src/HOL/Complex.thy
 author haftmann Sun Oct 19 18:05:26 2014 +0200 (2014-10-19) changeset 58709 efdc6c533bd3 parent 58146 d91c1e50b36e child 58740 cb9d84d3e7f2 permissions -rw-r--r--
prefer generic elimination rules for even/odd over specialized unfold rules for nat
     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 text {*

    14 We use the @{text codatatype} command to define the type of complex numbers. This allows us to use

    15 @{text primcorec} to define complex functions by defining their real and imaginary result

    16 separately.

    17 *}

    18

    19 codatatype complex = Complex (Re: real) (Im: real)

    20

    21 lemma complex_surj: "Complex (Re z) (Im z) = z"

    22   by (rule complex.collapse)

    23

    24 lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"

    25   by (rule complex.expand) simp

    26

    27 lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"

    28   by (auto intro: complex.expand)

    29

    30 subsection {* Addition and Subtraction *}

    31

    32 instantiation complex :: ab_group_add

    33 begin

    34

    35 primcorec zero_complex where

    36   "Re 0 = 0"

    37 | "Im 0 = 0"

    38

    39 primcorec plus_complex where

    40   "Re (x + y) = Re x + Re y"

    41 | "Im (x + y) = Im x + Im y"

    42

    43 primcorec uminus_complex where

    44   "Re (- x) = - Re x"

    45 | "Im (- x) = - Im x"

    46

    47 primcorec minus_complex where

    48   "Re (x - y) = Re x - Re y"

    49 | "Im (x - y) = Im x - Im y"

    50

    51 instance

    52   by intro_classes (simp_all add: complex_eq_iff)

    53

    54 end

    55

    56 subsection {* Multiplication and Division *}

    57

    58 instantiation complex :: field_inverse_zero

    59 begin

    60

    61 primcorec one_complex where

    62   "Re 1 = 1"

    63 | "Im 1 = 0"

    64

    65 primcorec times_complex where

    66   "Re (x * y) = Re x * Re y - Im x * Im y"

    67 | "Im (x * y) = Re x * Im y + Im x * Re y"

    68

    69 primcorec inverse_complex where

    70   "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"

    71 | "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"

    72

    73 definition "x / (y\<Colon>complex) = x * inverse y"

    74

    75 instance

    76   by intro_classes

    77      (simp_all add: complex_eq_iff divide_complex_def

    78       distrib_left distrib_right right_diff_distrib left_diff_distrib

    79       power2_eq_square add_divide_distrib [symmetric])

    80

    81 end

    82

    83 lemma Re_divide: "Re (x / y) = (Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"

    84   unfolding divide_complex_def by (simp add: add_divide_distrib)

    85

    86 lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"

    87   unfolding divide_complex_def times_complex.sel inverse_complex.sel

    88   by (simp_all add: divide_simps)

    89

    90 lemma Re_power2: "Re (x ^ 2) = (Re x)^2 - (Im x)^2"

    91   by (simp add: power2_eq_square)

    92

    93 lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x"

    94   by (simp add: power2_eq_square)

    95

    96 lemma Re_power_real: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n "

    97   by (induct n) simp_all

    98

    99 lemma Im_power_real: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0"

   100   by (induct n) simp_all

   101

   102 subsection {* Scalar Multiplication *}

   103

   104 instantiation complex :: real_field

   105 begin

   106

   107 primcorec scaleR_complex where

   108   "Re (scaleR r x) = r * Re x"

   109 | "Im (scaleR r x) = r * Im x"

   110

   111 instance

   112 proof

   113   fix a b :: real and x y :: complex

   114   show "scaleR a (x + y) = scaleR a x + scaleR a y"

   115     by (simp add: complex_eq_iff distrib_left)

   116   show "scaleR (a + b) x = scaleR a x + scaleR b x"

   117     by (simp add: complex_eq_iff distrib_right)

   118   show "scaleR a (scaleR b x) = scaleR (a * b) x"

   119     by (simp add: complex_eq_iff mult.assoc)

   120   show "scaleR 1 x = x"

   121     by (simp add: complex_eq_iff)

   122   show "scaleR a x * y = scaleR a (x * y)"

   123     by (simp add: complex_eq_iff algebra_simps)

