author | paulson <lp15@cam.ac.uk> |
Tue, 31 Mar 2015 15:00:03 +0100 | |
changeset 59862 | 44b3f4fa33ca |
parent 59751 | 916c0f6c83e3 |
child 59865 | 8a20dd967385 |
permissions | -rw-r--r-- |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
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 |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
10 |
imports Binomial Series Deriv NthRoot |
15131 | 11 |
begin |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
12 |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
13 |
lemma of_real_fact [simp]: "of_real (fact n) = fact n" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
14 |
by (metis of_nat_fact of_real_of_nat_eq) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
15 |
|
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
16 |
lemma real_fact_nat [simp]: "real (fact n :: nat) = fact n" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
17 |
by (simp add: real_of_nat_def) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
18 |
|
59731 | 19 |
lemma real_fact_int [simp]: "real (fact n :: int) = fact n" |
20 |
by (metis of_int_of_nat_eq of_nat_fact real_of_int_def) |
|
21 |
||
57025 | 22 |
lemma root_test_convergence: |
23 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
|
24 |
assumes f: "(\<lambda>n. root n (norm (f n))) ----> x" -- "could be weakened to lim sup" |
|
25 |
assumes "x < 1" |
|
26 |
shows "summable f" |
|
27 |
proof - |
|
28 |
have "0 \<le> x" |
|
29 |
by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1]) |
|
30 |
from `x < 1` obtain z where z: "x < z" "z < 1" |
|
31 |
by (metis dense) |
|
32 |
from f `x < z` |
|
33 |
have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially" |
|
34 |
by (rule order_tendstoD) |
|
35 |
then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially" |
|
36 |
using eventually_ge_at_top |
|
37 |
proof eventually_elim |
|
38 |
fix n assume less: "root n (norm (f n)) < z" and n: "1 \<le> n" |
|
39 |
from power_strict_mono[OF less, of n] n |
|
40 |
show "norm (f n) \<le> z ^ n" |
|
41 |
by simp |
|
42 |
qed |
|
43 |
then show "summable f" |
|
44 |
unfolding eventually_sequentially |
|
45 |
using z `0 \<le> x` by (auto intro!: summable_comparison_test[OF _ summable_geometric]) |
|
46 |
qed |
|
47 |
||
29164 | 48 |
subsection {* Properties of Power Series *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
49 |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
50 |
lemma lemma_realpow_diff: |
31017 | 51 |
fixes y :: "'a::monoid_mult" |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
52 |
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
|
53 |
proof - |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
54 |
assume "p \<le> n" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
55 |
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
|
56 |
thus ?thesis by (simp add: power_commutes) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
57 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
58 |
|
15229 | 59 |
lemma lemma_realpow_diff_sumr2: |
53079 | 60 |
fixes y :: "'a::{comm_ring,monoid_mult}" |
61 |
shows |
|
62 |
"x ^ (Suc n) - y ^ (Suc n) = |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
63 |
(x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))" |
54573 | 64 |
proof (induct n) |
65 |
case (Suc n) |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
66 |
have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x^n) - y * (y * y ^ n)" |
54573 | 67 |
by simp |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
68 |
also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x^n)" |
54573 | 69 |
by (simp add: algebra_simps) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
70 |
also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)" |
54573 | 71 |
by (simp only: Suc) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
72 |
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
|
73 |
by (simp only: mult.left_commute) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
74 |
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
|
75 |
by (simp add: field_simps Suc_diff_le setsum_left_distrib setsum_right_distrib) |
54573 | 76 |
finally show ?case . |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
77 |
qed simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
78 |
|
55832 | 79 |
corollary power_diff_sumr2: --{* @{text COMPLEX_POLYFUN} in HOL Light *} |
55734 | 80 |
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
|
81 |
shows "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)" |
55734 | 82 |
using lemma_realpow_diff_sumr2[of x "n - 1" y] |
83 |
by (cases "n = 0") (simp_all add: field_simps) |
|
84 |
||
15229 | 85 |
lemma lemma_realpow_rev_sumr: |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
86 |
"(\<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
|
87 |
(\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))" |
57129
7edb7550663e
introduce more powerful reindexing rules for big operators
hoelzl
parents:
57025
diff
changeset
|
88 |
by (subst nat_diff_setsum_reindex[symmetric]) simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
89 |
|
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
90 |
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
|
91 |
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
|
92 |
shows "n \<noteq> 0 \<Longrightarrow> x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
93 |
using lemma_realpow_diff_sumr2 [of x _ 1] |
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
94 |
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
|
95 |
|
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
96 |
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
|
97 |
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
|
98 |
shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
99 |
using lemma_realpow_diff_sumr2 [of 1 _ x] |
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
100 |
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
|
101 |
|
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
55417
diff
changeset
|
102 |
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
|
103 |
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
|
104 |
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
|
105 |
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
|
106 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
107 |
text{*Power series has a `circle` of convergence, i.e. if it sums for @{term |
53079 | 108 |
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
|
109 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
110 |
lemma powser_insidea: |
53599 | 111 |
fixes x z :: "'a::real_normed_div_algebra" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
112 |
assumes 1: "summable (\<lambda>n. f n * x^n)" |
53079 | 113 |
and 2: "norm z < norm x" |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
114 |
shows "summable (\<lambda>n. norm (f n * z ^ n))" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
115 |
proof - |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
116 |
from 2 have x_neq_0: "x \<noteq> 0" by clarsimp |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
117 |
from 1 have "(\<lambda>n. f n * x^n) ----> 0" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
118 |
by (rule summable_LIMSEQ_zero) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
119 |
hence "convergent (\<lambda>n. f n * x^n)" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
120 |
by (rule convergentI) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
121 |
hence "Cauchy (\<lambda>n. f n * x^n)" |
44726 | 122 |
by (rule convergent_Cauchy) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
123 |
hence "Bseq (\<lambda>n. f n * x^n)" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
124 |
by (rule Cauchy_Bseq) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
125 |
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
|
126 |
by (simp add: Bseq_def, safe) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
127 |
have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le> |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
128 |
K * norm (z ^ n) * inverse (norm (x^n))" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
129 |
proof (intro exI allI impI) |
53079 | 130 |
fix n::nat |
131 |
assume "0 \<le> n" |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
132 |
have "norm (norm (f n * z ^ n)) * norm (x^n) = |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
133 |
norm (f n * x^n) * norm (z ^ n)" |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
134 |
by (simp add: norm_mult abs_mult) |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
135 |
also have "\<dots> \<le> K * norm (z ^ n)" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
136 |
by (simp only: mult_right_mono 4 norm_ge_zero) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
137 |
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
|
138 |
by (simp add: x_neq_0) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
139 |
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
|
140 |
by (simp only: mult.assoc) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
141 |
finally show "norm (norm (f n * z ^ n)) \<le> |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
142 |
K * norm (z ^ n) * inverse (norm (x^n))" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
143 |
by (simp add: mult_le_cancel_right x_neq_0) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
144 |
qed |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
145 |
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
|
146 |
proof - |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
147 |
from 2 have "norm (norm (z * inverse x)) < 1" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
148 |
using x_neq_0 |
53599 | 149 |
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
|
150 |
hence "summable (\<lambda>n. norm (z * inverse x) ^ n)" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
151 |
by (rule summable_geometric) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
152 |
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
|
153 |
by (rule summable_mult) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
154 |
thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))" |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
155 |
using x_neq_0 |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
156 |
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
|
157 |
power_inverse norm_power mult.assoc) |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
158 |
qed |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
159 |
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
|
160 |
by (rule summable_comparison_test) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
161 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
162 |
|
15229 | 163 |
lemma powser_inside: |
53599 | 164 |
fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}" |
53079 | 165 |
shows |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
166 |
"summable (\<lambda>n. f n * (x^n)) \<Longrightarrow> norm z < norm x \<Longrightarrow> |
53079 | 167 |
summable (\<lambda>n. f n * (z ^ n))" |
168 |
by (rule powser_insidea [THEN summable_norm_cancel]) |
|
169 |
||
170 |
lemma sum_split_even_odd: |
|
171 |
fixes f :: "nat \<Rightarrow> real" |
|
172 |
shows |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
173 |
"(\<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
|
174 |
(\<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
|
175 |
proof (induct n) |
53079 | 176 |
case 0 |
177 |
then show ?case by simp |
|
178 |
next |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
179 |
case (Suc n) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
180 |
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
|
181 |
(\<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
|
182 |
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
|
183 |
also have "\<dots> = (\<Sum>i<Suc n. f (2 * i)) + (\<Sum>i<Suc n. g (2 * i + 1))" |
53079 | 184 |
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
|
185 |
finally show ?case . |
53079 | 186 |
qed |
187 |
||
188 |
lemma sums_if': |
|
189 |
fixes g :: "nat \<Rightarrow> real" |
|
190 |
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
|
191 |
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
|
192 |
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
|
193 |
proof (rule LIMSEQ_I) |
53079 | 194 |
fix r :: real |
195 |
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
|
196 |
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
|
197 |
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
|
198 |
|
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
199 |
let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)" |
53079 | 200 |
{ |
201 |
fix m |
|
202 |
assume "m \<ge> 2 * no" |
|
203 |
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
|
204 |
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
|
205 |
using sum_split_even_odd by auto |
53079 | 206 |
hence "(norm (?SUM (2 * (m div 2)) - x) < r)" |
207 |
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
|
208 |
moreover |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
209 |
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
|
210 |
proof (cases "even m") |
53079 | 211 |
case True |
58710
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58709
diff
changeset
|
212 |
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
|
213 |
next |
53079 | 214 |
case False |
58834 | 215 |
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
|
216 |
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
|
217 |
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
|
218 |
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
|
219 |
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
|
220 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
221 |
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
|
222 |
} |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
223 |
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
|
224 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
225 |
|
53079 | 226 |
lemma sums_if: |
227 |
fixes g :: "nat \<Rightarrow> real" |
|
228 |
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
|
229 |
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
|
230 |
proof - |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
231 |
let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)" |
53079 | 232 |
{ |
233 |
fix B T E |
|
234 |
have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)" |
|
235 |
by (cases B) auto |
|
236 |
} note if_sum = this |
|
237 |
have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" |
|
238 |
using sums_if'[OF `g sums x`] . |
|
41970 | 239 |
{ |
41550 | 240 |
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
|
241 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
242 |
have "?s sums y" using sums_if'[OF `f sums y`] . |
41970 | 243 |
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
|
244 |
have "(\<lambda> n. if even n then f (n div 2) else 0) sums y" |
57418 | 245 |
by (simp add: lessThan_Suc_eq_insert_0 image_iff setsum.reindex if_eq sums_def cong del: if_cong) |
53079 | 246 |
} |
247 |
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
|
248 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
249 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
250 |
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
|
251 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
252 |
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
|
253 |
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
|
254 |
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
|
255 |
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
|
256 |
((\<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
|
257 |
(is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)") |
53079 | 258 |
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
|
259 |
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
|
260 |
|
53079 | 261 |
show "\<forall>n. ?f n \<le> ?f (Suc n)" |
262 |
proof |
|
263 |
fix n |
|
264 |
show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto |
|
265 |
qed |
|
266 |
show "\<forall>n. ?g (Suc n) \<le> ?g n" |
|
267 |
proof |
|
268 |
fix n |
|
269 |
show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"] |
|
270 |
unfolding One_nat_def by auto |
|
271 |
qed |
|
272 |
show "\<forall>n. ?f n \<le> ?g n" |
|
273 |
proof |
|
274 |
fix n |
|
275 |
show "?f n \<le> ?g n" using fg_diff a_pos |
|
276 |
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
|
277 |
qed |
53079 | 278 |
show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff |
279 |
proof (rule LIMSEQ_I) |
|
280 |
fix r :: real |
|
281 |
assume "0 < r" |
|
282 |
with `a ----> 0`[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" |
|
283 |
by auto |
|
284 |
hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto |
|
285 |
thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto |
|
286 |
qed |
|
41970 | 287 |
qed |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
288 |
|
53079 | 289 |
lemma summable_Leibniz': |
290 |
fixes a :: "nat \<Rightarrow> real" |
|
291 |
assumes a_zero: "a ----> 0" |
|
292 |
and a_pos: "\<And> n. 0 \<le> a n" |
|
293 |
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
|
294 |
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
|
295 |
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
|
296 |
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
|
297 |
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
|
298 |
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
|
299 |
proof - |
53079 | 300 |
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
|
301 |
let ?P = "\<lambda>n. \<Sum>i<n. ?S i" |
53079 | 302 |
let ?f = "\<lambda>n. ?P (2 * n)" |
303 |
let ?g = "\<lambda>n. ?P (2 * n + 1)" |
|
304 |
obtain l :: real |
|
305 |
where below_l: "\<forall> n. ?f n \<le> l" |
|
306 |
and "?f ----> l" |
|
307 |
and above_l: "\<forall> n. l \<le> ?g n" |
|
308 |
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
|
309 |
using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast |
41970 | 310 |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
311 |
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
|
312 |
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
|
313 |
proof (rule LIMSEQ_I) |
53079 | 314 |
fix r :: real |
315 |
assume "0 < r" |
|
41970 | 316 |
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
|
317 |
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
|
318 |
|
41970 | 319 |
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
|
320 |
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
|
321 |
|
53079 | 322 |
{ |
323 |
fix n :: nat |
|
324 |
assume "n \<ge> (max (2 * f_no) (2 * g_no))" |
|
325 |
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
|
326 |
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
|
327 |
proof (cases "even n") |
53079 | 328 |
case True |
58710
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58709
diff
changeset
|
329 |
then have n_eq: "2 * (n div 2) = n" by (simp add: even_two_times_div_two) |
53079 | 330 |
with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" |
331 |
by auto |
|
332 |
from f[OF this] show ?thesis |
|
333 |
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
|
334 |
next |
53079 | 335 |
case False |
336 |
hence "even (n - 1)" by simp |
|
58710
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
58709
diff
changeset
|
337 |
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
|
338 |
by (simp add: even_two_times_div_two) |
53079 | 339 |
hence range_eq: "n - 1 + 1 = n" |
340 |
using odd_pos[OF False] by auto |
|
341 |
||
342 |
from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" |
|
343 |
by auto |
|
344 |
from g[OF this] show ?thesis |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
345 |
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
|
346 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
347 |
} |
53079 | 348 |
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
|
349 |
qed |
53079 | 350 |
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
|
351 |
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
|
352 |
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
|
353 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
354 |
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
|
355 |
|
53079 | 356 |
fix n |
357 |
show "suminf ?S \<le> ?g n" |
|
358 |
unfolding sums_unique[OF sums_l, symmetric] using above_l by auto |
|
359 |
show "?f n \<le> suminf ?S" |
|
360 |
unfolding sums_unique[OF sums_l, symmetric] using below_l by auto |
|
361 |
show "?g ----> suminf ?S" |
|
362 |
using `?g ----> l` `l = suminf ?S` by auto |
|
363 |
show "?f ----> suminf ?S" |
|
364 |
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
|
365 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
366 |
|
53079 | 367 |
theorem summable_Leibniz: |
368 |
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
|
369 |
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
|
370 |
shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable") |
53079 | 371 |
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
|
372 |
(\<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 | 373 |
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
|
374 |
(\<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
|
375 |
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
|
376 |
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
|
377 |
proof - |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
378 |
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
|
379 |
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
|
380 |
case True |
53079 | 381 |
hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" |
382 |
by auto |
|
383 |
{ |
|
384 |
fix n |
|
385 |
have "a (Suc n) \<le> a n" |
|
386 |
using ord[where n="Suc n" and m=n] by auto |
|
387 |
} note mono = this |
|
388 |
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
|
389 |
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
|
390 |
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
|
391 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
392 |
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
|
393 |
case False |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
394 |
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
|
395 |
have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto |
53079 | 396 |
hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" |
397 |
by auto |
|
398 |
{ |
|
399 |
fix n |
|
400 |
have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] |
|
401 |
by auto |
|
402 |
} note monotone = this |
|
403 |
note leibniz = |
|
404 |
summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", |
|
405 |
OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone] |
|
406 |
have "summable (\<lambda> n. (-1)^n * ?a n)" |
|
407 |
using leibniz(1) by auto |
|
408 |
then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" |
|
409 |
unfolding summable_def by auto |
|
410 |
from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l" |
|
411 |
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
|
412 |
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
|
413 |
moreover |
53079 | 414 |
have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>" |
415 |
unfolding minus_diff_minus by auto |
|
41970 | 416 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
417 |
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
|
418 |
have move_minus: "(\<Sum>n. - ((- 1) ^ n * a n)) = - (\<Sum>n. (- 1) ^ n * a n)" |
53079 | 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 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
421 |
have ?pos using `0 \<le> ?a 0` by auto |
53079 | 422 |
moreover have ?neg |
423 |
using leibniz(2,4) |
|
424 |
unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le |
|
425 |
by auto |
|
426 |
moreover have ?f and ?g |
|
427 |
using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel] |
|
428 |
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
|
429 |
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
|
430 |
qed |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
431 |
then show ?summable and ?pos and ?neg and ?f and ?g |
54573 | 432 |
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
|
433 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
434 |
|
29164 | 435 |
subsection {* Term-by-Term Differentiability of Power Series *} |
23043 | 436 |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
437 |
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
|
438 |
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
|
439 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
440 |
text{*Lemma about distributing negation over it*} |
53079 | 441 |
lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)" |
442 |
by (simp add: diffs_def) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
443 |
|
29163 | 444 |
lemma sums_Suc_imp: |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
445 |
"(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
|
446 |
using sums_Suc_iff[of f] by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
447 |
|
15229 | 448 |
lemma diffs_equiv: |
41970 | 449 |
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
|
450 |
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
|
451 |
(\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)" |
53079 | 452 |
unfolding diffs_def |
54573 | 453 |
by (simp add: summable_sums sums_Suc_imp) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
454 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
455 |
lemma lemma_termdiff1: |
31017 | 456 |
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
|
457 |
"(\<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
|
458 |
(\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))" |
53079 | 459 |
by (auto simp add: algebra_simps power_add [symmetric]) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
460 |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
461 |
lemma sumr_diff_mult_const2: |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
462 |
"setsum f {..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i<n. f i - r)" |
53079 | 463 |
by (simp add: setsum_subtractf) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
464 |
|
15229 | 465 |
lemma lemma_termdiff2: |
31017 | 466 |
fixes h :: "'a :: {field}" |
53079 | 467 |
assumes h: "h \<noteq> 0" |
468 |
shows |
|
469 |
"((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
|
470 |
h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p. |
53079 | 471 |
(z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs") |
472 |
apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h) |
|
473 |
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
|
474 |
apply (simp add: mult.assoc [symmetric]) |
53079 | 475 |
apply (cases "n", simp) |
476 |
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
|
477 |
right_diff_distrib [symmetric] mult.assoc |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
478 |
del: power_Suc setsum_lessThan_Suc of_nat_Suc) |
53079 | 479 |
apply (subst lemma_realpow_rev_sumr) |
480 |
apply (subst sumr_diff_mult_const2) |
|
481 |
apply simp |
|
482 |
apply (simp only: lemma_termdiff1 setsum_right_distrib) |
|
57418 | 483 |
apply (rule setsum.cong [OF refl]) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
484 |
apply (simp add: less_iff_Suc_add) |
53079 | 485 |
apply (clarify) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
486 |
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
|
487 |
del: setsum_lessThan_Suc power_Suc) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
488 |
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
|
489 |
apply (simp add: ac_simps) |
53079 | 490 |
done |
20860 | 491 |
|
492 |
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
|
493 |
fixes K :: "'a::linordered_semidom" |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
494 |
assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K" |
53079 | 495 |
and K: "0 \<le> K" |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
496 |
shows "setsum f {..<n-k} \<le> of_nat n * K" |
53079 | 497 |
apply (rule order_trans [OF setsum_mono]) |
498 |
apply (rule f, simp) |
|
499 |
apply (simp add: mult_right_mono K) |
|
500 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
501 |
|
15229 | 502 |
lemma lemma_termdiff3: |
31017 | 503 |
fixes h z :: "'a::{real_normed_field}" |
20860 | 504 |
assumes 1: "h \<noteq> 0" |
53079 | 505 |
and 2: "norm z \<le> K" |
506 |
and 3: "norm (z + h) \<le> K" |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
507 |
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
|
508 |
\<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h" |
20860 | 509 |
proof - |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
510 |
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
|
511 |
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
|
512 |
(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
|
513 |
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
|
514 |
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
|
515 |
proof (rule mult_right_mono [OF _ norm_ge_zero]) |
53079 | 516 |
from norm_ge_zero 2 have K: "0 \<le> K" |
517 |
by (rule order_trans) |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
518 |
have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n" |
20860 | 519 |
apply (erule subst) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
520 |
apply (simp only: norm_mult norm_power power_add) |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
521 |
apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K) |
20860 | 522 |
done |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
523 |
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
|
524 |
\<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))" |
20860 | 525 |
apply (intro |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
526 |
order_trans [OF norm_setsum] |
20860 | 527 |
real_setsum_nat_ivl_bounded2 |
528 |
mult_nonneg_nonneg |
|
47489 | 529 |
of_nat_0_le_iff |
20860 | 530 |
zero_le_power K) |
531 |
apply (rule le_Kn, simp) |
|
532 |
done |
|
533 |
qed |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
534 |
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
|
535 |
by (simp only: mult.assoc) |
20860 | 536 |
finally show ?thesis . |
537 |
qed |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
538 |
|
20860 | 539 |
lemma lemma_termdiff4: |
56167 | 540 |
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" |
20860 | 541 |
assumes k: "0 < (k::real)" |
53079 | 542 |
and le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h" |
20860 | 543 |
shows "f -- 0 --> 0" |
56167 | 544 |
proof (rule tendsto_norm_zero_cancel) |
545 |
show "(\<lambda>h. norm (f h)) -- 0 --> 0" |
|
546 |
proof (rule real_tendsto_sandwich) |
|
547 |
show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)" |
|
20860 | 548 |
by simp |
56167 | 549 |
show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)" |
550 |
using k by (auto simp add: eventually_at dist_norm le) |
|
551 |
show "(\<lambda>h. 0) -- (0::'a) --> (0::real)" |
|
552 |
by (rule tendsto_const) |
|
553 |
have "(\<lambda>h. K * norm h) -- (0::'a) --> K * norm (0::'a)" |
|
554 |
by (intro tendsto_intros) |
|
555 |
then show "(\<lambda>h. K * norm h) -- (0::'a) --> 0" |
|
556 |
by simp |
|
20860 | 557 |
qed |
558 |
qed |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
559 |
|
15229 | 560 |
lemma lemma_termdiff5: |
56167 | 561 |
fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach" |
20860 | 562 |
assumes k: "0 < (k::real)" |
563 |
assumes f: "summable f" |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
564 |
assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h" |
20860 | 565 |
shows "(\<lambda>h. suminf (g h)) -- 0 --> 0" |
566 |
proof (rule lemma_termdiff4 [OF k]) |
|
53079 | 567 |
fix h::'a |
568 |
assume "h \<noteq> 0" and "norm h < k" |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
569 |
hence A: "\<forall>n. norm (g h n) \<le> f n * norm h" |
20860 | 570 |
by (simp add: le) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
571 |
hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h" |
20860 | 572 |
by simp |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
573 |
moreover from f have B: "summable (\<lambda>n. f n * norm h)" |
20860 | 574 |
by (rule summable_mult2) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
575 |
ultimately have C: "summable (\<lambda>n. norm (g h n))" |
20860 | 576 |
by (rule summable_comparison_test) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
577 |
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
|
578 |
by (rule summable_norm) |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
579 |
also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)" |
56213 | 580 |
by (rule suminf_le) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
581 |
also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h" |
20860 | 582 |
by (rule suminf_mult2 [symmetric]) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
583 |
finally show "norm (suminf (g h)) \<le> suminf f * norm h" . |
20860 | 584 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
585 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
586 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
587 |
text{* FIXME: Long proofs*} |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
588 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
589 |
lemma termdiffs_aux: |
31017 | 590 |
fixes x :: "'a::{real_normed_field,banach}" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
591 |
assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)" |
53079 | 592 |
and 2: "norm x < norm K" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
593 |
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
|
594 |
- of_nat n * x ^ (n - Suc 0))) -- 0 --> 0" |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
595 |
proof - |
20860 | 596 |
from dense [OF 2] |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
597 |
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
|
598 |
from norm_ge_zero r1 have r: "0 < r" |
20860 | 599 |
by (rule order_le_less_trans) |
600 |
hence r_neq_0: "r \<noteq> 0" by simp |
|
601 |
show ?thesis |
|
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
602 |
proof (rule lemma_termdiff5) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
603 |
show "0 < r - norm x" using r1 by simp |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
604 |
from r r2 have "norm (of_real r::'a) < norm K" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
605 |
by simp |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
606 |
with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))" |
20860 | 607 |
by (rule powser_insidea) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
608 |
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
|
609 |
using r |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
610 |
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
|
611 |
hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))" |
20860 | 612 |
by (rule diffs_equiv [THEN sums_summable]) |
53079 | 613 |
also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) = |
614 |
(\<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
|
615 |
apply (rule ext) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
616 |
apply (simp add: diffs_def) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
617 |
apply (case_tac n, simp_all add: r_neq_0) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
618 |
done |
41970 | 619 |
finally have "summable |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
620 |
(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))" |
20860 | 621 |
by (rule diffs_equiv [THEN sums_summable]) |
622 |
also have |
|
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
623 |
"(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * |
20860 | 624 |
r ^ (n - Suc 0)) = |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
625 |
(\<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
|
626 |
apply (rule ext) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
627 |
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
|
628 |
apply (rename_tac nat) |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
629 |
apply (case_tac "nat", simp) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
630 |
apply (simp add: r_neq_0) |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
631 |
done |
53079 | 632 |
finally |
633 |
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
|
634 |
next |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
635 |
fix h::'a and n::nat |
20860 | 636 |
assume h: "h \<noteq> 0" |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
637 |
assume "norm h < r - norm x" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
638 |
hence "norm x + norm h < r" by simp |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
639 |
with norm_triangle_ineq have xh: "norm (x + h) < r" |
20860 | 640 |
by (rule order_le_less_trans) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
641 |
show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
642 |
\<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
|
643 |
apply (simp only: norm_mult mult.assoc) |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
644 |
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
|
645 |
apply (simp add: mult.assoc [symmetric]) |
54575 | 646 |
apply (metis h lemma_termdiff3 less_eq_real_def r1 xh) |
20860 | 647 |
done |
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
648 |
qed |
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset
|
649 |
qed |
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19765
diff
changeset
|
650 |
|
20860 | 651 |
lemma termdiffs: |
31017 | 652 |
fixes K x :: "'a::{real_normed_field,banach}" |
20860 | 653 |
assumes 1: "summable (\<lambda>n. c n * K ^ n)" |
54575 | 654 |
and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)" |
655 |
and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)" |
|
656 |
and 4: "norm x < norm K" |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
657 |
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
|
658 |
unfolding DERIV_def |
29163 | 659 |
proof (rule LIM_zero_cancel) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
660 |
show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x^n)) / h |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
661 |
- suminf (\<lambda>n. diffs c n * x^n)) -- 0 --> 0" |
20860 | 662 |
proof (rule LIM_equal2) |
29163 | 663 |
show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq) |
20860 | 664 |
next |
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
665 |
fix h :: 'a |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
666 |
assume "norm (h - 0) < norm K - norm x" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
667 |
hence "norm x + norm h < norm K" by simp |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
668 |
hence 5: "norm (x + h) < norm K" |
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset
|
669 |
by (rule norm_triangle_ineq [THEN order_le_less_trans]) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
670 |
have "summable (\<lambda>n. c n * x^n)" |
56167 | 671 |
and "summable (\<lambda>n. c n * (x + h) ^ n)" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
672 |
and "summable (\<lambda>n. diffs c n * x^n)" |
56167 | 673 |
