src/HOL/Complex.thy
author paulson <lp15@cam.ac.uk>
Wed Mar 18 17:23:22 2015 +0000 (2015-03-18)
changeset 59746 ddae5727c5a9
parent 59741 5b762cd73a8e
child 59862 44b3f4fa33ca
permissions -rw-r--r--
new HOL Light material about exp, sin, cos
     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 lemma Re_ii_times [simp]: "Re (ii*z) = - Im z"
   219   by simp
   220 
   221 lemma Im_ii_times [simp]: "Im (ii*z) = Re z"
   222   by simp
   223 
   224 lemma ii_times_eq_iff: "ii*w = z \<longleftrightarrow> w = -(ii*z)"
   225   by auto
   226 
   227 lemma divide_numeral_i [simp]: "z / (numeral n * ii) = -(ii*z) / numeral n"
   228   by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right)
   229 
   230 subsection {* Vector Norm *}
   231 
   232 instantiation complex :: real_normed_field
   233 begin
   234 
   235 definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
   236 
   237 abbreviation cmod :: "complex \<Rightarrow> real"
   238   where "cmod \<equiv> norm"
   239 
   240 definition complex_sgn_def:
   241   "sgn x = x /\<^sub>R cmod x"
   242 
   243 definition dist_complex_def:
   244   "dist x y = cmod (x - y)"
   245 
   246 definition open_complex_def:
   247   "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
   248 
   249 instance proof
   250   fix r :: real and x y :: complex and S :: "complex set"
   251   show "(norm x = 0) = (x = 0)"
   252     by (simp add: norm_complex_def complex_eq_iff)
   253   show "norm (x + y) \<le> norm x + norm y"
   254     by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)
   255   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
   256     by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult)
   257   show "norm (x * y) = norm x * norm y"
   258     by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
   259 qed (rule complex_sgn_def dist_complex_def open_complex_def)+
   260 
   261 end
   262 
   263 lemma norm_ii [simp]: "norm ii = 1"
   264   by (simp add: norm_complex_def)
   265 
   266 lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1"
   267   by (simp add: norm_complex_def)
   268 
   269 lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>"
   270   by (simp add: norm_mult cmod_unit_one)
   271 
   272 lemma complex_Re_le_cmod: "Re x \<le> cmod x"
   273   unfolding norm_complex_def
   274   by (rule real_sqrt_sum_squares_ge1)
   275 
   276 lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
   277   by (rule order_trans [OF _ norm_ge_zero]) simp
   278 
   279 lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a"
   280   by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp
   281 
   282 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
   283   by (simp add: norm_complex_def)
   284 
   285 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
   286   by (simp add: norm_complex_def)
   287 
   288 lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>"
   289   apply (subst complex_eq)
   290   apply (rule order_trans)
   291   apply (rule norm_triangle_ineq)
   292   apply (simp add: norm_mult)
   293   done
   294 
   295 lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>"
   296   by (simp add: norm_complex_def)
   297 
   298 lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>"
   299   by (simp add: norm_complex_def)
   300 
   301 lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2"
   302   by (simp add: norm_complex_def)
   303 
   304 lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z"
   305   using abs_Re_le_cmod[of z] by auto
   306 
   307 lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0"
   308   by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2])
   309      (auto simp add: norm_complex_def)
   310 
   311 lemma abs_sqrt_wlog:
   312   fixes x::"'a::linordered_idom"
   313   assumes "\<And>x::'a. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
   314 by (metis abs_ge_zero assms power2_abs)
   315 
   316 lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"
   317   unfolding norm_complex_def
   318   apply (rule abs_sqrt_wlog [where x="Re z"])
   319   apply (rule abs_sqrt_wlog [where x="Im z"])
   320   apply (rule power2_le_imp_le)
   321   apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])
   322   done
   323 
   324 lemma complex_unit_circle: "z \<noteq> 0 \<Longrightarrow> (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1"
   325   by (simp add: norm_complex_def divide_simps complex_eq_iff)
   326 
   327 
   328 text {* Properties of complex signum. *}
   329 
   330 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
   331   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)
   332 
   333 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
   334   by (simp add: complex_sgn_def divide_inverse)
   335 
   336 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
   337   by (simp add: complex_sgn_def divide_inverse)
   338 
   339 