   124   show "x * scaleR a y = scaleR a (x * y)"

   125     by (simp add: complex_eq_iff algebra_simps)

   126 qed

   127

   128 end

   129

   130 subsection {* Numerals, Arithmetic, and Embedding from Reals *}

   131

   132 abbreviation complex_of_real :: "real \<Rightarrow> complex"

   133   where "complex_of_real \<equiv> of_real"

   134

   135 declare [[coercion complex_of_real]]

   136 declare [[coercion "of_int :: int \<Rightarrow> complex"]]

   137 declare [[coercion "of_nat :: nat \<Rightarrow> complex"]]

   138

   139 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"

   140   by (induct n) simp_all

   141

   142 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"

   143   by (induct n) simp_all

   144

   145 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"

   146   by (cases z rule: int_diff_cases) simp

   147

   148 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"

   149   by (cases z rule: int_diff_cases) simp

   150

   151 lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"

   152   using complex_Re_of_int [of "numeral v"] by simp

   153

   154 lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"

   155   using complex_Im_of_int [of "numeral v"] by simp

   156

   157 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"

   158   by (simp add: of_real_def)

   159

   160 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"

   161   by (simp add: of_real_def)

   162

   163 subsection {* The Complex Number $i$ *}

   164

   165 primcorec "ii" :: complex  ("\<i>") where

   166   "Re ii = 0"

   167 | "Im ii = 1"

   168

   169 lemma Complex_eq[simp]: "Complex a b = a + \<i> * b"

   170   by (simp add: complex_eq_iff)

   171

   172 lemma complex_eq: "a = Re a + \<i> * Im a"

   173   by (simp add: complex_eq_iff)

   174

   175 lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))"

   176   by (simp add: fun_eq_iff complex_eq)

   177

   178 lemma i_squared [simp]: "ii * ii = -1"

   179   by (simp add: complex_eq_iff)

   180

   181 lemma power2_i [simp]: "ii\<^sup>2 = -1"

   182   by (simp add: power2_eq_square)

   183

   184 lemma inverse_i [simp]: "inverse ii = - ii"

   185   by (rule inverse_unique) simp

   186

   187 lemma divide_i [simp]: "x / ii = - ii * x"

   188   by (simp add: divide_complex_def)

   189

   190 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"

   191   by (simp add: mult.assoc [symmetric])

   192

   193 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"

   194   by (simp add: complex_eq_iff)

   195

   196 lemma complex_i_not_one [simp]: "ii \<noteq> 1"

   197   by (simp add: complex_eq_iff)

   198

   199 lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"

   200   by (simp add: complex_eq_iff)

   201

   202 lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> - numeral w"

   203   by (simp add: complex_eq_iff)

   204

   205 lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)"

   206   by (simp add: complex_eq_iff polar_Ex)

   207

   208 subsection {* Vector Norm *}

   209

   210 instantiation complex :: real_normed_field

   211 begin

   212

   213 definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"

   214

   215 abbreviation cmod :: "complex \<Rightarrow> real"

   216   where "cmod \<equiv> norm"

   217

   218 definition complex_sgn_def:

   219   "sgn x = x /\<^sub>R cmod x"

   220

   221 definition dist_complex_def:

   222   "dist x y = cmod (x - y)"

   223

   224 definition open_complex_def:

   225   "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"

   226

   227 instance proof

   228   fix r :: real and x y :: complex and S :: "complex set"

   229   show "(norm x = 0) = (x = 0)"

   230     by (simp add: norm_complex_def complex_eq_iff)

   231   show "norm (x + y) \<le> norm x + norm y"

   232     by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)

   233   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"

   234     by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult)

   235   show "norm (x * y) = norm x * norm y"

   236     by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps)

   237 qed (rule complex_sgn_def dist_complex_def open_complex_def)+

   238

   239 end

   240

   241 lemma norm_ii [simp]: "norm ii = 1"

   242   by (simp add: norm_complex_def)

   243

   244 lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1"

   245   by (simp add: norm_complex_def)

   246

   247 lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>"

   248   by (simp add: norm_mult cmod_unit_one)

   249

   250 lemma complex_Re_le_cmod: "Re x \<le> cmod x"

