src/HOL/Complex.thy
author haftmann
Sun Sep 21 16:56:11 2014 +0200 (2014-09-21)
changeset 58410 6d46ad54a2ab
parent 58146 d91c1e50b36e
child 58709 efdc6c533bd3
permissions -rw-r--r--
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
     1 (*  Title:       HOL/Complex.thy
     2     Author:      Jacques D. Fleuriot
     3     Copyright:   2001 University of Edinburgh
     4     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
     5 *)
     6 
     7 header {* Complex Numbers: Rectangular and Polar Representations *}
     8 
     9 theory Complex
    10 imports Transcendental
    11 begin
    12 
    13 text {*
    14 We use the @{text codatatype} command to define the type of complex numbers. This allows us to use
    15 @{text primcorec} to define complex functions by defining their real and imaginary result
    16 separately.
    17 *}
    18 
    19 codatatype complex = Complex (Re: real) (Im: real)
    20 
    21 lemma complex_surj: "Complex (Re z) (Im z) = z"
    22   by (rule complex.collapse)
    23 
    24 lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
    25   by (rule complex.expand) simp
    26 
    27 lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
    28   by (auto intro: complex.expand)
    29 
    30 subsection {* Addition and Subtraction *}
    31 
    32 instantiation complex :: ab_group_add
    33 begin
    34 
    35 primcorec zero_complex where
    36   "Re 0 = 0"
    37 | "Im 0 = 0"
    38 
    39 primcorec plus_complex where
    40   "Re (x + y) = Re x + Re y"
    41 | "Im (x + y) = Im x + Im y"
    42 
    43 primcorec uminus_complex where
    44   "Re (- x) = - Re x"
    45 | "Im (- x) = - Im x"
    46 
    47 primcorec minus_complex where
    48   "Re (x - y) = Re x - Re y"
    49 | "Im (x - y) = Im x - Im y"
    50 
    51 instance
    52   by intro_classes (simp_all add: complex_eq_iff)
    53 
    54 end
    55 
    56 subsection {* Multiplication and Division *}
    57 
    58 instantiation complex :: field_inverse_zero
    59 begin
    60 
    61 primcorec one_complex where
    62   "Re 1 = 1"
    63 | "Im 1 = 0"
    64 
    65 primcorec times_complex where
    66   "Re (x * y) = Re x * Re y - Im x * Im y"
    67 | "Im (x * y) = Re x * Im y + Im x * Re y"
    68 
    69 primcorec inverse_complex where
    70   "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
    71 | "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
    72 
    73 definition "x / (y\<Colon>complex) = x * inverse y"
    74 
    75 instance
    76   by intro_classes 
    77      (simp_all add: complex_eq_iff divide_complex_def
    78       distrib_left distrib_right right_diff_distrib left_diff_distrib
    79       power2_eq_square add_divide_distrib [symmetric])
    80 
    81 end
    82 
    83 lemma Re_divide: "Re (x / y) = (Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
    84   unfolding divide_complex_def by (simp add: add_divide_distrib)
    85 
    86 lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
    87   unfolding divide_complex_def times_complex.sel inverse_complex.sel
    88   by (simp_all add: divide_simps)
    89 
    90 lemma Re_power2: "Re (x ^ 2) = (Re x)^2 - (Im x)^2"
    91   by (simp add: power2_eq_square)
    92 
    93 lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x"
    94   by (simp add: power2_eq_square)
    95 
    96 lemma Re_power_real: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n "
    97   by (induct n) simp_all
    98 
    99 lemma Im_power_real: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0"
   100   by (induct n) simp_all
   101 
   102 subsection {* Scalar Multiplication *}
   103 
   104 instantiation complex :: real_field
   105 begin
   106 
   107 primcorec scaleR_complex where
   108   "Re (scaleR r x) = r * Re x"
   109 | "Im (scaleR r x) = r * Im x"
   110 
   111 instance
   112 proof
   113   fix a b :: real and x y :: complex
   114   show "scaleR a (x + y) = scaleR a x + scaleR a y"
   115     by (simp add: complex_eq_iff distrib_left)
   116   show "scaleR (a + b) x = scaleR a x + scaleR b x"
   117     by (simp add: complex_eq_iff distrib_right)
   118   show "scaleR a (scaleR b x) = scaleR (a * b) x"
   119     by (simp add: complex_eq_iff mult.assoc)
   120   show "scaleR 1 x = x"
   121     by (simp add: complex_eq_iff)
   122   show "scaleR a x * y = scaleR a (x * y)"
   123     by (simp add: complex_eq_iff algebra_simps)
   124   show "x * scaleR a y = scaleR a (x * y)"
