") |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
368 |
proof - |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
369 |
have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
370 |
proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)") |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
371 |
case True |
53079 | 372 |
hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" |
373 |
by auto |
|
374 |
{ |
|
375 |
fix n |
|
376 |
have "a (Suc n) \<le> a n" |
|
377 |
using ord[where n="Suc n" and m=n] by auto |
|
378 |
} note mono = this |
|
379 |
note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
380 |
from leibniz[OF mono] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
381 |
show ?thesis using `0 \<le> a 0` by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
382 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
383 |
let ?a = "\<lambda> n. - a n" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
384 |
case False |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
385 |
with monoseq_le[OF `monoseq a` `a ----> 0`] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
386 |
have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto |
53079 | 387 |
hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" |
388 |
by auto |
|
389 |
{ |
|
390 |
fix n |
|
391 |
have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] |
|
392 |
by auto |
|
393 |
} note monotone = this |
|
394 |
note leibniz = |
|
395 |
summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", |
|
396 |
OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone] |
|
397 |
have "summable (\<lambda> n. (-1)^n * ?a n)" |
|
398 |
using leibniz(1) by auto |
|
399 |
then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" |
|
400 |
unfolding summable_def by auto |
|
401 |
from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l" |
|
402 |
by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
403 |
hence ?summable unfolding summable_def by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
404 |
moreover |
53079 | 405 |
have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>" |
406 |
unfolding minus_diff_minus by auto |
|
41970 | 407 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
408 |
from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus] |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
409 |
have move_minus: "(\<Sum>n. - ((- 1) ^ n * a n)) = - (\<Sum>n. (- 1) ^ n * a n)" |
53079 | 410 |
by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
411 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
412 |
have ?pos using `0 \<le> ?a 0` by auto |
53079 | 413 |
moreover have ?neg |
414 |
using leibniz(2,4) |
|
415 |
unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le |
|
416 |
by auto |
|
417 |
moreover have ?f and ?g |
|
418 |
using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel] |
|
419 |
by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
420 |
ultimately show ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
421 |
qed |
54576 | 422 |
then show ?summable and ?pos and ?neg and ?f and ?g |
54573 | 423 |
by safe |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
424 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
425 |
|
29164 | 426 |
subsection {* Term-by-Term Differentiability of Power Series *} |
23043 | 427 |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
428 |
definition diffs :: "(nat \<Rightarrow> 'a::ring_1) \<Rightarrow> nat \<Rightarrow> 'a" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
429 |
where "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))" |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
430 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
431 |
text{*Lemma about distributing negation over it*} |
53079 | 432 |
lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)" |
433 |
by (simp add: diffs_def) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
434 |
|
29163 | 435 |
lemma sums_Suc_imp: |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
436 |
"(f::nat \<Rightarrow> 'a::real_normed_vector) 0 = 0 \<Longrightarrow> (\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
437 |
using sums_Suc_iff[of f] by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
438 |
|
15229 | 439 |
lemma diffs_equiv: |
41970 | 440 |
fixes x :: "'a::{real_normed_vector, ring_1}" |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
441 |
shows "summable (\<lambda>n. diffs c n * x^n) \<Longrightarrow> |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
442 |
(\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)" |
53079 | 443 |
unfolding diffs_def |
54573 | 444 |
by (simp add: summable_sums sums_Suc_imp) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
445 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
446 |
lemma lemma_termdiff1: |
31017 | 447 |
fixes z :: "'a :: {monoid_mult,comm_ring}" shows |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
448 |
"(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) = |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
449 |
(\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))" |
53079 | 450 |
by (auto simp add: algebra_simps power_add [symmetric]) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
451 |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
452 |
lemma sumr_diff_mult_const2: |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
453 |
"setsum f {..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i<n. f i - r)" |
53079 | 454 |
by (simp add: setsum_subtractf) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
455 |
|
15229 | 456 |
lemma lemma_termdiff2: |
31017 | 457 |
fixes h :: "'a :: {field}" |
53079 | 458 |
assumes h: "h \<noteq> 0" |
459 |
shows |
|
460 |
"((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) = |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
461 |
h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p. |
53079 | 462 |
(z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs") |
463 |
apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h) |
|
464 |
apply (simp add: right_diff_distrib diff_divide_distrib h) |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
465 |
apply (simp add: mult.assoc [symmetric]) |
53079 | 466 |
apply (cases "n", simp) |
467 |
apply (simp add: lemma_realpow_diff_sumr2 h |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
468 |
right_diff_distrib [symmetric] mult.assoc |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
469 |
del: power_Suc setsum_lessThan_Suc of_nat_Suc) |
53079 | 470 |
apply (subst lemma_realpow_rev_sumr) |
471 |
apply (subst sumr_diff_mult_const2) |
|
472 |
apply simp |
|
473 |
apply (simp only: lemma_termdiff1 setsum_right_distrib) |
|
57418 | 474 |
apply (rule setsum.cong [OF refl]) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
475 |
apply (simp add: less_iff_Suc_add) |
53079 | 476 |
apply (clarify) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
477 |
apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 ac_simps |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
478 |
del: setsum_lessThan_Suc power_Suc) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
479 |
apply (subst mult.assoc [symmetric], subst power_add [symmetric]) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
480 |
apply (simp add: ac_simps) |
53079 | 481 |
done |
20860 | 482 |
|
483 |
lemma real_setsum_nat_ivl_bounded2: |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
484 |
fixes K :: "'a::linordered_semidom" |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
485 |
assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K" |
53079 | 486 |
and K: "0 \<le> K" |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
487 |
shows "setsum f {..<n-k} \<le> of_nat n * K" |
53079 | 488 |
apply (rule order_trans [OF setsum_mono]) |
489 |
apply (rule f, simp) |
|
490 |
apply (simp add: mult_right_mono K) |
|
491 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
492 |
|
15229 | 493 |
lemma lemma_termdiff3: |
31017 | 494 |
fixes h z :: "'a::{real_normed_field}" |
20860 | 495 |
assumes 1: "h \<noteq> 0" |
53079 | 496 |
and 2: "norm z \<le> K" |
497 |
and 3: "norm (z + h) \<le> K" |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
498 |
shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
499 |
\<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h" |
20860 | 500 |
proof - |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
501 |
have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) = |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
502 |
norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
503 |
(z + h) ^ q * z ^ (n - 2 - q)) * norm h" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
504 |
by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
505 |
also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
506 |
proof (rule mult_right_mono [OF _ norm_ge_zero]) |
53079 | 507 |
from norm_ge_zero 2 have K: "0 \<le> K" |
508 |
by (rule order_trans) |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
509 |
have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n" |
20860 | 510 |
apply (erule subst) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
511 |
apply (simp only: norm_mult norm_power power_add) |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
512 |
apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K) |
20860 | 513 |
done |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
514 |
show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
515 |
\<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))" |
20860 | 516 |
apply (intro |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
517 |
order_trans [OF norm_setsum] |
20860 | 518 |
real_setsum_nat_ivl_bounded2 |
519 |
mult_nonneg_nonneg |
|
47489 | 520 |
of_nat_0_le_iff |
20860 | 521 |
zero_le_power K) |
522 |
apply (rule le_Kn, simp) |
|
523 |
done |
|
524 |
qed |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
525 |
also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
526 |
by (simp only: mult.assoc) |
20860 | 527 |
finally show ?thesis . |
528 |
qed |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
529 |
|
20860 | 530 |
lemma lemma_termdiff4: |
56167 | 531 |
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" |
20860 | 532 |
assumes k: "0 < (k::real)" |
53079 | 533 |
and le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h" |
20860 | 534 |
shows "f -- 0 --> 0" |
56167 | 535 |
proof (rule tendsto_norm_zero_cancel) |
536 |
show "(\<lambda>h. norm (f h)) -- 0 --> 0" |
|
537 |
proof (rule real_tendsto_sandwich) |
|
538 |
show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)" |
|
20860 | 539 |
by simp |
56167 | 540 |
show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)" |
541 |
using k by (auto simp add: eventually_at dist_norm le) |
|
542 |
show "(\<lambda>h. 0) -- (0::'a) --> (0::real)" |
|
543 |
by (rule tendsto_const) |
|
544 |
have "(\<lambda>h. K * norm h) -- (0::'a) --> K * norm (0::'a)" |
|
545 |
by (intro tendsto_intros) |
|
546 |
then show "(\<lambda>h. K * norm h) -- (0::'a) --> 0" |
|
547 |
by simp |
|
20860 | 548 |
qed |
549 |
qed |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
550 |
|
15229 | 551 |
lemma lemma_termdiff5: |
56167 | 552 |
fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach" |
20860 | 553 |
assumes k: "0 < (k::real)" |
554 |
assumes f: "summable f" |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
555 |
assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h" |
20860 | 556 |
shows "(\<lambda>h. suminf (g h)) -- 0 --> 0" |
557 |
proof (rule lemma_termdiff4 [OF k]) |
|
53079 | 558 |
fix h::'a |
559 |
assume "h \<noteq> 0" and "norm h < k" |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
560 |
hence A: "\<forall>n. norm (g h n) \<le> f n * norm h" |
20860 | 561 |
by (simp add: le) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
562 |
hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h" |
20860 | 563 |
by simp |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
564 |
moreover from f have B: "summable (\<lambda>n. f n * norm h)" |
20860 | 565 |
by (rule summable_mult2) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
566 |
ultimately have C: "summable (\<lambda>n. norm (g h n))" |
20860 | 567 |
by (rule summable_comparison_test) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
568 |
hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
569 |
by (rule summable_norm) |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
570 |
also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)" |
56213 | 571 |
by (rule suminf_le) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
572 |
also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h" |
20860 | 573 |
by (rule suminf_mult2 [symmetric]) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
574 |
finally show "norm (suminf (g h)) \<le> suminf f * norm h" . |
20860 | 575 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
576 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
577 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
578 |
text{* FIXME: Long proofs*} |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
579 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
580 |
lemma termdiffs_aux: |
31017 | 581 |
fixes x :: "'a::{real_normed_field,banach}" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
582 |
assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)" |
53079 | 583 |
and 2: "norm x < norm K" |
20860 | 584 |
shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
585 |
- of_nat n * x ^ (n - Suc 0))) -- 0 --> 0" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
586 |
proof - |
20860 | 587 |
from dense [OF 2] |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
588 |
obtain r where r1: "norm x < r" and r2: "r < norm K" by fast |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
589 |
from norm_ge_zero r1 have r: "0 < r" |
20860 | 590 |
by (rule order_le_less_trans) |
591 |
hence r_neq_0: "r \<noteq> 0" by simp |
|
592 |
show ?thesis |
|
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
593 |
proof (rule lemma_termdiff5) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
594 |
show "0 < r - norm x" using r1 by simp |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
595 |
from r r2 have "norm (of_real r::'a) < norm K" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
596 |
by simp |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
597 |
with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))" |
20860 | 598 |
by (rule powser_insidea) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
599 |
hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
600 |
using r |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
601 |
by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc) |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
602 |
hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))" |
20860 | 603 |
by (rule diffs_equiv [THEN sums_summable]) |
53079 | 604 |
also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) = |
605 |
(\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))" |
|
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
606 |
apply (rule ext) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
607 |
apply (simp add: diffs_def) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
608 |
apply (case_tac n, simp_all add: r_neq_0) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
609 |
done |
41970 | 610 |
finally have "summable |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
611 |
(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))" |
20860 | 612 |
by (rule diffs_equiv [THEN sums_summable]) |
613 |
also have |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
614 |
"(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * |
20860 | 615 |
r ^ (n - Suc 0)) = |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
616 |
(\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
617 |
apply (rule ext) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
618 |
apply (case_tac "n", simp) |
55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
54576
diff
changeset
|
619 |
apply (rename_tac nat) |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
620 |
apply (case_tac "nat", simp) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
621 |
apply (simp add: r_neq_0) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
622 |
done |
53079 | 623 |
finally |
624 |
show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" . |
|
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
625 |
next |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
626 |
fix h::'a and n::nat |
20860 | 627 |
assume h: "h \<noteq> 0" |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
628 |
assume "norm h < r - norm x" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
629 |
hence "norm x + norm h < r" by simp |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
630 |
with norm_triangle_ineq have xh: "norm (x + h) < r" |
20860 | 631 |
by (rule order_le_less_trans) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
632 |
show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
633 |
\<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
634 |
apply (simp only: norm_mult mult.assoc) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
635 |
apply (rule mult_left_mono [OF _ norm_ge_zero]) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
636 |
apply (simp add: mult.assoc [symmetric]) |
54575 | 637 |
apply (metis h lemma_termdiff3 less_eq_real_def r1 xh) |
20860 | 638 |
done |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
639 |
qed |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
640 |
qed |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19765
diff
changeset
|
641 |
|
20860 | 642 |
lemma termdiffs: |
31017 | 643 |
fixes K x :: "'a::{real_normed_field,banach}" |
20860 | 644 |
assumes 1: "summable (\<lambda>n. c n * K ^ n)" |
54575 | 645 |
and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)" |
646 |
and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)" |
|
647 |
and 4: "norm x < norm K" |
|
20860 | 648 |
shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
649 |
unfolding DERIV_def |
29163 | 650 |
proof (rule LIM_zero_cancel) |
20860 | 651 |
show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h |
652 |
- suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0" |
|
653 |
proof (rule LIM_equal2) |
|
29163 | 654 |
show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq) |
20860 | 655 |
next |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
656 |
fix h :: 'a |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
657 |
assume "norm (h - 0) < norm K - norm x" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
658 |
hence "norm x + norm h < norm K" by simp |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
659 |
hence 5: "norm (x + h) < norm K" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
660 |
by (rule norm_triangle_ineq [THEN order_le_less_trans]) |
56167 | 661 |
have "summable (\<lambda>n. c n * x ^ n)" |
662 |
and "summable (\<lambda>n. c n * (x + h) ^ n)" |
|
663 |
and "summable (\<lambda>n. diffs c n * x ^ n)" |
|
664 |
using 1 2 4 5 by (auto elim: powser_inside) |
|
665 |
then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h - (\<Sum>n. diffs c n * x ^ n) = |
|
666 |
(\<Sum>n. (c n * (x + h) ^ n - c n * x ^ n) / h - of_nat n * c n * x ^ (n - Suc 0))" |
|
667 |
by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums) |
|
668 |
then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h - (\<Sum>n. diffs c n * x ^ n) = |
|
669 |
(\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))" |
|
54575 | 670 |
by (simp add: algebra_simps) |
20860 | 671 |
next |
53079 | 672 |
show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0" |
673 |
by (rule termdiffs_aux [OF 3 4]) |
|
20860 | 674 |
qed |
675 |
qed |
|
676 |
||
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
677 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
678 |
subsection {* Derivability of power series *} |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
679 |
|
53079 | 680 |
lemma DERIV_series': |
681 |
fixes f :: "real \<Rightarrow> nat \<Rightarrow> real" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
682 |
assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)" |
53079 | 683 |
and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}" |
684 |
and "summable (f' x0)" |
|
685 |
and "summable L" |
|
686 |
and L_def: "\<And>n x y. \<lbrakk> x \<in> { a <..< b} ; y \<in> { a <..< b} \<rbrakk> \<Longrightarrow> \<bar>f x n - f y n\<bar> \<le> L n * \<bar>x - y\<bar>" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
687 |
shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
688 |
unfolding DERIV_def |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
689 |
proof (rule LIM_I) |
53079 | 690 |
fix r :: real |
691 |
assume "0 < r" hence "0 < r/3" by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
692 |
|
41970 | 693 |
obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
694 |
using suminf_exist_split[OF `0 < r/3` `summable L`] by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
695 |
|
41970 | 696 |
obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
697 |
using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
698 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
699 |
let ?N = "Suc (max N_L N_f')" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
700 |
have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
701 |
L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
702 |
|
53079 | 703 |
let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
704 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
705 |
let ?r = "r / (3 * real ?N)" |
56541 | 706 |
from `0 < r` have "0 < ?r" by simp |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
707 |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
708 |
let ?s = "\<lambda>n. SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
709 |
def S' \<equiv> "Min (?s ` {..< ?N })" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
710 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
711 |
have "0 < S'" unfolding S'_def |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
712 |
proof (rule iffD2[OF Min_gr_iff]) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
713 |
show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x" |
53079 | 714 |
proof |
715 |
fix x |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
716 |
assume "x \<in> ?s ` {..<?N}" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
717 |
then obtain n where "x = ?s n" and "n \<in> {..<?N}" |
53079 | 718 |
using image_iff[THEN iffD1] by blast |
41970 | 719 |
from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def] |
53079 | 720 |
obtain s where s_bound: "0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r)" |
721 |
by auto |
|
722 |
have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound) |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
723 |
thus "0 < x" unfolding `x = ?s n` . |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
724 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
725 |
qed auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
726 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
727 |
def S \<equiv> "min (min (x0 - a) (b - x0)) S'" |
53079 | 728 |
hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0" |
729 |
and "S \<le> S'" using x0_in_I and `0 < S'` |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
730 |
by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
731 |
|
53079 | 732 |
{ |
733 |
fix x |
|
734 |
assume "x \<noteq> 0" and "\<bar> x \<bar> < S" |
|
735 |
hence x_in_I: "x0 + x \<in> { a <..< b }" |
|
736 |
using S_a S_b by auto |
|
41970 | 737 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
738 |
note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
739 |
note div_smbl = summable_divide[OF diff_smbl] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
740 |
note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
741 |
note ign = summable_ignore_initial_segment[where k="?N"] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
742 |
note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
743 |
note div_shft_smbl = summable_divide[OF diff_shft_smbl] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
744 |
note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
745 |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
746 |
{ fix n |
41970 | 747 |
have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>" |
53079 | 748 |
using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] |
749 |
unfolding abs_divide . |
|
750 |
hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
751 |
using `x \<noteq> 0` by auto } |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
752 |
note 1 = this and 2 = summable_rabs_comparison_test[OF _ ign[OF `summable L`]] |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
753 |
then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" |
56213 | 754 |
by (metis (lifting) abs_idempotent order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF `summable L`]]]) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
755 |
then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") |
53079 | 756 |
using L_estimate by auto |
757 |
||
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
758 |
have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n \<bar>)" .. |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
759 |
also have "\<dots> < (\<Sum>n<?N. ?r)" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
760 |
proof (rule setsum_strict_mono) |
53079 | 761 |
fix n |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
762 |
assume "n \<in> {..< ?N}" |
53079 | 763 |
have "\<bar>x\<bar> < S" using `\<bar>x\<bar> < S` . |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
764 |
also have "S \<le> S'" using `S \<le> S'` . |
41970 | 765 |
also have "S' \<le> ?s n" unfolding S'_def |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
766 |
proof (rule Min_le_iff[THEN iffD2]) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
767 |
have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
768 |
using `n \<in> {..< ?N}` by auto |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
769 |
thus "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n" by blast |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
770 |
qed auto |
53079 | 771 |
finally have "\<bar>x\<bar> < ?s n" . |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
772 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
773 |
from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
774 |
have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" . |
53079 | 775 |
with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n` show "\<bar>?diff n x - f' x0 n\<bar> < ?r" |
776 |
by blast |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
777 |
qed auto |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
778 |
also have "\<dots> = of_nat (card {..<?N}) * ?r" |
53079 | 779 |
by (rule setsum_constant) |
780 |
also have "\<dots> = real ?N * ?r" |
|
781 |
unfolding real_eq_of_nat by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
782 |
also have "\<dots> = r/3" by auto |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
783 |
finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") . |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
784 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
785 |
from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]] |
53079 | 786 |
have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> = |
787 |
\<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>" |
|
788 |
unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] |
|
789 |
using suminf_divide[OF diff_smbl, symmetric] by auto |
|
790 |
also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>" |
|
791 |
unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] |
|
792 |
unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]] |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
793 |
apply (subst (5) add.commute) |
53079 | 794 |
by (rule abs_triangle_ineq) |
795 |
also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" |
|
796 |
using abs_triangle_ineq4 by auto |
|
41970 | 797 |
also have "\<dots> < r /3 + r/3 + r/3" |
36842 | 798 |
using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3` |
799 |
by (rule add_strict_mono [OF add_less_le_mono]) |
|
53079 | 800 |
finally have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
801 |
by auto |
53079 | 802 |
} |
803 |
thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow> |
|
804 |
norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" |
|
805 |
using `0 < S` unfolding real_norm_def diff_0_right by blast |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
806 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
807 |
|
53079 | 808 |
lemma DERIV_power_series': |
809 |
fixes f :: "nat \<Rightarrow> real" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
810 |
assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)" |
53079 | 811 |
and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
812 |
shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
813 |
(is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))") |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
814 |
proof - |
53079 | 815 |
{ |
816 |
fix R' |
|
817 |
assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'" |
|
818 |
hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" |
|
819 |
