author | huffman |
Wed, 07 Sep 2011 10:04:07 -0700 | |
changeset 44823 | 6ce95c8c0ba8 |
parent 44764 | 264436dd9491 |
child 44824 | 34b83d981380 |
permissions | -rw-r--r-- |
41959 | 1 |
(* Title: HOL/Complex.thy |
13957 | 2 |
Author: Jacques D. Fleuriot |
3 |
Copyright: 2001 University of Edinburgh |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
4 |
Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 |
13957 | 5 |
*) |
6 |
||
14377 | 7 |
header {* Complex Numbers: Rectangular and Polar Representations *} |
14373 | 8 |
|
15131 | 9 |
theory Complex |
28952
15a4b2cf8c34
made repository layout more coherent with logical distribution structure; stripped some $Id$s
haftmann
parents:
28944
diff
changeset
|
10 |
imports Transcendental |
15131 | 11 |
begin |
13957 | 12 |
|
14373 | 13 |
datatype complex = Complex real real |
13957 | 14 |
|
44724 | 15 |
primrec Re :: "complex \<Rightarrow> real" |
16 |
where Re: "Re (Complex x y) = x" |
|
14373 | 17 |
|
44724 | 18 |
primrec Im :: "complex \<Rightarrow> real" |
19 |
where Im: "Im (Complex x y) = y" |
|
14373 | 20 |
|
21 |
lemma complex_surj [simp]: "Complex (Re z) (Im z) = z" |
|
22 |
by (induct z) simp |
|
13957 | 23 |
|
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
24 |
lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y" |
25712 | 25 |
by (induct x, induct y) simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
26 |
|
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
27 |
lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y" |
25712 | 28 |
by (induct x, induct y) simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
29 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
30 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
31 |
subsection {* Addition and Subtraction *} |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
32 |
|
25599 | 33 |
instantiation complex :: ab_group_add |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
34 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
35 |
|
44724 | 36 |
definition complex_zero_def: |
37 |
"0 = Complex 0 0" |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
38 |
|
44724 | 39 |
definition complex_add_def: |
40 |
"x + y = Complex (Re x + Re y) (Im x + Im y)" |
|
23124 | 41 |
|
44724 | 42 |
definition complex_minus_def: |
43 |
"- x = Complex (- Re x) (- Im x)" |
|
14323 | 44 |
|
44724 | 45 |
definition complex_diff_def: |
46 |
"x - (y\<Colon>complex) = x + - y" |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
47 |
|
25599 | 48 |
lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" |
49 |
by (simp add: complex_zero_def) |
|
14323 | 50 |
|
14374 | 51 |
lemma complex_Re_zero [simp]: "Re 0 = 0" |
25599 | 52 |
by (simp add: complex_zero_def) |
14374 | 53 |
|
54 |
lemma complex_Im_zero [simp]: "Im 0 = 0" |
|
25599 | 55 |
by (simp add: complex_zero_def) |
56 |
||
25712 | 57 |
lemma complex_add [simp]: |
58 |
"Complex a b + Complex c d = Complex (a + c) (b + d)" |
|
59 |
by (simp add: complex_add_def) |
|
60 |
||
25599 | 61 |
lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y" |
62 |
by (simp add: complex_add_def) |
|
63 |
||
64 |
lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y" |
|
65 |
by (simp add: complex_add_def) |
|
14323 | 66 |
|
25712 | 67 |
lemma complex_minus [simp]: |
68 |
"- (Complex a b) = Complex (- a) (- b)" |
|
25599 | 69 |
by (simp add: complex_minus_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
70 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
71 |
lemma complex_Re_minus [simp]: "Re (- x) = - Re x" |
25599 | 72 |
by (simp add: complex_minus_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
73 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
74 |
lemma complex_Im_minus [simp]: "Im (- x) = - Im x" |
25599 | 75 |
by (simp add: complex_minus_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
76 |
|
23275 | 77 |
lemma complex_diff [simp]: |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
78 |
"Complex a b - Complex c d = Complex (a - c) (b - d)" |
25599 | 79 |
by (simp add: complex_diff_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
80 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
81 |
lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y" |
25599 | 82 |
by (simp add: complex_diff_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
83 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
84 |
lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y" |
25599 | 85 |
by (simp add: complex_diff_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
86 |
|
25712 | 87 |
instance |
88 |
by intro_classes (simp_all add: complex_add_def complex_diff_def) |
|
89 |
||
90 |
end |
|
91 |
||
92 |
||
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
93 |
subsection {* Multiplication and Division *} |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
94 |
|
36409 | 95 |
instantiation complex :: field_inverse_zero |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
96 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
97 |
|
44724 | 98 |
definition complex_one_def: |
99 |
"1 = Complex 1 0" |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
100 |
|
44724 | 101 |
definition complex_mult_def: |
102 |
"x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)" |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
103 |
|
44724 | 104 |
definition complex_inverse_def: |
105 |
"inverse x = |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
106 |
Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))" |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
107 |
|
44724 | 108 |
definition complex_divide_def: |
109 |
"x / (y\<Colon>complex) = x * inverse y" |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
110 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
111 |
lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)" |
25712 | 112 |
by (simp add: complex_one_def) |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
113 |
|
14374 | 114 |
lemma complex_Re_one [simp]: "Re 1 = 1" |
25712 | 115 |
by (simp add: complex_one_def) |
14323 | 116 |
|
14374 | 117 |
lemma complex_Im_one [simp]: "Im 1 = 0" |
25712 | 118 |
by (simp add: complex_one_def) |
14323 | 119 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
120 |
lemma complex_mult [simp]: |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
121 |
"Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)" |
25712 | 122 |
by (simp add: complex_mult_def) |
14323 | 123 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
124 |
lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y" |
25712 | 125 |
by (simp add: complex_mult_def) |
14323 | 126 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
127 |
lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y" |
25712 | 128 |
by (simp add: complex_mult_def) |
14323 | 129 |
|
14377 | 130 |
lemma complex_inverse [simp]: |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
131 |
"inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))" |
25712 | 132 |
by (simp add: complex_inverse_def) |
14335 | 133 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
134 |
lemma complex_Re_inverse: |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
135 |
"Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)" |
25712 | 136 |
by (simp add: complex_inverse_def) |
14323 | 137 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
138 |
lemma complex_Im_inverse: |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
139 |
"Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)" |
25712 | 140 |
by (simp add: complex_inverse_def) |
14335 | 141 |
|
25712 | 142 |
instance |
143 |
by intro_classes (simp_all add: complex_mult_def |
|
44724 | 144 |
right_distrib left_distrib right_diff_distrib left_diff_distrib |
145 |
complex_inverse_def complex_divide_def |
|
146 |
power2_eq_square add_divide_distrib [symmetric] |
|
147 |
complex_eq_iff) |
|
14335 | 148 |
|
25712 | 149 |
end |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
150 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
151 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
152 |
subsection {* Numerals and Arithmetic *} |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
153 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
154 |
instantiation complex :: number_ring |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
155 |
begin |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
156 |
|
44724 | 157 |
definition complex_number_of_def: |
158 |
"number_of w = (of_int w \<Colon> complex)" |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
159 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
160 |
instance |
25712 | 161 |
by intro_classes (simp only: complex_number_of_def) |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
162 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
163 |
end |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
164 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
165 |
lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n" |
44724 | 166 |
by (induct n) simp_all |
20556
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
167 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
168 |
lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0" |
44724 | 169 |
by (induct n) simp_all |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
170 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
171 |
lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z" |
44724 | 172 |
by (cases z rule: int_diff_cases) simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
173 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
174 |
lemma complex_Im_of_int [simp]: "Im (of_int z) = 0" |
44724 | 175 |
by (cases z rule: int_diff_cases) simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
176 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
177 |
lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v" |
44724 | 178 |
unfolding number_of_eq by (rule complex_Re_of_int) |
20556
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
179 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
180 |
lemma complex_Im_number_of [simp]: "Im (number_of v) = 0" |
44724 | 181 |
unfolding number_of_eq by (rule complex_Im_of_int) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
182 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
183 |
lemma Complex_eq_number_of [simp]: |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
184 |
"(Complex a b = number_of w) = (a = number_of w \<and> b = 0)" |
44724 | 185 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
186 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
187 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
188 |
subsection {* Scalar Multiplication *} |
20556
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
189 |
|
25712 | 190 |
instantiation complex :: real_field |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
191 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
192 |
|
44724 | 193 |
definition complex_scaleR_def: |
