src/HOL/Complex.thy
 author hoelzl Mon Jun 16 17:52:33 2014 +0200 (2014-06-16) changeset 57259 3a448982a74a parent 56889 48a745e1bde7 child 57512 cc97b347b301 permissions -rw-r--r--
add more derivative and continuity rules for complex-values functions
     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

    15 We use the @{text codatatype}-command to define the type of complex numbers. This might look strange

    16 at first, but allows us to use @{text primcorec} to define complex-functions by defining their

    17 real and imaginary result separate.

    18

    19 *}

    20

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

    22

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

    24   by (rule complex.collapse)

    25

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

    27   by (rule complex.expand) simp

    28

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

    30   by (auto intro: complex.expand)

    31

    32 subsection {* Addition and Subtraction *}

    33

    34 instantiation complex :: ab_group_add

    35 begin

    36

    37 primcorec zero_complex where

    38   "Re 0 = 0"

    39 | "Im 0 = 0"

    40

    41 primcorec plus_complex where

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

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

    44

    45 primcorec uminus_complex where

    46   "Re (- x) = - Re x"

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

    48

    49 primcorec minus_complex where

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

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

    52

    53 instance

    54   by intro_classes (simp_all add: complex_eq_iff)

    55

    56 end

    57

    58 subsection {* Multiplication and Division *}

    59

    60 instantiation complex :: field_inverse_zero

    61 begin

    62

    63 primcorec one_complex where

    64   "Re 1 = 1"

    65 | "Im 1 = 0"

    66

    67 primcorec times_complex where

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

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

    70

    71 primcorec inverse_complex where

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

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

    74

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

    76

    77 instance

    78   by intro_classes

    79      (simp_all add: complex_eq_iff divide_complex_def

    80       distrib_left distrib_right right_diff_distrib left_diff_distrib

    81       power2_eq_square add_divide_distrib [symmetric])

    82

    83 end

    84

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

    86   unfolding divide_complex_def by (simp add: add_divide_distrib)

    87

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

    89   unfolding divide_complex_def times_complex.sel inverse_complex.sel

    90   by (simp_all add: divide_simps)

    91

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

    93   by (simp add: power2_eq_square)

    94

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

    96   by (simp add: power2_eq_square)

    97

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

    99   by (induct n) simp_all

   100

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

   102   by (induct n) simp_all

   103

   104 subsection {* Scalar Multiplication *}

   105

   106 instantiation complex :: real_field

   107 begin

   108

   109 primcorec scaleR_complex where

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

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

   112

   113 instance

   114 proof

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

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

   117     by (simp add: complex_eq_iff distrib_left)

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

   119     by (simp add: complex_eq_iff distrib_right)

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

   121     by (simp add: complex_eq_iff mult_assoc)

   122   show "scaleR 1 x = x"

   123     by (simp add: complex_eq_iff)

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

   125     by (simp add: complex_eq_iff algebra_simps)

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

   127     by (simp add: complex_eq_iff algebra_simps)

   128 qed

   129

   130 end

   131

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

   133

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

   135   where "complex_of_real \<equiv> of_real"

   136

   137 declare [[coercion complex_of_real]]

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

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

   140

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

   142   by (induct n) simp_all

   143

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

   145   by (induct n) simp_all

   146

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

   148   by (cases z rule: int_diff_cases) simp

   149

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

   151   by (cases z rule: int_diff_cases) simp

   152

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

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

   155

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

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

   158

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

   160   by (simp add: of_real_def)

   161

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

   163   by (simp add: of_real_def)

   164

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

   166

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

   168   "Re ii = 0"

   169 | "Im ii = 1"

   170

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

   172   by (simp add: complex_eq_iff)

   173

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

   175   by (simp add: complex_eq_iff)

   176

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

   178   by (simp add: fun_eq_iff complex_eq)

   179

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

   181   by (simp add: complex_eq_iff)

   182

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

   184   by (simp add: power2_eq_square)

   185

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

   187   by (rule inverse_unique) simp

   188

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

   190   by (simp add: divide_complex_def)

   191

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

   193   by (simp add: mult_assoc [symmetric])

   194

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

   196   by (simp add: complex_eq_iff)

   197

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

   199   by (simp add: complex_eq_iff)

   200

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

   202   by (simp add: complex_eq_iff)

   203

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

   205   by (simp add: complex_eq_iff)

