author | wenzelm |
Sun, 18 May 2008 15:04:09 +0200 | |
changeset 26939 | 1035c89b4c02 |
parent 26311 | 81a0fc28b0de |
child 28944 | e27abf0db984 |
permissions | -rw-r--r-- |
13957 | 1 |
(* Title: Complex.thy |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
2 |
ID: $Id$ |
13957 | 3 |
Author: Jacques D. Fleuriot |
4 |
Copyright: 2001 University of Edinburgh |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
5 |
Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 |
13957 | 6 |
*) |
7 |
||
14377 | 8 |
header {* Complex Numbers: Rectangular and Polar Representations *} |
14373 | 9 |
|
15131 | 10 |
theory Complex |
25599 | 11 |
imports "../Real/Real" "../Hyperreal/Transcendental" |
15131 | 12 |
begin |
13957 | 13 |
|
14373 | 14 |
datatype complex = Complex real real |
13957 | 15 |
|
25712 | 16 |
primrec |
17 |
Re :: "complex \<Rightarrow> real" |
|
18 |
where |
|
19 |
Re: "Re (Complex x y) = x" |
|
14373 | 20 |
|
25712 | 21 |
primrec |
22 |
Im :: "complex \<Rightarrow> real" |
|
23 |
where |
|
24 |
Im: "Im (Complex x y) = y" |
|
14373 | 25 |
|
26 |
lemma complex_surj [simp]: "Complex (Re z) (Im z) = z" |
|
27 |
by (induct z) simp |
|
13957 | 28 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
29 |
lemma complex_equality [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y" |
25712 | 30 |
by (induct x, induct y) simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
31 |
|
25599 | 32 |
lemma expand_complex_eq: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y" |
25712 | 33 |
by (induct x, induct y) simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
34 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
35 |
lemmas complex_Re_Im_cancel_iff = expand_complex_eq |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
36 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
37 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
38 |
subsection {* Addition and Subtraction *} |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
39 |
|
25599 | 40 |
instantiation complex :: ab_group_add |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
41 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
42 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
43 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
44 |
complex_zero_def: "0 = Complex 0 0" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
45 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
46 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
47 |
complex_add_def: "x + y = Complex (Re x + Re y) (Im x + Im y)" |
23124 | 48 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
49 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
50 |
complex_minus_def: "- x = Complex (- Re x) (- Im x)" |
14323 | 51 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
52 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
53 |
complex_diff_def: "x - (y\<Colon>complex) = x + - y" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
54 |
|
25599 | 55 |
lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" |
56 |
by (simp add: complex_zero_def) |
|
14323 | 57 |
|
14374 | 58 |
lemma complex_Re_zero [simp]: "Re 0 = 0" |
25599 | 59 |
by (simp add: complex_zero_def) |
14374 | 60 |
|
61 |
lemma complex_Im_zero [simp]: "Im 0 = 0" |
|
25599 | 62 |
by (simp add: complex_zero_def) |
63 |
||
25712 | 64 |
lemma complex_add [simp]: |
65 |
"Complex a b + Complex c d = Complex (a + c) (b + d)" |
|
66 |
by (simp add: complex_add_def) |
|
67 |
||
25599 | 68 |
lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y" |
69 |
by (simp add: complex_add_def) |
|
70 |
||
71 |
lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y" |
|
72 |
by (simp add: complex_add_def) |
|
14323 | 73 |
|
25712 | 74 |
lemma complex_minus [simp]: |
75 |
"- (Complex a b) = Complex (- a) (- b)" |
|
25599 | 76 |
by (simp add: complex_minus_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
77 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
78 |
lemma complex_Re_minus [simp]: "Re (- x) = - Re x" |
25599 | 79 |
by (simp add: complex_minus_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_Im_minus [simp]: "Im (- x) = - Im x" |
25599 | 82 |
by (simp add: complex_minus_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
83 |
|
23275 | 84 |
lemma complex_diff [simp]: |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
85 |
"Complex a b - Complex c d = Complex (a - c) (b - d)" |
25599 | 86 |
by (simp add: complex_diff_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
87 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
88 |
lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y" |
25599 | 89 |
by (simp add: complex_diff_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
90 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
91 |
lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y" |
25599 | 92 |
by (simp add: complex_diff_def) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
93 |
|
25712 | 94 |
instance |
95 |
by intro_classes (simp_all add: complex_add_def complex_diff_def) |
|
96 |
||
97 |
end |
|
98 |
||
99 |
||
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
100 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
101 |
subsection {* Multiplication and Division *} |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
102 |
|
25712 | 103 |
instantiation complex :: "{field, division_by_zero}" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
104 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
105 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
106 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
107 |
complex_one_def: "1 = Complex 1 0" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
108 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
109 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
110 |
complex_mult_def: "x * y = |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
111 |
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
|
112 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
113 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
114 |
complex_inverse_def: "inverse x = |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
