src/HOL/Complex.thy
author paulson <lp15@cam.ac.uk>
Thu Mar 05 17:30:29 2015 +0000 (2015-03-05)
changeset 59613 7103019278f0
parent 59000 6eb0725503fc
child 59658 0cc388370041
permissions -rw-r--r--
The function frac. Various lemmas about limits, series, the exp function, etc.
     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 section {* 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 "of_real :: real \<Rightarrow> complex"]]
   136 declare [[coercion "of_rat :: rat \<Rightarrow> complex"]]
   137 declare [[coercion "of_int :: int \<Rightarrow> complex"]]
   138 declare [[coercion "of_nat :: nat \<Rightarrow> complex"]]
   139 
   140 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
   141   by (induct n) simp_all
   142 
   143 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
   144   by (induct n) simp_all
   145 
   146 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
   147   by (cases z rule: int_diff_cases) simp
   148 
   149 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
   150   by (cases z rule: int_diff_cases) simp
   151 
   152 lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"
   153   using complex_Re_of_int [of "numeral v"] by simp
   154 
   155 lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"
   156   using complex_Im_of_int [of "numeral v"] by simp
   157 
   158 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
   159   by (simp add: of_real_def)
   160 
   161 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
   162   by (simp add: of_real_def)
   163 
   164 lemma Re_divide_numeral [simp]: "Re (z / numeral w) = Re z / numeral w"
   165   by (simp add: Re_divide sqr_conv_mult)
   166 
   167 lemma Im_divide_numeral [simp]: "Im (z / numeral w) = Im z / numeral w"
   168   by (simp add: Im_divide sqr_conv_mult)
   169 
   170 subsection {* The Complex Number $i$ *}
   171 
   172 primcorec "ii" :: complex  ("\<i>") where
   173   "Re ii = 0"
   174 | "Im ii = 1"
   175 
   176 lemma Complex_eq[simp]: "Complex a b = a + \<i> * b"
   177   by (simp add: complex_eq_iff)
   178 
   179 lemma complex_eq: "a = Re a + \<i> * Im a"
   180   by (simp add: complex_eq_iff)
   181 
   182 lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))"
   183   by (simp add: fun_eq_iff complex_eq)
   184 
   185 lemma i_squared [simp]: "ii * ii = -1"
   186   by (simp add: complex_eq_iff)
   187 
   188 lemma power2_i [simp]: "ii\<^sup>2 = -1"
   189   by (simp add: power2_eq_square)
   190 
   191 lemma inverse_i [simp]: "inverse ii = - ii"
   192   by (rule inverse_unique) simp
   193 
   194 lemma divide_i [simp]: "x / ii = - ii * x"
   195   by (simp add: divide_complex_def)
   196 
   197 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
   198   by (simp add: mult.assoc [symmetric])
   199 
   200 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
   201   by (simp add: complex_eq_iff)
   202 
   203 lemma complex_i_not_one [simp]: "ii \<noteq> 1"
   204   by (simp add: complex_eq_iff)
   205 
   206 lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
   207   by (simp add: complex_eq_iff)
   208 
   209 lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> - numeral w"
   210   by (simp add: complex_eq_iff)
   211 
   212 lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)"
   213   by (simp add: complex_eq_iff polar_Ex)
   214 
   215 lemma i_even_power [simp]: "\<i> ^ (n * 2) = (-1) ^ n"
   216   by (metis mult.commute power2_i power_mult)
   217 
   218 subsection {* Vector Norm *}
   219 
   220 instantiation complex :: real_normed_field
   221 begin
   222 
   223 definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
   224 
   225 abbreviation cmod :: "complex \<Rightarrow> real"
   226   where "cmod \<equiv> norm"
   227 
   228 definition complex_sgn_def:
   229   "sgn x = x /\<^sub>R cmod x"
   230 
   231 definition dist_complex_def:
   232   "dist x y = cmod (x - y)"
   233 
   234 definition open_complex_def:
   235   "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
   236 
   237 instance proof
   238   fix r :: real and x y :: complex and S :: "complex set"
   239   show "(norm x = 0) = (x = 0)"
   240     by (simp add: norm_complex_def complex_eq_iff)
   241   show "norm (x + y) \<le> norm x + norm y"
   242     by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)
   243   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
   244     by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult)
   245   show "norm (x * y) = norm x * norm y"
   246     by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
   247 qed (rule complex_sgn_def dist_complex_def open_complex_def)+
   248 
   249 end
   250 
   251 lemma norm_ii [simp]: "norm ii = 1"
   252   by (simp add: norm_complex_def)
