src/HOL/Complex.thy
author huffman
Thu Aug 18 21:23:31 2011 -0700 (2011-08-18)
changeset 44290 23a5137162ea
parent 44127 7b57b9295d98
child 44291 dbd9965745fd
permissions -rw-r--r--
remove more bounded_linear locale interpretations (cf. f0de18b62d63)
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
haftmann@25712
    15
primrec
haftmann@25712
    16
  Re :: "complex \<Rightarrow> real"
haftmann@25712
    17
where
haftmann@25712
    18
  Re: "Re (Complex x y) = x"
paulson@14373
    19
haftmann@25712
    20
primrec
haftmann@25712
    21
  Im :: "complex \<Rightarrow> real"
haftmann@25712
    22
where
haftmann@25712
    23
  Im: "Im (Complex x y) = y"
paulson@14373
    24
paulson@14373
    25
lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
paulson@14373
    26
  by (induct z) simp
paulson@13957
    27
huffman@44065
    28
lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
haftmann@25712
    29
  by (induct x, induct y) simp
huffman@23125
    30
huffman@44065
    31
lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
haftmann@25712
    32
  by (induct x, induct y) simp
huffman@23125
    33
huffman@23125
    34
huffman@23125
    35
subsection {* Addition and Subtraction *}
huffman@23125
    36
haftmann@25599
    37
instantiation complex :: ab_group_add
haftmann@25571
    38
begin
haftmann@25571
    39
haftmann@25571
    40
definition
haftmann@25571
    41
  complex_zero_def: "0 = Complex 0 0"
haftmann@25571
    42
haftmann@25571
    43
definition
haftmann@25571
    44
  complex_add_def: "x + y = Complex (Re x + Re y) (Im x + Im y)"
huffman@23124
    45
haftmann@25571
    46
definition
haftmann@25571
    47
  complex_minus_def: "- x = Complex (- Re x) (- Im x)"
paulson@14323
    48
haftmann@25571
    49
definition
haftmann@25571
    50
  complex_diff_def: "x - (y\<Colon>complex) = x + - y"
haftmann@25571
    51
haftmann@25599
    52
lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
haftmann@25599
    53
  by (simp add: complex_zero_def)
paulson@14323
    54
paulson@14374
    55
lemma complex_Re_zero [simp]: "Re 0 = 0"
haftmann@25599
    56
  by (simp add: complex_zero_def)
paulson@14374
    57
paulson@14374
    58
lemma complex_Im_zero [simp]: "Im 0 = 0"
haftmann@25599
    59
  by (simp add: complex_zero_def)
haftmann@25599
    60
haftmann@25712
    61
lemma complex_add [simp]:
haftmann@25712
    62
  "Complex a b + Complex c d = Complex (a + c) (b + d)"
haftmann@25712
    63
  by (simp add: complex_add_def)
haftmann@25712
    64
haftmann@25599
    65
lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"
haftmann@25599
    66
  by (simp add: complex_add_def)
haftmann@25599
    67
haftmann@25599
    68
lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"
haftmann@25599
    69
  by (simp add: complex_add_def)
paulson@14323
    70
haftmann@25712
    71
lemma complex_minus [simp]:
haftmann@25712
    72
  "- (Complex a b) = Complex (- a) (- b)"
haftmann@25599
    73
  by (simp add: complex_minus_def)
huffman@23125
    74
huffman@23125
    75
lemma complex_Re_minus [simp]: "Re (- x) = - Re x"
haftmann@25599
    76
  by (simp add: complex_minus_def)
huffman@23125
    77
huffman@23125
    78
lemma complex_Im_minus [simp]: "Im (- x) = - Im x"
haftmann@25599
    79
  by (simp add: complex_minus_def)
huffman@23125
    80
huffman@23275
    81
lemma complex_diff [simp]:
huffman@23125
    82
  "Complex a b - Complex c d = Complex (a - c) (b - d)"
haftmann@25599
    83
  by (simp add: complex_diff_def)
huffman@23125
    84
huffman@23125
    85
lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"
haftmann@25599
    86
  by (simp add: complex_diff_def)
huffman@23125
    87
huffman@23125
    88
lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"
haftmann@25599
    89
  by (simp add: complex_diff_def)
huffman@23125
    90
haftmann@25712
    91
instance
haftmann@25712
    92
  by intro_classes (simp_all add: complex_add_def complex_diff_def)
haftmann@25712
    93
haftmann@25712
    94
end
haftmann@25712
    95
haftmann@25712
    96
huffman@23125
    97
huffman@23125
    98
subsection {* Multiplication and Division *}
huffman@23125
    99
haftmann@36409
   100
instantiation complex :: field_inverse_zero
haftmann@25571
   101
begin
haftmann@25571
   102
haftmann@25571
   103
definition
haftmann@25571
   104
  complex_one_def: "1 = Complex 1 0"
haftmann@25571
   105
haftmann@25571
   106
definition
haftmann@25571
   107
  complex_mult_def: "x * y =
haftmann@25571
   108
    Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