194 |
"scaleR r x = Complex (r * Re x) (r * Im x)" |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
195 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
196 |
lemma complex_scaleR [simp]: |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
197 |
"scaleR r (Complex a b) = Complex (r * a) (r * b)" |
25712 | 198 |
unfolding complex_scaleR_def by simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
199 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
200 |
lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x" |
25712 | 201 |
unfolding complex_scaleR_def by simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
202 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
203 |
lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x" |
25712 | 204 |
unfolding complex_scaleR_def by simp |
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset
|
205 |
|
25712 | 206 |
instance |
20556
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
207 |
proof |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
208 |
fix a b :: real and x y :: complex |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
209 |
show "scaleR a (x + y) = scaleR a x + scaleR a y" |
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
210 |
by (simp add: complex_eq_iff right_distrib) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
211 |
show "scaleR (a + b) x = scaleR a x + scaleR b x" |
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
212 |
by (simp add: complex_eq_iff left_distrib) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
213 |
show "scaleR a (scaleR b x) = scaleR (a * b) x" |
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
214 |
by (simp add: complex_eq_iff mult_assoc) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
215 |
show "scaleR 1 x = x" |
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
216 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
217 |
show "scaleR a x * y = scaleR a (x * y)" |
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
218 |
by (simp add: complex_eq_iff algebra_simps) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
219 |
show "x * scaleR a y = scaleR a (x * y)" |
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
220 |
by (simp add: complex_eq_iff algebra_simps) |
20556
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
221 |
qed |
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
222 |
|
25712 | 223 |
end |
224 |
||
20556
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
225 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
226 |
subsection{* Properties of Embedding from Reals *} |
14323 | 227 |
|
44724 | 228 |
abbreviation complex_of_real :: "real \<Rightarrow> complex" |
229 |
where "complex_of_real \<equiv> of_real" |
|
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
230 |
|
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
231 |
lemma complex_of_real_def: "complex_of_real r = Complex r 0" |
44724 | 232 |
by (simp add: of_real_def complex_scaleR_def) |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
233 |
|
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
234 |
lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" |
44724 | 235 |
by (simp add: complex_of_real_def) |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
236 |
|
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
237 |
lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" |
44724 | 238 |
by (simp add: complex_of_real_def) |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
239 |
|
14377 | 240 |
lemma Complex_add_complex_of_real [simp]: |
44724 | 241 |
shows "Complex x y + complex_of_real r = Complex (x+r) y" |
242 |
by (simp add: complex_of_real_def) |
|
14377 | 243 |
|
244 |
lemma complex_of_real_add_Complex [simp]: |
|
44724 | 245 |
shows "complex_of_real r + Complex x y = Complex (r+x) y" |
246 |
by (simp add: complex_of_real_def) |
|
14377 | 247 |
|
248 |
lemma Complex_mult_complex_of_real: |
|
44724 | 249 |
shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)" |
250 |
by (simp add: complex_of_real_def) |
|
14377 | 251 |
|
252 |
lemma complex_of_real_mult_Complex: |
|
44724 | 253 |
shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)" |
254 |
by (simp add: complex_of_real_def) |
|
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
255 |
|
14377 | 256 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
257 |
subsection {* Vector Norm *} |
14323 | 258 |
|
25712 | 259 |
instantiation complex :: real_normed_field |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
260 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
261 |
|
31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset
|
262 |
definition complex_norm_def: |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset
|
263 |
"norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
264 |
|
44724 | 265 |
abbreviation cmod :: "complex \<Rightarrow> real" |
266 |
where "cmod \<equiv> norm" |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
267 |
|
31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset
|
268 |
definition complex_sgn_def: |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset
|
269 |
"sgn x = x /\<^sub>R cmod x" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
270 |
|
31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset
|
271 |
definition dist_complex_def: |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset
|
272 |
"dist x y = cmod (x - y)" |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset
|
273 |
|
37767 | 274 |
definition open_complex_def: |
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31419
diff
changeset
|
275 |
"open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" |
31292 | 276 |
|
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
277 |
lemmas cmod_def = complex_norm_def |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
278 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
279 |
lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)" |
25712 | 280 |
by (simp add: complex_norm_def) |
22852 | 281 |
|
31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31292
diff
changeset
|
282 |
instance proof |
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31419
diff
changeset
|
283 |
fix r :: real and x y :: complex and S :: "complex set" |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
284 |
show "0 \<le> norm x" |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
285 |
by (induct x) simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
286 |
show "(norm x = 0) = (x = 0)" |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
287 |
by (induct x) simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
288 |
show "norm (x + y) \<le> norm x + norm y" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
289 |
by (induct x, induct y) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
290 |
(simp add: real_sqrt_sum_squares_triangle_ineq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
291 |
show "norm (scaleR r x) = \<bar>r\<bar> * norm x" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
292 |
by (induct x) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
293 |
(simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
294 |
show "norm (x * y) = norm x * norm y" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
295 |
by (induct x, induct y) |
29667 | 296 |
(simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps) |
31292 | 297 |
show "sgn x = x /\<^sub>R cmod x" |
298 |
by (rule complex_sgn_def) |
|
299 |
show "dist x y = cmod (x - y)" |
|
300 |
by (rule dist_complex_def) |
|
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31419
diff
changeset
|
301 |
show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31419
diff
changeset
|
302 |
by (rule open_complex_def) |
24520 | 303 |
qed |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
304 |
|
25712 | 305 |
end |
306 |
||
44761 | 307 |
lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1" |
44724 | 308 |
by simp |
14323 | 309 |
|
44761 | 310 |
lemma cmod_complex_polar: |
44724 | 311 |
"cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" |
312 |
by (simp add: norm_mult) |
|
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
313 |
|
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
314 |
lemma complex_Re_le_cmod: "Re x \<le> cmod x" |
44724 | 315 |
unfolding complex_norm_def |
316 |
by (rule real_sqrt_sum_squares_ge1) |
|
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
317 |
|
44761 | 318 |
lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x" |
44724 | 319 |
by (rule order_trans [OF _ norm_ge_zero], simp) |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
320 |
|
44761 | 321 |
lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \<le> cmod a" |
44724 | 322 |
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) |
14323 | 323 |
|
26117 | 324 |
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x" |
44724 | 325 |
by (cases x) simp |
26117 | 326 |
|
327 |
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x" |
|
44724 | 328 |
by (cases x) simp |
329 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
330 |
|
23123 | 331 |
subsection {* Completeness of the Complexes *} |
332 |
||
44290
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
333 |
lemma bounded_linear_Re: "bounded_linear Re" |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
334 |
by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
335 |
|
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
336 |
lemma bounded_linear_Im: "bounded_linear Im" |
44127 | 337 |
by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def) |
23123 | 338 |
|
44290
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
339 |
lemmas tendsto_Re [tendsto_intros] = |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
340 |
bounded_linear.tendsto [OF bounded_linear_Re] |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
341 |
|
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
342 |
lemmas tendsto_Im [tendsto_intros] = |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
343 |
bounded_linear.tendsto [OF bounded_linear_Im] |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
344 |
|
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
345 |
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re] |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
346 |
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im] |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
347 |
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re] |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
348 |
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im] |
23123 | 349 |
|
36825 | 350 |
lemma tendsto_Complex [tendsto_intros]: |
44724 | 351 |
assumes "(f ---> a) F" and "(g ---> b) F" |
352 |
shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F" |
|
36825 | 353 |
proof (rule tendstoI) |
354 |
fix r :: real assume "0 < r" |
|
355 |
hence "0 < r / sqrt 2" by (simp add: divide_pos_pos) |
|