115 |
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
|
116 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
117 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
118 |
complex_divide_def: "x / (y\<Colon>complex) = x * inverse y" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
119 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
120 |
lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)" |
25712 | 121 |
by (simp add: complex_one_def) |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
122 |
|
14374 | 123 |
lemma complex_Re_one [simp]: "Re 1 = 1" |
25712 | 124 |
by (simp add: complex_one_def) |
14323 | 125 |
|
14374 | 126 |
lemma complex_Im_one [simp]: "Im 1 = 0" |
25712 | 127 |
by (simp add: complex_one_def) |
14323 | 128 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
129 |
lemma complex_mult [simp]: |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
130 |
"Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)" |
25712 | 131 |
by (simp add: complex_mult_def) |
14323 | 132 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
133 |
lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y" |
25712 | 134 |
by (simp add: complex_mult_def) |
14323 | 135 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
136 |
lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y" |
25712 | 137 |
by (simp add: complex_mult_def) |
14323 | 138 |
|
14377 | 139 |
lemma complex_inverse [simp]: |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
140 |
"inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))" |
25712 | 141 |
by (simp add: complex_inverse_def) |
14335 | 142 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
143 |
lemma complex_Re_inverse: |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
144 |
"Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)" |
25712 | 145 |
by (simp add: complex_inverse_def) |
14323 | 146 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
147 |
lemma complex_Im_inverse: |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
148 |
"Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)" |
25712 | 149 |
by (simp add: complex_inverse_def) |
14335 | 150 |
|
25712 | 151 |
instance |
152 |
by intro_classes (simp_all add: complex_mult_def |
|
153 |
right_distrib left_distrib right_diff_distrib left_diff_distrib |
|
154 |
complex_inverse_def complex_divide_def |
|
155 |
power2_eq_square add_divide_distrib [symmetric] |
|
156 |
expand_complex_eq) |
|
14335 | 157 |
|
25712 | 158 |
end |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
159 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
160 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
161 |
subsection {* Exponentiation *} |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
162 |
|
25712 | 163 |
instantiation complex :: recpower |
164 |
begin |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
165 |
|
25712 | 166 |
primrec power_complex where |
167 |
complexpow_0: "z ^ 0 = (1\<Colon>complex)" |
|
168 |
| complexpow_Suc: "z ^ Suc n = (z\<Colon>complex) * z ^ n" |
|
169 |
||
170 |
instance by intro_classes simp_all |
|
171 |
||
172 |
end |
|
14335 | 173 |
|
14323 | 174 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
175 |
subsection {* Numerals and Arithmetic *} |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
176 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
177 |
instantiation complex :: number_ring |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
178 |
begin |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
179 |
|
25712 | 180 |
definition number_of_complex where |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
181 |
complex_number_of_def: "number_of w = (of_int w \<Colon> complex)" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
182 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
183 |
instance |
25712 | 184 |
by intro_classes (simp only: complex_number_of_def) |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
185 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
186 |
end |
23125
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 |
lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
189 |
by (induct n) simp_all |
20556
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
190 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
191 |
lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
192 |
by (induct n) simp_all |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
193 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
194 |
lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
195 |
by (cases z rule: int_diff_cases) simp |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
196 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
197 |
lemma complex_Im_of_int [simp]: "Im (of_int z) = 0" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
198 |
by (cases z rule: int_diff_cases) simp |
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_number_of [simp]: "Re (number_of v) = number_of v" |
25502 | 201 |
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
|
202 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
203 |
lemma complex_Im_number_of [simp]: "Im (number_of v) = 0" |
25502 | 204 |
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
|
205 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
206 |
lemma Complex_eq_number_of [simp]: |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
207 |
"(Complex a b = number_of w) = (a = number_of w \<and> b = 0)" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
208 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
209 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
210 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
211 |
subsection {* Scalar Multiplication *} |
20556
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
212 |
|
25712 | 213 |
instantiation complex :: real_field |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
214 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
215 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
216 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
217 |