   125     by (simp add: complex_eq_iff algebra_simps)
   126 qed
   127 
   128 end
   129 
   130 subsection {* Numerals, Arithmetic, and Embedding from Reals *}
   131 
   132 abbreviation complex_of_real :: "real \<Rightarrow> complex"
   133   where "complex_of_real \<equiv> of_real"
   134 
   135 declare [[coercion complex_of_real]]
   136 declare [[coercion "of_int :: int \<Rightarrow> complex"]]
   137 declare [[coercion "of_nat :: nat \<Rightarrow> complex"]]
   138 
   139 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
   140   by (induct n) simp_all
   141 
   142 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
   143   by (induct n) simp_all
   144 
   145 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
   146   by (cases z rule: int_diff_cases) simp
   147 
   148 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
   149   by (cases z rule: int_diff_cases) simp
   150 
   151 lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"
   152   using complex_Re_of_int [of "numeral v"] by simp
   153 
   154 lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"
   155   using complex_Im_of_int [of "numeral v"] by simp
   156 
   157 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
   158   by (simp add: of_real_def)
   159 
   160 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
   161   by (simp add: of_real_def)
   162 
   163 subsection {* The Complex Number $i$ *}
   164 
   165 primcorec "ii" :: complex  ("\<i>") where
   166   "Re ii = 0"
   167 | "Im ii = 1"
   168 
   169 lemma Complex_eq[simp]: "Complex a b = a + \<i> * b"
   170   by (simp add: complex_eq_iff)
   171 
   172 lemma complex_eq: "a = Re a + \<i> * Im a"
   173   by (simp add: complex_eq_iff)
   174 
   175 lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))"
   176   by (simp add: fun_eq_iff complex_eq)
   177 
   178 lemma i_squared [simp]: "ii * ii = -1"
   179   by (simp add: complex_eq_iff)
   180 
   181 lemma power2_i [simp]: "ii\<^sup>2 = -1"
   182   by (simp add: power2_eq_square)
   183 
   184 lemma inverse_i [simp]: "inverse ii = - ii"
   185   by (rule inverse_unique) simp
   186 
   187 lemma divide_i [simp]: "x / ii = - ii * x"
   188   by (simp add: divide_complex_def)
   189 
   190 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
   191   by (simp add: mult.assoc [symmetric])
   192 
   193 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
   194   by (simp add: complex_eq_iff)
   195 
   196 lemma complex_i_not_one [simp]: "ii \<noteq> 1"
   197   by (simp add: complex_eq_iff)
   198 
   199 lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
   200   by (simp add: complex_eq_iff)
   201 
   202 lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> - numeral w"
   203   by (simp add: complex_eq_iff)
   204 
   205 lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)"
   206   by (simp add: complex_eq_iff polar_Ex)
   207 
   208 subsection {* Vector Norm *}
   209 
   210 instantiation complex :: real_normed_field
   211 begin
   212 
   213 definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
   214 
   215 abbreviation cmod :: "complex \<Rightarrow> real"
   216   where "cmod \<equiv> norm"
   217 
   218 definition complex_sgn_def:
   219   "sgn x = x /\<^sub>R cmod x"
   220 
   221 definition dist_complex_def:
   222   "dist x y = cmod (x - y)"
   223 
   224 definition open_complex_def:
   225   "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
   226 
   227 instance proof
   228   fix r :: real and x y :: complex and S :: "complex set"
   229   show "(norm x = 0) = (x = 0)"
   230     by (simp add: norm_complex_def complex_eq_iff)
   231   show "norm (x + y) \<le> norm x + norm y"
   232     by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)
   233   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
   234     by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult)
   235   show "norm (x * y) = norm x * norm y"
   236     by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
   237 qed (rule complex_sgn_def dist_complex_def open_complex_def)+
   238 
   239 end
   240 
   241 lemma norm_ii [simp]: "norm ii = 1"
   242   by (simp add: norm_complex_def)
   243 
   244 lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1"
   245   by (simp add: norm_complex_def)
   246 
   247 lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>"
   248   by (simp add: norm_mult cmod_unit_one)
   249 
   250 lemma complex_Re_le_cmod: "Re x \<le> cmod x"