huffman@23125
   109
haftmann@25571
   110
definition
haftmann@25571
   111
  complex_inverse_def: "inverse x =
haftmann@25571
   112
    Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"
huffman@23125
   113
haftmann@25571
   114
definition
haftmann@25571
   115
  complex_divide_def: "x / (y\<Colon>complex) = x * inverse y"
haftmann@25571
   116
huffman@23125
   117
lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
haftmann@25712
   118
  by (simp add: complex_one_def)
huffman@22861
   119
paulson@14374
   120
lemma complex_Re_one [simp]: "Re 1 = 1"
haftmann@25712
   121
  by (simp add: complex_one_def)
paulson@14323
   122
paulson@14374
   123
lemma complex_Im_one [simp]: "Im 1 = 0"
haftmann@25712
   124
  by (simp add: complex_one_def)
paulson@14323
   125
huffman@23125
   126
lemma complex_mult [simp]:
huffman@23125
   127
  "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
haftmann@25712
   128
  by (simp add: complex_mult_def)
paulson@14323
   129
huffman@23125
   130
lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"
haftmann@25712
   131
  by (simp add: complex_mult_def)
paulson@14323
   132
huffman@23125
   133
lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"
haftmann@25712
   134
  by (simp add: complex_mult_def)
paulson@14323
   135
paulson@14377
   136
lemma complex_inverse [simp]:
huffman@23125
   137
  "inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))"
haftmann@25712
   138
  by (simp add: complex_inverse_def)
paulson@14335
   139
huffman@23125
   140
lemma complex_Re_inverse:
huffman@23125
   141
  "Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
haftmann@25712
   142
  by (simp add: complex_inverse_def)
paulson@14323
   143
huffman@23125
   144
lemma complex_Im_inverse:
huffman@23125
   145
  "Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
haftmann@25712
   146
  by (simp add: complex_inverse_def)
paulson@14335
   147
haftmann@25712
   148
instance
haftmann@25712
   149
  by intro_classes (simp_all add: complex_mult_def
haftmann@25712
   150
  right_distrib left_distrib right_diff_distrib left_diff_distrib
haftmann@25712
   151
  complex_inverse_def complex_divide_def
haftmann@25712
   152
  power2_eq_square add_divide_distrib [symmetric]
huffman@44065
   153
  complex_eq_iff)
paulson@14335
   154
haftmann@25712
   155
end
huffman@23125
   156
huffman@23125
   157
huffman@23125
   158
subsection {* Numerals and Arithmetic *}
huffman@23125
   159
haftmann@25571
   160
instantiation complex :: number_ring
haftmann@25571
   161
begin
huffman@23125
   162
haftmann@25712
   163
definition number_of_complex where
haftmann@25571
   164
  complex_number_of_def: "number_of w = (of_int w \<Colon> complex)"
haftmann@25571
   165
haftmann@25571
   166
instance
haftmann@25712
   167
  by intro_classes (simp only: complex_number_of_def)
haftmann@25571
   168
haftmann@25571
   169
end
huffman@23125
   170
huffman@23125
   171
lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
huffman@23125
   172
by (induct n) simp_all
huffman@20556
   173
huffman@23125
   174
lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
huffman@23125
   175
by (induct n) simp_all
huffman@23125
   176
huffman@23125
   177
lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
huffman@23125
   178
by (cases z rule: int_diff_cases) simp
huffman@23125
   179
huffman@23125
   180
lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
huffman@23125
   181
by (cases z rule: int_diff_cases) simp
huffman@23125
   182
huffman@23125
   183
lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v"
haftmann@25502
   184
unfolding number_of_eq by (rule complex_Re_of_int)
huffman@20556
   185
huffman@23125
   186
lemma complex_Im_number_of [simp]: "Im (number_of v) = 0"
haftmann@25502
   187
unfolding number_of_eq by (rule complex_Im_of_int)
huffman@23125
   188
huffman@23125
   189
lemma Complex_eq_number_of [simp]:
huffman@23125
   190
  "(Complex a b = number_of w) = (a = number_of w \<and> b = 0)"
huffman@44065
   191
by (simp add: complex_eq_iff)
huffman@23125
   192
huffman@23125
   193
huffman@23125
   194
subsection {* Scalar Multiplication *}
huffman@20556
   195
haftmann@25712
   196
instantiation complex :: real_field
haftmann@25571
   197
begin
haftmann@25571
   198
haftmann@25571
   199
definition
haftmann@25571
   200
  complex_scaleR_def: "scaleR r x = Complex (r * Re x) (r * Im x)"
haftmann@25571
   201
huffman@23125
   202
lemma complex_scaleR [simp]:
huffman@23125
   203
  "scaleR r (Complex a b) = Complex (r * a) (r * b)"
haftmann@25712
   204
  unfolding complex_scaleR_def by simp
huffman@23125
   205
huffman@23125
   206
lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"
haftmann@25712
   207
  unfolding complex_scaleR_def by simp
huffman@23125
   208
huffman@23125
   209
lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"
haftmann@25712
   210
  unfolding complex_scaleR_def by simp
huffman@22972
   211
haftmann@25712
   212
instance
huffman@20556
   213
proof
huffman@23125
   214
  fix a b :: real and x y :: complex
huffman@23125
   215
  show "scaleR a (x + y) = scaleR a x + scaleR a y"
huffman@44065
   216
    by (simp add: complex_eq_iff right_distrib)
huffman@23125
   217
  show "scaleR (a + b) x = scaleR a x + scaleR b x"
huffman@44065
   218
    by (simp add: complex_eq_iff left_distrib)
huffman@23125
   219
  show "scaleR a (scaleR b x) = scaleR (a * b) x"
huffman@44065
   220
    by (simp add: complex_eq_iff mult_assoc)
huffman@23125
   221
  show "scaleR 1 x = x"
huffman@44065
   222
    by (simp add: complex_eq_iff)
huffman@23125
   223
  show "scaleR a x * y = scaleR a (x * y)"
huffman@44065
   224
    by (simp add: complex_eq_iff algebra_simps)
huffman@23125
   225
  show "x * scaleR a y = scaleR a (x * y)"
huffman@44065
   226
    by (simp add: complex_eq_iff algebra_simps)
huffman@20556
   227
qed
huffman@20556
   228
haftmann@25712
   229
end
haftmann@25712
   230
huffman@20556
   231
huffman@23125
   232
subsection{* Properties of Embedding from Reals *}
paulson@14323
   233
huffman@20557
   234
abbreviation
huffman@23125
   235
  complex_of_real :: "real \<Rightarrow> complex" where
huffman@23125
   236
    "complex_of_real \<equiv> of_real"
huffman@20557
   237
huffman@20557
   238
lemma complex_of_real_def: "complex_of_real r = Complex r 0"
huffman@20557
   239
by (simp add: of_real_def complex_scaleR_def)
huffman@20557
   240
huffman@20557
   241
lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
huffman@20557
   242
by (simp add: complex_of_real_def)
huffman@20557
   243
huffman@20557
   244
lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
huffman@20557
   245
by (simp add: complex_of_real_def)
huffman@20557
   246
paulson@14377
   247
lemma Complex_add_complex_of_real [simp]:
paulson@14377
   248
     "Complex x y + complex_of_real r = Complex (x+r) y"
paulson@14377
   249
by (simp add: complex_of_real_def)
paulson@14377
   250
paulson@14377
   251
lemma complex_of_real_add_Complex [simp]:
paulson@14377
   252
     "complex_of_real r + Complex x y = Complex (r+x) y"
huffman@23125
   253
by (simp add: complex_of_real_def)
paulson@14377
   254
paulson@14377
   255
lemma Complex_mult_complex_of_real:
paulson@14377
   256
     "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
paulson@14377
   257
by (simp add: complex_of_real_def)
paulson@14377
   258
paulson@14377
   259
lemma complex_of_real_mult_Complex:
paulson@14377
   260
     "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
huffman@23125
   261
by (simp add: complex_of_real_def)
huffman@20557
   262
paulson@14377
   263
huffman@23125
   264
subsection {* Vector Norm *}
paulson@14323
   265
haftmann@25712
   266
instantiation complex :: real_normed_field
haftmann@25571
   267
begin
haftmann@25571
   268
huffman@31413
   269
definition complex_norm_def:
huffman@31413
   270
  "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
haftmann@25571
   271
huffman@20557
   272
abbreviation
huffman@22861
   273
  cmod :: "complex \<Rightarrow> real" where
haftmann@25712
   274
  "cmod \<equiv> norm"
haftmann@25571
   275
huffman@31413
   276
definition complex_sgn_def:
huffman@31413
   277
  "sgn x = x /\<^sub>R cmod x"
haftmann@25571
   278
huffman@31413
   279
definition dist_complex_def:
huffman@31413
   280
  "dist x y = cmod (x - y)"
huffman@31413
   281
haftmann@37767
   282
definition open_complex_def:
huffman@31492
   283
  "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31292
   284
huffman@20557
   285
lemmas cmod_def = complex_norm_def
huffman@20557
   286
huffman@23125
   287
lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
haftmann@25712
   288
  by (simp add: complex_norm_def)
huffman@22852
   289
huffman@31413
   290
instance proof
huffman@31492
   291
  fix r :: real and x y :: complex and S :: "complex set"
huffman@23125
   292
  show "0 \<le> norm x"
huffman@22861
   293
    by (induct x) simp
huffman@23125
   294
  show "(norm x = 0) = (x = 0)"