using 1 2 4 5 by (auto elim: powser_inside) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
674 |
then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) = |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
675 |
(\<Sum>n. (c n * (x + h) ^ n - c n * x^n) / h - of_nat n * c n * x ^ (n - Suc 0))" |
56167 | 676 |
by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
677 |
then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) = |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
678 |
(\<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))" |
54575 | 679 |
by (simp add: algebra_simps) |
20860 | 680 |
next |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
681 |
show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0" |
53079 | 682 |
by (rule termdiffs_aux [OF 3 4]) |
20860 | 683 |
qed |
684 |
qed |
|
685 |
||
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
686 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
687 |
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
|
688 |
|
53079 | 689 |
lemma DERIV_series': |
690 |
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
|
691 |
assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)" |
53079 | 692 |
and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}" |
693 |
and "summable (f' x0)" |
|
694 |
and "summable L" |
|
695 |
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
|
696 |
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
|
697 |
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
|
698 |
proof (rule LIM_I) |
53079 | 699 |
fix r :: real |
700 |
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
|
701 |
|
41970 | 702 |
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
|
703 |
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
|
704 |
|
41970 | 705 |
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
|
706 |
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
|
707 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
708 |
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
|
709 |
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
|
710 |
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
|
711 |
|
53079 | 712 |
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
|
713 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
714 |
let ?r = "r / (3 * real ?N)" |
56541 | 715 |
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
|
716 |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
717 |
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
|
718 |
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
|
719 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
720 |
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
|
721 |
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
|
722 |
show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x" |
53079 | 723 |
proof |
724 |
fix x |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
725 |
assume "x \<in> ?s ` {..<?N}" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
726 |
then obtain n where "x = ?s n" and "n \<in> {..<?N}" |
53079 | 727 |
using image_iff[THEN iffD1] by blast |
41970 | 728 |
from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def] |
53079 | 729 |
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)" |
730 |
by auto |
|
731 |
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
|
732 |
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
|
733 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
734 |
qed auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
735 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
736 |
def S \<equiv> "min (min (x0 - a) (b - x0)) S'" |
53079 | 737 |
hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0" |
738 |
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
|
739 |
by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
740 |
|
53079 | 741 |
{ |
742 |
fix x |
|
743 |
assume "x \<noteq> 0" and "\<bar> x \<bar> < S" |
|
744 |
hence x_in_I: "x0 + x \<in> { a <..< b }" |
|
745 |
using S_a S_b by auto |
|
41970 | 746 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
747 |
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
|
748 |
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
|
749 |
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
|
750 |
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
|
751 |
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
|
752 |
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
|
753 |
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
|
754 |
|
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
755 |
{ fix n |
41970 | 756 |
have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>" |
53079 | 757 |
using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] |
758 |
unfolding abs_divide . |
|
759 |
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
|
760 |
using `x \<noteq> 0` by auto } |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
761 |
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
|
762 |
then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" |
56213 | 763 |
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
|
764 |
then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") |
53079 | 765 |
using L_estimate by auto |
766 |
||
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
767 |
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
|
768 |
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
|
769 |
proof (rule setsum_strict_mono) |
53079 | 770 |
fix n |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
771 |
assume "n \<in> {..< ?N}" |
53079 | 772 |
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
|
773 |
also have "S \<le> S'" using `S \<le> S'` . |
41970 | 774 |
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
|
775 |
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
|
776 |
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
|
777 |
using `n \<in> {..< ?N}` by auto |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
778 |
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
|
779 |
qed auto |
53079 | 780 |
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
|
781 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
782 |
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
|
783 |
have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" . |
53079 | 784 |
with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n` show "\<bar>?diff n x - f' x0 n\<bar> < ?r" |
785 |
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
|
786 |
qed auto |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
787 |
also have "\<dots> = of_nat (card {..<?N}) * ?r" |
53079 | 788 |
by (rule setsum_constant) |
789 |
also have "\<dots> = real ?N * ?r" |
|
790 |
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
|
791 |
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
|
792 |
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
|
793 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
794 |
from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]] |
53079 | 795 |
have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> = |
796 |
\<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>" |
|
797 |
unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] |
|
798 |
using suminf_divide[OF diff_smbl, symmetric] by auto |
|
799 |
also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>" |
|
800 |
unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] |
|
801 |
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
|
802 |
apply (subst (5) add.commute) |
53079 | 803 |
by (rule abs_triangle_ineq) |
804 |
also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" |
|
805 |
using abs_triangle_ineq4 by auto |
|
41970 | 806 |
also have "\<dots> < r /3 + r/3 + r/3" |
36842 | 807 |
using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3` |
808 |
by (rule add_strict_mono [OF add_less_le_mono]) |
|
53079 | 809 |
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
|
810 |
by auto |
53079 | 811 |
} |
812 |
thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow> |
|
813 |
norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" |
|
814 |
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
|
815 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
816 |
|
53079 | 817 |
lemma DERIV_power_series': |
818 |
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
|
819 |
assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)" |
53079 | 820 |
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
|
821 |
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
|
822 |
(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
|
823 |
proof - |
53079 | 824 |
{ |
825 |
fix R' |
|
826 |
assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'" |
|
827 |
hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" |
|
828 |
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
|
829 |
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
|
830 |
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
|
831 |
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
|
832 |
proof - |
53079 | 833 |
have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" |
834 |
using `0 < R'` `0 < R` `R' < R` by auto |
|
835 |
hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" |
|
836 |
using `R' < R` by auto |
|
837 |
have "norm R' < norm ((R' + R) / 2)" |
|
838 |
using `0 < R'` `0 < R` `R' < R` by auto |
|
839 |
from powser_insidea[OF converges[OF in_Rball] this] show ?thesis |
|
840 |
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
|
841 |
qed |
53079 | 842 |
{ |
843 |
fix n x y |
|
844 |
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
|
845 |
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
|
846 |
proof - |
53079 | 847 |
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
|
848 |
(\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>" |
53079 | 849 |
unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult |
850 |
by auto |
|
41970 | 851 |
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
|
852 |
proof (rule mult_left_mono) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
853 |
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 | 854 |
by (rule setsum_abs) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
855 |
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
|
856 |
proof (rule setsum_mono) |
53079 | 857 |
fix p |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
858 |
assume "p \<in> {..<Suc n}" |
53079 | 859 |
hence "p \<le> n" by auto |
860 |
{ |
|
861 |
fix n |
|
862 |
fix x :: real |
|
863 |
assume "x \<in> {-R'<..<R'}" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
864 |
hence "\<bar>x\<bar> \<le> R'" by auto |
53079 | 865 |
hence "\<bar>x^n\<bar> \<le> R'^n" |
866 |
unfolding power_abs by (rule power_mono, auto) |
|
867 |
} |
|
868 |
from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'` |
|
869 |
have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" |
|
870 |
unfolding abs_mult by auto |
|
871 |
thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" |
|
872 |
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
|
873 |
qed |
53079 | 874 |
also have "\<dots> = real (Suc n) * R' ^ n" |
875 |
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
|
876 |
finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>" |
53079 | 877 |
unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] . |
878 |
show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" |
|
879 |
unfolding abs_mult[symmetric] by auto |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
880 |
qed |
53079 | 881 |
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
|
882 |
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
|
883 |
finally show ?thesis . |
53079 | 884 |
qed |
885 |
} |
|
886 |
{ |
|
887 |
fix n |
|
888 |
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
|
889 |
by (auto intro!: derivative_eq_intros simp del: power_Suc simp: real_of_nat_def) |
53079 | 890 |
} |
891 |
{ |
|
892 |
fix x |
|
893 |
assume "x \<in> {-R' <..< R'}" |
|
894 |
hence "R' \<in> {-R <..< R}" and "norm x < norm R'" |
|
895 |
using assms `R' < R` by auto |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
896 |
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
|
897 |
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
|
898 |
fix n |
53079 | 899 |
have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" |
900 |
by (rule mult_left_mono) auto |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
901 |
show "norm (f n * x^n) \<le> norm (f n * real (Suc n) * x^n)" |
53079 | 902 |
unfolding real_norm_def abs_mult |
903 |
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
|
904 |
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
|
905 |
from this[THEN summable_mult2[where c=x], unfolded mult.assoc, unfolded mult.commute] |
53079 | 906 |
show "summable (?f x)" by auto |
907 |
} |
|
908 |
show "summable (?f' x0)" |
|
909 |
using converges[OF `x0 \<in> {-R <..< R}`] . |
|
910 |
show "x0 \<in> {-R' <..< R'}" |
|
911 |
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
|
912 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
913 |
} 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
|
914 |
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
|
915 |
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
|
916 |
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
|
917 |
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
|
918 |
case True |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
919 |
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
|
920 |
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
|
921 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
922 |
case False |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
923 |
have "- ?R < 0" using assms by auto |
41970 | 924 |
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
|
925 |
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
|
926 |
qed |
53079 | 927 |
hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" |
928 |
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
|
929 |
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
|
930 |
show ?thesis . |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
931 |
qed |
29695 | 932 |
|
53079 | 933 |
|
29164 | 934 |
subsection {* Exponential Function *} |
23043 | 935 |
|
58656 | 936 |
definition exp :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
937 |
where "exp = (\<lambda>x. \<Sum>n. x^n /\<^sub>R fact n)" |
23043 | 938 |
|
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
939 |
lemma summable_exp_generic: |
31017 | 940 |
fixes x :: "'a::{real_normed_algebra_1,banach}" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
941 |
defines S_def: "S \<equiv> \<lambda>n. x^n /\<^sub>R fact n" |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
942 |
shows "summable S" |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
943 |
proof - |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
944 |
have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R (Suc n)" |
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents:
30082
diff
changeset
|
945 |
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
|
946 |
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
|
947 |
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
|
948 |
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
|
949 |
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
|
950 |
from r1 show ?thesis |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
951 |
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
|
952 |
fix n :: nat |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
953 |
assume n: "N \<le> n" |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
954 |
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
|
955 |
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
|
956 |
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
|
957 |
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
|
958 |
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
|
959 |
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
|
960 |
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
|
961 |
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
|
962 |
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
|
963 |
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
|
964 |
thus "norm (S (Suc n)) \<le> r * norm (S n)" |
35216 | 965 |
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
|
966 |
qed |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
967 |
qed |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
968 |
|
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
969 |
lemma summable_norm_exp: |
31017 | 970 |
fixes x :: "'a::{real_normed_algebra_1,banach}" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
971 |
shows "summable (\<lambda>n. norm (x^n /\<^sub>R fact n))" |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
972 |
proof (rule summable_norm_comparison_test [OF exI, rule_format]) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
973 |
show "summable (\<lambda>n. norm x^n /\<^sub>R fact n)" |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
974 |
by (rule summable_exp_generic) |
53079 | 975 |
fix n |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
976 |
show "norm (x^n /\<^sub>R fact n) \<le> norm x^n /\<^sub>R fact n" |
35216 | 977 |
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
|
978 |
qed |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
979 |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
980 |
lemma summable_exp: |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
981 |
fixes x :: "'a::{real_normed_field,banach}" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
982 |
shows "summable (\<lambda>n. inverse (fact n) * x^n)" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
983 |
using summable_exp_generic [where x=x] |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
984 |
by (simp add: scaleR_conv_of_real nonzero_of_real_inverse) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
985 |
|
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
986 |
lemma exp_converges: "(\<lambda>n. x^n /\<^sub>R fact n) sums exp x" |
53079 | 987 |
unfolding exp_def by (rule summable_exp_generic [THEN summable_sums]) |
23043 | 988 |
|
41970 | 989 |
lemma exp_fdiffs: |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
990 |
fixes XXX :: "'a::{real_normed_field,banach}" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
991 |
shows "diffs (\<lambda>n. inverse (fact n)) = (\<lambda>n. inverse (fact n :: 'a))" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
992 |
by (simp add: diffs_def mult_ac nonzero_inverse_mult_distrib nonzero_of_real_inverse |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
993 |
del: mult_Suc of_nat_Suc) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
994 |
|
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
995 |
lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))" |
53079 | 996 |
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
|
997 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
998 |
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)" |
53079 | 999 |
unfolding exp_def scaleR_conv_of_real |
1000 |
apply (rule DERIV_cong) |
|
1001 |
apply (rule termdiffs [where K="of_real (1 + norm x)"]) |
|
1002 |
apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs) |
|
1003 |
apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+ |
|
1004 |
apply (simp del: of_real_add) |
|
1005 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1006 |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1007 |
declare DERIV_exp[THEN DERIV_chain2, derivative_intros] |
51527 | 1008 |
|
58656 | 1009 |
lemma norm_exp: "norm (exp x) \<le> exp (norm x)" |
1010 |
proof - |
|
1011 |
from summable_norm[OF summable_norm_exp, of x] |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1012 |
have "norm (exp x) \<le> (\<Sum>n. inverse (fact n) * norm (x^n))" |
58656 | 1013 |
by (simp add: exp_def) |
1014 |
also have "\<dots> \<le> exp (norm x)" |
|
1015 |
using summable_exp_generic[of "norm x"] summable_norm_exp[of x] |
|
1016 |
by (auto simp: exp_def intro!: suminf_le norm_power_ineq) |
|
1017 |
finally show ?thesis . |
|
1018 |
qed |
|
1019 |
||
1020 |
lemma isCont_exp: |
|
1021 |
fixes x::"'a::{real_normed_field,banach}" |
|
1022 |
shows "isCont exp x" |
|
44311 | 1023 |
by (rule DERIV_exp [THEN DERIV_isCont]) |
1024 |
||
58656 | 1025 |
lemma isCont_exp' [simp]: |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1026 |
fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}" |
58656 | 1027 |
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a" |
44311 | 1028 |
by (rule isCont_o2 [OF _ isCont_exp]) |
1029 |
||
1030 |
lemma tendsto_exp [tendsto_intros]: |
|
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1031 |
fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}" |
58656 | 1032 |
shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F" |
44311 | 1033 |
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
|
1034 |
|
53079 | 1035 |
lemma continuous_exp [continuous_intros]: |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1036 |
fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}" |
58656 | 1037 |
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
|
1038 |
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
|
1039 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
1040 |
lemma continuous_on_exp [continuous_intros]: |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1041 |
fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}" |
58656 | 1042 |
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
|
1043 |
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
|
1044 |
|
53079 | 1045 |
|
29167 | 1046 |
subsubsection {* Properties of the Exponential Function *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1047 |
|
23278 | 1048 |
lemma powser_zero: |
31017 | 1049 |
fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}" |
23278 | 1050 |
shows "(\<Sum>n. f n * 0 ^ n) = f 0" |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1051 |
proof - |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
1052 |
have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)" |
56213 | 1053 |
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
|
1054 |
thus ?thesis unfolding One_nat_def by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1055 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1056 |
|
23278 | 1057 |
lemma exp_zero [simp]: "exp 0 = 1" |
53079 | 1058 |
unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero) |
23278 | 1059 |
|
58656 | 1060 |
lemma exp_series_add_commuting: |
1061 |
fixes x y :: "'a::{real_normed_algebra_1, banach}" |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1062 |
defines S_def: "S \<equiv> \<lambda>x n. x^n /\<^sub>R fact n" |
58656 | 1063 |
assumes comm: "x * y = y * x" |
56213 | 1064 |
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
|
1065 |
proof (induct n) |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1066 |
case 0 |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1067 |
show ?case |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1068 |
unfolding S_def by simp |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1069 |
next |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1070 |
case (Suc n) |
25062 | 1071 |
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
|
1072 |
unfolding S_def by (simp del: mult_Suc) |
25062 | 1073 |
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
|
1074 |
by simp |
58656 | 1075 |
have S_comm: "\<And>n. S x n * y = y * S x n" |
1076 |
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
|
1077 |
|
25062 | 1078 |
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
|
1079 |
by (simp only: times_S) |
56213 | 1080 |
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
|
1081 |
by (simp only: Suc) |
56213 | 1082 |
also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n-i)) |
1083 |
+ 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
|
1084 |
by (rule distrib_right) |
58656 | 1085 |
also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i)) |
1086 |
+ (\<Sum>i\<le>n. S x i * y * S y (n-i))" |
|
1087 |
by (simp add: setsum_right_distrib ac_simps S_comm) |
|
1088 |
also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i)) |
|
56213 | 1089 |
+ (\<Sum>i\<le>n. S x i * (y * S y (n-i)))" |
58656 | 1090 |
by (simp add: ac_simps) |
56213 | 1091 |
also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) |
1092 |
+ (\<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
|
1093 |
by (simp add: times_S Suc_diff_le) |
56213 | 1094 |
also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) = |
1095 |
(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))" |
|
1096 |
by (subst setsum_atMost_Suc_shift) simp |
|
1097 |
also have "(\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) = |
|
1098 |
(\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))" |
|
1099 |
by simp |
|
1100 |
also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) + |
|
1101 |
(\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) = |
|
1102 |
(\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))" |
|
57418 | 1103 |
by (simp only: setsum.distrib [symmetric] scaleR_left_distrib [symmetric] |
56213 | 1104 |
real_of_nat_add [symmetric]) simp |
1105 |
also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n-i))" |
|
23127 | 1106 |
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
|
1107 |
finally show |
56213 | 1108 |
"S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))" |
35216 | 1109 |
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
|
1110 |
qed |
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1111 |
|
58656 | 1112 |
lemma exp_add_commuting: "x * y = y * x \<Longrightarrow> exp (x + y) = exp x * exp y" |
53079 | 1113 |
unfolding exp_def |
58656 | 1114 |
by (simp only: Cauchy_product summable_norm_exp exp_series_add_commuting) |
1115 |
||
1116 |
lemma exp_add: |
|
1117 |
fixes x y::"'a::{real_normed_field,banach}" |
|
1118 |
shows "exp (x + y) = exp x * exp y" |
|
1119 |
by (rule exp_add_commuting) (simp add: ac_simps) |
|
1120 |
||
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1121 |
lemma exp_double: "exp(2 * z) = exp z ^ 2" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1122 |
by (simp add: exp_add_commuting mult_2 power2_eq_square) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1123 |
|
58656 | 1124 |
lemmas mult_exp_exp = exp_add [symmetric] |
29170 | 1125 |
|
23241 | 1126 |
lemma exp_of_real: "exp (of_real x) = of_real (exp x)" |
53079 | 1127 |
unfolding exp_def |
1128 |
apply (subst suminf_of_real) |
|
1129 |
apply (rule summable_exp_generic) |
|
1130 |
apply (simp add: scaleR_conv_of_real) |
|
1131 |
done |
|
23241 | 1132 |
|
59862 | 1133 |
corollary exp_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> exp z \<in> \<real>" |
1134 |
by (metis Reals_cases Reals_of_real exp_of_real) |
|
1135 |
||
29170 | 1136 |
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0" |
1137 |
proof |
|
58656 | 1138 |
have "exp x * exp (- x) = 1" by (simp add: exp_add_commuting[symmetric]) |
29170 | 1139 |
also assume "exp x = 0" |
1140 |
finally show "False" by simp |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1141 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1142 |
|
58656 | 1143 |
lemma exp_minus_inverse: |
1144 |
shows "exp x * exp (- x) = 1" |
|
1145 |
by (simp add: exp_add_commuting[symmetric]) |
|
1146 |
||
1147 |
lemma exp_minus: |
|
1148 |
fixes x :: "'a::{real_normed_field, banach}" |
|
1149 |
shows "exp (- x) = inverse (exp x)" |
|
1150 |
by (intro inverse_unique [symmetric] exp_minus_inverse) |
|
1151 |
||
1152 |
lemma exp_diff: |
|
1153 |
fixes x :: "'a::{real_normed_field, banach}" |
|
1154 |
shows "exp (x - y) = exp x / exp y" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
1155 |
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
|
1156 |
|
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1157 |
lemma exp_of_nat_mult: |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1158 |
fixes x :: "'a::{real_normed_field,banach}" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1159 |
shows "exp(of_nat n * x) = exp(x) ^ n" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1160 |
by (induct n) (auto simp add: distrib_left exp_add mult.commute) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1161 |
|
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1162 |
lemma exp_setsum: "finite I \<Longrightarrow> exp(setsum f I) = setprod (\<lambda>x. exp(f x)) I" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1163 |
by (induction I rule: finite_induct) (auto simp: exp_add_commuting mult.commute) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1164 |
|
29167 | 1165 |
|
1166 |
subsubsection {* Properties of the Exponential Function on Reals *} |
|
1167 |
||
29170 | 1168 |
text {* Comparisons of @{term "exp x"} with zero. *} |
29167 | 1169 |
|
1170 |
text{*Proof: because every exponential can be seen as a square.*} |
|
1171 |
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)" |
|
1172 |
proof - |
|
1173 |
have "0 \<le> exp (x/2) * exp (x/2)" by simp |
|
1174 |
thus ?thesis by (simp add: exp_add [symmetric]) |
|
1175 |
qed |
|
1176 |
||
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1177 |
lemma exp_gt_zero [simp]: "0 < exp (x::real)" |
53079 | 1178 |
by (simp add: order_less_le) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1179 |
|
29170 | 1180 |
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0" |
53079 | 1181 |
by (simp add: not_less) |
29170 | 1182 |
|
1183 |
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0" |
|
53079 | 1184 |
by (simp add: not_le) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1185 |
|
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1186 |
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x" |
53079 | 1187 |
by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1188 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1189 |
(*FIXME: superseded by exp_of_nat_mult*) |
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1190 |
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
|
1191 |
by (induct n) (auto simp add: real_of_nat_Suc distrib_left exp_add mult.commute) |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1192 |
|
29170 | 1193 |
text {* Strict monotonicity of exponential. *} |
1194 |
||
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1195 |
lemma exp_ge_add_one_self_aux: |
54575 | 1196 |
assumes "0 \<le> (x::real)" shows "1+x \<le> exp(x)" |
1197 |
using order_le_imp_less_or_eq [OF assms] |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1198 |
proof |
54575 | 1199 |
assume "0 < x" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1200 |
have "1+x \<le> (\<Sum>n<2. inverse (fact n) * x^n)" |
54575 | 1201 |
by (auto simp add: numeral_2_eq_2) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1202 |
also have "... \<le> (\<Sum>n. inverse (fact n) * x^n)" |
56213 | 1203 |
apply (rule setsum_le_suminf [OF summable_exp]) |
54575 | 1204 |
using `0 < x` |
1205 |
apply (auto simp add: zero_le_mult_iff) |
|
1206 |
done |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1207 |
finally show "1+x \<le> exp x" |
54575 | 1208 |
by (simp add: exp_def) |
1209 |
next |
|
1210 |
assume "0 = x" |
|
1211 |
then show "1 + x \<le> exp x" |
|
1212 |
by auto |
|
1213 |
qed |
|
29170 | 1214 |
|
1215 |
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x" |
|
1216 |
proof - |
|
1217 |
assume x: "0 < x" |
|
1218 |
hence "1 < 1 + x" by simp |
|
1219 |
also from x have "1 + x \<le> exp x" |
|
1220 |
by (simp add: exp_ge_add_one_self_aux) |
|
1221 |
finally show ?thesis . |
|
1222 |
qed |
|
1223 |
||
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1224 |
lemma exp_less_mono: |
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset
|
1225 |
fixes x y :: real |
53079 | 1226 |
assumes "x < y" |
1227 |
shows "exp x < exp y" |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1228 |
proof - |
29165
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset
|
1229 |
from `x < y` have "0 < y - x" by simp |
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset
|
1230 |
hence "1 < exp (y - x)" by (rule exp_gt_one) |
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset
|
1231 |
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
|
1232 |
thus "exp x < exp y" by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1233 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1234 |
|
53079 | 1235 |
lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y" |
54575 | 1236 |
unfolding linorder_not_le [symmetric] |
1237 |
by (auto simp add: order_le_less exp_less_mono) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1238 |
|
29170 | 1239 |
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y" |
53079 | 1240 |
by (auto intro: exp_less_mono exp_less_cancel) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1241 |
|
29170 | 1242 |
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y" |
53079 | 1243 |
by (auto simp add: linorder_not_less [symmetric]) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1244 |
|
29170 | 1245 |
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y" |
53079 | 1246 |
by (simp add: order_eq_iff) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1247 |
|
29170 | 1248 |
text {* Comparisons of @{term "exp x"} with one. *} |
1249 |
||
1250 |
lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x" |
|
1251 |
using exp_less_cancel_iff [where x=0 and y=x] by simp |
|
1252 |
||
1253 |
lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0" |
|
1254 |
using exp_less_cancel_iff [where x=x and y=0] by simp |
|
1255 |
||
1256 |
lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x" |
|
1257 |
using exp_le_cancel_iff [where x=0 and y=x] by simp |
|
1258 |
||
1259 |
lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0" |
|
1260 |
using exp_le_cancel_iff [where x=x and y=0] by simp |
|
1261 |
||
1262 |
lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0" |
|
1263 |
using exp_inj_iff [where x=x and y=0] by simp |
|
1264 |
||
53079 | 1265 |
lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y" |
44755 | 1266 |
proof (rule IVT) |
1267 |
assume "1 \<le> y" |
|
1268 |
hence "0 \<le> y - 1" by simp |
|
1269 |
hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux) |
|
1270 |
thus "y \<le> exp (y - 1)" by simp |
|
1271 |
qed (simp_all add: le_diff_eq) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1272 |
|
53079 | 1273 |
lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y" |
44755 | 1274 |
proof (rule linorder_le_cases [of 1 y]) |
53079 | 1275 |
assume "1 \<le> y" |
1276 |
thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total) |
|
44755 | 1277 |
next |
1278 |
assume "0 < y" and "y \<le> 1" |
|
1279 |
hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff) |
|
1280 |
then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total) |
|
1281 |
hence "exp (- x) = y" by (simp add: exp_minus) |
|
1282 |
thus "\<exists>x. exp x = y" .. |
|
1283 |
qed |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1284 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1285 |
|
29164 | 1286 |
subsection {* Natural Logarithm *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1287 |
|
53079 | 1288 |
definition ln :: "real \<Rightarrow> real" |
1289 |
where "ln x = (THE u. exp u = x)" |
|
23043 | 1290 |
|
1291 |
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
|
1292 |
by (simp add: ln_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1293 |
|
22654
c2b6b5a9e136
new simp rule exp_ln; new standard proof of DERIV_exp_ln_one; changed imports
huffman
parents:
22653
diff
changeset
|
1294 |
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
|
1295 |
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
|
1296 |
|
29171 | 1297 |
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
|
1298 |
by (metis exp_gt_zero exp_ln) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1299 |
|
29171 | 1300 |
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
|
1301 |
by (erule subst, rule ln_exp) |
29171 | 1302 |
|
1303 |
lemma ln_one [simp]: "ln 1 = 0" |
|
53079 | 1304 |
by (rule ln_unique) simp |
1305 |
||
1306 |
lemma ln_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y" |
|
1307 |
by (rule ln_unique) (simp add: exp_add) |
|
29171 | 1308 |
|
59746
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
1309 |
lemma ln_setprod: |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
1310 |
"\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> f i > 0\<rbrakk> \<Longrightarrow> ln(setprod f I) = setsum (\<lambda>x. ln(f x)) I" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
1311 |
by (induction I rule: finite_induct) (auto simp: ln_mult setprod_pos) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
1312 |
|
29171 | 1313 |
lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x" |
53079 | 1314 |
by (rule ln_unique) (simp add: exp_minus) |
1315 |
||
1316 |
lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y" |
|
1317 |
by (rule ln_unique) (simp add: exp_diff) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1318 |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1319 |
lemma ln_realpow: "0 < x \<Longrightarrow> ln (x^n) = real n * ln x" |
53079 | 1320 |
by (rule ln_unique) (simp add: exp_real_of_nat_mult) |
1321 |
||
1322 |
lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y" |
|
1323 |
by (subst exp_less_cancel_iff [symmetric]) simp |
|
1324 |
||
1325 |
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
|
1326 |
by (simp add: linorder_not_less [symmetric]) |
29171 | 1327 |
|
53079 | 1328 |
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
|
1329 |
by (simp add: order_eq_iff) |
29171 | 1330 |
|
1331 |
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
|
1332 |
apply (rule exp_le_cancel_iff [THEN iffD1]) |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1333 |
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
|
1334 |
done |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1335 |
|
29171 | 1336 |
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
|
1337 |
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
|
1338 |
|
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1339 |
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
|
1340 |
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
|
1341 |
|
53079 | 1342 |
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
|
1343 |
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
|
1344 |
|
53079 | 1345 |
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
|
1346 |
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
|
1347 |
|
53079 | 1348 |
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
|
1349 |
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
|
1350 |
|
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
1351 |
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
|
1352 |
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
|
1353 |
|
53079 | 1354 |
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
|
1355 |
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
|
1356 |
|
53079 | 1357 |
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
|
1358 |
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
|
1359 |
|
53079 | 1360 |
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
|
1361 |
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
|
1362 |
|
53079 | 1363 |
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
|