   251   unfolding norm_complex_def
   252   by (rule real_sqrt_sum_squares_ge1)
   253 
   254 lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
   255   by (rule order_trans [OF _ norm_ge_zero]) simp
   256 
   257 lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a"
   258   by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp
   259 
   260 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
   261   by (simp add: norm_complex_def)
   262 
   263 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
   264   by (simp add: norm_complex_def)
   265 
   266 lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>"
   267   apply (subst complex_eq)
   268   apply (rule order_trans)
   269   apply (rule norm_triangle_ineq)
   270   apply (simp add: norm_mult)
   271   done
   272 
   273 lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>"
   274   by (simp add: norm_complex_def)
   275 
   276 lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>"
   277   by (simp add: norm_complex_def)
   278 
   279 lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2"
   280   by (simp add: norm_complex_def)
   281 
   282 lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z"
   283   using abs_Re_le_cmod[of z] by auto
   284 
   285 lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0"
   286   by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2])
   287      (auto simp add: norm_complex_def)
   288 
   289 lemma abs_sqrt_wlog:
   290   fixes x::"'a::linordered_idom"
   291   assumes "\<And>x::'a. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
   292 by (metis abs_ge_zero assms power2_abs)
   293 
   294 lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"
   295   unfolding norm_complex_def
   296   apply (rule abs_sqrt_wlog [where x="Re z"])
   297   apply (rule abs_sqrt_wlog [where x="Im z"])
   298   apply (rule power2_le_imp_le)
   299   apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])
   300   done
   301 
   302 
   303 text {* Properties of complex signum. *}
   304 
   305 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
   306   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)
   307 
   308 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
   309   by (simp add: complex_sgn_def divide_inverse)
   310 
   311 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
   312   by (simp add: complex_sgn_def divide_inverse)
   313 
   314 
   315 subsection {* Completeness of the Complexes *}
   316 
   317 lemma bounded_linear_Re: "bounded_linear Re"
   318   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
   319 
   320 lemma bounded_linear_Im: "bounded_linear Im"
   321   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
   322 
   323 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
   324 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
   325 lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]
   326 lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]
   327 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
   328 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
   329 lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]
   330 lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]
   331 lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]
   332 lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]
   333 lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]
   334 lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]
   335 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]
   336 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]
   337 
   338 lemma tendsto_Complex [tendsto_intros]:
   339   "(f ---> a) F \<Longrightarrow> (g ---> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
   340   by (auto intro!: tendsto_intros)
   341 
   342 lemma tendsto_complex_iff:
   343   "(f ---> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) ---> Re x) F \<and> ((\<lambda>x. Im (f x)) ---> Im x) F)"
   344 proof safe
   345   assume "((\<lambda>x. Re (f x)) ---> Re x) F" "((\<lambda>x. Im (f x)) ---> Im x) F"
   346   from tendsto_Complex[OF this] show "(f ---> x) F"
   347     unfolding complex.collapse .
   348 qed (auto intro: tendsto_intros)
   349 
   350 lemma continuous_complex_iff: "continuous F f \<longleftrightarrow>
   351     continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))"
   352   unfolding continuous_def tendsto_complex_iff ..