   251   unfolding norm_complex_def

   252   by (rule real_sqrt_sum_squares_ge1)

   253

   254 lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"

   255   by (rule order_trans [OF _ norm_ge_zero]) simp

   256

   257 lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a"

   258   by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp

   259

   260 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"

   261   by (simp add: norm_complex_def)

   262

   263 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"

   264   by (simp add: norm_complex_def)

   265

   266 lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>"

   267   apply (subst complex_eq)

   268   apply (rule order_trans)

   269   apply (rule norm_triangle_ineq)

   270   apply (simp add: norm_mult)

   271   done

   272

   273 lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>"

   274   by (simp add: norm_complex_def)

   275

   276 lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>"

   277   by (simp add: norm_complex_def)

   278

   279 lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2"

   280   by (simp add: norm_complex_def)

   281

   282 lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z"

   283   using abs_Re_le_cmod[of z] by auto

   284

   285 lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0"

   286   by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2])

   287      (auto simp add: norm_complex_def)

   288

   289 lemma abs_sqrt_wlog:

   290   fixes x::"'a::linordered_idom"

   291   assumes "\<And>x::'a. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"

   292 by (metis abs_ge_zero assms power2_abs)

   293

   294 lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"

   295   unfolding norm_complex_def

   296   apply (rule abs_sqrt_wlog [where x="Re z"])

   297   apply (rule abs_sqrt_wlog [where x="Im z"])

   298   apply (rule power2_le_imp_le)

   299   apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])

   300   done

   301

   302

   303 text {* Properties of complex signum. *}

   304

   305 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"

   306   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)

   307

   308 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"

   309   by (simp add: complex_sgn_def divide_inverse)

   310

   311 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"

   312   by (simp add: complex_sgn_def divide_inverse)

   313

   314

   315 subsection {* Completeness of the Complexes *}

   316

   317 lemma bounded_linear_Re: "bounded_linear Re"

   318   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)

   319

   320 lemma bounded_linear_Im: "bounded_linear Im"

   321   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)

   322

   323 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]

   324 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]

   325 lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]

   326 lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]

   327 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]

   328 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]

   329 lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]

   330 lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]

   331 lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]

   332 lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]

   333 lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]

   334 lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]

   335 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]

   336 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]

   337

   338 lemma tendsto_Complex [tendsto_intros]:

   339   "(f ---> a) F \<Longrightarrow> (g ---> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"

   340   by (auto intro!: tendsto_intros)

   341

   342 lemma tendsto_complex_iff:

   343   "(f ---> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) ---> Re x) F \<and> ((\<lambda>x. Im (f x)) ---> Im x) F)"

   344 proof safe

   345   assume "((\<lambda>x. Re (f x)) ---> Re x) F" "((\<lambda>x. Im (f x)) ---> Im x) F"

   346   from tendsto_Complex[OF this] show "(f ---> x) F"

   347     unfolding complex.collapse .

   348 qed (auto intro: tendsto_intros)

   349

   350 lemma continuous_complex_iff: "continuous F f \<longleftrightarrow>

   351     continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))"

   352   unfolding continuous_def tendsto_complex_iff ..

   353

   354 lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow>

   355     ((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and>

   356     ((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F"

   357   unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff

   358   by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right)

   359

   360 lemma has_field_derivative_Re[derivative_intros]:

   361   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F"

   362   unfolding has_vector_derivative_complex_iff by safe

   363

   364 lemma has_field_derivative_Im[derivative_intros]:

   365   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F"

   366   unfolding has_vector_derivative_complex_iff by safe

   367

   368 instance complex :: banach

   369 proof

   370   fix X :: "nat \<Rightarrow> complex"

   371   assume X: "Cauchy X"

   372   then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"

   373     by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)

   374   then show "convergent X"

   375     unfolding complex.collapse by (rule convergentI)

   376 qed

   377

   378 declare

   379   DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]

   380

   381 subsection {* Complex Conjugation *}

   382

   383 primcorec cnj :: "complex \<Rightarrow> complex" where

   384   "Re (cnj z) = Re z"

   385 | "Im (cnj z) = - Im z"

   386

   387 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"

   388   by (simp add: complex_eq_iff)

   389

   390 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"

