src/HOL/Complex.thy
author huffman
Wed Sep 07 22:44:26 2011 -0700 (2011-09-07)
changeset 44828 3d6a79e0e1d0
parent 44827 4d1384a1fc82
child 44841 e55503200061
permissions -rw-r--r--
add some new lemmas about cis and rcis;
simplify some proofs;
wenzelm@41959
     1
(*  Title:       HOL/Complex.thy
paulson@13957
     2
    Author:      Jacques D. Fleuriot
paulson@13957
     3
    Copyright:   2001 University of Edinburgh
paulson@14387
     4
    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
paulson@13957
     5
*)
paulson@13957
     6
paulson@14377
     7
header {* Complex Numbers: Rectangular and Polar Representations *}
paulson@14373
     8
nipkow@15131
     9
theory Complex
haftmann@28952
    10
imports Transcendental
nipkow@15131
    11
begin
paulson@13957
    12
paulson@14373
    13
datatype complex = Complex real real
paulson@13957
    14
huffman@44724
    15
primrec Re :: "complex \<Rightarrow> real"
huffman@44724
    16
  where Re: "Re (Complex x y) = x"
paulson@14373
    17
huffman@44724
    18
primrec Im :: "complex \<Rightarrow> real"
huffman@44724
    19
  where Im: "Im (Complex x y) = y"
paulson@14373
    20
paulson@14373
    21
lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
paulson@14373
    22
  by (induct z) simp
paulson@13957
    23
huffman@44065
    24
lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
haftmann@25712
    25
  by (induct x, induct y) simp
huffman@23125
    26
huffman@44065
    27
lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
haftmann@25712
    28
  by (induct x, induct y) simp
huffman@23125
    29
huffman@23125
    30
huffman@23125
    31
subsection {* Addition and Subtraction *}
huffman@23125
    32
haftmann@25599
    33
instantiation complex :: ab_group_add
haftmann@25571
    34
begin
haftmann@25571
    35
huffman@44724
    36
definition complex_zero_def:
huffman@44724
    37
  "0 = Complex 0 0"
haftmann@25571
    38
huffman@44724
    39
definition complex_add_def:
huffman@44724
    40
  "x + y = Complex (Re x + Re y) (Im x + Im y)"
huffman@23124
    41
huffman@44724
    42
definition complex_minus_def:
huffman@44724
    43
  "- x = Complex (- Re x) (- Im x)"
paulson@14323
    44
huffman@44724
    45
definition complex_diff_def:
huffman@44724
    46
  "x - (y\<Colon>complex) = x + - y"
haftmann@25571
    47
haftmann@25599
    48
lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
haftmann@25599
    49
  by (simp add: complex_zero_def)
paulson@14323
    50
paulson@14374
    51
lemma complex_Re_zero [simp]: "Re 0 = 0"
haftmann@25599
    52
  by (simp add: complex_zero_def)
paulson@14374
    53
paulson@14374
    54
lemma complex_Im_zero [simp]: "Im 0 = 0"
haftmann@25599
    55
  by (simp add: complex_zero_def)
haftmann@25599
    56
haftmann@25712
    57
lemma complex_add [simp]:
haftmann@25712
    58
  "Complex a b + Complex c d = Complex (a + c) (b + d)"
haftmann@25712
    59
  by (simp add: complex_add_def)
haftmann@25712
    60
haftmann@25599
    61
lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"
haftmann@25599
    62
  by (simp add: complex_add_def)
haftmann@25599
    63
haftmann@25599
    64
lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"
haftmann@25599
    65
  by (simp add: complex_add_def)
paulson@14323
    66
haftmann@25712
    67
lemma complex_minus [simp]:
haftmann@25712
    68
  "- (Complex a b) = Complex (- a) (- b)"
haftmann@25599
    69
  by (simp add: complex_minus_def)
huffman@23125
    70
huffman@23125
    71
lemma complex_Re_minus [simp]: "Re (- x) = - Re x"
haftmann@25599
    72
  by (simp add: complex_minus_def)
huffman@23125
    73
huffman@23125
    74
lemma complex_Im_minus [simp]: "Im (- x) = - Im x"
haftmann@25599
    75
  by (simp add: complex_minus_def)
huffman@23125
    76
huffman@23275
    77
lemma complex_diff [simp]:
huffman@23125
    78
  "Complex a b - Complex c d = Complex (a - c) (b - d)"
haftmann@25599
    79
  by (simp add: complex_diff_def)
huffman@23125
    80
huffman@23125
    81
lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"
haftmann@25599
    82
  by (simp add: complex_diff_def)
huffman@23125
    83
huffman@23125
    84
lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"
haftmann@25599
    85
  by (simp add: complex_diff_def)
huffman@23125
    86
haftmann@25712
    87
instance
haftmann@25712
    88
  by intro_classes (simp_all add: complex_add_def complex_diff_def)
haftmann@25712
    89
haftmann@25712
    90
end
haftmann@25712
    91
haftmann@25712