   353 
   354 lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow>
   355     ((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and>
   356     ((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F"
   357   unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff
   358   by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right)
   359 
   360 lemma has_field_derivative_Re[derivative_intros]:
   361   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F"
   362   unfolding has_vector_derivative_complex_iff by safe
   363 
   364 lemma has_field_derivative_Im[derivative_intros]:
   365   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F"
   366   unfolding has_vector_derivative_complex_iff by safe
   367 
   368 instance complex :: banach
   369 proof
   370   fix X :: "nat \<Rightarrow> complex"
   371   assume X: "Cauchy X"
   372   then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
   373     by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)
   374   then show "convergent X"
   375     unfolding complex.collapse by (rule convergentI)
   376 qed
   377 
   378 declare
   379   DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]
   380 
   381 subsection {* Complex Conjugation *}
   382 
   383 primcorec cnj :: "complex \<Rightarrow> complex" where
   384   "Re (cnj z) = Re z"
   385 | "Im (cnj z) = - Im z"
   386 
   387 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
   388   by (simp add: complex_eq_iff)
   389 
   390 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
   391   by (simp add: complex_eq_iff)
   392 
   393 lemma complex_cnj_zero [simp]: "cnj 0 = 0"
   394   by (simp add: complex_eq_iff)
   395 
   396 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
   397   by (simp add: complex_eq_iff)
   398 
   399 lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"
   400   by (simp add: complex_eq_iff)
   401 
   402 lemma cnj_setsum [simp]: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"
   403   by (induct s rule: infinite_finite_induct) auto
   404 
   405 lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"
   406   by (simp add: complex_eq_iff)
   407 
   408 lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"
   409   by (simp add: complex_eq_iff)
   410 
   411 lemma complex_cnj_one [simp]: "cnj 1 = 1"
   412   by (simp add: complex_eq_iff)
   413 
   414 lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"
   415   by (simp add: complex_eq_iff)
   416 
   417 lemma cnj_setprod [simp]: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"
   418   by (induct s rule: infinite_finite_induct) auto
   419 
   420 lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"
   421   by (simp add: complex_eq_iff)
   422 
   423 lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"
   424   by (simp add: divide_complex_def)
   425 
   426 lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"
   427   by (induct n) simp_all
   428 
   429 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
   430   by (simp add: complex_eq_iff)
   431 
   432 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
   433   by (simp add: complex_eq_iff)
   434 
   435 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
   436   by (simp add: complex_eq_iff)
   437 
   438 lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
   439   by (simp add: complex_eq_iff)
   440 
   441 lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"
   442   by (simp add: complex_eq_iff)
   443 
   444 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
   445   by (simp add: norm_complex_def)
   446 
   447 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
   448   by (simp add: complex_eq_iff)
   449 
   450 lemma complex_cnj_i [simp]: "cnj ii = - ii"
   451   by (simp add: complex_eq_iff)
   452 
   453 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
   454   by (simp add: complex_eq_iff)
   455 
   456 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
   457   by (simp add: complex_eq_iff)
   458 
   459 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
   460   by (simp add: complex_eq_iff power2_eq_square)
   461 
   462 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
   463   by (simp add: norm_mult power2_eq_square)
   464 
   465 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
   466   by (simp add: norm_complex_def power2_eq_square)
   467 
   468 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
   469   by simp
   470 
   471 lemma bounded_linear_cnj: "bounded_linear cnj"
   472   using complex_cnj_add complex_cnj_scaleR
   473   by (rule bounded_linear_intro [where K=1], simp)
   474 
   475 lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]
   476 lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]
   477 lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]
   478 lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
   479 lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]
   480 
   481 lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"
   482   by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)
   483 
   484 lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
   485   by (simp add: sums_def lim_cnj cnj_setsum [symmetric] del: cnj_setsum)
   486 
   487 
   488 subsection{*Basic Lemmas*}
   489 
   490 lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
   491   by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)
   492 
   493 lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
   494   by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
   495 
   496 lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
   497 by (cases z)
   498    (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]
   499          simp del: of_real_power)
   500 
   501 lemma re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0"
   502   by (auto simp add: Re_divide)
   503   
   504 lemma im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0"
   505   by (auto simp add: Im_divide)
   506 
   507 lemma complex_div_gt_0: 
   508   "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"
   509 proof cases
   510   assume "b = 0" then show ?thesis by auto
   511 next
   512   assume "b \<noteq> 0"
   513   then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"
   514     by (simp add: complex_eq_iff sum_power2_gt_zero_iff)
   515   then show ?thesis
   516     by (simp add: Re_divide Im_divide zero_less_divide_iff)
   517 qed
   518 
   519 lemma re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0"
   520   and im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0"
   521   using complex_div_gt_0 by auto
   522 
   523 lemma re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"
   524   by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0)
   525 
   526 lemma im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"
   527   by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less)
   528 
   529 lemma re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"
   530   by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0)
   531 
   532 lemma im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"
   533   by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff)
   534 
   535 lemma re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"
   536   by (metis not_le re_complex_div_gt_0)
   537 
   538 lemma im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"
   539   by (metis im_complex_div_gt_0 not_le)
   540 
   541 lemma Re_setsum[simp]: "Re (setsum f s) = (\<Sum>x\<in>s. Re (f x))"
   542   by (induct s rule: infinite_finite_induct) auto
   543 
   544 lemma Im_setsum[simp]: "Im (setsum f s) = (\<Sum>x\<in>s. Im(f x))"
   545   by (induct s rule: infinite_finite_induct) auto
   546 
   547 lemma sums_complex_iff: "f sums x \<longleftrightarrow> ((\<lambda>x. Re (f x)) sums Re x) \<and> ((\<lambda>x. Im (f x)) sums Im x)"
   548   unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..