   253 
   254 lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1"
   255   by (simp add: norm_complex_def)
   256 
   257 lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>"
   258   by (simp add: norm_mult cmod_unit_one)
   259 
   260 lemma complex_Re_le_cmod: "Re x \<le> cmod x"
   261   unfolding norm_complex_def
   262   by (rule real_sqrt_sum_squares_ge1)
   263 
   264 lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
   265   by (rule order_trans [OF _ norm_ge_zero]) simp
   266 
   267 lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a"
   268   by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp
   269 
   270 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
   271   by (simp add: norm_complex_def)
   272 
   273 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
   274   by (simp add: norm_complex_def)
   275 
   276 lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>"
   277   apply (subst complex_eq)
   278   apply (rule order_trans)
   279   apply (rule norm_triangle_ineq)
   280   apply (simp add: norm_mult)
   281   done
   282 
   283 lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>"
   284   by (simp add: norm_complex_def)
   285 
   286 lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>"
   287   by (simp add: norm_complex_def)
   288 
   289 lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2"
   290   by (simp add: norm_complex_def)
   291 
   292 lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z"
   293   using abs_Re_le_cmod[of z] by auto
   294 
   295 lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0"
   296   by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2])
   297      (auto simp add: norm_complex_def)
   298 
   299 lemma abs_sqrt_wlog:
   300   fixes x::"'a::linordered_idom"
   301   assumes "\<And>x::'a. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
   302 by (metis abs_ge_zero assms power2_abs)
   303 
   304 lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"
   305   unfolding norm_complex_def
   306   apply (rule abs_sqrt_wlog [where x="Re z"])
   307   apply (rule abs_sqrt_wlog [where x="Im z"])
   308   apply (rule power2_le_imp_le)
   309   apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])
   310   done
   311 
   312 
   313 text {* Properties of complex signum. *}
   314 
   315 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
   316   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)
   317 
   318 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
   319   by (simp add: complex_sgn_def divide_inverse)
   320 
   321 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
   322   by (simp add: complex_sgn_def divide_inverse)
   323 
   324 
   325 subsection {* Completeness of the Complexes *}
   326 
   327 lemma bounded_linear_Re: "bounded_linear Re"
   328   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
   329 
   330 lemma bounded_linear_Im: "bounded_linear Im"
   331   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
   332 
   333 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
   334 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
   335 lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]
   336 lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]
   337 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
   338 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
   339 lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]
   340 lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]
   341 lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]
   342 lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]
   343 lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]
   344 lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]
   345 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]
   346 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]
   347 
   348 lemma tendsto_Complex [tendsto_intros]:
   349   "(f ---> a) F \<Longrightarrow> (g ---> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
   350   by (auto intro!: tendsto_intros)
   351 
   352 lemma tendsto_complex_iff:
   353   "(f ---> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) ---> Re x) F \<and> ((\<lambda>x. Im (f x)) ---> Im x) F)"
   354 proof safe
   355   assume "((\<lambda>x. Re (f x)) ---> Re x) F" "((\<lambda>x. Im (f x)) ---> Im x) F"
   356   from tendsto_Complex[OF this] show "(f ---> x) F"
   357     unfolding complex.collapse .
   358 qed (auto intro: tendsto_intros)
   359 
   360 lemma continuous_complex_iff: "continuous F f \<longleftrightarrow>
   361     continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))"
   362   unfolding continuous_def tendsto_complex_iff ..