   340 subsection {* Completeness of the Complexes *}
   341 
   342 lemma bounded_linear_Re: "bounded_linear Re"
   343   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
   344 
   345 lemma bounded_linear_Im: "bounded_linear Im"
   346   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
   347 
   348 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
   349 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
   350 lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]
   351 lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]
   352 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
   353 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
   354 lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]
   355 lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]
   356 lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]
   357 lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]
   358 lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]
   359 lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]
   360 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]
   361 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]
   362 
   363 lemma tendsto_Complex [tendsto_intros]:
   364   "(f ---> a) F \<Longrightarrow> (g ---> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
   365   by (auto intro!: tendsto_intros)
   366 
   367 lemma tendsto_complex_iff:
   368   "(f ---> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) ---> Re x) F \<and> ((\<lambda>x. Im (f x)) ---> Im x) F)"
   369 proof safe
   370   assume "((\<lambda>x. Re (f x)) ---> Re x) F" "((\<lambda>x. Im (f x)) ---> Im x) F"
   371   from tendsto_Complex[OF this] show "(f ---> x) F"
   372     unfolding complex.collapse .
   373 qed (auto intro: tendsto_intros)
   374 
   375 lemma continuous_complex_iff: "continuous F f \<longleftrightarrow>
   376     continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))"
   377   unfolding continuous_def tendsto_complex_iff ..
   378 
   379 lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow>
   380     ((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and>
   381     ((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F"
   382   unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff
   383   by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right)
   384 
   385 lemma has_field_derivative_Re[derivative_intros]:
   386   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F"
   387   unfolding has_vector_derivative_complex_iff by safe
   388 
   389 lemma has_field_derivative_Im[derivative_intros]:
   390   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F"
   391   unfolding has_vector_derivative_complex_iff by safe
   392 
   393 instance complex :: banach
   394 proof
   395   fix X :: "nat \<Rightarrow> complex"
   396   assume X: "Cauchy X"
   397   then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
   398     by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)
   399   then show "convergent X"
   400     unfolding complex.collapse by (rule convergentI)
   401 qed
   402 
   403 declare
   404   DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]
   405 
   406 subsection {* Complex Conjugation *}
   407 
   408 primcorec cnj :: "complex \<Rightarrow> complex" where
   409   "Re (cnj z) = Re z"
   410 | "Im (cnj z) = - Im z"
   411 
   412 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
   413   by (simp add: complex_eq_iff)
   414 
   415 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
   416   by (simp add: complex_eq_iff)
   417 
   418 lemma complex_cnj_zero [simp]: "cnj 0 = 0"
   419   by (simp add: complex_eq_iff)
   420 
   421 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
   422   by (simp add: complex_eq_iff)
   423 
   424 lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"
   425   by (simp add: complex_eq_iff)
   426 
   427 lemma cnj_setsum [simp]: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"
   428   by (induct s rule: infinite_finite_induct) auto
   429 
   430 lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"
   431   by (simp add: complex_eq_iff)
   432 
   433 lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"
   434   by (simp add: complex_eq_iff)
   435 
   436 lemma complex_cnj_one [simp]: "cnj 1 = 1"
   437   by (simp add: complex_eq_iff)
   438 
   439 lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"
   440   by (simp add: complex_eq_iff)
   441 
   442 lemma cnj_setprod [simp]: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"
   443   by (induct s rule: infinite_finite_induct) auto
   444 
   445 lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"
   446   by (simp add: complex_eq_iff)
   447 
   448 lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"
   449   by (simp add: divide_complex_def)
   450 
   451 lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"
   452   by (induct n) simp_all
