src/HOL/Complex.thy
 author huffman Sun Sep 04 21:03:54 2011 -0700 (2011-09-04) changeset 44715 1a17d8913976 parent 44712 1e490e891c88 child 44724 0b900a9d8023 permissions -rw-r--r--
```     1 (*  Title:       HOL/Complex.thy
```
```     2     Author:      Jacques D. Fleuriot
```
```     3     Copyright:   2001 University of Edinburgh
```
```     4     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
```
```     5 *)
```
```     6
```
```     7 header {* Complex Numbers: Rectangular and Polar Representations *}
```
```     8
```
```     9 theory Complex
```
```    10 imports Transcendental
```
```    11 begin
```
```    12
```
```    13 datatype complex = Complex real real
```
```    14
```
```    15 primrec
```
```    16   Re :: "complex \<Rightarrow> real"
```
```    17 where
```
```    18   Re: "Re (Complex x y) = x"
```
```    19
```
```    20 primrec
```
```    21   Im :: "complex \<Rightarrow> real"
```
```    22 where
```
```    23   Im: "Im (Complex x y) = y"
```
```    24
```
```    25 lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
```
```    26   by (induct z) simp
```
```    27
```
```    28 lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
```
```    29   by (induct x, induct y) simp
```
```    30
```
```    31 lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
```
```    32   by (induct x, induct y) simp
```
```    33
```
```    34
```
```    35 subsection {* Addition and Subtraction *}
```
```    36
```
```    37 instantiation complex :: ab_group_add
```
```    38 begin
```
```    39
```
```    40 definition
```
```    41   complex_zero_def: "0 = Complex 0 0"
```
```    42
```
```    43 definition
```
```    44   complex_add_def: "x + y = Complex (Re x + Re y) (Im x + Im y)"
```
```    45
```
```    46 definition
```
```    47   complex_minus_def: "- x = Complex (- Re x) (- Im x)"
```
```    48
```
```    49 definition
```
```    50   complex_diff_def: "x - (y\<Colon>complex) = x + - y"
```
```    51
```
```    52 lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
```
```    53   by (simp add: complex_zero_def)
```
```    54
```
```    55 lemma complex_Re_zero [simp]: "Re 0 = 0"
```
```    56   by (simp add: complex_zero_def)
```
```    57
```
```    58 lemma complex_Im_zero [simp]: "Im 0 = 0"
```
```    59   by (simp add: complex_zero_def)
```
```    60
```
```    61 lemma complex_add [simp]:
```
```    62   "Complex a b + Complex c d = Complex (a + c) (b + d)"
```
```    63   by (simp add: complex_add_def)
```
```    64
```
```    65 lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"
```
```    66   by (simp add: complex_add_def)
```
```    67
```
```    68 lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"
```
```    69   by (simp add: complex_add_def)
```
```    70
```
```    71 lemma complex_minus [simp]:
```
```    72   "- (Complex a b) = Complex (- a) (- b)"
```
```    73   by (simp add: complex_minus_def)
```
```    74
```
```    75 lemma complex_Re_minus [simp]: "Re (- x) = - Re x"
```
```    76   by (simp add: complex_minus_def)
```
```    77
```
```    78 lemma complex_Im_minus [simp]: "Im (- x) = - Im x"
```
```    79   by (simp add: complex_minus_def)
```
```    80
```
```    81 lemma complex_diff [simp]:
```
```    82   "Complex a b - Complex c d = Complex (a - c) (b - d)"
```
```    83   by (simp add: complex_diff_def)
```
```    84
```
```    85 lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"
```
```    86   by (simp add: complex_diff_def)
```
```    87
```
```    88 lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"
```
```    89   by (simp add: complex_diff_def)
```
```    90
```
```    91 instance
```
```    92   by intro_classes (simp_all add: complex_add_def complex_diff_def)
```
```    93
```
```    94 end
```
```    95
```
```    96
```
```    97
```
```    98 subsection {* Multiplication and Division *}
```
```    99
```
```   100 instantiation complex :: field_inverse_zero
```
```   101 begin
```
```   102
```
```   103 definition
```
```   104   complex_one_def: "1 = Complex 1 0"
```
```   105
```
```   106 definition
```
```   107   complex_mult_def: "x * y =
```
```   108     Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
```
```   109
```
```   110 definition
```
```   111   complex_inverse_def: "inverse x =
```
```   112     Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"
```
```   113
```
```   114 definition
```
```   115   complex_divide_def: "x / (y\<Colon>complex) = x * inverse y"
```
```   116
```
```   117 lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
