src/HOL/Complex.thy
 author blanchet Sun May 04 18:14:58 2014 +0200 (2014-05-04) changeset 56846 9df717fef2bb parent 56541 0e3abadbef39 child 56889 48a745e1bde7 permissions -rw-r--r--
renamed 'xxx_size' to 'size_xxx' for old datatype package
     1 (*  Title:       HOL/Complex.thy

     2     Author:      Jacques D. Fleuriot

     3     Copyright:   2001 University of Edinburgh

     4     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4

     5 *)

     6

     7 header {* Complex Numbers: Rectangular and Polar Representations *}

     8

     9 theory Complex

    10 imports Transcendental

    11 begin

    12

    13 datatype complex = Complex real real

    14

    15 primrec Re :: "complex \<Rightarrow> real"

    16   where Re: "Re (Complex x y) = x"

    17

    18 primrec Im :: "complex \<Rightarrow> real"

    19   where Im: "Im (Complex x y) = y"

    20

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

    22   by (induct z) simp

    23

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

    25   by (induct x, induct y) simp

    26

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

    28   by (induct x, induct y) simp

    29

    30

    31 subsection {* Addition and Subtraction *}

    32

    33 instantiation complex :: ab_group_add

    34 begin

    35

    36 definition complex_zero_def:

    37   "0 = Complex 0 0"

    38

    39 definition complex_add_def:

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

    41

    42 definition complex_minus_def:

    43   "- x = Complex (- Re x) (- Im x)"

    44

    45 definition complex_diff_def:

    46   "x - (y\<Colon>complex) = x + - y"

    47

    48 lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"

    49   by (simp add: complex_zero_def)

    50

    51 lemma complex_Re_zero [simp]: "Re 0 = 0"

    52   by (simp add: complex_zero_def)

    53

    54 lemma complex_Im_zero [simp]: "Im 0 = 0"

    55   by (simp add: complex_zero_def)

    56

    57 lemma complex_add [simp]:

    58   "Complex a b + Complex c d = Complex (a + c) (b + d)"

    59   by (simp add: complex_add_def)

    60

    61 lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"

    62   by (simp add: complex_add_def)

    63

    64 lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"

    65   by (simp add: complex_add_def)

    66

    67 lemma complex_minus [simp]:

    68   "- (Complex a b) = Complex (- a) (- b)"

    69   by (simp add: complex_minus_def)

    70

    71 lemma complex_Re_minus [simp]: "Re (- x) = - Re x"

    72   by (simp add: complex_minus_def)

    73

    74 lemma complex_Im_minus [simp]: "Im (- x) = - Im x"

    75   by (simp add: complex_minus_def)

    76

    77 lemma complex_diff [simp]:

    78   "Complex a b - Complex c d = Complex (a - c) (b - d)"

    79   by (simp add: complex_diff_def)

    80

    81 lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"

    82   by (simp add: complex_diff_def)

    83

    84 lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"

    85   by (simp add: complex_diff_def)

    86

    87 instance

    88   by intro_classes (simp_all add: complex_add_def complex_diff_def)

    89

    90 end

    91

    92

    93 subsection {* Multiplication and Division *}

    94

    95 instantiation complex :: field_inverse_zero

    96 begin

    97

    98 definition complex_one_def:

    99   "1 = Complex 1 0"

   100

   101 definition complex_mult_def:

   102   "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"

   103

   104 definition complex_inverse_def:

   105   "inverse x =

   106     Complex (Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)) (- Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2))"

   107

   108 definition complex_divide_def:

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

   110

   111 lemma Complex_eq_1 [simp]:

   112   "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"

   113   by (simp add: complex_one_def)

   114

   115 lemma Complex_eq_neg_1 [simp]:

   116   "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"

   117   by (simp add: complex_one_def)

   118

   119 lemma complex_Re_one [simp]: "Re 1 = 1"

   120   by (simp add: complex_one_def)

   121

   122 lemma complex_Im_one [simp]: "Im 1 = 0"

   123   by (simp add: complex_one_def)

   124

   125 lemma complex_mult [simp]:

   126   "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"

   127   by (simp add: complex_mult_def)

   128

   129 lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"

   130   by (simp add: complex_mult_def)

   131

   132 lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"

   133   by (simp add: complex_mult_def)