complex_scaleR_def: "scaleR r x = Complex (r * Re x) (r * Im x)" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
218 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
219 |
lemma complex_scaleR [simp]: |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
220 |
"scaleR r (Complex a b) = Complex (r * a) (r * b)" |
25712 | 221 |
unfolding complex_scaleR_def by simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
222 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
223 |
lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x" |
25712 | 224 |
unfolding complex_scaleR_def by simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
225 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
226 |
lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x" |
25712 | 227 |
unfolding complex_scaleR_def by simp |
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset
|
228 |
|
25712 | 229 |
instance |
20556
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
230 |
proof |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
231 |
fix a b :: real and x y :: complex |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
232 |
show "scaleR a (x + y) = scaleR a x + scaleR a y" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
233 |
by (simp add: expand_complex_eq right_distrib) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
234 |
show "scaleR (a + b) x = scaleR a x + scaleR b x" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
235 |
by (simp add: expand_complex_eq left_distrib) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
236 |
show "scaleR a (scaleR b x) = scaleR (a * b) x" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
237 |
by (simp add: expand_complex_eq mult_assoc) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
238 |
show "scaleR 1 x = x" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
239 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
240 |
show "scaleR a x * y = scaleR a (x * y)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23275
diff
changeset
|
241 |
by (simp add: expand_complex_eq ring_simps) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
242 |
show "x * scaleR a y = scaleR a (x * y)" |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23275
diff
changeset
|
243 |
by (simp add: expand_complex_eq ring_simps) |
20556
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
244 |
qed |
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
245 |
|
25712 | 246 |
end |
247 |
||
20556
2e8227b81bf1
add instance for real_algebra_1 and real_normed_div_algebra
huffman
parents:
20485
diff
changeset
|
248 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
249 |
subsection{* Properties of Embedding from Reals *} |
14323 | 250 |
|
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
251 |
abbreviation |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
252 |
complex_of_real :: "real \<Rightarrow> complex" where |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
253 |
"complex_of_real \<equiv> of_real" |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
254 |
|
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
255 |
lemma complex_of_real_def: "complex_of_real r = Complex r 0" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
256 |
by (simp add: of_real_def complex_scaleR_def) |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
257 |
|
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
258 |
lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
259 |
by (simp add: complex_of_real_def) |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
260 |
|
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
261 |
lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
262 |
by (simp add: complex_of_real_def) |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
263 |
|
14377 | 264 |
lemma Complex_add_complex_of_real [simp]: |
265 |
"Complex x y + complex_of_real r = Complex (x+r) y" |
|
266 |
by (simp add: complex_of_real_def) |
|
267 |
||
268 |
lemma complex_of_real_add_Complex [simp]: |
|
269 |
"complex_of_real r + Complex x y = Complex (r+x) y" |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
270 |
by (simp add: complex_of_real_def) |
14377 | 271 |
|
272 |
lemma Complex_mult_complex_of_real: |
|
273 |
"Complex x y * complex_of_real r = Complex (x*r) (y*r)" |
|
274 |
by (simp add: complex_of_real_def) |
|
275 |
||
276 |
lemma complex_of_real_mult_Complex: |
|
277 |
"complex_of_real r * Complex x y = Complex (r*x) (r*y)" |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
278 |
by (simp add: complex_of_real_def) |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
279 |
|
14377 | 280 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
281 |
subsection {* Vector Norm *} |
14323 | 282 |
|
25712 | 283 |
instantiation complex :: real_normed_field |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
284 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
285 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
286 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
287 |
complex_norm_def: "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
288 |
|
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
289 |
abbreviation |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
290 |
cmod :: "complex \<Rightarrow> real" where |
25712 | 291 |
"cmod \<equiv> norm" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
292 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
293 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
294 |
complex_sgn_def: "sgn x = x /\<^sub>R cmod x" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
295 |
|
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
296 |
lemmas cmod_def = complex_norm_def |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
297 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
298 |
lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)" |
25712 | 299 |
by (simp add: complex_norm_def) |
22852 | 300 |
|
25712 | 301 |
instance |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
302 |
proof |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
303 |
fix r :: real and x y :: complex |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