huffman@22861
   295
    by (induct x) simp
huffman@23125
   296
  show "norm (x + y) \<le> norm x + norm y"
huffman@23125
   297
    by (induct x, induct y)
huffman@23125
   298
       (simp add: real_sqrt_sum_squares_triangle_ineq)
huffman@23125
   299
  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
huffman@23125
   300
    by (induct x)
huffman@23125
   301
       (simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult)
huffman@23125
   302
  show "norm (x * y) = norm x * norm y"
huffman@23125
   303
    by (induct x, induct y)
nipkow@29667
   304
       (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
huffman@31292
   305
  show "sgn x = x /\<^sub>R cmod x"
huffman@31292
   306
    by (rule complex_sgn_def)
huffman@31292
   307
  show "dist x y = cmod (x - y)"
huffman@31292
   308
    by (rule dist_complex_def)
huffman@31492
   309
  show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31492
   310
    by (rule open_complex_def)
huffman@24520
   311
qed
huffman@20557
   312
haftmann@25712
   313
end
haftmann@25712
   314
huffman@22861
   315
lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1"
huffman@22861
   316
by simp
paulson@14323
   317
huffman@22861
   318
lemma cmod_complex_polar [simp]:
huffman@22861
   319
     "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
huffman@23125
   320
by (simp add: norm_mult)
huffman@22861
   321
huffman@22861
   322
lemma complex_Re_le_cmod: "Re x \<le> cmod x"
huffman@22861
   323
unfolding complex_norm_def
huffman@22861
   324
by (rule real_sqrt_sum_squares_ge1)
huffman@22861
   325
huffman@22861
   326
lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x"
huffman@22861
   327
by (rule order_trans [OF _ norm_ge_zero], simp)
huffman@22861
   328
huffman@22861
   329
lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a"
huffman@22861
   330
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
paulson@14323
   331
huffman@22861
   332
lemmas real_sum_squared_expand = power2_sum [where 'a=real]
paulson@14323
   333
chaieb@26117
   334
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
chaieb@26117
   335
by (cases x) simp
chaieb@26117
   336
chaieb@26117
   337
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
chaieb@26117
   338
by (cases x) simp
paulson@14354
   339
huffman@23123
   340
subsection {* Completeness of the Complexes *}
huffman@23123
   341
huffman@44290
   342
lemma bounded_linear_Re: "bounded_linear Re"
huffman@44290
   343
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@44290
   344
huffman@44290
   345
lemma bounded_linear_Im: "bounded_linear Im"
huffman@44127
   346
  by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
huffman@23123
   347
huffman@44290
   348
lemmas tendsto_Re [tendsto_intros] =
huffman@44290
   349
  bounded_linear.tendsto [OF bounded_linear_Re]
huffman@44290
   350
huffman@44290
   351
lemmas tendsto_Im [tendsto_intros] =
huffman@44290
   352
  bounded_linear.tendsto [OF bounded_linear_Im]
huffman@44290
   353
huffman@44290
   354
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
huffman@44290
   355
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
huffman@44290
   356
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
huffman@44290
   357
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
huffman@23123
   358
huffman@36825
   359
lemma tendsto_Complex [tendsto_intros]:
huffman@36825
   360
  assumes "(f ---> a) net" and "(g ---> b) net"
huffman@36825
   361
  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) net"
huffman@36825
   362
proof (rule tendstoI)
huffman@36825
   363
  fix r :: real assume "0 < r"
huffman@36825
   364
  hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
huffman@36825
   365
  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) net"
huffman@36825
   366
    using `(f ---> a) net` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   367
  moreover
huffman@36825
   368
  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) net"
huffman@36825
   369
    using `(g ---> b) net` and `0 < r / sqrt 2` by (rule tendstoD)
huffman@36825
   370
  ultimately
huffman@36825
   371
  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) net"
huffman@36825
   372
    by (rule eventually_elim2)
huffman@36825
   373
       (simp add: dist_norm real_sqrt_sum_squares_less)
huffman@36825
   374
qed
huffman@36825
   375
huffman@23123
   376
lemma LIMSEQ_Complex:
huffman@23123
   377
  "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b"