44724 | 356 |
have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F" |
357 |
using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD) |
|
36825 | 358 |
moreover |
44724 | 359 |
have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F" |
360 |
using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD) |
|
36825 | 361 |
ultimately |
44724 | 362 |
show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F" |
36825 | 363 |
by (rule eventually_elim2) |
364 |
(simp add: dist_norm real_sqrt_sum_squares_less) |
|
365 |
qed |
|
366 |
||
23123 | 367 |
instance complex :: banach |
368 |
proof |
|
369 |
fix X :: "nat \<Rightarrow> complex" |
|
370 |
assume X: "Cauchy X" |
|
44290
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
371 |
from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))" |
23123 | 372 |
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) |
44290
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
373 |
from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))" |
23123 | 374 |
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) |
375 |
have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" |
|
44748
7f6838b3474a
remove redundant lemma LIMSEQ_Complex in favor of tendsto_Complex
huffman
parents:
44724
diff
changeset
|
376 |
using tendsto_Complex [OF 1 2] by simp |
23123 | 377 |
thus "convergent X" |
378 |
by (rule convergentI) |
|
379 |
qed |
|
380 |
||
381 |
||
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
382 |
subsection {* The Complex Number @{term "\<i>"} *} |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
383 |
|
44724 | 384 |
definition "ii" :: complex ("\<i>") |
385 |
where i_def: "ii \<equiv> Complex 0 1" |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
386 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
387 |
lemma complex_Re_i [simp]: "Re ii = 0" |
44724 | 388 |
by (simp add: i_def) |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
389 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
390 |
lemma complex_Im_i [simp]: "Im ii = 1" |
44724 | 391 |
by (simp add: i_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
392 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
393 |
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)" |
44724 | 394 |
by (simp add: i_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
395 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
396 |
lemma complex_i_not_zero [simp]: "ii \<noteq> 0" |
44724 | 397 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
398 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
399 |
lemma complex_i_not_one [simp]: "ii \<noteq> 1" |
44724 | 400 |
by (simp add: complex_eq_iff) |
23124 | 401 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
402 |
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w" |
44724 | 403 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
404 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
405 |
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a" |
44724 | 406 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
407 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
408 |
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a" |
44724 | 409 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
410 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
411 |
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" |
44724 | 412 |
by (simp add: i_def complex_of_real_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
413 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
414 |
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" |
44724 | 415 |
by (simp add: i_def complex_of_real_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
416 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
417 |
lemma i_squared [simp]: "ii * ii = -1" |
44724 | 418 |
by (simp add: i_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
419 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
420 |
lemma power2_i [simp]: "ii\<twosuperior> = -1" |
44724 | 421 |
by (simp add: power2_eq_square) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
422 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
423 |
lemma inverse_i [simp]: "inverse ii = - ii" |
44724 | 424 |
by (rule inverse_unique, simp) |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
425 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
426 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
427 |
subsection {* Complex Conjugation *} |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
428 |
|
44724 | 429 |
definition cnj :: "complex \<Rightarrow> complex" where |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
430 |
"cnj z = Complex (Re z) (- Im z)" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
431 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
432 |
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)" |
44724 | 433 |
by (simp add: cnj_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
434 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
435 |
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" |
44724 | 436 |
by (simp add: cnj_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
437 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
438 |
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x" |
44724 | 439 |
by (simp add: cnj_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
440 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
441 |
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" |