304 |
show "0 \<le> norm x" |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
305 |
by (induct x) simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
306 |
show "(norm x = 0) = (x = 0)" |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
307 |
by (induct x) simp |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
308 |
show "norm (x + y) \<le> norm x + norm y" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
309 |
by (induct x, induct y) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
310 |
(simp add: real_sqrt_sum_squares_triangle_ineq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
311 |
show "norm (scaleR r x) = \<bar>r\<bar> * norm x" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
312 |
by (induct x) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
313 |
(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
|
314 |
show "norm (x * y) = norm x * norm y" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
315 |
by (induct x, induct y) |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23275
diff
changeset
|
316 |
(simp add: real_sqrt_mult [symmetric] power2_eq_square ring_simps) |
24506 | 317 |
show "sgn x = x /\<^sub>R cmod x" by(simp add: complex_sgn_def) |
24520 | 318 |
qed |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
319 |
|
25712 | 320 |
end |
321 |
||
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
322 |
lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1" |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
323 |
by simp |
14323 | 324 |
|
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
325 |
lemma cmod_complex_polar [simp]: |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
326 |
"cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r" |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
327 |
by (simp add: norm_mult) |
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
328 |
|
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
329 |
lemma complex_Re_le_cmod: "Re x \<le> cmod x" |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
330 |
unfolding complex_norm_def |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
331 |
by (rule real_sqrt_sum_squares_ge1) |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
332 |
|
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
333 |
lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x" |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
334 |
by (rule order_trans [OF _ norm_ge_zero], simp) |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
335 |
|
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
336 |
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a" |
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
337 |
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp) |
14323 | 338 |
|
22861
8ec47039614e
clean up complex norm proofs, remove redundant lemmas
huffman
parents:
22852
diff
changeset
|
339 |
lemmas real_sum_squared_expand = power2_sum [where 'a=real] |
14323 | 340 |
|
26117 | 341 |
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x" |
342 |
by (cases x) simp |
|
343 |
||
344 |
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x" |
|
345 |
by (cases x) simp |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
346 |
|
23123 | 347 |
subsection {* Completeness of the Complexes *} |
348 |
||
349 |
interpretation Re: bounded_linear ["Re"] |
|
350 |
apply (unfold_locales, simp, simp) |
|
351 |
apply (rule_tac x=1 in exI) |
|
352 |
apply (simp add: complex_norm_def) |
|
353 |
done |
|
354 |
||
355 |
interpretation Im: bounded_linear ["Im"] |
|
356 |
apply (unfold_locales, simp, simp) |
|
357 |
apply (rule_tac x=1 in exI) |
|
358 |
apply (simp add: complex_norm_def) |
|
359 |
done |
|
360 |
||
361 |
lemma LIMSEQ_Complex: |
|
362 |
"\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b" |
|
363 |
apply (rule LIMSEQ_I) |
|
364 |
apply (subgoal_tac "0 < r / sqrt 2") |
|
365 |
apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) |
|
366 |
apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe) |
|
367 |
apply (rename_tac M N, rule_tac x="max M N" in exI, safe) |
|
368 |
apply (simp add: real_sqrt_sum_squares_less) |
|
369 |
apply (simp add: divide_pos_pos) |
|
370 |
done |
|
371 |
||
372 |
instance complex :: banach |
|
373 |
proof |
|
374 |
fix X :: "nat \<Rightarrow> complex" |
|
375 |
assume X: "Cauchy X" |
|
376 |
from Re.Cauchy [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))" |
|
377 |
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) |
|
378 |
from Im.Cauchy [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))" |
|
379 |
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) |
|
380 |
have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))" |
|
381 |
using LIMSEQ_Complex [OF 1 2] by simp |
|
382 |
thus "convergent X" |
|
383 |
by (rule convergentI) |
|
384 |
qed |
|
385 |
||
386 |
||
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
387 |
subsection {* The Complex Number @{term "\<i>"} *} |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
388 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
389 |
definition |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
390 |
"ii" :: complex ("\<i>") where |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
391 |
i_def: "ii \<equiv> Complex 0 1" |
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_Re_i [simp]: "Re ii = 0" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
394 |
by (simp add: i_def) |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
395 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
396 |
lemma complex_Im_i [simp]: "Im ii = 1" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
397 |
by (simp add: i_def) |
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_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
400 |
by (simp add: i_def) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
401 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
402 |
lemma complex_i_not_zero [simp]: "ii \<noteq> 0" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
403 |
by (simp add: expand_complex_eq) |