huffman@36825
   378
by (rule tendsto_Complex)
huffman@23123
   379
huffman@23123
   380
instance complex :: banach
huffman@23123
   381
proof
huffman@23123
   382
  fix X :: "nat \<Rightarrow> complex"
huffman@23123
   383
  assume X: "Cauchy X"
huffman@44290
   384
  from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
huffman@23123
   385
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@44290
   386
  from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
huffman@23123
   387
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
huffman@23123
   388
  have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
huffman@23123
   389
    using LIMSEQ_Complex [OF 1 2] by simp
huffman@23123
   390
  thus "convergent X"
huffman@23123
   391
    by (rule convergentI)
huffman@23123
   392
qed
huffman@23123
   393
huffman@23123
   394
huffman@23125
   395
subsection {* The Complex Number @{term "\<i>"} *}
huffman@23125
   396
huffman@23125
   397
definition
huffman@23125
   398
  "ii" :: complex  ("\<i>") where
huffman@23125
   399
  i_def: "ii \<equiv> Complex 0 1"
huffman@23125
   400
huffman@23125
   401
lemma complex_Re_i [simp]: "Re ii = 0"
huffman@23125
   402
by (simp add: i_def)
paulson@14354
   403
huffman@23125
   404
lemma complex_Im_i [simp]: "Im ii = 1"
huffman@23125
   405
by (simp add: i_def)
huffman@23125
   406
huffman@23125
   407
lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
huffman@23125
   408
by (simp add: i_def)
huffman@23125
   409
huffman@23125
   410
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
huffman@44065
   411
by (simp add: complex_eq_iff)
huffman@23125
   412
huffman@23125
   413
lemma complex_i_not_one [simp]: "ii \<noteq> 1"
huffman@44065
   414
by (simp add: complex_eq_iff)
huffman@23124
   415
huffman@23125
   416
lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w"
huffman@44065
   417
by (simp add: complex_eq_iff)
huffman@23125
   418
huffman@23125
   419
lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
huffman@44065
   420
by (simp add: complex_eq_iff)
huffman@23125
   421
huffman@23125
   422
lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
huffman@44065
   423
by (simp add: complex_eq_iff)
huffman@23125
   424
huffman@23125
   425
lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
huffman@23125
   426
by (simp add: i_def complex_of_real_def)
huffman@23125
   427
huffman@23125
   428
lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
huffman@23125
   429
by (simp add: i_def complex_of_real_def)
huffman@23125
   430
huffman@23125
   431
lemma i_squared [simp]: "ii * ii = -1"
huffman@23125
   432
by (simp add: i_def)
huffman@23125
   433
huffman@23125
   434
lemma power2_i [simp]: "ii\<twosuperior> = -1"
huffman@23125
   435
by (simp add: power2_eq_square)
huffman@23125
   436
huffman@23125
   437
lemma inverse_i [simp]: "inverse ii = - ii"
huffman@23125
   438
by (rule inverse_unique, simp)
paulson@14354
   439
paulson@14354
   440
huffman@23125
   441
subsection {* Complex Conjugation *}
huffman@23125
   442
huffman@23125
   443
definition
huffman@23125
   444
  cnj :: "complex \<Rightarrow> complex" where
huffman@23125
   445
  "cnj z = Complex (Re z) (- Im z)"
huffman@23125
   446
huffman@23125
   447
lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
huffman@23125
   448
by (simp add: cnj_def)
huffman@23125
   449
huffman@23125
   450
lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
huffman@23125
   451
by (simp add: cnj_def)
huffman@23125
   452
huffman@23125
   453
lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
huffman@23125
   454
by (simp add: cnj_def)
huffman@23125
   455
huffman@23125
   456
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
huffman@44065
   457
by (simp add: complex_eq_iff)
huffman@23125
   458
huffman@23125
   459
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
huffman@23125
   460
by (simp add: cnj_def)
huffman@23125
   461
huffman@23125
   462
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
huffman@44065
   463
by (simp add: complex_eq_iff)
huffman@23125
   464
huffman@23125
   465
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
huffman@44065
   466
by (simp add: complex_eq_iff)
huffman@23125
   467
huffman@23125
   468
lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
huffman@44065
   469
by (simp add: complex_eq_iff)
huffman@23125
   470
huffman@23125
   471
lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
huffman@44065
   472