   391   by (simp add: complex_eq_iff)

   392

   393 lemma complex_cnj_zero [simp]: "cnj 0 = 0"

   394   by (simp add: complex_eq_iff)

   395

   396 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"

   397   by (simp add: complex_eq_iff)

   398

   399 lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"

   400   by (simp add: complex_eq_iff)

   401

   402 lemma cnj_setsum [simp]: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"

   403   by (induct s rule: infinite_finite_induct) auto

   404

   405 lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"

   406   by (simp add: complex_eq_iff)

   407

   408 lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"

   409   by (simp add: complex_eq_iff)

   410

   411 lemma complex_cnj_one [simp]: "cnj 1 = 1"

   412   by (simp add: complex_eq_iff)

   413

   414 lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"

   415   by (simp add: complex_eq_iff)

   416

   417 lemma cnj_setprod [simp]: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"

   418   by (induct s rule: infinite_finite_induct) auto

   419

   420 lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"

   421   by (simp add: complex_eq_iff)

   422

   423 lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"

   424   by (simp add: divide_complex_def)

   425

   426 lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"

   427   by (induct n) simp_all

   428

   429 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"

   430   by (simp add: complex_eq_iff)

   431

   432 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"

   433   by (simp add: complex_eq_iff)

   434

   435 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"

   436   by (simp add: complex_eq_iff)

   437

   438 lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"

   439   by (simp add: complex_eq_iff)

   440

   441 lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"

   442   by (simp add: complex_eq_iff)

   443

   444 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"

   445   by (simp add: norm_complex_def)

   446

   447 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"

   448   by (simp add: complex_eq_iff)

   449

   450 lemma complex_cnj_i [simp]: "cnj ii = - ii"

   451   by (simp add: complex_eq_iff)

   452

   453 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"

   454   by (simp add: complex_eq_iff)

   455

   456 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"

   457   by (simp add: complex_eq_iff)

   458

   459 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"

   460   by (simp add: complex_eq_iff power2_eq_square)

   461

   462 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"

   463   by (simp add: norm_mult power2_eq_square)

   464

   465 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"

   466   by (simp add: norm_complex_def power2_eq_square)

   467

   468 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"

   469   by simp

   470

   471 lemma bounded_linear_cnj: "bounded_linear cnj"

   472   using complex_cnj_add complex_cnj_scaleR

   473   by (rule bounded_linear_intro [where K=1], simp)

   474

   475 lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]

   476 lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]

   477 lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]

   478 lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]

   479 lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]

   480

   481 lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"

   482   by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)

   483

   484 lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"

   485   by (simp add: sums_def lim_cnj cnj_setsum [symmetric] del: cnj_setsum)

   486

   487

   488 subsection{*Basic Lemmas*}

   489

   490 lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"

   491   by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)

   492

   493 lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"

   494   by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)

   495

   496 lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"

   497 by (cases z)

   498    (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]

   499          simp del: of_real_power)

   500

   501 lemma re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0"

   502   by (auto simp add: Re_divide)

   503

   504 lemma im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0"

   505   by (auto simp add: Im_divide)

   506

   507 lemma complex_div_gt_0:

   508   "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"

   509 proof cases

   510   assume "b = 0" then show ?thesis by auto

   511 next

   512   assume "b \<noteq> 0"

   513   then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"

   514     by (simp add: complex_eq_iff sum_power2_gt_zero_iff)

   515   then show ?thesis

   516     by (simp add: Re_divide Im_divide zero_less_divide_iff)

   517 qed

   518

   519 lemma re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0"

   520   and im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0"

   521   using complex_div_gt_0 by auto

   522

   523 lemma re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"

   524   by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0)

   525

   526 lemma im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"

   527   by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less)

   528

   529 lemma re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"

   530   by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0)

   531

   532 lemma im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"

   533   by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff)

   534

   535 lemma re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"

   536   by (metis not_le re_complex_div_gt_0)

   537

   538 lemma im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"

   539   by (metis im_complex_div_gt_0 not_le)

   540

   541 lemma Re_setsum[simp]: "Re (setsum f s) = (\<Sum>x\<in>s. Re (f x))"

   542   by (induct s rule: infinite_finite_induct) auto

   543

   544 lemma Im_setsum[simp]: "Im (setsum f s) = (\<Sum>x\<in>s. Im(f x))"

   545   by (induct s rule: infinite_finite_induct) auto

   546

   547 lemma sums_complex_iff: "f sums x \<longleftrightarrow> ((\<lambda>x. Re (f x)) sums Re x) \<and> ((\<lambda>x. Im (f x)) sums Im x)"

   548   unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..