1364 |
by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1365 |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1366 |
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
|
1367 |
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
|
1368 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1369 |
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
|
1370 |
proof cases |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1371 |
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
|
1372 |
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
|
1373 |
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
|
1374 |
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
|
1375 |
by simp |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1376 |
next |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
1377 |
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
|
1378 |
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
|
1379 |
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
|
1380 |
(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
|
1381 |
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
|
1382 |
qed |
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
1383 |
|
45915 | 1384 |
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
|
1385 |
"(f ---> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F" |
45915 | 1386 |
by (rule isCont_tendsto_compose [OF isCont_ln]) |
1387 |
||
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
|
1388 |
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
|
1389 |
"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
|
1390 |
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
|
1391 |
|
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
|
1392 |
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
|
1393 |
"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
|
1394 |
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
|
1395 |
|
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
|
1396 |
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
|
1397 |
"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
|
1398 |
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
|
1399 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
1400 |
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
|
1401 |
"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
|
1402 |
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
|
1403 |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
1404 |
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
|
1405 |
apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"]) |
54576 | 1406 |
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
|
1407 |
done |
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
1408 |
|
53079 | 1409 |
lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x" |
33667 | 1410 |
by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse) |
1411 |
||
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1412 |
declare DERIV_ln_divide[THEN DERIV_chain2, derivative_intros] |
51527 | 1413 |
|
53079 | 1414 |
lemma ln_series: |
1415 |
assumes "0 < x" and "x < 2" |
|
1416 |
shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" |
|
1417 |
(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
|
1418 |
proof - |
53079 | 1419 |
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
|
1420 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1421 |
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
|
1422 |
proof (rule DERIV_isconst3[where x=x]) |
53079 | 1423 |
fix x :: real |
1424 |
assume "x \<in> {0 <..< 2}" |
|
1425 |
hence "0 < x" and "x < 2" by auto |
|
1426 |
have "norm (1 - x) < 1" |
|
1427 |
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
|
1428 |
have "1 / x = 1 / (1 - (1 - x))" by auto |
53079 | 1429 |
also have "\<dots> = (\<Sum> n. (1 - x)^n)" |
1430 |
using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique) |
|
1431 |
also have "\<dots> = suminf (?f' x)" |
|
1432 |
unfolding power_mult_distrib[symmetric] |
|
1433 |
by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto) |
|
1434 |
finally have "DERIV ln x :> suminf (?f' x)" |
|
1435 |
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
|
1436 |
moreover |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
1437 |
have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto |
53079 | 1438 |
have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> |
1439 |
(\<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
|
1440 |
proof (rule DERIV_power_series') |
53079 | 1441 |
show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" |
1442 |
using `0 < x` `x < 2` by auto |
|
1443 |
fix x :: real |
|
1444 |
assume "x \<in> {- 1<..<1}" |
|
1445 |
hence "norm (-x) < 1" by auto |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1446 |
show "summable (\<lambda>n. (- 1) ^ n * (1 / real (n + 1)) * real (Suc n) * x^n)" |
53079 | 1447 |
unfolding One_nat_def |
1448 |
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
|
1449 |
qed |
53079 | 1450 |
hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" |
1451 |
unfolding One_nat_def by auto |
|
1452 |
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
|
1453 |
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
|
1454 |
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
|
1455 |
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
|
1456 |
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
|
1457 |
qed (auto simp add: assms) |
44289 | 1458 |
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
|
1459 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
1460 |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1461 |
lemma exp_first_two_terms: |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1462 |
fixes x :: "'a::{real_normed_field,banach}" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1463 |
shows "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) * (x ^ (n+2)))" |
50326 | 1464 |
proof - |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1465 |
have "exp x = suminf (\<lambda>n. inverse(fact n) * (x^n))" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1466 |
by (simp add: exp_def scaleR_conv_of_real nonzero_of_real_inverse) |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1467 |
also from summable_exp have "... = (\<Sum> n. inverse(fact(n+2)) * (x ^ (n+2))) + |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1468 |
(\<Sum> n::nat<2. inverse(fact n) * (x^n))" (is "_ = _ + ?a") |
50326 | 1469 |
by (rule suminf_split_initial_segment) |
1470 |
also have "?a = 1 + x" |
|
1471 |
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
|
1472 |
finally show ?thesis |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
1473 |
by simp |
50326 | 1474 |
qed |
1475 |
||
53079 | 1476 |
lemma exp_bound: "0 <= (x::real) \<Longrightarrow> x <= 1 \<Longrightarrow> exp x <= 1 + x + x\<^sup>2" |
50326 | 1477 |
proof - |
1478 |
assume a: "0 <= x" |
|
1479 |
assume b: "x <= 1" |
|
53079 | 1480 |
{ |
1481 |
fix n :: nat |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1482 |
have "(2::nat) * 2 ^ n \<le> fact (n + 2)" |
53079 | 1483 |
by (induct n) simp_all |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1484 |
hence "real ((2::nat) * 2 ^ n) \<le> real_of_nat (fact (n + 2))" |
50326 | 1485 |
by (simp only: real_of_nat_le_iff) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1486 |
hence "((2::real) * 2 ^ n) \<le> fact (n + 2)" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1487 |
unfolding of_nat_fact real_of_nat_def |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1488 |
by (simp add: of_nat_mult of_nat_power) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1489 |
hence "inverse (fact (n + 2)) \<le> inverse ((2::real) * 2 ^ n)" |
50326 | 1490 |
by (rule le_imp_inverse_le) simp |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1491 |
hence "inverse (fact (n + 2)) \<le> 1/(2::real) * (1/2)^n" |
53079 | 1492 |
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
|
1493 |
hence "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)" |
50326 | 1494 |
by (rule mult_mono) |
56536 | 1495 |
(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
|
1496 |
hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1497 |
unfolding power_add by (simp add: ac_simps del: fact.simps) } |
50326 | 1498 |
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
|
1499 |
have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))" |
50326 | 1500 |
by (intro sums_mult geometric_sums, simp) |
53076 | 1501 |
hence aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2" |
50326 | 1502 |
by simp |
53079 | 1503 |
have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2" |
50326 | 1504 |
proof - |
53079 | 1505 |
have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= |
1506 |
suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))" |
|
56213 | 1507 |
apply (rule suminf_le) |
50326 | 1508 |
apply (rule allI, rule aux1) |
1509 |
apply (rule summable_exp [THEN summable_ignore_initial_segment]) |
|
1510 |
by (rule sums_summable, rule aux2) |
|
53076 | 1511 |
also have "... = x\<^sup>2" |
50326 | 1512 |
by (rule sums_unique [THEN sym], rule aux2) |
1513 |
finally show ?thesis . |
|
1514 |
qed |
|
1515 |
thus ?thesis unfolding exp_first_two_terms by auto |
|
1516 |
qed |
|
1517 |
||
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1518 |
corollary exp_half_le2: "exp(1/2) \<le> (2::real)" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1519 |
using exp_bound [of "1/2"] |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1520 |
by (simp add: field_simps) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1521 |
|
59741
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
1522 |
corollary exp_le: "exp 1 \<le> (3::real)" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
1523 |
using exp_bound [of 1] |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
1524 |
by (simp add: field_simps) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
1525 |
|
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1526 |
lemma exp_bound_half: "norm(z) \<le> 1/2 \<Longrightarrow> norm(exp z) \<le> 2" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1527 |
by (blast intro: order_trans intro!: exp_half_le2 norm_exp) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1528 |
|
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1529 |
lemma exp_bound_lemma: |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1530 |
assumes "norm(z) \<le> 1/2" shows "norm(exp z) \<le> 1 + 2 * norm(z)" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1531 |
proof - |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1532 |
have n: "(norm z)\<^sup>2 \<le> norm z * 1" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1533 |
unfolding power2_eq_square |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1534 |
apply (rule mult_left_mono) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1535 |
using assms |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1536 |
apply (auto simp: ) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1537 |
done |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1538 |
show ?thesis |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1539 |
apply (rule order_trans [OF norm_exp]) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1540 |
apply (rule order_trans [OF exp_bound]) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1541 |
using assms n |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1542 |
apply (auto simp: ) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1543 |
done |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1544 |
qed |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1545 |
|
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1546 |
lemma real_exp_bound_lemma: |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1547 |
fixes x :: real |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1548 |
shows "0 \<le> x \<Longrightarrow> x \<le> 1/2 \<Longrightarrow> exp(x) \<le> 1 + 2 * x" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1549 |
using exp_bound_lemma [of x] |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1550 |
by simp |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1551 |
|
53079 | 1552 |
lemma ln_one_minus_pos_upper_bound: "0 <= x \<Longrightarrow> x < 1 \<Longrightarrow> ln (1 - x) <= - x" |
50326 | 1553 |
proof - |
1554 |
assume a: "0 <= (x::real)" and b: "x < 1" |
|
53076 | 1555 |
have "(1 - x) * (1 + x + x\<^sup>2) = (1 - x^3)" |
50326 | 1556 |
by (simp add: algebra_simps power2_eq_square power3_eq_cube) |
1557 |
also have "... <= 1" |
|
1558 |
by (auto simp add: a) |
|
53076 | 1559 |
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
|
1560 |
moreover have c: "0 < 1 + x + x\<^sup>2" |
50326 | 1561 |
by (simp add: add_pos_nonneg a) |
53076 | 1562 |
ultimately have "1 - x <= 1 / (1 + x + x\<^sup>2)" |
50326 | 1563 |
by (elim mult_imp_le_div_pos) |
1564 |
also have "... <= 1 / exp x" |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1565 |
by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs |
54576 | 1566 |
real_sqrt_pow2_iff real_sqrt_power) |
50326 | 1567 |
also have "... = exp (-x)" |
1568 |
by (auto simp add: exp_minus divide_inverse) |
|
1569 |
finally have "1 - x <= exp (- x)" . |
|
1570 |
also have "1 - x = exp (ln (1 - x))" |
|
54576 | 1571 |
by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq) |
50326 | 1572 |
finally have "exp (ln (1 - x)) <= exp (- x)" . |
1573 |
thus ?thesis by (auto simp only: exp_le_cancel_iff) |
|
1574 |
qed |
|
1575 |
||
1576 |
lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x" |
|
1577 |
apply (case_tac "0 <= x") |
|
1578 |
apply (erule exp_ge_add_one_self_aux) |
|
1579 |
apply (case_tac "x <= -1") |
|
1580 |
apply (subgoal_tac "1 + x <= 0") |
|
1581 |
apply (erule order_trans) |
|
1582 |
apply simp |
|
1583 |
apply simp |
|
1584 |
apply (subgoal_tac "1 + x = exp(ln (1 + x))") |
|
1585 |
apply (erule ssubst) |
|
1586 |
apply (subst exp_le_cancel_iff) |
|
1587 |
apply (subgoal_tac "ln (1 - (- x)) <= - (- x)") |
|
1588 |
apply simp |
|
1589 |
apply (rule ln_one_minus_pos_upper_bound) |
|
1590 |
apply auto |
|
1591 |
done |
|
1592 |
||
53079 | 1593 |
lemma ln_one_plus_pos_lower_bound: "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> x - x\<^sup>2 <= ln (1 + x)" |
51527 | 1594 |
proof - |
1595 |
assume a: "0 <= x" and b: "x <= 1" |
|
53076 | 1596 |
have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)" |
51527 | 1597 |
by (rule exp_diff) |
53076 | 1598 |
also have "... <= (1 + x + x\<^sup>2) / exp (x \<^sup>2)" |
54576 | 1599 |
by (metis a b divide_right_mono exp_bound exp_ge_zero) |
53076 | 1600 |
also have "... <= (1 + x + x\<^sup>2) / (1 + x\<^sup>2)" |
56544 | 1601 |
by (simp add: a divide_left_mono add_pos_nonneg) |
51527 | 1602 |
also from a have "... <= 1 + x" |
1603 |
by (simp add: field_simps add_strict_increasing zero_le_mult_iff) |
|
53076 | 1604 |
finally have "exp (x - x\<^sup>2) <= 1 + x" . |
51527 | 1605 |
also have "... = exp (ln (1 + x))" |
1606 |
proof - |
|
1607 |
from a have "0 < 1 + x" by auto |
|
1608 |
thus ?thesis |
|
1609 |
by (auto simp only: exp_ln_iff [THEN sym]) |
|
1610 |
qed |
|
53076 | 1611 |
finally have "exp (x - x\<^sup>2) <= exp (ln (1 + x))" . |
54576 | 1612 |
thus ?thesis |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1613 |
by (metis exp_le_cancel_iff) |
51527 | 1614 |
qed |
1615 |
||
53079 | 1616 |
lemma ln_one_minus_pos_lower_bound: |
1617 |
"0 <= x \<Longrightarrow> x <= (1 / 2) \<Longrightarrow> - x - 2 * x\<^sup>2 <= ln (1 - x)" |
|
51527 | 1618 |
proof - |
1619 |
assume a: "0 <= x" and b: "x <= (1 / 2)" |
|
53079 | 1620 |
from b have c: "x < 1" by auto |
51527 | 1621 |
then have "ln (1 - x) = - ln (1 + x / (1 - x))" |
54576 | 1622 |
apply (subst ln_inverse [symmetric]) |
1623 |
apply (simp add: field_simps) |
|
1624 |
apply (rule arg_cong [where f=ln]) |
|
1625 |
apply (simp add: field_simps) |
|
1626 |
done |
|
51527 | 1627 |
also have "- (x / (1 - x)) <= ..." |
53079 | 1628 |
proof - |
51527 | 1629 |
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
|
1630 |
using a c by (intro ln_add_one_self_le_self) auto |
51527 | 1631 |
thus ?thesis |
1632 |
by auto |
|
1633 |
qed |
|
1634 |
also have "- (x / (1 - x)) = -x / (1 - x)" |
|
1635 |
by auto |
|
1636 |
finally have d: "- x / (1 - x) <= ln (1 - x)" . |
|
1637 |
have "0 < 1 - x" using a b by simp |
|
53076 | 1638 |
hence e: "-x - 2 * x\<^sup>2 <= - x / (1 - x)" |
51527 | 1639 |
using mult_right_le_one_le[of "x*x" "2*x"] a b |
53079 | 1640 |
by (simp add: field_simps power2_eq_square) |
53076 | 1641 |
from e d show "- x - 2 * x\<^sup>2 <= ln (1 - x)" |
51527 | 1642 |
by (rule order_trans) |
1643 |
qed |
|
1644 |
||
53079 | 1645 |
lemma ln_add_one_self_le_self2: "-1 < x \<Longrightarrow> ln(1 + x) <= x" |
51527 | 1646 |
apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp) |
1647 |
apply (subst ln_le_cancel_iff) |
|
1648 |
apply auto |
|
53079 | 1649 |
done |
51527 | 1650 |
|
1651 |
lemma abs_ln_one_plus_x_minus_x_bound_nonneg: |
|
53079 | 1652 |
"0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> abs(ln (1 + x) - x) <= x\<^sup>2" |
51527 | 1653 |
proof - |
1654 |
assume x: "0 <= x" |
|
1655 |
assume x1: "x <= 1" |
|
1656 |
from x have "ln (1 + x) <= x" |
|
1657 |
by (rule ln_add_one_self_le_self) |
|
53079 | 1658 |
then have "ln (1 + x) - x <= 0" |
51527 | 1659 |
by simp |
1660 |
then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)" |
|
1661 |
by (rule abs_of_nonpos) |
|
53079 | 1662 |
also have "... = x - ln (1 + x)" |
51527 | 1663 |
by simp |
53076 | 1664 |
also have "... <= x\<^sup>2" |
51527 | 1665 |
proof - |
53076 | 1666 |
from x x1 have "x - x\<^sup>2 <= ln (1 + x)" |
51527 | 1667 |
by (intro ln_one_plus_pos_lower_bound) |
1668 |
thus ?thesis |
|
1669 |
by simp |
|
1670 |
qed |
|
1671 |
finally show ?thesis . |
|
1672 |
qed |
|
1673 |
||
1674 |
lemma abs_ln_one_plus_x_minus_x_bound_nonpos: |
|
53079 | 1675 |
"-(1 / 2) <= x \<Longrightarrow> x <= 0 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2" |
51527 | 1676 |
proof - |
1677 |
assume a: "-(1 / 2) <= x" |
|
1678 |
assume b: "x <= 0" |
|
53079 | 1679 |
have "abs(ln (1 + x) - x) = x - ln(1 - (-x))" |
51527 | 1680 |
apply (subst abs_of_nonpos) |
1681 |
apply simp |
|
1682 |
apply (rule ln_add_one_self_le_self2) |
|
1683 |
using a apply auto |
|
1684 |
done |
|
53076 | 1685 |
also have "... <= 2 * x\<^sup>2" |
1686 |
apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 <= ln (1 - (-x))") |
|
51527 | 1687 |
apply (simp add: algebra_simps) |
1688 |
apply (rule ln_one_minus_pos_lower_bound) |
|
1689 |
using a b apply auto |
|
1690 |
done |
|
1691 |
finally show ?thesis . |
|
1692 |
qed |
|
1693 |
||
1694 |
lemma abs_ln_one_plus_x_minus_x_bound: |
|
53079 | 1695 |
"abs x <= 1 / 2 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2" |
51527 | 1696 |
apply (case_tac "0 <= x") |
1697 |
apply (rule order_trans) |
|
1698 |
apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg) |
|
1699 |
apply auto |
|
1700 |
apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos) |
|
1701 |
apply auto |
|
53079 | 1702 |
done |
1703 |
||
1704 |
lemma ln_x_over_x_mono: "exp 1 <= x \<Longrightarrow> x <= y \<Longrightarrow> (ln y / y) <= (ln x / x)" |
|
51527 | 1705 |
proof - |
1706 |
assume x: "exp 1 <= x" "x <= y" |
|
1707 |
moreover have "0 < exp (1::real)" by simp |
|
1708 |
ultimately have a: "0 < x" and b: "0 < y" |
|
1709 |
by (fast intro: less_le_trans order_trans)+ |
|
1710 |
have "x * ln y - x * ln x = x * (ln y - ln x)" |
|
1711 |
by (simp add: algebra_simps) |
|
1712 |
also have "... = x * ln(y / x)" |
|
1713 |
by (simp only: ln_div a b) |
|
1714 |
also have "y / x = (x + (y - x)) / x" |
|
1715 |
by simp |
|
1716 |
also have "... = 1 + (y - x) / x" |
|
1717 |
using x a by (simp add: field_simps) |
|
1718 |
also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)" |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1719 |
using x a |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56544
diff
changeset
|
1720 |
by (intro mult_left_mono ln_add_one_self_le_self) simp_all |
51527 | 1721 |
also have "... = y - x" using a by simp |
1722 |
also have "... = (y - x) * ln (exp 1)" by simp |
|
1723 |
also have "... <= (y - x) * ln x" |
|
1724 |
apply (rule mult_left_mono) |
|
1725 |
apply (subst ln_le_cancel_iff) |
|
1726 |
apply fact |
|
1727 |
apply (rule a) |
|
1728 |
apply (rule x) |
|
1729 |
using x apply simp |
|
1730 |
done |
|
1731 |
also have "... = y * ln x - x * ln x" |
|
1732 |
by (rule left_diff_distrib) |
|
1733 |
finally have "x * ln y <= y * ln x" |
|
1734 |
by arith |
|
1735 |
then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps) |
|
1736 |
also have "... = y * (ln x / x)" by simp |
|
1737 |
finally show ?thesis using b by (simp add: field_simps) |
|
1738 |
qed |
|
1739 |
||
53079 | 1740 |
lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1" |
51527 | 1741 |
using exp_ge_add_one_self[of "ln x"] by simp |
1742 |
||
1743 |
lemma ln_eq_minus_one: |
|
53079 | 1744 |
assumes "0 < x" "ln x = x - 1" |
1745 |
shows "x = 1" |
|
51527 | 1746 |
proof - |
53079 | 1747 |
let ?l = "\<lambda>y. ln y - y + 1" |
51527 | 1748 |
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
|
1749 |
by (auto intro!: derivative_eq_intros) |
51527 | 1750 |
|
1751 |
show ?thesis |
|
1752 |
proof (cases rule: linorder_cases) |
|
1753 |
assume "x < 1" |
|
1754 |
from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast |
|
1755 |
from `x < a` have "?l x < ?l a" |
|
1756 |
proof (rule DERIV_pos_imp_increasing, safe) |
|
53079 | 1757 |
fix y |
1758 |
assume "x \<le> y" "y \<le> a" |
|
51527 | 1759 |
with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y" |
1760 |
by (auto simp: field_simps) |
|
1761 |
with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z" |
|
1762 |
by auto |
|
1763 |
qed |
|
1764 |
also have "\<dots> \<le> 0" |
|
1765 |
using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps) |
|
1766 |
finally show "x = 1" using assms by auto |
|
1767 |
next |
|
1768 |
assume "1 < x" |
|
53079 | 1769 |
from dense[OF this] obtain a where "1 < a" "a < x" by blast |
51527 | 1770 |
from `a < x` have "?l x < ?l a" |
1771 |
proof (rule DERIV_neg_imp_decreasing, safe) |
|
53079 | 1772 |
fix y |
1773 |
assume "a \<le> y" "y \<le> x" |
|
51527 | 1774 |
with `1 < a` have "1 / y - 1 < 0" "0 < y" |
1775 |
by (auto simp: field_simps) |
|
1776 |
with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0" |
|
1777 |
by blast |
|
1778 |
qed |
|
1779 |
also have "\<dots> \<le> 0" |
|
1780 |
using ln_le_minus_one `1 < a` by (auto simp: field_simps) |
|
1781 |
finally show "x = 1" using assms by auto |
|
53079 | 1782 |
next |
1783 |
assume "x = 1" |
|
1784 |
then show ?thesis by simp |
|
1785 |
qed |
|
51527 | 1786 |
qed |
1787 |
||
50326 | 1788 |
lemma exp_at_bot: "(exp ---> (0::real)) at_bot" |
1789 |
unfolding tendsto_Zfun_iff |
|
1790 |
proof (rule ZfunI, simp add: eventually_at_bot_dense) |
|
1791 |
fix r :: real assume "0 < r" |
|
53079 | 1792 |
{ |
1793 |
fix x |
|
1794 |
assume "x < ln r" |
|
50326 | 1795 |
then have "exp x < exp (ln r)" |
1796 |
by simp |
|
1797 |
with `0 < r` have "exp x < r" |
|
53079 | 1798 |
by simp |
1799 |
} |
|
50326 | 1800 |
then show "\<exists>k. \<forall>n<k. exp n < r" by auto |
1801 |
qed |
|
1802 |
||
1803 |
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
|
1804 |
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
|
1805 |
(auto intro: eventually_gt_at_top) |
50326 | 1806 |
|
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1807 |
lemma lim_exp_minus_1: |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1808 |
fixes x :: "'a::{real_normed_field,banach}" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1809 |
shows "((\<lambda>z::'a. (exp(z) - 1) / z) ---> 1) (at 0)" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1810 |
proof - |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1811 |
have "((\<lambda>z::'a. exp(z) - 1) has_field_derivative 1) (at 0)" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1812 |
by (intro derivative_eq_intros | simp)+ |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1813 |
then show ?thesis |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1814 |
by (simp add: Deriv.DERIV_iff2) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1815 |
qed |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
1816 |
|
50326 | 1817 |
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
|
1818 |
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
|
1819 |
(auto simp: eventually_at_filter) |
50326 | 1820 |
|
1821 |
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
|
1822 |
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
|
1823 |
(auto intro: eventually_gt_at_top) |
50326 | 1824 |
|
50347 | 1825 |
lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) ---> (0::real)) at_top" |
1826 |
proof (induct k) |
|
53079 | 1827 |
case 0 |
50347 | 1828 |
show "((\<lambda>x. x ^ 0 / exp x) ---> (0::real)) at_top" |
1829 |
by (simp add: inverse_eq_divide[symmetric]) |
|
1830 |
(metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono |
|
1831 |
at_top_le_at_infinity order_refl) |
|
1832 |
next |
|
1833 |
case (Suc k) |
|
1834 |
show ?case |
|
1835 |
proof (rule lhospital_at_top_at_top) |
|
1836 |
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
|
1837 |
by eventually_elim (intro derivative_eq_intros, auto) |
50347 | 1838 |
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
|
1839 |
by eventually_elim auto |
50347 | 1840 |
show "eventually (\<lambda>x. exp x \<noteq> 0) at_top" |
1841 |
by auto |
|
1842 |
from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"] |
|
1843 |
show "((\<lambda>x. real (Suc k) * x ^ k / exp x) ---> 0) at_top" |
|
1844 |
by simp |
|
1845 |
qed (rule exp_at_top) |
|
1846 |
qed |
|
1847 |
||
51527 | 1848 |
|
53079 | 1849 |
definition powr :: "[real,real] => real" (infixr "powr" 80) |
1850 |
-- {*exponentation with real exponent*} |
|
1851 |
where "x powr a = exp(a * ln x)" |
|
1852 |
||
1853 |
definition log :: "[real,real] => real" |
|
1854 |
-- {*logarithm of @{term x} to base @{term a}*} |
|
1855 |
where "log a x = ln x / ln a" |
|
51527 | 1856 |
|
1857 |
||
1858 |
lemma tendsto_log [tendsto_intros]: |
|
1859 |
"\<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" |
|
1860 |
unfolding log_def by (intro tendsto_intros) auto |
|
1861 |
||
1862 |
lemma continuous_log: |
|
53079 | 1863 |
assumes "continuous F f" |
1864 |
and "continuous F g" |
|
1865 |
and "0 < f (Lim F (\<lambda>x. x))" |
|
1866 |
and "f (Lim F (\<lambda>x. x)) \<noteq> 1" |
|
1867 |
and "0 < g (Lim F (\<lambda>x. x))" |
|
51527 | 1868 |
shows "continuous F (\<lambda>x. log (f x) (g x))" |
1869 |
using assms unfolding continuous_def by (rule tendsto_log) |
|
1870 |
||
1871 |
lemma continuous_at_within_log[continuous_intros]: |
|
53079 | 1872 |
assumes "continuous (at a within s) f" |
1873 |
and "continuous (at a within s) g" |
|
1874 |
and "0 < f a" |
|
1875 |
and "f a \<noteq> 1" |
|
1876 |
and "0 < g a" |
|
51527 | 1877 |
shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))" |
1878 |
using assms unfolding continuous_within by (rule tendsto_log) |
|
1879 |
||
1880 |
lemma isCont_log[continuous_intros, simp]: |
|
1881 |
assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a" |
|
1882 |
shows "isCont (\<lambda>x. log (f x) (g x)) a" |
|
1883 |
using assms unfolding continuous_at by (rule tendsto_log) |
|
1884 |
||
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
1885 |
lemma continuous_on_log[continuous_intros]: |
53079 | 1886 |
assumes "continuous_on s f" "continuous_on s g" |
1887 |
and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x" |
|
51527 | 1888 |
shows "continuous_on s (\<lambda>x. log (f x) (g x))" |
1889 |
using assms unfolding continuous_on_def by (fast intro: tendsto_log) |
|
1890 |
||
1891 |
lemma powr_one_eq_one [simp]: "1 powr a = 1" |
|
53079 | 1892 |
by (simp add: powr_def) |
51527 | 1893 |
|
1894 |
lemma powr_zero_eq_one [simp]: "x powr 0 = 1" |
|
53079 | 1895 |
by (simp add: powr_def) |
51527 | 1896 |
|
1897 |
lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)" |
|
53079 | 1898 |
by (simp add: powr_def) |
51527 | 1899 |
declare powr_one_gt_zero_iff [THEN iffD2, simp] |
1900 |
||
53079 | 1901 |
lemma powr_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)" |
1902 |
by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left) |
|
51527 | 1903 |
|
1904 |
lemma powr_gt_zero [simp]: "0 < x powr a" |
|
53079 | 1905 |
by (simp add: powr_def) |
51527 | 1906 |
|
1907 |
lemma powr_ge_pzero [simp]: "0 <= x powr y" |
|
53079 | 1908 |
by (rule order_less_imp_le, rule powr_gt_zero) |
51527 | 1909 |
|
1910 |
lemma powr_not_zero [simp]: "x powr a \<noteq> 0" |
|
53079 | 1911 |
by (simp add: powr_def) |
1912 |
||
1913 |
lemma powr_divide: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)" |
|
1914 |
apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult) |
|
1915 |
apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse) |
|
1916 |
done |
|
51527 | 1917 |
|
1918 |
lemma powr_divide2: "x powr a / x powr b = x powr (a - b)" |
|
1919 |
apply (simp add: powr_def) |
|
1920 |
apply (subst exp_diff [THEN sym]) |
|
1921 |
apply (simp add: left_diff_distrib) |
|
53079 | 1922 |
done |
51527 | 1923 |
|
1924 |
lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)" |
|
53079 | 1925 |
by (simp add: powr_def exp_add [symmetric] distrib_right) |
1926 |
||
1927 |
lemma powr_mult_base: "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)" |
|
1928 |
using assms by (auto simp: powr_add) |
|
51527 | 1929 |
|
1930 |
lemma powr_powr: "(x powr a) powr b = x powr (a * b)" |
|
53079 | 1931 |
by (simp add: powr_def) |
51527 | 1932 |
|
1933 |
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
|
1934 |
by (simp add: powr_powr mult.commute) |
51527 | 1935 |
|
1936 |
lemma powr_minus: "x powr (-a) = inverse (x powr a)" |
|
53079 | 1937 |
by (simp add: powr_def exp_minus [symmetric]) |
51527 | 1938 |
|
1939 |
lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)" |
|
53079 | 1940 |
by (simp add: divide_inverse powr_minus) |
1941 |
||
58984
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
1942 |
lemma divide_powr_uminus: "a / b powr c = a * b powr (- c)" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
1943 |
by (simp add: powr_minus_divide) |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
1944 |
|
53079 | 1945 |
lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b" |
1946 |
by (simp add: powr_def) |
|
1947 |
||
1948 |
lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b" |
|
1949 |
by (simp add: powr_def) |
|
1950 |
||
1951 |
lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a < x powr b) = (a < b)" |
|
1952 |
by (blast intro: powr_less_cancel powr_less_mono) |
|
1953 |
||
1954 |
lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a \<le> x powr b) = (a \<le> b)" |
|
1955 |
by (simp add: linorder_not_less [symmetric]) |
|
51527 | 1956 |
|
1957 |
lemma log_ln: "ln x = log (exp(1)) x" |
|
53079 | 1958 |
by (simp add: log_def) |
1959 |
||
1960 |
lemma DERIV_log: |
|
1961 |
assumes "x > 0" |
|
1962 |
shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)" |
|
51527 | 1963 |
proof - |
1964 |
def lb \<equiv> "1 / ln b" |
|
1965 |
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
|
1966 |
using `x > 0` by (auto intro!: derivative_eq_intros) |
51527 | 1967 |
ultimately show ?thesis |
1968 |
by (simp add: log_def) |
|
1969 |
qed |
|
1970 |
||
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
1971 |
lemmas DERIV_log[THEN DERIV_chain2, derivative_intros] |
51527 | 1972 |
|
53079 | 1973 |
lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x" |
1974 |
by (simp add: powr_def log_def) |
|
1975 |
||
1976 |
lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y" |
|
1977 |
by (simp add: log_def powr_def) |
|
1978 |
||
1979 |
lemma log_mult: |
|
1980 |
"0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> |
|
1981 |
log a (x * y) = log a x + log a y" |
|
1982 |
by (simp add: log_def ln_mult divide_inverse distrib_right) |
|
1983 |
||
1984 |
lemma log_eq_div_ln_mult_log: |
|
1985 |
"0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> |
|
1986 |
log a x = (ln b/ln a) * log b x" |
|
1987 |
by (simp add: log_def divide_inverse) |
|
51527 | 1988 |
|
1989 |
text{*Base 10 logarithms*} |
|
53079 | 1990 |
lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x" |
1991 |
by (simp add: log_def) |
|
1992 |
||
1993 |
lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x" |
|
1994 |
by (simp add: log_def) |
|
51527 | 1995 |
|
1996 |
lemma log_one [simp]: "log a 1 = 0" |
|
53079 | 1997 |
by (simp add: log_def) |
51527 | 1998 |
|
1999 |
lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1" |
|
53079 | 2000 |
by (simp add: log_def) |
2001 |
||
2002 |
lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x" |
|
2003 |
apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1]) |
|
2004 |
apply (simp add: log_mult [symmetric]) |
|
2005 |
done |
|
2006 |
||
2007 |
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" |
|
2008 |
by (simp add: log_mult divide_inverse log_inverse) |
|
51527 | 2009 |
|
58984
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2010 |
lemma log_add_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log b x + y = log b (x * b powr y)" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2011 |
and add_log_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> y + log b x = log b (b powr y * x)" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2012 |
and log_minus_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log b x - y = log b (x * b powr -y)" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2013 |
and minus_log_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> y - log b x = log b (b powr y / x)" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2014 |
by (simp_all add: log_mult log_divide) |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2015 |
|
51527 | 2016 |
lemma log_less_cancel_iff [simp]: |
53079 | 2017 |
"1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y" |
2018 |
apply safe |
|
2019 |
apply (rule_tac [2] powr_less_cancel) |
|
2020 |
apply (drule_tac a = "log a x" in powr_less_mono, auto) |
|
2021 |
done |
|
2022 |
||
2023 |
lemma log_inj: |
|
2024 |
assumes "1 < b" |
|
2025 |
shows "inj_on (log b) {0 <..}" |
|
51527 | 2026 |
proof (rule inj_onI, simp) |
53079 | 2027 |
fix x y |
2028 |
assume pos: "0 < x" "0 < y" and *: "log b x = log b y" |
|
51527 | 2029 |
show "x = y" |
2030 |
proof (cases rule: linorder_cases) |
|
53079 | 2031 |
assume "x = y" |
2032 |
then show ?thesis by simp |
|
2033 |
next |
|
51527 | 2034 |
assume "x < y" hence "log b x < log b y" |
2035 |
using log_less_cancel_iff[OF `1 < b`] pos by simp |
|
53079 | 2036 |
then show ?thesis using * by simp |
51527 | 2037 |
next |
2038 |
assume "y < x" hence "log b y < log b x" |
|
2039 |
using log_less_cancel_iff[OF `1 < b`] pos by simp |
|
53079 | 2040 |
then show ?thesis using * by simp |
2041 |
qed |
|
51527 | 2042 |
qed |
2043 |
||
2044 |
lemma log_le_cancel_iff [simp]: |
|
53079 | 2045 |
"1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (log a x \<le> log a y) = (x \<le> y)" |
2046 |
by (simp add: linorder_not_less [symmetric]) |
|
51527 | 2047 |
|
2048 |
lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x" |
|
2049 |
using log_less_cancel_iff[of a 1 x] by simp |
|
2050 |
||
2051 |
lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x" |
|
2052 |
using log_le_cancel_iff[of a 1 x] by simp |
|
2053 |
||
2054 |
lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1" |
|
2055 |
using log_less_cancel_iff[of a x 1] by simp |
|
2056 |
||
2057 |
lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1" |
|
2058 |
using log_le_cancel_iff[of a x 1] by simp |
|
2059 |
||
2060 |
lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x" |
|
2061 |
using log_less_cancel_iff[of a a x] by simp |
|
2062 |
||
2063 |
lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x" |
|
2064 |
using log_le_cancel_iff[of a a x] by simp |
|
2065 |
||
2066 |
lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a" |
|
2067 |
using log_less_cancel_iff[of a x a] by simp |
|
2068 |
||
2069 |
lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a" |
|
2070 |
using log_le_cancel_iff[of a x a] by simp |
|
2071 |
||
58984
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2072 |
lemma le_log_iff: |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2073 |
assumes "1 < b" "x > 0" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2074 |
shows "y \<le> log b x \<longleftrightarrow> b powr y \<le> x" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2075 |
by (metis assms(1) assms(2) dual_order.strict_trans powr_le_cancel_iff powr_log_cancel |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2076 |
powr_one_eq_one powr_one_gt_zero_iff) |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2077 |
|
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2078 |
lemma less_log_iff: |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2079 |
assumes "1 < b" "x > 0" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2080 |
shows "y < log b x \<longleftrightarrow> b powr y < x" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2081 |
by (metis assms(1) assms(2) dual_order.strict_trans less_irrefl powr_less_cancel_iff |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2082 |
powr_log_cancel zero_less_one) |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2083 |
|
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2084 |
lemma |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2085 |
assumes "1 < b" "x > 0" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2086 |
shows log_less_iff: "log b x < y \<longleftrightarrow> x < b powr y" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2087 |
and log_le_iff: "log b x \<le> y \<longleftrightarrow> x \<le> b powr y" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2088 |
using le_log_iff[OF assms, of y] less_log_iff[OF assms, of y] |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2089 |
by auto |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2090 |
|
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2091 |
lemmas powr_le_iff = le_log_iff[symmetric] |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2092 |
and powr_less_iff = le_log_iff[symmetric] |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2093 |
and less_powr_iff = log_less_iff[symmetric] |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2094 |
and le_powr_iff = log_le_iff[symmetric] |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2095 |
|
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2096 |
lemma |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2097 |
floor_log_eq_powr_iff: "x > 0 \<Longrightarrow> b > 1 \<Longrightarrow> \<lfloor>log b x\<rfloor> = k \<longleftrightarrow> b powr k \<le> x \<and> x < b powr (k + 1)" |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2098 |
by (auto simp add: floor_eq_iff powr_le_iff less_powr_iff) |
ae0c56c485ae
added lemmas: convert between powr and log in comparisons, pull log out of addition/subtraction
immler
parents:
58981
diff
changeset
|
2099 |
|
51527 | 2100 |
lemma powr_realpow: "0 < x ==> x powr (real n) = x^n" |
53079 | 2101 |
apply (induct n) |
2102 |
apply simp |
|
51527 | 2103 |
apply (subgoal_tac "real(Suc n) = real n + 1") |
2104 |
apply (erule ssubst) |
|
2105 |
apply (subst powr_add, simp, simp) |
|
53079 | 2106 |
done |