   363 
   364 lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow>
   365     ((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and>
   366     ((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F"
   367   unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff
   368   by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right)
   369 
   370 lemma has_field_derivative_Re[derivative_intros]:
   371   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F"
   372   unfolding has_vector_derivative_complex_iff by safe
   373 
   374 lemma has_field_derivative_Im[derivative_intros]:
   375   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F"
   376   unfolding has_vector_derivative_complex_iff by safe
   377 
   378 instance complex :: banach
   379 proof
   380   fix X :: "nat \<Rightarrow> complex"
   381   assume X: "Cauchy X"
   382   then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
   383     by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)
   384   then show "convergent X"
   385     unfolding complex.collapse by (rule convergentI)
   386 qed
   387 
   388 declare
   389   DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]
   390 
   391 subsection {* Complex Conjugation *}
   392 
   393 primcorec cnj :: "complex \<Rightarrow> complex" where
   394   "Re (cnj z) = Re z"
   395 | "Im (cnj z) = - Im z"
   396 
   397 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
   398   by (simp add: complex_eq_iff)
   399 
   400 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
   401   by (simp add: complex_eq_iff)
   402 
   403 lemma complex_cnj_zero [simp]: "cnj 0 = 0"
   404   by (simp add: complex_eq_iff)
   405 
   406 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
   407   by (simp add: complex_eq_iff)
   408 
   409 lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"
   410   by (simp add: complex_eq_iff)
   411 
   412 lemma cnj_setsum [simp]: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"
   413   by (induct s rule: infinite_finite_induct) auto
   414 
   415 lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"
   416   by (simp add: complex_eq_iff)
   417 
   418 lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"
   419   by (simp add: complex_eq_iff)
   420 
   421 lemma complex_cnj_one [simp]: "cnj 1 = 1"
   422   by (simp add: complex_eq_iff)
   423 
   424 lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"
   425   by (simp add: complex_eq_iff)
   426 
   427 lemma cnj_setprod [simp]: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"
   428   by (induct s rule: infinite_finite_induct) auto
   429 
   430 lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"
   431   by (simp add: complex_eq_iff)
   432 
   433 lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"
   434   by (simp add: divide_complex_def)
   435 
   436 lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"
   437   by (induct n) simp_all
   438 
   439 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
   440   by (simp add: complex_eq_iff)
   441 
   442 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
   443   by (simp add: complex_eq_iff)
   444 
   445 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
   446   by (simp add: complex_eq_iff)
   447 
   448 lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
   449   by (simp add: complex_eq_iff)
   450 
   451 lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"
   452   by (simp add: complex_eq_iff)
   453 
   454 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
   455   by (simp add: norm_complex_def)
   456 
   457 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
   458   by (simp add: complex_eq_iff)
   459 
   460 lemma complex_cnj_i [simp]: "cnj ii = - ii"
   461   by (simp add: complex_eq_iff)
   462 
   463 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
   464   by (simp add: complex_eq_iff)
   465 
   466 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
   467   by (simp add: complex_eq_iff)
   468 
   469 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
   470   by (simp add: complex_eq_iff power2_eq_square)
   471 
   472 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
   473   by (simp add: norm_mult power2_eq_square)
   474 
   475 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
   476   by (simp add: norm_complex_def power2_eq_square)
   477 
   478 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
   479   by simp
   480 
   481 lemma bounded_linear_cnj: "bounded_linear cnj"
   482   using complex_cnj_add complex_cnj_scaleR
   483   by (rule bounded_linear_intro [where K=1], simp)
   484 
   485 lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]
   486 lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]
   487 lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]
   488 lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
   489 lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]
   490 
   491 lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"
   492   by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)
   493 
   494 lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
   495   by (simp add: sums_def lim_cnj cnj_setsum [symmetric] del: cnj_setsum)
   496 
   497 
   498 subsection{*Basic Lemmas*}
   499 
   500 lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
   501   by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)
   502 
   503 lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
   504   by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
   505 
   506 lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
   507 by (cases z)
   508    (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]
   509          simp del: of_real_power)
   510 
   511 lemma re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0"
   512   by (auto simp add: Re_divide)
   513 
   514 lemma im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0"
   515   by (auto simp add: Im_divide)
   516 
   517 lemma complex_div_gt_0:
   518   "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"
   519 proof cases
   520   assume "b = 0" then show ?thesis by auto
   521 next
   522   assume "b \<noteq> 0"
   523   then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"
   524     by (simp add: complex_eq_iff sum_power2_gt_zero_iff)
   525   then show ?thesis
   526     by (simp add: Re_divide Im_divide zero_less_divide_iff)
   527 qed
   528 
   529 lemma re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0"
   530   and im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0"
   531   using complex_div_gt_0 by auto
   532 
   533 lemma re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"
   534   by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0)
   535 
   536 lemma im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"
   537   by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less)
   538 
   539 lemma re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"
   540   by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0)
   541 
   542 lemma im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"
   543   by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff)
   544 
   545 lemma re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"
   546   by (metis not_le re_complex_div_gt_0)
   547 
   548 lemma im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"
   549   by (metis im_complex_div_gt_0 not_le)
   550 
   551 lemma Re_setsum[simp]: "Re (setsum f s) = (\<Sum>x\<in>s. Re (f x))"
   552   by (induct s rule: infinite_finite_induct) auto
   553 
   554 lemma Im_setsum[simp]: "Im (setsum f s) = (\<Sum>x\<in>s. Im(f x))"
   555   by (induct s rule: infinite_finite_induct) auto
   556 
   557 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)"
   558   unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..