```
```   118   by (simp add: complex_one_def)
```
```   119
```
```   120 lemma complex_Re_one [simp]: "Re 1 = 1"
```
```   121   by (simp add: complex_one_def)
```
```   122
```
```   123 lemma complex_Im_one [simp]: "Im 1 = 0"
```
```   124   by (simp add: complex_one_def)
```
```   125
```
```   126 lemma complex_mult [simp]:
```
```   127   "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
```
```   128   by (simp add: complex_mult_def)
```
```   129
```
```   130 lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"
```
```   131   by (simp add: complex_mult_def)
```
```   132
```
```   133 lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"
```
```   134   by (simp add: complex_mult_def)
```
```   135
```
```   136 lemma complex_inverse [simp]:
```
```   137   "inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))"
```
```   138   by (simp add: complex_inverse_def)
```
```   139
```
```   140 lemma complex_Re_inverse:
```
```   141   "Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
```
```   142   by (simp add: complex_inverse_def)
```
```   143
```
```   144 lemma complex_Im_inverse:
```
```   145   "Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
```
```   146   by (simp add: complex_inverse_def)
```
```   147
```
```   148 instance
```
```   149   by intro_classes (simp_all add: complex_mult_def
```
```   150   right_distrib left_distrib right_diff_distrib left_diff_distrib
```
```   151   complex_inverse_def complex_divide_def
```
```   152   power2_eq_square add_divide_distrib [symmetric]
```
```   153   complex_eq_iff)
```
```   154
```
```   155 end
```
```   156
```
```   157
```
```   158 subsection {* Numerals and Arithmetic *}
```
```   159
```
```   160 instantiation complex :: number_ring
```
```   161 begin
```
```   162
```
```   163 definition number_of_complex where
```
```   164   complex_number_of_def: "number_of w = (of_int w \<Colon> complex)"
```
```   165
```
```   166 instance
```
```   167   by intro_classes (simp only: complex_number_of_def)
```
```   168
```
```   169 end
```
```   170
```
```   171 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
```
```   172 by (induct n) simp_all
```
```   173
```
```   174 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
```
```   175 by (induct n) simp_all
```
```   176
```
```   177 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
```
```   178 by (cases z rule: int_diff_cases) simp
```
```   179
```
```   180 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
```
```   181 by (cases z rule: int_diff_cases) simp
```
```   182
```
```   183 lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v"
```
```   184 unfolding number_of_eq by (rule complex_Re_of_int)
```
```   185
```
```   186 lemma complex_Im_number_of [simp]: "Im (number_of v) = 0"
```
```   187 unfolding number_of_eq by (rule complex_Im_of_int)
```
```   188
```
```   189 lemma Complex_eq_number_of [simp]:
```
```   190   "(Complex a b = number_of w) = (a = number_of w \<and> b = 0)"
```
```   191 by (simp add: complex_eq_iff)
```
```   192
```
```   193
```
```   194 subsection {* Scalar Multiplication *}
```
```   195
```
```   196 instantiation complex :: real_field
```
```   197 begin
```
```   198
```
```   199 definition
```
```   200   complex_scaleR_def: "scaleR r x = Complex (r * Re x) (r * Im x)"
```
```   201
```
```   202 lemma complex_scaleR [simp]:
```
```   203   "scaleR r (Complex a b) = Complex (r * a) (r * b)"
```
```   204   unfolding complex_scaleR_def by simp
```
```   205
```
```   206 lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"
```
```   207   unfolding complex_scaleR_def by simp
```
```   208
```
```   209 lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"
```
```   210   unfolding complex_scaleR_def by simp
```
```   211
```
```   212 instance
```
```   213 proof
```
```   214   fix a b :: real and x y :: complex
```
```   215   show "scaleR a (x + y) = scaleR a x + scaleR a y"
```
```   216     by (simp add: complex_eq_iff right_distrib)
```
```   217   show "scaleR (a + b) x = scaleR a x + scaleR b x"
```
```   218     by (simp add: complex_eq_iff left_distrib)
```
```   219   show "scaleR a (scaleR b x) = scaleR (a * b) x"
```
```   220     by (simp add: complex_eq_iff mult_assoc)
```
```   221   show "scaleR 1 x = x"
```
```   222     by (simp add: complex_eq_iff)
```
```   223   show "scaleR a x * y = scaleR a (x * y)"
```
```   224     by (simp add: complex_eq_iff algebra_simps)
```