   134

   135 lemma complex_inverse [simp]:

   136   "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"

   137   by (simp add: complex_inverse_def)

   138

   139 lemma complex_Re_inverse:

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

   141   by (simp add: complex_inverse_def)

   142

   143 lemma complex_Im_inverse:

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

   145   by (simp add: complex_inverse_def)

   146

   147 instance

   148   by intro_classes (simp_all add: complex_mult_def

   149     distrib_left distrib_right right_diff_distrib left_diff_distrib

   150     complex_inverse_def complex_divide_def

   151     power2_eq_square add_divide_distrib [symmetric]

   152     complex_eq_iff)

   153

   154 end

   155

   156

   157 subsection {* Numerals and Arithmetic *}

   158

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

   160   by (induct n) simp_all

   161

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

   163   by (induct n) simp_all

   164

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

   166   by (cases z rule: int_diff_cases) simp

   167

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

   169   by (cases z rule: int_diff_cases) simp

   170

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

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

   173

   174 lemma complex_Re_neg_numeral [simp]: "Re (- numeral v) = - numeral v"

   175   using complex_Re_of_int [of "- numeral v"] by simp

   176

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

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

   179

   180 lemma complex_Im_neg_numeral [simp]: "Im (- numeral v) = 0"

   181   using complex_Im_of_int [of "- numeral v"] by simp

   182

   183 lemma Complex_eq_numeral [simp]:

   184   "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"

   185   by (simp add: complex_eq_iff)

   186

   187 lemma Complex_eq_neg_numeral [simp]:

   188   "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"

   189   by (simp add: complex_eq_iff)

   190

   191

   192 subsection {* Scalar Multiplication *}

   193

   194 instantiation complex :: real_field

   195 begin

   196

   197 definition complex_scaleR_def:

   198   "scaleR r x = Complex (r * Re x) (r * Im x)"

   199

   200 lemma complex_scaleR [simp]:

   201   "scaleR r (Complex a b) = Complex (r * a) (r * b)"

   202   unfolding complex_scaleR_def by simp

   203

   204 lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"

   205   unfolding complex_scaleR_def by simp

   206

   207 lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"

   208   unfolding complex_scaleR_def by simp

   209

   210 instance

   211 proof

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

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

   214     by (simp add: complex_eq_iff distrib_left)

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

   216     by (simp add: complex_eq_iff distrib_right)

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

   218     by (simp add: complex_eq_iff mult_assoc)

   219   show "scaleR 1 x = x"

   220     by (simp add: complex_eq_iff)

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

   222     by (simp add: complex_eq_iff algebra_simps)

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

   224     by (simp add: complex_eq_iff algebra_simps)

   225 qed

   226

   227 end

   228

   229

   230 subsection{* Properties of Embedding from Reals *}

   231

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

   233   where "complex_of_real \<equiv> of_real"

   234

   235 declare [[coercion complex_of_real]]

   236

   237 lemma complex_of_real_def: "complex_of_real r = Complex r 0"

   238   by (simp add: of_real_def complex_scaleR_def)

   239

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

   241   by (simp add: complex_of_real_def)

   242

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

   244   by (simp add: complex_of_real_def)

   245

   246 lemma Complex_add_complex_of_real [simp]:

   247   shows "Complex x y + complex_of_real r = Complex (x+r) y"

   248   by (simp add: complex_of_real_def)

   249

   250 lemma complex_of_real_add_Complex [simp]:

   251   shows "complex_of_real r + Complex x y = Complex (r+x) y"

   252   by (simp add: complex_of_real_def)

   253

   254 lemma Complex_mult_complex_of_real:

   255   shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"

   256   by (simp add: complex_of_real_def)

   257

   258 lemma complex_of_real_mult_Complex:

   259   shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"

   260   by (simp add: complex_of_real_def)

   261

   262 lemma complex_eq_cancel_iff2 [simp]:

   263   shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"

   264   by (simp add: complex_of_real_def)

   265

   266 lemma complex_split_polar:

   267      "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"

   268   by (simp add: complex_eq_iff polar_Ex)

   269

   270

   271 subsection {* Vector Norm *}

   272

   273 instantiation complex :: real_normed_field

   274 begin

   275

   276 definition complex_norm_def:

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

   278

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

   280   where "cmod \<equiv> norm"