51527 | 2107 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2108 |
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
|
2109 |
unfolding real_of_nat_numeral [symmetric] by (rule powr_realpow) |
52139 | 2110 |
|
57180 | 2111 |
lemma powr2_sqrt[simp]: "0 < x \<Longrightarrow> sqrt x powr 2 = x" |
2112 |
by(simp add: powr_realpow_numeral) |
|
2113 |
||
51527 | 2114 |
lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))" |
2115 |
apply (case_tac "x = 0", simp, simp) |
|
2116 |
apply (rule powr_realpow [THEN sym], simp) |
|
53079 | 2117 |
done |
51527 | 2118 |
|
2119 |
lemma powr_int: |
|
2120 |
assumes "x > 0" |
|
2121 |
shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))" |
|
53079 | 2122 |
proof (cases "i < 0") |
2123 |
case True |
|
51527 | 2124 |
have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps) |
2125 |
show ?thesis using `i < 0` `x > 0` by (simp add: r field_simps powr_realpow[symmetric]) |
|
53079 | 2126 |
next |
2127 |
case False |
|
2128 |
then show ?thesis by (simp add: assms powr_realpow[symmetric]) |
|
2129 |
qed |
|
51527 | 2130 |
|
58981 | 2131 |
lemma compute_powr[code]: |
2132 |
fixes i::real |
|
2133 |
shows "b powr i = |
|
2134 |
(if b \<le> 0 then Code.abort (STR ''op powr with nonpositive base'') (\<lambda>_. b powr i) |
|
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
58984
diff
changeset
|
2135 |
else if floor i = i then (if 0 \<le> i then b ^ nat(floor i) else 1 / b ^ nat(floor (- i))) |
58981 | 2136 |
else Code.abort (STR ''op powr with non-integer exponent'') (\<lambda>_. b powr i))" |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
58984
diff
changeset
|
2137 |
by (auto simp: powr_int) |
58981 | 2138 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2139 |
lemma powr_one: "0 < x \<Longrightarrow> x powr 1 = x" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2140 |
using powr_realpow [of x 1] by simp |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2141 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2142 |
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
|
2143 |
by (fact powr_realpow_numeral) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2144 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2145 |
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
|
2146 |
using powr_int [of x "- 1"] by simp |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2147 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
2148 |
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
|
2149 |
using powr_int [of x "- numeral n"] by simp |
51527 | 2150 |
|
53079 | 2151 |
lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)" |
51527 | 2152 |
by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr) |
2153 |
||
56483 | 2154 |
lemma ln_powr: "ln (x powr y) = y * ln x" |
2155 |
by (simp add: powr_def) |
|
2156 |
||
56952 | 2157 |
lemma ln_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> ln (root n b) = ln b / n" |
2158 |
by(simp add: root_powr_inverse ln_powr) |
|
2159 |
||
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2160 |
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
|
2161 |
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
|
2162 |
|
56952 | 2163 |
lemma log_root: "\<lbrakk> n > 0; a > 0 \<rbrakk> \<Longrightarrow> log b (root n a) = log b a / n" |
2164 |
by(simp add: log_def ln_root) |
|
2165 |
||
56483 | 2166 |
lemma log_powr: "log b (x powr y) = y * log b x" |
2167 |
by (simp add: log_def ln_powr) |
|
2168 |
||
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2169 |
lemma log_nat_power: "0 < x \<Longrightarrow> log b (x^n) = real n * log b x" |
56483 | 2170 |
by (simp add: log_powr powr_realpow [symmetric]) |
2171 |
||
2172 |
lemma log_base_change: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log b x = log a x / log a b" |
|
2173 |
by (simp add: log_def) |
|
2174 |
||
2175 |
lemma log_base_pow: "0 < a \<Longrightarrow> log (a ^ n) x = log a x / n" |
|
2176 |
by (simp add: log_def ln_realpow) |
|
2177 |
||
2178 |
lemma log_base_powr: "log (a powr b) x = log a x / b" |
|
2179 |
by (simp add: log_def ln_powr) |
|
51527 | 2180 |
|
56952 | 2181 |
lemma log_base_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> log (root n b) x = n * (log b x)" |
2182 |
by(simp add: log_def ln_root) |
|
2183 |
||
51527 | 2184 |
lemma ln_bound: "1 <= x ==> ln x <= x" |
2185 |
apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1") |
|
2186 |
apply simp |
|
2187 |
apply (rule ln_add_one_self_le_self, simp) |
|
53079 | 2188 |
done |
51527 | 2189 |
|
2190 |
lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b" |
|
53079 | 2191 |
apply (cases "x = 1", simp) |
2192 |
apply (cases "a = b", simp) |
|
51527 | 2193 |
apply (rule order_less_imp_le) |
2194 |
apply (rule powr_less_mono, auto) |
|
53079 | 2195 |
done |
51527 | 2196 |
|
2197 |
lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a" |
|
2198 |
apply (subst powr_zero_eq_one [THEN sym]) |
|
2199 |
apply (rule powr_mono, assumption+) |
|
53079 | 2200 |
done |
2201 |
||
2202 |
lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a < y powr a" |
|
51527 | 2203 |
apply (unfold powr_def) |
2204 |
apply (rule exp_less_mono) |
|
2205 |
apply (rule mult_strict_left_mono) |
|
2206 |
apply (subst ln_less_cancel_iff, assumption) |
|
2207 |
apply (rule order_less_trans) |
|
2208 |
prefer 2 |
|
2209 |
apply assumption+ |
|
53079 | 2210 |
done |
2211 |
||
2212 |
lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a < x powr a" |
|
51527 | 2213 |
apply (unfold powr_def) |
2214 |
apply (rule exp_less_mono) |
|
2215 |
apply (rule mult_strict_left_mono_neg) |
|
2216 |
apply (subst ln_less_cancel_iff) |
|
2217 |
apply assumption |
|
2218 |
apply (rule order_less_trans) |
|
2219 |
prefer 2 |
|
2220 |
apply assumption+ |
|
53079 | 2221 |
done |
51527 | 2222 |
|
2223 |
lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a" |
|
2224 |
apply (case_tac "a = 0", simp) |
|
2225 |
apply (case_tac "x = y", simp) |
|
54575 | 2226 |
apply (metis less_eq_real_def powr_less_mono2) |
53079 | 2227 |
done |
2228 |
||
2229 |
lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y" |
|
51527 | 2230 |
unfolding powr_def exp_inj_iff by simp |
2231 |
||
2232 |
lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a" |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2233 |
by (metis less_eq_real_def ln_less_self mult_imp_le_div_pos ln_powr mult.commute |
54575 | 2234 |
order.strict_trans2 powr_gt_zero zero_less_one) |
51527 | 2235 |
|
2236 |
lemma ln_powr_bound2: |
|
2237 |
assumes "1 < x" and "0 < a" |
|
2238 |
shows "(ln x) powr a <= (a powr a) * x" |
|
2239 |
proof - |
|
2240 |
from assms have "ln x <= (x powr (1 / a)) / (1 / a)" |
|
54575 | 2241 |
by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff) |
51527 | 2242 |
also have "... = a * (x powr (1 / a))" |
2243 |
by simp |
|
2244 |
finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a" |
|
54575 | 2245 |
by (metis assms less_imp_le ln_gt_zero powr_mono2) |
51527 | 2246 |
also have "... = (a powr a) * ((x powr (1 / a)) powr a)" |
54575 | 2247 |
by (metis assms(2) powr_mult powr_gt_zero) |
51527 | 2248 |
also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)" |
2249 |
by (rule powr_powr) |
|
54575 | 2250 |
also have "... = x" using assms |
2251 |
by auto |
|
51527 | 2252 |
finally show ?thesis . |
2253 |
qed |
|
2254 |
||
2255 |
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
|
2256 |
"\<lbrakk>(f ---> a) F; (g ---> b) F; a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x powr g x) ---> a powr b) F" |
51527 | 2257 |
unfolding powr_def by (intro tendsto_intros) |
2258 |
||
2259 |
lemma continuous_powr: |
|
53079 | 2260 |
assumes "continuous F f" |
2261 |
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
|
2262 |
and "f (Lim F (\<lambda>x. x)) \<noteq> 0" |
51527 | 2263 |
shows "continuous F (\<lambda>x. (f x) powr (g x))" |
2264 |
using assms unfolding continuous_def by (rule tendsto_powr) |
|
2265 |
||
2266 |
lemma continuous_at_within_powr[continuous_intros]: |
|
53079 | 2267 |
assumes "continuous (at a within s) f" |
2268 |
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
|
2269 |
and "f a \<noteq> 0" |
51527 | 2270 |
shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))" |
2271 |
using assms unfolding continuous_within by (rule tendsto_powr) |
|
2272 |
||
2273 |
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
|
2274 |
assumes "isCont f a" "isCont g a" "f a \<noteq> 0" |
51527 | 2275 |
shows "isCont (\<lambda>x. (f x) powr g x) a" |
2276 |
using assms unfolding continuous_at by (rule tendsto_powr) |
|
2277 |
||
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
2278 |
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
|
2279 |
assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> 0" |
51527 | 2280 |
shows "continuous_on s (\<lambda>x. (f x) powr (g x))" |
2281 |
using assms unfolding continuous_on_def by (fast intro: tendsto_powr) |
|
2282 |
||
2283 |
(* FIXME: generalize by replacing d by with g x and g ---> d? *) |
|
2284 |
lemma tendsto_zero_powrI: |
|
2285 |
assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> 0) F" |
|
53079 | 2286 |
and "0 < d" |
51527 | 2287 |
shows "((\<lambda>x. f x powr d) ---> 0) F" |
2288 |
proof (rule tendstoI) |
|
2289 |
fix e :: real assume "0 < e" |
|
2290 |
def Z \<equiv> "e powr (1 / d)" |
|
2291 |
with `0 < e` have "0 < Z" by simp |
|
2292 |
with assms have "eventually (\<lambda>x. 0 < f x \<and> dist (f x) 0 < Z) F" |
|
2293 |
by (intro eventually_conj tendstoD) |
|
2294 |
moreover |
|
2295 |
from assms have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> x powr d < Z powr d" |
|
2296 |
by (intro powr_less_mono2) (auto simp: dist_real_def) |
|
2297 |
with assms `0 < e` have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> dist (x powr d) 0 < e" |
|
2298 |
unfolding dist_real_def Z_def by (auto simp: powr_powr) |
|
2299 |
ultimately |
|
2300 |
show "eventually (\<lambda>x. dist (f x powr d) 0 < e) F" by (rule eventually_elim1) |
|
2301 |
qed |
|
2302 |
||
2303 |
lemma tendsto_neg_powr: |
|
53079 | 2304 |
assumes "s < 0" |
2305 |
and "LIM x F. f x :> at_top" |
|
51527 | 2306 |
shows "((\<lambda>x. f x powr s) ---> 0) F" |
2307 |
proof (rule tendstoI) |
|
2308 |
fix e :: real assume "0 < e" |
|
2309 |
def Z \<equiv> "e powr (1 / s)" |
|
2310 |
from assms have "eventually (\<lambda>x. Z < f x) F" |
|
2311 |
by (simp add: filterlim_at_top_dense) |
|
2312 |
moreover |
|
2313 |
from assms have "\<And>x. Z < x \<Longrightarrow> x powr s < Z powr s" |
|
2314 |
by (auto simp: Z_def intro!: powr_less_mono2_neg) |
|
2315 |
with assms `0 < e` have "\<And>x. Z < x \<Longrightarrow> dist (x powr s) 0 < e" |
|
2316 |
by (simp add: powr_powr Z_def dist_real_def) |
|
2317 |
ultimately |
|
2318 |
show "eventually (\<lambda>x. dist (f x powr s) 0 < e) F" by (rule eventually_elim1) |
|
2319 |
qed |
|
2320 |
||
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2321 |
(* 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
|
2322 |
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
|
2323 |
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
|
2324 |
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
|
2325 |
proof cases |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2326 |
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
|
2327 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2328 |
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
|
2329 |
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
|
2330 |
then have "((\<lambda>y. ln (1 + x * y) / y) ---> x) (at 0)" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2331 |
by (auto simp add: has_field_derivative_def field_has_derivative_at) |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2332 |
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
|
2333 |
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
|
2334 |
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
|
2335 |
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
|
2336 |
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
|
2337 |
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
|
2338 |
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
|
2339 |
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
|
2340 |
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
|
2341 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2342 |
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
|
2343 |
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
|
2344 |
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
|
2345 |
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
|
2346 |
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
|
2347 |
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
|
2348 |
done |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2349 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2350 |
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
|
2351 |
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
|
2352 |
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
|
2353 |
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
|
2354 |
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
|
2355 |
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
|
2356 |
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
|
2357 |
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
|
2358 |
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
|
2359 |
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
|
2360 |
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
|
2361 |
done |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2362 |
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
|
2363 |
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
|
2364 |
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
|
2365 |
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
|
2366 |
qed auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57180
diff
changeset
|
2367 |
|
29164 | 2368 |
subsection {* Sine and Cosine *} |
2369 |
||
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2370 |
definition sin_coeff :: "nat \<Rightarrow> real" where |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2371 |
"sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / (fact n))" |
31271 | 2372 |
|
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2373 |
definition cos_coeff :: "nat \<Rightarrow> real" where |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2374 |
"cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / (fact n) else 0)" |
31271 | 2375 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2376 |
definition sin :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2377 |
where "sin = (\<lambda>x. \<Sum>n. sin_coeff n *\<^sub>R x^n)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2378 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2379 |
definition cos :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2380 |
where "cos = (\<lambda>x. \<Sum>n. cos_coeff n *\<^sub>R x^n)" |
31271 | 2381 |
|
44319
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2382 |
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
|
2383 |
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
|
2384 |
|
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2385 |
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
|
2386 |
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
|
2387 |
|
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2388 |
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
|
2389 |
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
|
2390 |
by (simp del: mult_Suc) |
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2391 |
|
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2392 |
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
|
2393 |
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
|
2394 |
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
|
2395 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2396 |
lemma summable_norm_sin: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2397 |
fixes x :: "'a::{real_normed_algebra_1,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2398 |
shows "summable (\<lambda>n. norm (sin_coeff n *\<^sub>R x^n))" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2399 |
unfolding sin_coeff_def |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2400 |
apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]]) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2401 |
apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) |
53079 | 2402 |
done |
29164 | 2403 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2404 |
lemma summable_norm_cos: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2405 |
fixes x :: "'a::{real_normed_algebra_1,banach}" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2406 |
shows "summable (\<lambda>n. norm (cos_coeff n *\<^sub>R x^n))" |
53079 | 2407 |
unfolding cos_coeff_def |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2408 |
apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]]) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2409 |
apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) |
53079 | 2410 |
done |
29164 | 2411 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2412 |
lemma sin_converges: "(\<lambda>n. sin_coeff n *\<^sub>R x^n) sums sin(x)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2413 |
unfolding sin_def |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2414 |
by (metis (full_types) summable_norm_cancel summable_norm_sin summable_sums) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2415 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2416 |
lemma cos_converges: "(\<lambda>n. cos_coeff n *\<^sub>R x^n) sums cos(x)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2417 |
unfolding cos_def |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2418 |
by (metis (full_types) summable_norm_cancel summable_norm_cos summable_sums) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2419 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2420 |
lemma sin_of_real: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2421 |
fixes x::real |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2422 |
shows "sin (of_real x) = of_real (sin x)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2423 |
proof - |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2424 |
have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) = (\<lambda>n. sin_coeff n *\<^sub>R (of_real x)^n)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2425 |
proof |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2426 |
fix n |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2427 |
show "of_real (sin_coeff n *\<^sub>R x^n) = sin_coeff n *\<^sub>R of_real x^n" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2428 |
by (simp add: scaleR_conv_of_real) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2429 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2430 |
also have "... sums (sin (of_real x))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2431 |
by (rule sin_converges) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2432 |
finally have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) sums (sin (of_real x))" . |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2433 |
then show ?thesis |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2434 |
using sums_unique2 sums_of_real [OF sin_converges] |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2435 |
by blast |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2436 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2437 |
|
59862 | 2438 |
corollary sin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> sin z \<in> \<real>" |
2439 |
by (metis Reals_cases Reals_of_real sin_of_real) |
|
2440 |
||
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2441 |
lemma cos_of_real: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2442 |
fixes x::real |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2443 |
shows "cos (of_real x) = of_real (cos x)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2444 |
proof - |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2445 |
have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) = (\<lambda>n. cos_coeff n *\<^sub>R (of_real x)^n)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2446 |
proof |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2447 |
fix n |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2448 |
show "of_real (cos_coeff n *\<^sub>R x^n) = cos_coeff n *\<^sub>R of_real x^n" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2449 |
by (simp add: scaleR_conv_of_real) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2450 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2451 |
also have "... sums (cos (of_real x))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2452 |
by (rule cos_converges) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2453 |
finally have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) sums (cos (of_real x))" . |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2454 |
then show ?thesis |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2455 |
using sums_unique2 sums_of_real [OF cos_converges] |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2456 |
by blast |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2457 |
qed |
29164 | 2458 |
|
59862 | 2459 |
corollary cos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cos z \<in> \<real>" |
2460 |
by (metis Reals_cases Reals_of_real cos_of_real) |
|
2461 |
||
44319
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2462 |
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
|
2463 |
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
|
2464 |
|
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2465 |
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
|
2466 |
by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc) |
29164 | 2467 |
|
2468 |
text{*Now at last we can get the derivatives of exp, sin and cos*} |
|
2469 |
||
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2470 |
lemma DERIV_sin [simp]: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2471 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2472 |
shows "DERIV sin x :> cos(x)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2473 |
unfolding sin_def cos_def scaleR_conv_of_real |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2474 |
apply (rule DERIV_cong) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2475 |
apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"]) |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2476 |
apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2477 |
summable_minus_iff scaleR_conv_of_real [symmetric] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2478 |
summable_norm_sin [THEN summable_norm_cancel] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2479 |
summable_norm_cos [THEN summable_norm_cancel]) |
44319
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2480 |
done |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2481 |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
2482 |
declare DERIV_sin[THEN DERIV_chain2, derivative_intros] |
51527 | 2483 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2484 |
lemma DERIV_cos [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2485 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2486 |
shows "DERIV cos x :> -sin(x)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2487 |
unfolding sin_def cos_def scaleR_conv_of_real |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2488 |
apply (rule DERIV_cong) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2489 |
apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"]) |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2490 |
apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus |
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2491 |
diffs_sin_coeff diffs_cos_coeff |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2492 |
summable_minus_iff scaleR_conv_of_real [symmetric] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2493 |
summable_norm_sin [THEN summable_norm_cancel] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2494 |
summable_norm_cos [THEN summable_norm_cancel]) |
44319
806e0390de53
move sin_coeff and cos_coeff lemmas to Transcendental.thy; simplify some proofs
huffman
parents:
44318
diff
changeset
|
2495 |
done |
29164 | 2496 |
|
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
2497 |
declare DERIV_cos[THEN DERIV_chain2, derivative_intros] |
51527 | 2498 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2499 |
lemma isCont_sin: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2500 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2501 |
shows "isCont sin x" |
44311 | 2502 |
by (rule DERIV_sin [THEN DERIV_isCont]) |
2503 |
||
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2504 |
lemma isCont_cos: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2505 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2506 |
shows "isCont cos x" |
44311 | 2507 |
by (rule DERIV_cos [THEN DERIV_isCont]) |
2508 |
||
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2509 |
lemma isCont_sin' [simp]: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2510 |
fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2511 |
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a" |
44311 | 2512 |
by (rule isCont_o2 [OF _ isCont_sin]) |
2513 |
||
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2514 |
(*FIXME A CONTEXT FOR F WOULD BE BETTER*) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2515 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2516 |
lemma isCont_cos' [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2517 |
fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2518 |
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a" |
44311 | 2519 |
by (rule isCont_o2 [OF _ isCont_cos]) |
2520 |
||
2521 |
lemma tendsto_sin [tendsto_intros]: |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2522 |
fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2523 |
shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F" |
44311 | 2524 |
by (rule isCont_tendsto_compose [OF isCont_sin]) |
2525 |
||
2526 |
lemma tendsto_cos [tendsto_intros]: |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2527 |
fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2528 |
shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F" |
44311 | 2529 |
by (rule isCont_tendsto_compose [OF isCont_cos]) |
29164 | 2530 |
|
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
|
2531 |
lemma continuous_sin [continuous_intros]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2532 |
fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2533 |
shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (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
|
2534 |
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
|
2535 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
2536 |
lemma continuous_on_sin [continuous_intros]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2537 |
fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2538 |
shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (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
|
2539 |
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
|
2540 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2541 |
lemma continuous_within_sin: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2542 |
fixes z :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2543 |
shows "continuous (at z within s) sin" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2544 |
by (simp add: continuous_within tendsto_sin) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2545 |
|
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
|
2546 |
lemma continuous_cos [continuous_intros]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2547 |
fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2548 |
shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (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
|
2549 |
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
|
2550 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
2551 |
lemma continuous_on_cos [continuous_intros]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2552 |
fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2553 |
shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (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
|
2554 |
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
|
2555 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2556 |
lemma continuous_within_cos: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2557 |
fixes z :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2558 |
shows "continuous (at z within s) cos" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2559 |
by (simp add: continuous_within tendsto_cos) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2560 |
|
29164 | 2561 |
subsection {* Properties of Sine and Cosine *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2562 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2563 |
lemma sin_zero [simp]: "sin 0 = 0" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2564 |
unfolding sin_def sin_coeff_def by (simp add: scaleR_conv_of_real powser_zero) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2565 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2566 |
lemma cos_zero [simp]: "cos 0 = 1" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2567 |
unfolding cos_def cos_coeff_def by (simp add: scaleR_conv_of_real powser_zero) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2568 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2569 |
lemma DERIV_fun_sin: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2570 |
"DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. sin(g x)) x :> cos(g x) * m" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2571 |
by (auto intro!: derivative_intros) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2572 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2573 |
lemma DERIV_fun_cos: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2574 |
"DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. cos(g x)) x :> -sin(g x) * m" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2575 |
by (auto intro!: derivative_eq_intros simp: real_of_nat_def) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2576 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2577 |
subsection {*Deriving the Addition Formulas*} |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2578 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2579 |
text{*The the product of two cosine series*} |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2580 |
lemma cos_x_cos_y: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2581 |
fixes x :: "'a::{real_normed_field,banach}" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2582 |
shows "(\<lambda>p. \<Sum>n\<le>p. |
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2583 |
if even p \<and> even n |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2584 |
then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2585 |
sums (cos x * cos y)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2586 |
proof - |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2587 |
{ fix n p::nat |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2588 |
assume "n\<le>p" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2589 |
then have *: "even n \<Longrightarrow> even p \<Longrightarrow> (-1) ^ (n div 2) * (-1) ^ ((p - n) div 2) = (-1 :: real) ^ (p div 2)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2590 |
by (metis div_add power_add le_add_diff_inverse odd_add) |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2591 |
have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) = |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2592 |
(if even p \<and> even n then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2593 |
using `n\<le>p` |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2594 |
by (auto simp: * algebra_simps cos_coeff_def binomial_fact real_of_nat_def) |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2595 |
} |
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2596 |
then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> even n |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2597 |
then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) = |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2598 |
(\<lambda>p. \<Sum>n\<le>p. (cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2599 |
by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2600 |
also have "... = (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n *\<^sub>R x^n) * (cos_coeff (p - n) *\<^sub>R y^(p-n)))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2601 |
by (simp add: algebra_simps) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2602 |
also have "... sums (cos x * cos y)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2603 |
using summable_norm_cos |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2604 |
by (auto simp: cos_def scaleR_conv_of_real intro!: Cauchy_product_sums) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2605 |
finally show ?thesis . |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2606 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2607 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2608 |
text{*The product of two sine series*} |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2609 |
lemma sin_x_sin_y: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2610 |
fixes x :: "'a::{real_normed_field,banach}" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2611 |
shows "(\<lambda>p. \<Sum>n\<le>p. |
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2612 |
if even p \<and> odd n |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2613 |
then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2614 |
sums (sin x * sin y)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2615 |
proof - |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2616 |
{ fix n p::nat |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2617 |
assume "n\<le>p" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2618 |
{ assume np: "odd n" "even p" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2619 |
with `n\<le>p` have "n - Suc 0 + (p - Suc n) = p - Suc (Suc 0)" "Suc (Suc 0) \<le> p" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2620 |
by arith+ |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2621 |
moreover have "(p - Suc (Suc 0)) div 2 = p div 2 - Suc 0" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2622 |
by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2623 |
ultimately have *: "(-1) ^ ((n - Suc 0) div 2) * (-1) ^ ((p - Suc n) div 2) = - ((-1 :: real) ^ (p div 2))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2624 |
using np `n\<le>p` |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2625 |
apply (simp add: power_add [symmetric] div_add [symmetric] del: div_add) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2626 |
apply (metis (no_types) One_nat_def Suc_1 le_div_geq minus_minus mult.left_neutral mult_minus_left power.simps(2) zero_less_Suc) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2627 |
done |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2628 |
} then |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2629 |
have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) = |
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2630 |
(if even p \<and> odd n |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2631 |
then -((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2632 |
using `n\<le>p` |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2633 |
by (auto simp: algebra_simps sin_coeff_def binomial_fact real_of_nat_def) |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2634 |
} |
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2635 |
then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> odd n |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2636 |
then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) = |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2637 |
(\<lambda>p. \<Sum>n\<le>p. (sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2638 |
by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2639 |
also have "... = (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n *\<^sub>R x^n) * (sin_coeff (p - n) *\<^sub>R y^(p-n)))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2640 |
by (simp add: algebra_simps) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2641 |
also have "... sums (sin x * sin y)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2642 |
using summable_norm_sin |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2643 |
by (auto simp: sin_def scaleR_conv_of_real intro!: Cauchy_product_sums) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2644 |
finally show ?thesis . |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2645 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2646 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2647 |
lemma sums_cos_x_plus_y: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2648 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2649 |
shows |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2650 |
"(\<lambda>p. \<Sum>n\<le>p. if even p |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2651 |
then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2652 |
else 0) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2653 |
sums cos (x + y)" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2654 |
proof - |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2655 |
{ fix p::nat |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2656 |
have "(\<Sum>n\<le>p. if even p |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2657 |
then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2658 |
else 0) = |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2659 |
(if even p |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2660 |
then \<Sum>n\<le>p. ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2661 |
else 0)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2662 |
by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2663 |
also have "... = (if even p |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2664 |
then of_real ((-1) ^ (p div 2) / (fact p)) * (\<Sum>n\<le>p. (p choose n) *\<^sub>R (x^n) * y^(p-n)) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2665 |
else 0)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2666 |
by (auto simp: setsum_right_distrib field_simps scaleR_conv_of_real nonzero_of_real_divide) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2667 |
also have "... = cos_coeff p *\<^sub>R ((x + y) ^ p)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2668 |
by (simp add: cos_coeff_def binomial_ring [of x y] scaleR_conv_of_real real_of_nat_def atLeast0AtMost) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2669 |
finally have "(\<Sum>n\<le>p. if even p |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2670 |
then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2671 |
else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)" . |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2672 |
} |
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2673 |
then have "(\<lambda>p. \<Sum>n\<le>p. |