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 complex_i_not_one [simp]: "ii \<noteq> 1" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
406 |
by (simp add: expand_complex_eq) |
23124 | 407 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
408 |
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
409 |
by (simp add: expand_complex_eq) |
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_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
412 |
by (simp add: expand_complex_eq) |
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_mult_i [simp]: "Complex a b * ii = Complex (- b) a" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
415 |
by (simp add: expand_complex_eq) |
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_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
418 |
by (simp add: i_def complex_of_real_def) |
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 complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
421 |
by (simp add: i_def complex_of_real_def) |
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 i_squared [simp]: "ii * ii = -1" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
424 |
by (simp add: i_def) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
425 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
426 |
lemma power2_i [simp]: "ii\<twosuperior> = -1" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
427 |
by (simp add: power2_eq_square) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
428 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
429 |
lemma inverse_i [simp]: "inverse ii = - ii" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
430 |
by (rule inverse_unique, simp) |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
431 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
432 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
433 |
subsection {* Complex Conjugation *} |
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 |
definition |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
436 |
cnj :: "complex \<Rightarrow> complex" where |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
437 |
"cnj z = Complex (Re z) (- Im z)" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
438 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
439 |
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
440 |
by (simp add: cnj_def) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
441 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
442 |
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
443 |
by (simp add: cnj_def) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
444 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
445 |
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
446 |
by (simp add: cnj_def) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
447 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
448 |
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
449 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
450 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
451 |
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
452 |
by (simp add: cnj_def) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
453 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
454 |
lemma complex_cnj_zero [simp]: "cnj 0 = 0" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
455 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
456 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
457 |
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
458 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
459 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
460 |
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
461 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
462 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
463 |
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
464 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
465 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
466 |
lemma complex_cnj_minus: "cnj (- x) = - cnj x" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
467 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
468 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
469 |
lemma complex_cnj_one [simp]: "cnj 1 = 1" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
470 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
471 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
472 |
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
473 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
474 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
475 |
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
476 |
by (simp add: complex_inverse_def) |
14323 | 477 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
478 |
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
479 |
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
480 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
481 |
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
482 |
by (induct n, simp_all add: complex_cnj_mult) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
483 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
484 |
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
485 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
486 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
487 |
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
488 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
489 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
490 |
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
491 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
492 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
493 |
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
494 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
495 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
496 |
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
497 |
by (simp add: complex_norm_def) |
14323 | 498 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
499 |
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
500 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