    92
huffman@23125
    93
subsection {* Multiplication and Division *}
huffman@23125
    94
haftmann@36409
    95
instantiation complex :: field_inverse_zero
haftmann@25571
    96
begin
haftmann@25571
    97
huffman@44724
    98
definition complex_one_def:
huffman@44724
    99
  "1 = Complex 1 0"
haftmann@25571
   100
huffman@44724
   101
definition complex_mult_def:
huffman@44724
   102
  "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
huffman@23125
   103
huffman@44724
   104
definition complex_inverse_def:
huffman@44724
   105
  "inverse x =
haftmann@25571
   106
    Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"
huffman@23125
   107
huffman@44724
   108
definition complex_divide_def:
huffman@44724
   109
  "x / (y\<Colon>complex) = x * inverse y"
haftmann@25571
   110
huffman@23125
   111
lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
haftmann@25712
   112
  by (simp add: complex_one_def)
huffman@22861
   113
paulson@14374
   114
lemma complex_Re_one [simp]: "Re 1 = 1"
haftmann@25712
   115
  by (simp add: complex_one_def)
paulson@14323
   116
paulson@14374
   117
lemma complex_Im_one [simp]: "Im 1 = 0"
haftmann@25712
   118
  by (simp add: complex_one_def)
paulson@14323
   119
huffman@23125
   120
lemma complex_mult [simp]:
huffman@23125
   121
  "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
haftmann@25712
   122
  by (simp add: complex_mult_def)
paulson@14323
   123
huffman@23125
   124
lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"
haftmann@25712
   125
  by (simp add: complex_mult_def)
paulson@14323
   126
huffman@23125
   127
lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"
haftmann@25712
   128
  by (simp add: complex_mult_def)
paulson@14323
   129
paulson@14377
   130
lemma complex_inverse [simp]:
huffman@23125
   131
  "inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))"
haftmann@25712
   132
  by (simp add: complex_inverse_def)
paulson@14335
   133
huffman@23125
   134
lemma complex_Re_inverse:
huffman@23125
   135
  "Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
haftmann@25712
   136
  by (simp add: complex_inverse_def)
paulson@14323
   137
huffman@23125
   138
lemma complex_Im_inverse:
huffman@23125
   139
  "Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
haftmann@25712
   140
  by (simp add: complex_inverse_def)
paulson@14335
   141
haftmann@25712
   142
instance
haftmann@25712
   143
  by intro_classes (simp_all add: complex_mult_def
huffman@44724
   144
    right_distrib left_distrib right_diff_distrib left_diff_distrib
huffman@44724
   145
    complex_inverse_def complex_divide_def
huffman@44724
   146
    power2_eq_square add_divide_distrib [symmetric]
huffman@44724
   147
    complex_eq_iff)
paulson@14335
   148
haftmann@25712
   149
end
huffman@23125
   150
huffman@23125
   151
huffman@23125
   152
subsection {* Numerals and Arithmetic *}
huffman@23125
   153
haftmann@25571
   154
instantiation complex :: number_ring
haftmann@25571
   155
begin
huffman@23125
   156
huffman@44724
   157
definition complex_number_of_def:
huffman@44724
   158
  "number_of w = (of_int w \<Colon> complex)"
haftmann@25571
   159
haftmann@25571
   160
instance
haftmann@25712
   161
  by intro_classes (simp only: complex_number_of_def)
haftmann@25571
   162
haftmann@25571
   163
end
huffman@23125
   164
huffman@23125
   165
lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
huffman@44724
   166
  by (induct n) simp_all
huffman@20556
   167
huffman@23125
   168
lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
huffman@44724
   169
  by (induct n) simp_all
huffman@23125
   170
huffman@23125
   171
lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
huffman@44724
   172
  by (cases z rule: int_diff_cases) simp
huffman@23125
   173
huffman@23125
   174
lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
huffman@44724
   175
  by (cases z rule: int_diff_cases) simp
huffman@23125
   176
huffman@23125
   177
lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v"
huffman@44724
   178
  unfolding number_of_eq by (rule complex_Re_of_int)
huffman@20556
   179
huffman@23125
   180
lemma complex_Im_number_of [simp]: "Im (number_of v) = 0"
huffman@44724
   181
  unfolding number_of_eq by (rule complex_Im_of_int)
huffman@23125
   182
huffman@23125
   183
lemma Complex_eq_number_of [simp]:
huffman@23125
   184
  "(Complex a b = number_of w) = (a = number_of w \<and> b = 0)"
huffman@44724
   185