by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
820 |
have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
821 |
proof (rule DERIV_series') |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
822 |
show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
823 |
proof - |
53079 | 824 |
have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" |
825 |
using `0 < R'` `0 < R` `R' < R` by auto |
|
826 |
hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" |
|
827 |
using `R' < R` by auto |
|
828 |
have "norm R' < norm ((R' + R) / 2)" |
|
829 |
using `0 < R'` `0 < R` `R' < R` by auto |
|
830 |
from powser_insidea[OF converges[OF in_Rball] this] show ?thesis |
|
831 |
by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
832 |
qed |
53079 | 833 |
{ |
834 |
fix n x y |
|
835 |
assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
836 |
show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
837 |
proof - |
53079 | 838 |
have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> = |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
839 |
(\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>" |
53079 | 840 |
unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult |
841 |
by auto |
|
41970 | 842 |
also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
843 |
proof (rule mult_left_mono) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
844 |
have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)" |
53079 | 845 |
by (rule setsum_abs) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
846 |
also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
847 |
proof (rule setsum_mono) |
53079 | 848 |
fix p |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
849 |
assume "p \<in> {..<Suc n}" |
53079 | 850 |
hence "p \<le> n" by auto |
851 |
{ |
|
852 |
fix n |
|
853 |
fix x :: real |
|
854 |
assume "x \<in> {-R'<..<R'}" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
855 |
hence "\<bar>x\<bar> \<le> R'" by auto |
53079 | 856 |
hence "\<bar>x^n\<bar> \<le> R'^n" |
857 |
unfolding power_abs by (rule power_mono, auto) |
|
858 |
} |
|
859 |
from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'` |
|
860 |
have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" |
|
861 |
unfolding abs_mult by auto |
|
862 |
thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" |
|
863 |
unfolding power_add[symmetric] using `p \<le> n` by auto |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
864 |
qed |
53079 | 865 |
also have "\<dots> = real (Suc n) * R' ^ n" |
866 |
unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
867 |
finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>" |
53079 | 868 |
unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] . |
869 |
show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" |
|
870 |
unfolding abs_mult[symmetric] by auto |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
871 |
qed |
53079 | 872 |
also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
873 |
unfolding abs_mult mult.assoc[symmetric] by algebra |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
874 |
finally show ?thesis . |
53079 | 875 |
qed |
876 |
} |
|
877 |
{ |
|
878 |
fix n |
|
879 |
show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)" |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
880 |
by (auto intro!: derivative_eq_intros simp del: power_Suc simp: real_of_nat_def) |
53079 | 881 |
} |
882 |
{ |
|
883 |
fix x |
|
884 |
assume "x \<in> {-R' <..< R'}" |
|
885 |
hence "R' \<in> {-R <..< R}" and "norm x < norm R'" |
|
886 |
using assms `R' < R` by auto |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
887 |
have "summable (\<lambda> n. f n * x^n)" |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
888 |
proof (rule summable_comparison_test, intro exI allI impI) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
889 |
fix n |
53079 | 890 |
have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" |
891 |
by (rule mult_left_mono) auto |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
892 |
show "norm (f n * x ^ n) \<le> norm (f n * real (Suc n) * x ^ n)" |
53079 | 893 |
unfolding real_norm_def abs_mult |
894 |
by (rule mult_right_mono) (auto simp add: le[unfolded mult_1_right]) |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
895 |
qed (rule powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
896 |
from this[THEN summable_mult2[where c=x], unfolded mult.assoc, unfolded mult.commute] |
53079 | 897 |
show "summable (?f x)" by auto |
898 |
} |
|
899 |
show "summable (?f' x0)" |
|
900 |
using converges[OF `x0 \<in> {-R <..< R}`] . |
|
901 |
show "x0 \<in> {-R' <..< R'}" |
|
902 |
using `x0 \<in> {-R' <..< R'}` . |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
903 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
904 |
} note for_subinterval = this |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
905 |
let ?R = "(R + \<bar>x0\<bar>) / 2" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
906 |
have "\<bar>x0\<bar> < ?R" using assms by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
907 |
hence "- ?R < x0" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
908 |
proof (cases "x0 < 0") |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
909 |
case True |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
910 |
hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
911 |
thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
912 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
913 |
case False |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
914 |
have "- ?R < 0" using assms by auto |
41970 | 915 |
also have "\<dots> \<le> x0" using False by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
916 |
finally show ?thesis . |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
917 |
qed |
53079 | 918 |
hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" |
919 |
using assms by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
920 |
from for_subinterval[OF this] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
921 |
show ?thesis . |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
922 |
qed |
29695 | 923 |
|
53079 | 924 |
|
29164 | 925 |
subsection {* Exponential Function *} |
23043 | 926 |
|
58656 | 927 |
definition exp :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}" |
53079 | 928 |
where "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))" |
23043 | 929 |
|
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
930 |
lemma summable_exp_generic: |
31017 | 931 |
fixes x :: "'a::{real_normed_algebra_1,banach}" |
25062 | 932 |
defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)" |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
933 |
shows "summable S" |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
934 |
proof - |
25062 | 935 |
have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)" |
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents:
30082
diff
changeset
|
936 |
unfolding S_def by (simp del: mult_Suc) |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
937 |
obtain r :: real where r0: "0 < r" and r1: "r < 1" |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
938 |
using dense [OF zero_less_one] by fast |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
939 |
obtain N :: nat where N: "norm x < real N * r" |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
940 |
using reals_Archimedean3 [OF r0] by fast |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
941 |
from r1 show ?thesis |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
942 |
proof (rule summable_ratio_test [rule_format]) |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
943 |
fix n :: nat |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
944 |
assume n: "N \<le> n" |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
945 |
have "norm x \<le> real N * r" |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
946 |
using N by (rule order_less_imp_le) |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
947 |
also have "real N * r \<le> real (Suc n) * r" |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
948 |
using r0 n by (simp add: mult_right_mono) |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
949 |
finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)" |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
950 |
using norm_ge_zero by (rule mult_right_mono) |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
951 |
hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)" |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
952 |
by (rule order_trans [OF norm_mult_ineq]) |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
953 |
hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
954 |
by (simp add: pos_divide_le_eq ac_simps) |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
955 |
thus "norm (S (Suc n)) \<le> r * norm (S n)" |
35216 | 956 |
by (simp add: S_Suc inverse_eq_divide) |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
957 |
qed |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
958 |
qed |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
959 |
|
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
960 |
lemma summable_norm_exp: |
31017 | 961 |
fixes x :: "'a::{real_normed_algebra_1,banach}" |
25062 | 962 |
shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))" |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
963 |
proof (rule summable_norm_comparison_test [OF exI, rule_format]) |
25062 | 964 |
show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))" |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
965 |
by (rule summable_exp_generic) |
53079 | 966 |
fix n |
967 |
show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)" |
|
35216 | 968 |
by (simp add: norm_power_ineq) |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
969 |
qed |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
970 |
|
53079 | 971 |
lemma summable_exp: "summable (\<lambda>n. inverse (real (fact n)) * x ^ n)" |
972 |
using summable_exp_generic [where x=x] by simp |
|
23043 | 973 |
|
25062 | 974 |
lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x" |
53079 | 975 |
unfolding exp_def by (rule summable_exp_generic [THEN summable_sums]) |
23043 | 976 |
|
977 |
||
41970 | 978 |
lemma exp_fdiffs: |
53079 | 979 |
"diffs (\<lambda>n. inverse(real (fact n))) = (\<lambda>n. inverse(real (fact n)))" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
980 |
by (simp add: diffs_def mult.assoc [symmetric] real_of_nat_def of_nat_mult |
53079 | 981 |
del: mult_Suc of_nat_Suc) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
982 |
|
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
983 |
lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))" |
53079 | 984 |
by (simp add: diffs_def) |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
985 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
986 |
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)" |
53079 | 987 |
unfolding exp_def scaleR_conv_of_real |
988 |
apply (rule DERIV_cong) |
|
989 |
apply (rule termdiffs [where K="of_real (1 + norm x)"]) |
|
990 |
apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs) |
|
991 |
apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+ |
|
992 |
apply (simp del: of_real_add) |
|
993 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
994 |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
995 |
declare DERIV_exp[THEN DERIV_chain2, derivative_intros] |
51527 | 996 |
|
58656 | 997 |
lemma norm_exp: "norm (exp x) \<le> exp (norm x)" |
998 |
proof - |
|
999 |
from summable_norm[OF summable_norm_exp, of x] |
|
1000 |
have "norm (exp x) \<le> (\<Sum>n. inverse (real (fact n)) * norm (x ^ n))" |
|
1001 |
by (simp add: exp_def) |
|
1002 |
also have "\<dots> \<le> exp (norm x)" |
|
1003 |
using summable_exp_generic[of "norm x"] summable_norm_exp[of x] |
|
1004 |
by (auto simp: exp_def intro!: suminf_le norm_power_ineq) |
|
1005 |
finally show ?thesis . |
|
1006 |
qed |
|
1007 |
||
1008 |
lemma isCont_exp: |
|
1009 |
fixes x::"'a::{real_normed_field,banach}" |
|
1010 |
shows "isCont exp x" |
|
44311 | 1011 |
by (rule DERIV_exp [THEN DERIV_isCont]) |
1012 |
||
58656 | 1013 |
lemma isCont_exp' [simp]: |
1014 |
fixes f::"_ \<Rightarrow>'a::{real_normed_field,banach}" |
|
1015 |
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a" |
|
44311 | 1016 |
by (rule isCont_o2 [OF _ isCont_exp]) |
1017 |
||
1018 |
lemma tendsto_exp [tendsto_intros]: |
|
58656 | 1019 |
fixes f::"_ \<Rightarrow>'a::{real_normed_field,banach}" |
1020 |
shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F" |
|
44311 | 1021 |
by (rule isCont_tendsto_compose [OF isCont_exp]) |
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
1022 |
|
53079 | 1023 |
lemma continuous_exp [continuous_intros]: |
58656 | 1024 |
fixes f::"_ \<Rightarrow>'a::{real_normed_field,banach}" |
1025 |
shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))" |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1026 |
unfolding continuous_def by (rule tendsto_exp) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1027 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
1028 |
lemma continuous_on_exp [continuous_intros]: |
58656 | 1029 |
fixes f::"_ \<Rightarrow>'a::{real_normed_field,banach}" |
1030 |
shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))" |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1031 |
unfolding continuous_on_def by (auto intro: tendsto_exp) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1032 |
|
53079 | 1033 |
|
29167 | 1034 |
subsubsection {* Properties of the Exponential Function *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1035 |
|
23278 | 1036 |
lemma powser_zero: |
31017 | 1037 |
fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}" |
23278 | 1038 |
shows "(\<Sum>n. f n * 0 ^ n) = f 0" |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1039 |
proof - |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
1040 |
have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)" |
56213 | 1041 |
by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left) |
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset
|
1042 |
thus ?thesis unfolding One_nat_def by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1043 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1044 |
|
23278 | 1045 |
lemma exp_zero [simp]: "exp 0 = 1" |
53079 | 1046 |
unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero) |
23278 | 1047 |
|
58656 | 1048 |
lemma exp_series_add_commuting: |
1049 |
fixes x y :: "'a::{real_normed_algebra_1, banach}" |
|
25062 | 1050 |
defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)" |
58656 | 1051 |
assumes comm: "x * y = y * x" |
56213 | 1052 |
shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))" |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1053 |
proof (induct n) |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1054 |
case 0 |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1055 |
show ?case |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1056 |
unfolding S_def by simp |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1057 |
next |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1058 |
case (Suc n) |
25062 | 1059 |
have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)" |
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents:
30082
diff
changeset
|
1060 |
unfolding S_def by (simp del: mult_Suc) |
25062 | 1061 |
hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)" |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1062 |
by simp |
58656 | 1063 |
have S_comm: "\<And>n. S x n * y = y * S x n" |
1064 |
by (simp add: power_commuting_commutes comm S_def) |
|
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1065 |
|
25062 | 1066 |
have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n" |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1067 |
by (simp only: times_S) |
56213 | 1068 |
also have "\<dots> = (x + y) * (\<Sum>i\<le>n. S x i * S y (n-i))" |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1069 |
by (simp only: Suc) |
56213 | 1070 |
also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n-i)) |
1071 |
+ y * (\<Sum>i\<le>n. S x i * S y (n-i))" |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
47489
diff
changeset
|
1072 |
by (rule distrib_right) |
58656 | 1073 |
also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i)) |
1074 |
+ (\<Sum>i\<le>n. S x i * y * S y (n-i))" |
|
1075 |
by (simp add: setsum_right_distrib ac_simps S_comm) |
|
1076 |
also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i)) |
|
56213 | 1077 |
+ (\<Sum>i\<le>n. S x i * (y * S y (n-i)))" |
58656 | 1078 |
by (simp add: ac_simps) |
56213 | 1079 |
also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) |
1080 |
+ (\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))" |
|
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1081 |
by (simp add: times_S Suc_diff_le) |
56213 | 1082 |
also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) = |
1083 |
(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))" |
|
1084 |
by (subst setsum_atMost_Suc_shift) simp |
|
1085 |
also have "(\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) = |
|
1086 |
(\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))" |
|
1087 |
by simp |
|
1088 |
also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) + |
|
1089 |
(\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) = |
|
1090 |
(\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))" |
|
57418 | 1091 |
by (simp only: setsum.distrib [symmetric] scaleR_left_distrib [symmetric] |
56213 | 1092 |
real_of_nat_add [symmetric]) simp |
1093 |
also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n-i))" |
|
23127 | 1094 |
by (simp only: scaleR_right.setsum) |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1095 |
finally show |
56213 | 1096 |
"S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))" |
35216 | 1097 |
by (simp del: setsum_cl_ivl_Suc) |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1098 |
qed |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1099 |
|
58656 | 1100 |
lemma exp_add_commuting: "x * y = y * x \<Longrightarrow> exp (x + y) = exp x * exp y" |
53079 | 1101 |
unfolding exp_def |
58656 | 1102 |
by (simp only: Cauchy_product summable_norm_exp exp_series_add_commuting) |
1103 |
||
1104 |
lemma exp_add: |
|
1105 |
fixes x y::"'a::{real_normed_field,banach}" |
|
1106 |
shows "exp (x + y) = exp x * exp y" |
|
1107 |
by (rule exp_add_commuting) (simp add: ac_simps) |
|
1108 |
||
1109 |
lemmas mult_exp_exp = exp_add [symmetric] |
|
29170 | 1110 |
|
23241 | 1111 |
lemma exp_of_real: "exp (of_real x) = of_real (exp x)" |
53079 | 1112 |
unfolding exp_def |
1113 |
apply (subst suminf_of_real) |
|
1114 |
apply (rule summable_exp_generic) |
|
1115 |
apply (simp add: scaleR_conv_of_real) |
|
1116 |
done |
|
23241 | 1117 |
|
29170 | 1118 |
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0" |
1119 |
proof |
|
58656 | 1120 |
have "exp x * exp (- x) = 1" by (simp add: exp_add_commuting[symmetric]) |
29170 | 1121 |
also assume "exp x = 0" |
1122 |
finally show "False" by simp |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1123 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1124 |
|
58656 | 1125 |
lemma exp_minus_inverse: |
1126 |
shows "exp x * exp (- x) = 1" |
|
1127 |
by (simp add: exp_add_commuting[symmetric]) |
|
1128 |
||
1129 |
lemma exp_minus: |
|
1130 |
fixes x :: "'a::{real_normed_field, banach}" |
|
1131 |
shows "exp (- x) = inverse (exp x)" |
|
1132 |
by (intro inverse_unique [symmetric] exp_minus_inverse) |
|
1133 |
||
1134 |
lemma exp_diff: |
|
1135 |
fixes x :: "'a::{real_normed_field, banach}" |
|
1136 |
shows "exp (x - y) = exp x / exp y" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
1137 |
using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1138 |
|
29167 | 1139 |
|
1140 |
subsubsection {* Properties of the Exponential Function on Reals *} |
|
1141 |
||
29170 | 1142 |
text {* Comparisons of @{term "exp x"} with zero. *} |
29167 | 1143 |
|
1144 |
text{*Proof: because every exponential can be seen as a square.*} |
|
1145 |
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)" |
|
1146 |
proof - |
|
1147 |
have "0 \<le> exp (x/2) * exp (x/2)" by simp |
|
1148 |
thus ?thesis by (simp add: exp_add [symmetric]) |
|
1149 |
qed |
|
1150 |
||
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1151 |
lemma exp_gt_zero [simp]: "0 < exp (x::real)" |
53079 | 1152 |
by (simp add: order_less_le) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1153 |
|
29170 | 1154 |
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0" |
53079 | 1155 |
by (simp add: not_less) |
29170 | 1156 |
|
1157 |
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0" |
|
53079 | 1158 |
by (simp add: not_le) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1159 |
|
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1160 |
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x" |
53079 | 1161 |
by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1162 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1163 |
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
1164 |
by (induct n) (auto simp add: real_of_nat_Suc distrib_left exp_add mult.commute) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1165 |
|
29170 | 1166 |
text {* Strict monotonicity of exponential. *} |
1167 |
||
54575 | 1168 |
lemma exp_ge_add_one_self_aux: |
1169 |
assumes "0 \<le> (x::real)" shows "1+x \<le> exp(x)" |
|
1170 |
using order_le_imp_less_or_eq [OF assms] |
|
1171 |
proof |
|
1172 |
assume "0 < x" |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
1173 |
have "1+x \<le> (\<Sum>n<2. inverse (real (fact n)) * x ^ n)" |
54575 | 1174 |
by (auto simp add: numeral_2_eq_2) |
1175 |
also have "... \<le> (\<Sum>n. inverse (real (fact n)) * x ^ n)" |
|
56213 | 1176 |
apply (rule setsum_le_suminf [OF summable_exp]) |
54575 | 1177 |
using `0 < x` |
1178 |
apply (auto simp add: zero_le_mult_iff) |
|
1179 |
done |
|
1180 |
finally show "1+x \<le> exp x" |
|
1181 |
by (simp add: exp_def) |
|
1182 |
next |
|
1183 |
assume "0 = x" |
|
1184 |
then show "1 + x \<le> exp x" |
|
1185 |
by auto |
|
1186 |
qed |
|
29170 | 1187 |
|
1188 |
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x" |
|
1189 |
proof - |
|
1190 |
assume x: "0 < x" |
|
1191 |
hence "1 < 1 + x" by simp |
|
1192 |
also from x have "1 + x \<le> exp x" |
|
1193 |
by (simp add: exp_ge_add_one_self_aux) |
|
1194 |
finally show ?thesis . |
|
1195 |
qed |
|
1196 |
||
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1197 |
lemma exp_less_mono: |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1198 |
fixes x y :: real |
53079 | 1199 |
assumes "x < y" |
1200 |
shows "exp x < exp y" |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1201 |
proof - |
29165
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset
|
1202 |
from `x < y` have "0 < y - x" by simp |
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset
|
1203 |
hence "1 < exp (y - x)" by (rule exp_gt_one) |
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset
|
1204 |
hence "1 < exp y / exp x" by (simp only: exp_diff) |
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset
|
1205 |
thus "exp x < exp y" by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1206 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1207 |
|
53079 | 1208 |
lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y" |
54575 | 1209 |
unfolding linorder_not_le [symmetric] |
1210 |
by (auto simp add: order_le_less exp_less_mono) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1211 |
|
29170 | 1212 |
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y" |
53079 | 1213 |
by (auto intro: exp_less_mono exp_less_cancel) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1214 |
|
29170 | 1215 |
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y" |
53079 | 1216 |
by (auto simp add: linorder_not_less [symmetric]) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1217 |
|
29170 | 1218 |
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y" |
53079 | 1219 |
by (simp add: order_eq_iff) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1220 |
|
29170 | 1221 |
text {* Comparisons of @{term "exp x"} with one. *} |
1222 |
||
1223 |
lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x" |
|
1224 |
using exp_less_cancel_iff [where x=0 and y=x] by simp |
|
1225 |
||
1226 |
lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0" |
|
1227 |
using exp_less_cancel_iff [where x=x and y=0] by simp |
|
1228 |
||
1229 |
lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x" |
|
1230 |
using exp_le_cancel_iff [where x=0 and y=x] by simp |
|
1231 |
||
1232 |
lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0" |
|
1233 |
using exp_le_cancel_iff [where x=x and y=0] by simp |
|
1234 |
||
1235 |
lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0" |
|
1236 |
using exp_inj_iff [where x=x and y=0] by simp |
|
1237 |
||
53079 | 1238 |
lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y" |
44755 | 1239 |
proof (rule IVT) |
1240 |
assume "1 \<le> y" |
|
1241 |
hence "0 \<le> y - 1" by simp |
|
1242 |
hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux) |
|
1243 |
thus "y \<le> exp (y - 1)" by simp |
|
1244 |
qed (simp_all add: le_diff_eq) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1245 |
|
53079 | 1246 |
lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y" |
44755 | 1247 |
proof (rule linorder_le_cases [of 1 y]) |
53079 | 1248 |
assume "1 \<le> y" |
1249 |
thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total) |
|
44755 | 1250 |
next |
1251 |
assume "0 < y" and "y \<le> 1" |
|
1252 |
hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff) |
|
1253 |
then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total) |
|
1254 |
hence "exp (- x) = y" by (simp add: exp_minus) |
|
1255 |
thus "\<exists>x. exp x = y" .. |
|
1256 |
qed |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1257 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1258 |
|
29164 | 1259 |
subsection {* Natural Logarithm *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1260 |
|
53079 | 1261 |
definition ln :: "real \<Rightarrow> real" |
1262 |
where "ln x = (THE u. exp u = x)" |
|
23043 | 1263 |
|
1264 |
lemma ln_exp [simp]: "ln (exp x) = x" |
|
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1265 |
by (simp add: ln_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1266 |
|
22654
c2b6b5a9e136
new simp rule exp_ln; new standard proof of DERIV_exp_ln_one; changed imports
huffman
parents:
22653
diff
changeset
|
1267 |
lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1268 |
by (auto dest: exp_total) |
22654
c2b6b5a9e136
new simp rule exp_ln; new standard proof of DERIV_exp_ln_one; changed imports
huffman
parents:
22653
diff
changeset
|
1269 |
|
29171 | 1270 |
lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1271 |
by (metis exp_gt_zero exp_ln) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1272 |
|
29171 | 1273 |
lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1274 |
by (erule subst, rule ln_exp) |
29171 | 1275 |
|
1276 |
lemma ln_one [simp]: "ln 1 = 0" |
|
53079 | 1277 |
by (rule ln_unique) simp |
1278 |
||
1279 |
lemma ln_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y" |
|
1280 |
by (rule ln_unique) (simp add: exp_add) |
|
29171 | 1281 |
|
1282 |
lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x" |
|
53079 | 1283 |
by (rule ln_unique) (simp add: exp_minus) |
1284 |
||
1285 |
lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y" |
|
1286 |
by (rule ln_unique) (simp add: exp_diff) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1287 |
|
29171 | 1288 |
lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x" |
53079 | 1289 |
by (rule ln_unique) (simp add: exp_real_of_nat_mult) |
1290 |
||
1291 |
lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y" |
|
1292 |
by (subst exp_less_cancel_iff [symmetric]) simp |
|
1293 |
||
1294 |
lemma ln_le_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y" |
|
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1295 |
by (simp add: linorder_not_less [symmetric]) |
29171 | 1296 |
|
53079 | 1297 |
lemma ln_inj_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1298 |
by (simp add: order_eq_iff) |
29171 | 1299 |
|
1300 |
lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x" |
|
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1301 |
apply (rule exp_le_cancel_iff [THEN iffD1]) |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1302 |
apply (simp add: exp_ge_add_one_self_aux) |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1303 |
done |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1304 |
|
29171 | 1305 |
lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1306 |
by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1307 |
|
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1308 |
lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x" |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1309 |
using ln_le_cancel_iff [of 1 x] by simp |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1310 |
|
53079 | 1311 |
lemma ln_ge_zero_imp_ge_one: "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1312 |
using ln_le_cancel_iff [of 1 x] by simp |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1313 |
|
53079 | 1314 |
lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1315 |
using ln_le_cancel_iff [of 1 x] by simp |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1316 |
|
53079 | 1317 |
lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1318 |