   453 
   454 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
   455   by (simp add: complex_eq_iff)
   456 
   457 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
   458   by (simp add: complex_eq_iff)
   459 
   460 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
   461   by (simp add: complex_eq_iff)
   462 
   463 lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
   464   by (simp add: complex_eq_iff)
   465 
   466 lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"
   467   by (simp add: complex_eq_iff)
   468 
   469 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
   470   by (simp add: norm_complex_def)
   471 
   472 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
   473   by (simp add: complex_eq_iff)
   474 
   475 lemma complex_cnj_i [simp]: "cnj ii = - ii"
   476   by (simp add: complex_eq_iff)
   477 
   478 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
   479   by (simp add: complex_eq_iff)
   480 
   481 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
   482   by (simp add: complex_eq_iff)
   483 
   484 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
   485   by (simp add: complex_eq_iff power2_eq_square)
   486 
   487 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
   488   by (simp add: norm_mult power2_eq_square)
   489 
   490 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
   491   by (simp add: norm_complex_def power2_eq_square)
   492 
   493 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
   494   by simp
   495 
   496 lemma bounded_linear_cnj: "bounded_linear cnj"
   497   using complex_cnj_add complex_cnj_scaleR
   498   by (rule bounded_linear_intro [where K=1], simp)
   499 
   500 lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]
   501 lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]
   502 lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]
   503 lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
   504 lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]
   505 
   506 lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"
   507   by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)
   508 
   509 lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
   510   by (simp add: sums_def lim_cnj cnj_setsum [symmetric] del: cnj_setsum)
   511 
   512 
   513 subsection{*Basic Lemmas*}
   514 
   515 lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
   516   by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)
   517 
   518 lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
   519   by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
   520 
   521 lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
   522 by (cases z)
   523    (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]
   524          simp del: of_real_power)
   525 
   526 lemma Re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0"
   527   by (auto simp add: Re_divide)
   528 
   529 lemma Im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0"
   530   by (auto simp add: Im_divide)
   531 
   532 lemma complex_div_gt_0:
   533   "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"
   534 proof cases
   535   assume "b = 0" then show ?thesis by auto
   536 next
   537   assume "b \<noteq> 0"
   538   then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"
   539     by (simp add: complex_eq_iff sum_power2_gt_zero_iff)
   540   then show ?thesis
   541     by (simp add: Re_divide Im_divide zero_less_divide_iff)
   542 qed
   543 
   544 lemma Re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0"
   545   and Im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0"
   546   using complex_div_gt_0 by auto
   547 
   548 lemma Re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"
   549   by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0)
   550 
   551 lemma Im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"
   552   by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less)
   553 
   554 lemma Re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"
   555   by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0)
   556 
   557 lemma Im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"
   558   by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff)
   559 
   560 lemma Re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"
   561   by (metis not_le Re_complex_div_gt_0)
   562 
   563 lemma Im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"
   564   by (metis Im_complex_div_gt_0 not_le)
   565 
   566 lemma Re_setsum[simp]: "Re (setsum f s) = (\<Sum>x\<in>s. Re (f x))"
   567   by (induct s rule: infinite_finite_induct) auto
   568 
   569 lemma Im_setsum[simp]: "Im (setsum f s) = (\<Sum>x\<in>s. Im(f x))"
   570   by (induct s rule: infinite_finite_induct) auto
   571 
   572 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)"
   573   unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..