by (simp add: complex_eq_iff)
huffman@23125
   473
huffman@23125
   474
lemma complex_cnj_minus: "cnj (- x) = - cnj x"
huffman@44065
   475
by (simp add: complex_eq_iff)
huffman@23125
   476
huffman@23125
   477
lemma complex_cnj_one [simp]: "cnj 1 = 1"
huffman@44065
   478
by (simp add: complex_eq_iff)
huffman@23125
   479
huffman@23125
   480
lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
huffman@44065
   481
by (simp add: complex_eq_iff)
huffman@23125
   482
huffman@23125
   483
lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
huffman@23125
   484
by (simp add: complex_inverse_def)
paulson@14323
   485
huffman@23125
   486
lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
huffman@23125
   487
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
huffman@23125
   488
huffman@23125
   489
lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
huffman@23125
   490
by (induct n, simp_all add: complex_cnj_mult)
huffman@23125
   491
huffman@23125
   492
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
huffman@44065
   493
by (simp add: complex_eq_iff)
huffman@23125
   494
huffman@23125
   495
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
huffman@44065
   496
by (simp add: complex_eq_iff)
huffman@23125
   497
huffman@23125
   498
lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"
huffman@44065
   499
by (simp add: complex_eq_iff)
huffman@23125
   500
huffman@23125
   501
lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
huffman@44065
   502
by (simp add: complex_eq_iff)
huffman@23125
   503
huffman@23125
   504
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
huffman@23125
   505
by (simp add: complex_norm_def)
paulson@14323
   506
huffman@23125
   507
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
huffman@44065
   508
by (simp add: complex_eq_iff)
huffman@23125
   509
huffman@23125
   510
lemma complex_cnj_i [simp]: "cnj ii = - ii"
huffman@44065
   511
by (simp add: complex_eq_iff)
huffman@23125
   512
huffman@23125
   513
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
huffman@44065
   514
by (simp add: complex_eq_iff)
huffman@23125
   515
huffman@23125
   516
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
huffman@44065
   517
by (simp add: complex_eq_iff)
paulson@14354
   518
huffman@23125
   519
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
huffman@44065
   520
by (simp add: complex_eq_iff power2_eq_square)
huffman@23125
   521
huffman@23125
   522
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
huffman@23125
   523
by (simp add: norm_mult power2_eq_square)
huffman@23125
   524
huffman@44290
   525
lemma bounded_linear_cnj: "bounded_linear cnj"
huffman@44127
   526
  using complex_cnj_add complex_cnj_scaleR
huffman@44127
   527
  by (rule bounded_linear_intro [where K=1], simp)
paulson@14354
   528
huffman@44290
   529
lemmas tendsto_cnj [tendsto_intros] =
huffman@44290
   530
  bounded_linear.tendsto [OF bounded_linear_cnj]
huffman@44290
   531
huffman@44290
   532
lemmas isCont_cnj [simp] =
huffman@44290
   533
  bounded_linear.isCont [OF bounded_linear_cnj]
huffman@44290
   534
paulson@14354
   535
huffman@22972
   536
subsection{*The Functions @{term sgn} and @{term arg}*}
paulson@14323
   537
huffman@22972
   538
text {*------------ Argand -------------*}
huffman@20557
   539
wenzelm@21404
   540
definition
wenzelm@21404
   541
  arg :: "complex => real" where
huffman@20557
   542
  "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"
huffman@20557
   543
paulson@14374
   544
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
nipkow@24506
   545
by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute)
paulson@14323
   546
paulson@14323
   547
lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
huffman@20725
   548
by (simp add: i_def complex_of_real_def)
paulson@14323
   549
paulson@14374
   550
lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"
huffman@20725
   551
by (simp add: i_def complex_one_def)
paulson@14323
   552
paulson@14374
   553
lemma complex_eq_cancel_iff2 [simp]:
paulson@14377
   554
     "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
paulson@14377
   555
by (simp add: complex_of_real_def)
paulson@14323
   556
paulson@14374
   557
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
nipkow@24506
   558
by (simp add: complex_sgn_def divide_inverse)
paulson@14323
   559
paulson@14374
   560
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
nipkow@24506
   561
by (simp add: complex_sgn_def divide_inverse)