   206

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

   208   by (simp add: complex_eq_iff polar_Ex)

   209

   210 subsection {* Vector Norm *}

   211

   212 instantiation complex :: real_normed_field

   213 begin

   214

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

   216

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

   218   where "cmod \<equiv> norm"

   219

   220 definition complex_sgn_def:

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

   222

   223 definition dist_complex_def:

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

   225

   226 definition open_complex_def:

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

   228

   229 instance proof

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

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

   232     by (simp add: norm_complex_def complex_eq_iff)

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

   234     by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)

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

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

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

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

   239 qed (rule complex_sgn_def dist_complex_def open_complex_def)+

   240

   241 end

   242

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

   244   by (simp add: norm_complex_def)

   245

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

   247   by (simp add: norm_complex_def)

   248

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

   250   by (simp add: norm_mult cmod_unit_one)

   251

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

   253   unfolding norm_complex_def

   254   by (rule real_sqrt_sum_squares_ge1)

   255

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

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

   258

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

   260   by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp

   261

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

   263   by (simp add: norm_complex_def)

   264

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

   266   by (simp add: norm_complex_def)

   267

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

   269   apply (subst complex_eq)

   270   apply (rule order_trans)

   271   apply (rule norm_triangle_ineq)

   272   apply (simp add: norm_mult)

   273   done

   274

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

   276   by (simp add: norm_complex_def)

   277

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

   279   by (simp add: norm_complex_def)

   280

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

   282   by (simp add: norm_complex_def)

   283

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

   285   using abs_Re_le_cmod[of z] by auto

   286

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

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

   289      (auto simp add: norm_complex_def)

   290

   291 lemma abs_sqrt_wlog:

   292   fixes x::"'a::linordered_idom"

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

   294 by (metis abs_ge_zero assms power2_abs)

   295

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

   297   unfolding norm_complex_def

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

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

   300   apply (rule power2_le_imp_le)

   301   apply (simp_all add: power2_sum add_commute sum_squares_bound real_sqrt_mult [symmetric])

   302   done

   303

   304

   305 text {* Properties of complex signum. *}

   306

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

   308   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)

   309

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

   311   by (simp add: complex_sgn_def divide_inverse)

   312

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

   314   by (simp add: complex_sgn_def divide_inverse)

   315

   316

   317 subsection {* Completeness of the Complexes *}

   318

   319 lemma bounded_linear_Re: "bounded_linear Re"

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

   321

   322 lemma bounded_linear_Im: "bounded_linear Im"

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

   324

   325 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]

   326 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]

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

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

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

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

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

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

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

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

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

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

   337 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]

   338 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]

   339

   340 lemma tendsto_Complex [tendsto_intros]:

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

   342   by (auto intro!: tendsto_intros)

   343

   344 lemma tendsto_complex_iff:

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

   346 proof safe

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

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

   349     unfolding complex.collapse .

   350 qed (auto intro: tendsto_intros)

   351

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

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

   354   unfolding continuous_def tendsto_complex_iff ..

   355

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

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

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

   359   unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff

   360   by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right)

   361

   362 lemma has_field_derivative_Re[derivative_intros]:

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

   364   unfolding has_vector_derivative_complex_iff by safe

   365

   366 lemma has_field_derivative_Im[derivative_intros]:

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

   368   unfolding has_vector_derivative_complex_iff by safe

   369

   370 instance complex :: banach

   371 proof

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

   373   assume X: "Cauchy X"

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

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

   376   then show "convergent X"

   377     unfolding complex.collapse by (rule convergentI)

   378 qed

   379

   380 declare

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

   382

   383 subsection {* Complex Conjugation *}

   384

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

   386   "Re (cnj z) = Re z"

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

   388

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

   390   by (simp add: complex_eq_iff)

   391

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

   393   by (simp add: complex_eq_iff)

   394

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

   396   by (simp add: complex_eq_iff)

   397

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

   399   by (simp add: complex_eq_iff)

   400

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

   402   by (simp add: complex_eq_iff)

   403

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

   405   by (induct s rule: infinite_finite_induct) auto

   406

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

   408   by (simp add: complex_eq_iff)

   409

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

   411   by (simp add: complex_eq_iff)

   412

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

   414   by (simp add: complex_eq_iff)

   415

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

   417   by (simp add: complex_eq_iff)

   418

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

   420   by (induct s rule: infinite_finite_induct) auto

   421

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

   423   by (simp add: complex_eq_iff)