   549

   550 lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and>  summable (\<lambda>x. Im (f x))"

   551   unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)

   552

   553 lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"

   554   unfolding summable_complex_iff by simp

   555

   556 lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"

   557   unfolding summable_complex_iff by blast

   558

   559 lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"

   560   unfolding summable_complex_iff by blast

   561

   562 lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"

   563   by (auto simp: Reals_def complex_eq_iff)

   564

   565 lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"

   566   by (auto simp: complex_is_Real_iff complex_eq_iff)

   567

   568 lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm(z) = abs(Re z)"

   569   by (simp add: complex_is_Real_iff norm_complex_def)

   570

   571 lemma series_comparison_complex:

   572   fixes f:: "nat \<Rightarrow> 'a::banach"

   573   assumes sg: "summable g"

   574      and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"

   575      and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"

   576   shows "summable f"

   577 proof -

   578   have g: "\<And>n. cmod (g n) = Re (g n)" using assms

   579     by (metis abs_of_nonneg in_Reals_norm)

   580   show ?thesis

   581     apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])

   582     using sg

   583     apply (auto simp: summable_def)

   584     apply (rule_tac x="Re s" in exI)

   585     apply (auto simp: g sums_Re)

   586     apply (metis fg g)

   587     done

   588 qed

   589

   590 subsection{*Finally! Polar Form for Complex Numbers*}

   591

   592 subsubsection {* $\cos \theta + i \sin \theta$ *}

   593

   594 primcorec cis :: "real \<Rightarrow> complex" where

   595   "Re (cis a) = cos a"

   596 | "Im (cis a) = sin a"

   597

   598 lemma cis_zero [simp]: "cis 0 = 1"

   599   by (simp add: complex_eq_iff)

   600

   601 lemma norm_cis [simp]: "norm (cis a) = 1"

   602   by (simp add: norm_complex_def)

   603

   604 lemma sgn_cis [simp]: "sgn (cis a) = cis a"

   605   by (simp add: sgn_div_norm)

   606

   607 lemma cis_neq_zero [simp]: "cis a \<noteq> 0"

   608   by (metis norm_cis norm_zero zero_neq_one)

   609

   610 lemma cis_mult: "cis a * cis b = cis (a + b)"

   611   by (simp add: complex_eq_iff cos_add sin_add)

   612

   613 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"

   614   by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)

   615

   616 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"

   617   by (simp add: complex_eq_iff)

   618

   619 lemma cis_divide: "cis a / cis b = cis (a - b)"

   620   by (simp add: divide_complex_def cis_mult)

   621

   622 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"

   623   by (auto simp add: DeMoivre)

   624

   625 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"

   626   by (auto simp add: DeMoivre)

   627

   628 lemma cis_pi: "cis pi = -1"

   629   by (simp add: complex_eq_iff)

   630

   631 subsubsection {* $r(\cos \theta + i \sin \theta)$ *}

   632

   633 definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" where

   634   "rcis r a = complex_of_real r * cis a"

   635

   636 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"

   637   by (simp add: rcis_def)

   638

   639 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"

   640   by (simp add: rcis_def)

   641

   642 lemma rcis_Ex: "\<exists>r a. z = rcis r a"

   643   by (simp add: complex_eq_iff polar_Ex)

   644

   645 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"

   646   by (simp add: rcis_def norm_mult)

   647

   648 lemma cis_rcis_eq: "cis a = rcis 1 a"

   649   by (simp add: rcis_def)

   650

   651 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"

   652   by (simp add: rcis_def cis_mult)

   653

   654 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"

   655   by (simp add: rcis_def)

   656

   657 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"

   658   by (simp add: rcis_def)

   659

   660 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"

   661   by (simp add: rcis_def)

   662

   663 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"