```   225   show "x * scaleR a y = scaleR a (x * y)"
```
```   226     by (simp add: complex_eq_iff algebra_simps)
```
```   227 qed
```
```   228
```
```   229 end
```
```   230
```
```   231
```
```   232 subsection{* Properties of Embedding from Reals *}
```
```   233
```
```   234 abbreviation
```
```   235   complex_of_real :: "real \<Rightarrow> complex" where
```
```   236     "complex_of_real \<equiv> of_real"
```
```   237
```
```   238 lemma complex_of_real_def: "complex_of_real r = Complex r 0"
```
```   239 by (simp add: of_real_def complex_scaleR_def)
```
```   240
```
```   241 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
```
```   242 by (simp add: complex_of_real_def)
```
```   243
```
```   244 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
```
```   245 by (simp add: complex_of_real_def)
```
```   246
```
```   247 lemma Complex_add_complex_of_real [simp]:
```
```   248      "Complex x y + complex_of_real r = Complex (x+r) y"
```
```   249 by (simp add: complex_of_real_def)
```
```   250
```
```   251 lemma complex_of_real_add_Complex [simp]:
```
```   252      "complex_of_real r + Complex x y = Complex (r+x) y"
```
```   253 by (simp add: complex_of_real_def)
```
```   254
```
```   255 lemma Complex_mult_complex_of_real:
```
```   256      "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
```
```   257 by (simp add: complex_of_real_def)
```
```   258
```
```   259 lemma complex_of_real_mult_Complex:
```
```   260      "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
```
```   261 by (simp add: complex_of_real_def)
```
```   262
```
```   263
```
```   264 subsection {* Vector Norm *}
```
```   265
```
```   266 instantiation complex :: real_normed_field
```
```   267 begin
```
```   268
```
```   269 definition complex_norm_def:
```
```   270   "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
```
```   271
```
```   272 abbreviation
```
```   273   cmod :: "complex \<Rightarrow> real" where
```
```   274   "cmod \<equiv> norm"
```
```   275
```
```   276 definition complex_sgn_def:
```
```   277   "sgn x = x /\<^sub>R cmod x"
```
```   278
```
```   279 definition dist_complex_def:
```
```   280   "dist x y = cmod (x - y)"
```
```   281
```
```   282 definition open_complex_def:
```
```   283   "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   284
```
```   285 lemmas cmod_def = complex_norm_def
```
```   286
```
```   287 lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
```
```   288   by (simp add: complex_norm_def)
```
```   289
```
```   290 instance proof
```
```   291   fix r :: real and x y :: complex and S :: "complex set"
```
```   292   show "0 \<le> norm x"
```
```   293     by (induct x) simp
```
```   294   show "(norm x = 0) = (x = 0)"
```
```   295     by (induct x) simp
```
```   296   show "norm (x + y) \<le> norm x + norm y"
```
```   297     by (induct x, induct y)
```
```   298        (simp add: real_sqrt_sum_squares_triangle_ineq)
```
```   299   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
```
```   300     by (induct x)
```
```   301        (simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult)
```
```   302   show "norm (x * y) = norm x * norm y"
```
```   303     by (induct x, induct y)
```
```   304        (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
```
```   305   show "sgn x = x /\<^sub>R cmod x"
```
```   306     by (rule complex_sgn_def)
```
```   307   show "dist x y = cmod (x - y)"
```
```   308     by (rule dist_complex_def)
```
```   309   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   310     by (rule open_complex_def)
```
```   311 qed
```
```   312
```
```   313 end
```
```   314
```
```   315 lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1"
```
```   316 by simp
```
```   317
```
```   318 lemma cmod_complex_polar [simp]:
```
```   319      "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
```
```   320 by (simp add: norm_mult)
```
```   321
```
```   322 lemma complex_Re_le_cmod: "Re x \<le> cmod x"
```
```   323 unfolding complex_norm_def
```
```   324 by (rule real_sqrt_sum_squares_ge1)
```
```   325
```
```   326 lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x"
```
```   327 by (rule order_trans [OF _ norm_ge_zero], simp)
```
```   328
```
```   329 lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a"
```
```   330 by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
```
```   331
```
```   332 lemmas real_sum_squared_expand = power2_sum [where 'a=real]
```
```   333
```