using ln_less_cancel_iff [of x 1] by simp |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1319 |
|
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1320 |
lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x" |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1321 |
using ln_less_cancel_iff [of 1 x] by simp |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1322 |
|
53079 | 1323 |
lemma ln_gt_zero_imp_gt_one: "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1324 |
using ln_less_cancel_iff [of 1 x] by simp |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1325 |
|
53079 | 1326 |
lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1327 |
using ln_less_cancel_iff [of 1 x] by simp |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1328 |
|
53079 | 1329 |
lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1330 |
using ln_inj_iff [of x 1] by simp |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1331 |
|
53079 | 1332 |
lemma ln_less_zero: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1333 |
by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1334 |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1335 |
lemma ln_neg_is_const: "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1336 |
by (auto simp add: ln_def intro!: arg_cong[where f=The]) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1337 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1338 |
lemma isCont_ln: assumes "x \<noteq> 0" shows "isCont ln x" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1339 |
proof cases |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1340 |
assume "0 < x" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1341 |
moreover then have "isCont ln (exp (ln x))" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1342 |
by (intro isCont_inv_fun[where d="\<bar>x\<bar>" and f=exp]) auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1343 |
ultimately show ?thesis |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1344 |
by simp |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1345 |
next |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1346 |
assume "\<not> 0 < x" with `x \<noteq> 0` show "isCont ln x" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1347 |
unfolding isCont_def |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1348 |
by (subst filterlim_cong[OF _ refl, of _ "nhds (ln 0)" _ "\<lambda>_. ln 0"]) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1349 |
(auto simp: ln_neg_is_const not_less eventually_at dist_real_def |
58729
e8ecc79aee43
add tendsto_const and tendsto_ident_at as simp and intro rules
hoelzl
parents:
58710
diff
changeset
|
1350 |
intro!: exI[of _ "\<bar>x\<bar>"]) |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1351 |
qed |
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
1352 |
|
45915 | 1353 |
lemma tendsto_ln [tendsto_intros]: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1354 |
"(f ---> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F" |
45915 | 1355 |
by (rule isCont_tendsto_compose [OF isCont_ln]) |
1356 |
||
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1357 |
lemma continuous_ln: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1358 |
"continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. ln (f x))" |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1359 |
unfolding continuous_def by (rule tendsto_ln) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1360 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1361 |
lemma isCont_ln' [continuous_intros]: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1362 |
"continuous (at x) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x))" |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1363 |
unfolding continuous_at by (rule tendsto_ln) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1364 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1365 |
lemma continuous_within_ln [continuous_intros]: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1366 |
"continuous (at x within s) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x))" |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1367 |
unfolding continuous_within by (rule tendsto_ln) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1368 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
1369 |
lemma continuous_on_ln [continuous_intros]: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1370 |
"continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. f x \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x))" |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1371 |
unfolding continuous_on_def by (auto intro: tendsto_ln) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
1372 |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
1373 |
lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1374 |
apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"]) |
54576 | 1375 |
apply (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln) |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1376 |
done |
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
1377 |
|
53079 | 1378 |
lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x" |
33667 | 1379 |
by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse) |
1380 |
||
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1381 |
declare DERIV_ln_divide[THEN DERIV_chain2, derivative_intros] |
51527 | 1382 |
|
53079 | 1383 |
lemma ln_series: |
1384 |
assumes "0 < x" and "x < 2" |
|
1385 |
shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" |
|
1386 |
(is "ln x = suminf (?f (x - 1))") |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1387 |
proof - |
53079 | 1388 |
let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1389 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1390 |
have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1391 |
proof (rule DERIV_isconst3[where x=x]) |
53079 | 1392 |
fix x :: real |
1393 |
assume "x \<in> {0 <..< 2}" |
|
1394 |
hence "0 < x" and "x < 2" by auto |
|
1395 |
have "norm (1 - x) < 1" |
|
1396 |
using `0 < x` and `x < 2` by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1397 |
have "1 / x = 1 / (1 - (1 - x))" by auto |
53079 | 1398 |
also have "\<dots> = (\<Sum> n. (1 - x)^n)" |
1399 |
using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique) |
|
1400 |
also have "\<dots> = suminf (?f' x)" |
|
1401 |
unfolding power_mult_distrib[symmetric] |
|
1402 |
by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto) |
|
1403 |
finally have "DERIV ln x :> suminf (?f' x)" |
|
1404 |
using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1405 |
moreover |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1406 |
have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto |
53079 | 1407 |
have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> |
1408 |
(\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1409 |
proof (rule DERIV_power_series') |
53079 | 1410 |
show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" |
1411 |
using `0 < x` `x < 2` by auto |
|
1412 |
fix x :: real |
|
1413 |
assume "x \<in> {- 1<..<1}" |
|
1414 |
hence "norm (-x) < 1" by auto |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
1415 |
show "summable (\<lambda>n. (- 1) ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)" |
53079 | 1416 |
unfolding One_nat_def |
1417 |
by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`]) |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1418 |
qed |
53079 | 1419 |
hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" |
1420 |
unfolding One_nat_def by auto |
|
1421 |
hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1422 |
unfolding DERIV_def repos . |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1423 |
ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1424 |
by (rule DERIV_diff) |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1425 |
thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1426 |
qed (auto simp add: assms) |
44289 | 1427 |
thus ?thesis by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1428 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1429 |
|
50326 | 1430 |
lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) * (x ^ (n+2)))" |
1431 |
proof - |
|
53079 | 1432 |
have "exp x = suminf (\<lambda>n. inverse(fact n) * (x ^ n))" |
50326 | 1433 |
by (simp add: exp_def) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
1434 |
also from summable_exp have "... = (\<Sum> n. inverse(fact(n+2)) * (x ^ (n+2))) + |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
1435 |
(\<Sum> n::nat<2. inverse(fact n) * (x ^ n))" (is "_ = _ + ?a") |
50326 | 1436 |
by (rule suminf_split_initial_segment) |
1437 |
also have "?a = 1 + x" |
|
1438 |
by (simp add: numeral_2_eq_2) |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
1439 |
finally show ?thesis |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
1440 |
by simp |
50326 | 1441 |
qed |
1442 |
||
53079 | 1443 |
lemma exp_bound: "0 <= (x::real) \<Longrightarrow> x <= 1 \<Longrightarrow> exp x <= 1 + x + x\<^sup>2" |
50326 | 1444 |
proof - |
1445 |
assume a: "0 <= x" |
|
1446 |
assume b: "x <= 1" |
|
53079 | 1447 |
{ |
1448 |
fix n :: nat |
|
50326 | 1449 |
have "2 * 2 ^ n \<le> fact (n + 2)" |
53079 | 1450 |
by (induct n) simp_all |
50326 | 1451 |
hence "real ((2::nat) * 2 ^ n) \<le> real (fact (n + 2))" |
1452 |
by (simp only: real_of_nat_le_iff) |
|
1453 |
hence "2 * 2 ^ n \<le> real (fact (n + 2))" |
|
1454 |
by simp |
|
1455 |
hence "inverse (fact (n + 2)) \<le> inverse (2 * 2 ^ n)" |
|
1456 |
by (rule le_imp_inverse_le) simp |
|
1457 |
hence "inverse (fact (n + 2)) \<le> 1/2 * (1/2)^n" |
|
53079 | 1458 |
by (simp add: power_inverse) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
1459 |
hence "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)" |
50326 | 1460 |
by (rule mult_mono) |
56536 | 1461 |
(rule mult_mono, simp_all add: power_le_one a b) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
1462 |
hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1463 |
unfolding power_add by (simp add: ac_simps del: fact_Suc) } |
50326 | 1464 |
note aux1 = this |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
1465 |
have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))" |
50326 | 1466 |
by (intro sums_mult geometric_sums, simp) |
53076 | 1467 |
hence aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2" |
50326 | 1468 |
by simp |
53079 | 1469 |
have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2" |
50326 | 1470 |
proof - |
53079 | 1471 |
have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= |
1472 |
suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))" |
|
56213 | 1473 |
apply (rule suminf_le) |
50326 | 1474 |
apply (rule allI, rule aux1) |
1475 |
apply (rule summable_exp [THEN summable_ignore_initial_segment]) |
|
1476 |
by (rule sums_summable, rule aux2) |
|
53076 | 1477 |
also have "... = x\<^sup>2" |
50326 | 1478 |
by (rule sums_unique [THEN sym], rule aux2) |
1479 |
finally show ?thesis . |
|
1480 |
qed |
|
1481 |
thus ?thesis unfolding exp_first_two_terms by auto |
|
1482 |
qed |
|
1483 |
||
53079 | 1484 |
lemma ln_one_minus_pos_upper_bound: "0 <= x \<Longrightarrow> x < 1 \<Longrightarrow> ln (1 - x) <= - x" |
50326 | 1485 |
proof - |
1486 |
assume a: "0 <= (x::real)" and b: "x < 1" |
|
53076 | 1487 |
have "(1 - x) * (1 + x + x\<^sup>2) = (1 - x^3)" |
50326 | 1488 |
by (simp add: algebra_simps power2_eq_square power3_eq_cube) |
1489 |
also have "... <= 1" |
|
1490 |
by (auto simp add: a) |
|
53076 | 1491 |
finally have "(1 - x) * (1 + x + x\<^sup>2) <= 1" . |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
1492 |
moreover have c: "0 < 1 + x + x\<^sup>2" |
50326 | 1493 |
by (simp add: add_pos_nonneg a) |
53076 | 1494 |
ultimately have "1 - x <= 1 / (1 + x + x\<^sup>2)" |
50326 | 1495 |
by (elim mult_imp_le_div_pos) |
1496 |
also have "... <= 1 / exp x" |
|
54576 | 1497 |
by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs |
1498 |
real_sqrt_pow2_iff real_sqrt_power) |
|
50326 | 1499 |
also have "... = exp (-x)" |
1500 |
by (auto simp add: exp_minus divide_inverse) |
|
1501 |
finally have "1 - x <= exp (- x)" . |
|
1502 |
also have "1 - x = exp (ln (1 - x))" |
|
54576 | 1503 |
by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq) |
50326 | 1504 |
finally have "exp (ln (1 - x)) <= exp (- x)" . |
1505 |
thus ?thesis by (auto simp only: exp_le_cancel_iff) |
|
1506 |
qed |
|
1507 |
||
1508 |
lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x" |
|
1509 |
apply (case_tac "0 <= x") |
|
1510 |
apply (erule exp_ge_add_one_self_aux) |
|
1511 |
apply (case_tac "x <= -1") |
|
1512 |
apply (subgoal_tac "1 + x <= 0") |
|
1513 |
apply (erule order_trans) |
|
1514 |
apply simp |
|
1515 |
apply simp |
|
1516 |
apply (subgoal_tac "1 + x = exp(ln (1 + x))") |
|
1517 |
apply (erule ssubst) |
|
1518 |
apply (subst exp_le_cancel_iff) |
|
1519 |
apply (subgoal_tac "ln (1 - (- x)) <= - (- x)") |
|
1520 |
apply simp |
|
1521 |
apply (rule ln_one_minus_pos_upper_bound) |
|
1522 |
apply auto |
|
1523 |
done |
|
1524 |
||
53079 | 1525 |
lemma ln_one_plus_pos_lower_bound: "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> x - x\<^sup>2 <= ln (1 + x)" |
51527 | 1526 |
proof - |
1527 |
assume a: "0 <= x" and b: "x <= 1" |
|
53076 | 1528 |
have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)" |
51527 | 1529 |
by (rule exp_diff) |
53076 | 1530 |
also have "... <= (1 + x + x\<^sup>2) / exp (x \<^sup>2)" |
54576 | 1531 |
by (metis a b divide_right_mono exp_bound exp_ge_zero) |
53076 | 1532 |
also have "... <= (1 + x + x\<^sup>2) / (1 + x\<^sup>2)" |
56544 | 1533 |
by (simp add: a divide_left_mono add_pos_nonneg) |
51527 | 1534 |
also from a have "... <= 1 + x" |
1535 |
by (simp add: field_simps add_strict_increasing zero_le_mult_iff) |
|
53076 | 1536 |
finally have "exp (x - x\<^sup>2) <= 1 + x" . |
51527 | 1537 |
also have "... = exp (ln (1 + x))" |
1538 |
proof - |
|
1539 |
from a have "0 < 1 + x" by auto |
|
1540 |
thus ?thesis |
|
1541 |
by (auto simp only: exp_ln_iff [THEN sym]) |
|
1542 |
qed |
|
53076 | 1543 |
finally have "exp (x - x\<^sup>2) <= exp (ln (1 + x))" . |
54576 | 1544 |
thus ?thesis |
1545 |
by (metis exp_le_cancel_iff) |
|
51527 | 1546 |
qed |
1547 |
||
53079 | 1548 |
lemma ln_one_minus_pos_lower_bound: |
1549 |
"0 <= x \<Longrightarrow> x <= (1 / 2) \<Longrightarrow> - x - 2 * x\<^sup>2 <= ln (1 - x)" |
|
51527 | 1550 |
proof - |
1551 |
assume a: "0 <= x" and b: "x <= (1 / 2)" |
|
53079 | 1552 |
from b have c: "x < 1" by auto |
51527 | 1553 |
then have "ln (1 - x) = - ln (1 + x / (1 - x))" |
54576 | 1554 |
apply (subst ln_inverse [symmetric]) |
1555 |
apply (simp add: field_simps) |
|
1556 |
apply (rule arg_cong [where f=ln]) |
|
1557 |
apply (simp add: field_simps) |
|
1558 |
done |
|
51527 | 1559 |
also have "- (x / (1 - x)) <= ..." |
53079 | 1560 |
proof - |
51527 | 1561 |
have "ln (1 + x / (1 - x)) <= x / (1 - x)" |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1562 |
using a c by (intro ln_add_one_self_le_self) auto |
51527 | 1563 |
thus ?thesis |
1564 |
by auto |
|
1565 |
qed |
|
1566 |
also have "- (x / (1 - x)) = -x / (1 - x)" |
|
1567 |
by auto |
|
1568 |
finally have d: "- x / (1 - x) <= ln (1 - x)" . |
|
1569 |
have "0 < 1 - x" using a b by simp |
|
53076 | 1570 |
hence e: "-x - 2 * x\<^sup>2 <= - x / (1 - x)" |
51527 | 1571 |
using mult_right_le_one_le[of "x*x" "2*x"] a b |
53079 | 1572 |
by (simp add: field_simps power2_eq_square) |
53076 | 1573 |
from e d show "- x - 2 * x\<^sup>2 <= ln (1 - x)" |
51527 | 1574 |
by (rule order_trans) |
1575 |
qed |
|
1576 |
||
53079 | 1577 |
lemma ln_add_one_self_le_self2: "-1 < x \<Longrightarrow> ln(1 + x) <= x" |
51527 | 1578 |
apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp) |
1579 |
apply (subst ln_le_cancel_iff) |
|
1580 |
apply auto |
|
53079 | 1581 |
done |
51527 | 1582 |
|
1583 |
lemma abs_ln_one_plus_x_minus_x_bound_nonneg: |
|
53079 | 1584 |
"0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> abs(ln (1 + x) - x) <= x\<^sup>2" |
51527 | 1585 |
proof - |
1586 |
assume x: "0 <= x" |
|
1587 |
assume x1: "x <= 1" |
|
1588 |
from x have "ln (1 + x) <= x" |
|
1589 |
by (rule ln_add_one_self_le_self) |
|
53079 | 1590 |
then have "ln (1 + x) - x <= 0" |
51527 | 1591 |
by simp |
1592 |
then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)" |
|
1593 |
by (rule abs_of_nonpos) |
|
53079 | 1594 |
also have "... = x - ln (1 + x)" |
51527 | 1595 |
by simp |
53076 | 1596 |
also have "... <= x\<^sup>2" |
51527 | 1597 |
proof - |
53076 | 1598 |
from x x1 have "x - x\<^sup>2 <= ln (1 + x)" |
51527 | 1599 |
by (intro ln_one_plus_pos_lower_bound) |
1600 |
thus ?thesis |
|
1601 |
by simp |
|
1602 |
qed |
|
1603 |
finally show ?thesis . |
|
1604 |
qed |
|
1605 |
||
1606 |
lemma abs_ln_one_plus_x_minus_x_bound_nonpos: |
|
53079 | 1607 |
"-(1 / 2) <= x \<Longrightarrow> x <= 0 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2" |
51527 | 1608 |
proof - |
1609 |
assume a: "-(1 / 2) <= x" |
|
1610 |
assume b: "x <= 0" |
|
53079 | 1611 |
have "abs(ln (1 + x) - x) = x - ln(1 - (-x))" |
51527 | 1612 |
apply (subst abs_of_nonpos) |
1613 |
apply simp |
|
1614 |
apply (rule ln_add_one_self_le_self2) |
|
1615 |
using a apply auto |
|
1616 |
done |
|
53076 | 1617 |
also have "... <= 2 * x\<^sup>2" |
1618 |
apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 <= ln (1 - (-x))") |
|
51527 | 1619 |
apply (simp add: algebra_simps) |
1620 |
apply (rule ln_one_minus_pos_lower_bound) |
|
1621 |
using a b apply auto |
|
1622 |
done |
|
1623 |
finally show ?thesis . |
|
1624 |
qed |
|
1625 |
||
1626 |
lemma abs_ln_one_plus_x_minus_x_bound: |
|
53079 | 1627 |
"abs x <= 1 / 2 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2" |
51527 | 1628 |
apply (case_tac "0 <= x") |
1629 |
apply (rule order_trans) |
|
1630 |
apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg) |
|
1631 |
apply auto |
|
1632 |
apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos) |
|
1633 |
apply auto |
|
53079 | 1634 |
done |
1635 |
||
1636 |
lemma ln_x_over_x_mono: "exp 1 <= x \<Longrightarrow> x <= y \<Longrightarrow> (ln y / y) <= (ln x / x)" |
|
51527 | 1637 |
proof - |
1638 |
assume x: "exp 1 <= x" "x <= y" |
|
1639 |
moreover have "0 < exp (1::real)" by simp |
|
1640 |
ultimately have a: "0 < x" and b: "0 < y" |
|
1641 |
by (fast intro: less_le_trans order_trans)+ |
|
1642 |
have "x * ln y - x * ln x = x * (ln y - ln x)" |
|
1643 |
by (simp add: algebra_simps) |
|
1644 |
also have "... = x * ln(y / x)" |
|
1645 |
by (simp only: ln_div a b) |
|
1646 |
also have "y / x = (x + (y - x)) / x" |
|
1647 |
by simp |
|
1648 |
also have "... = 1 + (y - x) / x" |
|
1649 |
using x a by (simp add: field_simps) |
|
1650 |
also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)" |
|
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1651 |
using x a |
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1652 |
by (intro mult_left_mono ln_add_one_self_le_self) simp_all |
51527 | 1653 |
also have "... = y - x" using a by simp |
1654 |
also have "... = (y - x) * ln (exp 1)" by simp |
|
1655 |
also have "... <= (y - x) * ln x" |
|
1656 |
apply (rule mult_left_mono) |
|
1657 |
apply (subst ln_le_cancel_iff) |
|
1658 |
apply fact |
|
1659 |
apply (rule a) |
|
1660 |
apply (rule x) |
|
1661 |
using x apply simp |
|
1662 |
done |
|
1663 |
also have "... = y * ln x - x * ln x" |
|
1664 |
by (rule left_diff_distrib) |
|
1665 |
finally have "x * ln y <= y * ln x" |
|
1666 |
by arith |
|
1667 |
then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps) |
|
1668 |
also have "... = y * (ln x / x)" by simp |
|
1669 |
finally show ?thesis using b by (simp add: field_simps) |
|
1670 |
qed |
|
1671 |
||
53079 | 1672 |
lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1" |
51527 | 1673 |
using exp_ge_add_one_self[of "ln x"] by simp |
1674 |
||
1675 |
lemma ln_eq_minus_one: |
|
53079 | 1676 |
assumes "0 < x" "ln x = x - 1" |
1677 |
shows "x = 1" |
|
51527 | 1678 |
proof - |
53079 | 1679 |
let ?l = "\<lambda>y. ln y - y + 1" |
51527 | 1680 |
have D: "\<And>x. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1681 |
by (auto intro!: derivative_eq_intros) |
51527 | 1682 |
|
1683 |
show ?thesis |
|
1684 |
proof (cases rule: linorder_cases) |
|
1685 |
assume "x < 1" |
|
1686 |
from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast |
|
1687 |
from `x < a` have "?l x < ?l a" |
|
1688 |
proof (rule DERIV_pos_imp_increasing, safe) |
|
53079 | 1689 |
fix y |
1690 |
assume "x \<le> y" "y \<le> a" |
|
51527 | 1691 |
with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y" |
1692 |
by (auto simp: field_simps) |
|
1693 |
with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z" |
|
1694 |
by auto |
|
1695 |
qed |
|
1696 |
also have "\<dots> \<le> 0" |
|
1697 |
using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps) |
|
1698 |
finally show "x = 1" using assms by auto |
|
1699 |
next |
|
1700 |
assume "1 < x" |
|
53079 | 1701 |
from dense[OF this] obtain a where "1 < a" "a < x" by blast |
51527 | 1702 |
from `a < x` have "?l x < ?l a" |
1703 |
proof (rule DERIV_neg_imp_decreasing, safe) |
|
53079 | 1704 |
fix y |
1705 |
assume "a \<le> y" "y \<le> x" |
|
51527 | 1706 |
with `1 < a` have "1 / y - 1 < 0" "0 < y" |
1707 |
by (auto simp: field_simps) |
|
1708 |
with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0" |
|
1709 |
by blast |
|
1710 |
qed |
|
1711 |
also have "\<dots> \<le> 0" |
|
1712 |
using ln_le_minus_one `1 < a` by (auto simp: field_simps) |
|
1713 |
finally show "x = 1" using assms by auto |
|
53079 | 1714 |
next |
1715 |
assume "x = 1" |
|
1716 |
then show ?thesis by simp |
|
1717 |
qed |
|
51527 | 1718 |
qed |
1719 |
||
50326 | 1720 |
lemma exp_at_bot: "(exp ---> (0::real)) at_bot" |
1721 |
unfolding tendsto_Zfun_iff |
|
1722 |
proof (rule ZfunI, simp add: eventually_at_bot_dense) |
|
1723 |
fix r :: real assume "0 < r" |
|
53079 | 1724 |
{ |
1725 |
fix x |
|
1726 |
assume "x < ln r" |
|
50326 | 1727 |
then have "exp x < exp (ln r)" |
1728 |
by simp |
|
1729 |
with `0 < r` have "exp x < r" |
|
53079 | 1730 |
by simp |
1731 |
} |
|
50326 | 1732 |
then show "\<exists>k. \<forall>n<k. exp n < r" by auto |
1733 |
qed |
|
1734 |
||
1735 |
lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top" |
|
50346
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
1736 |
by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"]) |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
1737 |
(auto intro: eventually_gt_at_top) |
50326 | 1738 |
|
1739 |
lemma ln_at_0: "LIM x at_right 0. ln x :> at_bot" |
|
50346
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
1740 |
by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"]) |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51527
diff
changeset
|
1741 |
(auto simp: eventually_at_filter) |
50326 | 1742 |
|
1743 |
lemma ln_at_top: "LIM x at_top. ln x :> at_top" |
|
50346
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
1744 |
by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"]) |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
1745 |
(auto intro: eventually_gt_at_top) |
50326 | 1746 |
|
50347 | 1747 |
lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) ---> (0::real)) at_top" |
1748 |
proof (induct k) |
|
53079 | 1749 |
case 0 |
50347 | 1750 |
show "((\<lambda>x. x ^ 0 / exp x) ---> (0::real)) at_top" |
1751 |
by (simp add: inverse_eq_divide[symmetric]) |
|
1752 |
(metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono |
|
1753 |
at_top_le_at_infinity order_refl) |
|
1754 |
next |
|
1755 |
case (Suc k) |
|
1756 |
show ?case |
|
1757 |
proof (rule lhospital_at_top_at_top) |
|
1758 |
show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top" |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1759 |
by eventually_elim (intro derivative_eq_intros, auto) |
50347 | 1760 |
show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1761 |
by eventually_elim auto |
50347 | 1762 |
show "eventually (\<lambda>x. exp x \<noteq> 0) at_top" |
1763 |
by auto |
|
1764 |
from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"] |
|
1765 |
show "((\<lambda>x. real (Suc k) * x ^ k / exp x) ---> 0) at_top" |
|
1766 |
by simp |
|
1767 |
qed (rule exp_at_top) |
|
1768 |
qed |
|
1769 |
||
51527 | 1770 |
|
53079 | 1771 |
definition powr :: "[real,real] => real" (infixr "powr" 80) |
1772 |
-- {*exponentation with real exponent*} |
|
1773 |
where "x powr a = exp(a * ln x)" |
|
1774 |
||
1775 |
definition log :: "[real,real] => real" |
|
1776 |
-- {*logarithm of @{term x} to base @{term a}*} |
|
1777 |
where "log a x = ln x / ln a" |
|
51527 | 1778 |
|
1779 |
||
1780 |
lemma tendsto_log [tendsto_intros]: |
|
1781 |
"\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a; a \<noteq> 1; 0 < b\<rbrakk> \<Longrightarrow> ((\<lambda>x. log (f x) (g x)) ---> log a b) F" |
|
1782 |
unfolding log_def by (intro tendsto_intros) auto |
|
1783 |
||
1784 |
lemma continuous_log: |
|
53079 | 1785 |
assumes "continuous F f" |
1786 |
and "continuous F g" |
|
1787 |
and "0 < f (Lim F (\<lambda>x. x))" |
|
1788 |
and "f (Lim F (\<lambda>x. x)) \<noteq> 1" |
|
1789 |
and "0 < g (Lim F (\<lambda>x. x))" |
|
51527 | 1790 |
shows "continuous F (\<lambda>x. log (f x) (g x))" |
1791 |
using assms unfolding continuous_def by (rule tendsto_log) |
|
1792 |
||
1793 |
lemma continuous_at_within_log[continuous_intros]: |
|
53079 | 1794 |
assumes "continuous (at a within s) f" |
1795 |
and "continuous (at a within s) g" |
|
1796 |
and "0 < f a" |
|
1797 |
and "f a \<noteq> 1" |
|
1798 |
and "0 < g a" |
|
51527 | 1799 |
shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))" |
1800 |
using assms unfolding continuous_within by (rule tendsto_log) |
|
1801 |
||
1802 |
lemma isCont_log[continuous_intros, simp]: |
|
1803 |
assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a" |
|
1804 |
shows "isCont (\<lambda>x. log (f x) (g x)) a" |
|
1805 |
using assms unfolding continuous_at by (rule tendsto_log) |
|
1806 |
||
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
1807 |
lemma continuous_on_log[continuous_intros]: |
53079 | 1808 |
assumes "continuous_on s f" "continuous_on s g" |
1809 |
and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x" |
|
51527 | 1810 |
shows "continuous_on s (\<lambda>x. log (f x) (g x))" |
1811 |
using assms unfolding continuous_on_def by (fast intro: tendsto_log) |
|
1812 |
||
1813 |
lemma powr_one_eq_one [simp]: "1 powr a = 1" |
|
53079 | 1814 |
by (simp add: powr_def) |
51527 | 1815 |
|
1816 |
lemma powr_zero_eq_one [simp]: "x powr 0 = 1" |
|
53079 | 1817 |
by (simp add: powr_def) |
51527 | 1818 |
|
1819 |
lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)" |
|
53079 | 1820 |
by (simp add: powr_def) |
51527 | 1821 |
declare powr_one_gt_zero_iff [THEN iffD2, simp] |
1822 |
||
53079 | 1823 |
lemma powr_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)" |
1824 |
by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left) |
|
51527 | 1825 |
|
1826 |
lemma powr_gt_zero [simp]: "0 < x powr a" |
|
53079 | 1827 |
by (simp add: powr_def) |
51527 | 1828 |
|
1829 |
lemma powr_ge_pzero [simp]: "0 <= x powr y" |
|
53079 | 1830 |
by (rule order_less_imp_le, rule powr_gt_zero) |
51527 | 1831 |
|
1832 |
lemma powr_not_zero [simp]: "x powr a \<noteq> 0" |
|
53079 | 1833 |
by (simp add: powr_def) |
1834 |
||
1835 |
lemma powr_divide: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)" |
|
1836 |
apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult) |
|
1837 |
apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse) |
|
1838 |
done |
|
51527 | 1839 |
|
1840 |
lemma powr_divide2: "x powr a / x powr b = x powr (a - b)" |
|
1841 |
apply (simp add: powr_def) |
|
1842 |
apply (subst exp_diff [THEN sym]) |
|
1843 |
apply (simp add: left_diff_distrib) |
|
53079 | 1844 |
done |
51527 | 1845 |
|
1846 |
lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)" |
|
53079 | 1847 |
by (simp add: powr_def exp_add [symmetric] distrib_right) |
1848 |
||
1849 |
lemma powr_mult_base: "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)" |
|
1850 |
using assms by (auto simp: powr_add) |
|
51527 | 1851 |
|
1852 |
lemma powr_powr: "(x powr a) powr b = x powr (a * b)" |
|
53079 | 1853 |
by (simp add: powr_def) |
51527 | 1854 |
|
1855 |
lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a" |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
1856 |
by (simp add: powr_powr mult.commute) |
51527 | 1857 |
|
1858 |
lemma powr_minus: "x powr (-a) = inverse (x powr a)" |
|
53079 | 1859 |
by (simp add: powr_def exp_minus [symmetric]) |
51527 | 1860 |
|
1861 |
lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)" |
|
53079 | 1862 |
by (simp add: divide_inverse powr_minus) |
1863 |
||
1864 |
lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b" |
|
1865 |
by (simp add: powr_def) |
|
1866 |
||
1867 |
lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b" |
|
1868 |
by (simp add: powr_def) |
|
1869 |
||
1870 |
lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a < x powr b) = (a < b)" |
|
1871 |
by (blast intro: powr_less_cancel powr_less_mono) |
|
1872 |
||
1873 |
lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a \<le> x powr b) = (a \<le> b)" |
|
1874 |
by (simp add: linorder_not_less [symmetric]) |
|
51527 | 1875 |
|
1876 |
lemma log_ln: "ln x = log (exp(1)) x" |
|
53079 | 1877 |
by (simp add: log_def) |
1878 |
||
1879 |
lemma DERIV_log: |
|
1880 |
assumes "x > 0" |
|
1881 |
shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)" |
|
51527 | 1882 |
proof - |
1883 |
def lb \<equiv> "1 / ln b" |
|
1884 |
moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x" |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1885 |