   574 
   575 lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and>  summable (\<lambda>x. Im (f x))"
   576   unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)
   577 
   578 lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"
   579   unfolding summable_complex_iff by simp
   580 
   581 lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"
   582   unfolding summable_complex_iff by blast
   583 
   584 lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"
   585   unfolding summable_complex_iff by blast
   586 
   587 lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
   588   by (auto simp: Reals_def complex_eq_iff)
   589 
   590 lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
   591   by (auto simp: complex_is_Real_iff complex_eq_iff)
   592 
   593 lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm(z) = abs(Re z)"
   594   by (simp add: complex_is_Real_iff norm_complex_def)
   595 
   596 lemma series_comparison_complex:
   597   fixes f:: "nat \<Rightarrow> 'a::banach"
   598   assumes sg: "summable g"
   599      and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
   600      and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
   601   shows "summable f"
   602 proof -
   603   have g: "\<And>n. cmod (g n) = Re (g n)" using assms
   604     by (metis abs_of_nonneg in_Reals_norm)
   605   show ?thesis
   606     apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
   607     using sg
   608     apply (auto simp: summable_def)
   609     apply (rule_tac x="Re s" in exI)
   610     apply (auto simp: g sums_Re)
   611     apply (metis fg g)
   612     done
   613 qed
   614 
   615 subsection{*Polar Form for Complex Numbers*}
   616 
   617 lemma complex_unimodular_polar: "(norm z = 1) \<Longrightarrow> \<exists>x. z = Complex (cos x) (sin x)"
   618   using sincos_total_2pi [of "Re z" "Im z"]
   619   by auto (metis cmod_power2 complex_eq power_one)
   620 
   621 subsubsection {* $\cos \theta + i \sin \theta$ *}
   622 
   623 primcorec cis :: "real \<Rightarrow> complex" where
   624   "Re (cis a) = cos a"
   625 | "Im (cis a) = sin a"
   626 
   627 lemma cis_zero [simp]: "cis 0 = 1"
   628   by (simp add: complex_eq_iff)
   629 
   630 lemma norm_cis [simp]: "norm (cis a) = 1"
   631   by (simp add: norm_complex_def)
   632 
   633 lemma sgn_cis [simp]: "sgn (cis a) = cis a"
   634   by (simp add: sgn_div_norm)
   635 
   636 lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
   637   by (metis norm_cis norm_zero zero_neq_one)
   638 
   639 lemma cis_mult: "cis a * cis b = cis (a + b)"
   640   by (simp add: complex_eq_iff cos_add sin_add)
   641 
   642 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
   643   by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
   644 
   645 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
   646   by (simp add: complex_eq_iff)
   647 
   648 lemma cis_divide: "cis a / cis b = cis (a - b)"
   649   by (simp add: divide_complex_def cis_mult)
   650 
   651 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
   652   by (auto simp add: DeMoivre)
   653 
   654 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
   655   by (auto simp add: DeMoivre)
   656 
   657 lemma cis_pi: "cis pi = -1"
   658   by (simp add: complex_eq_iff)
   659 
   660 subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
   661 
   662 definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" where
   663   "rcis r a = complex_of_real r * cis a"
   664 
   665 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
   666   by (simp add: rcis_def)
   667 
   668 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
   669   by (simp add: rcis_def)
   670 
   671 lemma rcis_Ex: "\<exists>r a. z = rcis r a"
   672   by (simp add: complex_eq_iff polar_Ex)
   673 
   674 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
   675   by (simp add: rcis_def norm_mult)
   676 
   677 lemma cis_rcis_eq: "cis a = rcis 1 a"
   678   by (simp add: rcis_def)
   679 
   680 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
   681   by (simp add: rcis_def cis_mult)
   682 
   683 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
   684   by (simp add: rcis_def)
   685 
   686 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
   687   by (simp add: rcis_def)
   688 
   689 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
   690   by (simp add: rcis_def)
   691 
   692 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
   693   by (simp add: rcis_def power_mult_distrib DeMoivre)
   694 
   695 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
   696   by (simp add: divide_inverse rcis_def)
   697 
   698 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
   699   by (simp add: rcis_def cis_divide [symmetric])
   700 
   701 subsubsection {* Complex exponential *}
   702 
   703 abbreviation Exp :: "complex \<Rightarrow> complex"
   704   where "Exp \<equiv> exp"
   705 
   706 lemma cis_conv_exp: "cis b = exp (\<i> * b)"
   707 proof -
   708   { fix n :: nat
   709     have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"
   710       by (induct n)
   711          (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps
   712                         power2_eq_square real_of_nat_Suc add_nonneg_eq_0_iff
   713                         real_of_nat_def[symmetric])
   714     then have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =
   715         of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"
   716       by (simp add: field_simps) }
   717   then show ?thesis using sin_converges [of b] cos_converges [of b]
   718     by (auto simp add: cis.ctr exp_def simp del: of_real_mult
   719              intro!: sums_unique sums_add sums_mult sums_of_real)
   720 qed
   721 
   722 lemma Exp_eq_polar: "Exp z = exp (Re z) * cis (Im z)"
   723   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp
   724 
   725 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
   726   unfolding Exp_eq_polar by simp
   727 
   728 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
   729   unfolding Exp_eq_polar by simp
   730 
   731 lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1"
   732   by (simp add: norm_complex_def)
   733 
   734 lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)"
   735   by (simp add: cis.code cmod_complex_polar Exp_eq_polar)
   736 
   737 lemma complex_Exp_Ex: "\<exists>a r. z = complex_of_real r * Exp a"
   738   apply (insert rcis_Ex [of z])
   739   apply (auto simp add: Exp_eq_polar rcis_def mult.assoc [symmetric])
   740   apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
   741   done
   742 
   743 lemma Exp_two_pi_i [simp]: "Exp((2::complex) * complex_of_real pi * ii) = 1"
   744   by (simp add: Exp_eq_polar complex_eq_iff)
   745 
   746 subsubsection {* Complex argument *}
   747 
   748 definition arg :: "complex \<Rightarrow> real" where
   749   "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
   750 
   751 lemma arg_zero: "arg 0 = 0"
   752   by (simp add: arg_def)
   753 
   754 lemma arg_unique:
   755   assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
   756   shows "arg z = x"
   757 proof -
   758   from assms have "z \<noteq> 0" by auto
   759   have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
   760   proof
   761     fix a def d \<equiv> "a - x"
   762     assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
   763     from a assms have "- (2*pi) < d \<and> d < 2*pi"
   764       unfolding d_def by simp
   765     moreover from a assms have "cos a = cos x" and "sin a = sin x"
   766       by (simp_all add: complex_eq_iff)
   767     hence cos: "cos d = 1" unfolding d_def cos_diff by simp
   768     moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)
   769     ultimately have "d = 0"
   770       unfolding sin_zero_iff
   771       by (auto elim!: evenE dest!: less_2_cases)
   772     thus "a = x" unfolding d_def by simp
   773   qed (simp add: assms del: Re_sgn Im_sgn)
   774   with `z \<noteq> 0` show "arg z = x"
   775     unfolding arg_def by simp
   776 qed
   777 
   778 lemma arg_correct:
   779   assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
   780 proof (simp add: arg_def assms, rule someI_ex)
   781   obtain r a where z: "z = rcis r a" using rcis_Ex by fast
   782   with assms have "r \<noteq> 0" by auto
   783   def b \<equiv> "if 0 < r then a else a + pi"
   784   have b: "sgn z = cis b"
   785     unfolding z b_def rcis_def using `r \<noteq> 0`
   786     by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)
   787   have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
   788     by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)
   789   have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
   790     by (case_tac x rule: int_diff_cases)
   791        (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
   792   def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
   793   have "sgn z = cis c"
   794     unfolding b c_def
   795     by (simp add: cis_divide [symmetric] cis_2pi_int)
   796   moreover have "- pi < c \<and> c \<le> pi"
   797     using ceiling_correct [of "(b - pi) / (2*pi)"]
   798     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
   799   ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
   800 qed
   801 
   802 lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
   803   by (cases "z = 0") (simp_all add: arg_zero arg_correct)
   804 
   805 lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
   806   by (simp add: arg_correct)
   807 
   808 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
   809   by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
   810 
   811 lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0"
   812   using cis_arg [of y] by (simp add: complex_eq_iff)
   813 
   814 subsection {* Square root of complex numbers *}
   815 
   816 primcorec csqrt :: "complex \<Rightarrow> complex" where
   817   "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"
   818 | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"
   819 
   820 lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)"
   821   by (simp add: complex_eq_iff norm_complex_def)
   822 
   823 lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>"
   824   by (simp add: complex_eq_iff norm_complex_def)
   825 
   826 lemma csqrt_0 [simp]: "csqrt 0 = 0"
   827   by simp
   828 