   281

   282 definition complex_sgn_def:

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

   284

   285 definition dist_complex_def:

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

   287

   288 definition open_complex_def:

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

   290

   291 lemmas cmod_def = complex_norm_def

   292

   293 lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"

   294   by (simp add: complex_norm_def)

   295

   296 instance proof

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

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

   299     by (induct x) simp

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

   301     by (induct x, induct y)

   302        (simp add: real_sqrt_sum_squares_triangle_ineq)

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

   304     by (induct x)

   305        (simp add: power_mult_distrib distrib_left [symmetric] real_sqrt_mult)

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

   307     by (induct x, induct y)

   308        (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)

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

   310     by (rule complex_sgn_def)

   311   show "dist x y = cmod (x - y)"

   312     by (rule dist_complex_def)

   313   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"

   314     by (rule open_complex_def)

   315 qed

   316

   317 end

   318

   319 lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"

   320   by simp

   321

   322 lemma cmod_complex_polar:

   323   "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"

   324   by (simp add: norm_mult)

   325

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

   327   unfolding complex_norm_def

   328   by (rule real_sqrt_sum_squares_ge1)

   329

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

   331   by (rule order_trans [OF _ norm_ge_zero], simp)

   332

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

   334   by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)

   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

   343 lemma abs_sqrt_wlog:

   344   fixes x::"'a::linordered_idom"

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

   346 by (metis abs_ge_zero assms power2_abs)

   347

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

   349   unfolding complex_norm_def

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

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

   352   apply (rule power2_le_imp_le)

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

   354   done

   355

   356

   357 text {* Properties of complex signum. *}

   358

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

   360   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)

   361

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

   363   by (simp add: complex_sgn_def divide_inverse)

   364

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

   366   by (simp add: complex_sgn_def divide_inverse)

   367

   368

   369 subsection {* Completeness of the Complexes *}

   370

   371 lemma bounded_linear_Re: "bounded_linear Re"

   372   by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)

   373

   374 lemma bounded_linear_Im: "bounded_linear Im"

   375   by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)

   376

   377 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]

   378 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]

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

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

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

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

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

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

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

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

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

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

   389 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]

   390 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]

   391

   392 lemma tendsto_Complex [tendsto_intros]:

   393   assumes "(f ---> a) F" and "(g ---> b) F"

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

   395 proof (rule tendstoI)

   396   fix r :: real assume "0 < r"

   397   hence "0 < r / sqrt 2" by simp

   398   have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"

   399     using (f ---> a) F and 0 < r / sqrt 2 by (rule tendstoD)

   400   moreover

   401   have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"

   402     using (g ---> b) F and 0 < r / sqrt 2 by (rule tendstoD)

   403   ultimately

   404   show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"

   405     by (rule eventually_elim2)

   406        (simp add: dist_norm real_sqrt_sum_squares_less)

   407 qed

   408

   409

   410 lemma tendsto_complex_iff:

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

   412 proof -

   413   have f: "f = (\<lambda>x. Complex (Re (f x)) (Im (f x)))" and x: "x = Complex (Re x) (Im x)"

   414     by simp_all

   415   show ?thesis

   416     apply (subst f)

   417     apply (subst x)

   418     apply (intro iffI tendsto_Complex conjI)

   419     apply (simp_all add: tendsto_Re tendsto_Im)

   420     done

   421 qed

   422

   423 instance complex :: banach

   424 proof

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

   426   assume X: "Cauchy X"

   427   from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"

   428     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)

   429   from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"

   430     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)

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

   432     using tendsto_Complex [OF 1 2] by simp

   433   thus "convergent X"

   434     by (rule convergentI)

   435 qed

   436

   437 declare

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

   439

   440

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

   442

   443 definition "ii" :: complex  ("\<i>")

   444   where i_def: "ii \<equiv> Complex 0 1"

   445

   446 lemma complex_Re_i [simp]: "Re ii = 0"

   447   by (simp add: i_def)

   448

   449 lemma complex_Im_i [simp]: "Im ii = 1"

   450   by (simp add: i_def)

   451

   452 lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"

   453   by (simp add: i_def)

   454

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

   456   by (simp add: i_def)

   457

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

   459   by (simp add: complex_eq_iff)

   460

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

   462   by (simp add: complex_eq_iff)