using `x > 0` by (auto intro!: derivative_eq_intros) |
51527 | 1886 |
ultimately show ?thesis |
1887 |
by (simp add: log_def) |
|
1888 |
qed |
|
1889 |
||
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1890 |
lemmas DERIV_log[THEN DERIV_chain2, derivative_intros] |
51527 | 1891 |
|
53079 | 1892 |
lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x" |
1893 |
by (simp add: powr_def log_def) |
|
1894 |
||
1895 |
lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y" |
|
1896 |
by (simp add: log_def powr_def) |
|
1897 |
||
1898 |
lemma log_mult: |
|
1899 |
"0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> |
|
1900 |
log a (x * y) = log a x + log a y" |
|
1901 |
by (simp add: log_def ln_mult divide_inverse distrib_right) |
|
1902 |
||
1903 |
lemma log_eq_div_ln_mult_log: |
|
1904 |
"0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> |
|
1905 |
log a x = (ln b/ln a) * log b x" |
|
1906 |
by (simp add: log_def divide_inverse) |
|
51527 | 1907 |
|
1908 |
text{*Base 10 logarithms*} |
|
53079 | 1909 |
lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x" |
1910 |
by (simp add: log_def) |
|
1911 |
||
1912 |
lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x" |
|
1913 |
by (simp add: log_def) |
|
51527 | 1914 |
|
1915 |
lemma log_one [simp]: "log a 1 = 0" |
|
53079 | 1916 |
by (simp add: log_def) |
51527 | 1917 |
|
1918 |
lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1" |
|
53079 | 1919 |
by (simp add: log_def) |
1920 |
||
1921 |
lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x" |
|
1922 |
apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1]) |
|
1923 |
apply (simp add: log_mult [symmetric]) |
|
1924 |
done |
|
1925 |
||
1926 |
lemma log_divide: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a (x/y) = log a x - log a y" |
|
1927 |
by (simp add: log_mult divide_inverse log_inverse) |
|
51527 | 1928 |
|
1929 |
lemma log_less_cancel_iff [simp]: |
|
53079 | 1930 |
"1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y" |
1931 |
apply safe |
|
1932 |
apply (rule_tac [2] powr_less_cancel) |
|
1933 |
apply (drule_tac a = "log a x" in powr_less_mono, auto) |
|
1934 |
done |
|
1935 |
||
1936 |
lemma log_inj: |
|
1937 |
assumes "1 < b" |
|
1938 |
shows "inj_on (log b) {0 <..}" |
|
51527 | 1939 |
proof (rule inj_onI, simp) |
53079 | 1940 |
fix x y |
1941 |
assume pos: "0 < x" "0 < y" and *: "log b x = log b y" |
|
51527 | 1942 |
show "x = y" |
1943 |
proof (cases rule: linorder_cases) |
|
53079 | 1944 |
assume "x = y" |
1945 |
then show ?thesis by simp |
|
1946 |
next |
|
51527 | 1947 |
assume "x < y" hence "log b x < log b y" |
1948 |
using log_less_cancel_iff[OF `1 < b`] pos by simp |
|
53079 | 1949 |
then show ?thesis using * by simp |
51527 | 1950 |
next |
1951 |
assume "y < x" hence "log b y < log b x" |
|
1952 |
using log_less_cancel_iff[OF `1 < b`] pos by simp |
|
53079 | 1953 |
then show ?thesis using * by simp |
1954 |
qed |
|
51527 | 1955 |
qed |
1956 |
||
1957 |
lemma log_le_cancel_iff [simp]: |
|
53079 | 1958 |
"1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (log a x \<le> log a y) = (x \<le> y)" |
1959 |
by (simp add: linorder_not_less [symmetric]) |
|
51527 | 1960 |
|
1961 |
lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x" |
|
1962 |
using log_less_cancel_iff[of a 1 x] by simp |
|
1963 |
||
1964 |
lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x" |
|
1965 |
using log_le_cancel_iff[of a 1 x] by simp |
|
1966 |
||
1967 |
lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1" |
|
1968 |
using log_less_cancel_iff[of a x 1] by simp |
|
1969 |
||
1970 |
lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1" |
|
1971 |
using log_le_cancel_iff[of a x 1] by simp |
|
1972 |
||
1973 |
lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x" |
|
1974 |
using log_less_cancel_iff[of a a x] by simp |
|
1975 |
||
1976 |
lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x" |
|
1977 |
using log_le_cancel_iff[of a a x] by simp |
|
1978 |
||
1979 |
lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a" |
|
1980 |
using log_less_cancel_iff[of a x a] by simp |
|
1981 |
||
1982 |
lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a" |
|
1983 |
using log_le_cancel_iff[of a x a] by simp |
|
1984 |
||
1985 |
lemma powr_realpow: "0 < x ==> x powr (real n) = x^n" |
|
53079 | 1986 |
apply (induct n) |
1987 |
apply simp |
|
51527 | 1988 |
apply (subgoal_tac "real(Suc n) = real n + 1") |
1989 |
apply (erule ssubst) |
|
1990 |
apply (subst powr_add, simp, simp) |
|
53079 | 1991 |
done |
51527 | 1992 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
1993 |
lemma powr_realpow_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
1994 |
unfolding real_of_nat_numeral [symmetric] by (rule powr_realpow) |
52139 | 1995 |
|
57180 | 1996 |
lemma powr2_sqrt[simp]: "0 < x \<Longrightarrow> sqrt x powr 2 = x" |
1997 |
by(simp add: powr_realpow_numeral) |
|
1998 |
||
51527 | 1999 |
lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))" |
2000 |
apply (case_tac "x = 0", simp, simp) |
|
2001 |
apply (rule powr_realpow [THEN sym], simp) |
|
53079 | 2002 |
done |
51527 | 2003 |
|
2004 |
lemma powr_int: |
|
2005 |
assumes "x > 0" |
|
2006 |
shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))" |
|
53079 | 2007 |
proof (cases "i < 0") |
2008 |
case True |
|
51527 | 2009 |
have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps) |
2010 |
show ?thesis using `i < 0` `x > 0` by (simp add: r field_simps powr_realpow[symmetric]) |
|
53079 | 2011 |
next |
2012 |
case False |
|
2013 |
then show ?thesis by (simp add: assms powr_realpow[symmetric]) |
|
2014 |
qed |
|
51527 | 2015 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2016 |
lemma powr_one: "0 < x \<Longrightarrow> x powr 1 = x" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2017 |
using powr_realpow [of x 1] by simp |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2018 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2019 |
lemma powr_numeral: "0 < x \<Longrightarrow> x powr numeral n = x ^ numeral n" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2020 |
by (fact powr_realpow_numeral) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2021 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2022 |
lemma powr_neg_one: "0 < x \<Longrightarrow> x powr - 1 = 1 / x" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2023 |
using powr_int [of x "- 1"] by simp |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2024 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2025 |
lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2026 |
using powr_int [of x "- numeral n"] by simp |
51527 | 2027 |
|
53079 | 2028 |
lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)" |
51527 | 2029 |
by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr) |
2030 |
||
56483 | 2031 |
lemma ln_powr: "ln (x powr y) = y * ln x" |
2032 |
by (simp add: powr_def) |
|
2033 |
||
56952 | 2034 |
lemma ln_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> ln (root n b) = ln b / n" |
2035 |
by(simp add: root_powr_inverse ln_powr) |
|
2036 |
||
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2037 |
lemma ln_sqrt: "0 < x \<Longrightarrow> ln (sqrt x) = ln x / 2" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2038 |
by (simp add: ln_powr powr_numeral ln_powr[symmetric] mult.commute) |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2039 |
|
56952 | 2040 |
lemma log_root: "\<lbrakk> n > 0; a > 0 \<rbrakk> \<Longrightarrow> log b (root n a) = log b a / n" |
2041 |
by(simp add: log_def ln_root) |
|
2042 |
||
56483 | 2043 |
lemma log_powr: "log b (x powr y) = y * log b x" |
2044 |
by (simp add: log_def ln_powr) |
|
2045 |
||
2046 |
lemma log_nat_power: "0 < x \<Longrightarrow> log b (x ^ n) = real n * log b x" |
|
2047 |
by (simp add: log_powr powr_realpow [symmetric]) |
|
2048 |
||
2049 |
lemma log_base_change: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log b x = log a x / log a b" |
|
2050 |
by (simp add: log_def) |
|
2051 |
||
2052 |
lemma log_base_pow: "0 < a \<Longrightarrow> log (a ^ n) x = log a x / n" |
|
2053 |
by (simp add: log_def ln_realpow) |
|
2054 |
||
2055 |
lemma log_base_powr: "log (a powr b) x = log a x / b" |
|
2056 |
by (simp add: log_def ln_powr) |
|
51527 | 2057 |
|
56952 | 2058 |
lemma log_base_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> log (root n b) x = n * (log b x)" |
2059 |
by(simp add: log_def ln_root) |
|
2060 |
||
51527 | 2061 |
lemma ln_bound: "1 <= x ==> ln x <= x" |
2062 |
apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1") |
|
2063 |
apply simp |
|
2064 |
apply (rule ln_add_one_self_le_self, simp) |
|
53079 | 2065 |
done |
51527 | 2066 |
|
2067 |
lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b" |
|
53079 | 2068 |
apply (cases "x = 1", simp) |
2069 |
apply (cases "a = b", simp) |
|
51527 | 2070 |
apply (rule order_less_imp_le) |
2071 |
apply (rule powr_less_mono, auto) |
|
53079 | 2072 |
done |
51527 | 2073 |
|
2074 |
lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a" |
|
2075 |
apply (subst powr_zero_eq_one [THEN sym]) |
|
2076 |
apply (rule powr_mono, assumption+) |
|
53079 | 2077 |
done |
2078 |
||
2079 |
lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a < y powr a" |
|
51527 | 2080 |
apply (unfold powr_def) |
2081 |
apply (rule exp_less_mono) |
|
2082 |
apply (rule mult_strict_left_mono) |
|
2083 |
apply (subst ln_less_cancel_iff, assumption) |
|
2084 |
apply (rule order_less_trans) |
|
2085 |
prefer 2 |
|
2086 |
apply assumption+ |
|
53079 | 2087 |
done |
2088 |
||
2089 |
lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a < x powr a" |
|
51527 | 2090 |
apply (unfold powr_def) |
2091 |
apply (rule exp_less_mono) |
|
2092 |
apply (rule mult_strict_left_mono_neg) |
|
2093 |
apply (subst ln_less_cancel_iff) |
|
2094 |
apply assumption |
|
2095 |
apply (rule order_less_trans) |
|
2096 |
prefer 2 |
|
2097 |
apply assumption+ |
|
53079 | 2098 |
done |
51527 | 2099 |
|
2100 |
lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a" |
|
2101 |
apply (case_tac "a = 0", simp) |
|
2102 |
apply (case_tac "x = y", simp) |
|
54575 | 2103 |
apply (metis less_eq_real_def powr_less_mono2) |
53079 | 2104 |
done |
2105 |
||
2106 |
lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y" |
|
51527 | 2107 |
unfolding powr_def exp_inj_iff by simp |
2108 |
||
2109 |
lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a" |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2110 |
by (metis less_eq_real_def ln_less_self mult_imp_le_div_pos ln_powr mult.commute |
54575 | 2111 |
order.strict_trans2 powr_gt_zero zero_less_one) |
51527 | 2112 |
|
2113 |
lemma ln_powr_bound2: |
|
2114 |
assumes "1 < x" and "0 < a" |
|
2115 |
shows "(ln x) powr a <= (a powr a) * x" |
|
2116 |
proof - |
|
2117 |
from assms have "ln x <= (x powr (1 / a)) / (1 / a)" |
|
54575 | 2118 |
by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff) |
51527 | 2119 |
also have "... = a * (x powr (1 / a))" |
2120 |
by simp |
|
2121 |
finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a" |
|
54575 | 2122 |
by (metis assms less_imp_le ln_gt_zero powr_mono2) |
51527 | 2123 |
also have "... = (a powr a) * ((x powr (1 / a)) powr a)" |
54575 | 2124 |
by (metis assms(2) powr_mult powr_gt_zero) |
51527 | 2125 |
also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)" |
2126 |
by (rule powr_powr) |
|
54575 | 2127 |
also have "... = x" using assms |
2128 |
by auto |
|
51527 | 2129 |
finally show ?thesis . |
2130 |
qed |
|
2131 |
||
2132 |
lemma tendsto_powr [tendsto_intros]: |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2133 |
"\<lbrakk>(f ---> a) F; (g ---> b) F; a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x powr g x) ---> a powr b) F" |
51527 | 2134 |
unfolding powr_def by (intro tendsto_intros) |
2135 |
||
2136 |
lemma continuous_powr: |
|
53079 | 2137 |
assumes "continuous F f" |
2138 |
and "continuous F g" |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2139 |
and "f (Lim F (\<lambda>x. x)) \<noteq> 0" |
51527 | 2140 |
shows "continuous F (\<lambda>x. (f x) powr (g x))" |
2141 |
using assms unfolding continuous_def by (rule tendsto_powr) |
|
2142 |
||
2143 |
lemma continuous_at_within_powr[continuous_intros]: |
|
53079 | 2144 |
assumes "continuous (at a within s) f" |
2145 |
and "continuous (at a within s) g" |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2146 |
and "f a \<noteq> 0" |
51527 | 2147 |
shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))" |
2148 |
using assms unfolding continuous_within by (rule tendsto_powr) |
|
2149 |
||
2150 |
lemma isCont_powr[continuous_intros, simp]: |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2151 |
assumes "isCont f a" "isCont g a" "f a \<noteq> 0" |
51527 | 2152 |
shows "isCont (\<lambda>x. (f x) powr g x) a" |
2153 |
using assms unfolding continuous_at by (rule tendsto_powr) |
|
2154 |
||
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
2155 |
lemma continuous_on_powr[continuous_intros]: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2156 |
assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> 0" |
51527 | 2157 |
shows "continuous_on s (\<lambda>x. (f x) powr (g x))" |
2158 |
using assms unfolding continuous_on_def by (fast intro: tendsto_powr) |
|
2159 |
||
2160 |
(* FIXME: generalize by replacing d by with g x and g ---> d? *) |
|
2161 |
lemma tendsto_zero_powrI: |
|
2162 |
assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> 0) F" |
|
53079 | 2163 |
and "0 < d" |
51527 | 2164 |
shows "((\<lambda>x. f x powr d) ---> 0) F" |
2165 |
proof (rule tendstoI) |
|
2166 |
fix e :: real assume "0 < e" |
|
2167 |
def Z \<equiv> "e powr (1 / d)" |
|
2168 |
with `0 < e` have "0 < Z" by simp |
|
2169 |
with assms have "eventually (\<lambda>x. 0 < f x \<and> dist (f x) 0 < Z) F" |
|
2170 |
by (intro eventually_conj tendstoD) |
|
2171 |
moreover |
|
2172 |
from assms have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> x powr d < Z powr d" |
|
2173 |
by (intro powr_less_mono2) (auto simp: dist_real_def) |
|
2174 |
with assms `0 < e` have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> dist (x powr d) 0 < e" |
|
2175 |
unfolding dist_real_def Z_def by (auto simp: powr_powr) |
|
2176 |
ultimately |
|
2177 |
show "eventually (\<lambda>x. dist (f x powr d) 0 < e) F" by (rule eventually_elim1) |
|
2178 |
qed |
|
2179 |
||
2180 |
lemma tendsto_neg_powr: |
|
53079 | 2181 |
assumes "s < 0" |
2182 |
and "LIM x F. f x :> at_top" |
|
51527 | 2183 |
shows "((\<lambda>x. f x powr s) ---> 0) F" |
2184 |
proof (rule tendstoI) |
|
2185 |
fix e :: real assume "0 < e" |
|
2186 |
def Z \<equiv> "e powr (1 / s)" |
|
2187 |
from assms have "eventually (\<lambda>x. Z < f x) F" |
|
2188 |
by (simp add: filterlim_at_top_dense) |
|
2189 |
moreover |
|
2190 |
from assms have "\<And>x. Z < x \<Longrightarrow> x powr s < Z powr s" |
|
2191 |
by (auto simp: Z_def intro!: powr_less_mono2_neg) |
|
2192 |
with assms `0 < e` have "\<And>x. Z < x \<Longrightarrow> dist (x powr s) 0 < e" |
|
2193 |
by (simp add: powr_powr Z_def dist_real_def) |
|
2194 |
ultimately |
|
2195 |
show "eventually (\<lambda>x. dist (f x powr s) 0 < e) F" by (rule eventually_elim1) |
|
2196 |
qed |
|
2197 |
||
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2198 |
(* it is funny that this isn't in the library! It could go in Transcendental *) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2199 |
lemma tendsto_exp_limit_at_right: |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2200 |
fixes x :: real |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2201 |
shows "((\<lambda>y. (1 + x * y) powr (1 / y)) ---> exp x) (at_right 0)" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2202 |
proof cases |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2203 |
assume "x \<noteq> 0" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2204 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2205 |
have "((\<lambda>y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2206 |
by (auto intro!: derivative_eq_intros) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2207 |
then have "((\<lambda>y. ln (1 + x * y) / y) ---> x) (at 0)" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2208 |
by (auto simp add: has_field_derivative_def field_has_derivative_at) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2209 |
then have *: "((\<lambda>y. exp (ln (1 + x * y) / y)) ---> exp x) (at 0)" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2210 |
by (rule tendsto_intros) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2211 |
then show ?thesis |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2212 |
proof (rule filterlim_mono_eventually) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2213 |
show "eventually (\<lambda>xa. exp (ln (1 + x * xa) / xa) = (1 + x * xa) powr (1 / xa)) (at_right 0)" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2214 |
unfolding eventually_at_right[OF zero_less_one] |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2215 |
using `x \<noteq> 0` by (intro exI[of _ "1 / \<bar>x\<bar>"]) (auto simp: field_simps powr_def) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2216 |
qed (simp_all add: at_eq_sup_left_right) |
58729
e8ecc79aee43
add tendsto_const and tendsto_ident_at as simp and intro rules
hoelzl
parents:
58710
diff
changeset
|
2217 |
qed simp |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2218 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2219 |
lemma tendsto_exp_limit_at_top: |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2220 |
fixes x :: real |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2221 |
shows "((\<lambda>y. (1 + x / y) powr y) ---> exp x) at_top" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2222 |
apply (subst filterlim_at_top_to_right) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2223 |
apply (simp add: inverse_eq_divide) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2224 |
apply (rule tendsto_exp_limit_at_right) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2225 |
done |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2226 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2227 |
lemma tendsto_exp_limit_sequentially: |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2228 |
fixes x :: real |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2229 |
shows "(\<lambda>n. (1 + x / n) ^ n) ----> exp x" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2230 |
proof (rule filterlim_mono_eventually) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2231 |
from reals_Archimedean2 [of "abs x"] obtain n :: nat where *: "real n > abs x" .. |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2232 |
hence "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2233 |
apply (intro eventually_sequentiallyI [of n]) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2234 |
apply (case_tac "x \<ge> 0") |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2235 |
apply (rule add_pos_nonneg, auto intro: divide_nonneg_nonneg) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2236 |
apply (subgoal_tac "x / real xa > -1") |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2237 |
apply (auto simp add: field_simps) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2238 |
done |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2239 |
then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2240 |
by (rule eventually_elim1) (erule powr_realpow) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2241 |
show "(\<lambda>n. (1 + x / real n) powr real n) ----> exp x" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2242 |
by (rule filterlim_compose [OF tendsto_exp_limit_at_top filterlim_real_sequentially]) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2243 |
qed auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2244 |
|
29164 | 2245 |
subsection {* Sine and Cosine *} |
2246 |
||
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2247 |
definition sin_coeff :: "nat \<Rightarrow> real" where |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2248 |
"sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / real (fact n))" |
31271 | 2249 |
|
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2250 |
definition cos_coeff :: "nat \<Rightarrow> real" where |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2251 |
"cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / real (fact n) else 0)" |
31271 | 2252 |
|
53079 | 2253 |
definition sin :: "real \<Rightarrow> real" |
2254 |
where "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)" |
|
2255 |
||
2256 |
definition cos :: "real \<Rightarrow> real" |
|
2257 |
where "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)" |
|
31271 | 2258 |
|
44319
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2259 |
lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0" |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2260 |
unfolding sin_coeff_def by simp |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2261 |
|
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2262 |
lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1" |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2263 |
unfolding cos_coeff_def by simp |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2264 |
|
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2265 |
lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)" |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2266 |
unfolding cos_coeff_def sin_coeff_def |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2267 |
by (simp del: mult_Suc) |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2268 |
|
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2269 |
lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)" |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2270 |
unfolding cos_coeff_def sin_coeff_def |
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2271 |
by (simp del: mult_Suc) (auto elim: oddE) |
44319
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2272 |
|
31271 | 2273 |
lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)" |
53079 | 2274 |
unfolding sin_coeff_def |
2275 |
apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]]) |
|
2276 |
apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) |
|
2277 |
done |
|
29164 | 2278 |
|
31271 | 2279 |
lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)" |
53079 | 2280 |
unfolding cos_coeff_def |
2281 |
apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]]) |
|
2282 |
apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) |
|
2283 |
done |
|
29164 | 2284 |
|
31271 | 2285 |
lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)" |
53079 | 2286 |
unfolding sin_def by (rule summable_sin [THEN summable_sums]) |
29164 | 2287 |
|
31271 | 2288 |
lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)" |
53079 | 2289 |
unfolding cos_def by (rule summable_cos [THEN summable_sums]) |
29164 | 2290 |
|
44319
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2291 |
lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff" |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2292 |
by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc) |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2293 |
|
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2294 |
lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)" |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56409
diff
changeset
|
2295 |
by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc) |
29164 | 2296 |
|
2297 |
text{*Now at last we can get the derivatives of exp, sin and cos*} |
|
2298 |
||
2299 |
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)" |
|
44319
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2300 |
unfolding sin_def cos_def |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2301 |
apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"]) |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2302 |
apply (simp_all add: diffs_sin_coeff diffs_cos_coeff |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2303 |
summable_minus summable_sin summable_cos) |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2304 |
done |
29164 | 2305 |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
2306 |
declare DERIV_sin[THEN DERIV_chain2, derivative_intros] |
51527 | 2307 |
|
29164 | 2308 |
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)" |
44319
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2309 |
unfolding cos_def sin_def |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2310 |
apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"]) |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2311 |
apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2312 |
summable_minus summable_sin summable_cos suminf_minus) |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2313 |
done |
29164 | 2314 |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
2315 |
declare DERIV_cos[THEN DERIV_chain2, derivative_intros] |
51527 | 2316 |
|
44311 | 2317 |
lemma isCont_sin: "isCont sin x" |
2318 |
by (rule DERIV_sin [THEN DERIV_isCont]) |
|
2319 |
||
2320 |
lemma isCont_cos: "isCont cos x" |
|
2321 |
by (rule DERIV_cos [THEN DERIV_isCont]) |
|
2322 |
||
2323 |
lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a" |
|
2324 |
by (rule isCont_o2 [OF _ isCont_sin]) |
|
2325 |
||
2326 |
lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a" |
|
2327 |
by (rule isCont_o2 [OF _ isCont_cos]) |
|
2328 |
||
2329 |
lemma tendsto_sin [tendsto_intros]: |
|
2330 |
"(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F" |
|
2331 |
by (rule isCont_tendsto_compose [OF isCont_sin]) |
|
2332 |
||
2333 |
lemma tendsto_cos [tendsto_intros]: |
|
2334 |
"(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F" |
|
2335 |
by (rule isCont_tendsto_compose [OF isCont_cos]) |
|
29164 | 2336 |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2337 |
lemma continuous_sin [continuous_intros]: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2338 |
"continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2339 |
unfolding continuous_def by (rule tendsto_sin) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2340 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
2341 |
lemma continuous_on_sin [continuous_intros]: |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2342 |
"continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2343 |
unfolding continuous_on_def by (auto intro: tendsto_sin) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2344 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2345 |
lemma continuous_cos [continuous_intros]: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2346 |
"continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2347 |
unfolding continuous_def by (rule tendsto_cos) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2348 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
2349 |
lemma continuous_on_cos [continuous_intros]: |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2350 |
"continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2351 |
unfolding continuous_on_def by (auto intro: tendsto_cos) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
2352 |
|
29164 | 2353 |
subsection {* Properties of Sine and Cosine *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2354 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2355 |
lemma sin_zero [simp]: "sin 0 = 0" |
44311 | 2356 |
unfolding sin_def sin_coeff_def by (simp add: powser_zero) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2357 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2358 |
lemma cos_zero [simp]: "cos 0 = 1" |
44311 | 2359 |
unfolding cos_def cos_coeff_def by (simp add: powser_zero) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2360 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
2361 |
lemma sin_cos_squared_add [simp]: "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2362 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
2363 |
have "\<forall>x. DERIV (\<lambda>x. (sin x)\<^sup>2 + (cos x)\<^sup>2) x :> 0" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
2364 |
by (auto intro!: derivative_eq_intros) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
2365 |
hence "(sin x)\<^sup>2 + (cos x)\<^sup>2 = (sin 0)\<^sup>2 + (cos 0)\<^sup>2" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2366 |
by (rule DERIV_isconst_all) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
2367 |
thus "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1" by simp |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2368 |
qed |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2369 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
2370 |
lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2371 |
by (subst add.commute, rule sin_cos_squared_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2372 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2373 |
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2374 |
using sin_cos_squared_add2 [unfolded power2_eq_square] . |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2375 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
2376 |
lemma sin_squared_eq: "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2377 |
unfolding eq_diff_eq by (rule sin_cos_squared_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2378 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
2379 |
lemma cos_squared_eq: "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2380 |
unfolding eq_diff_eq by (rule sin_cos_squared_add2) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2381 |
|
15081 | 2382 |
lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2383 |