501 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
502 |
lemma complex_cnj_i [simp]: "cnj ii = - ii" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
503 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
504 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
505 |
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
506 |
by (simp add: expand_complex_eq) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
507 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
508 |
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
509 |
by (simp add: expand_complex_eq) |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
510 |
|
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
511 |
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
512 |
by (simp add: expand_complex_eq power2_eq_square) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
513 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
514 |
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
515 |
by (simp add: norm_mult power2_eq_square) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
516 |
|
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
517 |
interpretation cnj: bounded_linear ["cnj"] |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
518 |
apply (unfold_locales) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
519 |
apply (rule complex_cnj_add) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
520 |
apply (rule complex_cnj_scaleR) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
521 |
apply (rule_tac x=1 in exI, simp) |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
522 |
done |
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
523 |
|
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
524 |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset
|
525 |
subsection{*The Functions @{term sgn} and @{term arg}*} |
14323 | 526 |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22968
diff
changeset
|
527 |
text {*------------ Argand -------------*} |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
528 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
529 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
530 |
arg :: "complex => real" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
531 |
"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
|
532 |
|
14374 | 533 |
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)" |
24506 | 534 |
by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute) |
14323 | 535 |
|
536 |
lemma i_mult_eq: "ii * ii = complex_of_real (-1)" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
537 |
by (simp add: i_def complex_of_real_def) |
14323 | 538 |
|
14374 | 539 |
lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
540 |
by (simp add: i_def complex_one_def) |
14323 | 541 |
|
14374 | 542 |
lemma complex_eq_cancel_iff2 [simp]: |
14377 | 543 |
"(Complex x y = complex_of_real xa) = (x = xa & y = 0)" |
544 |
by (simp add: complex_of_real_def) |
|
14323 | 545 |
|
14374 | 546 |
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z" |
24506 | 547 |
by (simp add: complex_sgn_def divide_inverse) |
14323 | 548 |
|
14374 | 549 |
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z" |
24506 | 550 |
by (simp add: complex_sgn_def divide_inverse) |
14323 | 551 |
|
552 |
lemma complex_inverse_complex_split: |
|
553 |
"inverse(complex_of_real x + ii * complex_of_real y) = |
|
554 |
complex_of_real(x/(x ^ 2 + y ^ 2)) - |
|
555 |
ii * complex_of_real(y/(x ^ 2 + y ^ 2))" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
556 |
by (simp add: complex_of_real_def i_def diff_minus divide_inverse) |
14323 | 557 |
|
558 |
(*----------------------------------------------------------------------------*) |
|
559 |
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *) |
|
560 |
(* many of the theorems are not used - so should they be kept? *) |
|
561 |
(*----------------------------------------------------------------------------*) |
|
562 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
563 |
lemma cos_arg_i_mult_zero_pos: |
14377 | 564 |
"0 < y ==> cos (arg(Complex 0 y)) = 0" |
14373 | 565 |
apply (simp add: arg_def abs_if) |
14334 | 566 |
apply (rule_tac a = "pi/2" in someI2, auto) |
567 |
apply (rule order_less_trans [of _ 0], auto) |
|
14323 | 568 |
done |
569 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
570 |
lemma cos_arg_i_mult_zero_neg: |
14377 | 571 |
"y < 0 ==> cos (arg(Complex 0 y)) = 0" |
14373 | 572 |
apply (simp add: arg_def abs_if) |
14334 | 573 |
apply (rule_tac a = "- pi/2" in someI2, auto) |
574 |
apply (rule order_trans [of _ 0], auto) |
|
14323 | 575 |
done |
576 |
||
14374 | 577 |
lemma cos_arg_i_mult_zero [simp]: |
14377 | 578 |
"y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0" |
579 |
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg) |
|
14323 | 580 |
|
581 |
||
582 |
subsection{*Finally! Polar Form for Complex Numbers*} |
|
583 |
||
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
584 |
definition |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
585 |
|
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
586 |
(* abbreviation for (cos a + i sin a) *) |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
587 |
cis :: "real => complex" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
588 |
"cis a = Complex (cos a) (sin a)" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
589 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
590 |
definition |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
591 |
(* abbreviation for r*(cos a + i sin a) *) |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
592 |
rcis :: "[real, real] => complex" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
593 |
"rcis r a = complex_of_real r * cis a" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
594 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
595 |
definition |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
596 |
(* e ^ (x + iy) *) |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20763
diff
changeset
|
597 |
expi :: "complex => complex" where |
20557
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
598 |