  by (simp add: complex_eq_iff)
huffman@23125
   186
huffman@23125
   187
huffman@23125
   188
subsection {* Scalar Multiplication *}
huffman@20556
   189
haftmann@25712
   190
instantiation complex :: real_field
haftmann@25571
   191
begin
haftmann@25571
   192
huffman@44724
   193
definition complex_scaleR_def:
huffman@44724
   194
  "scaleR r x = Complex (r * Re x) (r * Im x)"
haftmann@25571
   195
huffman@23125
   196
lemma complex_scaleR [simp]:
huffman@23125
   197
  "scaleR r (Complex a b) = Complex (r * a) (r * b)"
haftmann@25712
   198
  unfolding complex_scaleR_def by simp
huffman@23125
   199
huffman@23125
   200
lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"
haftmann@25712
   201
  unfolding complex_scaleR_def by simp
huffman@23125
   202
huffman@23125
   203
lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"
haftmann@25712
   204
  unfolding complex_scaleR_def by simp
huffman@22972
   205
haftmann@25712
   206
instance
huffman@20556
   207
proof
huffman@23125
   208
  fix a b :: real and x y :: complex
huffman@23125
   209
  show "scaleR a (x + y) = scaleR a x + scaleR a y"
huffman@44065
   210
    by (simp add: complex_eq_iff right_distrib)
huffman@23125
   211
  show "scaleR (a + b) x = scaleR a x + scaleR b x"
huffman@44065
   212
    by (simp add: complex_eq_iff left_distrib)
huffman@23125
   213
  show "scaleR a (scaleR b x) = scaleR (a * b) x"
huffman@44065
   214
    by (simp add: complex_eq_iff mult_assoc)
huffman@23125
   215
  show "scaleR 1 x = x"
huffman@44065
   216
    by (simp add: complex_eq_iff)
huffman@23125
   217
  show "scaleR a x * y = scaleR a (x * y)"
huffman@44065
   218
    by (simp add: complex_eq_iff algebra_simps)
huffman@23125
   219
  show "x * scaleR a y = scaleR a (x * y)"
huffman@44065
   220
    by (simp add: complex_eq_iff algebra_simps)
huffman@20556
   221
qed
huffman@20556
   222
haftmann@25712
   223
end
haftmann@25712
   224
huffman@20556
   225
huffman@23125
   226
subsection{* Properties of Embedding from Reals *}
paulson@14323
   227
huffman@44724
   228
abbreviation complex_of_real :: "real \<Rightarrow> complex"
huffman@44724
   229
  where "complex_of_real \<equiv> of_real"
huffman@20557
   230
huffman@20557
   231
lemma complex_of_real_def: "complex_of_real r = Complex r 0"
huffman@44724
   232
  by (simp add: of_real_def complex_scaleR_def)
huffman@20557
   233
huffman@20557
   234
lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
huffman@44724
   235
  by (simp add: complex_of_real_def)
huffman@20557
   236
huffman@20557
   237
lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
huffman@44724
   238
  by (simp add: complex_of_real_def)
huffman@20557
   239
paulson@14377
   240
lemma Complex_add_complex_of_real [simp]:
huffman@44724
   241
  shows "Complex x y + complex_of_real r = Complex (x+r) y"
huffman@44724
   242
  by (simp add: complex_of_real_def)
paulson@14377
   243
paulson@14377
   244
lemma complex_of_real_add_Complex [simp]:
huffman@44724
   245
  shows "complex_of_real r + Complex x y = Complex (r+x) y"
huffman@44724
   246
  by (simp add: complex_of_real_def)
paulson@14377
   247
paulson@14377
   248
lemma Complex_mult_complex_of_real:
huffman@44724
   249
  shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
huffman@44724
   250
  by (simp add: complex_of_real_def)
paulson@14377
   251
paulson@14377
   252
lemma complex_of_real_mult_Complex:
huffman@44724
   253
  shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
huffman@44724
   254
  by (simp add: complex_of_real_def)
huffman@20557
   255
huffman@44827
   256
lemma complex_split_polar:
huffman@44827
   257
     "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
huffman@44827
   258
  by (simp add: complex_eq_iff polar_Ex)
huffman@44827
   259
paulson@14377
   260
huffman@23125
   261
subsection {* Vector Norm *}
paulson@14323
   262
haftmann@25712
   263
instantiation complex :: real_normed_field
haftmann@25571
   264
begin
haftmann@25571
   265
huffman@31413
   266
definition complex_norm_def:
huffman@31413
   267
  "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
haftmann@25571
   268
huffman@44724
   269
abbreviation cmod :: "complex \<Rightarrow> real"
huffman@44724
   270
  where "cmod \<equiv> norm"
haftmann@25571
   271
huffman@31413
   272
definition complex_sgn_def:
huffman@31413
   273
  "sgn x = x /\<^sub>R cmod x"
haftmann@25571
   274
huffman@31413
   275
definition dist_complex_def:
huffman@31413
   276