by (rule power2_le_imp_le, simp_all add: sin_squared_eq) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2384 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2385 |
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2386 |
using abs_sin_le_one [of x] unfolding abs_le_iff by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2387 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2388 |
lemma sin_le_one [simp]: "sin x \<le> 1" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2389 |
using abs_sin_le_one [of x] unfolding abs_le_iff by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2390 |
|
15081 | 2391 |
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2392 |
by (rule power2_le_imp_le, simp_all add: cos_squared_eq) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2393 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2394 |
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2395 |
using abs_cos_le_one [of x] unfolding abs_le_iff by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2396 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2397 |
lemma cos_le_one [simp]: "cos x \<le> 1" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2398 |
using abs_cos_le_one [of x] unfolding abs_le_iff by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2399 |
|
41970 | 2400 |
lemma DERIV_fun_pow: "DERIV g x :> m ==> |
53079 | 2401 |
DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
2402 |
by (auto intro!: derivative_eq_intros simp: real_of_nat_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2403 |
|
15229 | 2404 |
lemma DERIV_fun_exp: |
53079 | 2405 |
"DERIV g x :> m ==> DERIV (\<lambda>x. exp(g x)) x :> exp(g x) * m" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
2406 |
by (auto intro!: derivative_intros) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2407 |
|
15229 | 2408 |
lemma DERIV_fun_sin: |
53079 | 2409 |
"DERIV g x :> m ==> DERIV (\<lambda>x. sin(g x)) x :> cos(g x) * m" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
2410 |
by (auto intro!: derivative_intros) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2411 |
|
15229 | 2412 |
lemma DERIV_fun_cos: |
53079 | 2413 |
"DERIV g x :> m ==> DERIV (\<lambda>x. cos(g x)) x :> -sin(g x) * m" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
2414 |
by (auto intro!: derivative_eq_intros simp: real_of_nat_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2415 |
|
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2416 |
lemma sin_cos_add_lemma: |
53079 | 2417 |
"(sin (x + y) - (sin x * cos y + cos x * sin y))\<^sup>2 + |
2418 |
(cos (x + y) - (cos x * cos y - sin x * sin y))\<^sup>2 = 0" |
|
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2419 |
(is "?f x = 0") |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2420 |
proof - |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2421 |
have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
2422 |
by (auto intro!: derivative_eq_intros simp add: algebra_simps) |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2423 |
hence "?f x = ?f 0" |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2424 |
by (rule DERIV_isconst_all) |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2425 |
thus ?thesis by simp |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2426 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2427 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2428 |
lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2429 |
using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2430 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2431 |
lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2432 |
using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2433 |
|
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2434 |
lemma sin_cos_minus_lemma: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
2435 |
"(sin(-x) + sin(x))\<^sup>2 + (cos(-x) - cos(x))\<^sup>2 = 0" (is "?f x = 0") |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2436 |
proof - |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2437 |
have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0" |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
2438 |
by (auto intro!: derivative_eq_intros simp add: algebra_simps) |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2439 |
hence "?f x = ?f 0" |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2440 |
by (rule DERIV_isconst_all) |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2441 |
thus ?thesis by simp |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2442 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2443 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2444 |
lemma sin_minus [simp]: "sin (-x) = -sin(x)" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2445 |
using sin_cos_minus_lemma [where x=x] by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2446 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2447 |
lemma cos_minus [simp]: "cos (-x) = cos(x)" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2448 |
using sin_cos_minus_lemma [where x=x] by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2449 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2450 |
lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
2451 |
using sin_add [of x "- y"] by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2452 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2453 |
lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2454 |
by (simp add: sin_diff mult.commute) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2455 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2456 |
lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
2457 |
using cos_add [of x "- y"] by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2458 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2459 |
lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2460 |
by (simp add: cos_diff mult.commute) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2461 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2462 |
lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x" |
29165
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset
|
2463 |
using sin_add [where x=x and y=x] by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2464 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
2465 |
lemma cos_double: "cos(2* x) = ((cos x)\<^sup>2) - ((sin x)\<^sup>2)" |
29165
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset
|
2466 |
using cos_add [where x=x and y=x] |
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset
|
2467 |
by (simp add: power2_eq_square) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2468 |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2469 |
lemma sin_x_le_x: assumes x: "x \<ge> 0" shows "sin x \<le> x" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2470 |
proof - |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2471 |
let ?f = "\<lambda>x. x - sin x" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2472 |
from x have "?f x \<ge> ?f 0" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2473 |
apply (rule DERIV_nonneg_imp_nondecreasing) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2474 |
apply (intro allI impI exI[of _ "1 - cos x" for x]) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2475 |
apply (auto intro!: derivative_eq_intros simp: field_simps) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2476 |
done |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2477 |
thus "sin x \<le> x" by simp |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2478 |
qed |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2479 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2480 |
lemma sin_x_ge_neg_x: assumes x: "x \<ge> 0" shows "sin x \<ge> - x" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2481 |
proof - |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2482 |
let ?f = "\<lambda>x. x + sin x" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2483 |
from x have "?f x \<ge> ?f 0" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2484 |
apply (rule DERIV_nonneg_imp_nondecreasing) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2485 |
apply (intro allI impI exI[of _ "1 + cos x" for x]) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2486 |
apply (auto intro!: derivative_eq_intros simp: field_simps real_0_le_add_iff) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2487 |
done |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2488 |
thus "sin x \<ge> -x" by simp |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2489 |
qed |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2490 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2491 |
lemma abs_sin_x_le_abs_x: "\<bar>sin x\<bar> \<le> \<bar>x\<bar>" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2492 |
using sin_x_ge_neg_x [of x] sin_x_le_x [of x] sin_x_ge_neg_x [of "-x"] sin_x_le_x [of "-x"] |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2493 |
by (auto simp: abs_real_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2494 |
|
29164 | 2495 |
subsection {* The Constant Pi *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2496 |
|
53079 | 2497 |
definition pi :: real |
2498 |
where "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)" |
|
23043 | 2499 |
|
41970 | 2500 |
text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"}; |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2501 |
hence define pi.*} |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2502 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2503 |
lemma sin_paired: |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2504 |
"(\<lambda>n. (- 1) ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) sums sin x" |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2505 |
proof - |
31271 | 2506 |
have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x" |
44727
d45acd50a894
modify lemma sums_group, and shorten proofs that use it
huffman
parents:
44726
diff
changeset
|
2507 |
by (rule sin_converges [THEN sums_group], simp) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
2508 |
thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: ac_simps) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2509 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2510 |
|
44728 | 2511 |
lemma sin_gt_zero: |
53079 | 2512 |
assumes "0 < x" and "x < 2" |
2513 |
shows "0 < sin x" |
|
44728 | 2514 |
proof - |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2515 |
let ?f = "\<lambda>n. \<Sum>k = n*2..<n*2+2. (- 1) ^ k / real (fact (2*k+1)) * x^(2*k+1)" |
44728 | 2516 |
have pos: "\<forall>n. 0 < ?f n" |
2517 |
proof |
|
2518 |
fix n :: nat |
|
2519 |
let ?k2 = "real (Suc (Suc (4 * n)))" |
|
2520 |
let ?k3 = "real (Suc (Suc (Suc (4 * n))))" |
|
2521 |
have "x * x < ?k2 * ?k3" |
|
2522 |
using assms by (intro mult_strict_mono', simp_all) |
|
2523 |
hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)" |
|
2524 |
by (intro mult_strict_right_mono zero_less_power `0 < x`) |
|
2525 |
thus "0 < ?f n" |
|
2526 |
by (simp del: mult_Suc, |
|
56544 | 2527 |
simp add: less_divide_eq field_simps del: mult_Suc) |
44728 | 2528 |
qed |
2529 |
have sums: "?f sums sin x" |
|
2530 |
by (rule sin_paired [THEN sums_group], simp) |
|
2531 |
show "0 < sin x" |
|
2532 |
unfolding sums_unique [OF sums] |
|
2533 |
using sums_summable [OF sums] pos |
|
56213 | 2534 |
by (rule suminf_pos) |
44728 | 2535 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2536 |
|
53079 | 2537 |
lemma cos_double_less_one: "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1" |
2538 |
using sin_gt_zero [where x = x] by (auto simp add: cos_squared_eq cos_double) |
|
2539 |
||
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2540 |
lemma cos_paired: "(\<lambda>n. (- 1) ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x" |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2541 |
proof - |
31271 | 2542 |
have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x" |
44727
d45acd50a894
modify lemma sums_group, and shorten proofs that use it
huffman
parents:
44726
diff
changeset
|
2543 |
by (rule cos_converges [THEN sums_group], simp) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
2544 |
thus ?thesis unfolding cos_coeff_def by (simp add: ac_simps) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2545 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2546 |
|
53602 | 2547 |
lemmas realpow_num_eq_if = power_eq_if |
2548 |
||
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2549 |
lemma sumr_pos_lt_pair: |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2550 |
fixes f :: "nat \<Rightarrow> real" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2551 |
shows "\<lbrakk>summable f; |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2552 |
\<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk> |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2553 |
\<Longrightarrow> setsum f {..<k} < suminf f" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2554 |
unfolding One_nat_def |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2555 |
apply (subst suminf_split_initial_segment [where k="k"]) |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2556 |
apply assumption |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2557 |
apply simp |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2558 |
apply (drule_tac k="k" in summable_ignore_initial_segment) |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2559 |
apply (drule_tac k="Suc (Suc 0)" in sums_group [OF summable_sums], simp) |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2560 |
apply simp |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2561 |
apply (frule sums_unique) |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2562 |
apply (drule sums_summable) |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2563 |
apply simp |
56213 | 2564 |
apply (erule suminf_pos) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
2565 |
apply (simp add: ac_simps) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2566 |
done |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2567 |
|
53602 | 2568 |
lemma cos_two_less_zero [simp]: |
2569 |
"cos 2 < 0" |
|
2570 |
proof - |
|
2571 |
note fact_Suc [simp del] |
|
2572 |
from cos_paired |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2573 |
have "(\<lambda>n. - ((- 1) ^ n / real (fact (2 * n)) * 2 ^ (2 * n))) sums - cos 2" |
53602 | 2574 |
by (rule sums_minus) |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2575 |
then have *: "(\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n)))) sums - cos 2" |
53602 | 2576 |
by simp |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2577 |
then have **: "summable (\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n))))" |
53602 | 2578 |
by (rule sums_summable) |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2579 |
have "0 < (\<Sum>n<Suc (Suc (Suc 0)). - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n))))" |
53602 | 2580 |
by (simp add: fact_num_eq_if_nat realpow_num_eq_if) |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2581 |
moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n)))) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2582 |
< (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n))))" |
53602 | 2583 |
proof - |
2584 |
{ fix d |
|
2585 |
have "4 * real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))) |
|
2586 |
< real (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))) * |
|
2587 |
fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))" |
|
2588 |
by (simp only: real_of_nat_mult) (auto intro!: mult_strict_mono fact_less_mono_nat) |
|
2589 |
then have "4 * real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))) |
|
2590 |
< real (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))))" |
|
2591 |
by (simp only: fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"]) |
|
2592 |
then have "4 * inverse (real (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))))) |
|
2593 |
< inverse (real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))" |
|
2594 |
by (simp add: inverse_eq_divide less_divide_eq) |
|
2595 |
} |
|
2596 |
note *** = this |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
2597 |
have [simp]: "\<And>x y::real. 0 < x - y \<longleftrightarrow> y < x" by arith |
53602 | 2598 |
from ** show ?thesis by (rule sumr_pos_lt_pair) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2599 |
(simp add: divide_inverse mult.assoc [symmetric] ***) |
53602 | 2600 |
qed |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2601 |
ultimately have "0 < (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n))))" |
53602 | 2602 |
by (rule order_less_trans) |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2603 |
moreover from * have "- cos 2 = (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n))))" |
53602 | 2604 |
by (rule sums_unique) |
2605 |
ultimately have "0 < - cos 2" by simp |
|
2606 |
then show ?thesis by simp |
|
2607 |
qed |
|
23053 | 2608 |
|
2609 |
lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq] |
|
2610 |
lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le] |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2611 |
|
53079 | 2612 |
lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 \<and> cos x = 0" |
44730 | 2613 |
proof (rule ex_ex1I) |
2614 |
show "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0" |
|
2615 |
by (rule IVT2, simp_all) |
|
2616 |
next |
|
2617 |
fix x y |
|
2618 |
assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0" |
|
2619 |
assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0" |
|
56181
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56167
diff
changeset
|
2620 |
have [simp]: "\<forall>x. cos differentiable (at x)" |
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56167
diff
changeset
|
2621 |
unfolding real_differentiable_def by (auto intro: DERIV_cos) |
44730 | 2622 |
from x y show "x = y" |
2623 |
apply (cut_tac less_linear [of x y], auto) |
|
2624 |
apply (drule_tac f = cos in Rolle) |
|
2625 |
apply (drule_tac [5] f = cos in Rolle) |
|
2626 |
apply (auto dest!: DERIV_cos [THEN DERIV_unique]) |
|
2627 |
apply (metis order_less_le_trans less_le sin_gt_zero) |
|
2628 |
apply (metis order_less_le_trans less_le sin_gt_zero) |
|
2629 |
done |
|
2630 |
qed |
|
31880 | 2631 |
|
23053 | 2632 |
lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)" |
53079 | 2633 |
by (simp add: pi_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2634 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2635 |
lemma cos_pi_half [simp]: "cos (pi / 2) = 0" |
53079 | 2636 |
by (simp add: pi_half cos_is_zero [THEN theI']) |
23053 | 2637 |
|
2638 |
lemma pi_half_gt_zero [simp]: "0 < pi / 2" |
|
53079 | 2639 |
apply (rule order_le_neq_trans) |
2640 |
apply (simp add: pi_half cos_is_zero [THEN theI']) |
|
54575 | 2641 |
apply (metis cos_pi_half cos_zero zero_neq_one) |
53079 | 2642 |
done |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2643 |
|
23053 | 2644 |
lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric] |
2645 |
lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le] |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2646 |
|
23053 | 2647 |
lemma pi_half_less_two [simp]: "pi / 2 < 2" |
53079 | 2648 |
apply (rule order_le_neq_trans) |
2649 |
apply (simp add: pi_half cos_is_zero [THEN theI']) |
|
54575 | 2650 |
apply (metis cos_pi_half cos_two_neq_zero) |
53079 | 2651 |
done |
23053 | 2652 |
|
2653 |
lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq] |
|
2654 |
lemmas pi_half_le_two [simp] = pi_half_less_two [THEN order_less_imp_le] |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2655 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2656 |
lemma pi_gt_zero [simp]: "0 < pi" |
53079 | 2657 |
using pi_half_gt_zero by simp |
23053 | 2658 |
|
2659 |
lemma pi_ge_zero [simp]: "0 \<le> pi" |
|
53079 | 2660 |
by (rule pi_gt_zero [THEN order_less_imp_le]) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2661 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2662 |
lemma pi_neq_zero [simp]: "pi \<noteq> 0" |
53079 | 2663 |
by (rule pi_gt_zero [THEN less_imp_neq, symmetric]) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2664 |
|
23053 | 2665 |
lemma pi_not_less_zero [simp]: "\<not> pi < 0" |
53079 | 2666 |
by (simp add: linorder_not_less) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2667 |
|
29165
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset
|
2668 |
lemma minus_pi_half_less_zero: "-(pi/2) < 0" |
53079 | 2669 |
by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2670 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2671 |
lemma m2pi_less_pi: "- (2 * pi) < pi" |
53079 | 2672 |
by simp |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2673 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2674 |
lemma sin_pi_half [simp]: "sin(pi/2) = 1" |
53079 | 2675 |
using sin_cos_squared_add2 [where x = "pi/2"] |
2676 |
using sin_gt_zero [OF pi_half_gt_zero pi_half_less_two] |
|
2677 |
by (simp add: power2_eq_1_iff) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2678 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2679 |
lemma cos_pi [simp]: "cos pi = -1" |
53079 | 2680 |
using cos_add [where x = "pi/2" and y = "pi/2"] by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2681 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2682 |
lemma sin_pi [simp]: "sin pi = 0" |
53079 | 2683 |
using sin_add [where x = "pi/2" and y = "pi/2"] by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2684 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2685 |
lemma sin_cos_eq: "sin x = cos (pi/2 - x)" |
53079 | 2686 |
by (simp add: cos_diff) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2687 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2688 |
lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)" |
53079 | 2689 |
by (simp add: cos_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2690 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2691 |
lemma cos_sin_eq: "cos x = sin (pi/2 - x)" |
53079 | 2692 |
by (simp add: sin_diff) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2693 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2694 |
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x" |
53079 | 2695 |
by (simp add: sin_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2696 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2697 |
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x" |
53079 | 2698 |
by (simp add: sin_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2699 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2700 |
lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x" |
53079 | 2701 |
by (simp add: cos_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2702 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2703 |
lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x" |
53079 | 2704 |
by (simp add: sin_add cos_double) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2705 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2706 |
lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x" |
53079 | 2707 |
by (simp add: cos_add cos_double) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2708 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2709 |
lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n" |
53079 | 2710 |
by (induct n) (auto simp add: real_of_nat_Suc distrib_right) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2711 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
2712 |
lemma cos_npi2 [simp]: "cos (pi * real n) = (- 1) ^ n" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2713 |
by (metis cos_npi mult.commute) |
15383 | 2714 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2715 |
lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0" |
53079 | 2716 |
by (induct n) (auto simp add: real_of_nat_Suc distrib_right) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2717 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2718 |
lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2719 |
by (simp add: mult.commute [of pi]) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2720 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2721 |
lemma cos_two_pi [simp]: "cos (2 * pi) = 1" |
53079 | 2722 |
by (simp add: cos_double) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2723 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2724 |
lemma sin_two_pi [simp]: "sin (2 * pi) = 0" |
53079 | 2725 |
by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2726 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2727 |
lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x" |
54575 | 2728 |
by (metis sin_gt_zero order_less_trans pi_half_less_two) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2729 |
|
41970 | 2730 |
lemma sin_less_zero: |
53079 | 2731 |
assumes "- pi/2 < x" and "x < 0" |
2732 |
shows "sin x < 0" |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2733 |
proof - |
41970 | 2734 |
have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2735 |
thus ?thesis by simp |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2736 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2737 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2738 |
lemma pi_less_4: "pi < 4" |
53079 | 2739 |
using pi_half_less_two by auto |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2740 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2741 |
lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x" |
53079 | 2742 |
apply (cut_tac pi_less_4) |
2743 |
apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all) |
|
2744 |
apply (cut_tac cos_is_zero, safe) |
|
2745 |
apply (rename_tac y z) |
|
2746 |
apply (drule_tac x = y in spec) |
|
2747 |
apply (drule_tac x = "pi/2" in spec, simp) |
|
2748 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2749 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2750 |
lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x" |
53079 | 2751 |
apply (rule_tac x = x and y = 0 in linorder_cases) |
54575 | 2752 |
apply (metis cos_gt_zero cos_minus minus_less_iff neg_0_less_iff_less) |
53079 | 2753 |
apply (auto intro: cos_gt_zero) |
2754 |
done |
|
41970 | 2755 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2756 |
lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x" |
53079 | 2757 |
apply (auto simp add: order_le_less cos_gt_zero_pi) |
2758 |
apply (subgoal_tac "x = pi/2", auto) |
|
2759 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2760 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2761 |
lemma sin_gt_zero_pi: "[| 0 < x; x < pi |] ==> 0 < sin x" |
53079 | 2762 |
by (simp add: sin_cos_eq cos_gt_zero_pi) |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2763 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2764 |
lemma pi_ge_two: "2 \<le> pi" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2765 |
proof (rule ccontr) |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2766 |
assume "\<not> 2 \<le> pi" hence "pi < 2" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2767 |
have "\<exists>y > pi. y < 2 \<and> y < 2 * pi" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2768 |
proof (cases "2 < 2 * pi") |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2769 |
case True with dense[OF `pi < 2`] show ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2770 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2771 |
case False have "pi < 2 * pi" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2772 |
from dense[OF this] and False show ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2773 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2774 |
then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2775 |
hence "0 < sin y" using sin_gt_zero by auto |
41970 | 2776 |
moreover |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2777 |
have "sin y < 0" using sin_gt_zero_pi[of "y - pi"] `pi < y` and `y < 2 * pi` sin_periodic_pi[of "y - pi"] by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2778 |
ultimately show False by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2779 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2780 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2781 |
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x" |
53079 | 2782 |
by (auto simp add: order_le_less sin_gt_zero_pi) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2783 |
|
44745 | 2784 |
text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}. |
2785 |
It should be possible to factor out some of the common parts. *} |
|
2786 |
||
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2787 |
lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)" |
44745 | 2788 |
proof (rule ex_ex1I) |
2789 |
assume y: "-1 \<le> y" "y \<le> 1" |
|
2790 |
show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y" |
|
2791 |
by (rule IVT2, simp_all add: y) |
|
2792 |
next |
|
2793 |
fix a b |
|
2794 |
assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y" |
|
2795 |
assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y" |
|
56181
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56167
diff
changeset
|
2796 |
have [simp]: "\<forall>x. cos differentiable (at x)" |
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56167
diff
changeset
|
2797 |
unfolding real_differentiable_def by (auto intro: DERIV_cos) |
44745 | 2798 |
from a b show "a = b" |
2799 |
apply (cut_tac less_linear [of a b], auto) |
|
2800 |
apply (drule_tac f = cos in Rolle) |
|
2801 |
apply (drule_tac [5] f = cos in Rolle) |
|
2802 |