   664   by (simp add: rcis_def power_mult_distrib DeMoivre)

   665

   666 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"

   667   by (simp add: divide_inverse rcis_def)

   668

   669 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"

   670   by (simp add: rcis_def cis_divide [symmetric])

   671

   672 subsubsection {* Complex exponential *}

   673

   674 abbreviation expi :: "complex \<Rightarrow> complex"

   675   where "expi \<equiv> exp"

   676

   677 lemma cis_conv_exp: "cis b = exp (\<i> * b)"

   678 proof -

   679   { fix n :: nat

   680     have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"

   681       by (induct n)

   682          (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps

   683                         power2_eq_square real_of_nat_Suc add_nonneg_eq_0_iff

   684                         real_of_nat_def[symmetric])

   685     then have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =

   686         of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"

   687       by (simp add: field_simps) }

   688   then show ?thesis

   689     by (auto simp add: cis.ctr exp_def simp del: of_real_mult

   690              intro!: sums_unique sums_add sums_mult sums_of_real sin_converges cos_converges)

   691 qed

   692

   693 lemma expi_def: "expi z = exp (Re z) * cis (Im z)"

   694   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp

   695

   696 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"

   697   unfolding expi_def by simp

   698

   699 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"

   700   unfolding expi_def by simp

   701

   702 lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"

   703 apply (insert rcis_Ex [of z])

   704 apply (auto simp add: expi_def rcis_def mult.assoc [symmetric])

   705 apply (rule_tac x = "ii * complex_of_real a" in exI, auto)

   706 done

   707

   708 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"

   709   by (simp add: expi_def complex_eq_iff)

   710

   711 subsubsection {* Complex argument *}

   712

   713 definition arg :: "complex \<Rightarrow> real" where

   714   "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"

   715

   716 lemma arg_zero: "arg 0 = 0"

   717   by (simp add: arg_def)

   718

   719 lemma arg_unique:

   720   assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"

   721   shows "arg z = x"

   722 proof -

   723   from assms have "z \<noteq> 0" by auto

   724   have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"

   725   proof

   726     fix a def d \<equiv> "a - x"

   727     assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"

   728     from a assms have "- (2*pi) < d \<and> d < 2*pi"

   729       unfolding d_def by simp

   730     moreover from a assms have "cos a = cos x" and "sin a = sin x"

   731       by (simp_all add: complex_eq_iff)

   732     hence cos: "cos d = 1" unfolding d_def cos_diff by simp

   733     moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)

   734     ultimately have "d = 0"

   735       unfolding sin_zero_iff

   736       by (auto simp add: numeral_2_eq_2 less_Suc_eq elim!: evenE)

   737     thus "a = x" unfolding d_def by simp

   738   qed (simp add: assms del: Re_sgn Im_sgn)

   739   with z \<noteq> 0 show "arg z = x"

   740     unfolding arg_def by simp

   741 qed

   742

   743 lemma arg_correct:

   744   assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"

   745 proof (simp add: arg_def assms, rule someI_ex)

   746   obtain r a where z: "z = rcis r a" using rcis_Ex by fast

   747   with assms have "r \<noteq> 0" by auto

   748   def b \<equiv> "if 0 < r then a else a + pi"

   749   have b: "sgn z = cis b"

   750     unfolding z b_def rcis_def using r \<noteq> 0

   751     by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)

   752   have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"

   753     by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)

   754   have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"

   755     by (case_tac x rule: int_diff_cases)

   756        (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)

   757   def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"

   758   have "sgn z = cis c"

   759     unfolding b c_def

   760     by (simp add: cis_divide [symmetric] cis_2pi_int)

   761   moreover have "- pi < c \<and> c \<le> pi"

   762     using ceiling_correct [of "(b - pi) / (2*pi)"]

   763     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)

   764   ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast

   765 qed

   766

   767 lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"

   768   by (cases "z = 0") (simp_all add: arg_zero arg_correct)

   769

   770 lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"

   771   by (simp add: arg_correct)

   772

   773 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"

   774   by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)

   775

   776 lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0"

   777   using cis_arg [of y] by (simp add: complex_eq_iff)

   778

   779 subsection {* Square root of complex numbers *}

   780

   781 primcorec csqrt :: "complex \<Rightarrow> complex" where

   782   "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"