  "dist x y = cmod (x - y)"
huffman@31413
   277
haftmann@37767
   278
definition open_complex_def:
huffman@31492
   279
  "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31292
   280
huffman@20557
   281
lemmas cmod_def = complex_norm_def
huffman@20557
   282
huffman@23125
   283
lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
haftmann@25712
   284
  by (simp add: complex_norm_def)
huffman@22852
   285
huffman@31413
   286
instance proof
huffman@31492
   287
  fix r :: real and x y :: complex and S :: "complex set"
huffman@23125
   288
  show "0 \<le> norm x"
huffman@22861
   289
    by (induct x) simp
huffman@23125
   290
  show "(norm x = 0) = (x = 0)"
huffman@22861
   291
    by (induct x) simp
huffman@23125
   292
  show "norm (x + y) \<le> norm x + norm y"
huffman@23125
   293
    by (induct x, induct y)
huffman@23125
   294
       (simp add: real_sqrt_sum_squares_triangle_ineq)
huffman@23125
   295
  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
huffman@23125
   296
    by (induct x)
huffman@23125
   297
       (simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult)
huffman@23125
   298
  show "norm (x * y) = norm x * norm y"
huffman@23125
   299
    by (induct x, induct y)
nipkow@29667
   300
       (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
huffman@31292
   301
  show "sgn x = x /\<^sub>R cmod x"
huffman@31292
   302
    by (rule complex_sgn_def)
huffman@31292
   303
  show "dist x y = cmod (x - y)"
huffman@31292
   304
    by (rule dist_complex_def)
huffman@31492
   305
  show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31492
   306
    by (rule open_complex_def)
huffman@24520
   307
qed
huffman@20557
   308
haftmann@25712
   309
end
haftmann@25712
   310
huffman@44761
   311
lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"
huffman@44724
   312
  by simp
paulson@14323
   313
huffman@44761
   314
lemma cmod_complex_polar:
huffman@44724
   315
  "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
huffman@44724
   316
  by (simp add: norm_mult)
huffman@22861
   317
huffman@22861
   318
lemma complex_Re_le_cmod: "Re x \<le> cmod x"
huffman@44724
   319
  unfolding complex_norm_def
huffman@44724
   320
  by (rule real_sqrt_sum_squares_ge1)
huffman@22861
   321
huffman@44761
   322
lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
huffman@44724
   323
  by (rule order_trans [OF _ norm_ge_zero], simp)
huffman@22861
   324
huffman@44761
   325
lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \<le> cmod a"
huffman@44724
   326
  by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
paulson@14323
   327
chaieb@26117
   328
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
huffman@44724
   329
  by (cases x) simp
chaieb@26117
   330
chaieb@26117
   331
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
huffman@44724
   332
  by (cases x) simp
huffman@44724
   333
paulson@14354
   334
huffman@23123
   335
subsection {* Completeness of the Complexes *}
huffman@23123
   336
huffman@44290
   337
lemma bounded_linear_Re: "bounded_linear Re"
huffman@44290
   338
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@44290
   339
huffman@44290
   340
lemma bounded_linear_Im: "bounded_linear Im"
huffman@44127
   341
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@23123
   342
huffman@44290
   343
lemmas tendsto_Re [tendsto_intros] =
huffman@44290
   344
  bounded_linear.tendsto [OF bounded_linear_Re]
huffman@44290
   345
huffman@44290
   346
lemmas tendsto_Im [tendsto_intros] =
huffman@44290
   347
  bounded_linear.tendsto [OF bounded_linear_Im]
huffman@44290
   348
huffman@44290
   349
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
huffman@44290
   350
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
huffman@44290
   351
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
huffman@44290
   352
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
huffman@23123
   353
huffman@36825
   354
lemma tendsto_Complex [tendsto_intros]:
huffman@44724
   355
  assumes "(f ---> a) F" and "(g ---> b) F"
huffman@44724
   356
  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
huffman@36825
   357
proof (rule tendstoI)
huffman@36825
   358
  fix r :: real assume "0 < r"
huffman@36825
   359
  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
huffman@44724
   360
  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"
huffman@44724
   361
    using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   362