```   334 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
```
```   335 by (cases x) simp
```
```   336
```
```   337 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
```
```   338 by (cases x) simp
```
```   339
```
```   340 subsection {* Completeness of the Complexes *}
```
```   341
```
```   342 lemma bounded_linear_Re: "bounded_linear Re"
```
```   343   by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
```
```   344
```
```   345 lemma bounded_linear_Im: "bounded_linear Im"
```
```   346   by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
```
```   347
```
```   348 lemmas tendsto_Re [tendsto_intros] =
```
```   349   bounded_linear.tendsto [OF bounded_linear_Re]
```
```   350
```
```   351 lemmas tendsto_Im [tendsto_intros] =
```
```   352   bounded_linear.tendsto [OF bounded_linear_Im]
```
```   353
```
```   354 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
```
```   355 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
```
```   356 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
```
```   357 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
```
```   358
```
```   359 lemma tendsto_Complex [tendsto_intros]:
```
```   360   assumes "(f ---> a) net" and "(g ---> b) net"
```
```   361   shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) net"
```
```   362 proof (rule tendstoI)
```
```   363   fix r :: real assume "0 < r"
```
```   364   hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
```
```   365   have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) net"
```
```   366     using `(f ---> a) net` and `0 < r / sqrt 2` by (rule tendstoD)
```
```   367   moreover
```
```   368   have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) net"
```
```   369     using `(g ---> b) net` and `0 < r / sqrt 2` by (rule tendstoD)
```
```   370   ultimately
```
```   371   show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) net"
```
```   372     by (rule eventually_elim2)
```
```   373        (simp add: dist_norm real_sqrt_sum_squares_less)
```
```   374 qed
```
```   375
```
```   376 lemma LIMSEQ_Complex:
```
```   377   "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b"
```
```   378 by (rule tendsto_Complex)
```
```   379
```
```   380 instance complex :: banach
```
```   381 proof
```
```   382   fix X :: "nat \<Rightarrow> complex"
```
```   383   assume X: "Cauchy X"
```
```   384   from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
```
```   385     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   386   from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
```
```   387     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
```
```   388   have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
```
```   389     using LIMSEQ_Complex [OF 1 2] by simp
```
```   390   thus "convergent X"
```
```   391     by (rule convergentI)
```
```   392 qed
```
```   393
```
```   394
```
```   395 subsection {* The Complex Number @{term "\<i>"} *}
```
```   396
```
```   397 definition
```
```   398   "ii" :: complex  ("\<i>") where
```
```   399   i_def: "ii \<equiv> Complex 0 1"
```
```   400
```
```   401 lemma complex_Re_i [simp]: "Re ii = 0"
```
```   402 by (simp add: i_def)
```
```   403
```
```   404 lemma complex_Im_i [simp]: "Im ii = 1"
```
```   405 by (simp add: i_def)
```
```   406
```
```   407 lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
```
```   408 by (simp add: i_def)
```
```   409
```
```   410 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
```
```   411 by (simp add: complex_eq_iff)
```
```   412
```
```   413 lemma complex_i_not_one [simp]: "ii \<noteq> 1"
```
```   414 by (simp add: complex_eq_iff)
```
```   415
```
```   416 lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w"
```
```   417 by (simp add: complex_eq_iff)
```
```   418
```
```   419 lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
```
```   420 by (simp add: complex_eq_iff)
```
```   421
```
```   422 lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
```
```   423 by (simp add: complex_eq_iff)
```
```   424
```
```   425 lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
```
```   426 by (simp add: i_def complex_of_real_def)
```
```   427
```
```   428 lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
```
```   429 by (simp add: i_def complex_of_real_def)
```
```   430
```
```   431 lemma i_squared [simp]: "ii * ii = -1"
```
```   432 by (simp add: i_def)
```
```   433
```
```   434 lemma power2_i [simp]: "ii\<twosuperior> = -1"
```
```   435 by (simp add: power2_eq_square)