   549   
   550 lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and>  summable (\<lambda>x. Im (f x))"
   551   unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)
   552 
   553 lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"
   554   unfolding summable_complex_iff by simp
   555 
   556 lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"
   557   unfolding summable_complex_iff by blast
   558 
   559 lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"
   560   unfolding summable_complex_iff by blast
   561 
   562 lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
   563   by (auto simp: Reals_def complex_eq_iff)
   564 
   565 lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
   566   by (auto simp: complex_is_Real_iff complex_eq_iff)
   567 
   568 lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm(z) = abs(Re z)"
   569   by (simp add: complex_is_Real_iff norm_complex_def)
   570 
   571 lemma series_comparison_complex:
   572   fixes f:: "nat \<Rightarrow> 'a::banach"
   573   assumes sg: "summable g"
   574      and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
   575      and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
   576   shows "summable f"
   577 proof -
   578   have g: "\<And>n. cmod (g n) = Re (g n)" using assms
   579     by (metis abs_of_nonneg in_Reals_norm)
   580   show ?thesis
   581     apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
   582     using sg
   583     apply (auto simp: summable_def)
   584     apply (rule_tac x="Re s" in exI)
   585     apply (auto simp: g sums_Re)
   586     apply (metis fg g)
   587     done
   588 qed
   589 
   590 subsection{*Finally! Polar Form for Complex Numbers*}
   591 
   592 subsubsection {* $\cos \theta + i \sin \theta$ *}
   593 
   594 primcorec cis :: "real \<Rightarrow> complex" where
   595   "Re (cis a) = cos a"
   596 | "Im (cis a) = sin a"
   597 
   598 lemma cis_zero [simp]: "cis 0 = 1"
   599   by (simp add: complex_eq_iff)
   600 
   601 lemma norm_cis [simp]: "norm (cis a) = 1"
   602   by (simp add: norm_complex_def)
   603 
   604 lemma sgn_cis [simp]: "sgn (cis a) = cis a"
   605   by (simp add: sgn_div_norm)
   606 
   607 lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
   608   by (metis norm_cis norm_zero zero_neq_one)
   609 
   610 lemma cis_mult: "cis a * cis b = cis (a + b)"
   611   by (simp add: complex_eq_iff cos_add sin_add)
   612 
   613 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
   614   by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
   615 
   616 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
   617   by (simp add: complex_eq_iff)
   618 
   619 lemma cis_divide: "cis a / cis b = cis (a - b)"
   620   by (simp add: divide_complex_def cis_mult)
   621 
   622 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
   623   by (auto simp add: DeMoivre)
   624 
   625 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
   626   by (auto simp add: DeMoivre)
   627 
   628 lemma cis_pi: "cis pi = -1"
   629   by (simp add: complex_eq_iff)
   630 
   631 subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
   632 
   633 definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" where
   634   "rcis r a = complex_of_real r * cis a"
   635 
   636 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
   637   by (simp add: rcis_def)
   638 
   639 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
   640   by (simp add: rcis_def)
   641 
   642 lemma rcis_Ex: "\<exists>r a. z = rcis r a"
   643   by (simp add: complex_eq_iff polar_Ex)
   644 
   645 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
   646   by (simp add: rcis_def norm_mult)
   647 
   648 lemma cis_rcis_eq: "cis a = rcis 1 a"
   649   by (simp add: rcis_def)
   650 
   651 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
   652   by (simp add: rcis_def cis_mult)
   653 
   654 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
   655   by (simp add: rcis_def)
   656 
   657 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
   658   by (simp add: rcis_def)
   659 
   660 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
   661   by (simp add: rcis_def)
   662 
   663 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
   664   by (simp add: rcis_def power_mult_distrib DeMoivre)
   665 
   666 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
   667   by (simp add: divide_inverse rcis_def)
   668 
   669 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
   670   by (simp add: rcis_def cis_divide [symmetric])
   671 
   672 subsubsection {* Complex exponential *}
   673 
   674 abbreviation expi :: "complex \<Rightarrow> complex"
   675   where "expi \<equiv> exp"
   676 
   677 lemma cis_conv_exp: "cis b = exp (\<i> * b)"
   678 proof -
   679   { fix n :: nat
   680     have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"
   681       by (induct n)
   682          (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps
   683                         power2_eq_square real_of_nat_Suc add_nonneg_eq_0_iff
   684                         real_of_nat_def[symmetric])
   685     then have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =
   686         of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"
   687       by (simp add: field_simps) }
   688   then show ?thesis
   689     by (auto simp add: cis.ctr exp_def simp del: of_real_mult
   690              intro!: sums_unique sums_add sums_mult sums_of_real sin_converges cos_converges)
   691 qed
   692 
   693 lemma expi_def: "expi z = exp (Re z) * cis (Im z)"
   694   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp
   695 
   696 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
   697   unfolding expi_def by simp
   698 
   699 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
   700   unfolding expi_def by simp
   701 
   702 lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
   703 apply (insert rcis_Ex [of z])
   704 apply (auto simp add: expi_def rcis_def mult.assoc [symmetric])
   705 apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
   706 done
   707 
   708 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
   709   by (simp add: expi_def complex_eq_iff)
   710 
   711 subsubsection {* Complex argument *}
   712 
   713 definition arg :: "complex \<Rightarrow> real" where
   714   "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
   715 
   716 lemma arg_zero: "arg 0 = 0"
   717   by (simp add: arg_def)
   718 
   719 lemma arg_unique:
   720   assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
   721   shows "arg z = x"
   722 proof -
   723   from assms have "z \<noteq> 0" by auto
   724   have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
   725   proof
   726     fix a def d \<equiv> "a - x"
   727     assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
   728     from a assms have "- (2*pi) < d \<and> d < 2*pi"
   729       unfolding d_def by simp
   730     moreover from a assms have "cos a = cos x" and "sin a = sin x"
   731       by (simp_all add: complex_eq_iff)
   732     hence cos: "cos d = 1" unfolding d_def cos_diff by simp
   733     moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)
   734     ultimately have "d = 0"
   735       unfolding sin_zero_iff even_mult_two_ex
   736       by (auto simp add: numeral_2_eq_2 less_Suc_eq)
   737     thus "a = x" unfolding d_def by simp
   738   qed (simp add: assms del: Re_sgn Im_sgn)
   739   with `z \<noteq> 0` show "arg z = x"
   740     unfolding arg_def by simp
   741 qed
   742 
   743 lemma arg_correct:
   744   assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
   745 proof (simp add: arg_def assms, rule someI_ex)
   746   obtain r a where z: "z = rcis r a" using rcis_Ex by fast
   747   with assms have "r \<noteq> 0" by auto
   748   def b \<equiv> "if 0 < r then a else a + pi"
   749   have b: "sgn z = cis b"
   750     unfolding z b_def rcis_def using `r \<noteq> 0`
   751     by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)
   752   have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
   753     by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)
   754   have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
   755     by (case_tac x rule: int_diff_cases)
   756        (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
   757   def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
   758   have "sgn z = cis c"
   759     unfolding b c_def
   760     by (simp add: cis_divide [symmetric] cis_2pi_int)
   761   moreover have "- pi < c \<and> c \<le> pi"
   762     using ceiling_correct [of "(b - pi) / (2*pi)"]
   763     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
   764   ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
   765 qed
   766 
   767 lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
   768   by (cases "z = 0") (simp_all add: arg_zero arg_correct)
   769 
   770 lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
   771   by (simp add: arg_correct)
   772 
   773 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
   774   by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
   775 
   776 lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0"
   777   using cis_arg [of y] by (simp add: complex_eq_iff)
   778 
   779 subsection {* Square root of complex numbers *}
   780 
   781 primcorec csqrt :: "complex \<Rightarrow> complex" where
   782   "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"
   783 | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"
   784 
   785 lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)"
   786   by (simp add: complex_eq_iff norm_complex_def)
   787 
   788 lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>"
   789   by (simp add: complex_eq_iff norm_complex_def)
   790 
   791 lemma csqrt_0 [simp]: "csqrt 0 = 0"
   792   by simp
   793 
   794 lemma csqrt_1 [simp]: "csqrt 1 = 1"
   795   by simp
   796 