  moreover
huffman@44724
   363
  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"
huffman@44724
   364
    using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   365
  ultimately
huffman@44724
   366
  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"
huffman@36825
   367
    by (rule eventually_elim2)
huffman@36825
   368
       (simp add: dist_norm real_sqrt_sum_squares_less)
huffman@36825
   369
qed
huffman@36825
   370
huffman@23123
   371
instance complex :: banach
huffman@23123
   372
proof
huffman@23123
   373
  fix X :: "nat \<Rightarrow> complex"
huffman@23123
   374
  assume X: "Cauchy X"
huffman@44290
   375
  from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
huffman@23123
   376
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@44290
   377
  from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
huffman@23123
   378
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@23123
   379
  have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
huffman@44748
   380
    using tendsto_Complex [OF 1 2] by simp
huffman@23123
   381
  thus "convergent X"
huffman@23123
   382
    by (rule convergentI)
huffman@23123
   383
qed
huffman@23123
   384
huffman@23123
   385
huffman@44827
   386
subsection {* The Complex Number $i$ *}
huffman@23125
   387
huffman@44724
   388
definition "ii" :: complex  ("\<i>")
huffman@44724
   389
  where i_def: "ii \<equiv> Complex 0 1"
huffman@23125
   390
huffman@23125
   391
lemma complex_Re_i [simp]: "Re ii = 0"
huffman@44724
   392
  by (simp add: i_def)
paulson@14354
   393
huffman@23125
   394
lemma complex_Im_i [simp]: "Im ii = 1"
huffman@44724
   395
  by (simp add: i_def)
huffman@23125
   396
huffman@23125
   397
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
huffman@44724
   398
  by (simp add: i_def)
huffman@23125
   399
huffman@23125
   400
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
huffman@44724
   401
  by (simp add: complex_eq_iff)
huffman@23125
   402
huffman@23125
   403
lemma complex_i_not_one [simp]: "ii \<noteq> 1"
huffman@44724
   404
  by (simp add: complex_eq_iff)
huffman@23124
   405
huffman@23125
   406
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w"
huffman@44724
   407
  by (simp add: complex_eq_iff)
huffman@23125
   408
huffman@23125
   409
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
huffman@44724
   410
  by (simp add: complex_eq_iff)
huffman@23125
   411
huffman@23125
   412
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
huffman@44724
   413
  by (simp add: complex_eq_iff)
huffman@23125
   414
huffman@23125
   415
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
huffman@44724
   416
  by (simp add: i_def complex_of_real_def)
huffman@23125
   417
huffman@23125
   418
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
huffman@44724
   419
  by (simp add: i_def complex_of_real_def)
huffman@23125
   420
huffman@23125
   421
lemma i_squared [simp]: "ii * ii = -1"
huffman@44724
   422
  by (simp add: i_def)
huffman@23125
   423
huffman@23125
   424
lemma power2_i [simp]: "ii\<twosuperior> = -1"
huffman@44724
   425
  by (simp add: power2_eq_square)
huffman@23125
   426
huffman@23125
   427
lemma inverse_i [simp]: "inverse ii = - ii"
huffman@44724
   428
  by (rule inverse_unique, simp)
paulson@14354
   429
huffman@44827
   430
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
huffman@44827
   431
  by (simp add: mult_assoc [symmetric])
huffman@44827
   432
paulson@14354
   433
huffman@23125
   434
subsection {* Complex Conjugation *}
huffman@23125
   435
huffman@44724
   436
definition cnj :: "complex \<Rightarrow> complex" where
huffman@23125
   437
  "cnj z = Complex (Re z) (- Im z)"
huffman@23125
   438
huffman@23125
   439
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
huffman@44724
   440
  by (simp add: cnj_def)
huffman@23125
   441
huffman@23125
   442
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
huffman@44724
   443
  by (simp add: cnj_def)
huffman@23125
   444
huffman@23125
   445
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
huffman@44724
   446
  by (simp add: cnj_def)
huffman@23125
   447
huffman@23125
   448
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
huffman@44724
   449
  by (simp add: complex_eq_iff)
huffman@23125
   450
huffman@23125
   451
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
huffman@44724
   452
  by (simp add: cnj_def)
huffman@23125
   453
huffman@23125
   454