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2674 |
if even p |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2675 |
then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2676 |
else 0) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2677 |
= (\<lambda>p. cos_coeff p *\<^sub>R ((x+y)^p))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2678 |
by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2679 |
also have "... sums cos (x + y)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2680 |
by (rule cos_converges) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2681 |
finally show ?thesis . |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2682 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2683 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2684 |
theorem cos_add: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2685 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2686 |
shows "cos (x + y) = cos x * cos y - sin x * sin y" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2687 |
proof - |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2688 |
{ fix n p::nat |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2689 |
assume "n\<le>p" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2690 |
then have "(if even p \<and> even n |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2691 |
then ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) - |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2692 |
(if even p \<and> odd n |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2693 |
then - ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2694 |
= (if even p |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2695 |
then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2696 |
by simp |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2697 |
} |
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2698 |
then have "(\<lambda>p. \<Sum>n\<le>p. (if even p |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2699 |
then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2700 |
sums (cos x * cos y - sin x * sin y)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2701 |
using sums_diff [OF cos_x_cos_y [of x y] sin_x_sin_y [of x y]] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2702 |
by (simp add: setsum_subtractf [symmetric]) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2703 |
then show ?thesis |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2704 |
by (blast intro: sums_cos_x_plus_y sums_unique2) |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2705 |
qed |
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2706 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2707 |
lemma sin_minus_converges: "(\<lambda>n. - (sin_coeff n *\<^sub>R (-x)^n)) sums sin(x)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2708 |
proof - |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2709 |
have [simp]: "\<And>n. - (sin_coeff n *\<^sub>R (-x)^n) = (sin_coeff n *\<^sub>R x^n)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2710 |
by (auto simp: sin_coeff_def elim!: oddE) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2711 |
show ?thesis |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2712 |
by (simp add: sin_def summable_norm_sin [THEN summable_norm_cancel, THEN summable_sums]) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2713 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2714 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2715 |
lemma sin_minus [simp]: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2716 |
fixes x :: "'a::{real_normed_algebra_1,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2717 |
shows "sin (-x) = -sin(x)" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2718 |
using sin_minus_converges [of x] |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2719 |
by (auto simp: sin_def summable_norm_sin [THEN summable_norm_cancel] suminf_minus sums_iff equation_minus_iff) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2720 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2721 |
lemma cos_minus_converges: "(\<lambda>n. (cos_coeff n *\<^sub>R (-x)^n)) sums cos(x)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2722 |
proof - |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2723 |
have [simp]: "\<And>n. (cos_coeff n *\<^sub>R (-x)^n) = (cos_coeff n *\<^sub>R x^n)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2724 |
by (auto simp: Transcendental.cos_coeff_def elim!: evenE) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2725 |
show ?thesis |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2726 |
by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel, THEN summable_sums]) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2727 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2728 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2729 |
lemma cos_minus [simp]: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2730 |
fixes x :: "'a::{real_normed_algebra_1,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2731 |
shows "cos (-x) = cos(x)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2732 |
using cos_minus_converges [of x] |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2733 |
by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel] |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2734 |
suminf_minus sums_iff equation_minus_iff) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2735 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2736 |
lemma sin_cos_squared_add [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2737 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2738 |
shows "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2739 |
using cos_add [of x "-x"] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2740 |
by (simp add: power2_eq_square algebra_simps) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2741 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2742 |
lemma sin_cos_squared_add2 [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2743 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2744 |
shows "(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
|
2745 |
by (subst add.commute, rule sin_cos_squared_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2746 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2747 |
lemma sin_cos_squared_add3 [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2748 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2749 |
shows "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
|
2750 |
using sin_cos_squared_add2 [unfolded power2_eq_square] . |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2751 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2752 |
lemma sin_squared_eq: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2753 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2754 |
shows "(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
|
2755 |
unfolding eq_diff_eq by (rule sin_cos_squared_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2756 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2757 |
lemma cos_squared_eq: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2758 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2759 |
shows "(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
|
2760 |
unfolding eq_diff_eq by (rule sin_cos_squared_add2) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2761 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2762 |
lemma abs_sin_le_one [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2763 |
fixes x :: real |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2764 |
shows "\<bar>sin x\<bar> \<le> 1" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2765 |
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
|
2766 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2767 |
lemma sin_ge_minus_one [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2768 |
fixes x :: real |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2769 |
shows "-1 \<le> sin x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2770 |
using abs_sin_le_one [of x] unfolding abs_le_iff by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2771 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2772 |
lemma sin_le_one [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2773 |
fixes x :: real |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2774 |
shows "sin x \<le> 1" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2775 |
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
|
2776 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2777 |
lemma abs_cos_le_one [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2778 |
fixes x :: real |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2779 |
shows "\<bar>cos x\<bar> \<le> 1" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2780 |
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
|
2781 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2782 |
lemma cos_ge_minus_one [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2783 |
fixes x :: real |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2784 |
shows "-1 \<le> cos x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2785 |
using abs_cos_le_one [of x] unfolding abs_le_iff by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2786 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2787 |
lemma cos_le_one [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2788 |
fixes x :: real |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2789 |
shows "cos x \<le> 1" |
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset
|
2790 |
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
|
2791 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2792 |
lemma cos_diff: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2793 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2794 |
shows "cos (x - y) = cos x * cos y + sin x * sin y" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2795 |
using cos_add [of x "- y"] by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2796 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2797 |
lemma cos_double: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2798 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2799 |
shows "cos(2*x) = (cos x)\<^sup>2 - (sin x)\<^sup>2" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2800 |
using cos_add [where x=x and y=x] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2801 |
by (simp add: power2_eq_square) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2802 |
|
41970 | 2803 |
lemma DERIV_fun_pow: "DERIV g x :> m ==> |
53079 | 2804 |
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
|
2805 |
by (auto intro!: derivative_eq_intros simp: real_of_nat_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2806 |
|
15229 | 2807 |
lemma DERIV_fun_exp: |
53079 | 2808 |
"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
|
2809 |
by (auto intro!: derivative_intros) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2810 |
|
29164 | 2811 |
subsection {* The Constant Pi *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2812 |
|
53079 | 2813 |
definition pi :: real |
2814 |
where "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)" |
|
23043 | 2815 |
|
41970 | 2816 |
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
|
2817 |
hence define pi.*} |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2818 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2819 |
lemma sin_paired: |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2820 |
fixes x :: real |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2821 |
shows "(\<lambda>n. (- 1) ^ n / (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums sin x" |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2822 |
proof - |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2823 |
have "(\<lambda>n. \<Sum>k = n*2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2824 |
apply (rule sums_group) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2825 |
using sin_converges [of x, unfolded scaleR_conv_of_real] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2826 |
by auto |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
2827 |
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
|
2828 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2829 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2830 |
lemma sin_gt_zero_02: |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2831 |
fixes x :: real |
53079 | 2832 |
assumes "0 < x" and "x < 2" |
2833 |
shows "0 < sin x" |
|
44728 | 2834 |
proof - |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2835 |
let ?f = "\<lambda>n::nat. \<Sum>k = n*2..<n*2+2. (- 1) ^ k / (fact (2*k+1)) * x^(2*k+1)" |
44728 | 2836 |
have pos: "\<forall>n. 0 < ?f n" |
2837 |
proof |
|
2838 |
fix n :: nat |
|
2839 |
let ?k2 = "real (Suc (Suc (4 * n)))" |
|
2840 |
let ?k3 = "real (Suc (Suc (Suc (4 * n))))" |
|
2841 |
have "x * x < ?k2 * ?k3" |
|
2842 |
using assms by (intro mult_strict_mono', simp_all) |
|
2843 |
hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)" |
|
2844 |
by (intro mult_strict_right_mono zero_less_power `0 < x`) |
|
2845 |
thus "0 < ?f n" |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2846 |
by (simp add: real_of_nat_def divide_simps mult_ac del: mult_Suc) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2847 |
qed |
44728 | 2848 |
have sums: "?f sums sin x" |
2849 |
by (rule sin_paired [THEN sums_group], simp) |
|
2850 |
show "0 < sin x" |
|
2851 |
unfolding sums_unique [OF sums] |
|
2852 |
using sums_summable [OF sums] pos |
|
56213 | 2853 |
by (rule suminf_pos) |
44728 | 2854 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2855 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2856 |
lemma cos_double_less_one: |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2857 |
fixes x :: real |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2858 |
shows "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2859 |
using sin_gt_zero_02 [where x = x] by (auto simp: cos_squared_eq cos_double) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2860 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2861 |
lemma cos_paired: |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2862 |
fixes x :: real |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2863 |
shows "(\<lambda>n. (- 1) ^ n / (fact (2 * n)) * x ^ (2 * n)) sums cos x" |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2864 |
proof - |
31271 | 2865 |
have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2866 |
apply (rule sums_group) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2867 |
using cos_converges [of x, unfolded scaleR_conv_of_real] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2868 |
by auto |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
2869 |
thus ?thesis unfolding cos_coeff_def by (simp add: ac_simps) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2870 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2871 |
|
53602 | 2872 |
lemmas realpow_num_eq_if = power_eq_if |
2873 |
||
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2874 |
lemma sumr_pos_lt_pair: (*FIXME A MESS, BUT THE REAL MESS IS THE NEXT THEOREM*) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2875 |
fixes f :: "nat \<Rightarrow> real" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2876 |
shows "\<lbrakk>summable f; |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2877 |
\<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
|
2878 |
\<Longrightarrow> setsum f {..<k} < suminf f" |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2879 |
unfolding One_nat_def |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2880 |
apply (subst suminf_split_initial_segment [where k=k], assumption, simp) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2881 |
apply (drule_tac k=k in summable_ignore_initial_segment) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2882 |
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
|
2883 |
apply simp |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2884 |
apply (frule sums_unique) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2885 |
apply (drule sums_summable, simp) |
56213 | 2886 |
apply (erule suminf_pos) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
2887 |
apply (simp add: ac_simps) |
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2888 |
done |
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56181
diff
changeset
|
2889 |
|
53602 | 2890 |
lemma cos_two_less_zero [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2891 |
"cos 2 < (0::real)" |
53602 | 2892 |
proof - |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2893 |
note fact.simps(2) [simp del] |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2894 |
from sums_minus [OF cos_paired] |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2895 |
have *: "(\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / fact (2 * n))) sums - cos (2::real)" |
53602 | 2896 |
by simp |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2897 |
then have **: "summable (\<lambda>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" |
53602 | 2898 |
by (rule sums_summable) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2899 |
have "0 < (\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2900 |
by (simp add: fact_num_eq_if realpow_num_eq_if) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2901 |
moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n)))) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2902 |
< (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" |
53602 | 2903 |
proof - |
2904 |
{ fix d |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2905 |
have "(4::real) * (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2906 |
< (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))) * |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2907 |
fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2908 |
unfolding real_of_nat_mult |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2909 |
by (rule mult_strict_mono) (simp_all add: fact_less_mono) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2910 |
then have "(4::real) * (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2911 |
< (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))))" |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2912 |
by (simp only: fact.simps(2) [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"] real_of_nat_def of_nat_mult of_nat_fact) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2913 |
then have "(4::real) * inverse (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))) |
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2914 |
< inverse (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))" |
53602 | 2915 |
by (simp add: inverse_eq_divide less_divide_eq) |
2916 |
} |
|
2917 |
note *** = this |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
2918 |
have [simp]: "\<And>x y::real. 0 < x - y \<longleftrightarrow> y < x" by arith |
53602 | 2919 |
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
|
2920 |
(simp add: divide_inverse mult.assoc [symmetric] ***) |
53602 | 2921 |
qed |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2922 |
ultimately have "0 < (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" |
53602 | 2923 |
by (rule order_less_trans) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
2924 |
moreover from * have "- cos 2 = (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" |
53602 | 2925 |
by (rule sums_unique) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2926 |
ultimately have "(0::real) < - cos 2" by simp |
53602 | 2927 |
then show ?thesis by simp |
2928 |
qed |
|
23053 | 2929 |
|
2930 |
lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq] |
|
2931 |
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
|
2932 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2933 |
lemma cos_is_zero: "EX! x::real. 0 \<le> x & x \<le> 2 \<and> cos x = 0" |
44730 | 2934 |
proof (rule ex_ex1I) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2935 |
show "\<exists>x::real. 0 \<le> x & x \<le> 2 & cos x = 0" |
44730 | 2936 |
by (rule IVT2, simp_all) |
2937 |
next |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2938 |
fix x::real and y::real |
44730 | 2939 |
assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0" |
2940 |
assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0" |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2941 |
have [simp]: "\<forall>x::real. cos differentiable (at x)" |
56181
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56167
diff
changeset
|
2942 |
unfolding real_differentiable_def by (auto intro: DERIV_cos) |
44730 | 2943 |
from x y show "x = y" |
2944 |
apply (cut_tac less_linear [of x y], auto) |
|
2945 |
apply (drule_tac f = cos in Rolle) |
|
2946 |
apply (drule_tac [5] f = cos in Rolle) |
|
2947 |
apply (auto dest!: DERIV_cos [THEN DERIV_unique]) |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2948 |
apply (metis order_less_le_trans less_le sin_gt_zero_02) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2949 |
apply (metis order_less_le_trans less_le sin_gt_zero_02) |
44730 | 2950 |
done |
2951 |
qed |
|
31880 | 2952 |
|
23053 | 2953 |
lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)" |
53079 | 2954 |
by (simp add: pi_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2955 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2956 |
lemma cos_pi_half [simp]: "cos (pi / 2) = 0" |
53079 | 2957 |
by (simp add: pi_half cos_is_zero [THEN theI']) |
23053 | 2958 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
2959 |
lemma cos_of_real_pi_half [simp]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2960 |
fixes x :: "'a :: {real_field,banach,real_normed_algebra_1}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2961 |
shows "cos ((of_real pi / 2) :: 'a) = 0" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2962 |
by (metis cos_pi_half cos_of_real eq_numeral_simps(4) nonzero_of_real_divide of_real_0 of_real_numeral) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2963 |
|
23053 | 2964 |
lemma pi_half_gt_zero [simp]: "0 < pi / 2" |
53079 | 2965 |
apply (rule order_le_neq_trans) |
2966 |
apply (simp add: pi_half cos_is_zero [THEN theI']) |
|
54575 | 2967 |
apply (metis cos_pi_half cos_zero zero_neq_one) |
53079 | 2968 |
done |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2969 |
|
23053 | 2970 |
lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric] |
2971 |
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
|
2972 |
|
23053 | 2973 |
lemma pi_half_less_two [simp]: "pi / 2 < 2" |
53079 | 2974 |
apply (rule order_le_neq_trans) |
2975 |
apply (simp add: pi_half cos_is_zero [THEN theI']) |
|
54575 | 2976 |
apply (metis cos_pi_half cos_two_neq_zero) |
53079 | 2977 |
done |
23053 | 2978 |
|
2979 |
lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq] |
|
2980 |
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
|
2981 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2982 |
lemma pi_gt_zero [simp]: "0 < pi" |
53079 | 2983 |
using pi_half_gt_zero by simp |
23053 | 2984 |
|
2985 |
lemma pi_ge_zero [simp]: "0 \<le> pi" |
|
53079 | 2986 |
by (rule pi_gt_zero [THEN order_less_imp_le]) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2987 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2988 |
lemma pi_neq_zero [simp]: "pi \<noteq> 0" |
53079 | 2989 |
by (rule pi_gt_zero [THEN less_imp_neq, symmetric]) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2990 |
|
23053 | 2991 |
lemma pi_not_less_zero [simp]: "\<not> pi < 0" |
53079 | 2992 |
by (simp add: linorder_not_less) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2993 |
|
29165
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset
|
2994 |
lemma minus_pi_half_less_zero: "-(pi/2) < 0" |
53079 | 2995 |
by simp |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
2996 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
2997 |
lemma m2pi_less_pi: "- (2*pi) < pi" |
53079 | 2998 |
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
|
2999 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3000 |
lemma sin_pi_half [simp]: "sin(pi/2) = 1" |
53079 | 3001 |
using sin_cos_squared_add2 [where x = "pi/2"] |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3002 |
using sin_gt_zero_02 [OF pi_half_gt_zero pi_half_less_two] |
53079 | 3003 |
by (simp add: power2_eq_1_iff) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3004 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3005 |
lemma sin_of_real_pi_half [simp]: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3006 |
fixes x :: "'a :: {real_field,banach,real_normed_algebra_1}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3007 |
shows "sin ((of_real pi / 2) :: 'a) = 1" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3008 |
using sin_pi_half |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3009 |
by (metis sin_pi_half eq_numeral_simps(4) nonzero_of_real_divide of_real_1 of_real_numeral sin_of_real) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3010 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3011 |
lemma sin_cos_eq: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3012 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3013 |
shows "sin x = cos (of_real pi / 2 - x)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3014 |
by (simp add: cos_diff) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3015 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3016 |
lemma minus_sin_cos_eq: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3017 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3018 |
shows "-sin x = cos (x + of_real pi / 2)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3019 |
by (simp add: cos_add nonzero_of_real_divide) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3020 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3021 |
lemma cos_sin_eq: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3022 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3023 |
shows "cos x = sin (of_real pi / 2 - x)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3024 |
using sin_cos_eq [of "of_real pi / 2 - x"] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3025 |
by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3026 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3027 |
lemma sin_add: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3028 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3029 |
shows "sin (x + y) = sin x * cos y + cos x * sin y" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3030 |
using cos_add [of "of_real pi / 2 - x" "-y"] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3031 |
by (simp add: cos_sin_eq) (simp add: sin_cos_eq) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3032 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3033 |
lemma sin_diff: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3034 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3035 |
shows "sin (x - y) = sin x * cos y - cos x * sin y" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3036 |
using sin_add [of x "- y"] by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3037 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3038 |
lemma sin_double: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3039 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3040 |
shows "sin(2 * x) = 2 * sin x * cos x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3041 |
using sin_add [where x=x and y=x] by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3042 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3043 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3044 |
lemma cos_of_real_pi [simp]: "cos (of_real pi) = -1" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3045 |
using cos_add [where x = "pi/2" and y = "pi/2"] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3046 |
by (simp add: cos_of_real) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3047 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3048 |
lemma sin_of_real_pi [simp]: "sin (of_real pi) = 0" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3049 |
using sin_add [where x = "pi/2" and y = "pi/2"] |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3050 |
by (simp add: sin_of_real) |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3051 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3052 |
lemma cos_pi [simp]: "cos pi = -1" |
53079 | 3053 |
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
|
3054 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3055 |
lemma sin_pi [simp]: "sin pi = 0" |
53079 | 3056 |
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
|
3057 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3058 |
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x" |
53079 | 3059 |
by (simp add: sin_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3060 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3061 |
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x" |
53079 | 3062 |
by (simp add: sin_add) |
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 cos_periodic_pi [simp]: "cos (x + pi) = - cos x" |
53079 | 3065 |
by (simp add: cos_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3066 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3067 |
lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3068 |
by (simp add: sin_add sin_double cos_double) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3069 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3070 |
lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3071 |
by (simp add: cos_add sin_double cos_double) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3072 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3073 |
lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3074 |
by (induct n) (auto simp: real_of_nat_Suc distrib_right) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3075 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
3076 |
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
|
3077 |
by (metis cos_npi mult.commute) |
15383 | 3078 |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3079 |
lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3080 |
by (induct n) (auto simp: real_of_nat_Suc distrib_right) |
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 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
|
3083 |
by (simp add: mult.commute [of pi]) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3084 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3085 |
lemma cos_two_pi [simp]: "cos (2*pi) = 1" |
53079 | 3086 |
by (simp add: cos_double) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3087 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3088 |
lemma sin_two_pi [simp]: "sin (2*pi) = 0" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3089 |
by (simp add: sin_double) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3090 |
|
59741
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3091 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3092 |
lemma sin_times_sin: |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3093 |
fixes w :: "'a::{real_normed_field,banach}" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3094 |
shows "sin(w) * sin(z) = (cos(w - z) - cos(w + z)) / 2" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3095 |
by (simp add: cos_diff cos_add) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3096 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3097 |
lemma sin_times_cos: |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3098 |
fixes w :: "'a::{real_normed_field,banach}" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3099 |
shows "sin(w) * cos(z) = (sin(w + z) + sin(w - z)) / 2" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3100 |
by (simp add: sin_diff sin_add) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3101 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3102 |
lemma cos_times_sin: |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3103 |
fixes w :: "'a::{real_normed_field,banach}" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3104 |
shows "cos(w) * sin(z) = (sin(w + z) - sin(w - z)) / 2" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3105 |
by (simp add: sin_diff sin_add) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3106 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3107 |
lemma cos_times_cos: |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3108 |
fixes w :: "'a::{real_normed_field,banach}" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3109 |
shows "cos(w) * cos(z) = (cos(w - z) + cos(w + z)) / 2" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3110 |
by (simp add: cos_diff cos_add) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3111 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3112 |
lemma sin_plus_sin: (*FIXME field_inverse_zero should not be necessary*) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3113 |
fixes w :: "'a::{real_normed_field,banach,field_inverse_zero}" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3114 |
shows "sin(w) + sin(z) = 2 * sin((w + z) / 2) * cos((w - z) / 2)" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3115 |
apply (simp add: mult.assoc sin_times_cos) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3116 |
apply (simp add: field_simps) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3117 |
done |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3118 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3119 |
lemma sin_diff_sin: |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3120 |
fixes w :: "'a::{real_normed_field,banach,field_inverse_zero}" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3121 |
shows "sin(w) - sin(z) = 2 * sin((w - z) / 2) * cos((w + z) / 2)" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3122 |
apply (simp add: mult.assoc sin_times_cos) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3123 |
apply (simp add: field_simps) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3124 |
done |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3125 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3126 |
lemma cos_plus_cos: |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3127 |
fixes w :: "'a::{real_normed_field,banach,field_inverse_zero}" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3128 |
shows "cos(w) + cos(z) = 2 * cos((w + z) / 2) * cos((w - z) / 2)" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3129 |
apply (simp add: mult.assoc cos_times_cos) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3130 |
apply (simp add: field_simps) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3131 |
done |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3132 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3133 |
lemma cos_diff_cos: |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3134 |
fixes w :: "'a::{real_normed_field,banach,field_inverse_zero}" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3135 |
shows "cos(w) - cos(z) = 2 * sin((w + z) / 2) * sin((z - w) / 2)" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3136 |
apply (simp add: mult.assoc sin_times_sin) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3137 |
apply (simp add: field_simps) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3138 |
done |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3139 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3140 |
lemma cos_double_cos: |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3141 |
fixes z :: "'a::{real_normed_field,banach}" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3142 |
shows "cos(2 * z) = 2 * cos z ^ 2 - 1" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3143 |
by (simp add: cos_double sin_squared_eq) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3144 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3145 |
lemma cos_double_sin: |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3146 |
fixes z :: "'a::{real_normed_field,banach}" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3147 |
shows "cos(2 * z) = 1 - 2 * sin z ^ 2" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3148 |
by (simp add: cos_double sin_squared_eq) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3149 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3150 |
lemma sin_pi_minus [simp]: "sin (pi - x) = sin x" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3151 |
by (metis sin_minus sin_periodic_pi minus_minus uminus_add_conv_diff) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3152 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3153 |
lemma cos_pi_minus [simp]: "cos (pi - x) = -(cos x)" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3154 |
by (metis cos_minus cos_periodic_pi uminus_add_conv_diff) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3155 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3156 |
lemma sin_minus_pi [simp]: "sin (x - pi) = - (sin x)" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3157 |
by (simp add: sin_diff) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3158 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3159 |
lemma cos_minus_pi [simp]: "cos (x - pi) = -(cos x)" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3160 |
by (simp add: cos_diff) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3161 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3162 |
lemma sin_2pi_minus [simp]: "sin (2*pi - x) = -(sin x)" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3163 |
by (metis sin_periodic_pi2 add_diff_eq mult_2 sin_pi_minus) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3164 |
|
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3165 |
lemma cos_2pi_minus [simp]: "cos (2*pi - x) = cos x" |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3166 |
by (metis (no_types, hide_lams) cos_add cos_minus cos_two_pi sin_minus sin_two_pi |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3167 |
diff_0_right minus_diff_eq mult_1 mult_zero_left uminus_add_conv_diff) |
5b762cd73a8e
Lots of new material on complex-valued functions. Modified simplification of (x/n)^k
paulson <lp15@cam.ac.uk>
parents:
59731
diff
changeset
|
3168 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3169 |
lemma sin_gt_zero2: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < sin x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3170 |
by (metis sin_gt_zero_02 order_less_trans pi_half_less_two) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3171 |
|
41970 | 3172 |
lemma sin_less_zero: |
53079 | 3173 |
assumes "- pi/2 < x" and "x < 0" |
3174 |
shows "sin x < 0" |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3175 |
proof - |
41970 | 3176 |
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
|
3177 |
thus ?thesis by simp |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3178 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3179 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3180 |
lemma pi_less_4: "pi < 4" |
53079 | 3181 |
using pi_half_less_two by auto |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3182 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3183 |
lemma cos_gt_zero: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < cos x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3184 |