```
```   436
```
```   437 lemma inverse_i [simp]: "inverse ii = - ii"
```
```   438 by (rule inverse_unique, simp)
```
```   439
```
```   440
```
```   441 subsection {* Complex Conjugation *}
```
```   442
```
```   443 definition
```
```   444   cnj :: "complex \<Rightarrow> complex" where
```
```   445   "cnj z = Complex (Re z) (- Im z)"
```
```   446
```
```   447 lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
```
```   448 by (simp add: cnj_def)
```
```   449
```
```   450 lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
```
```   451 by (simp add: cnj_def)
```
```   452
```
```   453 lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
```
```   454 by (simp add: cnj_def)
```
```   455
```
```   456 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
```
```   457 by (simp add: complex_eq_iff)
```
```   458
```
```   459 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
```
```   460 by (simp add: cnj_def)
```
```   461
```
```   462 lemma complex_cnj_zero [simp]: "cnj 0 = 0"
```
```   463 by (simp add: complex_eq_iff)
```
```   464
```
```   465 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
```
```   466 by (simp add: complex_eq_iff)
```
```   467
```
```   468 lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
```
```   469 by (simp add: complex_eq_iff)
```
```   470
```
```   471 lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
```
```   472 by (simp add: complex_eq_iff)
```
```   473
```
```   474 lemma complex_cnj_minus: "cnj (- x) = - cnj x"
```
```   475 by (simp add: complex_eq_iff)
```
```   476
```
```   477 lemma complex_cnj_one [simp]: "cnj 1 = 1"
```
```   478 by (simp add: complex_eq_iff)
```
```   479
```
```   480 lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
```
```   481 by (simp add: complex_eq_iff)
```
```   482
```
```   483 lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
```
```   484 by (simp add: complex_inverse_def)
```
```   485
```
```   486 lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
```
```   487 by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
```
```   488
```
```   489 lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
```
```   490 by (induct n, simp_all add: complex_cnj_mult)
```
```   491
```
```   492 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
```
```   493 by (simp add: complex_eq_iff)
```
```   494
```
```   495 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
```
```   496 by (simp add: complex_eq_iff)
```
```   497
```
```   498 lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"
```
```   499 by (simp add: complex_eq_iff)
```
```   500
```
```   501 lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
```
```   502 by (simp add: complex_eq_iff)
```
```   503
```
```   504 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
```
```   505 by (simp add: complex_norm_def)
```
```   506
```
```   507 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
```
```   508 by (simp add: complex_eq_iff)
```
```   509
```
```   510 lemma complex_cnj_i [simp]: "cnj ii = - ii"
```
```   511 by (simp add: complex_eq_iff)
```
```   512
```
```   513 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
```
```   514 by (simp add: complex_eq_iff)
```
```   515
```
```   516 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
```
```   517 by (simp add: complex_eq_iff)
```
```   518
```
```   519 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
```
```   520 by (simp add: complex_eq_iff power2_eq_square)
```
```   521
```
```   522 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
```
```   523 by (simp add: norm_mult power2_eq_square)
```
```   524
```
```   525 lemma bounded_linear_cnj: "bounded_linear cnj"
```
```   526   using complex_cnj_add complex_cnj_scaleR
```
```   527   by (rule bounded_linear_intro [where K=1], simp)
```
```   528
```
```   529 lemmas tendsto_cnj [tendsto_intros] =
```
```   530   bounded_linear.tendsto [OF bounded_linear_cnj]
```
```   531
```
```   532 lemmas isCont_cnj [simp] =
```
```   533   bounded_linear.isCont [OF bounded_linear_cnj]
```
```   534
```
```   535
```
```   536 subsection{*The Functions @{term sgn} and @{term arg}*}
```
```   537
```
```   538 text {*------------ Argand -------------*}
```
```   539
```
```   540 definition
```
```   541   arg :: "complex => real" where
```
```   542   "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"
```
```   543
```
```   544 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
```