   463

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

   465   by (simp add: complex_eq_iff)

   466

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

   468   by (simp add: complex_eq_iff)

   469

   470 lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"

   471   by (simp add: complex_eq_iff)

   472

   473 lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"

   474   by (simp add: complex_eq_iff)

   475

   476 lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"

   477   by (simp add: i_def complex_of_real_def)

   478

   479 lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"

   480   by (simp add: i_def complex_of_real_def)

   481

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

   483   by (simp add: i_def)

   484

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

   486   by (simp add: power2_eq_square)

   487

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

   489   by (rule inverse_unique, simp)

   490

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

   492   by (simp add: mult_assoc [symmetric])

   493

   494

   495 subsection {* Complex Conjugation *}

   496

   497 definition cnj :: "complex \<Rightarrow> complex" where

   498   "cnj z = Complex (Re z) (- Im z)"

   499

   500 lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"

   501   by (simp add: cnj_def)

   502

   503 lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"

   504   by (simp add: cnj_def)

   505

   506 lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"

   507   by (simp add: cnj_def)

   508

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

   510   by (simp add: complex_eq_iff)

   511

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

   513   by (simp add: cnj_def)

   514

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

   516   by (simp add: complex_eq_iff)

   517

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

   519   by (simp add: complex_eq_iff)

   520

   521 lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"

   522   by (simp add: complex_eq_iff)

   523

   524 lemma cnj_setsum: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"

   525   by (induct s rule: infinite_finite_induct) (auto simp: complex_cnj_add)

   526

   527 lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"

   528   by (simp add: complex_eq_iff)

   529

   530 lemma complex_cnj_minus: "cnj (- x) = - cnj x"

   531   by (simp add: complex_eq_iff)

   532

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

   534   by (simp add: complex_eq_iff)

   535

   536 lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"

   537   by (simp add: complex_eq_iff)

   538

   539 lemma cnj_setprod: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"

   540   by (induct s rule: infinite_finite_induct) (auto simp: complex_cnj_mult)

   541

   542 lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"

   543   by (simp add: complex_inverse_def)

   544

   545 lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"

   546   by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)

   547

   548 lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"

   549   by (induct n, simp_all add: complex_cnj_mult)

   550

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

   552   by (simp add: complex_eq_iff)

   553

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

   555   by (simp add: complex_eq_iff)

   556

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

   558   by (simp add: complex_eq_iff)

   559

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

   561   by (simp add: complex_eq_iff)

   562

   563 lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"

   564   by (simp add: complex_eq_iff)

   565

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

   567   by (simp add: complex_norm_def)

   568

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

   570   by (simp add: complex_eq_iff)

   571

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

   573   by (simp add: complex_eq_iff)

   574

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

   576   by (simp add: complex_eq_iff)

   577

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

   579   by (simp add: complex_eq_iff)

   580

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

   582   by (simp add: complex_eq_iff power2_eq_square)

   583

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

   585   by (simp add: norm_mult power2_eq_square)

   586

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

   588   by (simp add: cmod_def power2_eq_square)

   589

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

   591   by simp

   592

   593 lemma bounded_linear_cnj: "bounded_linear cnj"

   594   using complex_cnj_add complex_cnj_scaleR

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

   596

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

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

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

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

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

   602

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

   604   by (simp add: tendsto_iff dist_complex_def Complex.complex_cnj_diff [symmetric])

   605

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

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

   608

   609

   610 subsection{*Basic Lemmas*}

   611

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

   613   by (metis complex_Im_zero complex_Re_zero complex_eqI sum_power2_eq_zero_iff)

   614

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

   616 by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)

   617

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

   619 apply (cases z, auto)

   620 by (metis complex_of_real_def of_real_add of_real_power power2_eq_square)

   621

   622 lemma complex_div_eq_0:

   623     "(Re(a / b) = 0 \<longleftrightarrow> Re(a * cnj b) = 0) & (Im(a / b) = 0 \<longleftrightarrow> Im(a * cnj b) = 0)"

   624 proof (cases "b=0")

   625   case True then show ?thesis by auto

   626 next

   627   case False

   628   show ?thesis

   629   proof (cases b)

   630     case (Complex x y)

   631     then have "x\<^sup>2 + y\<^sup>2 > 0"

   632       by (metis Complex_eq_0 False sum_power2_gt_zero_iff)