apply (auto dest!: DERIV_cos [THEN DERIV_unique]) |
|
2803 |
apply (metis order_less_le_trans less_le sin_gt_zero_pi) |
|
2804 |
apply (metis order_less_le_trans less_le sin_gt_zero_pi) |
|
2805 |
done |
|
2806 |
qed |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2807 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2808 |
lemma sin_total: |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2809 |
"[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)" |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2810 |
apply (rule ccontr) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2811 |
apply (subgoal_tac "\<forall>x. (- (pi/2) \<le> x & x \<le> pi/2 & (sin x = y)) = (0 \<le> (x + pi/2) & (x + pi/2) \<le> pi & (cos (x + pi/2) = -y))") |
18585 | 2812 |
apply (erule contrapos_np) |
45309
5885ec8eb6b0
removed ad-hoc simp rules sin_cos_eq[symmetric], minus_sin_cos_eq[symmetric], cos_sin_eq[symmetric]
huffman
parents:
45308
diff
changeset
|
2813 |
apply simp |
41970 | 2814 |
apply (cut_tac y="-y" in cos_total, simp) apply simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2815 |
apply (erule ex1E) |
15229 | 2816 |
apply (rule_tac a = "x - (pi/2)" in ex1I) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2817 |
apply (simp (no_asm) add: add.assoc) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2818 |
apply (rotate_tac 3) |
45309
5885ec8eb6b0
removed ad-hoc simp rules sin_cos_eq[symmetric], minus_sin_cos_eq[symmetric], cos_sin_eq[symmetric]
huffman
parents:
45308
diff
changeset
|
2819 |
apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all add: cos_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2820 |
done |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2821 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2822 |
lemma reals_Archimedean4: |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2823 |
"[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y" |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2824 |
apply (auto dest!: reals_Archimedean3) |
41970 | 2825 |
apply (drule_tac x = x in spec, clarify) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2826 |
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y") |
41970 | 2827 |
prefer 2 apply (erule LeastI) |
2828 |
apply (case_tac "LEAST m::nat. x < real m * y", simp) |
|
55417
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
54576
diff
changeset
|
2829 |
apply (rename_tac m) |
01fbfb60c33e
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents:
54576
diff
changeset
|
2830 |
apply (subgoal_tac "~ x < real m * y") |
41970 | 2831 |
prefer 2 apply (rule not_less_Least, simp, force) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2832 |
done |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2833 |
|
41970 | 2834 |
(* Pre Isabelle99-2 proof was simpler- numerals arithmetic |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2835 |
now causes some unwanted re-arrangements of literals! *) |
15229 | 2836 |
lemma cos_zero_lemma: |
41970 | 2837 |
"[| 0 \<le> x; cos x = 0 |] ==> |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2838 |
\<exists>n::nat. ~even n & x = real n * (pi/2)" |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2839 |
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe) |
41970 | 2840 |
apply (subgoal_tac "0 \<le> x - real n * pi & |
15086 | 2841 |
(x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ") |
29667 | 2842 |
apply (auto simp add: algebra_simps real_of_nat_Suc) |
2843 |
prefer 2 apply (simp add: cos_diff) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2844 |
apply (simp add: cos_diff) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2845 |
apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0") |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2846 |
apply (rule_tac [2] cos_total, safe) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2847 |
apply (drule_tac x = "x - real n * pi" in spec) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2848 |
apply (drule_tac x = "pi/2" in spec) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2849 |
apply (simp add: cos_diff) |
15229 | 2850 |
apply (rule_tac x = "Suc (2 * n)" in exI) |
29667 | 2851 |
apply (simp add: real_of_nat_Suc algebra_simps, auto) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2852 |
done |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2853 |
|
15229 | 2854 |
lemma sin_zero_lemma: |
41970 | 2855 |
"[| 0 \<le> x; sin x = 0 |] ==> |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2856 |
\<exists>n::nat. even n & x = real n * (pi/2)" |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2857 |
apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ") |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2858 |
apply (clarify, rule_tac x = "n - 1" in exI) |
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2859 |
apply (auto elim!: oddE simp add: real_of_nat_Suc field_simps)[1] |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2860 |
apply (rule cos_zero_lemma) |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2861 |
apply (auto simp add: cos_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2862 |
done |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2863 |
|
15229 | 2864 |
lemma cos_zero_iff: |
41970 | 2865 |
"(cos x = 0) = |
2866 |
((\<exists>n::nat. ~even n & (x = real n * (pi/2))) | |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2867 |
(\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))" |
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2868 |
proof - |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2869 |
{ fix n :: nat |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2870 |
assume "odd n" |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2871 |
then obtain m where "n = 2 * m + 1" .. |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2872 |
then have "cos (real n * pi / 2) = 0" |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2873 |
by (simp add: field_simps real_of_nat_Suc) (simp add: cos_add add_divide_distrib) |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2874 |
} note * = this |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2875 |
show ?thesis |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2876 |
apply (rule iffI) |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2877 |
apply (cut_tac linorder_linear [of 0 x], safe) |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2878 |
apply (drule cos_zero_lemma, assumption+) |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2879 |
apply (cut_tac x="-x" in cos_zero_lemma, simp, simp) |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2880 |
apply (auto dest: *) |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2881 |
done |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2882 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2883 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2884 |
(* ditto: but to a lesser extent *) |
15229 | 2885 |
lemma sin_zero_iff: |
41970 | 2886 |
"(sin x = 0) = |
2887 |
((\<exists>n::nat. even n & (x = real n * (pi/2))) | |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2888 |
(\<exists>n::nat. even n & (x = -(real n * (pi/2)))))" |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2889 |
apply (rule iffI) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2890 |
apply (cut_tac linorder_linear [of 0 x], safe) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2891 |
apply (drule sin_zero_lemma, assumption+) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2892 |
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe) |
41970 | 2893 |
apply (force simp add: minus_equation_iff [of x]) |
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
2894 |
apply (auto elim: evenE) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2895 |
done |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2896 |
|
53079 | 2897 |
lemma cos_monotone_0_pi: |
2898 |
assumes "0 \<le> y" and "y < x" and "x \<le> pi" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2899 |
shows "cos x < cos y" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2900 |
proof - |
33549 | 2901 |
have "- (x - y) < 0" using assms by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2902 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2903 |
from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]] |
53079 | 2904 |
obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" |
2905 |
by auto |
|
33549 | 2906 |
hence "0 < z" and "z < pi" using assms by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2907 |
hence "0 < sin z" using sin_gt_zero_pi by auto |
53079 | 2908 |
hence "cos x - cos y < 0" |
2909 |
unfolding cos_diff minus_mult_commute[symmetric] |
|
2910 |
using `- (x - y) < 0` by (rule mult_pos_neg2) |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2911 |
thus ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2912 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2913 |
|
53079 | 2914 |
lemma cos_monotone_0_pi': |
2915 |
assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" |
|
2916 |
shows "cos x \<le> cos y" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2917 |
proof (cases "y < x") |
53079 | 2918 |
case True |
2919 |
show ?thesis |
|
2920 |
using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2921 |
next |
53079 | 2922 |
case False |
2923 |
hence "y = x" using `y \<le> x` by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2924 |
thus ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2925 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2926 |
|
53079 | 2927 |
lemma cos_monotone_minus_pi_0: |
2928 |
assumes "-pi \<le> y" and "y < x" and "x \<le> 0" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2929 |
shows "cos y < cos x" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2930 |
proof - |
53079 | 2931 |
have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" |
2932 |
using assms by auto |
|
2933 |
from cos_monotone_0_pi[OF this] show ?thesis |
|
2934 |
unfolding cos_minus . |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2935 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2936 |
|
53079 | 2937 |
lemma cos_monotone_minus_pi_0': |
2938 |
assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0" |
|
2939 |
shows "cos y \<le> cos x" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2940 |
proof (cases "y < x") |
53079 | 2941 |
case True |
2942 |
show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] |
|
2943 |
by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2944 |
next |
53079 | 2945 |
case False |
2946 |
hence "y = x" using `y \<le> x` by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2947 |
thus ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2948 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2949 |
|
53079 | 2950 |
lemma sin_monotone_2pi': |
2951 |
assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2" |
|
2952 |
shows "sin y \<le> sin x" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2953 |
proof - |
33549 | 2954 |
have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi" |
2955 |
using pi_ge_two and assms by auto |
|
53079 | 2956 |
from cos_monotone_0_pi'[OF this] show ?thesis |
2957 |
unfolding minus_sin_cos_eq[symmetric] by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
2958 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2959 |
|
53079 | 2960 |
|
29164 | 2961 |
subsection {* Tangent *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2962 |
|
53079 | 2963 |
definition tan :: "real \<Rightarrow> real" |
2964 |
where "tan = (\<lambda>x. sin x / cos x)" |
|
23043 | 2965 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2966 |
lemma tan_zero [simp]: "tan 0 = 0" |
44311 | 2967 |
by (simp add: tan_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2968 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2969 |
lemma tan_pi [simp]: "tan pi = 0" |
44311 | 2970 |
by (simp add: tan_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2971 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2972 |
lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0" |
44311 | 2973 |
by (simp add: tan_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2974 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2975 |
lemma tan_minus [simp]: "tan (-x) = - tan x" |
44311 | 2976 |
by (simp add: tan_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2977 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2978 |
lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x" |
44311 | 2979 |
by (simp add: tan_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2980 |
|
41970 | 2981 |
lemma lemma_tan_add1: |
44311 | 2982 |
"\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)" |
2983 |
by (simp add: tan_def cos_add field_simps) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2984 |
|
41970 | 2985 |
lemma add_tan_eq: |
44311 | 2986 |
"\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)" |
2987 |
by (simp add: tan_def sin_add field_simps) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2988 |
|
15229 | 2989 |
lemma tan_add: |
41970 | 2990 |
"[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |] |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2991 |
==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))" |
44311 | 2992 |
by (simp add: add_tan_eq lemma_tan_add1, simp add: tan_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2993 |
|
15229 | 2994 |
lemma tan_double: |
41970 | 2995 |
"[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |] |
53076 | 2996 |
==> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)" |
44311 | 2997 |
using tan_add [of x x] by (simp add: power2_eq_square) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2998 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2999 |
lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x" |
53079 | 3000 |
by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) |
41970 | 3001 |
|
3002 |
lemma tan_less_zero: |
|
53079 | 3003 |
assumes lb: "- pi/2 < x" and "x < 0" |
3004 |
shows "tan x < 0" |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3005 |
proof - |
41970 | 3006 |
have "0 < tan (- x)" using assms by (simp only: tan_gt_zero) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3007 |
thus ?thesis by simp |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3008 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3009 |
|
44756
efcd71fbaeec
simplify proof of tan_half, removing unused assumptions
huffman
parents:
44755
diff
changeset
|
3010 |
lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)" |
efcd71fbaeec
simplify proof of tan_half, removing unused assumptions
huffman
parents:
44755
diff
changeset
|
3011 |
unfolding tan_def sin_double cos_double sin_squared_eq |
efcd71fbaeec
simplify proof of tan_half, removing unused assumptions
huffman
parents:
44755
diff
changeset
|
3012 |
by (simp add: power2_eq_square) |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3013 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3014 |
lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)" |
44311 | 3015 |
unfolding tan_def |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
3016 |
by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square) |
44311 | 3017 |
|
3018 |
lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x" |
|
3019 |
by (rule DERIV_tan [THEN DERIV_isCont]) |
|
3020 |
||
3021 |
lemma isCont_tan' [simp]: |
|
3022 |
"\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a" |
|
3023 |
by (rule isCont_o2 [OF _ isCont_tan]) |
|
3024 |
||
3025 |
lemma tendsto_tan [tendsto_intros]: |
|
3026 |
"\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F" |
|
3027 |
by (rule isCont_tendsto_compose [OF isCont_tan]) |
|
3028 |
||
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3029 |
lemma continuous_tan: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3030 |
"continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3031 |
unfolding continuous_def by (rule tendsto_tan) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3032 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3033 |
lemma isCont_tan'' [continuous_intros]: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3034 |
"continuous (at x) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. tan (f x))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3035 |
unfolding continuous_at by (rule tendsto_tan) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3036 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3037 |
lemma continuous_within_tan [continuous_intros]: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3038 |
"continuous (at x within s) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3039 |
unfolding continuous_within by (rule tendsto_tan) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3040 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
3041 |
lemma continuous_on_tan [continuous_intros]: |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3042 |
"continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3043 |
unfolding continuous_on_def by (auto intro: tendsto_tan) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3044 |
|
53079 | 3045 |
lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) -- pi/2 --> 0" |
44311 | 3046 |
by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3047 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3048 |
lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x" |
53079 | 3049 |
apply (cut_tac LIM_cos_div_sin) |
3050 |
apply (simp only: LIM_eq) |
|
3051 |
apply (drule_tac x = "inverse y" in spec, safe, force) |
|
3052 |
apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe) |
|
3053 |
apply (rule_tac x = "(pi/2) - e" in exI) |
|
3054 |
apply (simp (no_asm_simp)) |
|
3055 |
apply (drule_tac x = "(pi/2) - e" in spec) |
|
3056 |
apply (auto simp add: tan_def sin_diff cos_diff) |
|
3057 |
apply (rule inverse_less_iff_less [THEN iffD1]) |
|
3058 |
apply (auto simp add: divide_inverse) |
|
3059 |
apply (rule mult_pos_pos) |
|
3060 |
apply (subgoal_tac [3] "0 < sin e & 0 < cos e") |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
3061 |
apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult.commute) |
53079 | 3062 |
done |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3063 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3064 |
lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y" |
53079 | 3065 |
apply (frule order_le_imp_less_or_eq, safe) |
3066 |
prefer 2 apply force |
|
3067 |
apply (drule lemma_tan_total, safe) |
|
3068 |
apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl) |
|
3069 |
apply (auto intro!: DERIV_tan [THEN DERIV_isCont]) |
|
3070 |
apply (drule_tac y = xa in order_le_imp_less_or_eq) |
|
3071 |
apply (auto dest: cos_gt_zero) |
|
3072 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3073 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3074 |
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y" |
53079 | 3075 |
apply (cut_tac linorder_linear [of 0 y], safe) |
3076 |
apply (drule tan_total_pos) |
|
3077 |
apply (cut_tac [2] y="-y" in tan_total_pos, safe) |
|
3078 |
apply (rule_tac [3] x = "-x" in exI) |
|
3079 |
apply (auto del: exI intro!: exI) |
|
3080 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3081 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3082 |
lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y" |
53079 | 3083 |
apply (cut_tac y = y in lemma_tan_total1, auto) |
57492
74bf65a1910a
Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
57418
diff
changeset
|
3084 |
apply hypsubst_thin |
53079 | 3085 |
apply (cut_tac x = xa and y = y in linorder_less_linear, auto) |
3086 |
apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0") |
|
3087 |
apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0") |
|
3088 |
apply (rule_tac [4] Rolle) |
|
3089 |
apply (rule_tac [2] Rolle) |
|
3090 |
apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI |
|
56181
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56167
diff
changeset
|
3091 |
simp add: real_differentiable_def) |
53079 | 3092 |
txt{*Now, simulate TRYALL*} |
3093 |
apply (rule_tac [!] DERIV_tan asm_rl) |
|
3094 |
apply (auto dest!: DERIV_unique [OF _ DERIV_tan] |
|
3095 |
simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym]) |
|
3096 |
done |
|
3097 |
||
3098 |
lemma tan_monotone: |
|
3099 |
assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3100 |
shows "tan y < tan x" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3101 |
proof - |
53079 | 3102 |
have "\<forall>x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse ((cos x')\<^sup>2)" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3103 |
proof (rule allI, rule impI) |
53079 | 3104 |
fix x' :: real |
3105 |
assume "y \<le> x' \<and> x' \<le> x" |
|
33549 | 3106 |
hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3107 |
from cos_gt_zero_pi[OF this] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3108 |
have "cos x' \<noteq> 0" by auto |
53076 | 3109 |
thus "DERIV tan x' :> inverse ((cos x')\<^sup>2)" by (rule DERIV_tan) |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3110 |
qed |
41970 | 3111 |
from MVT2[OF `y < x` this] |
53079 | 3112 |
obtain z where "y < z" and "z < x" |
3113 |
and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto |
|
33549 | 3114 |
hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3115 |
hence "0 < cos z" using cos_gt_zero_pi by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3116 |
hence inv_pos: "0 < inverse ((cos z)\<^sup>2)" by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3117 |
have "0 < x - y" using `y < x` by auto |
56544 | 3118 |
with inv_pos have "0 < tan x - tan y" unfolding tan_diff by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3119 |
thus ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3120 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3121 |
|
53079 | 3122 |
lemma tan_monotone': |
3123 |
assumes "- (pi / 2) < y" |
|
3124 |
and "y < pi / 2" |
|
3125 |
and "- (pi / 2) < x" |
|
3126 |
and "x < pi / 2" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3127 |
shows "(y < x) = (tan y < tan x)" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3128 |
proof |
53079 | 3129 |
assume "y < x" |
3130 |
thus "tan y < tan x" |
|
3131 |
using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3132 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3133 |
assume "tan y < tan x" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3134 |
show "y < x" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3135 |
proof (rule ccontr) |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3136 |
assume "\<not> y < x" hence "x \<le> y" by auto |
41970 | 3137 |
hence "tan x \<le> tan y" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3138 |
proof (cases "x = y") |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3139 |
case True thus ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3140 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3141 |
case False hence "x < y" using `x \<le> y` by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3142 |
from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3143 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3144 |
thus False using `tan y < tan x` by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3145 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3146 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3147 |
|
53079 | 3148 |
lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)" |
3149 |
unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3150 |
|
41970 | 3151 |
lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3152 |
by (simp add: tan_def) |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3153 |
|
53079 | 3154 |
lemma tan_periodic_nat[simp]: |
3155 |
fixes n :: nat |
|
3156 |
shows "tan (x + real n * pi) = tan x" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3157 |
proof (induct n arbitrary: x) |
53079 | 3158 |
case 0 |
3159 |
then show ?case by simp |
|
3160 |
next |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3161 |
case (Suc n) |
53079 | 3162 |
have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" |
3163 |
unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3164 |
show ?case unfolding split_pi_off using Suc by auto |
53079 | 3165 |
qed |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3166 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3167 |
lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3168 |
proof (cases "0 \<le> i") |
53079 | 3169 |
case True |
3170 |
hence i_nat: "real i = real (nat i)" by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3171 |
show ?thesis unfolding i_nat by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3172 |
next |
53079 | 3173 |
case False |
3174 |
hence i_nat: "real i = - real (nat (-i))" by auto |
|
3175 |
have "tan x = tan (x + real i * pi - real i * pi)" |
|
3176 |
by auto |
|
3177 |
also have "\<dots> = tan (x + real i * pi)" |
|
3178 |
unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat) |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3179 |
finally show ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3180 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3181 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46240
diff
changeset
|
3182 |
lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46240
diff
changeset
|
3183 |
using tan_periodic_int[of _ "numeral n" ] unfolding real_numeral . |
23043 | 3184 |
|
3185 |
subsection {* Inverse Trigonometric Functions *} |
|
3186 |
||
53079 | 3187 |
definition arcsin :: "real => real" |
3188 |
where "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)" |
|
3189 |
||
3190 |
definition arccos :: "real => real" |
|
3191 |
where "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)" |
|
3192 |
||
3193 |
definition arctan :: "real => real" |
|
3194 |
where "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)" |
|
23043 | 3195 |
|
15229 | 3196 |
lemma arcsin: |
53079 | 3197 |
"-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> |
3198 |
-(pi/2) \<le> arcsin y & arcsin y \<le> pi/2 & sin(arcsin y) = y" |
|
3199 |
unfolding arcsin_def by (rule theI' [OF sin_total]) |
|
23011 | 3200 |
|
3201 |
lemma arcsin_pi: |
|
53079 | 3202 |
"-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y" |
3203 |
apply (drule (1) arcsin) |
|
3204 |
apply (force intro: order_trans) |
|
3205 |
done |
|
3206 |
||
3207 |
lemma sin_arcsin [simp]: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin(arcsin y) = y" |
|
3208 |
by (blast dest: arcsin) |
|
3209 |
||
3210 |
lemma arcsin_bounded: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2" |
|
3211 |
by (blast dest: arcsin) |
|
3212 |
||
3213 |
lemma arcsin_lbound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y" |
|
3214 |
by (blast dest: arcsin) |
|
3215 |
||
3216 |
lemma arcsin_ubound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2" |
|
3217 |
by (blast dest: arcsin) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3218 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3219 |
lemma arcsin_lt_bounded: |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3220 |
"[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2" |
53079 | 3221 |
apply (frule order_less_imp_le) |
3222 |
apply (frule_tac y = y in order_less_imp_le) |
|
3223 |
apply (frule arcsin_bounded) |
|
3224 |
apply (safe, simp) |
|
3225 |
apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq) |
|
3226 |
apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe) |
|
3227 |
apply (drule_tac [!] f = sin in arg_cong, auto) |
|
3228 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3229 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3230 |
lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x" |
53079 | 3231 |
apply (unfold arcsin_def) |
3232 |
apply (rule the1_equality) |
|
3233 |
apply (rule sin_total, auto) |
|
3234 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3235 |
|
22975 | 3236 |
lemma arccos: |
41970 | 3237 |
"[| -1 \<le> y; y \<le> 1 |] |
22975 | 3238 |
==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y" |
53079 | 3239 |
unfolding arccos_def by (rule theI' [OF cos_total]) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3240 |
|
22975 | 3241 |
lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y" |
53079 | 3242 |
by (blast dest: arccos) |
41970 | 3243 |
|
22975 | 3244 |
lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi" |
53079 | 3245 |
by (blast dest: arccos) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3246 |
|
22975 | 3247 |
lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y" |
53079 | 3248 |
by (blast dest: arccos) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3249 |
|
22975 | 3250 |
lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi" |
53079 | 3251 |
by (blast dest: arccos) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3252 |
|
22975 | 3253 |
lemma arccos_lt_bounded: |
41970 | 3254 |
"[| -1 < y; y < 1 |] |
22975 | 3255 |
==> 0 < arccos y & arccos y < pi" |
53079 | 3256 |
apply (frule order_less_imp_le) |
3257 |
apply (frule_tac y = y in order_less_imp_le) |
|
3258 |
apply (frule arccos_bounded, auto) |
|
3259 |
apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq) |
|
3260 |
apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto) |
|
3261 |
apply (drule_tac [!] f = cos in arg_cong, auto) |
|
3262 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3263 |
|
22975 | 3264 |
lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x" |
53079 | 3265 |
apply (simp add: arccos_def) |
3266 |
apply (auto intro!: the1_equality cos_total) |
|
3267 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3268 |