44724 | 442 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
443 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
444 |
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" |
44724 | 445 |
by (simp add: cnj_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
446 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
447 |
lemma complex_cnj_zero [simp]: "cnj 0 = 0" |
44724 | 448 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
449 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
450 |
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" |
44724 | 451 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
452 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
453 |
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" |
44724 | 454 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
455 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
456 |
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y" |
44724 | 457 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
458 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
459 |
lemma complex_cnj_minus: "cnj (- x) = - cnj x" |
44724 | 460 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
461 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
462 |
lemma complex_cnj_one [simp]: "cnj 1 = 1" |
44724 | 463 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
464 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
465 |
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" |
44724 | 466 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
467 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
468 |
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" |
44724 | 469 |
by (simp add: complex_inverse_def) |
14323 | 470 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
471 |
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y" |
44724 | 472 |
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
473 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
474 |
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" |
44724 | 475 |
by (induct n, simp_all add: complex_cnj_mult) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
476 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
477 |
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" |
44724 | 478 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
479 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
480 |
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" |
44724 | 481 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
482 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
483 |
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w" |
44724 | 484 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
485 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
486 |
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" |
44724 | 487 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
488 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
489 |
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" |
44724 | 490 |
by (simp add: complex_norm_def) |
14323 | 491 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
492 |
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" |
44724 | 493 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
494 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
495 |
lemma complex_cnj_i [simp]: "cnj ii = - ii" |
44724 | 496 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
497 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
498 |
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" |
44724 | 499 |
by (simp add: complex_eq_iff) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
500 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
501 |
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii" |
44724 | 502 |
by (simp add: complex_eq_iff) |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
503 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
504 |
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" |
44724 | 505 |
by (simp add: complex_eq_iff power2_eq_square) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
506 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
507 |
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>" |
44724 | 508 |
by (simp add: norm_mult power2_eq_square) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
509 |
|
44290
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
510 |
lemma bounded_linear_cnj: "bounded_linear cnj" |
44127 | 511 |
using complex_cnj_add complex_cnj_scaleR |
512 |
by (rule bounded_linear_intro [where K=1], simp) |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
513 |
|
44290
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
514 |
lemmas tendsto_cnj [tendsto_intros] = |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
515 |
bounded_linear.tendsto [OF bounded_linear_cnj] |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
516 |
|
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
517 |
lemmas isCont_cnj [simp] = |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
518 |
bounded_linear.isCont [OF bounded_linear_cnj] |
23a5137162ea
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
huffman
parents:
44127
diff
changeset
|
519 |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
520 |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset
|
521 |
subsection{*The Functions @{term sgn} and @{term arg}*} |
14323 | 522 |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset
|
523 |
