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