paulson@14323
   562
paulson@14323
   563
lemma complex_inverse_complex_split:
paulson@14323
   564
     "inverse(complex_of_real x + ii * complex_of_real y) =
paulson@14323
   565
      complex_of_real(x/(x ^ 2 + y ^ 2)) -
paulson@14323
   566
      ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
huffman@20725
   567
by (simp add: complex_of_real_def i_def diff_minus divide_inverse)
paulson@14323
   568
paulson@14323
   569
(*----------------------------------------------------------------------------*)
paulson@14323
   570
(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
paulson@14323
   571
(* many of the theorems are not used - so should they be kept?                *)
paulson@14323
   572
(*----------------------------------------------------------------------------*)
paulson@14323
   573
paulson@14354
   574
lemma cos_arg_i_mult_zero_pos:
paulson@14377
   575
   "0 < y ==> cos (arg(Complex 0 y)) = 0"
paulson@14373
   576
apply (simp add: arg_def abs_if)
paulson@14334
   577
apply (rule_tac a = "pi/2" in someI2, auto)
paulson@14334
   578
apply (rule order_less_trans [of _ 0], auto)
paulson@14323
   579
done
paulson@14323
   580
paulson@14354
   581
lemma cos_arg_i_mult_zero_neg:
paulson@14377
   582
   "y < 0 ==> cos (arg(Complex 0 y)) = 0"
paulson@14373
   583
apply (simp add: arg_def abs_if)
paulson@14334
   584
apply (rule_tac a = "- pi/2" in someI2, auto)
paulson@14334
   585
apply (rule order_trans [of _ 0], auto)
paulson@14323
   586
done
paulson@14323
   587
paulson@14374
   588
lemma cos_arg_i_mult_zero [simp]:
paulson@14377
   589
     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
paulson@14377
   590
by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
paulson@14323
   591
paulson@14323
   592
paulson@14323
   593
subsection{*Finally! Polar Form for Complex Numbers*}
paulson@14323
   594
huffman@20557
   595
definition
huffman@20557
   596
huffman@20557
   597
  (* abbreviation for (cos a + i sin a) *)
wenzelm@21404
   598
  cis :: "real => complex" where
huffman@20557
   599
  "cis a = Complex (cos a) (sin a)"
huffman@20557
   600
wenzelm@21404
   601
definition
huffman@20557
   602
  (* abbreviation for r*(cos a + i sin a) *)
wenzelm@21404
   603
  rcis :: "[real, real] => complex" where
huffman@20557
   604
  "rcis r a = complex_of_real r * cis a"
huffman@20557
   605
wenzelm@21404
   606
definition
huffman@20557
   607
  (* e ^ (x + iy) *)
wenzelm@21404
   608
  expi :: "complex => complex" where
huffman@20557
   609
  "expi z = complex_of_real(exp (Re z)) * cis (Im z)"
huffman@20557
   610
paulson@14374
   611
lemma complex_split_polar:
paulson@14377
   612
     "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
huffman@20725
   613
apply (induct z)
paulson@14377
   614
apply (auto simp add: polar_Ex complex_of_real_mult_Complex)
paulson@14323
   615
done
paulson@14323
   616
paulson@14354
   617
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
huffman@20725
   618
apply (induct z)
paulson@14377
   619
apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)
paulson@14323
   620
done
paulson@14323
   621
paulson@14374
   622
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
paulson@14373
   623
by (simp add: rcis_def cis_def)
paulson@14323
   624
paulson@14348
   625
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
paulson@14373
   626
by (simp add: rcis_def cis_def)
paulson@14323
   627
paulson@14377
   628
lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"
paulson@14377
   629
proof -
paulson@14377
   630
  have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)"
huffman@20725
   631
    by (simp only: power_mult_distrib right_distrib)
paulson@14377
   632
  thus ?thesis by simp
paulson@14377
   633
qed
paulson@14323
   634
paulson@14374
   635
lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
paulson@14377
   636
by (simp add: rcis_def cis_def sin_cos_squared_add2_mult)
paulson@14323
   637
huffman@23125
   638
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
huffman@23125
   639
by (simp add: cmod_def power2_eq_square)
huffman@23125
   640
paulson@14374
   641
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
huffman@23125
   642
by simp
paulson@14323
   643
paulson@14323
   644
paulson@14323
   645
(*---------------------------------------------------------------------------*)
paulson@14323
   646
(*  (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b)                      *)
paulson@14323
   647