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
huffman@44724
   455
  by (simp add: complex_eq_iff)
huffman@23125
   456
huffman@23125
   457
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
huffman@44724
   458
  by (simp add: complex_eq_iff)
huffman@23125
   459
huffman@23125
   460
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
huffman@44724
   461
  by (simp add: complex_eq_iff)
huffman@23125
   462
huffman@23125
   463
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
huffman@44724
   464
  by (simp add: complex_eq_iff)
huffman@23125
   465
huffman@23125
   466
lemma complex_cnj_minus: "cnj (- x) = - cnj x"
huffman@44724
   467
  by (simp add: complex_eq_iff)
huffman@23125
   468
huffman@23125
   469
lemma complex_cnj_one [simp]: "cnj 1 = 1"
huffman@44724
   470
  by (simp add: complex_eq_iff)
huffman@23125
   471
huffman@23125
   472
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
huffman@44724
   473
  by (simp add: complex_eq_iff)
huffman@23125
   474
huffman@23125
   475
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
huffman@44724
   476
  by (simp add: complex_inverse_def)
paulson@14323
   477
huffman@23125
   478
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
huffman@44724
   479
  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
huffman@23125
   480
huffman@23125
   481
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
huffman@44724
   482
  by (induct n, simp_all add: complex_cnj_mult)
huffman@23125
   483
huffman@23125
   484
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
huffman@44724
   485
  by (simp add: complex_eq_iff)
huffman@23125
   486
huffman@23125
   487
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
huffman@44724
   488
  by (simp add: complex_eq_iff)
huffman@23125
   489
huffman@23125
   490
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"
huffman@44724
   491
  by (simp add: complex_eq_iff)
huffman@23125
   492
huffman@23125
   493
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
huffman@44724
   494
  by (simp add: complex_eq_iff)
huffman@23125
   495
huffman@23125
   496
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
huffman@44724
   497
  by (simp add: complex_norm_def)
paulson@14323
   498
huffman@23125
   499
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
huffman@44724
   500
  by (simp add: complex_eq_iff)
huffman@23125
   501
huffman@23125
   502
lemma complex_cnj_i [simp]: "cnj ii = - ii"
huffman@44724
   503
  by (simp add: complex_eq_iff)
huffman@23125
   504
huffman@23125
   505
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
huffman@44724
   506
  by (simp add: complex_eq_iff)
huffman@23125
   507
huffman@23125
   508
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
huffman@44724
   509
  by (simp add: complex_eq_iff)
paulson@14354
   510
huffman@23125
   511
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
huffman@44724
   512
  by (simp add: complex_eq_iff power2_eq_square)
huffman@23125
   513
huffman@23125
   514
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
huffman@44724
   515
  by (simp add: norm_mult power2_eq_square)
huffman@23125
   516
huffman@44827
   517
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
huffman@44827
   518
  by (simp add: cmod_def power2_eq_square)
huffman@44827
   519
huffman@44827
   520
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
huffman@44827
   521
  by simp
huffman@44827
   522
huffman@44290
   523
lemma bounded_linear_cnj: "bounded_linear cnj"
huffman@44127
   524
  using complex_cnj_add complex_cnj_scaleR
huffman@44127
   525
  by (rule bounded_linear_intro [where K=1], simp)
paulson@14354
   526
huffman@44290
   527
lemmas tendsto_cnj [tendsto_intros] =
huffman@44290
   528
  bounded_linear.tendsto [OF bounded_linear_cnj]
huffman@44290
   529
huffman@44290
   530
lemmas isCont_cnj [simp] =
huffman@44290
   531
  bounded_linear.isCont [OF bounded_linear_cnj]
huffman@44290
   532
paulson@14354
   533
huffman@44827
   534
subsection {* Complex Signum and Argument *}
huffman@20557
   535
huffman@44724
   536
definition arg :: "complex => real" where
huffman@20557
   537
  "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"
huffman@20557
   538
paulson@14374
   539
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
huffman@44724
   540
  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
paulson@14323
   541
paulson@14374
   542
lemma complex_eq_cancel_iff2 [simp]:
huffman@44724
   543
  shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
huffman@44724
   544
  by (simp add: complex_of_real_def)
paulson@14323
   545
paulson@14374
   546
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
huffman@44724
   547
  by (simp add: complex_sgn_def divide_inverse)
paulson@14323
   548
paulson@14374
   549
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
huffman@44724
   550
  by (simp add: complex_sgn_def divide_inverse)
paulson@14323
   551
paulson@14323
   552
lemma complex_inverse_complex_split:
paulson@14323
   553
     "inverse(complex_of_real x + ii * complex_of_real y) =
paulson@14323
   554
      complex_of_real(x/(x ^ 2 + y ^ 2)) -
paulson@14323
   555
      ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
huffman@44724
   556
  by (simp add: complex_of_real_def i_def diff_minus divide_inverse)
paulson@14323
   557
paulson@14323
   558
(*----------------------------------------------------------------------------*)
paulson@14323
   559
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
paulson@14323
   560
(* many of the theorems are not used - so should they be kept?                *)
paulson@14323
   561
(*----------------------------------------------------------------------------*)
paulson@14323
   562
paulson@14354
   563
lemma cos_arg_i_mult_zero_pos:
paulson@14377
   564
   "0 < y ==> cos (arg(Complex 0 y)) = 0"
paulson@14373
   565
apply (simp add: arg_def abs_if)
paulson@14334
   566
apply (rule_tac a = "pi/2" in someI2, auto)
paulson@14334
   567
apply (rule order_less_trans [of _ 0], auto)
paulson@14323
   568
done
paulson@14323
   569
paulson@14354
   570
lemma cos_arg_i_mult_zero_neg:
paulson@14377
   571
   "y < 0 ==> cos (arg(Complex 0 y)) = 0"
paulson@14373
   572
apply (simp add: arg_def abs_if)
paulson@14334
   573
apply (rule_tac a = "- pi/2" in someI2, auto)
paulson@14334
   574
apply (rule order_trans [of _ 0], auto)
paulson@14323
   575
done
paulson@14323
   576
paulson@14374
   577
lemma cos_arg_i_mult_zero [simp]:
paulson@14377
   578
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
paulson@14377
   579
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
paulson@14323
   580
paulson@14323
   581
paulson@14323
   582
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
   583
huffman@44827
   584
subsubsection {* $\cos \theta + i \sin \theta$ *}
huffman@20557
   585
huffman@44715
   586
definition cis :: "real \<Rightarrow> complex" where
huffman@20557
   587
  "cis a = Complex (cos a) (sin a)"
huffman@20557
   588
huffman@44827
   589
lemma Re_cis [simp]: "Re (cis a) = cos a"
huffman@44827
   590
  by (simp add: cis_def)
huffman@44827
   591
huffman@44827
   592
lemma Im_cis [simp]: "Im (cis a) = sin a"
huffman@44827
   593
  by (simp add: cis_def)
huffman@44827
   594
huffman@44827
   595
lemma cis_zero [simp]: "cis 0 = 1"
huffman@44827
   596
  by (simp add: cis_def)
huffman@44827
   597
huffman@44828
   598
lemma norm_cis [simp]: "norm (cis a) = 1"
huffman@44828
   599
  by (simp add: cis_def)
huffman@44828
   600
huffman@44828
   601
lemma sgn_cis [simp]: "sgn (cis a) = cis a"
huffman@44828
   602
  by (simp add: sgn_div_norm)
huffman@44828
   603
huffman@44828
   604
lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
huffman@44828
   605
  by (metis norm_cis norm_zero zero_neq_one)
huffman@44828
   606
huffman@44827
   607
lemma cis_mult: "cis a * cis b = cis (a + b)"
huffman@44827
   608
  by (simp add: cis_def cos_add sin_add)
huffman@44827
   609
huffman@44827
   610
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
huffman@44827
   611
  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
huffman@44827
   612
huffman@44827
   613
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
huffman@44827
   614
  by (simp add: cis_def)
huffman@44827
   615
huffman@44827
   616
lemma cis_divide: "cis a / cis b = cis (a - b)"
huffman@44827
   617
  by (simp add: complex_divide_def cis_mult diff_minus)
huffman@44827
   618
huffman@44827
   619
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
huffman@44827
   620
  by (auto simp add: DeMoivre)
huffman@44827
   621
huffman@44827
   622
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
huffman@44827
   623
  by (auto simp add: DeMoivre)
huffman@44827
   624
huffman@44827
   625
subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
huffman@44715
   626
huffman@44715
   627
definition rcis :: "[real, real] \<Rightarrow> complex" where