   633     then have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)"

   634       by (metis add_divide_distrib)

   635     with Complex False show ?thesis

   636       by (auto simp: complex_divide_def)

   637   qed

   638 qed

   639

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

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

   642 using complex_div_eq_0 by auto

   643

   644

   645 lemma complex_div_gt_0:

   646     "(Re(a / b) > 0 \<longleftrightarrow> Re(a * cnj b) > 0) & (Im(a / b) > 0 \<longleftrightarrow> Im(a * cnj b) > 0)"

   647 proof (cases "b=0")

   648   case True then show ?thesis by auto

   649 next

   650   case False

   651   show ?thesis

   652   proof (cases b)

   653     case (Complex x y)

   654     then have "x\<^sup>2 + y\<^sup>2 > 0"

   655       by (metis Complex_eq_0 False sum_power2_gt_zero_iff)

   656     moreover have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)"

   657       by (metis add_divide_distrib)

   658     ultimately show ?thesis using Complex False 0 < x\<^sup>2 + y\<^sup>2

   659       apply (simp add: complex_divide_def  zero_less_divide_iff less_divide_eq)

   660       apply (metis less_divide_eq mult_zero_left diff_conv_add_uminus minus_divide_left)

   661       done

   662   qed

   663 qed

   664

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

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

   667 using complex_div_gt_0 by auto

   668

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

   670   by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0)

   671

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

   673   by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less)

   674

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

   676   by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0)

   677

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

   679   by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff)

   680

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

   682   by (metis not_le re_complex_div_gt_0)

   683

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

   685   by (metis im_complex_div_gt_0 not_le)

   686

   687 lemma Re_setsum: "Re(setsum f s) = setsum (%x. Re(f x)) s"

   688   by (induct s rule: infinite_finite_induct) auto

   689

   690 lemma Im_setsum: "Im(setsum f s) = setsum (%x. Im(f x)) s"

   691   by (induct s rule: infinite_finite_induct) auto

   692

   693 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)"

   694   unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..

   695

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

   697   unfolding summable_def sums_complex_iff[abs_def] by (metis Im.simps Re.simps)

   698

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

   700   unfolding summable_complex_iff by simp

   701

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

   703   unfolding summable_complex_iff by blast

   704

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

   706   unfolding summable_complex_iff by blast

   707

   708 lemma Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"

   709   by (induct s rule: infinite_finite_induct) auto

   710

   711 lemma Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"

   712   by (metis Complex_setsum')

   713

   714 lemma of_real_setsum: "of_real (setsum f s) = setsum (%x. of_real(f x)) s"

   715   by (induct s rule: infinite_finite_induct) auto

   716

   717 lemma of_real_setprod: "of_real (setprod f s) = setprod (%x. of_real(f x)) s"

   718   by (induct s rule: infinite_finite_induct) auto

   719

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

   721 by (metis Reals_cases Reals_of_real complex.exhaust complex.inject complex_cnj

   722           complex_of_real_def equal_neg_zero)

   723

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

   725   by (metis Reals_of_real complex_of_real_def)

   726

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

   728   by (metis Re_complex_of_real Reals_cases norm_of_real)

   729

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

   731   by (metis Complex_in_Reals Im_complex_of_real Reals_cases complex_surj)

   732

   733 lemma series_comparison_complex:

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

   735   assumes sg: "summable g"

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

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

   738   shows "summable f"

   739 proof -

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

   741     by (metis abs_of_nonneg in_Reals_norm)

   742   show ?thesis

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

   744     using sg

   745     apply (auto simp: summable_def)

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

   747     apply (auto simp: g sums_Re)

   748     apply (metis fg g)

   749     done

   750 qed

   751

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

   753

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

   755

   756 definition cis :: "real \<Rightarrow> complex" where

   757   "cis a = Complex (cos a) (sin a)"

   758

   759 lemma Re_cis [simp]: "Re (cis a) = cos a"

   760   by (simp add: cis_def)

   761

   762 lemma Im_cis [simp]: "Im (cis a) = sin a"

   763   by (simp add: cis_def)

   764

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

   766   by (simp add: cis_def)

   767

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

   769   by (simp add: cis_def)

   770

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

   772   by (simp add: sgn_div_norm)