"expi z = complex_of_real(exp (Re z)) * cis (Im z)" |
81dd3679f92c
complex_of_real abbreviates of_real::real=>complex;
huffman
parents:
20556
diff
changeset
|
599 |
|
14374 | 600 |
lemma complex_split_polar: |
14377 | 601 |
"\<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
|
602 |
apply (induct z) |
14377 | 603 |
apply (auto simp add: polar_Ex complex_of_real_mult_Complex) |
14323 | 604 |
done |
605 |
||
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
606 |
lemma rcis_Ex: "\<exists>r a. z = rcis r a" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
607 |
apply (induct z) |
14377 | 608 |
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex) |
14323 | 609 |
done |
610 |
||
14374 | 611 |
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a" |
14373 | 612 |
by (simp add: rcis_def cis_def) |
14323 | 613 |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
614 |
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a" |
14373 | 615 |
by (simp add: rcis_def cis_def) |
14323 | 616 |
|
14377 | 617 |
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>" |
618 |
proof - |
|
619 |
have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
620 |
by (simp only: power_mult_distrib right_distrib) |
14377 | 621 |
thus ?thesis by simp |
622 |
qed |
|
14323 | 623 |
|
14374 | 624 |
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r" |
14377 | 625 |
by (simp add: rcis_def cis_def sin_cos_squared_add2_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))" |
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
628 |
by (simp add: cmod_def power2_eq_square) |
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" |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
631 |
by simp |
14323 | 632 |
|
633 |
||
634 |
(*---------------------------------------------------------------------------*) |
|
635 |
(* (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b) *) |
|
636 |
(*---------------------------------------------------------------------------*) |
|
637 |
||
638 |
lemma cis_rcis_eq: "cis a = rcis 1 a" |
|
14373 | 639 |
by (simp add: rcis_def) |
14323 | 640 |
|
14374 | 641 |
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)" |
15013 | 642 |
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib |
643 |
complex_of_real_def) |
|
14323 | 644 |
|
645 |
lemma cis_mult: "cis a * cis b = cis (a + b)" |
|
14373 | 646 |
by (simp add: cis_rcis_eq rcis_mult) |
14323 | 647 |
|
14374 | 648 |
lemma cis_zero [simp]: "cis 0 = 1" |
14377 | 649 |
by (simp add: cis_def complex_one_def) |
14323 | 650 |
|
14374 | 651 |
lemma rcis_zero_mod [simp]: "rcis 0 a = 0" |
14373 | 652 |
by (simp add: rcis_def) |
14323 | 653 |
|
14374 | 654 |
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r" |
14373 | 655 |
by (simp add: rcis_def) |
14323 | 656 |
|
657 |
lemma complex_of_real_minus_one: |
|
658 |
"complex_of_real (-(1::real)) = -(1::complex)" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
659 |
by (simp add: complex_of_real_def complex_one_def) |
14323 | 660 |
|
14374 | 661 |
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x" |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
662 |
by (simp add: mult_assoc [symmetric]) |
14323 | 663 |
|
664 |
||
665 |
lemma cis_real_of_nat_Suc_mult: |
|
666 |
"cis (real (Suc n) * a) = cis a * cis (real n * a)" |
|
14377 | 667 |
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib) |
14323 | 668 |
|
669 |
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)" |
|
670 |
apply (induct_tac "n") |
|
671 |
apply (auto simp add: cis_real_of_nat_Suc_mult) |
|
672 |
done |
|
673 |
||
14374 | 674 |
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)" |
22890 | 675 |
by (simp add: rcis_def power_mult_distrib DeMoivre) |
14323 | 676 |
|
14374 | 677 |
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)" |
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
678 |
by (simp add: cis_def complex_inverse_complex_split diff_minus) |
14323 | 679 |
|
680 |
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)" |
|
22884 | 681 |
by (simp add: divide_inverse rcis_def) |
14323 | 682 |
|
683 |
lemma cis_divide: "cis a / cis b = cis (a - b)" |
|
14373 | 684 |
by (simp add: complex_divide_def cis_mult real_diff_def) |
14323 | 685 |
|
14354
988aa4648597
types complex and hcomplex are now instances of class ringpower:
paulson
parents:
14353
diff
changeset
|
686 |
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)" |
14373 | 687 |
apply (simp add: complex_divide_def) |
688 |
apply (case_tac "r2=0", simp) |
|
689 |
apply (simp add: rcis_inverse rcis_mult real_diff_def) |
|
14323 | 690 |
done |
691 |
||
14374 | 692 |
lemma Re_cis [simp]: "Re(cis a) = cos a" |
14373 | 693 |
by (simp add: cis_def) |
14323 | 694 |
|
14374 | 695 |
lemma Im_cis [simp]: "Im(cis a) = sin a" |
14373 | 696 |
by (simp add: cis_def) |
14323 | 697 |
|
698 |
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)" |
|
14334 | 699 |
by (auto simp add: DeMoivre) |
14323 | 700 |
|
701 |
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)" |
|
14334 | 702 |
by (auto simp add: DeMoivre) |
14323 | 703 |
|
704 |
lemma expi_add: "expi(a + b) = expi(a) * expi(b)" |
|
20725
72e20198f834
instance complex :: real_normed_field; cleaned up
huffman
parents:
20560
diff
changeset
|
705 |
by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac) |
14323 | 706 |
|
14374 | 707 |
lemma expi_zero [simp]: "expi (0::complex) = 1" |
14373 | 708 |
by (simp add: expi_def) |
14323 | 709 |
|
14374 | 710 |
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a" |
14373 | 711 |
apply (insert rcis_Ex [of z]) |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
712 |
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric]) |
14334 | 713 |
apply (rule_tac x = "ii * complex_of_real a" in exI, auto) |
14323 | 714 |
done |
715 |
||
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
716 |
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1" |
23125
6f7b5b96241f
cleaned up proofs; reorganized sections; removed redundant lemmas
huffman
parents:
23124
diff
changeset
|
717 |
by (simp add: expi_def cis_def) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset
|
718 |
|
13957 | 719 |
end |