   424

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

   426   by (simp add: divide_complex_def)

   427

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

   429   by (induct n) simp_all

   430

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

   432   by (simp add: complex_eq_iff)

   433

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

   435   by (simp add: complex_eq_iff)

   436

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

   438   by (simp add: complex_eq_iff)

   439

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

   441   by (simp add: complex_eq_iff)

   442

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

   444   by (simp add: complex_eq_iff)

   445

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

   447   by (simp add: norm_complex_def)

   448

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

   450   by (simp add: complex_eq_iff)

   451

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

   453   by (simp add: complex_eq_iff)

   454

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

   456   by (simp add: complex_eq_iff)

   457

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

   459   by (simp add: complex_eq_iff)

   460

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

   462   by (simp add: complex_eq_iff power2_eq_square)

   463

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

   465   by (simp add: norm_mult power2_eq_square)

   466

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

   468   by (simp add: norm_complex_def power2_eq_square)

   469

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

   471   by simp

   472

   473 lemma bounded_linear_cnj: "bounded_linear cnj"

   474   using complex_cnj_add complex_cnj_scaleR

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

   476

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

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

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

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

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

   482

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

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

   485

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

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

   488

   489

   490 subsection{*Basic Lemmas*}

   491

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

   493   by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)

   494

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

   496   by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)

   497

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

   499 by (cases z)

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

   501          simp del: of_real_power)

   502

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

   504   by (auto simp add: Re_divide)

   505

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

   507   by (auto simp add: Im_divide)

   508

   509 lemma complex_div_gt_0:

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

   511 proof cases

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

   513 next

   514   assume "b \<noteq> 0"

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

   516     by (simp add: complex_eq_iff sum_power2_gt_zero_iff)

   517   then show ?thesis

   518     by (simp add: Re_divide Im_divide zero_less_divide_iff)

   519 qed

   520

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

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

   523   using complex_div_gt_0 by auto

   524

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

   526   by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0)

   527

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

   529   by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less)

   530

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

   532   by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0)

   533

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

   535   by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff)

   536

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

   538   by (metis not_le re_complex_div_gt_0)

   539

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

   541   by (metis im_complex_div_gt_0 not_le)

   542

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

   544   by (induct s rule: infinite_finite_induct) auto

   545

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

   547   by (induct s rule: infinite_finite_induct) auto

   548

   549 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)"

   550   unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..

   551

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

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

   554

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

   556   unfolding summable_complex_iff by simp

   557

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

   559   unfolding summable_complex_iff by blast

   560

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

   562   unfolding summable_complex_iff by blast

   563

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

   565   by (auto simp: Reals_def complex_eq_iff)

   566

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

   568   by (auto simp: complex_is_Real_iff complex_eq_iff)

   569

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

   571   by (simp add: complex_is_Real_iff norm_complex_def)

   572

   573 lemma series_comparison_complex:

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

   575   assumes sg: "summable g"

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

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

   578   shows "summable f"

   579 proof -

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

   581     by (metis abs_of_nonneg in_Reals_norm)

   582   show ?thesis

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

   584     using sg

   585     apply (auto simp: summable_def)

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

   587     apply (auto simp: g sums_Re)

   588     apply (metis fg g)

   589     done

   590 qed

   591

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

   593

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

   595

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

   597   "Re (cis a) = cos a"

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

   599

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

   601   by (simp add: complex_eq_iff)

   602

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

   604   by (simp add: norm_complex_def)

   605

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

   607   by (simp add: sgn_div_norm)

   608

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

   610   by (metis norm_cis norm_zero zero_neq_one)

   611

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

   613   by (simp add: complex_eq_iff cos_add sin_add)

   614

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

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

   617

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

   619   by (simp add: complex_eq_iff)

   620

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

   622   by (simp add: divide_complex_def cis_mult)

   623

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

   625   by (auto simp add: DeMoivre)

   626

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

   628   by (auto simp add: DeMoivre)

   629

   630 lemma cis_pi: "cis pi = -1"

   631   by (simp add: complex_eq_iff)

   632

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

   634

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

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

   637

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

   639   by (simp add: rcis_def)

   640

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

   642   by (simp add: rcis_def)

   643

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

   645   by (simp add: complex_eq_iff polar_Ex)

   646

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

   648   by (simp add: rcis_def norm_mult)

   649

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

   651   by (simp add: rcis_def)