huffman@20557
   628
  "rcis r a = complex_of_real r * cis a"
huffman@20557
   629
huffman@44827
   630
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
huffman@44828
   631
  by (simp add: rcis_def)
huffman@44827
   632
huffman@44827
   633
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
huffman@44828
   634
  by (simp add: rcis_def)
huffman@44827
   635
huffman@44827
   636
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
huffman@44828
   637
  by (simp add: complex_eq_iff polar_Ex)
huffman@44827
   638
huffman@44827
   639
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
huffman@44828
   640
  by (simp add: rcis_def norm_mult)
huffman@44827
   641
huffman@44827
   642
lemma cis_rcis_eq: "cis a = rcis 1 a"
huffman@44827
   643
  by (simp add: rcis_def)
huffman@44827
   644
huffman@44827
   645
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
huffman@44828
   646
  by (simp add: rcis_def cis_mult)
huffman@44827
   647
huffman@44827
   648
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
huffman@44827
   649
  by (simp add: rcis_def)
huffman@44827
   650
huffman@44827
   651
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
huffman@44827
   652
  by (simp add: rcis_def)
huffman@44827
   653
huffman@44828
   654
lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
huffman@44828
   655
  by (simp add: rcis_def)
huffman@44828
   656
huffman@44827
   657
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
huffman@44827
   658
  by (simp add: rcis_def power_mult_distrib DeMoivre)
huffman@44827
   659
huffman@44827
   660
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
huffman@44827
   661
  by (simp add: divide_inverse rcis_def)
huffman@44827
   662
huffman@44827
   663
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
huffman@44828
   664
  by (simp add: rcis_def cis_divide [symmetric])
huffman@44827
   665
huffman@44827
   666
subsubsection {* Complex exponential *}
huffman@44827
   667
huffman@44291
   668
abbreviation expi :: "complex \<Rightarrow> complex"
huffman@44291
   669
  where "expi \<equiv> exp"
huffman@44291
   670
huffman@44712
   671
lemma cis_conv_exp: "cis b = exp (Complex 0 b)"
huffman@44291
   672
proof (rule complex_eqI)
huffman@44291
   673
  { fix n have "Complex 0 b ^ n =
huffman@44291
   674
    real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"
huffman@44291
   675
      apply (induct n)
huffman@44291
   676
      apply (simp add: cos_coeff_def sin_coeff_def)
huffman@44291
   677
      apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)
huffman@44291
   678
      done } note * = this
huffman@44712
   679
  show "Re (cis b) = Re (exp (Complex 0 b))"
huffman@44291
   680
    unfolding exp_def cis_def cos_def
huffman@44291
   681
    by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],
huffman@44291
   682
      simp add: * mult_assoc [symmetric])
huffman@44712
   683
  show "Im (cis b) = Im (exp (Complex 0 b))"
huffman@44291
   684
    unfolding exp_def cis_def sin_def
huffman@44291
   685
    by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],
huffman@44291
   686
      simp add: * mult_assoc [symmetric])
huffman@44291
   687
qed
huffman@44291
   688
huffman@44291
   689
lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"
huffman@44712
   690
  unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp
huffman@20557
   691
huffman@44828
   692
lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
huffman@44828
   693
  unfolding expi_def by simp
huffman@44828
   694
huffman@44828
   695
lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
huffman@44828
   696
  unfolding expi_def by simp
huffman@44828
   697
paulson@14374
   698
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
paulson@14373
   699
apply (insert rcis_Ex [of z])
huffman@23125
   700
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
paulson@14334
   701
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
   702
done
paulson@14323
   703
paulson@14387
   704
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
huffman@44724
   705
  by (simp add: expi_def cis_def)
paulson@14387
   706
huffman@44065
   707
text {* Legacy theorem names *}
huffman@44065
   708
huffman@44065
   709
lemmas expand_complex_eq = complex_eq_iff
huffman@44065
   710
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
huffman@44065
   711
lemmas complex_equality = complex_eqI
huffman@44065
   712
paulson@13957
   713
end