   829 lemma csqrt_1 [simp]: "csqrt 1 = 1"
   830   by simp
   831 
   832 lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2"
   833   by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)
   834 
   835 lemma power2_csqrt[simp,algebra]: "(csqrt z)\<^sup>2 = z"
   836 proof cases
   837   assume "Im z = 0" then show ?thesis
   838     using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]
   839     by (cases "0::real" "Re z" rule: linorder_cases)
   840        (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)
   841 next
   842   assume "Im z \<noteq> 0"
   843   moreover
   844   have "cmod z * cmod z - Re z * Re z = Im z * Im z"
   845     by (simp add: norm_complex_def power2_eq_square)
   846   moreover
   847   have "\<bar>Re z\<bar> \<le> cmod z"
   848     by (simp add: norm_complex_def)
   849   ultimately show ?thesis
   850     by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq
   851                   field_simps real_sqrt_mult[symmetric] real_sqrt_divide)
   852 qed
   853 
   854 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
   855   by auto (metis power2_csqrt power_eq_0_iff)
   856 
   857 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
   858   by auto (metis power2_csqrt power2_eq_1_iff)
   859 
   860 lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"
   861   by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)
   862 
   863 lemma Re_csqrt: "0 \<le> Re (csqrt z)"
   864   by (metis csqrt_principal le_less)
   865 
   866 lemma csqrt_square:
   867   assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)"
   868   shows "csqrt (b^2) = b"
   869 proof -
   870   have "csqrt (b^2) = b \<or> csqrt (b^2) = - b"
   871     unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt)
   872   moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0"
   873     using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff)
   874   ultimately show ?thesis
   875     by auto
   876 qed
   877 
   878 lemma csqrt_unique:
   879     "w^2 = z \<Longrightarrow> (0 < Re w \<or> Re w = 0 \<and> 0 \<le> Im w) \<Longrightarrow> csqrt z = w"
   880   by (auto simp: csqrt_square)
   881 
   882 lemma csqrt_minus [simp]:
   883   assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)"
   884   shows "csqrt (- x) = \<i> * csqrt x"
   885 proof -
   886   have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x"
   887   proof (rule csqrt_square)
   888     have "Im (csqrt x) \<le> 0"
   889       using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)
   890     then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)"
   891       by (auto simp add: Re_csqrt simp del: csqrt.simps)
   892   qed
   893   also have "(\<i> * csqrt x)^2 = - x"
   894     by (simp add: power_mult_distrib)
   895   finally show ?thesis .
   896 qed
   897 
   898 text {* Legacy theorem names *}
   899 
   900 lemmas expand_complex_eq = complex_eq_iff
   901 lemmas complex_Re_Im_cancel_iff = complex_eq_iff
   902 lemmas complex_equality = complex_eqI
   903 lemmas cmod_def = norm_complex_def
   904 lemmas complex_norm_def = norm_complex_def
   905 lemmas complex_divide_def = divide_complex_def
   906 
   907 lemma legacy_Complex_simps:
   908   shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
   909     and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"
   910     and complex_minus: "- (Complex a b) = Complex (- a) (- b)"
   911     and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"
   912     and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"
   913     and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"
   914     and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
   915     and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"
   916     and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"
   917     and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"
   918     and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"
   919     and Complex_eq_i: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
   920     and i_mult_Complex: "ii * Complex a b = Complex (- b) a"
   921     and Complex_mult_i: "Complex a b * ii = Complex (- b) a"
   922     and i_complex_of_real: "ii * complex_of_real r = Complex 0 r"
   923     and complex_of_real_i: "complex_of_real r * ii = Complex 0 r"
   924     and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"
   925     and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"
   926     and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
   927     and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
   928     and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
   929     and complex_cn: "cnj (Complex a b) = Complex a (- b)"
   930     and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"
   931     and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"
   932     and complex_of_real_def: "complex_of_real r = Complex r 0"
   933     and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"
   934   by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)
   935 
   936 lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
   937   by (metis Reals_of_real complex_of_real_def)
   938 
   939 end