   797 lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2"
   798   by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)
   799 
   800 lemma power2_csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
   801 proof cases
   802   assume "Im z = 0" then show ?thesis
   803     using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]
   804     by (cases "0::real" "Re z" rule: linorder_cases)
   805        (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)
   806 next
   807   assume "Im z \<noteq> 0"
   808   moreover
   809   have "cmod z * cmod z - Re z * Re z = Im z * Im z"
   810     by (simp add: norm_complex_def power2_eq_square)
   811   moreover
   812   have "\<bar>Re z\<bar> \<le> cmod z"
   813     by (simp add: norm_complex_def)
   814   ultimately show ?thesis
   815     by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq
   816                   field_simps real_sqrt_mult[symmetric] real_sqrt_divide)
   817 qed
   818 
   819 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
   820   by auto (metis power2_csqrt power_eq_0_iff)
   821 
   822 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
   823   by auto (metis power2_csqrt power2_eq_1_iff)
   824 
   825 lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"
   826   by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)
   827 
   828 lemma Re_csqrt: "0 \<le> Re (csqrt z)"
   829   by (metis csqrt_principal le_less)
   830 
   831 lemma csqrt_square:
   832   assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)"
   833   shows "csqrt (b^2) = b"
   834 proof -
   835   have "csqrt (b^2) = b \<or> csqrt (b^2) = - b"
   836     unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt)
   837   moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0"
   838     using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff)
   839   ultimately show ?thesis
   840     by auto
   841 qed
   842 
   843 lemma csqrt_minus [simp]: 
   844   assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)"
   845   shows "csqrt (- x) = \<i> * csqrt x"
   846 proof -
   847   have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x"
   848   proof (rule csqrt_square)
   849     have "Im (csqrt x) \<le> 0"
   850       using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)
   851     then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)"
   852       by (auto simp add: Re_csqrt simp del: csqrt.simps)
   853   qed
   854   also have "(\<i> * csqrt x)^2 = - x"
   855     by (simp add: power2_csqrt power_mult_distrib)
   856   finally show ?thesis .
   857 qed
   858 
   859 text {* Legacy theorem names *}
   860 
   861 lemmas expand_complex_eq = complex_eq_iff
   862 lemmas complex_Re_Im_cancel_iff = complex_eq_iff
   863 lemmas complex_equality = complex_eqI
   864 lemmas cmod_def = norm_complex_def
   865 lemmas complex_norm_def = norm_complex_def
   866 lemmas complex_divide_def = divide_complex_def
   867 
   868 lemma legacy_Complex_simps:
   869   shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
   870     and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"
   871     and complex_minus: "- (Complex a b) = Complex (- a) (- b)"
   872     and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"
   873     and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"
   874     and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"
   875     and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
   876     and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"
   877     and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"
   878     and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"
   879     and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"
   880     and Complex_eq_i: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
   881     and i_mult_Complex: "ii * Complex a b = Complex (- b) a"
   882     and Complex_mult_i: "Complex a b * ii = Complex (- b) a"
   883     and i_complex_of_real: "ii * complex_of_real r = Complex 0 r"
   884     and complex_of_real_i: "complex_of_real r * ii = Complex 0 r"
   885     and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"
   886     and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"
   887     and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
   888     and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
   889     and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
   890     and complex_cn: "cnj (Complex a b) = Complex a (- b)"
   891     and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"
   892     and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"
   893     and complex_of_real_def: "complex_of_real r = Complex r 0"
   894     and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"
   895   by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)
   896 
   897 lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
   898   by (metis Reals_of_real complex_of_real_def)
   899 
   900 end