(*---------------------------------------------------------------------------*)
paulson@14323
   648
paulson@14323
   649
lemma cis_rcis_eq: "cis a = rcis 1 a"
paulson@14373
   650
by (simp add: rcis_def)
paulson@14323
   651
paulson@14374
   652
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
paulson@15013
   653
by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib
paulson@15013
   654
              complex_of_real_def)
paulson@14323
   655
paulson@14323
   656
lemma cis_mult: "cis a * cis b = cis (a + b)"
paulson@14373
   657
by (simp add: cis_rcis_eq rcis_mult)
paulson@14323
   658
paulson@14374
   659
lemma cis_zero [simp]: "cis 0 = 1"
paulson@14377
   660
by (simp add: cis_def complex_one_def)
paulson@14323
   661
paulson@14374
   662
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
paulson@14373
   663
by (simp add: rcis_def)
paulson@14323
   664
paulson@14374
   665
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
paulson@14373
   666
by (simp add: rcis_def)
paulson@14323
   667
paulson@14323
   668
lemma complex_of_real_minus_one:
paulson@14323
   669
   "complex_of_real (-(1::real)) = -(1::complex)"
huffman@20725
   670
by (simp add: complex_of_real_def complex_one_def)
paulson@14323
   671
paulson@14374
   672
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
huffman@23125
   673
by (simp add: mult_assoc [symmetric])
paulson@14323
   674
paulson@14323
   675
paulson@14323
   676
lemma cis_real_of_nat_Suc_mult:
paulson@14323
   677
   "cis (real (Suc n) * a) = cis a * cis (real n * a)"
paulson@14377
   678
by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib)
paulson@14323
   679
paulson@14323
   680
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
paulson@14323
   681
apply (induct_tac "n")
paulson@14323
   682
apply (auto simp add: cis_real_of_nat_Suc_mult)
paulson@14323
   683
done
paulson@14323
   684
paulson@14374
   685
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
huffman@22890
   686
by (simp add: rcis_def power_mult_distrib DeMoivre)
paulson@14323
   687
paulson@14374
   688
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
huffman@20725
   689
by (simp add: cis_def complex_inverse_complex_split diff_minus)
paulson@14323
   690
paulson@14323
   691
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
huffman@22884
   692
by (simp add: divide_inverse rcis_def)
paulson@14323
   693
paulson@14323
   694
lemma cis_divide: "cis a / cis b = cis (a - b)"
haftmann@37887
   695
by (simp add: complex_divide_def cis_mult diff_minus)
paulson@14323
   696
paulson@14354
   697
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
paulson@14373
   698
apply (simp add: complex_divide_def)
paulson@14373
   699
apply (case_tac "r2=0", simp)
haftmann@37887
   700
apply (simp add: rcis_inverse rcis_mult diff_minus)
paulson@14323
   701
done
paulson@14323
   702
paulson@14374
   703
lemma Re_cis [simp]: "Re(cis a) = cos a"
paulson@14373
   704
by (simp add: cis_def)
paulson@14323
   705
paulson@14374
   706
lemma Im_cis [simp]: "Im(cis a) = sin a"
paulson@14373
   707
by (simp add: cis_def)
paulson@14323
   708
paulson@14323
   709
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
paulson@14334
   710
by (auto simp add: DeMoivre)
paulson@14323
   711
paulson@14323
   712
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
paulson@14334
   713
by (auto simp add: DeMoivre)
paulson@14323
   714
paulson@14323
   715
lemma expi_add: "expi(a + b) = expi(a) * expi(b)"
huffman@20725
   716
by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac)
paulson@14323
   717
paulson@14374
   718
lemma expi_zero [simp]: "expi (0::complex) = 1"
paulson@14373
   719
by (simp add: expi_def)
paulson@14323
   720
paulson@14374
   721
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
paulson@14373
   722
apply (insert rcis_Ex [of z])
huffman@23125
   723
apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
paulson@14334
   724
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
paulson@14323
   725
done
paulson@14323
   726
paulson@14387
   727
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
huffman@23125
   728
by (simp add: expi_def cis_def)
paulson@14387
   729
huffman@44065
   730
text {* Legacy theorem names *}
huffman@44065
   731
huffman@44065
   732
lemmas expand_complex_eq = complex_eq_iff
huffman@44065
   733
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
huffman@44065
   734
lemmas complex_equality = complex_eqI
huffman@44065
   735
paulson@13957
   736
end