   559 
   560 lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and>  summable (\<lambda>x. Im (f x))"
   561   unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)
   562 
   563 lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"
   564   unfolding summable_complex_iff by simp
   565 
   566 lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"
   567   unfolding summable_complex_iff by blast
   568 
   569 lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"
   570   unfolding summable_complex_iff by blast
   571 
   572 lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
   573   by (auto simp: Reals_def complex_eq_iff)
   574 
   575 lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
   576   by (auto simp: complex_is_Real_iff complex_eq_iff)
   577 
   578 lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm(z) = abs(Re z)"
   579   by (simp add: complex_is_Real_iff norm_complex_def)
   580 
   581 lemma series_comparison_complex:
   582   fixes f:: "nat \<Rightarrow> 'a::banach"
   583   assumes sg: "summable g"
   584      and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
   585      and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
   586   shows "summable f"
   587 proof -
   588   have g: "\<And>n. cmod (g n) = Re (g n)" using assms
   589     by (metis abs_of_nonneg in_Reals_norm)
   590   show ?thesis
   591     apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
   592     using sg
   593     apply (auto simp: summable_def)
   594     apply (rule_tac x="Re s" in exI)
   595     apply (auto simp: g sums_Re)
   596     apply (metis fg g)
   597     done
   598 qed
   599 
   600 subsection{*Finally! Polar Form for Complex Numbers*}
   601 
   602 subsubsection {* $\cos \theta + i \sin \theta$ *}
   603 
   604 primcorec cis :: "real \<Rightarrow> complex" where
   605   "Re (cis a) = cos a"
   606 | "Im (cis a) = sin a"
   607 
   608 lemma cis_zero [simp]: "cis 0 = 1"
   609   by (simp add: complex_eq_iff)
   610 
   611 lemma norm_cis [simp]: "norm (cis a) = 1"
   612   by (simp add: norm_complex_def)
   613 
   614 lemma sgn_cis [simp]: "sgn (cis a) = cis a"
   615   by (simp add: sgn_div_norm)
   616 
   617 lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
   618   by (metis norm_cis norm_zero zero_neq_one)
   619 
   620 lemma cis_mult: "cis a * cis b = cis (a + b)"
   621   by (simp add: complex_eq_iff cos_add sin_add)
   622 
   623 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
   624   by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
   625 
   626 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
   627   by (simp add: complex_eq_iff)
   628 
   629 lemma cis_divide: "cis a / cis b = cis (a - b)"
   630   by (simp add: divide_complex_def cis_mult)
   631 
   632 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
   633   by (auto simp add: DeMoivre)
   634 
   635 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
   636   by (auto simp add: DeMoivre)
   637 
   638 lemma cis_pi: "cis pi = -1"
   639   by (simp add: complex_eq_iff)
   640 
   641 subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
   642 
   643 definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" where
   644   "rcis r a = complex_of_real r * cis a"
   645 
   646 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
   647   by (simp add: rcis_def)
   648 
   649 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
   650   by (simp add: rcis_def)
   651 
   652 lemma rcis_Ex: "\<exists>r a. z = rcis r a"
   653   by (simp add: complex_eq_iff polar_Ex)
   654 
   655 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
   656   by (simp add: rcis_def norm_mult)
   657 
   658 lemma cis_rcis_eq: "cis a = rcis 1 a"
   659   by (simp add: rcis_def)
   660 
   661 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
   662   by (simp add: rcis_def cis_mult)
   663 
   664 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
   665   by (simp add: rcis_def)
   666 
   667 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
   668   by (simp add: rcis_def)
   669 
   670 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
   671   by (simp add: rcis_def)
   672 
   673 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
   674   by (simp add: rcis_def power_mult_distrib DeMoivre)
   675 
   676 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
   677   by (simp add: divide_inverse rcis_def)
   678 
   679 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
   680   by (simp add: rcis_def cis_divide [symmetric])
   681 
   682 subsubsection {* Complex exponential *}
   683 
   684 abbreviation expi :: "complex \<Rightarrow> complex"
   685   where "expi \<equiv> exp"
   686 
   687 lemma cis_conv_exp: "cis b = exp (\<i> * b)"
   688 proof -
   689   { fix n :: nat
   690     have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"
   691       by (induct n)
   692          (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps
   693                         power2_eq_square real_of_nat_Suc add_nonneg_eq_0_iff
   694                         real_of_nat_def[symmetric])
   695     then have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =
   696         of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"
   697       by (simp add: field_simps) }
   698   then show ?thesis
   699     by (auto simp add: cis.ctr exp_def simp del: of_real_mult
   700              intro!: sums_unique sums_add sums_mult sums_of_real sin_converges cos_converges)
   701 qed
   702 
   703 lemma expi_def: "expi z = exp (Re z) * cis (Im z)"
   704   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp
   705 
   706 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
   707   unfolding expi_def by simp
   708 
   709 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
   710   unfolding expi_def by simp
   711 
   712 lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
   713 apply (insert rcis_Ex [of z])
   714 apply (auto simp add: expi_def rcis_def mult.assoc [symmetric])
   715 apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
   716 done
   717 
   718 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
   719   by (simp add: expi_def complex_eq_iff)
   720 
   721 subsubsection {* Complex argument *}
   722 
   723 definition arg :: "complex \<Rightarrow> real" where
   724   "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
   725 
   726 lemma arg_zero: "arg 0 = 0"
   727   by (simp add: arg_def)
   728 
   729 lemma arg_unique:
   730   assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
   731   shows "arg z = x"
   732 proof -
   733   from assms have "z \<noteq> 0" by auto
   734   have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
   735   proof
   736     fix a def d \<equiv> "a - x"
   737     assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
   738     from a assms have "- (2*pi) < d \<and> d < 2*pi"
   739       unfolding d_def by simp
   740     moreover from a assms have "cos a = cos x" and "sin a = sin x"
   741       by (simp_all add: complex_eq_iff)
   742     hence cos: "cos d = 1" unfolding d_def cos_diff by simp
   743     moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)
   744     ultimately have "d = 0"
   745       unfolding sin_zero_iff
   746       by (auto elim!: evenE dest!: less_2_cases)
   747     thus "a = x" unfolding d_def by simp
   748   qed (simp add: assms del: Re_sgn Im_sgn)
   749   with `z \<noteq> 0` show "arg z = x"
   750     unfolding arg_def by simp
   751 qed
   752 
   753 lemma arg_correct:
   754   assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
   755 proof (simp add: arg_def assms, rule someI_ex)
   756   obtain r a where z: "z = rcis r a" using rcis_Ex by fast
   757   with assms have "r \<noteq> 0" by auto
   758   def b \<equiv> "if 0 < r then a else a + pi"
   759   have b: "sgn z = cis b"
   760     unfolding z b_def rcis_def using `r \<noteq> 0`
   761     by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)
   762   have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
   763     by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)
   764   have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
   765     by (case_tac x rule: int_diff_cases)
   766        (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
   767   def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
   768   have "sgn z = cis c"
   769     unfolding b c_def
   770     by (simp add: cis_divide [symmetric] cis_2pi_int)
   771   moreover have "- pi < c \<and> c \<le> pi"
   772     using ceiling_correct [of "(b - pi) / (2*pi)"]
   773     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
   774   ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
   775 qed
   776 
   777 lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
   778   by (cases "z = 0") (simp_all add: arg_zero arg_correct)
   779 
   780 lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
   781   by (simp add: arg_correct)
   782 
   783 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
   784   by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
   785 
   786 lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0"
   787   using cis_arg [of y] by (simp add: complex_eq_iff)
   788 
   789 subsection {* Square root of complex numbers *}
   790 
   791 primcorec csqrt :: "complex \<Rightarrow> complex" where
   792   "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"
   793 | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"
   794 
   795 lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)"
   796   by (simp add: complex_eq_iff norm_complex_def)
   797 
   798 lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>"
   799   by (simp add: complex_eq_iff norm_complex_def)
   800 
   801 lemma csqrt_0 [simp]: "csqrt 0 = 0"
   802   by simp
   803 
   804 lemma csqrt_1 [simp]: "csqrt 1 = 1"
   805   by simp
   806 