text {*------------ Argand -------------*} |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
524 |
|
44724 | 525 |
definition arg :: "complex => real" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
526 |
"arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
527 |
|
14374 | 528 |
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" |
44724 | 529 |
by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute) |
14323 | 530 |
|
14374 | 531 |
lemma complex_eq_cancel_iff2 [simp]: |
44724 | 532 |
shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)" |
533 |
by (simp add: complex_of_real_def) |
|
14323 | 534 |
|
14374 | 535 |
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" |
44724 | 536 |
by (simp add: complex_sgn_def divide_inverse) |
14323 | 537 |
|
14374 | 538 |
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" |
44724 | 539 |
by (simp add: complex_sgn_def divide_inverse) |
14323 | 540 |
|
541 |
lemma complex_inverse_complex_split: |
|
542 |
"inverse(complex_of_real x + ii * complex_of_real y) = |
|
543 |
complex_of_real(x/(x ^ 2 + y ^ 2)) - |
|
544 |
ii * complex_of_real(y/(x ^ 2 + y ^ 2))" |
|
44724 | 545 |
by (simp add: complex_of_real_def i_def diff_minus divide_inverse) |
14323 | 546 |
|
547 |
(*----------------------------------------------------------------------------*) |
|
548 |
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) |
|
549 |
(* many of the theorems are not used - so should they be kept? *) |
|
550 |
(*----------------------------------------------------------------------------*) |
|
551 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
552 |
lemma cos_arg_i_mult_zero_pos: |
14377 | 553 |
"0 < y ==> cos (arg(Complex 0 y)) = 0" |
14373 | 554 |
apply (simp add: arg_def abs_if) |
14334 | 555 |
apply (rule_tac a = "pi/2" in someI2, auto) |
556 |
apply (rule order_less_trans [of _ 0], auto) |
|
14323 | 557 |
done |
558 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
559 |
lemma cos_arg_i_mult_zero_neg: |
14377 | 560 |
"y < 0 ==> cos (arg(Complex 0 y)) = 0" |
14373 | 561 |
apply (simp add: arg_def abs_if) |
14334 | 562 |
apply (rule_tac a = "- pi/2" in someI2, auto) |
563 |
apply (rule order_trans [of _ 0], auto) |
|
14323 | 564 |
done |
565 |
||
14374 | 566 |
lemma cos_arg_i_mult_zero [simp]: |
14377 | 567 |
"y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" |
568 |
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) |
|
14323 | 569 |
|
570 |
||
571 |
subsection{*Finally! Polar Form for Complex Numbers*} |
|
572 |
||
44715 | 573 |
text {* An abbreviation for @{text "cos a + i sin a"}. *} |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
574 |
|
44715 | 575 |
definition cis :: "real \<Rightarrow> complex" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
576 |
"cis a = Complex (cos a) (sin a)" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
577 |
|
44715 | 578 |
text {* An abbreviation for @{text "r(cos a + i sin a)"}. *} |
579 |
||
580 |
definition rcis :: "[real, real] \<Rightarrow> complex" where |
|
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
581 |
"rcis r a = complex_of_real r * cis a" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
582 |
|
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
583 |
abbreviation expi :: "complex \<Rightarrow> complex" |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
584 |
where "expi \<equiv> exp" |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
585 |
|
44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset
|
586 |
lemma cis_conv_exp: "cis b = exp (Complex 0 b)" |
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
587 |
proof (rule complex_eqI) |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
588 |
{ fix n have "Complex 0 b ^ n = |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
589 |
real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)" |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
590 |
apply (induct n) |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
591 |
apply (simp add: cos_coeff_def sin_coeff_def) |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
592 |
apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc) |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
593 |
done } note * = this |
44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset
|
594 |
show "Re (cis b) = Re (exp (Complex 0 b))" |
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
595 |
unfolding exp_def cis_def cos_def |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
596 |
by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic], |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
597 |
simp add: * mult_assoc [symmetric]) |
44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset
|
598 |
show "Im (cis b) = Im (exp (Complex 0 b))" |
44291
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
599 |
unfolding exp_def cis_def sin_def |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
600 |
by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic], |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
601 |
simp add: * mult_assoc [symmetric]) |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
602 |
qed |
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
603 |
|
dbd9965745fd
define complex exponential 'expi' as abbreviation for 'exp'
huffman
parents:
44290
diff
changeset
|
604 |
lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)" |
44712
1e490e891c88
replace lemma expi_imaginary with reoriented lemma cis_conv_exp
huffman