by (simp add: cos_sin_eq sin_gt_zero2) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3185 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3186 |
lemma cos_gt_zero_pi: "\<lbrakk>-(pi/2) < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < cos x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3187 |
using cos_gt_zero [of x] cos_gt_zero [of "-x"] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3188 |
by (cases rule: linorder_cases [of x 0]) auto |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3189 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3190 |
lemma cos_ge_zero: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2\<rbrakk> \<Longrightarrow> 0 \<le> cos x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3191 |
apply (auto simp: order_le_less cos_gt_zero_pi) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3192 |
by (metis cos_pi_half eq_divide_eq eq_numeral_simps(4)) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3193 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3194 |
lemma sin_gt_zero: "\<lbrakk>0 < x; x < pi \<rbrakk> \<Longrightarrow> 0 < sin x" |
53079 | 3195 |
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
|
3196 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3197 |
lemma sin_lt_zero: "pi < x \<Longrightarrow> x < 2*pi \<Longrightarrow> sin x < 0" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3198 |
using sin_gt_zero [of "x-pi"] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3199 |
by (simp add: sin_diff) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3200 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3201 |
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
|
3202 |
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
|
3203 |
assume "\<not> 2 \<le> pi" hence "pi < 2" by auto |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3204 |
have "\<exists>y > pi. y < 2 \<and> y < 2*pi" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3205 |
proof (cases "2 < 2*pi") |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3206 |
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
|
3207 |
next |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3208 |
case False have "pi < 2*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
|
3209 |
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
|
3210 |
qed |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3211 |
then obtain y where "pi < y" and "y < 2" and "y < 2*pi" by blast |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3212 |
hence "0 < sin y" using sin_gt_zero_02 by auto |
41970 | 3213 |
moreover |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3214 |
have "sin y < 0" using sin_gt_zero[of "y - pi"] `pi < y` and `y < 2*pi` sin_periodic_pi[of "y - 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
|
3215 |
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
|
3216 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3217 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3218 |
lemma sin_ge_zero: "\<lbrakk>0 \<le> x; x \<le> pi\<rbrakk> \<Longrightarrow> 0 \<le> sin x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3219 |
by (auto simp: order_le_less sin_gt_zero) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3220 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3221 |
lemma sin_le_zero: "pi \<le> x \<Longrightarrow> x < 2*pi \<Longrightarrow> sin x \<le> 0" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3222 |
using sin_ge_zero [of "x-pi"] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3223 |
by (simp add: sin_diff) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3224 |
|
44745 | 3225 |
text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}. |
3226 |
It should be possible to factor out some of the common parts. *} |
|
3227 |
||
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3228 |
lemma cos_total: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)" |
44745 | 3229 |
proof (rule ex_ex1I) |
3230 |
assume y: "-1 \<le> y" "y \<le> 1" |
|
3231 |
show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y" |
|
3232 |
by (rule IVT2, simp_all add: y) |
|
3233 |
next |
|
3234 |
fix a b |
|
3235 |
assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y" |
|
3236 |
assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y" |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3237 |
have [simp]: "\<forall>x::real. cos differentiable (at x)" |
56181
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56167
diff
changeset
|
3238 |
unfolding real_differentiable_def by (auto intro: DERIV_cos) |
44745 | 3239 |
from a b show "a = b" |
3240 |
apply (cut_tac less_linear [of a b], auto) |
|
3241 |
apply (drule_tac f = cos in Rolle) |
|
3242 |
apply (drule_tac [5] f = cos in Rolle) |
|
3243 |
apply (auto dest!: DERIV_cos [THEN DERIV_unique]) |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3244 |
apply (metis order_less_le_trans less_le sin_gt_zero) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3245 |
apply (metis order_less_le_trans less_le sin_gt_zero) |
44745 | 3246 |
done |
3247 |
qed |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3248 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3249 |
lemma sin_total: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3250 |
assumes y: "-1 \<le> y" "y \<le> 1" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3251 |
shows "\<exists>! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3252 |
proof - |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3253 |
from cos_total [OF y] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3254 |
obtain x where x: "0 \<le> x" "x \<le> pi" "cos x = y" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3255 |
and uniq: "\<And>x'. 0 \<le> x' \<Longrightarrow> x' \<le> pi \<Longrightarrow> cos x' = y \<Longrightarrow> x' = x " |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3256 |
by blast |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3257 |
show ?thesis |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3258 |
apply (simp add: sin_cos_eq) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3259 |
apply (rule ex1I [where a="pi/2 - x"]) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3260 |
apply (cut_tac [2] x'="pi/2 - xa" in uniq) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3261 |
using x |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3262 |
apply auto |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3263 |
done |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3264 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3265 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3266 |
lemma reals_Archimedean4: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3267 |
"\<lbrakk>0 < y; 0 \<le> x\<rbrakk> \<Longrightarrow> \<exists>n. real n * y \<le> x & x < real (Suc n) * y" |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3268 |
apply (auto dest!: reals_Archimedean3) |
41970 | 3269 |
apply (drule_tac x = x in spec, clarify) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3270 |
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y") |
41970 | 3271 |
prefer 2 apply (erule LeastI) |
3272 |
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
|
3273 |
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
|
3274 |
apply (subgoal_tac "~ x < real m * y") |
41970 | 3275 |
prefer 2 apply (rule not_less_Least, simp, force) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3276 |
done |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3277 |
|
15229 | 3278 |
lemma cos_zero_lemma: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3279 |
"\<lbrakk>0 \<le> x; cos x = 0\<rbrakk> \<Longrightarrow> |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3280 |
\<exists>n::nat. odd n & x = real n * (pi/2)" |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3281 |
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe) |
41970 | 3282 |
apply (subgoal_tac "0 \<le> x - real n * pi & |
15086 | 3283 |
(x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ") |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3284 |
apply (auto simp: algebra_simps real_of_nat_Suc) |
29667 | 3285 |
prefer 2 apply (simp add: cos_diff) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3286 |
apply (simp add: cos_diff) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3287 |
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
|
3288 |
apply (rule_tac [2] cos_total, safe) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3289 |
apply (drule_tac x = "x - real n * pi" in spec) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3290 |
apply (drule_tac x = "pi/2" in spec) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3291 |
apply (simp add: cos_diff) |
15229 | 3292 |
apply (rule_tac x = "Suc (2 * n)" in exI) |
29667 | 3293 |
apply (simp add: real_of_nat_Suc algebra_simps, auto) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3294 |
done |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3295 |
|
15229 | 3296 |
lemma sin_zero_lemma: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3297 |
"\<lbrakk>0 \<le> x; sin x = 0\<rbrakk> \<Longrightarrow> |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3298 |
\<exists>n::nat. even n & x = real n * (pi/2)" |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3299 |
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
|
3300 |
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
|
3301 |
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
|
3302 |
apply (rule cos_zero_lemma) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3303 |
apply (auto simp: cos_add) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3304 |
done |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3305 |
|
15229 | 3306 |
lemma cos_zero_iff: |
41970 | 3307 |
"(cos x = 0) = |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3308 |
((\<exists>n::nat. odd n & (x = real n * (pi/2))) | |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3309 |
(\<exists>n::nat. odd 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
|
3310 |
proof - |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3311 |
{ fix n :: nat |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3312 |
assume "odd n" |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3313 |
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
|
3314 |
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
|
3315 |
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
|
3316 |
} note * = this |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3317 |
show ?thesis |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3318 |
apply (rule iffI) |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3319 |
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
|
3320 |
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
|
3321 |
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
|
3322 |
apply (auto dest: *) |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3323 |
done |
efdc6c533bd3
prefer generic elimination rules for even/odd over specialized unfold rules for nat
haftmann
parents:
58656
diff
changeset
|
3324 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3325 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3326 |
(* ditto: but to a lesser extent *) |
15229 | 3327 |
lemma sin_zero_iff: |
41970 | 3328 |
"(sin x = 0) = |
3329 |
((\<exists>n::nat. even n & (x = real n * (pi/2))) | |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3330 |
(\<exists>n::nat. even n & (x = -(real n * (pi/2)))))" |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3331 |
apply (rule iffI) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3332 |
apply (cut_tac linorder_linear [of 0 x], safe) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3333 |
apply (drule sin_zero_lemma, assumption+) |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3334 |
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe) |
41970 | 3335 |
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
|
3336 |
apply (auto elim: evenE) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3337 |
done |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3338 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3339 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3340 |
lemma cos_zero_iff_int: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3341 |
"cos x = 0 \<longleftrightarrow> (\<exists>n::int. odd n & x = real n * (pi/2))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3342 |
proof safe |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3343 |
assume "cos x = 0" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3344 |
then show "\<exists>n::int. odd n & x = real n * (pi/2)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3345 |
apply (simp add: cos_zero_iff, safe) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3346 |
apply (metis even_int_iff real_of_int_of_nat_eq) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3347 |
apply (rule_tac x="- (int n)" in exI, simp) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3348 |
done |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3349 |
next |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3350 |
fix n::int |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3351 |
assume "odd n" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3352 |
then show "cos (real n * (pi / 2)) = 0" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3353 |
apply (simp add: cos_zero_iff) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3354 |
apply (case_tac n rule: int_cases2, simp) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3355 |
apply (rule disjI2) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3356 |
apply (rule_tac x="nat (-n)" in exI, simp) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3357 |
done |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3358 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3359 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3360 |
lemma sin_zero_iff_int: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3361 |
"sin x = 0 \<longleftrightarrow> (\<exists>n::int. even n & (x = real n * (pi/2)))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3362 |
proof safe |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3363 |
assume "sin x = 0" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3364 |
then show "\<exists>n::int. even n \<and> x = real n * (pi / 2)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3365 |
apply (simp add: sin_zero_iff, safe) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3366 |
apply (metis even_int_iff real_of_int_of_nat_eq) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3367 |
apply (rule_tac x="- (int n)" in exI, simp) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3368 |
done |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3369 |
next |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3370 |
fix n::int |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3371 |
assume "even n" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3372 |
then show "sin (real n * (pi / 2)) = 0" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3373 |
apply (simp add: sin_zero_iff) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3374 |
apply (case_tac n rule: int_cases2, simp) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3375 |
apply (rule disjI2) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3376 |
apply (rule_tac x="nat (-n)" in exI, simp) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3377 |
done |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3378 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3379 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3380 |
lemma sin_zero_iff_int2: "sin x = 0 \<longleftrightarrow> (\<exists>n::int. x = real n * pi)" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3381 |
apply (simp only: sin_zero_iff_int) |
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3382 |
apply (safe elim!: evenE) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3383 |
apply (simp_all add: field_simps) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3384 |
using dvd_triv_left by fastforce |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3385 |
|
53079 | 3386 |
lemma cos_monotone_0_pi: |
3387 |
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
|
3388 |
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
|
3389 |
proof - |
33549 | 3390 |
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
|
3391 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3392 |
from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]] |
53079 | 3393 |
obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" |
3394 |
by auto |
|
33549 | 3395 |
hence "0 < z" and "z < pi" using assms by auto |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3396 |
hence "0 < sin z" using sin_gt_zero by auto |
53079 | 3397 |
hence "cos x - cos y < 0" |
3398 |
unfolding cos_diff minus_mult_commute[symmetric] |
|
3399 |
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
|
3400 |
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
|
3401 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3402 |
|
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3403 |
lemma cos_monotone_0_pi_le: |
53079 | 3404 |
assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" |
3405 |
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
|
3406 |
proof (cases "y < x") |
53079 | 3407 |
case True |
3408 |
show ?thesis |
|
3409 |
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
|
3410 |
next |
53079 | 3411 |
case False |
3412 |
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
|
3413 |
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
|
3414 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3415 |
|
53079 | 3416 |
lemma cos_monotone_minus_pi_0: |
3417 |
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
|
3418 |
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
|
3419 |
proof - |
53079 | 3420 |
have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" |
3421 |
using assms by auto |
|
3422 |
from cos_monotone_0_pi[OF this] show ?thesis |
|
3423 |
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
|
3424 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3425 |
|
53079 | 3426 |
lemma cos_monotone_minus_pi_0': |
3427 |
assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0" |
|
3428 |
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
|
3429 |
proof (cases "y < x") |
53079 | 3430 |
case True |
3431 |
show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] |
|
3432 |
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
|
3433 |
next |
53079 | 3434 |
case False |
3435 |
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
|
3436 |
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
|
3437 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3438 |
|
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3439 |
lemma sin_monotone_2pi: |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3440 |
assumes "- (pi/2) \<le> y" and "y < x" and "x \<le> pi/2" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3441 |
shows "sin y < sin x" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3442 |
apply (simp add: sin_cos_eq) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3443 |
apply (rule cos_monotone_0_pi) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3444 |
using assms |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3445 |
apply auto |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3446 |
done |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3447 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3448 |
lemma sin_monotone_2pi_le: |
53079 | 3449 |
assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2" |
3450 |
shows "sin y \<le> sin x" |
|
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3451 |
by (metis assms le_less sin_monotone_2pi) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3452 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3453 |
lemma sin_x_le_x: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3454 |
fixes x::real assumes x: "x \<ge> 0" shows "sin x \<le> x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3455 |
proof - |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3456 |
let ?f = "\<lambda>x. x - sin x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3457 |
from x have "?f x \<ge> ?f 0" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3458 |
apply (rule DERIV_nonneg_imp_nondecreasing) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3459 |
apply (intro allI impI exI[of _ "1 - cos x" for x]) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3460 |
apply (auto intro!: derivative_eq_intros simp: field_simps) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3461 |
done |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3462 |
thus "sin x \<le> x" 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
|
3463 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3464 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3465 |
lemma sin_x_ge_neg_x: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3466 |
fixes x::real assumes x: "x \<ge> 0" shows "sin x \<ge> - x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3467 |
proof - |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3468 |
let ?f = "\<lambda>x. x + sin x" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3469 |
from x have "?f x \<ge> ?f 0" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3470 |
apply (rule DERIV_nonneg_imp_nondecreasing) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3471 |
apply (intro allI impI exI[of _ "1 + cos x" for x]) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3472 |
apply (auto intro!: derivative_eq_intros simp: field_simps real_0_le_add_iff) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3473 |
done |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3474 |
thus "sin x \<ge> -x" by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3475 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3476 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3477 |
lemma abs_sin_x_le_abs_x: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3478 |
fixes x::real shows "\<bar>sin x\<bar> \<le> \<bar>x\<bar>" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3479 |
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"] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3480 |
by (auto simp: abs_real_def) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3481 |
|
53079 | 3482 |
|
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3483 |
subsection {* More Corollaries about Sine and Cosine *} |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3484 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3485 |
lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3486 |
proof - |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3487 |
have "sin ((real n + 1/2) * pi) = cos (real n * pi)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3488 |
by (auto simp: algebra_simps sin_add) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3489 |
thus ?thesis |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3490 |
by (simp add: real_of_nat_Suc distrib_right add_divide_distrib |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3491 |
mult.commute [of pi]) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3492 |
qed |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3493 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3494 |
lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3495 |
by (cases "even n") (simp_all add: cos_double mult.assoc) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3496 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3497 |
lemma cos_3over2_pi [simp]: "cos (3/2*pi) = 0" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3498 |
apply (subgoal_tac "cos (pi + pi/2) = 0", simp) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3499 |
apply (subst cos_add, simp) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3500 |
done |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3501 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3502 |
lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3503 |
by (auto simp: mult.assoc sin_double) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3504 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3505 |
lemma sin_3over2_pi [simp]: "sin (3/2*pi) = - 1" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3506 |
apply (subgoal_tac "sin (pi + pi/2) = - 1", simp) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3507 |
apply (subst sin_add, simp) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3508 |
done |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3509 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3510 |
lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3511 |
by (simp only: cos_add sin_add real_of_nat_Suc distrib_right distrib_left add_divide_distrib, auto) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3512 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3513 |
lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3514 |
by (auto intro!: derivative_eq_intros) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3515 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3516 |
lemma sin_zero_norm_cos_one: |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3517 |
fixes x :: "'a::{real_normed_field,banach}" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3518 |
assumes "sin x = 0" shows "norm (cos x) = 1" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3519 |
using sin_cos_squared_add [of x, unfolded assms] |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3520 |
by (simp add: square_norm_one) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3521 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3522 |
lemma sin_zero_abs_cos_one: "sin x = 0 \<Longrightarrow> \<bar>cos x\<bar> = (1::real)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3523 |
using sin_zero_norm_cos_one by fastforce |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3524 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3525 |
lemma cos_one_sin_zero: |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3526 |
fixes x :: "'a::{real_normed_field,banach}" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3527 |
assumes "cos x = 1" shows "sin x = 0" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3528 |
using sin_cos_squared_add [of x, unfolded assms] |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3529 |
by simp |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3530 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3531 |
lemma sin_times_pi_eq_0: "sin(x * pi) = 0 \<longleftrightarrow> x \<in> Ints" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3532 |
by (simp add: sin_zero_iff_int2) (metis Ints_cases Ints_real_of_int real_of_int_def) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3533 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3534 |
lemma cos_one_2pi: |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3535 |
"cos(x) = 1 \<longleftrightarrow> (\<exists>n::nat. x = n * 2*pi) | (\<exists>n::nat. x = -(n * 2*pi))" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3536 |
(is "?lhs = ?rhs") |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3537 |
proof |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3538 |
assume "cos(x) = 1" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3539 |
then have "sin x = 0" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3540 |
by (simp add: cos_one_sin_zero) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3541 |
then show ?rhs |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3542 |
proof (simp only: sin_zero_iff, elim exE disjE conjE) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3543 |
fix n::nat |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3544 |
assume n: "even n" "x = real n * (pi/2)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3545 |
then obtain m where m: "n = 2 * m" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3546 |
using dvdE by blast |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3547 |
then have me: "even m" using `?lhs` n |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3548 |
by (auto simp: field_simps) (metis one_neq_neg_one power_minus_odd power_one) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3549 |
show ?rhs |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3550 |
using m me n |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3551 |
by (auto simp: field_simps elim!: evenE) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3552 |
next |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3553 |
fix n::nat |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3554 |
assume n: "even n" "x = - (real n * (pi/2))" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3555 |
then obtain m where m: "n = 2 * m" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3556 |
using dvdE by blast |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3557 |
then have me: "even m" using `?lhs` n |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3558 |
by (auto simp: field_simps) (metis one_neq_neg_one power_minus_odd power_one) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3559 |
show ?rhs |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3560 |
using m me n |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3561 |
by (auto simp: field_simps elim!: evenE) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3562 |
qed |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3563 |
next |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3564 |
assume "?rhs" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3565 |
then show "cos x = 1" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3566 |
by (metis cos_2npi cos_minus mult.assoc mult.left_commute) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3567 |
qed |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3568 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3569 |
lemma cos_one_2pi_int: "cos(x) = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2*pi)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3570 |
apply auto --{*FIXME simproc bug*} |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3571 |
apply (auto simp: cos_one_2pi) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3572 |
apply (metis real_of_int_of_nat_eq) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3573 |
apply (metis mult_minus_right real_of_int_minus real_of_int_of_nat_eq) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3574 |
by (metis mult_minus_right of_int_of_nat real_of_int_def real_of_nat_def) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3575 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3576 |
lemma sin_cos_sqrt: "0 \<le> sin(x) \<Longrightarrow> (sin(x) = sqrt(1 - (cos(x) ^ 2)))" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3577 |
using sin_squared_eq real_sqrt_unique by fastforce |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3578 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3579 |
lemma sin_eq_0_pi: "-pi < x \<Longrightarrow> x < pi \<Longrightarrow> sin(x) = 0 \<Longrightarrow> x = 0" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3580 |
by (metis sin_gt_zero sin_minus minus_less_iff neg_0_less_iff_less not_less_iff_gr_or_eq) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3581 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3582 |
lemma cos_treble_cos: |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3583 |
fixes x :: "'a::{real_normed_field,banach}" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3584 |
shows "cos(3 * x) = 4 * cos(x) ^ 3 - 3 * cos x" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3585 |
proof - |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3586 |
have *: "(sin x * (sin x * 3)) = 3 - (cos x * (cos x * 3))" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3587 |
by (simp add: mult.assoc [symmetric] sin_squared_eq [unfolded power2_eq_square]) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3588 |
have "cos(3 * x) = cos(2*x + x)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3589 |
by simp |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3590 |
also have "... = 4 * cos(x) ^ 3 - 3 * cos x" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3591 |
apply (simp only: cos_add cos_double sin_double) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3592 |
apply (simp add: * field_simps power2_eq_square power3_eq_cube) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3593 |
done |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3594 |
finally show ?thesis . |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3595 |
qed |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3596 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3597 |
lemma cos_45: "cos (pi / 4) = sqrt 2 / 2" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3598 |
proof - |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3599 |
let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3600 |
have nonneg: "0 \<le> ?c" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3601 |
by (simp add: cos_ge_zero) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3602 |
have "0 = cos (pi / 4 + pi / 4)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3603 |
by simp |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3604 |
also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3605 |
by (simp only: cos_add power2_eq_square) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3606 |
also have "\<dots> = 2 * ?c\<^sup>2 - 1" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3607 |
by (simp add: sin_squared_eq) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3608 |
finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3609 |
by (simp add: power_divide) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3610 |
thus ?thesis |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3611 |
using nonneg by (rule power2_eq_imp_eq) simp |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3612 |
qed |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3613 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3614 |
lemma cos_30: "cos (pi / 6) = sqrt 3/2" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3615 |
proof - |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3616 |
let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3617 |
have pos_c: "0 < ?c" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3618 |
by (rule cos_gt_zero, simp, simp) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3619 |
have "0 = cos (pi / 6 + pi / 6 + pi / 6)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3620 |
by simp |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3621 |
also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3622 |
by (simp only: cos_add sin_add) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3623 |
also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3624 |
by (simp add: algebra_simps power2_eq_square) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3625 |
finally have "?c\<^sup>2 = (sqrt 3/2)\<^sup>2" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3626 |
using pos_c by (simp add: sin_squared_eq power_divide) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3627 |
thus ?thesis |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3628 |
using pos_c [THEN order_less_imp_le] |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3629 |
by (rule power2_eq_imp_eq) simp |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3630 |
qed |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3631 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3632 |
lemma sin_45: "sin (pi / 4) = sqrt 2 / 2" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3633 |
by (simp add: sin_cos_eq cos_45) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3634 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3635 |
lemma sin_60: "sin (pi / 3) = sqrt 3/2" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3636 |
by (simp add: sin_cos_eq cos_30) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3637 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3638 |
lemma cos_60: "cos (pi / 3) = 1 / 2" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3639 |
apply (rule power2_eq_imp_eq) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3640 |
apply (simp add: cos_squared_eq sin_60 power_divide) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3641 |
apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3642 |
done |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3643 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3644 |
lemma sin_30: "sin (pi / 6) = 1 / 2" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3645 |
by (simp add: sin_cos_eq cos_60) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3646 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3647 |
lemma cos_integer_2pi: "n \<in> Ints \<Longrightarrow> cos(2*pi * n) = 1" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3648 |
by (metis Ints_cases cos_one_2pi_int mult.assoc mult.commute real_of_int_def) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3649 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3650 |
lemma sin_integer_2pi: "n \<in> Ints \<Longrightarrow> sin(2*pi * n) = 0" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3651 |
by (metis sin_two_pi Ints_mult mult.assoc mult.commute sin_times_pi_eq_0) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3652 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3653 |
lemma cos_int_2npi [simp]: "cos (2 * real (n::int) * pi) = 1" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3654 |
by (simp add: cos_one_2pi_int) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3655 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3656 |
lemma sin_int_2npi [simp]: "sin (2 * real (n::int) * pi) = 0" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3657 |
by (metis Ints_real_of_int mult.assoc mult.commute sin_integer_2pi) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3658 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3659 |
lemma sincos_principal_value: "\<exists>y. (-pi < y \<and> y \<le> pi) \<and> (sin(y) = sin(x) \<and> cos(y) = cos(x))" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3660 |
apply (rule exI [where x="pi - (2*pi) * frac((pi - x) / (2*pi))"]) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3661 |
apply (auto simp: field_simps frac_lt_1) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3662 |
apply (simp_all add: frac_def divide_simps) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3663 |
apply (simp_all add: add_divide_distrib diff_divide_distrib) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3664 |
apply (simp_all add: sin_diff cos_diff mult.assoc [symmetric] cos_integer_2pi sin_integer_2pi) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3665 |
done |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3666 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3667 |
|
29164 | 3668 |
subsection {* Tangent *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3669 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3670 |
definition tan :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" |
53079 | 3671 |
where "tan = (\<lambda>x. sin x / cos x)" |
23043 | 3672 |
|
59862 | 3673 |
lemma tan_of_real: |
3674 |
fixes XXX :: "'a::{real_normed_field,banach,field_inverse_zero}" |
|
3675 |
shows "of_real(tan x) = (tan(of_real x) :: 'a)" |
|
3676 |