   807 lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2"
   808   by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)
   809 
   810 lemma power2_csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
   811 proof cases
   812   assume "Im z = 0" then show ?thesis
   813     using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]
   814     by (cases "0::real" "Re z" rule: linorder_cases)
   815        (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)
   816 next
   817   assume "Im z \<noteq> 0"
   818   moreover
   819   have "cmod z * cmod z - Re z * Re z = Im z * Im z"
   820     by (simp add: norm_complex_def power2_eq_square)
   821   moreover
   822   have "\<bar>Re z\<bar> \<le> cmod z"
   823     by (simp add: norm_complex_def)
   824   ultimately show ?thesis
   825     by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq
   826                   field_simps real_sqrt_mult[symmetric] real_sqrt_divide)
   827 qed
   828 
   829 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
   830   by auto (metis power2_csqrt power_eq_0_iff)
   831 
   832 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
   833   by auto (metis power2_csqrt power2_eq_1_iff)
   834 
   835 lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"
   836   by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)
   837 
   838 lemma Re_csqrt: "0 \<le> Re (csqrt z)"
   839   by (metis csqrt_principal le_less)
   840 
   841 lemma csqrt_square:
   842   assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)"
   843   shows "csqrt (b^2) = b"
   844 proof -
   845   have "csqrt (b^2) = b \<or> csqrt (b^2) = - b"
   846     unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt)
   847   moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0"
   848     using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff)
   849   ultimately show ?thesis
   850     by auto
   851 qed
   852 
   853 lemma csqrt_minus [simp]:
   854   assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)"
   855   shows "csqrt (- x) = \<i> * csqrt x"
   856 proof -
   857   have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x"
   858   proof (rule csqrt_square)
   859     have "Im (csqrt x) \<le> 0"
   860       using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)
   861     then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)"
   862       by (auto simp add: Re_csqrt simp del: csqrt.simps)
   863   qed
   864   also have "(\<i> * csqrt x)^2 = - x"
   865     by (simp add: power2_csqrt power_mult_distrib)
   866   finally show ?thesis .
   867 qed
   868 
   869 text {* Legacy theorem names *}
   870 
   871 lemmas expand_complex_eq = complex_eq_iff
   872 lemmas complex_Re_Im_cancel_iff = complex_eq_iff
   873 lemmas complex_equality = complex_eqI
   874 lemmas cmod_def = norm_complex_def
   875 lemmas complex_norm_def = norm_complex_def
   876 lemmas complex_divide_def = divide_complex_def
   877 
   878 lemma legacy_Complex_simps:
   879   shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
   880     and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"
   881     and complex_minus: "- (Complex a b) = Complex (- a) (- b)"
   882     and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"
   883     and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"
   884     and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"
   885     and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
   886     and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"
   887     and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"
   888     and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"
   889     and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"
   890     and Complex_eq_i: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
   891     and i_mult_Complex: "ii * Complex a b = Complex (- b) a"
   892     and Complex_mult_i: "Complex a b * ii = Complex (- b) a"
   893     and i_complex_of_real: "ii * complex_of_real r = Complex 0 r"
   894     and complex_of_real_i: "complex_of_real r * ii = Complex 0 r"
   895     and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"
   896     and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"
   897     and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
   898     and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
   899     and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
   900     and complex_cn: "cnj (Complex a b) = Complex a (- b)"
   901     and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"
   902     and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"
   903     and complex_of_real_def: "complex_of_real r = Complex r 0"
   904     and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"
   905   by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)
   906 
   907 lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
   908   by (metis Reals_of_real complex_of_real_def)
   909 
   910 end