parents:
44711
diff
changeset
|
605 |
unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
606 |
|
14374 | 607 |
lemma complex_split_polar: |
14377 | 608 |
"\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
609 |
apply (induct z) |
14377 | 610 |
apply (auto simp add: polar_Ex complex_of_real_mult_Complex) |
14323 | 611 |
done |
612 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
613 |
lemma rcis_Ex: "\<exists>r a. z = rcis r a" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
614 |
apply (induct z) |
14377 | 615 |
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) |
14323 | 616 |
done |
617 |
||
14374 | 618 |
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" |
44724 | 619 |
by (simp add: rcis_def cis_def) |
14323 | 620 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
621 |
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" |
44724 | 622 |
by (simp add: rcis_def cis_def) |
14323 | 623 |
|
14374 | 624 |
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" |
44761 | 625 |
by (simp add: rcis_def cis_def norm_mult) |
14323 | 626 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
627 |
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))" |
44724 | 628 |
by (simp add: cmod_def power2_eq_square) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
629 |
|
14374 | 630 |
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0" |
44724 | 631 |
by simp |
14323 | 632 |
|
633 |
lemma cis_rcis_eq: "cis a = rcis 1 a" |
|
44724 | 634 |
by (simp add: rcis_def) |
14323 | 635 |
|
14374 | 636 |
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" |
44724 | 637 |
by (simp add: rcis_def cis_def cos_add sin_add right_distrib |
638 |
right_diff_distrib complex_of_real_def) |
|
14323 | 639 |
|
640 |
lemma cis_mult: "cis a * cis b = cis (a + b)" |
|
44724 | 641 |
by (simp add: cis_rcis_eq rcis_mult) |
14323 | 642 |
|
14374 | 643 |
lemma cis_zero [simp]: "cis 0 = 1" |
44724 | 644 |
by (simp add: cis_def complex_one_def) |
14323 | 645 |
|
14374 | 646 |
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" |
44724 | 647 |
by (simp add: rcis_def) |
14323 | 648 |
|
14374 | 649 |
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" |
44724 | 650 |
by (simp add: rcis_def) |
14323 | 651 |
|
652 |
lemma complex_of_real_minus_one: |
|
653 |
"complex_of_real (-(1::real)) = -(1::complex)" |
|
44724 | 654 |
by (simp add: complex_of_real_def complex_one_def) |
14323 | 655 |
|
14374 | 656 |
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x" |
44724 | 657 |
by (simp add: mult_assoc [symmetric]) |
14323 | 658 |
|
659 |
||
660 |
lemma cis_real_of_nat_Suc_mult: |
|
661 |
"cis (real (Suc n) * a) = cis a * cis (real n * a)" |
|
44724 | 662 |
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) |
14323 | 663 |
|
664 |
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" |
|
665 |
apply (induct_tac "n") |
|
666 |
apply (auto simp add: cis_real_of_nat_Suc_mult) |
|
667 |
done |
|
668 |
||
14374 | 669 |
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" |
44724 | 670 |
by (simp add: rcis_def power_mult_distrib DeMoivre) |
14323 | 671 |
|
14374 | 672 |
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)" |
44724 | 673 |
by (simp add: cis_def complex_inverse_complex_split diff_minus) |
14323 | 674 |
|
675 |
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)" |
|
44724 | 676 |
by (simp add: divide_inverse rcis_def) |
14323 | 677 |
|
678 |
lemma cis_divide: "cis a / cis b = cis (a - b)" |
|
44724 | 679 |
by (simp add: complex_divide_def cis_mult diff_minus) |
14323 | 680 |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
681 |
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)" |
14373 | 682 |
apply (simp add: complex_divide_def) |
683 |
apply (case_tac "r2=0", simp) |
|
37887 | 684 |
apply (simp add: rcis_inverse rcis_mult diff_minus) |
14323 | 685 |
done |
686 |
||
14374 | 687 |
lemma Re_cis [simp]: "Re(cis a) = cos a" |
44724 | 688 |
by (simp add: cis_def) |
14323 | 689 |
|
14374 | 690 |
lemma Im_cis [simp]: "Im(cis a) = sin a" |
44724 | 691 |
by (simp add: cis_def) |
14323 | 692 |
|
693 |
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" |
|
44724 | 694 |
by (auto simp add: DeMoivre) |
14323 | 695 |
|
696 |
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" |
|
44724 | 697 |
by (auto simp add: DeMoivre) |
14323 | 698 |
|
14374 | 699 |
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" |
14373 | 700 |
apply (insert rcis_Ex [of z]) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
701 |
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) |
14334 | 702 |
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) |
14323 | 703 |
done |
704 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
705 |
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" |
44724 | 706 |
by (simp add: expi_def cis_def) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
707 |
|
44065
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
708 |
text {* Legacy theorem names *} |
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
709 |
|
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
710 |
lemmas expand_complex_eq = complex_eq_iff |
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
711 |
lemmas complex_Re_Im_cancel_iff = complex_eq_iff |
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
712 |
lemmas complex_equality = complex_eqI |
eb64ffccfc75
standard theorem naming scheme: complex_eqI, complex_eq_iff
huffman
parents:
41959
diff
changeset
|
713 |
|
13957 | 714 |
end |