by (simp add: tan_def sin_of_real cos_of_real) |
|
3677 |
||
3678 |
lemma tan_in_Reals [simp]: |
|
3679 |
fixes z :: "'a::{real_normed_field,banach,field_inverse_zero}" |
|
3680 |
shows "z \<in> \<real> \<Longrightarrow> tan z \<in> \<real>" |
|
3681 |
by (simp add: tan_def) |
|
3682 |
||
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3683 |
lemma tan_zero [simp]: "tan 0 = 0" |
44311 | 3684 |
by (simp add: tan_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3685 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3686 |
lemma tan_pi [simp]: "tan pi = 0" |
44311 | 3687 |
by (simp add: tan_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3688 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3689 |
lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0" |
44311 | 3690 |
by (simp add: tan_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3691 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3692 |
lemma tan_minus [simp]: "tan (-x) = - tan x" |
44311 | 3693 |
by (simp add: tan_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3694 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3695 |
lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x" |
44311 | 3696 |
by (simp add: tan_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3697 |
|
41970 | 3698 |
lemma lemma_tan_add1: |
44311 | 3699 |
"\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)" |
3700 |
by (simp add: tan_def cos_add field_simps) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3701 |
|
41970 | 3702 |
lemma add_tan_eq: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3703 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3704 |
shows "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)" |
44311 | 3705 |
by (simp add: tan_def sin_add field_simps) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3706 |
|
15229 | 3707 |
lemma tan_add: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3708 |
fixes x :: "'a::{real_normed_field,banach}" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3709 |
shows |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3710 |
"\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0\<rbrakk> |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3711 |
\<Longrightarrow> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3712 |
by (simp add: add_tan_eq lemma_tan_add1 field_simps) (simp add: tan_def) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3713 |
|
15229 | 3714 |
lemma tan_double: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3715 |
fixes x :: "'a::{real_normed_field,banach}" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3716 |
shows |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3717 |
"\<lbrakk>cos x \<noteq> 0; cos (2 * x) \<noteq> 0\<rbrakk> |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3718 |
\<Longrightarrow> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)" |
44311 | 3719 |
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
|
3720 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3721 |
lemma tan_gt_zero: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < tan x" |
53079 | 3722 |
by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) |
41970 | 3723 |
|
3724 |
lemma tan_less_zero: |
|
53079 | 3725 |
assumes lb: "- pi/2 < x" and "x < 0" |
3726 |
shows "tan x < 0" |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3727 |
proof - |
41970 | 3728 |
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
|
3729 |
thus ?thesis by simp |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3730 |
qed |
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3731 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3732 |
lemma tan_half: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3733 |
fixes x :: "'a::{real_normed_field,banach,field_inverse_zero}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3734 |
shows "tan x = sin (2 * x) / (cos (2 * x) + 1)" |
44756
efcd71fbaeec
simplify proof of tan_half, removing unused assumptions
huffman
parents:
44755
diff
changeset
|
3735 |
unfolding tan_def sin_double cos_double sin_squared_eq |
efcd71fbaeec
simplify proof of tan_half, removing unused assumptions
huffman
parents:
44755
diff
changeset
|
3736 |
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
|
3737 |
|
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3738 |
lemma tan_30: "tan (pi / 6) = 1 / sqrt 3" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3739 |
unfolding tan_def by (simp add: sin_30 cos_30) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3740 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3741 |
lemma tan_45: "tan (pi / 4) = 1" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3742 |
unfolding tan_def by (simp add: sin_45 cos_45) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3743 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3744 |
lemma tan_60: "tan (pi / 3) = sqrt 3" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3745 |
unfolding tan_def by (simp add: sin_60 cos_60) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3746 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3747 |
lemma DERIV_tan [simp]: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3748 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3749 |
shows "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)" |
44311 | 3750 |
unfolding tan_def |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
3751 |
by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square) |
44311 | 3752 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3753 |
lemma isCont_tan: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3754 |
fixes x :: "'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3755 |
shows "cos x \<noteq> 0 \<Longrightarrow> isCont tan x" |
44311 | 3756 |
by (rule DERIV_tan [THEN DERIV_isCont]) |
3757 |
||
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3758 |
lemma isCont_tan' [simp,continuous_intros]: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3759 |
fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3760 |
shows "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a" |
44311 | 3761 |
by (rule isCont_o2 [OF _ isCont_tan]) |
3762 |
||
3763 |
lemma tendsto_tan [tendsto_intros]: |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3764 |
fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3765 |
shows "\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F" |
44311 | 3766 |
by (rule isCont_tendsto_compose [OF isCont_tan]) |
3767 |
||
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
|
3768 |
lemma continuous_tan: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3769 |
fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3770 |
shows "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (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
|
3771 |
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
|
3772 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3773 |
lemma continuous_on_tan [continuous_intros]: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3774 |
fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3775 |
shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3776 |
unfolding continuous_on_def by (auto intro: tendsto_tan) |
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
|
3777 |
|
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
|
3778 |
lemma continuous_within_tan [continuous_intros]: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3779 |
fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
3780 |
shows |
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
|
3781 |
"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
|
3782 |
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
|
3783 |
|
53079 | 3784 |
lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) -- pi/2 --> 0" |
44311 | 3785 |
by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3786 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3787 |
lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x" |
53079 | 3788 |
apply (cut_tac LIM_cos_div_sin) |
3789 |
apply (simp only: LIM_eq) |
|
3790 |
apply (drule_tac x = "inverse y" in spec, safe, force) |
|
3791 |
apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe) |
|
3792 |
apply (rule_tac x = "(pi/2) - e" in exI) |
|
3793 |
apply (simp (no_asm_simp)) |
|
3794 |
apply (drule_tac x = "(pi/2) - e" in spec) |
|
3795 |
apply (auto simp add: tan_def sin_diff cos_diff) |
|
3796 |
apply (rule inverse_less_iff_less [THEN iffD1]) |
|
3797 |
apply (auto simp add: divide_inverse) |
|
3798 |
apply (rule mult_pos_pos) |
|
3799 |
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
|
3800 |
apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult.commute) |
53079 | 3801 |
done |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3802 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3803 |
lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y" |
53079 | 3804 |
apply (frule order_le_imp_less_or_eq, safe) |
3805 |
prefer 2 apply force |
|
3806 |
apply (drule lemma_tan_total, safe) |
|
3807 |
apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl) |
|
3808 |
apply (auto intro!: DERIV_tan [THEN DERIV_isCont]) |
|
3809 |
apply (drule_tac y = xa in order_le_imp_less_or_eq) |
|
3810 |
apply (auto dest: cos_gt_zero) |
|
3811 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3812 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3813 |
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y" |
53079 | 3814 |
apply (cut_tac linorder_linear [of 0 y], safe) |
3815 |
apply (drule tan_total_pos) |
|
3816 |
apply (cut_tac [2] y="-y" in tan_total_pos, safe) |
|
3817 |
apply (rule_tac [3] x = "-x" in exI) |
|
3818 |
apply (auto del: exI intro!: exI) |
|
3819 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3820 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3821 |
lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y" |
53079 | 3822 |
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
|
3823 |
apply hypsubst_thin |
53079 | 3824 |
apply (cut_tac x = xa and y = y in linorder_less_linear, auto) |
3825 |
apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0") |
|
3826 |
apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0") |
|
3827 |
apply (rule_tac [4] Rolle) |
|
3828 |
apply (rule_tac [2] Rolle) |
|
3829 |
apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI |
|
56181
2aa0b19e74f3
unify syntax for has_derivative and differentiable
hoelzl
parents:
56167
diff
changeset
|
3830 |
simp add: real_differentiable_def) |
53079 | 3831 |
txt{*Now, simulate TRYALL*} |
3832 |
apply (rule_tac [!] DERIV_tan asm_rl) |
|
3833 |
apply (auto dest!: DERIV_unique [OF _ DERIV_tan] |
|
3834 |
simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym]) |
|
3835 |
done |
|
3836 |
||
3837 |
lemma tan_monotone: |
|
3838 |
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
|
3839 |
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
|
3840 |
proof - |
53079 | 3841 |
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
|
3842 |
proof (rule allI, rule impI) |
53079 | 3843 |
fix x' :: real |
3844 |
assume "y \<le> x' \<and> x' \<le> x" |
|
33549 | 3845 |
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
|
3846 |
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
|
3847 |
have "cos x' \<noteq> 0" by auto |
53076 | 3848 |
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
|
3849 |
qed |
41970 | 3850 |
from MVT2[OF `y < x` this] |
53079 | 3851 |
obtain z where "y < z" and "z < x" |
3852 |
and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto |
|
33549 | 3853 |
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
|
3854 |
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
|
3855 |
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
|
3856 |
have "0 < x - y" using `y < x` by auto |
56544 | 3857 |
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
|
3858 |
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
|
3859 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3860 |
|
53079 | 3861 |
lemma tan_monotone': |
3862 |
assumes "- (pi / 2) < y" |
|
3863 |
and "y < pi / 2" |
|
3864 |
and "- (pi / 2) < x" |
|
3865 |
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
|
3866 |
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
|
3867 |
proof |
53079 | 3868 |
assume "y < x" |
3869 |
thus "tan y < tan x" |
|
3870 |
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
|
3871 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3872 |
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
|
3873 |
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
|
3874 |
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
|
3875 |
assume "\<not> y < x" hence "x \<le> y" by auto |
41970 | 3876 |
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
|
3877 |
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
|
3878 |
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
|
3879 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3880 |
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
|
3881 |
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
|
3882 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3883 |
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
|
3884 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3885 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3886 |
|
53079 | 3887 |
lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)" |
3888 |
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
|
3889 |
|
41970 | 3890 |
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
|
3891 |
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
|
3892 |
|
53079 | 3893 |
lemma tan_periodic_nat[simp]: |
3894 |
fixes n :: nat |
|
3895 |
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
|
3896 |
proof (induct n arbitrary: x) |
53079 | 3897 |
case 0 |
3898 |
then show ?case by simp |
|
3899 |
next |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3900 |
case (Suc n) |
53079 | 3901 |
have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" |
3902 |
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
|
3903 |
show ?case unfolding split_pi_off using Suc by auto |
53079 | 3904 |
qed |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3905 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3906 |
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
|
3907 |
proof (cases "0 \<le> i") |
53079 | 3908 |
case True |
3909 |
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
|
3910 |
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
|
3911 |
next |
53079 | 3912 |
case False |
3913 |
hence i_nat: "real i = - real (nat (-i))" by auto |
|
3914 |
have "tan x = tan (x + real i * pi - real i * pi)" |
|
3915 |
by auto |
|
3916 |
also have "\<dots> = tan (x + real i * pi)" |
|
3917 |
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
|
3918 |
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
|
3919 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
3920 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46240
diff
changeset
|
3921 |
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
|
3922 |
using tan_periodic_int[of _ "numeral n" ] unfolding real_numeral . |
23043 | 3923 |
|
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3924 |
lemma tan_minus_45: "tan (-(pi/4)) = -1" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3925 |
unfolding tan_def by (simp add: sin_45 cos_45) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3926 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3927 |
lemma tan_diff: |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3928 |
fixes x :: "'a::{real_normed_field,banach}" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3929 |
shows |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3930 |
"\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0; cos (x - y) \<noteq> 0\<rbrakk> |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3931 |
\<Longrightarrow> tan(x - y) = (tan(x) - tan(y))/(1 + tan(x) * tan(y))" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3932 |
using tan_add [of x "-y"] |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3933 |
by simp |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3934 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3935 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3936 |
lemma tan_pos_pi2_le: "0 \<le> x ==> x < pi/2 \<Longrightarrow> 0 \<le> tan x" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3937 |
using less_eq_real_def tan_gt_zero by auto |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3938 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3939 |
lemma cos_tan: "abs(x) < pi/2 \<Longrightarrow> cos(x) = 1 / sqrt(1 + tan(x) ^ 2)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3940 |
using cos_gt_zero_pi [of x] |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3941 |
by (simp add: divide_simps tan_def real_sqrt_divide abs_if split: split_if_asm) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3942 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3943 |
lemma sin_tan: "abs(x) < pi/2 \<Longrightarrow> sin(x) = tan(x) / sqrt(1 + tan(x) ^ 2)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3944 |
using cos_gt_zero [of "x"] cos_gt_zero [of "-x"] |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3945 |
by (force simp add: divide_simps tan_def real_sqrt_divide abs_if split: split_if_asm) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3946 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3947 |
lemma tan_mono_le: "-(pi/2) < x ==> x \<le> y ==> y < pi/2 \<Longrightarrow> tan(x) \<le> tan(y)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3948 |
using less_eq_real_def tan_monotone by auto |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3949 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3950 |
lemma tan_mono_lt_eq: "-(pi/2) < x ==> x < pi/2 ==> -(pi/2) < y ==> y < pi/2 |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3951 |
\<Longrightarrow> (tan(x) < tan(y) \<longleftrightarrow> x < y)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3952 |
using tan_monotone' by blast |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3953 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3954 |
lemma tan_mono_le_eq: "-(pi/2) < x ==> x < pi/2 ==> -(pi/2) < y ==> y < pi/2 |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3955 |
\<Longrightarrow> (tan(x) \<le> tan(y) \<longleftrightarrow> x \<le> y)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3956 |
by (meson tan_mono_le not_le tan_monotone) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3957 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3958 |
lemma tan_bound_pi2: "abs(x) < pi/4 \<Longrightarrow> abs(tan x) < 1" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3959 |
using tan_45 tan_monotone [of x "pi/4"] tan_monotone [of "-x" "pi/4"] |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3960 |
by (auto simp: abs_if split: split_if_asm) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3961 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3962 |
lemma tan_cot: "tan(pi/2 - x) = inverse(tan x)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
3963 |
by (simp add: tan_def sin_diff cos_diff) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
3964 |
|
23043 | 3965 |
subsection {* Inverse Trigonometric Functions *} |
3966 |
||
53079 | 3967 |
definition arcsin :: "real => real" |
3968 |
where "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)" |
|
3969 |
||
3970 |
definition arccos :: "real => real" |
|
3971 |
where "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)" |
|
3972 |
||
3973 |
definition arctan :: "real => real" |
|
3974 |
where "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)" |
|
23043 | 3975 |
|
15229 | 3976 |
lemma arcsin: |
53079 | 3977 |
"-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> |
3978 |
-(pi/2) \<le> arcsin y & arcsin y \<le> pi/2 & sin(arcsin y) = y" |
|
3979 |
unfolding arcsin_def by (rule theI' [OF sin_total]) |
|
23011 | 3980 |
|
3981 |
lemma arcsin_pi: |
|
53079 | 3982 |
"-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y" |
3983 |
apply (drule (1) arcsin) |
|
3984 |
apply (force intro: order_trans) |
|
3985 |
done |
|
3986 |
||
3987 |
lemma sin_arcsin [simp]: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin(arcsin y) = y" |
|
3988 |
by (blast dest: arcsin) |
|
3989 |
||
3990 |
lemma arcsin_bounded: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2" |
|
3991 |
by (blast dest: arcsin) |
|
3992 |
||
3993 |
lemma arcsin_lbound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y" |
|
3994 |
by (blast dest: arcsin) |
|
3995 |
||
3996 |
lemma arcsin_ubound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2" |
|
3997 |
by (blast dest: arcsin) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3998 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
3999 |
lemma arcsin_lt_bounded: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4000 |
"\<lbrakk>-1 < y; y < 1\<rbrakk> \<Longrightarrow> -(pi/2) < arcsin y & arcsin y < pi/2" |
53079 | 4001 |
apply (frule order_less_imp_le) |
4002 |
apply (frule_tac y = y in order_less_imp_le) |
|
4003 |
apply (frule arcsin_bounded) |
|
4004 |
apply (safe, simp) |
|
4005 |
apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq) |
|
4006 |
apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe) |
|
4007 |
apply (drule_tac [!] f = sin in arg_cong, auto) |
|
4008 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4009 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4010 |
lemma arcsin_sin: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2\<rbrakk> \<Longrightarrow> arcsin(sin x) = x" |
53079 | 4011 |
apply (unfold arcsin_def) |
4012 |
apply (rule the1_equality) |
|
4013 |
apply (rule sin_total, auto) |
|
4014 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4015 |
|
22975 | 4016 |
lemma arccos: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4017 |
"\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4018 |
\<Longrightarrow> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y" |
53079 | 4019 |
unfolding arccos_def by (rule theI' [OF cos_total]) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4020 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4021 |
lemma cos_arccos [simp]: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> cos(arccos y) = y" |
53079 | 4022 |
by (blast dest: arccos) |
41970 | 4023 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4024 |
lemma arccos_bounded: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> arccos y & arccos y \<le> pi" |
53079 | 4025 |
by (blast dest: arccos) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4026 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4027 |
lemma arccos_lbound: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> arccos y" |
53079 | 4028 |
by (blast dest: arccos) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4029 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4030 |
lemma arccos_ubound: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi" |
53079 | 4031 |
by (blast dest: arccos) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4032 |
|
22975 | 4033 |
lemma arccos_lt_bounded: |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4034 |
"\<lbrakk>-1 < y; y < 1\<rbrakk> |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4035 |
\<Longrightarrow> 0 < arccos y & arccos y < pi" |
53079 | 4036 |
apply (frule order_less_imp_le) |
4037 |
apply (frule_tac y = y in order_less_imp_le) |
|
4038 |
apply (frule arccos_bounded, auto) |
|
4039 |
apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq) |
|
4040 |
apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto) |
|
4041 |
apply (drule_tac [!] f = cos in arg_cong, auto) |
|
4042 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4043 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4044 |
lemma arccos_cos: "\<lbrakk>0 \<le> x; x \<le> pi\<rbrakk> \<Longrightarrow> arccos(cos x) = x" |
53079 | 4045 |
apply (simp add: arccos_def) |
4046 |
apply (auto intro!: the1_equality cos_total) |
|
4047 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4048 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4049 |
lemma arccos_cos2: "\<lbrakk>x \<le> 0; -pi \<le> x\<rbrakk> \<Longrightarrow> arccos(cos x) = -x" |
53079 | 4050 |
apply (simp add: arccos_def) |
4051 |
apply (auto intro!: the1_equality cos_total) |
|
4052 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4053 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
4054 |
lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)" |
53079 | 4055 |
apply (subgoal_tac "x\<^sup>2 \<le> 1") |
4056 |
apply (rule power2_eq_imp_eq) |
|
4057 |
apply (simp add: cos_squared_eq) |
|
4058 |
apply (rule cos_ge_zero) |
|
4059 |
apply (erule (1) arcsin_lbound) |
|
4060 |
apply (erule (1) arcsin_ubound) |
|
4061 |
apply simp |
|
4062 |
apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp) |
|
4063 |
apply (rule power_mono, simp, simp) |
|
4064 |
done |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
4065 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
4066 |
lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)" |
53079 | 4067 |
apply (subgoal_tac "x\<^sup>2 \<le> 1") |
4068 |
apply (rule power2_eq_imp_eq) |
|
4069 |
apply (simp add: sin_squared_eq) |
|
4070 |
apply (rule sin_ge_zero) |
|
4071 |
apply (erule (1) arccos_lbound) |
|
4072 |
apply (erule (1) arccos_ubound) |
|
4073 |
apply simp |
|
4074 |
apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp) |
|
4075 |
apply (rule power_mono, simp, simp) |
|
4076 |
done |
|
4077 |
||
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4078 |
lemma arccos_0 [simp]: "arccos 0 = pi/2" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4079 |
by (metis arccos_cos cos_gt_zero cos_pi cos_pi_half pi_gt_zero pi_half_ge_zero not_le not_zero_less_neg_numeral numeral_One) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4080 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4081 |
lemma arccos_1 [simp]: "arccos 1 = 0" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4082 |
using arccos_cos by force |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4083 |
|
53079 | 4084 |
lemma arctan [simp]: "- (pi/2) < arctan y & arctan y < pi/2 & tan (arctan y) = y" |
4085 |
unfolding arctan_def by (rule theI' [OF tan_total]) |
|
4086 |
||
4087 |
lemma tan_arctan: "tan (arctan y) = y" |
|
4088 |
by auto |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4089 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4090 |
lemma arctan_bounded: "- (pi/2) < arctan y & arctan y < pi/2" |
53079 | 4091 |
by (auto simp only: arctan) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4092 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4093 |
lemma arctan_lbound: "- (pi/2) < arctan y" |
53079 | 4094 |
by auto |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4095 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4096 |
lemma arctan_ubound: "arctan y < pi/2" |
53079 | 4097 |
by (auto simp only: arctan) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4098 |
|
44746 | 4099 |
lemma arctan_unique: |
53079 | 4100 |
assumes "-(pi/2) < x" |
4101 |
and "x < pi/2" |
|
4102 |
and "tan x = y" |
|
44746 | 4103 |
shows "arctan y = x" |
4104 |
using assms arctan [of y] tan_total [of y] by (fast elim: ex1E) |
|
4105 |
||
53079 | 4106 |
lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x" |
4107 |
by (rule arctan_unique) simp_all |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4108 |
|
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4109 |
lemma arctan_zero_zero [simp]: "arctan 0 = 0" |
53079 | 4110 |
by (rule arctan_unique) simp_all |
44746 | 4111 |
|
4112 |
lemma arctan_minus: "arctan (- x) = - arctan x" |
|
4113 |
apply (rule arctan_unique) |
|
4114 |
apply (simp only: neg_less_iff_less arctan_ubound) |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4115 |
apply (metis minus_less_iff arctan_lbound, simp) |
44746 | 4116 |
done |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4117 |
|
44725 | 4118 |
lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0" |
4119 |
by (intro less_imp_neq [symmetric] cos_gt_zero_pi |
|
4120 |
arctan_lbound arctan_ubound) |
|
4121 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
4122 |
lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)" |
44725 | 4123 |
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
|
4124 |
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
|
4125 |
show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp |
44725 | 4126 |
show "0 \<le> cos (arctan x)" |
4127 |
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
|
4128 |
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
|
4129 |
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
|
4130 |
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
|
4131 |
using `0 < 1 + x\<^sup>2` by (simp add: power_divide eq_divide_eq) |
44725 | 4132 |
qed |
4133 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
4134 |
lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)" |
44725 | 4135 |
using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]] |
4136 |
using tan_arctan [of x] unfolding tan_def cos_arctan |
|
4137 |
by (simp add: eq_divide_eq) |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4138 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4139 |
lemma tan_sec: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4140 |
fixes x :: "'a::{real_normed_field,banach,field_inverse_zero}" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4141 |
shows "cos x \<noteq> 0 \<Longrightarrow> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2" |
53079 | 4142 |
apply (rule power_inverse [THEN subst]) |
56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
56213
diff
changeset
|
4143 |
apply (rule_tac c1 = "(cos x)\<^sup>2" in mult_right_cancel [THEN iffD1]) |
53079 | 4144 |
apply (auto dest: field_power_not_zero |
4145 |
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
|
4146 |
mult.assoc power_inverse [symmetric]) |
53079 | 4147 |
done |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4148 |
|
44746 | 4149 |
lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y" |
4150 |
by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan) |
|
4151 |
||
4152 |
lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y" |
|
4153 |
by (simp only: not_less [symmetric] arctan_less_iff) |
|
4154 |
||
4155 |
lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y" |
|
4156 |
by (simp only: eq_iff [where 'a=real] arctan_le_iff) |
|
4157 |
||
4158 |
lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x" |
|
4159 |
using arctan_less_iff [of 0 x] by simp |
|
4160 |
||
4161 |
lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0" |
|
4162 |
using arctan_less_iff [of x 0] by simp |
|
4163 |
||
4164 |
lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x" |
|
4165 |
using arctan_le_iff [of 0 x] by simp |
|
4166 |
||
4167 |
lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0" |
|
4168 |
using arctan_le_iff [of x 0] by simp |
|
4169 |
||
4170 |
lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0" |
|
4171 |
using arctan_eq_iff [of x 0] by simp |
|
4172 |
||
51482
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4173 |
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
|
4174 |
proof - |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4175 |
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
|
4176 |
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
|
4177 |
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
|
4178 |
proof safe |
53079 | 4179 |
fix x :: real |
4180 |
assume "x \<in> {-1..1}" |
|
4181 |
then show "x \<in> sin ` {- pi / 2..pi / 2}" |
|
4182 |
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
|
4183 |
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
|
4184 |
qed simp |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4185 |
finally show ?thesis . |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4186 |
qed |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4187 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
4188 |
lemma continuous_on_arcsin [continuous_intros]: |
51482
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4189 |
"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
|
4190 |
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
|
4191 |
by (auto simp: comp_def subset_eq) |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4192 |
|
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4193 |
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
|
4194 |
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
|
4195 |
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
|
4196 |
|
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4197 |
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
|
4198 |
proof - |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4199 |
have "continuous_on (cos ` {0 .. pi}) arccos" |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
4200 |
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
|
4201 |
also have "cos ` {0 .. pi} = {-1 .. 1}" |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4202 |
proof safe |
53079 | 4203 |
fix x :: real |
4204 |
assume "x \<in> {-1..1}" |
|
4205 |
then show "x \<in> cos ` {0..pi}" |
|
4206 |
using arccos_lbound arccos_ubound |
|
4207 |
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
|
4208 |
qed simp |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4209 |
finally show ?thesis . |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4210 |
qed |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4211 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
4212 |
lemma continuous_on_arccos [continuous_intros]: |
51482
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4213 |
"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
|
4214 |
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
|
4215 |
by (auto simp: comp_def subset_eq) |
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4216 |
|
80efd8c49f52
arcsin and arccos are continuous on {0 .. 1} (including the endpoints)
hoelzl
parents:
51481
diff
changeset
|
4217 |
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
|
4218 |
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
|
4219 |
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
|
4220 |
|
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
4221 |
lemma isCont_arctan: "isCont arctan x" |
53079 | 4222 |
apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify) |
4223 |
apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify) |
|
4224 |
apply (subgoal_tac "isCont arctan (tan (arctan x))", simp) |
|
4225 |
apply (erule (1) isCont_inverse_function2 [where f=tan]) |
|
4226 |
apply (metis arctan_tan order_le_less_trans order_less_le_trans) |
|
4227 |
apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le) |
|
4228 |
done |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
4229 |
|
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
|
4230 |
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
|
4231 |
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
|
4232 |
|
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
|
4233 |
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
|
4234 |
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
|
4235 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56261
diff
changeset
|
4236 |
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
|
4237 |
unfolding continuous_on_def by (auto intro: tendsto_arctan) |
53079 | 4238 |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
4239 |
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
|
4240 |
"\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4241 |
apply (rule DERIV_inverse_function [where f=sin and a="-1" and b=1]) |
53079 | 4242 |
apply (rule DERIV_cong [OF DERIV_sin]) |
4243 |
apply (simp add: cos_arcsin) |
|
4244 |
apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp) |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4245 |
apply (rule power_strict_mono, simp, simp, simp, assumption, assumption) |
53079 | 4246 |
apply simp |
4247 |
apply (erule (1) isCont_arcsin) |
|
4248 |
done |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
4249 |
|
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
4250 |
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
|
4251 |
"\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4252 |
apply (rule DERIV_inverse_function [where f=cos and a="-1" and b=1]) |
53079 | 4253 |
apply (rule DERIV_cong [OF DERIV_cos]) |
4254 |
apply (simp add: sin_arccos) |
|
4255 |
apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp) |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4256 |
apply (rule power_strict_mono, simp, simp, simp, assumption, assumption) |
53079 | 4257 |
apply simp |
4258 |
apply (erule (1) isCont_arccos) |
|
4259 |
done |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
4260 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
4261 |
lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)" |
53079 | 4262 |
apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"]) |
4263 |
apply (rule DERIV_cong [OF DERIV_tan]) |
|
4264 |
apply (rule cos_arctan_not_zero) |
|
4265 |
apply (simp add: power_inverse tan_sec [symmetric]) |
|
4266 |
apply (subgoal_tac "0 < 1 + x\<^sup>2", simp) |
|
4267 |
apply (simp add: add_pos_nonneg) |
|
4268 |
apply (simp, simp, simp, rule isCont_arctan) |
|
4269 |
done |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
4270 |
|
31880 | 4271 |
declare |
56381
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
4272 |
DERIV_arcsin[THEN DERIV_chain2, derivative_intros] |
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
4273 |
DERIV_arccos[THEN DERIV_chain2, derivative_intros] |
0556204bc230
merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents:
56371
diff
changeset
|
4274 |
DERIV_arctan[THEN DERIV_chain2, derivative_intros] |
31880 | 4275 |
|
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
|
4276 |
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
|
4277 |
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
|
4278 |
(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
|
4279 |
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
|
4280 |
|
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
|
4281 |
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
|
4282 |
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
|
4283 |
(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
|
4284 |
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
|
4285 |
|
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
|
4286 |
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
|
4287 |
proof (rule tendstoI) |
53079 | 4288 |
fix e :: real |
4289 |
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
|
4290 |
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
|
4291 |
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
|
4292 |
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
|
4293 |
|
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
|
4294 |
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
|
4295 |
proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI) |
53079 | 4296 |
fix x |
4297 |
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
|
4298 |
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
|
4299 |
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
|
4300 |
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
|
4301 |
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
|
4302 |