```   545 by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute)
```
```   546
```
```   547 lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
```
```   548 by (simp add: i_def complex_of_real_def)
```
```   549
```
```   550 lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"
```
```   551 by (simp add: i_def complex_one_def)
```
```   552
```
```   553 lemma complex_eq_cancel_iff2 [simp]:
```
```   554      "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
```
```   555 by (simp add: complex_of_real_def)
```
```   556
```
```   557 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
```
```   558 by (simp add: complex_sgn_def divide_inverse)
```
```   559
```
```   560 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
```
```   561 by (simp add: complex_sgn_def divide_inverse)
```
```   562
```
```   563 lemma complex_inverse_complex_split:
```
```   564      "inverse(complex_of_real x + ii * complex_of_real y) =
```
```   565       complex_of_real(x/(x ^ 2 + y ^ 2)) -
```
```   566       ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
```
```   567 by (simp add: complex_of_real_def i_def diff_minus divide_inverse)
```
```   568
```
```   569 (*----------------------------------------------------------------------------*)
```
```   570 (* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
```
```   571 (* many of the theorems are not used - so should they be kept?                *)
```
```   572 (*----------------------------------------------------------------------------*)
```
```   573
```
```   574 lemma cos_arg_i_mult_zero_pos:
```
```   575    "0 < y ==> cos (arg(Complex 0 y)) = 0"
```
```   576 apply (simp add: arg_def abs_if)
```
```   577 apply (rule_tac a = "pi/2" in someI2, auto)
```
```   578 apply (rule order_less_trans [of _ 0], auto)
```
```   579 done
```
```   580
```
```   581 lemma cos_arg_i_mult_zero_neg:
```
```   582    "y < 0 ==> cos (arg(Complex 0 y)) = 0"
```
```   583 apply (simp add: arg_def abs_if)
```
```   584 apply (rule_tac a = "- pi/2" in someI2, auto)
```
```   585 apply (rule order_trans [of _ 0], auto)
```
```   586 done
```
```   587
```
```   588 lemma cos_arg_i_mult_zero [simp]:
```
```   589      "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
```
```   590 by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
```
```   591
```
```   592
```
```   593 subsection{*Finally! Polar Form for Complex Numbers*}
```
```   594
```
```   595 text {* An abbreviation for @{text "cos a + i sin a"}. *}
```
```   596
```
```   597 definition cis :: "real \<Rightarrow> complex" where
```
```   598   "cis a = Complex (cos a) (sin a)"
```
```   599
```
```   600 text {* An abbreviation for @{text "r(cos a + i sin a)"}. *}
```
```   601
```
```   602 definition rcis :: "[real, real] \<Rightarrow> complex" where
```
```   603   "rcis r a = complex_of_real r * cis a"
```
```   604
```
```   605 abbreviation expi :: "complex \<Rightarrow> complex"
```
```   606   where "expi \<equiv> exp"
```
```   607
```
```   608 lemma cis_conv_exp: "cis b = exp (Complex 0 b)"
```
```   609 proof (rule complex_eqI)
```
```   610   { fix n have "Complex 0 b ^ n =
```
```   611     real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"
```
```   612       apply (induct n)
```
```   613       apply (simp add: cos_coeff_def sin_coeff_def)
```
```   614       apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)
```
```   615       done } note * = this
```
```   616   show "Re (cis b) = Re (exp (Complex 0 b))"
```
```   617     unfolding exp_def cis_def cos_def
```
```   618     by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],
```
```   619       simp add: * mult_assoc [symmetric])
```
```   620   show "Im (cis b) = Im (exp (Complex 0 b))"
```
```   621     unfolding exp_def cis_def sin_def
```
```   622     by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],
```
```   623       simp add: * mult_assoc [symmetric])
```
```   624 qed
```
```   625
```
```   626 lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"
```
```   627   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp
```
```   628
```
```   629 lemma complex_split_polar:
```
```   630      "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
```
```   631 apply (induct z)
```
```   632 apply (auto simp add: polar_Ex complex_of_real_mult_Complex)
```
```   633 done
```
```   634
```
```   635 lemma rcis_Ex: "\<exists>r a. z = rcis r a"
```
```   636 apply (induct z)
```
```   637 apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)