   652

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

   654   by (simp add: rcis_def cis_mult)

   655

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

   657   by (simp add: rcis_def)

   658

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

   660   by (simp add: rcis_def)

   661

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

   663   by (simp add: rcis_def)

   664

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

   666   by (simp add: rcis_def power_mult_distrib DeMoivre)

   667

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

   669   by (simp add: divide_inverse rcis_def)

   670

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

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

   673

   674 subsubsection {* Complex exponential *}

   675

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

   677   where "expi \<equiv> exp"

   678

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

   680 proof -

   681   { fix n :: nat

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

   683       by (induct n)

   684          (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps

   685                         power2_eq_square real_of_nat_Suc add_nonneg_eq_0_iff

   686                         real_of_nat_def[symmetric])

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

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

   689       by (simp add: field_simps) }

   690   then show ?thesis

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

   692              intro!: sums_unique sums_add sums_mult sums_of_real sin_converges cos_converges)

   693 qed

   694

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

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

   697

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

   699   unfolding expi_def by simp

   700

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

   702   unfolding expi_def by simp

   703

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

   705 apply (insert rcis_Ex [of z])

   706 apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])

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

   708 done

   709

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

   711   by (simp add: expi_def complex_eq_iff)

   712

   713 subsubsection {* Complex argument *}

   714

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

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

   717

   718 lemma arg_zero: "arg 0 = 0"

   719   by (simp add: arg_def)

   720

   721 lemma arg_unique:

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

   723   shows "arg z = x"

   724 proof -

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

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

   727   proof

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

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

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

   731       unfolding d_def by simp

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

   733       by (simp_all add: complex_eq_iff)

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

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

   736     ultimately have "d = 0"

   737       unfolding sin_zero_iff even_mult_two_ex

   738       by (auto simp add: numeral_2_eq_2 less_Suc_eq)

   739     thus "a = x" unfolding d_def by simp

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

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

   742     unfolding arg_def by simp

   743 qed

   744

   745 lemma arg_correct:

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

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

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

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

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

   751   have b: "sgn z = cis b"

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

   753     by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)

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

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

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

   757     by (case_tac x rule: int_diff_cases)

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

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

   760   have "sgn z = cis c"

   761     unfolding b c_def

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

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

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

   765     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)

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

   767 qed

   768

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

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

   771

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

   773   by (simp add: arg_correct)

   774

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

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

   777

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

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

   780

   781 subsection {* Square root of complex numbers *}

   782

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

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

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

   786

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

   788   by (simp add: complex_eq_iff norm_complex_def)

   789

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

   791   by (simp add: complex_eq_iff norm_complex_def)

   792

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

   794   by simp

   795

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

   797   by simp

   798

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

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

   801

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

   803 proof cases

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

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

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

   807        (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)

   808 next

   809   assume "Im z \<noteq> 0"

   810   moreover

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

   812     by (simp add: norm_complex_def power2_eq_square)

   813   moreover

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

   815     by (simp add: norm_complex_def)

   816   ultimately show ?thesis

   817     by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq

   818                   field_simps real_sqrt_mult[symmetric] real_sqrt_divide)

   819 qed

   820

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

   822   by auto (metis power2_csqrt power_eq_0_iff)

   823

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

   825   by auto (metis power2_csqrt power2_eq_1_iff)

   826

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

   828   by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)

   829

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

   831   by (metis csqrt_principal le_less)

   832

   833 lemma csqrt_square:

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

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

   836 proof -

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

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

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

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

   841   ultimately show ?thesis

   842     by auto

   843 qed

   844

   845 lemma csqrt_minus [simp]:

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

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

   848 proof -

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

   850   proof (rule csqrt_square)

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

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

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

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

   855   qed

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

   857     by (simp add: power2_csqrt power_mult_distrib)

   858   finally show ?thesis .

   859 qed

   860

   861 text {* Legacy theorem names *}

   862

   863 lemmas expand_complex_eq = complex_eq_iff

   864 lemmas complex_Re_Im_cancel_iff = complex_eq_iff

   865 lemmas complex_equality = complex_eqI

   866 lemmas cmod_def = norm_complex_def

   867 lemmas complex_norm_def = norm_complex_def

   868 lemmas complex_divide_def = divide_complex_def

   869

   870 lemma legacy_Complex_simps:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   898

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

   900   by (metis Reals_of_real complex_of_real_def)

   901

   902 end