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
|
4303 |
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
|
4304 |
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
|
4305 |
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
|
4306 |
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
|
4307 |
|
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
|
4308 |
lemma tendsto_arctan_at_bot: "(arctan ---> - (pi/2)) at_bot" |
53079 | 4309 |
unfolding filterlim_at_bot_mirror arctan_minus |
4310 |
by (intro tendsto_minus tendsto_arctan_at_top) |
|
4311 |
||
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
|
4312 |
|
59746
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4313 |
subsection{* Prove Totality of the Trigonometric Functions *} |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4314 |
|
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4315 |
lemma sin_mono_less_eq: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2; -(pi/2) \<le> y; y \<le> pi/2\<rbrakk> |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4316 |
\<Longrightarrow> (sin(x) < sin(y) \<longleftrightarrow> x < y)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4317 |
by (metis not_less_iff_gr_or_eq sin_monotone_2pi) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4318 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4319 |
lemma sin_mono_le_eq: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2; -(pi/2) \<le> y; y \<le> pi/2\<rbrakk> |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4320 |
\<Longrightarrow> (sin(x) \<le> sin(y) \<longleftrightarrow> x \<le> y)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4321 |
by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4322 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4323 |
lemma sin_inj_pi: "-(pi/2) \<le> x ==> x \<le> pi/2 ==> |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4324 |
-(pi/2) \<le> y ==> y \<le> pi/2 ==> sin(x) = sin(y) \<Longrightarrow> x = y" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4325 |
by (metis arcsin_sin) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4326 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4327 |
lemma cos_mono_lt_eq: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4328 |
\<Longrightarrow> (cos(x) < cos(y) \<longleftrightarrow> y < x)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4329 |
by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4330 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4331 |
lemma cos_mono_le_eq: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4332 |
\<Longrightarrow> (cos(x) \<le> cos(y) \<longleftrightarrow> y \<le> x)" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4333 |
by (metis arccos_cos cos_monotone_0_pi_le eq_iff linear) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4334 |
|
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4335 |
lemma cos_inj_pi: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi ==> cos(x) = cos(y) |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4336 |
\<Longrightarrow> x = y" |
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4337 |
by (metis arccos_cos) |
59746
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4338 |
|
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4339 |
lemma arccos_le_pi2: "\<lbrakk>0 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi/2" |
59751
916c0f6c83e3
New material for complex sin, cos, tan, Ln, also some reorganisation
paulson <lp15@cam.ac.uk>
parents:
59746
diff
changeset
|
4340 |
by (metis (mono_tags) arccos_0 arccos cos_le_one cos_monotone_0_pi_le |
59746
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4341 |
cos_pi cos_pi_half pi_half_ge_zero antisym_conv less_eq_neg_nonpos linear minus_minus order.trans order_refl) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4342 |
|
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4343 |
lemma sincos_total_pi_half: |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4344 |
assumes "0 \<le> x" "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4345 |
shows "\<exists>t. 0 \<le> t \<and> t \<le> pi/2 \<and> x = cos t \<and> y = sin t" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4346 |
proof - |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4347 |
have x1: "x \<le> 1" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4348 |
using assms |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4349 |
by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4350 |
moreover with assms have ax: "0 \<le> arccos x" "cos(arccos x) = x" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4351 |
by (auto simp: arccos) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4352 |
moreover have "y = sqrt (1 - x\<^sup>2)" using assms |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4353 |
by (metis abs_of_nonneg add.commute add_diff_cancel real_sqrt_abs) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4354 |
ultimately show ?thesis using assms arccos_le_pi2 [of x] |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4355 |
by (rule_tac x="arccos x" in exI) (auto simp: sin_arccos) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4356 |
qed |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4357 |
|
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4358 |
lemma sincos_total_pi: |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4359 |
assumes "0 \<le> y" and "x\<^sup>2 + y\<^sup>2 = 1" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4360 |
shows "\<exists>t. 0 \<le> t \<and> t \<le> pi \<and> x = cos t \<and> y = sin t" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4361 |
proof (cases rule: le_cases [of 0 x]) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4362 |
case le from sincos_total_pi_half [OF le] |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4363 |
show ?thesis |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4364 |
by (metis pi_ge_two pi_half_le_two add.commute add_le_cancel_left add_mono assms) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4365 |
next |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4366 |
case ge |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4367 |
then have "0 \<le> -x" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4368 |
by simp |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4369 |
then obtain t where "t\<ge>0" "t \<le> pi/2" "-x = cos t" "y = sin t" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4370 |
using sincos_total_pi_half assms |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4371 |
apply auto |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4372 |
by (metis `0 \<le> - x` power2_minus) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4373 |
then show ?thesis |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4374 |
by (rule_tac x="pi-t" in exI, auto) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4375 |
qed |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4376 |
|
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4377 |
lemma sincos_total_2pi_le: |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4378 |
assumes "x\<^sup>2 + y\<^sup>2 = 1" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4379 |
shows "\<exists>t. 0 \<le> t \<and> t \<le> 2*pi \<and> x = cos t \<and> y = sin t" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4380 |
proof (cases rule: le_cases [of 0 y]) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4381 |
case le from sincos_total_pi [OF le] |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4382 |
show ?thesis |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4383 |
by (metis assms le_add_same_cancel1 mult.commute mult_2_right order.trans) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4384 |
next |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4385 |
case ge |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4386 |
then have "0 \<le> -y" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4387 |
by simp |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4388 |
then obtain t where "t\<ge>0" "t \<le> pi" "x = cos t" "-y = sin t" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4389 |
using sincos_total_pi assms |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4390 |
apply auto |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4391 |
by (metis `0 \<le> - y` power2_minus) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4392 |
then show ?thesis |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4393 |
by (rule_tac x="2*pi-t" in exI, auto) |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4394 |
qed |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4395 |
|
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4396 |
lemma sincos_total_2pi: |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4397 |
assumes "x\<^sup>2 + y\<^sup>2 = 1" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4398 |
obtains t where "0 \<le> t" "t < 2*pi" "x = cos t" "y = sin t" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4399 |
proof - |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4400 |
from sincos_total_2pi_le [OF assms] |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4401 |
obtain t where t: "0 \<le> t" "t \<le> 2*pi" "x = cos t" "y = sin t" |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4402 |
by blast |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4403 |
show ?thesis |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4404 |
apply (cases "t = 2*pi") |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4405 |
using t that |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4406 |
apply force+ |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4407 |
done |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4408 |
qed |
ddae5727c5a9
new HOL Light material about exp, sin, cos
paulson <lp15@cam.ac.uk>
parents:
59741
diff
changeset
|
4409 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4410 |
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
|
4411 |
|
44746 | 4412 |
lemma arctan_one: "arctan 1 = pi / 4" |
4413 |
by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi) |
|
4414 |
||
53079 | 4415 |
lemma tan_total_pi4: |
4416 |
assumes "\<bar>x\<bar> < 1" |
|
4417 |
shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x" |
|
44746 | 4418 |
proof |
4419 |
show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x" |
|
4420 |
unfolding arctan_one [symmetric] arctan_minus [symmetric] |
|
4421 |
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
|
4422 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4423 |
|
53079 | 4424 |
lemma arctan_add: |
4425 |
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
|
4426 |
shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))" |
44746 | 4427 |
proof (rule arctan_unique [symmetric]) |
4428 |
have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y" |
|
4429 |
unfolding arctan_one [symmetric] arctan_minus [symmetric] |
|
4430 |
unfolding arctan_le_iff arctan_less_iff using assms by auto |
|
4431 |
from add_le_less_mono [OF this] |
|
4432 |
show 1: "- (pi / 2) < arctan x + arctan y" by simp |
|
4433 |
have "arctan x \<le> pi / 4" and "arctan y < pi / 4" |
|
4434 |
unfolding arctan_one [symmetric] |
|
4435 |
unfolding arctan_le_iff arctan_less_iff using assms by auto |
|
4436 |
from add_le_less_mono [OF this] |
|
4437 |
show 2: "arctan x + arctan y < pi / 2" by simp |
|
4438 |
show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)" |
|
4439 |
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
|
4440 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4441 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4442 |
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
|
4443 |
proof - |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4444 |
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
|
4445 |
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
|
4446 |
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
|
4447 |
moreover |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4448 |
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
|
4449 |
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
|
4450 |
have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto |
41970 | 4451 |
moreover |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4452 |
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
|
4453 |
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
|
4454 |
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
|
4455 |
ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto |
44746 | 4456 |
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
|
4457 |
qed |
44746 | 4458 |
|
53079 | 4459 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4460 |
subsection {* Introducing the inverse tangent power series *} |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4461 |
|
53079 | 4462 |
lemma monoseq_arctan_series: |
4463 |
fixes x :: real |
|
4464 |
assumes "\<bar>x\<bar> \<le> 1" |
|
4465 |
shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a") |
|
4466 |
proof (cases "x = 0") |
|
4467 |
case True |
|
4468 |
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
|
4469 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4470 |
case False |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4471 |
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
|
4472 |
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
|
4473 |
proof - |
53079 | 4474 |
{ |
4475 |
fix n |
|
4476 |
fix x :: real |
|
4477 |
assume "0 \<le> x" and "x \<le> 1" |
|
4478 |
have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le> |
|
4479 |
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
|
4480 |
proof (rule mult_mono) |
53079 | 4481 |
show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))" |
4482 |
by (rule frac_le) simp_all |
|
4483 |
show "0 \<le> 1 / real (Suc (n * 2))" |
|
4484 |
by auto |
|
4485 |
show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)" |
|
4486 |
by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`) |
|
4487 |
show "0 \<le> x ^ Suc (Suc n * 2)" |
|
4488 |
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
|
4489 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4490 |
} note mono = this |
41970 | 4491 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4492 |
show ?thesis |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4493 |
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
|
4494 |
case True from mono[OF this `x \<le> 1`, THEN allI] |
53079 | 4495 |
show ?thesis unfolding Suc_eq_plus1[symmetric] |
4496 |
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
|
4497 |
next |
53079 | 4498 |
case False |
4499 |
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
|
4500 |
from mono[OF this] |
53079 | 4501 |
have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge> |
4502 |
1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using `0 \<le> -x` by auto |
|
31790 | 4503 |
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
|
4504 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4505 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4506 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4507 |
|
53079 | 4508 |
lemma zeroseq_arctan_series: |
4509 |
fixes x :: real |
|
4510 |
assumes "\<bar>x\<bar> \<le> 1" |
|
4511 |
shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0") |
|
4512 |
proof (cases "x = 0") |
|
4513 |
case True |
|
4514 |
thus ?thesis |
|
58729
e8ecc79aee43
add tendsto_const and tendsto_ident_at as simp and intro rules
hoelzl
parents:
58710
diff
changeset
|
4515 |
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
|
4516 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4517 |
case False |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4518 |
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
|
4519 |
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
|
4520 |
proof (cases "\<bar>x\<bar> < 1") |
53079 | 4521 |
case True |
4522 |
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
|
4523 |
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
|
4524 |
have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0" |
31790 | 4525 |
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
|
4526 |
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
|
4527 |
next |
53079 | 4528 |
case False |
4529 |
hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto |
|
4530 |
hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x" |
|
4531 |
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
|
4532 |
from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]] |
31790 | 4533 |
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
|
4534 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4535 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4536 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4537 |
text{*FIXME: generalise from the reals via type classes?*} |
53079 | 4538 |
lemma summable_arctan_series: |
4539 |
fixes x :: real and n :: nat |
|
4540 |
assumes "\<bar>x\<bar> \<le> 1" |
|
4541 |
shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" |
|
4542 |
(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
|
4543 |
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
|
4544 |
|
53079 | 4545 |
lemma less_one_imp_sqr_less_one: |
4546 |
fixes x :: real |
|
4547 |
assumes "\<bar>x\<bar> < 1" |
|
4548 |
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
|
4549 |
proof - |
54573 | 4550 |
have "\<bar>x\<^sup>2\<bar> < 1" |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
4551 |
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
|
4552 |
thus ?thesis using zero_le_power2 by auto |
41970 | 4553 |
qed |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4554 |
|
53079 | 4555 |
lemma DERIV_arctan_series: |
4556 |
assumes "\<bar> x \<bar> < 1" |
|
4557 |
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))" |
|
4558 |
(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
|
4559 |
proof - |
53079 | 4560 |
let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0" |
4561 |
||
4562 |
have n_even: "\<And>n :: nat. even n \<Longrightarrow> 2 * (n div 2) = n" |
|
4563 |
by presburger |
|
4564 |
then have if_eq: "\<And>n x'. ?f n * real (Suc n) * x'^n = |
|
4565 |
(if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)" |
|
4566 |
by auto |
|
4567 |
||
4568 |
{ |
|
4569 |
fix x :: real |
|
4570 |
assume "\<bar>x\<bar> < 1" |
|
4571 |
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
|
4572 |
have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)" |
53076 | 4573 |
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
|
4574 |
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
|
4575 |
} 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
|
4576 |
|
53079 | 4577 |
{ |
4578 |
fix f :: "nat \<Rightarrow> real" |
|
4579 |
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
|
4580 |
proof |
53079 | 4581 |
fix x :: real |
4582 |
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
|
4583 |
from sums_if[OF sums_zero this] |
53079 | 4584 |
show "(\<lambda>n. if even n then f (n div 2) else 0) sums x" |
4585 |
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
|
4586 |
next |
53079 | 4587 |
fix x :: real |
4588 |
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
|
4589 |
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
|
4590 |
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
|
4591 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4592 |
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
|
4593 |
} 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
|
4594 |
|
53079 | 4595 |
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
|
4596 |
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
|
4597 |
by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4598 |
|
53079 | 4599 |
{ |
4600 |
fix x :: real |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
4601 |
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
|
4602 |
(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
|
4603 |
using n_even by auto |
53079 | 4604 |
have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)" by auto |
4605 |
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
|
4606 |
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
|
4607 |
by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4608 |
} 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
|
4609 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4610 |
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
|
4611 |
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
|
4612 |
show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto |
53079 | 4613 |
{ |
4614 |
fix x' :: real |
|
4615 |
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
|
4616 |
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
|
4617 |
then |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
4618 |
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
|
4619 |
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
|
4620 |
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
|
4621 |
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
|
4622 |
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
|
4623 |
apply (rule sums_if) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
4624 |
apply (rule sums_zero) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
4625 |
apply (rule summable_sums) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
4626 |
apply (rule *) |
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
57514
diff
changeset
|
4627 |
done |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4628 |
} |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4629 |
qed auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4630 |
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
|
4631 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4632 |
|
53079 | 4633 |
lemma arctan_series: |
4634 |
assumes "\<bar> x \<bar> \<le> 1" |
|
4635 |
shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" |
|
4636 |
(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
|
4637 |
proof - |
53079 | 4638 |
let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)" |
4639 |
||
4640 |
{ |
|
4641 |
fix r x :: real |
|
4642 |
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
|
4643 |
have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto |
53079 | 4644 |
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
|
4645 |
} 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
|
4646 |
|
53079 | 4647 |
{ |
4648 |
fix x :: real |
|
4649 |
assume "\<bar>x\<bar> \<le> 1" |
|
4650 |
note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] |
|
4651 |
} note arctan_series_borders = this |
|
4652 |
||
4653 |
{ |
|
4654 |
fix x :: real |
|
4655 |
assume "\<bar>x\<bar> < 1" |
|
4656 |
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
|
4657 |
proof - |
53079 | 4658 |
obtain r where "\<bar>x\<bar> < r" and "r < 1" |
4659 |
using dense[OF `\<bar>x\<bar> < 1`] by blast |
|
4660 |
hence "0 < r" and "-r < x" and "x < r" by auto |
|
4661 |
||
4662 |
have suminf_eq_arctan_bounded: "\<And>x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow> |
|
4663 |
suminf (?c x) - arctan x = suminf (?c a) - arctan a" |
|
4664 |
proof - |
|
4665 |
fix x a b |
|
4666 |
assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b" |
|
4667 |
hence "\<bar>x\<bar> < r" by auto |
|
4668 |
show "suminf (?c x) - arctan x = suminf (?c a) - arctan a" |
|
4669 |
proof (rule DERIV_isconst2[of "a" "b"]) |
|
4670 |
show "a < b" and "a \<le> x" and "x \<le> b" |
|
4671 |
using `a < b` `a \<le> x` `x \<le> b` by auto |
|
4672 |
have "\<forall>x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0" |
|
4673 |
proof (rule allI, rule impI) |
|
4674 |
fix x |
|
4675 |
assume "-r < x \<and> x < r" |
|
4676 |
hence "\<bar>x\<bar> < r" by auto |
|
4677 |
hence "\<bar>x\<bar> < 1" using `r < 1` by auto |
|
4678 |
have "\<bar> - (x\<^sup>2) \<bar> < 1" |
|
4679 |
using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto |
|
4680 |
hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))" |
|
4681 |
unfolding real_norm_def[symmetric] by (rule geometric_sums) |
|
4682 |
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
|
4683 |
unfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2] by auto |
53079 | 4684 |
hence suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)" |
4685 |
using sums_unique unfolding inverse_eq_divide by auto |
|
4686 |
have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))" |
|
4687 |
unfolding suminf_c'_eq_geom |
|
4688 |
by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`]) |
|
56261 | 4689 |
from DERIV_diff [OF this DERIV_arctan] |
53079 | 4690 |
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
|
4691 |
by auto |
53079 | 4692 |
qed |
4693 |
hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0" |
|
4694 |
using `-r < a` `b < r` by auto |
|
4695 |
thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0" |
|
4696 |
using `\<bar>x\<bar> < r` by auto |
|
4697 |
show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y" |
|
4698 |
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
|
4699 |
qed |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4700 |
qed |
53079 | 4701 |
|
4702 |
have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0" |
|
4703 |
unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero |
|
4704 |
by auto |
|
4705 |
||
4706 |
have "suminf (?c x) - arctan x = 0" |
|
4707 |
proof (cases "x = 0") |
|
4708 |
case True |
|
4709 |
thus ?thesis using suminf_arctan_zero by auto |
|
4710 |
next |
|
4711 |
case False |
|
4712 |
hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto |
|
4713 |
have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0" |
|
59647
c6f413b660cf
clarified Drule.gen_all: observe context more carefully;
wenzelm
parents:
59613
diff
changeset
|
4714 |
by (rule suminf_eq_arctan_bounded[where x1="0" and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>", symmetric]) |
53079 | 4715 |
(simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less) |
4716 |
moreover |
|
4717 |
have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)" |
|
59647
c6f413b660cf
clarified Drule.gen_all: observe context more carefully;
wenzelm
parents:
59613
diff
changeset
|
4718 |
by (rule suminf_eq_arctan_bounded[where x1="x" and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>"]) |
54573 | 4719 |
(simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less) |
53079 | 4720 |
ultimately |
4721 |
show ?thesis using suminf_arctan_zero by auto |
|
4722 |
qed |
|
4723 |
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
|
4724 |
qed |
53079 | 4725 |
} 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
|
4726 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4727 |
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
|
4728 |
proof (cases "\<bar>x\<bar> < 1") |
53079 | 4729 |
case True |
4730 |
thus ?thesis by (rule when_less_one) |
|
4731 |
next |
|
4732 |
case False |
|
4733 |
hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto |
|
4734 |
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
|
4735 |
let ?diff = "\<lambda> x n. \<bar> arctan x - (\<Sum> i<n. ?c x i)\<bar>" |
53079 | 4736 |
{ |
4737 |
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
|
4738 |
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
|
4739 |
moreover |
53079 | 4740 |
{ |
4741 |
fix x :: real |
|
4742 |
assume "0 < x" and "x < 1" |
|
4743 |
hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto |
|
4744 |
from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)" |
|
4745 |
by auto |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
4746 |
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 | 4747 |
have "0 < 1 / real (n*2+1) * x^(n*2+1)" |
4748 |
by (rule mult_pos_pos, auto simp only: zero_less_power[OF `0 < x`], auto) |
|
4749 |
hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)" |
|
4750 |
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
|
4751 |
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
|
4752 |
proof (cases "even n") |
53079 | 4753 |
case True |
4754 |
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
|
4755 |
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
|
4756 |
then have "2 * m = n" .. |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
4757 |
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
|
4758 |
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 | 4759 |
by auto |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
4760 |
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
|
4761 |
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
|
4762 |
finally show ?thesis . |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
4763 |
next |
53079 | 4764 |
case False |
4765 |
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
|
4766 |
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
|
4767 |
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
|
4768 |
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
|
4769 |
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
|
4770 |
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 | 4771 |
by auto |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
4772 |
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
|
4773 |
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
|
4774 |
finally show ?thesis . |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32047
diff
changeset
|
4775 |
qed |
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4776 |
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
|
4777 |
} |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4778 |
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
|
4779 |
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
|
4780 |
unfolding diff_conv_add_uminus divide_inverse |
53079 | 4781 |
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
|
4782 |
isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53602
diff
changeset
|
4783 |
simp del: add_uminus_conv_diff) |
53079 | 4784 |
ultimately have "0 \<le> ?a 1 n - ?diff 1 n" |
4785 |
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
|
4786 |
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
|
4787 |
} |
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset
|
4788 |
have "?a 1 ----> 0" |
44568
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44319
diff
changeset
|
4789 |
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
|
4790 |
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
|
4791 |
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
|
4792 |
proof (rule LIMSEQ_I) |
53079 | 4793 |
fix r :: real |
4794 |
assume "0 < r" |
|
4795 |
obtain N :: nat where N_I: "\<And>n. N \<le> n \<Longrightarrow> ?a 1 n < r" |
|
4796 |
using LIMSEQ_D[OF `?a 1 ----> 0` `0 < r`] by auto |
|
4797 |
{ |
|
4798 |
fix n |
|
4799 |
assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this] |
|
4800 |
have "norm (?diff 1 n - 0) < r" by auto |
|
4801 |
} |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4802 |
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
|
4803 |
qed |
44710 | 4804 |
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
|
4805 |
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
|
4806 |
hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique) |
41970 | 4807 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4808 |
show ?thesis |
53079 | 4809 |
proof (cases "x = 1") |
4810 |
case True |
|
4811 |
then show ?thesis by (simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`) |
|
4812 |
next |
|
4813 |
case False |
|
4814 |
hence "x = -1" using `\<bar>x\<bar> = 1` by auto |
|
41970 | 4815 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4816 |
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
|
4817 |
have "- (2 * pi) < 0" using pi_gt_zero by auto |
41970 | 4818 |
|
53079 | 4819 |
have c_minus_minus: "\<And>i. ?c (- 1) i = - ?c 1 i" |
4820 |
unfolding One_nat_def by auto |
|
4821 |
||
4822 |
have "arctan (- 1) = arctan (tan (-(pi / 4)))" |
|
4823 |
unfolding tan_45 tan_minus .. |
|
4824 |
also have "\<dots> = - (pi / 4)" |
|
4825 |
by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero]) |
|
4826 |
also have "\<dots> = - (arctan (tan (pi / 4)))" |
|
4827 |
unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero]) |
|
4828 |
also have "\<dots> = - (arctan 1)" |
|
4829 |
unfolding tan_45 .. |
|
4830 |
also have "\<dots> = - (\<Sum> i. ?c 1 i)" |
|
4831 |
using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto |
|
4832 |
also have "\<dots> = (\<Sum> i. ?c (- 1) i)" |
|
4833 |
using suminf_minus[OF sums_summable[OF `(?c 1) sums (arctan 1)`]] |
|
4834 |
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
|
4835 |
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
|
4836 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4837 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4838 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4839 |
|
53079 | 4840 |
lemma arctan_half: |
4841 |
fixes x :: real |
|
53076 | 4842 |
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
|
4843 |
proof - |
53079 | 4844 |
obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x" |
4845 |
using tan_total by blast |
|
4846 |
hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2" |
|
4847 |
by auto |
|
4848 |
||
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4849 |
have "0 < cos y" using cos_gt_zero_pi[OF low high] . |
53079 | 4850 |
hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y" |
4851 |
by auto |
|
4852 |
||
4853 |
have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2" |
|
4854 |
unfolding tan_def power_divide .. |
|
4855 |
also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2" |
|
4856 |
using `cos y \<noteq> 0` by auto |
|
4857 |
also have "\<dots> = 1 / (cos y)\<^sup>2" |
|
4858 |
unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 .. |
|
53076 | 4859 |
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
|
4860 |
|
53079 | 4861 |
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
|
4862 |
unfolding tan_def using `cos y \<noteq> 0` by (simp add: field_simps) |
53079 | 4863 |
also have "\<dots> = tan y / (1 + 1 / cos y)" |
4864 |
using `cos y \<noteq> 0` unfolding add_divide_distrib by auto |
|
4865 |
also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))" |
|
4866 |
unfolding cos_sqrt .. |
|
4867 |
also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))" |
|
4868 |
unfolding real_sqrt_divide by auto |
|
4869 |
finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))" |
|
4870 |
unfolding `1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2` . |
|
4871 |
||
4872 |
have "arctan x = y" |
|
4873 |
using arctan_tan low high y_eq by auto |
|
4874 |
also have "\<dots> = 2 * (arctan (tan (y/2)))" |
|
4875 |
using arctan_tan[OF low2 high2] by auto |
|
4876 |
also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" |
|
4877 |
unfolding tan_half by auto |
|
4878 |
finally show ?thesis |
|
4879 |
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
|
4880 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4881 |
|
53079 | 4882 |
lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y" |
4883 |
by (simp only: arctan_less_iff) |
|
4884 |
||
4885 |
lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y" |
|
4886 |
by (simp only: arctan_le_iff) |
|
44746 | 4887 |
|
4888 |
lemma arctan_inverse: |
|
53079 | 4889 |
assumes "x \<noteq> 0" |
4890 |
shows "arctan (1 / x) = sgn x * pi / 2 - arctan x" |
|
44746 | 4891 |
proof (rule arctan_unique) |
4892 |
show "- (pi / 2) < sgn x * pi / 2 - arctan x" |
|
4893 |
using arctan_bounded [of x] assms |
|
4894 |
unfolding sgn_real_def |
|
4895 |
apply (auto simp add: algebra_simps) |
|
4896 |
apply (drule zero_less_arctan_iff [THEN iffD2]) |
|
4897 |
apply arith |
|
4898 |
done |
|
4899 |
show "sgn x * pi / 2 - arctan x < pi / 2" |
|
4900 |
using arctan_bounded [of "- x"] assms |
|
4901 |
unfolding sgn_real_def arctan_minus |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
4902 |
by (auto simp add: algebra_simps) |
44746 | 4903 |
show "tan (sgn x * pi / 2 - arctan x) = 1 / x" |
4904 |
unfolding tan_inverse [of "arctan x", unfolded tan_arctan] |
|
4905 |
unfolding sgn_real_def |
|
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56409
diff
changeset
|
4906 |
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
|
4907 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4908 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset
|
4909 |
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
|
4910 |
proof - |
44746 | 4911 |
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
|
4912 |
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
|
4913 |
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
|
4914 |
qed |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4915 |
|
53079 | 4916 |
|
22978
1cd8cc21a7c3
clean up polar_Ex proofs; remove unnecessary lemmas
huffman
parents:
22977
diff
changeset
|
4917 |
subsection {* Existence of Polar Coordinates *} |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4918 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
4919 |
lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1" |
53079 | 4920 |
apply (rule power2_le_imp_le [OF _ zero_le_one]) |
4921 |
apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero) |
|
4922 |
done |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4923 |
|
22978
1cd8cc21a7c3
clean up polar_Ex proofs; remove unnecessary lemmas
huffman
parents:
22977
diff
changeset
|
4924 |
lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y" |
53079 | 4925 |
by (simp add: abs_le_iff) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4926 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52139
diff
changeset
|
4927 |
lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)" |
53079 | 4928 |
by (simp add: sin_arccos abs_le_iff) |
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4929 |
|
22978
1cd8cc21a7c3
clean up polar_Ex proofs; remove unnecessary lemmas
huffman
parents:
22977
diff
changeset
|
4930 |
lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one] |
15228 | 4931 |
|
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset
|
4932 |
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
|
4933 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59647
diff
changeset
|
4934 |
lemma polar_Ex: "\<exists>r::real. \<exists>a. x = r * cos a & y = r * sin a" |
54573 | 4935 |
proof - |
4936 |
have polar_ex1: "\<And>y. 0 < y \<Longrightarrow> \<exists>r a. x = r * cos a & y = r * sin a" |
|
4937 |
apply (rule_tac x = "sqrt (x\<^sup>2 + y\<^sup>2)" in exI) |
|
4938 |
apply (rule_tac x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))" in exI) |
|
4939 |
apply (simp add: cos_arccos_lemma1 sin_arccos_lemma1 power_divide |
|
4940 |
real_sqrt_mult [symmetric] right_diff_distrib) |
|
4941 |
done |
|
4942 |
show ?thesis |
|
4943 |
proof (cases "0::real" y rule: linorder_cases) |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
4944 |
case less |
54573 | 4945 |
then show ?thesis by (rule polar_ex1) |
4946 |
next |
|
4947 |
case equal |
|
4948 |
then show ?thesis |
|
4949 |
by (force simp add: intro!: cos_zero sin_zero) |
|
4950 |
next |
|
4951 |
case greater |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
4952 |
then show ?thesis |
54573 | 4953 |
using polar_ex1 [where y="-y"] |
4954 |
by auto (metis cos_minus minus_minus minus_mult_right sin_minus) |
|
4955 |
qed |
|
4956 |
qed |
|
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset
|
4957 |
|
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset
|
4958 |
end |