|
22975 | 3269 |
lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x" |
53079 | 3270 |
apply (simp add: arccos_def) |
3271 |
apply (auto intro!: the1_equality cos_total) |
|
3272 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3273 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3274 |
lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)" |
53079 | 3275 |
apply (subgoal_tac "x\<^sup>2 \<le> 1") |
3276 |
apply (rule power2_eq_imp_eq) |
|
3277 |
apply (simp add: cos_squared_eq) |
|
3278 |
apply (rule cos_ge_zero) |
|
3279 |
apply (erule (1) arcsin_lbound) |
|
3280 |
apply (erule (1) arcsin_ubound) |
|
3281 |
apply simp |
|
3282 |
apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp) |
|
3283 |
apply (rule power_mono, simp, simp) |
|
3284 |
done |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
3285 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3286 |
lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)" |
53079 | 3287 |
apply (subgoal_tac "x\<^sup>2 \<le> 1") |
3288 |
apply (rule power2_eq_imp_eq) |
|
3289 |
apply (simp add: sin_squared_eq) |
|
3290 |
apply (rule sin_ge_zero) |
|
3291 |
apply (erule (1) arccos_lbound) |
|
3292 |
apply (erule (1) arccos_ubound) |
|
3293 |
apply simp |
|
3294 |
apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp) |
|
3295 |
apply (rule power_mono, simp, simp) |
|
3296 |
done |
|
3297 |
||
3298 |
lemma arctan [simp]: "- (pi/2) < arctan y & arctan y < pi/2 & tan (arctan y) = y" |
|
3299 |
unfolding arctan_def by (rule theI' [OF tan_total]) |
|
3300 |
||
3301 |
lemma tan_arctan: "tan (arctan y) = y" |
|
3302 |
by auto |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3303 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3304 |
lemma arctan_bounded: "- (pi/2) < arctan y & arctan y < pi/2" |
53079 | 3305 |
by (auto simp only: arctan) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3306 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3307 |
lemma arctan_lbound: "- (pi/2) < arctan y" |
53079 | 3308 |
by auto |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3309 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3310 |
lemma arctan_ubound: "arctan y < pi/2" |
53079 | 3311 |
by (auto simp only: arctan) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3312 |
|
44746 | 3313 |
lemma arctan_unique: |
53079 | 3314 |
assumes "-(pi/2) < x" |
3315 |
and "x < pi/2" |
|
3316 |
and "tan x = y" |
|
44746 | 3317 |
shows "arctan y = x" |
3318 |
using assms arctan [of y] tan_total [of y] by (fast elim: ex1E) |
|
3319 |
||
53079 | 3320 |
lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x" |
3321 |
by (rule arctan_unique) simp_all |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3322 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3323 |
lemma arctan_zero_zero [simp]: "arctan 0 = 0" |
53079 | 3324 |
by (rule arctan_unique) simp_all |
44746 | 3325 |
|
3326 |
lemma arctan_minus: "arctan (- x) = - arctan x" |
|
3327 |
apply (rule arctan_unique) |
|
3328 |
apply (simp only: neg_less_iff_less arctan_ubound) |
|
3329 |
apply (metis minus_less_iff arctan_lbound) |
|
3330 |
apply simp |
|
3331 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3332 |
|
44725 | 3333 |
lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0" |
3334 |
by (intro less_imp_neq [symmetric] cos_gt_zero_pi |
|
3335 |
arctan_lbound arctan_ubound) |
|
3336 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3337 |
lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)" |
44725 | 3338 |
proof (rule power2_eq_imp_eq) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3339 |
have "0 < 1 + x\<^sup>2" by (simp add: add_pos_nonneg) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3340 |
show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp |
44725 | 3341 |
show "0 \<le> cos (arctan x)" |
3342 |
by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3343 |
have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
47489
diff
changeset
|
3344 |
unfolding tan_def by (simp add: distrib_left power_divide) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3345 |
thus "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3346 |
using `0 < 1 + x\<^sup>2` by (simp add: power_divide eq_divide_eq) |
44725 | 3347 |
qed |
3348 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3349 |
lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)" |
44725 | 3350 |
using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]] |
3351 |
using tan_arctan [of x] unfolding tan_def cos_arctan |
|
3352 |
by (simp add: eq_divide_eq) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3353 |
|
53076 | 3354 |
lemma tan_sec: "cos x \<noteq> 0 ==> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2" |
53079 | 3355 |
apply (rule power_inverse [THEN subst]) |
56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
56213
diff
changeset
|
3356 |
apply (rule_tac c1 = "(cos x)\<^sup>2" in mult_right_cancel [THEN iffD1]) |
53079 | 3357 |
apply (auto dest: field_power_not_zero |
3358 |
simp add: power_mult_distrib distrib_right power_divide tan_def |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
3359 |
mult.assoc power_inverse [symmetric]) |
53079 | 3360 |
done |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3361 |
|
44746 | 3362 |
lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y" |
3363 |
by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan) |
|
3364 |
||
3365 |
lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y" |
|
3366 |
by (simp only: not_less [symmetric] arctan_less_iff) |
|
3367 |
||
3368 |
lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y" |
|
3369 |
by (simp only: eq_iff [where 'a=real] arctan_le_iff) |
|
3370 |
||
3371 |
lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x" |
|
3372 |
using arctan_less_iff [of 0 x] by simp |
|
3373 |
||
3374 |
lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0" |
|
3375 |
using arctan_less_iff [of x 0] by simp |
|
3376 |
||
3377 |
lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x" |
|
3378 |
using arctan_le_iff [of 0 x] by simp |
|
3379 |
||
3380 |
lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0" |
|
3381 |
using arctan_le_iff [of x 0] by simp |
|
3382 |
||
3383 |
lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0" |
|
3384 |
using arctan_eq_iff [of x 0] by simp |
|
3385 |
||
51482
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3386 |
lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin" |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3387 |
proof - |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3388 |
have "continuous_on (sin ` {- pi / 2 .. pi / 2}) arcsin" |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
3389 |
by (rule continuous_on_inv) (auto intro: continuous_intros simp: arcsin_sin) |
51482
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3390 |
also have "sin ` {- pi / 2 .. pi / 2} = {-1 .. 1}" |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3391 |
proof safe |
53079 | 3392 |
fix x :: real |
3393 |
assume "x \<in> {-1..1}" |
|
3394 |
then show "x \<in> sin ` {- pi / 2..pi / 2}" |
|
3395 |
using arcsin_lbound arcsin_ubound |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56409
diff
changeset
|
3396 |
by (intro image_eqI[where x="arcsin x"]) auto |
51482
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3397 |
qed simp |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3398 |
finally show ?thesis . |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3399 |
qed |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3400 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
3401 |
lemma continuous_on_arcsin [continuous_intros]: |
51482
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3402 |
"continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. -1 \<le> f x \<and> f x \<le> 1) \<Longrightarrow> continuous_on s (\<lambda>x. arcsin (f x))" |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3403 |
using continuous_on_compose[of s f, OF _ continuous_on_subset[OF continuous_on_arcsin']] |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3404 |
by (auto simp: comp_def subset_eq) |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3405 |
|
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3406 |
lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x" |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3407 |
using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"] |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3408 |
by (auto simp: continuous_on_eq_continuous_at subset_eq) |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3409 |
|
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3410 |
lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos" |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3411 |
proof - |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3412 |
have "continuous_on (cos ` {0 .. pi}) arccos" |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
3413 |
by (rule continuous_on_inv) (auto intro: continuous_intros simp: arccos_cos) |
51482
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3414 |
also have "cos ` {0 .. pi} = {-1 .. 1}" |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3415 |
proof safe |
53079 | 3416 |
fix x :: real |
3417 |
assume "x \<in> {-1..1}" |
|
3418 |
then show "x \<in> cos ` {0..pi}" |
|
3419 |
using arccos_lbound arccos_ubound |
|
3420 |
by (intro image_eqI[where x="arccos x"]) auto |
|
51482
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3421 |
qed simp |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3422 |
finally show ?thesis . |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3423 |
qed |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3424 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
3425 |
lemma continuous_on_arccos [continuous_intros]: |
51482
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3426 |
"continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. -1 \<le> f x \<and> f x \<le> 1) \<Longrightarrow> continuous_on s (\<lambda>x. arccos (f x))" |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3427 |
using continuous_on_compose[of s f, OF _ continuous_on_subset[OF continuous_on_arccos']] |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3428 |
by (auto simp: comp_def subset_eq) |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3429 |
|
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3430 |
lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x" |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3431 |
using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"] |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
3432 |
by (auto simp: continuous_on_eq_continuous_at subset_eq) |
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
3433 |
|
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
3434 |
lemma isCont_arctan: "isCont arctan x" |
53079 | 3435 |
apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify) |
3436 |
apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify) |
|
3437 |
apply (subgoal_tac "isCont arctan (tan (arctan x))", simp) |
|
3438 |
apply (erule (1) isCont_inverse_function2 [where f=tan]) |
|
3439 |
apply (metis arctan_tan order_le_less_trans order_less_le_trans) |
|
3440 |
apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le) |
|
3441 |
done |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
3442 |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3443 |
lemma tendsto_arctan [tendsto_intros]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) ---> arctan x) F" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3444 |
by (rule isCont_tendsto_compose [OF isCont_arctan]) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3445 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3446 |
lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3447 |
unfolding continuous_def by (rule tendsto_arctan) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3448 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
3449 |
lemma continuous_on_arctan [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))" |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51477
diff
changeset
|
3450 |
unfolding continuous_on_def by (auto intro: tendsto_arctan) |
53079 | 3451 |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
3452 |
lemma DERIV_arcsin: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3453 |
"\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))" |
53079 | 3454 |
apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"]) |
3455 |
apply (rule DERIV_cong [OF DERIV_sin]) |
|
3456 |
apply (simp add: cos_arcsin) |
|
3457 |
apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp) |
|
3458 |
apply (rule power_strict_mono, simp, simp, simp) |
|
3459 |
apply assumption |
|
3460 |
apply assumption |
|
3461 |
apply simp |
|
3462 |
apply (erule (1) isCont_arcsin) |
|
3463 |
done |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
3464 |
|
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
3465 |
lemma DERIV_arccos: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3466 |
"\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))" |
53079 | 3467 |
apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"]) |
3468 |
apply (rule DERIV_cong [OF DERIV_cos]) |
|
3469 |
apply (simp add: sin_arccos) |
|
3470 |
apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp) |
|
3471 |
apply (rule power_strict_mono, simp, simp, simp) |
|
3472 |
apply assumption |
|
3473 |
apply assumption |
|
3474 |
apply simp |
|
3475 |
apply (erule (1) isCont_arccos) |
|
3476 |
done |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
3477 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3478 |
lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)" |
53079 | 3479 |
apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"]) |
3480 |
apply (rule DERIV_cong [OF DERIV_tan]) |
|
3481 |
apply (rule cos_arctan_not_zero) |
|
3482 |
apply (simp add: power_inverse tan_sec [symmetric]) |
|
3483 |
apply (subgoal_tac "0 < 1 + x\<^sup>2", simp) |
|
3484 |
apply (simp add: add_pos_nonneg) |
|
3485 |
apply (simp, simp, simp, rule isCont_arctan) |
|
3486 |
done |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
3487 |
|
31880 | 3488 |
declare |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
3489 |
DERIV_arcsin[THEN DERIV_chain2, derivative_intros] |
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
3490 |
DERIV_arccos[THEN DERIV_chain2, derivative_intros] |
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
3491 |
DERIV_arctan[THEN DERIV_chain2, derivative_intros] |
31880 | 3492 |
|
50346
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3493 |
lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- pi/2))" |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3494 |
by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. - pi/2 < x \<and> x < pi/2" and P="\<lambda>x. True" and g=arctan]) |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56409
diff
changeset
|
3495 |
(auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1 |
50346
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3496 |
intro!: tan_monotone exI[of _ "pi/2"]) |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3497 |
|
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3498 |
lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))" |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3499 |
by (rule filterlim_at_top_at_left[where Q="\<lambda>x. - pi/2 < x \<and> x < pi/2" and P="\<lambda>x. True" and g=arctan]) |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56409
diff
changeset
|
3500 |
(auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1 |
50346
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3501 |
intro!: tan_monotone exI[of _ "pi/2"]) |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3502 |
|
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3503 |
lemma tendsto_arctan_at_top: "(arctan ---> (pi/2)) at_top" |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3504 |
proof (rule tendstoI) |
53079 | 3505 |
fix e :: real |
3506 |
assume "0 < e" |
|
50346
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3507 |
def y \<equiv> "pi/2 - min (pi/2) e" |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3508 |
then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y" |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3509 |
using `0 < e` by auto |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3510 |
|
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3511 |
show "eventually (\<lambda>x. dist (arctan x) (pi / 2) < e) at_top" |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3512 |
proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI) |
53079 | 3513 |
fix x |
3514 |
assume "tan y < x" |
|
50346
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3515 |
then have "arctan (tan y) < arctan x" |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3516 |
by (simp add: arctan_less_iff) |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3517 |
with y have "y < arctan x" |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3518 |
by (subst (asm) arctan_tan) simp_all |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3519 |
with arctan_ubound[of x, arith] y `0 < e` |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3520 |
show "dist (arctan x) (pi / 2) < e" |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3521 |
by (simp add: dist_real_def) |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3522 |
qed |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3523 |
qed |
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3524 |
|
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3525 |
lemma tendsto_arctan_at_bot: "(arctan ---> - (pi/2)) at_bot" |
53079 | 3526 |
unfolding filterlim_at_bot_mirror arctan_minus |
3527 |
by (intro tendsto_minus tendsto_arctan_at_top) |
|
3528 |
||
50346
a75c6429c3c3
add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents:
50326
diff
changeset
|
3529 |
|
23043 | 3530 |
subsection {* More Theorems about Sin and Cos *} |
3531 |
||
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3532 |
lemma cos_45: "cos (pi / 4) = sqrt 2 / 2" |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3533 |
proof - |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3534 |
let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)" |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3535 |
have nonneg: "0 \<le> ?c" |
45308
2e84e5f0463b
extend cancellation simproc patterns to cover terms like '- (2 * pi) < pi'
huffman
parents:
44756
diff
changeset
|
3536 |
by (simp add: cos_ge_zero) |
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3537 |
have "0 = cos (pi / 4 + pi / 4)" |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3538 |
by simp |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3539 |
also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2" |
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3540 |
by (simp only: cos_add power2_eq_square) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3541 |
also have "\<dots> = 2 * ?c\<^sup>2 - 1" |
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3542 |
by (simp add: sin_squared_eq) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3543 |
finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2" |
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3544 |
by (simp add: power_divide) |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3545 |
thus ?thesis |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3546 |
using nonneg by (rule power2_eq_imp_eq) simp |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3547 |
qed |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3548 |
|
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3549 |
lemma cos_30: "cos (pi / 6) = sqrt 3 / 2" |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3550 |
proof - |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3551 |
let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)" |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3552 |
have pos_c: "0 < ?c" |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3553 |
by (rule cos_gt_zero, simp, simp) |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3554 |
have "0 = cos (pi / 6 + pi / 6 + pi / 6)" |
23066
26a9157b620a
new field_combine_numerals simproc, which uses fractions as coefficients
huffman
parents:
23053
diff
changeset
|
3555 |
by simp |
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3556 |
also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s" |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3557 |
by (simp only: cos_add sin_add) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3558 |
also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)" |
29667 | 3559 |
by (simp add: algebra_simps power2_eq_square) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
3560 |
finally have "?c\<^sup>2 = (sqrt 3 / 2)\<^sup>2" |
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3561 |
using pos_c by (simp add: sin_squared_eq power_divide) |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3562 |
thus ?thesis |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3563 |
using pos_c [THEN order_less_imp_le] |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3564 |
by (rule power2_eq_imp_eq) simp |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3565 |
qed |
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3566 |
|
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3567 |
lemma sin_45: "sin (pi / 4) = sqrt 2 / 2" |
53079 | 3568 |
by (simp add: sin_cos_eq cos_45) |
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3569 |
|
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3570 |
lemma sin_60: "sin (pi / 3) = sqrt 3 / 2" |
53079 | 3571 |
by (simp add: sin_cos_eq cos_30) |
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3572 |
|
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3573 |
lemma cos_60: "cos (pi / 3) = 1 / 2" |
53079 | 3574 |
apply (rule power2_eq_imp_eq) |
3575 |
apply (simp add: cos_squared_eq sin_60 power_divide) |
|
3576 |
apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all) |
|
3577 |
done |
|
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3578 |
|
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3579 |
lemma sin_30: "sin (pi / 6) = 1 / 2" |
53079 | 3580 |
by (simp add: sin_cos_eq cos_60) |
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3581 |
|
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3582 |
lemma tan_30: "tan (pi / 6) = 1 / sqrt 3" |
53079 | 3583 |
unfolding tan_def by (simp add: sin_30 cos_30) |
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3584 |
|
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3585 |
lemma tan_45: "tan (pi / 4) = 1" |
53079 | 3586 |
unfolding tan_def by (simp add: sin_45 cos_45) |
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3587 |
|
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3588 |
lemma tan_60: "tan (pi / 3) = sqrt 3" |
53079 | 3589 |
unfolding tan_def by (simp add: sin_60 cos_60) |
23052
0e36f0dbfa1c
add lemmas for sin,cos,tan of 30,45,60 degrees; cleaned up
huffman
parents:
23049
diff
changeset
|
3590 |
|
15383 | 3591 |
lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n" |
3592 |
proof - |
|
3593 |
have "sin ((real n + 1/2) * pi) = cos (real n * pi)" |
|
29667 | 3594 |
by (auto simp add: algebra_simps sin_add) |
15383 | 3595 |
thus ?thesis |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
47489
diff
changeset
|
3596 |
by (simp add: real_of_nat_Suc distrib_right add_divide_distrib |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
3597 |
mult.commute [of pi]) |
15383 | 3598 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3599 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3600 |
lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1" |
58740 | 3601 |
by (cases "even n") (simp_all add: cos_double mult.assoc) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3602 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3603 |
lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0" |
53079 | 3604 |
apply (subgoal_tac "cos (pi + pi/2) = 0", simp) |
3605 |
apply (subst cos_add, simp) |
|
3606 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3607 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3608 |
lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
3609 |
by (auto simp add: mult.assoc) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3610 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3611 |
lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1" |
53079 | 3612 |
apply (subgoal_tac "sin (pi + pi/2) = - 1", simp) |
3613 |
apply (subst sin_add, simp) |
|
3614 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3615 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3616 |
lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0" |
53079 | 3617 |
apply (simp only: cos_add sin_add real_of_nat_Suc distrib_right distrib_left add_divide_distrib) |
3618 |
apply auto |
|
3619 |
done |
|
3620 |
||
3621 |
lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)" |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
3622 |
by (auto intro!: derivative_eq_intros) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3623 |
|
15081 | 3624 |
lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1" |
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3625 |
by (auto simp add: sin_zero_iff elim: evenE) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3626 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3627 |
lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0" |
53079 | 3628 |
using sin_cos_squared_add3 [where x = x] by auto |
3629 |
||
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3630 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3631 |
subsection {* Machins formula *} |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3632 |
|
44746 | 3633 |
lemma arctan_one: "arctan 1 = pi / 4" |
3634 |
by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi) |
|
3635 |
||
53079 | 3636 |
lemma tan_total_pi4: |
3637 |
assumes "\<bar>x\<bar> < 1" |
|
3638 |
shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x" |
|
44746 | 3639 |
proof |
3640 |
show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x" |
|
3641 |
unfolding arctan_one [symmetric] arctan_minus [symmetric] |
|
3642 |
unfolding arctan_less_iff using assms by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3643 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3644 |
|
53079 | 3645 |
lemma arctan_add: |
3646 |
assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3647 |
shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))" |
44746 | 3648 |
proof (rule arctan_unique [symmetric]) |
3649 |
have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y" |
|
3650 |
unfolding arctan_one [symmetric] arctan_minus [symmetric] |
|
3651 |
unfolding arctan_le_iff arctan_less_iff using assms by auto |
|
3652 |
from add_le_less_mono [OF this] |
|
3653 |
show 1: "- (pi / 2) < arctan x + arctan y" by simp |
|
3654 |
have "arctan x \<le> pi / 4" and "arctan y < pi / 4" |
|
3655 |
unfolding arctan_one [symmetric] |
|
3656 |
unfolding arctan_le_iff arctan_less_iff using assms by auto |
|
3657 |
from add_le_less_mono [OF this] |
|
3658 |
show 2: "arctan x + arctan y < pi / 2" by simp |
|
3659 |
show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)" |
|
3660 |
using cos_gt_zero_pi [OF 1 2] by (simp add: tan_add) |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3661 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3662 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3663 |
theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3664 |
proof - |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3665 |
have "\<bar>1 / 5\<bar> < (1 :: real)" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3666 |
from arctan_add[OF less_imp_le[OF this] this] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3667 |
have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3668 |
moreover |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3669 |
have "\<bar>5 / 12\<bar> < (1 :: real)" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3670 |
from arctan_add[OF less_imp_le[OF this] this] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3671 |
have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto |
41970 | 3672 |
moreover |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3673 |
have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3674 |
from arctan_add[OF this] |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3675 |
have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3676 |
ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto |
44746 | 3677 |
thus ?thesis unfolding arctan_one by algebra |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3678 |
qed |
44746 | 3679 |
|
53079 | 3680 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3681 |
subsection {* Introducing the arcus tangens power series *} |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3682 |
|
53079 | 3683 |
lemma monoseq_arctan_series: |
3684 |
fixes x :: real |
|
3685 |
assumes "\<bar>x\<bar> \<le> 1" |
|
3686 |
shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a") |
|
3687 |
proof (cases "x = 0") |
|
3688 |
case True |
|
3689 |
thus ?thesis unfolding monoseq_def One_nat_def by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3690 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3691 |
case False |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3692 |
have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3693 |