```
```   638 done
```
```   639
```
```   640 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
```
```   641 by (simp add: rcis_def cis_def)
```
```   642
```
```   643 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
```
```   644 by (simp add: rcis_def cis_def)
```
```   645
```
```   646 lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"
```
```   647 proof -
```
```   648   have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)"
```
```   649     by (simp only: power_mult_distrib right_distrib)
```
```   650   thus ?thesis by simp
```
```   651 qed
```
```   652
```
```   653 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
```
```   654 by (simp add: rcis_def cis_def sin_cos_squared_add2_mult)
```
```   655
```
```   656 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
```
```   657 by (simp add: cmod_def power2_eq_square)
```
```   658
```
```   659 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
```
```   660 by simp
```
```   661
```
```   662 lemma cis_rcis_eq: "cis a = rcis 1 a"
```
```   663 by (simp add: rcis_def)
```
```   664
```
```   665 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
```
```   666 by (simp add: rcis_def cis_def cos_add sin_add right_distrib right_diff_distrib
```
```   667               complex_of_real_def)
```
```   668
```
```   669 lemma cis_mult: "cis a * cis b = cis (a + b)"
```
```   670 by (simp add: cis_rcis_eq rcis_mult)
```
```   671
```
```   672 lemma cis_zero [simp]: "cis 0 = 1"
```
```   673 by (simp add: cis_def complex_one_def)
```
```   674
```
```   675 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
```
```   676 by (simp add: rcis_def)
```
```   677
```
```   678 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
```
```   679 by (simp add: rcis_def)
```
```   680
```
```   681 lemma complex_of_real_minus_one:
```
```   682    "complex_of_real (-(1::real)) = -(1::complex)"
```
```   683 by (simp add: complex_of_real_def complex_one_def)
```
```   684
```
```   685 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
```
```   686 by (simp add: mult_assoc [symmetric])
```
```   687
```
```   688
```
```   689 lemma cis_real_of_nat_Suc_mult:
```
```   690    "cis (real (Suc n) * a) = cis a * cis (real n * a)"
```
```   691 by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib)
```
```   692
```
```   693 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
```
```   694 apply (induct_tac "n")
```
```   695 apply (auto simp add: cis_real_of_nat_Suc_mult)
```
```   696 done
```
```   697
```
```   698 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
```
```   699 by (simp add: rcis_def power_mult_distrib DeMoivre)
```
```   700
```
```   701 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
```
```   702 by (simp add: cis_def complex_inverse_complex_split diff_minus)
```
```   703
```
```   704 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
```
```   705 by (simp add: divide_inverse rcis_def)
```
```   706
```
```   707 lemma cis_divide: "cis a / cis b = cis (a - b)"
```
```   708 by (simp add: complex_divide_def cis_mult diff_minus)
```
```   709
```
```   710 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
```
```   711 apply (simp add: complex_divide_def)
```
```   712 apply (case_tac "r2=0", simp)
```
```   713 apply (simp add: rcis_inverse rcis_mult diff_minus)
```
```   714 done
```
```   715
```
```   716 lemma Re_cis [simp]: "Re(cis a) = cos a"
```
```   717 by (simp add: cis_def)
```
```   718
```
```   719 lemma Im_cis [simp]: "Im(cis a) = sin a"
```
```   720 by (simp add: cis_def)
```
```   721
```
```   722 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
```
```   723 by (auto simp add: DeMoivre)
```
```   724
```
```   725 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
```
```   726 by (auto simp add: DeMoivre)
```
```   727
```
```   728 lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
```
```   729 apply (insert rcis_Ex [of z])
```
```   730 apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
```
```   731 apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
```
```   732 done
```
```   733
```
```   734 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
```
```   735 by (simp add: expi_def cis_def)
```
```   736
```
```   737 text {* Legacy theorem names *}
```
```   738
```
```   739 lemmas expand_complex_eq = complex_eq_iff
```
```   740 lemmas complex_Re_Im_cancel_iff = complex_eq_iff
```
```   741 lemmas complex_equality = complex_eqI
```
```   742
```
```   743 end
```