   783 | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"

   784

   785 lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)"

   786   by (simp add: complex_eq_iff norm_complex_def)

   787

   788 lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>"

   789   by (simp add: complex_eq_iff norm_complex_def)

   790

   791 lemma csqrt_0 [simp]: "csqrt 0 = 0"

   792   by simp

   793

   794 lemma csqrt_1 [simp]: "csqrt 1 = 1"

   795   by simp

   796

   797 lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2"

   798   by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)

   799

   800 lemma power2_csqrt[algebra]: "(csqrt z)\<^sup>2 = z"

   801 proof cases

   802   assume "Im z = 0" then show ?thesis

   803     using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]

   804     by (cases "0::real" "Re z" rule: linorder_cases)

   805        (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)

   806 next

   807   assume "Im z \<noteq> 0"

   808   moreover

   809   have "cmod z * cmod z - Re z * Re z = Im z * Im z"

   810     by (simp add: norm_complex_def power2_eq_square)

   811   moreover

   812   have "\<bar>Re z\<bar> \<le> cmod z"

   813     by (simp add: norm_complex_def)

   814   ultimately show ?thesis

   815     by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq

   816                   field_simps real_sqrt_mult[symmetric] real_sqrt_divide)

   817 qed

   818

   819 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"

   820   by auto (metis power2_csqrt power_eq_0_iff)

   821

   822 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"

   823   by auto (metis power2_csqrt power2_eq_1_iff)

   824

   825 lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"

   826   by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)

   827

   828 lemma Re_csqrt: "0 \<le> Re (csqrt z)"

   829   by (metis csqrt_principal le_less)

   830

   831 lemma csqrt_square:

   832   assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)"

   833   shows "csqrt (b^2) = b"

   834 proof -

   835   have "csqrt (b^2) = b \<or> csqrt (b^2) = - b"

   836     unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt)

   837   moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0"

   838     using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff)

   839   ultimately show ?thesis

   840     by auto

   841 qed

   842

   843 lemma csqrt_minus [simp]:

   844   assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)"

   845   shows "csqrt (- x) = \<i> * csqrt x"

   846 proof -

   847   have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x"

   848   proof (rule csqrt_square)

   849     have "Im (csqrt x) \<le> 0"

   850       using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)

   851     then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)"

   852       by (auto simp add: Re_csqrt simp del: csqrt.simps)

   853   qed

   854   also have "(\<i> * csqrt x)^2 = - x"

   855     by (simp add: power2_csqrt power_mult_distrib)

   856   finally show ?thesis .

   857 qed

   858

   859 text {* Legacy theorem names *}

   860

   861 lemmas expand_complex_eq = complex_eq_iff

   862 lemmas complex_Re_Im_cancel_iff = complex_eq_iff

   863 lemmas complex_equality = complex_eqI

   864 lemmas cmod_def = norm_complex_def

   865 lemmas complex_norm_def = norm_complex_def

   866 lemmas complex_divide_def = divide_complex_def

   867

   868 lemma legacy_Complex_simps:

   869   shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"

   870     and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"

   871     and complex_minus: "- (Complex a b) = Complex (- a) (- b)"

   872     and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"

   873     and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"

   874     and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"

   875     and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"

   876     and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"

   877     and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"

   878     and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"

   879     and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"

   880     and Complex_eq_i: "(Complex x y = ii) = (x = 0 \<and> y = 1)"

   881     and i_mult_Complex: "ii * Complex a b = Complex (- b) a"

   882     and Complex_mult_i: "Complex a b * ii = Complex (- b) a"

   883     and i_complex_of_real: "ii * complex_of_real r = Complex 0 r"

   884     and complex_of_real_i: "complex_of_real r * ii = Complex 0 r"

   885     and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"

   886     and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"

   887     and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"

   888     and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"

   889     and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"

   890     and complex_cn: "cnj (Complex a b) = Complex a (- b)"

   891     and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"

   892     and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"

   893     and complex_of_real_def: "complex_of_real r = Complex r 0"

   894     and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"

   895   by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)

   896

   897 lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"

   898   by (metis Reals_of_real complex_of_real_def)

   899

   900 end