   773

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

   775   by (metis norm_cis norm_zero zero_neq_one)

   776

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

   778   by (simp add: cis_def cos_add sin_add)

   779

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

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

   782

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

   784   by (simp add: cis_def)

   785

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

   787   by (simp add: complex_divide_def cis_mult)

   788

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

   790   by (auto simp add: DeMoivre)

   791

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

   793   by (auto simp add: DeMoivre)

   794

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

   796

   797 definition rcis :: "[real, real] \<Rightarrow> complex" where

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

   799

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

   801   by (simp add: rcis_def)

   802

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

   804   by (simp add: rcis_def)

   805

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

   807   by (simp add: complex_eq_iff polar_Ex)

   808

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

   810   by (simp add: rcis_def norm_mult)

   811

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

   813   by (simp add: rcis_def)

   814

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

   816   by (simp add: rcis_def cis_mult)

   817

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

   819   by (simp add: rcis_def)

   820

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

   822   by (simp add: rcis_def)

   823

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

   825   by (simp add: rcis_def)

   826

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

   828   by (simp add: rcis_def power_mult_distrib DeMoivre)

   829

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

   831   by (simp add: divide_inverse rcis_def)

   832

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

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

   835

   836 subsubsection {* Complex exponential *}

   837

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

   839   where "expi \<equiv> exp"

   840

   841 lemma cis_conv_exp: "cis b = exp (Complex 0 b)"

   842 proof (rule complex_eqI)

   843   { fix n have "Complex 0 b ^ n =

   844     real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"

   845       apply (induct n)

   846       apply (simp add: cos_coeff_def sin_coeff_def)

   847       apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)

   848       done } note * = this

   849   show "Re (cis b) = Re (exp (Complex 0 b))"

   850     unfolding exp_def cis_def cos_def

   851     by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],

   852       simp add: * mult_assoc [symmetric])

   853   show "Im (cis b) = Im (exp (Complex 0 b))"

   854     unfolding exp_def cis_def sin_def

   855     by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],

   856       simp add: * mult_assoc [symmetric])

   857 qed

   858

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

   860   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp

   861

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

   863   unfolding expi_def by simp

   864

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

   866   unfolding expi_def by simp

   867

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

   869 apply (insert rcis_Ex [of z])

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

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

   872 done

   873

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

   875   by (simp add: expi_def cis_def)

   876

   877 subsubsection {* Complex argument *}

   878

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

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

   881

   882 lemma arg_zero: "arg 0 = 0"

   883   by (simp add: arg_def)

   884

   885 lemma of_nat_less_of_int_iff: (* TODO: move *)

   886   "(of_nat n :: 'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"

   887   by (metis of_int_of_nat_eq of_int_less_iff)

   888

   889 lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *)

   890   "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"

   891   using of_nat_less_of_int_iff [of n "numeral w", where 'a=real]

   892   by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric])

   893

   894 lemma arg_unique:

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

   896   shows "arg z = x"

   897 proof -

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

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

   900   proof

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

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

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

   904       unfolding d_def by simp

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

   906       by (simp_all add: complex_eq_iff)

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

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

   909     ultimately have "d = 0"

   910       unfolding sin_zero_iff even_mult_two_ex

   911       by (auto simp add: numeral_2_eq_2 less_Suc_eq)

   912     thus "a = x" unfolding d_def by simp

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

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

   915     unfolding arg_def by simp

   916 qed

   917

   918 lemma arg_correct:

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

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

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

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

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

   924   have b: "sgn z = cis b"

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

   926     by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)

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

   928     by (induct_tac n, simp_all add: distrib_left cis_mult [symmetric],

   929       simp add: cis_def)

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

   931     by (case_tac x rule: int_diff_cases,

   932       simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)

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

   934   have "sgn z = cis c"

   935     unfolding b c_def

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

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

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

   939     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)

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

   941 qed

   942

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

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

   945

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

   947   by (simp add: arg_correct)

   948

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

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

   951

   952 lemma cos_arg_i_mult_zero [simp]:

   953      "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"

   954   using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)

   955

   956 text {* Legacy theorem names *}

   957

   958 lemmas expand_complex_eq = complex_eq_iff

   959 lemmas complex_Re_Im_cancel_iff = complex_eq_iff

   960 lemmas complex_equality = complex_eqI

   961

   962 end