show "monoseq ?a" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3694 |
proof - |
53079 | 3695 |
{ |
3696 |
fix n |
|
3697 |
fix x :: real |
|
3698 |
assume "0 \<le> x" and "x \<le> 1" |
|
3699 |
have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le> |
|
3700 |
1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3701 |
proof (rule mult_mono) |
53079 | 3702 |
show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))" |
3703 |
by (rule frac_le) simp_all |
|
3704 |
show "0 \<le> 1 / real (Suc (n * 2))" |
|
3705 |
by auto |
|
3706 |
show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)" |
|
3707 |
by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`) |
|
3708 |
show "0 \<le> x ^ Suc (Suc n * 2)" |
|
3709 |
by (rule zero_le_power) (simp add: `0 \<le> x`) |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3710 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3711 |
} note mono = this |
41970 | 3712 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3713 |
show ?thesis |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3714 |
proof (cases "0 \<le> x") |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3715 |
case True from mono[OF this `x \<le> 1`, THEN allI] |
53079 | 3716 |
show ?thesis unfolding Suc_eq_plus1[symmetric] |
3717 |
by (rule mono_SucI2) |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3718 |
next |
53079 | 3719 |
case False |
3720 |
hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3721 |
from mono[OF this] |
53079 | 3722 |
have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge> |
3723 |
1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using `0 \<le> -x` by auto |
|
31790 | 3724 |
thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI]) |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3725 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3726 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3727 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3728 |
|
53079 | 3729 |
lemma zeroseq_arctan_series: |
3730 |
fixes x :: real |
|
3731 |
assumes "\<bar>x\<bar> \<le> 1" |
|
3732 |
shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0") |
|
3733 |
proof (cases "x = 0") |
|
3734 |
case True |
|
3735 |
thus ?thesis |
|
58729
e8ecc79aee43
add tendsto_const and tendsto_ident_at as simp and intro rules
hoelzl
parents:
58710
diff
changeset
|
3736 |
unfolding One_nat_def by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3737 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3738 |
case False |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3739 |
have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3740 |
show "?a ----> 0" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3741 |
proof (cases "\<bar>x\<bar> < 1") |
53079 | 3742 |
case True |
3743 |
hence "norm x < 1" by auto |
|
44568
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44319
diff
changeset
|
3744 |
from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]] |
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset
|
3745 |
have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0" |
31790 | 3746 |
unfolding inverse_eq_divide Suc_eq_plus1 by simp |
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset
|
3747 |
then show ?thesis using pos2 by (rule LIMSEQ_linear) |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3748 |
next |
53079 | 3749 |
case False |
3750 |
hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto |
|
3751 |
hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x" |
|
3752 |
unfolding One_nat_def by auto |
|
44568
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44319
diff
changeset
|
3753 |
from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]] |
31790 | 3754 |
show ?thesis unfolding n_eq Suc_eq_plus1 by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3755 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3756 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3757 |
|
53079 | 3758 |
lemma summable_arctan_series: |
3759 |
fixes x :: real and n :: nat |
|
3760 |
assumes "\<bar>x\<bar> \<le> 1" |
|
3761 |
shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" |
|
3762 |
(is "summable (?c x)") |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3763 |
by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms]) |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3764 |
|
53079 | 3765 |
lemma less_one_imp_sqr_less_one: |
3766 |
fixes x :: real |
|
3767 |
assumes "\<bar>x\<bar> < 1" |
|
3768 |
shows "x\<^sup>2 < 1" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3769 |
proof - |
54573 | 3770 |
have "\<bar>x\<^sup>2\<bar> < 1" |
3771 |
by (metis abs_power2 assms pos2 power2_abs power_0 power_strict_decreasing zero_eq_power2 zero_less_abs_iff) |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3772 |
thus ?thesis using zero_le_power2 by auto |
41970 | 3773 |
qed |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3774 |
|
53079 | 3775 |
lemma DERIV_arctan_series: |
3776 |
assumes "\<bar> x \<bar> < 1" |
|
3777 |
shows "DERIV (\<lambda> x'. \<Sum> k. (-1)^k * (1 / real (k*2+1) * x' ^ (k*2+1))) x :> (\<Sum> k. (-1)^k * x^(k*2))" |
|
3778 |
(is "DERIV ?arctan _ :> ?Int") |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3779 |
proof - |
53079 | 3780 |
let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0" |
3781 |
||
3782 |
have n_even: "\<And>n :: nat. even n \<Longrightarrow> 2 * (n div 2) = n" |
|
3783 |
by presburger |
|
3784 |
then have if_eq: "\<And>n x'. ?f n * real (Suc n) * x'^n = |
|
3785 |
(if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)" |
|
3786 |
by auto |
|
3787 |
||
3788 |
{ |
|
3789 |
fix x :: real |
|
3790 |
assume "\<bar>x\<bar> < 1" |
|
3791 |
hence "x\<^sup>2 < 1" by (rule less_one_imp_sqr_less_one) |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3792 |
have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)" |
53076 | 3793 |
by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x\<^sup>2 < 1` order_less_imp_le[OF `x\<^sup>2 < 1`]) |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3794 |
hence "summable (\<lambda> n. (- 1) ^ n * x^(2*n))" unfolding power_mult . |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3795 |
} note summable_Integral = this |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3796 |
|
53079 | 3797 |
{ |
3798 |
fix f :: "nat \<Rightarrow> real" |
|
3799 |
have "\<And>x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3800 |
proof |
53079 | 3801 |
fix x :: real |
3802 |
assume "f sums x" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3803 |
from sums_if[OF sums_zero this] |
53079 | 3804 |
show "(\<lambda>n. if even n then f (n div 2) else 0) sums x" |
3805 |
by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3806 |
next |
53079 | 3807 |
fix x :: real |
3808 |
assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x" |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
3809 |
from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult.commute]] |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3810 |
show "f sums x" unfolding sums_def by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3811 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3812 |
hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" .. |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3813 |
} note sums_even = this |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3814 |
|
53079 | 3815 |
have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int" |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3816 |
unfolding if_eq mult.commute[of _ 2] suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * x ^ (2 * n)", symmetric] |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3817 |
by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3818 |
|
53079 | 3819 |
{ |
3820 |
fix x :: real |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3821 |
have if_eq': "\<And>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n = |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3822 |
(if even n then (- 1) ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3823 |
using n_even by auto |
53079 | 3824 |
have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)" by auto |
3825 |
have "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x" |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3826 |
unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric] |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3827 |
by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3828 |
} note arctan_eq = this |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3829 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3830 |
have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3831 |
proof (rule DERIV_power_series') |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3832 |
show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto |
53079 | 3833 |
{ |
3834 |
fix x' :: real |
|
3835 |
assume x'_bounds: "x' \<in> {- 1 <..< 1}" |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3836 |
then have "\<bar>x'\<bar> < 1" by auto |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3837 |
then |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3838 |
have *: "summable (\<lambda>n. (- 1) ^ n * x' ^ (2 * n))" |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3839 |
by (rule summable_Integral) |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3840 |
let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3841 |
show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3842 |
apply (rule sums_summable [where l="0 + ?S"]) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3843 |
apply (rule sums_if) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3844 |
apply (rule sums_zero) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3845 |
apply (rule summable_sums) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3846 |
apply (rule *) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3847 |
done |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3848 |
} |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3849 |
qed auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3850 |
thus ?thesis unfolding Int_eq arctan_eq . |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3851 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3852 |
|
53079 | 3853 |
lemma arctan_series: |
3854 |
assumes "\<bar> x \<bar> \<le> 1" |
|
3855 |
shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" |
|
3856 |
(is "_ = suminf (\<lambda> n. ?c x n)") |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3857 |
proof - |
53079 | 3858 |
let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)" |
3859 |
||
3860 |
{ |
|
3861 |
fix r x :: real |
|
3862 |
assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3863 |
have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto |
53079 | 3864 |
from DERIV_arctan_series[OF this] have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" . |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3865 |
} note DERIV_arctan_suminf = this |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3866 |
|
53079 | 3867 |
{ |
3868 |
fix x :: real |
|
3869 |
assume "\<bar>x\<bar> \<le> 1" |
|
3870 |
note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] |
|
3871 |
} note arctan_series_borders = this |
|
3872 |
||
3873 |
{ |
|
3874 |
fix x :: real |
|
3875 |
assume "\<bar>x\<bar> < 1" |
|
3876 |
have "arctan x = (\<Sum>k. ?c x k)" |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3877 |
proof - |
53079 | 3878 |
obtain r where "\<bar>x\<bar> < r" and "r < 1" |
3879 |
using dense[OF `\<bar>x\<bar> < 1`] by blast |
|
3880 |
hence "0 < r" and "-r < x" and "x < r" by auto |
|
3881 |
||
3882 |
have suminf_eq_arctan_bounded: "\<And>x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow> |
|
3883 |
suminf (?c x) - arctan x = suminf (?c a) - arctan a" |
|
3884 |
proof - |
|
3885 |
fix x a b |
|
3886 |
assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b" |
|
3887 |
hence "\<bar>x\<bar> < r" by auto |
|
3888 |
show "suminf (?c x) - arctan x = suminf (?c a) - arctan a" |
|
3889 |
proof (rule DERIV_isconst2[of "a" "b"]) |
|
3890 |
show "a < b" and "a \<le> x" and "x \<le> b" |
|
3891 |
using `a < b` `a \<le> x` `x \<le> b` by auto |
|
3892 |
have "\<forall>x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0" |
|
3893 |
proof (rule allI, rule impI) |
|
3894 |
fix x |
|
3895 |
assume "-r < x \<and> x < r" |
|
3896 |
hence "\<bar>x\<bar> < r" by auto |
|
3897 |
hence "\<bar>x\<bar> < 1" using `r < 1` by auto |
|
3898 |
have "\<bar> - (x\<^sup>2) \<bar> < 1" |
|
3899 |
using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto |
|
3900 |
hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))" |
|
3901 |
unfolding real_norm_def[symmetric] by (rule geometric_sums) |
|
3902 |
hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))" |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
3903 |
unfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2] by auto |
53079 | 3904 |
hence suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)" |
3905 |
using sums_unique unfolding inverse_eq_divide by auto |
|
3906 |
have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))" |
|
3907 |
unfolding suminf_c'_eq_geom |
|
3908 |
by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`]) |
|
56261 | 3909 |
from DERIV_diff [OF this DERIV_arctan] |
53079 | 3910 |
show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
3911 |
by auto |
53079 | 3912 |
qed |
3913 |
hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0" |
|
3914 |
using `-r < a` `b < r` by auto |
|
3915 |
thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0" |
|
3916 |
using `\<bar>x\<bar> < r` by auto |
|
3917 |
show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y" |
|
3918 |
using DERIV_in_rball DERIV_isCont by auto |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3919 |
qed |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3920 |
qed |
53079 | 3921 |
|
3922 |
have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0" |
|
3923 |
unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero |
|
3924 |
by auto |
|
3925 |
||
3926 |
have "suminf (?c x) - arctan x = 0" |
|
3927 |
proof (cases "x = 0") |
|
3928 |
case True |
|
3929 |
thus ?thesis using suminf_arctan_zero by auto |
|
3930 |
next |
|
3931 |
case False |
|
3932 |
hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto |
|
3933 |
have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0" |
|
3934 |
by (rule suminf_eq_arctan_bounded[where x="0" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>", symmetric]) |
|
3935 |
(simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less) |
|
3936 |
moreover |
|
3937 |
have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)" |
|
3938 |
by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"]) |
|
54573 | 3939 |
(simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less) |
53079 | 3940 |
ultimately |
3941 |
show ?thesis using suminf_arctan_zero by auto |
|
3942 |
qed |
|
3943 |
thus ?thesis by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3944 |
qed |
53079 | 3945 |
} note when_less_one = this |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3946 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3947 |
show "arctan x = suminf (\<lambda> n. ?c x n)" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3948 |
proof (cases "\<bar>x\<bar> < 1") |
53079 | 3949 |
case True |
3950 |
thus ?thesis by (rule when_less_one) |
|
3951 |
next |
|
3952 |
case False |
|
3953 |
hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto |
|
3954 |
let ?a = "\<lambda>x n. \<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>" |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
3955 |
let ?diff = "\<lambda> x n. \<bar> arctan x - (\<Sum> i<n. ?c x i)\<bar>" |
53079 | 3956 |
{ |
3957 |
fix n :: nat |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3958 |
have "0 < (1 :: real)" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3959 |
moreover |
53079 | 3960 |
{ |
3961 |
fix x :: real |
|
3962 |
assume "0 < x" and "x < 1" |
|
3963 |
hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto |
|
3964 |
from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)" |
|
3965 |
by auto |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3966 |
note bounds = mp[OF arctan_series_borders(2)[OF `\<bar>x\<bar> \<le> 1`] this, unfolded when_less_one[OF `\<bar>x\<bar> < 1`, symmetric], THEN spec] |
53079 | 3967 |
have "0 < 1 / real (n*2+1) * x^(n*2+1)" |
3968 |
by (rule mult_pos_pos, auto simp only: zero_less_power[OF `0 < x`], auto) |
|
3969 |
hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)" |
|
3970 |
by (rule abs_of_pos) |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3971 |
have "?diff x n \<le> ?a x n" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3972 |
proof (cases "even n") |
53079 | 3973 |
case True |
3974 |
hence sgn_pos: "(-1)^n = (1::real)" by auto |
|
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3975 |
from `even n` obtain m where "n = 2 * m" .. |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3976 |
then have "2 * m = n" .. |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3977 |
from bounds[of m, unfolded this atLeastAtMost_iff] |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
3978 |
have "\<bar>arctan x - (\<Sum>i<n. (?c x i))\<bar> \<le> (\<Sum>i<n + 1. (?c x i)) - (\<Sum>i<n. (?c x i))" |
53079 | 3979 |
by auto |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3980 |
also have "\<dots> = ?c x n" unfolding One_nat_def by auto |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3981 |
also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3982 |
finally show ?thesis . |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3983 |
next |
53079 | 3984 |
case False |
3985 |
hence sgn_neg: "(-1)^n = (-1::real)" by auto |
|
58709
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3986 |
from `odd n` obtain m where "n = 2 * m + 1" .. |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3987 |
then have m_def: "2 * m + 1 = n" .. |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3988 |
hence m_plus: "2 * (m + 1) = n + 1" by auto |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3989 |
from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2] |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
3990 |
have "\<bar>arctan x - (\<Sum>i<n. (?c x i))\<bar> \<le> (\<Sum>i<n. (?c x i)) - (\<Sum>i<n+1. (?c x i))" |
53079 | 3991 |
by auto |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3992 |
also have "\<dots> = - ?c x n" unfolding One_nat_def by auto |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3993 |
also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3994 |
finally show ?thesis . |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
3995 |
qed |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3996 |
hence "0 \<le> ?a x n - ?diff x n" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3997 |
} |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3998 |
hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3999 |
moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
4000 |
unfolding diff_conv_add_uminus divide_inverse |
53079 | 4001 |
by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
4002 |
isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
4003 |
simp del: add_uminus_conv_diff) |
53079 | 4004 |
ultimately have "0 \<le> ?a 1 n - ?diff 1 n" |
4005 |
by (rule LIM_less_bound) |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4006 |
hence "?diff 1 n \<le> ?a 1 n" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4007 |
} |
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset
|
4008 |
have "?a 1 ----> 0" |
44568
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44319
diff
changeset
|
4009 |
unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def |
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44319
diff
changeset
|
4010 |
by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat) |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4011 |
have "?diff 1 ----> 0" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4012 |
proof (rule LIMSEQ_I) |
53079 | 4013 |
fix r :: real |
4014 |
assume "0 < r" |
|
4015 |
obtain N :: nat where N_I: "\<And>n. N \<le> n \<Longrightarrow> ?a 1 n < r" |
|
4016 |
using LIMSEQ_D[OF `?a 1 ----> 0` `0 < r`] by auto |
|
4017 |
{ |
|
4018 |
fix n |
|
4019 |
assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this] |
|
4020 |
have "norm (?diff 1 n - 0) < r" by auto |
|
4021 |
} |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4022 |
thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4023 |
qed |
44710 | 4024 |
from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus] |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4025 |
have "(?c 1) sums (arctan 1)" unfolding sums_def by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4026 |
hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique) |
41970 | 4027 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4028 |
show ?thesis |
53079 | 4029 |
proof (cases "x = 1") |
4030 |
case True |
|
4031 |
then show ?thesis by (simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`) |
|
4032 |
next |
|
4033 |
case False |
|
4034 |
hence "x = -1" using `\<bar>x\<bar> = 1` by auto |
|
41970 | 4035 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4036 |
have "- (pi / 2) < 0" using pi_gt_zero by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4037 |
have "- (2 * pi) < 0" using pi_gt_zero by auto |
41970 | 4038 |
|
53079 | 4039 |
have c_minus_minus: "\<And>i. ?c (- 1) i = - ?c 1 i" |
4040 |
unfolding One_nat_def by auto |
|
4041 |
||
4042 |
have "arctan (- 1) = arctan (tan (-(pi / 4)))" |
|
4043 |
unfolding tan_45 tan_minus .. |
|
4044 |
also have "\<dots> = - (pi / 4)" |
|
4045 |
by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero]) |
|
4046 |
also have "\<dots> = - (arctan (tan (pi / 4)))" |
|
4047 |
unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero]) |
|
4048 |
also have "\<dots> = - (arctan 1)" |
|
4049 |
unfolding tan_45 .. |
|
4050 |
also have "\<dots> = - (\<Sum> i. ?c 1 i)" |
|
4051 |
using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto |
|
4052 |
also have "\<dots> = (\<Sum> i. ?c (- 1) i)" |
|
4053 |
using suminf_minus[OF sums_summable[OF `(?c 1) sums (arctan 1)`]] |
|
4054 |
unfolding c_minus_minus by auto |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4055 |
finally show ?thesis using `x = -1` by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4056 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4057 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4058 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4059 |
|
53079 | 4060 |
lemma arctan_half: |
4061 |
fixes x :: real |
|
53076 | 4062 |
shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))" |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4063 |
proof - |
53079 | 4064 |
obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x" |
4065 |
using tan_total by blast |
|
4066 |
hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2" |
|
4067 |
by auto |
|
4068 |
||
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4069 |
have "0 < cos y" using cos_gt_zero_pi[OF low high] . |
53079 | 4070 |
hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y" |
4071 |
by auto |
|
4072 |
||
4073 |
have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2" |
|
4074 |
unfolding tan_def power_divide .. |
|
4075 |
also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2" |
|
4076 |
using `cos y \<noteq> 0` by auto |
|
4077 |
also have "\<dots> = 1 / (cos y)\<^sup>2" |
|
4078 |
unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 .. |
|
53076 | 4079 |
finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" . |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4080 |
|
53079 | 4081 |
have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)" |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
4082 |
unfolding tan_def using `cos y \<noteq> 0` by (simp add: field_simps) |
53079 | 4083 |
also have "\<dots> = tan y / (1 + 1 / cos y)" |
4084 |
using `cos y \<noteq> 0` unfolding add_divide_distrib by auto |
|
4085 |
also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))" |
|
4086 |
unfolding cos_sqrt .. |
|
4087 |
also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))" |
|
4088 |
unfolding real_sqrt_divide by auto |
|
4089 |
finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))" |
|
4090 |
unfolding `1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2` . |
|
4091 |
||
4092 |
have "arctan x = y" |
|
4093 |
using arctan_tan low high y_eq by auto |
|
4094 |
also have "\<dots> = 2 * (arctan (tan (y/2)))" |
|
4095 |
using arctan_tan[OF low2 high2] by auto |
|
4096 |
also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" |
|
4097 |
unfolding tan_half by auto |
|
4098 |
finally show ?thesis |
|
4099 |
unfolding eq `tan y = x` . |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4100 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4101 |
|
53079 | 4102 |
lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y" |
4103 |
by (simp only: arctan_less_iff) |
|
4104 |
||
4105 |
lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y" |
|
4106 |
by (simp only: arctan_le_iff) |
|
44746 | 4107 |
|
4108 |
lemma arctan_inverse: |
|
53079 | 4109 |
assumes "x \<noteq> 0" |
4110 |
shows "arctan (1 / x) = sgn x * pi / 2 - arctan x" |
|
44746 | 4111 |
proof (rule arctan_unique) |
4112 |
show "- (pi / 2) < sgn x * pi / 2 - arctan x" |
|
4113 |
using arctan_bounded [of x] assms |
|
4114 |
unfolding sgn_real_def |
|
4115 |
apply (auto simp add: algebra_simps) |
|
4116 |
apply (drule zero_less_arctan_iff [THEN iffD2]) |
|
4117 |
apply arith |
|
4118 |
done |
|
4119 |
show "sgn x * pi / 2 - arctan x < pi / 2" |
|
4120 |
using arctan_bounded [of "- x"] assms |
|
4121 |
unfolding sgn_real_def arctan_minus |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
4122 |
by (auto simp add: algebra_simps) |
44746 | 4123 |
show "tan (sgn x * pi / 2 - arctan x) = 1 / x" |
4124 |
unfolding tan_inverse [of "arctan x", unfolded tan_arctan] |
|
4125 |
unfolding sgn_real_def |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56409
diff
changeset
|
4126 |
by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff) |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4127 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4128 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4129 |
theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM") |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4130 |
proof - |
44746 | 4131 |
have "pi / 4 = arctan 1" using arctan_one by auto |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4132 |
also have "\<dots> = ?SUM" using arctan_series[of 1] by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4133 |
finally show ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4134 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4135 |
|
53079 | 4136 |
|
22978
1cd8cc21a7c3
clean up polar_Ex proofs; remove unnecessary lemmas
huffman
parents:
22977
diff
changeset
|
4137 |
subsection {* Existence of Polar Coordinates *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4138 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
4139 |
lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1" |
53079 | 4140 |
apply (rule power2_le_imp_le [OF _ zero_le_one]) |
4141 |
apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero) |
|
4142 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4143 |
|
22978
1cd8cc21a7c3
clean up polar_Ex proofs; remove unnecessary lemmas
huffman
parents:
22977
diff
changeset
|
4144 |
lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y" |
53079 | 4145 |
by (simp add: abs_le_iff) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4146 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
4147 |
lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)" |
53079 | 4148 |
by (simp add: sin_arccos abs_le_iff) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4149 |
|
22978
1cd8cc21a7c3
clean up polar_Ex proofs; remove unnecessary lemmas
huffman
parents:
22977
diff
changeset
|
4150 |
lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one] |
15228 | 4151 |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
4152 |
lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one] |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4153 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4154 |
lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a" |
54573 | 4155 |
proof - |
4156 |
have polar_ex1: "\<And>y. 0 < y \<Longrightarrow> \<exists>r a. x = r * cos a & y = r * sin a" |
|
4157 |
apply (rule_tac x = "sqrt (x\<^sup>2 + y\<^sup>2)" in exI) |
|
4158 |
apply (rule_tac x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))" in exI) |
|
4159 |
apply (simp add: cos_arccos_lemma1 sin_arccos_lemma1 power_divide |
|
4160 |
real_sqrt_mult [symmetric] right_diff_distrib) |
|
4161 |
done |
|
4162 |
show ?thesis |
|
4163 |
proof (cases "0::real" y rule: linorder_cases) |
|
4164 |
case less |
|
4165 |
then show ?thesis by (rule polar_ex1) |
|
4166 |
next |
|
4167 |
case equal |
|
4168 |
then show ?thesis |
|
4169 |
by (force simp add: intro!: cos_zero sin_zero) |
|
4170 |
next |
|
4171 |
case greater |
|
4172 |
then show ?thesis |
|
4173 |
using polar_ex1 [where y="-y"] |
|
4174 |
by auto (metis cos_minus minus_minus minus_mult_right sin_minus) |
|
4175 |
qed |
|
4176 |
qed |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4177 |
|
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset
|
4178 |
end |